*> \brief \b DBBCSD
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DBBCSD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dbbcsd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dbbcsd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dbbcsd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q,
*                          THETA, PHI, U1, LDU1, U2, LDU2, V1T, LDV1T,
*                          V2T, LDV2T, B11D, B11E, B12D, B12E, B21D, B21E,
*                          B22D, B22E, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS
*       INTEGER            INFO, LDU1, LDU2, LDV1T, LDV2T, LWORK, M, P, Q
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   B11D( * ), B11E( * ), B12D( * ), B12E( * ),
*      $                   B21D( * ), B21E( * ), B22D( * ), B22E( * ),
*      $                   PHI( * ), THETA( * ), WORK( * )
*       DOUBLE PRECISION   U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ),
*      $                   V2T( LDV2T, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DBBCSD computes the CS decomposition of an orthogonal matrix in
*> bidiagonal-block form,
*>
*>
*>     [ B11 | B12 0  0 ]
*>     [  0  |  0 -I  0 ]
*> X = [----------------]
*>     [ B21 | B22 0  0 ]
*>     [  0  |  0  0  I ]
*>
*>                               [  C | -S  0  0 ]
*>                   [ U1 |    ] [  0 |  0 -I  0 ] [ V1 |    ]**T
*>                 = [---------] [---------------] [---------]   .
*>                   [    | U2 ] [  S |  C  0  0 ] [    | V2 ]
*>                               [  0 |  0  0  I ]
*>
*> X is M-by-M, its top-left block is P-by-Q, and Q must be no larger
*> than P, M-P, or M-Q. (If Q is not the smallest index, then X must be
*> transposed and/or permuted. This can be done in constant time using
*> the TRANS and SIGNS options. See DORCSD for details.)
*>
*> The bidiagonal matrices B11, B12, B21, and B22 are represented
*> implicitly by angles THETA(1:Q) and PHI(1:Q-1).
*>
*> The orthogonal matrices U1, U2, V1T, and V2T are input/output.
*> The input matrices are pre- or post-multiplied by the appropriate
*> singular vector matrices.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBU1
*> \verbatim
*>          JOBU1 is CHARACTER
*>          = 'Y':      U1 is updated;
*>          otherwise:  U1 is not updated.
*> \endverbatim
*>
*> \param[in] JOBU2
*> \verbatim
*>          JOBU2 is CHARACTER
*>          = 'Y':      U2 is updated;
*>          otherwise:  U2 is not updated.
*> \endverbatim
*>
*> \param[in] JOBV1T
*> \verbatim
*>          JOBV1T is CHARACTER
*>          = 'Y':      V1T is updated;
*>          otherwise:  V1T is not updated.
*> \endverbatim
*>
*> \param[in] JOBV2T
*> \verbatim
*>          JOBV2T is CHARACTER
*>          = 'Y':      V2T is updated;
*>          otherwise:  V2T is not updated.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER
*>          = 'T':      X, U1, U2, V1T, and V2T are stored in row-major
*>                      order;
*>          otherwise:  X, U1, U2, V1T, and V2T are stored in column-
*>                      major order.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows and columns in X, the orthogonal matrix in
*>          bidiagonal-block form.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*>          P is INTEGER
*>          The number of rows in the top-left block of X. 0 <= P <= M.
*> \endverbatim
*>
*> \param[in] Q
*> \verbatim
*>          Q is INTEGER
*>          The number of columns in the top-left block of X.
*>          0 <= Q <= MIN(P,M-P,M-Q).
*> \endverbatim
*>
*> \param[in,out] THETA
*> \verbatim
*>          THETA is DOUBLE PRECISION array, dimension (Q)
*>          On entry, the angles THETA(1),...,THETA(Q) that, along with
*>          PHI(1), ...,PHI(Q-1), define the matrix in bidiagonal-block
*>          form. On exit, the angles whose cosines and sines define the
*>          diagonal blocks in the CS decomposition.
*> \endverbatim
*>
*> \param[in,out] PHI
*> \verbatim
*>          PHI is DOUBLE PRECISION array, dimension (Q-1)
*>          The angles PHI(1),...,PHI(Q-1) that, along with THETA(1),...,
*>          THETA(Q), define the matrix in bidiagonal-block form.
*> \endverbatim
*>
*> \param[in,out] U1
*> \verbatim
*>          U1 is DOUBLE PRECISION array, dimension (LDU1,P)
*>          On entry, a P-by-P matrix. On exit, U1 is postmultiplied
*>          by the left singular vector matrix common to [ B11 ; 0 ] and
*>          [ B12 0 0 ; 0 -I 0 0 ].
*> \endverbatim
*>
*> \param[in] LDU1
*> \verbatim
*>          LDU1 is INTEGER
*>          The leading dimension of the array U1, LDU1 >= MAX(1,P).
*> \endverbatim
*>
*> \param[in,out] U2
*> \verbatim
*>          U2 is DOUBLE PRECISION array, dimension (LDU2,M-P)
*>          On entry, an (M-P)-by-(M-P) matrix. On exit, U2 is
*>          postmultiplied by the left singular vector matrix common to
*>          [ B21 ; 0 ] and [ B22 0 0 ; 0 0 I ].
*> \endverbatim
*>
*> \param[in] LDU2
*> \verbatim
*>          LDU2 is INTEGER
*>          The leading dimension of the array U2, LDU2 >= MAX(1,M-P).
*> \endverbatim
*>
*> \param[in,out] V1T
*> \verbatim
*>          V1T is DOUBLE PRECISION array, dimension (LDV1T,Q)
*>          On entry, a Q-by-Q matrix. On exit, V1T is premultiplied
*>          by the transpose of the right singular vector
*>          matrix common to [ B11 ; 0 ] and [ B21 ; 0 ].
*> \endverbatim
*>
*> \param[in] LDV1T
*> \verbatim
*>          LDV1T is INTEGER
*>          The leading dimension of the array V1T, LDV1T >= MAX(1,Q).
*> \endverbatim
*>
*> \param[in,out] V2T
*> \verbatim
*>          V2T is DOUBLE PRECISION array, dimension (LDV2T,M-Q)
*>          On entry, an (M-Q)-by-(M-Q) matrix. On exit, V2T is
*>          premultiplied by the transpose of the right
*>          singular vector matrix common to [ B12 0 0 ; 0 -I 0 ] and
*>          [ B22 0 0 ; 0 0 I ].
*> \endverbatim
*>
*> \param[in] LDV2T
*> \verbatim
*>          LDV2T is INTEGER
*>          The leading dimension of the array V2T, LDV2T >= MAX(1,M-Q).
*> \endverbatim
*>
*> \param[out] B11D
*> \verbatim
*>          B11D is DOUBLE PRECISION array, dimension (Q)
*>          When DBBCSD converges, B11D contains the cosines of THETA(1),
*>          ..., THETA(Q). If DBBCSD fails to converge, then B11D
*>          contains the diagonal of the partially reduced top-left
*>          block.
*> \endverbatim
*>
*> \param[out] B11E
*> \verbatim
*>          B11E is DOUBLE PRECISION array, dimension (Q-1)
*>          When DBBCSD converges, B11E contains zeros. If DBBCSD fails
*>          to converge, then B11E contains the superdiagonal of the
*>          partially reduced top-left block.
*> \endverbatim
*>
*> \param[out] B12D
*> \verbatim
*>          B12D is DOUBLE PRECISION array, dimension (Q)
*>          When DBBCSD converges, B12D contains the negative sines of
*>          THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then
*>          B12D contains the diagonal of the partially reduced top-right
*>          block.
*> \endverbatim
*>
*> \param[out] B12E
*> \verbatim
*>          B12E is DOUBLE PRECISION array, dimension (Q-1)
*>          When DBBCSD converges, B12E contains zeros. If DBBCSD fails
*>          to converge, then B12E contains the subdiagonal of the
*>          partially reduced top-right block.
*> \endverbatim
*>
*> \param[out] B21D
*> \verbatim
*>          B21D is DOUBLE PRECISION  array, dimension (Q)
*>          When DBBCSD converges, B21D contains the negative sines of
*>          THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then
*>          B21D contains the diagonal of the partially reduced bottom-left
*>          block.
*> \endverbatim
*>
*> \param[out] B21E
*> \verbatim
*>          B21E is DOUBLE PRECISION  array, dimension (Q-1)
*>          When DBBCSD converges, B21E contains zeros. If DBBCSD fails
*>          to converge, then B21E contains the subdiagonal of the
*>          partially reduced bottom-left block.
*> \endverbatim
*>
*> \param[out] B22D
*> \verbatim
*>          B22D is DOUBLE PRECISION  array, dimension (Q)
*>          When DBBCSD converges, B22D contains the negative sines of
*>          THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then
*>          B22D contains the diagonal of the partially reduced bottom-right
*>          block.
*> \endverbatim
*>
*> \param[out] B22E
*> \verbatim
*>          B22E is DOUBLE PRECISION  array, dimension (Q-1)
*>          When DBBCSD converges, B22E contains zeros. If DBBCSD fails
*>          to converge, then B22E contains the subdiagonal of the
*>          partially reduced bottom-right block.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= MAX(1,8*Q).
*>
*>          If LWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the optimal size of the WORK array,
*>          returns this value as the first entry of the work array, and
*>          no error message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if DBBCSD did not converge, INFO specifies the number
*>                of nonzero entries in PHI, and B11D, B11E, etc.,
*>                contain the partially reduced matrix.
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  TOLMUL  DOUBLE PRECISION, default = MAX(10,MIN(100,EPS**(-1/8)))
*>          TOLMUL controls the convergence criterion of the QR loop.
*>          Angles THETA(i), PHI(i) are rounded to 0 or PI/2 when they
*>          are within TOLMUL*EPS of either bound.
*> \endverbatim
*
*> \par References:
*  ================
*>
*>  [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
*>      Algorithms, 50(1):33-65, 2009.
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup bbcsd
*
*  =====================================================================
      SUBROUTINE DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P,
     $                   Q,
     $                   THETA, PHI, U1, LDU1, U2, LDU2, V1T, LDV1T,
     $                   V2T, LDV2T, B11D, B11E, B12D, B12E, B21D, B21E,
     $                   B22D, B22E, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS
      INTEGER            INFO, LDU1, LDU2, LDV1T, LDV2T, LWORK, M, P, Q
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   B11D( * ), B11E( * ), B12D( * ), B12E( * ),
     $                   B21D( * ), B21E( * ), B22D( * ), B22E( * ),
     $                   PHI( * ), THETA( * ), WORK( * )
      DOUBLE PRECISION   U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ),
     $                   V2T( LDV2T, * )
*     ..
*
*  ===================================================================
*
*     .. Parameters ..
      INTEGER            MAXITR
      PARAMETER          ( MAXITR = 6 )
      DOUBLE PRECISION   HUNDRED, MEIGHTH, ONE, TEN, ZERO
      PARAMETER          ( HUNDRED = 100.0D0, MEIGHTH = -0.125D0,
     $                     ONE = 1.0D0, TEN = 10.0D0, ZERO = 0.0D0 )
      DOUBLE PRECISION   NEGONE
      PARAMETER          ( NEGONE = -1.0D0 )
      DOUBLE PRECISION   PIOVER2
      PARAMETER ( PIOVER2 = 1.57079632679489661923132169163975144210D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            COLMAJOR, LQUERY, RESTART11, RESTART12,
     $                   RESTART21, RESTART22, WANTU1, WANTU2, WANTV1T,
     $                   WANTV2T
      INTEGER            I, IMIN, IMAX, ITER, IU1CS, IU1SN, IU2CS,
     $                   IU2SN, IV1TCS, IV1TSN, IV2TCS, IV2TSN, J,
     $                   LWORKMIN, LWORKOPT, MAXIT, MINI
      DOUBLE PRECISION   B11BULGE, B12BULGE, B21BULGE, B22BULGE, DUMMY,
     $                   EPS, MU, NU, R, SIGMA11, SIGMA21,
     $                   TEMP, THETAMAX, THETAMIN, THRESH, TOL, TOLMUL,
     $                   UNFL, X1, X2, Y1, Y2
*
*     .. External Subroutines ..
      EXTERNAL           DLASR, DSCAL, DSWAP, DLARTGP, DLARTGS,
     $                   DLAS2,
     $                   XERBLA
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      LOGICAL            LSAME
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, ATAN2, COS, MAX, MIN, SIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test input arguments
*
      INFO = 0
      LQUERY = LWORK .EQ. -1
      WANTU1 = LSAME( JOBU1, 'Y' )
      WANTU2 = LSAME( JOBU2, 'Y' )
      WANTV1T = LSAME( JOBV1T, 'Y' )
      WANTV2T = LSAME( JOBV2T, 'Y' )
      COLMAJOR = .NOT. LSAME( TRANS, 'T' )
*
      IF( M .LT. 0 ) THEN
         INFO = -6
      ELSE IF( P .LT. 0 .OR. P .GT. M ) THEN
         INFO = -7
      ELSE IF( Q .LT. 0 .OR. Q .GT. M ) THEN
         INFO = -8
      ELSE IF( Q .GT. P .OR. Q .GT. M-P .OR. Q .GT. M-Q ) THEN
         INFO = -8
      ELSE IF( WANTU1 .AND. LDU1 .LT. P ) THEN
         INFO = -12
      ELSE IF( WANTU2 .AND. LDU2 .LT. M-P ) THEN
         INFO = -14
      ELSE IF( WANTV1T .AND. LDV1T .LT. Q ) THEN
         INFO = -16
      ELSE IF( WANTV2T .AND. LDV2T .LT. M-Q ) THEN
         INFO = -18
      END IF
*
*     Quick return if Q = 0
*
      IF( INFO .EQ. 0 .AND. Q .EQ. 0 ) THEN
         LWORKMIN = 1
         WORK(1) = LWORKMIN
         RETURN
      END IF
*
*     Compute workspace
*
      IF( INFO .EQ. 0 ) THEN
         IU1CS = 1
         IU1SN = IU1CS + Q
         IU2CS = IU1SN + Q
         IU2SN = IU2CS + Q
         IV1TCS = IU2SN + Q
         IV1TSN = IV1TCS + Q
         IV2TCS = IV1TSN + Q
         IV2TSN = IV2TCS + Q
         LWORKOPT = IV2TSN + Q - 1
         LWORKMIN = LWORKOPT
         WORK(1) = LWORKOPT
         IF( LWORK .LT. LWORKMIN .AND. .NOT. LQUERY ) THEN
            INFO = -28
         END IF
      END IF
*
      IF( INFO .NE. 0 ) THEN
         CALL XERBLA( 'DBBCSD', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Get machine constants
*
      EPS = DLAMCH( 'Epsilon' )
      UNFL = DLAMCH( 'Safe minimum' )
      TOLMUL = MAX( TEN, MIN( HUNDRED, EPS**MEIGHTH ) )
      TOL = TOLMUL*EPS
      THRESH = MAX( TOL, MAXITR*Q*Q*UNFL )
*
*     Test for negligible sines or cosines
*
      DO I = 1, Q
         IF( THETA(I) .LT. THRESH ) THEN
            THETA(I) = ZERO
         ELSE IF( THETA(I) .GT. PIOVER2-THRESH ) THEN
            THETA(I) = PIOVER2
         END IF
      END DO
      DO I = 1, Q-1
         IF( PHI(I) .LT. THRESH ) THEN
            PHI(I) = ZERO
         ELSE IF( PHI(I) .GT. PIOVER2-THRESH ) THEN
            PHI(I) = PIOVER2
         END IF
      END DO
*
*     Initial deflation
*
      IMAX = Q
      DO WHILE( IMAX .GT. 1 )
         IF( PHI(IMAX-1) .NE. ZERO ) THEN
            EXIT
         END IF
         IMAX = IMAX - 1
      END DO
      IMIN = IMAX - 1
      IF  ( IMIN .GT. 1 ) THEN
         DO WHILE( PHI(IMIN-1) .NE. ZERO )
            IMIN = IMIN - 1
            IF  ( IMIN .LE. 1 ) EXIT
         END DO
      END IF
*
*     Initialize iteration counter
*
      MAXIT = MAXITR*Q*Q
      ITER = 0
*
*     Begin main iteration loop
*
      DO WHILE( IMAX .GT. 1 )
*
*        Compute the matrix entries
*
         B11D(IMIN) = COS( THETA(IMIN) )
         B21D(IMIN) = -SIN( THETA(IMIN) )
         DO I = IMIN, IMAX - 1
            B11E(I) = -SIN( THETA(I) ) * SIN( PHI(I) )
            B11D(I+1) = COS( THETA(I+1) ) * COS( PHI(I) )
            B12D(I) = SIN( THETA(I) ) * COS( PHI(I) )
            B12E(I) = COS( THETA(I+1) ) * SIN( PHI(I) )
            B21E(I) = -COS( THETA(I) ) * SIN( PHI(I) )
            B21D(I+1) = -SIN( THETA(I+1) ) * COS( PHI(I) )
            B22D(I) = COS( THETA(I) ) * COS( PHI(I) )
            B22E(I) = -SIN( THETA(I+1) ) * SIN( PHI(I) )
         END DO
         B12D(IMAX) = SIN( THETA(IMAX) )
         B22D(IMAX) = COS( THETA(IMAX) )
*
*        Abort if not converging; otherwise, increment ITER
*
         IF( ITER .GT. MAXIT ) THEN
            INFO = 0
            DO I = 1, Q
               IF( PHI(I) .NE. ZERO )
     $            INFO = INFO + 1
            END DO
            RETURN
         END IF
*
         ITER = ITER + IMAX - IMIN
*
*        Compute shifts
*
         THETAMAX = THETA(IMIN)
         THETAMIN = THETA(IMIN)
         DO I = IMIN+1, IMAX
            IF( THETA(I) > THETAMAX )
     $         THETAMAX = THETA(I)
            IF( THETA(I) < THETAMIN )
     $         THETAMIN = THETA(I)
         END DO
*
         IF( THETAMAX .GT. PIOVER2 - THRESH ) THEN
*
*           Zero on diagonals of B11 and B22; induce deflation with a
*           zero shift
*
            MU = ZERO
            NU = ONE
*
         ELSE IF( THETAMIN .LT. THRESH ) THEN
*
*           Zero on diagonals of B12 and B22; induce deflation with a
*           zero shift
*
            MU = ONE
            NU = ZERO
*
         ELSE
*
*           Compute shifts for B11 and B21 and use the lesser
*
            CALL DLAS2( B11D(IMAX-1), B11E(IMAX-1), B11D(IMAX),
     $                  SIGMA11,
     $                  DUMMY )
            CALL DLAS2( B21D(IMAX-1), B21E(IMAX-1), B21D(IMAX),
     $                  SIGMA21,
     $                  DUMMY )
*
            IF( SIGMA11 .LE. SIGMA21 ) THEN
               MU = SIGMA11
               NU = SQRT( ONE - MU**2 )
               IF( MU .LT. THRESH ) THEN
                  MU = ZERO
                  NU = ONE
               END IF
            ELSE
               NU = SIGMA21
               MU = SQRT( 1.0 - NU**2 )
               IF( NU .LT. THRESH ) THEN
                  MU = ONE
                  NU = ZERO
               END IF
            END IF
         END IF
*
*        Rotate to produce bulges in B11 and B21
*
         IF( MU .LE. NU ) THEN
            CALL DLARTGS( B11D(IMIN), B11E(IMIN), MU,
     $                    WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1) )
         ELSE
            CALL DLARTGS( B21D(IMIN), B21E(IMIN), NU,
     $                    WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1) )
         END IF
*
         TEMP = WORK(IV1TCS+IMIN-1)*B11D(IMIN) +
     $          WORK(IV1TSN+IMIN-1)*B11E(IMIN)
         B11E(IMIN) = WORK(IV1TCS+IMIN-1)*B11E(IMIN) -
     $                WORK(IV1TSN+IMIN-1)*B11D(IMIN)
         B11D(IMIN) = TEMP
         B11BULGE = WORK(IV1TSN+IMIN-1)*B11D(IMIN+1)
         B11D(IMIN+1) = WORK(IV1TCS+IMIN-1)*B11D(IMIN+1)
         TEMP = WORK(IV1TCS+IMIN-1)*B21D(IMIN) +
     $          WORK(IV1TSN+IMIN-1)*B21E(IMIN)
         B21E(IMIN) = WORK(IV1TCS+IMIN-1)*B21E(IMIN) -
     $                WORK(IV1TSN+IMIN-1)*B21D(IMIN)
         B21D(IMIN) = TEMP
         B21BULGE = WORK(IV1TSN+IMIN-1)*B21D(IMIN+1)
         B21D(IMIN+1) = WORK(IV1TCS+IMIN-1)*B21D(IMIN+1)
*
*        Compute THETA(IMIN)
*
         THETA( IMIN ) = ATAN2( SQRT( B21D(IMIN)**2+B21BULGE**2 ),
     $                   SQRT( B11D(IMIN)**2+B11BULGE**2 ) )
*
*        Chase the bulges in B11(IMIN+1,IMIN) and B21(IMIN+1,IMIN)
*
         IF( B11D(IMIN)**2+B11BULGE**2 .GT. THRESH**2 ) THEN
            CALL DLARTGP( B11BULGE, B11D(IMIN), WORK(IU1SN+IMIN-1),
     $                    WORK(IU1CS+IMIN-1), R )
         ELSE IF( MU .LE. NU ) THEN
            CALL DLARTGS( B11E( IMIN ), B11D( IMIN + 1 ), MU,
     $                    WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1) )
         ELSE
            CALL DLARTGS( B12D( IMIN ), B12E( IMIN ), NU,
     $                    WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1) )
         END IF
         IF( B21D(IMIN)**2+B21BULGE**2 .GT. THRESH**2 ) THEN
            CALL DLARTGP( B21BULGE, B21D(IMIN), WORK(IU2SN+IMIN-1),
     $                    WORK(IU2CS+IMIN-1), R )
         ELSE IF( NU .LT. MU ) THEN
            CALL DLARTGS( B21E( IMIN ), B21D( IMIN + 1 ), NU,
     $                    WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1) )
         ELSE
            CALL DLARTGS( B22D(IMIN), B22E(IMIN), MU,
     $                    WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1) )
         END IF
         WORK(IU2CS+IMIN-1) = -WORK(IU2CS+IMIN-1)
         WORK(IU2SN+IMIN-1) = -WORK(IU2SN+IMIN-1)
*
         TEMP = WORK(IU1CS+IMIN-1)*B11E(IMIN) +
     $          WORK(IU1SN+IMIN-1)*B11D(IMIN+1)
         B11D(IMIN+1) = WORK(IU1CS+IMIN-1)*B11D(IMIN+1) -
     $                  WORK(IU1SN+IMIN-1)*B11E(IMIN)
         B11E(IMIN) = TEMP
         IF( IMAX .GT. IMIN+1 ) THEN
            B11BULGE = WORK(IU1SN+IMIN-1)*B11E(IMIN+1)
            B11E(IMIN+1) = WORK(IU1CS+IMIN-1)*B11E(IMIN+1)
         END IF
         TEMP = WORK(IU1CS+IMIN-1)*B12D(IMIN) +
     $          WORK(IU1SN+IMIN-1)*B12E(IMIN)
         B12E(IMIN) = WORK(IU1CS+IMIN-1)*B12E(IMIN) -
     $                WORK(IU1SN+IMIN-1)*B12D(IMIN)
         B12D(IMIN) = TEMP
         B12BULGE = WORK(IU1SN+IMIN-1)*B12D(IMIN+1)
         B12D(IMIN+1) = WORK(IU1CS+IMIN-1)*B12D(IMIN+1)
         TEMP = WORK(IU2CS+IMIN-1)*B21E(IMIN) +
     $          WORK(IU2SN+IMIN-1)*B21D(IMIN+1)
         B21D(IMIN+1) = WORK(IU2CS+IMIN-1)*B21D(IMIN+1) -
     $                  WORK(IU2SN+IMIN-1)*B21E(IMIN)
         B21E(IMIN) = TEMP
         IF( IMAX .GT. IMIN+1 ) THEN
            B21BULGE = WORK(IU2SN+IMIN-1)*B21E(IMIN+1)
            B21E(IMIN+1) = WORK(IU2CS+IMIN-1)*B21E(IMIN+1)
         END IF
         TEMP = WORK(IU2CS+IMIN-1)*B22D(IMIN) +
     $          WORK(IU2SN+IMIN-1)*B22E(IMIN)
         B22E(IMIN) = WORK(IU2CS+IMIN-1)*B22E(IMIN) -
     $                WORK(IU2SN+IMIN-1)*B22D(IMIN)
         B22D(IMIN) = TEMP
         B22BULGE = WORK(IU2SN+IMIN-1)*B22D(IMIN+1)
         B22D(IMIN+1) = WORK(IU2CS+IMIN-1)*B22D(IMIN+1)
*
*        Inner loop: chase bulges from B11(IMIN,IMIN+2),
*        B12(IMIN,IMIN+1), B21(IMIN,IMIN+2), and B22(IMIN,IMIN+1) to
*        bottom-right
*
         DO I = IMIN+1, IMAX-1
*
*           Compute PHI(I-1)
*
            X1 = SIN(THETA(I-1))*B11E(I-1) + COS(THETA(I-1))*B21E(I-1)
            X2 = SIN(THETA(I-1))*B11BULGE + COS(THETA(I-1))*B21BULGE
            Y1 = SIN(THETA(I-1))*B12D(I-1) + COS(THETA(I-1))*B22D(I-1)
            Y2 = SIN(THETA(I-1))*B12BULGE + COS(THETA(I-1))*B22BULGE
*
            PHI(I-1) = ATAN2( SQRT(X1**2+X2**2), SQRT(Y1**2+Y2**2) )
*
*           Determine if there are bulges to chase or if a new direct
*           summand has been reached
*
            RESTART11 = B11E(I-1)**2 + B11BULGE**2 .LE. THRESH**2
            RESTART21 = B21E(I-1)**2 + B21BULGE**2 .LE. THRESH**2
            RESTART12 = B12D(I-1)**2 + B12BULGE**2 .LE. THRESH**2
            RESTART22 = B22D(I-1)**2 + B22BULGE**2 .LE. THRESH**2
*
*           If possible, chase bulges from B11(I-1,I+1), B12(I-1,I),
*           B21(I-1,I+1), and B22(I-1,I). If necessary, restart bulge-
*           chasing by applying the original shift again.
*
            IF( .NOT. RESTART11 .AND. .NOT. RESTART21 ) THEN
               CALL DLARTGP( X2, X1, WORK(IV1TSN+I-1),
     $                       WORK(IV1TCS+I-1),
     $                       R )
            ELSE IF( .NOT. RESTART11 .AND. RESTART21 ) THEN
               CALL DLARTGP( B11BULGE, B11E(I-1), WORK(IV1TSN+I-1),
     $                       WORK(IV1TCS+I-1), R )
            ELSE IF( RESTART11 .AND. .NOT. RESTART21 ) THEN
               CALL DLARTGP( B21BULGE, B21E(I-1), WORK(IV1TSN+I-1),
     $                       WORK(IV1TCS+I-1), R )
            ELSE IF( MU .LE. NU ) THEN
               CALL DLARTGS( B11D(I), B11E(I), MU, WORK(IV1TCS+I-1),
     $                       WORK(IV1TSN+I-1) )
            ELSE
               CALL DLARTGS( B21D(I), B21E(I), NU, WORK(IV1TCS+I-1),
     $                       WORK(IV1TSN+I-1) )
            END IF
            WORK(IV1TCS+I-1) = -WORK(IV1TCS+I-1)
            WORK(IV1TSN+I-1) = -WORK(IV1TSN+I-1)
            IF( .NOT. RESTART12 .AND. .NOT. RESTART22 ) THEN
               CALL DLARTGP( Y2, Y1, WORK(IV2TSN+I-1-1),
     $                       WORK(IV2TCS+I-1-1), R )
            ELSE IF( .NOT. RESTART12 .AND. RESTART22 ) THEN
               CALL DLARTGP( B12BULGE, B12D(I-1), WORK(IV2TSN+I-1-1),
     $                       WORK(IV2TCS+I-1-1), R )
            ELSE IF( RESTART12 .AND. .NOT. RESTART22 ) THEN
               CALL DLARTGP( B22BULGE, B22D(I-1), WORK(IV2TSN+I-1-1),
     $                       WORK(IV2TCS+I-1-1), R )
            ELSE IF( NU .LT. MU ) THEN
               CALL DLARTGS( B12E(I-1), B12D(I), NU,
     $                       WORK(IV2TCS+I-1-1),
     $                       WORK(IV2TSN+I-1-1) )
            ELSE
               CALL DLARTGS( B22E(I-1), B22D(I), MU,
     $                       WORK(IV2TCS+I-1-1),
     $                       WORK(IV2TSN+I-1-1) )
            END IF
*
            TEMP = WORK(IV1TCS+I-1)*B11D(I) + WORK(IV1TSN+I-1)*B11E(I)
            B11E(I) = WORK(IV1TCS+I-1)*B11E(I) -
     $                WORK(IV1TSN+I-1)*B11D(I)
            B11D(I) = TEMP
            B11BULGE = WORK(IV1TSN+I-1)*B11D(I+1)
            B11D(I+1) = WORK(IV1TCS+I-1)*B11D(I+1)
            TEMP = WORK(IV1TCS+I-1)*B21D(I) + WORK(IV1TSN+I-1)*B21E(I)
            B21E(I) = WORK(IV1TCS+I-1)*B21E(I) -
     $                WORK(IV1TSN+I-1)*B21D(I)
            B21D(I) = TEMP
            B21BULGE = WORK(IV1TSN+I-1)*B21D(I+1)
            B21D(I+1) = WORK(IV1TCS+I-1)*B21D(I+1)
            TEMP = WORK(IV2TCS+I-1-1)*B12E(I-1) +
     $             WORK(IV2TSN+I-1-1)*B12D(I)
            B12D(I) = WORK(IV2TCS+I-1-1)*B12D(I) -
     $                WORK(IV2TSN+I-1-1)*B12E(I-1)
            B12E(I-1) = TEMP
            B12BULGE = WORK(IV2TSN+I-1-1)*B12E(I)
            B12E(I) = WORK(IV2TCS+I-1-1)*B12E(I)
            TEMP = WORK(IV2TCS+I-1-1)*B22E(I-1) +
     $             WORK(IV2TSN+I-1-1)*B22D(I)
            B22D(I) = WORK(IV2TCS+I-1-1)*B22D(I) -
     $                WORK(IV2TSN+I-1-1)*B22E(I-1)
            B22E(I-1) = TEMP
            B22BULGE = WORK(IV2TSN+I-1-1)*B22E(I)
            B22E(I) = WORK(IV2TCS+I-1-1)*B22E(I)
*
*           Compute THETA(I)
*
            X1 = COS(PHI(I-1))*B11D(I) + SIN(PHI(I-1))*B12E(I-1)
            X2 = COS(PHI(I-1))*B11BULGE + SIN(PHI(I-1))*B12BULGE
            Y1 = COS(PHI(I-1))*B21D(I) + SIN(PHI(I-1))*B22E(I-1)
            Y2 = COS(PHI(I-1))*B21BULGE + SIN(PHI(I-1))*B22BULGE
*
            THETA(I) = ATAN2( SQRT(Y1**2+Y2**2), SQRT(X1**2+X2**2) )
*
*           Determine if there are bulges to chase or if a new direct
*           summand has been reached
*
            RESTART11 =   B11D(I)**2 + B11BULGE**2 .LE. THRESH**2
            RESTART12 = B12E(I-1)**2 + B12BULGE**2 .LE. THRESH**2
            RESTART21 =   B21D(I)**2 + B21BULGE**2 .LE. THRESH**2
            RESTART22 = B22E(I-1)**2 + B22BULGE**2 .LE. THRESH**2
*
*           If possible, chase bulges from B11(I+1,I), B12(I+1,I-1),
*           B21(I+1,I), and B22(I+1,I-1). If necessary, restart bulge-
*           chasing by applying the original shift again.
*
            IF( .NOT. RESTART11 .AND. .NOT. RESTART12 ) THEN
               CALL DLARTGP( X2, X1, WORK(IU1SN+I-1),
     $                       WORK(IU1CS+I-1),
     $                       R )
            ELSE IF( .NOT. RESTART11 .AND. RESTART12 ) THEN
               CALL DLARTGP( B11BULGE, B11D(I), WORK(IU1SN+I-1),
     $                       WORK(IU1CS+I-1), R )
            ELSE IF( RESTART11 .AND. .NOT. RESTART12 ) THEN
               CALL DLARTGP( B12BULGE, B12E(I-1), WORK(IU1SN+I-1),
     $                       WORK(IU1CS+I-1), R )
            ELSE IF( MU .LE. NU ) THEN
               CALL DLARTGS( B11E(I), B11D(I+1), MU, WORK(IU1CS+I-1),
     $                       WORK(IU1SN+I-1) )
            ELSE
               CALL DLARTGS( B12D(I), B12E(I), NU, WORK(IU1CS+I-1),
     $                       WORK(IU1SN+I-1) )
            END IF
            IF( .NOT. RESTART21 .AND. .NOT. RESTART22 ) THEN
               CALL DLARTGP( Y2, Y1, WORK(IU2SN+I-1),
     $                       WORK(IU2CS+I-1),
     $                       R )
            ELSE IF( .NOT. RESTART21 .AND. RESTART22 ) THEN
               CALL DLARTGP( B21BULGE, B21D(I), WORK(IU2SN+I-1),
     $                       WORK(IU2CS+I-1), R )
            ELSE IF( RESTART21 .AND. .NOT. RESTART22 ) THEN
               CALL DLARTGP( B22BULGE, B22E(I-1), WORK(IU2SN+I-1),
     $                       WORK(IU2CS+I-1), R )
            ELSE IF( NU .LT. MU ) THEN
               CALL DLARTGS( B21E(I), B21E(I+1), NU, WORK(IU2CS+I-1),
     $                       WORK(IU2SN+I-1) )
            ELSE
               CALL DLARTGS( B22D(I), B22E(I), MU, WORK(IU2CS+I-1),
     $                       WORK(IU2SN+I-1) )
            END IF
            WORK(IU2CS+I-1) = -WORK(IU2CS+I-1)
            WORK(IU2SN+I-1) = -WORK(IU2SN+I-1)
*
            TEMP = WORK(IU1CS+I-1)*B11E(I) + WORK(IU1SN+I-1)*B11D(I+1)
            B11D(I+1) = WORK(IU1CS+I-1)*B11D(I+1) -
     $                  WORK(IU1SN+I-1)*B11E(I)
            B11E(I) = TEMP
            IF( I .LT. IMAX - 1 ) THEN
               B11BULGE = WORK(IU1SN+I-1)*B11E(I+1)
               B11E(I+1) = WORK(IU1CS+I-1)*B11E(I+1)
            END IF
            TEMP = WORK(IU2CS+I-1)*B21E(I) + WORK(IU2SN+I-1)*B21D(I+1)
            B21D(I+1) = WORK(IU2CS+I-1)*B21D(I+1) -
     $                  WORK(IU2SN+I-1)*B21E(I)
            B21E(I) = TEMP
            IF( I .LT. IMAX - 1 ) THEN
               B21BULGE = WORK(IU2SN+I-1)*B21E(I+1)
               B21E(I+1) = WORK(IU2CS+I-1)*B21E(I+1)
            END IF
            TEMP = WORK(IU1CS+I-1)*B12D(I) + WORK(IU1SN+I-1)*B12E(I)
            B12E(I) = WORK(IU1CS+I-1)*B12E(I) - WORK(IU1SN+I-1)*B12D(I)
            B12D(I) = TEMP
            B12BULGE = WORK(IU1SN+I-1)*B12D(I+1)
            B12D(I+1) = WORK(IU1CS+I-1)*B12D(I+1)
            TEMP = WORK(IU2CS+I-1)*B22D(I) + WORK(IU2SN+I-1)*B22E(I)
            B22E(I) = WORK(IU2CS+I-1)*B22E(I) - WORK(IU2SN+I-1)*B22D(I)
            B22D(I) = TEMP
            B22BULGE = WORK(IU2SN+I-1)*B22D(I+1)
            B22D(I+1) = WORK(IU2CS+I-1)*B22D(I+1)
*
         END DO
*
*        Compute PHI(IMAX-1)
*
         X1 = SIN(THETA(IMAX-1))*B11E(IMAX-1) +
     $        COS(THETA(IMAX-1))*B21E(IMAX-1)
         Y1 = SIN(THETA(IMAX-1))*B12D(IMAX-1) +
     $        COS(THETA(IMAX-1))*B22D(IMAX-1)
         Y2 = SIN(THETA(IMAX-1))*B12BULGE + COS(THETA(IMAX-1))*B22BULGE
*
         PHI(IMAX-1) = ATAN2( ABS(X1), SQRT(Y1**2+Y2**2) )
*
*        Chase bulges from B12(IMAX-1,IMAX) and B22(IMAX-1,IMAX)
*
         RESTART12 = B12D(IMAX-1)**2 + B12BULGE**2 .LE. THRESH**2
         RESTART22 = B22D(IMAX-1)**2 + B22BULGE**2 .LE. THRESH**2
*
         IF( .NOT. RESTART12 .AND. .NOT. RESTART22 ) THEN
            CALL DLARTGP( Y2, Y1, WORK(IV2TSN+IMAX-1-1),
     $                    WORK(IV2TCS+IMAX-1-1), R )
         ELSE IF( .NOT. RESTART12 .AND. RESTART22 ) THEN
            CALL DLARTGP( B12BULGE, B12D(IMAX-1),
     $                    WORK(IV2TSN+IMAX-1-1),
     $                    WORK(IV2TCS+IMAX-1-1), R )
         ELSE IF( RESTART12 .AND. .NOT. RESTART22 ) THEN
            CALL DLARTGP( B22BULGE, B22D(IMAX-1),
     $                    WORK(IV2TSN+IMAX-1-1),
     $                    WORK(IV2TCS+IMAX-1-1), R )
         ELSE IF( NU .LT. MU ) THEN
            CALL DLARTGS( B12E(IMAX-1), B12D(IMAX), NU,
     $                    WORK(IV2TCS+IMAX-1-1), WORK(IV2TSN+IMAX-1-1) )
         ELSE
            CALL DLARTGS( B22E(IMAX-1), B22D(IMAX), MU,
     $                    WORK(IV2TCS+IMAX-1-1), WORK(IV2TSN+IMAX-1-1) )
         END IF
*
         TEMP = WORK(IV2TCS+IMAX-1-1)*B12E(IMAX-1) +
     $          WORK(IV2TSN+IMAX-1-1)*B12D(IMAX)
         B12D(IMAX) = WORK(IV2TCS+IMAX-1-1)*B12D(IMAX) -
     $                WORK(IV2TSN+IMAX-1-1)*B12E(IMAX-1)
         B12E(IMAX-1) = TEMP
         TEMP = WORK(IV2TCS+IMAX-1-1)*B22E(IMAX-1) +
     $          WORK(IV2TSN+IMAX-1-1)*B22D(IMAX)
         B22D(IMAX) = WORK(IV2TCS+IMAX-1-1)*B22D(IMAX) -
     $                WORK(IV2TSN+IMAX-1-1)*B22E(IMAX-1)
         B22E(IMAX-1) = TEMP
*
*        Update singular vectors
*
         IF( WANTU1 ) THEN
            IF( COLMAJOR ) THEN
               CALL DLASR( 'R', 'V', 'F', P, IMAX-IMIN+1,
     $                     WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1),
     $                     U1(1,IMIN), LDU1 )
            ELSE
               CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, P,
     $                     WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1),
     $                     U1(IMIN,1), LDU1 )
            END IF
         END IF
         IF( WANTU2 ) THEN
            IF( COLMAJOR ) THEN
               CALL DLASR( 'R', 'V', 'F', M-P, IMAX-IMIN+1,
     $                     WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1),
     $                     U2(1,IMIN), LDU2 )
            ELSE
               CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, M-P,
     $                     WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1),
     $                     U2(IMIN,1), LDU2 )
            END IF
         END IF
         IF( WANTV1T ) THEN
            IF( COLMAJOR ) THEN
               CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, Q,
     $                     WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1),
     $                     V1T(IMIN,1), LDV1T )
            ELSE
               CALL DLASR( 'R', 'V', 'F', Q, IMAX-IMIN+1,
     $                     WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1),
     $                     V1T(1,IMIN), LDV1T )
            END IF
         END IF
         IF( WANTV2T ) THEN
            IF( COLMAJOR ) THEN
               CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, M-Q,
     $                     WORK(IV2TCS+IMIN-1), WORK(IV2TSN+IMIN-1),
     $                     V2T(IMIN,1), LDV2T )
            ELSE
               CALL DLASR( 'R', 'V', 'F', M-Q, IMAX-IMIN+1,
     $                     WORK(IV2TCS+IMIN-1), WORK(IV2TSN+IMIN-1),
     $                     V2T(1,IMIN), LDV2T )
            END IF
         END IF
*
*        Fix signs on B11(IMAX-1,IMAX) and B21(IMAX-1,IMAX)
*
         IF( B11E(IMAX-1)+B21E(IMAX-1) .GT. 0 ) THEN
            B11D(IMAX) = -B11D(IMAX)
            B21D(IMAX) = -B21D(IMAX)
            IF( WANTV1T ) THEN
               IF( COLMAJOR ) THEN
                  CALL DSCAL( Q, NEGONE, V1T(IMAX,1), LDV1T )
               ELSE
                  CALL DSCAL( Q, NEGONE, V1T(1,IMAX), 1 )
               END IF
            END IF
         END IF
*
*        Compute THETA(IMAX)
*
         X1 = COS(PHI(IMAX-1))*B11D(IMAX) +
     $        SIN(PHI(IMAX-1))*B12E(IMAX-1)
         Y1 = COS(PHI(IMAX-1))*B21D(IMAX) +
     $        SIN(PHI(IMAX-1))*B22E(IMAX-1)
*
         THETA(IMAX) = ATAN2( ABS(Y1), ABS(X1) )
*
*        Fix signs on B11(IMAX,IMAX), B12(IMAX,IMAX-1), B21(IMAX,IMAX),
*        and B22(IMAX,IMAX-1)
*
         IF( B11D(IMAX)+B12E(IMAX-1) .LT. 0 ) THEN
            B12D(IMAX) = -B12D(IMAX)
            IF( WANTU1 ) THEN
               IF( COLMAJOR ) THEN
                  CALL DSCAL( P, NEGONE, U1(1,IMAX), 1 )
               ELSE
                  CALL DSCAL( P, NEGONE, U1(IMAX,1), LDU1 )
               END IF
            END IF
         END IF
         IF( B21D(IMAX)+B22E(IMAX-1) .GT. 0 ) THEN
            B22D(IMAX) = -B22D(IMAX)
            IF( WANTU2 ) THEN
               IF( COLMAJOR ) THEN
                  CALL DSCAL( M-P, NEGONE, U2(1,IMAX), 1 )
               ELSE
                  CALL DSCAL( M-P, NEGONE, U2(IMAX,1), LDU2 )
               END IF
            END IF
         END IF
*
*        Fix signs on B12(IMAX,IMAX) and B22(IMAX,IMAX)
*
         IF( B12D(IMAX)+B22D(IMAX) .LT. 0 ) THEN
            IF( WANTV2T ) THEN
               IF( COLMAJOR ) THEN
                  CALL DSCAL( M-Q, NEGONE, V2T(IMAX,1), LDV2T )
               ELSE
                  CALL DSCAL( M-Q, NEGONE, V2T(1,IMAX), 1 )
               END IF
            END IF
         END IF
*
*        Test for negligible sines or cosines
*
         DO I = IMIN, IMAX
            IF( THETA(I) .LT. THRESH ) THEN
               THETA(I) = ZERO
            ELSE IF( THETA(I) .GT. PIOVER2-THRESH ) THEN
               THETA(I) = PIOVER2
            END IF
         END DO
         DO I = IMIN, IMAX-1
            IF( PHI(I) .LT. THRESH ) THEN
               PHI(I) = ZERO
            ELSE IF( PHI(I) .GT. PIOVER2-THRESH ) THEN
               PHI(I) = PIOVER2
            END IF
         END DO
*
*        Deflate
*
         IF (IMAX .GT. 1) THEN
            DO WHILE( PHI(IMAX-1) .EQ. ZERO )
               IMAX = IMAX - 1
               IF (IMAX .LE. 1) EXIT
            END DO
         END IF
         IF( IMIN .GT. IMAX - 1 )
     $      IMIN = IMAX - 1
         IF (IMIN .GT. 1) THEN
            DO WHILE (PHI(IMIN-1) .NE. ZERO)
                IMIN = IMIN - 1
                IF (IMIN .LE. 1) EXIT
            END DO
         END IF
*
*        Repeat main iteration loop
*
      END DO
*
*     Postprocessing: order THETA from least to greatest
*
      DO I = 1, Q
*
         MINI = I
         THETAMIN = THETA(I)
         DO J = I+1, Q
            IF( THETA(J) .LT. THETAMIN ) THEN
               MINI = J
               THETAMIN = THETA(J)
            END IF
         END DO
*
         IF( MINI .NE. I ) THEN
            THETA(MINI) = THETA(I)
            THETA(I) = THETAMIN
            IF( COLMAJOR ) THEN
               IF( WANTU1 )
     $            CALL DSWAP( P, U1(1,I), 1, U1(1,MINI), 1 )
               IF( WANTU2 )
     $            CALL DSWAP( M-P, U2(1,I), 1, U2(1,MINI), 1 )
               IF( WANTV1T )
     $            CALL DSWAP( Q, V1T(I,1), LDV1T, V1T(MINI,1),
     $                        LDV1T )
               IF( WANTV2T )
     $            CALL DSWAP( M-Q, V2T(I,1), LDV2T, V2T(MINI,1),
     $               LDV2T )
            ELSE
               IF( WANTU1 )
     $            CALL DSWAP( P, U1(I,1), LDU1, U1(MINI,1), LDU1 )
               IF( WANTU2 )
     $            CALL DSWAP( M-P, U2(I,1), LDU2, U2(MINI,1), LDU2 )
               IF( WANTV1T )
     $            CALL DSWAP( Q, V1T(1,I), 1, V1T(1,MINI), 1 )
               IF( WANTV2T )
     $            CALL DSWAP( M-Q, V2T(1,I), 1, V2T(1,MINI), 1 )
            END IF
         END IF
*
      END DO
*
      RETURN
*
*     End of DBBCSD
*
      END

*> \brief \b DBDSDC
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DBDSDC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dbdsdc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dbdsdc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dbdsdc.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DBDSDC( UPLO, COMPQ, N, D, E, U, LDU, VT, LDVT, Q, IQ,
*                          WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          COMPQ, UPLO
*       INTEGER            INFO, LDU, LDVT, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IQ( * ), IWORK( * )
*       DOUBLE PRECISION   D( * ), E( * ), Q( * ), U( LDU, * ),
*      $                   VT( LDVT, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DBDSDC computes the singular value decomposition (SVD) of a real
*> N-by-N (upper or lower) bidiagonal matrix B:  B = U * S * VT,
*> using a divide and conquer method, where S is a diagonal matrix
*> with non-negative diagonal elements (the singular values of B), and
*> U and VT are orthogonal matrices of left and right singular vectors,
*> respectively. DBDSDC can be used to compute all singular values,
*> and optionally, singular vectors or singular vectors in compact form.
*>
*> The code currently calls DLASDQ if singular values only are desired.
*> However, it can be slightly modified to compute singular values
*> using the divide and conquer method.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  B is upper bidiagonal.
*>          = 'L':  B is lower bidiagonal.
*> \endverbatim
*>
*> \param[in] COMPQ
*> \verbatim
*>          COMPQ is CHARACTER*1
*>          Specifies whether singular vectors are to be computed
*>          as follows:
*>          = 'N':  Compute singular values only;
*>          = 'P':  Compute singular values and compute singular
*>                  vectors in compact form;
*>          = 'I':  Compute singular values and singular vectors.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix B.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, the n diagonal elements of the bidiagonal matrix B.
*>          On exit, if INFO=0, the singular values of B.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          On entry, the elements of E contain the offdiagonal
*>          elements of the bidiagonal matrix whose SVD is desired.
*>          On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*>          U is DOUBLE PRECISION array, dimension (LDU,N)
*>          If  COMPQ = 'I', then:
*>             On exit, if INFO = 0, U contains the left singular vectors
*>             of the bidiagonal matrix.
*>          For other values of COMPQ, U is not referenced.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>          The leading dimension of the array U.  LDU >= 1.
*>          If singular vectors are desired, then LDU >= max( 1, N ).
*> \endverbatim
*>
*> \param[out] VT
*> \verbatim
*>          VT is DOUBLE PRECISION array, dimension (LDVT,N)
*>          If  COMPQ = 'I', then:
*>             On exit, if INFO = 0, VT**T contains the right singular
*>             vectors of the bidiagonal matrix.
*>          For other values of COMPQ, VT is not referenced.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*>          LDVT is INTEGER
*>          The leading dimension of the array VT.  LDVT >= 1.
*>          If singular vectors are desired, then LDVT >= max( 1, N ).
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ)
*>          If  COMPQ = 'P', then:
*>             On exit, if INFO = 0, Q and IQ contain the left
*>             and right singular vectors in a compact form,
*>             requiring O(N log N) space instead of 2*N**2.
*>             In particular, Q contains all the DOUBLE PRECISION data in
*>             LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1))))
*>             words of memory, where SMLSIZ is returned by ILAENV and
*>             is equal to the maximum size of the subproblems at the
*>             bottom of the computation tree (usually about 25).
*>          For other values of COMPQ, Q is not referenced.
*> \endverbatim
*>
*> \param[out] IQ
*> \verbatim
*>          IQ is INTEGER array, dimension (LDIQ)
*>          If  COMPQ = 'P', then:
*>             On exit, if INFO = 0, Q and IQ contain the left
*>             and right singular vectors in a compact form,
*>             requiring O(N log N) space instead of 2*N**2.
*>             In particular, IQ contains all INTEGER data in
*>             LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1))))
*>             words of memory, where SMLSIZ is returned by ILAENV and
*>             is equal to the maximum size of the subproblems at the
*>             bottom of the computation tree (usually about 25).
*>          For other values of COMPQ, IQ is not referenced.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          If COMPQ = 'N' then LWORK >= (4 * N).
*>          If COMPQ = 'P' then LWORK >= (6 * N).
*>          If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N).
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (8*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  The algorithm failed to compute a singular value.
*>                The update process of divide and conquer failed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup bdsdc
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Huan Ren, Computer Science Division, University of
*>     California at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE DBDSDC( UPLO, COMPQ, N, D, E, U, LDU, VT, LDVT, Q,
     $                   IQ,
     $                   WORK, IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          COMPQ, UPLO
      INTEGER            INFO, LDU, LDVT, N
*     ..
*     .. Array Arguments ..
      INTEGER            IQ( * ), IWORK( * )
      DOUBLE PRECISION   D( * ), E( * ), Q( * ), U( LDU, * ),
     $                   VT( LDVT, * ), WORK( * )
*     ..
*
*  =====================================================================
*  Changed dimension statement in comment describing E from (N) to
*  (N-1).  Sven, 17 Feb 05.
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            DIFL, DIFR, GIVCOL, GIVNUM, GIVPTR, I, IC,
     $                   ICOMPQ, IERR, II, IS, IU, IUPLO, IVT, J, K, KK,
     $                   MLVL, NM1, NSIZE, PERM, POLES, QSTART, SMLSIZ,
     $                   SMLSZP, SQRE, START, WSTART, Z
      DOUBLE PRECISION   CS, EPS, ORGNRM, P, R, SN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANST
      EXTERNAL           LSAME, ILAENV, DLAMCH, DLANST
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLARTG, DLASCL, DLASD0, DLASDA,
     $                   DLASDQ,
     $                   DLASET, DLASR, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, INT, LOG, SIGN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IUPLO = 0
      IF( LSAME( UPLO, 'U' ) )
     $   IUPLO = 1
      IF( LSAME( UPLO, 'L' ) )
     $   IUPLO = 2
      IF( LSAME( COMPQ, 'N' ) ) THEN
         ICOMPQ = 0
      ELSE IF( LSAME( COMPQ, 'P' ) ) THEN
         ICOMPQ = 1
      ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
         ICOMPQ = 2
      ELSE
         ICOMPQ = -1
      END IF
      IF( IUPLO.EQ.0 ) THEN
         INFO = -1
      ELSE IF( ICOMPQ.LT.0 ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( ( LDU.LT.1 ) .OR. ( ( ICOMPQ.EQ.2 ) .AND. ( LDU.LT.
     $         N ) ) ) THEN
         INFO = -7
      ELSE IF( ( LDVT.LT.1 ) .OR. ( ( ICOMPQ.EQ.2 ) .AND. ( LDVT.LT.
     $         N ) ) ) THEN
         INFO = -9
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DBDSDC', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
      SMLSIZ = ILAENV( 9, 'DBDSDC', ' ', 0, 0, 0, 0 )
      IF( N.EQ.1 ) THEN
         IF( ICOMPQ.EQ.1 ) THEN
            Q( 1 ) = SIGN( ONE, D( 1 ) )
            Q( 1+SMLSIZ*N ) = ONE
         ELSE IF( ICOMPQ.EQ.2 ) THEN
            U( 1, 1 ) = SIGN( ONE, D( 1 ) )
            VT( 1, 1 ) = ONE
         END IF
         D( 1 ) = ABS( D( 1 ) )
         RETURN
      END IF
      NM1 = N - 1
*
*     If matrix lower bidiagonal, rotate to be upper bidiagonal
*     by applying Givens rotations on the left
*
      WSTART = 1
      QSTART = 3
      IF( ICOMPQ.EQ.1 ) THEN
         CALL DCOPY( N, D, 1, Q( 1 ), 1 )
         CALL DCOPY( N-1, E, 1, Q( N+1 ), 1 )
      END IF
      IF( IUPLO.EQ.2 ) THEN
         QSTART = 5
         IF( ICOMPQ .EQ. 2 ) WSTART = 2*N - 1
         DO 10 I = 1, N - 1
            CALL DLARTG( D( I ), E( I ), CS, SN, R )
            D( I ) = R
            E( I ) = SN*D( I+1 )
            D( I+1 ) = CS*D( I+1 )
            IF( ICOMPQ.EQ.1 ) THEN
               Q( I+2*N ) = CS
               Q( I+3*N ) = SN
            ELSE IF( ICOMPQ.EQ.2 ) THEN
               WORK( I ) = CS
               WORK( NM1+I ) = -SN
            END IF
   10    CONTINUE
      END IF
*
*     If ICOMPQ = 0, use DLASDQ to compute the singular values.
*
      IF( ICOMPQ.EQ.0 ) THEN
*        Ignore WSTART, instead using WORK( 1 ), since the two vectors
*        for CS and -SN above are added only if ICOMPQ == 2,
*        and adding them exceeds documented WORK size of 4*n.
         CALL DLASDQ( 'U', 0, N, 0, 0, 0, D, E, VT, LDVT, U, LDU, U,
     $                LDU, WORK( 1 ), INFO )
         GO TO 40
      END IF
*
*     If N is smaller than the minimum divide size SMLSIZ, then solve
*     the problem with another solver.
*
      IF( N.LE.SMLSIZ ) THEN
         IF( ICOMPQ.EQ.2 ) THEN
            CALL DLASET( 'A', N, N, ZERO, ONE, U, LDU )
            CALL DLASET( 'A', N, N, ZERO, ONE, VT, LDVT )
            CALL DLASDQ( 'U', 0, N, N, N, 0, D, E, VT, LDVT, U, LDU,
     $                   U,
     $                   LDU, WORK( WSTART ), INFO )
         ELSE IF( ICOMPQ.EQ.1 ) THEN
            IU = 1
            IVT = IU + N
            CALL DLASET( 'A', N, N, ZERO, ONE,
     $                   Q( IU+( QSTART-1 )*N ),
     $                   N )
            CALL DLASET( 'A', N, N, ZERO, ONE,
     $                   Q( IVT+( QSTART-1 )*N ),
     $                   N )
            CALL DLASDQ( 'U', 0, N, N, N, 0, D, E,
     $                   Q( IVT+( QSTART-1 )*N ), N,
     $                   Q( IU+( QSTART-1 )*N ), N,
     $                   Q( IU+( QSTART-1 )*N ), N, WORK( WSTART ),
     $                   INFO )
         END IF
         GO TO 40
      END IF
*
      IF( ICOMPQ.EQ.2 ) THEN
         CALL DLASET( 'A', N, N, ZERO, ONE, U, LDU )
         CALL DLASET( 'A', N, N, ZERO, ONE, VT, LDVT )
      END IF
*
*     Scale.
*
      ORGNRM = DLANST( 'M', N, D, E )
      IF( ORGNRM.EQ.ZERO )
     $   RETURN
      CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, IERR )
      CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, NM1, 1, E, NM1, IERR )
*
      EPS = (0.9D+0)*DLAMCH( 'Epsilon' )
*
      MLVL = INT( LOG( DBLE( N ) / DBLE( SMLSIZ+1 ) ) / LOG( TWO ) ) + 1
      SMLSZP = SMLSIZ + 1
*
      IF( ICOMPQ.EQ.1 ) THEN
         IU = 1
         IVT = 1 + SMLSIZ
         DIFL = IVT + SMLSZP
         DIFR = DIFL + MLVL
         Z = DIFR + MLVL*2
         IC = Z + MLVL
         IS = IC + 1
         POLES = IS + 1
         GIVNUM = POLES + 2*MLVL
*
         K = 1
         GIVPTR = 2
         PERM = 3
         GIVCOL = PERM + MLVL
      END IF
*
      DO 20 I = 1, N
         IF( ABS( D( I ) ).LT.EPS ) THEN
            D( I ) = SIGN( EPS, D( I ) )
         END IF
   20 CONTINUE
*
      START = 1
      SQRE = 0
*
      DO 30 I = 1, NM1
         IF( ( ABS( E( I ) ).LT.EPS ) .OR. ( I.EQ.NM1 ) ) THEN
*
*           Subproblem found. First determine its size and then
*           apply divide and conquer on it.
*
            IF( I.LT.NM1 ) THEN
*
*              A subproblem with E(I) small for I < NM1.
*
               NSIZE = I - START + 1
            ELSE IF( ABS( E( I ) ).GE.EPS ) THEN
*
*              A subproblem with E(NM1) not too small but I = NM1.
*
               NSIZE = N - START + 1
            ELSE
*
*              A subproblem with E(NM1) small. This implies an
*              1-by-1 subproblem at D(N). Solve this 1-by-1 problem
*              first.
*
               NSIZE = I - START + 1
               IF( ICOMPQ.EQ.2 ) THEN
                  U( N, N ) = SIGN( ONE, D( N ) )
                  VT( N, N ) = ONE
               ELSE IF( ICOMPQ.EQ.1 ) THEN
                  Q( N+( QSTART-1 )*N ) = SIGN( ONE, D( N ) )
                  Q( N+( SMLSIZ+QSTART-1 )*N ) = ONE
               END IF
               D( N ) = ABS( D( N ) )
            END IF
            IF( ICOMPQ.EQ.2 ) THEN
               CALL DLASD0( NSIZE, SQRE, D( START ), E( START ),
     $                      U( START, START ), LDU, VT( START, START ),
     $                      LDVT, SMLSIZ, IWORK, WORK( WSTART ), INFO )
            ELSE
               CALL DLASDA( ICOMPQ, SMLSIZ, NSIZE, SQRE, D( START ),
     $                      E( START ), Q( START+( IU+QSTART-2 )*N ), N,
     $                      Q( START+( IVT+QSTART-2 )*N ),
     $                      IQ( START+K*N ), Q( START+( DIFL+QSTART-2 )*
     $                      N ), Q( START+( DIFR+QSTART-2 )*N ),
     $                      Q( START+( Z+QSTART-2 )*N ),
     $                      Q( START+( POLES+QSTART-2 )*N ),
     $                      IQ( START+GIVPTR*N ), IQ( START+GIVCOL*N ),
     $                      N, IQ( START+PERM*N ),
     $                      Q( START+( GIVNUM+QSTART-2 )*N ),
     $                      Q( START+( IC+QSTART-2 )*N ),
     $                      Q( START+( IS+QSTART-2 )*N ),
     $                      WORK( WSTART ), IWORK, INFO )
            END IF
            IF( INFO.NE.0 ) THEN
               RETURN
            END IF
            START = I + 1
         END IF
   30 CONTINUE
*
*     Unscale
*
      CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, IERR )
   40 CONTINUE
*
*     Use Selection Sort to minimize swaps of singular vectors
*
      DO 60 II = 2, N
         I = II - 1
         KK = I
         P = D( I )
         DO 50 J = II, N
            IF( D( J ).GT.P ) THEN
               KK = J
               P = D( J )
            END IF
   50    CONTINUE
         IF( KK.NE.I ) THEN
            D( KK ) = D( I )
            D( I ) = P
            IF( ICOMPQ.EQ.1 ) THEN
               IQ( I ) = KK
            ELSE IF( ICOMPQ.EQ.2 ) THEN
               CALL DSWAP( N, U( 1, I ), 1, U( 1, KK ), 1 )
               CALL DSWAP( N, VT( I, 1 ), LDVT, VT( KK, 1 ), LDVT )
            END IF
         ELSE IF( ICOMPQ.EQ.1 ) THEN
            IQ( I ) = I
         END IF
   60 CONTINUE
*
*     If ICOMPQ = 1, use IQ(N,1) as the indicator for UPLO
*
      IF( ICOMPQ.EQ.1 ) THEN
         IF( IUPLO.EQ.1 ) THEN
            IQ( N ) = 1
         ELSE
            IQ( N ) = 0
         END IF
      END IF
*
*     If B is lower bidiagonal, update U by those Givens rotations
*     which rotated B to be upper bidiagonal
*
      IF( ( IUPLO.EQ.2 ) .AND. ( ICOMPQ.EQ.2 ) )
     $   CALL DLASR( 'L', 'V', 'B', N, N, WORK( 1 ), WORK( N ), U,
     $               LDU )
*
      RETURN
*
*     End of DBDSDC
*
      END
*> \brief \b DBDSQR
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DBDSQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dbdsqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dbdsqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dbdsqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U,
*                          LDU, C, LDC, WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   C( LDC, * ), D( * ), E( * ), U( LDU, * ),
*      $                   VT( LDVT, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DBDSQR computes the singular values and, optionally, the right and/or
*> left singular vectors from the singular value decomposition (SVD) of
*> a real N-by-N (upper or lower) bidiagonal matrix B using the implicit
*> zero-shift QR algorithm.  The SVD of B has the form
*>
*>    B = Q * S * P**T
*>
*> where S is the diagonal matrix of singular values, Q is an orthogonal
*> matrix of left singular vectors, and P is an orthogonal matrix of
*> right singular vectors.  If left singular vectors are requested, this
*> subroutine actually returns U*Q instead of Q, and, if right singular
*> vectors are requested, this subroutine returns P**T*VT instead of
*> P**T, for given real input matrices U and VT.  When U and VT are the
*> orthogonal matrices that reduce a general matrix A to bidiagonal
*> form:  A = U*B*VT, as computed by DGEBRD, then
*>
*>    A = (U*Q) * S * (P**T*VT)
*>
*> is the SVD of A.  Optionally, the subroutine may also compute Q**T*C
*> for a given real input matrix C.
*>
*> See "Computing  Small Singular Values of Bidiagonal Matrices With
*> Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan,
*> LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11,
*> no. 5, pp. 873-912, Sept 1990) and
*> "Accurate singular values and differential qd algorithms," by
*> B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics
*> Department, University of California at Berkeley, July 1992
*> for a detailed description of the algorithm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  B is upper bidiagonal;
*>          = 'L':  B is lower bidiagonal.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix B.  N >= 0.
*> \endverbatim
*>
*> \param[in] NCVT
*> \verbatim
*>          NCVT is INTEGER
*>          The number of columns of the matrix VT. NCVT >= 0.
*> \endverbatim
*>
*> \param[in] NRU
*> \verbatim
*>          NRU is INTEGER
*>          The number of rows of the matrix U. NRU >= 0.
*> \endverbatim
*>
*> \param[in] NCC
*> \verbatim
*>          NCC is INTEGER
*>          The number of columns of the matrix C. NCC >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, the n diagonal elements of the bidiagonal matrix B.
*>          On exit, if INFO=0, the singular values of B in decreasing
*>          order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          On entry, the N-1 offdiagonal elements of the bidiagonal
*>          matrix B.
*>          On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
*>          will contain the diagonal and superdiagonal elements of a
*>          bidiagonal matrix orthogonally equivalent to the one given
*>          as input.
*> \endverbatim
*>
*> \param[in,out] VT
*> \verbatim
*>          VT is DOUBLE PRECISION array, dimension (LDVT, NCVT)
*>          On entry, an N-by-NCVT matrix VT.
*>          On exit, VT is overwritten by P**T * VT.
*>          Not referenced if NCVT = 0.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*>          LDVT is INTEGER
*>          The leading dimension of the array VT.
*>          LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.
*> \endverbatim
*>
*> \param[in,out] U
*> \verbatim
*>          U is DOUBLE PRECISION array, dimension (LDU, N)
*>          On entry, an NRU-by-N matrix U.
*>          On exit, U is overwritten by U * Q.
*>          Not referenced if NRU = 0.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>          The leading dimension of the array U.  LDU >= max(1,NRU).
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC, NCC)
*>          On entry, an N-by-NCC matrix C.
*>          On exit, C is overwritten by Q**T * C.
*>          Not referenced if NCC = 0.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C.
*>          LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (LWORK)
*>          LWORK = 4*N, if NCVT = NRU = NCC = 0, and
*>          LWORK = 4*(N-1), otherwise
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  If INFO = -i, the i-th argument had an illegal value
*>          > 0:
*>             if NCVT = NRU = NCC = 0,
*>                = 1, a split was marked by a positive value in E
*>                = 2, current block of Z not diagonalized after 30*N
*>                     iterations (in inner while loop)
*>                = 3, termination criterion of outer while loop not met
*>                     (program created more than N unreduced blocks)
*>             else NCVT = NRU = NCC = 0,
*>                   the algorithm did not converge; D and E contain the
*>                   elements of a bidiagonal matrix which is orthogonally
*>                   similar to the input matrix B;  if INFO = i, i
*>                   elements of E have not converged to zero.
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  TOLMUL  DOUBLE PRECISION, default = max(10,min(100,EPS**(-1/8)))
*>          TOLMUL controls the convergence criterion of the QR loop.
*>          If it is positive, TOLMUL*EPS is the desired relative
*>             precision in the computed singular values.
*>          If it is negative, abs(TOLMUL*EPS*sigma_max) is the
*>             desired absolute accuracy in the computed singular
*>             values (corresponds to relative accuracy
*>             abs(TOLMUL*EPS) in the largest singular value.
*>          abs(TOLMUL) should be between 1 and 1/EPS, and preferably
*>             between 10 (for fast convergence) and .1/EPS
*>             (for there to be some accuracy in the results).
*>          Default is to lose at either one eighth or 2 of the
*>             available decimal digits in each computed singular value
*>             (whichever is smaller).
*>
*>  MAXITR  INTEGER, default = 6
*>          MAXITR controls the maximum number of passes of the
*>          algorithm through its inner loop. The algorithms stops
*>          (and so fails to converge) if the number of passes
*>          through the inner loop exceeds MAXITR*N**2.
*>
*> \endverbatim
*
*> \par Note:
*  ===========
*>
*> \verbatim
*>  Bug report from Cezary Dendek.
*>  On March 23rd 2017, the INTEGER variable MAXIT = MAXITR*N**2 is
*>  removed since it can overflow pretty easily (for N larger or equal
*>  than 18,919). We instead use MAXITDIVN = MAXITR*N.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup bdsqr
*
*  =====================================================================
      SUBROUTINE DBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U,
     $                   LDU, C, LDC, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   C( LDC, * ), D( * ), E( * ), U( LDU, * ),
     $                   VT( LDVT, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D0 )
      DOUBLE PRECISION   NEGONE
      PARAMETER          ( NEGONE = -1.0D0 )
      DOUBLE PRECISION   HNDRTH
      PARAMETER          ( HNDRTH = 0.01D0 )
      DOUBLE PRECISION   TEN
      PARAMETER          ( TEN = 10.0D0 )
      DOUBLE PRECISION   HNDRD
      PARAMETER          ( HNDRD = 100.0D0 )
      DOUBLE PRECISION   MEIGTH
      PARAMETER          ( MEIGTH = -0.125D0 )
      INTEGER            MAXITR
      PARAMETER          ( MAXITR = 6 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER, ROTATE
      INTEGER            I, IDIR, ISUB, ITER, ITERDIVN, J, LL, LLL, M,
     $                   MAXITDIVN, NM1, NM12, NM13, OLDLL, OLDM
      DOUBLE PRECISION   ABSE, ABSS, COSL, COSR, CS, EPS, F, G, H, MU,
     $                   OLDCS, OLDSN, R, SHIFT, SIGMN, SIGMX, SINL,
     $                   SINR, SLL, SMAX, SMIN, SMINOA,
     $                   SN, THRESH, TOL, TOLMUL, UNFL
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARTG, DLAS2, DLASQ1, DLASR, DLASV2,
     $                   DROT,
     $                   DSCAL, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, MAX, MIN, SIGN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      LOWER = LSAME( UPLO, 'L' )
      IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LOWER ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NCVT.LT.0 ) THEN
         INFO = -3
      ELSE IF( NRU.LT.0 ) THEN
         INFO = -4
      ELSE IF( NCC.LT.0 ) THEN
         INFO = -5
      ELSE IF( ( NCVT.EQ.0 .AND. LDVT.LT.1 ) .OR.
     $         ( NCVT.GT.0 .AND. LDVT.LT.MAX( 1, N ) ) ) THEN
         INFO = -9
      ELSE IF( LDU.LT.MAX( 1, NRU ) ) THEN
         INFO = -11
      ELSE IF( ( NCC.EQ.0 .AND. LDC.LT.1 ) .OR.
     $         ( NCC.GT.0 .AND. LDC.LT.MAX( 1, N ) ) ) THEN
         INFO = -13
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DBDSQR', -INFO )
         RETURN
      END IF
      IF( N.EQ.0 )
     $   RETURN
      IF( N.EQ.1 )
     $   GO TO 160
*
*     ROTATE is true if any singular vectors desired, false otherwise
*
      ROTATE = ( NCVT.GT.0 ) .OR. ( NRU.GT.0 ) .OR. ( NCC.GT.0 )
*
*     If no singular vectors desired, use qd algorithm
*
      IF( .NOT.ROTATE ) THEN
         CALL DLASQ1( N, D, E, WORK, INFO )
*
*     If INFO equals 2, dqds didn't finish, try to finish
*
         IF( INFO .NE. 2 ) RETURN
         INFO = 0
      END IF
*
      NM1 = N - 1
      NM12 = NM1 + NM1
      NM13 = NM12 + NM1
      IDIR = 0
*
*     Get machine constants
*
      EPS = DLAMCH( 'Epsilon' )
      UNFL = DLAMCH( 'Safe minimum' )
*
*     If matrix lower bidiagonal, rotate to be upper bidiagonal
*     by applying Givens rotations on the left
*
      IF( LOWER ) THEN
         DO 10 I = 1, N - 1
            CALL DLARTG( D( I ), E( I ), CS, SN, R )
            D( I ) = R
            E( I ) = SN*D( I+1 )
            D( I+1 ) = CS*D( I+1 )
            WORK( I ) = CS
            WORK( NM1+I ) = SN
   10    CONTINUE
*
*        Update singular vectors if desired
*
         IF( NRU.GT.0 )
     $      CALL DLASR( 'R', 'V', 'F', NRU, N, WORK( 1 ), WORK( N ),
     $                  U,
     $                  LDU )
         IF( NCC.GT.0 )
     $      CALL DLASR( 'L', 'V', 'F', N, NCC, WORK( 1 ), WORK( N ),
     $                  C,
     $                  LDC )
      END IF
*
*     Compute singular values to relative accuracy TOL
*     (By setting TOL to be negative, algorithm will compute
*     singular values to absolute accuracy ABS(TOL)*norm(input matrix))
*
      TOLMUL = MAX( TEN, MIN( HNDRD, EPS**MEIGTH ) )
      TOL = TOLMUL*EPS
*
*     Compute approximate maximum, minimum singular values
*
      SMAX = ZERO
      DO 20 I = 1, N
         SMAX = MAX( SMAX, ABS( D( I ) ) )
   20 CONTINUE
      DO 30 I = 1, N - 1
         SMAX = MAX( SMAX, ABS( E( I ) ) )
   30 CONTINUE
      SMIN = ZERO
      IF( TOL.GE.ZERO ) THEN
*
*        Relative accuracy desired
*
         SMINOA = ABS( D( 1 ) )
         IF( SMINOA.EQ.ZERO )
     $      GO TO 50
         MU = SMINOA
         DO 40 I = 2, N
            MU = ABS( D( I ) )*( MU / ( MU+ABS( E( I-1 ) ) ) )
            SMINOA = MIN( SMINOA, MU )
            IF( SMINOA.EQ.ZERO )
     $         GO TO 50
   40    CONTINUE
   50    CONTINUE
         SMINOA = SMINOA / SQRT( DBLE( N ) )
         THRESH = MAX( TOL*SMINOA, MAXITR*(N*(N*UNFL)) )
      ELSE
*
*        Absolute accuracy desired
*
         THRESH = MAX( ABS( TOL )*SMAX, MAXITR*(N*(N*UNFL)) )
      END IF
*
*     Prepare for main iteration loop for the singular values
*     (MAXIT is the maximum number of passes through the inner
*     loop permitted before nonconvergence signalled.)
*
      MAXITDIVN = MAXITR*N
      ITERDIVN = 0
      ITER = -1
      OLDLL = -1
      OLDM = -1
*
*     M points to last element of unconverged part of matrix
*
      M = N
*
*     Begin main iteration loop
*
   60 CONTINUE
*
*     Check for convergence or exceeding iteration count
*
      IF( M.LE.1 )
     $   GO TO 160
*
      IF( ITER.GE.N ) THEN
         ITER = ITER - N
         ITERDIVN = ITERDIVN + 1
         IF( ITERDIVN.GE.MAXITDIVN )
     $      GO TO 200
      END IF
*
*     Find diagonal block of matrix to work on
*
      IF( TOL.LT.ZERO .AND. ABS( D( M ) ).LE.THRESH )
     $   D( M ) = ZERO
      SMAX = ABS( D( M ) )
      DO 70 LLL = 1, M - 1
         LL = M - LLL
         ABSS = ABS( D( LL ) )
         ABSE = ABS( E( LL ) )
         IF( TOL.LT.ZERO .AND. ABSS.LE.THRESH )
     $      D( LL ) = ZERO
         IF( ABSE.LE.THRESH )
     $      GO TO 80
         SMAX = MAX( SMAX, ABSS, ABSE )
   70 CONTINUE
      LL = 0
      GO TO 90
   80 CONTINUE
      E( LL ) = ZERO
*
*     Matrix splits since E(LL) = 0
*
      IF( LL.EQ.M-1 ) THEN
*
*        Convergence of bottom singular value, return to top of loop
*
         M = M - 1
         GO TO 60
      END IF
   90 CONTINUE
      LL = LL + 1
*
*     E(LL) through E(M-1) are nonzero, E(LL-1) is zero
*
      IF( LL.EQ.M-1 ) THEN
*
*        2 by 2 block, handle separately
*
         CALL DLASV2( D( M-1 ), E( M-1 ), D( M ), SIGMN, SIGMX, SINR,
     $                COSR, SINL, COSL )
         D( M-1 ) = SIGMX
         E( M-1 ) = ZERO
         D( M ) = SIGMN
*
*        Compute singular vectors, if desired
*
         IF( NCVT.GT.0 )
     $      CALL DROT( NCVT, VT( M-1, 1 ), LDVT, VT( M, 1 ), LDVT,
     $                 COSR,
     $                 SINR )
         IF( NRU.GT.0 )
     $      CALL DROT( NRU, U( 1, M-1 ), 1, U( 1, M ), 1, COSL,
     $                 SINL )
         IF( NCC.GT.0 )
     $      CALL DROT( NCC, C( M-1, 1 ), LDC, C( M, 1 ), LDC, COSL,
     $                 SINL )
         M = M - 2
         GO TO 60
      END IF
*
*     If working on new submatrix, choose shift direction
*     (from larger end diagonal element towards smaller)
*
      IF( LL.GT.OLDM .OR. M.LT.OLDLL ) THEN
         IF( ABS( D( LL ) ).GE.ABS( D( M ) ) ) THEN
*
*           Chase bulge from top (big end) to bottom (small end)
*
            IDIR = 1
         ELSE
*
*           Chase bulge from bottom (big end) to top (small end)
*
            IDIR = 2
         END IF
      END IF
*
*     Apply convergence tests
*
      IF( IDIR.EQ.1 ) THEN
*
*        Run convergence test in forward direction
*        First apply standard test to bottom of matrix
*
         IF( ABS( E( M-1 ) ).LE.ABS( TOL )*ABS( D( M ) ) .OR.
     $       ( TOL.LT.ZERO .AND. ABS( E( M-1 ) ).LE.THRESH ) ) THEN
            E( M-1 ) = ZERO
            GO TO 60
         END IF
*
         IF( TOL.GE.ZERO ) THEN
*
*           If relative accuracy desired,
*           apply convergence criterion forward
*
            MU = ABS( D( LL ) )
            SMIN = MU
            DO 100 LLL = LL, M - 1
               IF( ABS( E( LLL ) ).LE.TOL*MU ) THEN
                  E( LLL ) = ZERO
                  GO TO 60
               END IF
               MU = ABS( D( LLL+1 ) )*( MU / ( MU+ABS( E( LLL ) ) ) )
               SMIN = MIN( SMIN, MU )
  100       CONTINUE
         END IF
*
      ELSE
*
*        Run convergence test in backward direction
*        First apply standard test to top of matrix
*
         IF( ABS( E( LL ) ).LE.ABS( TOL )*ABS( D( LL ) ) .OR.
     $       ( TOL.LT.ZERO .AND. ABS( E( LL ) ).LE.THRESH ) ) THEN
            E( LL ) = ZERO
            GO TO 60
         END IF
*
         IF( TOL.GE.ZERO ) THEN
*
*           If relative accuracy desired,
*           apply convergence criterion backward
*
            MU = ABS( D( M ) )
            SMIN = MU
            DO 110 LLL = M - 1, LL, -1
               IF( ABS( E( LLL ) ).LE.TOL*MU ) THEN
                  E( LLL ) = ZERO
                  GO TO 60
               END IF
               MU = ABS( D( LLL ) )*( MU / ( MU+ABS( E( LLL ) ) ) )
               SMIN = MIN( SMIN, MU )
  110       CONTINUE
         END IF
      END IF
      OLDLL = LL
      OLDM = M
*
*     Compute shift.  First, test if shifting would ruin relative
*     accuracy, and if so set the shift to zero.
*
      IF( TOL.GE.ZERO .AND. N*TOL*( SMIN / SMAX ).LE.
     $    MAX( EPS, HNDRTH*TOL ) ) THEN
*
*        Use a zero shift to avoid loss of relative accuracy
*
         SHIFT = ZERO
      ELSE
*
*        Compute the shift from 2-by-2 block at end of matrix
*
         IF( IDIR.EQ.1 ) THEN
            SLL = ABS( D( LL ) )
            CALL DLAS2( D( M-1 ), E( M-1 ), D( M ), SHIFT, R )
         ELSE
            SLL = ABS( D( M ) )
            CALL DLAS2( D( LL ), E( LL ), D( LL+1 ), SHIFT, R )
         END IF
*
*        Test if shift negligible, and if so set to zero
*
         IF( SLL.GT.ZERO ) THEN
            IF( ( SHIFT / SLL )**2.LT.EPS )
     $         SHIFT = ZERO
         END IF
      END IF
*
*     Increment iteration count
*
      ITER = ITER + M - LL
*
*     If SHIFT = 0, do simplified QR iteration
*
      IF( SHIFT.EQ.ZERO ) THEN
         IF( IDIR.EQ.1 ) THEN
*
*           Chase bulge from top to bottom
*           Save cosines and sines for later singular vector updates
*
            CS = ONE
            OLDCS = ONE
            DO 120 I = LL, M - 1
               CALL DLARTG( D( I )*CS, E( I ), CS, SN, R )
               IF( I.GT.LL )
     $            E( I-1 ) = OLDSN*R
               CALL DLARTG( OLDCS*R, D( I+1 )*SN, OLDCS, OLDSN,
     $                      D( I ) )
               WORK( I-LL+1 ) = CS
               WORK( I-LL+1+NM1 ) = SN
               WORK( I-LL+1+NM12 ) = OLDCS
               WORK( I-LL+1+NM13 ) = OLDSN
  120       CONTINUE
            H = D( M )*CS
            D( M ) = H*OLDCS
            E( M-1 ) = H*OLDSN
*
*           Update singular vectors
*
            IF( NCVT.GT.0 )
     $         CALL DLASR( 'L', 'V', 'F', M-LL+1, NCVT, WORK( 1 ),
     $                     WORK( N ), VT( LL, 1 ), LDVT )
            IF( NRU.GT.0 )
     $         CALL DLASR( 'R', 'V', 'F', NRU, M-LL+1,
     $                     WORK( NM12+1 ),
     $                     WORK( NM13+1 ), U( 1, LL ), LDU )
            IF( NCC.GT.0 )
     $         CALL DLASR( 'L', 'V', 'F', M-LL+1, NCC,
     $                     WORK( NM12+1 ),
     $                     WORK( NM13+1 ), C( LL, 1 ), LDC )
*
*           Test convergence
*
            IF( ABS( E( M-1 ) ).LE.THRESH )
     $         E( M-1 ) = ZERO
*
         ELSE
*
*           Chase bulge from bottom to top
*           Save cosines and sines for later singular vector updates
*
            CS = ONE
            OLDCS = ONE
            DO 130 I = M, LL + 1, -1
               CALL DLARTG( D( I )*CS, E( I-1 ), CS, SN, R )
               IF( I.LT.M )
     $            E( I ) = OLDSN*R
               CALL DLARTG( OLDCS*R, D( I-1 )*SN, OLDCS, OLDSN,
     $                      D( I ) )
               WORK( I-LL ) = CS
               WORK( I-LL+NM1 ) = -SN
               WORK( I-LL+NM12 ) = OLDCS
               WORK( I-LL+NM13 ) = -OLDSN
  130       CONTINUE
            H = D( LL )*CS
            D( LL ) = H*OLDCS
            E( LL ) = H*OLDSN
*
*           Update singular vectors
*
            IF( NCVT.GT.0 )
     $         CALL DLASR( 'L', 'V', 'B', M-LL+1, NCVT,
     $                     WORK( NM12+1 ),
     $                     WORK( NM13+1 ), VT( LL, 1 ), LDVT )
            IF( NRU.GT.0 )
     $         CALL DLASR( 'R', 'V', 'B', NRU, M-LL+1, WORK( 1 ),
     $                     WORK( N ), U( 1, LL ), LDU )
            IF( NCC.GT.0 )
     $         CALL DLASR( 'L', 'V', 'B', M-LL+1, NCC, WORK( 1 ),
     $                     WORK( N ), C( LL, 1 ), LDC )
*
*           Test convergence
*
            IF( ABS( E( LL ) ).LE.THRESH )
     $         E( LL ) = ZERO
         END IF
      ELSE
*
*        Use nonzero shift
*
         IF( IDIR.EQ.1 ) THEN
*
*           Chase bulge from top to bottom
*           Save cosines and sines for later singular vector updates
*
            F = ( ABS( D( LL ) )-SHIFT )*
     $          ( SIGN( ONE, D( LL ) )+SHIFT / D( LL ) )
            G = E( LL )
            DO 140 I = LL, M - 1
               CALL DLARTG( F, G, COSR, SINR, R )
               IF( I.GT.LL )
     $            E( I-1 ) = R
               F = COSR*D( I ) + SINR*E( I )
               E( I ) = COSR*E( I ) - SINR*D( I )
               G = SINR*D( I+1 )
               D( I+1 ) = COSR*D( I+1 )
               CALL DLARTG( F, G, COSL, SINL, R )
               D( I ) = R
               F = COSL*E( I ) + SINL*D( I+1 )
               D( I+1 ) = COSL*D( I+1 ) - SINL*E( I )
               IF( I.LT.M-1 ) THEN
                  G = SINL*E( I+1 )
                  E( I+1 ) = COSL*E( I+1 )
               END IF
               WORK( I-LL+1 ) = COSR
               WORK( I-LL+1+NM1 ) = SINR
               WORK( I-LL+1+NM12 ) = COSL
               WORK( I-LL+1+NM13 ) = SINL
  140       CONTINUE
            E( M-1 ) = F
*
*           Update singular vectors
*
            IF( NCVT.GT.0 )
     $         CALL DLASR( 'L', 'V', 'F', M-LL+1, NCVT, WORK( 1 ),
     $                     WORK( N ), VT( LL, 1 ), LDVT )
            IF( NRU.GT.0 )
     $         CALL DLASR( 'R', 'V', 'F', NRU, M-LL+1,
     $                     WORK( NM12+1 ),
     $                     WORK( NM13+1 ), U( 1, LL ), LDU )
            IF( NCC.GT.0 )
     $         CALL DLASR( 'L', 'V', 'F', M-LL+1, NCC,
     $                     WORK( NM12+1 ),
     $                     WORK( NM13+1 ), C( LL, 1 ), LDC )
*
*           Test convergence
*
            IF( ABS( E( M-1 ) ).LE.THRESH )
     $         E( M-1 ) = ZERO
*
         ELSE
*
*           Chase bulge from bottom to top
*           Save cosines and sines for later singular vector updates
*
            F = ( ABS( D( M ) )-SHIFT )*( SIGN( ONE, D( M ) )+SHIFT /
     $          D( M ) )
            G = E( M-1 )
            DO 150 I = M, LL + 1, -1
               CALL DLARTG( F, G, COSR, SINR, R )
               IF( I.LT.M )
     $            E( I ) = R
               F = COSR*D( I ) + SINR*E( I-1 )
               E( I-1 ) = COSR*E( I-1 ) - SINR*D( I )
               G = SINR*D( I-1 )
               D( I-1 ) = COSR*D( I-1 )
               CALL DLARTG( F, G, COSL, SINL, R )
               D( I ) = R
               F = COSL*E( I-1 ) + SINL*D( I-1 )
               D( I-1 ) = COSL*D( I-1 ) - SINL*E( I-1 )
               IF( I.GT.LL+1 ) THEN
                  G = SINL*E( I-2 )
                  E( I-2 ) = COSL*E( I-2 )
               END IF
               WORK( I-LL ) = COSR
               WORK( I-LL+NM1 ) = -SINR
               WORK( I-LL+NM12 ) = COSL
               WORK( I-LL+NM13 ) = -SINL
  150       CONTINUE
            E( LL ) = F
*
*           Test convergence
*
            IF( ABS( E( LL ) ).LE.THRESH )
     $         E( LL ) = ZERO
*
*           Update singular vectors if desired
*
            IF( NCVT.GT.0 )
     $         CALL DLASR( 'L', 'V', 'B', M-LL+1, NCVT,
     $                     WORK( NM12+1 ),
     $                     WORK( NM13+1 ), VT( LL, 1 ), LDVT )
            IF( NRU.GT.0 )
     $         CALL DLASR( 'R', 'V', 'B', NRU, M-LL+1, WORK( 1 ),
     $                     WORK( N ), U( 1, LL ), LDU )
            IF( NCC.GT.0 )
     $         CALL DLASR( 'L', 'V', 'B', M-LL+1, NCC, WORK( 1 ),
     $                     WORK( N ), C( LL, 1 ), LDC )
         END IF
      END IF
*
*     QR iteration finished, go back and check convergence
*
      GO TO 60
*
*     All singular values converged, so make them positive
*
  160 CONTINUE
      DO 170 I = 1, N
         IF( D( I ).EQ.ZERO ) THEN
*
*           Avoid -ZERO
*
            D( I ) = ZERO
         END IF
         IF( D( I ).LT.ZERO ) THEN
            D( I ) = -D( I )
*
*           Change sign of singular vectors, if desired
*
            IF( NCVT.GT.0 )
     $         CALL DSCAL( NCVT, NEGONE, VT( I, 1 ), LDVT )
         END IF
  170 CONTINUE
*
*     Sort the singular values into decreasing order (insertion sort on
*     singular values, but only one transposition per singular vector)
*
      DO 190 I = 1, N - 1
*
*        Scan for smallest D(I)
*
         ISUB = 1
         SMIN = D( 1 )
         DO 180 J = 2, N + 1 - I
            IF( D( J ).LE.SMIN ) THEN
               ISUB = J
               SMIN = D( J )
            END IF
  180    CONTINUE
         IF( ISUB.NE.N+1-I ) THEN
*
*           Swap singular values and vectors
*
            D( ISUB ) = D( N+1-I )
            D( N+1-I ) = SMIN
            IF( NCVT.GT.0 )
     $         CALL DSWAP( NCVT, VT( ISUB, 1 ), LDVT, VT( N+1-I, 1 ),
     $                     LDVT )
            IF( NRU.GT.0 )
     $         CALL DSWAP( NRU, U( 1, ISUB ), 1, U( 1, N+1-I ), 1 )
            IF( NCC.GT.0 )
     $         CALL DSWAP( NCC, C( ISUB, 1 ), LDC, C( N+1-I, 1 ),
     $                     LDC )
         END IF
  190 CONTINUE
      GO TO 220
*
*     Maximum number of iterations exceeded, failure to converge
*
  200 CONTINUE
      INFO = 0
      DO 210 I = 1, N - 1
         IF( E( I ).NE.ZERO )
     $      INFO = INFO + 1
  210 CONTINUE
  220 CONTINUE
      RETURN
*
*     End of DBDSQR
*
      END
*> \brief \b DDISNA
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DDISNA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ddisna.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ddisna.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ddisna.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DDISNA( JOB, M, N, D, SEP, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOB
*       INTEGER            INFO, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), SEP( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DDISNA computes the reciprocal condition numbers for the eigenvectors
*> of a real symmetric or complex Hermitian matrix or for the left or
*> right singular vectors of a general m-by-n matrix. The reciprocal
*> condition number is the 'gap' between the corresponding eigenvalue or
*> singular value and the nearest other one.
*>
*> The bound on the error, measured by angle in radians, in the I-th
*> computed vector is given by
*>
*>        DLAMCH( 'E' ) * ( ANORM / SEP( I ) )
*>
*> where ANORM = 2-norm(A) = max( abs( D(j) ) ).  SEP(I) is not allowed
*> to be smaller than DLAMCH( 'E' )*ANORM in order to limit the size of
*> the error bound.
*>
*> DDISNA may also be used to compute error bounds for eigenvectors of
*> the generalized symmetric definite eigenproblem.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOB
*> \verbatim
*>          JOB is CHARACTER*1
*>          Specifies for which problem the reciprocal condition numbers
*>          should be computed:
*>          = 'E':  the eigenvectors of a symmetric/Hermitian matrix;
*>          = 'L':  the left singular vectors of a general matrix;
*>          = 'R':  the right singular vectors of a general matrix.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          If JOB = 'L' or 'R', the number of columns of the matrix,
*>          in which case N >= 0. Ignored if JOB = 'E'.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (M) if JOB = 'E'
*>                              dimension (min(M,N)) if JOB = 'L' or 'R'
*>          The eigenvalues (if JOB = 'E') or singular values (if JOB =
*>          'L' or 'R') of the matrix, in either increasing or decreasing
*>          order. If singular values, they must be non-negative.
*> \endverbatim
*>
*> \param[out] SEP
*> \verbatim
*>          SEP is DOUBLE PRECISION array, dimension (M) if JOB = 'E'
*>                               dimension (min(M,N)) if JOB = 'L' or 'R'
*>          The reciprocal condition numbers of the vectors.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup disna
*
*  =====================================================================
      SUBROUTINE DDISNA( JOB, M, N, D, SEP, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOB
      INTEGER            INFO, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), SEP( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            DECR, EIGEN, INCR, LEFT, RIGHT, SING
      INTEGER            I, K
      DOUBLE PRECISION   ANORM, EPS, NEWGAP, OLDGAP, SAFMIN, THRESH
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      EIGEN = LSAME( JOB, 'E' )
      LEFT = LSAME( JOB, 'L' )
      RIGHT = LSAME( JOB, 'R' )
      SING = LEFT .OR. RIGHT
      IF( EIGEN ) THEN
         K = M
      ELSE IF( SING ) THEN
         K = MIN( M, N )
      END IF
      IF( .NOT.EIGEN .AND. .NOT.SING ) THEN
         INFO = -1
      ELSE IF( M.LT.0 ) THEN
         INFO = -2
      ELSE IF( K.LT.0 ) THEN
         INFO = -3
      ELSE
         INCR = .TRUE.
         DECR = .TRUE.
         DO 10 I = 1, K - 1
            IF( INCR )
     $         INCR = INCR .AND. D( I ).LE.D( I+1 )
            IF( DECR )
     $         DECR = DECR .AND. D( I ).GE.D( I+1 )
   10    CONTINUE
         IF( SING .AND. K.GT.0 ) THEN
            IF( INCR )
     $         INCR = INCR .AND. ZERO.LE.D( 1 )
            IF( DECR )
     $         DECR = DECR .AND. D( K ).GE.ZERO
         END IF
         IF( .NOT.( INCR .OR. DECR ) )
     $      INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DDISNA', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( K.EQ.0 )
     $   RETURN
*
*     Compute reciprocal condition numbers
*
      IF( K.EQ.1 ) THEN
         SEP( 1 ) = DLAMCH( 'O' )
      ELSE
         OLDGAP = ABS( D( 2 )-D( 1 ) )
         SEP( 1 ) = OLDGAP
         DO 20 I = 2, K - 1
            NEWGAP = ABS( D( I+1 )-D( I ) )
            SEP( I ) = MIN( OLDGAP, NEWGAP )
            OLDGAP = NEWGAP
   20    CONTINUE
         SEP( K ) = OLDGAP
      END IF
      IF( SING ) THEN
         IF( ( LEFT .AND. M.GT.N ) .OR. ( RIGHT .AND. M.LT.N ) ) THEN
            IF( INCR )
     $         SEP( 1 ) = MIN( SEP( 1 ), D( 1 ) )
            IF( DECR )
     $         SEP( K ) = MIN( SEP( K ), D( K ) )
         END IF
      END IF
*
*     Ensure that reciprocal condition numbers are not less than
*     threshold, in order to limit the size of the error bound
*
      EPS = DLAMCH( 'E' )
      SAFMIN = DLAMCH( 'S' )
      ANORM = MAX( ABS( D( 1 ) ), ABS( D( K ) ) )
      IF( ANORM.EQ.ZERO ) THEN
         THRESH = EPS
      ELSE
         THRESH = MAX( EPS*ANORM, SAFMIN )
      END IF
      DO 30 I = 1, K
         SEP( I ) = MAX( SEP( I ), THRESH )
   30 CONTINUE
*
      RETURN
*
*     End of DDISNA
*
      END
*> \brief \b DGBBRD
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGBBRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbbrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbbrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbbrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q,
*                          LDQ, PT, LDPT, C, LDC, WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          VECT
*       INTEGER            INFO, KL, KU, LDAB, LDC, LDPT, LDQ, M, N, NCC
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * ), C( LDC, * ), D( * ), E( * ),
*      $                   PT( LDPT, * ), Q( LDQ, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGBBRD reduces a real general m-by-n band matrix A to upper
*> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
*>
*> The routine computes B, and optionally forms Q or P**T, or computes
*> Q**T*C for a given matrix C.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] VECT
*> \verbatim
*>          VECT is CHARACTER*1
*>          Specifies whether or not the matrices Q and P**T are to be
*>          formed.
*>          = 'N': do not form Q or P**T;
*>          = 'Q': form Q only;
*>          = 'P': form P**T only;
*>          = 'B': form both.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NCC
*> \verbatim
*>          NCC is INTEGER
*>          The number of columns of the matrix C.  NCC >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*>          KL is INTEGER
*>          The number of subdiagonals of the matrix A. KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*>          KU is INTEGER
*>          The number of superdiagonals of the matrix A. KU >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          On entry, the m-by-n band matrix A, stored in rows 1 to
*>          KL+KU+1. The j-th column of A is stored in the j-th column of
*>          the array AB as follows:
*>          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
*>          On exit, A is overwritten by values generated during the
*>          reduction.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array A. LDAB >= KL+KU+1.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (min(M,N))
*>          The diagonal elements of the bidiagonal matrix B.
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (min(M,N)-1)
*>          The superdiagonal elements of the bidiagonal matrix B.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ,M)
*>          If VECT = 'Q' or 'B', the m-by-m orthogonal matrix Q.
*>          If VECT = 'N' or 'P', the array Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q.
*>          LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] PT
*> \verbatim
*>          PT is DOUBLE PRECISION array, dimension (LDPT,N)
*>          If VECT = 'P' or 'B', the n-by-n orthogonal matrix P'.
*>          If VECT = 'N' or 'Q', the array PT is not referenced.
*> \endverbatim
*>
*> \param[in] LDPT
*> \verbatim
*>          LDPT is INTEGER
*>          The leading dimension of the array PT.
*>          LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,NCC)
*>          On entry, an m-by-ncc matrix C.
*>          On exit, C is overwritten by Q**T*C.
*>          C is not referenced if NCC = 0.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C.
*>          LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (2*max(M,N))
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gbbrd
*
*  =====================================================================
      SUBROUTINE DGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q,
     $                   LDQ, PT, LDPT, C, LDC, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          VECT
      INTEGER            INFO, KL, KU, LDAB, LDC, LDPT, LDQ, M, N, NCC
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * ), C( LDC, * ), D( * ), E( * ),
     $                   PT( LDPT, * ), Q( LDQ, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            WANTB, WANTC, WANTPT, WANTQ
      INTEGER            I, INCA, J, J1, J2, KB, KB1, KK, KLM, KLU1,
     $                   KUN, L, MINMN, ML, ML0, MN, MU, MU0, NR, NRT
      DOUBLE PRECISION   RA, RB, RC, RS
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARGV, DLARTG, DLARTV, DLASET, DROT,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      WANTB = LSAME( VECT, 'B' )
      WANTQ = LSAME( VECT, 'Q' ) .OR. WANTB
      WANTPT = LSAME( VECT, 'P' ) .OR. WANTB
      WANTC = NCC.GT.0
      KLU1 = KL + KU + 1
      INFO = 0
      IF( .NOT.WANTQ .AND.
     $    .NOT.WANTPT .AND.
     $    .NOT.LSAME( VECT, 'N' ) )
     $     THEN
         INFO = -1
      ELSE IF( M.LT.0 ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( NCC.LT.0 ) THEN
         INFO = -4
      ELSE IF( KL.LT.0 ) THEN
         INFO = -5
      ELSE IF( KU.LT.0 ) THEN
         INFO = -6
      ELSE IF( LDAB.LT.KLU1 ) THEN
         INFO = -8
      ELSE IF( LDQ.LT.1 .OR. WANTQ .AND. LDQ.LT.MAX( 1, M ) ) THEN
         INFO = -12
      ELSE IF( LDPT.LT.1 .OR. WANTPT .AND. LDPT.LT.MAX( 1, N ) ) THEN
         INFO = -14
      ELSE IF( LDC.LT.1 .OR. WANTC .AND. LDC.LT.MAX( 1, M ) ) THEN
         INFO = -16
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGBBRD', -INFO )
         RETURN
      END IF
*
*     Initialize Q and P**T to the unit matrix, if needed
*
      IF( WANTQ )
     $   CALL DLASET( 'Full', M, M, ZERO, ONE, Q, LDQ )
      IF( WANTPT )
     $   CALL DLASET( 'Full', N, N, ZERO, ONE, PT, LDPT )
*
*     Quick return if possible.
*
      IF( M.EQ.0 .OR. N.EQ.0 )
     $   RETURN
*
      MINMN = MIN( M, N )
*
      IF( KL+KU.GT.1 ) THEN
*
*        Reduce to upper bidiagonal form if KU > 0; if KU = 0, reduce
*        first to lower bidiagonal form and then transform to upper
*        bidiagonal
*
         IF( KU.GT.0 ) THEN
            ML0 = 1
            MU0 = 2
         ELSE
            ML0 = 2
            MU0 = 1
         END IF
*
*        Wherever possible, plane rotations are generated and applied in
*        vector operations of length NR over the index set J1:J2:KLU1.
*
*        The sines of the plane rotations are stored in WORK(1:max(m,n))
*        and the cosines in WORK(max(m,n)+1:2*max(m,n)).
*
         MN = MAX( M, N )
         KLM = MIN( M-1, KL )
         KUN = MIN( N-1, KU )
         KB = KLM + KUN
         KB1 = KB + 1
         INCA = KB1*LDAB
         NR = 0
         J1 = KLM + 2
         J2 = 1 - KUN
*
         DO 90 I = 1, MINMN
*
*           Reduce i-th column and i-th row of matrix to bidiagonal form
*
            ML = KLM + 1
            MU = KUN + 1
            DO 80 KK = 1, KB
               J1 = J1 + KB
               J2 = J2 + KB
*
*              generate plane rotations to annihilate nonzero elements
*              which have been created below the band
*
               IF( NR.GT.0 )
     $            CALL DLARGV( NR, AB( KLU1, J1-KLM-1 ), INCA,
     $                         WORK( J1 ), KB1, WORK( MN+J1 ), KB1 )
*
*              apply plane rotations from the left
*
               DO 10 L = 1, KB
                  IF( J2-KLM+L-1.GT.N ) THEN
                     NRT = NR - 1
                  ELSE
                     NRT = NR
                  END IF
                  IF( NRT.GT.0 )
     $               CALL DLARTV( NRT, AB( KLU1-L, J1-KLM+L-1 ),
     $                            INCA,
     $                            AB( KLU1-L+1, J1-KLM+L-1 ), INCA,
     $                            WORK( MN+J1 ), WORK( J1 ), KB1 )
   10          CONTINUE
*
               IF( ML.GT.ML0 ) THEN
                  IF( ML.LE.M-I+1 ) THEN
*
*                    generate plane rotation to annihilate a(i+ml-1,i)
*                    within the band, and apply rotation from the left
*
                     CALL DLARTG( AB( KU+ML-1, I ), AB( KU+ML, I ),
     $                            WORK( MN+I+ML-1 ), WORK( I+ML-1 ),
     $                            RA )
                     AB( KU+ML-1, I ) = RA
                     IF( I.LT.N )
     $                  CALL DROT( MIN( KU+ML-2, N-I ),
     $                             AB( KU+ML-2, I+1 ), LDAB-1,
     $                             AB( KU+ML-1, I+1 ), LDAB-1,
     $                             WORK( MN+I+ML-1 ), WORK( I+ML-1 ) )
                  END IF
                  NR = NR + 1
                  J1 = J1 - KB1
               END IF
*
               IF( WANTQ ) THEN
*
*                 accumulate product of plane rotations in Q
*
                  DO 20 J = J1, J2, KB1
                     CALL DROT( M, Q( 1, J-1 ), 1, Q( 1, J ), 1,
     $                          WORK( MN+J ), WORK( J ) )
   20             CONTINUE
               END IF
*
               IF( WANTC ) THEN
*
*                 apply plane rotations to C
*
                  DO 30 J = J1, J2, KB1
                     CALL DROT( NCC, C( J-1, 1 ), LDC, C( J, 1 ),
     $                          LDC,
     $                          WORK( MN+J ), WORK( J ) )
   30             CONTINUE
               END IF
*
               IF( J2+KUN.GT.N ) THEN
*
*                 adjust J2 to keep within the bounds of the matrix
*
                  NR = NR - 1
                  J2 = J2 - KB1
               END IF
*
               DO 40 J = J1, J2, KB1
*
*                 create nonzero element a(j-1,j+ku) above the band
*                 and store it in WORK(n+1:2*n)
*
                  WORK( J+KUN ) = WORK( J )*AB( 1, J+KUN )
                  AB( 1, J+KUN ) = WORK( MN+J )*AB( 1, J+KUN )
   40          CONTINUE
*
*              generate plane rotations to annihilate nonzero elements
*              which have been generated above the band
*
               IF( NR.GT.0 )
     $            CALL DLARGV( NR, AB( 1, J1+KUN-1 ), INCA,
     $                         WORK( J1+KUN ), KB1, WORK( MN+J1+KUN ),
     $                         KB1 )
*
*              apply plane rotations from the right
*
               DO 50 L = 1, KB
                  IF( J2+L-1.GT.M ) THEN
                     NRT = NR - 1
                  ELSE
                     NRT = NR
                  END IF
                  IF( NRT.GT.0 )
     $               CALL DLARTV( NRT, AB( L+1, J1+KUN-1 ), INCA,
     $                            AB( L, J1+KUN ), INCA,
     $                            WORK( MN+J1+KUN ), WORK( J1+KUN ),
     $                            KB1 )
   50          CONTINUE
*
               IF( ML.EQ.ML0 .AND. MU.GT.MU0 ) THEN
                  IF( MU.LE.N-I+1 ) THEN
*
*                    generate plane rotation to annihilate a(i,i+mu-1)
*                    within the band, and apply rotation from the right
*
                     CALL DLARTG( AB( KU-MU+3, I+MU-2 ),
     $                            AB( KU-MU+2, I+MU-1 ),
     $                            WORK( MN+I+MU-1 ), WORK( I+MU-1 ),
     $                            RA )
                     AB( KU-MU+3, I+MU-2 ) = RA
                     CALL DROT( MIN( KL+MU-2, M-I ),
     $                          AB( KU-MU+4, I+MU-2 ), 1,
     $                          AB( KU-MU+3, I+MU-1 ), 1,
     $                          WORK( MN+I+MU-1 ), WORK( I+MU-1 ) )
                  END IF
                  NR = NR + 1
                  J1 = J1 - KB1
               END IF
*
               IF( WANTPT ) THEN
*
*                 accumulate product of plane rotations in P**T
*
                  DO 60 J = J1, J2, KB1
                     CALL DROT( N, PT( J+KUN-1, 1 ), LDPT,
     $                          PT( J+KUN, 1 ), LDPT, WORK( MN+J+KUN ),
     $                          WORK( J+KUN ) )
   60             CONTINUE
               END IF
*
               IF( J2+KB.GT.M ) THEN
*
*                 adjust J2 to keep within the bounds of the matrix
*
                  NR = NR - 1
                  J2 = J2 - KB1
               END IF
*
               DO 70 J = J1, J2, KB1
*
*                 create nonzero element a(j+kl+ku,j+ku-1) below the
*                 band and store it in WORK(1:n)
*
                  WORK( J+KB ) = WORK( J+KUN )*AB( KLU1, J+KUN )
                  AB( KLU1, J+KUN ) = WORK( MN+J+KUN )*AB( KLU1, J+KUN )
   70          CONTINUE
*
               IF( ML.GT.ML0 ) THEN
                  ML = ML - 1
               ELSE
                  MU = MU - 1
               END IF
   80       CONTINUE
   90    CONTINUE
      END IF
*
      IF( KU.EQ.0 .AND. KL.GT.0 ) THEN
*
*        A has been reduced to lower bidiagonal form
*
*        Transform lower bidiagonal form to upper bidiagonal by applying
*        plane rotations from the left, storing diagonal elements in D
*        and off-diagonal elements in E
*
         DO 100 I = 1, MIN( M-1, N )
            CALL DLARTG( AB( 1, I ), AB( 2, I ), RC, RS, RA )
            D( I ) = RA
            IF( I.LT.N ) THEN
               E( I ) = RS*AB( 1, I+1 )
               AB( 1, I+1 ) = RC*AB( 1, I+1 )
            END IF
            IF( WANTQ )
     $         CALL DROT( M, Q( 1, I ), 1, Q( 1, I+1 ), 1, RC, RS )
            IF( WANTC )
     $         CALL DROT( NCC, C( I, 1 ), LDC, C( I+1, 1 ), LDC, RC,
     $                    RS )
  100    CONTINUE
         IF( M.LE.N )
     $      D( M ) = AB( 1, M )
      ELSE IF( KU.GT.0 ) THEN
*
*        A has been reduced to upper bidiagonal form
*
         IF( M.LT.N ) THEN
*
*           Annihilate a(m,m+1) by applying plane rotations from the
*           right, storing diagonal elements in D and off-diagonal
*           elements in E
*
            RB = AB( KU, M+1 )
            DO 110 I = M, 1, -1
               CALL DLARTG( AB( KU+1, I ), RB, RC, RS, RA )
               D( I ) = RA
               IF( I.GT.1 ) THEN
                  RB = -RS*AB( KU, I )
                  E( I-1 ) = RC*AB( KU, I )
               END IF
               IF( WANTPT )
     $            CALL DROT( N, PT( I, 1 ), LDPT, PT( M+1, 1 ), LDPT,
     $                       RC, RS )
  110       CONTINUE
         ELSE
*
*           Copy off-diagonal elements to E and diagonal elements to D
*
            DO 120 I = 1, MINMN - 1
               E( I ) = AB( KU, I+1 )
  120       CONTINUE
            DO 130 I = 1, MINMN
               D( I ) = AB( KU+1, I )
  130       CONTINUE
         END IF
      ELSE
*
*        A is diagonal. Set elements of E to zero and copy diagonal
*        elements to D.
*
         DO 140 I = 1, MINMN - 1
            E( I ) = ZERO
  140    CONTINUE
         DO 150 I = 1, MINMN
            D( I ) = AB( 1, I )
  150    CONTINUE
      END IF
      RETURN
*
*     End of DGBBRD
*
      END
*> \brief \b DGBCON
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGBCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbcon.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGBCON( NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND,
*                          WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          NORM
*       INTEGER            INFO, KL, KU, LDAB, N
*       DOUBLE PRECISION   ANORM, RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), IWORK( * )
*       DOUBLE PRECISION   AB( LDAB, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGBCON estimates the reciprocal of the condition number of a real
*> general band matrix A, in either the 1-norm or the infinity-norm,
*> using the LU factorization computed by DGBTRF.
*>
*> An estimate is obtained for norm(inv(A)), and the reciprocal of the
*> condition number is computed as
*>    RCOND = 1 / ( norm(A) * norm(inv(A)) ).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NORM
*> \verbatim
*>          NORM is CHARACTER*1
*>          Specifies whether the 1-norm condition number or the
*>          infinity-norm condition number is required:
*>          = '1' or 'O':  1-norm;
*>          = 'I':         Infinity-norm.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*>          KL is INTEGER
*>          The number of subdiagonals within the band of A.  KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*>          KU is INTEGER
*>          The number of superdiagonals within the band of A.  KU >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          Details of the LU factorization of the band matrix A, as
*>          computed by DGBTRF.  U is stored as an upper triangular band
*>          matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
*>          the multipliers used during the factorization are stored in
*>          rows KL+KU+2 to 2*KL+KU+1.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          The pivot indices; for 1 <= i <= N, row i of the matrix was
*>          interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*>          ANORM is DOUBLE PRECISION
*>          If NORM = '1' or 'O', the 1-norm of the original matrix A.
*>          If NORM = 'I', the infinity-norm of the original matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The reciprocal of the condition number of the matrix A,
*>          computed as RCOND = 1/(norm(A) * norm(inv(A))).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gbcon
*
*  =====================================================================
      SUBROUTINE DGBCON( NORM, N, KL, KU, AB, LDAB, IPIV, ANORM,
     $                   RCOND,
     $                   WORK, IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          NORM
      INTEGER            INFO, KL, KU, LDAB, N
      DOUBLE PRECISION   ANORM, RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), IWORK( * )
      DOUBLE PRECISION   AB( LDAB, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LNOTI, ONENRM
      CHARACTER          NORMIN
      INTEGER            IX, J, JP, KASE, KASE1, KD, LM
      DOUBLE PRECISION   AINVNM, SCALE, SMLNUM, T
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX
      DOUBLE PRECISION   DDOT, DLAMCH
      EXTERNAL           LSAME, IDAMAX, DDOT, DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DLACN2, DLATBS, DRSCL,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' )
      IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( KL.LT.0 ) THEN
         INFO = -3
      ELSE IF( KU.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDAB.LT.2*KL+KU+1 ) THEN
         INFO = -6
      ELSE IF( ANORM.LT.ZERO ) THEN
         INFO = -8
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGBCON', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      RCOND = ZERO
      IF( N.EQ.0 ) THEN
         RCOND = ONE
         RETURN
      ELSE IF( ANORM.EQ.ZERO ) THEN
         RETURN
      END IF
*
      SMLNUM = DLAMCH( 'Safe minimum' )
*
*     Estimate the norm of inv(A).
*
      AINVNM = ZERO
      NORMIN = 'N'
      IF( ONENRM ) THEN
         KASE1 = 1
      ELSE
         KASE1 = 2
      END IF
      KD = KL + KU + 1
      LNOTI = KL.GT.0
      KASE = 0
   10 CONTINUE
      CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
      IF( KASE.NE.0 ) THEN
         IF( KASE.EQ.KASE1 ) THEN
*
*           Multiply by inv(L).
*
            IF( LNOTI ) THEN
               DO 20 J = 1, N - 1
                  LM = MIN( KL, N-J )
                  JP = IPIV( J )
                  T = WORK( JP )
                  IF( JP.NE.J ) THEN
                     WORK( JP ) = WORK( J )
                     WORK( J ) = T
                  END IF
                  CALL DAXPY( LM, -T, AB( KD+1, J ), 1, WORK( J+1 ),
     $                        1 )
   20          CONTINUE
            END IF
*
*           Multiply by inv(U).
*
            CALL DLATBS( 'Upper', 'No transpose', 'Non-unit', NORMIN,
     $                   N,
     $                   KL+KU, AB, LDAB, WORK, SCALE, WORK( 2*N+1 ),
     $                   INFO )
         ELSE
*
*           Multiply by inv(U**T).
*
            CALL DLATBS( 'Upper', 'Transpose', 'Non-unit', NORMIN, N,
     $                   KL+KU, AB, LDAB, WORK, SCALE, WORK( 2*N+1 ),
     $                   INFO )
*
*           Multiply by inv(L**T).
*
            IF( LNOTI ) THEN
               DO 30 J = N - 1, 1, -1
                  LM = MIN( KL, N-J )
                  WORK( J ) = WORK( J ) - DDOT( LM, AB( KD+1, J ), 1,
     $                        WORK( J+1 ), 1 )
                  JP = IPIV( J )
                  IF( JP.NE.J ) THEN
                     T = WORK( JP )
                     WORK( JP ) = WORK( J )
                     WORK( J ) = T
                  END IF
   30          CONTINUE
            END IF
         END IF
*
*        Divide X by 1/SCALE if doing so will not cause overflow.
*
         NORMIN = 'Y'
         IF( SCALE.NE.ONE ) THEN
            IX = IDAMAX( N, WORK, 1 )
            IF( SCALE.LT.ABS( WORK( IX ) )*SMLNUM .OR. SCALE.EQ.ZERO )
     $         GO TO 40
            CALL DRSCL( N, SCALE, WORK, 1 )
         END IF
         GO TO 10
      END IF
*
*     Compute the estimate of the reciprocal condition number.
*
      IF( AINVNM.NE.ZERO )
     $   RCOND = ( ONE / AINVNM ) / ANORM
*
   40 CONTINUE
      RETURN
*
*     End of DGBCON
*
      END
*> \brief \b DGBEQU
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGBEQU + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbequ.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbequ.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbequ.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGBEQU( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
*                          AMAX, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, KL, KU, LDAB, M, N
*       DOUBLE PRECISION   AMAX, COLCND, ROWCND
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * ), C( * ), R( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGBEQU computes row and column scalings intended to equilibrate an
*> M-by-N band matrix A and reduce its condition number.  R returns the
*> row scale factors and C the column scale factors, chosen to try to
*> make the largest element in each row and column of the matrix B with
*> elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.
*>
*> R(i) and C(j) are restricted to be between SMLNUM = smallest safe
*> number and BIGNUM = largest safe number.  Use of these scaling
*> factors is not guaranteed to reduce the condition number of A but
*> works well in practice.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*>          KL is INTEGER
*>          The number of subdiagonals within the band of A.  KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*>          KU is INTEGER
*>          The number of superdiagonals within the band of A.  KU >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          The band matrix A, stored in rows 1 to KL+KU+1.  The j-th
*>          column of A is stored in the j-th column of the array AB as
*>          follows:
*>          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KL+KU+1.
*> \endverbatim
*>
*> \param[out] R
*> \verbatim
*>          R is DOUBLE PRECISION array, dimension (M)
*>          If INFO = 0, or INFO > M, R contains the row scale factors
*>          for A.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, C contains the column scale factors for A.
*> \endverbatim
*>
*> \param[out] ROWCND
*> \verbatim
*>          ROWCND is DOUBLE PRECISION
*>          If INFO = 0 or INFO > M, ROWCND contains the ratio of the
*>          smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
*>          AMAX is neither too large nor too small, it is not worth
*>          scaling by R.
*> \endverbatim
*>
*> \param[out] COLCND
*> \verbatim
*>          COLCND is DOUBLE PRECISION
*>          If INFO = 0, COLCND contains the ratio of the smallest
*>          C(i) to the largest C(i).  If COLCND >= 0.1, it is not
*>          worth scaling by C.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*>          AMAX is DOUBLE PRECISION
*>          Absolute value of largest matrix element.  If AMAX is very
*>          close to overflow or very close to underflow, the matrix
*>          should be scaled.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, and i is
*>                <= M:  the i-th row of A is exactly zero
*>                >  M:  the (i-M)-th column of A is exactly zero
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gbequ
*
*  =====================================================================
      SUBROUTINE DGBEQU( M, N, KL, KU, AB, LDAB, R, C, ROWCND,
     $                   COLCND,
     $                   AMAX, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, KL, KU, LDAB, M, N
      DOUBLE PRECISION   AMAX, COLCND, ROWCND
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * ), C( * ), R( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J, KD
      DOUBLE PRECISION   BIGNUM, RCMAX, RCMIN, SMLNUM
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( KL.LT.0 ) THEN
         INFO = -3
      ELSE IF( KU.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDAB.LT.KL+KU+1 ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGBEQU', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 ) THEN
         ROWCND = ONE
         COLCND = ONE
         AMAX = ZERO
         RETURN
      END IF
*
*     Get machine constants.
*
      SMLNUM = DLAMCH( 'S' )
      BIGNUM = ONE / SMLNUM
*
*     Compute row scale factors.
*
      DO 10 I = 1, M
         R( I ) = ZERO
   10 CONTINUE
*
*     Find the maximum element in each row.
*
      KD = KU + 1
      DO 30 J = 1, N
         DO 20 I = MAX( J-KU, 1 ), MIN( J+KL, M )
            R( I ) = MAX( R( I ), ABS( AB( KD+I-J, J ) ) )
   20    CONTINUE
   30 CONTINUE
*
*     Find the maximum and minimum scale factors.
*
      RCMIN = BIGNUM
      RCMAX = ZERO
      DO 40 I = 1, M
         RCMAX = MAX( RCMAX, R( I ) )
         RCMIN = MIN( RCMIN, R( I ) )
   40 CONTINUE
      AMAX = RCMAX
*
      IF( RCMIN.EQ.ZERO ) THEN
*
*        Find the first zero scale factor and return an error code.
*
         DO 50 I = 1, M
            IF( R( I ).EQ.ZERO ) THEN
               INFO = I
               RETURN
            END IF
   50    CONTINUE
      ELSE
*
*        Invert the scale factors.
*
         DO 60 I = 1, M
            R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
   60    CONTINUE
*
*        Compute ROWCND = min(R(I)) / max(R(I))
*
         ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
      END IF
*
*     Compute column scale factors
*
      DO 70 J = 1, N
         C( J ) = ZERO
   70 CONTINUE
*
*     Find the maximum element in each column,
*     assuming the row scaling computed above.
*
      KD = KU + 1
      DO 90 J = 1, N
         DO 80 I = MAX( J-KU, 1 ), MIN( J+KL, M )
            C( J ) = MAX( C( J ), ABS( AB( KD+I-J, J ) )*R( I ) )
   80    CONTINUE
   90 CONTINUE
*
*     Find the maximum and minimum scale factors.
*
      RCMIN = BIGNUM
      RCMAX = ZERO
      DO 100 J = 1, N
         RCMIN = MIN( RCMIN, C( J ) )
         RCMAX = MAX( RCMAX, C( J ) )
  100 CONTINUE
*
      IF( RCMIN.EQ.ZERO ) THEN
*
*        Find the first zero scale factor and return an error code.
*
         DO 110 J = 1, N
            IF( C( J ).EQ.ZERO ) THEN
               INFO = M + J
               RETURN
            END IF
  110    CONTINUE
      ELSE
*
*        Invert the scale factors.
*
         DO 120 J = 1, N
            C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
  120    CONTINUE
*
*        Compute COLCND = min(C(J)) / max(C(J))
*
         COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
      END IF
*
      RETURN
*
*     End of DGBEQU
*
      END
*> \brief \b DGBEQUB
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGBEQUB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbequb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbequb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbequb.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGBEQUB( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
*                           AMAX, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, KL, KU, LDAB, M, N
*       DOUBLE PRECISION   AMAX, COLCND, ROWCND
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * ), C( * ), R( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGBEQUB computes row and column scalings intended to equilibrate an
*> M-by-N matrix A and reduce its condition number.  R returns the row
*> scale factors and C the column scale factors, chosen to try to make
*> the largest element in each row and column of the matrix B with
*> elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
*> the radix.
*>
*> R(i) and C(j) are restricted to be a power of the radix between
*> SMLNUM = smallest safe number and BIGNUM = largest safe number.  Use
*> of these scaling factors is not guaranteed to reduce the condition
*> number of A but works well in practice.
*>
*> This routine differs from DGEEQU by restricting the scaling factors
*> to a power of the radix.  Barring over- and underflow, scaling by
*> these factors introduces no additional rounding errors.  However, the
*> scaled entries' magnitudes are no longer approximately 1 but lie
*> between sqrt(radix) and 1/sqrt(radix).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*>          KL is INTEGER
*>          The number of subdiagonals within the band of A.  KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*>          KU is INTEGER
*>          The number of superdiagonals within the band of A.  KU >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
*>          The j-th column of A is stored in the j-th column of the
*>          array AB as follows:
*>          AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array A.  LDAB >= max(1,M).
*> \endverbatim
*>
*> \param[out] R
*> \verbatim
*>          R is DOUBLE PRECISION array, dimension (M)
*>          If INFO = 0 or INFO > M, R contains the row scale factors
*>          for A.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0,  C contains the column scale factors for A.
*> \endverbatim
*>
*> \param[out] ROWCND
*> \verbatim
*>          ROWCND is DOUBLE PRECISION
*>          If INFO = 0 or INFO > M, ROWCND contains the ratio of the
*>          smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
*>          AMAX is neither too large nor too small, it is not worth
*>          scaling by R.
*> \endverbatim
*>
*> \param[out] COLCND
*> \verbatim
*>          COLCND is DOUBLE PRECISION
*>          If INFO = 0, COLCND contains the ratio of the smallest
*>          C(i) to the largest C(i).  If COLCND >= 0.1, it is not
*>          worth scaling by C.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*>          AMAX is DOUBLE PRECISION
*>          Absolute value of largest matrix element.  If AMAX is very
*>          close to overflow or very close to underflow, the matrix
*>          should be scaled.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i,  and i is
*>                <= M:  the i-th row of A is exactly zero
*>                >  M:  the (i-M)-th column of A is exactly zero
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gbequb
*
*  =====================================================================
      SUBROUTINE DGBEQUB( M, N, KL, KU, AB, LDAB, R, C, ROWCND,
     $                    COLCND,
     $                    AMAX, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, KL, KU, LDAB, M, N
      DOUBLE PRECISION   AMAX, COLCND, ROWCND
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * ), C( * ), R( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J, KD
      DOUBLE PRECISION   BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX, LOGRDX
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, LOG
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( KL.LT.0 ) THEN
         INFO = -3
      ELSE IF( KU.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDAB.LT.KL+KU+1 ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGBEQUB', -INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( M.EQ.0 .OR. N.EQ.0 ) THEN
         ROWCND = ONE
         COLCND = ONE
         AMAX = ZERO
         RETURN
      END IF
*
*     Get machine constants.  Assume SMLNUM is a power of the radix.
*
      SMLNUM = DLAMCH( 'S' )
      BIGNUM = ONE / SMLNUM
      RADIX = DLAMCH( 'B' )
      LOGRDX = LOG(RADIX)
*
*     Compute row scale factors.
*
      DO 10 I = 1, M
         R( I ) = ZERO
   10 CONTINUE
*
*     Find the maximum element in each row.
*
      KD = KU + 1
      DO 30 J = 1, N
         DO 20 I = MAX( J-KU, 1 ), MIN( J+KL, M )
            R( I ) = MAX( R( I ), ABS( AB( KD+I-J, J ) ) )
   20    CONTINUE
   30 CONTINUE
      DO I = 1, M
         IF( R( I ).GT.ZERO ) THEN
            R( I ) = RADIX**INT( LOG( R( I ) ) / LOGRDX )
         END IF
      END DO
*
*     Find the maximum and minimum scale factors.
*
      RCMIN = BIGNUM
      RCMAX = ZERO
      DO 40 I = 1, M
         RCMAX = MAX( RCMAX, R( I ) )
         RCMIN = MIN( RCMIN, R( I ) )
   40 CONTINUE
      AMAX = RCMAX
*
      IF( RCMIN.EQ.ZERO ) THEN
*
*        Find the first zero scale factor and return an error code.
*
         DO 50 I = 1, M
            IF( R( I ).EQ.ZERO ) THEN
               INFO = I
               RETURN
            END IF
   50    CONTINUE
      ELSE
*
*        Invert the scale factors.
*
         DO 60 I = 1, M
            R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
   60    CONTINUE
*
*        Compute ROWCND = min(R(I)) / max(R(I)).
*
         ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
      END IF
*
*     Compute column scale factors.
*
      DO 70 J = 1, N
         C( J ) = ZERO
   70 CONTINUE
*
*     Find the maximum element in each column,
*     assuming the row scaling computed above.
*
      DO 90 J = 1, N
         DO 80 I = MAX( J-KU, 1 ), MIN( J+KL, M )
            C( J ) = MAX( C( J ), ABS( AB( KD+I-J, J ) )*R( I ) )
   80    CONTINUE
         IF( C( J ).GT.ZERO ) THEN
            C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX )
         END IF
   90 CONTINUE
*
*     Find the maximum and minimum scale factors.
*
      RCMIN = BIGNUM
      RCMAX = ZERO
      DO 100 J = 1, N
         RCMIN = MIN( RCMIN, C( J ) )
         RCMAX = MAX( RCMAX, C( J ) )
  100 CONTINUE
*
      IF( RCMIN.EQ.ZERO ) THEN
*
*        Find the first zero scale factor and return an error code.
*
         DO 110 J = 1, N
            IF( C( J ).EQ.ZERO ) THEN
               INFO = M + J
               RETURN
            END IF
  110    CONTINUE
      ELSE
*
*        Invert the scale factors.
*
         DO 120 J = 1, N
            C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
  120    CONTINUE
*
*        Compute COLCND = min(C(J)) / max(C(J)).
*
         COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
      END IF
*
      RETURN
*
*     End of DGBEQUB
*
      END
*> \brief \b DGBRFS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGBRFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbrfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbrfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbrfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
*                          IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK,
*                          INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANS
*       INTEGER            INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), IWORK( * )
*       DOUBLE PRECISION   AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
*      $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGBRFS improves the computed solution to a system of linear
*> equations when the coefficient matrix is banded, and provides
*> error bounds and backward error estimates for the solution.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the form of the system of equations:
*>          = 'N':  A * X = B     (No transpose)
*>          = 'T':  A**T * X = B  (Transpose)
*>          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*>          KL is INTEGER
*>          The number of subdiagonals within the band of A.  KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*>          KU is INTEGER
*>          The number of superdiagonals within the band of A.  KU >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices B and X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          The original band matrix A, stored in rows 1 to KL+KU+1.
*>          The j-th column of A is stored in the j-th column of the
*>          array AB as follows:
*>          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KL+KU+1.
*> \endverbatim
*>
*> \param[in] AFB
*> \verbatim
*>          AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
*>          Details of the LU factorization of the band matrix A, as
*>          computed by DGBTRF.  U is stored as an upper triangular band
*>          matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
*>          the multipliers used during the factorization are stored in
*>          rows KL+KU+2 to 2*KL+KU+1.
*> \endverbatim
*>
*> \param[in] LDAFB
*> \verbatim
*>          LDAFB is INTEGER
*>          The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          The pivot indices from DGBTRF; for 1<=i<=N, row i of the
*>          matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          On entry, the solution matrix X, as computed by DGBTRS.
*>          On exit, the improved solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The estimated forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).  The estimate is as reliable as
*>          the estimate for RCOND, and is almost always a slight
*>          overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in
*>          any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  ITMAX is the maximum number of steps of iterative refinement.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gbrfs
*
*  =====================================================================
      SUBROUTINE DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
     $                   LDAFB,
     $                   IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS
      INTEGER            INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), IWORK( * )
      DOUBLE PRECISION   AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
     $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            ITMAX
      PARAMETER          ( ITMAX = 5 )
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
      DOUBLE PRECISION   TWO
      PARAMETER          ( TWO = 2.0D+0 )
      DOUBLE PRECISION   THREE
      PARAMETER          ( THREE = 3.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOTRAN
      CHARACTER          TRANST
      INTEGER            COUNT, I, J, K, KASE, KK, NZ
      DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DGBMV, DGBTRS, DLACN2,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NOTRAN = LSAME( TRANS, 'N' )
      IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
     $    LSAME( TRANS, 'C' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( KL.LT.0 ) THEN
         INFO = -3
      ELSE IF( KU.LT.0 ) THEN
         INFO = -4
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDAB.LT.KL+KU+1 ) THEN
         INFO = -7
      ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
         INFO = -9
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -12
      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
         INFO = -14
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGBRFS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
         DO 10 J = 1, NRHS
            FERR( J ) = ZERO
            BERR( J ) = ZERO
   10    CONTINUE
         RETURN
      END IF
*
      IF( NOTRAN ) THEN
         TRANST = 'T'
      ELSE
         TRANST = 'N'
      END IF
*
*     NZ = maximum number of nonzero elements in each row of A, plus 1
*
      NZ = MIN( KL+KU+2, N+1 )
      EPS = DLAMCH( 'Epsilon' )
      SAFMIN = DLAMCH( 'Safe minimum' )
      SAFE1 = NZ*SAFMIN
      SAFE2 = SAFE1 / EPS
*
*     Do for each right hand side
*
      DO 140 J = 1, NRHS
*
         COUNT = 1
         LSTRES = THREE
   20    CONTINUE
*
*        Loop until stopping criterion is satisfied.
*
*        Compute residual R = B - op(A) * X,
*        where op(A) = A, A**T, or A**H, depending on TRANS.
*
         CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
         CALL DGBMV( TRANS, N, N, KL, KU, -ONE, AB, LDAB, X( 1, J ),
     $               1,
     $               ONE, WORK( N+1 ), 1 )
*
*        Compute componentwise relative backward error from formula
*
*        max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
*
*        where abs(Z) is the componentwise absolute value of the matrix
*        or vector Z.  If the i-th component of the denominator is less
*        than SAFE2, then SAFE1 is added to the i-th components of the
*        numerator and denominator before dividing.
*
         DO 30 I = 1, N
            WORK( I ) = ABS( B( I, J ) )
   30    CONTINUE
*
*        Compute abs(op(A))*abs(X) + abs(B).
*
         IF( NOTRAN ) THEN
            DO 50 K = 1, N
               KK = KU + 1 - K
               XK = ABS( X( K, J ) )
               DO 40 I = MAX( 1, K-KU ), MIN( N, K+KL )
                  WORK( I ) = WORK( I ) + ABS( AB( KK+I, K ) )*XK
   40          CONTINUE
   50       CONTINUE
         ELSE
            DO 70 K = 1, N
               S = ZERO
               KK = KU + 1 - K
               DO 60 I = MAX( 1, K-KU ), MIN( N, K+KL )
                  S = S + ABS( AB( KK+I, K ) )*ABS( X( I, J ) )
   60          CONTINUE
               WORK( K ) = WORK( K ) + S
   70       CONTINUE
         END IF
         S = ZERO
         DO 80 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
            ELSE
               S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
     $             ( WORK( I )+SAFE1 ) )
            END IF
   80    CONTINUE
         BERR( J ) = S
*
*        Test stopping criterion. Continue iterating if
*           1) The residual BERR(J) is larger than machine epsilon, and
*           2) BERR(J) decreased by at least a factor of 2 during the
*              last iteration, and
*           3) At most ITMAX iterations tried.
*
         IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
     $       COUNT.LE.ITMAX ) THEN
*
*           Update solution and try again.
*
            CALL DGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV,
     $                   WORK( N+1 ), N, INFO )
            CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
            LSTRES = BERR( J )
            COUNT = COUNT + 1
            GO TO 20
         END IF
*
*        Bound error from formula
*
*        norm(X - XTRUE) / norm(X) .le. FERR =
*        norm( abs(inv(op(A)))*
*           ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
*
*        where
*          norm(Z) is the magnitude of the largest component of Z
*          inv(op(A)) is the inverse of op(A)
*          abs(Z) is the componentwise absolute value of the matrix or
*             vector Z
*          NZ is the maximum number of nonzeros in any row of A, plus 1
*          EPS is machine epsilon
*
*        The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
*        is incremented by SAFE1 if the i-th component of
*        abs(op(A))*abs(X) + abs(B) is less than SAFE2.
*
*        Use DLACN2 to estimate the infinity-norm of the matrix
*           inv(op(A)) * diag(W),
*        where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
*
         DO 90 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
            ELSE
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
            END IF
   90    CONTINUE
*
         KASE = 0
  100    CONTINUE
         CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
     $                FERR( J ),
     $                KASE, ISAVE )
         IF( KASE.NE.0 ) THEN
            IF( KASE.EQ.1 ) THEN
*
*              Multiply by diag(W)*inv(op(A)**T).
*
               CALL DGBTRS( TRANST, N, KL, KU, 1, AFB, LDAFB, IPIV,
     $                      WORK( N+1 ), N, INFO )
               DO 110 I = 1, N
                  WORK( N+I ) = WORK( N+I )*WORK( I )
  110          CONTINUE
            ELSE
*
*              Multiply by inv(op(A))*diag(W).
*
               DO 120 I = 1, N
                  WORK( N+I ) = WORK( N+I )*WORK( I )
  120          CONTINUE
               CALL DGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV,
     $                      WORK( N+1 ), N, INFO )
            END IF
            GO TO 100
         END IF
*
*        Normalize error.
*
         LSTRES = ZERO
         DO 130 I = 1, N
            LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
  130    CONTINUE
         IF( LSTRES.NE.ZERO )
     $      FERR( J ) = FERR( J ) / LSTRES
*
  140 CONTINUE
*
      RETURN
*
*     End of DGBRFS
*
      END
*> \brief <b> DGBSV computes the solution to system of linear equations A * X = B for GB matrices</b> (simple driver)
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGBSV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbsv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbsv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbsv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGBSV( N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, KL, KU, LDAB, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGBSV computes the solution to a real system of linear equations
*> A * X = B, where A is a band matrix of order N with KL subdiagonals
*> and KU superdiagonals, and X and B are N-by-NRHS matrices.
*>
*> The LU decomposition with partial pivoting and row interchanges is
*> used to factor A as A = L * U, where L is a product of permutation
*> and unit lower triangular matrices with KL subdiagonals, and U is
*> upper triangular with KL+KU superdiagonals.  The factored form of A
*> is then used to solve the system of equations A * X = B.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of linear equations, i.e., the order of the
*>          matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*>          KL is INTEGER
*>          The number of subdiagonals within the band of A.  KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*>          KU is INTEGER
*>          The number of superdiagonals within the band of A.  KU >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          On entry, the matrix A in band storage, in rows KL+1 to
*>          2*KL+KU+1; rows 1 to KL of the array need not be set.
*>          The j-th column of A is stored in the j-th column of the
*>          array AB as follows:
*>          AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL)
*>          On exit, details of the factorization: U is stored as an
*>          upper triangular band matrix with KL+KU superdiagonals in
*>          rows 1 to KL+KU+1, and the multipliers used during the
*>          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
*>          See below for further details.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          The pivot indices that define the permutation matrix P;
*>          row i of the matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the N-by-NRHS right hand side matrix B.
*>          On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, U(i,i) is exactly zero.  The factorization
*>                has been completed, but the factor U is exactly
*>                singular, and the solution has not been computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gbsv
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The band storage scheme is illustrated by the following example, when
*>  M = N = 6, KL = 2, KU = 1:
*>
*>  On entry:                       On exit:
*>
*>      *    *    *    +    +    +       *    *    *   u14  u25  u36
*>      *    *    +    +    +    +       *    *   u13  u24  u35  u46
*>      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
*>     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
*>     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
*>     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *
*>
*>  Array elements marked * are not used by the routine; elements marked
*>  + need not be set on entry, but are required by the routine to store
*>  elements of U because of fill-in resulting from the row interchanges.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGBSV( N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB,
     $                  INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, KL, KU, LDAB, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. External Subroutines ..
      EXTERNAL           DGBTRF, DGBTRS, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF( N.LT.0 ) THEN
         INFO = -1
      ELSE IF( KL.LT.0 ) THEN
         INFO = -2
      ELSE IF( KU.LT.0 ) THEN
         INFO = -3
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDAB.LT.2*KL+KU+1 ) THEN
         INFO = -6
      ELSE IF( LDB.LT.MAX( N, 1 ) ) THEN
         INFO = -9
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGBSV ', -INFO )
         RETURN
      END IF
*
*     Compute the LU factorization of the band matrix A.
*
      CALL DGBTRF( N, N, KL, KU, AB, LDAB, IPIV, INFO )
      IF( INFO.EQ.0 ) THEN
*
*        Solve the system A*X = B, overwriting B with X.
*
         CALL DGBTRS( 'No transpose', N, KL, KU, NRHS, AB, LDAB,
     $                IPIV,
     $                B, LDB, INFO )
      END IF
      RETURN
*
*     End of DGBSV
*
      END
*> \brief <b> DGBSVX computes the solution to system of linear equations A * X = B for GB matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGBSVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbsvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbsvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbsvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
*                          LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
*                          RCOND, FERR, BERR, WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          EQUED, FACT, TRANS
*       INTEGER            INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
*       DOUBLE PRECISION   RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), IWORK( * )
*       DOUBLE PRECISION   AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
*      $                   BERR( * ), C( * ), FERR( * ), R( * ),
*      $                   WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGBSVX uses the LU factorization to compute the solution to a real
*> system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
*> where A is a band matrix of order N with KL subdiagonals and KU
*> superdiagonals, and X and B are N-by-NRHS matrices.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
*  =================
*>
*> \verbatim
*>
*> The following steps are performed by this subroutine:
*>
*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
*>    the system:
*>       TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
*>       TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
*>       TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
*>    Whether or not the system will be equilibrated depends on the
*>    scaling of the matrix A, but if equilibration is used, A is
*>    overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
*>    or diag(C)*B (if TRANS = 'T' or 'C').
*>
*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
*>    matrix A (after equilibration if FACT = 'E') as
*>       A = L * U,
*>    where L is a product of permutation and unit lower triangular
*>    matrices with KL subdiagonals, and U is upper triangular with
*>    KL+KU superdiagonals.
*>
*> 3. If some U(i,i)=0, so that U is exactly singular, then the routine
*>    returns with INFO = i. Otherwise, the factored form of A is used
*>    to estimate the condition number of the matrix A.  If the
*>    reciprocal of the condition number is less than machine precision,
*>    INFO = N+1 is returned as a warning, but the routine still goes on
*>    to solve for X and compute error bounds as described below.
*>
*> 4. The system of equations is solved for X using the factored form
*>    of A.
*>
*> 5. Iterative refinement is applied to improve the computed solution
*>    matrix and calculate error bounds and backward error estimates
*>    for it.
*>
*> 6. If equilibration was used, the matrix X is premultiplied by
*>    diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
*>    that it solves the original system before equilibration.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] FACT
*> \verbatim
*>          FACT is CHARACTER*1
*>          Specifies whether or not the factored form of the matrix A is
*>          supplied on entry, and if not, whether the matrix A should be
*>          equilibrated before it is factored.
*>          = 'F':  On entry, AFB and IPIV contain the factored form of
*>                  A.  If EQUED is not 'N', the matrix A has been
*>                  equilibrated with scaling factors given by R and C.
*>                  AB, AFB, and IPIV are not modified.
*>          = 'N':  The matrix A will be copied to AFB and factored.
*>          = 'E':  The matrix A will be equilibrated if necessary, then
*>                  copied to AFB and factored.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the form of the system of equations.
*>          = 'N':  A * X = B     (No transpose)
*>          = 'T':  A**T * X = B  (Transpose)
*>          = 'C':  A**H * X = B  (Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of linear equations, i.e., the order of the
*>          matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*>          KL is INTEGER
*>          The number of subdiagonals within the band of A.  KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*>          KU is INTEGER
*>          The number of superdiagonals within the band of A.  KU >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices B and X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
*>          The j-th column of A is stored in the j-th column of the
*>          array AB as follows:
*>          AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
*>
*>          If FACT = 'F' and EQUED is not 'N', then A must have been
*>          equilibrated by the scaling factors in R and/or C.  AB is not
*>          modified if FACT = 'F' or 'N', or if FACT = 'E' and
*>          EQUED = 'N' on exit.
*>
*>          On exit, if EQUED .ne. 'N', A is scaled as follows:
*>          EQUED = 'R':  A := diag(R) * A
*>          EQUED = 'C':  A := A * diag(C)
*>          EQUED = 'B':  A := diag(R) * A * diag(C).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KL+KU+1.
*> \endverbatim
*>
*> \param[in,out] AFB
*> \verbatim
*>          AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
*>          If FACT = 'F', then AFB is an input argument and on entry
*>          contains details of the LU factorization of the band matrix
*>          A, as computed by DGBTRF.  U is stored as an upper triangular
*>          band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
*>          and the multipliers used during the factorization are stored
*>          in rows KL+KU+2 to 2*KL+KU+1.  If EQUED .ne. 'N', then AFB is
*>          the factored form of the equilibrated matrix A.
*>
*>          If FACT = 'N', then AFB is an output argument and on exit
*>          returns details of the LU factorization of A.
*>
*>          If FACT = 'E', then AFB is an output argument and on exit
*>          returns details of the LU factorization of the equilibrated
*>          matrix A (see the description of AB for the form of the
*>          equilibrated matrix).
*> \endverbatim
*>
*> \param[in] LDAFB
*> \verbatim
*>          LDAFB is INTEGER
*>          The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.
*> \endverbatim
*>
*> \param[in,out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          If FACT = 'F', then IPIV is an input argument and on entry
*>          contains the pivot indices from the factorization A = L*U
*>          as computed by DGBTRF; row i of the matrix was interchanged
*>          with row IPIV(i).
*>
*>          If FACT = 'N', then IPIV is an output argument and on exit
*>          contains the pivot indices from the factorization A = L*U
*>          of the original matrix A.
*>
*>          If FACT = 'E', then IPIV is an output argument and on exit
*>          contains the pivot indices from the factorization A = L*U
*>          of the equilibrated matrix A.
*> \endverbatim
*>
*> \param[in,out] EQUED
*> \verbatim
*>          EQUED is CHARACTER*1
*>          Specifies the form of equilibration that was done.
*>          = 'N':  No equilibration (always true if FACT = 'N').
*>          = 'R':  Row equilibration, i.e., A has been premultiplied by
*>                  diag(R).
*>          = 'C':  Column equilibration, i.e., A has been postmultiplied
*>                  by diag(C).
*>          = 'B':  Both row and column equilibration, i.e., A has been
*>                  replaced by diag(R) * A * diag(C).
*>          EQUED is an input argument if FACT = 'F'; otherwise, it is an
*>          output argument.
*> \endverbatim
*>
*> \param[in,out] R
*> \verbatim
*>          R is DOUBLE PRECISION array, dimension (N)
*>          The row scale factors for A.  If EQUED = 'R' or 'B', A is
*>          multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
*>          is not accessed.  R is an input argument if FACT = 'F';
*>          otherwise, R is an output argument.  If FACT = 'F' and
*>          EQUED = 'R' or 'B', each element of R must be positive.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (N)
*>          The column scale factors for A.  If EQUED = 'C' or 'B', A is
*>          multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
*>          is not accessed.  C is an input argument if FACT = 'F';
*>          otherwise, C is an output argument.  If FACT = 'F' and
*>          EQUED = 'C' or 'B', each element of C must be positive.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the right hand side matrix B.
*>          On exit,
*>          if EQUED = 'N', B is not modified;
*>          if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
*>          diag(R)*B;
*>          if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
*>          overwritten by diag(C)*B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
*>          to the original system of equations.  Note that A and B are
*>          modified on exit if EQUED .ne. 'N', and the solution to the
*>          equilibrated system is inv(diag(C))*X if TRANS = 'N' and
*>          EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
*>          and EQUED = 'R' or 'B'.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The estimate of the reciprocal condition number of the matrix
*>          A after equilibration (if done).  If RCOND is less than the
*>          machine precision (in particular, if RCOND = 0), the matrix
*>          is singular to working precision.  This condition is
*>          indicated by a return code of INFO > 0.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The estimated forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).  The estimate is as reliable as
*>          the estimate for RCOND, and is almost always a slight
*>          overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in
*>          any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,3*N))
*>          On exit, WORK(1) contains the reciprocal pivot growth
*>          factor norm(A)/norm(U). The "max absolute element" norm is
*>          used. If WORK(1) is much less than 1, then the stability
*>          of the LU factorization of the (equilibrated) matrix A
*>          could be poor. This also means that the solution X, condition
*>          estimator RCOND, and forward error bound FERR could be
*>          unreliable. If factorization fails with 0<INFO<=N, then
*>          WORK(1) contains the reciprocal pivot growth factor for the
*>          leading INFO columns of A.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, and i is
*>                <= N:  U(i,i) is exactly zero.  The factorization
*>                       has been completed, but the factor U is exactly
*>                       singular, so the solution and error bounds
*>                       could not be computed. RCOND = 0 is returned.
*>                = N+1: U is nonsingular, but RCOND is less than machine
*>                       precision, meaning that the matrix is singular
*>                       to working precision.  Nevertheless, the
*>                       solution and error bounds are computed because
*>                       there are a number of situations where the
*>                       computed solution can be more accurate than the
*>                       value of RCOND would suggest.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gbsvx
*
*  =====================================================================
      SUBROUTINE DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
     $                   LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
     $                   RCOND, FERR, BERR, WORK, IWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          EQUED, FACT, TRANS
      INTEGER            INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), IWORK( * )
      DOUBLE PRECISION   AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
     $                   BERR( * ), C( * ), FERR( * ), R( * ),
     $                   WORK( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
      CHARACTER          NORM
      INTEGER            I, INFEQU, J, J1, J2
      DOUBLE PRECISION   AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
     $                   ROWCND, RPVGRW, SMLNUM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANGB, DLANTB
      EXTERNAL           LSAME, DLAMCH, DLANGB, DLANTB
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGBCON, DGBEQU, DGBRFS, DGBTRF,
     $                   DGBTRS,
     $                   DLACPY, DLAQGB, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      NOFACT = LSAME( FACT, 'N' )
      EQUIL = LSAME( FACT, 'E' )
      NOTRAN = LSAME( TRANS, 'N' )
      IF( NOFACT .OR. EQUIL ) THEN
         EQUED = 'N'
         ROWEQU = .FALSE.
         COLEQU = .FALSE.
      ELSE
         ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
         COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
         SMLNUM = DLAMCH( 'Safe minimum' )
         BIGNUM = ONE / SMLNUM
      END IF
*
*     Test the input parameters.
*
      IF( .NOT.NOFACT .AND.
     $    .NOT.EQUIL .AND.
     $    .NOT.LSAME( FACT, 'F' ) )
     $     THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
     $         LSAME( TRANS, 'C' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( KL.LT.0 ) THEN
         INFO = -4
      ELSE IF( KU.LT.0 ) THEN
         INFO = -5
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -6
      ELSE IF( LDAB.LT.KL+KU+1 ) THEN
         INFO = -8
      ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
         INFO = -10
      ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
     $         ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
         INFO = -12
      ELSE
         IF( ROWEQU ) THEN
            RCMIN = BIGNUM
            RCMAX = ZERO
            DO 10 J = 1, N
               RCMIN = MIN( RCMIN, R( J ) )
               RCMAX = MAX( RCMAX, R( J ) )
   10       CONTINUE
            IF( RCMIN.LE.ZERO ) THEN
               INFO = -13
            ELSE IF( N.GT.0 ) THEN
               ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
            ELSE
               ROWCND = ONE
            END IF
         END IF
         IF( COLEQU .AND. INFO.EQ.0 ) THEN
            RCMIN = BIGNUM
            RCMAX = ZERO
            DO 20 J = 1, N
               RCMIN = MIN( RCMIN, C( J ) )
               RCMAX = MAX( RCMAX, C( J ) )
   20       CONTINUE
            IF( RCMIN.LE.ZERO ) THEN
               INFO = -14
            ELSE IF( N.GT.0 ) THEN
               COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
            ELSE
               COLCND = ONE
            END IF
         END IF
         IF( INFO.EQ.0 ) THEN
            IF( LDB.LT.MAX( 1, N ) ) THEN
               INFO = -16
            ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
               INFO = -18
            END IF
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGBSVX', -INFO )
         RETURN
      END IF
*
      IF( EQUIL ) THEN
*
*        Compute row and column scalings to equilibrate the matrix A.
*
         CALL DGBEQU( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
     $                AMAX, INFEQU )
         IF( INFEQU.EQ.0 ) THEN
*
*           Equilibrate the matrix.
*
            CALL DLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND,
     $                   COLCND,
     $                   AMAX, EQUED )
            ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
            COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
         END IF
      END IF
*
*     Scale the right hand side.
*
      IF( NOTRAN ) THEN
         IF( ROWEQU ) THEN
            DO 40 J = 1, NRHS
               DO 30 I = 1, N
                  B( I, J ) = R( I )*B( I, J )
   30          CONTINUE
   40       CONTINUE
         END IF
      ELSE IF( COLEQU ) THEN
         DO 60 J = 1, NRHS
            DO 50 I = 1, N
               B( I, J ) = C( I )*B( I, J )
   50       CONTINUE
   60    CONTINUE
      END IF
*
      IF( NOFACT .OR. EQUIL ) THEN
*
*        Compute the LU factorization of the band matrix A.
*
         DO 70 J = 1, N
            J1 = MAX( J-KU, 1 )
            J2 = MIN( J+KL, N )
            CALL DCOPY( J2-J1+1, AB( KU+1-J+J1, J ), 1,
     $                  AFB( KL+KU+1-J+J1, J ), 1 )
   70    CONTINUE
*
         CALL DGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO )
*
*        Return if INFO is non-zero.
*
         IF( INFO.GT.0 ) THEN
*
*           Compute the reciprocal pivot growth factor of the
*           leading rank-deficient INFO columns of A.
*
            ANORM = ZERO
            DO 90 J = 1, INFO
               DO 80 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 )
                  ANORM = MAX( ANORM, ABS( AB( I, J ) ) )
   80          CONTINUE
   90       CONTINUE
            RPVGRW = DLANTB( 'M', 'U', 'N', INFO, MIN( INFO-1,
     $                       KL+KU ),
     $                       AFB( MAX( 1, KL+KU+2-INFO ), 1 ), LDAFB,
     $                       WORK )
            IF( RPVGRW.EQ.ZERO ) THEN
               RPVGRW = ONE
            ELSE
               RPVGRW = ANORM / RPVGRW
            END IF
            WORK( 1 ) = RPVGRW
            RCOND = ZERO
            RETURN
         END IF
      END IF
*
*     Compute the norm of the matrix A and the
*     reciprocal pivot growth factor RPVGRW.
*
      IF( NOTRAN ) THEN
         NORM = '1'
      ELSE
         NORM = 'I'
      END IF
      ANORM = DLANGB( NORM, N, KL, KU, AB, LDAB, WORK )
      RPVGRW = DLANTB( 'M', 'U', 'N', N, KL+KU, AFB, LDAFB, WORK )
      IF( RPVGRW.EQ.ZERO ) THEN
         RPVGRW = ONE
      ELSE
         RPVGRW = DLANGB( 'M', N, KL, KU, AB, LDAB, WORK ) / RPVGRW
      END IF
*
*     Compute the reciprocal of the condition number of A.
*
      CALL DGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND,
     $             WORK, IWORK, INFO )
*
*     Compute the solution matrix X.
*
      CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
      CALL DGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX,
     $             INFO )
*
*     Use iterative refinement to improve the computed solution and
*     compute error bounds and backward error estimates for it.
*
      CALL DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
     $             IPIV,
     $             B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
*     Transform the solution matrix X to a solution of the original
*     system.
*
      IF( NOTRAN ) THEN
         IF( COLEQU ) THEN
            DO 110 J = 1, NRHS
               DO 100 I = 1, N
                  X( I, J ) = C( I )*X( I, J )
  100          CONTINUE
  110       CONTINUE
            DO 120 J = 1, NRHS
               FERR( J ) = FERR( J ) / COLCND
  120       CONTINUE
         END IF
      ELSE IF( ROWEQU ) THEN
         DO 140 J = 1, NRHS
            DO 130 I = 1, N
               X( I, J ) = R( I )*X( I, J )
  130       CONTINUE
  140    CONTINUE
         DO 150 J = 1, NRHS
            FERR( J ) = FERR( J ) / ROWCND
  150    CONTINUE
      END IF
*
*     Set INFO = N+1 if the matrix is singular to working precision.
*
      IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
     $   INFO = N + 1
*
      WORK( 1 ) = RPVGRW
      RETURN
*
*     End of DGBSVX
*
      END
*> \brief \b DGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGBTF2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbtf2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbtf2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbtf2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, KL, KU, LDAB, M, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   AB( LDAB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGBTF2 computes an LU factorization of a real m-by-n band matrix A
*> using partial pivoting with row interchanges.
*>
*> This is the unblocked version of the algorithm, calling Level 2 BLAS.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*>          KL is INTEGER
*>          The number of subdiagonals within the band of A.  KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*>          KU is INTEGER
*>          The number of superdiagonals within the band of A.  KU >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          On entry, the matrix A in band storage, in rows KL+1 to
*>          2*KL+KU+1; rows 1 to KL of the array need not be set.
*>          The j-th column of A is stored in the j-th column of the
*>          array AB as follows:
*>          AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)
*>
*>          On exit, details of the factorization: U is stored as an
*>          upper triangular band matrix with KL+KU superdiagonals in
*>          rows 1 to KL+KU+1, and the multipliers used during the
*>          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
*>          See below for further details.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (min(M,N))
*>          The pivot indices; for 1 <= i <= min(M,N), row i of the
*>          matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*>          > 0: if INFO = +i, U(i,i) is exactly zero. The factorization
*>               has been completed, but the factor U is exactly
*>               singular, and division by zero will occur if it is used
*>               to solve a system of equations.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gbtf2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The band storage scheme is illustrated by the following example, when
*>  M = N = 6, KL = 2, KU = 1:
*>
*>  On entry:                       On exit:
*>
*>      *    *    *    +    +    +       *    *    *   u14  u25  u36
*>      *    *    +    +    +    +       *    *   u13  u24  u35  u46
*>      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
*>     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
*>     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
*>     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *
*>
*>  Array elements marked * are not used by the routine; elements marked
*>  + need not be set on entry, but are required by the routine to store
*>  elements of U, because of fill-in resulting from the row
*>  interchanges.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, KL, KU, LDAB, M, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   AB( LDAB, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J, JP, JU, KM, KV
*     ..
*     .. External Functions ..
      INTEGER            IDAMAX
      EXTERNAL           IDAMAX
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGER, DSCAL, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     KV is the number of superdiagonals in the factor U, allowing for
*     fill-in.
*
      KV = KU + KL
*
*     Test the input parameters.
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( KL.LT.0 ) THEN
         INFO = -3
      ELSE IF( KU.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDAB.LT.KL+KV+1 ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGBTF2', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 )
     $   RETURN
*
*     Gaussian elimination with partial pivoting
*
*     Set fill-in elements in columns KU+2 to KV to zero.
*
      DO 20 J = KU + 2, MIN( KV, N )
         DO 10 I = KV - J + 2, KL
            AB( I, J ) = ZERO
   10    CONTINUE
   20 CONTINUE
*
*     JU is the index of the last column affected by the current stage
*     of the factorization.
*
      JU = 1
*
      DO 40 J = 1, MIN( M, N )
*
*        Set fill-in elements in column J+KV to zero.
*
         IF( J+KV.LE.N ) THEN
            DO 30 I = 1, KL
               AB( I, J+KV ) = ZERO
   30       CONTINUE
         END IF
*
*        Find pivot and test for singularity. KM is the number of
*        subdiagonal elements in the current column.
*
         KM = MIN( KL, M-J )
         JP = IDAMAX( KM+1, AB( KV+1, J ), 1 )
         IPIV( J ) = JP + J - 1
         IF( AB( KV+JP, J ).NE.ZERO ) THEN
            JU = MAX( JU, MIN( J+KU+JP-1, N ) )
*
*           Apply interchange to columns J to JU.
*
            IF( JP.NE.1 )
     $         CALL DSWAP( JU-J+1, AB( KV+JP, J ), LDAB-1,
     $                     AB( KV+1, J ), LDAB-1 )
*
            IF( KM.GT.0 ) THEN
*
*              Compute multipliers.
*
               CALL DSCAL( KM, ONE / AB( KV+1, J ), AB( KV+2, J ),
     $                     1 )
*
*              Update trailing submatrix within the band.
*
               IF( JU.GT.J )
     $            CALL DGER( KM, JU-J, -ONE, AB( KV+2, J ), 1,
     $                       AB( KV, J+1 ), LDAB-1, AB( KV+1, J+1 ),
     $                       LDAB-1 )
            END IF
         ELSE
*
*           If pivot is zero, set INFO to the index of the pivot
*           unless a zero pivot has already been found.
*
            IF( INFO.EQ.0 )
     $         INFO = J
         END IF
   40 CONTINUE
      RETURN
*
*     End of DGBTF2
*
      END
*> \brief \b DGBTRF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGBTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbtrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbtrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbtrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGBTRF( M, N, KL, KU, AB, LDAB, IPIV, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, KL, KU, LDAB, M, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   AB( LDAB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGBTRF computes an LU factorization of a real m-by-n band matrix A
*> using partial pivoting with row interchanges.
*>
*> This is the blocked version of the algorithm, calling Level 3 BLAS.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*>          KL is INTEGER
*>          The number of subdiagonals within the band of A.  KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*>          KU is INTEGER
*>          The number of superdiagonals within the band of A.  KU >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          On entry, the matrix A in band storage, in rows KL+1 to
*>          2*KL+KU+1; rows 1 to KL of the array need not be set.
*>          The j-th column of A is stored in the j-th column of the
*>          array AB as follows:
*>          AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)
*>
*>          On exit, details of the factorization: U is stored as an
*>          upper triangular band matrix with KL+KU superdiagonals in
*>          rows 1 to KL+KU+1, and the multipliers used during the
*>          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
*>          See below for further details.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (min(M,N))
*>          The pivot indices; for 1 <= i <= min(M,N), row i of the
*>          matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*>          > 0: if INFO = +i, U(i,i) is exactly zero. The factorization
*>               has been completed, but the factor U is exactly
*>               singular, and division by zero will occur if it is used
*>               to solve a system of equations.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gbtrf
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The band storage scheme is illustrated by the following example, when
*>  M = N = 6, KL = 2, KU = 1:
*>
*>  On entry:                       On exit:
*>
*>      *    *    *    +    +    +       *    *    *   u14  u25  u36
*>      *    *    +    +    +    +       *    *   u13  u24  u35  u46
*>      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
*>     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
*>     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
*>     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *
*>
*>  Array elements marked * are not used by the routine; elements marked
*>  + need not be set on entry, but are required by the routine to store
*>  elements of U because of fill-in resulting from the row interchanges.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGBTRF( M, N, KL, KU, AB, LDAB, IPIV, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, KL, KU, LDAB, M, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   AB( LDAB, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
      INTEGER            NBMAX, LDWORK
      PARAMETER          ( NBMAX = 64, LDWORK = NBMAX+1 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, I2, I3, II, IP, J, J2, J3, JB, JJ, JM, JP,
     $                   JU, K2, KM, KV, NB, NW
      DOUBLE PRECISION   TEMP
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   WORK13( LDWORK, NBMAX ),
     $                   WORK31( LDWORK, NBMAX )
*     ..
*     .. External Functions ..
      INTEGER            IDAMAX, ILAENV
      EXTERNAL           IDAMAX, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGBTF2, DGEMM, DGER, DLASWP,
     $                   DSCAL,
     $                   DSWAP, DTRSM, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     KV is the number of superdiagonals in the factor U, allowing for
*     fill-in
*
      KV = KU + KL
*
*     Test the input parameters.
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( KL.LT.0 ) THEN
         INFO = -3
      ELSE IF( KU.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDAB.LT.KL+KV+1 ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGBTRF', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 )
     $   RETURN
*
*     Determine the block size for this environment
*
      NB = ILAENV( 1, 'DGBTRF', ' ', M, N, KL, KU )
*
*     The block size must not exceed the limit set by the size of the
*     local arrays WORK13 and WORK31.
*
      NB = MIN( NB, NBMAX )
*
      IF( NB.LE.1 .OR. NB.GT.KL ) THEN
*
*        Use unblocked code
*
         CALL DGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO )
      ELSE
*
*        Use blocked code
*
*        Zero the superdiagonal elements of the work array WORK13
*
         DO 20 J = 1, NB
            DO 10 I = 1, J - 1
               WORK13( I, J ) = ZERO
   10       CONTINUE
   20    CONTINUE
*
*        Zero the subdiagonal elements of the work array WORK31
*
         DO 40 J = 1, NB
            DO 30 I = J + 1, NB
               WORK31( I, J ) = ZERO
   30       CONTINUE
   40    CONTINUE
*
*        Gaussian elimination with partial pivoting
*
*        Set fill-in elements in columns KU+2 to KV to zero
*
         DO 60 J = KU + 2, MIN( KV, N )
            DO 50 I = KV - J + 2, KL
               AB( I, J ) = ZERO
   50       CONTINUE
   60    CONTINUE
*
*        JU is the index of the last column affected by the current
*        stage of the factorization
*
         JU = 1
*
         DO 180 J = 1, MIN( M, N ), NB
            JB = MIN( NB, MIN( M, N )-J+1 )
*
*           The active part of the matrix is partitioned
*
*              A11   A12   A13
*              A21   A22   A23
*              A31   A32   A33
*
*           Here A11, A21 and A31 denote the current block of JB columns
*           which is about to be factorized. The number of rows in the
*           partitioning are JB, I2, I3 respectively, and the numbers
*           of columns are JB, J2, J3. The superdiagonal elements of A13
*           and the subdiagonal elements of A31 lie outside the band.
*
            I2 = MIN( KL-JB, M-J-JB+1 )
            I3 = MIN( JB, M-J-KL+1 )
*
*           J2 and J3 are computed after JU has been updated.
*
*           Factorize the current block of JB columns
*
            DO 80 JJ = J, J + JB - 1
*
*              Set fill-in elements in column JJ+KV to zero
*
               IF( JJ+KV.LE.N ) THEN
                  DO 70 I = 1, KL
                     AB( I, JJ+KV ) = ZERO
   70             CONTINUE
               END IF
*
*              Find pivot and test for singularity. KM is the number of
*              subdiagonal elements in the current column.
*
               KM = MIN( KL, M-JJ )
               JP = IDAMAX( KM+1, AB( KV+1, JJ ), 1 )
               IPIV( JJ ) = JP + JJ - J
               IF( AB( KV+JP, JJ ).NE.ZERO ) THEN
                  JU = MAX( JU, MIN( JJ+KU+JP-1, N ) )
                  IF( JP.NE.1 ) THEN
*
*                    Apply interchange to columns J to J+JB-1
*
                     IF( JP+JJ-1.LT.J+KL ) THEN
*
                        CALL DSWAP( JB, AB( KV+1+JJ-J, J ), LDAB-1,
     $                              AB( KV+JP+JJ-J, J ), LDAB-1 )
                     ELSE
*
*                       The interchange affects columns J to JJ-1 of A31
*                       which are stored in the work array WORK31
*
                        CALL DSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1,
     $                              WORK31( JP+JJ-J-KL, 1 ), LDWORK )
                        CALL DSWAP( J+JB-JJ, AB( KV+1, JJ ), LDAB-1,
     $                              AB( KV+JP, JJ ), LDAB-1 )
                     END IF
                  END IF
*
*                 Compute multipliers
*
                  CALL DSCAL( KM, ONE / AB( KV+1, JJ ), AB( KV+2,
     $                        JJ ),
     $                        1 )
*
*                 Update trailing submatrix within the band and within
*                 the current block. JM is the index of the last column
*                 which needs to be updated.
*
                  JM = MIN( JU, J+JB-1 )
                  IF( JM.GT.JJ )
     $               CALL DGER( KM, JM-JJ, -ONE, AB( KV+2, JJ ), 1,
     $                          AB( KV, JJ+1 ), LDAB-1,
     $                          AB( KV+1, JJ+1 ), LDAB-1 )
               ELSE
*
*                 If pivot is zero, set INFO to the index of the pivot
*                 unless a zero pivot has already been found.
*
                  IF( INFO.EQ.0 )
     $               INFO = JJ
               END IF
*
*              Copy current column of A31 into the work array WORK31
*
               NW = MIN( JJ-J+1, I3 )
               IF( NW.GT.0 )
     $            CALL DCOPY( NW, AB( KV+KL+1-JJ+J, JJ ), 1,
     $                        WORK31( 1, JJ-J+1 ), 1 )
   80       CONTINUE
            IF( J+JB.LE.N ) THEN
*
*              Apply the row interchanges to the other blocks.
*
               J2 = MIN( JU-J+1, KV ) - JB
               J3 = MAX( 0, JU-J-KV+1 )
*
*              Use DLASWP to apply the row interchanges to A12, A22, and
*              A32.
*
               CALL DLASWP( J2, AB( KV+1-JB, J+JB ), LDAB-1, 1, JB,
     $                      IPIV( J ), 1 )
*
*              Adjust the pivot indices.
*
               DO 90 I = J, J + JB - 1
                  IPIV( I ) = IPIV( I ) + J - 1
   90          CONTINUE
*
*              Apply the row interchanges to A13, A23, and A33
*              columnwise.
*
               K2 = J - 1 + JB + J2
               DO 110 I = 1, J3
                  JJ = K2 + I
                  DO 100 II = J + I - 1, J + JB - 1
                     IP = IPIV( II )
                     IF( IP.NE.II ) THEN
                        TEMP = AB( KV+1+II-JJ, JJ )
                        AB( KV+1+II-JJ, JJ ) = AB( KV+1+IP-JJ, JJ )
                        AB( KV+1+IP-JJ, JJ ) = TEMP
                     END IF
  100             CONTINUE
  110          CONTINUE
*
*              Update the relevant part of the trailing submatrix
*
               IF( J2.GT.0 ) THEN
*
*                 Update A12
*
                  CALL DTRSM( 'Left', 'Lower', 'No transpose',
     $                        'Unit',
     $                        JB, J2, ONE, AB( KV+1, J ), LDAB-1,
     $                        AB( KV+1-JB, J+JB ), LDAB-1 )
*
                  IF( I2.GT.0 ) THEN
*
*                    Update A22
*
                     CALL DGEMM( 'No transpose', 'No transpose', I2,
     $                           J2,
     $                           JB, -ONE, AB( KV+1+JB, J ), LDAB-1,
     $                           AB( KV+1-JB, J+JB ), LDAB-1, ONE,
     $                           AB( KV+1, J+JB ), LDAB-1 )
                  END IF
*
                  IF( I3.GT.0 ) THEN
*
*                    Update A32
*
                     CALL DGEMM( 'No transpose', 'No transpose', I3,
     $                           J2,
     $                           JB, -ONE, WORK31, LDWORK,
     $                           AB( KV+1-JB, J+JB ), LDAB-1, ONE,
     $                           AB( KV+KL+1-JB, J+JB ), LDAB-1 )
                  END IF
               END IF
*
               IF( J3.GT.0 ) THEN
*
*                 Copy the lower triangle of A13 into the work array
*                 WORK13
*
                  DO 130 JJ = 1, J3
                     DO 120 II = JJ, JB
                        WORK13( II, JJ ) = AB( II-JJ+1, JJ+J+KV-1 )
  120                CONTINUE
  130             CONTINUE
*
*                 Update A13 in the work array
*
                  CALL DTRSM( 'Left', 'Lower', 'No transpose',
     $                        'Unit',
     $                        JB, J3, ONE, AB( KV+1, J ), LDAB-1,
     $                        WORK13, LDWORK )
*
                  IF( I2.GT.0 ) THEN
*
*                    Update A23
*
                     CALL DGEMM( 'No transpose', 'No transpose', I2,
     $                           J3,
     $                           JB, -ONE, AB( KV+1+JB, J ), LDAB-1,
     $                           WORK13, LDWORK, ONE, AB( 1+JB, J+KV ),
     $                           LDAB-1 )
                  END IF
*
                  IF( I3.GT.0 ) THEN
*
*                    Update A33
*
                     CALL DGEMM( 'No transpose', 'No transpose', I3,
     $                           J3,
     $                           JB, -ONE, WORK31, LDWORK, WORK13,
     $                           LDWORK, ONE, AB( 1+KL, J+KV ), LDAB-1 )
                  END IF
*
*                 Copy the lower triangle of A13 back into place
*
                  DO 150 JJ = 1, J3
                     DO 140 II = JJ, JB
                        AB( II-JJ+1, JJ+J+KV-1 ) = WORK13( II, JJ )
  140                CONTINUE
  150             CONTINUE
               END IF
            ELSE
*
*              Adjust the pivot indices.
*
               DO 160 I = J, J + JB - 1
                  IPIV( I ) = IPIV( I ) + J - 1
  160          CONTINUE
            END IF
*
*           Partially undo the interchanges in the current block to
*           restore the upper triangular form of A31 and copy the upper
*           triangle of A31 back into place
*
            DO 170 JJ = J + JB - 1, J, -1
               JP = IPIV( JJ ) - JJ + 1
               IF( JP.NE.1 ) THEN
*
*                 Apply interchange to columns J to JJ-1
*
                  IF( JP+JJ-1.LT.J+KL ) THEN
*
*                    The interchange does not affect A31
*
                     CALL DSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1,
     $                           AB( KV+JP+JJ-J, J ), LDAB-1 )
                  ELSE
*
*                    The interchange does affect A31
*
                     CALL DSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1,
     $                           WORK31( JP+JJ-J-KL, 1 ), LDWORK )
                  END IF
               END IF
*
*              Copy the current column of A31 back into place
*
               NW = MIN( I3, JJ-J+1 )
               IF( NW.GT.0 )
     $            CALL DCOPY( NW, WORK31( 1, JJ-J+1 ), 1,
     $                        AB( KV+KL+1-JJ+J, JJ ), 1 )
  170       CONTINUE
  180    CONTINUE
      END IF
*
      RETURN
*
*     End of DGBTRF
*
      END
*> \brief \b DGBTRS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGBTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbtrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbtrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbtrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGBTRS( TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB,
*                          INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANS
*       INTEGER            INFO, KL, KU, LDAB, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGBTRS solves a system of linear equations
*>    A * X = B  or  A**T * X = B
*> with a general band matrix A using the LU factorization computed
*> by DGBTRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the form of the system of equations.
*>          = 'N':  A * X = B  (No transpose)
*>          = 'T':  A**T* X = B  (Transpose)
*>          = 'C':  A**T* X = B  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*>          KL is INTEGER
*>          The number of subdiagonals within the band of A.  KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*>          KU is INTEGER
*>          The number of superdiagonals within the band of A.  KU >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          Details of the LU factorization of the band matrix A, as
*>          computed by DGBTRF.  U is stored as an upper triangular band
*>          matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
*>          the multipliers used during the factorization are stored in
*>          rows KL+KU+2 to 2*KL+KU+1.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          The pivot indices; for 1 <= i <= N, row i of the matrix was
*>          interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the right hand side matrix B.
*>          On exit, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gbtrs
*
*  =====================================================================
      SUBROUTINE DGBTRS( TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B,
     $                   LDB,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS
      INTEGER            INFO, KL, KU, LDAB, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LNOTI, NOTRAN
      INTEGER            I, J, KD, L, LM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMV, DGER, DSWAP, DTBSV, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NOTRAN = LSAME( TRANS, 'N' )
      IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
     $    LSAME( TRANS, 'C' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( KL.LT.0 ) THEN
         INFO = -3
      ELSE IF( KU.LT.0 ) THEN
         INFO = -4
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDAB.LT.( 2*KL+KU+1 ) ) THEN
         INFO = -7
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -10
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGBTRS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 )
     $   RETURN
*
      KD = KU + KL + 1
      LNOTI = KL.GT.0
*
      IF( NOTRAN ) THEN
*
*        Solve  A*X = B.
*
*        Solve L*X = B, overwriting B with X.
*
*        L is represented as a product of permutations and unit lower
*        triangular matrices L = P(1) * L(1) * ... * P(n-1) * L(n-1),
*        where each transformation L(i) is a rank-one modification of
*        the identity matrix.
*
         IF( LNOTI ) THEN
            DO 10 J = 1, N - 1
               LM = MIN( KL, N-J )
               L = IPIV( J )
               IF( L.NE.J )
     $            CALL DSWAP( NRHS, B( L, 1 ), LDB, B( J, 1 ), LDB )
               CALL DGER( LM, NRHS, -ONE, AB( KD+1, J ), 1, B( J,
     $                    1 ),
     $                    LDB, B( J+1, 1 ), LDB )
   10       CONTINUE
         END IF
*
         DO 20 I = 1, NRHS
*
*           Solve U*X = B, overwriting B with X.
*
            CALL DTBSV( 'Upper', 'No transpose', 'Non-unit', N,
     $                  KL+KU,
     $                  AB, LDAB, B( 1, I ), 1 )
   20    CONTINUE
*
      ELSE
*
*        Solve A**T*X = B.
*
         DO 30 I = 1, NRHS
*
*           Solve U**T*X = B, overwriting B with X.
*
            CALL DTBSV( 'Upper', 'Transpose', 'Non-unit', N, KL+KU,
     $                  AB,
     $                  LDAB, B( 1, I ), 1 )
   30    CONTINUE
*
*        Solve L**T*X = B, overwriting B with X.
*
         IF( LNOTI ) THEN
            DO 40 J = N - 1, 1, -1
               LM = MIN( KL, N-J )
               CALL DGEMV( 'Transpose', LM, NRHS, -ONE, B( J+1, 1 ),
     $                     LDB, AB( KD+1, J ), 1, ONE, B( J, 1 ), LDB )
               L = IPIV( J )
               IF( L.NE.J )
     $            CALL DSWAP( NRHS, B( L, 1 ), LDB, B( J, 1 ), LDB )
   40       CONTINUE
         END IF
      END IF
      RETURN
*
*     End of DGBTRS
*
      END
*> \brief \b DGEBAK
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEBAK + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgebak.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgebak.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgebak.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEBAK( JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV,
*                          INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOB, SIDE
*       INTEGER            IHI, ILO, INFO, LDV, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   SCALE( * ), V( LDV, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEBAK forms the right or left eigenvectors of a real general matrix
*> by backward transformation on the computed eigenvectors of the
*> balanced matrix output by DGEBAL.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOB
*> \verbatim
*>          JOB is CHARACTER*1
*>          Specifies the type of backward transformation required:
*>          = 'N': do nothing, return immediately;
*>          = 'P': do backward transformation for permutation only;
*>          = 'S': do backward transformation for scaling only;
*>          = 'B': do backward transformations for both permutation and
*>                 scaling.
*>          JOB must be the same as the argument JOB supplied to DGEBAL.
*> \endverbatim
*>
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'R':  V contains right eigenvectors;
*>          = 'L':  V contains left eigenvectors.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of rows of the matrix V.  N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*>          ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*>          IHI is INTEGER
*>          The integers ILO and IHI determined by DGEBAL.
*>          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*> \endverbatim
*>
*> \param[in] SCALE
*> \verbatim
*>          SCALE is DOUBLE PRECISION array, dimension (N)
*>          Details of the permutation and scaling factors, as returned
*>          by DGEBAL.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of columns of the matrix V.  M >= 0.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension (LDV,M)
*>          On entry, the matrix of right or left eigenvectors to be
*>          transformed, as returned by DHSEIN or DTREVC.
*>          On exit, V is overwritten by the transformed eigenvectors.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of the array V. LDV >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gebak
*
*  =====================================================================
      SUBROUTINE DGEBAK( JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOB, SIDE
      INTEGER            IHI, ILO, INFO, LDV, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   SCALE( * ), V( LDV, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LEFTV, RIGHTV
      INTEGER            I, II, K
      DOUBLE PRECISION   S
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Decode and Test the input parameters
*
      RIGHTV = LSAME( SIDE, 'R' )
      LEFTV = LSAME( SIDE, 'L' )
*
      INFO = 0
      IF( .NOT.LSAME( JOB, 'N' ) .AND.
     $    .NOT.LSAME( JOB, 'P' ) .AND.
     $    .NOT.LSAME( JOB, 'S' ) .AND.
     $                .NOT.LSAME( JOB, 'B' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
         INFO = -4
      ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
         INFO = -5
      ELSE IF( M.LT.0 ) THEN
         INFO = -7
      ELSE IF( LDV.LT.MAX( 1, N ) ) THEN
         INFO = -9
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEBAK', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
      IF( M.EQ.0 )
     $   RETURN
      IF( LSAME( JOB, 'N' ) )
     $   RETURN
*
      IF( ILO.EQ.IHI )
     $   GO TO 30
*
*     Backward balance
*
      IF( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'B' ) ) THEN
*
         IF( RIGHTV ) THEN
            DO 10 I = ILO, IHI
               S = SCALE( I )
               CALL DSCAL( M, S, V( I, 1 ), LDV )
   10       CONTINUE
         END IF
*
         IF( LEFTV ) THEN
            DO 20 I = ILO, IHI
               S = ONE / SCALE( I )
               CALL DSCAL( M, S, V( I, 1 ), LDV )
   20       CONTINUE
         END IF
*
      END IF
*
*     Backward permutation
*
*     For  I = ILO-1 step -1 until 1,
*              IHI+1 step 1 until N do --
*
   30 CONTINUE
      IF( LSAME( JOB, 'P' ) .OR. LSAME( JOB, 'B' ) ) THEN
         IF( RIGHTV ) THEN
            DO 40 II = 1, N
               I = II
               IF( I.GE.ILO .AND. I.LE.IHI )
     $            GO TO 40
               IF( I.LT.ILO )
     $            I = ILO - II
               K = INT( SCALE( I ) )
               IF( K.EQ.I )
     $            GO TO 40
               CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV )
   40       CONTINUE
         END IF
*
         IF( LEFTV ) THEN
            DO 50 II = 1, N
               I = II
               IF( I.GE.ILO .AND. I.LE.IHI )
     $            GO TO 50
               IF( I.LT.ILO )
     $            I = ILO - II
               K = INT( SCALE( I ) )
               IF( K.EQ.I )
     $            GO TO 50
               CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV )
   50       CONTINUE
         END IF
      END IF
*
      RETURN
*
*     End of DGEBAK
*
      END
*> \brief \b DGEBAL
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEBAL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgebal.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgebal.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgebal.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOB
*       INTEGER            IHI, ILO, INFO, LDA, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), SCALE( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEBAL balances a general real matrix A.  This involves, first,
*> permuting A by a similarity transformation to isolate eigenvalues
*> in the first 1 to ILO-1 and last IHI+1 to N elements on the
*> diagonal; and second, applying a diagonal similarity transformation
*> to rows and columns ILO to IHI to make the rows and columns as
*> close in norm as possible.  Both steps are optional.
*>
*> Balancing may reduce the 1-norm of the matrix, and improve the
*> accuracy of the computed eigenvalues and/or eigenvectors.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOB
*> \verbatim
*>          JOB is CHARACTER*1
*>          Specifies the operations to be performed on A:
*>          = 'N':  none:  simply set ILO = 1, IHI = N, SCALE(I) = 1.0
*>                  for i = 1,...,N;
*>          = 'P':  permute only;
*>          = 'S':  scale only;
*>          = 'B':  both permute and scale.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the input matrix A.
*>          On exit,  A is overwritten by the balanced matrix.
*>          If JOB = 'N', A is not referenced.
*>          See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] ILO
*> \verbatim
*>          ILO is INTEGER
*> \endverbatim
*> \param[out] IHI
*> \verbatim
*>          IHI is INTEGER
*>          ILO and IHI are set to integers such that on exit
*>          A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
*>          If JOB = 'N' or 'S', ILO = 1 and IHI = N.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*>          SCALE is DOUBLE PRECISION array, dimension (N)
*>          Details of the permutations and scaling factors applied to
*>          A.  If P(j) is the index of the row and column interchanged
*>          with row and column j and D(j) is the scaling factor
*>          applied to row and column j, then
*>          SCALE(j) = P(j)    for j = 1,...,ILO-1
*>                   = D(j)    for j = ILO,...,IHI
*>                   = P(j)    for j = IHI+1,...,N.
*>          The order in which the interchanges are made is N to IHI+1,
*>          then 1 to ILO-1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gebal
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The permutations consist of row and column interchanges which put
*>  the matrix in the form
*>
*>             ( T1   X   Y  )
*>     P A P = (  0   B   Z  )
*>             (  0   0   T2 )
*>
*>  where T1 and T2 are upper triangular matrices whose eigenvalues lie
*>  along the diagonal.  The column indices ILO and IHI mark the starting
*>  and ending columns of the submatrix B. Balancing consists of applying
*>  a diagonal similarity transformation inv(D) * B * D to make the
*>  1-norms of each row of B and its corresponding column nearly equal.
*>  The output matrix is
*>
*>     ( T1     X*D          Y    )
*>     (  0  inv(D)*B*D  inv(D)*Z ).
*>     (  0      0           T2   )
*>
*>  Information about the permutations P and the diagonal matrix D is
*>  returned in the vector SCALE.
*>
*>  This subroutine is based on the EISPACK routine BALANC.
*>
*>  Modified by Tzu-Yi Chen, Computer Science Division, University of
*>    California at Berkeley, USA
*>
*>  Refactored by Evert Provoost, Department of Computer Science,
*>    KU Leuven, Belgium
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOB
      INTEGER            IHI, ILO, INFO, LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), SCALE( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
      DOUBLE PRECISION   SCLFAC
      PARAMETER          ( SCLFAC = 2.0D+0 )
      DOUBLE PRECISION   FACTOR
      PARAMETER          ( FACTOR = 0.95D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOCONV, CANSWAP
      INTEGER            I, ICA, IRA, J, K, L
      DOUBLE PRECISION   C, CA, F, G, R, RA, S, SFMAX1, SFMAX2, SFMIN1,
     $                   SFMIN2
*     ..
*     .. External Functions ..
      LOGICAL            DISNAN, LSAME
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH, DNRM2
      EXTERNAL           DISNAN, LSAME, IDAMAX, DLAMCH,
     $                   DNRM2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     Test the input parameters
*
      INFO = 0
      IF( .NOT.LSAME( JOB, 'N' ) .AND.
     $    .NOT.LSAME( JOB, 'P' ) .AND.
     $    .NOT.LSAME( JOB, 'S' ) .AND.
     $                .NOT.LSAME( JOB, 'B' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEBAL', -INFO )
         RETURN
      END IF
*
*     Quick returns.
*
      IF( N.EQ.0 ) THEN
         ILO = 1
         IHI = 0
         RETURN
      END IF
*
      IF( LSAME( JOB, 'N' ) ) THEN
         DO I = 1, N
            SCALE( I ) = ONE
         END DO
         ILO = 1
         IHI = N
         RETURN
      END IF
*
*     Permutation to isolate eigenvalues if possible.
*
      K = 1
      L = N
*
      IF( .NOT.LSAME( JOB, 'S' ) ) THEN
*
*        Row and column exchange.
*
         NOCONV = .TRUE.
         DO WHILE( NOCONV )
*
*           Search for rows isolating an eigenvalue and push them down.
*
            NOCONV = .FALSE.
            DO I = L, 1, -1
               CANSWAP = .TRUE.
               DO J = 1, L
                  IF( I.NE.J .AND. A( I, J ).NE.ZERO ) THEN
                     CANSWAP = .FALSE.
                     EXIT
                  END IF
               END DO
*
               IF( CANSWAP ) THEN
                  SCALE( L ) = I
                  IF( I.NE.L ) THEN
                     CALL DSWAP( L, A( 1, I ), 1, A( 1, L ), 1 )
                     CALL DSWAP( N-K+1, A( I, K ), LDA, A( L, K ),
     $                           LDA )
                  END IF
                  NOCONV = .TRUE.
*
                  IF( L.EQ.1 ) THEN
                     ILO = 1
                     IHI = 1
                     RETURN
                  END IF
*
                  L = L - 1
               END IF
            END DO
*
         END DO

         NOCONV = .TRUE.
         DO WHILE( NOCONV )
*
*           Search for columns isolating an eigenvalue and push them left.
*
            NOCONV = .FALSE.
            DO J = K, L
               CANSWAP = .TRUE.
               DO I = K, L
                  IF( I.NE.J .AND. A( I, J ).NE.ZERO ) THEN
                     CANSWAP = .FALSE.
                     EXIT
                  END IF
               END DO
*
               IF( CANSWAP ) THEN
                  SCALE( K ) = J
                  IF( J.NE.K ) THEN
                     CALL DSWAP( L, A( 1, J ), 1, A( 1, K ), 1 )
                     CALL DSWAP( N-K+1, A( J, K ), LDA, A( K, K ),
     $                           LDA )
                  END IF
                  NOCONV = .TRUE.
*
                  K = K + 1
               END IF
            END DO
*
         END DO
*
      END IF
*
*     Initialize SCALE for non-permuted submatrix.
*
      DO I = K, L
         SCALE( I ) = ONE
      END DO
*
*     If we only had to permute, we are done.
*
      IF( LSAME( JOB, 'P' ) ) THEN
         ILO = K
         IHI = L
         RETURN
      END IF
*
*     Balance the submatrix in rows K to L.
*
*     Iterative loop for norm reduction.
*
      SFMIN1 = DLAMCH( 'S' ) / DLAMCH( 'P' )
      SFMAX1 = ONE / SFMIN1
      SFMIN2 = SFMIN1*SCLFAC
      SFMAX2 = ONE / SFMIN2
*
      NOCONV = .TRUE.
      DO WHILE( NOCONV )
         NOCONV = .FALSE.
*
         DO I = K, L
*
            C = DNRM2( L-K+1, A( K, I ), 1 )
            R = DNRM2( L-K+1, A( I, K ), LDA )
            ICA = IDAMAX( L, A( 1, I ), 1 )
            CA = ABS( A( ICA, I ) )
            IRA = IDAMAX( N-K+1, A( I, K ), LDA )
            RA = ABS( A( I, IRA+K-1 ) )
*
*           Guard against zero C or R due to underflow.
*
            IF( C.EQ.ZERO .OR. R.EQ.ZERO ) CYCLE
*
*           Exit if NaN to avoid infinite loop
*
            IF( DISNAN( C+CA+R+RA ) ) THEN
               INFO = -3
               CALL XERBLA( 'DGEBAL', -INFO )
               RETURN
            END IF
*
            G = R / SCLFAC
            F = ONE
            S = C + R
*
            DO WHILE( C.LT.G .AND. MAX( F, C, CA ).LT.SFMAX2 .AND.
     $                MIN( R, G, RA ).GT.SFMIN2 )
               F = F*SCLFAC
               C = C*SCLFAC
               CA = CA*SCLFAC
               R = R / SCLFAC
               G = G / SCLFAC
               RA = RA / SCLFAC
            END DO
*
            G = C / SCLFAC
*
            DO WHILE( G.GE.R .AND. MAX( R, RA ).LT.SFMAX2 .AND.
     $                MIN( F, C, G, CA ).GT.SFMIN2 )
               F = F / SCLFAC
               C = C / SCLFAC
               G = G / SCLFAC
               CA = CA / SCLFAC
               R = R*SCLFAC
               RA = RA*SCLFAC
            END DO
*
*           Now balance.
*
            IF( ( C+R ).GE.FACTOR*S ) CYCLE
            IF( F.LT.ONE .AND. SCALE( I ).LT.ONE ) THEN
               IF( F*SCALE( I ).LE.SFMIN1 ) CYCLE
            END IF
            IF( F.GT.ONE .AND. SCALE( I ).GT.ONE ) THEN
               IF( SCALE( I ).GE.SFMAX1 / F ) CYCLE
            END IF
            G = ONE / F
            SCALE( I ) = SCALE( I )*F
            NOCONV = .TRUE.
*
            CALL DSCAL( N-K+1, G, A( I, K ), LDA )
            CALL DSCAL( L, F, A( 1, I ), 1 )
*
         END DO
*
      END DO
*
      ILO = K
      IHI = L
*
      RETURN
*
*     End of DGEBAL
*
      END
*> \brief \b DGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEBD2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgebd2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgebd2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgebd2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAUP( * ),
*      $                   TAUQ( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEBD2 reduces a real general m by n matrix A to upper or lower
*> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
*>
*> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows in the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns in the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the m by n general matrix to be reduced.
*>          On exit,
*>          if m >= n, the diagonal and the first superdiagonal are
*>            overwritten with the upper bidiagonal matrix B; the
*>            elements below the diagonal, with the array TAUQ, represent
*>            the orthogonal matrix Q as a product of elementary
*>            reflectors, and the elements above the first superdiagonal,
*>            with the array TAUP, represent the orthogonal matrix P as
*>            a product of elementary reflectors;
*>          if m < n, the diagonal and the first subdiagonal are
*>            overwritten with the lower bidiagonal matrix B; the
*>            elements below the first subdiagonal, with the array TAUQ,
*>            represent the orthogonal matrix Q as a product of
*>            elementary reflectors, and the elements above the diagonal,
*>            with the array TAUP, represent the orthogonal matrix P as
*>            a product of elementary reflectors.
*>          See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (min(M,N))
*>          The diagonal elements of the bidiagonal matrix B:
*>          D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (min(M,N)-1)
*>          The off-diagonal elements of the bidiagonal matrix B:
*>          if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
*>          if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
*> \endverbatim
*>
*> \param[out] TAUQ
*> \verbatim
*>          TAUQ is DOUBLE PRECISION array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors which
*>          represent the orthogonal matrix Q. See Further Details.
*> \endverbatim
*>
*> \param[out] TAUP
*> \verbatim
*>          TAUP is DOUBLE PRECISION array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors which
*>          represent the orthogonal matrix P. See Further Details.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (max(M,N))
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit.
*>          < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gebd2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrices Q and P are represented as products of elementary
*>  reflectors:
*>
*>  If m >= n,
*>
*>     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)
*>
*>  Each H(i) and G(i) has the form:
*>
*>     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T
*>
*>  where tauq and taup are real scalars, and v and u are real vectors;
*>  v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
*>  u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
*>  tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*>  If m < n,
*>
*>     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)
*>
*>  Each H(i) and G(i) has the form:
*>
*>     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T
*>
*>  where tauq and taup are real scalars, and v and u are real vectors;
*>  v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
*>  u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
*>  tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*>  The contents of A on exit are illustrated by the following examples:
*>
*>  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
*>
*>    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
*>    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
*>    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
*>    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
*>    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
*>    (  v1  v2  v3  v4  v5 )
*>
*>  where d and e denote diagonal and off-diagonal elements of B, vi
*>  denotes an element of the vector defining H(i), and ui an element of
*>  the vector defining G(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAUP( * ),
     $                   TAUQ( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF1F, DLARFG, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      END IF
      IF( INFO.LT.0 ) THEN
         CALL XERBLA( 'DGEBD2', -INFO )
         RETURN
      END IF
*
      IF( M.GE.N ) THEN
*
*        Reduce to upper bidiagonal form
*
         DO 10 I = 1, N
*
*           Generate elementary reflector H(i) to annihilate A(i+1:m,i)
*
            CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
     $                   TAUQ( I ) )
            D( I ) = A( I, I )
*
*           Apply H(i) to A(i:m,i+1:n) from the left
*
            IF( I.LT.N )
     $         CALL DLARF1F( 'Left', M-I+1, N-I, A( I, I ), 1,
     $                     TAUQ( I ),
     $                     A( I, I+1 ), LDA, WORK )
*
            IF( I.LT.N ) THEN
*
*              Generate elementary reflector G(i) to annihilate
*              A(i,i+2:n)
*
               CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ),
     $                      LDA, TAUP( I ) )
               E( I ) = A( I, I+1 )
*
*              Apply G(i) to A(i+1:m,i+1:n) from the right
*
               CALL DLARF1F( 'Right', M-I, N-I, A( I, I+1 ), LDA,
     $                     TAUP( I ), A( I+1, I+1 ), LDA, WORK )
            ELSE
               TAUP( I ) = ZERO
            END IF
   10    CONTINUE
      ELSE
*
*        Reduce to lower bidiagonal form
*
         DO 20 I = 1, M
*
*           Generate elementary reflector G(i) to annihilate A(i,i+1:n)
*
            CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ),
     $                   LDA,
     $                   TAUP( I ) )
            D( I ) = A( I, I )
*
*           Apply G(i) to A(i+1:m,i:n) from the right
*
            IF( I.LT.M )
     $         CALL DLARF1F( 'Right', M-I, N-I+1, A( I, I ), LDA,
     $                     TAUP( I ), A( I+1, I ), LDA, WORK )
*
            IF( I.LT.M ) THEN
*
*              Generate elementary reflector H(i) to annihilate
*              A(i+2:m,i)
*
               CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ),
     $                      1,
     $                      TAUQ( I ) )
               E( I ) = A( I+1, I )
*
*              Apply H(i) to A(i+1:m,i+1:n) from the left
*
               CALL DLARF1F( 'Left', M-I, N-I, A( I+1, I ), 1,
     $                     TAUQ( I ),
     $                     A( I+1, I+1 ), LDA, WORK )
            ELSE
               TAUQ( I ) = ZERO
            END IF
   20    CONTINUE
      END IF
      RETURN
*
*     End of DGEBD2
*
      END
*> \brief \b DGEBRD
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEBRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgebrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgebrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgebrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
*                          INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAUP( * ),
*      $                   TAUQ( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEBRD reduces a general real M-by-N matrix A to upper or lower
*> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
*>
*> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows in the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns in the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N general matrix to be reduced.
*>          On exit,
*>          if m >= n, the diagonal and the first superdiagonal are
*>            overwritten with the upper bidiagonal matrix B; the
*>            elements below the diagonal, with the array TAUQ, represent
*>            the orthogonal matrix Q as a product of elementary
*>            reflectors, and the elements above the first superdiagonal,
*>            with the array TAUP, represent the orthogonal matrix P as
*>            a product of elementary reflectors;
*>          if m < n, the diagonal and the first subdiagonal are
*>            overwritten with the lower bidiagonal matrix B; the
*>            elements below the first subdiagonal, with the array TAUQ,
*>            represent the orthogonal matrix Q as a product of
*>            elementary reflectors, and the elements above the diagonal,
*>            with the array TAUP, represent the orthogonal matrix P as
*>            a product of elementary reflectors.
*>          See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (min(M,N))
*>          The diagonal elements of the bidiagonal matrix B:
*>          D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (min(M,N)-1)
*>          The off-diagonal elements of the bidiagonal matrix B:
*>          if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
*>          if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
*> \endverbatim
*>
*> \param[out] TAUQ
*> \verbatim
*>          TAUQ is DOUBLE PRECISION array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors which
*>          represent the orthogonal matrix Q. See Further Details.
*> \endverbatim
*>
*> \param[out] TAUP
*> \verbatim
*>          TAUP is DOUBLE PRECISION array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors which
*>          represent the orthogonal matrix P. See Further Details.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The length of the array WORK.
*>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MAX(M,N), otherwise.
*>          For optimum performance LWORK >= (M+N)*NB, where NB
*>          is the optimal blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gebrd
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrices Q and P are represented as products of elementary
*>  reflectors:
*>
*>  If m >= n,
*>
*>     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)
*>
*>  Each H(i) and G(i) has the form:
*>
*>     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T
*>
*>  where tauq and taup are real scalars, and v and u are real vectors;
*>  v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
*>  u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
*>  tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*>  If m < n,
*>
*>     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)
*>
*>  Each H(i) and G(i) has the form:
*>
*>     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T
*>
*>  where tauq and taup are real scalars, and v and u are real vectors;
*>  v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
*>  u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
*>  tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*>  The contents of A on exit are illustrated by the following examples:
*>
*>  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
*>
*>    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
*>    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
*>    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
*>    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
*>    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
*>    (  v1  v2  v3  v4  v5 )
*>
*>  where d and e denote diagonal and off-diagonal elements of B, vi
*>  denotes an element of the vector defining H(i), and ui an element of
*>  the vector defining G(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAUP( * ),
     $                   TAUQ( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IINFO, J, LDWRKX, LDWRKY, LWKMIN, LWKOPT,
     $                   MINMN, NB, NBMIN, NX, WS
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEBD2, DGEMM, DLABRD, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE, MAX, MIN
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INFO = 0
      MINMN = MIN( M, N )
      IF( MINMN.EQ.0 ) THEN
         LWKMIN = 1
         LWKOPT = 1
      ELSE
         LWKMIN = MAX( M, N )
         NB = MAX( 1, ILAENV( 1, 'DGEBRD', ' ', M, N, -1, -1 ) )
         LWKOPT = ( M+N )*NB
      ENDIF
      WORK( 1 ) = DBLE( LWKOPT )
*
      LQUERY = ( LWORK.EQ.-1 )
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
         INFO = -10
      END IF
      IF( INFO.LT.0 ) THEN
         CALL XERBLA( 'DGEBRD', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( MINMN.EQ.0 ) THEN
         WORK( 1 ) = 1
         RETURN
      END IF
*
      WS = MAX( M, N )
      LDWRKX = M
      LDWRKY = N
*
      IF( NB.GT.1 .AND. NB.LT.MINMN ) THEN
*
*        Set the crossover point NX.
*
         NX = MAX( NB, ILAENV( 3, 'DGEBRD', ' ', M, N, -1, -1 ) )
*
*        Determine when to switch from blocked to unblocked code.
*
         IF( NX.LT.MINMN ) THEN
            WS = LWKOPT
            IF( LWORK.LT.WS ) THEN
*
*              Not enough work space for the optimal NB, consider using
*              a smaller block size.
*
               NBMIN = ILAENV( 2, 'DGEBRD', ' ', M, N, -1, -1 )
               IF( LWORK.GE.( M+N )*NBMIN ) THEN
                  NB = LWORK / ( M+N )
               ELSE
                  NB = 1
                  NX = MINMN
               END IF
            END IF
         END IF
      ELSE
         NX = MINMN
      END IF
*
      DO 30 I = 1, MINMN - NX, NB
*
*        Reduce rows and columns i:i+nb-1 to bidiagonal form and return
*        the matrices X and Y which are needed to update the unreduced
*        part of the matrix
*
         CALL DLABRD( M-I+1, N-I+1, NB, A( I, I ), LDA, D( I ),
     $                E( I ),
     $                TAUQ( I ), TAUP( I ), WORK, LDWRKX,
     $                WORK( LDWRKX*NB+1 ), LDWRKY )
*
*        Update the trailing submatrix A(i+nb:m,i+nb:n), using an update
*        of the form  A := A - V*Y**T - X*U**T
*
         CALL DGEMM( 'No transpose', 'Transpose', M-I-NB+1, N-I-NB+1,
     $               NB, -ONE, A( I+NB, I ), LDA,
     $               WORK( LDWRKX*NB+NB+1 ), LDWRKY, ONE,
     $               A( I+NB, I+NB ), LDA )
         CALL DGEMM( 'No transpose', 'No transpose', M-I-NB+1,
     $               N-I-NB+1,
     $               NB, -ONE, WORK( NB+1 ), LDWRKX, A( I, I+NB ), LDA,
     $               ONE, A( I+NB, I+NB ), LDA )
*
*        Copy diagonal and off-diagonal elements of B back into A
*
         IF( M.GE.N ) THEN
            DO 10 J = I, I + NB - 1
               A( J, J ) = D( J )
               A( J, J+1 ) = E( J )
   10       CONTINUE
         ELSE
            DO 20 J = I, I + NB - 1
               A( J, J ) = D( J )
               A( J+1, J ) = E( J )
   20       CONTINUE
         END IF
   30 CONTINUE
*
*     Use unblocked code to reduce the remainder of the matrix
*
      CALL DGEBD2( M-I+1, N-I+1, A( I, I ), LDA, D( I ), E( I ),
     $             TAUQ( I ), TAUP( I ), WORK, IINFO )
      WORK( 1 ) = WS
      RETURN
*
*     End of DGEBRD
*
      END
*> \brief \b DGECON
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGECON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgecon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgecon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgecon.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGECON( NORM, N, A, LDA, ANORM, RCOND, WORK, IWORK,
*                          INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          NORM
*       INTEGER            INFO, LDA, N
*       DOUBLE PRECISION   ANORM, RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGECON estimates the reciprocal of the condition number of a general
*> real matrix A, in either the 1-norm or the infinity-norm, using
*> the LU factorization computed by DGETRF.
*>
*> An estimate is obtained for norm(inv(A)), and the reciprocal of the
*> condition number is computed as
*>    RCOND = 1 / ( norm(A) * norm(inv(A)) ).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NORM
*> \verbatim
*>          NORM is CHARACTER*1
*>          Specifies whether the 1-norm condition number or the
*>          infinity-norm condition number is required:
*>          = '1' or 'O':  1-norm;
*>          = 'I':         Infinity-norm.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The factors L and U from the factorization A = P*L*U
*>          as computed by DGETRF.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*>          ANORM is DOUBLE PRECISION
*>          If NORM = '1' or 'O', the 1-norm of the original matrix A.
*>          If NORM = 'I', the infinity-norm of the original matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The reciprocal of the condition number of the matrix A,
*>          computed as RCOND = 1/(norm(A) * norm(inv(A))).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>                NaNs are illegal values for ANORM, and they propagate to
*>                the output parameter RCOND.
*>                Infinity is illegal for ANORM, and it propagates to the output
*>                parameter RCOND as 0.
*>          = 1:  if RCOND = NaN, or
*>                   RCOND = Inf, or
*>                   the computed norm of the inverse of A is 0.
*>                In the latter, RCOND = 0 is returned.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gecon
*
*  =====================================================================
      SUBROUTINE DGECON( NORM, N, A, LDA, ANORM, RCOND, WORK, IWORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          NORM
      INTEGER            INFO, LDA, N
      DOUBLE PRECISION   ANORM, RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ONENRM
      CHARACTER          NORMIN
      INTEGER            IX, KASE, KASE1
      DOUBLE PRECISION   AINVNM, SCALE, SL, SMLNUM, SU, HUGEVAL
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, IDAMAX, DLAMCH, DISNAN
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACN2, DLATRS, DRSCL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. Executable Statements ..
*
      HUGEVAL = DLAMCH( 'Overflow' )
*
*     Test the input parameters.
*
      INFO = 0
      ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' )
      IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      ELSE IF( ANORM.LT.ZERO ) THEN
         INFO = -5
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGECON', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      RCOND = ZERO
      IF( N.EQ.0 ) THEN
         RCOND = ONE
         RETURN
      ELSE IF( ANORM.EQ.ZERO ) THEN
         RETURN
      ELSE IF( DISNAN( ANORM ) ) THEN
         RCOND = ANORM
         INFO = -5
         RETURN
      ELSE IF( ANORM.GT.HUGEVAL ) THEN
         INFO = -5
         RETURN
      END IF
*
      SMLNUM = DLAMCH( 'Safe minimum' )
*
*     Estimate the norm of inv(A).
*
      AINVNM = ZERO
      NORMIN = 'N'
      IF( ONENRM ) THEN
         KASE1 = 1
      ELSE
         KASE1 = 2
      END IF
      KASE = 0
   10 CONTINUE
      CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
      IF( KASE.NE.0 ) THEN
         IF( KASE.EQ.KASE1 ) THEN
*
*           Multiply by inv(L).
*
            CALL DLATRS( 'Lower', 'No transpose', 'Unit', NORMIN, N,
     $                   A,
     $                   LDA, WORK, SL, WORK( 2*N+1 ), INFO )
*
*           Multiply by inv(U).
*
            CALL DLATRS( 'Upper', 'No transpose', 'Non-unit', NORMIN,
     $                   N,
     $                   A, LDA, WORK, SU, WORK( 3*N+1 ), INFO )
         ELSE
*
*           Multiply by inv(U**T).
*
            CALL DLATRS( 'Upper', 'Transpose', 'Non-unit', NORMIN, N,
     $                   A,
     $                   LDA, WORK, SU, WORK( 3*N+1 ), INFO )
*
*           Multiply by inv(L**T).
*
            CALL DLATRS( 'Lower', 'Transpose', 'Unit', NORMIN, N, A,
     $                   LDA, WORK, SL, WORK( 2*N+1 ), INFO )
         END IF
*
*        Divide X by 1/(SL*SU) if doing so will not cause overflow.
*
         SCALE = SL*SU
         NORMIN = 'Y'
         IF( SCALE.NE.ONE ) THEN
            IX = IDAMAX( N, WORK, 1 )
            IF( SCALE.LT.ABS( WORK( IX ) )*SMLNUM .OR. SCALE.EQ.ZERO )
     $         GO TO 20
            CALL DRSCL( N, SCALE, WORK, 1 )
         END IF
         GO TO 10
      END IF
*
*     Compute the estimate of the reciprocal condition number.
*
      IF( AINVNM.NE.ZERO ) THEN
         RCOND = ( ONE / AINVNM ) / ANORM
      ELSE
         INFO = 1
         RETURN
      END IF
*
*     Check for NaNs and Infs
*
      IF( DISNAN( RCOND ) .OR. RCOND.GT.HUGEVAL )
     $   INFO = 1
*
   20 CONTINUE
      RETURN
*
*     End of DGECON
*
      END
*> \brief \b DGEEQU
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEEQU + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeequ.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeequ.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeequ.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEEQU( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
*                          INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, M, N
*       DOUBLE PRECISION   AMAX, COLCND, ROWCND
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), C( * ), R( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEEQU computes row and column scalings intended to equilibrate an
*> M-by-N matrix A and reduce its condition number.  R returns the row
*> scale factors and C the column scale factors, chosen to try to make
*> the largest element in each row and column of the matrix B with
*> elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.
*>
*> R(i) and C(j) are restricted to be between SMLNUM = smallest safe
*> number and BIGNUM = largest safe number.  Use of these scaling
*> factors is not guaranteed to reduce the condition number of A but
*> works well in practice.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The M-by-N matrix whose equilibration factors are
*>          to be computed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] R
*> \verbatim
*>          R is DOUBLE PRECISION array, dimension (M)
*>          If INFO = 0 or INFO > M, R contains the row scale factors
*>          for A.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0,  C contains the column scale factors for A.
*> \endverbatim
*>
*> \param[out] ROWCND
*> \verbatim
*>          ROWCND is DOUBLE PRECISION
*>          If INFO = 0 or INFO > M, ROWCND contains the ratio of the
*>          smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
*>          AMAX is neither too large nor too small, it is not worth
*>          scaling by R.
*> \endverbatim
*>
*> \param[out] COLCND
*> \verbatim
*>          COLCND is DOUBLE PRECISION
*>          If INFO = 0, COLCND contains the ratio of the smallest
*>          C(i) to the largest C(i).  If COLCND >= 0.1, it is not
*>          worth scaling by C.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*>          AMAX is DOUBLE PRECISION
*>          Absolute value of largest matrix element.  If AMAX is very
*>          close to overflow or very close to underflow, the matrix
*>          should be scaled.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i,  and i is
*>                <= M:  the i-th row of A is exactly zero
*>                >  M:  the (i-M)-th column of A is exactly zero
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geequ
*
*  =====================================================================
      SUBROUTINE DGEEQU( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
      DOUBLE PRECISION   AMAX, COLCND, ROWCND
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( * ), R( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J
      DOUBLE PRECISION   BIGNUM, RCMAX, RCMIN, SMLNUM
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEEQU', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 ) THEN
         ROWCND = ONE
         COLCND = ONE
         AMAX = ZERO
         RETURN
      END IF
*
*     Get machine constants.
*
      SMLNUM = DLAMCH( 'S' )
      BIGNUM = ONE / SMLNUM
*
*     Compute row scale factors.
*
      DO 10 I = 1, M
         R( I ) = ZERO
   10 CONTINUE
*
*     Find the maximum element in each row.
*
      DO 30 J = 1, N
         DO 20 I = 1, M
            R( I ) = MAX( R( I ), ABS( A( I, J ) ) )
   20    CONTINUE
   30 CONTINUE
*
*     Find the maximum and minimum scale factors.
*
      RCMIN = BIGNUM
      RCMAX = ZERO
      DO 40 I = 1, M
         RCMAX = MAX( RCMAX, R( I ) )
         RCMIN = MIN( RCMIN, R( I ) )
   40 CONTINUE
      AMAX = RCMAX
*
      IF( RCMIN.EQ.ZERO ) THEN
*
*        Find the first zero scale factor and return an error code.
*
         DO 50 I = 1, M
            IF( R( I ).EQ.ZERO ) THEN
               INFO = I
               RETURN
            END IF
   50    CONTINUE
      ELSE
*
*        Invert the scale factors.
*
         DO 60 I = 1, M
            R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
   60    CONTINUE
*
*        Compute ROWCND = min(R(I)) / max(R(I))
*
         ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
      END IF
*
*     Compute column scale factors
*
      DO 70 J = 1, N
         C( J ) = ZERO
   70 CONTINUE
*
*     Find the maximum element in each column,
*     assuming the row scaling computed above.
*
      DO 90 J = 1, N
         DO 80 I = 1, M
            C( J ) = MAX( C( J ), ABS( A( I, J ) )*R( I ) )
   80    CONTINUE
   90 CONTINUE
*
*     Find the maximum and minimum scale factors.
*
      RCMIN = BIGNUM
      RCMAX = ZERO
      DO 100 J = 1, N
         RCMIN = MIN( RCMIN, C( J ) )
         RCMAX = MAX( RCMAX, C( J ) )
  100 CONTINUE
*
      IF( RCMIN.EQ.ZERO ) THEN
*
*        Find the first zero scale factor and return an error code.
*
         DO 110 J = 1, N
            IF( C( J ).EQ.ZERO ) THEN
               INFO = M + J
               RETURN
            END IF
  110    CONTINUE
      ELSE
*
*        Invert the scale factors.
*
         DO 120 J = 1, N
            C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
  120    CONTINUE
*
*        Compute COLCND = min(C(J)) / max(C(J))
*
         COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
      END IF
*
      RETURN
*
*     End of DGEEQU
*
      END
*> \brief \b DGEEQUB
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEEQUB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeequb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeequb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeequb.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEEQUB( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
*                           INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, M, N
*       DOUBLE PRECISION   AMAX, COLCND, ROWCND
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), C( * ), R( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEEQUB computes row and column scalings intended to equilibrate an
*> M-by-N matrix A and reduce its condition number.  R returns the row
*> scale factors and C the column scale factors, chosen to try to make
*> the largest element in each row and column of the matrix B with
*> elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
*> the radix.
*>
*> R(i) and C(j) are restricted to be a power of the radix between
*> SMLNUM = smallest safe number and BIGNUM = largest safe number.  Use
*> of these scaling factors is not guaranteed to reduce the condition
*> number of A but works well in practice.
*>
*> This routine differs from DGEEQU by restricting the scaling factors
*> to a power of the radix.  Barring over- and underflow, scaling by
*> these factors introduces no additional rounding errors.  However, the
*> scaled entries' magnitudes are no longer approximately 1 but lie
*> between sqrt(radix) and 1/sqrt(radix).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The M-by-N matrix whose equilibration factors are
*>          to be computed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] R
*> \verbatim
*>          R is DOUBLE PRECISION array, dimension (M)
*>          If INFO = 0 or INFO > M, R contains the row scale factors
*>          for A.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0,  C contains the column scale factors for A.
*> \endverbatim
*>
*> \param[out] ROWCND
*> \verbatim
*>          ROWCND is DOUBLE PRECISION
*>          If INFO = 0 or INFO > M, ROWCND contains the ratio of the
*>          smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
*>          AMAX is neither too large nor too small, it is not worth
*>          scaling by R.
*> \endverbatim
*>
*> \param[out] COLCND
*> \verbatim
*>          COLCND is DOUBLE PRECISION
*>          If INFO = 0, COLCND contains the ratio of the smallest
*>          C(i) to the largest C(i).  If COLCND >= 0.1, it is not
*>          worth scaling by C.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*>          AMAX is DOUBLE PRECISION
*>          Absolute value of largest matrix element.  If AMAX is very
*>          close to overflow or very close to underflow, the matrix
*>          should be scaled.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i,  and i is
*>                <= M:  the i-th row of A is exactly zero
*>                >  M:  the (i-M)-th column of A is exactly zero
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geequb
*
*  =====================================================================
      SUBROUTINE DGEEQUB( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
     $                    INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
      DOUBLE PRECISION   AMAX, COLCND, ROWCND
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( * ), R( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J
      DOUBLE PRECISION   BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX, LOGRDX
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, LOG
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEEQUB', -INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( M.EQ.0 .OR. N.EQ.0 ) THEN
         ROWCND = ONE
         COLCND = ONE
         AMAX = ZERO
         RETURN
      END IF
*
*     Get machine constants.  Assume SMLNUM is a power of the radix.
*
      SMLNUM = DLAMCH( 'S' )
      BIGNUM = ONE / SMLNUM
      RADIX = DLAMCH( 'B' )
      LOGRDX = LOG( RADIX )
*
*     Compute row scale factors.
*
      DO 10 I = 1, M
         R( I ) = ZERO
   10 CONTINUE
*
*     Find the maximum element in each row.
*
      DO 30 J = 1, N
         DO 20 I = 1, M
            R( I ) = MAX( R( I ), ABS( A( I, J ) ) )
   20    CONTINUE
   30 CONTINUE
      DO I = 1, M
         IF( R( I ).GT.ZERO ) THEN
            R( I ) = RADIX**INT( LOG( R( I ) ) / LOGRDX )
         END IF
      END DO
*
*     Find the maximum and minimum scale factors.
*
      RCMIN = BIGNUM
      RCMAX = ZERO
      DO 40 I = 1, M
         RCMAX = MAX( RCMAX, R( I ) )
         RCMIN = MIN( RCMIN, R( I ) )
   40 CONTINUE
      AMAX = RCMAX
*
      IF( RCMIN.EQ.ZERO ) THEN
*
*        Find the first zero scale factor and return an error code.
*
         DO 50 I = 1, M
            IF( R( I ).EQ.ZERO ) THEN
               INFO = I
               RETURN
            END IF
   50    CONTINUE
      ELSE
*
*        Invert the scale factors.
*
         DO 60 I = 1, M
            R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
   60    CONTINUE
*
*        Compute ROWCND = min(R(I)) / max(R(I)).
*
         ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
      END IF
*
*     Compute column scale factors
*
      DO 70 J = 1, N
         C( J ) = ZERO
   70 CONTINUE
*
*     Find the maximum element in each column,
*     assuming the row scaling computed above.
*
      DO 90 J = 1, N
         DO 80 I = 1, M
            C( J ) = MAX( C( J ), ABS( A( I, J ) )*R( I ) )
   80    CONTINUE
         IF( C( J ).GT.ZERO ) THEN
            C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX )
         END IF
   90 CONTINUE
*
*     Find the maximum and minimum scale factors.
*
      RCMIN = BIGNUM
      RCMAX = ZERO
      DO 100 J = 1, N
         RCMIN = MIN( RCMIN, C( J ) )
         RCMAX = MAX( RCMAX, C( J ) )
  100 CONTINUE
*
      IF( RCMIN.EQ.ZERO ) THEN
*
*        Find the first zero scale factor and return an error code.
*
         DO 110 J = 1, N
            IF( C( J ).EQ.ZERO ) THEN
               INFO = M + J
               RETURN
            END IF
  110    CONTINUE
      ELSE
*
*        Invert the scale factors.
*
         DO 120 J = 1, N
            C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
  120    CONTINUE
*
*        Compute COLCND = min(C(J)) / max(C(J)).
*
         COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
      END IF
*
      RETURN
*
*     End of DGEEQUB
*
      END
*> \brief <b> DGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEES + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgees.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgees.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgees.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, WR, WI,
*                         VS, LDVS, WORK, LWORK, BWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBVS, SORT
*       INTEGER            INFO, LDA, LDVS, LWORK, N, SDIM
*       ..
*       .. Array Arguments ..
*       LOGICAL            BWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ),
*      $                   WR( * )
*       ..
*       .. Function Arguments ..
*       LOGICAL            SELECT
*       EXTERNAL           SELECT
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEES computes for an N-by-N real nonsymmetric matrix A, the
*> eigenvalues, the real Schur form T, and, optionally, the matrix of
*> Schur vectors Z.  This gives the Schur factorization A = Z*T*(Z**T).
*>
*> Optionally, it also orders the eigenvalues on the diagonal of the
*> real Schur form so that selected eigenvalues are at the top left.
*> The leading columns of Z then form an orthonormal basis for the
*> invariant subspace corresponding to the selected eigenvalues.
*>
*> A matrix is in real Schur form if it is upper quasi-triangular with
*> 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in the
*> form
*>         [  a  b  ]
*>         [  c  a  ]
*>
*> where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBVS
*> \verbatim
*>          JOBVS is CHARACTER*1
*>          = 'N': Schur vectors are not computed;
*>          = 'V': Schur vectors are computed.
*> \endverbatim
*>
*> \param[in] SORT
*> \verbatim
*>          SORT is CHARACTER*1
*>          Specifies whether or not to order the eigenvalues on the
*>          diagonal of the Schur form.
*>          = 'N': Eigenvalues are not ordered;
*>          = 'S': Eigenvalues are ordered (see SELECT).
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*>          SELECT is a LOGICAL FUNCTION of two DOUBLE PRECISION arguments
*>          SELECT must be declared EXTERNAL in the calling subroutine.
*>          If SORT = 'S', SELECT is used to select eigenvalues to sort
*>          to the top left of the Schur form.
*>          If SORT = 'N', SELECT is not referenced.
*>          An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
*>          SELECT(WR(j),WI(j)) is true; i.e., if either one of a complex
*>          conjugate pair of eigenvalues is selected, then both complex
*>          eigenvalues are selected.
*>          Note that a selected complex eigenvalue may no longer
*>          satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
*>          ordering may change the value of complex eigenvalues
*>          (especially if the eigenvalue is ill-conditioned); in this
*>          case INFO is set to N+2 (see INFO below).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the N-by-N matrix A.
*>          On exit, A has been overwritten by its real Schur form T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] SDIM
*> \verbatim
*>          SDIM is INTEGER
*>          If SORT = 'N', SDIM = 0.
*>          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
*>                         for which SELECT is true. (Complex conjugate
*>                         pairs for which SELECT is true for either
*>                         eigenvalue count as 2.)
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*>          WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*>          WI is DOUBLE PRECISION array, dimension (N)
*>          WR and WI contain the real and imaginary parts,
*>          respectively, of the computed eigenvalues in the same order
*>          that they appear on the diagonal of the output Schur form T.
*>          Complex conjugate pairs of eigenvalues will appear
*>          consecutively with the eigenvalue having the positive
*>          imaginary part first.
*> \endverbatim
*>
*> \param[out] VS
*> \verbatim
*>          VS is DOUBLE PRECISION array, dimension (LDVS,N)
*>          If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
*>          vectors.
*>          If JOBVS = 'N', VS is not referenced.
*> \endverbatim
*>
*> \param[in] LDVS
*> \verbatim
*>          LDVS is INTEGER
*>          The leading dimension of the array VS.  LDVS >= 1; if
*>          JOBVS = 'V', LDVS >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) contains the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.  LWORK >= max(1,3*N).
*>          For good performance, LWORK must generally be larger.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] BWORK
*> \verbatim
*>          BWORK is LOGICAL array, dimension (N)
*>          Not referenced if SORT = 'N'.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value.
*>          > 0: if INFO = i, and i is
*>             <= N: the QR algorithm failed to compute all the
*>                   eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI
*>                   contain those eigenvalues which have converged; if
*>                   JOBVS = 'V', VS contains the matrix which reduces A
*>                   to its partially converged Schur form.
*>             = N+1: the eigenvalues could not be reordered because some
*>                   eigenvalues were too close to separate (the problem
*>                   is very ill-conditioned);
*>             = N+2: after reordering, roundoff changed values of some
*>                   complex eigenvalues so that leading eigenvalues in
*>                   the Schur form no longer satisfy SELECT=.TRUE.  This
*>                   could also be caused by underflow due to scaling.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gees
*
*  =====================================================================
      SUBROUTINE DGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, WR, WI,
     $                  VS, LDVS, WORK, LWORK, BWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBVS, SORT
      INTEGER            INFO, LDA, LDVS, LWORK, N, SDIM
*     ..
*     .. Array Arguments ..
      LOGICAL            BWORK( * )
      DOUBLE PRECISION   A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ),
     $                   WR( * )
*     ..
*     .. Function Arguments ..
      LOGICAL            SELECT
      EXTERNAL           SELECT
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            CURSL, LASTSL, LQUERY, LST2SL, SCALEA, WANTST,
     $                   WANTVS
      INTEGER            HSWORK, I, I1, I2, IBAL, ICOND, IERR, IEVAL,
     $                   IHI, ILO, INXT, IP, ITAU, IWRK, MAXWRK, MINWRK
      DOUBLE PRECISION   ANRM, BIGNUM, CSCALE, EPS, S, SEP, SMLNUM
*     ..
*     .. Local Arrays ..
      INTEGER            IDUM( 1 )
      DOUBLE PRECISION   DUM( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGEBAK, DGEBAL, DGEHRD, DHSEQR,
     $                   DLACPY,
     $                   DLASCL, DORGHR, DSWAP, DTRSEN, XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           LSAME, ILAENV, DLAMCH, DLANGE
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      WANTVS = LSAME( JOBVS, 'V' )
      WANTST = LSAME( SORT, 'S' )
      IF( ( .NOT.WANTVS ) .AND. ( .NOT.LSAME( JOBVS, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( ( .NOT.WANTST ) .AND.
     $         ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -6
      ELSE IF( LDVS.LT.1 .OR. ( WANTVS .AND. LDVS.LT.N ) ) THEN
         INFO = -11
      END IF
*
*     Compute workspace
*      (Note: Comments in the code beginning "Workspace:" describe the
*       minimal amount of workspace needed at that point in the code,
*       as well as the preferred amount for good performance.
*       NB refers to the optimal block size for the immediately
*       following subroutine, as returned by ILAENV.
*       HSWORK refers to the workspace preferred by DHSEQR, as
*       calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
*       the worst case.)
*
      IF( INFO.EQ.0 ) THEN
         IF( N.EQ.0 ) THEN
            MINWRK = 1
            MAXWRK = 1
         ELSE
            MAXWRK = 2*N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
            MINWRK = 3*N
*
            CALL DHSEQR( 'S', JOBVS, N, 1, N, A, LDA, WR, WI, VS,
     $                   LDVS,
     $             WORK, -1, IEVAL )
            HSWORK = INT( WORK( 1 ) )
*
            IF( .NOT.WANTVS ) THEN
               MAXWRK = MAX( MAXWRK, N + HSWORK )
            ELSE
               MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
     $                       'DORGHR', ' ', N, 1, N, -1 ) )
               MAXWRK = MAX( MAXWRK, N + HSWORK )
            END IF
         END IF
         WORK( 1 ) = MAXWRK
*
         IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
            INFO = -13
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEES ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 ) THEN
         SDIM = 0
         RETURN
      END IF
*
*     Get machine constants
*
      EPS = DLAMCH( 'P' )
      SMLNUM = DLAMCH( 'S' )
      BIGNUM = ONE / SMLNUM
      SMLNUM = SQRT( SMLNUM ) / EPS
      BIGNUM = ONE / SMLNUM
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      ANRM = DLANGE( 'M', N, N, A, LDA, DUM )
      SCALEA = .FALSE.
      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
         SCALEA = .TRUE.
         CSCALE = SMLNUM
      ELSE IF( ANRM.GT.BIGNUM ) THEN
         SCALEA = .TRUE.
         CSCALE = BIGNUM
      END IF
      IF( SCALEA )
     $   CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
*
*     Permute the matrix to make it more nearly triangular
*     (Workspace: need N)
*
      IBAL = 1
      CALL DGEBAL( 'P', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR )
*
*     Reduce to upper Hessenberg form
*     (Workspace: need 3*N, prefer 2*N+N*NB)
*
      ITAU = N + IBAL
      IWRK = N + ITAU
      CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
     $             LWORK-IWRK+1, IERR )
*
      IF( WANTVS ) THEN
*
*        Copy Householder vectors to VS
*
         CALL DLACPY( 'L', N, N, A, LDA, VS, LDVS )
*
*        Generate orthogonal matrix in VS
*        (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*
         CALL DORGHR( N, ILO, IHI, VS, LDVS, WORK( ITAU ),
     $                WORK( IWRK ),
     $                LWORK-IWRK+1, IERR )
      END IF
*
      SDIM = 0
*
*     Perform QR iteration, accumulating Schur vectors in VS if desired
*     (Workspace: need N+1, prefer N+HSWORK (see comments) )
*
      IWRK = ITAU
      CALL DHSEQR( 'S', JOBVS, N, ILO, IHI, A, LDA, WR, WI, VS, LDVS,
     $             WORK( IWRK ), LWORK-IWRK+1, IEVAL )
      IF( IEVAL.GT.0 )
     $   INFO = IEVAL
*
*     Sort eigenvalues if desired
*
      IF( WANTST .AND. INFO.EQ.0 ) THEN
         IF( SCALEA ) THEN
            CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WR, N, IERR )
            CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WI, N, IERR )
         END IF
         DO 10 I = 1, N
            BWORK( I ) = SELECT( WR( I ), WI( I ) )
   10    CONTINUE
*
*        Reorder eigenvalues and transform Schur vectors
*        (Workspace: none needed)
*
         CALL DTRSEN( 'N', JOBVS, BWORK, N, A, LDA, VS, LDVS, WR, WI,
     $                SDIM, S, SEP, WORK( IWRK ), LWORK-IWRK+1, IDUM, 1,
     $                ICOND )
         IF( ICOND.GT.0 )
     $      INFO = N + ICOND
      END IF
*
      IF( WANTVS ) THEN
*
*        Undo balancing
*        (Workspace: need N)
*
         CALL DGEBAK( 'P', 'R', N, ILO, IHI, WORK( IBAL ), N, VS,
     $                LDVS,
     $                IERR )
      END IF
*
      IF( SCALEA ) THEN
*
*        Undo scaling for the Schur form of A
*
         CALL DLASCL( 'H', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR )
         CALL DCOPY( N, A, LDA+1, WR, 1 )
         IF( CSCALE.EQ.SMLNUM ) THEN
*
*           If scaling back towards underflow, adjust WI if an
*           offdiagonal element of a 2-by-2 block in the Schur form
*           underflows.
*
            IF( IEVAL.GT.0 ) THEN
               I1 = IEVAL + 1
               I2 = IHI - 1
               CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI,
     $                      MAX( ILO-1, 1 ), IERR )
            ELSE IF( WANTST ) THEN
               I1 = 1
               I2 = N - 1
            ELSE
               I1 = ILO
               I2 = IHI - 1
            END IF
            INXT = I1 - 1
            DO 20 I = I1, I2
               IF( I.LT.INXT )
     $            GO TO 20
               IF( WI( I ).EQ.ZERO ) THEN
                  INXT = I + 1
               ELSE
                  IF( A( I+1, I ).EQ.ZERO ) THEN
                     WI( I ) = ZERO
                     WI( I+1 ) = ZERO
                  ELSE IF( A( I+1, I ).NE.ZERO .AND. A( I, I+1 ).EQ.
     $                     ZERO ) THEN
                     WI( I ) = ZERO
                     WI( I+1 ) = ZERO
                     IF( I.GT.1 )
     $                  CALL DSWAP( I-1, A( 1, I ), 1, A( 1, I+1 ),
     $                              1 )
                     IF( N.GT.I+1 )
     $                  CALL DSWAP( N-I-1, A( I, I+2 ), LDA,
     $                              A( I+1, I+2 ), LDA )
                     IF( WANTVS ) THEN
                        CALL DSWAP( N, VS( 1, I ), 1, VS( 1, I+1 ),
     $                              1 )
                     END IF
                     A( I, I+1 ) = A( I+1, I )
                     A( I+1, I ) = ZERO
                  END IF
                  INXT = I + 2
               END IF
   20       CONTINUE
         END IF
*
*        Undo scaling for the imaginary part of the eigenvalues
*
         CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-IEVAL, 1,
     $                WI( IEVAL+1 ), MAX( N-IEVAL, 1 ), IERR )
      END IF
*
      IF( WANTST .AND. INFO.EQ.0 ) THEN
*
*        Check if reordering successful
*
         LASTSL = .TRUE.
         LST2SL = .TRUE.
         SDIM = 0
         IP = 0
         DO 30 I = 1, N
            CURSL = SELECT( WR( I ), WI( I ) )
            IF( WI( I ).EQ.ZERO ) THEN
               IF( CURSL )
     $            SDIM = SDIM + 1
               IP = 0
               IF( CURSL .AND. .NOT.LASTSL )
     $            INFO = N + 2
            ELSE
               IF( IP.EQ.1 ) THEN
*
*                 Last eigenvalue of conjugate pair
*
                  CURSL = CURSL .OR. LASTSL
                  LASTSL = CURSL
                  IF( CURSL )
     $               SDIM = SDIM + 2
                  IP = -1
                  IF( CURSL .AND. .NOT.LST2SL )
     $               INFO = N + 2
               ELSE
*
*                 First eigenvalue of conjugate pair
*
                  IP = 1
               END IF
            END IF
            LST2SL = LASTSL
            LASTSL = CURSL
   30    CONTINUE
      END IF
*
      WORK( 1 ) = MAXWRK
      RETURN
*
*     End of DGEES
*
      END
*> \brief <b> DGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEESX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeesx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeesx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeesx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM,
*                          WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK,
*                          IWORK, LIWORK, BWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBVS, SENSE, SORT
*       INTEGER            INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM
*       DOUBLE PRECISION   RCONDE, RCONDV
*       ..
*       .. Array Arguments ..
*       LOGICAL            BWORK( * )
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ),
*      $                   WR( * )
*       ..
*       .. Function Arguments ..
*       LOGICAL            SELECT
*       EXTERNAL           SELECT
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEESX computes for an N-by-N real nonsymmetric matrix A, the
*> eigenvalues, the real Schur form T, and, optionally, the matrix of
*> Schur vectors Z.  This gives the Schur factorization A = Z*T*(Z**T).
*>
*> Optionally, it also orders the eigenvalues on the diagonal of the
*> real Schur form so that selected eigenvalues are at the top left;
*> computes a reciprocal condition number for the average of the
*> selected eigenvalues (RCONDE); and computes a reciprocal condition
*> number for the right invariant subspace corresponding to the
*> selected eigenvalues (RCONDV).  The leading columns of Z form an
*> orthonormal basis for this invariant subspace.
*>
*> For further explanation of the reciprocal condition numbers RCONDE
*> and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
*> these quantities are called s and sep respectively).
*>
*> A real matrix is in real Schur form if it is upper quasi-triangular
*> with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in
*> the form
*>           [  a  b  ]
*>           [  c  a  ]
*>
*> where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBVS
*> \verbatim
*>          JOBVS is CHARACTER*1
*>          = 'N': Schur vectors are not computed;
*>          = 'V': Schur vectors are computed.
*> \endverbatim
*>
*> \param[in] SORT
*> \verbatim
*>          SORT is CHARACTER*1
*>          Specifies whether or not to order the eigenvalues on the
*>          diagonal of the Schur form.
*>          = 'N': Eigenvalues are not ordered;
*>          = 'S': Eigenvalues are ordered (see SELECT).
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*>          SELECT is a LOGICAL FUNCTION of two DOUBLE PRECISION arguments
*>          SELECT must be declared EXTERNAL in the calling subroutine.
*>          If SORT = 'S', SELECT is used to select eigenvalues to sort
*>          to the top left of the Schur form.
*>          If SORT = 'N', SELECT is not referenced.
*>          An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
*>          SELECT(WR(j),WI(j)) is true; i.e., if either one of a
*>          complex conjugate pair of eigenvalues is selected, then both
*>          are.  Note that a selected complex eigenvalue may no longer
*>          satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
*>          ordering may change the value of complex eigenvalues
*>          (especially if the eigenvalue is ill-conditioned); in this
*>          case INFO may be set to N+3 (see INFO below).
*> \endverbatim
*>
*> \param[in] SENSE
*> \verbatim
*>          SENSE is CHARACTER*1
*>          Determines which reciprocal condition numbers are computed.
*>          = 'N': None are computed;
*>          = 'E': Computed for average of selected eigenvalues only;
*>          = 'V': Computed for selected right invariant subspace only;
*>          = 'B': Computed for both.
*>          If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, N)
*>          On entry, the N-by-N matrix A.
*>          On exit, A is overwritten by its real Schur form T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] SDIM
*> \verbatim
*>          SDIM is INTEGER
*>          If SORT = 'N', SDIM = 0.
*>          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
*>                         for which SELECT is true. (Complex conjugate
*>                         pairs for which SELECT is true for either
*>                         eigenvalue count as 2.)
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*>          WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*>          WI is DOUBLE PRECISION array, dimension (N)
*>          WR and WI contain the real and imaginary parts, respectively,
*>          of the computed eigenvalues, in the same order that they
*>          appear on the diagonal of the output Schur form T.  Complex
*>          conjugate pairs of eigenvalues appear consecutively with the
*>          eigenvalue having the positive imaginary part first.
*> \endverbatim
*>
*> \param[out] VS
*> \verbatim
*>          VS is DOUBLE PRECISION array, dimension (LDVS,N)
*>          If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
*>          vectors.
*>          If JOBVS = 'N', VS is not referenced.
*> \endverbatim
*>
*> \param[in] LDVS
*> \verbatim
*>          LDVS is INTEGER
*>          The leading dimension of the array VS.  LDVS >= 1, and if
*>          JOBVS = 'V', LDVS >= N.
*> \endverbatim
*>
*> \param[out] RCONDE
*> \verbatim
*>          RCONDE is DOUBLE PRECISION
*>          If SENSE = 'E' or 'B', RCONDE contains the reciprocal
*>          condition number for the average of the selected eigenvalues.
*>          Not referenced if SENSE = 'N' or 'V'.
*> \endverbatim
*>
*> \param[out] RCONDV
*> \verbatim
*>          RCONDV is DOUBLE PRECISION
*>          If SENSE = 'V' or 'B', RCONDV contains the reciprocal
*>          condition number for the selected right invariant subspace.
*>          Not referenced if SENSE = 'N' or 'E'.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.  LWORK >= max(1,3*N).
*>          Also, if SENSE = 'E' or 'V' or 'B',
*>          LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of
*>          selected eigenvalues computed by this routine.  Note that
*>          N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only
*>          returned if LWORK < max(1,3*N), but if SENSE = 'E' or 'V' or
*>          'B' this may not be large enough.
*>          For good performance, LWORK must generally be larger.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates upper bounds on the optimal sizes of the
*>          arrays WORK and IWORK, returns these values as the first
*>          entries of the WORK and IWORK arrays, and no error messages
*>          related to LWORK or LIWORK are issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK.
*>          LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).
*>          Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is
*>          only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this
*>          may not be large enough.
*>
*>          If LIWORK = -1, then a workspace query is assumed; the
*>          routine only calculates upper bounds on the optimal sizes of
*>          the arrays WORK and IWORK, returns these values as the first
*>          entries of the WORK and IWORK arrays, and no error messages
*>          related to LWORK or LIWORK are issued by XERBLA.
*> \endverbatim
*>
*> \param[out] BWORK
*> \verbatim
*>          BWORK is LOGICAL array, dimension (N)
*>          Not referenced if SORT = 'N'.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value.
*>          > 0: if INFO = i, and i is
*>             <= N: the QR algorithm failed to compute all the
*>                   eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI
*>                   contain those eigenvalues which have converged; if
*>                   JOBVS = 'V', VS contains the transformation which
*>                   reduces A to its partially converged Schur form.
*>             = N+1: the eigenvalues could not be reordered because some
*>                   eigenvalues were too close to separate (the problem
*>                   is very ill-conditioned);
*>             = N+2: after reordering, roundoff changed values of some
*>                   complex eigenvalues so that leading eigenvalues in
*>                   the Schur form no longer satisfy SELECT=.TRUE.  This
*>                   could also be caused by underflow due to scaling.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geesx
*
*  =====================================================================
      SUBROUTINE DGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM,
     $                   WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK,
     $                   IWORK, LIWORK, BWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBVS, SENSE, SORT
      INTEGER            INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM
      DOUBLE PRECISION   RCONDE, RCONDV
*     ..
*     .. Array Arguments ..
      LOGICAL            BWORK( * )
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ),
     $                   WR( * )
*     ..
*     .. Function Arguments ..
      LOGICAL            SELECT
      EXTERNAL           SELECT
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            CURSL, LASTSL, LQUERY, LST2SL, SCALEA, WANTSB,
     $                   WANTSE, WANTSN, WANTST, WANTSV, WANTVS
      INTEGER            HSWORK, I, I1, I2, IBAL, ICOND, IERR, IEVAL,
     $                   IHI, ILO, INXT, IP, ITAU, IWRK, LIWRK, LWRK,
     $                   MAXWRK, MINWRK
      DOUBLE PRECISION   ANRM, BIGNUM, CSCALE, EPS, SMLNUM
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   DUM( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGEBAK, DGEBAL, DGEHRD, DHSEQR,
     $                   DLACPY,
     $                   DLASCL, DORGHR, DSWAP, DTRSEN, XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           LSAME, ILAENV, DLAMCH, DLANGE
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      WANTVS = LSAME( JOBVS, 'V' )
      WANTST = LSAME( SORT, 'S' )
      WANTSN = LSAME( SENSE, 'N' )
      WANTSE = LSAME( SENSE, 'E' )
      WANTSV = LSAME( SENSE, 'V' )
      WANTSB = LSAME( SENSE, 'B' )
      LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
      IF( ( .NOT.WANTVS ) .AND. ( .NOT.LSAME( JOBVS, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( ( .NOT.WANTST ) .AND.
     $         ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
         INFO = -2
      ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSV .OR. WANTSB ) .OR.
     $         ( .NOT.WANTST .AND. .NOT.WANTSN ) ) THEN
         INFO = -4
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDVS.LT.1 .OR. ( WANTVS .AND. LDVS.LT.N ) ) THEN
         INFO = -12
      END IF
*
*     Compute workspace
*      (Note: Comments in the code beginning "RWorkspace:" describe the
*       minimal amount of real workspace needed at that point in the
*       code, as well as the preferred amount for good performance.
*       IWorkspace refers to integer workspace.
*       NB refers to the optimal block size for the immediately
*       following subroutine, as returned by ILAENV.
*       HSWORK refers to the workspace preferred by DHSEQR, as
*       calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
*       the worst case.
*       If SENSE = 'E', 'V' or 'B', then the amount of workspace needed
*       depends on SDIM, which is computed by the routine DTRSEN later
*       in the code.)
*
      IF( INFO.EQ.0 ) THEN
         LIWRK = 1
         IF( N.EQ.0 ) THEN
            MINWRK = 1
            LWRK = 1
         ELSE
            MAXWRK = 2*N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
            MINWRK = 3*N
*
            CALL DHSEQR( 'S', JOBVS, N, 1, N, A, LDA, WR, WI, VS,
     $                   LDVS,
     $             WORK, -1, IEVAL )
            HSWORK = INT( WORK( 1 ) )
*
            IF( .NOT.WANTVS ) THEN
               MAXWRK = MAX( MAXWRK, N + HSWORK )
            ELSE
               MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
     $                       'DORGHR', ' ', N, 1, N, -1 ) )
               MAXWRK = MAX( MAXWRK, N + HSWORK )
            END IF
            LWRK = MAXWRK
            IF( .NOT.WANTSN )
     $         LWRK = MAX( LWRK, N + ( N*N )/2 )
            IF( WANTSV .OR. WANTSB )
     $         LIWRK = ( N*N )/4
         END IF
         IWORK( 1 ) = LIWRK
         WORK( 1 ) = LWRK
*
         IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
            INFO = -16
         ELSE IF( LIWORK.LT.1 .AND. .NOT.LQUERY ) THEN
            INFO = -18
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEESX', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 ) THEN
         SDIM = 0
         RETURN
      END IF
*
*     Get machine constants
*
      EPS = DLAMCH( 'P' )
      SMLNUM = DLAMCH( 'S' )
      BIGNUM = ONE / SMLNUM
      SMLNUM = SQRT( SMLNUM ) / EPS
      BIGNUM = ONE / SMLNUM
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      ANRM = DLANGE( 'M', N, N, A, LDA, DUM )
      SCALEA = .FALSE.
      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
         SCALEA = .TRUE.
         CSCALE = SMLNUM
      ELSE IF( ANRM.GT.BIGNUM ) THEN
         SCALEA = .TRUE.
         CSCALE = BIGNUM
      END IF
      IF( SCALEA )
     $   CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
*
*     Permute the matrix to make it more nearly triangular
*     (RWorkspace: need N)
*
      IBAL = 1
      CALL DGEBAL( 'P', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR )
*
*     Reduce to upper Hessenberg form
*     (RWorkspace: need 3*N, prefer 2*N+N*NB)
*
      ITAU = N + IBAL
      IWRK = N + ITAU
      CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
     $             LWORK-IWRK+1, IERR )
*
      IF( WANTVS ) THEN
*
*        Copy Householder vectors to VS
*
         CALL DLACPY( 'L', N, N, A, LDA, VS, LDVS )
*
*        Generate orthogonal matrix in VS
*        (RWorkspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*
         CALL DORGHR( N, ILO, IHI, VS, LDVS, WORK( ITAU ),
     $                WORK( IWRK ),
     $                LWORK-IWRK+1, IERR )
      END IF
*
      SDIM = 0
*
*     Perform QR iteration, accumulating Schur vectors in VS if desired
*     (RWorkspace: need N+1, prefer N+HSWORK (see comments) )
*
      IWRK = ITAU
      CALL DHSEQR( 'S', JOBVS, N, ILO, IHI, A, LDA, WR, WI, VS, LDVS,
     $             WORK( IWRK ), LWORK-IWRK+1, IEVAL )
      IF( IEVAL.GT.0 )
     $   INFO = IEVAL
*
*     Sort eigenvalues if desired
*
      IF( WANTST .AND. INFO.EQ.0 ) THEN
         IF( SCALEA ) THEN
            CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WR, N, IERR )
            CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WI, N, IERR )
         END IF
         DO 10 I = 1, N
            BWORK( I ) = SELECT( WR( I ), WI( I ) )
   10    CONTINUE
*
*        Reorder eigenvalues, transform Schur vectors, and compute
*        reciprocal condition numbers
*        (RWorkspace: if SENSE is not 'N', need N+2*SDIM*(N-SDIM)
*                     otherwise, need N )
*        (IWorkspace: if SENSE is 'V' or 'B', need SDIM*(N-SDIM)
*                     otherwise, need 0 )
*
         CALL DTRSEN( SENSE, JOBVS, BWORK, N, A, LDA, VS, LDVS, WR,
     $                WI,
     $                SDIM, RCONDE, RCONDV, WORK( IWRK ), LWORK-IWRK+1,
     $                IWORK, LIWORK, ICOND )
         IF( .NOT.WANTSN )
     $      MAXWRK = MAX( MAXWRK, N+2*SDIM*( N-SDIM ) )
         IF( ICOND.EQ.-15 ) THEN
*
*           Not enough real workspace
*
            INFO = -16
         ELSE IF( ICOND.EQ.-17 ) THEN
*
*           Not enough integer workspace
*
            INFO = -18
         ELSE IF( ICOND.GT.0 ) THEN
*
*           DTRSEN failed to reorder or to restore standard Schur form
*
            INFO = ICOND + N
         END IF
      END IF
*
      IF( WANTVS ) THEN
*
*        Undo balancing
*        (RWorkspace: need N)
*
         CALL DGEBAK( 'P', 'R', N, ILO, IHI, WORK( IBAL ), N, VS,
     $                LDVS,
     $                IERR )
      END IF
*
      IF( SCALEA ) THEN
*
*        Undo scaling for the Schur form of A
*
         CALL DLASCL( 'H', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR )
         CALL DCOPY( N, A, LDA+1, WR, 1 )
         IF( ( WANTSV .OR. WANTSB ) .AND. INFO.EQ.0 ) THEN
            DUM( 1 ) = RCONDV
            CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1,
     $                   IERR )
            RCONDV = DUM( 1 )
         END IF
         IF( CSCALE.EQ.SMLNUM ) THEN
*
*           If scaling back towards underflow, adjust WI if an
*           offdiagonal element of a 2-by-2 block in the Schur form
*           underflows.
*
            IF( IEVAL.GT.0 ) THEN
               I1 = IEVAL + 1
               I2 = IHI - 1
               CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N,
     $                      IERR )
            ELSE IF( WANTST ) THEN
               I1 = 1
               I2 = N - 1
            ELSE
               I1 = ILO
               I2 = IHI - 1
            END IF
            INXT = I1 - 1
            DO 20 I = I1, I2
               IF( I.LT.INXT )
     $            GO TO 20
               IF( WI( I ).EQ.ZERO ) THEN
                  INXT = I + 1
               ELSE
                  IF( A( I+1, I ).EQ.ZERO ) THEN
                     WI( I ) = ZERO
                     WI( I+1 ) = ZERO
                  ELSE IF( A( I+1, I ).NE.ZERO .AND. A( I, I+1 ).EQ.
     $                     ZERO ) THEN
                     WI( I ) = ZERO
                     WI( I+1 ) = ZERO
                     IF( I.GT.1 )
     $                  CALL DSWAP( I-1, A( 1, I ), 1, A( 1, I+1 ),
     $                              1 )
                     IF( N.GT.I+1 )
     $                  CALL DSWAP( N-I-1, A( I, I+2 ), LDA,
     $                              A( I+1, I+2 ), LDA )
                     IF( WANTVS ) THEN
                       CALL DSWAP( N, VS( 1, I ), 1, VS( 1, I+1 ),
     $                             1 )
                     END IF
                     A( I, I+1 ) = A( I+1, I )
                     A( I+1, I ) = ZERO
                  END IF
                  INXT = I + 2
               END IF
   20       CONTINUE
         END IF
         CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-IEVAL, 1,
     $                WI( IEVAL+1 ), MAX( N-IEVAL, 1 ), IERR )
      END IF
*
      IF( WANTST .AND. INFO.EQ.0 ) THEN
*
*        Check if reordering successful
*
         LASTSL = .TRUE.
         LST2SL = .TRUE.
         SDIM = 0
         IP = 0
         DO 30 I = 1, N
            CURSL = SELECT( WR( I ), WI( I ) )
            IF( WI( I ).EQ.ZERO ) THEN
               IF( CURSL )
     $            SDIM = SDIM + 1
               IP = 0
               IF( CURSL .AND. .NOT.LASTSL )
     $            INFO = N + 2
            ELSE
               IF( IP.EQ.1 ) THEN
*
*                 Last eigenvalue of conjugate pair
*
                  CURSL = CURSL .OR. LASTSL
                  LASTSL = CURSL
                  IF( CURSL )
     $               SDIM = SDIM + 2
                  IP = -1
                  IF( CURSL .AND. .NOT.LST2SL )
     $               INFO = N + 2
               ELSE
*
*                 First eigenvalue of conjugate pair
*
                  IP = 1
               END IF
            END IF
            LST2SL = LASTSL
            LASTSL = CURSL
   30    CONTINUE
      END IF
*
      WORK( 1 ) = MAXWRK
      IF( WANTSV .OR. WANTSB ) THEN
         IWORK( 1 ) = MAX( 1, SDIM*( N-SDIM ) )
      ELSE
         IWORK( 1 ) = 1
      END IF
*
      RETURN
*
*     End of DGEESX
*
      END
*> \brief <b> DGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEEV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeev.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeev.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeev.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR,
*                         LDVR, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBVL, JOBVR
*       INTEGER            INFO, LDA, LDVL, LDVR, LWORK, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
*      $                   WI( * ), WORK( * ), WR( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEEV computes for an N-by-N real nonsymmetric matrix A, the
*> eigenvalues and, optionally, the left and/or right eigenvectors.
*>
*> The right eigenvector v(j) of A satisfies
*>                  A * v(j) = lambda(j) * v(j)
*> where lambda(j) is its eigenvalue.
*> The left eigenvector u(j) of A satisfies
*>               u(j)**H * A = lambda(j) * u(j)**H
*> where u(j)**H denotes the conjugate-transpose of u(j).
*>
*> The computed eigenvectors are normalized to have Euclidean norm
*> equal to 1 and largest component real.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBVL
*> \verbatim
*>          JOBVL is CHARACTER*1
*>          = 'N': left eigenvectors of A are not computed;
*>          = 'V': left eigenvectors of A are computed.
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*>          JOBVR is CHARACTER*1
*>          = 'N': right eigenvectors of A are not computed;
*>          = 'V': right eigenvectors of A are computed.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the N-by-N matrix A.
*>          On exit, A has been overwritten.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*>          WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*>          WI is DOUBLE PRECISION array, dimension (N)
*>          WR and WI contain the real and imaginary parts,
*>          respectively, of the computed eigenvalues.  Complex
*>          conjugate pairs of eigenvalues appear consecutively
*>          with the eigenvalue having the positive imaginary part
*>          first.
*> \endverbatim
*>
*> \param[out] VL
*> \verbatim
*>          VL is DOUBLE PRECISION array, dimension (LDVL,N)
*>          If JOBVL = 'V', the left eigenvectors u(j) are stored one
*>          after another in the columns of VL, in the same order
*>          as their eigenvalues.
*>          If JOBVL = 'N', VL is not referenced.
*>          If the j-th eigenvalue is real, then u(j) = VL(:,j),
*>          the j-th column of VL.
*>          If the j-th and (j+1)-st eigenvalues form a complex
*>          conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
*>          u(j+1) = VL(:,j) - i*VL(:,j+1).
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*>          LDVL is INTEGER
*>          The leading dimension of the array VL.  LDVL >= 1; if
*>          JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*>          VR is DOUBLE PRECISION array, dimension (LDVR,N)
*>          If JOBVR = 'V', the right eigenvectors v(j) are stored one
*>          after another in the columns of VR, in the same order
*>          as their eigenvalues.
*>          If JOBVR = 'N', VR is not referenced.
*>          If the j-th eigenvalue is real, then v(j) = VR(:,j),
*>          the j-th column of VR.
*>          If the j-th and (j+1)-st eigenvalues form a complex
*>          conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
*>          v(j+1) = VR(:,j) - i*VR(:,j+1).
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*>          LDVR is INTEGER
*>          The leading dimension of the array VR.  LDVR >= 1; if
*>          JOBVR = 'V', LDVR >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.  LWORK >= max(1,3*N), and
*>          if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N.  For good
*>          performance, LWORK must generally be larger.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = i, the QR algorithm failed to compute all the
*>                eigenvalues, and no eigenvectors have been computed;
*>                elements i+1:N of WR and WI contain eigenvalues which
*>                have converged.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*
*  @precisions fortran d -> s
*
*> \ingroup geev
*
*  =====================================================================
      SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL,
     $                  VR,
     $                  LDVR, WORK, LWORK, INFO )
      implicit none
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBVL, JOBVR
      INTEGER            INFO, LDA, LDVL, LDVR, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
     $                   WI( * ), WORK( * ), WR( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, SCALEA, WANTVL, WANTVR
      CHARACTER          SIDE
      INTEGER            HSWORK, I, IBAL, IERR, IHI, ILO, ITAU, IWRK, K,
     $                   LWORK_TREVC, MAXWRK, MINWRK, NOUT
      DOUBLE PRECISION   ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM,
     $                   SN
*     ..
*     .. Local Arrays ..
      LOGICAL            SELECT( 1 )
      DOUBLE PRECISION   DUM( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLACPY,
     $                   DLARTG,
     $                   DLASCL, DORGHR, DROT, DSCAL, DTREVC3, XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX, ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE, DLAPY2, DNRM2
      EXTERNAL           LSAME, IDAMAX, ILAENV, DLAMCH, DLANGE,
     $                   DLAPY2,
     $                   DNRM2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      WANTVL = LSAME( JOBVL, 'V' )
      WANTVR = LSAME( JOBVR, 'V' )
      IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( ( .NOT.WANTVR ) .AND.
     $         ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
         INFO = -9
      ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
         INFO = -11
      END IF
*
*     Compute workspace
*      (Note: Comments in the code beginning "Workspace:" describe the
*       minimal amount of workspace needed at that point in the code,
*       as well as the preferred amount for good performance.
*       NB refers to the optimal block size for the immediately
*       following subroutine, as returned by ILAENV.
*       HSWORK refers to the workspace preferred by DHSEQR, as
*       calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
*       the worst case.)
*
      IF( INFO.EQ.0 ) THEN
         IF( N.EQ.0 ) THEN
            MINWRK = 1
            MAXWRK = 1
         ELSE
            MAXWRK = 2*N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
            IF( WANTVL ) THEN
               MINWRK = 4*N
               MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
     $                       'DORGHR', ' ', N, 1, N, -1 ) )
               CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL,
     $                      LDVL,
     $                      WORK, -1, INFO )
               HSWORK = INT( WORK(1) )
               MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
               CALL DTREVC3( 'L', 'B', SELECT, N, A, LDA,
     $                       VL, LDVL, VR, LDVR, N, NOUT,
     $                       WORK, -1, IERR )
               LWORK_TREVC = INT( WORK(1) )
               MAXWRK = MAX( MAXWRK, N + LWORK_TREVC )
               MAXWRK = MAX( MAXWRK, 4*N )
            ELSE IF( WANTVR ) THEN
               MINWRK = 4*N
               MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
     $                       'DORGHR', ' ', N, 1, N, -1 ) )
               CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR,
     $                      LDVR,
     $                      WORK, -1, INFO )
               HSWORK = INT( WORK(1) )
               MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
               CALL DTREVC3( 'R', 'B', SELECT, N, A, LDA,
     $                       VL, LDVL, VR, LDVR, N, NOUT,
     $                       WORK, -1, IERR )
               LWORK_TREVC = INT( WORK(1) )
               MAXWRK = MAX( MAXWRK, N + LWORK_TREVC )
               MAXWRK = MAX( MAXWRK, 4*N )
            ELSE
               MINWRK = 3*N
               CALL DHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR,
     $                      LDVR,
     $                      WORK, -1, INFO )
               HSWORK = INT( WORK(1) )
               MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
            END IF
            MAXWRK = MAX( MAXWRK, MINWRK )
         END IF
         WORK( 1 ) = MAXWRK
*
         IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
            INFO = -13
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEEV ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Get machine constants
*
      EPS = DLAMCH( 'P' )
      SMLNUM = DLAMCH( 'S' )
      BIGNUM = ONE / SMLNUM
      SMLNUM = SQRT( SMLNUM ) / EPS
      BIGNUM = ONE / SMLNUM
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      ANRM = DLANGE( 'M', N, N, A, LDA, DUM )
      SCALEA = .FALSE.
      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
         SCALEA = .TRUE.
         CSCALE = SMLNUM
      ELSE IF( ANRM.GT.BIGNUM ) THEN
         SCALEA = .TRUE.
         CSCALE = BIGNUM
      END IF
      IF( SCALEA )
     $   CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
*
*     Balance the matrix
*     (Workspace: need N)
*
      IBAL = 1
      CALL DGEBAL( 'B', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR )
*
*     Reduce to upper Hessenberg form
*     (Workspace: need 3*N, prefer 2*N+N*NB)
*
      ITAU = IBAL + N
      IWRK = ITAU + N
      CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
     $             LWORK-IWRK+1, IERR )
*
      IF( WANTVL ) THEN
*
*        Want left eigenvectors
*        Copy Householder vectors to VL
*
         SIDE = 'L'
         CALL DLACPY( 'L', N, N, A, LDA, VL, LDVL )
*
*        Generate orthogonal matrix in VL
*        (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*
         CALL DORGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ),
     $                WORK( IWRK ),
     $                LWORK-IWRK+1, IERR )
*
*        Perform QR iteration, accumulating Schur vectors in VL
*        (Workspace: need N+1, prefer N+HSWORK (see comments) )
*
         IWRK = ITAU
         CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VL,
     $                LDVL,
     $                WORK( IWRK ), LWORK-IWRK+1, INFO )
*
         IF( WANTVR ) THEN
*
*           Want left and right eigenvectors
*           Copy Schur vectors to VR
*
            SIDE = 'B'
            CALL DLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
         END IF
*
      ELSE IF( WANTVR ) THEN
*
*        Want right eigenvectors
*        Copy Householder vectors to VR
*
         SIDE = 'R'
         CALL DLACPY( 'L', N, N, A, LDA, VR, LDVR )
*
*        Generate orthogonal matrix in VR
*        (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*
         CALL DORGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ),
     $                WORK( IWRK ),
     $                LWORK-IWRK+1, IERR )
*
*        Perform QR iteration, accumulating Schur vectors in VR
*        (Workspace: need N+1, prefer N+HSWORK (see comments) )
*
         IWRK = ITAU
         CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VR,
     $                LDVR,
     $                WORK( IWRK ), LWORK-IWRK+1, INFO )
*
      ELSE
*
*        Compute eigenvalues only
*        (Workspace: need N+1, prefer N+HSWORK (see comments) )
*
         IWRK = ITAU
         CALL DHSEQR( 'E', 'N', N, ILO, IHI, A, LDA, WR, WI, VR,
     $                LDVR,
     $                WORK( IWRK ), LWORK-IWRK+1, INFO )
      END IF
*
*     If INFO .NE. 0 from DHSEQR, then quit
*
      IF( INFO.NE.0 )
     $   GO TO 50
*
      IF( WANTVL .OR. WANTVR ) THEN
*
*        Compute left and/or right eigenvectors
*        (Workspace: need 4*N, prefer N + N + 2*N*NB)
*
         CALL DTREVC3( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR,
     $                 LDVR,
     $                 N, NOUT, WORK( IWRK ), LWORK-IWRK+1, IERR )
      END IF
*
      IF( WANTVL ) THEN
*
*        Undo balancing of left eigenvectors
*        (Workspace: need N)
*
         CALL DGEBAK( 'B', 'L', N, ILO, IHI, WORK( IBAL ), N, VL,
     $                LDVL,
     $                IERR )
*
*        Normalize left eigenvectors and make largest component real
*
         DO 20 I = 1, N
            IF( WI( I ).EQ.ZERO ) THEN
               SCL = ONE / DNRM2( N, VL( 1, I ), 1 )
               CALL DSCAL( N, SCL, VL( 1, I ), 1 )
            ELSE IF( WI( I ).GT.ZERO ) THEN
               SCL = ONE / DLAPY2( DNRM2( N, VL( 1, I ), 1 ),
     $               DNRM2( N, VL( 1, I+1 ), 1 ) )
               CALL DSCAL( N, SCL, VL( 1, I ), 1 )
               CALL DSCAL( N, SCL, VL( 1, I+1 ), 1 )
               DO 10 K = 1, N
                  WORK( IWRK+K-1 ) = VL( K, I )**2 + VL( K, I+1 )**2
   10          CONTINUE
               K = IDAMAX( N, WORK( IWRK ), 1 )
               CALL DLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R )
               CALL DROT( N, VL( 1, I ), 1, VL( 1, I+1 ), 1, CS, SN )
               VL( K, I+1 ) = ZERO
            END IF
   20    CONTINUE
      END IF
*
      IF( WANTVR ) THEN
*
*        Undo balancing of right eigenvectors
*        (Workspace: need N)
*
         CALL DGEBAK( 'B', 'R', N, ILO, IHI, WORK( IBAL ), N, VR,
     $                LDVR,
     $                IERR )
*
*        Normalize right eigenvectors and make largest component real
*
         DO 40 I = 1, N
            IF( WI( I ).EQ.ZERO ) THEN
               SCL = ONE / DNRM2( N, VR( 1, I ), 1 )
               CALL DSCAL( N, SCL, VR( 1, I ), 1 )
            ELSE IF( WI( I ).GT.ZERO ) THEN
               SCL = ONE / DLAPY2( DNRM2( N, VR( 1, I ), 1 ),
     $               DNRM2( N, VR( 1, I+1 ), 1 ) )
               CALL DSCAL( N, SCL, VR( 1, I ), 1 )
               CALL DSCAL( N, SCL, VR( 1, I+1 ), 1 )
               DO 30 K = 1, N
                  WORK( IWRK+K-1 ) = VR( K, I )**2 + VR( K, I+1 )**2
   30          CONTINUE
               K = IDAMAX( N, WORK( IWRK ), 1 )
               CALL DLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R )
               CALL DROT( N, VR( 1, I ), 1, VR( 1, I+1 ), 1, CS, SN )
               VR( K, I+1 ) = ZERO
            END IF
   40    CONTINUE
      END IF
*
*     Undo scaling if necessary
*
   50 CONTINUE
      IF( SCALEA ) THEN
         CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1,
     $                WR( INFO+1 ),
     $                MAX( N-INFO, 1 ), IERR )
         CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1,
     $                WI( INFO+1 ),
     $                MAX( N-INFO, 1 ), IERR )
         IF( INFO.GT.0 ) THEN
            CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N,
     $                   IERR )
            CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N,
     $                   IERR )
         END IF
      END IF
*
      WORK( 1 ) = MAXWRK
      RETURN
*
*     End of DGEEV
*
      END
*> \brief <b> DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEEVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeevx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeevx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeevx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI,
*                          VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,
*                          RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          BALANC, JOBVL, JOBVR, SENSE
*       INTEGER            IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
*       DOUBLE PRECISION   ABNRM
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), RCONDE( * ), RCONDV( * ),
*      $                   SCALE( * ), VL( LDVL, * ), VR( LDVR, * ),
*      $                   WI( * ), WORK( * ), WR( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEEVX computes for an N-by-N real nonsymmetric matrix A, the
*> eigenvalues and, optionally, the left and/or right eigenvectors.
*>
*> Optionally also, it computes a balancing transformation to improve
*> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
*> SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
*> (RCONDE), and reciprocal condition numbers for the right
*> eigenvectors (RCONDV).
*>
*> The right eigenvector v(j) of A satisfies
*>                  A * v(j) = lambda(j) * v(j)
*> where lambda(j) is its eigenvalue.
*> The left eigenvector u(j) of A satisfies
*>               u(j)**H * A = lambda(j) * u(j)**H
*> where u(j)**H denotes the conjugate-transpose of u(j).
*>
*> The computed eigenvectors are normalized to have Euclidean norm
*> equal to 1 and largest component real.
*>
*> Balancing a matrix means permuting the rows and columns to make it
*> more nearly upper triangular, and applying a diagonal similarity
*> transformation D * A * D**(-1), where D is a diagonal matrix, to
*> make its rows and columns closer in norm and the condition numbers
*> of its eigenvalues and eigenvectors smaller.  The computed
*> reciprocal condition numbers correspond to the balanced matrix.
*> Permuting rows and columns will not change the condition numbers
*> (in exact arithmetic) but diagonal scaling will.  For further
*> explanation of balancing, see section 4.10.2 of the LAPACK
*> Users' Guide.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] BALANC
*> \verbatim
*>          BALANC is CHARACTER*1
*>          Indicates how the input matrix should be diagonally scaled
*>          and/or permuted to improve the conditioning of its
*>          eigenvalues.
*>          = 'N': Do not diagonally scale or permute;
*>          = 'P': Perform permutations to make the matrix more nearly
*>                 upper triangular. Do not diagonally scale;
*>          = 'S': Diagonally scale the matrix, i.e. replace A by
*>                 D*A*D**(-1), where D is a diagonal matrix chosen
*>                 to make the rows and columns of A more equal in
*>                 norm. Do not permute;
*>          = 'B': Both diagonally scale and permute A.
*>
*>          Computed reciprocal condition numbers will be for the matrix
*>          after balancing and/or permuting. Permuting does not change
*>          condition numbers (in exact arithmetic), but balancing does.
*> \endverbatim
*>
*> \param[in] JOBVL
*> \verbatim
*>          JOBVL is CHARACTER*1
*>          = 'N': left eigenvectors of A are not computed;
*>          = 'V': left eigenvectors of A are computed.
*>          If SENSE = 'E' or 'B', JOBVL must = 'V'.
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*>          JOBVR is CHARACTER*1
*>          = 'N': right eigenvectors of A are not computed;
*>          = 'V': right eigenvectors of A are computed.
*>          If SENSE = 'E' or 'B', JOBVR must = 'V'.
*> \endverbatim
*>
*> \param[in] SENSE
*> \verbatim
*>          SENSE is CHARACTER*1
*>          Determines which reciprocal condition numbers are computed.
*>          = 'N': None are computed;
*>          = 'E': Computed for eigenvalues only;
*>          = 'V': Computed for right eigenvectors only;
*>          = 'B': Computed for eigenvalues and right eigenvectors.
*>
*>          If SENSE = 'E' or 'B', both left and right eigenvectors
*>          must also be computed (JOBVL = 'V' and JOBVR = 'V').
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the N-by-N matrix A.
*>          On exit, A has been overwritten.  If JOBVL = 'V' or
*>          JOBVR = 'V', A contains the real Schur form of the balanced
*>          version of the input matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*>          WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*>          WI is DOUBLE PRECISION array, dimension (N)
*>          WR and WI contain the real and imaginary parts,
*>          respectively, of the computed eigenvalues.  Complex
*>          conjugate pairs of eigenvalues will appear consecutively
*>          with the eigenvalue having the positive imaginary part
*>          first.
*> \endverbatim
*>
*> \param[out] VL
*> \verbatim
*>          VL is DOUBLE PRECISION array, dimension (LDVL,N)
*>          If JOBVL = 'V', the left eigenvectors u(j) are stored one
*>          after another in the columns of VL, in the same order
*>          as their eigenvalues.
*>          If JOBVL = 'N', VL is not referenced.
*>          If the j-th eigenvalue is real, then u(j) = VL(:,j),
*>          the j-th column of VL.
*>          If the j-th and (j+1)-st eigenvalues form a complex
*>          conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
*>          u(j+1) = VL(:,j) - i*VL(:,j+1).
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*>          LDVL is INTEGER
*>          The leading dimension of the array VL.  LDVL >= 1; if
*>          JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*>          VR is DOUBLE PRECISION array, dimension (LDVR,N)
*>          If JOBVR = 'V', the right eigenvectors v(j) are stored one
*>          after another in the columns of VR, in the same order
*>          as their eigenvalues.
*>          If JOBVR = 'N', VR is not referenced.
*>          If the j-th eigenvalue is real, then v(j) = VR(:,j),
*>          the j-th column of VR.
*>          If the j-th and (j+1)-st eigenvalues form a complex
*>          conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
*>          v(j+1) = VR(:,j) - i*VR(:,j+1).
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*>          LDVR is INTEGER
*>          The leading dimension of the array VR.  LDVR >= 1, and if
*>          JOBVR = 'V', LDVR >= N.
*> \endverbatim
*>
*> \param[out] ILO
*> \verbatim
*>          ILO is INTEGER
*> \endverbatim
*>
*> \param[out] IHI
*> \verbatim
*>          IHI is INTEGER
*>          ILO and IHI are integer values determined when A was
*>          balanced.  The balanced A(i,j) = 0 if I > J and
*>          J = 1,...,ILO-1 or I = IHI+1,...,N.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*>          SCALE is DOUBLE PRECISION array, dimension (N)
*>          Details of the permutations and scaling factors applied
*>          when balancing A.  If P(j) is the index of the row and column
*>          interchanged with row and column j, and D(j) is the scaling
*>          factor applied to row and column j, then
*>          SCALE(J) = P(J),    for J = 1,...,ILO-1
*>                   = D(J),    for J = ILO,...,IHI
*>                   = P(J)     for J = IHI+1,...,N.
*>          The order in which the interchanges are made is N to IHI+1,
*>          then 1 to ILO-1.
*> \endverbatim
*>
*> \param[out] ABNRM
*> \verbatim
*>          ABNRM is DOUBLE PRECISION
*>          The one-norm of the balanced matrix (the maximum
*>          of the sum of absolute values of elements of any column).
*> \endverbatim
*>
*> \param[out] RCONDE
*> \verbatim
*>          RCONDE is DOUBLE PRECISION array, dimension (N)
*>          RCONDE(j) is the reciprocal condition number of the j-th
*>          eigenvalue.
*> \endverbatim
*>
*> \param[out] RCONDV
*> \verbatim
*>          RCONDV is DOUBLE PRECISION array, dimension (N)
*>          RCONDV(j) is the reciprocal condition number of the j-th
*>          right eigenvector.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.   If SENSE = 'N' or 'E',
*>          LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V',
*>          LWORK >= 3*N.  If SENSE = 'V' or 'B', LWORK >= N*(N+6).
*>          For good performance, LWORK must generally be larger.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (2*N-2)
*>          If SENSE = 'N' or 'E', not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = i, the QR algorithm failed to compute all the
*>                eigenvalues, and no eigenvectors or condition numbers
*>                have been computed; elements 1:ILO-1 and i+1:N of WR
*>                and WI contain eigenvalues which have converged.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*
*  @precisions fortran d -> s
*
*> \ingroup geevx
*
*  =====================================================================
      SUBROUTINE DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR,
     $                   WI,
     $                   VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,
     $                   RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
      implicit none
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          BALANC, JOBVL, JOBVR, SENSE
      INTEGER            IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
      DOUBLE PRECISION   ABNRM
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), RCONDE( * ), RCONDV( * ),
     $                   SCALE( * ), VL( LDVL, * ), VR( LDVR, * ),
     $                   WI( * ), WORK( * ), WR( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, SCALEA, WANTVL, WANTVR, WNTSNB, WNTSNE,
     $                   WNTSNN, WNTSNV
      CHARACTER          JOB, SIDE
      INTEGER            HSWORK, I, ICOND, IERR, ITAU, IWRK, K,
     $                   LWORK_TREVC, MAXWRK, MINWRK, NOUT
      DOUBLE PRECISION   ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM,
     $                   SN
*     ..
*     .. Local Arrays ..
      LOGICAL            SELECT( 1 )
      DOUBLE PRECISION   DUM( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLACPY,
     $                   DLARTG,
     $                   DLASCL, DORGHR, DROT, DSCAL, DTREVC3, DTRSNA,
     $                   XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX, ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE, DLAPY2, DNRM2
      EXTERNAL           LSAME, IDAMAX, ILAENV, DLAMCH, DLANGE,
     $                   DLAPY2,
     $                   DNRM2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      WANTVL = LSAME( JOBVL, 'V' )
      WANTVR = LSAME( JOBVR, 'V' )
      WNTSNN = LSAME( SENSE, 'N' )
      WNTSNE = LSAME( SENSE, 'E' )
      WNTSNV = LSAME( SENSE, 'V' )
      WNTSNB = LSAME( SENSE, 'B' )
      IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'S' )
     $      .OR.
     $                  LSAME( BALANC, 'P' ) .OR.
     $                  LSAME( BALANC, 'B' ) ) )
     $     THEN
         INFO = -1
      ELSE IF( ( .NOT.WANTVL ) .AND.
     $         ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
         INFO = -2
      ELSE IF( ( .NOT.WANTVR ) .AND.
     $         ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
         INFO = -3
      ELSE IF( .NOT.( WNTSNN .OR. WNTSNE .OR. WNTSNB .OR. WNTSNV ) .OR.
     $         ( ( WNTSNE .OR. WNTSNB ) .AND. .NOT.( WANTVL .AND.
     $         WANTVR ) ) ) THEN
         INFO = -4
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
         INFO = -11
      ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
         INFO = -13
      END IF
*
*     Compute workspace
*      (Note: Comments in the code beginning "Workspace:" describe the
*       minimal amount of workspace needed at that point in the code,
*       as well as the preferred amount for good performance.
*       NB refers to the optimal block size for the immediately
*       following subroutine, as returned by ILAENV.
*       HSWORK refers to the workspace preferred by DHSEQR, as
*       calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
*       the worst case.)
*
      IF( INFO.EQ.0 ) THEN
         IF( N.EQ.0 ) THEN
            MINWRK = 1
            MAXWRK = 1
         ELSE
            MAXWRK = N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
*
            IF( WANTVL ) THEN
               CALL DTREVC3( 'L', 'B', SELECT, N, A, LDA,
     $                       VL, LDVL, VR, LDVR,
     $                       N, NOUT, WORK, -1, IERR )
               LWORK_TREVC = INT( WORK(1) )
               MAXWRK = MAX( MAXWRK, N + LWORK_TREVC )
               CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL,
     $                      LDVL,
     $                WORK, -1, INFO )
            ELSE IF( WANTVR ) THEN
               CALL DTREVC3( 'R', 'B', SELECT, N, A, LDA,
     $                       VL, LDVL, VR, LDVR,
     $                       N, NOUT, WORK, -1, IERR )
               LWORK_TREVC = INT( WORK(1) )
               MAXWRK = MAX( MAXWRK, N + LWORK_TREVC )
               CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR,
     $                      LDVR,
     $                WORK, -1, INFO )
            ELSE
               IF( WNTSNN ) THEN
                  CALL DHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR,
     $                LDVR, WORK, -1, INFO )
               ELSE
                  CALL DHSEQR( 'S', 'N', N, 1, N, A, LDA, WR, WI, VR,
     $                LDVR, WORK, -1, INFO )
               END IF
            END IF
            HSWORK = INT( WORK(1) )
*
            IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN
               MINWRK = 2*N
               IF( .NOT.WNTSNN )
     $            MINWRK = MAX( MINWRK, N*N+6*N )
               MAXWRK = MAX( MAXWRK, HSWORK )
               IF( .NOT.WNTSNN )
     $            MAXWRK = MAX( MAXWRK, N*N + 6*N )
            ELSE
               MINWRK = 3*N
               IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) )
     $            MINWRK = MAX( MINWRK, N*N + 6*N )
               MAXWRK = MAX( MAXWRK, HSWORK )
               MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1,
     $                       'DORGHR',
     $                       ' ', N, 1, N, -1 ) )
               IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) )
     $            MAXWRK = MAX( MAXWRK, N*N + 6*N )
               MAXWRK = MAX( MAXWRK, 3*N )
            END IF
            MAXWRK = MAX( MAXWRK, MINWRK )
         END IF
         WORK( 1 ) = MAXWRK
*
         IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
            INFO = -21
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEEVX', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Get machine constants
*
      EPS = DLAMCH( 'P' )
      SMLNUM = DLAMCH( 'S' )
      BIGNUM = ONE / SMLNUM
      SMLNUM = SQRT( SMLNUM ) / EPS
      BIGNUM = ONE / SMLNUM
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      ICOND = 0
      ANRM = DLANGE( 'M', N, N, A, LDA, DUM )
      SCALEA = .FALSE.
      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
         SCALEA = .TRUE.
         CSCALE = SMLNUM
      ELSE IF( ANRM.GT.BIGNUM ) THEN
         SCALEA = .TRUE.
         CSCALE = BIGNUM
      END IF
      IF( SCALEA )
     $   CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
*
*     Balance the matrix and compute ABNRM
*
      CALL DGEBAL( BALANC, N, A, LDA, ILO, IHI, SCALE, IERR )
      ABNRM = DLANGE( '1', N, N, A, LDA, DUM )
      IF( SCALEA ) THEN
         DUM( 1 ) = ABNRM
         CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR )
         ABNRM = DUM( 1 )
      END IF
*
*     Reduce to upper Hessenberg form
*     (Workspace: need 2*N, prefer N+N*NB)
*
      ITAU = 1
      IWRK = ITAU + N
      CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
     $             LWORK-IWRK+1, IERR )
*
      IF( WANTVL ) THEN
*
*        Want left eigenvectors
*        Copy Householder vectors to VL
*
         SIDE = 'L'
         CALL DLACPY( 'L', N, N, A, LDA, VL, LDVL )
*
*        Generate orthogonal matrix in VL
*        (Workspace: need 2*N-1, prefer N+(N-1)*NB)
*
         CALL DORGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ),
     $                WORK( IWRK ),
     $                LWORK-IWRK+1, IERR )
*
*        Perform QR iteration, accumulating Schur vectors in VL
*        (Workspace: need 1, prefer HSWORK (see comments) )
*
         IWRK = ITAU
         CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VL,
     $                LDVL,
     $                WORK( IWRK ), LWORK-IWRK+1, INFO )
*
         IF( WANTVR ) THEN
*
*           Want left and right eigenvectors
*           Copy Schur vectors to VR
*
            SIDE = 'B'
            CALL DLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
         END IF
*
      ELSE IF( WANTVR ) THEN
*
*        Want right eigenvectors
*        Copy Householder vectors to VR
*
         SIDE = 'R'
         CALL DLACPY( 'L', N, N, A, LDA, VR, LDVR )
*
*        Generate orthogonal matrix in VR
*        (Workspace: need 2*N-1, prefer N+(N-1)*NB)
*
         CALL DORGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ),
     $                WORK( IWRK ),
     $                LWORK-IWRK+1, IERR )
*
*        Perform QR iteration, accumulating Schur vectors in VR
*        (Workspace: need 1, prefer HSWORK (see comments) )
*
         IWRK = ITAU
         CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VR,
     $                LDVR,
     $                WORK( IWRK ), LWORK-IWRK+1, INFO )
*
      ELSE
*
*        Compute eigenvalues only
*        If condition numbers desired, compute Schur form
*
         IF( WNTSNN ) THEN
            JOB = 'E'
         ELSE
            JOB = 'S'
         END IF
*
*        (Workspace: need 1, prefer HSWORK (see comments) )
*
         IWRK = ITAU
         CALL DHSEQR( JOB, 'N', N, ILO, IHI, A, LDA, WR, WI, VR,
     $                LDVR,
     $                WORK( IWRK ), LWORK-IWRK+1, INFO )
      END IF
*
*     If INFO .NE. 0 from DHSEQR, then quit
*
      IF( INFO.NE.0 )
     $   GO TO 50
*
      IF( WANTVL .OR. WANTVR ) THEN
*
*        Compute left and/or right eigenvectors
*        (Workspace: need 3*N, prefer N + 2*N*NB)
*
         CALL DTREVC3( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR,
     $                 LDVR,
     $                 N, NOUT, WORK( IWRK ), LWORK-IWRK+1, IERR )
      END IF
*
*     Compute condition numbers if desired
*     (Workspace: need N*N+6*N unless SENSE = 'E')
*
      IF( .NOT.WNTSNN ) THEN
         CALL DTRSNA( SENSE, 'A', SELECT, N, A, LDA, VL, LDVL, VR,
     $                LDVR,
     $                RCONDE, RCONDV, N, NOUT, WORK( IWRK ), N, IWORK,
     $                ICOND )
      END IF
*
      IF( WANTVL ) THEN
*
*        Undo balancing of left eigenvectors
*
         CALL DGEBAK( BALANC, 'L', N, ILO, IHI, SCALE, N, VL, LDVL,
     $                IERR )
*
*        Normalize left eigenvectors and make largest component real
*
         DO 20 I = 1, N
            IF( WI( I ).EQ.ZERO ) THEN
               SCL = ONE / DNRM2( N, VL( 1, I ), 1 )
               CALL DSCAL( N, SCL, VL( 1, I ), 1 )
            ELSE IF( WI( I ).GT.ZERO ) THEN
               SCL = ONE / DLAPY2( DNRM2( N, VL( 1, I ), 1 ),
     $               DNRM2( N, VL( 1, I+1 ), 1 ) )
               CALL DSCAL( N, SCL, VL( 1, I ), 1 )
               CALL DSCAL( N, SCL, VL( 1, I+1 ), 1 )
               DO 10 K = 1, N
                  WORK( K ) = VL( K, I )**2 + VL( K, I+1 )**2
   10          CONTINUE
               K = IDAMAX( N, WORK, 1 )
               CALL DLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R )
               CALL DROT( N, VL( 1, I ), 1, VL( 1, I+1 ), 1, CS, SN )
               VL( K, I+1 ) = ZERO
            END IF
   20    CONTINUE
      END IF
*
      IF( WANTVR ) THEN
*
*        Undo balancing of right eigenvectors
*
         CALL DGEBAK( BALANC, 'R', N, ILO, IHI, SCALE, N, VR, LDVR,
     $                IERR )
*
*        Normalize right eigenvectors and make largest component real
*
         DO 40 I = 1, N
            IF( WI( I ).EQ.ZERO ) THEN
               SCL = ONE / DNRM2( N, VR( 1, I ), 1 )
               CALL DSCAL( N, SCL, VR( 1, I ), 1 )
            ELSE IF( WI( I ).GT.ZERO ) THEN
               SCL = ONE / DLAPY2( DNRM2( N, VR( 1, I ), 1 ),
     $               DNRM2( N, VR( 1, I+1 ), 1 ) )
               CALL DSCAL( N, SCL, VR( 1, I ), 1 )
               CALL DSCAL( N, SCL, VR( 1, I+1 ), 1 )
               DO 30 K = 1, N
                  WORK( K ) = VR( K, I )**2 + VR( K, I+1 )**2
   30          CONTINUE
               K = IDAMAX( N, WORK, 1 )
               CALL DLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R )
               CALL DROT( N, VR( 1, I ), 1, VR( 1, I+1 ), 1, CS, SN )
               VR( K, I+1 ) = ZERO
            END IF
   40    CONTINUE
      END IF
*
*     Undo scaling if necessary
*
   50 CONTINUE
      IF( SCALEA ) THEN
         CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1,
     $                WR( INFO+1 ),
     $                MAX( N-INFO, 1 ), IERR )
         CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1,
     $                WI( INFO+1 ),
     $                MAX( N-INFO, 1 ), IERR )
         IF( INFO.EQ.0 ) THEN
            IF( ( WNTSNV .OR. WNTSNB ) .AND. ICOND.EQ.0 )
     $         CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, RCONDV, N,
     $                      IERR )
         ELSE
            CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N,
     $                   IERR )
            CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N,
     $                   IERR )
         END IF
      END IF
*
      WORK( 1 ) = MAXWRK
      RETURN
*
*     End of DGEEVX
*
      END
*> \brief <b> DGEGS computes the eigenvalues, real Schur form, and, optionally, the left and/or right Schur vectors of a real matrix pair (A,B)</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEGS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgegs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgegs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgegs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
*                         ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
*                         LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBVSL, JOBVSR
*       INTEGER            INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
*      $                   B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
*      $                   VSR( LDVSR, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DGGES.
*>
*> DGEGS computes the eigenvalues, real Schur form, and, optionally,
*> left and or/right Schur vectors of a real matrix pair (A,B).
*> Given two square matrices A and B, the generalized real Schur
*> factorization has the form
*>
*>   A = Q*S*Z**T,  B = Q*T*Z**T
*>
*> where Q and Z are orthogonal matrices, T is upper triangular, and S
*> is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal
*> blocks, the 2-by-2 blocks corresponding to complex conjugate pairs
*> of eigenvalues of (A,B).  The columns of Q are the left Schur vectors
*> and the columns of Z are the right Schur vectors.
*>
*> If only the eigenvalues of (A,B) are needed, the driver routine
*> DGEGV should be used instead.  See DGEGV for a description of the
*> eigenvalues of the generalized nonsymmetric eigenvalue problem
*> (GNEP).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBVSL
*> \verbatim
*>          JOBVSL is CHARACTER*1
*>          = 'N':  do not compute the left Schur vectors;
*>          = 'V':  compute the left Schur vectors (returned in VSL).
*> \endverbatim
*>
*> \param[in] JOBVSR
*> \verbatim
*>          JOBVSR is CHARACTER*1
*>          = 'N':  do not compute the right Schur vectors;
*>          = 'V':  compute the right Schur vectors (returned in VSR).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A, B, VSL, and VSR.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, N)
*>          On entry, the matrix A.
*>          On exit, the upper quasi-triangular matrix S from the
*>          generalized real Schur factorization.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB, N)
*>          On entry, the matrix B.
*>          On exit, the upper triangular matrix T from the generalized
*>          real Schur factorization.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*>          ALPHAR is DOUBLE PRECISION array, dimension (N)
*>          The real parts of each scalar alpha defining an eigenvalue
*>          of GNEP.
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*>          ALPHAI is DOUBLE PRECISION array, dimension (N)
*>          The imaginary parts of each scalar alpha defining an
*>          eigenvalue of GNEP.  If ALPHAI(j) is zero, then the j-th
*>          eigenvalue is real; if positive, then the j-th and (j+1)-st
*>          eigenvalues are a complex conjugate pair, with
*>          ALPHAI(j+1) = -ALPHAI(j).
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION array, dimension (N)
*>          The scalars beta that define the eigenvalues of GNEP.
*>          Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
*>          beta = BETA(j) represent the j-th eigenvalue of the matrix
*>          pair (A,B), in one of the forms lambda = alpha/beta or
*>          mu = beta/alpha.  Since either lambda or mu may overflow,
*>          they should not, in general, be computed.
*> \endverbatim
*>
*> \param[out] VSL
*> \verbatim
*>          VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
*>          If JOBVSL = 'V', the matrix of left Schur vectors Q.
*>          Not referenced if JOBVSL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSL
*> \verbatim
*>          LDVSL is INTEGER
*>          The leading dimension of the matrix VSL. LDVSL >=1, and
*>          if JOBVSL = 'V', LDVSL >= N.
*> \endverbatim
*>
*> \param[out] VSR
*> \verbatim
*>          VSR is DOUBLE PRECISION array, dimension (LDVSR,N)
*>          If JOBVSR = 'V', the matrix of right Schur vectors Z.
*>          Not referenced if JOBVSR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSR
*> \verbatim
*>          LDVSR is INTEGER
*>          The leading dimension of the matrix VSR. LDVSR >= 1, and
*>          if JOBVSR = 'V', LDVSR >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.  LWORK >= max(1,4*N).
*>          For good performance, LWORK must generally be larger.
*>          To compute the optimal value of LWORK, call ILAENV to get
*>          blocksizes (for DGEQRF, DORMQR, and DORGQR.)  Then compute:
*>          NB  -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR
*>          The optimal LWORK is  2*N + N*(NB+1).
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          = 1,...,N:
*>                The QZ iteration failed.  (A,B) are not in Schur
*>                form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
*>                be correct for j=INFO+1,...,N.
*>          > N:  errors that usually indicate LAPACK problems:
*>                =N+1: error return from DGGBAL
*>                =N+2: error return from DGEQRF
*>                =N+3: error return from DORMQR
*>                =N+4: error return from DORGQR
*>                =N+5: error return from DGGHRD
*>                =N+6: error return from DHGEQZ (other than failed
*>                                                iteration)
*>                =N+7: error return from DGGBAK (computing VSL)
*>                =N+8: error return from DGGBAK (computing VSR)
*>                =N+9: error return from DLASCL (various places)
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleGEeigen
*
*  =====================================================================
      SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
     $                  ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
     $                  LWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBVSL, JOBVSR
      INTEGER            INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
     $                   B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
     $                   VSR( LDVSR, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ILASCL, ILBSCL, ILVSL, ILVSR, LQUERY
      INTEGER            ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
     $                   IRIGHT, IROWS, ITAU, IWORK, LOPT, LWKMIN,
     $                   LWKOPT, NB, NB1, NB2, NB3
      DOUBLE PRECISION   ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
     $                   SAFMIN, SMLNUM
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLACPY,
     $                   DLASCL, DLASET, DORGQR, DORMQR, XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           LSAME, ILAENV, DLAMCH, DLANGE
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          INT, MAX
*     ..
*     .. Executable Statements ..
*
*     Decode the input arguments
*
      IF( LSAME( JOBVSL, 'N' ) ) THEN
         IJOBVL = 1
         ILVSL = .FALSE.
      ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
         IJOBVL = 2
         ILVSL = .TRUE.
      ELSE
         IJOBVL = -1
         ILVSL = .FALSE.
      END IF
*
      IF( LSAME( JOBVSR, 'N' ) ) THEN
         IJOBVR = 1
         ILVSR = .FALSE.
      ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
         IJOBVR = 2
         ILVSR = .TRUE.
      ELSE
         IJOBVR = -1
         ILVSR = .FALSE.
      END IF
*
*     Test the input arguments
*
      LWKMIN = MAX( 4*N, 1 )
      LWKOPT = LWKMIN
      WORK( 1 ) = LWKOPT
      LQUERY = ( LWORK.EQ.-1 )
      INFO = 0
      IF( IJOBVL.LE.0 ) THEN
         INFO = -1
      ELSE IF( IJOBVR.LE.0 ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
         INFO = -12
      ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
         INFO = -14
      ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
         INFO = -16
      END IF
*
      IF( INFO.EQ.0 ) THEN
         NB1 = ILAENV( 1, 'DGEQRF', ' ', N, N, -1, -1 )
         NB2 = ILAENV( 1, 'DORMQR', ' ', N, N, N, -1 )
         NB3 = ILAENV( 1, 'DORGQR', ' ', N, N, N, -1 )
         NB = MAX( NB1, NB2, NB3 )
         LOPT = 2*N + N*( NB+1 )
         WORK( 1 ) = LOPT
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEGS ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Get machine constants
*
      EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
      SAFMIN = DLAMCH( 'S' )
      SMLNUM = N*SAFMIN / EPS
      BIGNUM = ONE / SMLNUM
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
      ILASCL = .FALSE.
      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
         ANRMTO = SMLNUM
         ILASCL = .TRUE.
      ELSE IF( ANRM.GT.BIGNUM ) THEN
         ANRMTO = BIGNUM
         ILASCL = .TRUE.
      END IF
*
      IF( ILASCL ) THEN
         CALL DLASCL( 'G', -1, -1, ANRM, ANRMTO, N, N, A, LDA,
     $                IINFO )
         IF( IINFO.NE.0 ) THEN
            INFO = N + 9
            RETURN
         END IF
      END IF
*
*     Scale B if max element outside range [SMLNUM,BIGNUM]
*
      BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
      ILBSCL = .FALSE.
      IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
         BNRMTO = SMLNUM
         ILBSCL = .TRUE.
      ELSE IF( BNRM.GT.BIGNUM ) THEN
         BNRMTO = BIGNUM
         ILBSCL = .TRUE.
      END IF
*
      IF( ILBSCL ) THEN
         CALL DLASCL( 'G', -1, -1, BNRM, BNRMTO, N, N, B, LDB,
     $                IINFO )
         IF( IINFO.NE.0 ) THEN
            INFO = N + 9
            RETURN
         END IF
      END IF
*
*     Permute the matrix to make it more nearly triangular
*     Workspace layout:  (2*N words -- "work..." not actually used)
*        left_permutation, right_permutation, work...
*
      ILEFT = 1
      IRIGHT = N + 1
      IWORK = IRIGHT + N
      CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
     $             WORK( IRIGHT ), WORK( IWORK ), IINFO )
      IF( IINFO.NE.0 ) THEN
         INFO = N + 1
         GO TO 10
      END IF
*
*     Reduce B to triangular form, and initialize VSL and/or VSR
*     Workspace layout:  ("work..." must have at least N words)
*        left_permutation, right_permutation, tau, work...
*
      IROWS = IHI + 1 - ILO
      ICOLS = N + 1 - ILO
      ITAU = IWORK
      IWORK = ITAU + IROWS
      CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
     $             WORK( IWORK ), LWORK+1-IWORK, IINFO )
      IF( IINFO.GE.0 )
     $   LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
      IF( IINFO.NE.0 ) THEN
         INFO = N + 2
         GO TO 10
      END IF
*
      CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
     $             WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
     $             LWORK+1-IWORK, IINFO )
      IF( IINFO.GE.0 )
     $   LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
      IF( IINFO.NE.0 ) THEN
         INFO = N + 3
         GO TO 10
      END IF
*
      IF( ILVSL ) THEN
         CALL DLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL )
         CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
     $                VSL( ILO+1, ILO ), LDVSL )
         CALL DORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
     $                WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
     $                IINFO )
         IF( IINFO.GE.0 )
     $      LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
         IF( IINFO.NE.0 ) THEN
            INFO = N + 4
            GO TO 10
         END IF
      END IF
*
      IF( ILVSR )
     $   CALL DLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR )
*
*     Reduce to generalized Hessenberg form
*
      CALL DGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
     $             LDVSL, VSR, LDVSR, IINFO )
      IF( IINFO.NE.0 ) THEN
         INFO = N + 5
         GO TO 10
      END IF
*
*     Perform QZ algorithm, computing Schur vectors if desired
*     Workspace layout:  ("work..." must have at least 1 word)
*        left_permutation, right_permutation, work...
*
      IWORK = ITAU
      CALL DHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
     $             ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
     $             WORK( IWORK ), LWORK+1-IWORK, IINFO )
      IF( IINFO.GE.0 )
     $   LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
      IF( IINFO.NE.0 ) THEN
         IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
            INFO = IINFO
         ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
            INFO = IINFO - N
         ELSE
            INFO = N + 6
         END IF
         GO TO 10
      END IF
*
*     Apply permutation to VSL and VSR
*
      IF( ILVSL ) THEN
         CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
     $                WORK( IRIGHT ), N, VSL, LDVSL, IINFO )
         IF( IINFO.NE.0 ) THEN
            INFO = N + 7
            GO TO 10
         END IF
      END IF
      IF( ILVSR ) THEN
         CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
     $                WORK( IRIGHT ), N, VSR, LDVSR, IINFO )
         IF( IINFO.NE.0 ) THEN
            INFO = N + 8
            GO TO 10
         END IF
      END IF
*
*     Undo scaling
*
      IF( ILASCL ) THEN
         CALL DLASCL( 'H', -1, -1, ANRMTO, ANRM, N, N, A, LDA,
     $                IINFO )
         IF( IINFO.NE.0 ) THEN
            INFO = N + 9
            RETURN
         END IF
         CALL DLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAR, N,
     $                IINFO )
         IF( IINFO.NE.0 ) THEN
            INFO = N + 9
            RETURN
         END IF
         CALL DLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAI, N,
     $                IINFO )
         IF( IINFO.NE.0 ) THEN
            INFO = N + 9
            RETURN
         END IF
      END IF
*
      IF( ILBSCL ) THEN
         CALL DLASCL( 'U', -1, -1, BNRMTO, BNRM, N, N, B, LDB,
     $                IINFO )
         IF( IINFO.NE.0 ) THEN
            INFO = N + 9
            RETURN
         END IF
         CALL DLASCL( 'G', -1, -1, BNRMTO, BNRM, N, 1, BETA, N,
     $                IINFO )
         IF( IINFO.NE.0 ) THEN
            INFO = N + 9
            RETURN
         END IF
      END IF
*
   10 CONTINUE
      WORK( 1 ) = LWKOPT
*
      RETURN
*
*     End of DGEGS
*
      END
*> \brief <b> DGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a real matrix pair (A,B).</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEGV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgegv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgegv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgegv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
*                         BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBVL, JOBVR
*       INTEGER            INFO, LDA, LDB, LDVL, LDVR, LWORK, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
*      $                   B( LDB, * ), BETA( * ), VL( LDVL, * ),
*      $                   VR( LDVR, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DGGEV.
*>
*> DGEGV computes the eigenvalues and, optionally, the left and/or right
*> eigenvectors of a real matrix pair (A,B).
*> Given two square matrices A and B,
*> the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
*> eigenvalues lambda and corresponding (non-zero) eigenvectors x such
*> that
*>
*>    A*x = lambda*B*x.
*>
*> An alternate form is to find the eigenvalues mu and corresponding
*> eigenvectors y such that
*>
*>    mu*A*y = B*y.
*>
*> These two forms are equivalent with mu = 1/lambda and x = y if
*> neither lambda nor mu is zero.  In order to deal with the case that
*> lambda or mu is zero or small, two values alpha and beta are returned
*> for each eigenvalue, such that lambda = alpha/beta and
*> mu = beta/alpha.
*>
*> The vectors x and y in the above equations are right eigenvectors of
*> the matrix pair (A,B).  Vectors u and v satisfying
*>
*>    u**H*A = lambda*u**H*B  or  mu*v**H*A = v**H*B
*>
*> are left eigenvectors of (A,B).
*>
*> Note: this routine performs "full balancing" on A and B
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBVL
*> \verbatim
*>          JOBVL is CHARACTER*1
*>          = 'N':  do not compute the left generalized eigenvectors;
*>          = 'V':  compute the left generalized eigenvectors (returned
*>                  in VL).
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*>          JOBVR is CHARACTER*1
*>          = 'N':  do not compute the right generalized eigenvectors;
*>          = 'V':  compute the right generalized eigenvectors (returned
*>                  in VR).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A, B, VL, and VR.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, N)
*>          On entry, the matrix A.
*>          If JOBVL = 'V' or JOBVR = 'V', then on exit A
*>          contains the real Schur form of A from the generalized Schur
*>          factorization of the pair (A,B) after balancing.
*>          If no eigenvectors were computed, then only the diagonal
*>          blocks from the Schur form will be correct.  See DGGHRD and
*>          DHGEQZ for details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB, N)
*>          On entry, the matrix B.
*>          If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
*>          upper triangular matrix obtained from B in the generalized
*>          Schur factorization of the pair (A,B) after balancing.
*>          If no eigenvectors were computed, then only those elements of
*>          B corresponding to the diagonal blocks from the Schur form of
*>          A will be correct.  See DGGHRD and DHGEQZ for details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*>          ALPHAR is DOUBLE PRECISION array, dimension (N)
*>          The real parts of each scalar alpha defining an eigenvalue of
*>          GNEP.
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*>          ALPHAI is DOUBLE PRECISION array, dimension (N)
*>          The imaginary parts of each scalar alpha defining an
*>          eigenvalue of GNEP.  If ALPHAI(j) is zero, then the j-th
*>          eigenvalue is real; if positive, then the j-th and
*>          (j+1)-st eigenvalues are a complex conjugate pair, with
*>          ALPHAI(j+1) = -ALPHAI(j).
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION array, dimension (N)
*>          The scalars beta that define the eigenvalues of GNEP.
*>
*>          Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
*>          beta = BETA(j) represent the j-th eigenvalue of the matrix
*>          pair (A,B), in one of the forms lambda = alpha/beta or
*>          mu = beta/alpha.  Since either lambda or mu may overflow,
*>          they should not, in general, be computed.
*> \endverbatim
*>
*> \param[out] VL
*> \verbatim
*>          VL is DOUBLE PRECISION array, dimension (LDVL,N)
*>          If JOBVL = 'V', the left eigenvectors u(j) are stored
*>          in the columns of VL, in the same order as their eigenvalues.
*>          If the j-th eigenvalue is real, then u(j) = VL(:,j).
*>          If the j-th and (j+1)-st eigenvalues form a complex conjugate
*>          pair, then
*>             u(j) = VL(:,j) + i*VL(:,j+1)
*>          and
*>            u(j+1) = VL(:,j) - i*VL(:,j+1).
*>
*>          Each eigenvector is scaled so that its largest component has
*>          abs(real part) + abs(imag. part) = 1, except for eigenvectors
*>          corresponding to an eigenvalue with alpha = beta = 0, which
*>          are set to zero.
*>          Not referenced if JOBVL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*>          LDVL is INTEGER
*>          The leading dimension of the matrix VL. LDVL >= 1, and
*>          if JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*>          VR is DOUBLE PRECISION array, dimension (LDVR,N)
*>          If JOBVR = 'V', the right eigenvectors x(j) are stored
*>          in the columns of VR, in the same order as their eigenvalues.
*>          If the j-th eigenvalue is real, then x(j) = VR(:,j).
*>          If the j-th and (j+1)-st eigenvalues form a complex conjugate
*>          pair, then
*>            x(j) = VR(:,j) + i*VR(:,j+1)
*>          and
*>            x(j+1) = VR(:,j) - i*VR(:,j+1).
*>
*>          Each eigenvector is scaled so that its largest component has
*>          abs(real part) + abs(imag. part) = 1, except for eigenvalues
*>          corresponding to an eigenvalue with alpha = beta = 0, which
*>          are set to zero.
*>          Not referenced if JOBVR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*>          LDVR is INTEGER
*>          The leading dimension of the matrix VR. LDVR >= 1, and
*>          if JOBVR = 'V', LDVR >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.  LWORK >= max(1,8*N).
*>          For good performance, LWORK must generally be larger.
*>          To compute the optimal value of LWORK, call ILAENV to get
*>          blocksizes (for DGEQRF, DORMQR, and DORGQR.)  Then compute:
*>          NB  -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR;
*>          The optimal LWORK is:
*>              2*N + MAX( 6*N, N*(NB+1) ).
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          = 1,...,N:
*>                The QZ iteration failed.  No eigenvectors have been
*>                calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
*>                should be correct for j=INFO+1,...,N.
*>          > N:  errors that usually indicate LAPACK problems:
*>                =N+1: error return from DGGBAL
*>                =N+2: error return from DGEQRF
*>                =N+3: error return from DORMQR
*>                =N+4: error return from DORGQR
*>                =N+5: error return from DGGHRD
*>                =N+6: error return from DHGEQZ (other than failed
*>                                                iteration)
*>                =N+7: error return from DTGEVC
*>                =N+8: error return from DGGBAK (computing VL)
*>                =N+9: error return from DGGBAK (computing VR)
*>                =N+10: error return from DLASCL (various calls)
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleGEeigen
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  Balancing
*>  ---------
*>
*>  This driver calls DGGBAL to both permute and scale rows and columns
*>  of A and B.  The permutations PL and PR are chosen so that PL*A*PR
*>  and PL*B*R will be upper triangular except for the diagonal blocks
*>  A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
*>  possible.  The diagonal scaling matrices DL and DR are chosen so
*>  that the pair  DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
*>  one (except for the elements that start out zero.)
*>
*>  After the eigenvalues and eigenvectors of the balanced matrices
*>  have been computed, DGGBAK transforms the eigenvectors back to what
*>  they would have been (in perfect arithmetic) if they had not been
*>  balanced.
*>
*>  Contents of A and B on Exit
*>  -------- -- - --- - -- ----
*>
*>  If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
*>  both), then on exit the arrays A and B will contain the real Schur
*>  form[*] of the "balanced" versions of A and B.  If no eigenvectors
*>  are computed, then only the diagonal blocks will be correct.
*>
*>  [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations",
*>      by Golub & van Loan, pub. by Johns Hopkins U. Press.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR,
     $                  ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, LWORK,
     $                  INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBVL, JOBVR
      INTEGER            INFO, LDA, LDB, LDVL, LDVR, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
     $                   B( LDB, * ), BETA( * ), VL( LDVL, * ),
     $                   VR( LDVR, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ILIMIT, ILV, ILVL, ILVR, LQUERY
      CHARACTER          CHTEMP
      INTEGER            ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
     $                   IN, IRIGHT, IROWS, ITAU, IWORK, JC, JR, LOPT,
     $                   LWKMIN, LWKOPT, NB, NB1, NB2, NB3
      DOUBLE PRECISION   ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM,
     $                   BNRM1, BNRM2, EPS, ONEPLS, SAFMAX, SAFMIN,
     $                   SALFAI, SALFAR, SBETA, SCALE, TEMP
*     ..
*     .. Local Arrays ..
      LOGICAL            LDUMMA( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLACPY,
     $                   DLASCL, DLASET, DORGQR, DORMQR, DTGEVC, XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           LSAME, ILAENV, DLAMCH, DLANGE
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, INT, MAX
*     ..
*     .. Executable Statements ..
*
*     Decode the input arguments
*
      IF( LSAME( JOBVL, 'N' ) ) THEN
         IJOBVL = 1
         ILVL = .FALSE.
      ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
         IJOBVL = 2
         ILVL = .TRUE.
      ELSE
         IJOBVL = -1
         ILVL = .FALSE.
      END IF
*
      IF( LSAME( JOBVR, 'N' ) ) THEN
         IJOBVR = 1
         ILVR = .FALSE.
      ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
         IJOBVR = 2
         ILVR = .TRUE.
      ELSE
         IJOBVR = -1
         ILVR = .FALSE.
      END IF
      ILV = ILVL .OR. ILVR
*
*     Test the input arguments
*
      LWKMIN = MAX( 8*N, 1 )
      LWKOPT = LWKMIN
      WORK( 1 ) = LWKOPT
      LQUERY = ( LWORK.EQ.-1 )
      INFO = 0
      IF( IJOBVL.LE.0 ) THEN
         INFO = -1
      ELSE IF( IJOBVR.LE.0 ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
         INFO = -12
      ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
         INFO = -14
      ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
         INFO = -16
      END IF
*
      IF( INFO.EQ.0 ) THEN
         NB1 = ILAENV( 1, 'DGEQRF', ' ', N, N, -1, -1 )
         NB2 = ILAENV( 1, 'DORMQR', ' ', N, N, N, -1 )
         NB3 = ILAENV( 1, 'DORGQR', ' ', N, N, N, -1 )
         NB = MAX( NB1, NB2, NB3 )
         LOPT = 2*N + MAX( 6*N, N*( NB+1 ) )
         WORK( 1 ) = LOPT
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEGV ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Get machine constants
*
      EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
      SAFMIN = DLAMCH( 'S' )
      SAFMIN = SAFMIN + SAFMIN
      SAFMAX = ONE / SAFMIN
      ONEPLS = ONE + ( 4*EPS )
*
*     Scale A
*
      ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
      ANRM1 = ANRM
      ANRM2 = ONE
      IF( ANRM.LT.ONE ) THEN
         IF( SAFMAX*ANRM.LT.ONE ) THEN
            ANRM1 = SAFMIN
            ANRM2 = SAFMAX*ANRM
         END IF
      END IF
*
      IF( ANRM.GT.ZERO ) THEN
         CALL DLASCL( 'G', -1, -1, ANRM, ONE, N, N, A, LDA, IINFO )
         IF( IINFO.NE.0 ) THEN
            INFO = N + 10
            RETURN
         END IF
      END IF
*
*     Scale B
*
      BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
      BNRM1 = BNRM
      BNRM2 = ONE
      IF( BNRM.LT.ONE ) THEN
         IF( SAFMAX*BNRM.LT.ONE ) THEN
            BNRM1 = SAFMIN
            BNRM2 = SAFMAX*BNRM
         END IF
      END IF
*
      IF( BNRM.GT.ZERO ) THEN
         CALL DLASCL( 'G', -1, -1, BNRM, ONE, N, N, B, LDB, IINFO )
         IF( IINFO.NE.0 ) THEN
            INFO = N + 10
            RETURN
         END IF
      END IF
*
*     Permute the matrix to make it more nearly triangular
*     Workspace layout:  (8*N words -- "work" requires 6*N words)
*        left_permutation, right_permutation, work...
*
      ILEFT = 1
      IRIGHT = N + 1
      IWORK = IRIGHT + N
      CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
     $             WORK( IRIGHT ), WORK( IWORK ), IINFO )
      IF( IINFO.NE.0 ) THEN
         INFO = N + 1
         GO TO 120
      END IF
*
*     Reduce B to triangular form, and initialize VL and/or VR
*     Workspace layout:  ("work..." must have at least N words)
*        left_permutation, right_permutation, tau, work...
*
      IROWS = IHI + 1 - ILO
      IF( ILV ) THEN
         ICOLS = N + 1 - ILO
      ELSE
         ICOLS = IROWS
      END IF
      ITAU = IWORK
      IWORK = ITAU + IROWS
      CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
     $             WORK( IWORK ), LWORK+1-IWORK, IINFO )
      IF( IINFO.GE.0 )
     $   LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
      IF( IINFO.NE.0 ) THEN
         INFO = N + 2
         GO TO 120
      END IF
*
      CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
     $             WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
     $             LWORK+1-IWORK, IINFO )
      IF( IINFO.GE.0 )
     $   LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
      IF( IINFO.NE.0 ) THEN
         INFO = N + 3
         GO TO 120
      END IF
*
      IF( ILVL ) THEN
         CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
         CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
     $                VL( ILO+1, ILO ), LDVL )
         CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
     $                WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
     $                IINFO )
         IF( IINFO.GE.0 )
     $      LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
         IF( IINFO.NE.0 ) THEN
            INFO = N + 4
            GO TO 120
         END IF
      END IF
*
      IF( ILVR )
     $   CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
*
*     Reduce to generalized Hessenberg form
*
      IF( ILV ) THEN
*
*        Eigenvectors requested -- work on whole matrix.
*
         CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
     $                LDVL, VR, LDVR, IINFO )
      ELSE
         CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
     $                B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IINFO )
      END IF
      IF( IINFO.NE.0 ) THEN
         INFO = N + 5
         GO TO 120
      END IF
*
*     Perform QZ algorithm
*     Workspace layout:  ("work..." must have at least 1 word)
*        left_permutation, right_permutation, work...
*
      IWORK = ITAU
      IF( ILV ) THEN
         CHTEMP = 'S'
      ELSE
         CHTEMP = 'E'
      END IF
      CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
     $             ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR,
     $             WORK( IWORK ), LWORK+1-IWORK, IINFO )
      IF( IINFO.GE.0 )
     $   LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
      IF( IINFO.NE.0 ) THEN
         IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
            INFO = IINFO
         ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
            INFO = IINFO - N
         ELSE
            INFO = N + 6
         END IF
         GO TO 120
      END IF
*
      IF( ILV ) THEN
*
*        Compute Eigenvectors  (DTGEVC requires 6*N words of workspace)
*
         IF( ILVL ) THEN
            IF( ILVR ) THEN
               CHTEMP = 'B'
            ELSE
               CHTEMP = 'L'
            END IF
         ELSE
            CHTEMP = 'R'
         END IF
*
         CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
     $                VR, LDVR, N, IN, WORK( IWORK ), IINFO )
         IF( IINFO.NE.0 ) THEN
            INFO = N + 7
            GO TO 120
         END IF
*
*        Undo balancing on VL and VR, rescale
*
         IF( ILVL ) THEN
            CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
     $                   WORK( IRIGHT ), N, VL, LDVL, IINFO )
            IF( IINFO.NE.0 ) THEN
               INFO = N + 8
               GO TO 120
            END IF
            DO 50 JC = 1, N
               IF( ALPHAI( JC ).LT.ZERO )
     $            GO TO 50
               TEMP = ZERO
               IF( ALPHAI( JC ).EQ.ZERO ) THEN
                  DO 10 JR = 1, N
                     TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
   10             CONTINUE
               ELSE
                  DO 20 JR = 1, N
                     TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
     $                      ABS( VL( JR, JC+1 ) ) )
   20             CONTINUE
               END IF
               IF( TEMP.LT.SAFMIN )
     $            GO TO 50
               TEMP = ONE / TEMP
               IF( ALPHAI( JC ).EQ.ZERO ) THEN
                  DO 30 JR = 1, N
                     VL( JR, JC ) = VL( JR, JC )*TEMP
   30             CONTINUE
               ELSE
                  DO 40 JR = 1, N
                     VL( JR, JC ) = VL( JR, JC )*TEMP
                     VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
   40             CONTINUE
               END IF
   50       CONTINUE
         END IF
         IF( ILVR ) THEN
            CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
     $                   WORK( IRIGHT ), N, VR, LDVR, IINFO )
            IF( IINFO.NE.0 ) THEN
               INFO = N + 9
               GO TO 120
            END IF
            DO 100 JC = 1, N
               IF( ALPHAI( JC ).LT.ZERO )
     $            GO TO 100
               TEMP = ZERO
               IF( ALPHAI( JC ).EQ.ZERO ) THEN
                  DO 60 JR = 1, N
                     TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
   60             CONTINUE
               ELSE
                  DO 70 JR = 1, N
                     TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
     $                      ABS( VR( JR, JC+1 ) ) )
   70             CONTINUE
               END IF
               IF( TEMP.LT.SAFMIN )
     $            GO TO 100
               TEMP = ONE / TEMP
               IF( ALPHAI( JC ).EQ.ZERO ) THEN
                  DO 80 JR = 1, N
                     VR( JR, JC ) = VR( JR, JC )*TEMP
   80             CONTINUE
               ELSE
                  DO 90 JR = 1, N
                     VR( JR, JC ) = VR( JR, JC )*TEMP
                     VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
   90             CONTINUE
               END IF
  100       CONTINUE
         END IF
*
*        End of eigenvector calculation
*
      END IF
*
*     Undo scaling in alpha, beta
*
*     Note: this does not give the alpha and beta for the unscaled
*     problem.
*
*     Un-scaling is limited to avoid underflow in alpha and beta
*     if they are significant.
*
      DO 110 JC = 1, N
         ABSAR = ABS( ALPHAR( JC ) )
         ABSAI = ABS( ALPHAI( JC ) )
         ABSB = ABS( BETA( JC ) )
         SALFAR = ANRM*ALPHAR( JC )
         SALFAI = ANRM*ALPHAI( JC )
         SBETA = BNRM*BETA( JC )
         ILIMIT = .FALSE.
         SCALE = ONE
*
*        Check for significant underflow in ALPHAI
*
         IF( ABS( SALFAI ).LT.SAFMIN .AND. ABSAI.GE.
     $       MAX( SAFMIN, EPS*ABSAR, EPS*ABSB ) ) THEN
            ILIMIT = .TRUE.
            SCALE = ( ONEPLS*SAFMIN / ANRM1 ) /
     $              MAX( ONEPLS*SAFMIN, ANRM2*ABSAI )
*
         ELSE IF( SALFAI.EQ.ZERO ) THEN
*
*           If insignificant underflow in ALPHAI, then make the
*           conjugate eigenvalue real.
*
            IF( ALPHAI( JC ).LT.ZERO .AND. JC.GT.1 ) THEN
               ALPHAI( JC-1 ) = ZERO
            ELSE IF( ALPHAI( JC ).GT.ZERO .AND. JC.LT.N ) THEN
               ALPHAI( JC+1 ) = ZERO
            END IF
         END IF
*
*        Check for significant underflow in ALPHAR
*
         IF( ABS( SALFAR ).LT.SAFMIN .AND. ABSAR.GE.
     $       MAX( SAFMIN, EPS*ABSAI, EPS*ABSB ) ) THEN
            ILIMIT = .TRUE.
            SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / ANRM1 ) /
     $              MAX( ONEPLS*SAFMIN, ANRM2*ABSAR ) )
         END IF
*
*        Check for significant underflow in BETA
*
         IF( ABS( SBETA ).LT.SAFMIN .AND. ABSB.GE.
     $       MAX( SAFMIN, EPS*ABSAR, EPS*ABSAI ) ) THEN
            ILIMIT = .TRUE.
            SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / BNRM1 ) /
     $              MAX( ONEPLS*SAFMIN, BNRM2*ABSB ) )
         END IF
*
*        Check for possible overflow when limiting scaling
*
         IF( ILIMIT ) THEN
            TEMP = ( SCALE*SAFMIN )*MAX( ABS( SALFAR ), ABS( SALFAI ),
     $             ABS( SBETA ) )
            IF( TEMP.GT.ONE )
     $         SCALE = SCALE / TEMP
            IF( SCALE.LT.ONE )
     $         ILIMIT = .FALSE.
         END IF
*
*        Recompute un-scaled ALPHAR, ALPHAI, BETA if necessary.
*
         IF( ILIMIT ) THEN
            SALFAR = ( SCALE*ALPHAR( JC ) )*ANRM
            SALFAI = ( SCALE*ALPHAI( JC ) )*ANRM
            SBETA = ( SCALE*BETA( JC ) )*BNRM
         END IF
         ALPHAR( JC ) = SALFAR
         ALPHAI( JC ) = SALFAI
         BETA( JC ) = SBETA
  110 CONTINUE
*
  120 CONTINUE
      WORK( 1 ) = LWKOPT
*
      RETURN
*
*     End of DGEGV
*
      END
*> \brief \b DGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEHD2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgehd2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgehd2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgehd2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            IHI, ILO, INFO, LDA, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEHD2 reduces a real general matrix A to upper Hessenberg form H by
*> an orthogonal similarity transformation:  Q**T * A * Q = H .
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*>          ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*>          IHI is INTEGER
*>
*>          It is assumed that A is already upper triangular in rows
*>          and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
*>          set by a previous call to DGEBAL; otherwise they should be
*>          set to 1 and N respectively. See Further Details.
*>          1 <= ILO <= IHI <= max(1,N).
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the n by n general matrix to be reduced.
*>          On exit, the upper triangle and the first subdiagonal of A
*>          are overwritten with the upper Hessenberg matrix H, and the
*>          elements below the first subdiagonal, with the array TAU,
*>          represent the orthogonal matrix Q as a product of elementary
*>          reflectors. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (N-1)
*>          The scalar factors of the elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gehd2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix Q is represented as a product of (ihi-ilo) elementary
*>  reflectors
*>
*>     Q = H(ilo) H(ilo+1) . . . H(ihi-1).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
*>  exit in A(i+2:ihi,i), and tau in TAU(i).
*>
*>  The contents of A are illustrated by the following example, with
*>  n = 7, ilo = 2 and ihi = 6:
*>
*>  on entry,                        on exit,
*>
*>  ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
*>  (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
*>  (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
*>  (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
*>  (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
*>  (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
*>  (                         a )    (                          a )
*>
*>  where a denotes an element of the original matrix A, h denotes a
*>  modified element of the upper Hessenberg matrix H, and vi denotes an
*>  element of the vector defining H(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            IHI, ILO, INFO, LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF1F, DLARFG, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INFO = 0
      IF( N.LT.0 ) THEN
         INFO = -1
      ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
         INFO = -2
      ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEHD2', -INFO )
         RETURN
      END IF
*
      DO 10 I = ILO, IHI - 1
*
*        Compute elementary reflector H(i) to annihilate A(i+2:ihi,i)
*
         CALL DLARFG( IHI-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
     $                TAU( I ) )
*
*        Apply H(i) to A(1:ihi,i+1:ihi) from the right
*
         CALL DLARF1F( 'Right', IHI, IHI-I, A( I+1, I ), 1, TAU( I ),
     $               A( 1, I+1 ), LDA, WORK )
*
*        Apply H(i) to A(i+1:ihi,i+1:n) from the left
*
         CALL DLARF1F( 'Left', IHI-I, N-I, A( I+1, I ), 1, TAU( I ),
     $               A( I+1, I+1 ), LDA, WORK )
*
   10 CONTINUE
*
      RETURN
*
*     End of DGEHD2
*
      END
*> \brief \b DGEHRD
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEHRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgehrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgehrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgehrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            IHI, ILO, INFO, LDA, LWORK, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION  A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEHRD reduces a real general matrix A to upper Hessenberg form H by
*> an orthogonal similarity transformation:  Q**T * A * Q = H .
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*>          ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*>          IHI is INTEGER
*>
*>          It is assumed that A is already upper triangular in rows
*>          and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
*>          set by a previous call to DGEBAL; otherwise they should be
*>          set to 1 and N respectively. See Further Details.
*>          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the N-by-N general matrix to be reduced.
*>          On exit, the upper triangle and the first subdiagonal of A
*>          are overwritten with the upper Hessenberg matrix H, and the
*>          elements below the first subdiagonal, with the array TAU,
*>          represent the orthogonal matrix Q as a product of elementary
*>          reflectors. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (N-1)
*>          The scalar factors of the elementary reflectors (see Further
*>          Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
*>          zero.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The length of the array WORK.  LWORK >= max(1,N).
*>          For good performance, LWORK should generally be larger.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gehrd
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix Q is represented as a product of (ihi-ilo) elementary
*>  reflectors
*>
*>     Q = H(ilo) H(ilo+1) . . . H(ihi-1).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
*>  exit in A(i+2:ihi,i), and tau in TAU(i).
*>
*>  The contents of A are illustrated by the following example, with
*>  n = 7, ilo = 2 and ihi = 6:
*>
*>  on entry,                        on exit,
*>
*>  ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
*>  (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
*>  (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
*>  (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
*>  (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
*>  (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
*>  (                         a )    (                          a )
*>
*>  where a denotes an element of the original matrix A, h denotes a
*>  modified element of the upper Hessenberg matrix H, and vi denotes an
*>  element of the vector defining H(i).
*>
*>  This file is a slight modification of LAPACK-3.0's DGEHRD
*>  subroutine incorporating improvements proposed by Quintana-Orti and
*>  Van de Geijn (2006). (See DLAHR2.)
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            IHI, ILO, INFO, LDA, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            NBMAX, LDT, TSIZE
      PARAMETER          ( NBMAX = 64, LDT = NBMAX+1,
     $                     TSIZE = LDT*NBMAX )
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0,
     $                     ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IB, IINFO, IWT, J, LDWORK, LWKOPT, NB,
     $                   NBMIN, NH, NX
      DOUBLE PRECISION   EI
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DGEHD2, DGEMM, DLAHR2, DLARFB,
     $                   DTRMM,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      IF( N.LT.0 ) THEN
         INFO = -1
      ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
         INFO = -2
      ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
         INFO = -8
      END IF
*
      NH = IHI - ILO + 1
      IF( INFO.EQ.0 ) THEN
*
*        Compute the workspace requirements
*
         IF( NH.LE.1 ) THEN
            LWKOPT = 1
         ELSE
            NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI,
     $                              -1 ) )
            LWKOPT = N*NB + TSIZE
         ENDIF
         WORK( 1 ) = LWKOPT
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEHRD', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Set elements 1:ILO-1 and IHI:N-1 of TAU to zero
*
      DO 10 I = 1, ILO - 1
         TAU( I ) = ZERO
   10 CONTINUE
      DO 20 I = MAX( 1, IHI ), N - 1
         TAU( I ) = ZERO
   20 CONTINUE
*
*     Quick return if possible
*
      IF( NH.LE.1 ) THEN
         WORK( 1 ) = 1
         RETURN
      END IF
*
*     Determine the block size
*
      NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI, -1 ) )
      NBMIN = 2
      IF( NB.GT.1 .AND. NB.LT.NH ) THEN
*
*        Determine when to cross over from blocked to unblocked code
*        (last block is always handled by unblocked code)
*
         NX = MAX( NB, ILAENV( 3, 'DGEHRD', ' ', N, ILO, IHI, -1 ) )
         IF( NX.LT.NH ) THEN
*
*           Determine if workspace is large enough for blocked code
*
            IF( LWORK.LT.LWKOPT ) THEN
*
*              Not enough workspace to use optimal NB:  determine the
*              minimum value of NB, and reduce NB or force use of
*              unblocked code
*
               NBMIN = MAX( 2, ILAENV( 2, 'DGEHRD', ' ', N, ILO, IHI,
     $                 -1 ) )
               IF( LWORK.GE.(N*NBMIN + TSIZE) ) THEN
                  NB = (LWORK-TSIZE) / N
               ELSE
                  NB = 1
               END IF
            END IF
         END IF
      END IF
      LDWORK = N
*
      IF( NB.LT.NBMIN .OR. NB.GE.NH ) THEN
*
*        Use unblocked code below
*
         I = ILO
*
      ELSE
*
*        Use blocked code
*
         IWT = 1 + N*NB
         DO 40 I = ILO, IHI - 1 - NX, NB
            IB = MIN( NB, IHI-I )
*
*           Reduce columns i:i+ib-1 to Hessenberg form, returning the
*           matrices V and T of the block reflector H = I - V*T*V**T
*           which performs the reduction, and also the matrix Y = A*V*T
*
            CALL DLAHR2( IHI, I, IB, A( 1, I ), LDA, TAU( I ),
     $                   WORK( IWT ), LDT, WORK, LDWORK )
*
*           Apply the block reflector H to A(1:ihi,i+ib:ihi) from the
*           right, computing  A := A - Y * V**T. V(i+ib,ib-1) must be set
*           to 1
*
            EI = A( I+IB, I+IB-1 )
            A( I+IB, I+IB-1 ) = ONE
            CALL DGEMM( 'No transpose', 'Transpose',
     $                  IHI, IHI-I-IB+1,
     $                  IB, -ONE, WORK, LDWORK, A( I+IB, I ), LDA, ONE,
     $                  A( 1, I+IB ), LDA )
            A( I+IB, I+IB-1 ) = EI
*
*           Apply the block reflector H to A(1:i,i+1:i+ib-1) from the
*           right
*
            CALL DTRMM( 'Right', 'Lower', 'Transpose',
     $                  'Unit', I, IB-1,
     $                  ONE, A( I+1, I ), LDA, WORK, LDWORK )
            DO 30 J = 0, IB-2
               CALL DAXPY( I, -ONE, WORK( LDWORK*J+1 ), 1,
     $                     A( 1, I+J+1 ), 1 )
   30       CONTINUE
*
*           Apply the block reflector H to A(i+1:ihi,i+ib:n) from the
*           left
*
            CALL DLARFB( 'Left', 'Transpose', 'Forward',
     $                   'Columnwise',
     $                   IHI-I, N-I-IB+1, IB, A( I+1, I ), LDA,
     $                   WORK( IWT ), LDT, A( I+1, I+IB ), LDA,
     $                   WORK, LDWORK )
   40    CONTINUE
      END IF
*
*     Use unblocked code to reduce the rest of the matrix
*
      CALL DGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO )
*
      WORK( 1 ) = LWKOPT
*
      RETURN
*
*     End of DGEHRD
*
      END
*> \brief \b DGEJSV
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEJSV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgejsv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgejsv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgejsv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
*                          M, N, A, LDA, SVA, U, LDU, V, LDV,
*                          WORK, LWORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       IMPLICIT    NONE
*       INTEGER     INFO, LDA, LDU, LDV, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION A( LDA, * ), SVA( N ), U( LDU, * ), V( LDV, * ),
*      $            WORK( LWORK )
*       INTEGER     IWORK( * )
*       CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEJSV computes the singular value decomposition (SVD) of a real M-by-N
*> matrix [A], where M >= N. The SVD of [A] is written as
*>
*>              [A] = [U] * [SIGMA] * [V]^t,
*>
*> where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N
*> diagonal elements, [U] is an M-by-N (or M-by-M) orthonormal matrix, and
*> [V] is an N-by-N orthogonal matrix. The diagonal elements of [SIGMA] are
*> the singular values of [A]. The columns of [U] and [V] are the left and
*> the right singular vectors of [A], respectively. The matrices [U] and [V]
*> are computed and stored in the arrays U and V, respectively. The diagonal
*> of [SIGMA] is computed and stored in the array SVA.
*> DGEJSV can sometimes compute tiny singular values and their singular vectors much
*> more accurately than other SVD routines, see below under Further Details.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBA
*> \verbatim
*>          JOBA is CHARACTER*1
*>        Specifies the level of accuracy:
*>       = 'C': This option works well (high relative accuracy) if A = B * D,
*>             with well-conditioned B and arbitrary diagonal matrix D.
*>             The accuracy cannot be spoiled by COLUMN scaling. The
*>             accuracy of the computed output depends on the condition of
*>             B, and the procedure aims at the best theoretical accuracy.
*>             The relative error max_{i=1:N}|d sigma_i| / sigma_i is
*>             bounded by f(M,N)*epsilon* cond(B), independent of D.
*>             The input matrix is preprocessed with the QRF with column
*>             pivoting. This initial preprocessing and preconditioning by
*>             a rank revealing QR factorization is common for all values of
*>             JOBA. Additional actions are specified as follows:
*>       = 'E': Computation as with 'C' with an additional estimate of the
*>             condition number of B. It provides a realistic error bound.
*>       = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings
*>             D1, D2, and well-conditioned matrix C, this option gives
*>             higher accuracy than the 'C' option. If the structure of the
*>             input matrix is not known, and relative accuracy is
*>             desirable, then this option is advisable. The input matrix A
*>             is preprocessed with QR factorization with FULL (row and
*>             column) pivoting.
*>       = 'G': Computation as with 'F' with an additional estimate of the
*>             condition number of B, where A=D*B. If A has heavily weighted
*>             rows, then using this condition number gives too pessimistic
*>             error bound.
*>       = 'A': Small singular values are the noise and the matrix is treated
*>             as numerically rank deficient. The error in the computed
*>             singular values is bounded by f(m,n)*epsilon*||A||.
*>             The computed SVD A = U * S * V^t restores A up to
*>             f(m,n)*epsilon*||A||.
*>             This gives the procedure the licence to discard (set to zero)
*>             all singular values below N*epsilon*||A||.
*>       = 'R': Similar as in 'A'. Rank revealing property of the initial
*>             QR factorization is used do reveal (using triangular factor)
*>             a gap sigma_{r+1} < epsilon * sigma_r in which case the
*>             numerical RANK is declared to be r. The SVD is computed with
*>             absolute error bounds, but more accurately than with 'A'.
*> \endverbatim
*>
*> \param[in] JOBU
*> \verbatim
*>          JOBU is CHARACTER*1
*>        Specifies whether to compute the columns of U:
*>       = 'U': N columns of U are returned in the array U.
*>       = 'F': full set of M left sing. vectors is returned in the array U.
*>       = 'W': U may be used as workspace of length M*N. See the description
*>             of U.
*>       = 'N': U is not computed.
*> \endverbatim
*>
*> \param[in] JOBV
*> \verbatim
*>          JOBV is CHARACTER*1
*>        Specifies whether to compute the matrix V:
*>       = 'V': N columns of V are returned in the array V; Jacobi rotations
*>             are not explicitly accumulated.
*>       = 'J': N columns of V are returned in the array V, but they are
*>             computed as the product of Jacobi rotations. This option is
*>             allowed only if JOBU .NE. 'N', i.e. in computing the full SVD.
*>       = 'W': V may be used as workspace of length N*N. See the description
*>             of V.
*>       = 'N': V is not computed.
*> \endverbatim
*>
*> \param[in] JOBR
*> \verbatim
*>          JOBR is CHARACTER*1
*>        Specifies the RANGE for the singular values. Issues the licence to
*>        set to zero small positive singular values if they are outside
*>        specified range. If A .NE. 0 is scaled so that the largest singular
*>        value of c*A is around DSQRT(BIG), BIG=SLAMCH('O'), then JOBR issues
*>        the licence to kill columns of A whose norm in c*A is less than
*>        DSQRT(SFMIN) (for JOBR = 'R'), or less than SMALL=SFMIN/EPSLN,
*>        where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E').
*>       = 'N': Do not kill small columns of c*A. This option assumes that
*>             BLAS and QR factorizations and triangular solvers are
*>             implemented to work in that range. If the condition of A
*>             is greater than BIG, use DGESVJ.
*>       = 'R': RESTRICTED range for sigma(c*A) is [DSQRT(SFMIN), DSQRT(BIG)]
*>             (roughly, as described above). This option is recommended.
*>                                            ~~~~~~~~~~~~~~~~~~~~~~~~~~~
*>        For computing the singular values in the FULL range [SFMIN,BIG]
*>        use DGESVJ.
*> \endverbatim
*>
*> \param[in] JOBT
*> \verbatim
*>          JOBT is CHARACTER*1
*>        If the matrix is square then the procedure may determine to use
*>        transposed A if A^t seems to be better with respect to convergence.
*>        If the matrix is not square, JOBT is ignored. This is subject to
*>        changes in the future.
*>        The decision is based on two values of entropy over the adjoint
*>        orbit of A^t * A. See the descriptions of WORK(6) and WORK(7).
*>       = 'T': transpose if entropy test indicates possibly faster
*>        convergence of Jacobi process if A^t is taken as input. If A is
*>        replaced with A^t, then the row pivoting is included automatically.
*>       = 'N': do not speculate.
*>        This option can be used to compute only the singular values, or the
*>        full SVD (U, SIGMA and V). For only one set of singular vectors
*>        (U or V), the caller should provide both U and V, as one of the
*>        matrices is used as workspace if the matrix A is transposed.
*>        The implementer can easily remove this constraint and make the
*>        code more complicated. See the descriptions of U and V.
*> \endverbatim
*>
*> \param[in] JOBP
*> \verbatim
*>          JOBP is CHARACTER*1
*>        Issues the licence to introduce structured perturbations to drown
*>        denormalized numbers. This licence should be active if the
*>        denormals are poorly implemented, causing slow computation,
*>        especially in cases of fast convergence (!). For details see [1,2].
*>        For the sake of simplicity, this perturbations are included only
*>        when the full SVD or only the singular values are requested. The
*>        implementer/user can easily add the perturbation for the cases of
*>        computing one set of singular vectors.
*>       = 'P': introduce perturbation
*>       = 'N': do not perturb
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>         The number of rows of the input matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>         The number of columns of the input matrix A. M >= N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] SVA
*> \verbatim
*>          SVA is DOUBLE PRECISION array, dimension (N)
*>          On exit,
*>          - For WORK(1)/WORK(2) = ONE: The singular values of A. During the
*>            computation SVA contains Euclidean column norms of the
*>            iterated matrices in the array A.
*>          - For WORK(1) .NE. WORK(2): The singular values of A are
*>            (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if
*>            sigma_max(A) overflows or if small singular values have been
*>            saved from underflow by scaling the input matrix A.
*>          - If JOBR='R' then some of the singular values may be returned
*>            as exact zeros obtained by "set to zero" because they are
*>            below the numerical rank threshold or are denormalized numbers.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*>          U is DOUBLE PRECISION array, dimension ( LDU, N ) or ( LDU, M )
*>          If JOBU = 'U', then U contains on exit the M-by-N matrix of
*>                         the left singular vectors.
*>          If JOBU = 'F', then U contains on exit the M-by-M matrix of
*>                         the left singular vectors, including an ONB
*>                         of the orthogonal complement of the Range(A).
*>          If JOBU = 'W'  .AND. (JOBV = 'V' .AND. JOBT = 'T' .AND. M = N),
*>                         then U is used as workspace if the procedure
*>                         replaces A with A^t. In that case, [V] is computed
*>                         in U as left singular vectors of A^t and then
*>                         copied back to the V array. This 'W' option is just
*>                         a reminder to the caller that in this case U is
*>                         reserved as workspace of length N*N.
*>          If JOBU = 'N'  U is not referenced, unless JOBT='T'.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>          The leading dimension of the array U,  LDU >= 1.
*>          IF  JOBU = 'U' or 'F' or 'W',  then LDU >= M.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension ( LDV, N )
*>          If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of
*>                         the right singular vectors;
*>          If JOBV = 'W', AND (JOBU = 'U' AND JOBT = 'T' AND M = N),
*>                         then V is used as workspace if the procedure
*>                         replaces A with A^t. In that case, [U] is computed
*>                         in V as right singular vectors of A^t and then
*>                         copied back to the U array. This 'W' option is just
*>                         a reminder to the caller that in this case V is
*>                         reserved as workspace of length N*N.
*>          If JOBV = 'N'  V is not referenced, unless JOBT='T'.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of the array V,  LDV >= 1.
*>          If JOBV = 'V' or 'J' or 'W', then LDV >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(7,LWORK))
*>          On exit, if N > 0 .AND. M > 0 (else not referenced),
*>          WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such
*>                    that SCALE*SVA(1:N) are the computed singular values
*>                    of A. (See the description of SVA().)
*>          WORK(2) = See the description of WORK(1).
*>          WORK(3) = SCONDA is an estimate for the condition number of
*>                    column equilibrated A. (If JOBA = 'E' or 'G')
*>                    SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1).
*>                    It is computed using DPOCON. It holds
*>                    N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
*>                    where R is the triangular factor from the QRF of A.
*>                    However, if R is truncated and the numerical rank is
*>                    determined to be strictly smaller than N, SCONDA is
*>                    returned as -1, thus indicating that the smallest
*>                    singular values might be lost.
*>
*>          If full SVD is needed, the following two condition numbers are
*>          useful for the analysis of the algorithm. They are provided for
*>          a developer/implementer who is familiar with the details of
*>          the method.
*>
*>          WORK(4) = an estimate of the scaled condition number of the
*>                    triangular factor in the first QR factorization.
*>          WORK(5) = an estimate of the scaled condition number of the
*>                    triangular factor in the second QR factorization.
*>          The following two parameters are computed if JOBT = 'T'.
*>          They are provided for a developer/implementer who is familiar
*>          with the details of the method.
*>
*>          WORK(6) = the entropy of A^t*A :: this is the Shannon entropy
*>                    of diag(A^t*A) / Trace(A^t*A) taken as point in the
*>                    probability simplex.
*>          WORK(7) = the entropy of A*A^t.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          Length of WORK to confirm proper allocation of work space.
*>          LWORK depends on the job:
*>
*>          If only SIGMA is needed (JOBU = 'N', JOBV = 'N') and
*>            -> .. no scaled condition estimate required (JOBE = 'N'):
*>               LWORK >= max(2*M+N,4*N+1,7). This is the minimal requirement.
*>               ->> For optimal performance (blocked code) the optimal value
*>               is LWORK >= max(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal
*>               block size for DGEQP3 and DGEQRF.
*>               In general, optimal LWORK is computed as
*>               LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), 7).
*>            -> .. an estimate of the scaled condition number of A is
*>               required (JOBA='E', 'G'). In this case, LWORK is the maximum
*>               of the above and N*N+4*N, i.e. LWORK >= max(2*M+N,N*N+4*N,7).
*>               ->> For optimal performance (blocked code) the optimal value
*>               is LWORK >= max(2*M+N,3*N+(N+1)*NB, N*N+4*N, 7).
*>               In general, the optimal length LWORK is computed as
*>               LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF),
*>                                                     N+N*N+LWORK(DPOCON),7).
*>
*>          If SIGMA and the right singular vectors are needed (JOBV = 'V'),
*>            -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
*>            -> For optimal performance, LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
*>               where NB is the optimal block size for DGEQP3, DGEQRF, DGELQF,
*>               DORMLQ. In general, the optimal length LWORK is computed as
*>               LWORK >= max(2*M+N,N+LWORK(DGEQP3), N+LWORK(DPOCON),
*>                       N+LWORK(DGELQF), 2*N+LWORK(DGEQRF), N+LWORK(DORMLQ)).
*>
*>          If SIGMA and the left singular vectors are needed
*>            -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
*>            -> For optimal performance:
*>               if JOBU = 'U' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
*>               if JOBU = 'F' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,N+M*NB,7),
*>               where NB is the optimal block size for DGEQP3, DGEQRF, DORMQR.
*>               In general, the optimal length LWORK is computed as
*>               LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DPOCON),
*>                        2*N+LWORK(DGEQRF), N+LWORK(DORMQR)).
*>               Here LWORK(DORMQR) equals N*NB (for JOBU = 'U') or
*>               M*NB (for JOBU = 'F').
*>
*>          If the full SVD is needed: (JOBU = 'U' or JOBU = 'F') and
*>            -> if JOBV = 'V'
*>               the minimal requirement is LWORK >= max(2*M+N,6*N+2*N*N).
*>            -> if JOBV = 'J' the minimal requirement is
*>               LWORK >= max(2*M+N, 4*N+N*N,2*N+N*N+6).
*>            -> For optimal performance, LWORK should be additionally
*>               larger than N+M*NB, where NB is the optimal block size
*>               for DORMQR.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(3,M+3*N)).
*>          On exit,
*>          IWORK(1) = the numerical rank determined after the initial
*>                     QR factorization with pivoting. See the descriptions
*>                     of JOBA and JOBR.
*>          IWORK(2) = the number of the computed nonzero singular values
*>          IWORK(3) = if nonzero, a warning message:
*>                     If IWORK(3) = 1 then some of the column norms of A
*>                     were denormalized floats. The requested high accuracy
*>                     is not warranted by the data.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>           < 0:  if INFO = -i, then the i-th argument had an illegal value.
*>           = 0:  successful exit;
*>           > 0:  DGEJSV  did not converge in the maximal allowed number
*>                 of sweeps. The computed values may be inaccurate.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gejsv
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  DGEJSV implements a preconditioned Jacobi SVD algorithm. It uses DGEQP3,
*>  DGEQRF, and DGELQF as preprocessors and preconditioners. Optionally, an
*>  additional row pivoting can be used as a preprocessor, which in some
*>  cases results in much higher accuracy. An example is matrix A with the
*>  structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned
*>  diagonal matrices and C is well-conditioned matrix. In that case, complete
*>  pivoting in the first QR factorizations provides accuracy dependent on the
*>  condition number of C, and independent of D1, D2. Such higher accuracy is
*>  not completely understood theoretically, but it works well in practice.
*>  Further, if A can be written as A = B*D, with well-conditioned B and some
*>  diagonal D, then the high accuracy is guaranteed, both theoretically and
*>  in software, independent of D. For more details see [1], [2].
*>     The computational range for the singular values can be the full range
*>  ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS
*>  & LAPACK routines called by DGEJSV are implemented to work in that range.
*>  If that is not the case, then the restriction for safe computation with
*>  the singular values in the range of normalized IEEE numbers is that the
*>  spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not
*>  overflow. This code (DGEJSV) is best used in this restricted range,
*>  meaning that singular values of magnitude below ||A||_2 / DLAMCH('O') are
*>  returned as zeros. See JOBR for details on this.
*>     Further, this implementation is somewhat slower than the one described
*>  in [1,2] due to replacement of some non-LAPACK components, and because
*>  the choice of some tuning parameters in the iterative part (DGESVJ) is
*>  left to the implementer on a particular machine.
*>     The rank revealing QR factorization (in this code: DGEQP3) should be
*>  implemented as in [3]. We have a new version of DGEQP3 under development
*>  that is more robust than the current one in LAPACK, with a cleaner cut in
*>  rank deficient cases. It will be available in the SIGMA library [4].
*>  If M is much larger than N, it is obvious that the initial QRF with
*>  column pivoting can be preprocessed by the QRF without pivoting. That
*>  well known trick is not used in DGEJSV because in some cases heavy row
*>  weighting can be treated with complete pivoting. The overhead in cases
*>  M much larger than N is then only due to pivoting, but the benefits in
*>  terms of accuracy have prevailed. The implementer/user can incorporate
*>  this extra QRF step easily. The implementer can also improve data movement
*>  (matrix transpose, matrix copy, matrix transposed copy) - this
*>  implementation of DGEJSV uses only the simplest, naive data movement.
*> \endverbatim
*
*> \par Contributors:
*  ==================
*>
*>  Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
*
*> \par References:
*  ================
*>
*> \verbatim
*>
*> [1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
*>     SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
*>     LAPACK Working note 169.
*> [2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
*>     SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
*>     LAPACK Working note 170.
*> [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR
*>     factorization software - a case study.
*>     ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28.
*>     LAPACK Working note 176.
*> [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
*>     QSVD, (H,K)-SVD computations.
*>     Department of Mathematics, University of Zagreb, 2008.
*> \endverbatim
*
*>  \par Bugs, examples and comments:
*   =================================
*>
*>  Please report all bugs and send interesting examples and/or comments to
*>  drmac@math.hr. Thank you.
*>
*  =====================================================================
      SUBROUTINE DGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
     $                   M, N, A, LDA, SVA, U, LDU, V, LDV,
     $                   WORK, LWORK, IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      IMPLICIT    NONE
      INTEGER     INFO, LDA, LDU, LDV, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION A( LDA, * ), SVA( N ), U( LDU, * ), V( LDV, * ),
     $            WORK( LWORK )
      INTEGER     IWORK( * )
      CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
*     ..
*
*  ===========================================================================
*
*     .. Local Parameters ..
      DOUBLE PRECISION   ZERO,  ONE
      PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION AAPP, AAQQ, AATMAX, AATMIN, BIG, BIG1, COND_OK,
     $        CONDR1, CONDR2, ENTRA,  ENTRAT, EPSLN,  MAXPRJ, SCALEM,
     $        SCONDA, SFMIN,  SMALL,  TEMP1,  USCAL1, USCAL2, XSC
      INTEGER IERR,   N1,     NR,     NUMRANK,        p, q,   WARNING
      LOGICAL ALMORT, DEFR,   ERREST, GOSCAL, JRACC,  KILL,   LSVEC,
     $        L2ABER, L2KILL, L2PERT, L2RANK, L2TRAN,
     $        NOSCAL, ROWPIV, RSVEC,  TRANSP
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC DABS, DLOG, MAX, MIN, DBLE, IDNINT, DSIGN, DSQRT
*     ..
*     .. External Functions ..
      DOUBLE PRECISION  DLAMCH, DNRM2
      INTEGER   IDAMAX
      LOGICAL   LSAME
      EXTERNAL  IDAMAX, LSAME, DLAMCH, DNRM2
*     ..
*     .. External Subroutines ..
      EXTERNAL  DCOPY,  DGELQF, DGEQP3, DGEQRF, DLACPY,
     $                   DLASCL,
     $          DLASET, DLASSQ, DLASWP, DORGQR, DORMLQ,
     $          DORMQR, DPOCON, DSCAL,  DSWAP,  DTRSM,  XERBLA
*
      EXTERNAL  DGESVJ
*     ..
*
*     Test the input arguments
*
      LSVEC  = LSAME( JOBU, 'U' ) .OR. LSAME( JOBU, 'F' )
      JRACC  = LSAME( JOBV, 'J' )
      RSVEC  = LSAME( JOBV, 'V' ) .OR. JRACC
      ROWPIV = LSAME( JOBA, 'F' ) .OR. LSAME( JOBA, 'G' )
      L2RANK = LSAME( JOBA, 'R' )
      L2ABER = LSAME( JOBA, 'A' )
      ERREST = LSAME( JOBA, 'E' ) .OR. LSAME( JOBA, 'G' )
      L2TRAN = LSAME( JOBT, 'T' )
      L2KILL = LSAME( JOBR, 'R' )
      DEFR   = LSAME( JOBR, 'N' )
      L2PERT = LSAME( JOBP, 'P' )
*
      IF ( .NOT.(ROWPIV .OR. L2RANK .OR. L2ABER .OR.
     $     ERREST .OR. LSAME( JOBA, 'C' ) )) THEN
         INFO = - 1
      ELSE IF ( .NOT.( LSVEC  .OR. LSAME( JOBU, 'N' ) .OR.
     $                             LSAME( JOBU, 'W' )) ) THEN
         INFO = - 2
      ELSE IF ( .NOT.( RSVEC .OR. LSAME( JOBV, 'N' ) .OR.
     $   LSAME( JOBV, 'W' )) .OR. ( JRACC .AND. (.NOT.LSVEC) ) ) THEN
         INFO = - 3
      ELSE IF ( .NOT. ( L2KILL .OR. DEFR ) )    THEN
         INFO = - 4
      ELSE IF ( .NOT. ( L2TRAN .OR. LSAME( JOBT, 'N' ) ) ) THEN
         INFO = - 5
      ELSE IF ( .NOT. ( L2PERT .OR. LSAME( JOBP, 'N' ) ) ) THEN
         INFO = - 6
      ELSE IF ( M .LT. 0 ) THEN
         INFO = - 7
      ELSE IF ( ( N .LT. 0 ) .OR. ( N .GT. M ) ) THEN
         INFO = - 8
      ELSE IF ( LDA .LT. M ) THEN
         INFO = - 10
      ELSE IF ( LSVEC .AND. ( LDU .LT. M ) ) THEN
         INFO = - 13
      ELSE IF ( RSVEC .AND. ( LDV .LT. N ) ) THEN
         INFO = - 15
      ELSE IF ( (.NOT.(LSVEC .OR. RSVEC .OR. ERREST).AND.
     &                           (LWORK .LT. MAX(7,4*N+1,2*M+N))) .OR.
     & (.NOT.(LSVEC .OR. RSVEC) .AND. ERREST .AND.
     &                         (LWORK .LT. MAX(7,4*N+N*N,2*M+N))) .OR.
     & (LSVEC .AND. (.NOT.RSVEC) .AND. (LWORK .LT. MAX(7,2*M+N,4*N+1)))
     & .OR.
     & (RSVEC .AND. (.NOT.LSVEC) .AND. (LWORK .LT. MAX(7,2*M+N,4*N+1)))
     & .OR.
     & (LSVEC .AND. RSVEC .AND. (.NOT.JRACC) .AND.
     &                          (LWORK.LT.MAX(2*M+N,6*N+2*N*N)))
     & .OR. (LSVEC .AND. RSVEC .AND. JRACC .AND.
     &                          LWORK.LT.MAX(2*M+N,4*N+N*N,2*N+N*N+6)))
     &   THEN
         INFO = - 17
      ELSE
*        #:)
         INFO = 0
      END IF
*
      IF ( INFO .NE. 0 ) THEN
*       #:(
         CALL XERBLA( 'DGEJSV', - INFO )
         RETURN
      END IF
*
*     Quick return for void matrix (Y3K safe)
* #:)
      IF ( ( M .EQ. 0 ) .OR. ( N .EQ. 0 ) ) THEN
         IWORK(1:3) = 0
         WORK(1:7) = 0
         RETURN
      ENDIF
*
*     Determine whether the matrix U should be M x N or M x M
*
      IF ( LSVEC ) THEN
         N1 = N
         IF ( LSAME( JOBU, 'F' ) ) N1 = M
      END IF
*
*     Set numerical parameters
*
*!    NOTE: Make sure DLAMCH() does not fail on the target architecture.
*
      EPSLN = DLAMCH('Epsilon')
      SFMIN = DLAMCH('SafeMinimum')
      SMALL = SFMIN / EPSLN
      BIG   = DLAMCH('O')
*     BIG   = ONE / SFMIN
*
*     Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N
*
*(!)  If necessary, scale SVA() to protect the largest norm from
*     overflow. It is possible that this scaling pushes the smallest
*     column norm left from the underflow threshold (extreme case).
*
      SCALEM  = ONE / DSQRT(DBLE(M)*DBLE(N))
      NOSCAL  = .TRUE.
      GOSCAL  = .TRUE.
      DO 1874 p = 1, N
         AAPP = ZERO
         AAQQ = ONE
         CALL DLASSQ( M, A(1,p), 1, AAPP, AAQQ )
         IF ( AAPP .GT. BIG ) THEN
            INFO = - 9
            CALL XERBLA( 'DGEJSV', -INFO )
            RETURN
         END IF
         AAQQ = DSQRT(AAQQ)
         IF ( ( AAPP .LT. (BIG / AAQQ) ) .AND. NOSCAL  ) THEN
            SVA(p)  = AAPP * AAQQ
         ELSE
            NOSCAL  = .FALSE.
            SVA(p)  = AAPP * ( AAQQ * SCALEM )
            IF ( GOSCAL ) THEN
               GOSCAL = .FALSE.
               CALL DSCAL( p-1, SCALEM, SVA, 1 )
            END IF
         END IF
 1874 CONTINUE
*
      IF ( NOSCAL ) SCALEM = ONE
*
      AAPP = ZERO
      AAQQ = BIG
      DO 4781 p = 1, N
         AAPP = MAX( AAPP, SVA(p) )
         IF ( SVA(p) .NE. ZERO ) AAQQ = MIN( AAQQ, SVA(p) )
 4781 CONTINUE
*
*     Quick return for zero M x N matrix
* #:)
      IF ( AAPP .EQ. ZERO ) THEN
         IF ( LSVEC ) CALL DLASET( 'G', M, N1, ZERO, ONE, U, LDU )
         IF ( RSVEC ) CALL DLASET( 'G', N, N,  ZERO, ONE, V, LDV )
         WORK(1) = ONE
         WORK(2) = ONE
         IF ( ERREST ) WORK(3) = ONE
         IF ( LSVEC .AND. RSVEC ) THEN
            WORK(4) = ONE
            WORK(5) = ONE
         END IF
         IF ( L2TRAN ) THEN
            WORK(6) = ZERO
            WORK(7) = ZERO
         END IF
         IWORK(1) = 0
         IWORK(2) = 0
         IWORK(3) = 0
         RETURN
      END IF
*
*     Issue warning if denormalized column norms detected. Override the
*     high relative accuracy request. Issue licence to kill columns
*     (set them to zero) whose norm is less than sigma_max / BIG (roughly).
* #:(
      WARNING = 0
      IF ( AAQQ .LE. SFMIN ) THEN
         L2RANK = .TRUE.
         L2KILL = .TRUE.
         WARNING = 1
      END IF
*
*     Quick return for one-column matrix
* #:)
      IF ( N .EQ. 1 ) THEN
*
         IF ( LSVEC ) THEN
            CALL DLASCL( 'G',0,0,SVA(1),SCALEM, M,1,A(1,1),LDA,IERR )
            CALL DLACPY( 'A', M, 1, A, LDA, U, LDU )
*           computing all M left singular vectors of the M x 1 matrix
            IF ( N1 .NE. N  ) THEN
               CALL DGEQRF( M, N, U,LDU, WORK, WORK(N+1),LWORK-N,
     $                      IERR )
               CALL DORGQR( M,N1,1, U,LDU,WORK,WORK(N+1),LWORK-N,
     $                      IERR )
               CALL DCOPY( M, A(1,1), 1, U(1,1), 1 )
            END IF
         END IF
         IF ( RSVEC ) THEN
             V(1,1) = ONE
         END IF
         IF ( SVA(1) .LT. (BIG*SCALEM) ) THEN
            SVA(1)  = SVA(1) / SCALEM
            SCALEM  = ONE
         END IF
         WORK(1) = ONE / SCALEM
         WORK(2) = ONE
         IF ( SVA(1) .NE. ZERO ) THEN
            IWORK(1) = 1
            IF ( ( SVA(1) / SCALEM) .GE. SFMIN ) THEN
               IWORK(2) = 1
            ELSE
               IWORK(2) = 0
            END IF
         ELSE
            IWORK(1) = 0
            IWORK(2) = 0
         END IF
         IWORK(3) = 0
         IF ( ERREST ) WORK(3) = ONE
         IF ( LSVEC .AND. RSVEC ) THEN
            WORK(4) = ONE
            WORK(5) = ONE
         END IF
         IF ( L2TRAN ) THEN
            WORK(6) = ZERO
            WORK(7) = ZERO
         END IF
         RETURN
*
      END IF
*
      TRANSP = .FALSE.
      L2TRAN = L2TRAN .AND. ( M .EQ. N )
*
      AATMAX = -ONE
      AATMIN =  BIG
      IF ( ROWPIV .OR. L2TRAN ) THEN
*
*     Compute the row norms, needed to determine row pivoting sequence
*     (in the case of heavily row weighted A, row pivoting is strongly
*     advised) and to collect information needed to compare the
*     structures of A * A^t and A^t * A (in the case L2TRAN.EQ..TRUE.).
*
         IF ( L2TRAN ) THEN
            DO 1950 p = 1, M
               XSC   = ZERO
               TEMP1 = ONE
               CALL DLASSQ( N, A(p,1), LDA, XSC, TEMP1 )
*              DLASSQ gets both the ell_2 and the ell_infinity norm
*              in one pass through the vector
               WORK(M+N+p)  = XSC * SCALEM
               WORK(N+p)    = XSC * (SCALEM*DSQRT(TEMP1))
               AATMAX = MAX( AATMAX, WORK(N+p) )
               IF (WORK(N+p) .NE. ZERO) AATMIN = MIN(AATMIN,WORK(N+p))
 1950       CONTINUE
         ELSE
            DO 1904 p = 1, M
               WORK(M+N+p) = SCALEM*DABS( A(p,IDAMAX(N,A(p,1),LDA)) )
               AATMAX = MAX( AATMAX, WORK(M+N+p) )
               AATMIN = MIN( AATMIN, WORK(M+N+p) )
 1904       CONTINUE
         END IF
*
      END IF
*
*     For square matrix A try to determine whether A^t  would be  better
*     input for the preconditioned Jacobi SVD, with faster convergence.
*     The decision is based on an O(N) function of the vector of column
*     and row norms of A, based on the Shannon entropy. This should give
*     the right choice in most cases when the difference actually matters.
*     It may fail and pick the slower converging side.
*
      ENTRA  = ZERO
      ENTRAT = ZERO
      IF ( L2TRAN ) THEN
*
         XSC   = ZERO
         TEMP1 = ONE
         CALL DLASSQ( N, SVA, 1, XSC, TEMP1 )
         TEMP1 = ONE / TEMP1
*
         ENTRA = ZERO
         DO 1113 p = 1, N
            BIG1  = ( ( SVA(p) / XSC )**2 ) * TEMP1
            IF ( BIG1 .NE. ZERO ) ENTRA = ENTRA + BIG1 * DLOG(BIG1)
 1113    CONTINUE
         ENTRA = - ENTRA / DLOG(DBLE(N))
*
*        Now, SVA().^2/Trace(A^t * A) is a point in the probability simplex.
*        It is derived from the diagonal of  A^t * A.  Do the same with the
*        diagonal of A * A^t, compute the entropy of the corresponding
*        probability distribution. Note that A * A^t and A^t * A have the
*        same trace.
*
         ENTRAT = ZERO
         DO 1114 p = N+1, N+M
            BIG1 = ( ( WORK(p) / XSC )**2 ) * TEMP1
            IF ( BIG1 .NE. ZERO ) ENTRAT = ENTRAT + BIG1 * DLOG(BIG1)
 1114    CONTINUE
         ENTRAT = - ENTRAT / DLOG(DBLE(M))
*
*        Analyze the entropies and decide A or A^t. Smaller entropy
*        usually means better input for the algorithm.
*
         TRANSP = ( ENTRAT .LT. ENTRA )
*
*        If A^t is better than A, transpose A.
*
         IF ( TRANSP ) THEN
*           In an optimal implementation, this trivial transpose
*           should be replaced with faster transpose.
            DO 1115 p = 1, N - 1
               DO 1116 q = p + 1, N
                   TEMP1 = A(q,p)
                  A(q,p) = A(p,q)
                  A(p,q) = TEMP1
 1116          CONTINUE
 1115       CONTINUE
            DO 1117 p = 1, N
               WORK(M+N+p) = SVA(p)
               SVA(p)      = WORK(N+p)
 1117       CONTINUE
            TEMP1  = AAPP
            AAPP   = AATMAX
            AATMAX = TEMP1
            TEMP1  = AAQQ
            AAQQ   = AATMIN
            AATMIN = TEMP1
            KILL   = LSVEC
            LSVEC  = RSVEC
            RSVEC  = KILL
            IF ( LSVEC ) N1 = N
*
            ROWPIV = .TRUE.
         END IF
*
      END IF
*     END IF L2TRAN
*
*     Scale the matrix so that its maximal singular value remains less
*     than DSQRT(BIG) -- the matrix is scaled so that its maximal column
*     has Euclidean norm equal to DSQRT(BIG/N). The only reason to keep
*     DSQRT(BIG) instead of BIG is the fact that DGEJSV uses LAPACK and
*     BLAS routines that, in some implementations, are not capable of
*     working in the full interval [SFMIN,BIG] and that they may provoke
*     overflows in the intermediate results. If the singular values spread
*     from SFMIN to BIG, then DGESVJ will compute them. So, in that case,
*     one should use DGESVJ instead of DGEJSV.
*
      BIG1   = DSQRT( BIG )
      TEMP1  = DSQRT( BIG / DBLE(N) )
*
      CALL DLASCL( 'G', 0, 0, AAPP, TEMP1, N, 1, SVA, N, IERR )
      IF ( AAQQ .GT. (AAPP * SFMIN) ) THEN
          AAQQ = ( AAQQ / AAPP ) * TEMP1
      ELSE
          AAQQ = ( AAQQ * TEMP1 ) / AAPP
      END IF
      TEMP1 = TEMP1 * SCALEM
      CALL DLASCL( 'G', 0, 0, AAPP, TEMP1, M, N, A, LDA, IERR )
*
*     To undo scaling at the end of this procedure, multiply the
*     computed singular values with USCAL2 / USCAL1.
*
      USCAL1 = TEMP1
      USCAL2 = AAPP
*
      IF ( L2KILL ) THEN
*        L2KILL enforces computation of nonzero singular values in
*        the restricted range of condition number of the initial A,
*        sigma_max(A) / sigma_min(A) approx. DSQRT(BIG)/DSQRT(SFMIN).
         XSC = DSQRT( SFMIN )
      ELSE
         XSC = SMALL
*
*        Now, if the condition number of A is too big,
*        sigma_max(A) / sigma_min(A) .GT. DSQRT(BIG/N) * EPSLN / SFMIN,
*        as a precaution measure, the full SVD is computed using DGESVJ
*        with accumulated Jacobi rotations. This provides numerically
*        more robust computation, at the cost of slightly increased run
*        time. Depending on the concrete implementation of BLAS and LAPACK
*        (i.e. how they behave in presence of extreme ill-conditioning) the
*        implementor may decide to remove this switch.
         IF ( ( AAQQ.LT.DSQRT(SFMIN) ) .AND. LSVEC .AND. RSVEC ) THEN
            JRACC = .TRUE.
         END IF
*
      END IF
      IF ( AAQQ .LT. XSC ) THEN
         DO 700 p = 1, N
            IF ( SVA(p) .LT. XSC ) THEN
               CALL DLASET( 'A', M, 1, ZERO, ZERO, A(1,p), LDA )
               SVA(p) = ZERO
            END IF
 700     CONTINUE
      END IF
*
*     Preconditioning using QR factorization with pivoting
*
      IF ( ROWPIV ) THEN
*        Optional row permutation (Bjoerck row pivoting):
*        A result by Cox and Higham shows that the Bjoerck's
*        row pivoting combined with standard column pivoting
*        has similar effect as Powell-Reid complete pivoting.
*        The ell-infinity norms of A are made nonincreasing.
         DO 1952 p = 1, M - 1
            q = IDAMAX( M-p+1, WORK(M+N+p), 1 ) + p - 1
            IWORK(2*N+p) = q
            IF ( p .NE. q ) THEN
               TEMP1       = WORK(M+N+p)
               WORK(M+N+p) = WORK(M+N+q)
               WORK(M+N+q) = TEMP1
            END IF
 1952    CONTINUE
         CALL DLASWP( N, A, LDA, 1, M-1, IWORK(2*N+1), 1 )
      END IF
*
*     End of the preparation phase (scaling, optional sorting and
*     transposing, optional flushing of small columns).
*
*     Preconditioning
*
*     If the full SVD is needed, the right singular vectors are computed
*     from a matrix equation, and for that we need theoretical analysis
*     of the Businger-Golub pivoting. So we use DGEQP3 as the first RR QRF.
*     In all other cases the first RR QRF can be chosen by other criteria
*     (eg speed by replacing global with restricted window pivoting, such
*     as in SGEQPX from TOMS # 782). Good results will be obtained using
*     SGEQPX with properly (!) chosen numerical parameters.
*     Any improvement of DGEQP3 improves overall performance of DGEJSV.
*
*     A * P1 = Q1 * [ R1^t 0]^t:
      DO 1963 p = 1, N
*        .. all columns are free columns
         IWORK(p) = 0
 1963 CONTINUE
      CALL DGEQP3( M,N,A,LDA, IWORK,WORK, WORK(N+1),LWORK-N, IERR )
*
*     The upper triangular matrix R1 from the first QRF is inspected for
*     rank deficiency and possibilities for deflation, or possible
*     ill-conditioning. Depending on the user specified flag L2RANK,
*     the procedure explores possibilities to reduce the numerical
*     rank by inspecting the computed upper triangular factor. If
*     L2RANK or L2ABER are up, then DGEJSV will compute the SVD of
*     A + dA, where ||dA|| <= f(M,N)*EPSLN.
*
      NR = 1
      IF ( L2ABER ) THEN
*        Standard absolute error bound suffices. All sigma_i with
*        sigma_i < N*EPSLN*||A|| are flushed to zero. This is an
*        aggressive enforcement of lower numerical rank by introducing a
*        backward error of the order of N*EPSLN*||A||.
         TEMP1 = DSQRT(DBLE(N))*EPSLN
         DO 3001 p = 2, N
            IF ( DABS(A(p,p)) .GE. (TEMP1*DABS(A(1,1))) ) THEN
               NR = NR + 1
            ELSE
               GO TO 3002
            END IF
 3001    CONTINUE
 3002    CONTINUE
      ELSE IF ( L2RANK ) THEN
*        .. similarly as above, only slightly more gentle (less aggressive).
*        Sudden drop on the diagonal of R1 is used as the criterion for
*        close-to-rank-deficient.
         TEMP1 = DSQRT(SFMIN)
         DO 3401 p = 2, N
            IF ( ( DABS(A(p,p)) .LT. (EPSLN*DABS(A(p-1,p-1))) ) .OR.
     $           ( DABS(A(p,p)) .LT. SMALL ) .OR.
     $           ( L2KILL .AND. (DABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3402
            NR = NR + 1
 3401    CONTINUE
 3402    CONTINUE
*
      ELSE
*        The goal is high relative accuracy. However, if the matrix
*        has high scaled condition number the relative accuracy is in
*        general not feasible. Later on, a condition number estimator
*        will be deployed to estimate the scaled condition number.
*        Here we just remove the underflowed part of the triangular
*        factor. This prevents the situation in which the code is
*        working hard to get the accuracy not warranted by the data.
         TEMP1  = DSQRT(SFMIN)
         DO 3301 p = 2, N
            IF ( ( DABS(A(p,p)) .LT. SMALL ) .OR.
     $          ( L2KILL .AND. (DABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3302
            NR = NR + 1
 3301    CONTINUE
 3302    CONTINUE
*
      END IF
*
      ALMORT = .FALSE.
      IF ( NR .EQ. N ) THEN
         MAXPRJ = ONE
         DO 3051 p = 2, N
            TEMP1  = DABS(A(p,p)) / SVA(IWORK(p))
            MAXPRJ = MIN( MAXPRJ, TEMP1 )
 3051    CONTINUE
         IF ( MAXPRJ**2 .GE. ONE - DBLE(N)*EPSLN ) ALMORT = .TRUE.
      END IF
*
*
      SCONDA = - ONE
      CONDR1 = - ONE
      CONDR2 = - ONE
*
      IF ( ERREST ) THEN
         IF ( N .EQ. NR ) THEN
            IF ( RSVEC ) THEN
*              .. V is available as workspace
               CALL DLACPY( 'U', N, N, A, LDA, V, LDV )
               DO 3053 p = 1, N
                  TEMP1 = SVA(IWORK(p))
                  CALL DSCAL( p, ONE/TEMP1, V(1,p), 1 )
 3053          CONTINUE
               CALL DPOCON( 'U', N, V, LDV, ONE, TEMP1,
     $              WORK(N+1), IWORK(2*N+M+1), IERR )
            ELSE IF ( LSVEC ) THEN
*              .. U is available as workspace
               CALL DLACPY( 'U', N, N, A, LDA, U, LDU )
               DO 3054 p = 1, N
                  TEMP1 = SVA(IWORK(p))
                  CALL DSCAL( p, ONE/TEMP1, U(1,p), 1 )
 3054          CONTINUE
               CALL DPOCON( 'U', N, U, LDU, ONE, TEMP1,
     $              WORK(N+1), IWORK(2*N+M+1), IERR )
            ELSE
               CALL DLACPY( 'U', N, N, A, LDA, WORK(N+1), N )
               DO 3052 p = 1, N
                  TEMP1 = SVA(IWORK(p))
                  CALL DSCAL( p, ONE/TEMP1, WORK(N+(p-1)*N+1), 1 )
 3052          CONTINUE
*           .. the columns of R are scaled to have unit Euclidean lengths.
               CALL DPOCON( 'U', N, WORK(N+1), N, ONE, TEMP1,
     $              WORK(N+N*N+1), IWORK(2*N+M+1), IERR )
            END IF
            SCONDA = ONE / DSQRT(TEMP1)
*           SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1).
*           N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
         ELSE
            SCONDA = - ONE
         END IF
      END IF
*
      L2PERT = L2PERT .AND. ( DABS( A(1,1)/A(NR,NR) ) .GT. DSQRT(BIG1) )
*     If there is no violent scaling, artificial perturbation is not needed.
*
*     Phase 3:
*
      IF ( .NOT. ( RSVEC .OR. LSVEC ) ) THEN
*
*         Singular Values only
*
*         .. transpose A(1:NR,1:N)
         DO 1946 p = 1, MIN( N-1, NR )
            CALL DCOPY( N-p, A(p,p+1), LDA, A(p+1,p), 1 )
 1946    CONTINUE
*
*        The following two DO-loops introduce small relative perturbation
*        into the strict upper triangle of the lower triangular matrix.
*        Small entries below the main diagonal are also changed.
*        This modification is useful if the computing environment does not
*        provide/allow FLUSH TO ZERO underflow, for it prevents many
*        annoying denormalized numbers in case of strongly scaled matrices.
*        The perturbation is structured so that it does not introduce any
*        new perturbation of the singular values, and it does not destroy
*        the job done by the preconditioner.
*        The licence for this perturbation is in the variable L2PERT, which
*        should be .FALSE. if FLUSH TO ZERO underflow is active.
*
         IF ( .NOT. ALMORT ) THEN
*
            IF ( L2PERT ) THEN
*              XSC = DSQRT(SMALL)
               XSC = EPSLN / DBLE(N)
               DO 4947 q = 1, NR
                  TEMP1 = XSC*DABS(A(q,q))
                  DO 4949 p = 1, N
                     IF ( ( (p.GT.q) .AND. (DABS(A(p,q)).LE.TEMP1) )
     $                    .OR. ( p .LT. q ) )
     $                     A(p,q) = DSIGN( TEMP1, A(p,q) )
 4949             CONTINUE
 4947          CONTINUE
            ELSE
               CALL DLASET( 'U', NR-1,NR-1, ZERO,ZERO, A(1,2),LDA )
            END IF
*
*            .. second preconditioning using the QR factorization
*
            CALL DGEQRF( N,NR, A,LDA, WORK, WORK(N+1),LWORK-N, IERR )
*
*           .. and transpose upper to lower triangular
            DO 1948 p = 1, NR - 1
               CALL DCOPY( NR-p, A(p,p+1), LDA, A(p+1,p), 1 )
 1948       CONTINUE
*
         END IF
*
*           Row-cyclic Jacobi SVD algorithm with column pivoting
*
*           .. again some perturbation (a "background noise") is added
*           to drown denormals
            IF ( L2PERT ) THEN
*              XSC = DSQRT(SMALL)
               XSC = EPSLN / DBLE(N)
               DO 1947 q = 1, NR
                  TEMP1 = XSC*DABS(A(q,q))
                  DO 1949 p = 1, NR
                     IF ( ( (p.GT.q) .AND. (DABS(A(p,q)).LE.TEMP1) )
     $                       .OR. ( p .LT. q ) )
     $                   A(p,q) = DSIGN( TEMP1, A(p,q) )
 1949             CONTINUE
 1947          CONTINUE
            ELSE
               CALL DLASET( 'U', NR-1, NR-1, ZERO, ZERO, A(1,2),
     $                      LDA )
            END IF
*
*           .. and one-sided Jacobi rotations are started on a lower
*           triangular matrix (plus perturbation which is ignored in
*           the part which destroys triangular form (confusing?!))
*
            CALL DGESVJ( 'L', 'NoU', 'NoV', NR, NR, A, LDA, SVA,
     $                      N, V, LDV, WORK, LWORK, INFO )
*
            SCALEM  = WORK(1)
            NUMRANK = IDNINT(WORK(2))
*
*
      ELSE IF ( RSVEC .AND. ( .NOT. LSVEC ) ) THEN
*
*        -> Singular Values and Right Singular Vectors <-
*
         IF ( ALMORT ) THEN
*
*           .. in this case NR equals N
            DO 1998 p = 1, NR
               CALL DCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
 1998       CONTINUE
            CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2),
     $                   LDV )
*
            CALL DGESVJ( 'L','U','N', N, NR, V,LDV, SVA, NR, A,LDA,
     $                  WORK, LWORK, INFO )
            SCALEM  = WORK(1)
            NUMRANK = IDNINT(WORK(2))

         ELSE
*
*        .. two more QR factorizations ( one QRF is not enough, two require
*        accumulated product of Jacobi rotations, three are perfect )
*
            CALL DLASET( 'Lower', NR-1, NR-1, ZERO, ZERO, A(2,1),
     $                   LDA )
            CALL DGELQF( NR, N, A, LDA, WORK, WORK(N+1), LWORK-N,
     $                   IERR)
            CALL DLACPY( 'Lower', NR, NR, A, LDA, V, LDV )
            CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2),
     $                   LDV )
            CALL DGEQRF( NR, NR, V, LDV, WORK(N+1), WORK(2*N+1),
     $                   LWORK-2*N, IERR )
            DO 8998 p = 1, NR
               CALL DCOPY( NR-p+1, V(p,p), LDV, V(p,p), 1 )
 8998       CONTINUE
            CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2),
     $                   LDV )
*
            CALL DGESVJ( 'Lower', 'U','N', NR, NR, V,LDV, SVA, NR, U,
     $                  LDU, WORK(N+1), LWORK, INFO )
            SCALEM  = WORK(N+1)
            NUMRANK = IDNINT(WORK(N+2))
            IF ( NR .LT. N ) THEN
               CALL DLASET( 'A',N-NR, NR, ZERO,ZERO, V(NR+1,1),
     $                      LDV )
               CALL DLASET( 'A',NR, N-NR, ZERO,ZERO, V(1,NR+1),
     $                      LDV )
               CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE, V(NR+1,NR+1),
     $                      LDV )
            END IF
*
         CALL DORMLQ( 'Left', 'Transpose', N, N, NR, A, LDA, WORK,
     $               V, LDV, WORK(N+1), LWORK-N, IERR )
*
         END IF
*
         DO 8991 p = 1, N
            CALL DCOPY( N, V(p,1), LDV, A(IWORK(p),1), LDA )
 8991    CONTINUE
         CALL DLACPY( 'All', N, N, A, LDA, V, LDV )
*
         IF ( TRANSP ) THEN
            CALL DLACPY( 'All', N, N, V, LDV, U, LDU )
         END IF
*
      ELSE IF ( LSVEC .AND. ( .NOT. RSVEC ) ) THEN
*
*        .. Singular Values and Left Singular Vectors                 ..
*
*        .. second preconditioning step to avoid need to accumulate
*        Jacobi rotations in the Jacobi iterations.
         DO 1965 p = 1, NR
            CALL DCOPY( N-p+1, A(p,p), LDA, U(p,p), 1 )
 1965    CONTINUE
         CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )
*
         CALL DGEQRF( N, NR, U, LDU, WORK(N+1), WORK(2*N+1),
     $              LWORK-2*N, IERR )
*
         DO 1967 p = 1, NR - 1
            CALL DCOPY( NR-p, U(p,p+1), LDU, U(p+1,p), 1 )
 1967    CONTINUE
         CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )
*
         CALL DGESVJ( 'Lower', 'U', 'N', NR,NR, U, LDU, SVA, NR, A,
     $        LDA, WORK(N+1), LWORK-N, INFO )
         SCALEM  = WORK(N+1)
         NUMRANK = IDNINT(WORK(N+2))
*
         IF ( NR .LT. M ) THEN
            CALL DLASET( 'A',  M-NR, NR,ZERO, ZERO, U(NR+1,1), LDU )
            IF ( NR .LT. N1 ) THEN
               CALL DLASET( 'A',NR, N1-NR, ZERO, ZERO, U(1,NR+1),
     $                      LDU )
               CALL DLASET( 'A',M-NR,N1-NR,ZERO,ONE,U(NR+1,NR+1),
     $                      LDU )
            END IF
         END IF
*
         CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,
     $               LDU, WORK(N+1), LWORK-N, IERR )
*
         IF ( ROWPIV )
     $       CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
*
         DO 1974 p = 1, N1
            XSC = ONE / DNRM2( M, U(1,p), 1 )
            CALL DSCAL( M, XSC, U(1,p), 1 )
 1974    CONTINUE
*
         IF ( TRANSP ) THEN
            CALL DLACPY( 'All', N, N, U, LDU, V, LDV )
         END IF
*
      ELSE
*
*        .. Full SVD ..
*
         IF ( .NOT. JRACC ) THEN
*
         IF ( .NOT. ALMORT ) THEN
*
*           Second Preconditioning Step (QRF [with pivoting])
*           Note that the composition of TRANSPOSE, QRF and TRANSPOSE is
*           equivalent to an LQF CALL. Since in many libraries the QRF
*           seems to be better optimized than the LQF, we do explicit
*           transpose and use the QRF. This is subject to changes in an
*           optimized implementation of DGEJSV.
*
            DO 1968 p = 1, NR
               CALL DCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
 1968       CONTINUE
*
*           .. the following two loops perturb small entries to avoid
*           denormals in the second QR factorization, where they are
*           as good as zeros. This is done to avoid painfully slow
*           computation with denormals. The relative size of the perturbation
*           is a parameter that can be changed by the implementer.
*           This perturbation device will be obsolete on machines with
*           properly implemented arithmetic.
*           To switch it off, set L2PERT=.FALSE. To remove it from  the
*           code, remove the action under L2PERT=.TRUE., leave the ELSE part.
*           The following two loops should be blocked and fused with the
*           transposed copy above.
*
            IF ( L2PERT ) THEN
               XSC = DSQRT(SMALL)
               DO 2969 q = 1, NR
                  TEMP1 = XSC*DABS( V(q,q) )
                  DO 2968 p = 1, N
                     IF ( ( p .GT. q ) .AND. ( DABS(V(p,q)) .LE. TEMP1 )
     $                   .OR. ( p .LT. q ) )
     $                   V(p,q) = DSIGN( TEMP1, V(p,q) )
                     IF ( p .LT. q ) V(p,q) = - V(p,q)
 2968             CONTINUE
 2969          CONTINUE
            ELSE
               CALL DLASET( 'U', NR-1, NR-1, ZERO, ZERO, V(1,2),
     $                      LDV )
            END IF
*
*           Estimate the row scaled condition number of R1
*           (If R1 is rectangular, N > NR, then the condition number
*           of the leading NR x NR submatrix is estimated.)
*
            CALL DLACPY( 'L', NR, NR, V, LDV, WORK(2*N+1), NR )
            DO 3950 p = 1, NR
               TEMP1 = DNRM2(NR-p+1,WORK(2*N+(p-1)*NR+p),1)
               CALL DSCAL(NR-p+1,ONE/TEMP1,WORK(2*N+(p-1)*NR+p),1)
 3950       CONTINUE
            CALL DPOCON('Lower',NR,WORK(2*N+1),NR,ONE,TEMP1,
     $                   WORK(2*N+NR*NR+1),IWORK(M+2*N+1),IERR)
            CONDR1 = ONE / DSQRT(TEMP1)
*           .. here need a second opinion on the condition number
*           .. then assume worst case scenario
*           R1 is OK for inverse <=> CONDR1 .LT. DBLE(N)
*           more conservative    <=> CONDR1 .LT. DSQRT(DBLE(N))
*
            COND_OK = DSQRT(DBLE(NR))
*[TP]       COND_OK is a tuning parameter.

            IF ( CONDR1 .LT. COND_OK ) THEN
*              .. the second QRF without pivoting. Note: in an optimized
*              implementation, this QRF should be implemented as the QRF
*              of a lower triangular matrix.
*              R1^t = Q2 * R2
               CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),
     $              LWORK-2*N, IERR )
*
               IF ( L2PERT ) THEN
                  XSC = DSQRT(SMALL)/EPSLN
                  DO 3959 p = 2, NR
                     DO 3958 q = 1, p - 1
                        TEMP1 = XSC * MIN(DABS(V(p,p)),DABS(V(q,q)))
                        IF ( DABS(V(q,p)) .LE. TEMP1 )
     $                     V(q,p) = DSIGN( TEMP1, V(q,p) )
 3958                CONTINUE
 3959             CONTINUE
               END IF
*
               IF ( NR .NE. N )
     $         CALL DLACPY( 'A', N, NR, V, LDV, WORK(2*N+1), N )
*              .. save ...
*
*           .. this transposed copy should be better than naive
               DO 1969 p = 1, NR - 1
                  CALL DCOPY( NR-p, V(p,p+1), LDV, V(p+1,p), 1 )
 1969          CONTINUE
*
               CONDR2 = CONDR1
*
            ELSE
*
*              .. ill-conditioned case: second QRF with pivoting
*              Note that windowed pivoting would be equally good
*              numerically, and more run-time efficient. So, in
*              an optimal implementation, the next call to DGEQP3
*              should be replaced with eg. CALL SGEQPX (ACM TOMS #782)
*              with properly (carefully) chosen parameters.
*
*              R1^t * P2 = Q2 * R2
               DO 3003 p = 1, NR
                  IWORK(N+p) = 0
 3003          CONTINUE
               CALL DGEQP3( N, NR, V, LDV, IWORK(N+1), WORK(N+1),
     $                  WORK(2*N+1), LWORK-2*N, IERR )
**               CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),
**     $              LWORK-2*N, IERR )
               IF ( L2PERT ) THEN
                  XSC = DSQRT(SMALL)
                  DO 3969 p = 2, NR
                     DO 3968 q = 1, p - 1
                        TEMP1 = XSC * MIN(DABS(V(p,p)),DABS(V(q,q)))
                        IF ( DABS(V(q,p)) .LE. TEMP1 )
     $                     V(q,p) = DSIGN( TEMP1, V(q,p) )
 3968                CONTINUE
 3969             CONTINUE
               END IF
*
               CALL DLACPY( 'A', N, NR, V, LDV, WORK(2*N+1), N )
*
               IF ( L2PERT ) THEN
                  XSC = DSQRT(SMALL)
                  DO 8970 p = 2, NR
                     DO 8971 q = 1, p - 1
                        TEMP1 = XSC * MIN(DABS(V(p,p)),DABS(V(q,q)))
                        V(p,q) = - DSIGN( TEMP1, V(q,p) )
 8971                CONTINUE
 8970             CONTINUE
               ELSE
                  CALL DLASET( 'L',NR-1,NR-1,ZERO,ZERO,V(2,1),LDV )
               END IF
*              Now, compute R2 = L3 * Q3, the LQ factorization.
               CALL DGELQF( NR, NR, V, LDV, WORK(2*N+N*NR+1),
     $               WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, IERR )
*              .. and estimate the condition number
               CALL DLACPY( 'L',NR,NR,V,LDV,WORK(2*N+N*NR+NR+1),NR )
               DO 4950 p = 1, NR
                  TEMP1 = DNRM2( p, WORK(2*N+N*NR+NR+p), NR )
                  CALL DSCAL( p, ONE/TEMP1, WORK(2*N+N*NR+NR+p), NR )
 4950          CONTINUE
               CALL DPOCON( 'L',NR,WORK(2*N+N*NR+NR+1),NR,ONE,TEMP1,
     $              WORK(2*N+N*NR+NR+NR*NR+1),IWORK(M+2*N+1),IERR )
               CONDR2 = ONE / DSQRT(TEMP1)
*
               IF ( CONDR2 .GE. COND_OK ) THEN
*                 .. save the Householder vectors used for Q3
*                 (this overwrites the copy of R2, as it will not be
*                 needed in this branch, but it does not overwrite the
*                 Huseholder vectors of Q2.).
                  CALL DLACPY( 'U', NR, NR, V, LDV, WORK(2*N+1), N )
*                 .. and the rest of the information on Q3 is in
*                 WORK(2*N+N*NR+1:2*N+N*NR+N)
               END IF
*
            END IF
*
            IF ( L2PERT ) THEN
               XSC = DSQRT(SMALL)
               DO 4968 q = 2, NR
                  TEMP1 = XSC * V(q,q)
                  DO 4969 p = 1, q - 1
*                    V(p,q) = - DSIGN( TEMP1, V(q,p) )
                     V(p,q) = - DSIGN( TEMP1, V(p,q) )
 4969             CONTINUE
 4968          CONTINUE
            ELSE
               CALL DLASET( 'U', NR-1,NR-1, ZERO,ZERO, V(1,2), LDV )
            END IF
*
*        Second preconditioning finished; continue with Jacobi SVD
*        The input matrix is lower triangular.
*
*        Recover the right singular vectors as solution of a well
*        conditioned triangular matrix equation.
*
            IF ( CONDR1 .LT. COND_OK ) THEN
*
               CALL DGESVJ( 'L','U','N',NR,NR,V,LDV,SVA,NR,U,
     $              LDU,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,INFO )
               SCALEM  = WORK(2*N+N*NR+NR+1)
               NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2))
               DO 3970 p = 1, NR
                  CALL DCOPY( NR, V(1,p), 1, U(1,p), 1 )
                  CALL DSCAL( NR, SVA(p),    V(1,p), 1 )
 3970          CONTINUE

*        .. pick the right matrix equation and solve it
*
               IF ( NR .EQ. N ) THEN
* :))             .. best case, R1 is inverted. The solution of this matrix
*                 equation is Q2*V2 = the product of the Jacobi rotations
*                 used in DGESVJ, premultiplied with the orthogonal matrix
*                 from the second QR factorization.
                  CALL DTRSM( 'L','U','N','N', NR,NR,ONE, A,LDA, V,
     $                        LDV )
               ELSE
*                 .. R1 is well conditioned, but non-square. Transpose(R2)
*                 is inverted to get the product of the Jacobi rotations
*                 used in DGESVJ. The Q-factor from the second QR
*                 factorization is then built in explicitly.
                  CALL DTRSM('L','U','T','N',NR,NR,ONE,WORK(2*N+1),
     $                 N,V,LDV)
                  IF ( NR .LT. N ) THEN
                    CALL DLASET('A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV)
                    CALL DLASET('A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV)
                    CALL DLASET('A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),
     $                           LDV)
                  END IF
                  CALL DORMQR('L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),
     $                 V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR)
               END IF
*
            ELSE IF ( CONDR2 .LT. COND_OK ) THEN
*
* :)           .. the input matrix A is very likely a relative of
*              the Kahan matrix :)
*              The matrix R2 is inverted. The solution of the matrix equation
*              is Q3^T*V3 = the product of the Jacobi rotations (applied to
*              the lower triangular L3 from the LQ factorization of
*              R2=L3*Q3), pre-multiplied with the transposed Q3.
               CALL DGESVJ( 'L', 'U', 'N', NR, NR, V, LDV, SVA, NR,
     $                      U,
     $              LDU, WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO )
               SCALEM  = WORK(2*N+N*NR+NR+1)
               NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2))
               DO 3870 p = 1, NR
                  CALL DCOPY( NR, V(1,p), 1, U(1,p), 1 )
                  CALL DSCAL( NR, SVA(p),    U(1,p), 1 )
 3870          CONTINUE
               CALL DTRSM('L','U','N','N',NR,NR,ONE,WORK(2*N+1),N,U,
     $                     LDU)
*              .. apply the permutation from the second QR factorization
               DO 873 q = 1, NR
                  DO 872 p = 1, NR
                     WORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)
 872              CONTINUE
                  DO 874 p = 1, NR
                     U(p,q) = WORK(2*N+N*NR+NR+p)
 874              CONTINUE
 873           CONTINUE
               IF ( NR .LT. N ) THEN
                  CALL DLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )
                  CALL DLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )
                  CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),
     $                         LDV )
               END IF
               CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),
     $              V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
            ELSE
*              Last line of defense.
* #:(          This is a rather pathological case: no scaled condition
*              improvement after two pivoted QR factorizations. Other
*              possibility is that the rank revealing QR factorization
*              or the condition estimator has failed, or the COND_OK
*              is set very close to ONE (which is unnecessary). Normally,
*              this branch should never be executed, but in rare cases of
*              failure of the RRQR or condition estimator, the last line of
*              defense ensures that DGEJSV completes the task.
*              Compute the full SVD of L3 using DGESVJ with explicit
*              accumulation of Jacobi rotations.
               CALL DGESVJ( 'L', 'U', 'V', NR, NR, V, LDV, SVA, NR,
     $                      U,
     $              LDU, WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO )
               SCALEM  = WORK(2*N+N*NR+NR+1)
               NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2))
               IF ( NR .LT. N ) THEN
                  CALL DLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )
                  CALL DLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )
                  CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),
     $                         LDV )
               END IF
               CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),
     $              V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
*
               CALL DORMLQ( 'L', 'T', NR, NR, NR, WORK(2*N+1), N,
     $              WORK(2*N+N*NR+1), U, LDU, WORK(2*N+N*NR+NR+1),
     $              LWORK-2*N-N*NR-NR, IERR )
               DO 773 q = 1, NR
                  DO 772 p = 1, NR
                     WORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)
 772              CONTINUE
                  DO 774 p = 1, NR
                     U(p,q) = WORK(2*N+N*NR+NR+p)
 774              CONTINUE
 773           CONTINUE
*
            END IF
*
*           Permute the rows of V using the (column) permutation from the
*           first QRF. Also, scale the columns to make them unit in
*           Euclidean norm. This applies to all cases.
*
            TEMP1 = DSQRT(DBLE(N)) * EPSLN
            DO 1972 q = 1, N
               DO 972 p = 1, N
                  WORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)
  972          CONTINUE
               DO 973 p = 1, N
                  V(p,q) = WORK(2*N+N*NR+NR+p)
  973          CONTINUE
               XSC = ONE / DNRM2( N, V(1,q), 1 )
               IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
     $           CALL DSCAL( N, XSC, V(1,q), 1 )
 1972       CONTINUE
*           At this moment, V contains the right singular vectors of A.
*           Next, assemble the left singular vector matrix U (M x N).
            IF ( NR .LT. M ) THEN
               CALL DLASET( 'A', M-NR, NR, ZERO, ZERO, U(NR+1,1),
     $                      LDU )
               IF ( NR .LT. N1 ) THEN
                  CALL DLASET('A',NR,N1-NR,ZERO,ZERO,U(1,NR+1),LDU)
                  CALL DLASET('A',M-NR,N1-NR,ZERO,ONE,U(NR+1,NR+1),
     $                         LDU)
               END IF
            END IF
*
*           The Q matrix from the first QRF is built into the left singular
*           matrix U. This applies to all cases.
*
            CALL DORMQR( 'Left', 'No_Tr', M, N1, N, A, LDA, WORK, U,
     $           LDU, WORK(N+1), LWORK-N, IERR )

*           The columns of U are normalized. The cost is O(M*N) flops.
            TEMP1 = DSQRT(DBLE(M)) * EPSLN
            DO 1973 p = 1, NR
               XSC = ONE / DNRM2( M, U(1,p), 1 )
               IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
     $          CALL DSCAL( M, XSC, U(1,p), 1 )
 1973       CONTINUE
*
*           If the initial QRF is computed with row pivoting, the left
*           singular vectors must be adjusted.
*
            IF ( ROWPIV )
     $          CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
*
         ELSE
*
*        .. the initial matrix A has almost orthogonal columns and
*        the second QRF is not needed
*
            CALL DLACPY( 'Upper', N, N, A, LDA, WORK(N+1), N )
            IF ( L2PERT ) THEN
               XSC = DSQRT(SMALL)
               DO 5970 p = 2, N
                  TEMP1 = XSC * WORK( N + (p-1)*N + p )
                  DO 5971 q = 1, p - 1
                     WORK(N+(q-1)*N+p)=-DSIGN(TEMP1,WORK(N+(p-1)*N+q))
 5971             CONTINUE
 5970          CONTINUE
            ELSE
               CALL DLASET( 'Lower',N-1,N-1,ZERO,ZERO,WORK(N+2),N )
            END IF
*
            CALL DGESVJ( 'Upper', 'U', 'N', N, N, WORK(N+1), N, SVA,
     $           N, U, LDU, WORK(N+N*N+1), LWORK-N-N*N, INFO )
*
            SCALEM  = WORK(N+N*N+1)
            NUMRANK = IDNINT(WORK(N+N*N+2))
            DO 6970 p = 1, N
               CALL DCOPY( N, WORK(N+(p-1)*N+1), 1, U(1,p), 1 )
               CALL DSCAL( N, SVA(p), WORK(N+(p-1)*N+1), 1 )
 6970       CONTINUE
*
            CALL DTRSM( 'Left', 'Upper', 'NoTrans', 'No UD', N, N,
     $           ONE, A, LDA, WORK(N+1), N )
            DO 6972 p = 1, N
               CALL DCOPY( N, WORK(N+p), N, V(IWORK(p),1), LDV )
 6972       CONTINUE
            TEMP1 = DSQRT(DBLE(N))*EPSLN
            DO 6971 p = 1, N
               XSC = ONE / DNRM2( N, V(1,p), 1 )
               IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
     $            CALL DSCAL( N, XSC, V(1,p), 1 )
 6971       CONTINUE
*
*           Assemble the left singular vector matrix U (M x N).
*
            IF ( N .LT. M ) THEN
               CALL DLASET( 'A',  M-N, N, ZERO, ZERO, U(N+1,1), LDU )
               IF ( N .LT. N1 ) THEN
                  CALL DLASET( 'A',N,  N1-N, ZERO, ZERO,  U(1,N+1),
     $                         LDU )
                  CALL DLASET( 'A',M-N,N1-N, ZERO, ONE,U(N+1,N+1),
     $                         LDU )
               END IF
            END IF
            CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,
     $           LDU, WORK(N+1), LWORK-N, IERR )
            TEMP1 = DSQRT(DBLE(M))*EPSLN
            DO 6973 p = 1, N1
               XSC = ONE / DNRM2( M, U(1,p), 1 )
               IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
     $            CALL DSCAL( M, XSC, U(1,p), 1 )
 6973       CONTINUE
*
            IF ( ROWPIV )
     $         CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
*
         END IF
*
*        end of the  >> almost orthogonal case <<  in the full SVD
*
         ELSE
*
*        This branch deploys a preconditioned Jacobi SVD with explicitly
*        accumulated rotations. It is included as optional, mainly for
*        experimental purposes. It does perform well, and can also be used.
*        In this implementation, this branch will be automatically activated
*        if the  condition number sigma_max(A) / sigma_min(A) is predicted
*        to be greater than the overflow threshold. This is because the
*        a posteriori computation of the singular vectors assumes robust
*        implementation of BLAS and some LAPACK procedures, capable of working
*        in presence of extreme values. Since that is not always the case, ...
*
         DO 7968 p = 1, NR
            CALL DCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
 7968    CONTINUE
*
         IF ( L2PERT ) THEN
            XSC = DSQRT(SMALL/EPSLN)
            DO 5969 q = 1, NR
               TEMP1 = XSC*DABS( V(q,q) )
               DO 5968 p = 1, N
                  IF ( ( p .GT. q ) .AND. ( DABS(V(p,q)) .LE. TEMP1 )
     $                .OR. ( p .LT. q ) )
     $                V(p,q) = DSIGN( TEMP1, V(p,q) )
                  IF ( p .LT. q ) V(p,q) = - V(p,q)
 5968          CONTINUE
 5969       CONTINUE
         ELSE
            CALL DLASET( 'U', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )
         END IF

         CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),
     $        LWORK-2*N, IERR )
         CALL DLACPY( 'L', N, NR, V, LDV, WORK(2*N+1), N )
*
         DO 7969 p = 1, NR
            CALL DCOPY( NR-p+1, V(p,p), LDV, U(p,p), 1 )
 7969    CONTINUE

         IF ( L2PERT ) THEN
            XSC = DSQRT(SMALL/EPSLN)
            DO 9970 q = 2, NR
               DO 9971 p = 1, q - 1
                  TEMP1 = XSC * MIN(DABS(U(p,p)),DABS(U(q,q)))
                  U(p,q) = - DSIGN( TEMP1, U(q,p) )
 9971          CONTINUE
 9970       CONTINUE
         ELSE
            CALL DLASET('U', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )
         END IF

         CALL DGESVJ( 'G', 'U', 'V', NR, NR, U, LDU, SVA,
     $        N, V, LDV, WORK(2*N+N*NR+1), LWORK-2*N-N*NR, INFO )
         SCALEM  = WORK(2*N+N*NR+1)
         NUMRANK = IDNINT(WORK(2*N+N*NR+2))

         IF ( NR .LT. N ) THEN
            CALL DLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )
            CALL DLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )
            CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )
         END IF

         CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),
     $        V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
*
*           Permute the rows of V using the (column) permutation from the
*           first QRF. Also, scale the columns to make them unit in
*           Euclidean norm. This applies to all cases.
*
            TEMP1 = DSQRT(DBLE(N)) * EPSLN
            DO 7972 q = 1, N
               DO 8972 p = 1, N
                  WORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)
 8972          CONTINUE
               DO 8973 p = 1, N
                  V(p,q) = WORK(2*N+N*NR+NR+p)
 8973          CONTINUE
               XSC = ONE / DNRM2( N, V(1,q), 1 )
               IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
     $           CALL DSCAL( N, XSC, V(1,q), 1 )
 7972       CONTINUE
*
*           At this moment, V contains the right singular vectors of A.
*           Next, assemble the left singular vector matrix U (M x N).
*
         IF ( NR .LT. M ) THEN
            CALL DLASET( 'A',  M-NR, NR, ZERO, ZERO, U(NR+1,1), LDU )
            IF ( NR .LT. N1 ) THEN
               CALL DLASET( 'A',NR,  N1-NR, ZERO, ZERO,  U(1,NR+1),
     $                      LDU )
               CALL DLASET( 'A',M-NR,N1-NR, ZERO, ONE,U(NR+1,NR+1),
     $                      LDU )
            END IF
         END IF
*
         CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,
     $        LDU, WORK(N+1), LWORK-N, IERR )
*
            IF ( ROWPIV )
     $         CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
*
*
         END IF
         IF ( TRANSP ) THEN
*           .. swap U and V because the procedure worked on A^t
            DO 6974 p = 1, N
               CALL DSWAP( N, U(1,p), 1, V(1,p), 1 )
 6974       CONTINUE
         END IF
*
      END IF
*     end of the full SVD
*
*     Undo scaling, if necessary (and possible)
*
      IF ( USCAL2 .LE. (BIG/SVA(1))*USCAL1 ) THEN
         CALL DLASCL( 'G', 0, 0, USCAL1, USCAL2, NR, 1, SVA, N,
     $                IERR )
         USCAL1 = ONE
         USCAL2 = ONE
      END IF
*
      IF ( NR .LT. N ) THEN
         DO 3004 p = NR+1, N
            SVA(p) = ZERO
 3004    CONTINUE
      END IF
*
      WORK(1) = USCAL2 * SCALEM
      WORK(2) = USCAL1
      IF ( ERREST ) WORK(3) = SCONDA
      IF ( LSVEC .AND. RSVEC ) THEN
         WORK(4) = CONDR1
         WORK(5) = CONDR2
      END IF
      IF ( L2TRAN ) THEN
         WORK(6) = ENTRA
         WORK(7) = ENTRAT
      END IF
*
      IWORK(1) = NR
      IWORK(2) = NUMRANK
      IWORK(3) = WARNING
*
      RETURN
*     ..
*     .. END OF DGEJSV
*     ..
      END
*
*> \brief \b DGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGELQ2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelq2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelq2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelq2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGELQ2( M, N, A, LDA, TAU, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGELQ2 computes an LQ factorization of a real m-by-n matrix A:
*>
*>    A = ( L 0 ) *  Q
*>
*> where:
*>
*>    Q is a n-by-n orthogonal matrix;
*>    L is a lower-triangular m-by-m matrix;
*>    0 is a m-by-(n-m) zero matrix, if m < n.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the m by n matrix A.
*>          On exit, the elements on and below the diagonal of the array
*>          contain the m by min(m,n) lower trapezoidal matrix L (L is
*>          lower triangular if m <= n); the elements above the diagonal,
*>          with the array TAU, represent the orthogonal matrix Q as a
*>          product of elementary reflectors (see Further Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (M)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gelq2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix Q is represented as a product of elementary reflectors
*>
*>     Q = H(k) . . . H(2) H(1), where k = min(m,n).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
*>  and tau in TAU(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGELQ2( M, N, A, LDA, TAU, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, K
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF1F, DLARFG, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGELQ2', -INFO )
         RETURN
      END IF
*
      K = MIN( M, N )
*
      DO 10 I = 1, K
*
*        Generate elementary reflector H(i) to annihilate A(i,i+1:n)
*
         CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA,
     $                TAU( I ) )
         IF( I.LT.M ) THEN
*
*           Apply H(i) to A(i+1:m,i:n) from the right
*
            CALL DLARF1F( 'Right', M-I, N-I+1, A( I, I ), LDA,
     $                  TAU( I ),
     $                  A( I+1, I ), LDA, WORK )
         END IF
   10 CONTINUE
      RETURN
*
*     End of DGELQ2
*
      END
*> \brief \b DGELQF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGELQF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelqf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelqf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelqf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGELQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGELQF computes an LQ factorization of a real M-by-N matrix A:
*>
*>    A = ( L 0 ) *  Q
*>
*> where:
*>
*>    Q is a N-by-N orthogonal matrix;
*>    L is a lower-triangular M-by-M matrix;
*>    0 is a M-by-(N-M) zero matrix, if M < N.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit, the elements on and below the diagonal of the array
*>          contain the m-by-min(m,n) lower trapezoidal matrix L (L is
*>          lower triangular if m <= n); the elements above the diagonal,
*>          with the array TAU, represent the orthogonal matrix Q as a
*>          product of elementary reflectors (see Further Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= M, otherwise.
*>          For optimum performance LWORK >= M*NB, where NB is the
*>          optimal blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gelqf
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix Q is represented as a product of elementary reflectors
*>
*>     Q = H(k) . . . H(2) H(1), where k = min(m,n).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
*>  and tau in TAU(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGELQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB,
     $                   NBMIN, NX
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGELQ2, DLARFB, DLARFT, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      K = MIN( M, N )
      NB = ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
      LQUERY = ( LWORK.EQ.-1 )
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      ELSE IF( .NOT.LQUERY ) THEN
         IF( LWORK.LE.0 .OR. ( N.GT.0 .AND. LWORK.LT.MAX( 1, M ) ) )
     $      INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGELQF', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         IF( K.EQ.0 ) THEN
            LWKOPT = 1
         ELSE
            LWKOPT = M*NB
         END IF
         WORK( 1 ) = LWKOPT
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( K.EQ.0 ) THEN
         WORK( 1 ) = 1
         RETURN
      END IF
*
      NBMIN = 2
      NX = 0
      IWS = M
      IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
*        Determine when to cross over from blocked to unblocked code.
*
         NX = MAX( 0, ILAENV( 3, 'DGELQF', ' ', M, N, -1, -1 ) )
         IF( NX.LT.K ) THEN
*
*           Determine if workspace is large enough for blocked code.
*
            LDWORK = M
            IWS = LDWORK*NB
            IF( LWORK.LT.IWS ) THEN
*
*              Not enough workspace to use optimal NB:  reduce NB and
*              determine the minimum value of NB.
*
               NB = LWORK / LDWORK
               NBMIN = MAX( 2, ILAENV( 2, 'DGELQF', ' ', M, N, -1,
     $                 -1 ) )
            END IF
         END IF
      END IF
*
      IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
*        Use blocked code initially
*
         DO 10 I = 1, K - NX, NB
            IB = MIN( K-I+1, NB )
*
*           Compute the LQ factorization of the current block
*           A(i:i+ib-1,i:n)
*
            CALL DGELQ2( IB, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
     $                   IINFO )
            IF( I+IB.LE.M ) THEN
*
*              Form the triangular factor of the block reflector
*              H = H(i) H(i+1) . . . H(i+ib-1)
*
               CALL DLARFT( 'Forward', 'Rowwise', N-I+1, IB, A( I,
     $                      I ),
     $                      LDA, TAU( I ), WORK, LDWORK )
*
*              Apply H to A(i+ib:m,i:n) from the right
*
               CALL DLARFB( 'Right', 'No transpose', 'Forward',
     $                      'Rowwise', M-I-IB+1, N-I+1, IB, A( I, I ),
     $                      LDA, WORK, LDWORK, A( I+IB, I ), LDA,
     $                      WORK( IB+1 ), LDWORK )
            END IF
   10    CONTINUE
      ELSE
         I = 1
      END IF
*
*     Use unblocked code to factor the last or only block.
*
      IF( I.LE.K )
     $   CALL DGELQ2( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
     $                IINFO )
*
      WORK( 1 ) = IWS
      RETURN
*
*     End of DGELQF
*
      END
*> \brief <b> DGELS solves overdetermined or underdetermined systems for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGELS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgels.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgels.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgels.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK,
*                         INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANS
*       INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGELS solves overdetermined or underdetermined real linear systems
*> involving an M-by-N matrix A, or its transpose, using a QR or LQ
*> factorization of A.
*>
*> It is assumed that A has full rank, and only a rudimentary protection
*> against rank-deficient matrices is provided. This subroutine only detects
*> exact rank-deficiency, where a diagonal element of the triangular factor
*> of A is exactly zero.
*>
*> It is conceivable for one (or more) of the diagonal elements of the triangular
*> factor of A to be subnormally tiny numbers without this subroutine signalling
*> an error. The solutions computed for such almost-rank-deficient matrices may
*> be less accurate due to a loss of numerical precision.
*>
*> The following options are provided:
*>
*> 1. If TRANS = 'N' and m >= n:  find the least squares solution of
*>    an overdetermined system, i.e., solve the least squares problem
*>                 minimize || B - A*X ||.
*>
*> 2. If TRANS = 'N' and m < n:  find the minimum norm solution of
*>    an underdetermined system A * X = B.
*>
*> 3. If TRANS = 'T' and m >= n:  find the minimum norm solution of
*>    an underdetermined system A**T * X = B.
*>
*> 4. If TRANS = 'T' and m < n:  find the least squares solution of
*>    an overdetermined system, i.e., solve the least squares problem
*>                 minimize || B - A**T * X ||.
*>
*> Several right hand side vectors b and solution vectors x can be
*> handled in a single call; they are stored as the columns of the
*> M-by-NRHS right hand side matrix B and the N-by-NRHS solution
*> matrix X.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N': the linear system involves A;
*>          = 'T': the linear system involves A**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of
*>          columns of the matrices B and X. NRHS >=0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit,
*>            if M >= N, A is overwritten by details of its QR
*>                       factorization as returned by DGEQRF;
*>            if M <  N, A is overwritten by details of its LQ
*>                       factorization as returned by DGELQF.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the matrix B of right hand side vectors, stored
*>          columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
*>          if TRANS = 'T'.
*>          On exit, if INFO = 0, B is overwritten by the solution
*>          vectors, stored columnwise:
*>          if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
*>          squares solution vectors; the residual sum of squares for the
*>          solution in each column is given by the sum of squares of
*>          elements N+1 to M in that column;
*>          if TRANS = 'N' and m < n, rows 1 to N of B contain the
*>          minimum norm solution vectors;
*>          if TRANS = 'T' and m >= n, rows 1 to M of B contain the
*>          minimum norm solution vectors;
*>          if TRANS = 'T' and m < n, rows 1 to M of B contain the
*>          least squares solution vectors; the residual sum of squares
*>          for the solution in each column is given by the sum of
*>          squares of elements M+1 to N in that column.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= MAX(1,M,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          LWORK >= max( 1, MN + max( MN, NRHS ) ).
*>          For optimal performance,
*>          LWORK >= max( 1, MN + max( MN, NRHS )*NB ).
*>          where MN = min(M,N) and NB is the optimum block size.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO =  i, the i-th diagonal element of the
*>                triangular factor of A is exactly zero, so that A does not have
*>                full rank; the least squares solution could not be
*>                computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gels
*
*  =====================================================================
      SUBROUTINE DGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK,
     $                  LWORK,
     $                  INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS
      INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, TPSD
      INTEGER            BROW, I, IASCL, IBSCL, J, MN, NB, SCLLEN, WSIZE
      DOUBLE PRECISION   ANRM, BIGNUM, BNRM, SMLNUM
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   RWORK( 1 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           LSAME, ILAENV, DLAMCH, DLANGE
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGELQF, DGEQRF, DLASCL, DLASET, DORMLQ,
     $                   DORMQR,
     $                   DTRTRS, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments.
*
      INFO = 0
      MN = MIN( M, N )
      LQUERY = ( LWORK.EQ.-1 )
      IF( .NOT.( LSAME( TRANS, 'N' ) .OR.
     $    LSAME( TRANS, 'T' ) ) ) THEN
         INFO = -1
      ELSE IF( M.LT.0 ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -6
      ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
         INFO = -8
      ELSE IF( LWORK.LT.MAX( 1, MN+MAX( MN, NRHS ) ) .AND. .NOT.LQUERY )
     $          THEN
         INFO = -10
      END IF
*
*     Figure out optimal block size
*
      IF( INFO.EQ.0 .OR. INFO.EQ.-10 ) THEN
*
         TPSD = .TRUE.
         IF( LSAME( TRANS, 'N' ) )
     $      TPSD = .FALSE.
*
         IF( M.GE.N ) THEN
            NB = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
            IF( TPSD ) THEN
               NB = MAX( NB, ILAENV( 1, 'DORMQR', 'LN', M, NRHS, N,
     $              -1 ) )
            ELSE
               NB = MAX( NB, ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N,
     $              -1 ) )
            END IF
         ELSE
            NB = ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
            IF( TPSD ) THEN
               NB = MAX( NB, ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M,
     $              -1 ) )
            ELSE
               NB = MAX( NB, ILAENV( 1, 'DORMLQ', 'LN', N, NRHS, M,
     $              -1 ) )
            END IF
         END IF
*
         WSIZE = MAX( 1, MN+MAX( MN, NRHS )*NB )
         WORK( 1 ) = DBLE( WSIZE )
*
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGELS ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( MIN( M, N, NRHS ).EQ.0 ) THEN
         CALL DLASET( 'Full', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
         RETURN
      END IF
*
*     Get machine parameters
*
      SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' )
      BIGNUM = ONE / SMLNUM
*
*     Scale A, B if max element outside range [SMLNUM,BIGNUM]
*
      ANRM = DLANGE( 'M', M, N, A, LDA, RWORK )
      IASCL = 0
      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
         IASCL = 1
      ELSE IF( ANRM.GT.BIGNUM ) THEN
*
*        Scale matrix norm down to BIGNUM
*
         CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
         IASCL = 2
      ELSE IF( ANRM.EQ.ZERO ) THEN
*
*        Matrix all zero. Return zero solution.
*
         CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
         GO TO 50
      END IF
*
      BROW = M
      IF( TPSD )
     $   BROW = N
      BNRM = DLANGE( 'M', BROW, NRHS, B, LDB, RWORK )
      IBSCL = 0
      IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, BROW, NRHS, B, LDB,
     $                INFO )
         IBSCL = 1
      ELSE IF( BNRM.GT.BIGNUM ) THEN
*
*        Scale matrix norm down to BIGNUM
*
         CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, BROW, NRHS, B, LDB,
     $                INFO )
         IBSCL = 2
      END IF
*
      IF( M.GE.N ) THEN
*
*        compute QR factorization of A
*
         CALL DGEQRF( M, N, A, LDA, WORK( 1 ), WORK( MN+1 ),
     $                LWORK-MN,
     $                INFO )
*
*        workspace at least N, optimally N*NB
*
         IF( .NOT.TPSD ) THEN
*
*           Least-Squares Problem min || A * X - B ||
*
*           B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS)
*
            CALL DORMQR( 'Left', 'Transpose', M, NRHS, N, A, LDA,
     $                   WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
     $                   INFO )
*
*           workspace at least NRHS, optimally NRHS*NB
*
*           B(1:N,1:NRHS) := inv(R) * B(1:N,1:NRHS)
*
            CALL DTRTRS( 'Upper', 'No transpose', 'Non-unit', N,
     $                   NRHS,
     $                   A, LDA, B, LDB, INFO )
*
            IF( INFO.GT.0 ) THEN
               RETURN
            END IF
*
            SCLLEN = N
*
         ELSE
*
*           Underdetermined system of equations A**T * X = B
*
*           B(1:N,1:NRHS) := inv(R**T) * B(1:N,1:NRHS)
*
            CALL DTRTRS( 'Upper', 'Transpose', 'Non-unit', N, NRHS,
     $                   A, LDA, B, LDB, INFO )
*
            IF( INFO.GT.0 ) THEN
               RETURN
            END IF
*
*           B(N+1:M,1:NRHS) = ZERO
*
            DO 20 J = 1, NRHS
               DO 10 I = N + 1, M
                  B( I, J ) = ZERO
   10          CONTINUE
   20       CONTINUE
*
*           B(1:M,1:NRHS) := Q(1:N,:) * B(1:N,1:NRHS)
*
            CALL DORMQR( 'Left', 'No transpose', M, NRHS, N, A, LDA,
     $                   WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
     $                   INFO )
*
*           workspace at least NRHS, optimally NRHS*NB
*
            SCLLEN = M
*
         END IF
*
      ELSE
*
*        Compute LQ factorization of A
*
         CALL DGELQF( M, N, A, LDA, WORK( 1 ), WORK( MN+1 ),
     $                LWORK-MN,
     $                INFO )
*
*        workspace at least M, optimally M*NB.
*
         IF( .NOT.TPSD ) THEN
*
*           underdetermined system of equations A * X = B
*
*           B(1:M,1:NRHS) := inv(L) * B(1:M,1:NRHS)
*
            CALL DTRTRS( 'Lower', 'No transpose', 'Non-unit', M,
     $                   NRHS,
     $                   A, LDA, B, LDB, INFO )
*
            IF( INFO.GT.0 ) THEN
               RETURN
            END IF
*
*           B(M+1:N,1:NRHS) = 0
*
            DO 40 J = 1, NRHS
               DO 30 I = M + 1, N
                  B( I, J ) = ZERO
   30          CONTINUE
   40       CONTINUE
*
*           B(1:N,1:NRHS) := Q(1:N,:)**T * B(1:M,1:NRHS)
*
            CALL DORMLQ( 'Left', 'Transpose', N, NRHS, M, A, LDA,
     $                   WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
     $                   INFO )
*
*           workspace at least NRHS, optimally NRHS*NB
*
            SCLLEN = N
*
         ELSE
*
*           overdetermined system min || A**T * X - B ||
*
*           B(1:N,1:NRHS) := Q * B(1:N,1:NRHS)
*
            CALL DORMLQ( 'Left', 'No transpose', N, NRHS, M, A, LDA,
     $                   WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
     $                   INFO )
*
*           workspace at least NRHS, optimally NRHS*NB
*
*           B(1:M,1:NRHS) := inv(L**T) * B(1:M,1:NRHS)
*
            CALL DTRTRS( 'Lower', 'Transpose', 'Non-unit', M, NRHS,
     $                   A, LDA, B, LDB, INFO )
*
            IF( INFO.GT.0 ) THEN
               RETURN
            END IF
*
            SCLLEN = M
*
         END IF
*
      END IF
*
*     Undo scaling
*
      IF( IASCL.EQ.1 ) THEN
         CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, SCLLEN, NRHS, B, LDB,
     $                INFO )
      ELSE IF( IASCL.EQ.2 ) THEN
         CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, SCLLEN, NRHS, B, LDB,
     $                INFO )
      END IF
      IF( IBSCL.EQ.1 ) THEN
         CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, SCLLEN, NRHS, B, LDB,
     $                INFO )
      ELSE IF( IBSCL.EQ.2 ) THEN
         CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, SCLLEN, NRHS, B, LDB,
     $                INFO )
      END IF
*
   50 CONTINUE
      WORK( 1 ) = DBLE( WSIZE )
*
      RETURN
*
*     End of DGELS
*
      END
*> \brief <b> DGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGELSD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelsd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelsd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelsd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
*                          WORK, LWORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
*       DOUBLE PRECISION   RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGELSD computes the minimum-norm solution to a real linear least
*> squares problem:
*>     minimize 2-norm(| b - A*x |)
*> using the singular value decomposition (SVD) of A. A is an M-by-N
*> matrix which may be rank-deficient.
*>
*> Several right hand side vectors b and solution vectors x can be
*> handled in a single call; they are stored as the columns of the
*> M-by-NRHS right hand side matrix B and the N-by-NRHS solution
*> matrix X.
*>
*> The problem is solved in three steps:
*> (1) Reduce the coefficient matrix A to bidiagonal form with
*>     Householder transformations, reducing the original problem
*>     into a "bidiagonal least squares problem" (BLS)
*> (2) Solve the BLS using a divide and conquer approach.
*> (3) Apply back all the Householder transformations to solve
*>     the original least squares problem.
*>
*> The effective rank of A is determined by treating as zero those
*> singular values which are less than RCOND times the largest singular
*> value.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit, A has been destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the M-by-NRHS right hand side matrix B.
*>          On exit, B is overwritten by the N-by-NRHS solution
*>          matrix X.  If m >= n and RANK = n, the residual
*>          sum-of-squares for the solution in the i-th column is given
*>          by the sum of squares of elements n+1:m in that column.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,max(M,N)).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (min(M,N))
*>          The singular values of A in decreasing order.
*>          The condition number of A in the 2-norm = S(1)/S(min(m,n)).
*> \endverbatim
*>
*> \param[in] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          RCOND is used to determine the effective rank of A.
*>          Singular values S(i) <= RCOND*S(1) are treated as zero.
*>          If RCOND < 0, machine precision is used instead.
*> \endverbatim
*>
*> \param[out] RANK
*> \verbatim
*>          RANK is INTEGER
*>          The effective rank of A, i.e., the number of singular values
*>          which are greater than RCOND*S(1).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK must be at least 1.
*>          The exact minimum amount of workspace needed depends on M,
*>          N and NRHS. As long as LWORK is at least
*>              12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,
*>          if M is greater than or equal to N or
*>              12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,
*>          if M is less than N, the code will execute correctly.
*>          SMLSIZ is returned by ILAENV and is equal to the maximum
*>          size of the subproblems at the bottom of the computation
*>          tree (usually about 25), and
*>             NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
*>          For good performance, LWORK should generally be larger.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*>          LIWORK >= max(1, 3 * MINMN * NLVL + 11 * MINMN),
*>          where MINMN = MIN( M,N ).
*>          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  the algorithm for computing the SVD failed to converge;
*>                if INFO = i, i off-diagonal elements of an intermediate
*>                bidiagonal form did not converge to zero.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gelsd
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Ren-Cang Li, Computer Science Division, University of
*>       California at Berkeley, USA \n
*>     Osni Marques, LBNL/NERSC, USA \n
*
*  =====================================================================
      SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
     $                   WORK, LWORK, IWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ,
     $                   LDWORK, LIWORK, MAXMN, MAXWRK, MINMN, MINWRK,
     $                   MM, MNTHR, NLVL, NWORK, SMLSIZ, WLALSD
      DOUBLE PRECISION   ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEBRD, DGELQF, DGEQRF, DLACPY,
     $                   DLALSD,
     $                   DLASCL, DLASET, DORMBR, DORMLQ, DORMQR, XERBLA
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           ILAENV, DLAMCH, DLANGE
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE, INT, LOG, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments.
*
      INFO = 0
      MINMN = MIN( M, N )
      MAXMN = MAX( M, N )
      MNTHR = ILAENV( 6, 'DGELSD', ' ', M, N, NRHS, -1 )
      LQUERY = ( LWORK.EQ.-1 )
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN
         INFO = -7
      END IF
*
      SMLSIZ = ILAENV( 9, 'DGELSD', ' ', 0, 0, 0, 0 )
*
*     Compute workspace.
*     (Note: Comments in the code beginning "Workspace:" describe the
*     minimal amount of workspace needed at that point in the code,
*     as well as the preferred amount for good performance.
*     NB refers to the optimal block size for the immediately
*     following subroutine, as returned by ILAENV.)
*
      MINWRK = 1
      LIWORK = 1
      MINMN = MAX( 1, MINMN )
      NLVL = MAX( INT( LOG( DBLE( MINMN ) / DBLE( SMLSIZ+1 ) ) /
     $       LOG( TWO ) ) + 1, 0 )
*
      IF( INFO.EQ.0 ) THEN
         MAXWRK = 1
         LIWORK = 3*MINMN*NLVL + 11*MINMN
         MM = M
         IF( M.GE.N .AND. M.GE.MNTHR ) THEN
*
*           Path 1a - overdetermined, with many more rows than columns.
*
            MM = N
            MAXWRK = MAX( MAXWRK, N+N*ILAENV( 1, 'DGEQRF', ' ', M, N,
     $               -1, -1 ) )
            MAXWRK = MAX( MAXWRK, N+NRHS*
     $               ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N, -1 ) )
         END IF
         IF( M.GE.N ) THEN
*
*           Path 1 - overdetermined or exactly determined.
*
            MAXWRK = MAX( MAXWRK, 3*N+( MM+N )*
     $               ILAENV( 1, 'DGEBRD', ' ', MM, N, -1, -1 ) )
            MAXWRK = MAX( MAXWRK, 3*N+NRHS*
     $               ILAENV( 1, 'DORMBR', 'QLT', MM, NRHS, N, -1 ) )
            MAXWRK = MAX( MAXWRK, 3*N+( N-1 )*
     $               ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, N, -1 ) )
            WLALSD = 9*N+2*N*SMLSIZ+8*N*NLVL+N*NRHS+(SMLSIZ+1)**2
            MAXWRK = MAX( MAXWRK, 3*N+WLALSD )
            MINWRK = MAX( 3*N+MM, 3*N+NRHS, 3*N+WLALSD )
         END IF
         IF( N.GT.M ) THEN
            WLALSD = 9*M+2*M*SMLSIZ+8*M*NLVL+M*NRHS+(SMLSIZ+1)**2
            IF( N.GE.MNTHR ) THEN
*
*              Path 2a - underdetermined, with many more columns
*              than rows.
*
               MAXWRK = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1,
     $                                -1 )
               MAXWRK = MAX( MAXWRK, M*M+4*M+2*M*
     $                  ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
               MAXWRK = MAX( MAXWRK, M*M+4*M+NRHS*
     $                  ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, M,
     $                          -1 ) )
               MAXWRK = MAX( MAXWRK, M*M+4*M+( M-1 )*
     $                  ILAENV( 1, 'DORMBR', 'PLN', M, NRHS, M,
     $                          -1 ) )
               IF( NRHS.GT.1 ) THEN
                  MAXWRK = MAX( MAXWRK, M*M+M+M*NRHS )
               ELSE
                  MAXWRK = MAX( MAXWRK, M*M+2*M )
               END IF
               MAXWRK = MAX( MAXWRK, M+NRHS*
     $                  ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M, -1 ) )
               MAXWRK = MAX( MAXWRK, M*M+4*M+WLALSD )
!     XXX: Ensure the Path 2a case below is triggered.  The workspace
!     calculation should use queries for all routines eventually.
               MAXWRK = MAX( MAXWRK,
     $              4*M+M*M+MAX( M, 2*M-4, NRHS, N-3*M ) )
            ELSE
*
*              Path 2 - remaining underdetermined cases.
*
               MAXWRK = 3*M + ( N+M )*ILAENV( 1, 'DGEBRD', ' ', M, N,
     $                  -1, -1 )
               MAXWRK = MAX( MAXWRK, 3*M+NRHS*
     $                  ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, N,
     $                          -1 ) )
               MAXWRK = MAX( MAXWRK, 3*M+M*
     $                  ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, M,
     $                          -1 ) )
               MAXWRK = MAX( MAXWRK, 3*M+WLALSD )
            END IF
            MINWRK = MAX( 3*M+NRHS, 3*M+M, 3*M+WLALSD )
         END IF
         MINWRK = MIN( MINWRK, MAXWRK )
         WORK( 1 ) = MAXWRK
         IWORK( 1 ) = LIWORK

         IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
            INFO = -12
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGELSD', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         GO TO 10
      END IF
*
*     Quick return if possible.
*
      IF( M.EQ.0 .OR. N.EQ.0 ) THEN
         RANK = 0
         RETURN
      END IF
*
*     Get machine parameters.
*
      EPS = DLAMCH( 'P' )
      SFMIN = DLAMCH( 'S' )
      SMLNUM = SFMIN / EPS
      BIGNUM = ONE / SMLNUM
*
*     Scale A if max entry outside range [SMLNUM,BIGNUM].
*
      ANRM = DLANGE( 'M', M, N, A, LDA, WORK )
      IASCL = 0
      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
*
*        Scale matrix norm up to SMLNUM.
*
         CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
         IASCL = 1
      ELSE IF( ANRM.GT.BIGNUM ) THEN
*
*        Scale matrix norm down to BIGNUM.
*
         CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
         IASCL = 2
      ELSE IF( ANRM.EQ.ZERO ) THEN
*
*        Matrix all zero. Return zero solution.
*
         CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
         CALL DLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 )
         RANK = 0
         GO TO 10
      END IF
*
*     Scale B if max entry outside range [SMLNUM,BIGNUM].
*
      BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK )
      IBSCL = 0
      IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
*
*        Scale matrix norm up to SMLNUM.
*
         CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB,
     $                INFO )
         IBSCL = 1
      ELSE IF( BNRM.GT.BIGNUM ) THEN
*
*        Scale matrix norm down to BIGNUM.
*
         CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB,
     $                INFO )
         IBSCL = 2
      END IF
*
*     If M < N make sure certain entries of B are zero.
*
      IF( M.LT.N )
     $   CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
*
*     Overdetermined case.
*
      IF( M.GE.N ) THEN
*
*        Path 1 - overdetermined or exactly determined.
*
         MM = M
         IF( M.GE.MNTHR ) THEN
*
*           Path 1a - overdetermined, with many more rows than columns.
*
            MM = N
            ITAU = 1
            NWORK = ITAU + N
*
*           Compute A=Q*R.
*           (Workspace: need 2*N, prefer N+N*NB)
*
            CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
     $                   LWORK-NWORK+1, INFO )
*
*           Multiply B by transpose(Q).
*           (Workspace: need N+NRHS, prefer N+NRHS*NB)
*
            CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAU ),
     $                   B,
     $                   LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
*
*           Zero out below R.
*
            IF( N.GT.1 ) THEN
               CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ),
     $                      LDA )
            END IF
         END IF
*
         IE = 1
         ITAUQ = IE + N
         ITAUP = ITAUQ + N
         NWORK = ITAUP + N
*
*        Bidiagonalize R in A.
*        (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB)
*
         CALL DGEBRD( MM, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
     $                WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
     $                INFO )
*
*        Multiply B by transpose of left bidiagonalizing vectors of R.
*        (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB)
*
         CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA,
     $                WORK( ITAUQ ),
     $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
*
*        Solve the bidiagonal least squares problem.
*
         CALL DLALSD( 'U', SMLSIZ, N, NRHS, S, WORK( IE ), B, LDB,
     $                RCOND, RANK, WORK( NWORK ), IWORK, INFO )
         IF( INFO.NE.0 ) THEN
            GO TO 10
         END IF
*
*        Multiply B by right bidiagonalizing vectors of R.
*
         CALL DORMBR( 'P', 'L', 'N', N, NRHS, N, A, LDA,
     $                WORK( ITAUP ),
     $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
*
      ELSE IF( N.GE.MNTHR .AND. LWORK.GE.4*M+M*M+
     $         MAX( M, 2*M-4, NRHS, N-3*M, WLALSD ) ) THEN
*
*        Path 2a - underdetermined, with many more columns than rows
*        and sufficient workspace for an efficient algorithm.
*
         LDWORK = M
         IF( LWORK.GE.MAX( 4*M+M*LDA+MAX( M, 2*M-4, NRHS, N-3*M ),
     $       M*LDA+M+M*NRHS, 4*M+M*LDA+WLALSD ) )LDWORK = LDA
         ITAU = 1
         NWORK = M + 1
*
*        Compute A=L*Q.
*        (Workspace: need 2*M, prefer M+M*NB)
*
         CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
     $                LWORK-NWORK+1, INFO )
         IL = NWORK
*
*        Copy L to WORK(IL), zeroing out above its diagonal.
*
         CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK )
         CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, WORK( IL+LDWORK ),
     $                LDWORK )
         IE = IL + LDWORK*M
         ITAUQ = IE + M
         ITAUP = ITAUQ + M
         NWORK = ITAUP + M
*
*        Bidiagonalize L in WORK(IL).
*        (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB)
*
         CALL DGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ),
     $                WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
     $                LWORK-NWORK+1, INFO )
*
*        Multiply B by transpose of left bidiagonalizing vectors of L.
*        (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
*
         CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK,
     $                WORK( ITAUQ ), B, LDB, WORK( NWORK ),
     $                LWORK-NWORK+1, INFO )
*
*        Solve the bidiagonal least squares problem.
*
         CALL DLALSD( 'U', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB,
     $                RCOND, RANK, WORK( NWORK ), IWORK, INFO )
         IF( INFO.NE.0 ) THEN
            GO TO 10
         END IF
*
*        Multiply B by right bidiagonalizing vectors of L.
*
         CALL DORMBR( 'P', 'L', 'N', M, NRHS, M, WORK( IL ), LDWORK,
     $                WORK( ITAUP ), B, LDB, WORK( NWORK ),
     $                LWORK-NWORK+1, INFO )
*
*        Zero out below first M rows of B.
*
         CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
         NWORK = ITAU + M
*
*        Multiply transpose(Q) by B.
*        (Workspace: need M+NRHS, prefer M+NRHS*NB)
*
         CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B,
     $                LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
*
      ELSE
*
*        Path 2 - remaining underdetermined cases.
*
         IE = 1
         ITAUQ = IE + M
         ITAUP = ITAUQ + M
         NWORK = ITAUP + M
*
*        Bidiagonalize A.
*        (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
*
         CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
     $                WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
     $                INFO )
*
*        Multiply B by transpose of left bidiagonalizing vectors.
*        (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB)
*
         CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA,
     $                WORK( ITAUQ ),
     $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
*
*        Solve the bidiagonal least squares problem.
*
         CALL DLALSD( 'L', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB,
     $                RCOND, RANK, WORK( NWORK ), IWORK, INFO )
         IF( INFO.NE.0 ) THEN
            GO TO 10
         END IF
*
*        Multiply B by right bidiagonalizing vectors of A.
*
         CALL DORMBR( 'P', 'L', 'N', N, NRHS, M, A, LDA,
     $                WORK( ITAUP ),
     $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
*
      END IF
*
*     Undo scaling.
*
      IF( IASCL.EQ.1 ) THEN
         CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB,
     $                INFO )
         CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
     $                INFO )
      ELSE IF( IASCL.EQ.2 ) THEN
         CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB,
     $                INFO )
         CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
     $                INFO )
      END IF
      IF( IBSCL.EQ.1 ) THEN
         CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB,
     $                INFO )
      ELSE IF( IBSCL.EQ.2 ) THEN
         CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB,
     $                INFO )
      END IF
*
   10 CONTINUE
      WORK( 1 ) = MAXWRK
      IWORK( 1 ) = LIWORK
      RETURN
*
*     End of DGELSD
*
      END
*> \brief <b> DGELSS solves overdetermined or underdetermined systems for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGELSS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelss.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelss.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelss.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
*                          WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
*       DOUBLE PRECISION   RCOND
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGELSS computes the minimum norm solution to a real linear least
*> squares problem:
*>
*> Minimize 2-norm(| b - A*x |).
*>
*> using the singular value decomposition (SVD) of A. A is an M-by-N
*> matrix which may be rank-deficient.
*>
*> Several right hand side vectors b and solution vectors x can be
*> handled in a single call; they are stored as the columns of the
*> M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix
*> X.
*>
*> The effective rank of A is determined by treating as zero those
*> singular values which are less than RCOND times the largest singular
*> value.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit, the first min(m,n) rows of A are overwritten with
*>          its right singular vectors, stored rowwise.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the M-by-NRHS right hand side matrix B.
*>          On exit, B is overwritten by the N-by-NRHS solution
*>          matrix X.  If m >= n and RANK = n, the residual
*>          sum-of-squares for the solution in the i-th column is given
*>          by the sum of squares of elements n+1:m in that column.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,max(M,N)).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (min(M,N))
*>          The singular values of A in decreasing order.
*>          The condition number of A in the 2-norm = S(1)/S(min(m,n)).
*> \endverbatim
*>
*> \param[in] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          RCOND is used to determine the effective rank of A.
*>          Singular values S(i) <= RCOND*S(1) are treated as zero.
*>          If RCOND < 0, machine precision is used instead.
*> \endverbatim
*>
*> \param[out] RANK
*> \verbatim
*>          RANK is INTEGER
*>          The effective rank of A, i.e., the number of singular values
*>          which are greater than RCOND*S(1).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= 1, and also:
*>          LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS )
*>          For good performance, LWORK should generally be larger.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  the algorithm for computing the SVD failed to converge;
*>                if INFO = i, i off-diagonal elements of an intermediate
*>                bidiagonal form did not converge to zero.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gelss
*
*  =====================================================================
      SUBROUTINE DGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
     $                   WORK, LWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            BDSPAC, BL, CHUNK, I, IASCL, IBSCL, IE, IL,
     $                   ITAU, ITAUP, ITAUQ, IWORK, LDWORK, MAXMN,
     $                   MAXWRK, MINMN, MINWRK, MM, MNTHR
      INTEGER            LWORK_DGEQRF, LWORK_DORMQR, LWORK_DGEBRD,
     $                   LWORK_DORMBR, LWORK_DORGBR, LWORK_DORMLQ,
     $                   LWORK_DGELQF
      DOUBLE PRECISION   ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM, THR
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   DUM( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DBDSQR, DCOPY, DGEBRD, DGELQF, DGEMM,
     $                   DGEMV,
     $                   DGEQRF, DLACPY, DLASCL, DLASET, DORGBR,
     $                   DORMBR, DORMLQ, DORMQR, DRSCL, XERBLA
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           ILAENV, DLAMCH, DLANGE
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      MINMN = MIN( M, N )
      MAXMN = MAX( M, N )
      LQUERY = ( LWORK.EQ.-1 )
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN
         INFO = -7
      END IF
*
*     Compute workspace
*      (Note: Comments in the code beginning "Workspace:" describe the
*       minimal amount of workspace needed at that point in the code,
*       as well as the preferred amount for good performance.
*       NB refers to the optimal block size for the immediately
*       following subroutine, as returned by ILAENV.)
*
      IF( INFO.EQ.0 ) THEN
         MINWRK = 1
         MAXWRK = 1
         IF( MINMN.GT.0 ) THEN
            MM = M
            MNTHR = ILAENV( 6, 'DGELSS', ' ', M, N, NRHS, -1 )
            IF( M.GE.N .AND. M.GE.MNTHR ) THEN
*
*              Path 1a - overdetermined, with many more rows than
*                        columns
*
*              Compute space needed for DGEQRF
               CALL DGEQRF( M, N, A, LDA, DUM(1), DUM(1), -1, INFO )
               LWORK_DGEQRF = INT( DUM(1) )
*              Compute space needed for DORMQR
               CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, DUM(1), B,
     $                   LDB, DUM(1), -1, INFO )
               LWORK_DORMQR = INT( DUM(1) )
               MM = N
               MAXWRK = MAX( MAXWRK, N + LWORK_DGEQRF )
               MAXWRK = MAX( MAXWRK, N + LWORK_DORMQR )
            END IF
            IF( M.GE.N ) THEN
*
*              Path 1 - overdetermined or exactly determined
*
*              Compute workspace needed for DBDSQR
*
               BDSPAC = MAX( 1, 5*N )
*              Compute space needed for DGEBRD
               CALL DGEBRD( MM, N, A, LDA, S, DUM(1), DUM(1),
     $                      DUM(1), DUM(1), -1, INFO )
               LWORK_DGEBRD = INT( DUM(1) )
*              Compute space needed for DORMBR
               CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA,
     $                      DUM(1),
     $                B, LDB, DUM(1), -1, INFO )
               LWORK_DORMBR = INT( DUM(1) )
*              Compute space needed for DORGBR
               CALL DORGBR( 'P', N, N, N, A, LDA, DUM(1),
     $                   DUM(1), -1, INFO )
               LWORK_DORGBR = INT( DUM(1) )
*              Compute total workspace needed
               MAXWRK = MAX( MAXWRK, 3*N + LWORK_DGEBRD )
               MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORMBR )
               MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR )
               MAXWRK = MAX( MAXWRK, BDSPAC )
               MAXWRK = MAX( MAXWRK, N*NRHS )
               MINWRK = MAX( 3*N + MM, 3*N + NRHS, BDSPAC )
               MAXWRK = MAX( MINWRK, MAXWRK )
            END IF
            IF( N.GT.M ) THEN
*
*              Compute workspace needed for DBDSQR
*
               BDSPAC = MAX( 1, 5*M )
               MINWRK = MAX( 3*M+NRHS, 3*M+N, BDSPAC )
               IF( N.GE.MNTHR ) THEN
*
*                 Path 2a - underdetermined, with many more columns
*                 than rows
*
*                 Compute space needed for DGELQF
                  CALL DGELQF( M, N, A, LDA, DUM(1), DUM(1),
     $                -1, INFO )
                  LWORK_DGELQF = INT( DUM(1) )
*                 Compute space needed for DGEBRD
                  CALL DGEBRD( M, M, A, LDA, S, DUM(1), DUM(1),
     $                      DUM(1), DUM(1), -1, INFO )
                  LWORK_DGEBRD = INT( DUM(1) )
*                 Compute space needed for DORMBR
                  CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA,
     $                DUM(1), B, LDB, DUM(1), -1, INFO )
                  LWORK_DORMBR = INT( DUM(1) )
*                 Compute space needed for DORGBR
                  CALL DORGBR( 'P', M, M, M, A, LDA, DUM(1),
     $                   DUM(1), -1, INFO )
                  LWORK_DORGBR = INT( DUM(1) )
*                 Compute space needed for DORMLQ
                  CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, DUM(1),
     $                 B, LDB, DUM(1), -1, INFO )
                  LWORK_DORMLQ = INT( DUM(1) )
*                 Compute total workspace needed
                  MAXWRK = M + LWORK_DGELQF
                  MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_DGEBRD )
                  MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_DORMBR )
                  MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_DORGBR )
                  MAXWRK = MAX( MAXWRK, M*M + M + BDSPAC )
                  IF( NRHS.GT.1 ) THEN
                     MAXWRK = MAX( MAXWRK, M*M + M + M*NRHS )
                  ELSE
                     MAXWRK = MAX( MAXWRK, M*M + 2*M )
                  END IF
                  MAXWRK = MAX( MAXWRK, M + LWORK_DORMLQ )
               ELSE
*
*                 Path 2 - underdetermined
*
*                 Compute space needed for DGEBRD
                  CALL DGEBRD( M, N, A, LDA, S, DUM(1), DUM(1),
     $                      DUM(1), DUM(1), -1, INFO )
                  LWORK_DGEBRD = INT( DUM(1) )
*                 Compute space needed for DORMBR
                  CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, A, LDA,
     $                DUM(1), B, LDB, DUM(1), -1, INFO )
                  LWORK_DORMBR = INT( DUM(1) )
*                 Compute space needed for DORGBR
                  CALL DORGBR( 'P', M, N, M, A, LDA, DUM(1),
     $                   DUM(1), -1, INFO )
                  LWORK_DORGBR = INT( DUM(1) )
                  MAXWRK = 3*M + LWORK_DGEBRD
                  MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORMBR )
                  MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR )
                  MAXWRK = MAX( MAXWRK, BDSPAC )
                  MAXWRK = MAX( MAXWRK, N*NRHS )
               END IF
            END IF
            MAXWRK = MAX( MINWRK, MAXWRK )
         END IF
         WORK( 1 ) = MAXWRK
*
         IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY )
     $      INFO = -12
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGELSS', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 ) THEN
         RANK = 0
         RETURN
      END IF
*
*     Get machine parameters
*
      EPS = DLAMCH( 'P' )
      SFMIN = DLAMCH( 'S' )
      SMLNUM = SFMIN / EPS
      BIGNUM = ONE / SMLNUM
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      ANRM = DLANGE( 'M', M, N, A, LDA, WORK )
      IASCL = 0
      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
         IASCL = 1
      ELSE IF( ANRM.GT.BIGNUM ) THEN
*
*        Scale matrix norm down to BIGNUM
*
         CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
         IASCL = 2
      ELSE IF( ANRM.EQ.ZERO ) THEN
*
*        Matrix all zero. Return zero solution.
*
         CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
         CALL DLASET( 'F', MINMN, 1, ZERO, ZERO, S, MINMN )
         RANK = 0
         GO TO 70
      END IF
*
*     Scale B if max element outside range [SMLNUM,BIGNUM]
*
      BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK )
      IBSCL = 0
      IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB,
     $                INFO )
         IBSCL = 1
      ELSE IF( BNRM.GT.BIGNUM ) THEN
*
*        Scale matrix norm down to BIGNUM
*
         CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB,
     $                INFO )
         IBSCL = 2
      END IF
*
*     Overdetermined case
*
      IF( M.GE.N ) THEN
*
*        Path 1 - overdetermined or exactly determined
*
         MM = M
         IF( M.GE.MNTHR ) THEN
*
*           Path 1a - overdetermined, with many more rows than columns
*
            MM = N
            ITAU = 1
            IWORK = ITAU + N
*
*           Compute A=Q*R
*           (Workspace: need 2*N, prefer N+N*NB)
*
            CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
     $                   LWORK-IWORK+1, INFO )
*
*           Multiply B by transpose(Q)
*           (Workspace: need N+NRHS, prefer N+NRHS*NB)
*
            CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAU ),
     $                   B,
     $                   LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
*
*           Zero out below R
*
            IF( N.GT.1 )
     $         CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ),
     $                      LDA )
         END IF
*
         IE = 1
         ITAUQ = IE + N
         ITAUP = ITAUQ + N
         IWORK = ITAUP + N
*
*        Bidiagonalize R in A
*        (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB)
*
         CALL DGEBRD( MM, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
     $                WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
     $                INFO )
*
*        Multiply B by transpose of left bidiagonalizing vectors of R
*        (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB)
*
         CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA,
     $                WORK( ITAUQ ),
     $                B, LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
*
*        Generate right bidiagonalizing vectors of R in A
*        (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
*
         CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
     $                WORK( IWORK ), LWORK-IWORK+1, INFO )
         IWORK = IE + N
*
*        Perform bidiagonal QR iteration
*          multiply B by transpose of left singular vectors
*          compute right singular vectors in A
*        (Workspace: need BDSPAC)
*
         CALL DBDSQR( 'U', N, N, 0, NRHS, S, WORK( IE ), A, LDA, DUM,
     $                1, B, LDB, WORK( IWORK ), INFO )
         IF( INFO.NE.0 )
     $      GO TO 70
*
*        Multiply B by reciprocals of singular values
*
         THR = MAX( RCOND*S( 1 ), SFMIN )
         IF( RCOND.LT.ZERO )
     $      THR = MAX( EPS*S( 1 ), SFMIN )
         RANK = 0
         DO 10 I = 1, N
            IF( S( I ).GT.THR ) THEN
               CALL DRSCL( NRHS, S( I ), B( I, 1 ), LDB )
               RANK = RANK + 1
            ELSE
               CALL DLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ),
     $                      LDB )
            END IF
   10    CONTINUE
*
*        Multiply B by right singular vectors
*        (Workspace: need N, prefer N*NRHS)
*
         IF( LWORK.GE.LDB*NRHS .AND. NRHS.GT.1 ) THEN
            CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, A, LDA, B, LDB,
     $                  ZERO,
     $                  WORK, LDB )
            CALL DLACPY( 'G', N, NRHS, WORK, LDB, B, LDB )
         ELSE IF( NRHS.GT.1 ) THEN
            CHUNK = LWORK / N
            DO 20 I = 1, NRHS, CHUNK
               BL = MIN( NRHS-I+1, CHUNK )
               CALL DGEMM( 'T', 'N', N, BL, N, ONE, A, LDA, B( 1,
     $                     I ),
     $                     LDB, ZERO, WORK, N )
               CALL DLACPY( 'G', N, BL, WORK, N, B( 1, I ), LDB )
   20       CONTINUE
         ELSE IF( NRHS.EQ.1 ) THEN
            CALL DGEMV( 'T', N, N, ONE, A, LDA, B, 1, ZERO, WORK, 1 )
            CALL DCOPY( N, WORK, 1, B, 1 )
         END IF
*
      ELSE IF( N.GE.MNTHR .AND. LWORK.GE.4*M+M*M+
     $         MAX( M, 2*M-4, NRHS, N-3*M ) ) THEN
*
*        Path 2a - underdetermined, with many more columns than rows
*        and sufficient workspace for an efficient algorithm
*
         LDWORK = M
         IF( LWORK.GE.MAX( 4*M+M*LDA+MAX( M, 2*M-4, NRHS, N-3*M ),
     $       M*LDA+M+M*NRHS ) )LDWORK = LDA
         ITAU = 1
         IWORK = M + 1
*
*        Compute A=L*Q
*        (Workspace: need 2*M, prefer M+M*NB)
*
         CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
     $                LWORK-IWORK+1, INFO )
         IL = IWORK
*
*        Copy L to WORK(IL), zeroing out above it
*
         CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK )
         CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, WORK( IL+LDWORK ),
     $                LDWORK )
         IE = IL + LDWORK*M
         ITAUQ = IE + M
         ITAUP = ITAUQ + M
         IWORK = ITAUP + M
*
*        Bidiagonalize L in WORK(IL)
*        (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB)
*
         CALL DGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ),
     $                WORK( ITAUQ ), WORK( ITAUP ), WORK( IWORK ),
     $                LWORK-IWORK+1, INFO )
*
*        Multiply B by transpose of left bidiagonalizing vectors of L
*        (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
*
         CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK,
     $                WORK( ITAUQ ), B, LDB, WORK( IWORK ),
     $                LWORK-IWORK+1, INFO )
*
*        Generate right bidiagonalizing vectors of R in WORK(IL)
*        (Workspace: need M*M+5*M-1, prefer M*M+4*M+(M-1)*NB)
*
         CALL DORGBR( 'P', M, M, M, WORK( IL ), LDWORK,
     $                WORK( ITAUP ),
     $                WORK( IWORK ), LWORK-IWORK+1, INFO )
         IWORK = IE + M
*
*        Perform bidiagonal QR iteration,
*           computing right singular vectors of L in WORK(IL) and
*           multiplying B by transpose of left singular vectors
*        (Workspace: need M*M+M+BDSPAC)
*
         CALL DBDSQR( 'U', M, M, 0, NRHS, S, WORK( IE ), WORK( IL ),
     $                LDWORK, A, LDA, B, LDB, WORK( IWORK ), INFO )
         IF( INFO.NE.0 )
     $      GO TO 70
*
*        Multiply B by reciprocals of singular values
*
         THR = MAX( RCOND*S( 1 ), SFMIN )
         IF( RCOND.LT.ZERO )
     $      THR = MAX( EPS*S( 1 ), SFMIN )
         RANK = 0
         DO 30 I = 1, M
            IF( S( I ).GT.THR ) THEN
               CALL DRSCL( NRHS, S( I ), B( I, 1 ), LDB )
               RANK = RANK + 1
            ELSE
               CALL DLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ),
     $                      LDB )
            END IF
   30    CONTINUE
         IWORK = IE
*
*        Multiply B by right singular vectors of L in WORK(IL)
*        (Workspace: need M*M+2*M, prefer M*M+M+M*NRHS)
*
         IF( LWORK.GE.LDB*NRHS+IWORK-1 .AND. NRHS.GT.1 ) THEN
            CALL DGEMM( 'T', 'N', M, NRHS, M, ONE, WORK( IL ),
     $                  LDWORK,
     $                  B, LDB, ZERO, WORK( IWORK ), LDB )
            CALL DLACPY( 'G', M, NRHS, WORK( IWORK ), LDB, B, LDB )
         ELSE IF( NRHS.GT.1 ) THEN
            CHUNK = ( LWORK-IWORK+1 ) / M
            DO 40 I = 1, NRHS, CHUNK
               BL = MIN( NRHS-I+1, CHUNK )
               CALL DGEMM( 'T', 'N', M, BL, M, ONE, WORK( IL ),
     $                     LDWORK,
     $                     B( 1, I ), LDB, ZERO, WORK( IWORK ), M )
               CALL DLACPY( 'G', M, BL, WORK( IWORK ), M, B( 1, I ),
     $                      LDB )
   40       CONTINUE
         ELSE IF( NRHS.EQ.1 ) THEN
            CALL DGEMV( 'T', M, M, ONE, WORK( IL ), LDWORK, B( 1,
     $                  1 ),1, ZERO, WORK( IWORK ), 1 )
            CALL DCOPY( M, WORK( IWORK ), 1, B( 1, 1 ), 1 )
         END IF
*
*        Zero out below first M rows of B
*
         CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
         IWORK = ITAU + M
*
*        Multiply transpose(Q) by B
*        (Workspace: need M+NRHS, prefer M+NRHS*NB)
*
         CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B,
     $                LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
*
      ELSE
*
*        Path 2 - remaining underdetermined cases
*
         IE = 1
         ITAUQ = IE + M
         ITAUP = ITAUQ + M
         IWORK = ITAUP + M
*
*        Bidiagonalize A
*        (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
*
         CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
     $                WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
     $                INFO )
*
*        Multiply B by transpose of left bidiagonalizing vectors
*        (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB)
*
         CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA,
     $                WORK( ITAUQ ),
     $                B, LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
*
*        Generate right bidiagonalizing vectors in A
*        (Workspace: need 4*M, prefer 3*M+M*NB)
*
         CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
     $                WORK( IWORK ), LWORK-IWORK+1, INFO )
         IWORK = IE + M
*
*        Perform bidiagonal QR iteration,
*           computing right singular vectors of A in A and
*           multiplying B by transpose of left singular vectors
*        (Workspace: need BDSPAC)
*
         CALL DBDSQR( 'L', M, N, 0, NRHS, S, WORK( IE ), A, LDA, DUM,
     $                1, B, LDB, WORK( IWORK ), INFO )
         IF( INFO.NE.0 )
     $      GO TO 70
*
*        Multiply B by reciprocals of singular values
*
         THR = MAX( RCOND*S( 1 ), SFMIN )
         IF( RCOND.LT.ZERO )
     $      THR = MAX( EPS*S( 1 ), SFMIN )
         RANK = 0
         DO 50 I = 1, M
            IF( S( I ).GT.THR ) THEN
               CALL DRSCL( NRHS, S( I ), B( I, 1 ), LDB )
               RANK = RANK + 1
            ELSE
               CALL DLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ),
     $                      LDB )
            END IF
   50    CONTINUE
*
*        Multiply B by right singular vectors of A
*        (Workspace: need N, prefer N*NRHS)
*
         IF( LWORK.GE.LDB*NRHS .AND. NRHS.GT.1 ) THEN
            CALL DGEMM( 'T', 'N', N, NRHS, M, ONE, A, LDA, B, LDB,
     $                  ZERO,
     $                  WORK, LDB )
            CALL DLACPY( 'F', N, NRHS, WORK, LDB, B, LDB )
         ELSE IF( NRHS.GT.1 ) THEN
            CHUNK = LWORK / N
            DO 60 I = 1, NRHS, CHUNK
               BL = MIN( NRHS-I+1, CHUNK )
               CALL DGEMM( 'T', 'N', N, BL, M, ONE, A, LDA, B( 1,
     $                     I ),
     $                     LDB, ZERO, WORK, N )
               CALL DLACPY( 'F', N, BL, WORK, N, B( 1, I ), LDB )
   60       CONTINUE
         ELSE IF( NRHS.EQ.1 ) THEN
            CALL DGEMV( 'T', M, N, ONE, A, LDA, B, 1, ZERO, WORK, 1 )
            CALL DCOPY( N, WORK, 1, B, 1 )
         END IF
      END IF
*
*     Undo scaling
*
      IF( IASCL.EQ.1 ) THEN
         CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB,
     $                INFO )
         CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
     $                INFO )
      ELSE IF( IASCL.EQ.2 ) THEN
         CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB,
     $                INFO )
         CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
     $                INFO )
      END IF
      IF( IBSCL.EQ.1 ) THEN
         CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB,
     $                INFO )
      ELSE IF( IBSCL.EQ.2 ) THEN
         CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB,
     $                INFO )
      END IF
*
   70 CONTINUE
      WORK( 1 ) = MAXWRK
      RETURN
*
*     End of DGELSS
*
      END
*> \brief <b> DGELSX solves overdetermined or underdetermined systems for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGELSX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelsx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelsx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelsx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
*                          WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LDB, M, N, NRHS, RANK
*       DOUBLE PRECISION   RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            JPVT( * )
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DGELSY.
*>
*> DGELSX computes the minimum-norm solution to a real linear least
*> squares problem:
*>     minimize || A * X - B ||
*> using a complete orthogonal factorization of A.  A is an M-by-N
*> matrix which may be rank-deficient.
*>
*> Several right hand side vectors b and solution vectors x can be
*> handled in a single call; they are stored as the columns of the
*> M-by-NRHS right hand side matrix B and the N-by-NRHS solution
*> matrix X.
*>
*> The routine first computes a QR factorization with column pivoting:
*>     A * P = Q * [ R11 R12 ]
*>                 [  0  R22 ]
*> with R11 defined as the largest leading submatrix whose estimated
*> condition number is less than 1/RCOND.  The order of R11, RANK,
*> is the effective rank of A.
*>
*> Then, R22 is considered to be negligible, and R12 is annihilated
*> by orthogonal transformations from the right, arriving at the
*> complete orthogonal factorization:
*>    A * P = Q * [ T11 0 ] * Z
*>                [  0  0 ]
*> The minimum-norm solution is then
*>    X = P * Z**T [ inv(T11)*Q1**T*B ]
*>                 [        0         ]
*> where Q1 consists of the first RANK columns of Q.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of
*>          columns of matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit, A has been overwritten by details of its
*>          complete orthogonal factorization.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the M-by-NRHS right hand side matrix B.
*>          On exit, the N-by-NRHS solution matrix X.
*>          If m >= n and RANK = n, the residual sum-of-squares for
*>          the solution in the i-th column is given by the sum of
*>          squares of elements N+1:M in that column.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,M,N).
*> \endverbatim
*>
*> \param[in,out] JPVT
*> \verbatim
*>          JPVT is INTEGER array, dimension (N)
*>          On entry, if JPVT(i) .ne. 0, the i-th column of A is an
*>          initial column, otherwise it is a free column.  Before
*>          the QR factorization of A, all initial columns are
*>          permuted to the leading positions; only the remaining
*>          free columns are moved as a result of column pivoting
*>          during the factorization.
*>          On exit, if JPVT(i) = k, then the i-th column of A*P
*>          was the k-th column of A.
*> \endverbatim
*>
*> \param[in] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          RCOND is used to determine the effective rank of A, which
*>          is defined as the order of the largest leading triangular
*>          submatrix R11 in the QR factorization with pivoting of A,
*>          whose estimated condition number < 1/RCOND.
*> \endverbatim
*>
*> \param[out] RANK
*> \verbatim
*>          RANK is INTEGER
*>          The effective rank of A, i.e., the order of the submatrix
*>          R11.  This is the same as the order of the submatrix T11
*>          in the complete orthogonal factorization of A.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension
*>                      (max( min(M,N)+3*N, 2*min(M,N)+NRHS )),
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleGEsolve
*
*  =====================================================================
      SUBROUTINE DGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND,
     $                   RANK, WORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, M, N, NRHS, RANK
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            JPVT( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            IMAX, IMIN
      PARAMETER          ( IMAX = 1, IMIN = 2 )
      DOUBLE PRECISION   ZERO, ONE, DONE, NTDONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, DONE = ZERO,
     $                   NTDONE = ONE )
*     ..
*     .. Local Scalars ..
      INTEGER            I, IASCL, IBSCL, ISMAX, ISMIN, J, K, MN
      DOUBLE PRECISION   ANRM, BIGNUM, BNRM, C1, C2, S1, S2, SMAX,
     $                   SMAXPR, SMIN, SMINPR, SMLNUM, T1, T2
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           DLAMCH, DLANGE
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEQPF, DLAIC1, DLASCL, DLASET, DLATZM, DORM2R,
     $                   DTRSM, DTZRQF, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. Executable Statements ..
*
      MN = MIN( M, N )
      ISMIN = MN + 1
      ISMAX = 2*MN + 1
*
*     Test the input arguments.
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
         INFO = -7
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGELSX', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( MIN( M, N, NRHS ).EQ.0 ) THEN
         RANK = 0
         RETURN
      END IF
*
*     Get machine parameters
*
      SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' )
      BIGNUM = ONE / SMLNUM
*
*     Scale A, B if max elements outside range [SMLNUM,BIGNUM]
*
      ANRM = DLANGE( 'M', M, N, A, LDA, WORK )
      IASCL = 0
      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
         IASCL = 1
      ELSE IF( ANRM.GT.BIGNUM ) THEN
*
*        Scale matrix norm down to BIGNUM
*
         CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
         IASCL = 2
      ELSE IF( ANRM.EQ.ZERO ) THEN
*
*        Matrix all zero. Return zero solution.
*
         CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
         RANK = 0
         GO TO 100
      END IF
*
      BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK )
      IBSCL = 0
      IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB,
     $                INFO )
         IBSCL = 1
      ELSE IF( BNRM.GT.BIGNUM ) THEN
*
*        Scale matrix norm down to BIGNUM
*
         CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB,
     $                INFO )
         IBSCL = 2
      END IF
*
*     Compute QR factorization with column pivoting of A:
*        A * P = Q * R
*
      CALL DGEQPF( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ),
     $             INFO )
*
*     workspace 3*N. Details of Householder rotations stored
*     in WORK(1:MN).
*
*     Determine RANK using incremental condition estimation
*
      WORK( ISMIN ) = ONE
      WORK( ISMAX ) = ONE
      SMAX = ABS( A( 1, 1 ) )
      SMIN = SMAX
      IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN
         RANK = 0
         CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
         GO TO 100
      ELSE
         RANK = 1
      END IF
*
   10 CONTINUE
      IF( RANK.LT.MN ) THEN
         I = RANK + 1
         CALL DLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ),
     $                A( I, I ), SMINPR, S1, C1 )
         CALL DLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ),
     $                A( I, I ), SMAXPR, S2, C2 )
*
         IF( SMAXPR*RCOND.LE.SMINPR ) THEN
            DO 20 I = 1, RANK
               WORK( ISMIN+I-1 ) = S1*WORK( ISMIN+I-1 )
               WORK( ISMAX+I-1 ) = S2*WORK( ISMAX+I-1 )
   20       CONTINUE
            WORK( ISMIN+RANK ) = C1
            WORK( ISMAX+RANK ) = C2
            SMIN = SMINPR
            SMAX = SMAXPR
            RANK = RANK + 1
            GO TO 10
         END IF
      END IF
*
*     Logically partition R = [ R11 R12 ]
*                             [  0  R22 ]
*     where R11 = R(1:RANK,1:RANK)
*
*     [R11,R12] = [ T11, 0 ] * Y
*
      IF( RANK.LT.N )
     $   CALL DTZRQF( RANK, N, A, LDA, WORK( MN+1 ), INFO )
*
*     Details of Householder rotations stored in WORK(MN+1:2*MN)
*
*     B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS)
*
      CALL DORM2R( 'Left', 'Transpose', M, NRHS, MN, A, LDA,
     $             WORK( 1 ), B, LDB, WORK( 2*MN+1 ), INFO )
*
*     workspace NRHS
*
*     B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS)
*
      CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK,
     $            NRHS, ONE, A, LDA, B, LDB )
*
      DO 40 I = RANK + 1, N
         DO 30 J = 1, NRHS
            B( I, J ) = ZERO
   30    CONTINUE
   40 CONTINUE
*
*     B(1:N,1:NRHS) := Y**T * B(1:N,1:NRHS)
*
      IF( RANK.LT.N ) THEN
         DO 50 I = 1, RANK
            CALL DLATZM( 'Left', N-RANK+1, NRHS, A( I, RANK+1 ), LDA,
     $                   WORK( MN+I ), B( I, 1 ), B( RANK+1, 1 ), LDB,
     $                   WORK( 2*MN+1 ) )
   50    CONTINUE
      END IF
*
*     workspace NRHS
*
*     B(1:N,1:NRHS) := P * B(1:N,1:NRHS)
*
      DO 90 J = 1, NRHS
         DO 60 I = 1, N
            WORK( 2*MN+I ) = NTDONE
   60    CONTINUE
         DO 80 I = 1, N
            IF( WORK( 2*MN+I ).EQ.NTDONE ) THEN
               IF( JPVT( I ).NE.I ) THEN
                  K = I
                  T1 = B( K, J )
                  T2 = B( JPVT( K ), J )
   70             CONTINUE
                  B( JPVT( K ), J ) = T1
                  WORK( 2*MN+K ) = DONE
                  T1 = T2
                  K = JPVT( K )
                  T2 = B( JPVT( K ), J )
                  IF( JPVT( K ).NE.I )
     $               GO TO 70
                  B( I, J ) = T1
                  WORK( 2*MN+K ) = DONE
               END IF
            END IF
   80    CONTINUE
   90 CONTINUE
*
*     Undo scaling
*
      IF( IASCL.EQ.1 ) THEN
         CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB,
     $                INFO )
         CALL DLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA,
     $                INFO )
      ELSE IF( IASCL.EQ.2 ) THEN
         CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB,
     $                INFO )
         CALL DLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA,
     $                INFO )
      END IF
      IF( IBSCL.EQ.1 ) THEN
         CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB,
     $                INFO )
      ELSE IF( IBSCL.EQ.2 ) THEN
         CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB,
     $                INFO )
      END IF
*
  100 CONTINUE
*
      RETURN
*
*     End of DGELSX
*
      END
*> \brief <b> DGELSY solves overdetermined or underdetermined systems for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGELSY + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelsy.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelsy.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelsy.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
*                          WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
*       DOUBLE PRECISION   RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            JPVT( * )
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGELSY computes the minimum-norm solution to a real linear least
*> squares problem:
*>     minimize || A * X - B ||
*> using a complete orthogonal factorization of A.  A is an M-by-N
*> matrix which may be rank-deficient.
*>
*> Several right hand side vectors b and solution vectors x can be
*> handled in a single call; they are stored as the columns of the
*> M-by-NRHS right hand side matrix B and the N-by-NRHS solution
*> matrix X.
*>
*> The routine first computes a QR factorization with column pivoting:
*>     A * P = Q * [ R11 R12 ]
*>                 [  0  R22 ]
*> with R11 defined as the largest leading submatrix whose estimated
*> condition number is less than 1/RCOND.  The order of R11, RANK,
*> is the effective rank of A.
*>
*> Then, R22 is considered to be negligible, and R12 is annihilated
*> by orthogonal transformations from the right, arriving at the
*> complete orthogonal factorization:
*>    A * P = Q * [ T11 0 ] * Z
*>                [  0  0 ]
*> The minimum-norm solution is then
*>    X = P * Z**T [ inv(T11)*Q1**T*B ]
*>                 [        0         ]
*> where Q1 consists of the first RANK columns of Q.
*>
*> This routine is basically identical to the original xGELSX except
*> three differences:
*>   o The call to the subroutine xGEQPF has been substituted by the
*>     the call to the subroutine xGEQP3. This subroutine is a Blas-3
*>     version of the QR factorization with column pivoting.
*>   o Matrix B (the right hand side) is updated with Blas-3.
*>   o The permutation of matrix B (the right hand side) is faster and
*>     more simple.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of
*>          columns of matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit, A has been overwritten by details of its
*>          complete orthogonal factorization.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the M-by-NRHS right hand side matrix B.
*>          On exit, the N-by-NRHS solution matrix X.
*>          If M = 0 or N = 0, B is not referenced.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,M,N).
*> \endverbatim
*>
*> \param[in,out] JPVT
*> \verbatim
*>          JPVT is INTEGER array, dimension (N)
*>          On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
*>          to the front of AP, otherwise column i is a free column.
*>          On exit, if JPVT(i) = k, then the i-th column of AP
*>          was the k-th column of A.
*> \endverbatim
*>
*> \param[in] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          RCOND is used to determine the effective rank of A, which
*>          is defined as the order of the largest leading triangular
*>          submatrix R11 in the QR factorization with pivoting of A,
*>          whose estimated condition number < 1/RCOND.
*> \endverbatim
*>
*> \param[out] RANK
*> \verbatim
*>          RANK is INTEGER
*>          The effective rank of A, i.e., the order of the submatrix
*>          R11.  This is the same as the order of the submatrix T11
*>          in the complete orthogonal factorization of A.
*>          If NRHS = 0, RANK = 0 on output.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          The unblocked strategy requires that:
*>             LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ),
*>          where MN = min( M, N ).
*>          The block algorithm requires that:
*>             LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ),
*>          where NB is an upper bound on the blocksize returned
*>          by ILAENV for the routines DGEQP3, DTZRZF, STZRQF, DORMQR,
*>          and DORMRZ.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: If INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gelsy
*
*> \par Contributors:
*  ==================
*>
*>    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA \n
*>    E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain \n
*>    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain \n
*>
*  =====================================================================
      SUBROUTINE DGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND,
     $                   RANK,
     $                   WORK, LWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            JPVT( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            IMAX, IMIN
      PARAMETER          ( IMAX = 1, IMIN = 2 )
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IASCL, IBSCL, ISMAX, ISMIN, J, LWKMIN,
     $                   LWKOPT, MN, NB, NB1, NB2, NB3, NB4
      DOUBLE PRECISION   ANRM, BIGNUM, BNRM, C1, C2, S1, S2, SMAX,
     $                   SMAXPR, SMIN, SMINPR, SMLNUM, WSIZE
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           ILAENV, DLAMCH, DLANGE
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGEQP3, DLAIC1, DLASCL,
     $                   DLASET,
     $                   DORMQR, DORMRZ, DTRSM, DTZRZF, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. Executable Statements ..
*
      MN = MIN( M, N )
      ISMIN = MN + 1
      ISMAX = 2*MN + 1
*
*     Test the input arguments.
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
         INFO = -7
      END IF
*
*     Figure out optimal block size
*
      IF( INFO.EQ.0 ) THEN
         IF( MN.EQ.0 .OR. NRHS.EQ.0 ) THEN
            LWKMIN = 1
            LWKOPT = 1
         ELSE
            NB1 = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
            NB2 = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )
            NB3 = ILAENV( 1, 'DORMQR', ' ', M, N, NRHS, -1 )
            NB4 = ILAENV( 1, 'DORMRQ', ' ', M, N, NRHS, -1 )
            NB = MAX( NB1, NB2, NB3, NB4 )
            LWKMIN = MN + MAX( 2*MN, N + 1, MN + NRHS )
            LWKOPT = MAX( LWKMIN,
     $                    MN + 2*N + NB*( N + 1 ), 2*MN + NB*NRHS )
         END IF
         WORK( 1 ) = LWKOPT
*
         IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
            INFO = -12
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGELSY', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( MN.EQ.0 .OR. NRHS.EQ.0 ) THEN
         RANK = 0
         RETURN
      END IF
*
*     Get machine parameters
*
      SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' )
      BIGNUM = ONE / SMLNUM
*
*     Scale A, B if max entries outside range [SMLNUM,BIGNUM]
*
      ANRM = DLANGE( 'M', M, N, A, LDA, WORK )
      IASCL = 0
      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
         IASCL = 1
      ELSE IF( ANRM.GT.BIGNUM ) THEN
*
*        Scale matrix norm down to BIGNUM
*
         CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
         IASCL = 2
      ELSE IF( ANRM.EQ.ZERO ) THEN
*
*        Matrix all zero. Return zero solution.
*
         CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
         RANK = 0
         GO TO 70
      END IF
*
      BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK )
      IBSCL = 0
      IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB,
     $                INFO )
         IBSCL = 1
      ELSE IF( BNRM.GT.BIGNUM ) THEN
*
*        Scale matrix norm down to BIGNUM
*
         CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB,
     $                INFO )
         IBSCL = 2
      END IF
*
*     Compute QR factorization with column pivoting of A:
*        A * P = Q * R
*
      CALL DGEQP3( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ),
     $             LWORK-MN, INFO )
      WSIZE = MN + WORK( MN+1 )
*
*     workspace: MN+2*N+NB*(N+1).
*     Details of Householder rotations stored in WORK(1:MN).
*
*     Determine RANK using incremental condition estimation
*
      WORK( ISMIN ) = ONE
      WORK( ISMAX ) = ONE
      SMAX = ABS( A( 1, 1 ) )
      SMIN = SMAX
      IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN
         RANK = 0
         CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
         GO TO 70
      ELSE
         RANK = 1
      END IF
*
   10 CONTINUE
      IF( RANK.LT.MN ) THEN
         I = RANK + 1
         CALL DLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ),
     $                A( I, I ), SMINPR, S1, C1 )
         CALL DLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ),
     $                A( I, I ), SMAXPR, S2, C2 )
*
         IF( SMAXPR*RCOND.LE.SMINPR ) THEN
            DO 20 I = 1, RANK
               WORK( ISMIN+I-1 ) = S1*WORK( ISMIN+I-1 )
               WORK( ISMAX+I-1 ) = S2*WORK( ISMAX+I-1 )
   20       CONTINUE
            WORK( ISMIN+RANK ) = C1
            WORK( ISMAX+RANK ) = C2
            SMIN = SMINPR
            SMAX = SMAXPR
            RANK = RANK + 1
            GO TO 10
         END IF
      END IF
*
*     workspace: 3*MN.
*
*     Logically partition R = [ R11 R12 ]
*                             [  0  R22 ]
*     where R11 = R(1:RANK,1:RANK)
*
*     [R11,R12] = [ T11, 0 ] * Y
*
      IF( RANK.LT.N )
     $   CALL DTZRZF( RANK, N, A, LDA, WORK( MN+1 ), WORK( 2*MN+1 ),
     $                LWORK-2*MN, INFO )
*
*     workspace: 2*MN.
*     Details of Householder rotations stored in WORK(MN+1:2*MN)
*
*     B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS)
*
      CALL DORMQR( 'Left', 'Transpose', M, NRHS, MN, A, LDA,
     $             WORK( 1 ),
     $             B, LDB, WORK( 2*MN+1 ), LWORK-2*MN, INFO )
      WSIZE = MAX( WSIZE, 2*MN+WORK( 2*MN+1 ) )
*
*     workspace: 2*MN+NB*NRHS.
*
*     B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS)
*
      CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK,
     $            NRHS, ONE, A, LDA, B, LDB )
*
      DO 40 J = 1, NRHS
         DO 30 I = RANK + 1, N
            B( I, J ) = ZERO
   30    CONTINUE
   40 CONTINUE
*
*     B(1:N,1:NRHS) := Y**T * B(1:N,1:NRHS)
*
      IF( RANK.LT.N ) THEN
         CALL DORMRZ( 'Left', 'Transpose', N, NRHS, RANK, N-RANK, A,
     $                LDA, WORK( MN+1 ), B, LDB, WORK( 2*MN+1 ),
     $                LWORK-2*MN, INFO )
      END IF
*
*     workspace: 2*MN+NRHS.
*
*     B(1:N,1:NRHS) := P * B(1:N,1:NRHS)
*
      DO 60 J = 1, NRHS
         DO 50 I = 1, N
            WORK( JPVT( I ) ) = B( I, J )
   50    CONTINUE
         CALL DCOPY( N, WORK( 1 ), 1, B( 1, J ), 1 )
   60 CONTINUE
*
*     workspace: N.
*
*     Undo scaling
*
      IF( IASCL.EQ.1 ) THEN
         CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB,
     $                INFO )
         CALL DLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA,
     $                INFO )
      ELSE IF( IASCL.EQ.2 ) THEN
         CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB,
     $                INFO )
         CALL DLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA,
     $                INFO )
      END IF
      IF( IBSCL.EQ.1 ) THEN
         CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB,
     $                INFO )
      ELSE IF( IBSCL.EQ.2 ) THEN
         CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB,
     $                INFO )
      END IF
*
   70 CONTINUE
      WORK( 1 ) = LWKOPT
*
      RETURN
*
*     End of DGELSY
*
      END
*> \brief \b DGEMQRT
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEMQRT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgemqrt.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgemqrt.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgemqrt.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEMQRT( SIDE, TRANS, M, N, K, NB, V, LDV, T, LDT,
*                          C, LDC, WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER SIDE, TRANS
*       INTEGER   INFO, K, LDV, LDC, M, N, NB, LDT
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION V( LDV, * ), C( LDC, * ), T( LDT, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEMQRT overwrites the general real M-by-N matrix C with
*>
*>                 SIDE = 'L'     SIDE = 'R'
*> TRANS = 'N':      Q C            C Q
*> TRANS = 'T':   Q**T C            C Q**T
*>
*> where Q is a real orthogonal matrix defined as the product of K
*> elementary reflectors:
*>
*>       Q = H(1) H(2) . . . H(K) = I - V T V**T
*>
*> generated using the compact WY representation as returned by DGEQRT.
*>
*> Q is of order M if SIDE = 'L' and of order N  if SIDE = 'R'.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': apply Q or Q**T from the Left;
*>          = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N':  No transpose, apply Q;
*>          = 'C':  Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines
*>          the matrix Q.
*>          If SIDE = 'L', M >= K >= 0;
*>          if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*>          NB is INTEGER
*>          The block size used for the storage of T.  K >= NB >= 1.
*>          This must be the same value of NB used to generate T
*>          in DGEQRT.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension (LDV,K)
*>          The i-th column must contain the vector which defines the
*>          elementary reflector H(i), for i = 1,2,...,k, as returned by
*>          DGEQRT in the first K columns of its array argument A.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of the array V.
*>          If SIDE = 'L', LDA >= max(1,M);
*>          if SIDE = 'R', LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,K)
*>          The upper triangular factors of the block reflectors
*>          as returned by DGEQRT, stored as a NB-by-N matrix.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T.  LDT >= NB.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the M-by-N matrix C.
*>          On exit, C is overwritten by Q C, Q**T C, C Q**T or C Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array. The dimension of
*>          WORK is N*NB if SIDE = 'L', or  M*NB if SIDE = 'R'.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gemqrt
*
*  =====================================================================
      SUBROUTINE DGEMQRT( SIDE, TRANS, M, N, K, NB, V, LDV, T, LDT,
     $                   C, LDC, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER SIDE, TRANS
      INTEGER   INFO, K, LDV, LDC, M, N, NB, LDT
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION V( LDV, * ), C( LDC, * ), T( LDT, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     ..
*     .. Local Scalars ..
      LOGICAL            LEFT, RIGHT, TRAN, NOTRAN
      INTEGER            I, IB, LDWORK, KF, Q
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA, DLARFB
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     .. Test the input arguments ..
*
      INFO   = 0
      LEFT   = LSAME( SIDE,  'L' )
      RIGHT  = LSAME( SIDE,  'R' )
      TRAN   = LSAME( TRANS, 'T' )
      NOTRAN = LSAME( TRANS, 'N' )
*
      IF( LEFT ) THEN
         LDWORK = MAX( 1, N )
         Q = M
      ELSE IF ( RIGHT ) THEN
         LDWORK = MAX( 1, M )
         Q = N
      END IF
      IF( .NOT.LEFT .AND. .NOT.RIGHT ) THEN
         INFO = -1
      ELSE IF( .NOT.TRAN .AND. .NOT.NOTRAN ) THEN
         INFO = -2
      ELSE IF( M.LT.0 ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( K.LT.0 .OR. K.GT.Q ) THEN
         INFO = -5
      ELSE IF( NB.LT.1 .OR. (NB.GT.K .AND. K.GT.0)) THEN
         INFO = -6
      ELSE IF( LDV.LT.MAX( 1, Q ) ) THEN
         INFO = -8
      ELSE IF( LDT.LT.NB ) THEN
         INFO = -10
      ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
         INFO = -12
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEMQRT', -INFO )
         RETURN
      END IF
*
*     .. Quick return if possible ..
*
      IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) RETURN
*
      IF( LEFT .AND. TRAN ) THEN
*
         DO I = 1, K, NB
            IB = MIN( NB, K-I+1 )
            CALL DLARFB( 'L', 'T', 'F', 'C', M-I+1, N, IB,
     $                   V( I, I ), LDV, T( 1, I ), LDT,
     $                   C( I, 1 ), LDC, WORK, LDWORK )
         END DO
*
      ELSE IF( RIGHT .AND. NOTRAN ) THEN
*
         DO I = 1, K, NB
            IB = MIN( NB, K-I+1 )
            CALL DLARFB( 'R', 'N', 'F', 'C', M, N-I+1, IB,
     $                   V( I, I ), LDV, T( 1, I ), LDT,
     $                   C( 1, I ), LDC, WORK, LDWORK )
         END DO
*
      ELSE IF( LEFT .AND. NOTRAN ) THEN
*
         KF = ((K-1)/NB)*NB+1
         DO I = KF, 1, -NB
            IB = MIN( NB, K-I+1 )
            CALL DLARFB( 'L', 'N', 'F', 'C', M-I+1, N, IB,
     $                   V( I, I ), LDV, T( 1, I ), LDT,
     $                   C( I, 1 ), LDC, WORK, LDWORK )
         END DO
*
      ELSE IF( RIGHT .AND. TRAN ) THEN
*
         KF = ((K-1)/NB)*NB+1
         DO I = KF, 1, -NB
            IB = MIN( NB, K-I+1 )
            CALL DLARFB( 'R', 'T', 'F', 'C', M, N-I+1, IB,
     $                   V( I, I ), LDV, T( 1, I ), LDT,
     $                   C( 1, I ), LDC, WORK, LDWORK )
         END DO
*
      END IF
*
      RETURN
*
*     End of DGEMQRT
*
      END
*> \brief \b DGEQL2 computes the QL factorization of a general rectangular matrix using an unblocked algorithm.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQL2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeql2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeql2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeql2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEQL2( M, N, A, LDA, TAU, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEQL2 computes a QL factorization of a real m by n matrix A:
*> A = Q * L.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the m by n matrix A.
*>          On exit, if m >= n, the lower triangle of the subarray
*>          A(m-n+1:m,1:n) contains the n by n lower triangular matrix L;
*>          if m <= n, the elements on and below the (n-m)-th
*>          superdiagonal contain the m by n lower trapezoidal matrix L;
*>          the remaining elements, with the array TAU, represent the
*>          orthogonal matrix Q as a product of elementary reflectors
*>          (see Further Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geql2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix Q is represented as a product of elementary reflectors
*>
*>     Q = H(k) . . . H(2) H(1), where k = min(m,n).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
*>  A(1:m-k+i-1,n-k+i), and tau in TAU(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGEQL2( M, N, A, LDA, TAU, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, K
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF1L, DLARFG, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEQL2', -INFO )
         RETURN
      END IF
*
      K = MIN( M, N )
*
      DO 10 I = K, 1, -1
*
*        Generate elementary reflector H(i) to annihilate
*        A(1:m-k+i-1,n-k+i)
*
         CALL DLARFG( M-K+I, A( M-K+I, N-K+I ), A( 1, N-K+I ), 1,
     $                TAU( I ) )
*
*        Apply H(i) to A(1:m-k+i,1:n-k+i-1) from the left
*
         CALL DLARF1L( 'Left', M-K+I, N-K+I-1, A( 1, N-K+I ), 1,
     $               TAU( I ),
     $               A, LDA, WORK )
   10 CONTINUE
      RETURN
*
*     End of DGEQL2
*
      END
*> \brief \b DGEQLF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQLF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqlf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqlf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqlf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEQLF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEQLF computes a QL factorization of a real M-by-N matrix A:
*> A = Q * L.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit,
*>          if m >= n, the lower triangle of the subarray
*>          A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L;
*>          if m <= n, the elements on and below the (n-m)-th
*>          superdiagonal contain the M-by-N lower trapezoidal matrix L;
*>          the remaining elements, with the array TAU, represent the
*>          orthogonal matrix Q as a product of elementary reflectors
*>          (see Further Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= N, otherwise.
*>          For optimum performance LWORK >= N*NB, where NB is the
*>          optimal blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geqlf
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix Q is represented as a product of elementary reflectors
*>
*>     Q = H(k) . . . H(2) H(1), where k = min(m,n).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
*>  A(1:m-k+i-1,n-k+i), and tau in TAU(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGEQLF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IB, IINFO, IWS, K, KI, KK, LDWORK, LWKOPT,
     $                   MU, NB, NBMIN, NU, NX
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEQL2, DLARFB, DLARFT, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      END IF
*
      IF( INFO.EQ.0 ) THEN
         K = MIN( M, N )
         IF( K.EQ.0 ) THEN
            LWKOPT = 1
         ELSE
            NB = ILAENV( 1, 'DGEQLF', ' ', M, N, -1, -1 )
            LWKOPT = N*NB
         END IF
         WORK( 1 ) = LWKOPT
*
         IF( .NOT.LQUERY ) THEN
            IF( LWORK.LE.0 .OR. ( M.GT.0 .AND. LWORK.LT.MAX( 1, N ) ) )
     $         INFO = -7
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEQLF', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( K.EQ.0 ) THEN
         RETURN
      END IF
*
      NBMIN = 2
      NX = 1
      IWS = N
      IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
*        Determine when to cross over from blocked to unblocked code.
*
         NX = MAX( 0, ILAENV( 3, 'DGEQLF', ' ', M, N, -1, -1 ) )
         IF( NX.LT.K ) THEN
*
*           Determine if workspace is large enough for blocked code.
*
            LDWORK = N
            IWS = LDWORK*NB
            IF( LWORK.LT.IWS ) THEN
*
*              Not enough workspace to use optimal NB:  reduce NB and
*              determine the minimum value of NB.
*
               NB = LWORK / LDWORK
               NBMIN = MAX( 2, ILAENV( 2, 'DGEQLF', ' ', M, N, -1,
     $                 -1 ) )
            END IF
         END IF
      END IF
*
      IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
*        Use blocked code initially.
*        The last kk columns are handled by the block method.
*
         KI = ( ( K-NX-1 ) / NB )*NB
         KK = MIN( K, KI+NB )
*
         DO 10 I = K - KK + KI + 1, K - KK + 1, -NB
            IB = MIN( K-I+1, NB )
*
*           Compute the QL factorization of the current block
*           A(1:m-k+i+ib-1,n-k+i:n-k+i+ib-1)
*
            CALL DGEQL2( M-K+I+IB-1, IB, A( 1, N-K+I ), LDA,
     $                   TAU( I ),
     $                   WORK, IINFO )
            IF( N-K+I.GT.1 ) THEN
*
*              Form the triangular factor of the block reflector
*              H = H(i+ib-1) . . . H(i+1) H(i)
*
               CALL DLARFT( 'Backward', 'Columnwise', M-K+I+IB-1, IB,
     $                      A( 1, N-K+I ), LDA, TAU( I ), WORK, LDWORK )
*
*              Apply H**T to A(1:m-k+i+ib-1,1:n-k+i-1) from the left
*
               CALL DLARFB( 'Left', 'Transpose', 'Backward',
     $                      'Columnwise', M-K+I+IB-1, N-K+I-1, IB,
     $                      A( 1, N-K+I ), LDA, WORK, LDWORK, A, LDA,
     $                      WORK( IB+1 ), LDWORK )
            END IF
   10    CONTINUE
         MU = M - K + I + NB - 1
         NU = N - K + I + NB - 1
      ELSE
         MU = M
         NU = N
      END IF
*
*     Use unblocked code to factor the last or only block
*
      IF( MU.GT.0 .AND. NU.GT.0 )
     $   CALL DGEQL2( MU, NU, A, LDA, TAU, WORK, IINFO )
*
      WORK( 1 ) = IWS
      RETURN
*
*     End of DGEQLF
*
      END
*> \brief \b DGEQP3
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQP3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqp3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqp3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqp3.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEQP3( M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       INTEGER            JPVT( * )
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEQP3 computes a QR factorization with column pivoting of a
*> matrix A:  A*P = Q*R  using Level 3 BLAS.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit, the upper triangle of the array contains the
*>          min(M,N)-by-N upper trapezoidal matrix R; the elements below
*>          the diagonal, together with the array TAU, represent the
*>          orthogonal matrix Q as a product of min(M,N) elementary
*>          reflectors.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] JPVT
*> \verbatim
*>          JPVT is INTEGER array, dimension (N)
*>          On entry, if JPVT(J).ne.0, the J-th column of A is permuted
*>          to the front of A*P (a leading column); if JPVT(J)=0,
*>          the J-th column of A is a free column.
*>          On exit, if JPVT(J)=K, then the J-th column of A*P was the
*>          the K-th column of A.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO=0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= 3*N+1.
*>          For optimal performance LWORK >= 2*N+( N+1 )*NB, where NB
*>          is the optimal blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit.
*>          < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geqp3
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix Q is represented as a product of elementary reflectors
*>
*>     Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real/complex vector
*>  with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
*>  A(i+1:m,i), and tau in TAU(i).
*> \endverbatim
*
*> \par Contributors:
*  ==================
*>
*>    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
*>    X. Sun, Computer Science Dept., Duke University, USA
*>
*  =====================================================================
      SUBROUTINE DGEQP3( M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      INTEGER            JPVT( * )
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            INB, INBMIN, IXOVER
      PARAMETER          ( INB = 1, INBMIN = 2, IXOVER = 3 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            FJB, IWS, J, JB, LWKOPT, MINMN, MINWS, NA, NB,
     $                   NBMIN, NFXD, NX, SM, SMINMN, SN, TOPBMN
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEQRF, DLAQP2, DLAQPS, DORMQR, DSWAP,
     $                   XERBLA
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      DOUBLE PRECISION   DNRM2
      EXTERNAL           ILAENV, DNRM2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          INT, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test input arguments
*  ====================
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      END IF
*
      IF( INFO.EQ.0 ) THEN
         MINMN = MIN( M, N )
         IF( MINMN.EQ.0 ) THEN
            IWS = 1
            LWKOPT = 1
         ELSE
            IWS = 3*N + 1
            NB = ILAENV( INB, 'DGEQRF', ' ', M, N, -1, -1 )
            LWKOPT = 2*N + ( N + 1 )*NB
         END IF
         WORK( 1 ) = LWKOPT
*
         IF( ( LWORK.LT.IWS ) .AND. .NOT.LQUERY ) THEN
            INFO = -8
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEQP3', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Move initial columns up front.
*
      NFXD = 1
      DO 10 J = 1, N
         IF( JPVT( J ).NE.0 ) THEN
            IF( J.NE.NFXD ) THEN
               CALL DSWAP( M, A( 1, J ), 1, A( 1, NFXD ), 1 )
               JPVT( J ) = JPVT( NFXD )
               JPVT( NFXD ) = J
            ELSE
               JPVT( J ) = J
            END IF
            NFXD = NFXD + 1
         ELSE
            JPVT( J ) = J
         END IF
   10 CONTINUE
      NFXD = NFXD - 1
*
*     Factorize fixed columns
*  =======================
*
*     Compute the QR factorization of fixed columns and update
*     remaining columns.
*
      IF( NFXD.GT.0 ) THEN
         NA = MIN( M, NFXD )
*CC      CALL DGEQR2( M, NA, A, LDA, TAU, WORK, INFO )
         CALL DGEQRF( M, NA, A, LDA, TAU, WORK, LWORK, INFO )
         IWS = MAX( IWS, INT( WORK( 1 ) ) )
         IF( NA.LT.N ) THEN
*CC         CALL DORM2R( 'Left', 'Transpose', M, N-NA, NA, A, LDA,
*CC  $                   TAU, A( 1, NA+1 ), LDA, WORK, INFO )
            CALL DORMQR( 'Left', 'Transpose', M, N-NA, NA, A, LDA,
     $                   TAU,
     $                   A( 1, NA+1 ), LDA, WORK, LWORK, INFO )
            IWS = MAX( IWS, INT( WORK( 1 ) ) )
         END IF
      END IF
*
*     Factorize free columns
*  ======================
*
      IF( NFXD.LT.MINMN ) THEN
*
         SM = M - NFXD
         SN = N - NFXD
         SMINMN = MINMN - NFXD
*
*        Determine the block size.
*
         NB = ILAENV( INB, 'DGEQRF', ' ', SM, SN, -1, -1 )
         NBMIN = 2
         NX = 0
*
         IF( ( NB.GT.1 ) .AND. ( NB.LT.SMINMN ) ) THEN
*
*           Determine when to cross over from blocked to unblocked code.
*
            NX = MAX( 0, ILAENV( IXOVER, 'DGEQRF', ' ', SM, SN, -1,
     $           -1 ) )
*
*
            IF( NX.LT.SMINMN ) THEN
*
*              Determine if workspace is large enough for blocked code.
*
               MINWS = 2*SN + ( SN+1 )*NB
               IWS = MAX( IWS, MINWS )
               IF( LWORK.LT.MINWS ) THEN
*
*                 Not enough workspace to use optimal NB: Reduce NB and
*                 determine the minimum value of NB.
*
                  NB = ( LWORK-2*SN ) / ( SN+1 )
                  NBMIN = MAX( 2, ILAENV( INBMIN, 'DGEQRF', ' ', SM,
     $                         SN,
     $                    -1, -1 ) )
*
*
               END IF
            END IF
         END IF
*
*        Initialize partial column norms. The first N elements of work
*        store the exact column norms.
*
         DO 20 J = NFXD + 1, N
            WORK( J ) = DNRM2( SM, A( NFXD+1, J ), 1 )
            WORK( N+J ) = WORK( J )
   20    CONTINUE
*
         IF( ( NB.GE.NBMIN ) .AND. ( NB.LT.SMINMN ) .AND.
     $       ( NX.LT.SMINMN ) ) THEN
*
*           Use blocked code initially.
*
            J = NFXD + 1
*
*           Compute factorization: while loop.
*
*
            TOPBMN = MINMN - NX
   30       CONTINUE
            IF( J.LE.TOPBMN ) THEN
               JB = MIN( NB, TOPBMN-J+1 )
*
*              Factorize JB columns among columns J:N.
*
               CALL DLAQPS( M, N-J+1, J-1, JB, FJB, A( 1, J ), LDA,
     $                      JPVT( J ), TAU( J ), WORK( J ), WORK( N+J ),
     $                      WORK( 2*N+1 ), WORK( 2*N+JB+1 ), N-J+1 )
*
               J = J + FJB
               GO TO 30
            END IF
         ELSE
            J = NFXD + 1
         END IF
*
*        Use unblocked code to factor the last or only block.
*
*
         IF( J.LE.MINMN )
     $      CALL DLAQP2( M, N-J+1, J-1, A( 1, J ), LDA, JPVT( J ),
     $                   TAU( J ), WORK( J ), WORK( N+J ),
     $                   WORK( 2*N+1 ) )
*
      END IF
*
      WORK( 1 ) = IWS
      RETURN
*
*     End of DGEQP3
*
      END
*> \brief \b DGEQPF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQPF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqpf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqpf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqpf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEQPF( M, N, A, LDA, JPVT, TAU, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, M, N
*       ..
*       .. Array Arguments ..
*       INTEGER            JPVT( * )
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DGEQP3.
*>
*> DGEQPF computes a QR factorization with column pivoting of a
*> real M-by-N matrix A: A*P = Q*R.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A. N >= 0
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit, the upper triangle of the array contains the
*>          min(M,N)-by-N upper triangular matrix R; the elements
*>          below the diagonal, together with the array TAU,
*>          represent the orthogonal matrix Q as a product of
*>          min(m,n) elementary reflectors.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] JPVT
*> \verbatim
*>          JPVT is INTEGER array, dimension (N)
*>          On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
*>          to the front of A*P (a leading column); if JPVT(i) = 0,
*>          the i-th column of A is a free column.
*>          On exit, if JPVT(i) = k, then the i-th column of A*P
*>          was the k-th column of A.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleGEcomputational
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix Q is represented as a product of elementary reflectors
*>
*>     Q = H(1) H(2) . . . H(n)
*>
*>  Each H(i) has the form
*>
*>     H = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i).
*>
*>  The matrix P is represented in jpvt as follows: If
*>     jpvt(j) = i
*>  then the jth column of P is the ith canonical unit vector.
*>
*>  Partial column norm updating strategy modified by
*>    Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
*>    University of Zagreb, Croatia.
*>  -- April 2011                                                      --
*>  For more details see LAPACK Working Note 176.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGEQPF( M, N, A, LDA, JPVT, TAU, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
*     ..
*     .. Array Arguments ..
      INTEGER            JPVT( * )
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, ITEMP, J, MA, MN, PVT
      DOUBLE PRECISION   AII, TEMP, TEMP2, TOL3Z
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEQR2, DLARF, DLARFG, DORM2R, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, SQRT
*     ..
*     .. External Functions ..
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH, DNRM2
      EXTERNAL           IDAMAX, DLAMCH, DNRM2
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEQPF', -INFO )
         RETURN
      END IF
*
      MN = MIN( M, N )
      TOL3Z = SQRT(DLAMCH('Epsilon'))
*
*     Move initial columns up front
*
      ITEMP = 1
      DO 10 I = 1, N
         IF( JPVT( I ).NE.0 ) THEN
            IF( I.NE.ITEMP ) THEN
               CALL DSWAP( M, A( 1, I ), 1, A( 1, ITEMP ), 1 )
               JPVT( I ) = JPVT( ITEMP )
               JPVT( ITEMP ) = I
            ELSE
               JPVT( I ) = I
            END IF
            ITEMP = ITEMP + 1
         ELSE
            JPVT( I ) = I
         END IF
   10 CONTINUE
      ITEMP = ITEMP - 1
*
*     Compute the QR factorization and update remaining columns
*
      IF( ITEMP.GT.0 ) THEN
         MA = MIN( ITEMP, M )
         CALL DGEQR2( M, MA, A, LDA, TAU, WORK, INFO )
         IF( MA.LT.N ) THEN
            CALL DORM2R( 'Left', 'Transpose', M, N-MA, MA, A, LDA, TAU,
     $                   A( 1, MA+1 ), LDA, WORK, INFO )
         END IF
      END IF
*
      IF( ITEMP.LT.MN ) THEN
*
*        Initialize partial column norms. The first n elements of
*        work store the exact column norms.
*
         DO 20 I = ITEMP + 1, N
            WORK( I ) = DNRM2( M-ITEMP, A( ITEMP+1, I ), 1 )
            WORK( N+I ) = WORK( I )
   20    CONTINUE
*
*        Compute factorization
*
         DO 40 I = ITEMP + 1, MN
*
*           Determine ith pivot column and swap if necessary
*
            PVT = ( I-1 ) + IDAMAX( N-I+1, WORK( I ), 1 )
*
            IF( PVT.NE.I ) THEN
               CALL DSWAP( M, A( 1, PVT ), 1, A( 1, I ), 1 )
               ITEMP = JPVT( PVT )
               JPVT( PVT ) = JPVT( I )
               JPVT( I ) = ITEMP
               WORK( PVT ) = WORK( I )
               WORK( N+PVT ) = WORK( N+I )
            END IF
*
*           Generate elementary reflector H(i)
*
            IF( I.LT.M ) THEN
               CALL DLARFG( M-I+1, A( I, I ), A( I+1, I ), 1,
     $                      TAU( I ) )
            ELSE
               CALL DLARFG( 1, A( M, M ), A( M, M ), 1, TAU( M ) )
            END IF
*
            IF( I.LT.N ) THEN
*
*              Apply H(i) to A(i:m,i+1:n) from the left
*
               AII = A( I, I )
               A( I, I ) = ONE
               CALL DLARF( 'LEFT', M-I+1, N-I, A( I, I ), 1,
     $                     TAU( I ), A( I, I+1 ), LDA, WORK( 2*N+1 ) )
               A( I, I ) = AII
            END IF
*
*           Update partial column norms
*
            DO 30 J = I + 1, N
               IF( WORK( J ).NE.ZERO ) THEN
*
*                 NOTE: The following 4 lines follow from the analysis in
*                 Lapack Working Note 176.
*
                  TEMP = ABS( A( I, J ) ) / WORK( J )
                  TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) )
                  TEMP2 = TEMP*( WORK( J ) / WORK( N+J ) )**2
                  IF( TEMP2 .LE. TOL3Z ) THEN
                     IF( M-I.GT.0 ) THEN
                        WORK( J ) = DNRM2( M-I, A( I+1, J ), 1 )
                        WORK( N+J ) = WORK( J )
                     ELSE
                        WORK( J ) = ZERO
                        WORK( N+J ) = ZERO
                     END IF
                  ELSE
                     WORK( J ) = WORK( J )*SQRT( TEMP )
                  END IF
               END IF
   30       CONTINUE
*
   40    CONTINUE
      END IF
      RETURN
*
*     End of DGEQPF
*
      END
*> \brief \b DGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQR2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqr2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqr2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqr2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEQR2( M, N, A, LDA, TAU, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEQR2 computes a QR factorization of a real m-by-n matrix A:
*>
*>    A = Q * ( R ),
*>            ( 0 )
*>
*> where:
*>
*>    Q is a m-by-m orthogonal matrix;
*>    R is an upper-triangular n-by-n matrix;
*>    0 is a (m-n)-by-n zero matrix, if m > n.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the m by n matrix A.
*>          On exit, the elements on and above the diagonal of the array
*>          contain the min(m,n) by n upper trapezoidal matrix R (R is
*>          upper triangular if m >= n); the elements below the diagonal,
*>          with the array TAU, represent the orthogonal matrix Q as a
*>          product of elementary reflectors (see Further Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geqr2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix Q is represented as a product of elementary reflectors
*>
*>     Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
*>  and tau in TAU(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGEQR2( M, N, A, LDA, TAU, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, K
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF1F, DLARFG, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEQR2', -INFO )
         RETURN
      END IF
*
      K = MIN( M, N )
*
      DO 10 I = 1, K
*
*        Generate elementary reflector H(i) to annihilate A(i+1:m,i)
*
         CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
     $                TAU( I ) )
         IF( I.LT.N ) THEN
*
*           Apply H(i) to A(i:m,i+1:n) from the left
*
            CALL DLARF1F( 'Left', M-I+1, N-I, A( I, I ), 1, TAU( I ),
     $                  A( I, I+1 ), LDA, WORK )
         END IF
   10 CONTINUE
      RETURN
*
*     End of DGEQR2
*
      END
*> \brief \b DGEQR2P computes the QR factorization of a general rectangular matrix with non-negative diagonal elements using an unblocked algorithm.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQR2P + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqr2p.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqr2p.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqr2p.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEQR2P( M, N, A, LDA, TAU, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEQR2P computes a QR factorization of a real m-by-n matrix A:
*>
*>    A = Q * ( R ),
*>            ( 0 )
*>
*> where:
*>
*>    Q is a m-by-m orthogonal matrix;
*>    R is an upper-triangular n-by-n matrix with nonnegative diagonal
*>    entries;
*>    0 is a (m-n)-by-n zero matrix, if m > n.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the m by n matrix A.
*>          On exit, the elements on and above the diagonal of the array
*>          contain the min(m,n) by n upper trapezoidal matrix R (R is
*>          upper triangular if m >= n). The diagonal entries of R are
*>          nonnegative; the elements below the diagonal,
*>          with the array TAU, represent the orthogonal matrix Q as a
*>          product of elementary reflectors (see Further Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geqr2p
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix Q is represented as a product of elementary reflectors
*>
*>     Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
*>  and tau in TAU(i).
*>
*> See Lapack Working Note 203 for details
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGEQR2P( M, N, A, LDA, TAU, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, K
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF1F, DLARFGP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEQR2P', -INFO )
         RETURN
      END IF
*
      K = MIN( M, N )
*
      DO 10 I = 1, K
*
*        Generate elementary reflector H(i) to annihilate A(i+1:m,i)
*
         CALL DLARFGP( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
     $                TAU( I ) )
         IF( I.LT.N ) THEN
*
*           Apply H(i) to A(i:m,i+1:n) from the left
*
            CALL DLARF1F( 'Left', M-I+1, N-I, A( I, I ), 1, TAU( I ),
     $                  A( I, I+1 ), LDA, WORK )
         END IF
   10 CONTINUE
      RETURN
*
*     End of DGEQR2P
*
      END
*> \brief \b DGEQRF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEQRF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEQRF computes a QR factorization of a real M-by-N matrix A:
*>
*>    A = Q * ( R ),
*>            ( 0 )
*>
*> where:
*>
*>    Q is a M-by-M orthogonal matrix;
*>    R is an upper-triangular N-by-N matrix;
*>    0 is a (M-N)-by-N zero matrix, if M > N.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit, the elements on and above the diagonal of the array
*>          contain the min(M,N)-by-N upper trapezoidal matrix R (R is
*>          upper triangular if m >= n); the elements below the diagonal,
*>          with the array TAU, represent the orthogonal matrix Q as a
*>          product of min(m,n) elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= N, otherwise.
*>          For optimum performance LWORK >= N*NB, where NB is
*>          the optimal blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geqrf
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix Q is represented as a product of elementary reflectors
*>
*>     Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
*>  and tau in TAU(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGEQRF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB,
     $                   NBMIN, NX
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEQR2, DLARFB, DLARFT, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      K = MIN( M, N )
      INFO = 0
      NB = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
      LQUERY = ( LWORK.EQ.-1 )
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      ELSE IF( .NOT.LQUERY ) THEN
         IF( LWORK.LE.0 .OR. ( M.GT.0 .AND. LWORK.LT.MAX( 1, N ) ) )
     $      INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEQRF', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         IF( K.EQ.0 ) THEN
            LWKOPT = 1
         ELSE
            LWKOPT = N*NB
         END IF
         WORK( 1 ) = LWKOPT
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( K.EQ.0 ) THEN
         WORK( 1 ) = 1
         RETURN
      END IF
*
      NBMIN = 2
      NX = 0
      IWS = N
      IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
*        Determine when to cross over from blocked to unblocked code.
*
         NX = MAX( 0, ILAENV( 3, 'DGEQRF', ' ', M, N, -1, -1 ) )
         IF( NX.LT.K ) THEN
*
*           Determine if workspace is large enough for blocked code.
*
            LDWORK = N
            IWS = LDWORK*NB
            IF( LWORK.LT.IWS ) THEN
*
*              Not enough workspace to use optimal NB:  reduce NB and
*              determine the minimum value of NB.
*
               NB = LWORK / LDWORK
               NBMIN = MAX( 2, ILAENV( 2, 'DGEQRF', ' ', M, N, -1,
     $                 -1 ) )
            END IF
         END IF
      END IF
*
      IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
*        Use blocked code initially
*
         DO 10 I = 1, K - NX, NB
            IB = MIN( K-I+1, NB )
*
*           Compute the QR factorization of the current block
*           A(i:m,i:i+ib-1)
*
            CALL DGEQR2( M-I+1, IB, A( I, I ), LDA, TAU( I ), WORK,
     $                   IINFO )
            IF( I+IB.LE.N ) THEN
*
*              Form the triangular factor of the block reflector
*              H = H(i) H(i+1) . . . H(i+ib-1)
*
               CALL DLARFT( 'Forward', 'Columnwise', M-I+1, IB,
     $                      A( I, I ), LDA, TAU( I ), WORK, LDWORK )
*
*              Apply H**T to A(i:m,i+ib:n) from the left
*
               CALL DLARFB( 'Left', 'Transpose', 'Forward',
     $                      'Columnwise', M-I+1, N-I-IB+1, IB,
     $                      A( I, I ), LDA, WORK, LDWORK, A( I, I+IB ),
     $                      LDA, WORK( IB+1 ), LDWORK )
            END IF
   10    CONTINUE
      ELSE
         I = 1
      END IF
*
*     Use unblocked code to factor the last or only block.
*
      IF( I.LE.K )
     $   CALL DGEQR2( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
     $                IINFO )
*
      WORK( 1 ) = IWS
      RETURN
*
*     End of DGEQRF
*
      END
*> \brief \b DGEQRFP
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQRFP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqrfp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqrfp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqrfp.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEQRFP( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEQR2P computes a QR factorization of a real M-by-N matrix A:
*>
*>    A = Q * ( R ),
*>            ( 0 )
*>
*> where:
*>
*>    Q is a M-by-M orthogonal matrix;
*>    R is an upper-triangular N-by-N matrix with nonnegative diagonal
*>    entries;
*>    0 is a (M-N)-by-N zero matrix, if M > N.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit, the elements on and above the diagonal of the array
*>          contain the min(M,N)-by-N upper trapezoidal matrix R (R is
*>          upper triangular if m >= n). The diagonal entries of R
*>          are nonnegative; the elements below the diagonal,
*>          with the array TAU, represent the orthogonal matrix Q as a
*>          product of min(m,n) elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= N, otherwise.
*>          For optimum performance LWORK >= N*NB, where NB is
*>          the optimal blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geqrfp
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix Q is represented as a product of elementary reflectors
*>
*>     Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
*>  and tau in TAU(i).
*>
*> See Lapack Working Note 203 for details
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGEQRFP( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IB, IINFO, IWS, K, LDWORK, LWKMIN, LWKOPT,
     $                   NB, NBMIN, NX
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEQR2P, DLARFB, DLARFT, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      NB = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
      K = MIN( M, N )
      IF( K.EQ.0 ) THEN
         LWKMIN = 1
         LWKOPT = 1
      ELSE
         LWKMIN = N
         LWKOPT = N*NB
      END IF
      WORK( 1 ) = LWKOPT
*
      LQUERY = ( LWORK.EQ.-1 )
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
         INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEQRFP', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( K.EQ.0 ) THEN
         WORK( 1 ) = 1
         RETURN
      END IF
*
      NBMIN = 2
      NX = 0
      IWS = LWKMIN
      IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
*        Determine when to cross over from blocked to unblocked code.
*
         NX = MAX( 0, ILAENV( 3, 'DGEQRF', ' ', M, N, -1, -1 ) )
         IF( NX.LT.K ) THEN
*
*           Determine if workspace is large enough for blocked code.
*
            LDWORK = N
            IWS = LDWORK*NB
            IF( LWORK.LT.IWS ) THEN
*
*              Not enough workspace to use optimal NB:  reduce NB and
*              determine the minimum value of NB.
*
               NB = LWORK / LDWORK
               NBMIN = MAX( 2, ILAENV( 2, 'DGEQRF', ' ', M, N, -1,
     $                 -1 ) )
            END IF
         END IF
      END IF
*
      IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
*        Use blocked code initially
*
         DO 10 I = 1, K - NX, NB
            IB = MIN( K-I+1, NB )
*
*           Compute the QR factorization of the current block
*           A(i:m,i:i+ib-1)
*
            CALL DGEQR2P( M-I+1, IB, A( I, I ), LDA, TAU( I ), WORK,
     $                   IINFO )
            IF( I+IB.LE.N ) THEN
*
*              Form the triangular factor of the block reflector
*              H = H(i) H(i+1) . . . H(i+ib-1)
*
               CALL DLARFT( 'Forward', 'Columnwise', M-I+1, IB,
     $                      A( I, I ), LDA, TAU( I ), WORK, LDWORK )
*
*              Apply H**T to A(i:m,i+ib:n) from the left
*
               CALL DLARFB( 'Left', 'Transpose', 'Forward',
     $                      'Columnwise', M-I+1, N-I-IB+1, IB,
     $                      A( I, I ), LDA, WORK, LDWORK, A( I, I+IB ),
     $                      LDA, WORK( IB+1 ), LDWORK )
            END IF
   10    CONTINUE
      ELSE
         I = 1
      END IF
*
*     Use unblocked code to factor the last or only block.
*
      IF( I.LE.K )
     $   CALL DGEQR2P( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
     $                IINFO )
*
      WORK( 1 ) = IWS
      RETURN
*
*     End of DGEQRFP
*
      END
*> \brief \b DGEQRT
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQRT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqrt.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqrt.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqrt.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEQRT( M, N, NB, A, LDA, T, LDT, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER INFO, LDA, LDT, M, N, NB
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION A( LDA, * ), T( LDT, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEQRT computes a blocked QR factorization of a real M-by-N matrix A
*> using the compact WY representation of Q.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*>          NB is INTEGER
*>          The block size to be used in the blocked QR.  MIN(M,N) >= NB >= 1.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit, the elements on and above the diagonal of the array
*>          contain the min(M,N)-by-N upper trapezoidal matrix R (R is
*>          upper triangular if M >= N); the elements below the diagonal
*>          are the columns of V.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,MIN(M,N))
*>          The upper triangular block reflectors stored in compact form
*>          as a sequence of upper triangular blocks.  See below
*>          for further details.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T.  LDT >= NB.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (NB*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geqrt
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix V stores the elementary reflectors H(i) in the i-th column
*>  below the diagonal. For example, if M=5 and N=3, the matrix V is
*>
*>               V = (  1       )
*>                   ( v1  1    )
*>                   ( v1 v2  1 )
*>                   ( v1 v2 v3 )
*>                   ( v1 v2 v3 )
*>
*>  where the vi's represent the vectors which define H(i), which are returned
*>  in the matrix A.  The 1's along the diagonal of V are not stored in A.
*>
*>  Let K=MIN(M,N).  The number of blocks is B = ceiling(K/NB), where each
*>  block is of order NB except for the last block, which is of order
*>  IB = K - (B-1)*NB.  For each of the B blocks, a upper triangular block
*>  reflector factor is computed: T1, T2, ..., TB.  The NB-by-NB (and IB-by-IB
*>  for the last block) T's are stored in the NB-by-K matrix T as
*>
*>               T = (T1 T2 ... TB).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGEQRT( M, N, NB, A, LDA, T, LDT, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER INFO, LDA, LDT, M, N, NB
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION A( LDA, * ), T( LDT, * ), WORK( * )
*     ..
*
* =====================================================================
*
*     ..
*     .. Local Scalars ..
      INTEGER    I, IB, IINFO, K
      LOGICAL    USE_RECURSIVE_QR
      PARAMETER( USE_RECURSIVE_QR=.TRUE. )
*     ..
*     .. External Subroutines ..
      EXTERNAL   DGEQRT2, DGEQRT3, DLARFB, XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NB.LT.1 .OR. ( NB.GT.MIN(M,N) .AND. MIN(M,N).GT.0 ) )THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -5
      ELSE IF( LDT.LT.NB ) THEN
         INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEQRT', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      K = MIN( M, N )
      IF( K.EQ.0 ) RETURN
*
*     Blocked loop of length K
*
      DO I = 1, K,  NB
         IB = MIN( K-I+1, NB )
*
*     Compute the QR factorization of the current block A(I:M,I:I+IB-1)
*
         IF( USE_RECURSIVE_QR ) THEN
            CALL DGEQRT3( M-I+1, IB, A(I,I), LDA, T(1,I), LDT,
     $                    IINFO )
         ELSE
            CALL DGEQRT2( M-I+1, IB, A(I,I), LDA, T(1,I), LDT,
     $                    IINFO )
         END IF
         IF( I+IB.LE.N ) THEN
*
*     Update by applying H**T to A(I:M,I+IB:N) from the left
*
            CALL DLARFB( 'L', 'T', 'F', 'C', M-I+1, N-I-IB+1, IB,
     $                   A( I, I ), LDA, T( 1, I ), LDT,
     $                   A( I, I+IB ), LDA, WORK , N-I-IB+1 )
         END IF
      END DO
      RETURN
*
*     End of DGEQRT
*
      END
*> \brief \b DGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQRT2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqrt2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqrt2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqrt2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEQRT2( M, N, A, LDA, T, LDT, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER   INFO, LDA, LDT, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), T( LDT, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEQRT2 computes a QR factorization of a real M-by-N matrix A,
*> using the compact WY representation of Q.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= N.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the real M-by-N matrix A.  On exit, the elements on and
*>          above the diagonal contain the N-by-N upper triangular matrix R; the
*>          elements below the diagonal are the columns of V.  See below for
*>          further details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,N)
*>          The N-by-N upper triangular factor of the block reflector.
*>          The elements on and above the diagonal contain the block
*>          reflector T; the elements below the diagonal are not used.
*>          See below for further details.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T.  LDT >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geqrt2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix V stores the elementary reflectors H(i) in the i-th column
*>  below the diagonal. For example, if M=5 and N=3, the matrix V is
*>
*>               V = (  1       )
*>                   ( v1  1    )
*>                   ( v1 v2  1 )
*>                   ( v1 v2 v3 )
*>                   ( v1 v2 v3 )
*>
*>  where the vi's represent the vectors which define H(i), which are returned
*>  in the matrix A.  The 1's along the diagonal of V are not stored in A.  The
*>  block reflector H is then given by
*>
*>               H = I - V * T * V**T
*>
*>  where V**T is the transpose of V.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGEQRT2( M, N, A, LDA, T, LDT, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER   INFO, LDA, LDT, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), T( LDT, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION  ONE, ZERO
      PARAMETER( ONE = 1.0D+00, ZERO = 0.0D+00 )
*     ..
*     .. Local Scalars ..
      INTEGER   I, K
      DOUBLE PRECISION   AII, ALPHA
*     ..
*     .. External Subroutines ..
      EXTERNAL  DLARFG, DGEMV, DGER, DTRMV, XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( M.LT.N ) THEN
         INFO = -1
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEQRT2', -INFO )
         RETURN
      END IF
*
      K = MIN( M, N )
*
      DO I = 1, K
*
*        Generate elem. refl. H(i) to annihilate A(i+1:m,i), tau(I) -> T(I,1)
*
         CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
     $                T( I, 1 ) )
         IF( I.LT.N ) THEN
*
*           Apply H(i) to A(I:M,I+1:N) from the left
*
            AII = A( I, I )
            A( I, I ) = ONE
*
*           W(1:N-I) := A(I:M,I+1:N)^H * A(I:M,I) [W = T(:,N)]
*
            CALL DGEMV( 'T',M-I+1, N-I, ONE, A( I, I+1 ), LDA,
     $                  A( I, I ), 1, ZERO, T( 1, N ), 1 )
*
*           A(I:M,I+1:N) = A(I:m,I+1:N) + alpha*A(I:M,I)*W(1:N-1)^H
*
            ALPHA = -(T( I, 1 ))
            CALL DGER( M-I+1, N-I, ALPHA, A( I, I ), 1,
     $           T( 1, N ), 1, A( I, I+1 ), LDA )
            A( I, I ) = AII
         END IF
      END DO
*
      DO I = 2, N
         AII = A( I, I )
         A( I, I ) = ONE
*
*        T(1:I-1,I) := alpha * A(I:M,1:I-1)**T * A(I:M,I)
*
         ALPHA = -T( I, 1 )
         CALL DGEMV( 'T', M-I+1, I-1, ALPHA, A( I, 1 ), LDA,
     $               A( I, I ), 1, ZERO, T( 1, I ), 1 )
         A( I, I ) = AII
*
*        T(1:I-1,I) := T(1:I-1,1:I-1) * T(1:I-1,I)
*
         CALL DTRMV( 'U', 'N', 'N', I-1, T, LDT, T( 1, I ), 1 )
*
*           T(I,I) = tau(I)
*
            T( I, I ) = T( I, 1 )
            T( I, 1) = ZERO
      END DO

*
*     End of DGEQRT2
*
      END
*> \brief \b DGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQRT3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqrt3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqrt3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqrt3.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       RECURSIVE SUBROUTINE DGEQRT3( M, N, A, LDA, T, LDT, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER   INFO, LDA, M, N, LDT
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), T( LDT, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEQRT3 recursively computes a QR factorization of a real M-by-N
*> matrix A, using the compact WY representation of Q.
*>
*> Based on the algorithm of Elmroth and Gustavson,
*> IBM J. Res. Develop. Vol 44 No. 4 July 2000.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= N.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the real M-by-N matrix A.  On exit, the elements on and
*>          above the diagonal contain the N-by-N upper triangular matrix R; the
*>          elements below the diagonal are the columns of V.  See below for
*>          further details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,N)
*>          The N-by-N upper triangular factor of the block reflector.
*>          The elements on and above the diagonal contain the block
*>          reflector T; the elements below the diagonal are not used.
*>          See below for further details.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T.  LDT >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geqrt3
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix V stores the elementary reflectors H(i) in the i-th column
*>  below the diagonal. For example, if M=5 and N=3, the matrix V is
*>
*>               V = (  1       )
*>                   ( v1  1    )
*>                   ( v1 v2  1 )
*>                   ( v1 v2 v3 )
*>                   ( v1 v2 v3 )
*>
*>  where the vi's represent the vectors which define H(i), which are returned
*>  in the matrix A.  The 1's along the diagonal of V are not stored in A.  The
*>  block reflector H is then given by
*>
*>               H = I - V * T * V**T
*>
*>  where V**T is the transpose of V.
*>
*>  For details of the algorithm, see Elmroth and Gustavson (cited above).
*> \endverbatim
*>
*  =====================================================================
      RECURSIVE SUBROUTINE DGEQRT3( M, N, A, LDA, T, LDT, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER   INFO, LDA, M, N, LDT
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), T( LDT, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER ( ONE = 1.0D+00 )
*     ..
*     .. Local Scalars ..
      INTEGER   I, I1, J, J1, N1, N2, IINFO
*     ..
*     .. External Subroutines ..
      EXTERNAL  DLARFG, DTRMM, DGEMM, XERBLA
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      IF( N .LT. 0 ) THEN
         INFO = -2
      ELSE IF( M .LT. N ) THEN
         INFO = -1
      ELSE IF( LDA .LT. MAX( 1, M ) ) THEN
         INFO = -4
      ELSE IF( LDT .LT. MAX( 1, N ) ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGEQRT3', -INFO )
         RETURN
      END IF
*
      IF( N.EQ.1 ) THEN
*
*        Compute Householder transform when N=1
*
         CALL DLARFG( M, A(1,1), A( MIN( 2, M ), 1 ), 1, T(1,1) )
*
      ELSE
*
*        Otherwise, split A into blocks...
*
         N1 = N/2
         N2 = N-N1
         J1 = MIN( N1+1, N )
         I1 = MIN( N+1, M )
*
*        Compute A(1:M,1:N1) <- (Y1,R1,T1), where Q1 = I - Y1 T1 Y1^H
*
         CALL DGEQRT3( M, N1, A, LDA, T, LDT, IINFO )
*
*        Compute A(1:M,J1:N) = Q1^H A(1:M,J1:N) [workspace: T(1:N1,J1:N)]
*
         DO J=1,N2
            DO I=1,N1
               T( I, J+N1 ) = A( I, J+N1 )
            END DO
         END DO
         CALL DTRMM( 'L', 'L', 'T', 'U', N1, N2, ONE,
     &               A, LDA, T( 1, J1 ), LDT )
*
         CALL DGEMM( 'T', 'N', N1, N2, M-N1, ONE, A( J1, 1 ), LDA,
     &               A( J1, J1 ), LDA, ONE, T( 1, J1 ), LDT)
*
         CALL DTRMM( 'L', 'U', 'T', 'N', N1, N2, ONE,
     &               T, LDT, T( 1, J1 ), LDT )
*
         CALL DGEMM( 'N', 'N', M-N1, N2, N1, -ONE, A( J1, 1 ), LDA,
     &               T( 1, J1 ), LDT, ONE, A( J1, J1 ), LDA )
*
         CALL DTRMM( 'L', 'L', 'N', 'U', N1, N2, ONE,
     &               A, LDA, T( 1, J1 ), LDT )
*
         DO J=1,N2
            DO I=1,N1
               A( I, J+N1 ) = A( I, J+N1 ) - T( I, J+N1 )
            END DO
         END DO
*
*        Compute A(J1:M,J1:N) <- (Y2,R2,T2) where Q2 = I - Y2 T2 Y2^H
*
         CALL DGEQRT3( M-N1, N2, A( J1, J1 ), LDA,
     &                T( J1, J1 ), LDT, IINFO )
*
*        Compute T3 = T(1:N1,J1:N) = -T1 Y1^H Y2 T2
*
         DO I=1,N1
            DO J=1,N2
               T( I, J+N1 ) = (A( J+N1, I ))
            END DO
         END DO
*
         CALL DTRMM( 'R', 'L', 'N', 'U', N1, N2, ONE,
     &               A( J1, J1 ), LDA, T( 1, J1 ), LDT )
*
         CALL DGEMM( 'T', 'N', N1, N2, M-N, ONE, A( I1, 1 ), LDA,
     &               A( I1, J1 ), LDA, ONE, T( 1, J1 ), LDT )
*
         CALL DTRMM( 'L', 'U', 'N', 'N', N1, N2, -ONE, T, LDT,
     &               T( 1, J1 ), LDT )
*
         CALL DTRMM( 'R', 'U', 'N', 'N', N1, N2, ONE,
     &               T( J1, J1 ), LDT, T( 1, J1 ), LDT )
*
*        Y = (Y1,Y2); R = [ R1  A(1:N1,J1:N) ];  T = [T1 T3]
*                         [  0        R2     ]       [ 0 T2]
*
      END IF
*
      RETURN
*
*     End of DGEQRT3
*
      END
*> \brief \b DGERFS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGERFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgerfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgerfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgerfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
*                          X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANS
*       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
*      $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGERFS improves the computed solution to a system of linear
*> equations and provides error bounds and backward error estimates for
*> the solution.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the form of the system of equations:
*>          = 'N':  A * X = B     (No transpose)
*>          = 'T':  A**T * X = B  (Transpose)
*>          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices B and X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The original N-by-N matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] AF
*> \verbatim
*>          AF is DOUBLE PRECISION array, dimension (LDAF,N)
*>          The factors L and U from the factorization A = P*L*U
*>          as computed by DGETRF.
*> \endverbatim
*>
*> \param[in] LDAF
*> \verbatim
*>          LDAF is INTEGER
*>          The leading dimension of the array AF.  LDAF >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          The pivot indices from DGETRF; for 1<=i<=N, row i of the
*>          matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          On entry, the solution matrix X, as computed by DGETRS.
*>          On exit, the improved solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The estimated forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).  The estimate is as reliable as
*>          the estimate for RCOND, and is almost always a slight
*>          overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in
*>          any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  ITMAX is the maximum number of steps of iterative refinement.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gerfs
*
*  =====================================================================
      SUBROUTINE DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B,
     $                   LDB,
     $                   X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS
      INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
     $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            ITMAX
      PARAMETER          ( ITMAX = 5 )
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
      DOUBLE PRECISION   TWO
      PARAMETER          ( TWO = 2.0D+0 )
      DOUBLE PRECISION   THREE
      PARAMETER          ( THREE = 3.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOTRAN
      CHARACTER          TRANST
      INTEGER            COUNT, I, J, K, KASE, NZ
      DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DGEMV, DGETRS, DLACN2,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NOTRAN = LSAME( TRANS, 'N' )
      IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
     $    LSAME( TRANS, 'C' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -10
      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
         INFO = -12
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGERFS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
         DO 10 J = 1, NRHS
            FERR( J ) = ZERO
            BERR( J ) = ZERO
   10    CONTINUE
         RETURN
      END IF
*
      IF( NOTRAN ) THEN
         TRANST = 'T'
      ELSE
         TRANST = 'N'
      END IF
*
*     NZ = maximum number of nonzero elements in each row of A, plus 1
*
      NZ = N + 1
      EPS = DLAMCH( 'Epsilon' )
      SAFMIN = DLAMCH( 'Safe minimum' )
      SAFE1 = NZ*SAFMIN
      SAFE2 = SAFE1 / EPS
*
*     Do for each right hand side
*
      DO 140 J = 1, NRHS
*
         COUNT = 1
         LSTRES = THREE
   20    CONTINUE
*
*        Loop until stopping criterion is satisfied.
*
*        Compute residual R = B - op(A) * X,
*        where op(A) = A, A**T, or A**H, depending on TRANS.
*
         CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
         CALL DGEMV( TRANS, N, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
     $               WORK( N+1 ), 1 )
*
*        Compute componentwise relative backward error from formula
*
*        max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
*
*        where abs(Z) is the componentwise absolute value of the matrix
*        or vector Z.  If the i-th component of the denominator is less
*        than SAFE2, then SAFE1 is added to the i-th components of the
*        numerator and denominator before dividing.
*
         DO 30 I = 1, N
            WORK( I ) = ABS( B( I, J ) )
   30    CONTINUE
*
*        Compute abs(op(A))*abs(X) + abs(B).
*
         IF( NOTRAN ) THEN
            DO 50 K = 1, N
               XK = ABS( X( K, J ) )
               DO 40 I = 1, N
                  WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
   40          CONTINUE
   50       CONTINUE
         ELSE
            DO 70 K = 1, N
               S = ZERO
               DO 60 I = 1, N
                  S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
   60          CONTINUE
               WORK( K ) = WORK( K ) + S
   70       CONTINUE
         END IF
         S = ZERO
         DO 80 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
            ELSE
               S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
     $             ( WORK( I )+SAFE1 ) )
            END IF
   80    CONTINUE
         BERR( J ) = S
*
*        Test stopping criterion. Continue iterating if
*           1) The residual BERR(J) is larger than machine epsilon, and
*           2) BERR(J) decreased by at least a factor of 2 during the
*              last iteration, and
*           3) At most ITMAX iterations tried.
*
         IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
     $       COUNT.LE.ITMAX ) THEN
*
*           Update solution and try again.
*
            CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
     $                   INFO )
            CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
            LSTRES = BERR( J )
            COUNT = COUNT + 1
            GO TO 20
         END IF
*
*        Bound error from formula
*
*        norm(X - XTRUE) / norm(X) .le. FERR =
*        norm( abs(inv(op(A)))*
*           ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
*
*        where
*          norm(Z) is the magnitude of the largest component of Z
*          inv(op(A)) is the inverse of op(A)
*          abs(Z) is the componentwise absolute value of the matrix or
*             vector Z
*          NZ is the maximum number of nonzeros in any row of A, plus 1
*          EPS is machine epsilon
*
*        The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
*        is incremented by SAFE1 if the i-th component of
*        abs(op(A))*abs(X) + abs(B) is less than SAFE2.
*
*        Use DLACN2 to estimate the infinity-norm of the matrix
*           inv(op(A)) * diag(W),
*        where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
*
         DO 90 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
            ELSE
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
            END IF
   90    CONTINUE
*
         KASE = 0
  100    CONTINUE
         CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
     $                FERR( J ),
     $                KASE, ISAVE )
         IF( KASE.NE.0 ) THEN
            IF( KASE.EQ.1 ) THEN
*
*              Multiply by diag(W)*inv(op(A)**T).
*
               CALL DGETRS( TRANST, N, 1, AF, LDAF, IPIV,
     $                      WORK( N+1 ),
     $                      N, INFO )
               DO 110 I = 1, N
                  WORK( N+I ) = WORK( I )*WORK( N+I )
  110          CONTINUE
            ELSE
*
*              Multiply by inv(op(A))*diag(W).
*
               DO 120 I = 1, N
                  WORK( N+I ) = WORK( I )*WORK( N+I )
  120          CONTINUE
               CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK( N+1 ),
     $                      N,
     $                      INFO )
            END IF
            GO TO 100
         END IF
*
*        Normalize error.
*
         LSTRES = ZERO
         DO 130 I = 1, N
            LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
  130    CONTINUE
         IF( LSTRES.NE.ZERO )
     $      FERR( J ) = FERR( J ) / LSTRES
*
  140 CONTINUE
*
      RETURN
*
*     End of DGERFS
*
      END
*> \brief \b DGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGERQ2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgerq2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgerq2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgerq2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGERQ2( M, N, A, LDA, TAU, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGERQ2 computes an RQ factorization of a real m by n matrix A:
*> A = R * Q.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the m by n matrix A.
*>          On exit, if m <= n, the upper triangle of the subarray
*>          A(1:m,n-m+1:n) contains the m by m upper triangular matrix R;
*>          if m >= n, the elements on and above the (m-n)-th subdiagonal
*>          contain the m by n upper trapezoidal matrix R; the remaining
*>          elements, with the array TAU, represent the orthogonal matrix
*>          Q as a product of elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (M)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gerq2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix Q is represented as a product of elementary reflectors
*>
*>     Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
*>  A(m-k+i,1:n-k+i-1), and tau in TAU(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGERQ2( M, N, A, LDA, TAU, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, K
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF1L, DLARFG, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGERQ2', -INFO )
         RETURN
      END IF
*
      K = MIN( M, N )
*
      DO 10 I = K, 1, -1
*
*        Generate elementary reflector H(i) to annihilate
*        A(m-k+i,1:n-k+i-1)
*
         CALL DLARFG( N-K+I, A( M-K+I, N-K+I ), A( M-K+I, 1 ), LDA,
     $                TAU( I ) )
*
*        Apply H(i) to A(1:m-k+i-1,1:n-k+i) from the right
*
         CALL DLARF1L( 'Right', M-K+I-1, N-K+I, A( M-K+I, 1 ), LDA,
     $               TAU( I ), A, LDA, WORK )
   10 CONTINUE
      RETURN
*
*     End of DGERQ2
*
      END
*> \brief \b DGERQF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGERQF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgerqf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgerqf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgerqf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGERQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGERQF computes an RQ factorization of a real M-by-N matrix A:
*> A = R * Q.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit,
*>          if m <= n, the upper triangle of the subarray
*>          A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R;
*>          if m >= n, the elements on and above the (m-n)-th subdiagonal
*>          contain the M-by-N upper trapezoidal matrix R;
*>          the remaining elements, with the array TAU, represent the
*>          orthogonal matrix Q as a product of min(m,n) elementary
*>          reflectors (see Further Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= M, otherwise.
*>          For optimum performance LWORK >= M*NB, where NB is
*>          the optimal blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gerqf
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix Q is represented as a product of elementary reflectors
*>
*>     Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
*>  A(m-k+i,1:n-k+i-1), and tau in TAU(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGERQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IB, IINFO, IWS, K, KI, KK, LDWORK, LWKOPT,
     $                   MU, NB, NBMIN, NU, NX
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGERQ2, DLARFB, DLARFT, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      END IF
*
      IF( INFO.EQ.0 ) THEN
         K = MIN( M, N )
         IF( K.EQ.0 ) THEN
            LWKOPT = 1
         ELSE
            NB = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )
            LWKOPT = M*NB
         END IF
         WORK( 1 ) = LWKOPT
*
         IF( .NOT.LQUERY ) THEN
            IF( LWORK.LE.0 .OR. ( N.GT.0 .AND. LWORK.LT.MAX( 1, M ) ) )
     $         INFO = -7
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGERQF', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( K.EQ.0 ) THEN
         RETURN
      END IF
*
      NBMIN = 2
      NX = 1
      IWS = M
      IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
*        Determine when to cross over from blocked to unblocked code.
*
         NX = MAX( 0, ILAENV( 3, 'DGERQF', ' ', M, N, -1, -1 ) )
         IF( NX.LT.K ) THEN
*
*           Determine if workspace is large enough for blocked code.
*
            LDWORK = M
            IWS = LDWORK*NB
            IF( LWORK.LT.IWS ) THEN
*
*              Not enough workspace to use optimal NB:  reduce NB and
*              determine the minimum value of NB.
*
               NB = LWORK / LDWORK
               NBMIN = MAX( 2, ILAENV( 2, 'DGERQF', ' ', M, N, -1,
     $                 -1 ) )
            END IF
         END IF
      END IF
*
      IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
*        Use blocked code initially.
*        The last kk rows are handled by the block method.
*
         KI = ( ( K-NX-1 ) / NB )*NB
         KK = MIN( K, KI+NB )
*
         DO 10 I = K - KK + KI + 1, K - KK + 1, -NB
            IB = MIN( K-I+1, NB )
*
*           Compute the RQ factorization of the current block
*           A(m-k+i:m-k+i+ib-1,1:n-k+i+ib-1)
*
            CALL DGERQ2( IB, N-K+I+IB-1, A( M-K+I, 1 ), LDA,
     $                   TAU( I ),
     $                   WORK, IINFO )
            IF( M-K+I.GT.1 ) THEN
*
*              Form the triangular factor of the block reflector
*              H = H(i+ib-1) . . . H(i+1) H(i)
*
               CALL DLARFT( 'Backward', 'Rowwise', N-K+I+IB-1, IB,
     $                      A( M-K+I, 1 ), LDA, TAU( I ), WORK, LDWORK )
*
*              Apply H to A(1:m-k+i-1,1:n-k+i+ib-1) from the right
*
               CALL DLARFB( 'Right', 'No transpose', 'Backward',
     $                      'Rowwise', M-K+I-1, N-K+I+IB-1, IB,
     $                      A( M-K+I, 1 ), LDA, WORK, LDWORK, A, LDA,
     $                      WORK( IB+1 ), LDWORK )
            END IF
   10    CONTINUE
         MU = M - K + I + NB - 1
         NU = N - K + I + NB - 1
      ELSE
         MU = M
         NU = N
      END IF
*
*     Use unblocked code to factor the last or only block
*
      IF( MU.GT.0 .AND. NU.GT.0 )
     $   CALL DGERQ2( MU, NU, A, LDA, TAU, WORK, IINFO )
*
      WORK( 1 ) = IWS
      RETURN
*
*     End of DGERQF
*
      END
*> \brief \b DGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGESC2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgesc2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgesc2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgesc2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGESC2( N, A, LDA, RHS, IPIV, JPIV, SCALE )
*
*       .. Scalar Arguments ..
*       INTEGER            LDA, N
*       DOUBLE PRECISION   SCALE
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), JPIV( * )
*       DOUBLE PRECISION   A( LDA, * ), RHS( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGESC2 solves a system of linear equations
*>
*>           A * X = scale* RHS
*>
*> with a general N-by-N matrix A using the LU factorization with
*> complete pivoting computed by DGETC2.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the  LU part of the factorization of the n-by-n
*>          matrix A computed by DGETC2:  A = P * L * U * Q
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1, N).
*> \endverbatim
*>
*> \param[in,out] RHS
*> \verbatim
*>          RHS is DOUBLE PRECISION array, dimension (N).
*>          On entry, the right hand side vector b.
*>          On exit, the solution vector X.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N).
*>          The pivot indices; for 1 <= i <= N, row i of the
*>          matrix has been interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in] JPIV
*> \verbatim
*>          JPIV is INTEGER array, dimension (N).
*>          The pivot indices; for 1 <= j <= N, column j of the
*>          matrix has been interchanged with column JPIV(j).
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*>          SCALE is DOUBLE PRECISION
*>          On exit, SCALE contains the scale factor. SCALE is chosen
*>          0 <= SCALE <= 1 to prevent overflow in the solution.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gesc2
*
*> \par Contributors:
*  ==================
*>
*>     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*>     Umea University, S-901 87 Umea, Sweden.
*
*  =====================================================================
      SUBROUTINE DGESC2( N, A, LDA, RHS, IPIV, JPIV, SCALE )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            LDA, N
      DOUBLE PRECISION   SCALE
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), JPIV( * )
      DOUBLE PRECISION   A( LDA, * ), RHS( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, TWO
      PARAMETER          ( ONE = 1.0D+0, TWO = 2.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J
      DOUBLE PRECISION   BIGNUM, EPS, SMLNUM, TEMP
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASWP, DSCAL
*     ..
*     .. External Functions ..
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           IDAMAX, DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS
*     ..
*     .. Executable Statements ..
*
*      Set constant to control overflow
*
      EPS = DLAMCH( 'P' )
      SMLNUM = DLAMCH( 'S' ) / EPS
      BIGNUM = ONE / SMLNUM
*
*     Apply permutations IPIV to RHS
*
      CALL DLASWP( 1, RHS, LDA, 1, N-1, IPIV, 1 )
*
*     Solve for L part
*
      DO 20 I = 1, N - 1
         DO 10 J = I + 1, N
            RHS( J ) = RHS( J ) - A( J, I )*RHS( I )
   10    CONTINUE
   20 CONTINUE
*
*     Solve for U part
*
      SCALE = ONE
*
*     Check for scaling
*
      I = IDAMAX( N, RHS, 1 )
      IF( TWO*SMLNUM*ABS( RHS( I ) ).GT.ABS( A( N, N ) ) ) THEN
         TEMP = ( ONE / TWO ) / ABS( RHS( I ) )
         CALL DSCAL( N, TEMP, RHS( 1 ), 1 )
         SCALE = SCALE*TEMP
      END IF
*
      DO 40 I = N, 1, -1
         TEMP = ONE / A( I, I )
         RHS( I ) = RHS( I )*TEMP
         DO 30 J = I + 1, N
            RHS( I ) = RHS( I ) - RHS( J )*( A( I, J )*TEMP )
   30    CONTINUE
   40 CONTINUE
*
*     Apply permutations JPIV to the solution (RHS)
*
      CALL DLASWP( 1, RHS, LDA, 1, N-1, JPIV, -1 )
      RETURN
*
*     End of DGESC2
*
      END
*> \brief \b DGESDD
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGESDD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgesdd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgesdd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgesdd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT,
*                          WORK, LWORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ
*       INTEGER            INFO, LDA, LDU, LDVT, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), S( * ), U( LDU, * ),
*      $                   VT( LDVT, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGESDD computes the singular value decomposition (SVD) of a real
*> M-by-N matrix A, optionally computing the left and right singular
*> vectors.  If singular vectors are desired, it uses a
*> divide-and-conquer algorithm.
*>
*> The SVD is written
*>
*>      A = U * SIGMA * transpose(V)
*>
*> where SIGMA is an M-by-N matrix which is zero except for its
*> min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
*> V is an N-by-N orthogonal matrix.  The diagonal elements of SIGMA
*> are the singular values of A; they are real and non-negative, and
*> are returned in descending order.  The first min(m,n) columns of
*> U and V are the left and right singular vectors of A.
*>
*> Note that the routine returns VT = V**T, not V.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          Specifies options for computing all or part of the matrix U:
*>          = 'A':  all M columns of U and all N rows of V**T are
*>                  returned in the arrays U and VT;
*>          = 'S':  the first min(M,N) columns of U and the first
*>                  min(M,N) rows of V**T are returned in the arrays U
*>                  and VT;
*>          = 'O':  If M >= N, the first N columns of U are overwritten
*>                  on the array A and all rows of V**T are returned in
*>                  the array VT;
*>                  otherwise, all columns of U are returned in the
*>                  array U and the first M rows of V**T are overwritten
*>                  in the array A;
*>          = 'N':  no columns of U or rows of V**T are computed.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the input matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the input matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit,
*>          if JOBZ = 'O',  A is overwritten with the first N columns
*>                          of U (the left singular vectors, stored
*>                          columnwise) if M >= N;
*>                          A is overwritten with the first M rows
*>                          of V**T (the right singular vectors, stored
*>                          rowwise) otherwise.
*>          if JOBZ .ne. 'O', the contents of A are destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (min(M,N))
*>          The singular values of A, sorted so that S(i) >= S(i+1).
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*>          U is DOUBLE PRECISION array, dimension (LDU,UCOL)
*>          UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N;
*>          UCOL = min(M,N) if JOBZ = 'S'.
*>          If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M
*>          orthogonal matrix U;
*>          if JOBZ = 'S', U contains the first min(M,N) columns of U
*>          (the left singular vectors, stored columnwise);
*>          if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>          The leading dimension of the array U.  LDU >= 1; if
*>          JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.
*> \endverbatim
*>
*> \param[out] VT
*> \verbatim
*>          VT is DOUBLE PRECISION array, dimension (LDVT,N)
*>          If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
*>          N-by-N orthogonal matrix V**T;
*>          if JOBZ = 'S', VT contains the first min(M,N) rows of
*>          V**T (the right singular vectors, stored rowwise);
*>          if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*>          LDVT is INTEGER
*>          The leading dimension of the array VT.  LDVT >= 1;
*>          if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
*>          if JOBZ = 'S', LDVT >= min(M,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK;
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= 1.
*>          If LWORK = -1, a workspace query is assumed.  The optimal
*>          size for the WORK array is calculated and stored in WORK(1),
*>          and no other work except argument checking is performed.
*>
*>          Let mx = max(M,N) and mn = min(M,N).
*>          If JOBZ = 'N', LWORK >= 3*mn + max( mx, 7*mn ).
*>          If JOBZ = 'O', LWORK >= 3*mn + max( mx, 5*mn*mn + 4*mn ).
*>          If JOBZ = 'S', LWORK >= 4*mn*mn + 7*mn.
*>          If JOBZ = 'A', LWORK >= 4*mn*mn + 6*mn + mx.
*>          These are not tight minimums in all cases; see comments inside code.
*>          For good performance, LWORK should generally be larger;
*>          a query is recommended.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (8*min(M,N))
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          <  0:  if INFO = -i, the i-th argument had an illegal value.
*>          = -4:  if A had a NAN entry.
*>          >  0:  DBDSDC did not converge, updating process failed.
*>          =  0:  successful exit.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gesdd
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Huan Ren, Computer Science Division, University of
*>     California at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT,
     $                   WORK, LWORK, IWORK, INFO )
      implicit none
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ
      INTEGER            INFO, LDA, LDU, LDVT, LWORK, M, N
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), S( * ), U( LDU, * ),
     $                   VT( LDVT, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, WNTQA, WNTQAS, WNTQN, WNTQO, WNTQS
      INTEGER            BDSPAC, BLK, CHUNK, I, IE, IERR, IL,
     $                   IR, ISCL, ITAU, ITAUP, ITAUQ, IU, IVT, LDWKVT,
     $                   LDWRKL, LDWRKR, LDWRKU, MAXWRK, MINMN, MINWRK,
     $                   MNTHR, NWORK, WRKBL
      INTEGER            LWORK_DGEBRD_MN, LWORK_DGEBRD_MM,
     $                   LWORK_DGEBRD_NN, LWORK_DGELQF_MN,
     $                   LWORK_DGEQRF_MN,
     $                   LWORK_DORGBR_P_MM, LWORK_DORGBR_Q_NN,
     $                   LWORK_DORGLQ_MN, LWORK_DORGLQ_NN,
     $                   LWORK_DORGQR_MM, LWORK_DORGQR_MN,
     $                   LWORK_DORMBR_PRT_MM, LWORK_DORMBR_QLN_MM,
     $                   LWORK_DORMBR_PRT_MN, LWORK_DORMBR_QLN_MN,
     $                   LWORK_DORMBR_PRT_NN, LWORK_DORMBR_QLN_NN
      DOUBLE PRECISION   ANRM, BIGNUM, EPS, SMLNUM
*     ..
*     .. Local Arrays ..
      INTEGER            IDUM( 1 )
      DOUBLE PRECISION   DUM( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DBDSDC, DGEBRD, DGELQF, DGEMM, DGEQRF,
     $                   DLACPY,
     $                   DLASCL, DLASET, DORGBR, DORGLQ, DORGQR, DORMBR,
     $                   XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      DOUBLE PRECISION   DLAMCH, DLANGE, DROUNDUP_LWORK
      EXTERNAL           DLAMCH, DLANGE, LSAME, DISNAN, 
     $                   DROUNDUP_LWORK
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          INT, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO   = 0
      MINMN  = MIN( M, N )
      WNTQA  = LSAME( JOBZ, 'A' )
      WNTQS  = LSAME( JOBZ, 'S' )
      WNTQAS = WNTQA .OR. WNTQS
      WNTQO  = LSAME( JOBZ, 'O' )
      WNTQN  = LSAME( JOBZ, 'N' )
      LQUERY = ( LWORK.EQ.-1 )
*
      IF( .NOT.( WNTQA .OR. WNTQS .OR. WNTQO .OR. WNTQN ) ) THEN
         INFO = -1
      ELSE IF( M.LT.0 ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -5
      ELSE IF( LDU.LT.1 .OR. ( WNTQAS .AND. LDU.LT.M ) .OR.
     $         ( WNTQO .AND. M.LT.N .AND. LDU.LT.M ) ) THEN
         INFO = -8
      ELSE IF( LDVT.LT.1 .OR. ( WNTQA .AND. LDVT.LT.N ) .OR.
     $         ( WNTQS .AND. LDVT.LT.MINMN ) .OR.
     $         ( WNTQO .AND. M.GE.N .AND. LDVT.LT.N ) ) THEN
         INFO = -10
      END IF
*
*     Compute workspace
*       Note: Comments in the code beginning "Workspace:" describe the
*       minimal amount of workspace allocated at that point in the code,
*       as well as the preferred amount for good performance.
*       NB refers to the optimal block size for the immediately
*       following subroutine, as returned by ILAENV.
*
      IF( INFO.EQ.0 ) THEN
         MINWRK = 1
         MAXWRK = 1
         BDSPAC = 0
         MNTHR  = INT( MINMN*11.0D0 / 6.0D0 )
         IF( M.GE.N .AND. MINMN.GT.0 ) THEN
*
*           Compute space needed for DBDSDC
*
            IF( WNTQN ) THEN
*              dbdsdc needs only 4*N (or 6*N for uplo=L for LAPACK <= 3.6)
*              keep 7*N for backwards compatibility.
               BDSPAC = 7*N
            ELSE
               BDSPAC = 3*N*N + 4*N
            END IF
*
*           Compute space preferred for each routine
            CALL DGEBRD( M, N, DUM(1), M, DUM(1), DUM(1), DUM(1),
     $                   DUM(1), DUM(1), -1, IERR )
            LWORK_DGEBRD_MN = INT( DUM(1) )
*
            CALL DGEBRD( N, N, DUM(1), N, DUM(1), DUM(1), DUM(1),
     $                   DUM(1), DUM(1), -1, IERR )
            LWORK_DGEBRD_NN = INT( DUM(1) )
*
            CALL DGEQRF( M, N, DUM(1), M, DUM(1), DUM(1), -1, IERR )
            LWORK_DGEQRF_MN = INT( DUM(1) )
*
            CALL DORGBR( 'Q', N, N, N, DUM(1), N, DUM(1), DUM(1), -1,
     $                   IERR )
            LWORK_DORGBR_Q_NN = INT( DUM(1) )
*
            CALL DORGQR( M, M, N, DUM(1), M, DUM(1), DUM(1), -1,
     $                   IERR )
            LWORK_DORGQR_MM = INT( DUM(1) )
*
            CALL DORGQR( M, N, N, DUM(1), M, DUM(1), DUM(1), -1,
     $                   IERR )
            LWORK_DORGQR_MN = INT( DUM(1) )
*
            CALL DORMBR( 'P', 'R', 'T', N, N, N, DUM(1), N,
     $                   DUM(1), DUM(1), N, DUM(1), -1, IERR )
            LWORK_DORMBR_PRT_NN = INT( DUM(1) )
*
            CALL DORMBR( 'Q', 'L', 'N', N, N, N, DUM(1), N,
     $                   DUM(1), DUM(1), N, DUM(1), -1, IERR )
            LWORK_DORMBR_QLN_NN = INT( DUM(1) )
*
            CALL DORMBR( 'Q', 'L', 'N', M, N, N, DUM(1), M,
     $                   DUM(1), DUM(1), M, DUM(1), -1, IERR )
            LWORK_DORMBR_QLN_MN = INT( DUM(1) )
*
            CALL DORMBR( 'Q', 'L', 'N', M, M, N, DUM(1), M,
     $                   DUM(1), DUM(1), M, DUM(1), -1, IERR )
            LWORK_DORMBR_QLN_MM = INT( DUM(1) )
*
            IF( M.GE.MNTHR ) THEN
               IF( WNTQN ) THEN
*
*                 Path 1 (M >> N, JOBZ='N')
*
                  WRKBL = N + LWORK_DGEQRF_MN
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
                  MAXWRK = MAX( WRKBL, BDSPAC + N )
                  MINWRK = BDSPAC + N
               ELSE IF( WNTQO ) THEN
*
*                 Path 2 (M >> N, JOBZ='O')
*
                  WRKBL = N + LWORK_DGEQRF_MN
                  WRKBL = MAX( WRKBL,   N + LWORK_DORGQR_MN )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_NN )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
                  WRKBL = MAX( WRKBL, 3*N + BDSPAC )
                  MAXWRK = WRKBL + 2*N*N
                  MINWRK = BDSPAC + 2*N*N + 3*N
               ELSE IF( WNTQS ) THEN
*
*                 Path 3 (M >> N, JOBZ='S')
*
                  WRKBL = N + LWORK_DGEQRF_MN
                  WRKBL = MAX( WRKBL,   N + LWORK_DORGQR_MN )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_NN )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
                  WRKBL = MAX( WRKBL, 3*N + BDSPAC )
                  MAXWRK = WRKBL + N*N
                  MINWRK = BDSPAC + N*N + 3*N
               ELSE IF( WNTQA ) THEN
*
*                 Path 4 (M >> N, JOBZ='A')
*
                  WRKBL = N + LWORK_DGEQRF_MN
                  WRKBL = MAX( WRKBL,   N + LWORK_DORGQR_MM )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_NN )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
                  WRKBL = MAX( WRKBL, 3*N + BDSPAC )
                  MAXWRK = WRKBL + N*N
                  MINWRK = N*N + MAX( 3*N + BDSPAC, N + M )
               END IF
            ELSE
*
*              Path 5 (M >= N, but not much larger)
*
               WRKBL = 3*N + LWORK_DGEBRD_MN
               IF( WNTQN ) THEN
*                 Path 5n (M >= N, jobz='N')
                  MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
                  MINWRK = 3*N + MAX( M, BDSPAC )
               ELSE IF( WNTQO ) THEN
*                 Path 5o (M >= N, jobz='O')
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_MN )
                  WRKBL = MAX( WRKBL, 3*N + BDSPAC )
                  MAXWRK = WRKBL + M*N
                  MINWRK = 3*N + MAX( M, N*N + BDSPAC )
               ELSE IF( WNTQS ) THEN
*                 Path 5s (M >= N, jobz='S')
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_MN )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
                  MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
                  MINWRK = 3*N + MAX( M, BDSPAC )
               ELSE IF( WNTQA ) THEN
*                 Path 5a (M >= N, jobz='A')
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_MM )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
                  MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
                  MINWRK = 3*N + MAX( M, BDSPAC )
               END IF
            END IF
         ELSE IF( MINMN.GT.0 ) THEN
*
*           Compute space needed for DBDSDC
*
            IF( WNTQN ) THEN
*              dbdsdc needs only 4*N (or 6*N for uplo=L for LAPACK <= 3.6)
*              keep 7*N for backwards compatibility.
               BDSPAC = 7*M
            ELSE
               BDSPAC = 3*M*M + 4*M
            END IF
*
*           Compute space preferred for each routine
            CALL DGEBRD( M, N, DUM(1), M, DUM(1), DUM(1), DUM(1),
     $                   DUM(1), DUM(1), -1, IERR )
            LWORK_DGEBRD_MN = INT( DUM(1) )
*
            CALL DGEBRD( M, M, A, M, S, DUM(1), DUM(1),
     $                   DUM(1), DUM(1), -1, IERR )
            LWORK_DGEBRD_MM = INT( DUM(1) )
*
            CALL DGELQF( M, N, A, M, DUM(1), DUM(1), -1, IERR )
            LWORK_DGELQF_MN = INT( DUM(1) )
*
            CALL DORGLQ( N, N, M, DUM(1), N, DUM(1), DUM(1), -1,
     $                   IERR )
            LWORK_DORGLQ_NN = INT( DUM(1) )
*
            CALL DORGLQ( M, N, M, A, M, DUM(1), DUM(1), -1, IERR )
            LWORK_DORGLQ_MN = INT( DUM(1) )
*
            CALL DORGBR( 'P', M, M, M, A, N, DUM(1), DUM(1), -1,
     $                   IERR )
            LWORK_DORGBR_P_MM = INT( DUM(1) )
*
            CALL DORMBR( 'P', 'R', 'T', M, M, M, DUM(1), M,
     $                   DUM(1), DUM(1), M, DUM(1), -1, IERR )
            LWORK_DORMBR_PRT_MM = INT( DUM(1) )
*
            CALL DORMBR( 'P', 'R', 'T', M, N, M, DUM(1), M,
     $                   DUM(1), DUM(1), M, DUM(1), -1, IERR )
            LWORK_DORMBR_PRT_MN = INT( DUM(1) )
*
            CALL DORMBR( 'P', 'R', 'T', N, N, M, DUM(1), N,
     $                   DUM(1), DUM(1), N, DUM(1), -1, IERR )
            LWORK_DORMBR_PRT_NN = INT( DUM(1) )
*
            CALL DORMBR( 'Q', 'L', 'N', M, M, M, DUM(1), M,
     $                   DUM(1), DUM(1), M, DUM(1), -1, IERR )
            LWORK_DORMBR_QLN_MM = INT( DUM(1) )
*
            IF( N.GE.MNTHR ) THEN
               IF( WNTQN ) THEN
*
*                 Path 1t (N >> M, JOBZ='N')
*
                  WRKBL = M + LWORK_DGELQF_MN
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
                  MAXWRK = MAX( WRKBL, BDSPAC + M )
                  MINWRK = BDSPAC + M
               ELSE IF( WNTQO ) THEN
*
*                 Path 2t (N >> M, JOBZ='O')
*
                  WRKBL = M + LWORK_DGELQF_MN
                  WRKBL = MAX( WRKBL,   M + LWORK_DORGLQ_MN )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MM )
                  WRKBL = MAX( WRKBL, 3*M + BDSPAC )
                  MAXWRK = WRKBL + 2*M*M
                  MINWRK = BDSPAC + 2*M*M + 3*M
               ELSE IF( WNTQS ) THEN
*
*                 Path 3t (N >> M, JOBZ='S')
*
                  WRKBL = M + LWORK_DGELQF_MN
                  WRKBL = MAX( WRKBL,   M + LWORK_DORGLQ_MN )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MM )
                  WRKBL = MAX( WRKBL, 3*M + BDSPAC )
                  MAXWRK = WRKBL + M*M
                  MINWRK = BDSPAC + M*M + 3*M
               ELSE IF( WNTQA ) THEN
*
*                 Path 4t (N >> M, JOBZ='A')
*
                  WRKBL = M + LWORK_DGELQF_MN
                  WRKBL = MAX( WRKBL,   M + LWORK_DORGLQ_NN )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MM )
                  WRKBL = MAX( WRKBL, 3*M + BDSPAC )
                  MAXWRK = WRKBL + M*M
                  MINWRK = M*M + MAX( 3*M + BDSPAC, M + N )
               END IF
            ELSE
*
*              Path 5t (N > M, but not much larger)
*
               WRKBL = 3*M + LWORK_DGEBRD_MN
               IF( WNTQN ) THEN
*                 Path 5tn (N > M, jobz='N')
                  MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
                  MINWRK = 3*M + MAX( N, BDSPAC )
               ELSE IF( WNTQO ) THEN
*                 Path 5to (N > M, jobz='O')
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MN )
                  WRKBL = MAX( WRKBL, 3*M + BDSPAC )
                  MAXWRK = WRKBL + M*N
                  MINWRK = 3*M + MAX( N, M*M + BDSPAC )
               ELSE IF( WNTQS ) THEN
*                 Path 5ts (N > M, jobz='S')
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MN )
                  MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
                  MINWRK = 3*M + MAX( N, BDSPAC )
               ELSE IF( WNTQA ) THEN
*                 Path 5ta (N > M, jobz='A')
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_NN )
                  MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
                  MINWRK = 3*M + MAX( N, BDSPAC )
               END IF
            END IF
         END IF

         MAXWRK = MAX( MAXWRK, MINWRK )
         WORK( 1 ) = DROUNDUP_LWORK( MAXWRK )
*
         IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
            INFO = -12
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGESDD', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 ) THEN
         RETURN
      END IF
*
*     Get machine constants
*
      EPS = DLAMCH( 'P' )
      SMLNUM = SQRT( DLAMCH( 'S' ) ) / EPS
      BIGNUM = ONE / SMLNUM
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      ANRM = DLANGE( 'M', M, N, A, LDA, DUM )
      IF( DISNAN( ANRM ) ) THEN
          INFO = -4
          RETURN
      END IF
      ISCL = 0
      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
         ISCL = 1
         CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, IERR )
      ELSE IF( ANRM.GT.BIGNUM ) THEN
         ISCL = 1
         CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, IERR )
      END IF
*
      IF( M.GE.N ) THEN
*
*        A has at least as many rows as columns. If A has sufficiently
*        more rows than columns, first reduce using the QR
*        decomposition (if sufficient workspace available)
*
         IF( M.GE.MNTHR ) THEN
*
            IF( WNTQN ) THEN
*
*              Path 1 (M >> N, JOBZ='N')
*              No singular vectors to be computed
*
               ITAU = 1
               NWORK = ITAU + N
*
*              Compute A=Q*R
*              Workspace: need   N [tau] + N    [work]
*              Workspace: prefer N [tau] + N*NB [work]
*
               CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
     $                      WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Zero out below R
*
               CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ),
     $                      LDA )
               IE = 1
               ITAUQ = IE + N
               ITAUP = ITAUQ + N
               NWORK = ITAUP + N
*
*              Bidiagonalize R in A
*              Workspace: need   3*N [e, tauq, taup] + N      [work]
*              Workspace: prefer 3*N [e, tauq, taup] + 2*N*NB [work]
*
               CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
     $                      WORK( ITAUQ ),
     $                      WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
     $                      IERR )
               NWORK = IE + N
*
*              Perform bidiagonal SVD, computing singular values only
*              Workspace: need   N [e] + BDSPAC
*
               CALL DBDSDC( 'U', 'N', N, S, WORK( IE ), DUM, 1, DUM,
     $                      1,
     $                      DUM, IDUM, WORK( NWORK ), IWORK, INFO )
*
            ELSE IF( WNTQO ) THEN
*
*              Path 2 (M >> N, JOBZ = 'O')
*              N left singular vectors to be overwritten on A and
*              N right singular vectors to be computed in VT
*
               IR = 1
*
*              WORK(IR) is LDWRKR by N
*
               IF( LWORK .GE. LDA*N + N*N + 3*N + BDSPAC ) THEN
                  LDWRKR = LDA
               ELSE
                  LDWRKR = ( LWORK - N*N - 3*N - BDSPAC ) / N
               END IF
               ITAU = IR + LDWRKR*N
               NWORK = ITAU + N
*
*              Compute A=Q*R
*              Workspace: need   N*N [R] + N [tau] + N    [work]
*              Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
               CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
     $                      WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Copy R to WORK(IR), zeroing out below it
*
               CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR )
               CALL DLASET( 'L', N - 1, N - 1, ZERO, ZERO,
     $                      WORK(IR+1),
     $                      LDWRKR )
*
*              Generate Q in A
*              Workspace: need   N*N [R] + N [tau] + N    [work]
*              Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
               CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
               IE = ITAU
               ITAUQ = IE + N
               ITAUP = ITAUQ + N
               NWORK = ITAUP + N
*
*              Bidiagonalize R in WORK(IR)
*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + N      [work]
*              Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + 2*N*NB [work]
*
               CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ),
     $                      WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              WORK(IU) is N by N
*
               IU = NWORK
               NWORK = IU + N*N
*
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in WORK(IU) and computing right
*              singular vectors of bidiagonal matrix in VT
*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + N*N [U] + BDSPAC
*
               CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ),
     $                      N,
     $                      VT, LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
     $                      INFO )
*
*              Overwrite WORK(IU) by left singular vectors of R
*              and VT by right singular vectors of R
*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + N*N [U] + N    [work]
*              Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ),
     $                      LDWRKR,
     $                      WORK( ITAUQ ), WORK( IU ), N, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
               CALL DORMBR( 'P', 'R', 'T', N, N, N, WORK( IR ),
     $                      LDWRKR,
     $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Multiply Q in A by left singular vectors of R in
*              WORK(IU), storing result in WORK(IR) and copying to A
*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + N*N [U]
*              Workspace: prefer M*N [R] + 3*N [e, tauq, taup] + N*N [U]
*
               DO 10 I = 1, M, LDWRKR
                  CHUNK = MIN( M - I + 1, LDWRKR )
                  CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ),
     $                        LDA, WORK( IU ), N, ZERO, WORK( IR ),
     $                        LDWRKR )
                  CALL DLACPY( 'F', CHUNK, N, WORK( IR ), LDWRKR,
     $                         A( I, 1 ), LDA )
   10          CONTINUE
*
            ELSE IF( WNTQS ) THEN
*
*              Path 3 (M >> N, JOBZ='S')
*              N left singular vectors to be computed in U and
*              N right singular vectors to be computed in VT
*
               IR = 1
*
*              WORK(IR) is N by N
*
               LDWRKR = N
               ITAU = IR + LDWRKR*N
               NWORK = ITAU + N
*
*              Compute A=Q*R
*              Workspace: need   N*N [R] + N [tau] + N    [work]
*              Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
               CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
     $                      WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Copy R to WORK(IR), zeroing out below it
*
               CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR )
               CALL DLASET( 'L', N - 1, N - 1, ZERO, ZERO,
     $                      WORK(IR+1),
     $                      LDWRKR )
*
*              Generate Q in A
*              Workspace: need   N*N [R] + N [tau] + N    [work]
*              Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
               CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
               IE = ITAU
               ITAUQ = IE + N
               ITAUP = ITAUQ + N
               NWORK = ITAUP + N
*
*              Bidiagonalize R in WORK(IR)
*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + N      [work]
*              Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + 2*N*NB [work]
*
               CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ),
     $                      WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagoal matrix in U and computing right singular
*              vectors of bidiagonal matrix in VT
*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + BDSPAC
*
               CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
     $                      LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
     $                      INFO )
*
*              Overwrite U by left singular vectors of R and VT
*              by right singular vectors of R
*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + N    [work]
*              Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + N*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ),
     $                      LDWRKR,
     $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
               CALL DORMBR( 'P', 'R', 'T', N, N, N, WORK( IR ),
     $                      LDWRKR,
     $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Multiply Q in A by left singular vectors of R in
*              WORK(IR), storing result in U
*              Workspace: need   N*N [R]
*
               CALL DLACPY( 'F', N, N, U, LDU, WORK( IR ), LDWRKR )
               CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA,
     $                     WORK( IR ),
     $                     LDWRKR, ZERO, U, LDU )
*
            ELSE IF( WNTQA ) THEN
*
*              Path 4 (M >> N, JOBZ='A')
*              M left singular vectors to be computed in U and
*              N right singular vectors to be computed in VT
*
               IU = 1
*
*              WORK(IU) is N by N
*
               LDWRKU = N
               ITAU = IU + LDWRKU*N
               NWORK = ITAU + N
*
*              Compute A=Q*R, copying result to U
*              Workspace: need   N*N [U] + N [tau] + N    [work]
*              Workspace: prefer N*N [U] + N [tau] + N*NB [work]
*
               CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
     $                      WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
               CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
*              Generate Q in U
*              Workspace: need   N*N [U] + N [tau] + M    [work]
*              Workspace: prefer N*N [U] + N [tau] + M*NB [work]
               CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
*              Produce R in A, zeroing out other entries
*
               CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ),
     $                      LDA )
               IE = ITAU
               ITAUQ = IE + N
               ITAUP = ITAUQ + N
               NWORK = ITAUP + N
*
*              Bidiagonalize R in A
*              Workspace: need   N*N [U] + 3*N [e, tauq, taup] + N      [work]
*              Workspace: prefer N*N [U] + 3*N [e, tauq, taup] + 2*N*NB [work]
*
               CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
     $                      WORK( ITAUQ ),
     $                      WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
     $                      IERR )
*
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in WORK(IU) and computing right
*              singular vectors of bidiagonal matrix in VT
*              Workspace: need   N*N [U] + 3*N [e, tauq, taup] + BDSPAC
*
               CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ),
     $                      N,
     $                      VT, LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
     $                      INFO )
*
*              Overwrite WORK(IU) by left singular vectors of R and VT
*              by right singular vectors of R
*              Workspace: need   N*N [U] + 3*N [e, tauq, taup] + N    [work]
*              Workspace: prefer N*N [U] + 3*N [e, tauq, taup] + N*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', N, N, N, A, LDA,
     $                      WORK( ITAUQ ), WORK( IU ), LDWRKU,
     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
               CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
     $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Multiply Q in U by left singular vectors of R in
*              WORK(IU), storing result in A
*              Workspace: need   N*N [U]
*
               CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU,
     $                     WORK( IU ),
     $                     LDWRKU, ZERO, A, LDA )
*
*              Copy left singular vectors of A from A to U
*
               CALL DLACPY( 'F', M, N, A, LDA, U, LDU )
*
            END IF
*
         ELSE
*
*           M .LT. MNTHR
*
*           Path 5 (M >= N, but not much larger)
*           Reduce to bidiagonal form without QR decomposition
*
            IE = 1
            ITAUQ = IE + N
            ITAUP = ITAUQ + N
            NWORK = ITAUP + N
*
*           Bidiagonalize A
*           Workspace: need   3*N [e, tauq, taup] + M        [work]
*           Workspace: prefer 3*N [e, tauq, taup] + (M+N)*NB [work]
*
            CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
     $                   WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
     $                   IERR )
            IF( WNTQN ) THEN
*
*              Path 5n (M >= N, JOBZ='N')
*              Perform bidiagonal SVD, only computing singular values
*              Workspace: need   3*N [e, tauq, taup] + BDSPAC
*
               CALL DBDSDC( 'U', 'N', N, S, WORK( IE ), DUM, 1, DUM,
     $                      1,
     $                      DUM, IDUM, WORK( NWORK ), IWORK, INFO )
            ELSE IF( WNTQO ) THEN
*              Path 5o (M >= N, JOBZ='O')
               IU = NWORK
               IF( LWORK .GE. M*N + 3*N + BDSPAC ) THEN
*
*                 WORK( IU ) is M by N
*
                  LDWRKU = M
                  NWORK = IU + LDWRKU*N
                  CALL DLASET( 'F', M, N, ZERO, ZERO, WORK( IU ),
     $                         LDWRKU )
*                 IR is unused; silence compile warnings
                  IR = -1
               ELSE
*
*                 WORK( IU ) is N by N
*
                  LDWRKU = N
                  NWORK = IU + LDWRKU*N
*
*                 WORK(IR) is LDWRKR by N
*
                  IR = NWORK
                  LDWRKR = ( LWORK - N*N - 3*N ) / N
               END IF
               NWORK = IU + LDWRKU*N
*
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in WORK(IU) and computing right
*              singular vectors of bidiagonal matrix in VT
*              Workspace: need   3*N [e, tauq, taup] + N*N [U] + BDSPAC
*
               CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ),
     $                      LDWRKU, VT, LDVT, DUM, IDUM, WORK( NWORK ),
     $                      IWORK, INFO )
*
*              Overwrite VT by right singular vectors of A
*              Workspace: need   3*N [e, tauq, taup] + N*N [U] + N    [work]
*              Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
*
               CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
     $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
               IF( LWORK .GE. M*N + 3*N + BDSPAC ) THEN
*
*                 Path 5o-fast
*                 Overwrite WORK(IU) by left singular vectors of A
*                 Workspace: need   3*N [e, tauq, taup] + M*N [U] + N    [work]
*                 Workspace: prefer 3*N [e, tauq, taup] + M*N [U] + N*NB [work]
*
                  CALL DORMBR( 'Q', 'L', 'N', M, N, N, A, LDA,
     $                         WORK( ITAUQ ), WORK( IU ), LDWRKU,
     $                         WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
*                 Copy left singular vectors of A from WORK(IU) to A
*
                  CALL DLACPY( 'F', M, N, WORK( IU ), LDWRKU, A,
     $                         LDA )
               ELSE
*
*                 Path 5o-slow
*                 Generate Q in A
*                 Workspace: need   3*N [e, tauq, taup] + N*N [U] + N    [work]
*                 Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
*
                  CALL DORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
     $                         WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
*                 Multiply Q in A by left singular vectors of
*                 bidiagonal matrix in WORK(IU), storing result in
*                 WORK(IR) and copying to A
*                 Workspace: need   3*N [e, tauq, taup] + N*N [U] + NB*N [R]
*                 Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + M*N  [R]
*
                  DO 20 I = 1, M, LDWRKR
                     CHUNK = MIN( M - I + 1, LDWRKR )
                     CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I,
     $                           1 ),
     $                           LDA, WORK( IU ), LDWRKU, ZERO,
     $                           WORK( IR ), LDWRKR )
                     CALL DLACPY( 'F', CHUNK, N, WORK( IR ), LDWRKR,
     $                            A( I, 1 ), LDA )
   20             CONTINUE
               END IF
*
            ELSE IF( WNTQS ) THEN
*
*              Path 5s (M >= N, JOBZ='S')
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in U and computing right singular
*              vectors of bidiagonal matrix in VT
*              Workspace: need   3*N [e, tauq, taup] + BDSPAC
*
               CALL DLASET( 'F', M, N, ZERO, ZERO, U, LDU )
               CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
     $                      LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
     $                      INFO )
*
*              Overwrite U by left singular vectors of A and VT
*              by right singular vectors of A
*              Workspace: need   3*N [e, tauq, taup] + N    [work]
*              Workspace: prefer 3*N [e, tauq, taup] + N*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', M, N, N, A, LDA,
     $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
               CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
     $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
            ELSE IF( WNTQA ) THEN
*
*              Path 5a (M >= N, JOBZ='A')
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in U and computing right singular
*              vectors of bidiagonal matrix in VT
*              Workspace: need   3*N [e, tauq, taup] + BDSPAC
*
               CALL DLASET( 'F', M, M, ZERO, ZERO, U, LDU )
               CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
     $                      LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
     $                      INFO )
*
*              Set the right corner of U to identity matrix
*
               IF( M.GT.N ) THEN
                  CALL DLASET( 'F', M - N, M - N, ZERO, ONE, U(N+1,
     $                         N+1),
     $                         LDU )
               END IF
*
*              Overwrite U by left singular vectors of A and VT
*              by right singular vectors of A
*              Workspace: need   3*N [e, tauq, taup] + M    [work]
*              Workspace: prefer 3*N [e, tauq, taup] + M*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
     $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
               CALL DORMBR( 'P', 'R', 'T', N, N, M, A, LDA,
     $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
            END IF
*
         END IF
*
      ELSE
*
*        A has more columns than rows. If A has sufficiently more
*        columns than rows, first reduce using the LQ decomposition (if
*        sufficient workspace available)
*
         IF( N.GE.MNTHR ) THEN
*
            IF( WNTQN ) THEN
*
*              Path 1t (N >> M, JOBZ='N')
*              No singular vectors to be computed
*
               ITAU = 1
               NWORK = ITAU + M
*
*              Compute A=L*Q
*              Workspace: need   M [tau] + M [work]
*              Workspace: prefer M [tau] + M*NB [work]
*
               CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
     $                      WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Zero out above L
*
               CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ),
     $                      LDA )
               IE = 1
               ITAUQ = IE + M
               ITAUP = ITAUQ + M
               NWORK = ITAUP + M
*
*              Bidiagonalize L in A
*              Workspace: need   3*M [e, tauq, taup] + M      [work]
*              Workspace: prefer 3*M [e, tauq, taup] + 2*M*NB [work]
*
               CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
     $                      WORK( ITAUQ ),
     $                      WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
     $                      IERR )
               NWORK = IE + M
*
*              Perform bidiagonal SVD, computing singular values only
*              Workspace: need   M [e] + BDSPAC
*
               CALL DBDSDC( 'U', 'N', M, S, WORK( IE ), DUM, 1, DUM,
     $                      1,
     $                      DUM, IDUM, WORK( NWORK ), IWORK, INFO )
*
            ELSE IF( WNTQO ) THEN
*
*              Path 2t (N >> M, JOBZ='O')
*              M right singular vectors to be overwritten on A and
*              M left singular vectors to be computed in U
*
               IVT = 1
*
*              WORK(IVT) is M by M
*              WORK(IL)  is M by M; it is later resized to M by chunk for gemm
*
               IL = IVT + M*M
               IF( LWORK .GE. M*N + M*M + 3*M + BDSPAC ) THEN
                  LDWRKL = M
                  CHUNK = N
               ELSE
                  LDWRKL = M
                  CHUNK = ( LWORK - M*M ) / M
               END IF
               ITAU = IL + LDWRKL*M
               NWORK = ITAU + M
*
*              Compute A=L*Q
*              Workspace: need   M*M [VT] + M*M [L] + M [tau] + M    [work]
*              Workspace: prefer M*M [VT] + M*M [L] + M [tau] + M*NB [work]
*
               CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
     $                      WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Copy L to WORK(IL), zeroing about above it
*
               CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWRKL )
               CALL DLASET( 'U', M - 1, M - 1, ZERO, ZERO,
     $                      WORK( IL + LDWRKL ), LDWRKL )
*
*              Generate Q in A
*              Workspace: need   M*M [VT] + M*M [L] + M [tau] + M    [work]
*              Workspace: prefer M*M [VT] + M*M [L] + M [tau] + M*NB [work]
*
               CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
               IE = ITAU
               ITAUQ = IE + M
               ITAUP = ITAUQ + M
               NWORK = ITAUP + M
*
*              Bidiagonalize L in WORK(IL)
*              Workspace: need   M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M      [work]
*              Workspace: prefer M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + 2*M*NB [work]
*
               CALL DGEBRD( M, M, WORK( IL ), LDWRKL, S, WORK( IE ),
     $                      WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in U, and computing right singular
*              vectors of bidiagonal matrix in WORK(IVT)
*              Workspace: need   M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + BDSPAC
*
               CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU,
     $                      WORK( IVT ), M, DUM, IDUM, WORK( NWORK ),
     $                      IWORK, INFO )
*
*              Overwrite U by left singular vectors of L and WORK(IVT)
*              by right singular vectors of L
*              Workspace: need   M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M    [work]
*              Workspace: prefer M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ),
     $                      LDWRKL,
     $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
               CALL DORMBR( 'P', 'R', 'T', M, M, M, WORK( IL ),
     $                      LDWRKL,
     $                      WORK( ITAUP ), WORK( IVT ), M,
     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
*              Multiply right singular vectors of L in WORK(IVT) by Q
*              in A, storing result in WORK(IL) and copying to A
*              Workspace: need   M*M [VT] + M*M [L]
*              Workspace: prefer M*M [VT] + M*N [L]
*              At this point, L is resized as M by chunk.
*
               DO 30 I = 1, N, CHUNK
                  BLK = MIN( N - I + 1, CHUNK )
                  CALL DGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IVT ),
     $                        M,
     $                        A( 1, I ), LDA, ZERO, WORK( IL ), LDWRKL )
                  CALL DLACPY( 'F', M, BLK, WORK( IL ), LDWRKL,
     $                         A( 1, I ), LDA )
   30          CONTINUE
*
            ELSE IF( WNTQS ) THEN
*
*              Path 3t (N >> M, JOBZ='S')
*              M right singular vectors to be computed in VT and
*              M left singular vectors to be computed in U
*
               IL = 1
*
*              WORK(IL) is M by M
*
               LDWRKL = M
               ITAU = IL + LDWRKL*M
               NWORK = ITAU + M
*
*              Compute A=L*Q
*              Workspace: need   M*M [L] + M [tau] + M    [work]
*              Workspace: prefer M*M [L] + M [tau] + M*NB [work]
*
               CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
     $                      WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Copy L to WORK(IL), zeroing out above it
*
               CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWRKL )
               CALL DLASET( 'U', M - 1, M - 1, ZERO, ZERO,
     $                      WORK( IL + LDWRKL ), LDWRKL )
*
*              Generate Q in A
*              Workspace: need   M*M [L] + M [tau] + M    [work]
*              Workspace: prefer M*M [L] + M [tau] + M*NB [work]
*
               CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
               IE = ITAU
               ITAUQ = IE + M
               ITAUP = ITAUQ + M
               NWORK = ITAUP + M
*
*              Bidiagonalize L in WORK(IU).
*              Workspace: need   M*M [L] + 3*M [e, tauq, taup] + M      [work]
*              Workspace: prefer M*M [L] + 3*M [e, tauq, taup] + 2*M*NB [work]
*
               CALL DGEBRD( M, M, WORK( IL ), LDWRKL, S, WORK( IE ),
     $                      WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in U and computing right singular
*              vectors of bidiagonal matrix in VT
*              Workspace: need   M*M [L] + 3*M [e, tauq, taup] + BDSPAC
*
               CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU, VT,
     $                      LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
     $                      INFO )
*
*              Overwrite U by left singular vectors of L and VT
*              by right singular vectors of L
*              Workspace: need   M*M [L] + 3*M [e, tauq, taup] + M    [work]
*              Workspace: prefer M*M [L] + 3*M [e, tauq, taup] + M*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ),
     $                      LDWRKL,
     $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
               CALL DORMBR( 'P', 'R', 'T', M, M, M, WORK( IL ),
     $                      LDWRKL,
     $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Multiply right singular vectors of L in WORK(IL) by
*              Q in A, storing result in VT
*              Workspace: need   M*M [L]
*
               CALL DLACPY( 'F', M, M, VT, LDVT, WORK( IL ), LDWRKL )
               CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IL ),
     $                     LDWRKL,
     $                     A, LDA, ZERO, VT, LDVT )
*
            ELSE IF( WNTQA ) THEN
*
*              Path 4t (N >> M, JOBZ='A')
*              N right singular vectors to be computed in VT and
*              M left singular vectors to be computed in U
*
               IVT = 1
*
*              WORK(IVT) is M by M
*
               LDWKVT = M
               ITAU = IVT + LDWKVT*M
               NWORK = ITAU + M
*
*              Compute A=L*Q, copying result to VT
*              Workspace: need   M*M [VT] + M [tau] + M    [work]
*              Workspace: prefer M*M [VT] + M [tau] + M*NB [work]
*
               CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
     $                      WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
               CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
*              Generate Q in VT
*              Workspace: need   M*M [VT] + M [tau] + N    [work]
*              Workspace: prefer M*M [VT] + M [tau] + N*NB [work]
*
               CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
*              Produce L in A, zeroing out other entries
*
               CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ),
     $                      LDA )
               IE = ITAU
               ITAUQ = IE + M
               ITAUP = ITAUQ + M
               NWORK = ITAUP + M
*
*              Bidiagonalize L in A
*              Workspace: need   M*M [VT] + 3*M [e, tauq, taup] + M      [work]
*              Workspace: prefer M*M [VT] + 3*M [e, tauq, taup] + 2*M*NB [work]
*
               CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
     $                      WORK( ITAUQ ),
     $                      WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
     $                      IERR )
*
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in U and computing right singular
*              vectors of bidiagonal matrix in WORK(IVT)
*              Workspace: need   M*M [VT] + 3*M [e, tauq, taup] + BDSPAC
*
               CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU,
     $                      WORK( IVT ), LDWKVT, DUM, IDUM,
     $                      WORK( NWORK ), IWORK, INFO )
*
*              Overwrite U by left singular vectors of L and WORK(IVT)
*              by right singular vectors of L
*              Workspace: need   M*M [VT] + 3*M [e, tauq, taup]+ M    [work]
*              Workspace: prefer M*M [VT] + 3*M [e, tauq, taup]+ M*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', M, M, M, A, LDA,
     $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
               CALL DORMBR( 'P', 'R', 'T', M, M, M, A, LDA,
     $                      WORK( ITAUP ), WORK( IVT ), LDWKVT,
     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
*              Multiply right singular vectors of L in WORK(IVT) by
*              Q in VT, storing result in A
*              Workspace: need   M*M [VT]
*
               CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IVT ),
     $                     LDWKVT,
     $                     VT, LDVT, ZERO, A, LDA )
*
*              Copy right singular vectors of A from A to VT
*
               CALL DLACPY( 'F', M, N, A, LDA, VT, LDVT )
*
            END IF
*
         ELSE
*
*           N .LT. MNTHR
*
*           Path 5t (N > M, but not much larger)
*           Reduce to bidiagonal form without LQ decomposition
*
            IE = 1
            ITAUQ = IE + M
            ITAUP = ITAUQ + M
            NWORK = ITAUP + M
*
*           Bidiagonalize A
*           Workspace: need   3*M [e, tauq, taup] + N        [work]
*           Workspace: prefer 3*M [e, tauq, taup] + (M+N)*NB [work]
*
            CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
     $                   WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
     $                   IERR )
            IF( WNTQN ) THEN
*
*              Path 5tn (N > M, JOBZ='N')
*              Perform bidiagonal SVD, only computing singular values
*              Workspace: need   3*M [e, tauq, taup] + BDSPAC
*
               CALL DBDSDC( 'L', 'N', M, S, WORK( IE ), DUM, 1, DUM,
     $                      1,
     $                      DUM, IDUM, WORK( NWORK ), IWORK, INFO )
            ELSE IF( WNTQO ) THEN
*              Path 5to (N > M, JOBZ='O')
               LDWKVT = M
               IVT = NWORK
               IF( LWORK .GE. M*N + 3*M + BDSPAC ) THEN
*
*                 WORK( IVT ) is M by N
*
                  CALL DLASET( 'F', M, N, ZERO, ZERO, WORK( IVT ),
     $                         LDWKVT )
                  NWORK = IVT + LDWKVT*N
*                 IL is unused; silence compile warnings
                  IL = -1
               ELSE
*
*                 WORK( IVT ) is M by M
*
                  NWORK = IVT + LDWKVT*M
                  IL = NWORK
*
*                 WORK(IL) is M by CHUNK
*
                  CHUNK = ( LWORK - M*M - 3*M ) / M
               END IF
*
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in U and computing right singular
*              vectors of bidiagonal matrix in WORK(IVT)
*              Workspace: need   3*M [e, tauq, taup] + M*M [VT] + BDSPAC
*
               CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU,
     $                      WORK( IVT ), LDWKVT, DUM, IDUM,
     $                      WORK( NWORK ), IWORK, INFO )
*
*              Overwrite U by left singular vectors of A
*              Workspace: need   3*M [e, tauq, taup] + M*M [VT] + M    [work]
*              Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
     $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
               IF( LWORK .GE. M*N + 3*M + BDSPAC ) THEN
*
*                 Path 5to-fast
*                 Overwrite WORK(IVT) by left singular vectors of A
*                 Workspace: need   3*M [e, tauq, taup] + M*N [VT] + M    [work]
*                 Workspace: prefer 3*M [e, tauq, taup] + M*N [VT] + M*NB [work]
*
                  CALL DORMBR( 'P', 'R', 'T', M, N, M, A, LDA,
     $                         WORK( ITAUP ), WORK( IVT ), LDWKVT,
     $                         WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
*                 Copy right singular vectors of A from WORK(IVT) to A
*
                  CALL DLACPY( 'F', M, N, WORK( IVT ), LDWKVT, A,
     $                         LDA )
               ELSE
*
*                 Path 5to-slow
*                 Generate P**T in A
*                 Workspace: need   3*M [e, tauq, taup] + M*M [VT] + M    [work]
*                 Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*NB [work]
*
                  CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
     $                         WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
*                 Multiply Q in A by right singular vectors of
*                 bidiagonal matrix in WORK(IVT), storing result in
*                 WORK(IL) and copying to A
*                 Workspace: need   3*M [e, tauq, taup] + M*M [VT] + M*NB [L]
*                 Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*N  [L]
*
                  DO 40 I = 1, N, CHUNK
                     BLK = MIN( N - I + 1, CHUNK )
                     CALL DGEMM( 'N', 'N', M, BLK, M, ONE,
     $                           WORK( IVT ),
     $                           LDWKVT, A( 1, I ), LDA, ZERO,
     $                           WORK( IL ), M )
                     CALL DLACPY( 'F', M, BLK, WORK( IL ), M, A( 1,
     $                            I ),
     $                            LDA )
   40             CONTINUE
               END IF
            ELSE IF( WNTQS ) THEN
*
*              Path 5ts (N > M, JOBZ='S')
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in U and computing right singular
*              vectors of bidiagonal matrix in VT
*              Workspace: need   3*M [e, tauq, taup] + BDSPAC
*
               CALL DLASET( 'F', M, N, ZERO, ZERO, VT, LDVT )
               CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU, VT,
     $                      LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
     $                      INFO )
*
*              Overwrite U by left singular vectors of A and VT
*              by right singular vectors of A
*              Workspace: need   3*M [e, tauq, taup] + M    [work]
*              Workspace: prefer 3*M [e, tauq, taup] + M*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
     $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
               CALL DORMBR( 'P', 'R', 'T', M, N, M, A, LDA,
     $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
            ELSE IF( WNTQA ) THEN
*
*              Path 5ta (N > M, JOBZ='A')
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in U and computing right singular
*              vectors of bidiagonal matrix in VT
*              Workspace: need   3*M [e, tauq, taup] + BDSPAC
*
               CALL DLASET( 'F', N, N, ZERO, ZERO, VT, LDVT )
               CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU, VT,
     $                      LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
     $                      INFO )
*
*              Set the right corner of VT to identity matrix
*
               IF( N.GT.M ) THEN
                  CALL DLASET( 'F', N-M, N-M, ZERO, ONE, VT(M+1,M+1),
     $                         LDVT )
               END IF
*
*              Overwrite U by left singular vectors of A and VT
*              by right singular vectors of A
*              Workspace: need   3*M [e, tauq, taup] + N    [work]
*              Workspace: prefer 3*M [e, tauq, taup] + N*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
     $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
               CALL DORMBR( 'P', 'R', 'T', N, N, M, A, LDA,
     $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
            END IF
*
         END IF
*
      END IF
*
*     Undo scaling if necessary
*
      IF( ISCL.EQ.1 ) THEN
         IF( ANRM.GT.BIGNUM )
     $      CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
     $                   IERR )
         IF( ANRM.LT.SMLNUM )
     $      CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
     $                   IERR )
      END IF
*
*     Return optimal workspace in WORK(1)
*
      WORK( 1 ) = DROUNDUP_LWORK( MAXWRK )
*
      RETURN
*
*     End of DGESDD
*
      END
*> \addtogroup gesv
*>
*> \brief <b> DGESV computes the solution to system of linear equations A * X = B for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGESV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgesv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgesv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgesv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGESV( N, NRHS, A, LDA, IPIV, B, LDB, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGESV computes the solution to a real system of linear equations
*>    A * X = B,
*> where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
*>
*> The LU decomposition with partial pivoting and row interchanges is
*> used to factor A as
*>    A = P * L * U,
*> where P is a permutation matrix, L is unit lower triangular, and U is
*> upper triangular.  The factored form of A is then used to solve the
*> system of equations A * X = B.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of linear equations, i.e., the order of the
*>          matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the N-by-N coefficient matrix A.
*>          On exit, the factors L and U from the factorization
*>          A = P*L*U; the unit diagonal elements of L are not stored.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          The pivot indices that define the permutation matrix P;
*>          row i of the matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the N-by-NRHS matrix of right hand side matrix B.
*>          On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, U(i,i) is exactly zero.  The factorization
*>                has been completed, but the factor U is exactly
*>                singular, so the solution could not be computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gesv
*
*  =====================================================================
      SUBROUTINE DGESV( N, NRHS, A, LDA, IPIV, B, LDB, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. External Subroutines ..
      EXTERNAL           DGETRF, DGETRS, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF( N.LT.0 ) THEN
         INFO = -1
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGESV ', -INFO )
         RETURN
      END IF
*
*     Compute the LU factorization of A.
*
      CALL DGETRF( N, N, A, LDA, IPIV, INFO )
      IF( INFO.EQ.0 ) THEN
*
*        Solve the system A*X = B, overwriting B with X.
*
         CALL DGETRS( 'No transpose', N, NRHS, A, LDA, IPIV, B, LDB,
     $                INFO )
      END IF
      RETURN
*
*     End of DGESV
*
      END
*> \brief <b> DGESVD computes the singular value decomposition (SVD) for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGESVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgesvd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgesvd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgesvd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT,
*                          WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBU, JOBVT
*       INTEGER            INFO, LDA, LDU, LDVT, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), S( * ), U( LDU, * ),
*      $                   VT( LDVT, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGESVD computes the singular value decomposition (SVD) of a real
*> M-by-N matrix A, optionally computing the left and/or right singular
*> vectors. The SVD is written
*>
*>      A = U * SIGMA * transpose(V)
*>
*> where SIGMA is an M-by-N matrix which is zero except for its
*> min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
*> V is an N-by-N orthogonal matrix.  The diagonal elements of SIGMA
*> are the singular values of A; they are real and non-negative, and
*> are returned in descending order.  The first min(m,n) columns of
*> U and V are the left and right singular vectors of A.
*>
*> Note that the routine returns V**T, not V.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBU
*> \verbatim
*>          JOBU is CHARACTER*1
*>          Specifies options for computing all or part of the matrix U:
*>          = 'A':  all M columns of U are returned in array U:
*>          = 'S':  the first min(m,n) columns of U (the left singular
*>                  vectors) are returned in the array U;
*>          = 'O':  the first min(m,n) columns of U (the left singular
*>                  vectors) are overwritten on the array A;
*>          = 'N':  no columns of U (no left singular vectors) are
*>                  computed.
*> \endverbatim
*>
*> \param[in] JOBVT
*> \verbatim
*>          JOBVT is CHARACTER*1
*>          Specifies options for computing all or part of the matrix
*>          V**T:
*>          = 'A':  all N rows of V**T are returned in the array VT;
*>          = 'S':  the first min(m,n) rows of V**T (the right singular
*>                  vectors) are returned in the array VT;
*>          = 'O':  the first min(m,n) rows of V**T (the right singular
*>                  vectors) are overwritten on the array A;
*>          = 'N':  no rows of V**T (no right singular vectors) are
*>                  computed.
*>
*>          JOBVT and JOBU cannot both be 'O'.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the input matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the input matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit,
*>          if JOBU = 'O',  A is overwritten with the first min(m,n)
*>                          columns of U (the left singular vectors,
*>                          stored columnwise);
*>          if JOBVT = 'O', A is overwritten with the first min(m,n)
*>                          rows of V**T (the right singular vectors,
*>                          stored rowwise);
*>          if JOBU .ne. 'O' and JOBVT .ne. 'O', the contents of A
*>                          are destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (min(M,N))
*>          The singular values of A, sorted so that S(i) >= S(i+1).
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*>          U is DOUBLE PRECISION array, dimension (LDU,UCOL)
*>          (LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'.
*>          If JOBU = 'A', U contains the M-by-M orthogonal matrix U;
*>          if JOBU = 'S', U contains the first min(m,n) columns of U
*>          (the left singular vectors, stored columnwise);
*>          if JOBU = 'N' or 'O', U is not referenced.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>          The leading dimension of the array U.  LDU >= 1; if
*>          JOBU = 'S' or 'A', LDU >= M.
*> \endverbatim
*>
*> \param[out] VT
*> \verbatim
*>          VT is DOUBLE PRECISION array, dimension (LDVT,N)
*>          If JOBVT = 'A', VT contains the N-by-N orthogonal matrix
*>          V**T;
*>          if JOBVT = 'S', VT contains the first min(m,n) rows of
*>          V**T (the right singular vectors, stored rowwise);
*>          if JOBVT = 'N' or 'O', VT is not referenced.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*>          LDVT is INTEGER
*>          The leading dimension of the array VT.  LDVT >= 1; if
*>          JOBVT = 'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK;
*>          if INFO > 0, WORK(2:MIN(M,N)) contains the unconverged
*>          superdiagonal elements of an upper bidiagonal matrix B
*>          whose diagonal is in S (not necessarily sorted). B
*>          satisfies A = U * B * VT, so it has the same singular values
*>          as A, and singular vectors related by U and VT.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          LWORK >= MAX(1,5*MIN(M,N)) for the paths (see comments inside code):
*>             - PATH 1  (M much larger than N, JOBU='N')
*>             - PATH 1t (N much larger than M, JOBVT='N')
*>          LWORK >= MAX(1,3*MIN(M,N) + MAX(M,N),5*MIN(M,N)) for the other paths
*>          For good performance, LWORK should generally be larger.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if DBDSQR did not converge, INFO specifies how many
*>                superdiagonals of an intermediate bidiagonal form B
*>                did not converge to zero. See the description of WORK
*>                above for details.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gesvd
*
*  =====================================================================
      SUBROUTINE DGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU,
     $                   VT, LDVT, WORK, LWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBU, JOBVT
      INTEGER            INFO, LDA, LDU, LDVT, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), S( * ), U( LDU, * ),
     $                   VT( LDVT, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, WNTUA, WNTUAS, WNTUN, WNTUO, WNTUS,
     $                   WNTVA, WNTVAS, WNTVN, WNTVO, WNTVS
      INTEGER            BDSPAC, BLK, CHUNK, I, IE, IERR, IR, ISCL,
     $                   ITAU, ITAUP, ITAUQ, IU, IWORK, LDWRKR, LDWRKU,
     $                   MAXWRK, MINMN, MINWRK, MNTHR, NCU, NCVT, NRU,
     $                   NRVT, WRKBL
      INTEGER            LWORK_DGEQRF, LWORK_DORGQR_N, LWORK_DORGQR_M,
     $                   LWORK_DGEBRD, LWORK_DORGBR_P, LWORK_DORGBR_Q,
     $                   LWORK_DGELQF, LWORK_DORGLQ_N, LWORK_DORGLQ_M
      DOUBLE PRECISION   ANRM, BIGNUM, EPS, SMLNUM
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   DUM( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DBDSQR, DGEBRD, DGELQF, DGEMM, DGEQRF,
     $                   DLACPY,
     $                   DLASCL, DLASET, DORGBR, DORGLQ, DORGQR, DORMBR,
     $                   XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           LSAME, ILAENV, DLAMCH, DLANGE
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      MINMN = MIN( M, N )
      WNTUA = LSAME( JOBU, 'A' )
      WNTUS = LSAME( JOBU, 'S' )
      WNTUAS = WNTUA .OR. WNTUS
      WNTUO = LSAME( JOBU, 'O' )
      WNTUN = LSAME( JOBU, 'N' )
      WNTVA = LSAME( JOBVT, 'A' )
      WNTVS = LSAME( JOBVT, 'S' )
      WNTVAS = WNTVA .OR. WNTVS
      WNTVO = LSAME( JOBVT, 'O' )
      WNTVN = LSAME( JOBVT, 'N' )
      LQUERY = ( LWORK.EQ.-1 )
*
      IF( .NOT.( WNTUA .OR. WNTUS .OR. WNTUO .OR. WNTUN ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( WNTVA .OR. WNTVS .OR. WNTVO .OR. WNTVN ) .OR.
     $         ( WNTVO .AND. WNTUO ) ) THEN
         INFO = -2
      ELSE IF( M.LT.0 ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -6
      ELSE IF( LDU.LT.1 .OR. ( WNTUAS .AND. LDU.LT.M ) ) THEN
         INFO = -9
      ELSE IF( LDVT.LT.1 .OR. ( WNTVA .AND. LDVT.LT.N ) .OR.
     $         ( WNTVS .AND. LDVT.LT.MINMN ) ) THEN
         INFO = -11
      END IF
*
*     Compute workspace
*      (Note: Comments in the code beginning "Workspace:" describe the
*       minimal amount of workspace needed at that point in the code,
*       as well as the preferred amount for good performance.
*       NB refers to the optimal block size for the immediately
*       following subroutine, as returned by ILAENV.)
*
      IF( INFO.EQ.0 ) THEN
         MINWRK = 1
         MAXWRK = 1
         IF( M.GE.N .AND. MINMN.GT.0 ) THEN
*
*           Compute space needed for DBDSQR
*
            MNTHR = ILAENV( 6, 'DGESVD', JOBU // JOBVT, M, N, 0, 0 )
            BDSPAC = 5*N
*           Compute space needed for DGEQRF
            CALL DGEQRF( M, N, A, LDA, DUM(1), DUM(1), -1, IERR )
            LWORK_DGEQRF = INT( DUM(1) )
*           Compute space needed for DORGQR
            CALL DORGQR( M, N, N, A, LDA, DUM(1), DUM(1), -1, IERR )
            LWORK_DORGQR_N = INT( DUM(1) )
            CALL DORGQR( M, M, N, A, LDA, DUM(1), DUM(1), -1, IERR )
            LWORK_DORGQR_M = INT( DUM(1) )
*           Compute space needed for DGEBRD
            CALL DGEBRD( N, N, A, LDA, S, DUM(1), DUM(1),
     $                   DUM(1), DUM(1), -1, IERR )
            LWORK_DGEBRD = INT( DUM(1) )
*           Compute space needed for DORGBR P
            CALL DORGBR( 'P', N, N, N, A, LDA, DUM(1),
     $                   DUM(1), -1, IERR )
            LWORK_DORGBR_P = INT( DUM(1) )
*           Compute space needed for DORGBR Q
            CALL DORGBR( 'Q', N, N, N, A, LDA, DUM(1),
     $                   DUM(1), -1, IERR )
            LWORK_DORGBR_Q = INT( DUM(1) )
*
            IF( M.GE.MNTHR ) THEN
               IF( WNTUN ) THEN
*
*                 Path 1 (M much larger than N, JOBU='N')
*
                  MAXWRK = N + LWORK_DGEQRF
                  MAXWRK = MAX( MAXWRK, 3*N + LWORK_DGEBRD )
                  IF( WNTVO .OR. WNTVAS )
     $               MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR_P )
                  MAXWRK = MAX( MAXWRK, BDSPAC )
                  MINWRK = MAX( 4*N, BDSPAC )
               ELSE IF( WNTUO .AND. WNTVN ) THEN
*
*                 Path 2 (M much larger than N, JOBU='O', JOBVT='N')
*
                  WRKBL = N + LWORK_DGEQRF
                  WRKBL = MAX( WRKBL, N + LWORK_DORGQR_N )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = MAX( N*N + WRKBL, N*N + M*N + N )
                  MINWRK = MAX( 3*N + M, BDSPAC )
               ELSE IF( WNTUO .AND. WNTVAS ) THEN
*
*                 Path 3 (M much larger than N, JOBU='O', JOBVT='S' or
*                 'A')
*
                  WRKBL = N + LWORK_DGEQRF
                  WRKBL = MAX( WRKBL, N + LWORK_DORGQR_N )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_P )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = MAX( N*N + WRKBL, N*N + M*N + N )
                  MINWRK = MAX( 3*N + M, BDSPAC )
               ELSE IF( WNTUS .AND. WNTVN ) THEN
*
*                 Path 4 (M much larger than N, JOBU='S', JOBVT='N')
*
                  WRKBL = N + LWORK_DGEQRF
                  WRKBL = MAX( WRKBL, N + LWORK_DORGQR_N )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = N*N + WRKBL
                  MINWRK = MAX( 3*N + M, BDSPAC )
               ELSE IF( WNTUS .AND. WNTVO ) THEN
*
*                 Path 5 (M much larger than N, JOBU='S', JOBVT='O')
*
                  WRKBL = N + LWORK_DGEQRF
                  WRKBL = MAX( WRKBL, N + LWORK_DORGQR_N )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_P )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = 2*N*N + WRKBL
                  MINWRK = MAX( 3*N + M, BDSPAC )
               ELSE IF( WNTUS .AND. WNTVAS ) THEN
*
*                 Path 6 (M much larger than N, JOBU='S', JOBVT='S' or
*                 'A')
*
                  WRKBL = N + LWORK_DGEQRF
                  WRKBL = MAX( WRKBL, N + LWORK_DORGQR_N )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_P )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = N*N + WRKBL
                  MINWRK = MAX( 3*N + M, BDSPAC )
               ELSE IF( WNTUA .AND. WNTVN ) THEN
*
*                 Path 7 (M much larger than N, JOBU='A', JOBVT='N')
*
                  WRKBL = N + LWORK_DGEQRF
                  WRKBL = MAX( WRKBL, N + LWORK_DORGQR_M )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = N*N + WRKBL
                  MINWRK = MAX( 3*N + M, BDSPAC )
               ELSE IF( WNTUA .AND. WNTVO ) THEN
*
*                 Path 8 (M much larger than N, JOBU='A', JOBVT='O')
*
                  WRKBL = N + LWORK_DGEQRF
                  WRKBL = MAX( WRKBL, N + LWORK_DORGQR_M )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_P )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = 2*N*N + WRKBL
                  MINWRK = MAX( 3*N + M, BDSPAC )
               ELSE IF( WNTUA .AND. WNTVAS ) THEN
*
*                 Path 9 (M much larger than N, JOBU='A', JOBVT='S' or
*                 'A')
*
                  WRKBL = N + LWORK_DGEQRF
                  WRKBL = MAX( WRKBL, N + LWORK_DORGQR_M )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_P )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = N*N + WRKBL
                  MINWRK = MAX( 3*N + M, BDSPAC )
               END IF
            ELSE
*
*              Path 10 (M at least N, but not much larger)
*
               CALL DGEBRD( M, N, A, LDA, S, DUM(1), DUM(1),
     $                   DUM(1), DUM(1), -1, IERR )
               LWORK_DGEBRD = INT( DUM(1) )
               MAXWRK = 3*N + LWORK_DGEBRD
               IF( WNTUS .OR. WNTUO ) THEN
                  CALL DORGBR( 'Q', M, N, N, A, LDA, DUM(1),
     $                   DUM(1), -1, IERR )
                  LWORK_DORGBR_Q = INT( DUM(1) )
                  MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR_Q )
               END IF
               IF( WNTUA ) THEN
                  CALL DORGBR( 'Q', M, M, N, A, LDA, DUM(1),
     $                   DUM(1), -1, IERR )
                  LWORK_DORGBR_Q = INT( DUM(1) )
                  MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR_Q )
               END IF
               IF( .NOT.WNTVN ) THEN
                 MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR_P )
               END IF
               MAXWRK = MAX( MAXWRK, BDSPAC )
               MINWRK = MAX( 3*N + M, BDSPAC )
            END IF
         ELSE IF( MINMN.GT.0 ) THEN
*
*           Compute space needed for DBDSQR
*
            MNTHR = ILAENV( 6, 'DGESVD', JOBU // JOBVT, M, N, 0, 0 )
            BDSPAC = 5*M
*           Compute space needed for DGELQF
            CALL DGELQF( M, N, A, LDA, DUM(1), DUM(1), -1, IERR )
            LWORK_DGELQF = INT( DUM(1) )
*           Compute space needed for DORGLQ
            CALL DORGLQ( N, N, M, DUM(1), N, DUM(1), DUM(1), -1,
     $                   IERR )
            LWORK_DORGLQ_N = INT( DUM(1) )
            CALL DORGLQ( M, N, M, A, LDA, DUM(1), DUM(1), -1, IERR )
            LWORK_DORGLQ_M = INT( DUM(1) )
*           Compute space needed for DGEBRD
            CALL DGEBRD( M, M, A, LDA, S, DUM(1), DUM(1),
     $                   DUM(1), DUM(1), -1, IERR )
            LWORK_DGEBRD = INT( DUM(1) )
*            Compute space needed for DORGBR P
            CALL DORGBR( 'P', M, M, M, A, N, DUM(1),
     $                   DUM(1), -1, IERR )
            LWORK_DORGBR_P = INT( DUM(1) )
*           Compute space needed for DORGBR Q
            CALL DORGBR( 'Q', M, M, M, A, N, DUM(1),
     $                   DUM(1), -1, IERR )
            LWORK_DORGBR_Q = INT( DUM(1) )
            IF( N.GE.MNTHR ) THEN
               IF( WNTVN ) THEN
*
*                 Path 1t(N much larger than M, JOBVT='N')
*
                  MAXWRK = M + LWORK_DGELQF
                  MAXWRK = MAX( MAXWRK, 3*M + LWORK_DGEBRD )
                  IF( WNTUO .OR. WNTUAS )
     $               MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR_Q )
                  MAXWRK = MAX( MAXWRK, BDSPAC )
                  MINWRK = MAX( 4*M, BDSPAC )
               ELSE IF( WNTVO .AND. WNTUN ) THEN
*
*                 Path 2t(N much larger than M, JOBU='N', JOBVT='O')
*
                  WRKBL = M + LWORK_DGELQF
                  WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_M )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = MAX( M*M + WRKBL, M*M + M*N + M )
                  MINWRK = MAX( 3*M + N, BDSPAC )
               ELSE IF( WNTVO .AND. WNTUAS ) THEN
*
*                 Path 3t(N much larger than M, JOBU='S' or 'A',
*                 JOBVT='O')
*
                  WRKBL = M + LWORK_DGELQF
                  WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_M )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_Q )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = MAX( M*M + WRKBL, M*M + M*N + M )
                  MINWRK = MAX( 3*M + N, BDSPAC )
               ELSE IF( WNTVS .AND. WNTUN ) THEN
*
*                 Path 4t(N much larger than M, JOBU='N', JOBVT='S')
*
                  WRKBL = M + LWORK_DGELQF
                  WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_M )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = M*M + WRKBL
                  MINWRK = MAX( 3*M + N, BDSPAC )
               ELSE IF( WNTVS .AND. WNTUO ) THEN
*
*                 Path 5t(N much larger than M, JOBU='O', JOBVT='S')
*
                  WRKBL = M + LWORK_DGELQF
                  WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_M )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_Q )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = 2*M*M + WRKBL
                  MINWRK = MAX( 3*M + N, BDSPAC )
               ELSE IF( WNTVS .AND. WNTUAS ) THEN
*
*                 Path 6t(N much larger than M, JOBU='S' or 'A',
*                 JOBVT='S')
*
                  WRKBL = M + LWORK_DGELQF
                  WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_M )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_Q )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = M*M + WRKBL
                  MINWRK = MAX( 3*M + N, BDSPAC )
               ELSE IF( WNTVA .AND. WNTUN ) THEN
*
*                 Path 7t(N much larger than M, JOBU='N', JOBVT='A')
*
                  WRKBL = M + LWORK_DGELQF
                  WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_N )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = M*M + WRKBL
                  MINWRK = MAX( 3*M + N, BDSPAC )
               ELSE IF( WNTVA .AND. WNTUO ) THEN
*
*                 Path 8t(N much larger than M, JOBU='O', JOBVT='A')
*
                  WRKBL = M + LWORK_DGELQF
                  WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_N )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_Q )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = 2*M*M + WRKBL
                  MINWRK = MAX( 3*M + N, BDSPAC )
               ELSE IF( WNTVA .AND. WNTUAS ) THEN
*
*                 Path 9t(N much larger than M, JOBU='S' or 'A',
*                 JOBVT='A')
*
                  WRKBL = M + LWORK_DGELQF
                  WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_N )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_Q )
                  WRKBL = MAX( WRKBL, BDSPAC )
                  MAXWRK = M*M + WRKBL
                  MINWRK = MAX( 3*M + N, BDSPAC )
               END IF
            ELSE
*
*              Path 10t(N greater than M, but not much larger)
*
               CALL DGEBRD( M, N, A, LDA, S, DUM(1), DUM(1),
     $                   DUM(1), DUM(1), -1, IERR )
               LWORK_DGEBRD = INT( DUM(1) )
               MAXWRK = 3*M + LWORK_DGEBRD
               IF( WNTVS .OR. WNTVO ) THEN
*                Compute space needed for DORGBR P
                 CALL DORGBR( 'P', M, N, M, A, N, DUM(1),
     $                   DUM(1), -1, IERR )
                 LWORK_DORGBR_P = INT( DUM(1) )
                 MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR_P )
               END IF
               IF( WNTVA ) THEN
                 CALL DORGBR( 'P', N, N, M, A, N, DUM(1),
     $                   DUM(1), -1, IERR )
                 LWORK_DORGBR_P = INT( DUM(1) )
                 MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR_P )
               END IF
               IF( .NOT.WNTUN ) THEN
                  MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR_Q )
               END IF
               MAXWRK = MAX( MAXWRK, BDSPAC )
               MINWRK = MAX( 3*M + N, BDSPAC )
            END IF
         END IF
         MAXWRK = MAX( MAXWRK, MINWRK )
         WORK( 1 ) = MAXWRK
*
         IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
            INFO = -13
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGESVD', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 ) THEN
         RETURN
      END IF
*
*     Get machine constants
*
      EPS = DLAMCH( 'P' )
      SMLNUM = SQRT( DLAMCH( 'S' ) ) / EPS
      BIGNUM = ONE / SMLNUM
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      ANRM = DLANGE( 'M', M, N, A, LDA, DUM )
      ISCL = 0
      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
         ISCL = 1
         CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, IERR )
      ELSE IF( ANRM.GT.BIGNUM ) THEN
         ISCL = 1
         CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, IERR )
      END IF
*
      IF( M.GE.N ) THEN
*
*        A has at least as many rows as columns. If A has sufficiently
*        more rows than columns, first reduce using the QR
*        decomposition (if sufficient workspace available)
*
         IF( M.GE.MNTHR ) THEN
*
            IF( WNTUN ) THEN
*
*              Path 1 (M much larger than N, JOBU='N')
*              No left singular vectors to be computed
*
               ITAU = 1
               IWORK = ITAU + N
*
*              Compute A=Q*R
*              (Workspace: need 2*N, prefer N + N*NB)
*
               CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
     $                      WORK( IWORK ),
     $                      LWORK-IWORK+1, IERR )
*
*              Zero out below R
*
               IF( N .GT. 1 ) THEN
                  CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ),
     $                         LDA )
               END IF
               IE = 1
               ITAUQ = IE + N
               ITAUP = ITAUQ + N
               IWORK = ITAUP + N
*
*              Bidiagonalize R in A
*              (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
               CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
     $                      WORK( ITAUQ ),
     $                      WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
     $                      IERR )
               NCVT = 0
               IF( WNTVO .OR. WNTVAS ) THEN
*
*                 If right singular vectors desired, generate P'.
*                 (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
                  CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
                  NCVT = N
               END IF
               IWORK = IE + N
*
*              Perform bidiagonal QR iteration, computing right
*              singular vectors of A in A if desired
*              (Workspace: need BDSPAC)
*
               CALL DBDSQR( 'U', N, NCVT, 0, 0, S, WORK( IE ), A,
     $                      LDA,
     $                      DUM, 1, DUM, 1, WORK( IWORK ), INFO )
*
*              If right singular vectors desired in VT, copy them there
*
               IF( WNTVAS )
     $            CALL DLACPY( 'F', N, N, A, LDA, VT, LDVT )
*
            ELSE IF( WNTUO .AND. WNTVN ) THEN
*
*              Path 2 (M much larger than N, JOBU='O', JOBVT='N')
*              N left singular vectors to be overwritten on A and
*              no right singular vectors to be computed
*
               IF( LWORK.GE.N*N+MAX( 4*N, BDSPAC ) ) THEN
*
*                 Sufficient workspace for a fast algorithm
*
                  IR = 1
                  IF( LWORK.GE.MAX( WRKBL, LDA*N + N ) + LDA*N ) THEN
*
*                    WORK(IU) is LDA by N, WORK(IR) is LDA by N
*
                     LDWRKU = LDA
                     LDWRKR = LDA
                  ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N + N ) + N*N ) THEN
*
*                    WORK(IU) is LDA by N, WORK(IR) is N by N
*
                     LDWRKU = LDA
                     LDWRKR = N
                  ELSE
*
*                    WORK(IU) is LDWRKU by N, WORK(IR) is N by N
*
                     LDWRKU = ( LWORK-N*N-N ) / N
                     LDWRKR = N
                  END IF
                  ITAU = IR + LDWRKR*N
                  IWORK = ITAU + N
*
*                 Compute A=Q*R
*                 (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
                  CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                 Copy R to WORK(IR) and zero out below it
*
                  CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ),
     $                         LDWRKR )
                  CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
     $                         WORK( IR+1 ),
     $                         LDWRKR )
*
*                 Generate Q in A
*                 (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
                  CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
                  IE = ITAU
                  ITAUQ = IE + N
                  ITAUP = ITAUQ + N
                  IWORK = ITAUP + N
*
*                 Bidiagonalize R in WORK(IR)
*                 (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
*
                  CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S,
     $                         WORK( IE ),
     $                         WORK( ITAUQ ), WORK( ITAUP ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                 Generate left vectors bidiagonalizing R
*                 (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
*
                  CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
     $                         WORK( ITAUQ ), WORK( IWORK ),
     $                         LWORK-IWORK+1, IERR )
                  IWORK = IE + N
*
*                 Perform bidiagonal QR iteration, computing left
*                 singular vectors of R in WORK(IR)
*                 (Workspace: need N*N + BDSPAC)
*
                  CALL DBDSQR( 'U', N, 0, N, 0, S, WORK( IE ), DUM,
     $                         1,
     $                         WORK( IR ), LDWRKR, DUM, 1,
     $                         WORK( IWORK ), INFO )
                  IU = IE + N
*
*                 Multiply Q in A by left singular vectors of R in
*                 WORK(IR), storing result in WORK(IU) and copying to A
*                 (Workspace: need N*N + 2*N, prefer N*N + M*N + N)
*
                  DO 10 I = 1, M, LDWRKU
                     CHUNK = MIN( M-I+1, LDWRKU )
                     CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I,
     $                           1 ),
     $                           LDA, WORK( IR ), LDWRKR, ZERO,
     $                           WORK( IU ), LDWRKU )
                     CALL DLACPY( 'F', CHUNK, N, WORK( IU ), LDWRKU,
     $                            A( I, 1 ), LDA )
   10             CONTINUE
*
               ELSE
*
*                 Insufficient workspace for a fast algorithm
*
                  IE = 1
                  ITAUQ = IE + N
                  ITAUP = ITAUQ + N
                  IWORK = ITAUP + N
*
*                 Bidiagonalize A
*                 (Workspace: need 3*N + M, prefer 3*N + (M + N)*NB)
*
                  CALL DGEBRD( M, N, A, LDA, S, WORK( IE ),
     $                         WORK( ITAUQ ), WORK( ITAUP ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                 Generate left vectors bidiagonalizing A
*                 (Workspace: need 4*N, prefer 3*N + N*NB)
*
                  CALL DORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
                  IWORK = IE + N
*
*                 Perform bidiagonal QR iteration, computing left
*                 singular vectors of A in A
*                 (Workspace: need BDSPAC)
*
                  CALL DBDSQR( 'U', N, 0, M, 0, S, WORK( IE ), DUM,
     $                         1,
     $                         A, LDA, DUM, 1, WORK( IWORK ), INFO )
*
               END IF
*
            ELSE IF( WNTUO .AND. WNTVAS ) THEN
*
*              Path 3 (M much larger than N, JOBU='O', JOBVT='S' or 'A')
*              N left singular vectors to be overwritten on A and
*              N right singular vectors to be computed in VT
*
               IF( LWORK.GE.N*N+MAX( 4*N, BDSPAC ) ) THEN
*
*                 Sufficient workspace for a fast algorithm
*
                  IR = 1
                  IF( LWORK.GE.MAX( WRKBL, LDA*N + N ) + LDA*N ) THEN
*
*                    WORK(IU) is LDA by N and WORK(IR) is LDA by N
*
                     LDWRKU = LDA
                     LDWRKR = LDA
                  ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N + N ) + N*N ) THEN
*
*                    WORK(IU) is LDA by N and WORK(IR) is N by N
*
                     LDWRKU = LDA
                     LDWRKR = N
                  ELSE
*
*                    WORK(IU) is LDWRKU by N and WORK(IR) is N by N
*
                     LDWRKU = ( LWORK-N*N-N ) / N
                     LDWRKR = N
                  END IF
                  ITAU = IR + LDWRKR*N
                  IWORK = ITAU + N
*
*                 Compute A=Q*R
*                 (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
                  CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                 Copy R to VT, zeroing out below it
*
                  CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
                  IF( N.GT.1 )
     $               CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
     $                            VT( 2, 1 ), LDVT )
*
*                 Generate Q in A
*                 (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
                  CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
                  IE = ITAU
                  ITAUQ = IE + N
                  ITAUP = ITAUQ + N
                  IWORK = ITAUP + N
*
*                 Bidiagonalize R in VT, copying result to WORK(IR)
*                 (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
*
                  CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ),
     $                         WORK( ITAUQ ), WORK( ITAUP ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
                  CALL DLACPY( 'L', N, N, VT, LDVT, WORK( IR ),
     $                         LDWRKR )
*
*                 Generate left vectors bidiagonalizing R in WORK(IR)
*                 (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
*
                  CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
     $                         WORK( ITAUQ ), WORK( IWORK ),
     $                         LWORK-IWORK+1, IERR )
*
*                 Generate right vectors bidiagonalizing R in VT
*                 (Workspace: need N*N + 4*N-1, prefer N*N + 3*N + (N-1)*NB)
*
                  CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
                  IWORK = IE + N
*
*                 Perform bidiagonal QR iteration, computing left
*                 singular vectors of R in WORK(IR) and computing right
*                 singular vectors of R in VT
*                 (Workspace: need N*N + BDSPAC)
*
                  CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ), VT,
     $                         LDVT,
     $                         WORK( IR ), LDWRKR, DUM, 1,
     $                         WORK( IWORK ), INFO )
                  IU = IE + N
*
*                 Multiply Q in A by left singular vectors of R in
*                 WORK(IR), storing result in WORK(IU) and copying to A
*                 (Workspace: need N*N + 2*N, prefer N*N + M*N + N)
*
                  DO 20 I = 1, M, LDWRKU
                     CHUNK = MIN( M-I+1, LDWRKU )
                     CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I,
     $                           1 ),
     $                           LDA, WORK( IR ), LDWRKR, ZERO,
     $                           WORK( IU ), LDWRKU )
                     CALL DLACPY( 'F', CHUNK, N, WORK( IU ), LDWRKU,
     $                            A( I, 1 ), LDA )
   20             CONTINUE
*
               ELSE
*
*                 Insufficient workspace for a fast algorithm
*
                  ITAU = 1
                  IWORK = ITAU + N
*
*                 Compute A=Q*R
*                 (Workspace: need 2*N, prefer N + N*NB)
*
                  CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                 Copy R to VT, zeroing out below it
*
                  CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
                  IF( N.GT.1 )
     $               CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
     $                            VT( 2, 1 ), LDVT )
*
*                 Generate Q in A
*                 (Workspace: need 2*N, prefer N + N*NB)
*
                  CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
                  IE = ITAU
                  ITAUQ = IE + N
                  ITAUP = ITAUQ + N
                  IWORK = ITAUP + N
*
*                 Bidiagonalize R in VT
*                 (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
                  CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ),
     $                         WORK( ITAUQ ), WORK( ITAUP ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                 Multiply Q in A by left vectors bidiagonalizing R
*                 (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
                  CALL DORMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT,
     $                         WORK( ITAUQ ), A, LDA, WORK( IWORK ),
     $                         LWORK-IWORK+1, IERR )
*
*                 Generate right vectors bidiagonalizing R in VT
*                 (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
                  CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
                  IWORK = IE + N
*
*                 Perform bidiagonal QR iteration, computing left
*                 singular vectors of A in A and computing right
*                 singular vectors of A in VT
*                 (Workspace: need BDSPAC)
*
                  CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), VT,
     $                         LDVT,
     $                         A, LDA, DUM, 1, WORK( IWORK ), INFO )
*
               END IF
*
            ELSE IF( WNTUS ) THEN
*
               IF( WNTVN ) THEN
*
*                 Path 4 (M much larger than N, JOBU='S', JOBVT='N')
*                 N left singular vectors to be computed in U and
*                 no right singular vectors to be computed
*
                  IF( LWORK.GE.N*N+MAX( 4*N, BDSPAC ) ) THEN
*
*                    Sufficient workspace for a fast algorithm
*
                     IR = 1
                     IF( LWORK.GE.WRKBL+LDA*N ) THEN
*
*                       WORK(IR) is LDA by N
*
                        LDWRKR = LDA
                     ELSE
*
*                       WORK(IR) is N by N
*
                        LDWRKR = N
                     END IF
                     ITAU = IR + LDWRKR*N
                     IWORK = ITAU + N
*
*                    Compute A=Q*R
*                    (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
                     CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Copy R to WORK(IR), zeroing out below it
*
                     CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ),
     $                            LDWRKR )
                     CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
     $                            WORK( IR+1 ), LDWRKR )
*
*                    Generate Q in A
*                    (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
                     CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IE = ITAU
                     ITAUQ = IE + N
                     ITAUP = ITAUQ + N
                     IWORK = ITAUP + N
*
*                    Bidiagonalize R in WORK(IR)
*                    (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
*
                     CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S,
     $                            WORK( IE ), WORK( ITAUQ ),
     $                            WORK( ITAUP ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
*
*                    Generate left vectors bidiagonalizing R in WORK(IR)
*                    (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
*
                     CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
     $                            WORK( ITAUQ ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
                     IWORK = IE + N
*
*                    Perform bidiagonal QR iteration, computing left
*                    singular vectors of R in WORK(IR)
*                    (Workspace: need N*N + BDSPAC)
*
                     CALL DBDSQR( 'U', N, 0, N, 0, S, WORK( IE ),
     $                            DUM,
     $                            1, WORK( IR ), LDWRKR, DUM, 1,
     $                            WORK( IWORK ), INFO )
*
*                    Multiply Q in A by left singular vectors of R in
*                    WORK(IR), storing result in U
*                    (Workspace: need N*N)
*
                     CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA,
     $                           WORK( IR ), LDWRKR, ZERO, U, LDU )
*
                  ELSE
*
*                    Insufficient workspace for a fast algorithm
*
                     ITAU = 1
                     IWORK = ITAU + N
*
*                    Compute A=Q*R, copying result to U
*                    (Workspace: need 2*N, prefer N + N*NB)
*
                     CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
*                    Generate Q in U
*                    (Workspace: need 2*N, prefer N + N*NB)
*
                     CALL DORGQR( M, N, N, U, LDU, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IE = ITAU
                     ITAUQ = IE + N
                     ITAUP = ITAUQ + N
                     IWORK = ITAUP + N
*
*                    Zero out below R in A
*
                     IF( N .GT. 1 ) THEN
                        CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
     $                               A( 2, 1 ), LDA )
                     END IF
*
*                    Bidiagonalize R in A
*                    (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
                     CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
     $                            WORK( ITAUQ ), WORK( ITAUP ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Multiply Q in U by left vectors bidiagonalizing R
*                    (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
                     CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
     $                            WORK( ITAUQ ), U, LDU, WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
                     IWORK = IE + N
*
*                    Perform bidiagonal QR iteration, computing left
*                    singular vectors of A in U
*                    (Workspace: need BDSPAC)
*
                     CALL DBDSQR( 'U', N, 0, M, 0, S, WORK( IE ),
     $                            DUM,
     $                            1, U, LDU, DUM, 1, WORK( IWORK ),
     $                            INFO )
*
                  END IF
*
               ELSE IF( WNTVO ) THEN
*
*                 Path 5 (M much larger than N, JOBU='S', JOBVT='O')
*                 N left singular vectors to be computed in U and
*                 N right singular vectors to be overwritten on A
*
                  IF( LWORK.GE.2*N*N+MAX( 4*N, BDSPAC ) ) THEN
*
*                    Sufficient workspace for a fast algorithm
*
                     IU = 1
                     IF( LWORK.GE.WRKBL+2*LDA*N ) THEN
*
*                       WORK(IU) is LDA by N and WORK(IR) is LDA by N
*
                        LDWRKU = LDA
                        IR = IU + LDWRKU*N
                        LDWRKR = LDA
                     ELSE IF( LWORK.GE.WRKBL+( LDA + N )*N ) THEN
*
*                       WORK(IU) is LDA by N and WORK(IR) is N by N
*
                        LDWRKU = LDA
                        IR = IU + LDWRKU*N
                        LDWRKR = N
                     ELSE
*
*                       WORK(IU) is N by N and WORK(IR) is N by N
*
                        LDWRKU = N
                        IR = IU + LDWRKU*N
                        LDWRKR = N
                     END IF
                     ITAU = IR + LDWRKR*N
                     IWORK = ITAU + N
*
*                    Compute A=Q*R
*                    (Workspace: need 2*N*N + 2*N, prefer 2*N*N + N + N*NB)
*
                     CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Copy R to WORK(IU), zeroing out below it
*
                     CALL DLACPY( 'U', N, N, A, LDA, WORK( IU ),
     $                            LDWRKU )
                     CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
     $                            WORK( IU+1 ), LDWRKU )
*
*                    Generate Q in A
*                    (Workspace: need 2*N*N + 2*N, prefer 2*N*N + N + N*NB)
*
                     CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IE = ITAU
                     ITAUQ = IE + N
                     ITAUP = ITAUQ + N
                     IWORK = ITAUP + N
*
*                    Bidiagonalize R in WORK(IU), copying result to
*                    WORK(IR)
*                    (Workspace: need 2*N*N + 4*N,
*                                prefer 2*N*N+3*N+2*N*NB)
*
                     CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S,
     $                            WORK( IE ), WORK( ITAUQ ),
     $                            WORK( ITAUP ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'U', N, N, WORK( IU ), LDWRKU,
     $                            WORK( IR ), LDWRKR )
*
*                    Generate left bidiagonalizing vectors in WORK(IU)
*                    (Workspace: need 2*N*N + 4*N, prefer 2*N*N + 3*N + N*NB)
*
                     CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
     $                            WORK( ITAUQ ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
*
*                    Generate right bidiagonalizing vectors in WORK(IR)
*                    (Workspace: need 2*N*N + 4*N-1,
*                                prefer 2*N*N+3*N+(N-1)*NB)
*
                     CALL DORGBR( 'P', N, N, N, WORK( IR ), LDWRKR,
     $                            WORK( ITAUP ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
                     IWORK = IE + N
*
*                    Perform bidiagonal QR iteration, computing left
*                    singular vectors of R in WORK(IU) and computing
*                    right singular vectors of R in WORK(IR)
*                    (Workspace: need 2*N*N + BDSPAC)
*
                     CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ),
     $                            WORK( IR ), LDWRKR, WORK( IU ),
     $                            LDWRKU, DUM, 1, WORK( IWORK ), INFO )
*
*                    Multiply Q in A by left singular vectors of R in
*                    WORK(IU), storing result in U
*                    (Workspace: need N*N)
*
                     CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA,
     $                           WORK( IU ), LDWRKU, ZERO, U, LDU )
*
*                    Copy right singular vectors of R to A
*                    (Workspace: need N*N)
*
                     CALL DLACPY( 'F', N, N, WORK( IR ), LDWRKR, A,
     $                            LDA )
*
                  ELSE
*
*                    Insufficient workspace for a fast algorithm
*
                     ITAU = 1
                     IWORK = ITAU + N
*
*                    Compute A=Q*R, copying result to U
*                    (Workspace: need 2*N, prefer N + N*NB)
*
                     CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
*                    Generate Q in U
*                    (Workspace: need 2*N, prefer N + N*NB)
*
                     CALL DORGQR( M, N, N, U, LDU, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IE = ITAU
                     ITAUQ = IE + N
                     ITAUP = ITAUQ + N
                     IWORK = ITAUP + N
*
*                    Zero out below R in A
*
                     IF( N .GT. 1 ) THEN
                        CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
     $                               A( 2, 1 ), LDA )
                     END IF
*
*                    Bidiagonalize R in A
*                    (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
                     CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
     $                            WORK( ITAUQ ), WORK( ITAUP ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Multiply Q in U by left vectors bidiagonalizing R
*                    (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
                     CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
     $                            WORK( ITAUQ ), U, LDU, WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
*
*                    Generate right vectors bidiagonalizing R in A
*                    (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
                     CALL DORGBR( 'P', N, N, N, A, LDA,
     $                            WORK( ITAUP ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IWORK = IE + N
*
*                    Perform bidiagonal QR iteration, computing left
*                    singular vectors of A in U and computing right
*                    singular vectors of A in A
*                    (Workspace: need BDSPAC)
*
                     CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), A,
     $                            LDA, U, LDU, DUM, 1, WORK( IWORK ),
     $                            INFO )
*
                  END IF
*
               ELSE IF( WNTVAS ) THEN
*
*                 Path 6 (M much larger than N, JOBU='S', JOBVT='S'
*                         or 'A')
*                 N left singular vectors to be computed in U and
*                 N right singular vectors to be computed in VT
*
                  IF( LWORK.GE.N*N+MAX( 4*N, BDSPAC ) ) THEN
*
*                    Sufficient workspace for a fast algorithm
*
                     IU = 1
                     IF( LWORK.GE.WRKBL+LDA*N ) THEN
*
*                       WORK(IU) is LDA by N
*
                        LDWRKU = LDA
                     ELSE
*
*                       WORK(IU) is N by N
*
                        LDWRKU = N
                     END IF
                     ITAU = IU + LDWRKU*N
                     IWORK = ITAU + N
*
*                    Compute A=Q*R
*                    (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
                     CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Copy R to WORK(IU), zeroing out below it
*
                     CALL DLACPY( 'U', N, N, A, LDA, WORK( IU ),
     $                            LDWRKU )
                     CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
     $                            WORK( IU+1 ), LDWRKU )
*
*                    Generate Q in A
*                    (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
                     CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IE = ITAU
                     ITAUQ = IE + N
                     ITAUP = ITAUQ + N
                     IWORK = ITAUP + N
*
*                    Bidiagonalize R in WORK(IU), copying result to VT
*                    (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
*
                     CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S,
     $                            WORK( IE ), WORK( ITAUQ ),
     $                            WORK( ITAUP ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'U', N, N, WORK( IU ), LDWRKU, VT,
     $                            LDVT )
*
*                    Generate left bidiagonalizing vectors in WORK(IU)
*                    (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
*
                     CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
     $                            WORK( ITAUQ ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
*
*                    Generate right bidiagonalizing vectors in VT
*                    (Workspace: need N*N + 4*N-1,
*                                prefer N*N+3*N+(N-1)*NB)
*
                     CALL DORGBR( 'P', N, N, N, VT, LDVT,
     $                            WORK( ITAUP ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IWORK = IE + N
*
*                    Perform bidiagonal QR iteration, computing left
*                    singular vectors of R in WORK(IU) and computing
*                    right singular vectors of R in VT
*                    (Workspace: need N*N + BDSPAC)
*
                     CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ), VT,
     $                            LDVT, WORK( IU ), LDWRKU, DUM, 1,
     $                            WORK( IWORK ), INFO )
*
*                    Multiply Q in A by left singular vectors of R in
*                    WORK(IU), storing result in U
*                    (Workspace: need N*N)
*
                     CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA,
     $                           WORK( IU ), LDWRKU, ZERO, U, LDU )
*
                  ELSE
*
*                    Insufficient workspace for a fast algorithm
*
                     ITAU = 1
                     IWORK = ITAU + N
*
*                    Compute A=Q*R, copying result to U
*                    (Workspace: need 2*N, prefer N + N*NB)
*
                     CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
*                    Generate Q in U
*                    (Workspace: need 2*N, prefer N + N*NB)
*
                     CALL DORGQR( M, N, N, U, LDU, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Copy R to VT, zeroing out below it
*
                     CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
                     IF( N.GT.1 )
     $                  CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
     $                               VT( 2, 1 ), LDVT )
                     IE = ITAU
                     ITAUQ = IE + N
                     ITAUP = ITAUQ + N
                     IWORK = ITAUP + N
*
*                    Bidiagonalize R in VT
*                    (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
                     CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ),
     $                            WORK( ITAUQ ), WORK( ITAUP ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Multiply Q in U by left bidiagonalizing vectors
*                    in VT
*                    (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
                     CALL DORMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT,
     $                            WORK( ITAUQ ), U, LDU, WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
*
*                    Generate right bidiagonalizing vectors in VT
*                    (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
                     CALL DORGBR( 'P', N, N, N, VT, LDVT,
     $                            WORK( ITAUP ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IWORK = IE + N
*
*                    Perform bidiagonal QR iteration, computing left
*                    singular vectors of A in U and computing right
*                    singular vectors of A in VT
*                    (Workspace: need BDSPAC)
*
                     CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), VT,
     $                            LDVT, U, LDU, DUM, 1, WORK( IWORK ),
     $                            INFO )
*
                  END IF
*
               END IF
*
            ELSE IF( WNTUA ) THEN
*
               IF( WNTVN ) THEN
*
*                 Path 7 (M much larger than N, JOBU='A', JOBVT='N')
*                 M left singular vectors to be computed in U and
*                 no right singular vectors to be computed
*
                  IF( LWORK.GE.N*N+MAX( N+M, 4*N, BDSPAC ) ) THEN
*
*                    Sufficient workspace for a fast algorithm
*
                     IR = 1
                     IF( LWORK.GE.WRKBL+LDA*N ) THEN
*
*                       WORK(IR) is LDA by N
*
                        LDWRKR = LDA
                     ELSE
*
*                       WORK(IR) is N by N
*
                        LDWRKR = N
                     END IF
                     ITAU = IR + LDWRKR*N
                     IWORK = ITAU + N
*
*                    Compute A=Q*R, copying result to U
*                    (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
                     CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
*                    Copy R to WORK(IR), zeroing out below it
*
                     CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ),
     $                            LDWRKR )
                     CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
     $                            WORK( IR+1 ), LDWRKR )
*
*                    Generate Q in U
*                    (Workspace: need N*N + N + M, prefer N*N + N + M*NB)
*
                     CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IE = ITAU
                     ITAUQ = IE + N
                     ITAUP = ITAUQ + N
                     IWORK = ITAUP + N
*
*                    Bidiagonalize R in WORK(IR)
*                    (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
*
                     CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S,
     $                            WORK( IE ), WORK( ITAUQ ),
     $                            WORK( ITAUP ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
*
*                    Generate left bidiagonalizing vectors in WORK(IR)
*                    (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
*
                     CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
     $                            WORK( ITAUQ ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
                     IWORK = IE + N
*
*                    Perform bidiagonal QR iteration, computing left
*                    singular vectors of R in WORK(IR)
*                    (Workspace: need N*N + BDSPAC)
*
                     CALL DBDSQR( 'U', N, 0, N, 0, S, WORK( IE ),
     $                            DUM,
     $                            1, WORK( IR ), LDWRKR, DUM, 1,
     $                            WORK( IWORK ), INFO )
*
*                    Multiply Q in U by left singular vectors of R in
*                    WORK(IR), storing result in A
*                    (Workspace: need N*N)
*
                     CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU,
     $                           WORK( IR ), LDWRKR, ZERO, A, LDA )
*
*                    Copy left singular vectors of A from A to U
*
                     CALL DLACPY( 'F', M, N, A, LDA, U, LDU )
*
                  ELSE
*
*                    Insufficient workspace for a fast algorithm
*
                     ITAU = 1
                     IWORK = ITAU + N
*
*                    Compute A=Q*R, copying result to U
*                    (Workspace: need 2*N, prefer N + N*NB)
*
                     CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
*                    Generate Q in U
*                    (Workspace: need N + M, prefer N + M*NB)
*
                     CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IE = ITAU
                     ITAUQ = IE + N
                     ITAUP = ITAUQ + N
                     IWORK = ITAUP + N
*
*                    Zero out below R in A
*
                     IF( N .GT. 1 ) THEN
                        CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
     $                                A( 2, 1 ), LDA )
                     END IF
*
*                    Bidiagonalize R in A
*                    (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
                     CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
     $                            WORK( ITAUQ ), WORK( ITAUP ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Multiply Q in U by left bidiagonalizing vectors
*                    in A
*                    (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
                     CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
     $                            WORK( ITAUQ ), U, LDU, WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
                     IWORK = IE + N
*
*                    Perform bidiagonal QR iteration, computing left
*                    singular vectors of A in U
*                    (Workspace: need BDSPAC)
*
                     CALL DBDSQR( 'U', N, 0, M, 0, S, WORK( IE ),
     $                            DUM,
     $                            1, U, LDU, DUM, 1, WORK( IWORK ),
     $                            INFO )
*
                  END IF
*
               ELSE IF( WNTVO ) THEN
*
*                 Path 8 (M much larger than N, JOBU='A', JOBVT='O')
*                 M left singular vectors to be computed in U and
*                 N right singular vectors to be overwritten on A
*
                  IF( LWORK.GE.2*N*N+MAX( N+M, 4*N, BDSPAC ) ) THEN
*
*                    Sufficient workspace for a fast algorithm
*
                     IU = 1
                     IF( LWORK.GE.WRKBL+2*LDA*N ) THEN
*
*                       WORK(IU) is LDA by N and WORK(IR) is LDA by N
*
                        LDWRKU = LDA
                        IR = IU + LDWRKU*N
                        LDWRKR = LDA
                     ELSE IF( LWORK.GE.WRKBL+( LDA + N )*N ) THEN
*
*                       WORK(IU) is LDA by N and WORK(IR) is N by N
*
                        LDWRKU = LDA
                        IR = IU + LDWRKU*N
                        LDWRKR = N
                     ELSE
*
*                       WORK(IU) is N by N and WORK(IR) is N by N
*
                        LDWRKU = N
                        IR = IU + LDWRKU*N
                        LDWRKR = N
                     END IF
                     ITAU = IR + LDWRKR*N
                     IWORK = ITAU + N
*
*                    Compute A=Q*R, copying result to U
*                    (Workspace: need 2*N*N + 2*N, prefer 2*N*N + N + N*NB)
*
                     CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
*                    Generate Q in U
*                    (Workspace: need 2*N*N + N + M, prefer 2*N*N + N + M*NB)
*
                     CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Copy R to WORK(IU), zeroing out below it
*
                     CALL DLACPY( 'U', N, N, A, LDA, WORK( IU ),
     $                            LDWRKU )
                     CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
     $                            WORK( IU+1 ), LDWRKU )
                     IE = ITAU
                     ITAUQ = IE + N
                     ITAUP = ITAUQ + N
                     IWORK = ITAUP + N
*
*                    Bidiagonalize R in WORK(IU), copying result to
*                    WORK(IR)
*                    (Workspace: need 2*N*N + 4*N,
*                                prefer 2*N*N+3*N+2*N*NB)
*
                     CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S,
     $                            WORK( IE ), WORK( ITAUQ ),
     $                            WORK( ITAUP ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'U', N, N, WORK( IU ), LDWRKU,
     $                            WORK( IR ), LDWRKR )
*
*                    Generate left bidiagonalizing vectors in WORK(IU)
*                    (Workspace: need 2*N*N + 4*N, prefer 2*N*N + 3*N + N*NB)
*
                     CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
     $                            WORK( ITAUQ ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
*
*                    Generate right bidiagonalizing vectors in WORK(IR)
*                    (Workspace: need 2*N*N + 4*N-1,
*                                prefer 2*N*N+3*N+(N-1)*NB)
*
                     CALL DORGBR( 'P', N, N, N, WORK( IR ), LDWRKR,
     $                            WORK( ITAUP ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
                     IWORK = IE + N
*
*                    Perform bidiagonal QR iteration, computing left
*                    singular vectors of R in WORK(IU) and computing
*                    right singular vectors of R in WORK(IR)
*                    (Workspace: need 2*N*N + BDSPAC)
*
                     CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ),
     $                            WORK( IR ), LDWRKR, WORK( IU ),
     $                            LDWRKU, DUM, 1, WORK( IWORK ), INFO )
*
*                    Multiply Q in U by left singular vectors of R in
*                    WORK(IU), storing result in A
*                    (Workspace: need N*N)
*
                     CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU,
     $                           WORK( IU ), LDWRKU, ZERO, A, LDA )
*
*                    Copy left singular vectors of A from A to U
*
                     CALL DLACPY( 'F', M, N, A, LDA, U, LDU )
*
*                    Copy right singular vectors of R from WORK(IR) to A
*
                     CALL DLACPY( 'F', N, N, WORK( IR ), LDWRKR, A,
     $                            LDA )
*
                  ELSE
*
*                    Insufficient workspace for a fast algorithm
*
                     ITAU = 1
                     IWORK = ITAU + N
*
*                    Compute A=Q*R, copying result to U
*                    (Workspace: need 2*N, prefer N + N*NB)
*
                     CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
*                    Generate Q in U
*                    (Workspace: need N + M, prefer N + M*NB)
*
                     CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IE = ITAU
                     ITAUQ = IE + N
                     ITAUP = ITAUQ + N
                     IWORK = ITAUP + N
*
*                    Zero out below R in A
*
                     IF( N .GT. 1 ) THEN
                        CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
     $                                A( 2, 1 ), LDA )
                     END IF
*
*                    Bidiagonalize R in A
*                    (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
                     CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
     $                            WORK( ITAUQ ), WORK( ITAUP ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Multiply Q in U by left bidiagonalizing vectors
*                    in A
*                    (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
                     CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
     $                            WORK( ITAUQ ), U, LDU, WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
*
*                    Generate right bidiagonalizing vectors in A
*                    (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
                     CALL DORGBR( 'P', N, N, N, A, LDA,
     $                            WORK( ITAUP ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IWORK = IE + N
*
*                    Perform bidiagonal QR iteration, computing left
*                    singular vectors of A in U and computing right
*                    singular vectors of A in A
*                    (Workspace: need BDSPAC)
*
                     CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), A,
     $                            LDA, U, LDU, DUM, 1, WORK( IWORK ),
     $                            INFO )
*
                  END IF
*
               ELSE IF( WNTVAS ) THEN
*
*                 Path 9 (M much larger than N, JOBU='A', JOBVT='S'
*                         or 'A')
*                 M left singular vectors to be computed in U and
*                 N right singular vectors to be computed in VT
*
                  IF( LWORK.GE.N*N+MAX( N+M, 4*N, BDSPAC ) ) THEN
*
*                    Sufficient workspace for a fast algorithm
*
                     IU = 1
                     IF( LWORK.GE.WRKBL+LDA*N ) THEN
*
*                       WORK(IU) is LDA by N
*
                        LDWRKU = LDA
                     ELSE
*
*                       WORK(IU) is N by N
*
                        LDWRKU = N
                     END IF
                     ITAU = IU + LDWRKU*N
                     IWORK = ITAU + N
*
*                    Compute A=Q*R, copying result to U
*                    (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
                     CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
*                    Generate Q in U
*                    (Workspace: need N*N + N + M, prefer N*N + N + M*NB)
*
                     CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Copy R to WORK(IU), zeroing out below it
*
                     CALL DLACPY( 'U', N, N, A, LDA, WORK( IU ),
     $                            LDWRKU )
                     CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
     $                            WORK( IU+1 ), LDWRKU )
                     IE = ITAU
                     ITAUQ = IE + N
                     ITAUP = ITAUQ + N
                     IWORK = ITAUP + N
*
*                    Bidiagonalize R in WORK(IU), copying result to VT
*                    (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
*
                     CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S,
     $                            WORK( IE ), WORK( ITAUQ ),
     $                            WORK( ITAUP ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'U', N, N, WORK( IU ), LDWRKU, VT,
     $                            LDVT )
*
*                    Generate left bidiagonalizing vectors in WORK(IU)
*                    (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
*
                     CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
     $                            WORK( ITAUQ ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
*
*                    Generate right bidiagonalizing vectors in VT
*                    (Workspace: need N*N + 4*N-1,
*                                prefer N*N+3*N+(N-1)*NB)
*
                     CALL DORGBR( 'P', N, N, N, VT, LDVT,
     $                            WORK( ITAUP ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IWORK = IE + N
*
*                    Perform bidiagonal QR iteration, computing left
*                    singular vectors of R in WORK(IU) and computing
*                    right singular vectors of R in VT
*                    (Workspace: need N*N + BDSPAC)
*
                     CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ), VT,
     $                            LDVT, WORK( IU ), LDWRKU, DUM, 1,
     $                            WORK( IWORK ), INFO )
*
*                    Multiply Q in U by left singular vectors of R in
*                    WORK(IU), storing result in A
*                    (Workspace: need N*N)
*
                     CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU,
     $                           WORK( IU ), LDWRKU, ZERO, A, LDA )
*
*                    Copy left singular vectors of A from A to U
*
                     CALL DLACPY( 'F', M, N, A, LDA, U, LDU )
*
                  ELSE
*
*                    Insufficient workspace for a fast algorithm
*
                     ITAU = 1
                     IWORK = ITAU + N
*
*                    Compute A=Q*R, copying result to U
*                    (Workspace: need 2*N, prefer N + N*NB)
*
                     CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
*                    Generate Q in U
*                    (Workspace: need N + M, prefer N + M*NB)
*
                     CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Copy R from A to VT, zeroing out below it
*
                     CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
                     IF( N.GT.1 )
     $                  CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
     $                               VT( 2, 1 ), LDVT )
                     IE = ITAU
                     ITAUQ = IE + N
                     ITAUP = ITAUQ + N
                     IWORK = ITAUP + N
*
*                    Bidiagonalize R in VT
*                    (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
                     CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ),
     $                            WORK( ITAUQ ), WORK( ITAUP ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Multiply Q in U by left bidiagonalizing vectors
*                    in VT
*                    (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
                     CALL DORMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT,
     $                            WORK( ITAUQ ), U, LDU, WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
*
*                    Generate right bidiagonalizing vectors in VT
*                    (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
                     CALL DORGBR( 'P', N, N, N, VT, LDVT,
     $                            WORK( ITAUP ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IWORK = IE + N
*
*                    Perform bidiagonal QR iteration, computing left
*                    singular vectors of A in U and computing right
*                    singular vectors of A in VT
*                    (Workspace: need BDSPAC)
*
                     CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), VT,
     $                            LDVT, U, LDU, DUM, 1, WORK( IWORK ),
     $                            INFO )
*
                  END IF
*
               END IF
*
            END IF
*
         ELSE
*
*           M .LT. MNTHR
*
*           Path 10 (M at least N, but not much larger)
*           Reduce to bidiagonal form without QR decomposition
*
            IE = 1
            ITAUQ = IE + N
            ITAUP = ITAUQ + N
            IWORK = ITAUP + N
*
*           Bidiagonalize A
*           (Workspace: need 3*N + M, prefer 3*N + (M + N)*NB)
*
            CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
     $                   WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
     $                   IERR )
            IF( WNTUAS ) THEN
*
*              If left singular vectors desired in U, copy result to U
*              and generate left bidiagonalizing vectors in U
*              (Workspace: need 3*N + NCU, prefer 3*N + NCU*NB)
*
               CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
               IF( WNTUS )
     $            NCU = N
               IF( WNTUA )
     $            NCU = M
               CALL DORGBR( 'Q', M, NCU, N, U, LDU, WORK( ITAUQ ),
     $                      WORK( IWORK ), LWORK-IWORK+1, IERR )
            END IF
            IF( WNTVAS ) THEN
*
*              If right singular vectors desired in VT, copy result to
*              VT and generate right bidiagonalizing vectors in VT
*              (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
               CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
               CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
     $                      WORK( IWORK ), LWORK-IWORK+1, IERR )
            END IF
            IF( WNTUO ) THEN
*
*              If left singular vectors desired in A, generate left
*              bidiagonalizing vectors in A
*              (Workspace: need 4*N, prefer 3*N + N*NB)
*
               CALL DORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
     $                      WORK( IWORK ), LWORK-IWORK+1, IERR )
            END IF
            IF( WNTVO ) THEN
*
*              If right singular vectors desired in A, generate right
*              bidiagonalizing vectors in A
*              (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
               CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
     $                      WORK( IWORK ), LWORK-IWORK+1, IERR )
            END IF
            IWORK = IE + N
            IF( WNTUAS .OR. WNTUO )
     $         NRU = M
            IF( WNTUN )
     $         NRU = 0
            IF( WNTVAS .OR. WNTVO )
     $         NCVT = N
            IF( WNTVN )
     $         NCVT = 0
            IF( ( .NOT.WNTUO ) .AND. ( .NOT.WNTVO ) ) THEN
*
*              Perform bidiagonal QR iteration, if desired, computing
*              left singular vectors in U and computing right singular
*              vectors in VT
*              (Workspace: need BDSPAC)
*
               CALL DBDSQR( 'U', N, NCVT, NRU, 0, S, WORK( IE ), VT,
     $                      LDVT, U, LDU, DUM, 1, WORK( IWORK ), INFO )
            ELSE IF( ( .NOT.WNTUO ) .AND. WNTVO ) THEN
*
*              Perform bidiagonal QR iteration, if desired, computing
*              left singular vectors in U and computing right singular
*              vectors in A
*              (Workspace: need BDSPAC)
*
               CALL DBDSQR( 'U', N, NCVT, NRU, 0, S, WORK( IE ), A,
     $                      LDA,
     $                      U, LDU, DUM, 1, WORK( IWORK ), INFO )
            ELSE
*
*              Perform bidiagonal QR iteration, if desired, computing
*              left singular vectors in A and computing right singular
*              vectors in VT
*              (Workspace: need BDSPAC)
*
               CALL DBDSQR( 'U', N, NCVT, NRU, 0, S, WORK( IE ), VT,
     $                      LDVT, A, LDA, DUM, 1, WORK( IWORK ), INFO )
            END IF
*
         END IF
*
      ELSE
*
*        A has more columns than rows. If A has sufficiently more
*        columns than rows, first reduce using the LQ decomposition (if
*        sufficient workspace available)
*
         IF( N.GE.MNTHR ) THEN
*
            IF( WNTVN ) THEN
*
*              Path 1t(N much larger than M, JOBVT='N')
*              No right singular vectors to be computed
*
               ITAU = 1
               IWORK = ITAU + M
*
*              Compute A=L*Q
*              (Workspace: need 2*M, prefer M + M*NB)
*
               CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
     $                      WORK( IWORK ),
     $                      LWORK-IWORK+1, IERR )
*
*              Zero out above L
*
               CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ),
     $                      LDA )
               IE = 1
               ITAUQ = IE + M
               ITAUP = ITAUQ + M
               IWORK = ITAUP + M
*
*              Bidiagonalize L in A
*              (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
               CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
     $                      WORK( ITAUQ ),
     $                      WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
     $                      IERR )
               IF( WNTUO .OR. WNTUAS ) THEN
*
*                 If left singular vectors desired, generate Q
*                 (Workspace: need 4*M, prefer 3*M + M*NB)
*
                  CALL DORGBR( 'Q', M, M, M, A, LDA, WORK( ITAUQ ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
               END IF
               IWORK = IE + M
               NRU = 0
               IF( WNTUO .OR. WNTUAS )
     $            NRU = M
*
*              Perform bidiagonal QR iteration, computing left singular
*              vectors of A in A if desired
*              (Workspace: need BDSPAC)
*
               CALL DBDSQR( 'U', M, 0, NRU, 0, S, WORK( IE ), DUM, 1,
     $                      A,
     $                      LDA, DUM, 1, WORK( IWORK ), INFO )
*
*              If left singular vectors desired in U, copy them there
*
               IF( WNTUAS )
     $            CALL DLACPY( 'F', M, M, A, LDA, U, LDU )
*
            ELSE IF( WNTVO .AND. WNTUN ) THEN
*
*              Path 2t(N much larger than M, JOBU='N', JOBVT='O')
*              M right singular vectors to be overwritten on A and
*              no left singular vectors to be computed
*
               IF( LWORK.GE.M*M+MAX( 4*M, BDSPAC ) ) THEN
*
*                 Sufficient workspace for a fast algorithm
*
                  IR = 1
                  IF( LWORK.GE.MAX( WRKBL, LDA*N + M ) + LDA*M ) THEN
*
*                    WORK(IU) is LDA by N and WORK(IR) is LDA by M
*
                     LDWRKU = LDA
                     CHUNK = N
                     LDWRKR = LDA
                  ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N + M ) + M*M ) THEN
*
*                    WORK(IU) is LDA by N and WORK(IR) is M by M
*
                     LDWRKU = LDA
                     CHUNK = N
                     LDWRKR = M
                  ELSE
*
*                    WORK(IU) is M by CHUNK and WORK(IR) is M by M
*
                     LDWRKU = M
                     CHUNK = ( LWORK-M*M-M ) / M
                     LDWRKR = M
                  END IF
                  ITAU = IR + LDWRKR*M
                  IWORK = ITAU + M
*
*                 Compute A=L*Q
*                 (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
                  CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                 Copy L to WORK(IR) and zero out above it
*
                  CALL DLACPY( 'L', M, M, A, LDA, WORK( IR ),
     $                         LDWRKR )
                  CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
     $                         WORK( IR+LDWRKR ), LDWRKR )
*
*                 Generate Q in A
*                 (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
                  CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
                  IE = ITAU
                  ITAUQ = IE + M
                  ITAUP = ITAUQ + M
                  IWORK = ITAUP + M
*
*                 Bidiagonalize L in WORK(IR)
*                 (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
*
                  CALL DGEBRD( M, M, WORK( IR ), LDWRKR, S,
     $                         WORK( IE ),
     $                         WORK( ITAUQ ), WORK( ITAUP ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                 Generate right vectors bidiagonalizing L
*                 (Workspace: need M*M + 4*M-1, prefer M*M + 3*M + (M-1)*NB)
*
                  CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
     $                         WORK( ITAUP ), WORK( IWORK ),
     $                         LWORK-IWORK+1, IERR )
                  IWORK = IE + M
*
*                 Perform bidiagonal QR iteration, computing right
*                 singular vectors of L in WORK(IR)
*                 (Workspace: need M*M + BDSPAC)
*
                  CALL DBDSQR( 'U', M, M, 0, 0, S, WORK( IE ),
     $                         WORK( IR ), LDWRKR, DUM, 1, DUM, 1,
     $                         WORK( IWORK ), INFO )
                  IU = IE + M
*
*                 Multiply right singular vectors of L in WORK(IR) by Q
*                 in A, storing result in WORK(IU) and copying to A
*                 (Workspace: need M*M + 2*M, prefer M*M + M*N + M)
*
                  DO 30 I = 1, N, CHUNK
                     BLK = MIN( N-I+1, CHUNK )
                     CALL DGEMM( 'N', 'N', M, BLK, M, ONE,
     $                           WORK( IR ),
     $                           LDWRKR, A( 1, I ), LDA, ZERO,
     $                           WORK( IU ), LDWRKU )
                     CALL DLACPY( 'F', M, BLK, WORK( IU ), LDWRKU,
     $                            A( 1, I ), LDA )
   30             CONTINUE
*
               ELSE
*
*                 Insufficient workspace for a fast algorithm
*
                  IE = 1
                  ITAUQ = IE + M
                  ITAUP = ITAUQ + M
                  IWORK = ITAUP + M
*
*                 Bidiagonalize A
*                 (Workspace: need 3*M + N, prefer 3*M + (M + N)*NB)
*
                  CALL DGEBRD( M, N, A, LDA, S, WORK( IE ),
     $                         WORK( ITAUQ ), WORK( ITAUP ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                 Generate right vectors bidiagonalizing A
*                 (Workspace: need 4*M, prefer 3*M + M*NB)
*
                  CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
                  IWORK = IE + M
*
*                 Perform bidiagonal QR iteration, computing right
*                 singular vectors of A in A
*                 (Workspace: need BDSPAC)
*
                  CALL DBDSQR( 'L', M, N, 0, 0, S, WORK( IE ), A,
     $                         LDA,
     $                         DUM, 1, DUM, 1, WORK( IWORK ), INFO )
*
               END IF
*
            ELSE IF( WNTVO .AND. WNTUAS ) THEN
*
*              Path 3t(N much larger than M, JOBU='S' or 'A', JOBVT='O')
*              M right singular vectors to be overwritten on A and
*              M left singular vectors to be computed in U
*
               IF( LWORK.GE.M*M+MAX( 4*M, BDSPAC ) ) THEN
*
*                 Sufficient workspace for a fast algorithm
*
                  IR = 1
                  IF( LWORK.GE.MAX( WRKBL, LDA*N + M ) + LDA*M ) THEN
*
*                    WORK(IU) is LDA by N and WORK(IR) is LDA by M
*
                     LDWRKU = LDA
                     CHUNK = N
                     LDWRKR = LDA
                  ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N + M ) + M*M ) THEN
*
*                    WORK(IU) is LDA by N and WORK(IR) is M by M
*
                     LDWRKU = LDA
                     CHUNK = N
                     LDWRKR = M
                  ELSE
*
*                    WORK(IU) is M by CHUNK and WORK(IR) is M by M
*
                     LDWRKU = M
                     CHUNK = ( LWORK-M*M-M ) / M
                     LDWRKR = M
                  END IF
                  ITAU = IR + LDWRKR*M
                  IWORK = ITAU + M
*
*                 Compute A=L*Q
*                 (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
                  CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                 Copy L to U, zeroing about above it
*
                  CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
                  CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, U( 1, 2 ),
     $                         LDU )
*
*                 Generate Q in A
*                 (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
                  CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
                  IE = ITAU
                  ITAUQ = IE + M
                  ITAUP = ITAUQ + M
                  IWORK = ITAUP + M
*
*                 Bidiagonalize L in U, copying result to WORK(IR)
*                 (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
*
                  CALL DGEBRD( M, M, U, LDU, S, WORK( IE ),
     $                         WORK( ITAUQ ), WORK( ITAUP ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
                  CALL DLACPY( 'U', M, M, U, LDU, WORK( IR ),
     $                         LDWRKR )
*
*                 Generate right vectors bidiagonalizing L in WORK(IR)
*                 (Workspace: need M*M + 4*M-1, prefer M*M + 3*M + (M-1)*NB)
*
                  CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
     $                         WORK( ITAUP ), WORK( IWORK ),
     $                         LWORK-IWORK+1, IERR )
*
*                 Generate left vectors bidiagonalizing L in U
*                 (Workspace: need M*M + 4*M, prefer M*M + 3*M + M*NB)
*
                  CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
                  IWORK = IE + M
*
*                 Perform bidiagonal QR iteration, computing left
*                 singular vectors of L in U, and computing right
*                 singular vectors of L in WORK(IR)
*                 (Workspace: need M*M + BDSPAC)
*
                  CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
     $                         WORK( IR ), LDWRKR, U, LDU, DUM, 1,
     $                         WORK( IWORK ), INFO )
                  IU = IE + M
*
*                 Multiply right singular vectors of L in WORK(IR) by Q
*                 in A, storing result in WORK(IU) and copying to A
*                 (Workspace: need M*M + 2*M, prefer M*M + M*N + M))
*
                  DO 40 I = 1, N, CHUNK
                     BLK = MIN( N-I+1, CHUNK )
                     CALL DGEMM( 'N', 'N', M, BLK, M, ONE,
     $                           WORK( IR ),
     $                           LDWRKR, A( 1, I ), LDA, ZERO,
     $                           WORK( IU ), LDWRKU )
                     CALL DLACPY( 'F', M, BLK, WORK( IU ), LDWRKU,
     $                            A( 1, I ), LDA )
   40             CONTINUE
*
               ELSE
*
*                 Insufficient workspace for a fast algorithm
*
                  ITAU = 1
                  IWORK = ITAU + M
*
*                 Compute A=L*Q
*                 (Workspace: need 2*M, prefer M + M*NB)
*
                  CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                 Copy L to U, zeroing out above it
*
                  CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
                  CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, U( 1, 2 ),
     $                         LDU )
*
*                 Generate Q in A
*                 (Workspace: need 2*M, prefer M + M*NB)
*
                  CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
                  IE = ITAU
                  ITAUQ = IE + M
                  ITAUP = ITAUQ + M
                  IWORK = ITAUP + M
*
*                 Bidiagonalize L in U
*                 (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
                  CALL DGEBRD( M, M, U, LDU, S, WORK( IE ),
     $                         WORK( ITAUQ ), WORK( ITAUP ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                 Multiply right vectors bidiagonalizing L by Q in A
*                 (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
                  CALL DORMBR( 'P', 'L', 'T', M, N, M, U, LDU,
     $                         WORK( ITAUP ), A, LDA, WORK( IWORK ),
     $                         LWORK-IWORK+1, IERR )
*
*                 Generate left vectors bidiagonalizing L in U
*                 (Workspace: need 4*M, prefer 3*M + M*NB)
*
                  CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
     $                         WORK( IWORK ), LWORK-IWORK+1, IERR )
                  IWORK = IE + M
*
*                 Perform bidiagonal QR iteration, computing left
*                 singular vectors of A in U and computing right
*                 singular vectors of A in A
*                 (Workspace: need BDSPAC)
*
                  CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), A,
     $                         LDA,
     $                         U, LDU, DUM, 1, WORK( IWORK ), INFO )
*
               END IF
*
            ELSE IF( WNTVS ) THEN
*
               IF( WNTUN ) THEN
*
*                 Path 4t(N much larger than M, JOBU='N', JOBVT='S')
*                 M right singular vectors to be computed in VT and
*                 no left singular vectors to be computed
*
                  IF( LWORK.GE.M*M+MAX( 4*M, BDSPAC ) ) THEN
*
*                    Sufficient workspace for a fast algorithm
*
                     IR = 1
                     IF( LWORK.GE.WRKBL+LDA*M ) THEN
*
*                       WORK(IR) is LDA by M
*
                        LDWRKR = LDA
                     ELSE
*
*                       WORK(IR) is M by M
*
                        LDWRKR = M
                     END IF
                     ITAU = IR + LDWRKR*M
                     IWORK = ITAU + M
*
*                    Compute A=L*Q
*                    (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
                     CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Copy L to WORK(IR), zeroing out above it
*
                     CALL DLACPY( 'L', M, M, A, LDA, WORK( IR ),
     $                            LDWRKR )
                     CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
     $                            WORK( IR+LDWRKR ), LDWRKR )
*
*                    Generate Q in A
*                    (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
                     CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IE = ITAU
                     ITAUQ = IE + M
                     ITAUP = ITAUQ + M
                     IWORK = ITAUP + M
*
*                    Bidiagonalize L in WORK(IR)
*                    (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
*
                     CALL DGEBRD( M, M, WORK( IR ), LDWRKR, S,
     $                            WORK( IE ), WORK( ITAUQ ),
     $                            WORK( ITAUP ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
*
*                    Generate right vectors bidiagonalizing L in
*                    WORK(IR)
*                    (Workspace: need M*M + 4*M, prefer M*M + 3*M + (M-1)*NB)
*
                     CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
     $                            WORK( ITAUP ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
                     IWORK = IE + M
*
*                    Perform bidiagonal QR iteration, computing right
*                    singular vectors of L in WORK(IR)
*                    (Workspace: need M*M + BDSPAC)
*
                     CALL DBDSQR( 'U', M, M, 0, 0, S, WORK( IE ),
     $                            WORK( IR ), LDWRKR, DUM, 1, DUM, 1,
     $                            WORK( IWORK ), INFO )
*
*                    Multiply right singular vectors of L in WORK(IR) by
*                    Q in A, storing result in VT
*                    (Workspace: need M*M)
*
                     CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IR ),
     $                           LDWRKR, A, LDA, ZERO, VT, LDVT )
*
                  ELSE
*
*                    Insufficient workspace for a fast algorithm
*
                     ITAU = 1
                     IWORK = ITAU + M
*
*                    Compute A=L*Q
*                    (Workspace: need 2*M, prefer M + M*NB)
*
                     CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Copy result to VT
*
                     CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
*                    Generate Q in VT
*                    (Workspace: need 2*M, prefer M + M*NB)
*
                     CALL DORGLQ( M, N, M, VT, LDVT, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IE = ITAU
                     ITAUQ = IE + M
                     ITAUP = ITAUQ + M
                     IWORK = ITAUP + M
*
*                    Zero out above L in A
*
                     CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1,
     $                            2 ),
     $                            LDA )
*
*                    Bidiagonalize L in A
*                    (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
                     CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
     $                            WORK( ITAUQ ), WORK( ITAUP ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Multiply right vectors bidiagonalizing L by Q in VT
*                    (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
                     CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA,
     $                            WORK( ITAUP ), VT, LDVT,
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IWORK = IE + M
*
*                    Perform bidiagonal QR iteration, computing right
*                    singular vectors of A in VT
*                    (Workspace: need BDSPAC)
*
                     CALL DBDSQR( 'U', M, N, 0, 0, S, WORK( IE ), VT,
     $                            LDVT, DUM, 1, DUM, 1, WORK( IWORK ),
     $                            INFO )
*
                  END IF
*
               ELSE IF( WNTUO ) THEN
*
*                 Path 5t(N much larger than M, JOBU='O', JOBVT='S')
*                 M right singular vectors to be computed in VT and
*                 M left singular vectors to be overwritten on A
*
                  IF( LWORK.GE.2*M*M+MAX( 4*M, BDSPAC ) ) THEN
*
*                    Sufficient workspace for a fast algorithm
*
                     IU = 1
                     IF( LWORK.GE.WRKBL+2*LDA*M ) THEN
*
*                       WORK(IU) is LDA by M and WORK(IR) is LDA by M
*
                        LDWRKU = LDA
                        IR = IU + LDWRKU*M
                        LDWRKR = LDA
                     ELSE IF( LWORK.GE.WRKBL+( LDA + M )*M ) THEN
*
*                       WORK(IU) is LDA by M and WORK(IR) is M by M
*
                        LDWRKU = LDA
                        IR = IU + LDWRKU*M
                        LDWRKR = M
                     ELSE
*
*                       WORK(IU) is M by M and WORK(IR) is M by M
*
                        LDWRKU = M
                        IR = IU + LDWRKU*M
                        LDWRKR = M
                     END IF
                     ITAU = IR + LDWRKR*M
                     IWORK = ITAU + M
*
*                    Compute A=L*Q
*                    (Workspace: need 2*M*M + 2*M, prefer 2*M*M + M + M*NB)
*
                     CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Copy L to WORK(IU), zeroing out below it
*
                     CALL DLACPY( 'L', M, M, A, LDA, WORK( IU ),
     $                            LDWRKU )
                     CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
     $                            WORK( IU+LDWRKU ), LDWRKU )
*
*                    Generate Q in A
*                    (Workspace: need 2*M*M + 2*M, prefer 2*M*M + M + M*NB)
*
                     CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IE = ITAU
                     ITAUQ = IE + M
                     ITAUP = ITAUQ + M
                     IWORK = ITAUP + M
*
*                    Bidiagonalize L in WORK(IU), copying result to
*                    WORK(IR)
*                    (Workspace: need 2*M*M + 4*M,
*                                prefer 2*M*M+3*M+2*M*NB)
*
                     CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S,
     $                            WORK( IE ), WORK( ITAUQ ),
     $                            WORK( ITAUP ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'L', M, M, WORK( IU ), LDWRKU,
     $                            WORK( IR ), LDWRKR )
*
*                    Generate right bidiagonalizing vectors in WORK(IU)
*                    (Workspace: need 2*M*M + 4*M-1,
*                                prefer 2*M*M+3*M+(M-1)*NB)
*
                     CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
     $                            WORK( ITAUP ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
*
*                    Generate left bidiagonalizing vectors in WORK(IR)
*                    (Workspace: need 2*M*M + 4*M, prefer 2*M*M + 3*M + M*NB)
*
                     CALL DORGBR( 'Q', M, M, M, WORK( IR ), LDWRKR,
     $                            WORK( ITAUQ ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
                     IWORK = IE + M
*
*                    Perform bidiagonal QR iteration, computing left
*                    singular vectors of L in WORK(IR) and computing
*                    right singular vectors of L in WORK(IU)
*                    (Workspace: need 2*M*M + BDSPAC)
*
                     CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
     $                            WORK( IU ), LDWRKU, WORK( IR ),
     $                            LDWRKR, DUM, 1, WORK( IWORK ), INFO )
*
*                    Multiply right singular vectors of L in WORK(IU) by
*                    Q in A, storing result in VT
*                    (Workspace: need M*M)
*
                     CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IU ),
     $                           LDWRKU, A, LDA, ZERO, VT, LDVT )
*
*                    Copy left singular vectors of L to A
*                    (Workspace: need M*M)
*
                     CALL DLACPY( 'F', M, M, WORK( IR ), LDWRKR, A,
     $                            LDA )
*
                  ELSE
*
*                    Insufficient workspace for a fast algorithm
*
                     ITAU = 1
                     IWORK = ITAU + M
*
*                    Compute A=L*Q, copying result to VT
*                    (Workspace: need 2*M, prefer M + M*NB)
*
                     CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
*                    Generate Q in VT
*                    (Workspace: need 2*M, prefer M + M*NB)
*
                     CALL DORGLQ( M, N, M, VT, LDVT, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IE = ITAU
                     ITAUQ = IE + M
                     ITAUP = ITAUQ + M
                     IWORK = ITAUP + M
*
*                    Zero out above L in A
*
                     CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1,
     $                            2 ),
     $                            LDA )
*
*                    Bidiagonalize L in A
*                    (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
                     CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
     $                            WORK( ITAUQ ), WORK( ITAUP ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Multiply right vectors bidiagonalizing L by Q in VT
*                    (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
                     CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA,
     $                            WORK( ITAUP ), VT, LDVT,
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Generate left bidiagonalizing vectors of L in A
*                    (Workspace: need 4*M, prefer 3*M + M*NB)
*
                     CALL DORGBR( 'Q', M, M, M, A, LDA,
     $                            WORK( ITAUQ ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IWORK = IE + M
*
*                    Perform bidiagonal QR iteration, compute left
*                    singular vectors of A in A and compute right
*                    singular vectors of A in VT
*                    (Workspace: need BDSPAC)
*
                     CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), VT,
     $                            LDVT, A, LDA, DUM, 1, WORK( IWORK ),
     $                            INFO )
*
                  END IF
*
               ELSE IF( WNTUAS ) THEN
*
*                 Path 6t(N much larger than M, JOBU='S' or 'A',
*                         JOBVT='S')
*                 M right singular vectors to be computed in VT and
*                 M left singular vectors to be computed in U
*
                  IF( LWORK.GE.M*M+MAX( 4*M, BDSPAC ) ) THEN
*
*                    Sufficient workspace for a fast algorithm
*
                     IU = 1
                     IF( LWORK.GE.WRKBL+LDA*M ) THEN
*
*                       WORK(IU) is LDA by N
*
                        LDWRKU = LDA
                     ELSE
*
*                       WORK(IU) is LDA by M
*
                        LDWRKU = M
                     END IF
                     ITAU = IU + LDWRKU*M
                     IWORK = ITAU + M
*
*                    Compute A=L*Q
*                    (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
                     CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Copy L to WORK(IU), zeroing out above it
*
                     CALL DLACPY( 'L', M, M, A, LDA, WORK( IU ),
     $                            LDWRKU )
                     CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
     $                            WORK( IU+LDWRKU ), LDWRKU )
*
*                    Generate Q in A
*                    (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
                     CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IE = ITAU
                     ITAUQ = IE + M
                     ITAUP = ITAUQ + M
                     IWORK = ITAUP + M
*
*                    Bidiagonalize L in WORK(IU), copying result to U
*                    (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
*
                     CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S,
     $                            WORK( IE ), WORK( ITAUQ ),
     $                            WORK( ITAUP ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'L', M, M, WORK( IU ), LDWRKU, U,
     $                            LDU )
*
*                    Generate right bidiagonalizing vectors in WORK(IU)
*                    (Workspace: need M*M + 4*M-1,
*                                prefer M*M+3*M+(M-1)*NB)
*
                     CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
     $                            WORK( ITAUP ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
*
*                    Generate left bidiagonalizing vectors in U
*                    (Workspace: need M*M + 4*M, prefer M*M + 3*M + M*NB)
*
                     CALL DORGBR( 'Q', M, M, M, U, LDU,
     $                            WORK( ITAUQ ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IWORK = IE + M
*
*                    Perform bidiagonal QR iteration, computing left
*                    singular vectors of L in U and computing right
*                    singular vectors of L in WORK(IU)
*                    (Workspace: need M*M + BDSPAC)
*
                     CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
     $                            WORK( IU ), LDWRKU, U, LDU, DUM, 1,
     $                            WORK( IWORK ), INFO )
*
*                    Multiply right singular vectors of L in WORK(IU) by
*                    Q in A, storing result in VT
*                    (Workspace: need M*M)
*
                     CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IU ),
     $                           LDWRKU, A, LDA, ZERO, VT, LDVT )
*
                  ELSE
*
*                    Insufficient workspace for a fast algorithm
*
                     ITAU = 1
                     IWORK = ITAU + M
*
*                    Compute A=L*Q, copying result to VT
*                    (Workspace: need 2*M, prefer M + M*NB)
*
                     CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
*                    Generate Q in VT
*                    (Workspace: need 2*M, prefer M + M*NB)
*
                     CALL DORGLQ( M, N, M, VT, LDVT, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Copy L to U, zeroing out above it
*
                     CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
                     CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, U( 1,
     $                            2 ),
     $                            LDU )
                     IE = ITAU
                     ITAUQ = IE + M
                     ITAUP = ITAUQ + M
                     IWORK = ITAUP + M
*
*                    Bidiagonalize L in U
*                    (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
                     CALL DGEBRD( M, M, U, LDU, S, WORK( IE ),
     $                            WORK( ITAUQ ), WORK( ITAUP ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Multiply right bidiagonalizing vectors in U by Q
*                    in VT
*                    (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
                     CALL DORMBR( 'P', 'L', 'T', M, N, M, U, LDU,
     $                            WORK( ITAUP ), VT, LDVT,
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Generate left bidiagonalizing vectors in U
*                    (Workspace: need 4*M, prefer 3*M + M*NB)
*
                     CALL DORGBR( 'Q', M, M, M, U, LDU,
     $                            WORK( ITAUQ ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IWORK = IE + M
*
*                    Perform bidiagonal QR iteration, computing left
*                    singular vectors of A in U and computing right
*                    singular vectors of A in VT
*                    (Workspace: need BDSPAC)
*
                     CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), VT,
     $                            LDVT, U, LDU, DUM, 1, WORK( IWORK ),
     $                            INFO )
*
                  END IF
*
               END IF
*
            ELSE IF( WNTVA ) THEN
*
               IF( WNTUN ) THEN
*
*                 Path 7t(N much larger than M, JOBU='N', JOBVT='A')
*                 N right singular vectors to be computed in VT and
*                 no left singular vectors to be computed
*
                  IF( LWORK.GE.M*M+MAX( N + M, 4*M, BDSPAC ) ) THEN
*
*                    Sufficient workspace for a fast algorithm
*
                     IR = 1
                     IF( LWORK.GE.WRKBL+LDA*M ) THEN
*
*                       WORK(IR) is LDA by M
*
                        LDWRKR = LDA
                     ELSE
*
*                       WORK(IR) is M by M
*
                        LDWRKR = M
                     END IF
                     ITAU = IR + LDWRKR*M
                     IWORK = ITAU + M
*
*                    Compute A=L*Q, copying result to VT
*                    (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
                     CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
*                    Copy L to WORK(IR), zeroing out above it
*
                     CALL DLACPY( 'L', M, M, A, LDA, WORK( IR ),
     $                            LDWRKR )
                     CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
     $                            WORK( IR+LDWRKR ), LDWRKR )
*
*                    Generate Q in VT
*                    (Workspace: need M*M + M + N, prefer M*M + M + N*NB)
*
                     CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IE = ITAU
                     ITAUQ = IE + M
                     ITAUP = ITAUQ + M
                     IWORK = ITAUP + M
*
*                    Bidiagonalize L in WORK(IR)
*                    (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
*
                     CALL DGEBRD( M, M, WORK( IR ), LDWRKR, S,
     $                            WORK( IE ), WORK( ITAUQ ),
     $                            WORK( ITAUP ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
*
*                    Generate right bidiagonalizing vectors in WORK(IR)
*                    (Workspace: need M*M + 4*M-1,
*                                prefer M*M+3*M+(M-1)*NB)
*
                     CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
     $                            WORK( ITAUP ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
                     IWORK = IE + M
*
*                    Perform bidiagonal QR iteration, computing right
*                    singular vectors of L in WORK(IR)
*                    (Workspace: need M*M + BDSPAC)
*
                     CALL DBDSQR( 'U', M, M, 0, 0, S, WORK( IE ),
     $                            WORK( IR ), LDWRKR, DUM, 1, DUM, 1,
     $                            WORK( IWORK ), INFO )
*
*                    Multiply right singular vectors of L in WORK(IR) by
*                    Q in VT, storing result in A
*                    (Workspace: need M*M)
*
                     CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IR ),
     $                           LDWRKR, VT, LDVT, ZERO, A, LDA )
*
*                    Copy right singular vectors of A from A to VT
*
                     CALL DLACPY( 'F', M, N, A, LDA, VT, LDVT )
*
                  ELSE
*
*                    Insufficient workspace for a fast algorithm
*
                     ITAU = 1
                     IWORK = ITAU + M
*
*                    Compute A=L*Q, copying result to VT
*                    (Workspace: need 2*M, prefer M + M*NB)
*
                     CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
*                    Generate Q in VT
*                    (Workspace: need M + N, prefer M + N*NB)
*
                     CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IE = ITAU
                     ITAUQ = IE + M
                     ITAUP = ITAUQ + M
                     IWORK = ITAUP + M
*
*                    Zero out above L in A
*
                     CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1,
     $                            2 ),
     $                            LDA )
*
*                    Bidiagonalize L in A
*                    (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
                     CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
     $                            WORK( ITAUQ ), WORK( ITAUP ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Multiply right bidiagonalizing vectors in A by Q
*                    in VT
*                    (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
                     CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA,
     $                            WORK( ITAUP ), VT, LDVT,
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IWORK = IE + M
*
*                    Perform bidiagonal QR iteration, computing right
*                    singular vectors of A in VT
*                    (Workspace: need BDSPAC)
*
                     CALL DBDSQR( 'U', M, N, 0, 0, S, WORK( IE ), VT,
     $                            LDVT, DUM, 1, DUM, 1, WORK( IWORK ),
     $                            INFO )
*
                  END IF
*
               ELSE IF( WNTUO ) THEN
*
*                 Path 8t(N much larger than M, JOBU='O', JOBVT='A')
*                 N right singular vectors to be computed in VT and
*                 M left singular vectors to be overwritten on A
*
                  IF( LWORK.GE.2*M*M+MAX( N + M, 4*M, BDSPAC ) ) THEN
*
*                    Sufficient workspace for a fast algorithm
*
                     IU = 1
                     IF( LWORK.GE.WRKBL+2*LDA*M ) THEN
*
*                       WORK(IU) is LDA by M and WORK(IR) is LDA by M
*
                        LDWRKU = LDA
                        IR = IU + LDWRKU*M
                        LDWRKR = LDA
                     ELSE IF( LWORK.GE.WRKBL+( LDA + M )*M ) THEN
*
*                       WORK(IU) is LDA by M and WORK(IR) is M by M
*
                        LDWRKU = LDA
                        IR = IU + LDWRKU*M
                        LDWRKR = M
                     ELSE
*
*                       WORK(IU) is M by M and WORK(IR) is M by M
*
                        LDWRKU = M
                        IR = IU + LDWRKU*M
                        LDWRKR = M
                     END IF
                     ITAU = IR + LDWRKR*M
                     IWORK = ITAU + M
*
*                    Compute A=L*Q, copying result to VT
*                    (Workspace: need 2*M*M + 2*M, prefer 2*M*M + M + M*NB)
*
                     CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
*                    Generate Q in VT
*                    (Workspace: need 2*M*M + M + N, prefer 2*M*M + M + N*NB)
*
                     CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Copy L to WORK(IU), zeroing out above it
*
                     CALL DLACPY( 'L', M, M, A, LDA, WORK( IU ),
     $                            LDWRKU )
                     CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
     $                            WORK( IU+LDWRKU ), LDWRKU )
                     IE = ITAU
                     ITAUQ = IE + M
                     ITAUP = ITAUQ + M
                     IWORK = ITAUP + M
*
*                    Bidiagonalize L in WORK(IU), copying result to
*                    WORK(IR)
*                    (Workspace: need 2*M*M + 4*M,
*                                prefer 2*M*M+3*M+2*M*NB)
*
                     CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S,
     $                            WORK( IE ), WORK( ITAUQ ),
     $                            WORK( ITAUP ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'L', M, M, WORK( IU ), LDWRKU,
     $                            WORK( IR ), LDWRKR )
*
*                    Generate right bidiagonalizing vectors in WORK(IU)
*                    (Workspace: need 2*M*M + 4*M-1,
*                                prefer 2*M*M+3*M+(M-1)*NB)
*
                     CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
     $                            WORK( ITAUP ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
*
*                    Generate left bidiagonalizing vectors in WORK(IR)
*                    (Workspace: need 2*M*M + 4*M, prefer 2*M*M + 3*M + M*NB)
*
                     CALL DORGBR( 'Q', M, M, M, WORK( IR ), LDWRKR,
     $                            WORK( ITAUQ ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
                     IWORK = IE + M
*
*                    Perform bidiagonal QR iteration, computing left
*                    singular vectors of L in WORK(IR) and computing
*                    right singular vectors of L in WORK(IU)
*                    (Workspace: need 2*M*M + BDSPAC)
*
                     CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
     $                            WORK( IU ), LDWRKU, WORK( IR ),
     $                            LDWRKR, DUM, 1, WORK( IWORK ), INFO )
*
*                    Multiply right singular vectors of L in WORK(IU) by
*                    Q in VT, storing result in A
*                    (Workspace: need M*M)
*
                     CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IU ),
     $                           LDWRKU, VT, LDVT, ZERO, A, LDA )
*
*                    Copy right singular vectors of A from A to VT
*
                     CALL DLACPY( 'F', M, N, A, LDA, VT, LDVT )
*
*                    Copy left singular vectors of A from WORK(IR) to A
*
                     CALL DLACPY( 'F', M, M, WORK( IR ), LDWRKR, A,
     $                            LDA )
*
                  ELSE
*
*                    Insufficient workspace for a fast algorithm
*
                     ITAU = 1
                     IWORK = ITAU + M
*
*                    Compute A=L*Q, copying result to VT
*                    (Workspace: need 2*M, prefer M + M*NB)
*
                     CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
*                    Generate Q in VT
*                    (Workspace: need M + N, prefer M + N*NB)
*
                     CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IE = ITAU
                     ITAUQ = IE + M
                     ITAUP = ITAUQ + M
                     IWORK = ITAUP + M
*
*                    Zero out above L in A
*
                     CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1,
     $                            2 ),
     $                            LDA )
*
*                    Bidiagonalize L in A
*                    (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
                     CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
     $                            WORK( ITAUQ ), WORK( ITAUP ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Multiply right bidiagonalizing vectors in A by Q
*                    in VT
*                    (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
                     CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA,
     $                            WORK( ITAUP ), VT, LDVT,
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Generate left bidiagonalizing vectors in A
*                    (Workspace: need 4*M, prefer 3*M + M*NB)
*
                     CALL DORGBR( 'Q', M, M, M, A, LDA,
     $                            WORK( ITAUQ ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IWORK = IE + M
*
*                    Perform bidiagonal QR iteration, computing left
*                    singular vectors of A in A and computing right
*                    singular vectors of A in VT
*                    (Workspace: need BDSPAC)
*
                     CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), VT,
     $                            LDVT, A, LDA, DUM, 1, WORK( IWORK ),
     $                            INFO )
*
                  END IF
*
               ELSE IF( WNTUAS ) THEN
*
*                 Path 9t(N much larger than M, JOBU='S' or 'A',
*                         JOBVT='A')
*                 N right singular vectors to be computed in VT and
*                 M left singular vectors to be computed in U
*
                  IF( LWORK.GE.M*M+MAX( N + M, 4*M, BDSPAC ) ) THEN
*
*                    Sufficient workspace for a fast algorithm
*
                     IU = 1
                     IF( LWORK.GE.WRKBL+LDA*M ) THEN
*
*                       WORK(IU) is LDA by M
*
                        LDWRKU = LDA
                     ELSE
*
*                       WORK(IU) is M by M
*
                        LDWRKU = M
                     END IF
                     ITAU = IU + LDWRKU*M
                     IWORK = ITAU + M
*
*                    Compute A=L*Q, copying result to VT
*                    (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
                     CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
*                    Generate Q in VT
*                    (Workspace: need M*M + M + N, prefer M*M + M + N*NB)
*
                     CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Copy L to WORK(IU), zeroing out above it
*
                     CALL DLACPY( 'L', M, M, A, LDA, WORK( IU ),
     $                            LDWRKU )
                     CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
     $                            WORK( IU+LDWRKU ), LDWRKU )
                     IE = ITAU
                     ITAUQ = IE + M
                     ITAUP = ITAUQ + M
                     IWORK = ITAUP + M
*
*                    Bidiagonalize L in WORK(IU), copying result to U
*                    (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
*
                     CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S,
     $                            WORK( IE ), WORK( ITAUQ ),
     $                            WORK( ITAUP ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'L', M, M, WORK( IU ), LDWRKU, U,
     $                            LDU )
*
*                    Generate right bidiagonalizing vectors in WORK(IU)
*                    (Workspace: need M*M + 4*M, prefer M*M + 3*M + (M-1)*NB)
*
                     CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
     $                            WORK( ITAUP ), WORK( IWORK ),
     $                            LWORK-IWORK+1, IERR )
*
*                    Generate left bidiagonalizing vectors in U
*                    (Workspace: need M*M + 4*M, prefer M*M + 3*M + M*NB)
*
                     CALL DORGBR( 'Q', M, M, M, U, LDU,
     $                            WORK( ITAUQ ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IWORK = IE + M
*
*                    Perform bidiagonal QR iteration, computing left
*                    singular vectors of L in U and computing right
*                    singular vectors of L in WORK(IU)
*                    (Workspace: need M*M + BDSPAC)
*
                     CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
     $                            WORK( IU ), LDWRKU, U, LDU, DUM, 1,
     $                            WORK( IWORK ), INFO )
*
*                    Multiply right singular vectors of L in WORK(IU) by
*                    Q in VT, storing result in A
*                    (Workspace: need M*M)
*
                     CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IU ),
     $                           LDWRKU, VT, LDVT, ZERO, A, LDA )
*
*                    Copy right singular vectors of A from A to VT
*
                     CALL DLACPY( 'F', M, N, A, LDA, VT, LDVT )
*
                  ELSE
*
*                    Insufficient workspace for a fast algorithm
*
                     ITAU = 1
                     IWORK = ITAU + M
*
*                    Compute A=L*Q, copying result to VT
*                    (Workspace: need 2*M, prefer M + M*NB)
*
                     CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
*                    Generate Q in VT
*                    (Workspace: need M + N, prefer M + N*NB)
*
                     CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Copy L to U, zeroing out above it
*
                     CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
                     CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, U( 1,
     $                            2 ),
     $                            LDU )
                     IE = ITAU
                     ITAUQ = IE + M
                     ITAUP = ITAUQ + M
                     IWORK = ITAUP + M
*
*                    Bidiagonalize L in U
*                    (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
                     CALL DGEBRD( M, M, U, LDU, S, WORK( IE ),
     $                            WORK( ITAUQ ), WORK( ITAUP ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Multiply right bidiagonalizing vectors in U by Q
*                    in VT
*                    (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
                     CALL DORMBR( 'P', 'L', 'T', M, N, M, U, LDU,
     $                            WORK( ITAUP ), VT, LDVT,
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
*
*                    Generate left bidiagonalizing vectors in U
*                    (Workspace: need 4*M, prefer 3*M + M*NB)
*
                     CALL DORGBR( 'Q', M, M, M, U, LDU,
     $                            WORK( ITAUQ ),
     $                            WORK( IWORK ), LWORK-IWORK+1, IERR )
                     IWORK = IE + M
*
*                    Perform bidiagonal QR iteration, computing left
*                    singular vectors of A in U and computing right
*                    singular vectors of A in VT
*                    (Workspace: need BDSPAC)
*
                     CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), VT,
     $                            LDVT, U, LDU, DUM, 1, WORK( IWORK ),
     $                            INFO )
*
                  END IF
*
               END IF
*
            END IF
*
         ELSE
*
*           N .LT. MNTHR
*
*           Path 10t(N greater than M, but not much larger)
*           Reduce to bidiagonal form without LQ decomposition
*
            IE = 1
            ITAUQ = IE + M
            ITAUP = ITAUQ + M
            IWORK = ITAUP + M
*
*           Bidiagonalize A
*           (Workspace: need 3*M + N, prefer 3*M + (M + N)*NB)
*
            CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
     $                   WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
     $                   IERR )
            IF( WNTUAS ) THEN
*
*              If left singular vectors desired in U, copy result to U
*              and generate left bidiagonalizing vectors in U
*              (Workspace: need 4*M-1, prefer 3*M + (M-1)*NB)
*
               CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
               CALL DORGBR( 'Q', M, M, N, U, LDU, WORK( ITAUQ ),
     $                      WORK( IWORK ), LWORK-IWORK+1, IERR )
            END IF
            IF( WNTVAS ) THEN
*
*              If right singular vectors desired in VT, copy result to
*              VT and generate right bidiagonalizing vectors in VT
*              (Workspace: need 3*M + NRVT, prefer 3*M + NRVT*NB)
*
               CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
               IF( WNTVA )
     $            NRVT = N
               IF( WNTVS )
     $            NRVT = M
               CALL DORGBR( 'P', NRVT, N, M, VT, LDVT, WORK( ITAUP ),
     $                      WORK( IWORK ), LWORK-IWORK+1, IERR )
            END IF
            IF( WNTUO ) THEN
*
*              If left singular vectors desired in A, generate left
*              bidiagonalizing vectors in A
*              (Workspace: need 4*M-1, prefer 3*M + (M-1)*NB)
*
               CALL DORGBR( 'Q', M, M, N, A, LDA, WORK( ITAUQ ),
     $                      WORK( IWORK ), LWORK-IWORK+1, IERR )
            END IF
            IF( WNTVO ) THEN
*
*              If right singular vectors desired in A, generate right
*              bidiagonalizing vectors in A
*              (Workspace: need 4*M, prefer 3*M + M*NB)
*
               CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
     $                      WORK( IWORK ), LWORK-IWORK+1, IERR )
            END IF
            IWORK = IE + M
            IF( WNTUAS .OR. WNTUO )
     $         NRU = M
            IF( WNTUN )
     $         NRU = 0
            IF( WNTVAS .OR. WNTVO )
     $         NCVT = N
            IF( WNTVN )
     $         NCVT = 0
            IF( ( .NOT.WNTUO ) .AND. ( .NOT.WNTVO ) ) THEN
*
*              Perform bidiagonal QR iteration, if desired, computing
*              left singular vectors in U and computing right singular
*              vectors in VT
*              (Workspace: need BDSPAC)
*
               CALL DBDSQR( 'L', M, NCVT, NRU, 0, S, WORK( IE ), VT,
     $                      LDVT, U, LDU, DUM, 1, WORK( IWORK ), INFO )
            ELSE IF( ( .NOT.WNTUO ) .AND. WNTVO ) THEN
*
*              Perform bidiagonal QR iteration, if desired, computing
*              left singular vectors in U and computing right singular
*              vectors in A
*              (Workspace: need BDSPAC)
*
               CALL DBDSQR( 'L', M, NCVT, NRU, 0, S, WORK( IE ), A,
     $                      LDA,
     $                      U, LDU, DUM, 1, WORK( IWORK ), INFO )
            ELSE
*
*              Perform bidiagonal QR iteration, if desired, computing
*              left singular vectors in A and computing right singular
*              vectors in VT
*              (Workspace: need BDSPAC)
*
               CALL DBDSQR( 'L', M, NCVT, NRU, 0, S, WORK( IE ), VT,
     $                      LDVT, A, LDA, DUM, 1, WORK( IWORK ), INFO )
            END IF
*
         END IF
*
      END IF
*
*     If DBDSQR failed to converge, copy unconverged superdiagonals
*     to WORK( 2:MINMN )
*
      IF( INFO.NE.0 ) THEN
         IF( IE.GT.2 ) THEN
            DO 50 I = 1, MINMN - 1
               WORK( I+1 ) = WORK( I+IE-1 )
   50       CONTINUE
         END IF
         IF( IE.LT.2 ) THEN
            DO 60 I = MINMN - 1, 1, -1
               WORK( I+1 ) = WORK( I+IE-1 )
   60       CONTINUE
         END IF
      END IF
*
*     Undo scaling if necessary
*
      IF( ISCL.EQ.1 ) THEN
         IF( ANRM.GT.BIGNUM )
     $      CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
     $                   IERR )
         IF( INFO.NE.0 .AND. ANRM.GT.BIGNUM )
     $      CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN-1, 1,
     $                   WORK( 2 ),
     $                   MINMN, IERR )
         IF( ANRM.LT.SMLNUM )
     $      CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
     $                   IERR )
         IF( INFO.NE.0 .AND. ANRM.LT.SMLNUM )
     $      CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN-1, 1,
     $                   WORK( 2 ),
     $                   MINMN, IERR )
      END IF
*
*     Return optimal workspace in WORK(1)
*
      WORK( 1 ) = MAXWRK
*
      RETURN
*
*     End of DGESVD
*
      END
*> \brief \b DGESVJ
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGESVJ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgesvj.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgesvj.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgesvj.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V,
*                          LDV, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LDV, LWORK, M, MV, N
*       CHARACTER*1        JOBA, JOBU, JOBV
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), SVA( N ), V( LDV, * ),
*      $                   WORK( LWORK )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGESVJ computes the singular value decomposition (SVD) of a real
*> M-by-N matrix A, where M >= N. The SVD of A is written as
*>                                    [++]   [xx]   [x0]   [xx]
*>              A = U * SIGMA * V^t,  [++] = [xx] * [ox] * [xx]
*>                                    [++]   [xx]
*> where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
*> matrix, and V is an N-by-N orthogonal matrix. The diagonal elements
*> of SIGMA are the singular values of A. The columns of U and V are the
*> left and the right singular vectors of A, respectively.
*> DGESVJ can sometimes compute tiny singular values and their singular vectors much
*> more accurately than other SVD routines, see below under Further Details.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBA
*> \verbatim
*>          JOBA is CHARACTER*1
*>          Specifies the structure of A.
*>          = 'L': The input matrix A is lower triangular;
*>          = 'U': The input matrix A is upper triangular;
*>          = 'G': The input matrix A is general M-by-N matrix, M >= N.
*> \endverbatim
*>
*> \param[in] JOBU
*> \verbatim
*>          JOBU is CHARACTER*1
*>          Specifies whether to compute the left singular vectors
*>          (columns of U):
*>          = 'U': The left singular vectors corresponding to the nonzero
*>                 singular values are computed and returned in the leading
*>                 columns of A. See more details in the description of A.
*>                 The default numerical orthogonality threshold is set to
*>                 approximately TOL=CTOL*EPS, CTOL=DSQRT(M), EPS=DLAMCH('E').
*>          = 'C': Analogous to JOBU='U', except that user can control the
*>                 level of numerical orthogonality of the computed left
*>                 singular vectors. TOL can be set to TOL = CTOL*EPS, where
*>                 CTOL is given on input in the array WORK.
*>                 No CTOL smaller than ONE is allowed. CTOL greater
*>                 than 1 / EPS is meaningless. The option 'C'
*>                 can be used if M*EPS is satisfactory orthogonality
*>                 of the computed left singular vectors, so CTOL=M could
*>                 save few sweeps of Jacobi rotations.
*>                 See the descriptions of A and WORK(1).
*>          = 'N': The matrix U is not computed. However, see the
*>                 description of A.
*> \endverbatim
*>
*> \param[in] JOBV
*> \verbatim
*>          JOBV is CHARACTER*1
*>          Specifies whether to compute the right singular vectors, that
*>          is, the matrix V:
*>          = 'V':  the matrix V is computed and returned in the array V
*>          = 'A':  the Jacobi rotations are applied to the MV-by-N
*>                  array V. In other words, the right singular vector
*>                  matrix V is not computed explicitly, instead it is
*>                  applied to an MV-by-N matrix initially stored in the
*>                  first MV rows of V.
*>          = 'N':  the matrix V is not computed and the array V is not
*>                  referenced
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the input matrix A. 1/DLAMCH('E') > M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the input matrix A.
*>          M >= N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit :
*>          If JOBU = 'U' .OR. JOBU = 'C' :
*>                 If INFO = 0 :
*>                 RANKA orthonormal columns of U are returned in the
*>                 leading RANKA columns of the array A. Here RANKA <= N
*>                 is the number of computed singular values of A that are
*>                 above the underflow threshold DLAMCH('S'). The singular
*>                 vectors corresponding to underflowed or zero singular
*>                 values are not computed. The value of RANKA is returned
*>                 in the array WORK as RANKA=NINT(WORK(2)). Also see the
*>                 descriptions of SVA and WORK. The computed columns of U
*>                 are mutually numerically orthogonal up to approximately
*>                 TOL=DSQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU = 'C'),
*>                 see the description of JOBU.
*>                 If INFO > 0 :
*>                 the procedure DGESVJ did not converge in the given number
*>                 of iterations (sweeps). In that case, the computed
*>                 columns of U may not be orthogonal up to TOL. The output
*>                 U (stored in A), SIGMA (given by the computed singular
*>                 values in SVA(1:N)) and V is still a decomposition of the
*>                 input matrix A in the sense that the residual
*>                 ||A-SCALE*U*SIGMA*V^T||_2 / ||A||_2 is small.
*>
*>          If JOBU = 'N' :
*>                 If INFO = 0 :
*>                 Note that the left singular vectors are 'for free' in the
*>                 one-sided Jacobi SVD algorithm. However, if only the
*>                 singular values are needed, the level of numerical
*>                 orthogonality of U is not an issue and iterations are
*>                 stopped when the columns of the iterated matrix are
*>                 numerically orthogonal up to approximately M*EPS. Thus,
*>                 on exit, A contains the columns of U scaled with the
*>                 corresponding singular values.
*>                 If INFO > 0 :
*>                 the procedure DGESVJ did not converge in the given number
*>                 of iterations (sweeps).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] SVA
*> \verbatim
*>          SVA is DOUBLE PRECISION array, dimension (N)
*>          On exit :
*>          If INFO = 0 :
*>          depending on the value SCALE = WORK(1), we have:
*>                 If SCALE = ONE :
*>                 SVA(1:N) contains the computed singular values of A.
*>                 During the computation SVA contains the Euclidean column
*>                 norms of the iterated matrices in the array A.
*>                 If SCALE .NE. ONE :
*>                 The singular values of A are SCALE*SVA(1:N), and this
*>                 factored representation is due to the fact that some of the
*>                 singular values of A might underflow or overflow.
*>          If INFO > 0 :
*>          the procedure DGESVJ did not converge in the given number of
*>          iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.
*> \endverbatim
*>
*> \param[in] MV
*> \verbatim
*>          MV is INTEGER
*>          If JOBV = 'A', then the product of Jacobi rotations in DGESVJ
*>          is applied to the first MV rows of V. See the description of JOBV.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension (LDV,N)
*>          If JOBV = 'V', then V contains on exit the N-by-N matrix of
*>                         the right singular vectors;
*>          If JOBV = 'A', then V contains the product of the computed right
*>                         singular vector matrix and the initial matrix in
*>                         the array V.
*>          If JOBV = 'N', then V is not referenced.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of the array V, LDV >= 1.
*>          If JOBV = 'V', then LDV >= max(1,N).
*>          If JOBV = 'A', then LDV >= max(1,MV) .
*> \endverbatim
*>
*> \param[in,out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On entry :
*>          If JOBU = 'C' :
*>          WORK(1) = CTOL, where CTOL defines the threshold for convergence.
*>                    The process stops if all columns of A are mutually
*>                    orthogonal up to CTOL*EPS, EPS=DLAMCH('E').
*>                    It is required that CTOL >= ONE, i.e. it is not
*>                    allowed to force the routine to obtain orthogonality
*>                    below EPS.
*>          On exit :
*>          WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
*>                    are the computed singular values of A.
*>                    (See description of SVA().)
*>          WORK(2) = NINT(WORK(2)) is the number of the computed nonzero
*>                    singular values.
*>          WORK(3) = NINT(WORK(3)) is the number of the computed singular
*>                    values that are larger than the underflow threshold.
*>          WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi
*>                    rotations needed for numerical convergence.
*>          WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
*>                    This is useful information in cases when DGESVJ did
*>                    not converge, as it can be used to estimate whether
*>                    the output is still useful and for post festum analysis.
*>          WORK(6) = the largest absolute value over all sines of the
*>                    Jacobi rotation angles in the last sweep. It can be
*>                    useful for a post festum analysis.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The length of the array WORK.
*>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MAX(6,M+N), otherwise.
*>
*>          If on entry LWORK = -1, then a workspace query is assumed and
*>          no computation is done; WORK(1) is set to the minial (and optimal)
*>          length of WORK.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, then the i-th argument had an illegal value
*>          > 0:  DGESVJ did not converge in the maximal allowed number (30)
*>                of sweeps. The output may still be useful. See the
*>                description of WORK.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gesvj
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane
*>  rotations. The rotations are implemented as fast scaled rotations of
*>  Anda and Park [1]. In the case of underflow of the Jacobi angle, a
*>  modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses
*>  column interchanges of de Rijk [2]. The relative accuracy of the computed
*>  singular values and the accuracy of the computed singular vectors (in
*>  angle metric) is as guaranteed by the theory of Demmel and Veselic [3].
*>  The condition number that determines the accuracy in the full rank case
*>  is essentially min_{D=diag} kappa(A*D), where kappa(.) is the
*>  spectral condition number. The best performance of this Jacobi SVD
*>  procedure is achieved if used in an  accelerated version of Drmac and
*>  Veselic [5,6], and it is the kernel routine in the SIGMA library [7].
*>  Some tuning parameters (marked with [TP]) are available for the
*>  implementer.
*>  The computational range for the nonzero singular values is the  machine
*>  number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even
*>  denormalized singular values can be computed with the corresponding
*>  gradual loss of accurate digits.
*> \endverbatim
*
*> \par Contributors:
*  ==================
*>
*> \verbatim
*>
*>  ============
*>
*>  Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
*> \endverbatim
*
*> \par References:
*  ================
*>
*> \verbatim
*>
*> [1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling.
*>     SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.
*> [2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the
*>     singular value decomposition on a vector computer.
*>     SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
*> [3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
*> [4] Z. Drmac: Implementation of Jacobi rotations for accurate singular
*>     value computation in floating point arithmetic.
*>     SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
*> [5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
*>     SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
*>     LAPACK Working note 169.
*> [6] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
*>     SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
*>     LAPACK Working note 170.
*> [7] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
*>     QSVD, (H,K)-SVD computations.
*>     Department of Mathematics, University of Zagreb, 2008.
*> \endverbatim
*
*>  \par Bugs, examples and comments:
*   =================================
*>
*> \verbatim
*>  ===========================
*>  Please report all bugs and send interesting test examples and comments to
*>  drmac@math.hr. Thank you.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V,
     $                   LDV, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDV, LWORK, M, MV, N
      CHARACTER*1        JOBA, JOBU, JOBV
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), SVA( N ), V( LDV, * ),
     $                   WORK( LWORK )
*     ..
*
*  =====================================================================
*
*     .. Local Parameters ..
      DOUBLE PRECISION   ZERO, HALF, ONE
      PARAMETER          ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0)
      INTEGER            NSWEEP
      PARAMETER          ( NSWEEP = 30 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG,
     $                   BIGTHETA, CS, CTOL, EPSLN, LARGE, MXAAPQ,
     $                   MXSINJ, ROOTBIG, ROOTEPS, ROOTSFMIN, ROOTTOL,
     $                   SKL, SFMIN, SMALL, SN, T, TEMP1, THETA,
     $                   THSIGN, TOL
      INTEGER            BLSKIP, EMPTSW, i, ibr, IERR, igl, IJBLSK, ir1,
     $                   ISWROT, jbc, jgl, KBL, LKAHEAD, MVL, N2, N34,
     $                   N4, NBL, NOTROT, p, PSKIPPED, q, ROWSKIP,
     $                   SWBAND, MINMN, LWMIN
      LOGICAL            APPLV, GOSCALE, LOWER, LQUERY, LSVEC, NOSCALE,
     $                   ROTOK, RSVEC, UCTOL, UPPER
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   FASTR( 5 )
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DABS, MAX, MIN, DBLE, DSIGN, DSQRT
*     ..
*     .. External Functions ..
*     ..
*     from BLAS
      DOUBLE PRECISION   DDOT, DNRM2
      EXTERNAL           DDOT, DNRM2
      INTEGER            IDAMAX
      EXTERNAL           IDAMAX
*     from LAPACK
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
*     ..
*     from BLAS
      EXTERNAL           DAXPY, DCOPY, DROTM, DSCAL, DSWAP
*     from LAPACK
      EXTERNAL           DLASCL, DLASET, DLASSQ, XERBLA
*
      EXTERNAL           DGSVJ0, DGSVJ1
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      LSVEC = LSAME( JOBU, 'U' )
      UCTOL = LSAME( JOBU, 'C' )
      RSVEC = LSAME( JOBV, 'V' )
      APPLV = LSAME( JOBV, 'A' )
      UPPER = LSAME( JOBA, 'U' )
      LOWER = LSAME( JOBA, 'L' )
*
      MINMN = MIN( M, N )
      IF( MINMN.EQ.0 ) THEN
         LWMIN = 1
      ELSE
         LWMIN = MAX( 6, M+N )
      END IF
*
      LQUERY = ( LWORK.EQ.-1 )
      IF( .NOT.( UPPER .OR. LOWER .OR. LSAME( JOBA, 'G' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( LSVEC .OR.
     $         UCTOL .OR.
     $         LSAME( JOBU, 'N' ) ) ) THEN
         INFO = -2
      ELSE IF( .NOT.( RSVEC .OR.
     $         APPLV .OR.
     $         LSAME( JOBV, 'N' ) ) ) THEN
         INFO = -3
      ELSE IF( M.LT.0 ) THEN
         INFO = -4
      ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN
         INFO = -5
      ELSE IF( LDA.LT.M ) THEN
         INFO = -7
      ELSE IF( MV.LT.0 ) THEN
         INFO = -9
      ELSE IF( ( RSVEC .AND. ( LDV.LT.N ) ) .OR.
     $         ( APPLV .AND. ( LDV.LT.MV ) ) ) THEN
         INFO = -11
      ELSE IF( UCTOL .AND. ( WORK( 1 ).LE.ONE ) ) THEN
         INFO = -12
      ELSE IF( LWORK.LT.LWMIN .AND. ( .NOT.LQUERY ) ) THEN
         INFO = -13
      ELSE
         INFO = 0
      END IF
*
*     #:(
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGESVJ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         WORK( 1 ) = LWMIN
         RETURN
      END IF
*
* #:) Quick return for void matrix
*
      IF( MINMN.EQ.0 ) RETURN
*
*     Set numerical parameters
*     The stopping criterion for Jacobi rotations is
*
*     max_{i<>j}|A(:,i)^T * A(:,j)|/(||A(:,i)||*||A(:,j)||) < CTOL*EPS
*
*     where EPS is the round-off and CTOL is defined as follows:
*
      IF( UCTOL ) THEN
*        ... user controlled
         CTOL = WORK( 1 )
      ELSE
*        ... default
         IF( LSVEC .OR. RSVEC .OR. APPLV ) THEN
            CTOL = DSQRT( DBLE( M ) )
         ELSE
            CTOL = DBLE( M )
         END IF
      END IF
*     ... and the machine dependent parameters are
*[!]  (Make sure that DLAMCH() works properly on the target machine.)
*
      EPSLN = DLAMCH( 'Epsilon' )
      ROOTEPS = DSQRT( EPSLN )
      SFMIN = DLAMCH( 'SafeMinimum' )
      ROOTSFMIN = DSQRT( SFMIN )
      SMALL = SFMIN / EPSLN
      BIG = DLAMCH( 'Overflow' )
*     BIG         = ONE    / SFMIN
      ROOTBIG = ONE / ROOTSFMIN
      LARGE = BIG / DSQRT( DBLE( M*N ) )
      BIGTHETA = ONE / ROOTEPS
*
      TOL = CTOL*EPSLN
      ROOTTOL = DSQRT( TOL )
*
      IF( DBLE( M )*EPSLN.GE.ONE ) THEN
         INFO = -4
         CALL XERBLA( 'DGESVJ', -INFO )
         RETURN
      END IF
*
*     Initialize the right singular vector matrix.
*
      IF( RSVEC ) THEN
         MVL = N
         CALL DLASET( 'A', MVL, N, ZERO, ONE, V, LDV )
      ELSE IF( APPLV ) THEN
         MVL = MV
      END IF
      RSVEC = RSVEC .OR. APPLV
*
*     Initialize SVA( 1:N ) = ( ||A e_i||_2, i = 1:N )
*(!)  If necessary, scale A to protect the largest singular value
*     from overflow. It is possible that saving the largest singular
*     value destroys the information about the small ones.
*     This initial scaling is almost minimal in the sense that the
*     goal is to make sure that no column norm overflows, and that
*     DSQRT(N)*max_i SVA(i) does not overflow. If INFinite entries
*     in A are detected, the procedure returns with INFO=-6.
*
      SKL= ONE / DSQRT( DBLE( M )*DBLE( N ) )
      NOSCALE = .TRUE.
      GOSCALE = .TRUE.
*
      IF( LOWER ) THEN
*        the input matrix is M-by-N lower triangular (trapezoidal)
         DO 1874 p = 1, N
            AAPP = ZERO
            AAQQ = ONE
            CALL DLASSQ( M-p+1, A( p, p ), 1, AAPP, AAQQ )
            IF( AAPP.GT.BIG ) THEN
               INFO = -6
               CALL XERBLA( 'DGESVJ', -INFO )
               RETURN
            END IF
            AAQQ = DSQRT( AAQQ )
            IF( ( AAPP.LT.( BIG / AAQQ ) ) .AND. NOSCALE ) THEN
               SVA( p ) = AAPP*AAQQ
            ELSE
               NOSCALE = .FALSE.
               SVA( p ) = AAPP*( AAQQ*SKL)
               IF( GOSCALE ) THEN
                  GOSCALE = .FALSE.
                  DO 1873 q = 1, p - 1
                     SVA( q ) = SVA( q )*SKL
 1873             CONTINUE
               END IF
            END IF
 1874    CONTINUE
      ELSE IF( UPPER ) THEN
*        the input matrix is M-by-N upper triangular (trapezoidal)
         DO 2874 p = 1, N
            AAPP = ZERO
            AAQQ = ONE
            CALL DLASSQ( p, A( 1, p ), 1, AAPP, AAQQ )
            IF( AAPP.GT.BIG ) THEN
               INFO = -6
               CALL XERBLA( 'DGESVJ', -INFO )
               RETURN
            END IF
            AAQQ = DSQRT( AAQQ )
            IF( ( AAPP.LT.( BIG / AAQQ ) ) .AND. NOSCALE ) THEN
               SVA( p ) = AAPP*AAQQ
            ELSE
               NOSCALE = .FALSE.
               SVA( p ) = AAPP*( AAQQ*SKL)
               IF( GOSCALE ) THEN
                  GOSCALE = .FALSE.
                  DO 2873 q = 1, p - 1
                     SVA( q ) = SVA( q )*SKL
 2873             CONTINUE
               END IF
            END IF
 2874    CONTINUE
      ELSE
*        the input matrix is M-by-N general dense
         DO 3874 p = 1, N
            AAPP = ZERO
            AAQQ = ONE
            CALL DLASSQ( M, A( 1, p ), 1, AAPP, AAQQ )
            IF( AAPP.GT.BIG ) THEN
               INFO = -6
               CALL XERBLA( 'DGESVJ', -INFO )
               RETURN
            END IF
            AAQQ = DSQRT( AAQQ )
            IF( ( AAPP.LT.( BIG / AAQQ ) ) .AND. NOSCALE ) THEN
               SVA( p ) = AAPP*AAQQ
            ELSE
               NOSCALE = .FALSE.
               SVA( p ) = AAPP*( AAQQ*SKL)
               IF( GOSCALE ) THEN
                  GOSCALE = .FALSE.
                  DO 3873 q = 1, p - 1
                     SVA( q ) = SVA( q )*SKL
 3873             CONTINUE
               END IF
            END IF
 3874    CONTINUE
      END IF
*
      IF( NOSCALE )SKL= ONE
*
*     Move the smaller part of the spectrum from the underflow threshold
*(!)  Start by determining the position of the nonzero entries of the
*     array SVA() relative to ( SFMIN, BIG ).
*
      AAPP = ZERO
      AAQQ = BIG
      DO 4781 p = 1, N
         IF( SVA( p ).NE.ZERO )AAQQ = MIN( AAQQ, SVA( p ) )
         AAPP = MAX( AAPP, SVA( p ) )
 4781 CONTINUE
*
* #:) Quick return for zero matrix
*
      IF( AAPP.EQ.ZERO ) THEN
         IF( LSVEC )CALL DLASET( 'G', M, N, ZERO, ONE, A, LDA )
         WORK( 1 ) = ONE
         WORK( 2 ) = ZERO
         WORK( 3 ) = ZERO
         WORK( 4 ) = ZERO
         WORK( 5 ) = ZERO
         WORK( 6 ) = ZERO
         RETURN
      END IF
*
* #:) Quick return for one-column matrix
*
      IF( N.EQ.1 ) THEN
         IF( LSVEC )CALL DLASCL( 'G', 0, 0, SVA( 1 ), SKL, M, 1,
     $                           A( 1, 1 ), LDA, IERR )
         WORK( 1 ) = ONE / SKL
         IF( SVA( 1 ).GE.SFMIN ) THEN
            WORK( 2 ) = ONE
         ELSE
            WORK( 2 ) = ZERO
         END IF
         WORK( 3 ) = ZERO
         WORK( 4 ) = ZERO
         WORK( 5 ) = ZERO
         WORK( 6 ) = ZERO
         RETURN
      END IF
*
*     Protect small singular values from underflow, and try to
*     avoid underflows/overflows in computing Jacobi rotations.
*
      SN = DSQRT( SFMIN / EPSLN )
      TEMP1 = DSQRT( BIG / DBLE( N ) )
      IF( ( AAPP.LE.SN ) .OR. ( AAQQ.GE.TEMP1 ) .OR.
     $    ( ( SN.LE.AAQQ ) .AND. ( AAPP.LE.TEMP1 ) ) ) THEN
         TEMP1 = MIN( BIG, TEMP1 / AAPP )
*         AAQQ  = AAQQ*TEMP1
*         AAPP  = AAPP*TEMP1
      ELSE IF( ( AAQQ.LE.SN ) .AND. ( AAPP.LE.TEMP1 ) ) THEN
         TEMP1 = MIN( SN / AAQQ, BIG / ( AAPP*DSQRT( DBLE( N ) ) ) )
*         AAQQ  = AAQQ*TEMP1
*         AAPP  = AAPP*TEMP1
      ELSE IF( ( AAQQ.GE.SN ) .AND. ( AAPP.GE.TEMP1 ) ) THEN
         TEMP1 = MAX( SN / AAQQ, TEMP1 / AAPP )
*         AAQQ  = AAQQ*TEMP1
*         AAPP  = AAPP*TEMP1
      ELSE IF( ( AAQQ.LE.SN ) .AND. ( AAPP.GE.TEMP1 ) ) THEN
         TEMP1 = MIN( SN / AAQQ, BIG / ( DSQRT( DBLE( N ) )*AAPP ) )
*         AAQQ  = AAQQ*TEMP1
*         AAPP  = AAPP*TEMP1
      ELSE
         TEMP1 = ONE
      END IF
*
*     Scale, if necessary
*
      IF( TEMP1.NE.ONE ) THEN
         CALL DLASCL( 'G', 0, 0, ONE, TEMP1, N, 1, SVA, N, IERR )
      END IF
      SKL= TEMP1*SKL
      IF( SKL.NE.ONE ) THEN
         CALL DLASCL( JOBA, 0, 0, ONE, SKL, M, N, A, LDA, IERR )
         SKL= ONE / SKL
      END IF
*
*     Row-cyclic Jacobi SVD algorithm with column pivoting
*
      EMPTSW = ( N*( N-1 ) ) / 2
      NOTROT = 0
      FASTR( 1 ) = ZERO
*
*     A is represented in factored form A = A * diag(WORK), where diag(WORK)
*     is initialized to identity. WORK is updated during fast scaled
*     rotations.
*
      DO 1868 q = 1, N
         WORK( q ) = ONE
 1868 CONTINUE
*
*
      SWBAND = 3
*[TP] SWBAND is a tuning parameter [TP]. It is meaningful and effective
*     if DGESVJ is used as a computational routine in the preconditioned
*     Jacobi SVD algorithm DGESVJ. For sweeps i=1:SWBAND the procedure
*     works on pivots inside a band-like region around the diagonal.
*     The boundaries are determined dynamically, based on the number of
*     pivots above a threshold.
*
      KBL = MIN( 8, N )
*[TP] KBL is a tuning parameter that defines the tile size in the
*     tiling of the p-q loops of pivot pairs. In general, an optimal
*     value of KBL depends on the matrix dimensions and on the
*     parameters of the computer's memory.
*
      NBL = N / KBL
      IF( ( NBL*KBL ).NE.N )NBL = NBL + 1
*
      BLSKIP = KBL**2
*[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.
*
      ROWSKIP = MIN( 5, KBL )
*[TP] ROWSKIP is a tuning parameter.
*
      LKAHEAD = 1
*[TP] LKAHEAD is a tuning parameter.
*
*     Quasi block transformations, using the lower (upper) triangular
*     structure of the input matrix. The quasi-block-cycling usually
*     invokes cubic convergence. Big part of this cycle is done inside
*     canonical subspaces of dimensions less than M.
*
      IF( ( LOWER .OR. UPPER ) .AND. ( N.GT.MAX( 64, 4*KBL ) ) ) THEN
*[TP] The number of partition levels and the actual partition are
*     tuning parameters.
         N4 = N / 4
         N2 = N / 2
         N34 = 3*N4
         IF( APPLV ) THEN
            q = 0
         ELSE
            q = 1
         END IF
*
         IF( LOWER ) THEN
*
*     This works very well on lower triangular matrices, in particular
*     in the framework of the preconditioned Jacobi SVD (xGEJSV).
*     The idea is simple:
*     [+ 0 0 0]   Note that Jacobi transformations of [0 0]
*     [+ + 0 0]                                       [0 0]
*     [+ + x 0]   actually work on [x 0]              [x 0]
*     [+ + x x]                    [x x].             [x x]
*
            CALL DGSVJ0( JOBV, M-N34, N-N34, A( N34+1, N34+1 ), LDA,
     $                   WORK( N34+1 ), SVA( N34+1 ), MVL,
     $                   V( N34*q+1, N34+1 ), LDV, EPSLN, SFMIN, TOL,
     $                   2, WORK( N+1 ), LWORK-N, IERR )
*
            CALL DGSVJ0( JOBV, M-N2, N34-N2, A( N2+1, N2+1 ), LDA,
     $                   WORK( N2+1 ), SVA( N2+1 ), MVL,
     $                   V( N2*q+1, N2+1 ), LDV, EPSLN, SFMIN, TOL, 2,
     $                   WORK( N+1 ), LWORK-N, IERR )
*
            CALL DGSVJ1( JOBV, M-N2, N-N2, N4, A( N2+1, N2+1 ), LDA,
     $                   WORK( N2+1 ), SVA( N2+1 ), MVL,
     $                   V( N2*q+1, N2+1 ), LDV, EPSLN, SFMIN, TOL, 1,
     $                   WORK( N+1 ), LWORK-N, IERR )
*
            CALL DGSVJ0( JOBV, M-N4, N2-N4, A( N4+1, N4+1 ), LDA,
     $                   WORK( N4+1 ), SVA( N4+1 ), MVL,
     $                   V( N4*q+1, N4+1 ), LDV, EPSLN, SFMIN, TOL, 1,
     $                   WORK( N+1 ), LWORK-N, IERR )
*
            CALL DGSVJ0( JOBV, M, N4, A, LDA, WORK, SVA, MVL, V, LDV,
     $                   EPSLN, SFMIN, TOL, 1, WORK( N+1 ), LWORK-N,
     $                   IERR )
*
            CALL DGSVJ1( JOBV, M, N2, N4, A, LDA, WORK, SVA, MVL, V,
     $                   LDV, EPSLN, SFMIN, TOL, 1, WORK( N+1 ),
     $                   LWORK-N, IERR )
*
*
         ELSE IF( UPPER ) THEN
*
*
            CALL DGSVJ0( JOBV, N4, N4, A, LDA, WORK, SVA, MVL, V,
     $                   LDV,
     $                   EPSLN, SFMIN, TOL, 2, WORK( N+1 ), LWORK-N,
     $                   IERR )
*
            CALL DGSVJ0( JOBV, N2, N4, A( 1, N4+1 ), LDA,
     $                   WORK( N4+1 ),
     $                   SVA( N4+1 ), MVL, V( N4*q+1, N4+1 ), LDV,
     $                   EPSLN, SFMIN, TOL, 1, WORK( N+1 ), LWORK-N,
     $                   IERR )
*
            CALL DGSVJ1( JOBV, N2, N2, N4, A, LDA, WORK, SVA, MVL, V,
     $                   LDV, EPSLN, SFMIN, TOL, 1, WORK( N+1 ),
     $                   LWORK-N, IERR )
*
            CALL DGSVJ0( JOBV, N2+N4, N4, A( 1, N2+1 ), LDA,
     $                   WORK( N2+1 ), SVA( N2+1 ), MVL,
     $                   V( N2*q+1, N2+1 ), LDV, EPSLN, SFMIN, TOL, 1,
     $                   WORK( N+1 ), LWORK-N, IERR )

         END IF
*
      END IF
*
*     .. Row-cyclic pivot strategy with de Rijk's pivoting ..
*
      DO 1993 i = 1, NSWEEP
*
*     .. go go go ...
*
         MXAAPQ = ZERO
         MXSINJ = ZERO
         ISWROT = 0
*
         NOTROT = 0
         PSKIPPED = 0
*
*     Each sweep is unrolled using KBL-by-KBL tiles over the pivot pairs
*     1 <= p < q <= N. This is the first step toward a blocked implementation
*     of the rotations. New implementation, based on block transformations,
*     is under development.
*
         DO 2000 ibr = 1, NBL
*
            igl = ( ibr-1 )*KBL + 1
*
            DO 1002 ir1 = 0, MIN( LKAHEAD, NBL-ibr )
*
               igl = igl + ir1*KBL
*
               DO 2001 p = igl, MIN( igl+KBL-1, N-1 )
*
*     .. de Rijk's pivoting
*
                  q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
                  IF( p.NE.q ) THEN
                     CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
                     IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1,
     $                                      V( 1, q ), 1 )
                     TEMP1 = SVA( p )
                     SVA( p ) = SVA( q )
                     SVA( q ) = TEMP1
                     TEMP1 = WORK( p )
                     WORK( p ) = WORK( q )
                     WORK( q ) = TEMP1
                  END IF
*
                  IF( ir1.EQ.0 ) THEN
*
*        Column norms are periodically updated by explicit
*        norm computation.
*        Caveat:
*        Unfortunately, some BLAS implementations compute DNRM2(M,A(1,p),1)
*        as DSQRT(DDOT(M,A(1,p),1,A(1,p),1)), which may cause the result to
*        overflow for ||A(:,p)||_2 > DSQRT(overflow_threshold), and to
*        underflow for ||A(:,p)||_2 < DSQRT(underflow_threshold).
*        Hence, DNRM2 cannot be trusted, not even in the case when
*        the true norm is far from the under(over)flow boundaries.
*        If properly implemented DNRM2 is available, the IF-THEN-ELSE
*        below should read "AAPP = DNRM2( M, A(1,p), 1 ) * WORK(p)".
*
                     IF( ( SVA( p ).LT.ROOTBIG ) .AND.
     $                   ( SVA( p ).GT.ROOTSFMIN ) ) THEN
                        SVA( p ) = DNRM2( M, A( 1, p ), 1 )*WORK( p )
                     ELSE
                        TEMP1 = ZERO
                        AAPP = ONE
                        CALL DLASSQ( M, A( 1, p ), 1, TEMP1, AAPP )
                        SVA( p ) = TEMP1*DSQRT( AAPP )*WORK( p )
                     END IF
                     AAPP = SVA( p )
                  ELSE
                     AAPP = SVA( p )
                  END IF
*
                  IF( AAPP.GT.ZERO ) THEN
*
                     PSKIPPED = 0
*
                     DO 2002 q = p + 1, MIN( igl+KBL-1, N )
*
                        AAQQ = SVA( q )
*
                        IF( AAQQ.GT.ZERO ) THEN
*
                           AAPP0 = AAPP
                           IF( AAQQ.GE.ONE ) THEN
                              ROTOK = ( SMALL*AAPP ).LE.AAQQ
                              IF( AAPP.LT.( BIG / AAQQ ) ) THEN
                                 AAPQ = ( DDOT( M, A( 1, p ), 1,
     $                                    A( 1,
     $                                  q ), 1 )*WORK( p )*WORK( q ) /
     $                                  AAQQ ) / AAPP
                              ELSE
                                 CALL DCOPY( M, A( 1, p ), 1,
     $                                       WORK( N+1 ), 1 )
                                 CALL DLASCL( 'G', 0, 0, AAPP,
     $                                        WORK( p ), M, 1,
     $                                        WORK( N+1 ), LDA, IERR )
                                 AAPQ = DDOT( M, WORK( N+1 ), 1,
     $                                  A( 1, q ), 1 )*WORK( q ) / AAQQ
                              END IF
                           ELSE
                              ROTOK = AAPP.LE.( AAQQ / SMALL )
                              IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
                                 AAPQ = ( DDOT( M, A( 1, p ), 1,
     $                                    A( 1,
     $                                  q ), 1 )*WORK( p )*WORK( q ) /
     $                                  AAQQ ) / AAPP
                              ELSE
                                 CALL DCOPY( M, A( 1, q ), 1,
     $                                       WORK( N+1 ), 1 )
                                 CALL DLASCL( 'G', 0, 0, AAQQ,
     $                                        WORK( q ), M, 1,
     $                                        WORK( N+1 ), LDA, IERR )
                                 AAPQ = DDOT( M, WORK( N+1 ), 1,
     $                                  A( 1, p ), 1 )*WORK( p ) / AAPP
                              END IF
                           END IF
*
                           MXAAPQ = MAX( MXAAPQ, DABS( AAPQ ) )
*
*        TO rotate or NOT to rotate, THAT is the question ...
*
                           IF( DABS( AAPQ ).GT.TOL ) THEN
*
*           .. rotate
*[RTD]      ROTATED = ROTATED + ONE
*
                              IF( ir1.EQ.0 ) THEN
                                 NOTROT = 0
                                 PSKIPPED = 0
                                 ISWROT = ISWROT + 1
                              END IF
*
                              IF( ROTOK ) THEN
*
                                 AQOAP = AAQQ / AAPP
                                 APOAQ = AAPP / AAQQ
                                 THETA = -HALF*DABS(AQOAP-APOAQ)/AAPQ
*
                                 IF( DABS( THETA ).GT.BIGTHETA ) THEN
*
                                    T = HALF / THETA
                                    FASTR( 3 ) = T*WORK( p ) / WORK( q )
                                    FASTR( 4 ) = -T*WORK( q ) /
     $                                           WORK( p )
                                    CALL DROTM( M, A( 1, p ), 1,
     $                                          A( 1, q ), 1, FASTR )
                                    IF( RSVEC )CALL DROTM( MVL,
     $                                              V( 1, p ), 1,
     $                                              V( 1, q ), 1,
     $                                              FASTR )
                                    SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
     $                                         ONE+T*APOAQ*AAPQ ) )
                                    AAPP = AAPP*DSQRT( MAX( ZERO,
     $                                     ONE-T*AQOAP*AAPQ ) )
                                    MXSINJ = MAX( MXSINJ, DABS( T ) )
*
                                 ELSE
*
*                 .. choose correct signum for THETA and rotate
*
                                    THSIGN = -DSIGN( ONE, AAPQ )
                                    T = ONE / ( THETA+THSIGN*
     $                                  DSQRT( ONE+THETA*THETA ) )
                                    CS = DSQRT( ONE / ( ONE+T*T ) )
                                    SN = T*CS
*
                                    MXSINJ = MAX( MXSINJ, DABS( SN ) )
                                    SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
     $                                         ONE+T*APOAQ*AAPQ ) )
                                    AAPP = AAPP*DSQRT( MAX( ZERO,
     $                                     ONE-T*AQOAP*AAPQ ) )
*
                                    APOAQ = WORK( p ) / WORK( q )
                                    AQOAP = WORK( q ) / WORK( p )
                                    IF( WORK( p ).GE.ONE ) THEN
                                       IF( WORK( q ).GE.ONE ) THEN
                                          FASTR( 3 ) = T*APOAQ
                                          FASTR( 4 ) = -T*AQOAP
                                          WORK( p ) = WORK( p )*CS
                                          WORK( q ) = WORK( q )*CS
                                          CALL DROTM( M, A( 1, p ),
     $                                                1,
     $                                                A( 1, q ), 1,
     $                                                FASTR )
                                          IF( RSVEC )CALL DROTM( MVL,
     $                                        V( 1, p ), 1, V( 1, q ),
     $                                        1, FASTR )
                                       ELSE
                                          CALL DAXPY( M, -T*AQOAP,
     $                                                A( 1, q ), 1,
     $                                                A( 1, p ), 1 )
                                          CALL DAXPY( M, CS*SN*APOAQ,
     $                                                A( 1, p ), 1,
     $                                                A( 1, q ), 1 )
                                          WORK( p ) = WORK( p )*CS
                                          WORK( q ) = WORK( q ) / CS
                                          IF( RSVEC ) THEN
                                             CALL DAXPY( MVL,
     $                                                   -T*AQOAP,
     $                                                   V( 1, q ), 1,
     $                                                   V( 1, p ), 1 )
                                             CALL DAXPY( MVL,
     $                                                   CS*SN*APOAQ,
     $                                                   V( 1, p ), 1,
     $                                                   V( 1, q ), 1 )
                                          END IF
                                       END IF
                                    ELSE
                                       IF( WORK( q ).GE.ONE ) THEN
                                          CALL DAXPY( M, T*APOAQ,
     $                                                A( 1, p ), 1,
     $                                                A( 1, q ), 1 )
                                          CALL DAXPY( M,
     $                                                -CS*SN*AQOAP,
     $                                                A( 1, q ), 1,
     $                                                A( 1, p ), 1 )
                                          WORK( p ) = WORK( p ) / CS
                                          WORK( q ) = WORK( q )*CS
                                          IF( RSVEC ) THEN
                                             CALL DAXPY( MVL,
     $                                                   T*APOAQ,
     $                                                   V( 1, p ), 1,
     $                                                   V( 1, q ), 1 )
                                             CALL DAXPY( MVL,
     $                                                   -CS*SN*AQOAP,
     $                                                   V( 1, q ), 1,
     $                                                   V( 1, p ), 1 )
                                          END IF
                                       ELSE
                                          IF( WORK( p ).GE.WORK( q ) )
     $                                        THEN
                                             CALL DAXPY( M, -T*AQOAP,
     $                                                   A( 1, q ), 1,
     $                                                   A( 1, p ), 1 )
                                             CALL DAXPY( M,
     $                                                   CS*SN*APOAQ,
     $                                                   A( 1, p ), 1,
     $                                                   A( 1, q ), 1 )
                                             WORK( p ) = WORK( p )*CS
                                             WORK( q ) = WORK( q ) / CS
                                             IF( RSVEC ) THEN
                                                CALL DAXPY( MVL,
     $                                               -T*AQOAP,
     $                                               V( 1, q ), 1,
     $                                               V( 1, p ), 1 )
                                                CALL DAXPY( MVL,
     $                                               CS*SN*APOAQ,
     $                                               V( 1, p ), 1,
     $                                               V( 1, q ), 1 )
                                             END IF
                                          ELSE
                                             CALL DAXPY( M, T*APOAQ,
     $                                                   A( 1, p ), 1,
     $                                                   A( 1, q ), 1 )
                                             CALL DAXPY( M,
     $                                                   -CS*SN*AQOAP,
     $                                                   A( 1, q ), 1,
     $                                                   A( 1, p ), 1 )
                                             WORK( p ) = WORK( p ) / CS
                                             WORK( q ) = WORK( q )*CS
                                             IF( RSVEC ) THEN
                                                CALL DAXPY( MVL,
     $                                               T*APOAQ, V( 1, p ),
     $                                               1, V( 1, q ), 1 )
                                                CALL DAXPY( MVL,
     $                                               -CS*SN*AQOAP,
     $                                               V( 1, q ), 1,
     $                                               V( 1, p ), 1 )
                                             END IF
                                          END IF
                                       END IF
                                    END IF
                                 END IF
*
                              ELSE
*              .. have to use modified Gram-Schmidt like transformation
                                 CALL DCOPY( M, A( 1, p ), 1,
     $                                       WORK( N+1 ), 1 )
                                 CALL DLASCL( 'G', 0, 0, AAPP, ONE,
     $                                        M,
     $                                        1, WORK( N+1 ), LDA,
     $                                        IERR )
                                 CALL DLASCL( 'G', 0, 0, AAQQ, ONE,
     $                                        M,
     $                                        1, A( 1, q ), LDA, IERR )
                                 TEMP1 = -AAPQ*WORK( p ) / WORK( q )
                                 CALL DAXPY( M, TEMP1, WORK( N+1 ),
     $                                       1,
     $                                       A( 1, q ), 1 )
                                 CALL DLASCL( 'G', 0, 0, ONE, AAQQ,
     $                                        M,
     $                                        1, A( 1, q ), LDA, IERR )
                                 SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
     $                                      ONE-AAPQ*AAPQ ) )
                                 MXSINJ = MAX( MXSINJ, SFMIN )
                              END IF
*           END IF ROTOK THEN ... ELSE
*
*           In the case of cancellation in updating SVA(q), SVA(p)
*           recompute SVA(q), SVA(p).
*
                              IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
     $                            THEN
                                 IF( ( AAQQ.LT.ROOTBIG ) .AND.
     $                               ( AAQQ.GT.ROOTSFMIN ) ) THEN
                                    SVA( q ) = DNRM2( M, A( 1, q ),
     $                                   1 )*
     $                                         WORK( q )
                                 ELSE
                                    T = ZERO
                                    AAQQ = ONE
                                    CALL DLASSQ( M, A( 1, q ), 1, T,
     $                                           AAQQ )
                                    SVA( q ) = T*DSQRT( AAQQ )*WORK( q )
                                 END IF
                              END IF
                              IF( ( AAPP / AAPP0 ).LE.ROOTEPS ) THEN
                                 IF( ( AAPP.LT.ROOTBIG ) .AND.
     $                               ( AAPP.GT.ROOTSFMIN ) ) THEN
                                    AAPP = DNRM2( M, A( 1, p ), 1 )*
     $                                     WORK( p )
                                 ELSE
                                    T = ZERO
                                    AAPP = ONE
                                    CALL DLASSQ( M, A( 1, p ), 1, T,
     $                                           AAPP )
                                    AAPP = T*DSQRT( AAPP )*WORK( p )
                                 END IF
                                 SVA( p ) = AAPP
                              END IF
*
                           ELSE
*        A(:,p) and A(:,q) already numerically orthogonal
                              IF( ir1.EQ.0 )NOTROT = NOTROT + 1
*[RTD]      SKIPPED  = SKIPPED  + 1
                              PSKIPPED = PSKIPPED + 1
                           END IF
                        ELSE
*        A(:,q) is zero column
                           IF( ir1.EQ.0 )NOTROT = NOTROT + 1
                           PSKIPPED = PSKIPPED + 1
                        END IF
*
                        IF( ( i.LE.SWBAND ) .AND.
     $                      ( PSKIPPED.GT.ROWSKIP ) ) THEN
                           IF( ir1.EQ.0 )AAPP = -AAPP
                           NOTROT = 0
                           GO TO 2103
                        END IF
*
 2002                CONTINUE
*     END q-LOOP
*
 2103                CONTINUE
*     bailed out of q-loop
*
                     SVA( p ) = AAPP
*
                  ELSE
                     SVA( p ) = AAPP
                     IF( ( ir1.EQ.0 ) .AND. ( AAPP.EQ.ZERO ) )
     $                   NOTROT = NOTROT + MIN( igl+KBL-1, N ) - p
                  END IF
*
 2001          CONTINUE
*     end of the p-loop
*     end of doing the block ( ibr, ibr )
 1002       CONTINUE
*     end of ir1-loop
*
* ... go to the off diagonal blocks
*
            igl = ( ibr-1 )*KBL + 1
*
            DO 2010 jbc = ibr + 1, NBL
*
               jgl = ( jbc-1 )*KBL + 1
*
*        doing the block at ( ibr, jbc )
*
               IJBLSK = 0
               DO 2100 p = igl, MIN( igl+KBL-1, N )
*
                  AAPP = SVA( p )
                  IF( AAPP.GT.ZERO ) THEN
*
                     PSKIPPED = 0
*
                     DO 2200 q = jgl, MIN( jgl+KBL-1, N )
*
                        AAQQ = SVA( q )
                        IF( AAQQ.GT.ZERO ) THEN
                           AAPP0 = AAPP
*
*     .. M x 2 Jacobi SVD ..
*
*        Safe Gram matrix computation
*
                           IF( AAQQ.GE.ONE ) THEN
                              IF( AAPP.GE.AAQQ ) THEN
                                 ROTOK = ( SMALL*AAPP ).LE.AAQQ
                              ELSE
                                 ROTOK = ( SMALL*AAQQ ).LE.AAPP
                              END IF
                              IF( AAPP.LT.( BIG / AAQQ ) ) THEN
                                 AAPQ = ( DDOT( M, A( 1, p ), 1,
     $                                    A( 1,
     $                                  q ), 1 )*WORK( p )*WORK( q ) /
     $                                  AAQQ ) / AAPP
                              ELSE
                                 CALL DCOPY( M, A( 1, p ), 1,
     $                                       WORK( N+1 ), 1 )
                                 CALL DLASCL( 'G', 0, 0, AAPP,
     $                                        WORK( p ), M, 1,
     $                                        WORK( N+1 ), LDA, IERR )
                                 AAPQ = DDOT( M, WORK( N+1 ), 1,
     $                                  A( 1, q ), 1 )*WORK( q ) / AAQQ
                              END IF
                           ELSE
                              IF( AAPP.GE.AAQQ ) THEN
                                 ROTOK = AAPP.LE.( AAQQ / SMALL )
                              ELSE
                                 ROTOK = AAQQ.LE.( AAPP / SMALL )
                              END IF
                              IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
                                 AAPQ = ( DDOT( M, A( 1, p ), 1,
     $                                    A( 1,
     $                                  q ), 1 )*WORK( p )*WORK( q ) /
     $                                  AAQQ ) / AAPP
                              ELSE
                                 CALL DCOPY( M, A( 1, q ), 1,
     $                                       WORK( N+1 ), 1 )
                                 CALL DLASCL( 'G', 0, 0, AAQQ,
     $                                        WORK( q ), M, 1,
     $                                        WORK( N+1 ), LDA, IERR )
                                 AAPQ = DDOT( M, WORK( N+1 ), 1,
     $                                  A( 1, p ), 1 )*WORK( p ) / AAPP
                              END IF
                           END IF
*
                           MXAAPQ = MAX( MXAAPQ, DABS( AAPQ ) )
*
*        TO rotate or NOT to rotate, THAT is the question ...
*
                           IF( DABS( AAPQ ).GT.TOL ) THEN
                              NOTROT = 0
*[RTD]      ROTATED  = ROTATED + 1
                              PSKIPPED = 0
                              ISWROT = ISWROT + 1
*
                              IF( ROTOK ) THEN
*
                                 AQOAP = AAQQ / AAPP
                                 APOAQ = AAPP / AAQQ
                                 THETA = -HALF*DABS(AQOAP-APOAQ)/AAPQ
                                 IF( AAQQ.GT.AAPP0 )THETA = -THETA
*
                                 IF( DABS( THETA ).GT.BIGTHETA ) THEN
                                    T = HALF / THETA
                                    FASTR( 3 ) = T*WORK( p ) / WORK( q )
                                    FASTR( 4 ) = -T*WORK( q ) /
     $                                           WORK( p )
                                    CALL DROTM( M, A( 1, p ), 1,
     $                                          A( 1, q ), 1, FASTR )
                                    IF( RSVEC )CALL DROTM( MVL,
     $                                              V( 1, p ), 1,
     $                                              V( 1, q ), 1,
     $                                              FASTR )
                                    SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
     $                                         ONE+T*APOAQ*AAPQ ) )
                                    AAPP = AAPP*DSQRT( MAX( ZERO,
     $                                     ONE-T*AQOAP*AAPQ ) )
                                    MXSINJ = MAX( MXSINJ, DABS( T ) )
                                 ELSE
*
*                 .. choose correct signum for THETA and rotate
*
                                    THSIGN = -DSIGN( ONE, AAPQ )
                                    IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN
                                    T = ONE / ( THETA+THSIGN*
     $                                  DSQRT( ONE+THETA*THETA ) )
                                    CS = DSQRT( ONE / ( ONE+T*T ) )
                                    SN = T*CS
                                    MXSINJ = MAX( MXSINJ, DABS( SN ) )
                                    SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
     $                                         ONE+T*APOAQ*AAPQ ) )
                                    AAPP = AAPP*DSQRT( MAX( ZERO,
     $                                     ONE-T*AQOAP*AAPQ ) )
*
                                    APOAQ = WORK( p ) / WORK( q )
                                    AQOAP = WORK( q ) / WORK( p )
                                    IF( WORK( p ).GE.ONE ) THEN
*
                                       IF( WORK( q ).GE.ONE ) THEN
                                          FASTR( 3 ) = T*APOAQ
                                          FASTR( 4 ) = -T*AQOAP
                                          WORK( p ) = WORK( p )*CS
                                          WORK( q ) = WORK( q )*CS
                                          CALL DROTM( M, A( 1, p ),
     $                                                1,
     $                                                A( 1, q ), 1,
     $                                                FASTR )
                                          IF( RSVEC )CALL DROTM( MVL,
     $                                        V( 1, p ), 1, V( 1, q ),
     $                                        1, FASTR )
                                       ELSE
                                          CALL DAXPY( M, -T*AQOAP,
     $                                                A( 1, q ), 1,
     $                                                A( 1, p ), 1 )
                                          CALL DAXPY( M, CS*SN*APOAQ,
     $                                                A( 1, p ), 1,
     $                                                A( 1, q ), 1 )
                                          IF( RSVEC ) THEN
                                             CALL DAXPY( MVL,
     $                                                   -T*AQOAP,
     $                                                   V( 1, q ), 1,
     $                                                   V( 1, p ), 1 )
                                             CALL DAXPY( MVL,
     $                                                   CS*SN*APOAQ,
     $                                                   V( 1, p ), 1,
     $                                                   V( 1, q ), 1 )
                                          END IF
                                          WORK( p ) = WORK( p )*CS
                                          WORK( q ) = WORK( q ) / CS
                                       END IF
                                    ELSE
                                       IF( WORK( q ).GE.ONE ) THEN
                                          CALL DAXPY( M, T*APOAQ,
     $                                                A( 1, p ), 1,
     $                                                A( 1, q ), 1 )
                                          CALL DAXPY( M,
     $                                                -CS*SN*AQOAP,
     $                                                A( 1, q ), 1,
     $                                                A( 1, p ), 1 )
                                          IF( RSVEC ) THEN
                                             CALL DAXPY( MVL,
     $                                                   T*APOAQ,
     $                                                   V( 1, p ), 1,
     $                                                   V( 1, q ), 1 )
                                             CALL DAXPY( MVL,
     $                                                   -CS*SN*AQOAP,
     $                                                   V( 1, q ), 1,
     $                                                   V( 1, p ), 1 )
                                          END IF
                                          WORK( p ) = WORK( p ) / CS
                                          WORK( q ) = WORK( q )*CS
                                       ELSE
                                          IF( WORK( p ).GE.WORK( q ) )
     $                                        THEN
                                             CALL DAXPY( M, -T*AQOAP,
     $                                                   A( 1, q ), 1,
     $                                                   A( 1, p ), 1 )
                                             CALL DAXPY( M,
     $                                                   CS*SN*APOAQ,
     $                                                   A( 1, p ), 1,
     $                                                   A( 1, q ), 1 )
                                             WORK( p ) = WORK( p )*CS
                                             WORK( q ) = WORK( q ) / CS
                                             IF( RSVEC ) THEN
                                                CALL DAXPY( MVL,
     $                                               -T*AQOAP,
     $                                               V( 1, q ), 1,
     $                                               V( 1, p ), 1 )
                                                CALL DAXPY( MVL,
     $                                               CS*SN*APOAQ,
     $                                               V( 1, p ), 1,
     $                                               V( 1, q ), 1 )
                                             END IF
                                          ELSE
                                             CALL DAXPY( M, T*APOAQ,
     $                                                   A( 1, p ), 1,
     $                                                   A( 1, q ), 1 )
                                             CALL DAXPY( M,
     $                                                   -CS*SN*AQOAP,
     $                                                   A( 1, q ), 1,
     $                                                   A( 1, p ), 1 )
                                             WORK( p ) = WORK( p ) / CS
                                             WORK( q ) = WORK( q )*CS
                                             IF( RSVEC ) THEN
                                                CALL DAXPY( MVL,
     $                                               T*APOAQ, V( 1, p ),
     $                                               1, V( 1, q ), 1 )
                                                CALL DAXPY( MVL,
     $                                               -CS*SN*AQOAP,
     $                                               V( 1, q ), 1,
     $                                               V( 1, p ), 1 )
                                             END IF
                                          END IF
                                       END IF
                                    END IF
                                 END IF
*
                              ELSE
                                 IF( AAPP.GT.AAQQ ) THEN
                                    CALL DCOPY( M, A( 1, p ), 1,
     $                                          WORK( N+1 ), 1 )
                                    CALL DLASCL( 'G', 0, 0, AAPP,
     $                                           ONE,
     $                                           M, 1, WORK( N+1 ), LDA,
     $                                           IERR )
                                    CALL DLASCL( 'G', 0, 0, AAQQ,
     $                                           ONE,
     $                                           M, 1, A( 1, q ), LDA,
     $                                           IERR )
                                    TEMP1 = -AAPQ*WORK( p ) / WORK( q )
                                    CALL DAXPY( M, TEMP1,
     $                                          WORK( N+1 ),
     $                                          1, A( 1, q ), 1 )
                                    CALL DLASCL( 'G', 0, 0, ONE,
     $                                           AAQQ,
     $                                           M, 1, A( 1, q ), LDA,
     $                                           IERR )
                                    SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
     $                                         ONE-AAPQ*AAPQ ) )
                                    MXSINJ = MAX( MXSINJ, SFMIN )
                                 ELSE
                                    CALL DCOPY( M, A( 1, q ), 1,
     $                                          WORK( N+1 ), 1 )
                                    CALL DLASCL( 'G', 0, 0, AAQQ,
     $                                           ONE,
     $                                           M, 1, WORK( N+1 ), LDA,
     $                                           IERR )
                                    CALL DLASCL( 'G', 0, 0, AAPP,
     $                                           ONE,
     $                                           M, 1, A( 1, p ), LDA,
     $                                           IERR )
                                    TEMP1 = -AAPQ*WORK( q ) / WORK( p )
                                    CALL DAXPY( M, TEMP1,
     $                                          WORK( N+1 ),
     $                                          1, A( 1, p ), 1 )
                                    CALL DLASCL( 'G', 0, 0, ONE,
     $                                           AAPP,
     $                                           M, 1, A( 1, p ), LDA,
     $                                           IERR )
                                    SVA( p ) = AAPP*DSQRT( MAX( ZERO,
     $                                         ONE-AAPQ*AAPQ ) )
                                    MXSINJ = MAX( MXSINJ, SFMIN )
                                 END IF
                              END IF
*           END IF ROTOK THEN ... ELSE
*
*           In the case of cancellation in updating SVA(q)
*           .. recompute SVA(q)
                              IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
     $                            THEN
                                 IF( ( AAQQ.LT.ROOTBIG ) .AND.
     $                               ( AAQQ.GT.ROOTSFMIN ) ) THEN
                                    SVA( q ) = DNRM2( M, A( 1, q ),
     $                                   1 )*
     $                                         WORK( q )
                                 ELSE
                                    T = ZERO
                                    AAQQ = ONE
                                    CALL DLASSQ( M, A( 1, q ), 1, T,
     $                                           AAQQ )
                                    SVA( q ) = T*DSQRT( AAQQ )*WORK( q )
                                 END IF
                              END IF
                              IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN
                                 IF( ( AAPP.LT.ROOTBIG ) .AND.
     $                               ( AAPP.GT.ROOTSFMIN ) ) THEN
                                    AAPP = DNRM2( M, A( 1, p ), 1 )*
     $                                     WORK( p )
                                 ELSE
                                    T = ZERO
                                    AAPP = ONE
                                    CALL DLASSQ( M, A( 1, p ), 1, T,
     $                                           AAPP )
                                    AAPP = T*DSQRT( AAPP )*WORK( p )
                                 END IF
                                 SVA( p ) = AAPP
                              END IF
*              end of OK rotation
                           ELSE
                              NOTROT = NOTROT + 1
*[RTD]      SKIPPED  = SKIPPED  + 1
                              PSKIPPED = PSKIPPED + 1
                              IJBLSK = IJBLSK + 1
                           END IF
                        ELSE
                           NOTROT = NOTROT + 1
                           PSKIPPED = PSKIPPED + 1
                           IJBLSK = IJBLSK + 1
                        END IF
*
                        IF( ( i.LE.SWBAND ) .AND. ( IJBLSK.GE.BLSKIP ) )
     $                      THEN
                           SVA( p ) = AAPP
                           NOTROT = 0
                           GO TO 2011
                        END IF
                        IF( ( i.LE.SWBAND ) .AND.
     $                      ( PSKIPPED.GT.ROWSKIP ) ) THEN
                           AAPP = -AAPP
                           NOTROT = 0
                           GO TO 2203
                        END IF
*
 2200                CONTINUE
*        end of the q-loop
 2203                CONTINUE
*
                     SVA( p ) = AAPP
*
                  ELSE
*
                     IF( AAPP.EQ.ZERO )NOTROT = NOTROT +
     $                   MIN( jgl+KBL-1, N ) - jgl + 1
                     IF( AAPP.LT.ZERO )NOTROT = 0
*
                  END IF
*
 2100          CONTINUE
*     end of the p-loop
 2010       CONTINUE
*     end of the jbc-loop
 2011       CONTINUE
*2011 bailed out of the jbc-loop
            DO 2012 p = igl, MIN( igl+KBL-1, N )
               SVA( p ) = DABS( SVA( p ) )
 2012       CONTINUE
***
 2000    CONTINUE
*2000 :: end of the ibr-loop
*
*     .. update SVA(N)
         IF( ( SVA( N ).LT.ROOTBIG ) .AND. ( SVA( N ).GT.ROOTSFMIN ) )
     $       THEN
            SVA( N ) = DNRM2( M, A( 1, N ), 1 )*WORK( N )
         ELSE
            T = ZERO
            AAPP = ONE
            CALL DLASSQ( M, A( 1, N ), 1, T, AAPP )
            SVA( N ) = T*DSQRT( AAPP )*WORK( N )
         END IF
*
*     Additional steering devices
*
         IF( ( i.LT.SWBAND ) .AND. ( ( MXAAPQ.LE.ROOTTOL ) .OR.
     $       ( ISWROT.LE.N ) ) )SWBAND = i
*
         IF( ( i.GT.SWBAND+1 ) .AND. ( MXAAPQ.LT.DSQRT( DBLE( N ) )*
     $       TOL ) .AND. ( DBLE( N )*MXAAPQ*MXSINJ.LT.TOL ) ) THEN
            GO TO 1994
         END IF
*
         IF( NOTROT.GE.EMPTSW )GO TO 1994
*
 1993 CONTINUE
*     end i=1:NSWEEP loop
*
* #:( Reaching this point means that the procedure has not converged.
      INFO = NSWEEP - 1
      GO TO 1995
*
 1994 CONTINUE
* #:) Reaching this point means numerical convergence after the i-th
*     sweep.
*
      INFO = 0
* #:) INFO = 0 confirms successful iterations.
 1995 CONTINUE
*
*     Sort the singular values and find how many are above
*     the underflow threshold.
*
      N2 = 0
      N4 = 0
      DO 5991 p = 1, N - 1
         q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
         IF( p.NE.q ) THEN
            TEMP1 = SVA( p )
            SVA( p ) = SVA( q )
            SVA( q ) = TEMP1
            TEMP1 = WORK( p )
            WORK( p ) = WORK( q )
            WORK( q ) = TEMP1
            CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
            IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 )
         END IF
         IF( SVA( p ).NE.ZERO ) THEN
            N4 = N4 + 1
            IF( SVA( p )*SKL.GT.SFMIN )N2 = N2 + 1
         END IF
 5991 CONTINUE
      IF( SVA( N ).NE.ZERO ) THEN
         N4 = N4 + 1
         IF( SVA( N )*SKL.GT.SFMIN )N2 = N2 + 1
      END IF
*
*     Normalize the left singular vectors.
*
      IF( LSVEC .OR. UCTOL ) THEN
         DO 1998 p = 1, N2
            CALL DSCAL( M, WORK( p ) / SVA( p ), A( 1, p ), 1 )
 1998    CONTINUE
      END IF
*
*     Scale the product of Jacobi rotations (assemble the fast rotations).
*
      IF( RSVEC ) THEN
         IF( APPLV ) THEN
            DO 2398 p = 1, N
               CALL DSCAL( MVL, WORK( p ), V( 1, p ), 1 )
 2398       CONTINUE
         ELSE
            DO 2399 p = 1, N
               TEMP1 = ONE / DNRM2( MVL, V( 1, p ), 1 )
               CALL DSCAL( MVL, TEMP1, V( 1, p ), 1 )
 2399       CONTINUE
         END IF
      END IF
*
*     Undo scaling, if necessary (and possible).
      IF( ( ( SKL.GT.ONE ) .AND. ( SVA( 1 ).LT.( BIG / SKL) ) )
     $    .OR. ( ( SKL.LT.ONE ) .AND. ( SVA( MAX( N2, 1 ) ) .GT.
     $    ( SFMIN / SKL) ) ) ) THEN
         DO 2400 p = 1, N
            SVA( P ) = SKL*SVA( P )
 2400    CONTINUE
         SKL= ONE
      END IF
*
      WORK( 1 ) = SKL
*     The singular values of A are SKL*SVA(1:N). If SKL.NE.ONE
*     then some of the singular values may overflow or underflow and
*     the spectrum is given in this factored representation.
*
      WORK( 2 ) = DBLE( N4 )
*     N4 is the number of computed nonzero singular values of A.
*
      WORK( 3 ) = DBLE( N2 )
*     N2 is the number of singular values of A greater than SFMIN.
*     If N2<N, SVA(N2:N) contains ZEROS and/or denormalized numbers
*     that may carry some information.
*
      WORK( 4 ) = DBLE( i )
*     i is the index of the last sweep before declaring convergence.
*
      WORK( 5 ) = MXAAPQ
*     MXAAPQ is the largest absolute value of scaled pivots in the
*     last sweep
*
      WORK( 6 ) = MXSINJ
*     MXSINJ is the largest absolute value of the sines of Jacobi angles
*     in the last sweep
*
      RETURN
*     ..
*     .. END OF DGESVJ
*     ..
      END
*> \brief <b> DGESVX computes the solution to system of linear equations A * X = B for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGESVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgesvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgesvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgesvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
*                          EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
*                          WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          EQUED, FACT, TRANS
*       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
*       DOUBLE PRECISION   RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
*      $                   BERR( * ), C( * ), FERR( * ), R( * ),
*      $                   WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGESVX uses the LU factorization to compute the solution to a real
*> system of linear equations
*>    A * X = B,
*> where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
*  =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
*>    the system:
*>       TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
*>       TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
*>       TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
*>    Whether or not the system will be equilibrated depends on the
*>    scaling of the matrix A, but if equilibration is used, A is
*>    overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
*>    or diag(C)*B (if TRANS = 'T' or 'C').
*>
*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
*>    matrix A (after equilibration if FACT = 'E') as
*>       A = P * L * U,
*>    where P is a permutation matrix, L is a unit lower triangular
*>    matrix, and U is upper triangular.
*>
*> 3. If some U(i,i)=0, so that U is exactly singular, then the routine
*>    returns with INFO = i. Otherwise, the factored form of A is used
*>    to estimate the condition number of the matrix A.  If the
*>    reciprocal of the condition number is less than machine precision,
*>    INFO = N+1 is returned as a warning, but the routine still goes on
*>    to solve for X and compute error bounds as described below.
*>
*> 4. The system of equations is solved for X using the factored form
*>    of A.
*>
*> 5. Iterative refinement is applied to improve the computed solution
*>    matrix and calculate error bounds and backward error estimates
*>    for it.
*>
*> 6. If equilibration was used, the matrix X is premultiplied by
*>    diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
*>    that it solves the original system before equilibration.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] FACT
*> \verbatim
*>          FACT is CHARACTER*1
*>          Specifies whether or not the factored form of the matrix A is
*>          supplied on entry, and if not, whether the matrix A should be
*>          equilibrated before it is factored.
*>          = 'F':  On entry, AF and IPIV contain the factored form of A.
*>                  If EQUED is not 'N', the matrix A has been
*>                  equilibrated with scaling factors given by R and C.
*>                  A, AF, and IPIV are not modified.
*>          = 'N':  The matrix A will be copied to AF and factored.
*>          = 'E':  The matrix A will be equilibrated if necessary, then
*>                  copied to AF and factored.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the form of the system of equations:
*>          = 'N':  A * X = B     (No transpose)
*>          = 'T':  A**T * X = B  (Transpose)
*>          = 'C':  A**H * X = B  (Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of linear equations, i.e., the order of the
*>          matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices B and X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the N-by-N matrix A.  If FACT = 'F' and EQUED is
*>          not 'N', then A must have been equilibrated by the scaling
*>          factors in R and/or C.  A is not modified if FACT = 'F' or
*>          'N', or if FACT = 'E' and EQUED = 'N' on exit.
*>
*>          On exit, if EQUED .ne. 'N', A is scaled as follows:
*>          EQUED = 'R':  A := diag(R) * A
*>          EQUED = 'C':  A := A * diag(C)
*>          EQUED = 'B':  A := diag(R) * A * diag(C).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] AF
*> \verbatim
*>          AF is DOUBLE PRECISION array, dimension (LDAF,N)
*>          If FACT = 'F', then AF is an input argument and on entry
*>          contains the factors L and U from the factorization
*>          A = P*L*U as computed by DGETRF.  If EQUED .ne. 'N', then
*>          AF is the factored form of the equilibrated matrix A.
*>
*>          If FACT = 'N', then AF is an output argument and on exit
*>          returns the factors L and U from the factorization A = P*L*U
*>          of the original matrix A.
*>
*>          If FACT = 'E', then AF is an output argument and on exit
*>          returns the factors L and U from the factorization A = P*L*U
*>          of the equilibrated matrix A (see the description of A for
*>          the form of the equilibrated matrix).
*> \endverbatim
*>
*> \param[in] LDAF
*> \verbatim
*>          LDAF is INTEGER
*>          The leading dimension of the array AF.  LDAF >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          If FACT = 'F', then IPIV is an input argument and on entry
*>          contains the pivot indices from the factorization A = P*L*U
*>          as computed by DGETRF; row i of the matrix was interchanged
*>          with row IPIV(i).
*>
*>          If FACT = 'N', then IPIV is an output argument and on exit
*>          contains the pivot indices from the factorization A = P*L*U
*>          of the original matrix A.
*>
*>          If FACT = 'E', then IPIV is an output argument and on exit
*>          contains the pivot indices from the factorization A = P*L*U
*>          of the equilibrated matrix A.
*> \endverbatim
*>
*> \param[in,out] EQUED
*> \verbatim
*>          EQUED is CHARACTER*1
*>          Specifies the form of equilibration that was done.
*>          = 'N':  No equilibration (always true if FACT = 'N').
*>          = 'R':  Row equilibration, i.e., A has been premultiplied by
*>                  diag(R).
*>          = 'C':  Column equilibration, i.e., A has been postmultiplied
*>                  by diag(C).
*>          = 'B':  Both row and column equilibration, i.e., A has been
*>                  replaced by diag(R) * A * diag(C).
*>          EQUED is an input argument if FACT = 'F'; otherwise, it is an
*>          output argument.
*> \endverbatim
*>
*> \param[in,out] R
*> \verbatim
*>          R is DOUBLE PRECISION array, dimension (N)
*>          The row scale factors for A.  If EQUED = 'R' or 'B', A is
*>          multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
*>          is not accessed.  R is an input argument if FACT = 'F';
*>          otherwise, R is an output argument.  If FACT = 'F' and
*>          EQUED = 'R' or 'B', each element of R must be positive.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (N)
*>          The column scale factors for A.  If EQUED = 'C' or 'B', A is
*>          multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
*>          is not accessed.  C is an input argument if FACT = 'F';
*>          otherwise, C is an output argument.  If FACT = 'F' and
*>          EQUED = 'C' or 'B', each element of C must be positive.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the N-by-NRHS right hand side matrix B.
*>          On exit,
*>          if EQUED = 'N', B is not modified;
*>          if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
*>          diag(R)*B;
*>          if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
*>          overwritten by diag(C)*B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
*>          to the original system of equations.  Note that A and B are
*>          modified on exit if EQUED .ne. 'N', and the solution to the
*>          equilibrated system is inv(diag(C))*X if TRANS = 'N' and
*>          EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
*>          and EQUED = 'R' or 'B'.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The estimate of the reciprocal condition number of the matrix
*>          A after equilibration (if done).  If RCOND is less than the
*>          machine precision (in particular, if RCOND = 0), the matrix
*>          is singular to working precision.  This condition is
*>          indicated by a return code of INFO > 0.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The estimated forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).  The estimate is as reliable as
*>          the estimate for RCOND, and is almost always a slight
*>          overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in
*>          any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,4*N))
*>          On exit, WORK(1) contains the reciprocal pivot growth
*>          factor norm(A)/norm(U). The "max absolute element" norm is
*>          used. If WORK(1) is much less than 1, then the stability
*>          of the LU factorization of the (equilibrated) matrix A
*>          could be poor. This also means that the solution X, condition
*>          estimator RCOND, and forward error bound FERR could be
*>          unreliable. If factorization fails with 0<INFO<=N, then
*>          WORK(1) contains the reciprocal pivot growth factor for the
*>          leading INFO columns of A.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, and i is
*>                <= N:  U(i,i) is exactly zero.  The factorization has
*>                       been completed, but the factor U is exactly
*>                       singular, so the solution and error bounds
*>                       could not be computed. RCOND = 0 is returned.
*>                = N+1: U is nonsingular, but RCOND is less than machine
*>                       precision, meaning that the matrix is singular
*>                       to working precision.  Nevertheless, the
*>                       solution and error bounds are computed because
*>                       there are a number of situations where the
*>                       computed solution can be more accurate than the
*>                       value of RCOND would suggest.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gesvx
*
*  =====================================================================
      SUBROUTINE DGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF,
     $                   IPIV,
     $                   EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
     $                   WORK, IWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          EQUED, FACT, TRANS
      INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
     $                   BERR( * ), C( * ), FERR( * ), R( * ),
     $                   WORK( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
      CHARACTER          NORM
      INTEGER            I, INFEQU, J
      DOUBLE PRECISION   AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
     $                   ROWCND, RPVGRW, SMLNUM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANGE, DLANTR
      EXTERNAL           LSAME, DLAMCH, DLANGE, DLANTR
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGECON, DGEEQU, DGERFS, DGETRF, DGETRS,
     $                   DLACPY,
     $                   DLAQGE, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      NOFACT = LSAME( FACT, 'N' )
      EQUIL = LSAME( FACT, 'E' )
      NOTRAN = LSAME( TRANS, 'N' )
      IF( NOFACT .OR. EQUIL ) THEN
         EQUED = 'N'
         ROWEQU = .FALSE.
         COLEQU = .FALSE.
      ELSE
         ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
         COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
         SMLNUM = DLAMCH( 'Safe minimum' )
         BIGNUM = ONE / SMLNUM
      END IF
*
*     Test the input parameters.
*
      IF( .NOT.NOFACT .AND.
     $    .NOT.EQUIL .AND.
     $    .NOT.LSAME( FACT, 'F' ) )
     $     THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
     $         LSAME( TRANS, 'C' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -6
      ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
         INFO = -8
      ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
     $         ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
         INFO = -10
      ELSE
         IF( ROWEQU ) THEN
            RCMIN = BIGNUM
            RCMAX = ZERO
            DO 10 J = 1, N
               RCMIN = MIN( RCMIN, R( J ) )
               RCMAX = MAX( RCMAX, R( J ) )
   10       CONTINUE
            IF( RCMIN.LE.ZERO ) THEN
               INFO = -11
            ELSE IF( N.GT.0 ) THEN
               ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
            ELSE
               ROWCND = ONE
            END IF
         END IF
         IF( COLEQU .AND. INFO.EQ.0 ) THEN
            RCMIN = BIGNUM
            RCMAX = ZERO
            DO 20 J = 1, N
               RCMIN = MIN( RCMIN, C( J ) )
               RCMAX = MAX( RCMAX, C( J ) )
   20       CONTINUE
            IF( RCMIN.LE.ZERO ) THEN
               INFO = -12
            ELSE IF( N.GT.0 ) THEN
               COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
            ELSE
               COLCND = ONE
            END IF
         END IF
         IF( INFO.EQ.0 ) THEN
            IF( LDB.LT.MAX( 1, N ) ) THEN
               INFO = -14
            ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
               INFO = -16
            END IF
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGESVX', -INFO )
         RETURN
      END IF
*
      IF( EQUIL ) THEN
*
*        Compute row and column scalings to equilibrate the matrix A.
*
         CALL DGEEQU( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
     $                INFEQU )
         IF( INFEQU.EQ.0 ) THEN
*
*           Equilibrate the matrix.
*
            CALL DLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
     $                   EQUED )
            ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
            COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
         END IF
      END IF
*
*     Scale the right hand side.
*
      IF( NOTRAN ) THEN
         IF( ROWEQU ) THEN
            DO 40 J = 1, NRHS
               DO 30 I = 1, N
                  B( I, J ) = R( I )*B( I, J )
   30          CONTINUE
   40       CONTINUE
         END IF
      ELSE IF( COLEQU ) THEN
         DO 60 J = 1, NRHS
            DO 50 I = 1, N
               B( I, J ) = C( I )*B( I, J )
   50       CONTINUE
   60    CONTINUE
      END IF
*
      IF( NOFACT .OR. EQUIL ) THEN
*
*        Compute the LU factorization of A.
*
         CALL DLACPY( 'Full', N, N, A, LDA, AF, LDAF )
         CALL DGETRF( N, N, AF, LDAF, IPIV, INFO )
*
*        Return if INFO is non-zero.
*
         IF( INFO.GT.0 ) THEN
*
*           Compute the reciprocal pivot growth factor of the
*           leading rank-deficient INFO columns of A.
*
            RPVGRW = DLANTR( 'M', 'U', 'N', INFO, INFO, AF, LDAF,
     $               WORK )
            IF( RPVGRW.EQ.ZERO ) THEN
               RPVGRW = ONE
            ELSE
               RPVGRW = DLANGE( 'M', N, INFO, A, LDA, WORK ) / RPVGRW
            END IF
            WORK( 1 ) = RPVGRW
            RCOND = ZERO
            RETURN
         END IF
      END IF
*
*     Compute the norm of the matrix A and the
*     reciprocal pivot growth factor RPVGRW.
*
      IF( NOTRAN ) THEN
         NORM = '1'
      ELSE
         NORM = 'I'
      END IF
      ANORM = DLANGE( NORM, N, N, A, LDA, WORK )
      RPVGRW = DLANTR( 'M', 'U', 'N', N, N, AF, LDAF, WORK )
      IF( RPVGRW.EQ.ZERO ) THEN
         RPVGRW = ONE
      ELSE
         RPVGRW = DLANGE( 'M', N, N, A, LDA, WORK ) / RPVGRW
      END IF
*
*     Compute the reciprocal of the condition number of A.
*
      CALL DGECON( NORM, N, AF, LDAF, ANORM, RCOND, WORK, IWORK,
     $             INFO )
*
*     Compute the solution matrix X.
*
      CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
      CALL DGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
*
*     Use iterative refinement to improve the computed solution and
*     compute error bounds and backward error estimates for it.
*
      CALL DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
     $             LDX, FERR, BERR, WORK, IWORK, INFO )
*
*     Transform the solution matrix X to a solution of the original
*     system.
*
      IF( NOTRAN ) THEN
         IF( COLEQU ) THEN
            DO 80 J = 1, NRHS
               DO 70 I = 1, N
                  X( I, J ) = C( I )*X( I, J )
   70          CONTINUE
   80       CONTINUE
            DO 90 J = 1, NRHS
               FERR( J ) = FERR( J ) / COLCND
   90       CONTINUE
         END IF
      ELSE IF( ROWEQU ) THEN
         DO 110 J = 1, NRHS
            DO 100 I = 1, N
               X( I, J ) = R( I )*X( I, J )
  100       CONTINUE
  110    CONTINUE
         DO 120 J = 1, NRHS
            FERR( J ) = FERR( J ) / ROWCND
  120    CONTINUE
      END IF
*
      WORK( 1 ) = RPVGRW
*
*     Set INFO = N+1 if the matrix is singular to working precision.
*
      IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
     $   INFO = N + 1
      RETURN
*
*     End of DGESVX
*
      END
*> \brief \b DGETC2 computes the LU factorization with complete pivoting of the general n-by-n matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGETC2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgetc2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgetc2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgetc2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGETC2( N, A, LDA, IPIV, JPIV, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), JPIV( * )
*       DOUBLE PRECISION   A( LDA, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGETC2 computes an LU factorization with complete pivoting of the
*> n-by-n matrix A. The factorization has the form A = P * L * U * Q,
*> where P and Q are permutation matrices, L is lower triangular with
*> unit diagonal elements and U is upper triangular.
*>
*> This is the Level 2 BLAS algorithm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, N)
*>          On entry, the n-by-n matrix A to be factored.
*>          On exit, the factors L and U from the factorization
*>          A = P*L*U*Q; the unit diagonal elements of L are not stored.
*>          If U(k, k) appears to be less than SMIN, U(k, k) is given the
*>          value of SMIN, i.e., giving a nonsingular perturbed system.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension(N).
*>          The pivot indices; for 1 <= i <= N, row i of the
*>          matrix has been interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[out] JPIV
*> \verbatim
*>          JPIV is INTEGER array, dimension(N).
*>          The pivot indices; for 1 <= j <= N, column j of the
*>          matrix has been interchanged with column JPIV(j).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>           = 0: successful exit
*>           > 0: if INFO = k, U(k, k) is likely to produce overflow if
*>                we try to solve for x in Ax = b. So U is perturbed to
*>                avoid the overflow.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup getc2
*
*> \par Contributors:
*  ==================
*>
*>     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*>     Umea University, S-901 87 Umea, Sweden.
*
*  =====================================================================
      SUBROUTINE DGETC2( N, A, LDA, IPIV, JPIV, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), JPIV( * )
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, IP, IPV, J, JP, JPV
      DOUBLE PRECISION   BIGNUM, EPS, SMIN, SMLNUM, XMAX
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGER, DSWAP
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. Executable Statements ..
*
      INFO = 0
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Set constants to control overflow
*
      EPS = DLAMCH( 'P' )
      SMLNUM = DLAMCH( 'S' ) / EPS
      BIGNUM = ONE / SMLNUM
*
*     Handle the case N=1 by itself
*
      IF( N.EQ.1 ) THEN
         IPIV( 1 ) = 1
         JPIV( 1 ) = 1
         IF( ABS( A( 1, 1 ) ).LT.SMLNUM ) THEN
            INFO = 1
            A( 1, 1 ) = SMLNUM
         END IF
         RETURN
      END IF
*
*     Factorize A using complete pivoting.
*     Set pivots less than SMIN to SMIN.
*
      DO 40 I = 1, N - 1
*
*        Find max element in matrix A
*
         XMAX = ZERO
         DO 20 JP = I, N
            DO 10 IP = I, N
               IF( ABS( A( IP, JP ) ).GE.XMAX ) THEN
                  XMAX = ABS( A( IP, JP ) )
                  IPV = IP
                  JPV = JP
               END IF
   10       CONTINUE
   20    CONTINUE
         IF( I.EQ.1 )
     $      SMIN = MAX( EPS*XMAX, SMLNUM )
*
*        Swap rows
*
         IF( IPV.NE.I )
     $      CALL DSWAP( N, A( IPV, 1 ), LDA, A( I, 1 ), LDA )
         IPIV( I ) = IPV
*
*        Swap columns
*
         IF( JPV.NE.I )
     $      CALL DSWAP( N, A( 1, JPV ), 1, A( 1, I ), 1 )
         JPIV( I ) = JPV
*
*        Check for singularity
*
         IF( ABS( A( I, I ) ).LT.SMIN ) THEN
            INFO = I
            A( I, I ) = SMIN
         END IF
         DO 30 J = I + 1, N
            A( J, I ) = A( J, I ) / A( I, I )
   30    CONTINUE
         CALL DGER( N-I, N-I, -ONE, A( I+1, I ), 1, A( I, I+1 ), LDA,
     $              A( I+1, I+1 ), LDA )
   40 CONTINUE
*
      IF( ABS( A( N, N ) ).LT.SMIN ) THEN
         INFO = N
         A( N, N ) = SMIN
      END IF
*
*     Set last pivots to N
*
      IPIV( N ) = N
      JPIV( N ) = N
*
      RETURN
*
*     End of DGETC2
*
      END
*> \brief \b DGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting with row interchanges (unblocked algorithm).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGETF2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgetf2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgetf2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgetf2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGETF2( M, N, A, LDA, IPIV, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, M, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   A( LDA, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGETF2 computes an LU factorization of a general m-by-n matrix A
*> using partial pivoting with row interchanges.
*>
*> The factorization has the form
*>    A = P * L * U
*> where P is a permutation matrix, L is lower triangular with unit
*> diagonal elements (lower trapezoidal if m > n), and U is upper
*> triangular (upper trapezoidal if m < n).
*>
*> This is the right-looking Level 2 BLAS version of the algorithm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the m by n matrix to be factored.
*>          On exit, the factors L and U from the factorization
*>          A = P*L*U; the unit diagonal elements of L are not stored.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (min(M,N))
*>          The pivot indices; for 1 <= i <= min(M,N), row i of the
*>          matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -k, the k-th argument had an illegal value
*>          > 0: if INFO = k, U(k,k) is exactly zero. The factorization
*>               has been completed, but the factor U is exactly
*>               singular, and division by zero will occur if it is used
*>               to solve a system of equations.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup getf2
*
*  =====================================================================
      SUBROUTINE DGETF2( M, N, A, LDA, IPIV, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   SFMIN
      INTEGER            I, J, JP
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      INTEGER            IDAMAX
      EXTERNAL           DLAMCH, IDAMAX
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGER, DSCAL, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGETF2', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 )
     $   RETURN
*
*     Compute machine safe minimum
*
      SFMIN = DLAMCH('S')
*
      DO 10 J = 1, MIN( M, N )
*
*        Find pivot and test for singularity.
*
         JP = J - 1 + IDAMAX( M-J+1, A( J, J ), 1 )
         IPIV( J ) = JP
         IF( A( JP, J ).NE.ZERO ) THEN
*
*           Apply the interchange to columns 1:N.
*
            IF( JP.NE.J )
     $         CALL DSWAP( N, A( J, 1 ), LDA, A( JP, 1 ), LDA )
*
*           Compute elements J+1:M of J-th column.
*
            IF( J.LT.M ) THEN
               IF( ABS(A( J, J )) .GE. SFMIN ) THEN
                  CALL DSCAL( M-J, ONE / A( J, J ), A( J+1, J ), 1 )
               ELSE
                 DO 20 I = 1, M-J
                    A( J+I, J ) = A( J+I, J ) / A( J, J )
   20            CONTINUE
               END IF
            END IF
*
         ELSE IF( INFO.EQ.0 ) THEN
*
            INFO = J
         END IF
*
         IF( J.LT.MIN( M, N ) ) THEN
*
*           Update trailing submatrix.
*
            CALL DGER( M-J, N-J, -ONE, A( J+1, J ), 1, A( J, J+1 ),
     $                 LDA,
     $                 A( J+1, J+1 ), LDA )
         END IF
   10 CONTINUE
      RETURN
*
*     End of DGETF2
*
      END
*> \brief \b DGETRF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGETRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgetrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgetrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgetrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGETRF( M, N, A, LDA, IPIV, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, M, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   A( LDA, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGETRF computes an LU factorization of a general M-by-N matrix A
*> using partial pivoting with row interchanges.
*>
*> The factorization has the form
*>    A = P * L * U
*> where P is a permutation matrix, L is lower triangular with unit
*> diagonal elements (lower trapezoidal if m > n), and U is upper
*> triangular (upper trapezoidal if m < n).
*>
*> This is the right-looking Level 3 BLAS version of the algorithm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix to be factored.
*>          On exit, the factors L and U from the factorization
*>          A = P*L*U; the unit diagonal elements of L are not stored.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (min(M,N))
*>          The pivot indices; for 1 <= i <= min(M,N), row i of the
*>          matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
*>                has been completed, but the factor U is exactly
*>                singular, and division by zero will occur if it is used
*>                to solve a system of equations.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup getrf
*
*  =====================================================================
      SUBROUTINE DGETRF( M, N, A, LDA, IPIV, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, IINFO, J, JB, NB
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMM, DGETRF2, DLASWP, DTRSM,
     $                   XERBLA
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGETRF', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 )
     $   RETURN
*
*     Determine the block size for this environment.
*
      NB = ILAENV( 1, 'DGETRF', ' ', M, N, -1, -1 )
      IF( NB.LE.1 .OR. NB.GE.MIN( M, N ) ) THEN
*
*        Use unblocked code.
*
         CALL DGETRF2( M, N, A, LDA, IPIV, INFO )
      ELSE
*
*        Use blocked code.
*
         DO 20 J = 1, MIN( M, N ), NB
            JB = MIN( MIN( M, N )-J+1, NB )
*
*           Factor diagonal and subdiagonal blocks and test for exact
*           singularity.
*
            CALL DGETRF2( M-J+1, JB, A( J, J ), LDA, IPIV( J ),
     $                    IINFO )
*
*           Adjust INFO and the pivot indices.
*
            IF( INFO.EQ.0 .AND. IINFO.GT.0 )
     $         INFO = IINFO + J - 1
            DO 10 I = J, MIN( M, J+JB-1 )
               IPIV( I ) = J - 1 + IPIV( I )
   10       CONTINUE
*
*           Apply interchanges to columns 1:J-1.
*
            CALL DLASWP( J-1, A, LDA, J, J+JB-1, IPIV, 1 )
*
            IF( J+JB.LE.N ) THEN
*
*              Apply interchanges to columns J+JB:N.
*
               CALL DLASWP( N-J-JB+1, A( 1, J+JB ), LDA, J, J+JB-1,
     $                      IPIV, 1 )
*
*              Compute block row of U.
*
               CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Unit',
     $                     JB,
     $                     N-J-JB+1, ONE, A( J, J ), LDA, A( J, J+JB ),
     $                     LDA )
               IF( J+JB.LE.M ) THEN
*
*                 Update trailing submatrix.
*
                  CALL DGEMM( 'No transpose', 'No transpose',
     $                        M-J-JB+1,
     $                        N-J-JB+1, JB, -ONE, A( J+JB, J ), LDA,
     $                        A( J, J+JB ), LDA, ONE, A( J+JB, J+JB ),
     $                        LDA )
               END IF
            END IF
   20    CONTINUE
      END IF
      RETURN
*
*     End of DGETRF
*
      END
*> \brief \b DGETRF2
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*  Definition:
*  ===========
*
*       RECURSIVE SUBROUTINE DGETRF2( M, N, A, LDA, IPIV, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, M, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   A( LDA, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGETRF2 computes an LU factorization of a general M-by-N matrix A
*> using partial pivoting with row interchanges.
*>
*> The factorization has the form
*>    A = P * L * U
*> where P is a permutation matrix, L is lower triangular with unit
*> diagonal elements (lower trapezoidal if m > n), and U is upper
*> triangular (upper trapezoidal if m < n).
*>
*> This is the recursive version of the algorithm. It divides
*> the matrix into four submatrices:
*>
*>        [  A11 | A12  ]  where A11 is n1 by n1 and A22 is n2 by n2
*>    A = [ -----|----- ]  with n1 = min(m,n)/2
*>        [  A21 | A22  ]       n2 = n-n1
*>
*>                                       [ A11 ]
*> The subroutine calls itself to factor [ --- ],
*>                                       [ A12 ]
*>                 [ A12 ]
*> do the swaps on [ --- ], solve A12, update A22,
*>                 [ A22 ]
*>
*> then calls itself to factor A22 and do the swaps on A21.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix to be factored.
*>          On exit, the factors L and U from the factorization
*>          A = P*L*U; the unit diagonal elements of L are not stored.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (min(M,N))
*>          The pivot indices; for 1 <= i <= min(M,N), row i of the
*>          matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
*>                has been completed, but the factor U is exactly
*>                singular, and division by zero will occur if it is used
*>                to solve a system of equations.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup getrf2
*
*  =====================================================================
      RECURSIVE SUBROUTINE DGETRF2( M, N, A, LDA, IPIV, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   SFMIN, TEMP
      INTEGER            I, IINFO, N1, N2
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      INTEGER            IDAMAX
      EXTERNAL           DLAMCH, IDAMAX
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMM, DSCAL, DLASWP, DTRSM, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGETRF2', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 )
     $   RETURN

      IF ( M.EQ.1 ) THEN
*
*        Use unblocked code for one row case
*        Just need to handle IPIV and INFO
*
         IPIV( 1 ) = 1
         IF ( A(1,1).EQ.ZERO )
     $      INFO = 1
*
      ELSE IF( N.EQ.1 ) THEN
*
*        Use unblocked code for one column case
*
*
*        Compute machine safe minimum
*
         SFMIN = DLAMCH('S')
*
*        Find pivot and test for singularity
*
         I = IDAMAX( M, A( 1, 1 ), 1 )
         IPIV( 1 ) = I
         IF( A( I, 1 ).NE.ZERO ) THEN
*
*           Apply the interchange
*
            IF( I.NE.1 ) THEN
               TEMP = A( 1, 1 )
               A( 1, 1 ) = A( I, 1 )
               A( I, 1 ) = TEMP
            END IF
*
*           Compute elements 2:M of the column
*
            IF( ABS(A( 1, 1 )) .GE. SFMIN ) THEN
               CALL DSCAL( M-1, ONE / A( 1, 1 ), A( 2, 1 ), 1 )
            ELSE
               DO 10 I = 1, M-1
                  A( 1+I, 1 ) = A( 1+I, 1 ) / A( 1, 1 )
   10          CONTINUE
            END IF
*
         ELSE
            INFO = 1
         END IF
*
      ELSE
*
*        Use recursive code
*
         N1 = MIN( M, N ) / 2
         N2 = N-N1
*
*               [ A11 ]
*        Factor [ --- ]
*               [ A21 ]
*
         CALL DGETRF2( M, N1, A, LDA, IPIV, IINFO )

         IF ( INFO.EQ.0 .AND. IINFO.GT.0 )
     $      INFO = IINFO
*
*                              [ A12 ]
*        Apply interchanges to [ --- ]
*                              [ A22 ]
*
         CALL DLASWP( N2, A( 1, N1+1 ), LDA, 1, N1, IPIV, 1 )
*
*        Solve A12
*
         CALL DTRSM( 'L', 'L', 'N', 'U', N1, N2, ONE, A, LDA,
     $               A( 1, N1+1 ), LDA )
*
*        Update A22
*
         CALL DGEMM( 'N', 'N', M-N1, N2, N1, -ONE, A( N1+1, 1 ), LDA,
     $               A( 1, N1+1 ), LDA, ONE, A( N1+1, N1+1 ), LDA )
*
*        Factor A22
*
         CALL DGETRF2( M-N1, N2, A( N1+1, N1+1 ), LDA, IPIV( N1+1 ),
     $                 IINFO )
*
*        Adjust INFO and the pivot indices
*
         IF ( INFO.EQ.0 .AND. IINFO.GT.0 )
     $      INFO = IINFO + N1
         DO 20 I = N1+1, MIN( M, N )
            IPIV( I ) = IPIV( I ) + N1
   20    CONTINUE
*
*        Apply interchanges to A21
*
         CALL DLASWP( N1, A( 1, 1 ), LDA, N1+1, MIN( M, N), IPIV, 1 )
*
      END IF
      RETURN
*
*     End of DGETRF2
*
      END
*> \brief \b DGETRI
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGETRI + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgetri.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgetri.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgetri.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGETRI( N, A, LDA, IPIV, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LWORK, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   A( LDA, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGETRI computes the inverse of a matrix using the LU factorization
*> computed by DGETRF.
*>
*> This method inverts U and then computes inv(A) by solving the system
*> inv(A)*L = inv(U) for inv(A).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the factors L and U from the factorization
*>          A = P*L*U as computed by DGETRF.
*>          On exit, if INFO = 0, the inverse of the original matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          The pivot indices from DGETRF; for 1<=i<=N, row i of the
*>          matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO=0, then WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.  LWORK >= max(1,N).
*>          For optimal performance LWORK >= N*NB, where NB is
*>          the optimal blocksize returned by ILAENV.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, U(i,i) is exactly zero; the matrix is
*>                singular and its inverse could not be computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup getri
*
*  =====================================================================
      SUBROUTINE DGETRI( N, A, LDA, IPIV, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IWS, J, JB, JJ, JP, LDWORK, LWKOPT, NB,
     $                   NBMIN, NN
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMM, DGEMV, DSWAP, DTRSM, DTRTRI,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NB = ILAENV( 1, 'DGETRI', ' ', N, -1, -1, -1 )
      LWKOPT = MAX( 1, N*NB )
      WORK( 1 ) = LWKOPT
*
      LQUERY = ( LWORK.EQ.-1 )
      IF( N.LT.0 ) THEN
         INFO = -1
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -3
      ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGETRI', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Form inv(U).  If INFO > 0 from DTRTRI, then U is singular,
*     and the inverse is not computed.
*
      CALL DTRTRI( 'Upper', 'Non-unit', N, A, LDA, INFO )
      IF( INFO.GT.0 )
     $   RETURN
*
      NBMIN = 2
      LDWORK = N
      IF( NB.GT.1 .AND. NB.LT.N ) THEN
         IWS = MAX( LDWORK*NB, 1 )
         IF( LWORK.LT.IWS ) THEN
            NB = LWORK / LDWORK
            NBMIN = MAX( 2, ILAENV( 2, 'DGETRI', ' ', N, -1, -1,
     $                   -1 ) )
         END IF
      ELSE
         IWS = N
      END IF
*
*     Solve the equation inv(A)*L = inv(U) for inv(A).
*
      IF( NB.LT.NBMIN .OR. NB.GE.N ) THEN
*
*        Use unblocked code.
*
         DO 20 J = N, 1, -1
*
*           Copy current column of L to WORK and replace with zeros.
*
            DO 10 I = J + 1, N
               WORK( I ) = A( I, J )
               A( I, J ) = ZERO
   10       CONTINUE
*
*           Compute current column of inv(A).
*
            IF( J.LT.N )
     $         CALL DGEMV( 'No transpose', N, N-J, -ONE, A( 1, J+1 ),
     $                     LDA, WORK( J+1 ), 1, ONE, A( 1, J ), 1 )
   20    CONTINUE
      ELSE
*
*        Use blocked code.
*
         NN = ( ( N-1 ) / NB )*NB + 1
         DO 50 J = NN, 1, -NB
            JB = MIN( NB, N-J+1 )
*
*           Copy current block column of L to WORK and replace with
*           zeros.
*
            DO 40 JJ = J, J + JB - 1
               DO 30 I = JJ + 1, N
                  WORK( I+( JJ-J )*LDWORK ) = A( I, JJ )
                  A( I, JJ ) = ZERO
   30          CONTINUE
   40       CONTINUE
*
*           Compute current block column of inv(A).
*
            IF( J+JB.LE.N )
     $         CALL DGEMM( 'No transpose', 'No transpose', N, JB,
     $                     N-J-JB+1, -ONE, A( 1, J+JB ), LDA,
     $                     WORK( J+JB ), LDWORK, ONE, A( 1, J ), LDA )
            CALL DTRSM( 'Right', 'Lower', 'No transpose', 'Unit', N,
     $                  JB,
     $                  ONE, WORK( J ), LDWORK, A( 1, J ), LDA )
   50    CONTINUE
      END IF
*
*     Apply column interchanges.
*
      DO 60 J = N - 1, 1, -1
         JP = IPIV( J )
         IF( JP.NE.J )
     $      CALL DSWAP( N, A( 1, J ), 1, A( 1, JP ), 1 )
   60 CONTINUE
*
      WORK( 1 ) = IWS
      RETURN
*
*     End of DGETRI
*
      END
*> \brief \b DGETRS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGETRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgetrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgetrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgetrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGETRS( TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANS
*       INTEGER            INFO, LDA, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGETRS solves a system of linear equations
*>    A * X = B  or  A**T * X = B
*> with a general N-by-N matrix A using the LU factorization computed
*> by DGETRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the form of the system of equations:
*>          = 'N':  A * X = B  (No transpose)
*>          = 'T':  A**T* X = B  (Transpose)
*>          = 'C':  A**T* X = B  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The factors L and U from the factorization A = P*L*U
*>          as computed by DGETRF.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          The pivot indices from DGETRF; for 1<=i<=N, row i of the
*>          matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the right hand side matrix B.
*>          On exit, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup getrs
*
*  =====================================================================
      SUBROUTINE DGETRS( TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS
      INTEGER            INFO, LDA, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOTRAN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASWP, DTRSM, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NOTRAN = LSAME( TRANS, 'N' )
      IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
     $    LSAME( TRANS, 'C' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -8
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGETRS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 )
     $   RETURN
*
      IF( NOTRAN ) THEN
*
*        Solve A * X = B.
*
*        Apply row interchanges to the right hand sides.
*
         CALL DLASWP( NRHS, B, LDB, 1, N, IPIV, 1 )
*
*        Solve L*X = B, overwriting B with X.
*
         CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Unit', N,
     $               NRHS,
     $               ONE, A, LDA, B, LDB )
*
*        Solve U*X = B, overwriting B with X.
*
         CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', N,
     $               NRHS, ONE, A, LDA, B, LDB )
      ELSE
*
*        Solve A**T * X = B.
*
*        Solve U**T *X = B, overwriting B with X.
*
         CALL DTRSM( 'Left', 'Upper', 'Transpose', 'Non-unit', N,
     $               NRHS,
     $               ONE, A, LDA, B, LDB )
*
*        Solve L**T *X = B, overwriting B with X.
*
         CALL DTRSM( 'Left', 'Lower', 'Transpose', 'Unit', N, NRHS,
     $               ONE,
     $               A, LDA, B, LDB )
*
*        Apply row interchanges to the solution vectors.
*
         CALL DLASWP( NRHS, B, LDB, 1, N, IPIV, -1 )
      END IF
*
      RETURN
*
*     End of DGETRS
*
      END
*> \brief \b DGGBAK
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGBAK + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggbak.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggbak.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggbak.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGGBAK( JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V,
*                          LDV, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOB, SIDE
*       INTEGER            IHI, ILO, INFO, LDV, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   LSCALE( * ), RSCALE( * ), V( LDV, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGGBAK forms the right or left eigenvectors of a real generalized
*> eigenvalue problem A*x = lambda*B*x, by backward transformation on
*> the computed eigenvectors of the balanced pair of matrices output by
*> DGGBAL.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOB
*> \verbatim
*>          JOB is CHARACTER*1
*>          Specifies the type of backward transformation required:
*>          = 'N':  do nothing, return immediately;
*>          = 'P':  do backward transformation for permutation only;
*>          = 'S':  do backward transformation for scaling only;
*>          = 'B':  do backward transformations for both permutation and
*>                  scaling.
*>          JOB must be the same as the argument JOB supplied to DGGBAL.
*> \endverbatim
*>
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'R':  V contains right eigenvectors;
*>          = 'L':  V contains left eigenvectors.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of rows of the matrix V.  N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*>          ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*>          IHI is INTEGER
*>          The integers ILO and IHI determined by DGGBAL.
*>          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*> \endverbatim
*>
*> \param[in] LSCALE
*> \verbatim
*>          LSCALE is DOUBLE PRECISION array, dimension (N)
*>          Details of the permutations and/or scaling factors applied
*>          to the left side of A and B, as returned by DGGBAL.
*> \endverbatim
*>
*> \param[in] RSCALE
*> \verbatim
*>          RSCALE is DOUBLE PRECISION array, dimension (N)
*>          Details of the permutations and/or scaling factors applied
*>          to the right side of A and B, as returned by DGGBAL.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of columns of the matrix V.  M >= 0.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension (LDV,M)
*>          On entry, the matrix of right or left eigenvectors to be
*>          transformed, as returned by DTGEVC.
*>          On exit, V is overwritten by the transformed eigenvectors.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of the matrix V. LDV >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggbak
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  See R.C. Ward, Balancing the generalized eigenvalue problem,
*>                 SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGGBAK( JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M,
     $                   V,
     $                   LDV, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOB, SIDE
      INTEGER            IHI, ILO, INFO, LDV, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   LSCALE( * ), RSCALE( * ), V( LDV, * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            LEFTV, RIGHTV
      INTEGER            I, K
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, INT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      RIGHTV = LSAME( SIDE, 'R' )
      LEFTV = LSAME( SIDE, 'L' )
*
      INFO = 0
      IF( .NOT.LSAME( JOB, 'N' ) .AND.
     $    .NOT.LSAME( JOB, 'P' ) .AND.
     $    .NOT.LSAME( JOB, 'S' ) .AND.
     $                .NOT.LSAME( JOB, 'B' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( ILO.LT.1 ) THEN
         INFO = -4
      ELSE IF( N.EQ.0 .AND. IHI.EQ.0 .AND. ILO.NE.1 ) THEN
         INFO = -4
      ELSE IF( N.GT.0 .AND. ( IHI.LT.ILO .OR. IHI.GT.MAX( 1, N ) ) )
     $   THEN
         INFO = -5
      ELSE IF( N.EQ.0 .AND. ILO.EQ.1 .AND. IHI.NE.0 ) THEN
         INFO = -5
      ELSE IF( M.LT.0 ) THEN
         INFO = -8
      ELSE IF( LDV.LT.MAX( 1, N ) ) THEN
         INFO = -10
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGGBAK', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
      IF( M.EQ.0 )
     $   RETURN
      IF( LSAME( JOB, 'N' ) )
     $   RETURN
*
      IF( ILO.EQ.IHI )
     $   GO TO 30
*
*     Backward balance
*
      IF( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'B' ) ) THEN
*
*        Backward transformation on right eigenvectors
*
         IF( RIGHTV ) THEN
            DO 10 I = ILO, IHI
               CALL DSCAL( M, RSCALE( I ), V( I, 1 ), LDV )
   10       CONTINUE
         END IF
*
*        Backward transformation on left eigenvectors
*
         IF( LEFTV ) THEN
            DO 20 I = ILO, IHI
               CALL DSCAL( M, LSCALE( I ), V( I, 1 ), LDV )
   20       CONTINUE
         END IF
      END IF
*
*     Backward permutation
*
   30 CONTINUE
      IF( LSAME( JOB, 'P' ) .OR. LSAME( JOB, 'B' ) ) THEN
*
*        Backward permutation on right eigenvectors
*
         IF( RIGHTV ) THEN
            IF( ILO.EQ.1 )
     $         GO TO 50
*
            DO 40 I = ILO - 1, 1, -1
               K = INT(RSCALE( I ))
               IF( K.EQ.I )
     $            GO TO 40
               CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV )
   40       CONTINUE
*
   50       CONTINUE
            IF( IHI.EQ.N )
     $         GO TO 70
            DO 60 I = IHI + 1, N
               K = INT(RSCALE( I ))
               IF( K.EQ.I )
     $            GO TO 60
               CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV )
   60       CONTINUE
         END IF
*
*        Backward permutation on left eigenvectors
*
   70    CONTINUE
         IF( LEFTV ) THEN
            IF( ILO.EQ.1 )
     $         GO TO 90
            DO 80 I = ILO - 1, 1, -1
               K = INT(LSCALE( I ))
               IF( K.EQ.I )
     $            GO TO 80
               CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV )
   80       CONTINUE
*
   90       CONTINUE
            IF( IHI.EQ.N )
     $         GO TO 110
            DO 100 I = IHI + 1, N
               K = INT(LSCALE( I ))
               IF( K.EQ.I )
     $            GO TO 100
               CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV )
  100       CONTINUE
         END IF
      END IF
*
  110 CONTINUE
*
      RETURN
*
*     End of DGGBAK
*
      END
*> \brief \b DGGBAL
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGBAL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggbal.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggbal.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggbal.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE,
*                          RSCALE, WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOB
*       INTEGER            IHI, ILO, INFO, LDA, LDB, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), LSCALE( * ),
*      $                   RSCALE( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGGBAL balances a pair of general real matrices (A,B).  This
*> involves, first, permuting A and B by similarity transformations to
*> isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N
*> elements on the diagonal; and second, applying a diagonal similarity
*> transformation to rows and columns ILO to IHI to make the rows
*> and columns as close in norm as possible. Both steps are optional.
*>
*> Balancing may reduce the 1-norm of the matrices, and improve the
*> accuracy of the computed eigenvalues and/or eigenvectors in the
*> generalized eigenvalue problem A*x = lambda*B*x.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOB
*> \verbatim
*>          JOB is CHARACTER*1
*>          Specifies the operations to be performed on A and B:
*>          = 'N':  none:  simply set ILO = 1, IHI = N, LSCALE(I) = 1.0
*>                  and RSCALE(I) = 1.0 for i = 1,...,N.
*>          = 'P':  permute only;
*>          = 'S':  scale only;
*>          = 'B':  both permute and scale.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A and B.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the input matrix A.
*>          On exit,  A is overwritten by the balanced matrix.
*>          If JOB = 'N', A is not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,N)
*>          On entry, the input matrix B.
*>          On exit,  B is overwritten by the balanced matrix.
*>          If JOB = 'N', B is not referenced.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ILO
*> \verbatim
*>          ILO is INTEGER
*> \endverbatim
*>
*> \param[out] IHI
*> \verbatim
*>          IHI is INTEGER
*>          ILO and IHI are set to integers such that on exit
*>          A(i,j) = 0 and B(i,j) = 0 if i > j and
*>          j = 1,...,ILO-1 or i = IHI+1,...,N.
*>          If JOB = 'N' or 'S', ILO = 1 and IHI = N.
*> \endverbatim
*>
*> \param[out] LSCALE
*> \verbatim
*>          LSCALE is DOUBLE PRECISION array, dimension (N)
*>          Details of the permutations and scaling factors applied
*>          to the left side of A and B.  If P(j) is the index of the
*>          row interchanged with row j, and D(j)
*>          is the scaling factor applied to row j, then
*>            LSCALE(j) = P(j)    for J = 1,...,ILO-1
*>                      = D(j)    for J = ILO,...,IHI
*>                      = P(j)    for J = IHI+1,...,N.
*>          The order in which the interchanges are made is N to IHI+1,
*>          then 1 to ILO-1.
*> \endverbatim
*>
*> \param[out] RSCALE
*> \verbatim
*>          RSCALE is DOUBLE PRECISION array, dimension (N)
*>          Details of the permutations and scaling factors applied
*>          to the right side of A and B.  If P(j) is the index of the
*>          column interchanged with column j, and D(j)
*>          is the scaling factor applied to column j, then
*>            LSCALE(j) = P(j)    for J = 1,...,ILO-1
*>                      = D(j)    for J = ILO,...,IHI
*>                      = P(j)    for J = IHI+1,...,N.
*>          The order in which the interchanges are made is N to IHI+1,
*>          then 1 to ILO-1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (lwork)
*>          lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and
*>          at least 1 when JOB = 'N' or 'P'.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggbal
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  See R.C. WARD, Balancing the generalized eigenvalue problem,
*>                 SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE,
     $                   RSCALE, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOB
      INTEGER            IHI, ILO, INFO, LDA, LDB, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), LSCALE( * ),
     $                   RSCALE( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, HALF, ONE
      PARAMETER          ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 )
      DOUBLE PRECISION   THREE, SCLFAC
      PARAMETER          ( THREE = 3.0D+0, SCLFAC = 1.0D+1 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, ICAB, IFLOW, IP1, IR, IRAB, IT, J, JC, JP1,
     $                   K, KOUNT, L, LCAB, LM1, LRAB, LSFMAX, LSFMIN,
     $                   M, NR, NRP2
      DOUBLE PRECISION   ALPHA, BASL, BETA, CAB, CMAX, COEF, COEF2,
     $                   COEF5, COR, EW, EWC, GAMMA, PGAMMA, RAB, SFMAX,
     $                   SFMIN, SUM, T, TA, TB, TC
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX
      DOUBLE PRECISION   DDOT, DLAMCH
      EXTERNAL           LSAME, IDAMAX, DDOT, DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DSCAL, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, INT, LOG10, MAX, MIN, SIGN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INFO = 0
      IF( .NOT.LSAME( JOB, 'N' ) .AND.
     $    .NOT.LSAME( JOB, 'P' ) .AND.
     $    .NOT.LSAME( JOB, 'S' ) .AND.
     $                .NOT.LSAME( JOB, 'B' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGGBAL', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 ) THEN
         ILO = 1
         IHI = N
         RETURN
      END IF
*
      IF( N.EQ.1 ) THEN
         ILO = 1
         IHI = N
         LSCALE( 1 ) = ONE
         RSCALE( 1 ) = ONE
         RETURN
      END IF
*
      IF( LSAME( JOB, 'N' ) ) THEN
         ILO = 1
         IHI = N
         DO 10 I = 1, N
            LSCALE( I ) = ONE
            RSCALE( I ) = ONE
   10    CONTINUE
         RETURN
      END IF
*
      K = 1
      L = N
      IF( LSAME( JOB, 'S' ) )
     $   GO TO 190
*
      GO TO 30
*
*     Permute the matrices A and B to isolate the eigenvalues.
*
*     Find row with one nonzero in columns 1 through L
*
   20 CONTINUE
      L = LM1
      IF( L.NE.1 )
     $   GO TO 30
*
      RSCALE( 1 ) = ONE
      LSCALE( 1 ) = ONE
      GO TO 190
*
   30 CONTINUE
      LM1 = L - 1
      DO 80 I = L, 1, -1
         DO 40 J = 1, LM1
            JP1 = J + 1
            IF( A( I, J ).NE.ZERO .OR. B( I, J ).NE.ZERO )
     $         GO TO 50
   40    CONTINUE
         J = L
         GO TO 70
*
   50    CONTINUE
         DO 60 J = JP1, L
            IF( A( I, J ).NE.ZERO .OR. B( I, J ).NE.ZERO )
     $         GO TO 80
   60    CONTINUE
         J = JP1 - 1
*
   70    CONTINUE
         M = L
         IFLOW = 1
         GO TO 160
   80 CONTINUE
      GO TO 100
*
*     Find column with one nonzero in rows K through N
*
   90 CONTINUE
      K = K + 1
*
  100 CONTINUE
      DO 150 J = K, L
         DO 110 I = K, LM1
            IP1 = I + 1
            IF( A( I, J ).NE.ZERO .OR. B( I, J ).NE.ZERO )
     $         GO TO 120
  110    CONTINUE
         I = L
         GO TO 140
  120    CONTINUE
         DO 130 I = IP1, L
            IF( A( I, J ).NE.ZERO .OR. B( I, J ).NE.ZERO )
     $         GO TO 150
  130    CONTINUE
         I = IP1 - 1
  140    CONTINUE
         M = K
         IFLOW = 2
         GO TO 160
  150 CONTINUE
      GO TO 190
*
*     Permute rows M and I
*
  160 CONTINUE
      LSCALE( M ) = I
      IF( I.EQ.M )
     $   GO TO 170
      CALL DSWAP( N-K+1, A( I, K ), LDA, A( M, K ), LDA )
      CALL DSWAP( N-K+1, B( I, K ), LDB, B( M, K ), LDB )
*
*     Permute columns M and J
*
  170 CONTINUE
      RSCALE( M ) = J
      IF( J.EQ.M )
     $   GO TO 180
      CALL DSWAP( L, A( 1, J ), 1, A( 1, M ), 1 )
      CALL DSWAP( L, B( 1, J ), 1, B( 1, M ), 1 )
*
  180 CONTINUE
      GO TO ( 20, 90 )IFLOW
*
  190 CONTINUE
      ILO = K
      IHI = L
*
      IF( LSAME( JOB, 'P' ) ) THEN
         DO 195 I = ILO, IHI
            LSCALE( I ) = ONE
            RSCALE( I ) = ONE
  195    CONTINUE
         RETURN
      END IF
*
      IF( ILO.EQ.IHI )
     $   RETURN
*
*     Balance the submatrix in rows ILO to IHI.
*
      NR = IHI - ILO + 1
      DO 200 I = ILO, IHI
         RSCALE( I ) = ZERO
         LSCALE( I ) = ZERO
*
         WORK( I ) = ZERO
         WORK( I+N ) = ZERO
         WORK( I+2*N ) = ZERO
         WORK( I+3*N ) = ZERO
         WORK( I+4*N ) = ZERO
         WORK( I+5*N ) = ZERO
  200 CONTINUE
*
*     Compute right side vector in resulting linear equations
*
      BASL = LOG10( SCLFAC )
      DO 240 I = ILO, IHI
         DO 230 J = ILO, IHI
            TB = B( I, J )
            TA = A( I, J )
            IF( TA.EQ.ZERO )
     $         GO TO 210
            TA = LOG10( ABS( TA ) ) / BASL
  210       CONTINUE
            IF( TB.EQ.ZERO )
     $         GO TO 220
            TB = LOG10( ABS( TB ) ) / BASL
  220       CONTINUE
            WORK( I+4*N ) = WORK( I+4*N ) - TA - TB
            WORK( J+5*N ) = WORK( J+5*N ) - TA - TB
  230    CONTINUE
  240 CONTINUE
*
      COEF = ONE / DBLE( 2*NR )
      COEF2 = COEF*COEF
      COEF5 = HALF*COEF2
      NRP2 = NR + 2
      BETA = ZERO
      IT = 1
*
*     Start generalized conjugate gradient iteration
*
  250 CONTINUE
*
      GAMMA = DDOT( NR, WORK( ILO+4*N ), 1, WORK( ILO+4*N ), 1 ) +
     $        DDOT( NR, WORK( ILO+5*N ), 1, WORK( ILO+5*N ), 1 )
*
      EW = ZERO
      EWC = ZERO
      DO 260 I = ILO, IHI
         EW = EW + WORK( I+4*N )
         EWC = EWC + WORK( I+5*N )
  260 CONTINUE
*
      GAMMA = COEF*GAMMA - COEF2*( EW**2+EWC**2 ) - COEF5*( EW-EWC )**2
      IF( GAMMA.EQ.ZERO )
     $   GO TO 350
      IF( IT.NE.1 )
     $   BETA = GAMMA / PGAMMA
      T = COEF5*( EWC-THREE*EW )
      TC = COEF5*( EW-THREE*EWC )
*
      CALL DSCAL( NR, BETA, WORK( ILO ), 1 )
      CALL DSCAL( NR, BETA, WORK( ILO+N ), 1 )
*
      CALL DAXPY( NR, COEF, WORK( ILO+4*N ), 1, WORK( ILO+N ), 1 )
      CALL DAXPY( NR, COEF, WORK( ILO+5*N ), 1, WORK( ILO ), 1 )
*
      DO 270 I = ILO, IHI
         WORK( I ) = WORK( I ) + TC
         WORK( I+N ) = WORK( I+N ) + T
  270 CONTINUE
*
*     Apply matrix to vector
*
      DO 300 I = ILO, IHI
         KOUNT = 0
         SUM = ZERO
         DO 290 J = ILO, IHI
            IF( A( I, J ).EQ.ZERO )
     $         GO TO 280
            KOUNT = KOUNT + 1
            SUM = SUM + WORK( J )
  280       CONTINUE
            IF( B( I, J ).EQ.ZERO )
     $         GO TO 290
            KOUNT = KOUNT + 1
            SUM = SUM + WORK( J )
  290    CONTINUE
         WORK( I+2*N ) = DBLE( KOUNT )*WORK( I+N ) + SUM
  300 CONTINUE
*
      DO 330 J = ILO, IHI
         KOUNT = 0
         SUM = ZERO
         DO 320 I = ILO, IHI
            IF( A( I, J ).EQ.ZERO )
     $         GO TO 310
            KOUNT = KOUNT + 1
            SUM = SUM + WORK( I+N )
  310       CONTINUE
            IF( B( I, J ).EQ.ZERO )
     $         GO TO 320
            KOUNT = KOUNT + 1
            SUM = SUM + WORK( I+N )
  320    CONTINUE
         WORK( J+3*N ) = DBLE( KOUNT )*WORK( J ) + SUM
  330 CONTINUE
*
      SUM = DDOT( NR, WORK( ILO+N ), 1, WORK( ILO+2*N ), 1 ) +
     $      DDOT( NR, WORK( ILO ), 1, WORK( ILO+3*N ), 1 )
      ALPHA = GAMMA / SUM
*
*     Determine correction to current iteration
*
      CMAX = ZERO
      DO 340 I = ILO, IHI
         COR = ALPHA*WORK( I+N )
         IF( ABS( COR ).GT.CMAX )
     $      CMAX = ABS( COR )
         LSCALE( I ) = LSCALE( I ) + COR
         COR = ALPHA*WORK( I )
         IF( ABS( COR ).GT.CMAX )
     $      CMAX = ABS( COR )
         RSCALE( I ) = RSCALE( I ) + COR
  340 CONTINUE
      IF( CMAX.LT.HALF )
     $   GO TO 350
*
      CALL DAXPY( NR, -ALPHA, WORK( ILO+2*N ), 1, WORK( ILO+4*N ),
     $            1 )
      CALL DAXPY( NR, -ALPHA, WORK( ILO+3*N ), 1, WORK( ILO+5*N ),
     $            1 )
*
      PGAMMA = GAMMA
      IT = IT + 1
      IF( IT.LE.NRP2 )
     $   GO TO 250
*
*     End generalized conjugate gradient iteration
*
  350 CONTINUE
      SFMIN = DLAMCH( 'S' )
      SFMAX = ONE / SFMIN
      LSFMIN = INT( LOG10( SFMIN ) / BASL+ONE )
      LSFMAX = INT( LOG10( SFMAX ) / BASL )
      DO 360 I = ILO, IHI
         IRAB = IDAMAX( N-ILO+1, A( I, ILO ), LDA )
         RAB = ABS( A( I, IRAB+ILO-1 ) )
         IRAB = IDAMAX( N-ILO+1, B( I, ILO ), LDB )
         RAB = MAX( RAB, ABS( B( I, IRAB+ILO-1 ) ) )
         LRAB = INT( LOG10( RAB+SFMIN ) / BASL+ONE )
         IR = INT(LSCALE( I ) + SIGN( HALF, LSCALE( I ) ))
         IR = MIN( MAX( IR, LSFMIN ), LSFMAX, LSFMAX-LRAB )
         LSCALE( I ) = SCLFAC**IR
         ICAB = IDAMAX( IHI, A( 1, I ), 1 )
         CAB = ABS( A( ICAB, I ) )
         ICAB = IDAMAX( IHI, B( 1, I ), 1 )
         CAB = MAX( CAB, ABS( B( ICAB, I ) ) )
         LCAB = INT( LOG10( CAB+SFMIN ) / BASL+ONE )
         JC = INT(RSCALE( I ) + SIGN( HALF, RSCALE( I ) ))
         JC = MIN( MAX( JC, LSFMIN ), LSFMAX, LSFMAX-LCAB )
         RSCALE( I ) = SCLFAC**JC
  360 CONTINUE
*
*     Row scaling of matrices A and B
*
      DO 370 I = ILO, IHI
         CALL DSCAL( N-ILO+1, LSCALE( I ), A( I, ILO ), LDA )
         CALL DSCAL( N-ILO+1, LSCALE( I ), B( I, ILO ), LDB )
  370 CONTINUE
*
*     Column scaling of matrices A and B
*
      DO 380 J = ILO, IHI
         CALL DSCAL( IHI, RSCALE( J ), A( 1, J ), 1 )
         CALL DSCAL( IHI, RSCALE( J ), B( 1, J ), 1 )
  380 CONTINUE
*
      RETURN
*
*     End of DGGBAL
*
      END
*> \brief <b> DGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGES + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgges.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgges.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgges.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
*                         SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR,
*                         LDVSR, WORK, LWORK, BWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBVSL, JOBVSR, SORT
*       INTEGER            INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
*       ..
*       .. Array Arguments ..
*       LOGICAL            BWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
*      $                   B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
*      $                   VSR( LDVSR, * ), WORK( * )
*       ..
*       .. Function Arguments ..
*       LOGICAL            SELCTG
*       EXTERNAL           SELCTG
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B),
*> the generalized eigenvalues, the generalized real Schur form (S,T),
*> optionally, the left and/or right matrices of Schur vectors (VSL and
*> VSR). This gives the generalized Schur factorization
*>
*>          (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )
*>
*> Optionally, it also orders the eigenvalues so that a selected cluster
*> of eigenvalues appears in the leading diagonal blocks of the upper
*> quasi-triangular matrix S and the upper triangular matrix T.The
*> leading columns of VSL and VSR then form an orthonormal basis for the
*> corresponding left and right eigenspaces (deflating subspaces).
*>
*> (If only the generalized eigenvalues are needed, use the driver
*> DGGEV instead, which is faster.)
*>
*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
*> or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
*> usually represented as the pair (alpha,beta), as there is a
*> reasonable interpretation for beta=0 or both being zero.
*>
*> A pair of matrices (S,T) is in generalized real Schur form if T is
*> upper triangular with non-negative diagonal and S is block upper
*> triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
*> to real generalized eigenvalues, while 2-by-2 blocks of S will be
*> "standardized" by making the corresponding elements of T have the
*> form:
*>         [  a  0  ]
*>         [  0  b  ]
*>
*> and the pair of corresponding 2-by-2 blocks in S and T will have a
*> complex conjugate pair of generalized eigenvalues.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBVSL
*> \verbatim
*>          JOBVSL is CHARACTER*1
*>          = 'N':  do not compute the left Schur vectors;
*>          = 'V':  compute the left Schur vectors.
*> \endverbatim
*>
*> \param[in] JOBVSR
*> \verbatim
*>          JOBVSR is CHARACTER*1
*>          = 'N':  do not compute the right Schur vectors;
*>          = 'V':  compute the right Schur vectors.
*> \endverbatim
*>
*> \param[in] SORT
*> \verbatim
*>          SORT is CHARACTER*1
*>          Specifies whether or not to order the eigenvalues on the
*>          diagonal of the generalized Schur form.
*>          = 'N':  Eigenvalues are not ordered;
*>          = 'S':  Eigenvalues are ordered (see SELCTG);
*> \endverbatim
*>
*> \param[in] SELCTG
*> \verbatim
*>          SELCTG is a LOGICAL FUNCTION of three DOUBLE PRECISION arguments
*>          SELCTG must be declared EXTERNAL in the calling subroutine.
*>          If SORT = 'N', SELCTG is not referenced.
*>          If SORT = 'S', SELCTG is used to select eigenvalues to sort
*>          to the top left of the Schur form.
*>          An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
*>          SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
*>          one of a complex conjugate pair of eigenvalues is selected,
*>          then both complex eigenvalues are selected.
*>
*>          Note that in the ill-conditioned case, a selected complex
*>          eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
*>          BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
*>          in this case.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A, B, VSL, and VSR.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, N)
*>          On entry, the first of the pair of matrices.
*>          On exit, A has been overwritten by its generalized Schur
*>          form S.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB, N)
*>          On entry, the second of the pair of matrices.
*>          On exit, B has been overwritten by its generalized Schur
*>          form T.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] SDIM
*> \verbatim
*>          SDIM is INTEGER
*>          If SORT = 'N', SDIM = 0.
*>          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
*>          for which SELCTG is true.  (Complex conjugate pairs for which
*>          SELCTG is true for either eigenvalue count as 2.)
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*>          ALPHAR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*>          ALPHAI is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION array, dimension (N)
*>          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
*>          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i,
*>          and  BETA(j),j=1,...,N are the diagonals of the complex Schur
*>          form (S,T) that would result if the 2-by-2 diagonal blocks of
*>          the real Schur form of (A,B) were further reduced to
*>          triangular form using 2-by-2 complex unitary transformations.
*>          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
*>          positive, then the j-th and (j+1)-st eigenvalues are a
*>          complex conjugate pair, with ALPHAI(j+1) negative.
*>
*>          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
*>          may easily over- or underflow, and BETA(j) may even be zero.
*>          Thus, the user should avoid naively computing the ratio.
*>          However, ALPHAR and ALPHAI will be always less than and
*>          usually comparable with norm(A) in magnitude, and BETA always
*>          less than and usually comparable with norm(B).
*> \endverbatim
*>
*> \param[out] VSL
*> \verbatim
*>          VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
*>          If JOBVSL = 'V', VSL will contain the left Schur vectors.
*>          Not referenced if JOBVSL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSL
*> \verbatim
*>          LDVSL is INTEGER
*>          The leading dimension of the matrix VSL. LDVSL >=1, and
*>          if JOBVSL = 'V', LDVSL >= N.
*> \endverbatim
*>
*> \param[out] VSR
*> \verbatim
*>          VSR is DOUBLE PRECISION array, dimension (LDVSR,N)
*>          If JOBVSR = 'V', VSR will contain the right Schur vectors.
*>          Not referenced if JOBVSR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSR
*> \verbatim
*>          LDVSR is INTEGER
*>          The leading dimension of the matrix VSR. LDVSR >= 1, and
*>          if JOBVSR = 'V', LDVSR >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If N = 0, LWORK >= 1, else LWORK >= MAX(8*N,6*N+16).
*>          For good performance, LWORK must generally be larger.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] BWORK
*> \verbatim
*>          BWORK is LOGICAL array, dimension (N)
*>          Not referenced if SORT = 'N'.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          = 1,...,N:
*>                The QZ iteration failed.  (A,B) are not in Schur
*>                form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
*>                be correct for j=INFO+1,...,N.
*>          > N:  =N+1: other than QZ iteration failed in DHGEQZ.
*>                =N+2: after reordering, roundoff changed values of
*>                      some complex eigenvalues so that leading
*>                      eigenvalues in the Generalized Schur form no
*>                      longer satisfy SELCTG=.TRUE.  This could also
*>                      be caused due to scaling.
*>                =N+3: reordering failed in DTGSEN.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gges
*
*  =====================================================================
      SUBROUTINE DGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B,
     $                  LDB,
     $                  SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR,
     $                  LDVSR, WORK, LWORK, BWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBVSL, JOBVSR, SORT
      INTEGER            INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
*     ..
*     .. Array Arguments ..
      LOGICAL            BWORK( * )
      DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
     $                   B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
     $                   VSR( LDVSR, * ), WORK( * )
*     ..
*     .. Function Arguments ..
      LOGICAL            SELCTG
      EXTERNAL           SELCTG
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
     $                   LQUERY, LST2SL, WANTST
      INTEGER            I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT,
     $                   ILO, IP, IRIGHT, IROWS, ITAU, IWRK, MAXWRK,
     $                   MINWRK
      DOUBLE PRECISION   ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
     $                   PVSR, SAFMAX, SAFMIN, SMLNUM
*     ..
*     .. Local Arrays ..
      INTEGER            IDUM( 1 )
      DOUBLE PRECISION   DIF( 2 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ,
     $                   DLACPY,
     $                   DLASCL, DLASET, DORGQR, DORMQR, DTGSEN, XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           LSAME, ILAENV, DLAMCH, DLANGE
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Decode the input arguments
*
      IF( LSAME( JOBVSL, 'N' ) ) THEN
         IJOBVL = 1
         ILVSL = .FALSE.
      ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
         IJOBVL = 2
         ILVSL = .TRUE.
      ELSE
         IJOBVL = -1
         ILVSL = .FALSE.
      END IF
*
      IF( LSAME( JOBVSR, 'N' ) ) THEN
         IJOBVR = 1
         ILVSR = .FALSE.
      ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
         IJOBVR = 2
         ILVSR = .TRUE.
      ELSE
         IJOBVR = -1
         ILVSR = .FALSE.
      END IF
*
      WANTST = LSAME( SORT, 'S' )
*
*     Test the input arguments
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      IF( IJOBVL.LE.0 ) THEN
         INFO = -1
      ELSE IF( IJOBVR.LE.0 ) THEN
         INFO = -2
      ELSE IF( ( .NOT.WANTST ) .AND.
     $         ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -9
      ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
         INFO = -15
      ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
         INFO = -17
      END IF
*
*     Compute workspace
*      (Note: Comments in the code beginning "Workspace:" describe the
*       minimal amount of workspace needed at that point in the code,
*       as well as the preferred amount for good performance.
*       NB refers to the optimal block size for the immediately
*       following subroutine, as returned by ILAENV.)
*
      IF( INFO.EQ.0 ) THEN
         IF( N.GT.0 )THEN
            MINWRK = MAX( 8*N, 6*N + 16 )
            MAXWRK = MINWRK - N +
     $               N*ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 )
            MAXWRK = MAX( MAXWRK, MINWRK - N +
     $                    N*ILAENV( 1, 'DORMQR', ' ', N, 1, N, -1 ) )
            IF( ILVSL ) THEN
               MAXWRK = MAX( MAXWRK, MINWRK - N +
     $                       N*ILAENV( 1, 'DORGQR', ' ', N, 1, N,
     $                                 -1 ) )
            END IF
         ELSE
            MINWRK = 1
            MAXWRK = 1
         END IF
         WORK( 1 ) = MAXWRK
*
         IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY )
     $      INFO = -19
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGGES ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 ) THEN
         SDIM = 0
         RETURN
      END IF
*
*     Get machine constants
*
      EPS = DLAMCH( 'P' )
      SAFMIN = DLAMCH( 'S' )
      SAFMAX = ONE / SAFMIN
      SMLNUM = SQRT( SAFMIN ) / EPS
      BIGNUM = ONE / SMLNUM
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
      ILASCL = .FALSE.
      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
         ANRMTO = SMLNUM
         ILASCL = .TRUE.
      ELSE IF( ANRM.GT.BIGNUM ) THEN
         ANRMTO = BIGNUM
         ILASCL = .TRUE.
      END IF
      IF( ILASCL )
     $   CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
*
*     Scale B if max element outside range [SMLNUM,BIGNUM]
*
      BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
      ILBSCL = .FALSE.
      IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
         BNRMTO = SMLNUM
         ILBSCL = .TRUE.
      ELSE IF( BNRM.GT.BIGNUM ) THEN
         BNRMTO = BIGNUM
         ILBSCL = .TRUE.
      END IF
      IF( ILBSCL )
     $   CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
*
*     Permute the matrix to make it more nearly triangular
*     (Workspace: need 6*N + 2*N space for storing balancing factors)
*
      ILEFT = 1
      IRIGHT = N + 1
      IWRK = IRIGHT + N
      CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
     $             WORK( IRIGHT ), WORK( IWRK ), IERR )
*
*     Reduce B to triangular form (QR decomposition of B)
*     (Workspace: need N, prefer N*NB)
*
      IROWS = IHI + 1 - ILO
      ICOLS = N + 1 - ILO
      ITAU = IWRK
      IWRK = ITAU + IROWS
      CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
     $             WORK( IWRK ), LWORK+1-IWRK, IERR )
*
*     Apply the orthogonal transformation to matrix A
*     (Workspace: need N, prefer N*NB)
*
      CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
     $             WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
     $             LWORK+1-IWRK, IERR )
*
*     Initialize VSL
*     (Workspace: need N, prefer N*NB)
*
      IF( ILVSL ) THEN
         CALL DLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL )
         IF( IROWS.GT.1 ) THEN
            CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
     $                   VSL( ILO+1, ILO ), LDVSL )
         END IF
         CALL DORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
     $                WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
      END IF
*
*     Initialize VSR
*
      IF( ILVSR )
     $   CALL DLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR )
*
*     Reduce to generalized Hessenberg form
*     (Workspace: none needed)
*
      CALL DGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
     $             LDVSL, VSR, LDVSR, IERR )
*
*     Perform QZ algorithm, computing Schur vectors if desired
*     (Workspace: need N)
*
      IWRK = ITAU
      CALL DHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
     $             ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
     $             WORK( IWRK ), LWORK+1-IWRK, IERR )
      IF( IERR.NE.0 ) THEN
         IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
            INFO = IERR
         ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
            INFO = IERR - N
         ELSE
            INFO = N + 1
         END IF
         GO TO 50
      END IF
*
*     Sort eigenvalues ALPHA/BETA if desired
*     (Workspace: need 4*N+16 )
*
      SDIM = 0
      IF( WANTST ) THEN
*
*        Undo scaling on eigenvalues before SELCTGing
*
         IF( ILASCL ) THEN
            CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N,
     $                   IERR )
            CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N,
     $                   IERR )
         END IF
         IF( ILBSCL )
     $      CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N,
     $                   IERR )
*
*        Select eigenvalues
*
         DO 10 I = 1, N
            BWORK( I ) = SELCTG( ALPHAR( I ), ALPHAI( I ),
     $             BETA( I ) )
   10    CONTINUE
*
         CALL DTGSEN( 0, ILVSL, ILVSR, BWORK, N, A, LDA, B, LDB,
     $                ALPHAR,
     $                ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, SDIM, PVSL,
     $                PVSR, DIF, WORK( IWRK ), LWORK-IWRK+1, IDUM, 1,
     $                IERR )
         IF( IERR.EQ.1 )
     $      INFO = N + 3
*
      END IF
*
*     Apply back-permutation to VSL and VSR
*     (Workspace: none needed)
*
      IF( ILVSL )
     $   CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
     $                WORK( IRIGHT ), N, VSL, LDVSL, IERR )
*
      IF( ILVSR )
     $   CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
     $                WORK( IRIGHT ), N, VSR, LDVSR, IERR )
*
*     Check if unscaling would cause over/underflow, if so, rescale
*     (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
*     B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
*
      IF( ILASCL ) THEN
         DO 20 I = 1, N
            IF( ALPHAI( I ).NE.ZERO ) THEN
               IF( ( ALPHAR( I ) / SAFMAX ).GT.( ANRMTO / ANRM ) .OR.
     $             ( SAFMIN / ALPHAR( I ) ).GT.( ANRM / ANRMTO ) ) THEN
                  WORK( 1 ) = ABS( A( I, I ) / ALPHAR( I ) )
                  BETA( I ) = BETA( I )*WORK( 1 )
                  ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
                  ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
               ELSE IF( ( ALPHAI( I ) / SAFMAX ).GT.
     $                  ( ANRMTO / ANRM ) .OR.
     $                  ( SAFMIN / ALPHAI( I ) ).GT.( ANRM / ANRMTO ) )
     $                   THEN
                  WORK( 1 ) = ABS( A( I, I+1 ) / ALPHAI( I ) )
                  BETA( I ) = BETA( I )*WORK( 1 )
                  ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
                  ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
               END IF
            END IF
   20    CONTINUE
      END IF
*
      IF( ILBSCL ) THEN
         DO 30 I = 1, N
            IF( ALPHAI( I ).NE.ZERO ) THEN
               IF( ( BETA( I ) / SAFMAX ).GT.( BNRMTO / BNRM ) .OR.
     $             ( SAFMIN / BETA( I ) ).GT.( BNRM / BNRMTO ) ) THEN
                  WORK( 1 ) = ABS( B( I, I ) / BETA( I ) )
                  BETA( I ) = BETA( I )*WORK( 1 )
                  ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
                  ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
               END IF
            END IF
   30    CONTINUE
      END IF
*
*     Undo scaling
*
      IF( ILASCL ) THEN
         CALL DLASCL( 'H', 0, 0, ANRMTO, ANRM, N, N, A, LDA, IERR )
         CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N,
     $                IERR )
         CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N,
     $                IERR )
      END IF
*
      IF( ILBSCL ) THEN
         CALL DLASCL( 'U', 0, 0, BNRMTO, BNRM, N, N, B, LDB, IERR )
         CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
      END IF
*
      IF( WANTST ) THEN
*
*        Check if reordering is correct
*
         LASTSL = .TRUE.
         LST2SL = .TRUE.
         SDIM = 0
         IP = 0
         DO 40 I = 1, N
            CURSL = SELCTG( ALPHAR( I ), ALPHAI( I ), BETA( I ) )
            IF( ALPHAI( I ).EQ.ZERO ) THEN
               IF( CURSL )
     $            SDIM = SDIM + 1
               IP = 0
               IF( CURSL .AND. .NOT.LASTSL )
     $            INFO = N + 2
            ELSE
               IF( IP.EQ.1 ) THEN
*
*                 Last eigenvalue of conjugate pair
*
                  CURSL = CURSL .OR. LASTSL
                  LASTSL = CURSL
                  IF( CURSL )
     $               SDIM = SDIM + 2
                  IP = -1
                  IF( CURSL .AND. .NOT.LST2SL )
     $               INFO = N + 2
               ELSE
*
*                 First eigenvalue of conjugate pair
*
                  IP = 1
               END IF
            END IF
            LST2SL = LASTSL
            LASTSL = CURSL
   40    CONTINUE
*
      END IF
*
   50 CONTINUE
*
      WORK( 1 ) = MAXWRK
*
      RETURN
*
*     End of DGGES
*
      END
*> \brief <b> DGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGESX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggesx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggesx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggesx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA,
*                          B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL,
*                          VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK,
*                          LIWORK, BWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBVSL, JOBVSR, SENSE, SORT
*       INTEGER            INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N,
*      $                   SDIM
*       ..
*       .. Array Arguments ..
*       LOGICAL            BWORK( * )
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
*      $                   B( LDB, * ), BETA( * ), RCONDE( 2 ),
*      $                   RCONDV( 2 ), VSL( LDVSL, * ), VSR( LDVSR, * ),
*      $                   WORK( * )
*       ..
*       .. Function Arguments ..
*       LOGICAL            SELCTG
*       EXTERNAL           SELCTG
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGGESX computes for a pair of N-by-N real nonsymmetric matrices
*> (A,B), the generalized eigenvalues, the real Schur form (S,T), and,
*> optionally, the left and/or right matrices of Schur vectors (VSL and
*> VSR).  This gives the generalized Schur factorization
*>
*>      (A,B) = ( (VSL) S (VSR)**T, (VSL) T (VSR)**T )
*>
*> Optionally, it also orders the eigenvalues so that a selected cluster
*> of eigenvalues appears in the leading diagonal blocks of the upper
*> quasi-triangular matrix S and the upper triangular matrix T; computes
*> a reciprocal condition number for the average of the selected
*> eigenvalues (RCONDE); and computes a reciprocal condition number for
*> the right and left deflating subspaces corresponding to the selected
*> eigenvalues (RCONDV). The leading columns of VSL and VSR then form
*> an orthonormal basis for the corresponding left and right eigenspaces
*> (deflating subspaces).
*>
*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
*> or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
*> usually represented as the pair (alpha,beta), as there is a
*> reasonable interpretation for beta=0 or for both being zero.
*>
*> A pair of matrices (S,T) is in generalized real Schur form if T is
*> upper triangular with non-negative diagonal and S is block upper
*> triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
*> to real generalized eigenvalues, while 2-by-2 blocks of S will be
*> "standardized" by making the corresponding elements of T have the
*> form:
*>         [  a  0  ]
*>         [  0  b  ]
*>
*> and the pair of corresponding 2-by-2 blocks in S and T will have a
*> complex conjugate pair of generalized eigenvalues.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBVSL
*> \verbatim
*>          JOBVSL is CHARACTER*1
*>          = 'N':  do not compute the left Schur vectors;
*>          = 'V':  compute the left Schur vectors.
*> \endverbatim
*>
*> \param[in] JOBVSR
*> \verbatim
*>          JOBVSR is CHARACTER*1
*>          = 'N':  do not compute the right Schur vectors;
*>          = 'V':  compute the right Schur vectors.
*> \endverbatim
*>
*> \param[in] SORT
*> \verbatim
*>          SORT is CHARACTER*1
*>          Specifies whether or not to order the eigenvalues on the
*>          diagonal of the generalized Schur form.
*>          = 'N':  Eigenvalues are not ordered;
*>          = 'S':  Eigenvalues are ordered (see SELCTG).
*> \endverbatim
*>
*> \param[in] SELCTG
*> \verbatim
*>          SELCTG is a LOGICAL FUNCTION of three DOUBLE PRECISION arguments
*>          SELCTG must be declared EXTERNAL in the calling subroutine.
*>          If SORT = 'N', SELCTG is not referenced.
*>          If SORT = 'S', SELCTG is used to select eigenvalues to sort
*>          to the top left of the Schur form.
*>          An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
*>          SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
*>          one of a complex conjugate pair of eigenvalues is selected,
*>          then both complex eigenvalues are selected.
*>          Note that a selected complex eigenvalue may no longer satisfy
*>          SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering,
*>          since ordering may change the value of complex eigenvalues
*>          (especially if the eigenvalue is ill-conditioned), in this
*>          case INFO is set to N+3.
*> \endverbatim
*>
*> \param[in] SENSE
*> \verbatim
*>          SENSE is CHARACTER*1
*>          Determines which reciprocal condition numbers are computed.
*>          = 'N':  None are computed;
*>          = 'E':  Computed for average of selected eigenvalues only;
*>          = 'V':  Computed for selected deflating subspaces only;
*>          = 'B':  Computed for both.
*>          If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A, B, VSL, and VSR.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, N)
*>          On entry, the first of the pair of matrices.
*>          On exit, A has been overwritten by its generalized Schur
*>          form S.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB, N)
*>          On entry, the second of the pair of matrices.
*>          On exit, B has been overwritten by its generalized Schur
*>          form T.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] SDIM
*> \verbatim
*>          SDIM is INTEGER
*>          If SORT = 'N', SDIM = 0.
*>          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
*>          for which SELCTG is true.  (Complex conjugate pairs for which
*>          SELCTG is true for either eigenvalue count as 2.)
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*>          ALPHAR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*>          ALPHAI is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION array, dimension (N)
*>          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
*>          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i
*>          and BETA(j),j=1,...,N  are the diagonals of the complex Schur
*>          form (S,T) that would result if the 2-by-2 diagonal blocks of
*>          the real Schur form of (A,B) were further reduced to
*>          triangular form using 2-by-2 complex unitary transformations.
*>          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
*>          positive, then the j-th and (j+1)-st eigenvalues are a
*>          complex conjugate pair, with ALPHAI(j+1) negative.
*>
*>          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
*>          may easily over- or underflow, and BETA(j) may even be zero.
*>          Thus, the user should avoid naively computing the ratio.
*>          However, ALPHAR and ALPHAI will be always less than and
*>          usually comparable with norm(A) in magnitude, and BETA always
*>          less than and usually comparable with norm(B).
*> \endverbatim
*>
*> \param[out] VSL
*> \verbatim
*>          VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
*>          If JOBVSL = 'V', VSL will contain the left Schur vectors.
*>          Not referenced if JOBVSL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSL
*> \verbatim
*>          LDVSL is INTEGER
*>          The leading dimension of the matrix VSL. LDVSL >=1, and
*>          if JOBVSL = 'V', LDVSL >= N.
*> \endverbatim
*>
*> \param[out] VSR
*> \verbatim
*>          VSR is DOUBLE PRECISION array, dimension (LDVSR,N)
*>          If JOBVSR = 'V', VSR will contain the right Schur vectors.
*>          Not referenced if JOBVSR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSR
*> \verbatim
*>          LDVSR is INTEGER
*>          The leading dimension of the matrix VSR. LDVSR >= 1, and
*>          if JOBVSR = 'V', LDVSR >= N.
*> \endverbatim
*>
*> \param[out] RCONDE
*> \verbatim
*>          RCONDE is DOUBLE PRECISION array, dimension ( 2 )
*>          If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
*>          reciprocal condition numbers for the average of the selected
*>          eigenvalues.
*>          Not referenced if SENSE = 'N' or 'V'.
*> \endverbatim
*>
*> \param[out] RCONDV
*> \verbatim
*>          RCONDV is DOUBLE PRECISION array, dimension ( 2 )
*>          If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
*>          reciprocal condition numbers for the selected deflating
*>          subspaces.
*>          Not referenced if SENSE = 'N' or 'E'.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
*>          LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else
*>          LWORK >= max( 8*N, 6*N+16 ).
*>          Note that 2*SDIM*(N-SDIM) <= N*N/2.
*>          Note also that an error is only returned if
*>          LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B'
*>          this may not be large enough.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the bound on the optimal size of the WORK
*>          array and the minimum size of the IWORK array, returns these
*>          values as the first entries of the WORK and IWORK arrays, and
*>          no error message related to LWORK or LIWORK is issued by
*>          XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*>          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK.
*>          If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
*>          LIWORK >= N+6.
*>
*>          If LIWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the bound on the optimal size of the
*>          WORK array and the minimum size of the IWORK array, returns
*>          these values as the first entries of the WORK and IWORK
*>          arrays, and no error message related to LWORK or LIWORK is
*>          issued by XERBLA.
*> \endverbatim
*>
*> \param[out] BWORK
*> \verbatim
*>          BWORK is LOGICAL array, dimension (N)
*>          Not referenced if SORT = 'N'.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          = 1,...,N:
*>                The QZ iteration failed.  (A,B) are not in Schur
*>                form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
*>                be correct for j=INFO+1,...,N.
*>          > N:  =N+1: other than QZ iteration failed in DHGEQZ
*>                =N+2: after reordering, roundoff changed values of
*>                      some complex eigenvalues so that leading
*>                      eigenvalues in the Generalized Schur form no
*>                      longer satisfy SELCTG=.TRUE.  This could also
*>                      be caused due to scaling.
*>                =N+3: reordering failed in DTGSEN.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggesx
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  An approximate (asymptotic) bound on the average absolute error of
*>  the selected eigenvalues is
*>
*>       EPS * norm((A, B)) / RCONDE( 1 ).
*>
*>  An approximate (asymptotic) bound on the maximum angular error in
*>  the computed deflating subspaces is
*>
*>       EPS * norm((A, B)) / RCONDV( 2 ).
*>
*>  See LAPACK User's Guide, section 4.11 for more information.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A,
     $                   LDA,
     $                   B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL,
     $                   VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK,
     $                   LIWORK, BWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBVSL, JOBVSR, SENSE, SORT
      INTEGER            INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N,
     $                   SDIM
*     ..
*     .. Array Arguments ..
      LOGICAL            BWORK( * )
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
     $                   B( LDB, * ), BETA( * ), RCONDE( 2 ),
     $                   RCONDV( 2 ), VSL( LDVSL, * ), VSR( LDVSR, * ),
     $                   WORK( * )
*     ..
*     .. Function Arguments ..
      LOGICAL            SELCTG
      EXTERNAL           SELCTG
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
     $                   LQUERY, LST2SL, WANTSB, WANTSE, WANTSN, WANTST,
     $                   WANTSV
      INTEGER            I, ICOLS, IERR, IHI, IJOB, IJOBVL, IJOBVR,
     $                   ILEFT, ILO, IP, IRIGHT, IROWS, ITAU, IWRK,
     $                   LIWMIN, LWRK, MAXWRK, MINWRK
      DOUBLE PRECISION   ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PL,
     $                   PR, SAFMAX, SAFMIN, SMLNUM
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   DIF( 2 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ,
     $                   DLACPY,
     $                   DLASCL, DLASET, DORGQR, DORMQR, DTGSEN, XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           LSAME, ILAENV, DLAMCH, DLANGE
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Decode the input arguments
*
      IF( LSAME( JOBVSL, 'N' ) ) THEN
         IJOBVL = 1
         ILVSL = .FALSE.
      ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
         IJOBVL = 2
         ILVSL = .TRUE.
      ELSE
         IJOBVL = -1
         ILVSL = .FALSE.
      END IF
*
      IF( LSAME( JOBVSR, 'N' ) ) THEN
         IJOBVR = 1
         ILVSR = .FALSE.
      ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
         IJOBVR = 2
         ILVSR = .TRUE.
      ELSE
         IJOBVR = -1
         ILVSR = .FALSE.
      END IF
*
      WANTST = LSAME( SORT, 'S' )
      WANTSN = LSAME( SENSE, 'N' )
      WANTSE = LSAME( SENSE, 'E' )
      WANTSV = LSAME( SENSE, 'V' )
      WANTSB = LSAME( SENSE, 'B' )
      LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
      IF( WANTSN ) THEN
         IJOB = 0
      ELSE IF( WANTSE ) THEN
         IJOB = 1
      ELSE IF( WANTSV ) THEN
         IJOB = 2
      ELSE IF( WANTSB ) THEN
         IJOB = 4
      END IF
*
*     Test the input arguments
*
      INFO = 0
      IF( IJOBVL.LE.0 ) THEN
         INFO = -1
      ELSE IF( IJOBVR.LE.0 ) THEN
         INFO = -2
      ELSE IF( ( .NOT.WANTST ) .AND.
     $         ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
         INFO = -3
      ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSV .OR. WANTSB ) .OR.
     $         ( .NOT.WANTST .AND. .NOT.WANTSN ) ) THEN
         INFO = -5
      ELSE IF( N.LT.0 ) THEN
         INFO = -6
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -8
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -10
      ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
         INFO = -16
      ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
         INFO = -18
      END IF
*
*     Compute workspace
*      (Note: Comments in the code beginning "Workspace:" describe the
*       minimal amount of workspace needed at that point in the code,
*       as well as the preferred amount for good performance.
*       NB refers to the optimal block size for the immediately
*       following subroutine, as returned by ILAENV.)
*
      IF( INFO.EQ.0 ) THEN
         IF( N.GT.0) THEN
            MINWRK = MAX( 8*N, 6*N + 16 )
            MAXWRK = MINWRK - N +
     $               N*ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 )
            MAXWRK = MAX( MAXWRK, MINWRK - N +
     $               N*ILAENV( 1, 'DORMQR', ' ', N, 1, N, -1 ) )
            IF( ILVSL ) THEN
               MAXWRK = MAX( MAXWRK, MINWRK - N +
     $                  N*ILAENV( 1, 'DORGQR', ' ', N, 1, N, -1 ) )
            END IF
            LWRK = MAXWRK
            IF( IJOB.GE.1 )
     $         LWRK = MAX( LWRK, N*N/2 )
         ELSE
            MINWRK = 1
            MAXWRK = 1
            LWRK   = 1
         END IF
         WORK( 1 ) = LWRK
         IF( WANTSN .OR. N.EQ.0 ) THEN
            LIWMIN = 1
         ELSE
            LIWMIN = N + 6
         END IF
         IWORK( 1 ) = LIWMIN
*
         IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
            INFO = -22
         ELSE IF( LIWORK.LT.LIWMIN  .AND. .NOT.LQUERY ) THEN
            INFO = -24
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGGESX', -INFO )
         RETURN
      ELSE IF (LQUERY) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 ) THEN
         SDIM = 0
         RETURN
      END IF
*
*     Get machine constants
*
      EPS = DLAMCH( 'P' )
      SAFMIN = DLAMCH( 'S' )
      SAFMAX = ONE / SAFMIN
      SMLNUM = SQRT( SAFMIN ) / EPS
      BIGNUM = ONE / SMLNUM
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
      ILASCL = .FALSE.
      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
         ANRMTO = SMLNUM
         ILASCL = .TRUE.
      ELSE IF( ANRM.GT.BIGNUM ) THEN
         ANRMTO = BIGNUM
         ILASCL = .TRUE.
      END IF
      IF( ILASCL )
     $   CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
*
*     Scale B if max element outside range [SMLNUM,BIGNUM]
*
      BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
      ILBSCL = .FALSE.
      IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
         BNRMTO = SMLNUM
         ILBSCL = .TRUE.
      ELSE IF( BNRM.GT.BIGNUM ) THEN
         BNRMTO = BIGNUM
         ILBSCL = .TRUE.
      END IF
      IF( ILBSCL )
     $   CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
*
*     Permute the matrix to make it more nearly triangular
*     (Workspace: need 6*N + 2*N for permutation parameters)
*
      ILEFT = 1
      IRIGHT = N + 1
      IWRK = IRIGHT + N
      CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
     $             WORK( IRIGHT ), WORK( IWRK ), IERR )
*
*     Reduce B to triangular form (QR decomposition of B)
*     (Workspace: need N, prefer N*NB)
*
      IROWS = IHI + 1 - ILO
      ICOLS = N + 1 - ILO
      ITAU = IWRK
      IWRK = ITAU + IROWS
      CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
     $             WORK( IWRK ), LWORK+1-IWRK, IERR )
*
*     Apply the orthogonal transformation to matrix A
*     (Workspace: need N, prefer N*NB)
*
      CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
     $             WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
     $             LWORK+1-IWRK, IERR )
*
*     Initialize VSL
*     (Workspace: need N, prefer N*NB)
*
      IF( ILVSL ) THEN
         CALL DLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL )
         IF( IROWS.GT.1 ) THEN
            CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
     $                   VSL( ILO+1, ILO ), LDVSL )
         END IF
         CALL DORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
     $                WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
      END IF
*
*     Initialize VSR
*
      IF( ILVSR )
     $   CALL DLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR )
*
*     Reduce to generalized Hessenberg form
*     (Workspace: none needed)
*
      CALL DGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
     $             LDVSL, VSR, LDVSR, IERR )
*
      SDIM = 0
*
*     Perform QZ algorithm, computing Schur vectors if desired
*     (Workspace: need N)
*
      IWRK = ITAU
      CALL DHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
     $             ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
     $             WORK( IWRK ), LWORK+1-IWRK, IERR )
      IF( IERR.NE.0 ) THEN
         IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
            INFO = IERR
         ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
            INFO = IERR - N
         ELSE
            INFO = N + 1
         END IF
         GO TO 60
      END IF
*
*     Sort eigenvalues ALPHA/BETA and compute the reciprocal of
*     condition number(s)
*     (Workspace: If IJOB >= 1, need MAX( 8*(N+1), 2*SDIM*(N-SDIM) )
*                 otherwise, need 8*(N+1) )
*
      IF( WANTST ) THEN
*
*        Undo scaling on eigenvalues before SELCTGing
*
         IF( ILASCL ) THEN
            CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N,
     $                   IERR )
            CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N,
     $                   IERR )
         END IF
         IF( ILBSCL )
     $      CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N,
     $                   IERR )
*
*        Select eigenvalues
*
         DO 10 I = 1, N
            BWORK( I ) = SELCTG( ALPHAR( I ), ALPHAI( I ),
     $             BETA( I ) )
   10    CONTINUE
*
*        Reorder eigenvalues, transform Generalized Schur vectors, and
*        compute reciprocal condition numbers
*
         CALL DTGSEN( IJOB, ILVSL, ILVSR, BWORK, N, A, LDA, B, LDB,
     $                ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
     $                SDIM, PL, PR, DIF, WORK( IWRK ), LWORK-IWRK+1,
     $                IWORK, LIWORK, IERR )
*
         IF( IJOB.GE.1 )
     $      MAXWRK = MAX( MAXWRK, 2*SDIM*( N-SDIM ) )
         IF( IERR.EQ.-22 ) THEN
*
*            not enough real workspace
*
            INFO = -22
         ELSE
            IF( IJOB.EQ.1 .OR. IJOB.EQ.4 ) THEN
               RCONDE( 1 ) = PL
               RCONDE( 2 ) = PR
            END IF
            IF( IJOB.EQ.2 .OR. IJOB.EQ.4 ) THEN
               RCONDV( 1 ) = DIF( 1 )
               RCONDV( 2 ) = DIF( 2 )
            END IF
            IF( IERR.EQ.1 )
     $         INFO = N + 3
         END IF
*
      END IF
*
*     Apply permutation to VSL and VSR
*     (Workspace: none needed)
*
      IF( ILVSL )
     $   CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
     $                WORK( IRIGHT ), N, VSL, LDVSL, IERR )
*
      IF( ILVSR )
     $   CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
     $                WORK( IRIGHT ), N, VSR, LDVSR, IERR )
*
*     Check if unscaling would cause over/underflow, if so, rescale
*     (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
*     B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
*
      IF( ILASCL ) THEN
         DO 20 I = 1, N
            IF( ALPHAI( I ).NE.ZERO ) THEN
               IF( ( ALPHAR( I ) / SAFMAX ).GT.( ANRMTO / ANRM ) .OR.
     $             ( SAFMIN / ALPHAR( I ) ).GT.( ANRM / ANRMTO ) ) THEN
                  WORK( 1 ) = ABS( A( I, I ) / ALPHAR( I ) )
                  BETA( I ) = BETA( I )*WORK( 1 )
                  ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
                  ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
               ELSE IF( ( ALPHAI( I ) / SAFMAX ).GT.
     $                  ( ANRMTO / ANRM ) .OR.
     $                  ( SAFMIN / ALPHAI( I ) ).GT.( ANRM / ANRMTO ) )
     $                   THEN
                  WORK( 1 ) = ABS( A( I, I+1 ) / ALPHAI( I ) )
                  BETA( I ) = BETA( I )*WORK( 1 )
                  ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
                  ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
               END IF
            END IF
   20    CONTINUE
      END IF
*
      IF( ILBSCL ) THEN
         DO 30 I = 1, N
            IF( ALPHAI( I ).NE.ZERO ) THEN
               IF( ( BETA( I ) / SAFMAX ).GT.( BNRMTO / BNRM ) .OR.
     $             ( SAFMIN / BETA( I ) ).GT.( BNRM / BNRMTO ) ) THEN
                  WORK( 1 ) = ABS( B( I, I ) / BETA( I ) )
                  BETA( I ) = BETA( I )*WORK( 1 )
                  ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
                  ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
               END IF
            END IF
   30    CONTINUE
      END IF
*
*     Undo scaling
*
      IF( ILASCL ) THEN
         CALL DLASCL( 'H', 0, 0, ANRMTO, ANRM, N, N, A, LDA, IERR )
         CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N,
     $                IERR )
         CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N,
     $                IERR )
      END IF
*
      IF( ILBSCL ) THEN
         CALL DLASCL( 'U', 0, 0, BNRMTO, BNRM, N, N, B, LDB, IERR )
         CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
      END IF
*
      IF( WANTST ) THEN
*
*        Check if reordering is correct
*
         LASTSL = .TRUE.
         LST2SL = .TRUE.
         SDIM = 0
         IP = 0
         DO 50 I = 1, N
            CURSL = SELCTG( ALPHAR( I ), ALPHAI( I ), BETA( I ) )
            IF( ALPHAI( I ).EQ.ZERO ) THEN
               IF( CURSL )
     $            SDIM = SDIM + 1
               IP = 0
               IF( CURSL .AND. .NOT.LASTSL )
     $            INFO = N + 2
            ELSE
               IF( IP.EQ.1 ) THEN
*
*                 Last eigenvalue of conjugate pair
*
                  CURSL = CURSL .OR. LASTSL
                  LASTSL = CURSL
                  IF( CURSL )
     $               SDIM = SDIM + 2
                  IP = -1
                  IF( CURSL .AND. .NOT.LST2SL )
     $               INFO = N + 2
               ELSE
*
*                 First eigenvalue of conjugate pair
*
                  IP = 1
               END IF
            END IF
            LST2SL = LASTSL
            LASTSL = CURSL
   50    CONTINUE
*
      END IF
*
   60 CONTINUE
*
      WORK( 1 ) = MAXWRK
      IWORK( 1 ) = LIWMIN
*
      RETURN
*
*     End of DGGESX
*
      END
*> \brief <b> DGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGEV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggev.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggev.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggev.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
*                         BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBVL, JOBVR
*       INTEGER            INFO, LDA, LDB, LDVL, LDVR, LWORK, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
*      $                   B( LDB, * ), BETA( * ), VL( LDVL, * ),
*      $                   VR( LDVR, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGGEV computes for a pair of N-by-N real nonsymmetric matrices (A,B)
*> the generalized eigenvalues, and optionally, the left and/or right
*> generalized eigenvectors.
*>
*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar
*> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
*> singular. It is usually represented as the pair (alpha,beta), as
*> there is a reasonable interpretation for beta=0, and even for both
*> being zero.
*>
*> The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
*> of (A,B) satisfies
*>
*>                  A * v(j) = lambda(j) * B * v(j).
*>
*> The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
*> of (A,B) satisfies
*>
*>                  u(j)**H * A  = lambda(j) * u(j)**H * B .
*>
*> where u(j)**H is the conjugate-transpose of u(j).
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBVL
*> \verbatim
*>          JOBVL is CHARACTER*1
*>          = 'N':  do not compute the left generalized eigenvectors;
*>          = 'V':  compute the left generalized eigenvectors.
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*>          JOBVR is CHARACTER*1
*>          = 'N':  do not compute the right generalized eigenvectors;
*>          = 'V':  compute the right generalized eigenvectors.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A, B, VL, and VR.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, N)
*>          On entry, the matrix A in the pair (A,B).
*>          On exit, A has been overwritten.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB, N)
*>          On entry, the matrix B in the pair (A,B).
*>          On exit, B has been overwritten.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*>          ALPHAR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*>          ALPHAI is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION array, dimension (N)
*>          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
*>          be the generalized eigenvalues.  If ALPHAI(j) is zero, then
*>          the j-th eigenvalue is real; if positive, then the j-th and
*>          (j+1)-st eigenvalues are a complex conjugate pair, with
*>          ALPHAI(j+1) negative.
*>
*>          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
*>          may easily over- or underflow, and BETA(j) may even be zero.
*>          Thus, the user should avoid naively computing the ratio
*>          alpha/beta.  However, ALPHAR and ALPHAI will be always less
*>          than and usually comparable with norm(A) in magnitude, and
*>          BETA always less than and usually comparable with norm(B).
*> \endverbatim
*>
*> \param[out] VL
*> \verbatim
*>          VL is DOUBLE PRECISION array, dimension (LDVL,N)
*>          If JOBVL = 'V', the left eigenvectors u(j) are stored one
*>          after another in the columns of VL, in the same order as
*>          their eigenvalues. If the j-th eigenvalue is real, then
*>          u(j) = VL(:,j), the j-th column of VL. If the j-th and
*>          (j+1)-th eigenvalues form a complex conjugate pair, then
*>          u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
*>          Each eigenvector is scaled so the largest component has
*>          abs(real part)+abs(imag. part)=1.
*>          Not referenced if JOBVL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*>          LDVL is INTEGER
*>          The leading dimension of the matrix VL. LDVL >= 1, and
*>          if JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*>          VR is DOUBLE PRECISION array, dimension (LDVR,N)
*>          If JOBVR = 'V', the right eigenvectors v(j) are stored one
*>          after another in the columns of VR, in the same order as
*>          their eigenvalues. If the j-th eigenvalue is real, then
*>          v(j) = VR(:,j), the j-th column of VR. If the j-th and
*>          (j+1)-th eigenvalues form a complex conjugate pair, then
*>          v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
*>          Each eigenvector is scaled so the largest component has
*>          abs(real part)+abs(imag. part)=1.
*>          Not referenced if JOBVR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*>          LDVR is INTEGER
*>          The leading dimension of the matrix VR. LDVR >= 1, and
*>          if JOBVR = 'V', LDVR >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.  LWORK >= max(1,8*N).
*>          For good performance, LWORK must generally be larger.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          = 1,...,N:
*>                The QZ iteration failed.  No eigenvectors have been
*>                calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
*>                should be correct for j=INFO+1,...,N.
*>          > N:  =N+1: other than QZ iteration failed in DHGEQZ.
*>                =N+2: error return from DTGEVC.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggev
*
*  =====================================================================
      SUBROUTINE DGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR,
     $                  ALPHAI,
     $                  BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBVL, JOBVR
      INTEGER            INFO, LDA, LDB, LDVL, LDVR, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
     $                   B( LDB, * ), BETA( * ), VL( LDVL, * ),
     $                   VR( LDVR, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY
      CHARACTER          CHTEMP
      INTEGER            ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT, ILO,
     $                   IN, IRIGHT, IROWS, ITAU, IWRK, JC, JR, MAXWRK,
     $                   MINWRK
      DOUBLE PRECISION   ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
     $                   SMLNUM, TEMP
*     ..
*     .. Local Arrays ..
      LOGICAL            LDUMMA( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ,
     $                   DLACPY,
     $                   DLASCL, DLASET, DORGQR, DORMQR, DTGEVC, XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           LSAME, ILAENV, DLAMCH, DLANGE
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Decode the input arguments
*
      IF( LSAME( JOBVL, 'N' ) ) THEN
         IJOBVL = 1
         ILVL = .FALSE.
      ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
         IJOBVL = 2
         ILVL = .TRUE.
      ELSE
         IJOBVL = -1
         ILVL = .FALSE.
      END IF
*
      IF( LSAME( JOBVR, 'N' ) ) THEN
         IJOBVR = 1
         ILVR = .FALSE.
      ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
         IJOBVR = 2
         ILVR = .TRUE.
      ELSE
         IJOBVR = -1
         ILVR = .FALSE.
      END IF
      ILV = ILVL .OR. ILVR
*
*     Test the input arguments
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      IF( IJOBVL.LE.0 ) THEN
         INFO = -1
      ELSE IF( IJOBVR.LE.0 ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
         INFO = -12
      ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
         INFO = -14
      END IF
*
*     Compute workspace
*      (Note: Comments in the code beginning "Workspace:" describe the
*       minimal amount of workspace needed at that point in the code,
*       as well as the preferred amount for good performance.
*       NB refers to the optimal block size for the immediately
*       following subroutine, as returned by ILAENV. The workspace is
*       computed assuming ILO = 1 and IHI = N, the worst case.)
*
      IF( INFO.EQ.0 ) THEN
         MINWRK = MAX( 1, 8*N )
         MAXWRK = MAX( 1, N*( 7 +
     $                 ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 ) ) )
         MAXWRK = MAX( MAXWRK, N*( 7 +
     $                 ILAENV( 1, 'DORMQR', ' ', N, 1, N, 0 ) ) )
         IF( ILVL ) THEN
            MAXWRK = MAX( MAXWRK, N*( 7 +
     $                 ILAENV( 1, 'DORGQR', ' ', N, 1, N, -1 ) ) )
         END IF
         WORK( 1 ) = MAXWRK
*
         IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY )
     $      INFO = -16
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGGEV ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Get machine constants
*
      EPS = DLAMCH( 'P' )
      SMLNUM = DLAMCH( 'S' )
      BIGNUM = ONE / SMLNUM
      SMLNUM = SQRT( SMLNUM ) / EPS
      BIGNUM = ONE / SMLNUM
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
      ILASCL = .FALSE.
      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
         ANRMTO = SMLNUM
         ILASCL = .TRUE.
      ELSE IF( ANRM.GT.BIGNUM ) THEN
         ANRMTO = BIGNUM
         ILASCL = .TRUE.
      END IF
      IF( ILASCL )
     $   CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
*
*     Scale B if max element outside range [SMLNUM,BIGNUM]
*
      BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
      ILBSCL = .FALSE.
      IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
         BNRMTO = SMLNUM
         ILBSCL = .TRUE.
      ELSE IF( BNRM.GT.BIGNUM ) THEN
         BNRMTO = BIGNUM
         ILBSCL = .TRUE.
      END IF
      IF( ILBSCL )
     $   CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
*
*     Permute the matrices A, B to isolate eigenvalues if possible
*     (Workspace: need 6*N)
*
      ILEFT = 1
      IRIGHT = N + 1
      IWRK = IRIGHT + N
      CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
     $             WORK( IRIGHT ), WORK( IWRK ), IERR )
*
*     Reduce B to triangular form (QR decomposition of B)
*     (Workspace: need N, prefer N*NB)
*
      IROWS = IHI + 1 - ILO
      IF( ILV ) THEN
         ICOLS = N + 1 - ILO
      ELSE
         ICOLS = IROWS
      END IF
      ITAU = IWRK
      IWRK = ITAU + IROWS
      CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
     $             WORK( IWRK ), LWORK+1-IWRK, IERR )
*
*     Apply the orthogonal transformation to matrix A
*     (Workspace: need N, prefer N*NB)
*
      CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
     $             WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
     $             LWORK+1-IWRK, IERR )
*
*     Initialize VL
*     (Workspace: need N, prefer N*NB)
*
      IF( ILVL ) THEN
         CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
         IF( IROWS.GT.1 ) THEN
            CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
     $                   VL( ILO+1, ILO ), LDVL )
         END IF
         CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
     $                WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
      END IF
*
*     Initialize VR
*
      IF( ILVR )
     $   CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
*
*     Reduce to generalized Hessenberg form
*     (Workspace: none needed)
*
      IF( ILV ) THEN
*
*        Eigenvectors requested -- work on whole matrix.
*
         CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
     $                LDVL, VR, LDVR, IERR )
      ELSE
         CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
     $                B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR )
      END IF
*
*     Perform QZ algorithm (Compute eigenvalues, and optionally, the
*     Schur forms and Schur vectors)
*     (Workspace: need N)
*
      IWRK = ITAU
      IF( ILV ) THEN
         CHTEMP = 'S'
      ELSE
         CHTEMP = 'E'
      END IF
      CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
     $             ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR,
     $             WORK( IWRK ), LWORK+1-IWRK, IERR )
      IF( IERR.NE.0 ) THEN
         IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
            INFO = IERR
         ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
            INFO = IERR - N
         ELSE
            INFO = N + 1
         END IF
         GO TO 110
      END IF
*
*     Compute Eigenvectors
*     (Workspace: need 6*N)
*
      IF( ILV ) THEN
         IF( ILVL ) THEN
            IF( ILVR ) THEN
               CHTEMP = 'B'
            ELSE
               CHTEMP = 'L'
            END IF
         ELSE
            CHTEMP = 'R'
         END IF
         CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL,
     $                LDVL,
     $                VR, LDVR, N, IN, WORK( IWRK ), IERR )
         IF( IERR.NE.0 ) THEN
            INFO = N + 2
            GO TO 110
         END IF
*
*        Undo balancing on VL and VR and normalization
*        (Workspace: none needed)
*
         IF( ILVL ) THEN
            CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
     $                   WORK( IRIGHT ), N, VL, LDVL, IERR )
            DO 50 JC = 1, N
               IF( ALPHAI( JC ).LT.ZERO )
     $            GO TO 50
               TEMP = ZERO
               IF( ALPHAI( JC ).EQ.ZERO ) THEN
                  DO 10 JR = 1, N
                     TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
   10             CONTINUE
               ELSE
                  DO 20 JR = 1, N
                     TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
     $                      ABS( VL( JR, JC+1 ) ) )
   20             CONTINUE
               END IF
               IF( TEMP.LT.SMLNUM )
     $            GO TO 50
               TEMP = ONE / TEMP
               IF( ALPHAI( JC ).EQ.ZERO ) THEN
                  DO 30 JR = 1, N
                     VL( JR, JC ) = VL( JR, JC )*TEMP
   30             CONTINUE
               ELSE
                  DO 40 JR = 1, N
                     VL( JR, JC ) = VL( JR, JC )*TEMP
                     VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
   40             CONTINUE
               END IF
   50       CONTINUE
         END IF
         IF( ILVR ) THEN
            CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
     $                   WORK( IRIGHT ), N, VR, LDVR, IERR )
            DO 100 JC = 1, N
               IF( ALPHAI( JC ).LT.ZERO )
     $            GO TO 100
               TEMP = ZERO
               IF( ALPHAI( JC ).EQ.ZERO ) THEN
                  DO 60 JR = 1, N
                     TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
   60             CONTINUE
               ELSE
                  DO 70 JR = 1, N
                     TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
     $                      ABS( VR( JR, JC+1 ) ) )
   70             CONTINUE
               END IF
               IF( TEMP.LT.SMLNUM )
     $            GO TO 100
               TEMP = ONE / TEMP
               IF( ALPHAI( JC ).EQ.ZERO ) THEN
                  DO 80 JR = 1, N
                     VR( JR, JC ) = VR( JR, JC )*TEMP
   80             CONTINUE
               ELSE
                  DO 90 JR = 1, N
                     VR( JR, JC ) = VR( JR, JC )*TEMP
                     VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
   90             CONTINUE
               END IF
  100       CONTINUE
         END IF
*
*        End of eigenvector calculation
*
      END IF
*
*     Undo scaling if necessary
*
  110 CONTINUE
*
      IF( ILASCL ) THEN
         CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N,
     $                IERR )
         CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N,
     $                IERR )
      END IF
*
      IF( ILBSCL ) THEN
         CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
      END IF
*
      WORK( 1 ) = MAXWRK
      RETURN
*
*     End of DGGEV
*
      END
*> \brief <b> DGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGEVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggevx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggevx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggevx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
*                          ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
*                          IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
*                          RCONDV, WORK, LWORK, IWORK, BWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          BALANC, JOBVL, JOBVR, SENSE
*       INTEGER            IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
*       DOUBLE PRECISION   ABNRM, BBNRM
*       ..
*       .. Array Arguments ..
*       LOGICAL            BWORK( * )
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
*      $                   B( LDB, * ), BETA( * ), LSCALE( * ),
*      $                   RCONDE( * ), RCONDV( * ), RSCALE( * ),
*      $                   VL( LDVL, * ), VR( LDVR, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
*> the generalized eigenvalues, and optionally, the left and/or right
*> generalized eigenvectors.
*>
*> Optionally also, it computes a balancing transformation to improve
*> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
*> LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
*> the eigenvalues (RCONDE), and reciprocal condition numbers for the
*> right eigenvectors (RCONDV).
*>
*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar
*> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
*> singular. It is usually represented as the pair (alpha,beta), as
*> there is a reasonable interpretation for beta=0, and even for both
*> being zero.
*>
*> The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
*> of (A,B) satisfies
*>
*>                  A * v(j) = lambda(j) * B * v(j) .
*>
*> The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
*> of (A,B) satisfies
*>
*>                  u(j)**H * A  = lambda(j) * u(j)**H * B.
*>
*> where u(j)**H is the conjugate-transpose of u(j).
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] BALANC
*> \verbatim
*>          BALANC is CHARACTER*1
*>          Specifies the balance option to be performed.
*>          = 'N':  do not diagonally scale or permute;
*>          = 'P':  permute only;
*>          = 'S':  scale only;
*>          = 'B':  both permute and scale.
*>          Computed reciprocal condition numbers will be for the
*>          matrices after permuting and/or balancing. Permuting does
*>          not change condition numbers (in exact arithmetic), but
*>          balancing does.
*> \endverbatim
*>
*> \param[in] JOBVL
*> \verbatim
*>          JOBVL is CHARACTER*1
*>          = 'N':  do not compute the left generalized eigenvectors;
*>          = 'V':  compute the left generalized eigenvectors.
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*>          JOBVR is CHARACTER*1
*>          = 'N':  do not compute the right generalized eigenvectors;
*>          = 'V':  compute the right generalized eigenvectors.
*> \endverbatim
*>
*> \param[in] SENSE
*> \verbatim
*>          SENSE is CHARACTER*1
*>          Determines which reciprocal condition numbers are computed.
*>          = 'N': none are computed;
*>          = 'E': computed for eigenvalues only;
*>          = 'V': computed for eigenvectors only;
*>          = 'B': computed for eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A, B, VL, and VR.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, N)
*>          On entry, the matrix A in the pair (A,B).
*>          On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
*>          or both, then A contains the first part of the real Schur
*>          form of the "balanced" versions of the input A and B.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB, N)
*>          On entry, the matrix B in the pair (A,B).
*>          On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
*>          or both, then B contains the second part of the real Schur
*>          form of the "balanced" versions of the input A and B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*>          ALPHAR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*>          ALPHAI is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION array, dimension (N)
*>          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
*>          be the generalized eigenvalues.  If ALPHAI(j) is zero, then
*>          the j-th eigenvalue is real; if positive, then the j-th and
*>          (j+1)-st eigenvalues are a complex conjugate pair, with
*>          ALPHAI(j+1) negative.
*>
*>          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
*>          may easily over- or underflow, and BETA(j) may even be zero.
*>          Thus, the user should avoid naively computing the ratio
*>          ALPHA/BETA. However, ALPHAR and ALPHAI will be always less
*>          than and usually comparable with norm(A) in magnitude, and
*>          BETA always less than and usually comparable with norm(B).
*> \endverbatim
*>
*> \param[out] VL
*> \verbatim
*>          VL is DOUBLE PRECISION array, dimension (LDVL,N)
*>          If JOBVL = 'V', the left eigenvectors u(j) are stored one
*>          after another in the columns of VL, in the same order as
*>          their eigenvalues. If the j-th eigenvalue is real, then
*>          u(j) = VL(:,j), the j-th column of VL. If the j-th and
*>          (j+1)-th eigenvalues form a complex conjugate pair, then
*>          u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
*>          Each eigenvector will be scaled so the largest component have
*>          abs(real part) + abs(imag. part) = 1.
*>          Not referenced if JOBVL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*>          LDVL is INTEGER
*>          The leading dimension of the matrix VL. LDVL >= 1, and
*>          if JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*>          VR is DOUBLE PRECISION array, dimension (LDVR,N)
*>          If JOBVR = 'V', the right eigenvectors v(j) are stored one
*>          after another in the columns of VR, in the same order as
*>          their eigenvalues. If the j-th eigenvalue is real, then
*>          v(j) = VR(:,j), the j-th column of VR. If the j-th and
*>          (j+1)-th eigenvalues form a complex conjugate pair, then
*>          v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
*>          Each eigenvector will be scaled so the largest component have
*>          abs(real part) + abs(imag. part) = 1.
*>          Not referenced if JOBVR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*>          LDVR is INTEGER
*>          The leading dimension of the matrix VR. LDVR >= 1, and
*>          if JOBVR = 'V', LDVR >= N.
*> \endverbatim
*>
*> \param[out] ILO
*> \verbatim
*>          ILO is INTEGER
*> \endverbatim
*>
*> \param[out] IHI
*> \verbatim
*>          IHI is INTEGER
*>          ILO and IHI are integer values such that on exit
*>          A(i,j) = 0 and B(i,j) = 0 if i > j and
*>          j = 1,...,ILO-1 or i = IHI+1,...,N.
*>          If BALANC = 'N' or 'S', ILO = 1 and IHI = N.
*> \endverbatim
*>
*> \param[out] LSCALE
*> \verbatim
*>          LSCALE is DOUBLE PRECISION array, dimension (N)
*>          Details of the permutations and scaling factors applied
*>          to the left side of A and B.  If PL(j) is the index of the
*>          row interchanged with row j, and DL(j) is the scaling
*>          factor applied to row j, then
*>            LSCALE(j) = PL(j)  for j = 1,...,ILO-1
*>                      = DL(j)  for j = ILO,...,IHI
*>                      = PL(j)  for j = IHI+1,...,N.
*>          The order in which the interchanges are made is N to IHI+1,
*>          then 1 to ILO-1.
*> \endverbatim
*>
*> \param[out] RSCALE
*> \verbatim
*>          RSCALE is DOUBLE PRECISION array, dimension (N)
*>          Details of the permutations and scaling factors applied
*>          to the right side of A and B.  If PR(j) is the index of the
*>          column interchanged with column j, and DR(j) is the scaling
*>          factor applied to column j, then
*>            RSCALE(j) = PR(j)  for j = 1,...,ILO-1
*>                      = DR(j)  for j = ILO,...,IHI
*>                      = PR(j)  for j = IHI+1,...,N
*>          The order in which the interchanges are made is N to IHI+1,
*>          then 1 to ILO-1.
*> \endverbatim
*>
*> \param[out] ABNRM
*> \verbatim
*>          ABNRM is DOUBLE PRECISION
*>          The one-norm of the balanced matrix A.
*> \endverbatim
*>
*> \param[out] BBNRM
*> \verbatim
*>          BBNRM is DOUBLE PRECISION
*>          The one-norm of the balanced matrix B.
*> \endverbatim
*>
*> \param[out] RCONDE
*> \verbatim
*>          RCONDE is DOUBLE PRECISION array, dimension (N)
*>          If SENSE = 'E' or 'B', the reciprocal condition numbers of
*>          the eigenvalues, stored in consecutive elements of the array.
*>          For a complex conjugate pair of eigenvalues two consecutive
*>          elements of RCONDE are set to the same value. Thus RCONDE(j),
*>          RCONDV(j), and the j-th columns of VL and VR all correspond
*>          to the j-th eigenpair.
*>          If SENSE = 'N or 'V', RCONDE is not referenced.
*> \endverbatim
*>
*> \param[out] RCONDV
*> \verbatim
*>          RCONDV is DOUBLE PRECISION array, dimension (N)
*>          If SENSE = 'V' or 'B', the estimated reciprocal condition
*>          numbers of the eigenvectors, stored in consecutive elements
*>          of the array. For a complex eigenvector two consecutive
*>          elements of RCONDV are set to the same value. If the
*>          eigenvalues cannot be reordered to compute RCONDV(j),
*>          RCONDV(j) is set to 0; this can only occur when the true
*>          value would be very small anyway.
*>          If SENSE = 'N' or 'E', RCONDV is not referenced.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= max(1,2*N).
*>          If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
*>          LWORK >= max(1,6*N).
*>          If SENSE = 'E' or 'B', LWORK >= max(1,10*N).
*>          If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N+6)
*>          If SENSE = 'E', IWORK is not referenced.
*> \endverbatim
*>
*> \param[out] BWORK
*> \verbatim
*>          BWORK is LOGICAL array, dimension (N)
*>          If SENSE = 'N', BWORK is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          = 1,...,N:
*>                The QZ iteration failed.  No eigenvectors have been
*>                calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
*>                should be correct for j=INFO+1,...,N.
*>          > N:  =N+1: other than QZ iteration failed in DHGEQZ.
*>                =N+2: error return from DTGEVC.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggevx
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  Balancing a matrix pair (A,B) includes, first, permuting rows and
*>  columns to isolate eigenvalues, second, applying diagonal similarity
*>  transformation to the rows and columns to make the rows and columns
*>  as close in norm as possible. The computed reciprocal condition
*>  numbers correspond to the balanced matrix. Permuting rows and columns
*>  will not change the condition numbers (in exact arithmetic) but
*>  diagonal scaling will.  For further explanation of balancing, see
*>  section 4.11.1.2 of LAPACK Users' Guide.
*>
*>  An approximate error bound on the chordal distance between the i-th
*>  computed generalized eigenvalue w and the corresponding exact
*>  eigenvalue lambda is
*>
*>       chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
*>
*>  An approximate error bound for the angle between the i-th computed
*>  eigenvector VL(i) or VR(i) is given by
*>
*>       EPS * norm(ABNRM, BBNRM) / DIF(i).
*>
*>  For further explanation of the reciprocal condition numbers RCONDE
*>  and RCONDV, see section 4.11 of LAPACK User's Guide.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B,
     $                   LDB,
     $                   ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
     $                   IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
     $                   RCONDV, WORK, LWORK, IWORK, BWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          BALANC, JOBVL, JOBVR, SENSE
      INTEGER            IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
      DOUBLE PRECISION   ABNRM, BBNRM
*     ..
*     .. Array Arguments ..
      LOGICAL            BWORK( * )
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
     $                   B( LDB, * ), BETA( * ), LSCALE( * ),
     $                   RCONDE( * ), RCONDV( * ), RSCALE( * ),
     $                   VL( LDVL, * ), VR( LDVR, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY, NOSCL,
     $                   PAIR, WANTSB, WANTSE, WANTSN, WANTSV
      CHARACTER          CHTEMP
      INTEGER            I, ICOLS, IERR, IJOBVL, IJOBVR, IN, IROWS,
     $                   ITAU, IWRK, IWRK1, J, JC, JR, M, MAXWRK,
     $                   MINWRK, MM
      DOUBLE PRECISION   ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
     $                   SMLNUM, TEMP
*     ..
*     .. Local Arrays ..
      LOGICAL            LDUMMA( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ,
     $                   DLACPY,
     $                   DLASCL, DLASET, DORGQR, DORMQR, DTGEVC, DTGSNA,
     $                   XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           LSAME, ILAENV, DLAMCH, DLANGE
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Decode the input arguments
*
      IF( LSAME( JOBVL, 'N' ) ) THEN
         IJOBVL = 1
         ILVL = .FALSE.
      ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
         IJOBVL = 2
         ILVL = .TRUE.
      ELSE
         IJOBVL = -1
         ILVL = .FALSE.
      END IF
*
      IF( LSAME( JOBVR, 'N' ) ) THEN
         IJOBVR = 1
         ILVR = .FALSE.
      ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
         IJOBVR = 2
         ILVR = .TRUE.
      ELSE
         IJOBVR = -1
         ILVR = .FALSE.
      END IF
      ILV = ILVL .OR. ILVR
*
      NOSCL  = LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'P' )
      WANTSN = LSAME( SENSE, 'N' )
      WANTSE = LSAME( SENSE, 'E' )
      WANTSV = LSAME( SENSE, 'V' )
      WANTSB = LSAME( SENSE, 'B' )
*
*     Test the input arguments
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC,
     $    'S' ) .OR.
     $                      LSAME( BALANC, 'P' ) .OR.
     $                      LSAME( BALANC, 'B' ) ) )
     $     THEN
         INFO = -1
      ELSE IF( IJOBVL.LE.0 ) THEN
         INFO = -2
      ELSE IF( IJOBVR.LE.0 ) THEN
         INFO = -3
      ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSB .OR. WANTSV ) )
     $          THEN
         INFO = -4
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -9
      ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
         INFO = -14
      ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
         INFO = -16
      END IF
*
*     Compute workspace
*      (Note: Comments in the code beginning "Workspace:" describe the
*       minimal amount of workspace needed at that point in the code,
*       as well as the preferred amount for good performance.
*       NB refers to the optimal block size for the immediately
*       following subroutine, as returned by ILAENV. The workspace is
*       computed assuming ILO = 1 and IHI = N, the worst case.)
*
      IF( INFO.EQ.0 ) THEN
         IF( N.EQ.0 ) THEN
            MINWRK = 1
            MAXWRK = 1
         ELSE
            IF( NOSCL .AND. .NOT.ILV ) THEN
               MINWRK = 2*N
            ELSE
               MINWRK = 6*N
            END IF
            IF( WANTSE .OR. WANTSB ) THEN
               MINWRK = 10*N
            END IF
            IF( WANTSV .OR. WANTSB ) THEN
               MINWRK = MAX( MINWRK, 2*N*( N + 4 ) + 16 )
            END IF
            MAXWRK = MINWRK
            MAXWRK = MAX( MAXWRK,
     $                    N + N*ILAENV( 1, 'DGEQRF', ' ', N, 1, N,
     $                                  0 ) )
            MAXWRK = MAX( MAXWRK,
     $                    N + N*ILAENV( 1, 'DORMQR', ' ', N, 1, N,
     $                                  0 ) )
            IF( ILVL ) THEN
               MAXWRK = MAX( MAXWRK, N +
     $                       N*ILAENV( 1, 'DORGQR', ' ', N, 1, N,
     $                                 0 ) )
            END IF
         END IF
         WORK( 1 ) = MAXWRK
*
         IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
            INFO = -26
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGGEVX', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*
*     Get machine constants
*
      EPS = DLAMCH( 'P' )
      SMLNUM = DLAMCH( 'S' )
      BIGNUM = ONE / SMLNUM
      SMLNUM = SQRT( SMLNUM ) / EPS
      BIGNUM = ONE / SMLNUM
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
      ILASCL = .FALSE.
      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
         ANRMTO = SMLNUM
         ILASCL = .TRUE.
      ELSE IF( ANRM.GT.BIGNUM ) THEN
         ANRMTO = BIGNUM
         ILASCL = .TRUE.
      END IF
      IF( ILASCL )
     $   CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
*
*     Scale B if max element outside range [SMLNUM,BIGNUM]
*
      BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
      ILBSCL = .FALSE.
      IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
         BNRMTO = SMLNUM
         ILBSCL = .TRUE.
      ELSE IF( BNRM.GT.BIGNUM ) THEN
         BNRMTO = BIGNUM
         ILBSCL = .TRUE.
      END IF
      IF( ILBSCL )
     $   CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
*
*     Permute and/or balance the matrix pair (A,B)
*     (Workspace: need 6*N if BALANC = 'S' or 'B', 1 otherwise)
*
      CALL DGGBAL( BALANC, N, A, LDA, B, LDB, ILO, IHI, LSCALE,
     $             RSCALE,
     $             WORK, IERR )
*
*     Compute ABNRM and BBNRM
*
      ABNRM = DLANGE( '1', N, N, A, LDA, WORK( 1 ) )
      IF( ILASCL ) THEN
         WORK( 1 ) = ABNRM
         CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, 1, 1, WORK( 1 ), 1,
     $                IERR )
         ABNRM = WORK( 1 )
      END IF
*
      BBNRM = DLANGE( '1', N, N, B, LDB, WORK( 1 ) )
      IF( ILBSCL ) THEN
         WORK( 1 ) = BBNRM
         CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, 1, 1, WORK( 1 ), 1,
     $                IERR )
         BBNRM = WORK( 1 )
      END IF
*
*     Reduce B to triangular form (QR decomposition of B)
*     (Workspace: need N, prefer N*NB )
*
      IROWS = IHI + 1 - ILO
      IF( ILV .OR. .NOT.WANTSN ) THEN
         ICOLS = N + 1 - ILO
      ELSE
         ICOLS = IROWS
      END IF
      ITAU = 1
      IWRK = ITAU + IROWS
      CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
     $             WORK( IWRK ), LWORK+1-IWRK, IERR )
*
*     Apply the orthogonal transformation to A
*     (Workspace: need N, prefer N*NB)
*
      CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
     $             WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
     $             LWORK+1-IWRK, IERR )
*
*     Initialize VL and/or VR
*     (Workspace: need N, prefer N*NB)
*
      IF( ILVL ) THEN
         CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
         IF( IROWS.GT.1 ) THEN
            CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
     $                   VL( ILO+1, ILO ), LDVL )
         END IF
         CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
     $                WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
      END IF
*
      IF( ILVR )
     $   CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
*
*     Reduce to generalized Hessenberg form
*     (Workspace: none needed)
*
      IF( ILV .OR. .NOT.WANTSN ) THEN
*
*        Eigenvectors requested -- work on whole matrix.
*
         CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
     $                LDVL, VR, LDVR, IERR )
      ELSE
         CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
     $                B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR )
      END IF
*
*     Perform QZ algorithm (Compute eigenvalues, and optionally, the
*     Schur forms and Schur vectors)
*     (Workspace: need N)
*
      IF( ILV .OR. .NOT.WANTSN ) THEN
         CHTEMP = 'S'
      ELSE
         CHTEMP = 'E'
      END IF
*
      CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
     $             ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK,
     $             LWORK, IERR )
      IF( IERR.NE.0 ) THEN
         IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
            INFO = IERR
         ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
            INFO = IERR - N
         ELSE
            INFO = N + 1
         END IF
         GO TO 130
      END IF
*
*     Compute Eigenvectors and estimate condition numbers if desired
*     (Workspace: DTGEVC: need 6*N
*                 DTGSNA: need 2*N*(N+2)+16 if SENSE = 'V' or 'B',
*                         need N otherwise )
*
      IF( ILV .OR. .NOT.WANTSN ) THEN
         IF( ILV ) THEN
            IF( ILVL ) THEN
               IF( ILVR ) THEN
                  CHTEMP = 'B'
               ELSE
                  CHTEMP = 'L'
               END IF
            ELSE
               CHTEMP = 'R'
            END IF
*
            CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL,
     $                   LDVL, VR, LDVR, N, IN, WORK, IERR )
            IF( IERR.NE.0 ) THEN
               INFO = N + 2
               GO TO 130
            END IF
         END IF
*
         IF( .NOT.WANTSN ) THEN
*
*           compute eigenvectors (DTGEVC) and estimate condition
*           numbers (DTGSNA). Note that the definition of the condition
*           number is not invariant under transformation (u,v) to
*           (Q*u, Z*v), where (u,v) are eigenvectors of the generalized
*           Schur form (S,T), Q and Z are orthogonal matrices. In order
*           to avoid using extra 2*N*N workspace, we have to recalculate
*           eigenvectors and estimate one condition numbers at a time.
*
            PAIR = .FALSE.
            DO 20 I = 1, N
*
               IF( PAIR ) THEN
                  PAIR = .FALSE.
                  GO TO 20
               END IF
               MM = 1
               IF( I.LT.N ) THEN
                  IF( A( I+1, I ).NE.ZERO ) THEN
                     PAIR = .TRUE.
                     MM = 2
                  END IF
               END IF
*
               DO 10 J = 1, N
                  BWORK( J ) = .FALSE.
   10          CONTINUE
               IF( MM.EQ.1 ) THEN
                  BWORK( I ) = .TRUE.
               ELSE IF( MM.EQ.2 ) THEN
                  BWORK( I ) = .TRUE.
                  BWORK( I+1 ) = .TRUE.
               END IF
*
               IWRK = MM*N + 1
               IWRK1 = IWRK + MM*N
*
*              Compute a pair of left and right eigenvectors.
*              (compute workspace: need up to 4*N + 6*N)
*
               IF( WANTSE .OR. WANTSB ) THEN
                  CALL DTGEVC( 'B', 'S', BWORK, N, A, LDA, B, LDB,
     $                         WORK( 1 ), N, WORK( IWRK ), N, MM, M,
     $                         WORK( IWRK1 ), IERR )
                  IF( IERR.NE.0 ) THEN
                     INFO = N + 2
                     GO TO 130
                  END IF
               END IF
*
               CALL DTGSNA( SENSE, 'S', BWORK, N, A, LDA, B, LDB,
     $                      WORK( 1 ), N, WORK( IWRK ), N, RCONDE( I ),
     $                      RCONDV( I ), MM, M, WORK( IWRK1 ),
     $                      LWORK-IWRK1+1, IWORK, IERR )
*
   20       CONTINUE
         END IF
      END IF
*
*     Undo balancing on VL and VR and normalization
*     (Workspace: none needed)
*
      IF( ILVL ) THEN
         CALL DGGBAK( BALANC, 'L', N, ILO, IHI, LSCALE, RSCALE, N,
     $                VL,
     $                LDVL, IERR )
*
         DO 70 JC = 1, N
            IF( ALPHAI( JC ).LT.ZERO )
     $         GO TO 70
            TEMP = ZERO
            IF( ALPHAI( JC ).EQ.ZERO ) THEN
               DO 30 JR = 1, N
                  TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
   30          CONTINUE
            ELSE
               DO 40 JR = 1, N
                  TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
     $                   ABS( VL( JR, JC+1 ) ) )
   40          CONTINUE
            END IF
            IF( TEMP.LT.SMLNUM )
     $         GO TO 70
            TEMP = ONE / TEMP
            IF( ALPHAI( JC ).EQ.ZERO ) THEN
               DO 50 JR = 1, N
                  VL( JR, JC ) = VL( JR, JC )*TEMP
   50          CONTINUE
            ELSE
               DO 60 JR = 1, N
                  VL( JR, JC ) = VL( JR, JC )*TEMP
                  VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
   60          CONTINUE
            END IF
   70    CONTINUE
      END IF
      IF( ILVR ) THEN
         CALL DGGBAK( BALANC, 'R', N, ILO, IHI, LSCALE, RSCALE, N,
     $                VR,
     $                LDVR, IERR )
         DO 120 JC = 1, N
            IF( ALPHAI( JC ).LT.ZERO )
     $         GO TO 120
            TEMP = ZERO
            IF( ALPHAI( JC ).EQ.ZERO ) THEN
               DO 80 JR = 1, N
                  TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
   80          CONTINUE
            ELSE
               DO 90 JR = 1, N
                  TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
     $                   ABS( VR( JR, JC+1 ) ) )
   90          CONTINUE
            END IF
            IF( TEMP.LT.SMLNUM )
     $         GO TO 120
            TEMP = ONE / TEMP
            IF( ALPHAI( JC ).EQ.ZERO ) THEN
               DO 100 JR = 1, N
                  VR( JR, JC ) = VR( JR, JC )*TEMP
  100          CONTINUE
            ELSE
               DO 110 JR = 1, N
                  VR( JR, JC ) = VR( JR, JC )*TEMP
                  VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
  110          CONTINUE
            END IF
  120    CONTINUE
      END IF
*
*     Undo scaling if necessary
*
  130 CONTINUE
*
      IF( ILASCL ) THEN
         CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N,
     $                IERR )
         CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N,
     $                IERR )
      END IF
*
      IF( ILBSCL ) THEN
         CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
      END IF
*
      WORK( 1 ) = MAXWRK
      RETURN
*
*     End of DGGEVX
*
      END
*> \brief \b DGGGLM
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGGLM + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggglm.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggglm.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggglm.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK,
*                          INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LDB, LWORK, M, N, P
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), D( * ), WORK( * ),
*      $                   X( * ), Y( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGGGLM solves a general Gauss-Markov linear model (GLM) problem:
*>
*>         minimize || y ||_2   subject to   d = A*x + B*y
*>             x
*>
*> where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
*> given N-vector. It is assumed that M <= N <= M+P, and
*>
*>            rank(A) = M    and    rank( A B ) = N.
*>
*> Under these assumptions, the constrained equation is always
*> consistent, and there is a unique solution x and a minimal 2-norm
*> solution y, which is obtained using a generalized QR factorization
*> of the matrices (A, B) given by
*>
*>    A = Q*(R),   B = Q*T*Z.
*>          (0)
*>
*> In particular, if matrix B is square nonsingular, then the problem
*> GLM is equivalent to the following weighted linear least squares
*> problem
*>
*>              minimize || inv(B)*(d-A*x) ||_2
*>                  x
*>
*> where inv(B) denotes the inverse of B.
*>
*> Callers of this subroutine should note that the singularity/rank-deficiency checks
*> implemented in this subroutine are rudimentary. The DTRTRS subroutine called by this
*> subroutine only signals a failure due to singularity if the problem is exactly singular.
*>
*> It is conceivable for one (or more) of the factors involved in the generalized QR
*> factorization of the pair (A, B) to be subnormally close to singularity without this
*> subroutine signalling an error. The solutions computed for such almost-rank-deficient
*> problems may be less accurate due to a loss of numerical precision.
*> 
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of rows of the matrices A and B.  N >= 0.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of columns of the matrix A.  0 <= M <= N.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*>          P is INTEGER
*>          The number of columns of the matrix B.  P >= N-M.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,M)
*>          On entry, the N-by-M matrix A.
*>          On exit, the upper triangular part of the array A contains
*>          the M-by-M upper triangular matrix R.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,P)
*>          On entry, the N-by-P matrix B.
*>          On exit, if N <= P, the upper triangle of the subarray
*>          B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
*>          if N > P, the elements on and above the (N-P)th subdiagonal
*>          contain the N-by-P upper trapezoidal matrix T.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, D is the left hand side of the GLM equation.
*>          On exit, D is destroyed.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (M)
*> \endverbatim
*>
*> \param[out] Y
*> \verbatim
*>          Y is DOUBLE PRECISION array, dimension (P)
*>
*>          On exit, X and Y are the solutions of the GLM problem.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= max(1,N+M+P).
*>          For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB,
*>          where NB is an upper bound for the optimal blocksizes for
*>          DGEQRF, SGERQF, DORMQR and SORMRQ.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          = 1:  the upper triangular factor R associated with A in the
*>                generalized QR factorization of the pair (A, B) is exactly
*>                singular, so that rank(A) < M; the least squares
*>                solution could not be computed.
*>          = 2:  the bottom (N-M) by (N-M) part of the upper trapezoidal
*>                factor T associated with B in the generalized QR
*>                factorization of the pair (A, B) is exactly singular, so that
*>                rank( A B ) < N; the least squares solution could not
*>                be computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggglm
*
*  =====================================================================
      SUBROUTINE DGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK,
     $                   LWORK,
     $                   INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, LWORK, M, N, P
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), D( * ), WORK( * ),
     $                   X( * ), Y( * )
*     ..
*
*  ===================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3,
     $                   NB4, NP
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGEMV, DGGQRF, DORMQR, DORMRQ,
     $                   DTRTRS,
     $                   XERBLA
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          INT, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INFO = 0
      NP = MIN( N, P )
      LQUERY = ( LWORK.EQ.-1 )
      IF( N.LT.0 ) THEN
         INFO = -1
      ELSE IF( M.LT.0 .OR. M.GT.N ) THEN
         INFO = -2
      ELSE IF( P.LT.0 .OR. P.LT.N-M ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -7
      END IF
*
*     Calculate workspace
*
      IF( INFO.EQ.0) THEN
         IF( N.EQ.0 ) THEN
            LWKMIN = 1
            LWKOPT = 1
         ELSE
            NB1 = ILAENV( 1, 'DGEQRF', ' ', N, M, -1, -1 )
            NB2 = ILAENV( 1, 'DGERQF', ' ', N, M, -1, -1 )
            NB3 = ILAENV( 1, 'DORMQR', ' ', N, M, P, -1 )
            NB4 = ILAENV( 1, 'DORMRQ', ' ', N, M, P, -1 )
            NB = MAX( NB1, NB2, NB3, NB4 )
            LWKMIN = M + N + P
            LWKOPT = M + NP + MAX( N, P )*NB
         END IF
         WORK( 1 ) = LWKOPT
*
         IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
            INFO = -12
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGGGLM', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 ) THEN
         DO I = 1, M
            X(I) = ZERO
         END DO
         DO I = 1, P
            Y(I) = ZERO
         END DO
         RETURN
      END IF
*
*     Compute the GQR factorization of matrices A and B:
*
*          Q**T*A = ( R11 ) M,    Q**T*B*Z**T = ( T11   T12 ) M
*                   (  0  ) N-M                 (  0    T22 ) N-M
*                      M                         M+P-N  N-M
*
*     where R11 and T22 are upper triangular, and Q and Z are
*     orthogonal.
*
      CALL DGGQRF( N, M, P, A, LDA, WORK, B, LDB, WORK( M+1 ),
     $             WORK( M+NP+1 ), LWORK-M-NP, INFO )
      LOPT = INT( WORK( M+NP+1 ) )
*
*     Update left-hand-side vector d = Q**T*d = ( d1 ) M
*                                               ( d2 ) N-M
*
      CALL DORMQR( 'Left', 'Transpose', N, 1, M, A, LDA, WORK, D,
     $             MAX( 1, N ), WORK( M+NP+1 ), LWORK-M-NP, INFO )
      LOPT = MAX( LOPT, INT( WORK( M+NP+1 ) ) )
*
*     Solve T22*y2 = d2 for y2
*
      IF( N.GT.M ) THEN
         CALL DTRTRS( 'Upper', 'No transpose', 'Non unit', N-M, 1,
     $                B( M+1, M+P-N+1 ), LDB, D( M+1 ), N-M, INFO )
*
         IF( INFO.GT.0 ) THEN
            INFO = 1
            RETURN
         END IF
*
         CALL DCOPY( N-M, D( M+1 ), 1, Y( M+P-N+1 ), 1 )
      END IF
*
*     Set y1 = 0
*
      DO 10 I = 1, M + P - N
         Y( I ) = ZERO
   10 CONTINUE
*
*     Update d1 = d1 - T12*y2
*
      CALL DGEMV( 'No transpose', M, N-M, -ONE, B( 1, M+P-N+1 ), LDB,
     $            Y( M+P-N+1 ), 1, ONE, D, 1 )
*
*     Solve triangular system: R11*x = d1
*
      IF( M.GT.0 ) THEN
         CALL DTRTRS( 'Upper', 'No Transpose', 'Non unit', M, 1, A,
     $                LDA,
     $                D, M, INFO )
*
         IF( INFO.GT.0 ) THEN
            INFO = 2
            RETURN
         END IF
*
*        Copy D to X
*
         CALL DCOPY( M, D, 1, X, 1 )
      END IF
*
*     Backward transformation y = Z**T *y
*
      CALL DORMRQ( 'Left', 'Transpose', P, 1, NP,
     $             B( MAX( 1, N-P+1 ), 1 ), LDB, WORK( M+1 ), Y,
     $             MAX( 1, P ), WORK( M+NP+1 ), LWORK-M-NP, INFO )
      WORK( 1 ) = M + NP + MAX( LOPT, INT( WORK( M+NP+1 ) ) )
*
      RETURN
*
*     End of DGGGLM
*
      END
*> \brief \b DGGHRD
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGHRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgghrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgghrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgghrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q,
*                          LDQ, Z, LDZ, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          COMPQ, COMPZ
*       INTEGER            IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
*      $                   Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGGHRD reduces a pair of real matrices (A,B) to generalized upper
*> Hessenberg form using orthogonal transformations, where A is a
*> general matrix and B is upper triangular.  The form of the
*> generalized eigenvalue problem is
*>    A*x = lambda*B*x,
*> and B is typically made upper triangular by computing its QR
*> factorization and moving the orthogonal matrix Q to the left side
*> of the equation.
*>
*> This subroutine simultaneously reduces A to a Hessenberg matrix H:
*>    Q**T*A*Z = H
*> and transforms B to another upper triangular matrix T:
*>    Q**T*B*Z = T
*> in order to reduce the problem to its standard form
*>    H*y = lambda*T*y
*> where y = Z**T*x.
*>
*> The orthogonal matrices Q and Z are determined as products of Givens
*> rotations.  They may either be formed explicitly, or they may be
*> postmultiplied into input matrices Q1 and Z1, so that
*>
*>      Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T
*>
*>      Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T
*>
*> If Q1 is the orthogonal matrix from the QR factorization of B in the
*> original equation A*x = lambda*B*x, then DGGHRD reduces the original
*> problem to generalized Hessenberg form.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] COMPQ
*> \verbatim
*>          COMPQ is CHARACTER*1
*>          = 'N': do not compute Q;
*>          = 'I': Q is initialized to the unit matrix, and the
*>                 orthogonal matrix Q is returned;
*>          = 'V': Q must contain an orthogonal matrix Q1 on entry,
*>                 and the product Q1*Q is returned.
*> \endverbatim
*>
*> \param[in] COMPZ
*> \verbatim
*>          COMPZ is CHARACTER*1
*>          = 'N': do not compute Z;
*>          = 'I': Z is initialized to the unit matrix, and the
*>                 orthogonal matrix Z is returned;
*>          = 'V': Z must contain an orthogonal matrix Z1 on entry,
*>                 and the product Z1*Z is returned.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A and B.  N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*>          ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*>          IHI is INTEGER
*>
*>          ILO and IHI mark the rows and columns of A which are to be
*>          reduced.  It is assumed that A is already upper triangular
*>          in rows and columns 1:ILO-1 and IHI+1:N.  ILO and IHI are
*>          normally set by a previous call to DGGBAL; otherwise they
*>          should be set to 1 and N respectively.
*>          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, N)
*>          On entry, the N-by-N general matrix to be reduced.
*>          On exit, the upper triangle and the first subdiagonal of A
*>          are overwritten with the upper Hessenberg matrix H, and the
*>          rest is set to zero.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB, N)
*>          On entry, the N-by-N upper triangular matrix B.
*>          On exit, the upper triangular matrix T = Q**T B Z.  The
*>          elements below the diagonal are set to zero.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ, N)
*>          On entry, if COMPQ = 'V', the orthogonal matrix Q1,
*>          typically from the QR factorization of B.
*>          On exit, if COMPQ='I', the orthogonal matrix Q, and if
*>          COMPQ = 'V', the product Q1*Q.
*>          Not referenced if COMPQ='N'.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q.
*>          LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
*>          On entry, if COMPZ = 'V', the orthogonal matrix Z1.
*>          On exit, if COMPZ='I', the orthogonal matrix Z, and if
*>          COMPZ = 'V', the product Z1*Z.
*>          Not referenced if COMPZ='N'.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.
*>          LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gghrd
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  This routine reduces A to Hessenberg and B to triangular form by
*>  an unblocked reduction, as described in _Matrix_Computations_,
*>  by Golub and Van Loan (Johns Hopkins Press.)
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB,
     $                   Q,
     $                   LDQ, Z, LDZ, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          COMPQ, COMPZ
      INTEGER            IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
     $                   Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ILQ, ILZ
      INTEGER            ICOMPQ, ICOMPZ, JCOL, JROW
      DOUBLE PRECISION   C, S, TEMP
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARTG, DLASET, DROT, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Decode COMPQ
*
      IF( LSAME( COMPQ, 'N' ) ) THEN
         ILQ = .FALSE.
         ICOMPQ = 1
      ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
         ILQ = .TRUE.
         ICOMPQ = 2
      ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
         ILQ = .TRUE.
         ICOMPQ = 3
      ELSE
         ICOMPQ = 0
      END IF
*
*     Decode COMPZ
*
      IF( LSAME( COMPZ, 'N' ) ) THEN
         ILZ = .FALSE.
         ICOMPZ = 1
      ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
         ILZ = .TRUE.
         ICOMPZ = 2
      ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
         ILZ = .TRUE.
         ICOMPZ = 3
      ELSE
         ICOMPZ = 0
      END IF
*
*     Test the input parameters.
*
      INFO = 0
      IF( ICOMPQ.LE.0 ) THEN
         INFO = -1
      ELSE IF( ICOMPZ.LE.0 ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( ILO.LT.1 ) THEN
         INFO = -4
      ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -9
      ELSE IF( ( ILQ .AND. LDQ.LT.N ) .OR. LDQ.LT.1 ) THEN
         INFO = -11
      ELSE IF( ( ILZ .AND. LDZ.LT.N ) .OR. LDZ.LT.1 ) THEN
         INFO = -13
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGGHRD', -INFO )
         RETURN
      END IF
*
*     Initialize Q and Z if desired.
*
      IF( ICOMPQ.EQ.3 )
     $   CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
      IF( ICOMPZ.EQ.3 )
     $   CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
*
*     Quick return if possible
*
      IF( N.LE.1 )
     $   RETURN
*
*     Zero out lower triangle of B
*
      DO 20 JCOL = 1, N - 1
         DO 10 JROW = JCOL + 1, N
            B( JROW, JCOL ) = ZERO
   10    CONTINUE
   20 CONTINUE
*
*     Reduce A and B
*
      DO 40 JCOL = ILO, IHI - 2
*
         DO 30 JROW = IHI, JCOL + 2, -1
*
*           Step 1: rotate rows JROW-1, JROW to kill A(JROW,JCOL)
*
            TEMP = A( JROW-1, JCOL )
            CALL DLARTG( TEMP, A( JROW, JCOL ), C, S,
     $                   A( JROW-1, JCOL ) )
            A( JROW, JCOL ) = ZERO
            CALL DROT( N-JCOL, A( JROW-1, JCOL+1 ), LDA,
     $                 A( JROW, JCOL+1 ), LDA, C, S )
            CALL DROT( N+2-JROW, B( JROW-1, JROW-1 ), LDB,
     $                 B( JROW, JROW-1 ), LDB, C, S )
            IF( ILQ )
     $         CALL DROT( N, Q( 1, JROW-1 ), 1, Q( 1, JROW ), 1, C,
     $                    S )
*
*           Step 2: rotate columns JROW, JROW-1 to kill B(JROW,JROW-1)
*
            TEMP = B( JROW, JROW )
            CALL DLARTG( TEMP, B( JROW, JROW-1 ), C, S,
     $                   B( JROW, JROW ) )
            B( JROW, JROW-1 ) = ZERO
            CALL DROT( IHI, A( 1, JROW ), 1, A( 1, JROW-1 ), 1, C,
     $                 S )
            CALL DROT( JROW-1, B( 1, JROW ), 1, B( 1, JROW-1 ), 1, C,
     $                 S )
            IF( ILZ )
     $         CALL DROT( N, Z( 1, JROW ), 1, Z( 1, JROW-1 ), 1, C,
     $                    S )
   30    CONTINUE
   40 CONTINUE
*
      RETURN
*
*     End of DGGHRD
*
      END
*> \brief <b> DGGLSE solves overdetermined or underdetermined systems for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGLSE + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgglse.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgglse.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgglse.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK,
*                          INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LDB, LWORK, M, N, P
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( * ), D( * ),
*      $                   WORK( * ), X( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGGLSE solves the linear equality-constrained least squares (LSE)
*> problem:
*>
*>         minimize || c - A*x ||_2   subject to   B*x = d
*>
*> where A is an M-by-N matrix, B is a P-by-N matrix, c is a given
*> M-vector, and d is a given P-vector. It is assumed that
*> P <= N <= M+P, and
*>
*>          rank(B) = P and  rank( (A) ) = N.
*>                               ( (B) )
*>
*> These conditions ensure that the LSE problem has a unique solution,
*> which is obtained using a generalized RQ factorization of the
*> matrices (B, A) given by
*>
*>    B = (0 R)*Q,   A = Z*T*Q.
*>
*> Callers of this subroutine should note that the singularity/rank-deficiency checks
*> implemented in this subroutine are rudimentary. The DTRTRS subroutine called by this
*> subroutine only signals a failure due to singularity if the problem is exactly singular.
*>
*> It is conceivable for one (or more) of the factors involved in the generalized RQ
*> factorization of the pair (B, A) to be subnormally close to singularity without this
*> subroutine signalling an error. The solutions computed for such almost-rank-deficient
*> problems may be less accurate due to a loss of numerical precision.
*> 
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*>          P is INTEGER
*>          The number of rows of the matrix B. 0 <= P <= N <= M+P.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit, the elements on and above the diagonal of the array
*>          contain the min(M,N)-by-N upper trapezoidal matrix T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,N)
*>          On entry, the P-by-N matrix B.
*>          On exit, the upper triangle of the subarray B(1:P,N-P+1:N)
*>          contains the P-by-P upper triangular matrix R.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,P).
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (M)
*>          On entry, C contains the right hand side vector for the
*>          least squares part of the LSE problem.
*>          On exit, the residual sum of squares for the solution
*>          is given by the sum of squares of elements N-P+1 to M of
*>          vector C.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (P)
*>          On entry, D contains the right hand side vector for the
*>          constrained equation.
*>          On exit, D is destroyed.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (N)
*>          On exit, X is the solution of the LSE problem.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= max(1,M+N+P).
*>          For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB,
*>          where NB is an upper bound for the optimal blocksizes for
*>          DGEQRF, SGERQF, DORMQR and SORMRQ.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          = 1:  the upper triangular factor R associated with B in the
*>                generalized RQ factorization of the pair (B, A) is exactly
*>                singular, so that rank(B) < P; the least squares
*>                solution could not be computed.
*>          = 2:  the (N-P) by (N-P) part of the upper trapezoidal factor
*>                T associated with A in the generalized RQ factorization
*>                of the pair (B, A) is exactly singular, so that
*>                rank( (A) ) < N; the least squares solution could not
*>                    ( (B) )
*>                be computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gglse
*
*  =====================================================================
      SUBROUTINE DGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, WORK,
     $                   LWORK,
     $                   INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, LWORK, M, N, P
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( * ), D( * ),
     $                   WORK( * ), X( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            LOPT, LWKMIN, LWKOPT, MN, NB, NB1, NB2, NB3,
     $                   NB4, NR
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DGEMV, DGGRQF, DORMQR,
     $                   DORMRQ,
     $                   DTRMV, DTRTRS, XERBLA
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          INT, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INFO = 0
      MN = MIN( M, N )
      LQUERY = ( LWORK.EQ.-1 )
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( P.LT.0 .OR. P.GT.N .OR. P.LT.N-M ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
         INFO = -7
      END IF
*
*     Calculate workspace
*
      IF( INFO.EQ.0) THEN
         IF( N.EQ.0 ) THEN
            LWKMIN = 1
            LWKOPT = 1
         ELSE
            NB1 = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
            NB2 = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )
            NB3 = ILAENV( 1, 'DORMQR', ' ', M, N, P, -1 )
            NB4 = ILAENV( 1, 'DORMRQ', ' ', M, N, P, -1 )
            NB = MAX( NB1, NB2, NB3, NB4 )
            LWKMIN = M + N + P
            LWKOPT = P + MN + MAX( M, N )*NB
         END IF
         WORK( 1 ) = LWKOPT
*
         IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
            INFO = -12
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGGLSE', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Compute the GRQ factorization of matrices B and A:
*
*            B*Q**T = (  0  T12 ) P   Z**T*A*Q**T = ( R11 R12 ) N-P
*                        N-P  P                     (  0  R22 ) M+P-N
*                                                      N-P  P
*
*     where T12 and R11 are upper triangular, and Q and Z are
*     orthogonal.
*
      CALL DGGRQF( P, M, N, B, LDB, WORK, A, LDA, WORK( P+1 ),
     $             WORK( P+MN+1 ), LWORK-P-MN, INFO )
      LOPT = INT( WORK( P+MN+1 ) )
*
*     Update c = Z**T *c = ( c1 ) N-P
*                          ( c2 ) M+P-N
*
      CALL DORMQR( 'Left', 'Transpose', M, 1, MN, A, LDA,
     $             WORK( P+1 ),
     $             C, MAX( 1, M ), WORK( P+MN+1 ), LWORK-P-MN, INFO )
      LOPT = MAX( LOPT, INT( WORK( P+MN+1 ) ) )
*
*     Solve T12*x2 = d for x2
*
      IF( P.GT.0 ) THEN
         CALL DTRTRS( 'Upper', 'No transpose', 'Non-unit', P, 1,
     $                B( 1, N-P+1 ), LDB, D, P, INFO )
*
         IF( INFO.GT.0 ) THEN
            INFO = 1
            RETURN
         END IF
*
*        Put the solution in X
*
         CALL DCOPY( P, D, 1, X( N-P+1 ), 1 )
*
*        Update c1
*
         CALL DGEMV( 'No transpose', N-P, P, -ONE, A( 1, N-P+1 ),
     $               LDA,
     $               D, 1, ONE, C, 1 )
      END IF
*
*     Solve R11*x1 = c1 for x1
*
      IF( N.GT.P ) THEN
         CALL DTRTRS( 'Upper', 'No transpose', 'Non-unit', N-P, 1,
     $                A, LDA, C, N-P, INFO )
*
         IF( INFO.GT.0 ) THEN
            INFO = 2
            RETURN
         END IF
*
*        Put the solutions in X
*
         CALL DCOPY( N-P, C, 1, X, 1 )
      END IF
*
*     Compute the residual vector:
*
      IF( M.LT.N ) THEN
         NR = M + P - N
         IF( NR.GT.0 )
     $      CALL DGEMV( 'No transpose', NR, N-M, -ONE, A( N-P+1,
     $                  M+1 ),
     $                  LDA, D( NR+1 ), 1, ONE, C( N-P+1 ), 1 )
      ELSE
         NR = P
      END IF
      IF( NR.GT.0 ) THEN
         CALL DTRMV( 'Upper', 'No transpose', 'Non unit', NR,
     $               A( N-P+1, N-P+1 ), LDA, D, 1 )
         CALL DAXPY( NR, -ONE, D, 1, C( N-P+1 ), 1 )
      END IF
*
*     Backward transformation x = Q**T*x
*
      CALL DORMRQ( 'Left', 'Transpose', N, 1, P, B, LDB, WORK( 1 ),
     $             X,
     $             N, WORK( P+MN+1 ), LWORK-P-MN, INFO )
      WORK( 1 ) = P + MN + MAX( LOPT, INT( WORK( P+MN+1 ) ) )
*
      RETURN
*
*     End of DGGLSE
*
      END
*> \brief \b DGGQRF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGQRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggqrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggqrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggqrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK,
*                          LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LDB, LWORK, M, N, P
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
*      $                   WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGGQRF computes a generalized QR factorization of an N-by-M matrix A
*> and an N-by-P matrix B:
*>
*>             A = Q*R,        B = Q*T*Z,
*>
*> where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
*> matrix, and R and T assume one of the forms:
*>
*> if N >= M,  R = ( R11 ) M  ,   or if N < M,  R = ( R11  R12 ) N,
*>                 (  0  ) N-M                         N   M-N
*>                    M
*>
*> where R11 is upper triangular, and
*>
*> if N <= P,  T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
*>                  P-N  N                           ( T21 ) P
*>                                                      P
*>
*> where T12 or T21 is upper triangular.
*>
*> In particular, if B is square and nonsingular, the GQR factorization
*> of A and B implicitly gives the QR factorization of inv(B)*A:
*>
*>              inv(B)*A = Z**T*(inv(T)*R)
*>
*> where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
*> transpose of the matrix Z.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of rows of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of columns of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*>          P is INTEGER
*>          The number of columns of the matrix B.  P >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,M)
*>          On entry, the N-by-M matrix A.
*>          On exit, the elements on and above the diagonal of the array
*>          contain the min(N,M)-by-M upper trapezoidal matrix R (R is
*>          upper triangular if N >= M); the elements below the diagonal,
*>          with the array TAUA, represent the orthogonal matrix Q as a
*>          product of min(N,M) elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] TAUA
*> \verbatim
*>          TAUA is DOUBLE PRECISION array, dimension (min(N,M))
*>          The scalar factors of the elementary reflectors which
*>          represent the orthogonal matrix Q (see Further Details).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,P)
*>          On entry, the N-by-P matrix B.
*>          On exit, if N <= P, the upper triangle of the subarray
*>          B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
*>          if N > P, the elements on and above the (N-P)-th subdiagonal
*>          contain the N-by-P upper trapezoidal matrix T; the remaining
*>          elements, with the array TAUB, represent the orthogonal
*>          matrix Z as a product of elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] TAUB
*> \verbatim
*>          TAUB is DOUBLE PRECISION array, dimension (min(N,P))
*>          The scalar factors of the elementary reflectors which
*>          represent the orthogonal matrix Z (see Further Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= max(1,N,M,P).
*>          For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
*>          where NB1 is the optimal blocksize for the QR factorization
*>          of an N-by-M matrix, NB2 is the optimal blocksize for the
*>          RQ factorization of an N-by-P matrix, and NB3 is the optimal
*>          blocksize for a call of DORMQR.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggqrf
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix Q is represented as a product of elementary reflectors
*>
*>     Q = H(1) H(2) . . . H(k), where k = min(n,m).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - taua * v * v**T
*>
*>  where taua is a real scalar, and v is a real vector with
*>  v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
*>  and taua in TAUA(i).
*>  To form Q explicitly, use LAPACK subroutine DORGQR.
*>  To use Q to update another matrix, use LAPACK subroutine DORMQR.
*>
*>  The matrix Z is represented as a product of elementary reflectors
*>
*>     Z = H(1) H(2) . . . H(k), where k = min(n,p).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - taub * v * v**T
*>
*>  where taub is a real scalar, and v is a real vector with
*>  v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in
*>  B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
*>  To form Z explicitly, use LAPACK subroutine DORGRQ.
*>  To use Z to update another matrix, use LAPACK subroutine DORMRQ.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK,
     $                   LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, LWORK, M, N, P
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
     $                   WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            LOPT, LWKOPT, NB, NB1, NB2, NB3
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEQRF, DGERQF, DORMQR, XERBLA
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          INT, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INFO = 0
      NB1 = ILAENV( 1, 'DGEQRF', ' ', N, M, -1, -1 )
      NB2 = ILAENV( 1, 'DGERQF', ' ', N, P, -1, -1 )
      NB3 = ILAENV( 1, 'DORMQR', ' ', N, M, P, -1 )
      NB = MAX( NB1, NB2, NB3 )
      LWKOPT = MAX( 1, MAX( N, M, P )*NB )
      WORK( 1 ) = LWKOPT
      LQUERY = ( LWORK.EQ.-1 )
      IF( N.LT.0 ) THEN
         INFO = -1
      ELSE IF( M.LT.0 ) THEN
         INFO = -2
      ELSE IF( P.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -8
      ELSE IF( LWORK.LT.MAX( 1, N, M, P ) .AND. .NOT.LQUERY ) THEN
         INFO = -11
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGGQRF', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     QR factorization of N-by-M matrix A: A = Q*R
*
      CALL DGEQRF( N, M, A, LDA, TAUA, WORK, LWORK, INFO )
      LOPT = INT( WORK( 1 ) )
*
*     Update B := Q**T*B.
*
      CALL DORMQR( 'Left', 'Transpose', N, P, MIN( N, M ), A, LDA,
     $             TAUA,
     $             B, LDB, WORK, LWORK, INFO )
      LOPT = MAX( LOPT, INT( WORK( 1 ) ) )
*
*     RQ factorization of N-by-P matrix B: B = T*Z.
*
      CALL DGERQF( N, P, B, LDB, TAUB, WORK, LWORK, INFO )
*
      WORK( 1 ) = MAX( LOPT, INT( WORK( 1 ) ) )
*
      RETURN
*
*     End of DGGQRF
*
      END
*> \brief \b DGGRQF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGRQF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggrqf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggrqf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggrqf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,
*                          LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LDB, LWORK, M, N, P
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
*      $                   WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGGRQF computes a generalized RQ factorization of an M-by-N matrix A
*> and a P-by-N matrix B:
*>
*>             A = R*Q,        B = Z*T*Q,
*>
*> where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
*> matrix, and R and T assume one of the forms:
*>
*> if M <= N,  R = ( 0  R12 ) M,   or if M > N,  R = ( R11 ) M-N,
*>                  N-M  M                           ( R21 ) N
*>                                                      N
*>
*> where R12 or R21 is upper triangular, and
*>
*> if P >= N,  T = ( T11 ) N  ,   or if P < N,  T = ( T11  T12 ) P,
*>                 (  0  ) P-N                         P   N-P
*>                    N
*>
*> where T11 is upper triangular.
*>
*> In particular, if B is square and nonsingular, the GRQ factorization
*> of A and B implicitly gives the RQ factorization of A*inv(B):
*>
*>              A*inv(B) = (R*inv(T))*Z**T
*>
*> where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
*> transpose of the matrix Z.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*>          P is INTEGER
*>          The number of rows of the matrix B.  P >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit, if M <= N, the upper triangle of the subarray
*>          A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R;
*>          if M > N, the elements on and above the (M-N)-th subdiagonal
*>          contain the M-by-N upper trapezoidal matrix R; the remaining
*>          elements, with the array TAUA, represent the orthogonal
*>          matrix Q as a product of elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAUA
*> \verbatim
*>          TAUA is DOUBLE PRECISION array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors which
*>          represent the orthogonal matrix Q (see Further Details).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,N)
*>          On entry, the P-by-N matrix B.
*>          On exit, the elements on and above the diagonal of the array
*>          contain the min(P,N)-by-N upper trapezoidal matrix T (T is
*>          upper triangular if P >= N); the elements below the diagonal,
*>          with the array TAUB, represent the orthogonal matrix Z as a
*>          product of elementary reflectors (see Further Details).
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,P).
*> \endverbatim
*>
*> \param[out] TAUB
*> \verbatim
*>          TAUB is DOUBLE PRECISION array, dimension (min(P,N))
*>          The scalar factors of the elementary reflectors which
*>          represent the orthogonal matrix Z (see Further Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= max(1,N,M,P).
*>          For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
*>          where NB1 is the optimal blocksize for the RQ factorization
*>          of an M-by-N matrix, NB2 is the optimal blocksize for the
*>          QR factorization of a P-by-N matrix, and NB3 is the optimal
*>          blocksize for a call of DORMRQ.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INF0= -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggrqf
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix Q is represented as a product of elementary reflectors
*>
*>     Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - taua * v * v**T
*>
*>  where taua is a real scalar, and v is a real vector with
*>  v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
*>  A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
*>  To form Q explicitly, use LAPACK subroutine DORGRQ.
*>  To use Q to update another matrix, use LAPACK subroutine DORMRQ.
*>
*>  The matrix Z is represented as a product of elementary reflectors
*>
*>     Z = H(1) H(2) . . . H(k), where k = min(p,n).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - taub * v * v**T
*>
*>  where taub is a real scalar, and v is a real vector with
*>  v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
*>  and taub in TAUB(i).
*>  To form Z explicitly, use LAPACK subroutine DORGQR.
*>  To use Z to update another matrix, use LAPACK subroutine DORMQR.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,
     $                   LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, LWORK, M, N, P
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
     $                   WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            LOPT, LWKOPT, NB, NB1, NB2, NB3
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEQRF, DGERQF, DORMRQ, XERBLA
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          INT, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INFO = 0
      NB1 = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )
      NB2 = ILAENV( 1, 'DGEQRF', ' ', P, N, -1, -1 )
      NB3 = ILAENV( 1, 'DORMRQ', ' ', M, N, P, -1 )
      NB = MAX( NB1, NB2, NB3 )
      LWKOPT = MAX( 1, MAX( N, M, P )*NB )
      WORK( 1 ) = LWKOPT
      LQUERY = ( LWORK.EQ.-1 )
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( P.LT.0 ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
         INFO = -8
      ELSE IF( LWORK.LT.MAX( 1, M, P, N ) .AND. .NOT.LQUERY ) THEN
         INFO = -11
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGGRQF', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     RQ factorization of M-by-N matrix A: A = R*Q
*
      CALL DGERQF( M, N, A, LDA, TAUA, WORK, LWORK, INFO )
      LOPT = INT( WORK( 1 ) )
*
*     Update B := B*Q**T
*
      CALL DORMRQ( 'Right', 'Transpose', P, N, MIN( M, N ),
     $             A( MAX( 1, M-N+1 ), 1 ), LDA, TAUA, B, LDB, WORK,
     $             LWORK, INFO )
      LOPT = MAX( LOPT, INT( WORK( 1 ) ) )
*
*     QR factorization of P-by-N matrix B: B = Z*T
*
      CALL DGEQRF( P, N, B, LDB, TAUB, WORK, LWORK, INFO )
      WORK( 1 ) = MAX( LOPT, INT( WORK( 1 ) ) )
*
      RETURN
*
*     End of DGGRQF
*
      END
*> \brief <b> DGGSVD computes the singular value decomposition (SVD) for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGSVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggsvd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggsvd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggsvd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
*                          LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
*                          IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBQ, JOBU, JOBV
*       INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), ALPHA( * ), B( LDB, * ),
*      $                   BETA( * ), Q( LDQ, * ), U( LDU, * ),
*      $                   V( LDV, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DGGSVD3.
*>
*> DGGSVD computes the generalized singular value decomposition (GSVD)
*> of an M-by-N real matrix A and P-by-N real matrix B:
*>
*>       U**T*A*Q = D1*( 0 R ),    V**T*B*Q = D2*( 0 R )
*>
*> where U, V and Q are orthogonal matrices.
*> Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T,
*> then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and
*> D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the
*> following structures, respectively:
*>
*> If M-K-L >= 0,
*>
*>                     K  L
*>        D1 =     K ( I  0 )
*>                 L ( 0  C )
*>             M-K-L ( 0  0 )
*>
*>                   K  L
*>        D2 =   L ( 0  S )
*>             P-L ( 0  0 )
*>
*>                 N-K-L  K    L
*>   ( 0 R ) = K (  0   R11  R12 )
*>             L (  0    0   R22 )
*>
*> where
*>
*>   C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
*>   S = diag( BETA(K+1),  ... , BETA(K+L) ),
*>   C**2 + S**2 = I.
*>
*>   R is stored in A(1:K+L,N-K-L+1:N) on exit.
*>
*> If M-K-L < 0,
*>
*>                   K M-K K+L-M
*>        D1 =   K ( I  0    0   )
*>             M-K ( 0  C    0   )
*>
*>                     K M-K K+L-M
*>        D2 =   M-K ( 0  S    0  )
*>             K+L-M ( 0  0    I  )
*>               P-L ( 0  0    0  )
*>
*>                    N-K-L  K   M-K  K+L-M
*>   ( 0 R ) =     K ( 0    R11  R12  R13  )
*>               M-K ( 0     0   R22  R23  )
*>             K+L-M ( 0     0    0   R33  )
*>
*> where
*>
*>   C = diag( ALPHA(K+1), ... , ALPHA(M) ),
*>   S = diag( BETA(K+1),  ... , BETA(M) ),
*>   C**2 + S**2 = I.
*>
*>   (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
*>   ( 0  R22 R23 )
*>   in B(M-K+1:L,N+M-K-L+1:N) on exit.
*>
*> The routine computes C, S, R, and optionally the orthogonal
*> transformation matrices U, V and Q.
*>
*> In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
*> A and B implicitly gives the SVD of A*inv(B):
*>                      A*inv(B) = U*(D1*inv(D2))*V**T.
*> If ( A**T,B**T)**T  has orthonormal columns, then the GSVD of A and B is
*> also equal to the CS decomposition of A and B. Furthermore, the GSVD
*> can be used to derive the solution of the eigenvalue problem:
*>                      A**T*A x = lambda* B**T*B x.
*> In some literature, the GSVD of A and B is presented in the form
*>                  U**T*A*X = ( 0 D1 ),   V**T*B*X = ( 0 D2 )
*> where U and V are orthogonal and X is nonsingular, D1 and D2 are
*> ``diagonal''.  The former GSVD form can be converted to the latter
*> form by taking the nonsingular matrix X as
*>
*>                      X = Q*( I   0    )
*>                            ( 0 inv(R) ).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBU
*> \verbatim
*>          JOBU is CHARACTER*1
*>          = 'U':  Orthogonal matrix U is computed;
*>          = 'N':  U is not computed.
*> \endverbatim
*>
*> \param[in] JOBV
*> \verbatim
*>          JOBV is CHARACTER*1
*>          = 'V':  Orthogonal matrix V is computed;
*>          = 'N':  V is not computed.
*> \endverbatim
*>
*> \param[in] JOBQ
*> \verbatim
*>          JOBQ is CHARACTER*1
*>          = 'Q':  Orthogonal matrix Q is computed;
*>          = 'N':  Q is not computed.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrices A and B.  N >= 0.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*>          P is INTEGER
*>          The number of rows of the matrix B.  P >= 0.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*>          K is INTEGER
*> \endverbatim
*>
*> \param[out] L
*> \verbatim
*>          L is INTEGER
*>
*>          On exit, K and L specify the dimension of the subblocks
*>          described in Purpose.
*>          K + L = effective numerical rank of (A**T,B**T)**T.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit, A contains the triangular matrix R, or part of R.
*>          See Purpose for details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,N)
*>          On entry, the P-by-N matrix B.
*>          On exit, B contains the triangular matrix R if M-K-L < 0.
*>          See Purpose for details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,P).
*> \endverbatim
*>
*> \param[out] ALPHA
*> \verbatim
*>          ALPHA is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION array, dimension (N)
*>
*>          On exit, ALPHA and BETA contain the generalized singular
*>          value pairs of A and B;
*>            ALPHA(1:K) = 1,
*>            BETA(1:K)  = 0,
*>          and if M-K-L >= 0,
*>            ALPHA(K+1:K+L) = C,
*>            BETA(K+1:K+L)  = S,
*>          or if M-K-L < 0,
*>            ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
*>            BETA(K+1:M) =S, BETA(M+1:K+L) =1
*>          and
*>            ALPHA(K+L+1:N) = 0
*>            BETA(K+L+1:N)  = 0
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*>          U is DOUBLE PRECISION array, dimension (LDU,M)
*>          If JOBU = 'U', U contains the M-by-M orthogonal matrix U.
*>          If JOBU = 'N', U is not referenced.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>          The leading dimension of the array U. LDU >= max(1,M) if
*>          JOBU = 'U'; LDU >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension (LDV,P)
*>          If JOBV = 'V', V contains the P-by-P orthogonal matrix V.
*>          If JOBV = 'N', V is not referenced.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of the array V. LDV >= max(1,P) if
*>          JOBV = 'V'; LDV >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ,N)
*>          If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q.
*>          If JOBQ = 'N', Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q. LDQ >= max(1,N) if
*>          JOBQ = 'Q'; LDQ >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array,
*>                      dimension (max(3*N,M,P)+N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*>          On exit, IWORK stores the sorting information. More
*>          precisely, the following loop will sort ALPHA
*>             for I = K+1, min(M,K+L)
*>                 swap ALPHA(I) and ALPHA(IWORK(I))
*>             endfor
*>          such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = 1, the Jacobi-type procedure failed to
*>                converge.  For further details, see subroutine DTGSJA.
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  TOLA    DOUBLE PRECISION
*>  TOLB    DOUBLE PRECISION
*>          TOLA and TOLB are the thresholds to determine the effective
*>          rank of (A',B')**T. Generally, they are set to
*>                   TOLA = MAX(M,N)*norm(A)*MAZHEPS,
*>                   TOLB = MAX(P,N)*norm(B)*MAZHEPS.
*>          The size of TOLA and TOLB may affect the size of backward
*>          errors of the decomposition.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleOTHERsing
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Huan Ren, Computer Science Division, University of
*>     California at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE DGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
     $                   LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
     $                   IWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBQ, JOBU, JOBV
      INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), ALPHA( * ), B( LDB, * ),
     $                   BETA( * ), Q( LDQ, * ), U( LDU, * ),
     $                   V( LDV, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            WANTQ, WANTU, WANTV
      INTEGER            I, IBND, ISUB, J, NCYCLE
      DOUBLE PRECISION   ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           LSAME, DLAMCH, DLANGE
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGGSVP, DTGSJA, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      WANTU = LSAME( JOBU, 'U' )
      WANTV = LSAME( JOBV, 'V' )
      WANTQ = LSAME( JOBQ, 'Q' )
*
      INFO = 0
      IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
         INFO = -2
      ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
         INFO = -3
      ELSE IF( M.LT.0 ) THEN
         INFO = -4
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      ELSE IF( P.LT.0 ) THEN
         INFO = -6
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -10
      ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
         INFO = -12
      ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
         INFO = -16
      ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
         INFO = -18
      ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
         INFO = -20
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGGSVD', -INFO )
         RETURN
      END IF
*
*     Compute the Frobenius norm of matrices A and B
*
      ANORM = DLANGE( '1', M, N, A, LDA, WORK )
      BNORM = DLANGE( '1', P, N, B, LDB, WORK )
*
*     Get machine precision and set up threshold for determining
*     the effective numerical rank of the matrices A and B.
*
      ULP = DLAMCH( 'Precision' )
      UNFL = DLAMCH( 'Safe Minimum' )
      TOLA = MAX( M, N )*MAX( ANORM, UNFL )*ULP
      TOLB = MAX( P, N )*MAX( BNORM, UNFL )*ULP
*
*     Preprocessing
*
      CALL DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA,
     $             TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, WORK,
     $             WORK( N+1 ), INFO )
*
*     Compute the GSVD of two upper "triangular" matrices
*
      CALL DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB,
     $             TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ,
     $             WORK, NCYCLE, INFO )
*
*     Sort the singular values and store the pivot indices in IWORK
*     Copy ALPHA to WORK, then sort ALPHA in WORK
*
      CALL DCOPY( N, ALPHA, 1, WORK, 1 )
      IBND = MIN( L, M-K )
      DO 20 I = 1, IBND
*
*        Scan for largest ALPHA(K+I)
*
         ISUB = I
         SMAX = WORK( K+I )
         DO 10 J = I + 1, IBND
            TEMP = WORK( K+J )
            IF( TEMP.GT.SMAX ) THEN
               ISUB = J
               SMAX = TEMP
            END IF
   10    CONTINUE
         IF( ISUB.NE.I ) THEN
            WORK( K+ISUB ) = WORK( K+I )
            WORK( K+I ) = SMAX
            IWORK( K+I ) = K + ISUB
         ELSE
            IWORK( K+I ) = K + I
         END IF
   20 CONTINUE
*
      RETURN
*
*     End of DGGSVD
*
      END
*> \brief \b DGGSVP
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGSVP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggsvp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggsvp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggsvp.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
*                          TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
*                          IWORK, TAU, WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBQ, JOBU, JOBV
*       INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
*       DOUBLE PRECISION   TOLA, TOLB
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
*      $                   TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DGGSVP3.
*>
*> DGGSVP computes orthogonal matrices U, V and Q such that
*>
*>                    N-K-L  K    L
*>  U**T*A*Q =     K ( 0    A12  A13 )  if M-K-L >= 0;
*>                 L ( 0     0   A23 )
*>             M-K-L ( 0     0    0  )
*>
*>                  N-K-L  K    L
*>         =     K ( 0    A12  A13 )  if M-K-L < 0;
*>             M-K ( 0     0   A23 )
*>
*>                  N-K-L  K    L
*>  V**T*B*Q =   L ( 0     0   B13 )
*>             P-L ( 0     0    0  )
*>
*> where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
*> upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
*> otherwise A23 is (M-K)-by-L upper trapezoidal.  K+L = the effective
*> numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T.
*>
*> This decomposition is the preprocessing step for computing the
*> Generalized Singular Value Decomposition (GSVD), see subroutine
*> DGGSVD.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBU
*> \verbatim
*>          JOBU is CHARACTER*1
*>          = 'U':  Orthogonal matrix U is computed;
*>          = 'N':  U is not computed.
*> \endverbatim
*>
*> \param[in] JOBV
*> \verbatim
*>          JOBV is CHARACTER*1
*>          = 'V':  Orthogonal matrix V is computed;
*>          = 'N':  V is not computed.
*> \endverbatim
*>
*> \param[in] JOBQ
*> \verbatim
*>          JOBQ is CHARACTER*1
*>          = 'Q':  Orthogonal matrix Q is computed;
*>          = 'N':  Q is not computed.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*>          P is INTEGER
*>          The number of rows of the matrix B.  P >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrices A and B.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit, A contains the triangular (or trapezoidal) matrix
*>          described in the Purpose section.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,N)
*>          On entry, the P-by-N matrix B.
*>          On exit, B contains the triangular matrix described in
*>          the Purpose section.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,P).
*> \endverbatim
*>
*> \param[in] TOLA
*> \verbatim
*>          TOLA is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] TOLB
*> \verbatim
*>          TOLB is DOUBLE PRECISION
*>
*>          TOLA and TOLB are the thresholds to determine the effective
*>          numerical rank of matrix B and a subblock of A. Generally,
*>          they are set to
*>             TOLA = MAX(M,N)*norm(A)*MACHEPS,
*>             TOLB = MAX(P,N)*norm(B)*MACHEPS.
*>          The size of TOLA and TOLB may affect the size of backward
*>          errors of the decomposition.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*>          K is INTEGER
*> \endverbatim
*>
*> \param[out] L
*> \verbatim
*>          L is INTEGER
*>
*>          On exit, K and L specify the dimension of the subblocks
*>          described in Purpose section.
*>          K + L = effective numerical rank of (A**T,B**T)**T.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*>          U is DOUBLE PRECISION array, dimension (LDU,M)
*>          If JOBU = 'U', U contains the orthogonal matrix U.
*>          If JOBU = 'N', U is not referenced.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>          The leading dimension of the array U. LDU >= max(1,M) if
*>          JOBU = 'U'; LDU >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension (LDV,P)
*>          If JOBV = 'V', V contains the orthogonal matrix V.
*>          If JOBV = 'N', V is not referenced.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of the array V. LDV >= max(1,P) if
*>          JOBV = 'V'; LDV >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ,N)
*>          If JOBQ = 'Q', Q contains the orthogonal matrix Q.
*>          If JOBQ = 'N', Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q. LDQ >= max(1,N) if
*>          JOBQ = 'Q'; LDQ >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (max(3*N,M,P))
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleOTHERcomputational
*
*> \par Further Details:
*  =====================
*>
*>  The subroutine uses LAPACK subroutine DGEQPF for the QR factorization
*>  with column pivoting to detect the effective numerical rank of the
*>  a matrix. It may be replaced by a better rank determination strategy.
*>
*  =====================================================================
      SUBROUTINE DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
     $                   TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
     $                   IWORK, TAU, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBQ, JOBU, JOBV
      INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
      DOUBLE PRECISION   TOLA, TOLB
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
     $                   TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            FORWRD, WANTQ, WANTU, WANTV
      INTEGER            I, J
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEQPF, DGEQR2, DGERQ2, DLACPY, DLAPMT, DLASET,
     $                   DORG2R, DORM2R, DORMR2, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      WANTU = LSAME( JOBU, 'U' )
      WANTV = LSAME( JOBV, 'V' )
      WANTQ = LSAME( JOBQ, 'Q' )
      FORWRD = .TRUE.
*
      INFO = 0
      IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
         INFO = -2
      ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
         INFO = -3
      ELSE IF( M.LT.0 ) THEN
         INFO = -4
      ELSE IF( P.LT.0 ) THEN
         INFO = -5
      ELSE IF( N.LT.0 ) THEN
         INFO = -6
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -8
      ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
         INFO = -10
      ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
         INFO = -16
      ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
         INFO = -18
      ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
         INFO = -20
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGGSVP', -INFO )
         RETURN
      END IF
*
*     QR with column pivoting of B: B*P = V*( S11 S12 )
*                                           (  0   0  )
*
      DO 10 I = 1, N
         IWORK( I ) = 0
   10 CONTINUE
      CALL DGEQPF( P, N, B, LDB, IWORK, TAU, WORK, INFO )
*
*     Update A := A*P
*
      CALL DLAPMT( FORWRD, M, N, A, LDA, IWORK )
*
*     Determine the effective rank of matrix B.
*
      L = 0
      DO 20 I = 1, MIN( P, N )
         IF( ABS( B( I, I ) ).GT.TOLB )
     $      L = L + 1
   20 CONTINUE
*
      IF( WANTV ) THEN
*
*        Copy the details of V, and form V.
*
         CALL DLASET( 'Full', P, P, ZERO, ZERO, V, LDV )
         IF( P.GT.1 )
     $      CALL DLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ),
     $                   LDV )
         CALL DORG2R( P, P, MIN( P, N ), V, LDV, TAU, WORK, INFO )
      END IF
*
*     Clean up B
*
      DO 40 J = 1, L - 1
         DO 30 I = J + 1, L
            B( I, J ) = ZERO
   30    CONTINUE
   40 CONTINUE
      IF( P.GT.L )
     $   CALL DLASET( 'Full', P-L, N, ZERO, ZERO, B( L+1, 1 ), LDB )
*
      IF( WANTQ ) THEN
*
*        Set Q = I and Update Q := Q*P
*
         CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
         CALL DLAPMT( FORWRD, N, N, Q, LDQ, IWORK )
      END IF
*
      IF( P.GE.L .AND. N.NE.L ) THEN
*
*        RQ factorization of (S11 S12): ( S11 S12 ) = ( 0 S12 )*Z
*
         CALL DGERQ2( L, N, B, LDB, TAU, WORK, INFO )
*
*        Update A := A*Z**T
*
         CALL DORMR2( 'Right', 'Transpose', M, N, L, B, LDB, TAU, A,
     $                LDA, WORK, INFO )
*
         IF( WANTQ ) THEN
*
*           Update Q := Q*Z**T
*
            CALL DORMR2( 'Right', 'Transpose', N, N, L, B, LDB, TAU,
     $                   Q, LDQ, WORK, INFO )
         END IF
*
*        Clean up B
*
         CALL DLASET( 'Full', L, N-L, ZERO, ZERO, B, LDB )
         DO 60 J = N - L + 1, N
            DO 50 I = J - N + L + 1, L
               B( I, J ) = ZERO
   50       CONTINUE
   60    CONTINUE
*
      END IF
*
*     Let              N-L     L
*                A = ( A11    A12 ) M,
*
*     then the following does the complete QR decomposition of A11:
*
*              A11 = U*(  0  T12 )*P1**T
*                      (  0   0  )
*
      DO 70 I = 1, N - L
         IWORK( I ) = 0
   70 CONTINUE
      CALL DGEQPF( M, N-L, A, LDA, IWORK, TAU, WORK, INFO )
*
*     Determine the effective rank of A11
*
      K = 0
      DO 80 I = 1, MIN( M, N-L )
         IF( ABS( A( I, I ) ).GT.TOLA )
     $      K = K + 1
   80 CONTINUE
*
*     Update A12 := U**T*A12, where A12 = A( 1:M, N-L+1:N )
*
      CALL DORM2R( 'Left', 'Transpose', M, L, MIN( M, N-L ), A, LDA,
     $             TAU, A( 1, N-L+1 ), LDA, WORK, INFO )
*
      IF( WANTU ) THEN
*
*        Copy the details of U, and form U
*
         CALL DLASET( 'Full', M, M, ZERO, ZERO, U, LDU )
         IF( M.GT.1 )
     $      CALL DLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA,
     $                   U( 2, 1 ), LDU )
         CALL DORG2R( M, M, MIN( M, N-L ), U, LDU, TAU, WORK,
     $                INFO )
      END IF
*
      IF( WANTQ ) THEN
*
*        Update Q( 1:N, 1:N-L )  = Q( 1:N, 1:N-L )*P1
*
         CALL DLAPMT( FORWRD, N, N-L, Q, LDQ, IWORK )
      END IF
*
*     Clean up A: set the strictly lower triangular part of
*     A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
*
      DO 100 J = 1, K - 1
         DO 90 I = J + 1, K
            A( I, J ) = ZERO
   90    CONTINUE
  100 CONTINUE
      IF( M.GT.K )
     $   CALL DLASET( 'Full', M-K, N-L, ZERO, ZERO, A( K+1, 1 ),
     $                LDA )
*
      IF( N-L.GT.K ) THEN
*
*        RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
*
         CALL DGERQ2( K, N-L, A, LDA, TAU, WORK, INFO )
*
         IF( WANTQ ) THEN
*
*           Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**T
*
            CALL DORMR2( 'Right', 'Transpose', N, N-L, K, A, LDA,
     $                   TAU, Q, LDQ, WORK, INFO )
         END IF
*
*        Clean up A
*
         CALL DLASET( 'Full', K, N-L-K, ZERO, ZERO, A, LDA )
         DO 120 J = N - L - K + 1, N - L
            DO 110 I = J - N + L + K + 1, K
               A( I, J ) = ZERO
  110       CONTINUE
  120    CONTINUE
*
      END IF
*
      IF( M.GT.K ) THEN
*
*        QR factorization of A( K+1:M,N-L+1:N )
*
         CALL DGEQR2( M-K, L, A( K+1, N-L+1 ), LDA, TAU, WORK, INFO )
*
         IF( WANTU ) THEN
*
*           Update U(:,K+1:M) := U(:,K+1:M)*U1
*
            CALL DORM2R( 'Right', 'No transpose', M, M-K,
     $                   MIN( M-K, L ), A( K+1, N-L+1 ), LDA, TAU,
     $                   U( 1, K+1 ), LDU, WORK, INFO )
         END IF
*
*        Clean up
*
         DO 140 J = N - L + 1, N
            DO 130 I = J - N + K + L + 1, M
               A( I, J ) = ZERO
  130       CONTINUE
  140    CONTINUE
*
      END IF
*
      RETURN
*
*     End of DGGSVP
*
      END
*> \brief \b DGSVJ0 pre-processor for the routine dgesvj.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGSVJ0 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgsvj0.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgsvj0.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgsvj0.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGSVJ0( JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS,
*                          SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LDV, LWORK, M, MV, N, NSWEEP
*       DOUBLE PRECISION   EPS, SFMIN, TOL
*       CHARACTER*1        JOBV
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), SVA( N ), D( N ), V( LDV, * ),
*      $                   WORK( LWORK )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGSVJ0 is called from DGESVJ as a pre-processor and that is its main
*> purpose. It applies Jacobi rotations in the same way as DGESVJ does, but
*> it does not check convergence (stopping criterion). Few tuning
*> parameters (marked by [TP]) are available for the implementer.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBV
*> \verbatim
*>          JOBV is CHARACTER*1
*>          Specifies whether the output from this procedure is used
*>          to compute the matrix V:
*>          = 'V': the product of the Jacobi rotations is accumulated
*>                 by postmultiplying the N-by-N array V.
*>                (See the description of V.)
*>          = 'A': the product of the Jacobi rotations is accumulated
*>                 by postmultiplying the MV-by-N array V.
*>                (See the descriptions of MV and V.)
*>          = 'N': the Jacobi rotations are not accumulated.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the input matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the input matrix A.
*>          M >= N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, M-by-N matrix A, such that A*diag(D) represents
*>          the input matrix.
*>          On exit,
*>          A_onexit * D_onexit represents the input matrix A*diag(D)
*>          post-multiplied by a sequence of Jacobi rotations, where the
*>          rotation threshold and the total number of sweeps are given in
*>          TOL and NSWEEP, respectively.
*>          (See the descriptions of D, TOL and NSWEEP.)
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The array D accumulates the scaling factors from the fast scaled
*>          Jacobi rotations.
*>          On entry, A*diag(D) represents the input matrix.
*>          On exit, A_onexit*diag(D_onexit) represents the input matrix
*>          post-multiplied by a sequence of Jacobi rotations, where the
*>          rotation threshold and the total number of sweeps are given in
*>          TOL and NSWEEP, respectively.
*>          (See the descriptions of A, TOL and NSWEEP.)
*> \endverbatim
*>
*> \param[in,out] SVA
*> \verbatim
*>          SVA is DOUBLE PRECISION array, dimension (N)
*>          On entry, SVA contains the Euclidean norms of the columns of
*>          the matrix A*diag(D).
*>          On exit, SVA contains the Euclidean norms of the columns of
*>          the matrix onexit*diag(D_onexit).
*> \endverbatim
*>
*> \param[in] MV
*> \verbatim
*>          MV is INTEGER
*>          If JOBV = 'A', then MV rows of V are post-multiplied by a
*>                           sequence of Jacobi rotations.
*>          If JOBV = 'N',   then MV is not referenced.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension (LDV,N)
*>          If JOBV = 'V' then N rows of V are post-multiplied by a
*>                           sequence of Jacobi rotations.
*>          If JOBV = 'A' then MV rows of V are post-multiplied by a
*>                           sequence of Jacobi rotations.
*>          If JOBV = 'N',   then V is not referenced.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of the array V,  LDV >= 1.
*>          If JOBV = 'V', LDV >= N.
*>          If JOBV = 'A', LDV >= MV.
*> \endverbatim
*>
*> \param[in] EPS
*> \verbatim
*>          EPS is DOUBLE PRECISION
*>          EPS = DLAMCH('Epsilon')
*> \endverbatim
*>
*> \param[in] SFMIN
*> \verbatim
*>          SFMIN is DOUBLE PRECISION
*>          SFMIN = DLAMCH('Safe Minimum')
*> \endverbatim
*>
*> \param[in] TOL
*> \verbatim
*>          TOL is DOUBLE PRECISION
*>          TOL is the threshold for Jacobi rotations. For a pair
*>          A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
*>          applied only if DABS(COS(angle(A(:,p),A(:,q)))) > TOL.
*> \endverbatim
*>
*> \param[in] NSWEEP
*> \verbatim
*>          NSWEEP is INTEGER
*>          NSWEEP is the number of sweeps of Jacobi rotations to be
*>          performed.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          LWORK is the dimension of WORK. LWORK >= M.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, then the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gsvj0
*
*> \par Further Details:
*  =====================
*>
*> DGSVJ0 is used just to enable DGESVJ to call a simplified version of
*> itself to work on a submatrix of the original matrix.
*>
*> \par Contributors:
*  ==================
*>
*> Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
*>
*> \par Bugs, Examples and Comments:
*  =================================
*>
*> Please report all bugs and send interesting test examples and comments to
*> drmac@math.hr. Thank you.
*
*  =====================================================================
      SUBROUTINE DGSVJ0( JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS,
     $                   SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDV, LWORK, M, MV, N, NSWEEP
      DOUBLE PRECISION   EPS, SFMIN, TOL
      CHARACTER*1        JOBV
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), SVA( N ), D( N ), V( LDV, * ),
     $                   WORK( LWORK )
*     ..
*
*  =====================================================================
*
*     .. Local Parameters ..
      DOUBLE PRECISION   ZERO, HALF, ONE
      PARAMETER          ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0)
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG,
     $                   BIGTHETA, CS, MXAAPQ, MXSINJ, ROOTBIG, ROOTEPS,
     $                   ROOTSFMIN, ROOTTOL, SMALL, SN, T, TEMP1, THETA,
     $                   THSIGN
      INTEGER            BLSKIP, EMPTSW, i, ibr, IERR, igl, IJBLSK, ir1,
     $                   ISWROT, jbc, jgl, KBL, LKAHEAD, MVL, NBL,
     $                   NOTROT, p, PSKIPPED, q, ROWSKIP, SWBAND
      LOGICAL            APPLV, ROTOK, RSVEC
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   FASTR( 5 )
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DABS, MAX, DBLE, MIN, DSIGN, DSQRT
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DDOT, DNRM2
      INTEGER            IDAMAX
      LOGICAL            LSAME
      EXTERNAL           IDAMAX, LSAME, DDOT, DNRM2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DLASCL, DLASSQ, DROTM,
     $                   DSWAP,
     $                   XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      APPLV = LSAME( JOBV, 'A' )
      RSVEC = LSAME( JOBV, 'V' )
      IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( M.LT.0 ) THEN
         INFO = -2
      ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN
         INFO = -3
      ELSE IF( LDA.LT.M ) THEN
         INFO = -5
      ELSE IF( ( RSVEC.OR.APPLV ) .AND. ( MV.LT.0 ) ) THEN
         INFO = -8
      ELSE IF( ( RSVEC.AND.( LDV.LT.N ) ).OR.
     $         ( APPLV.AND.( LDV.LT.MV ) ) ) THEN
         INFO = -10
      ELSE IF( TOL.LE.EPS ) THEN
         INFO = -13
      ELSE IF( NSWEEP.LT.0 ) THEN
         INFO = -14
      ELSE IF( LWORK.LT.M ) THEN
         INFO = -16
      ELSE
         INFO = 0
      END IF
*
*     #:(
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGSVJ0', -INFO )
         RETURN
      END IF
*
      IF( RSVEC ) THEN
         MVL = N
      ELSE IF( APPLV ) THEN
         MVL = MV
      END IF
      RSVEC = RSVEC .OR. APPLV

      ROOTEPS = DSQRT( EPS )
      ROOTSFMIN = DSQRT( SFMIN )
      SMALL = SFMIN / EPS
      BIG = ONE / SFMIN
      ROOTBIG = ONE / ROOTSFMIN
      BIGTHETA = ONE / ROOTEPS
      ROOTTOL = DSQRT( TOL )
*
*     -#- Row-cyclic Jacobi SVD algorithm with column pivoting -#-
*
      EMPTSW = ( N*( N-1 ) ) / 2
      NOTROT = 0
      FASTR( 1 ) = ZERO
*
*     -#- Row-cyclic pivot strategy with de Rijk's pivoting -#-
*

      SWBAND = 0
*[TP] SWBAND is a tuning parameter. It is meaningful and effective
*     if SGESVJ is used as a computational routine in the preconditioned
*     Jacobi SVD algorithm SGESVJ. For sweeps i=1:SWBAND the procedure
*     ......

      KBL = MIN( 8, N )
*[TP] KBL is a tuning parameter that defines the tile size in the
*     tiling of the p-q loops of pivot pairs. In general, an optimal
*     value of KBL depends on the matrix dimensions and on the
*     parameters of the computer's memory.
*
      NBL = N / KBL
      IF( ( NBL*KBL ).NE.N )NBL = NBL + 1

      BLSKIP = ( KBL**2 ) + 1
*[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.

      ROWSKIP = MIN( 5, KBL )
*[TP] ROWSKIP is a tuning parameter.

      LKAHEAD = 1
*[TP] LKAHEAD is a tuning parameter.
      SWBAND = 0
      PSKIPPED = 0
*
      DO 1993 i = 1, NSWEEP
*     .. go go go ...
*
         MXAAPQ = ZERO
         MXSINJ = ZERO
         ISWROT = 0
*
         NOTROT = 0
         PSKIPPED = 0
*
         DO 2000 ibr = 1, NBL

            igl = ( ibr-1 )*KBL + 1
*
            DO 1002 ir1 = 0, MIN( LKAHEAD, NBL-ibr )
*
               igl = igl + ir1*KBL
*
               DO 2001 p = igl, MIN( igl+KBL-1, N-1 )

*     .. de Rijk's pivoting
                  q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
                  IF( p.NE.q ) THEN
                     CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
                     IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1,
     $                                      V( 1, q ), 1 )
                     TEMP1 = SVA( p )
                     SVA( p ) = SVA( q )
                     SVA( q ) = TEMP1
                     TEMP1 = D( p )
                     D( p ) = D( q )
                     D( q ) = TEMP1
                  END IF
*
                  IF( ir1.EQ.0 ) THEN
*
*        Column norms are periodically updated by explicit
*        norm computation.
*        Caveat:
*        Some BLAS implementations compute DNRM2(M,A(1,p),1)
*        as DSQRT(DDOT(M,A(1,p),1,A(1,p),1)), which may result in
*        overflow for ||A(:,p)||_2 > DSQRT(overflow_threshold), and
*        underflow for ||A(:,p)||_2 < DSQRT(underflow_threshold).
*        Hence, DNRM2 cannot be trusted, not even in the case when
*        the true norm is far from the under(over)flow boundaries.
*        If properly implemented DNRM2 is available, the IF-THEN-ELSE
*        below should read "AAPP = DNRM2( M, A(1,p), 1 ) * D(p)".
*
                     IF( ( SVA( p ).LT.ROOTBIG ) .AND.
     $                   ( SVA( p ).GT.ROOTSFMIN ) ) THEN
                        SVA( p ) = DNRM2( M, A( 1, p ), 1 )*D( p )
                     ELSE
                        TEMP1 = ZERO
                        AAPP = ONE
                        CALL DLASSQ( M, A( 1, p ), 1, TEMP1, AAPP )
                        SVA( p ) = TEMP1*DSQRT( AAPP )*D( p )
                     END IF
                     AAPP = SVA( p )
                  ELSE
                     AAPP = SVA( p )
                  END IF

*
                  IF( AAPP.GT.ZERO ) THEN
*
                     PSKIPPED = 0
*
                     DO 2002 q = p + 1, MIN( igl+KBL-1, N )
*
                        AAQQ = SVA( q )

                        IF( AAQQ.GT.ZERO ) THEN
*
                           AAPP0 = AAPP
                           IF( AAQQ.GE.ONE ) THEN
                              ROTOK = ( SMALL*AAPP ).LE.AAQQ
                              IF( AAPP.LT.( BIG / AAQQ ) ) THEN
                                 AAPQ = ( DDOT( M, A( 1, p ), 1,
     $                                    A( 1,
     $                                  q ), 1 )*D( p )*D( q ) / AAQQ )
     $                                  / AAPP
                              ELSE
                                 CALL DCOPY( M, A( 1, p ), 1, WORK,
     $                                       1 )
                                 CALL DLASCL( 'G', 0, 0, AAPP,
     $                                        D( p ),
     $                                        M, 1, WORK, LDA, IERR )
                                 AAPQ = DDOT( M, WORK, 1, A( 1, q ),
     $                                  1 )*D( q ) / AAQQ
                              END IF
                           ELSE
                              ROTOK = AAPP.LE.( AAQQ / SMALL )
                              IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
                                 AAPQ = ( DDOT( M, A( 1, p ), 1,
     $                                    A( 1,
     $                                  q ), 1 )*D( p )*D( q ) / AAQQ )
     $                                  / AAPP
                              ELSE
                                 CALL DCOPY( M, A( 1, q ), 1, WORK,
     $                                       1 )
                                 CALL DLASCL( 'G', 0, 0, AAQQ,
     $                                        D( q ),
     $                                        M, 1, WORK, LDA, IERR )
                                 AAPQ = DDOT( M, WORK, 1, A( 1, p ),
     $                                  1 )*D( p ) / AAPP
                              END IF
                           END IF
*
                           MXAAPQ = MAX( MXAAPQ, DABS( AAPQ ) )
*
*        TO rotate or NOT to rotate, THAT is the question ...
*
                           IF( DABS( AAPQ ).GT.TOL ) THEN
*
*           .. rotate
*           ROTATED = ROTATED + ONE
*
                              IF( ir1.EQ.0 ) THEN
                                 NOTROT = 0
                                 PSKIPPED = 0
                                 ISWROT = ISWROT + 1
                              END IF
*
                              IF( ROTOK ) THEN
*
                                 AQOAP = AAQQ / AAPP
                                 APOAQ = AAPP / AAQQ
                                 THETA = -HALF*DABS( AQOAP-APOAQ )/AAPQ
*
                                 IF( DABS( THETA ).GT.BIGTHETA ) THEN
*
                                    T = HALF / THETA
                                    FASTR( 3 ) = T*D( p ) / D( q )
                                    FASTR( 4 ) = -T*D( q ) / D( p )
                                    CALL DROTM( M, A( 1, p ), 1,
     $                                          A( 1, q ), 1, FASTR )
                                    IF( RSVEC )CALL DROTM( MVL,
     $                                              V( 1, p ), 1,
     $                                              V( 1, q ), 1,
     $                                              FASTR )
                                    SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
     $                                         ONE+T*APOAQ*AAPQ ) )
                                    AAPP = AAPP*DSQRT( MAX( ZERO,
     $                                     ONE-T*AQOAP*AAPQ ) )
                                    MXSINJ = MAX( MXSINJ, DABS( T ) )
*
                                 ELSE
*
*                 .. choose correct signum for THETA and rotate
*
                                    THSIGN = -DSIGN( ONE, AAPQ )
                                    T = ONE / ( THETA+THSIGN*
     $                                  DSQRT( ONE+THETA*THETA ) )
                                    CS = DSQRT( ONE / ( ONE+T*T ) )
                                    SN = T*CS
*
                                    MXSINJ = MAX( MXSINJ, DABS( SN ) )
                                    SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
     $                                         ONE+T*APOAQ*AAPQ ) )
                                    AAPP = AAPP*DSQRT( MAX( ZERO,
     $                                     ONE-T*AQOAP*AAPQ ) )
*
                                    APOAQ = D( p ) / D( q )
                                    AQOAP = D( q ) / D( p )
                                    IF( D( p ).GE.ONE ) THEN
                                       IF( D( q ).GE.ONE ) THEN
                                          FASTR( 3 ) = T*APOAQ
                                          FASTR( 4 ) = -T*AQOAP
                                          D( p ) = D( p )*CS
                                          D( q ) = D( q )*CS
                                          CALL DROTM( M, A( 1, p ),
     $                                                1,
     $                                                A( 1, q ), 1,
     $                                                FASTR )
                                          IF( RSVEC )CALL DROTM( MVL,
     $                                        V( 1, p ), 1, V( 1, q ),
     $                                        1, FASTR )
                                       ELSE
                                          CALL DAXPY( M, -T*AQOAP,
     $                                                A( 1, q ), 1,
     $                                                A( 1, p ), 1 )
                                          CALL DAXPY( M, CS*SN*APOAQ,
     $                                                A( 1, p ), 1,
     $                                                A( 1, q ), 1 )
                                          D( p ) = D( p )*CS
                                          D( q ) = D( q ) / CS
                                          IF( RSVEC ) THEN
                                             CALL DAXPY( MVL,
     $                                                   -T*AQOAP,
     $                                                   V( 1, q ), 1,
     $                                                   V( 1, p ), 1 )
                                             CALL DAXPY( MVL,
     $                                                   CS*SN*APOAQ,
     $                                                   V( 1, p ), 1,
     $                                                   V( 1, q ), 1 )
                                          END IF
                                       END IF
                                    ELSE
                                       IF( D( q ).GE.ONE ) THEN
                                          CALL DAXPY( M, T*APOAQ,
     $                                                A( 1, p ), 1,
     $                                                A( 1, q ), 1 )
                                          CALL DAXPY( M,
     $                                                -CS*SN*AQOAP,
     $                                                A( 1, q ), 1,
     $                                                A( 1, p ), 1 )
                                          D( p ) = D( p ) / CS
                                          D( q ) = D( q )*CS
                                          IF( RSVEC ) THEN
                                             CALL DAXPY( MVL,
     $                                                   T*APOAQ,
     $                                                   V( 1, p ), 1,
     $                                                   V( 1, q ), 1 )
                                             CALL DAXPY( MVL,
     $                                                   -CS*SN*AQOAP,
     $                                                   V( 1, q ), 1,
     $                                                   V( 1, p ), 1 )
                                          END IF
                                       ELSE
                                          IF( D( p ).GE.D( q ) ) THEN
                                             CALL DAXPY( M, -T*AQOAP,
     $                                                   A( 1, q ), 1,
     $                                                   A( 1, p ), 1 )
                                             CALL DAXPY( M,
     $                                                   CS*SN*APOAQ,
     $                                                   A( 1, p ), 1,
     $                                                   A( 1, q ), 1 )
                                             D( p ) = D( p )*CS
                                             D( q ) = D( q ) / CS
                                             IF( RSVEC ) THEN
                                                CALL DAXPY( MVL,
     $                                               -T*AQOAP,
     $                                               V( 1, q ), 1,
     $                                               V( 1, p ), 1 )
                                                CALL DAXPY( MVL,
     $                                               CS*SN*APOAQ,
     $                                               V( 1, p ), 1,
     $                                               V( 1, q ), 1 )
                                             END IF
                                          ELSE
                                             CALL DAXPY( M, T*APOAQ,
     $                                                   A( 1, p ), 1,
     $                                                   A( 1, q ), 1 )
                                             CALL DAXPY( M,
     $                                                   -CS*SN*AQOAP,
     $                                                   A( 1, q ), 1,
     $                                                   A( 1, p ), 1 )
                                             D( p ) = D( p ) / CS
                                             D( q ) = D( q )*CS
                                             IF( RSVEC ) THEN
                                                CALL DAXPY( MVL,
     $                                               T*APOAQ, V( 1, p ),
     $                                               1, V( 1, q ), 1 )
                                                CALL DAXPY( MVL,
     $                                               -CS*SN*AQOAP,
     $                                               V( 1, q ), 1,
     $                                               V( 1, p ), 1 )
                                             END IF
                                          END IF
                                       END IF
                                    END IF
                                 END IF
*
                              ELSE
*              .. have to use modified Gram-Schmidt like transformation
                                 CALL DCOPY( M, A( 1, p ), 1, WORK,
     $                                       1 )
                                 CALL DLASCL( 'G', 0, 0, AAPP, ONE,
     $                                        M,
     $                                        1, WORK, LDA, IERR )
                                 CALL DLASCL( 'G', 0, 0, AAQQ, ONE,
     $                                        M,
     $                                        1, A( 1, q ), LDA, IERR )
                                 TEMP1 = -AAPQ*D( p ) / D( q )
                                 CALL DAXPY( M, TEMP1, WORK, 1,
     $                                       A( 1, q ), 1 )
                                 CALL DLASCL( 'G', 0, 0, ONE, AAQQ,
     $                                        M,
     $                                        1, A( 1, q ), LDA, IERR )
                                 SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
     $                                      ONE-AAPQ*AAPQ ) )
                                 MXSINJ = MAX( MXSINJ, SFMIN )
                              END IF
*           END IF ROTOK THEN ... ELSE
*
*           In the case of cancellation in updating SVA(q), SVA(p)
*           recompute SVA(q), SVA(p).
                              IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
     $                            THEN
                                 IF( ( AAQQ.LT.ROOTBIG ) .AND.
     $                               ( AAQQ.GT.ROOTSFMIN ) ) THEN
                                    SVA( q ) = DNRM2( M, A( 1, q ),
     $                                   1 )*
     $                                         D( q )
                                 ELSE
                                    T = ZERO
                                    AAQQ = ONE
                                    CALL DLASSQ( M, A( 1, q ), 1, T,
     $                                           AAQQ )
                                    SVA( q ) = T*DSQRT( AAQQ )*D( q )
                                 END IF
                              END IF
                              IF( ( AAPP / AAPP0 ).LE.ROOTEPS ) THEN
                                 IF( ( AAPP.LT.ROOTBIG ) .AND.
     $                               ( AAPP.GT.ROOTSFMIN ) ) THEN
                                    AAPP = DNRM2( M, A( 1, p ), 1 )*
     $                                     D( p )
                                 ELSE
                                    T = ZERO
                                    AAPP = ONE
                                    CALL DLASSQ( M, A( 1, p ), 1, T,
     $                                           AAPP )
                                    AAPP = T*DSQRT( AAPP )*D( p )
                                 END IF
                                 SVA( p ) = AAPP
                              END IF
*
                           ELSE
*        A(:,p) and A(:,q) already numerically orthogonal
                              IF( ir1.EQ.0 )NOTROT = NOTROT + 1
                              PSKIPPED = PSKIPPED + 1
                           END IF
                        ELSE
*        A(:,q) is zero column
                           IF( ir1.EQ.0 )NOTROT = NOTROT + 1
                           PSKIPPED = PSKIPPED + 1
                        END IF
*
                        IF( ( i.LE.SWBAND ) .AND.
     $                      ( PSKIPPED.GT.ROWSKIP ) ) THEN
                           IF( ir1.EQ.0 )AAPP = -AAPP
                           NOTROT = 0
                           GO TO 2103
                        END IF
*
 2002                CONTINUE
*     END q-LOOP
*
 2103                CONTINUE
*     bailed out of q-loop

                     SVA( p ) = AAPP

                  ELSE
                     SVA( p ) = AAPP
                     IF( ( ir1.EQ.0 ) .AND. ( AAPP.EQ.ZERO ) )
     $                   NOTROT = NOTROT + MIN( igl+KBL-1, N ) - p
                  END IF
*
 2001          CONTINUE
*     end of the p-loop
*     end of doing the block ( ibr, ibr )
 1002       CONTINUE
*     end of ir1-loop
*
*........................................................
* ... go to the off diagonal blocks
*
            igl = ( ibr-1 )*KBL + 1
*
            DO 2010 jbc = ibr + 1, NBL
*
               jgl = ( jbc-1 )*KBL + 1
*
*        doing the block at ( ibr, jbc )
*
               IJBLSK = 0
               DO 2100 p = igl, MIN( igl+KBL-1, N )
*
                  AAPP = SVA( p )
*
                  IF( AAPP.GT.ZERO ) THEN
*
                     PSKIPPED = 0
*
                     DO 2200 q = jgl, MIN( jgl+KBL-1, N )
*
                        AAQQ = SVA( q )
*
                        IF( AAQQ.GT.ZERO ) THEN
                           AAPP0 = AAPP
*
*     -#- M x 2 Jacobi SVD -#-
*
*        -#- Safe Gram matrix computation -#-
*
                           IF( AAQQ.GE.ONE ) THEN
                              IF( AAPP.GE.AAQQ ) THEN
                                 ROTOK = ( SMALL*AAPP ).LE.AAQQ
                              ELSE
                                 ROTOK = ( SMALL*AAQQ ).LE.AAPP
                              END IF
                              IF( AAPP.LT.( BIG / AAQQ ) ) THEN
                                 AAPQ = ( DDOT( M, A( 1, p ), 1,
     $                                    A( 1,
     $                                  q ), 1 )*D( p )*D( q ) / AAQQ )
     $                                  / AAPP
                              ELSE
                                 CALL DCOPY( M, A( 1, p ), 1, WORK,
     $                                       1 )
                                 CALL DLASCL( 'G', 0, 0, AAPP,
     $                                        D( p ),
     $                                        M, 1, WORK, LDA, IERR )
                                 AAPQ = DDOT( M, WORK, 1, A( 1, q ),
     $                                  1 )*D( q ) / AAQQ
                              END IF
                           ELSE
                              IF( AAPP.GE.AAQQ ) THEN
                                 ROTOK = AAPP.LE.( AAQQ / SMALL )
                              ELSE
                                 ROTOK = AAQQ.LE.( AAPP / SMALL )
                              END IF
                              IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
                                 AAPQ = ( DDOT( M, A( 1, p ), 1,
     $                                    A( 1,
     $                                  q ), 1 )*D( p )*D( q ) / AAQQ )
     $                                  / AAPP
                              ELSE
                                 CALL DCOPY( M, A( 1, q ), 1, WORK,
     $                                       1 )
                                 CALL DLASCL( 'G', 0, 0, AAQQ,
     $                                        D( q ),
     $                                        M, 1, WORK, LDA, IERR )
                                 AAPQ = DDOT( M, WORK, 1, A( 1, p ),
     $                                  1 )*D( p ) / AAPP
                              END IF
                           END IF
*
                           MXAAPQ = MAX( MXAAPQ, DABS( AAPQ ) )
*
*        TO rotate or NOT to rotate, THAT is the question ...
*
                           IF( DABS( AAPQ ).GT.TOL ) THEN
                              NOTROT = 0
*           ROTATED  = ROTATED + 1
                              PSKIPPED = 0
                              ISWROT = ISWROT + 1
*
                              IF( ROTOK ) THEN
*
                                 AQOAP = AAQQ / AAPP
                                 APOAQ = AAPP / AAQQ
                                 THETA = -HALF*DABS( AQOAP-APOAQ )/AAPQ
                                 IF( AAQQ.GT.AAPP0 )THETA = -THETA
*
                                 IF( DABS( THETA ).GT.BIGTHETA ) THEN
                                    T = HALF / THETA
                                    FASTR( 3 ) = T*D( p ) / D( q )
                                    FASTR( 4 ) = -T*D( q ) / D( p )
                                    CALL DROTM( M, A( 1, p ), 1,
     $                                          A( 1, q ), 1, FASTR )
                                    IF( RSVEC )CALL DROTM( MVL,
     $                                              V( 1, p ), 1,
     $                                              V( 1, q ), 1,
     $                                              FASTR )
                                    SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
     $                                         ONE+T*APOAQ*AAPQ ) )
                                    AAPP = AAPP*DSQRT( MAX( ZERO,
     $                                     ONE-T*AQOAP*AAPQ ) )
                                    MXSINJ = MAX( MXSINJ, DABS( T ) )
                                 ELSE
*
*                 .. choose correct signum for THETA and rotate
*
                                    THSIGN = -DSIGN( ONE, AAPQ )
                                    IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN
                                    T = ONE / ( THETA+THSIGN*
     $                                  DSQRT( ONE+THETA*THETA ) )
                                    CS = DSQRT( ONE / ( ONE+T*T ) )
                                    SN = T*CS
                                    MXSINJ = MAX( MXSINJ, DABS( SN ) )
                                    SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
     $                                         ONE+T*APOAQ*AAPQ ) )
                                    AAPP = AAPP*DSQRT( MAX( ZERO,
     $                                     ONE-T*AQOAP*AAPQ ) )
*
                                    APOAQ = D( p ) / D( q )
                                    AQOAP = D( q ) / D( p )
                                    IF( D( p ).GE.ONE ) THEN
*
                                       IF( D( q ).GE.ONE ) THEN
                                          FASTR( 3 ) = T*APOAQ
                                          FASTR( 4 ) = -T*AQOAP
                                          D( p ) = D( p )*CS
                                          D( q ) = D( q )*CS
                                          CALL DROTM( M, A( 1, p ),
     $                                                1,
     $                                                A( 1, q ), 1,
     $                                                FASTR )
                                          IF( RSVEC )CALL DROTM( MVL,
     $                                        V( 1, p ), 1, V( 1, q ),
     $                                        1, FASTR )
                                       ELSE
                                          CALL DAXPY( M, -T*AQOAP,
     $                                                A( 1, q ), 1,
     $                                                A( 1, p ), 1 )
                                          CALL DAXPY( M, CS*SN*APOAQ,
     $                                                A( 1, p ), 1,
     $                                                A( 1, q ), 1 )
                                          IF( RSVEC ) THEN
                                             CALL DAXPY( MVL,
     $                                                   -T*AQOAP,
     $                                                   V( 1, q ), 1,
     $                                                   V( 1, p ), 1 )
                                             CALL DAXPY( MVL,
     $                                                   CS*SN*APOAQ,
     $                                                   V( 1, p ), 1,
     $                                                   V( 1, q ), 1 )
                                          END IF
                                          D( p ) = D( p )*CS
                                          D( q ) = D( q ) / CS
                                       END IF
                                    ELSE
                                       IF( D( q ).GE.ONE ) THEN
                                          CALL DAXPY( M, T*APOAQ,
     $                                                A( 1, p ), 1,
     $                                                A( 1, q ), 1 )
                                          CALL DAXPY( M,
     $                                                -CS*SN*AQOAP,
     $                                                A( 1, q ), 1,
     $                                                A( 1, p ), 1 )
                                          IF( RSVEC ) THEN
                                             CALL DAXPY( MVL,
     $                                                   T*APOAQ,
     $                                                   V( 1, p ), 1,
     $                                                   V( 1, q ), 1 )
                                             CALL DAXPY( MVL,
     $                                                   -CS*SN*AQOAP,
     $                                                   V( 1, q ), 1,
     $                                                   V( 1, p ), 1 )
                                          END IF
                                          D( p ) = D( p ) / CS
                                          D( q ) = D( q )*CS
                                       ELSE
                                          IF( D( p ).GE.D( q ) ) THEN
                                             CALL DAXPY( M, -T*AQOAP,
     $                                                   A( 1, q ), 1,
     $                                                   A( 1, p ), 1 )
                                             CALL DAXPY( M,
     $                                                   CS*SN*APOAQ,
     $                                                   A( 1, p ), 1,
     $                                                   A( 1, q ), 1 )
                                             D( p ) = D( p )*CS
                                             D( q ) = D( q ) / CS
                                             IF( RSVEC ) THEN
                                                CALL DAXPY( MVL,
     $                                               -T*AQOAP,
     $                                               V( 1, q ), 1,
     $                                               V( 1, p ), 1 )
                                                CALL DAXPY( MVL,
     $                                               CS*SN*APOAQ,
     $                                               V( 1, p ), 1,
     $                                               V( 1, q ), 1 )
                                             END IF
                                          ELSE
                                             CALL DAXPY( M, T*APOAQ,
     $                                                   A( 1, p ), 1,
     $                                                   A( 1, q ), 1 )
                                             CALL DAXPY( M,
     $                                                   -CS*SN*AQOAP,
     $                                                   A( 1, q ), 1,
     $                                                   A( 1, p ), 1 )
                                             D( p ) = D( p ) / CS
                                             D( q ) = D( q )*CS
                                             IF( RSVEC ) THEN
                                                CALL DAXPY( MVL,
     $                                               T*APOAQ, V( 1, p ),
     $                                               1, V( 1, q ), 1 )
                                                CALL DAXPY( MVL,
     $                                               -CS*SN*AQOAP,
     $                                               V( 1, q ), 1,
     $                                               V( 1, p ), 1 )
                                             END IF
                                          END IF
                                       END IF
                                    END IF
                                 END IF
*
                              ELSE
                                 IF( AAPP.GT.AAQQ ) THEN
                                    CALL DCOPY( M, A( 1, p ), 1,
     $                                          WORK,
     $                                          1 )
                                    CALL DLASCL( 'G', 0, 0, AAPP,
     $                                           ONE,
     $                                           M, 1, WORK, LDA, IERR )
                                    CALL DLASCL( 'G', 0, 0, AAQQ,
     $                                           ONE,
     $                                           M, 1, A( 1, q ), LDA,
     $                                           IERR )
                                    TEMP1 = -AAPQ*D( p ) / D( q )
                                    CALL DAXPY( M, TEMP1, WORK, 1,
     $                                          A( 1, q ), 1 )
                                    CALL DLASCL( 'G', 0, 0, ONE,
     $                                           AAQQ,
     $                                           M, 1, A( 1, q ), LDA,
     $                                           IERR )
                                    SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
     $                                         ONE-AAPQ*AAPQ ) )
                                    MXSINJ = MAX( MXSINJ, SFMIN )
                                 ELSE
                                    CALL DCOPY( M, A( 1, q ), 1,
     $                                          WORK,
     $                                          1 )
                                    CALL DLASCL( 'G', 0, 0, AAQQ,
     $                                           ONE,
     $                                           M, 1, WORK, LDA, IERR )
                                    CALL DLASCL( 'G', 0, 0, AAPP,
     $                                           ONE,
     $                                           M, 1, A( 1, p ), LDA,
     $                                           IERR )
                                    TEMP1 = -AAPQ*D( q ) / D( p )
                                    CALL DAXPY( M, TEMP1, WORK, 1,
     $                                          A( 1, p ), 1 )
                                    CALL DLASCL( 'G', 0, 0, ONE,
     $                                           AAPP,
     $                                           M, 1, A( 1, p ), LDA,
     $                                           IERR )
                                    SVA( p ) = AAPP*DSQRT( MAX( ZERO,
     $                                         ONE-AAPQ*AAPQ ) )
                                    MXSINJ = MAX( MXSINJ, SFMIN )
                                 END IF
                              END IF
*           END IF ROTOK THEN ... ELSE
*
*           In the case of cancellation in updating SVA(q)
*           .. recompute SVA(q)
                              IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
     $                            THEN
                                 IF( ( AAQQ.LT.ROOTBIG ) .AND.
     $                               ( AAQQ.GT.ROOTSFMIN ) ) THEN
                                    SVA( q ) = DNRM2( M, A( 1, q ),
     $                                   1 )*
     $                                         D( q )
                                 ELSE
                                    T = ZERO
                                    AAQQ = ONE
                                    CALL DLASSQ( M, A( 1, q ), 1, T,
     $                                           AAQQ )
                                    SVA( q ) = T*DSQRT( AAQQ )*D( q )
                                 END IF
                              END IF
                              IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN
                                 IF( ( AAPP.LT.ROOTBIG ) .AND.
     $                               ( AAPP.GT.ROOTSFMIN ) ) THEN
                                    AAPP = DNRM2( M, A( 1, p ), 1 )*
     $                                     D( p )
                                 ELSE
                                    T = ZERO
                                    AAPP = ONE
                                    CALL DLASSQ( M, A( 1, p ), 1, T,
     $                                           AAPP )
                                    AAPP = T*DSQRT( AAPP )*D( p )
                                 END IF
                                 SVA( p ) = AAPP
                              END IF
*              end of OK rotation
                           ELSE
                              NOTROT = NOTROT + 1
                              PSKIPPED = PSKIPPED + 1
                              IJBLSK = IJBLSK + 1
                           END IF
                        ELSE
                           NOTROT = NOTROT + 1
                           PSKIPPED = PSKIPPED + 1
                           IJBLSK = IJBLSK + 1
                        END IF
*
                        IF( ( i.LE.SWBAND ) .AND. ( IJBLSK.GE.BLSKIP ) )
     $                      THEN
                           SVA( p ) = AAPP
                           NOTROT = 0
                           GO TO 2011
                        END IF
                        IF( ( i.LE.SWBAND ) .AND.
     $                      ( PSKIPPED.GT.ROWSKIP ) ) THEN
                           AAPP = -AAPP
                           NOTROT = 0
                           GO TO 2203
                        END IF
*
 2200                CONTINUE
*        end of the q-loop
 2203                CONTINUE
*
                     SVA( p ) = AAPP
*
                  ELSE
                     IF( AAPP.EQ.ZERO )NOTROT = NOTROT +
     $                   MIN( jgl+KBL-1, N ) - jgl + 1
                     IF( AAPP.LT.ZERO )NOTROT = 0
                  END IF

 2100          CONTINUE
*     end of the p-loop
 2010       CONTINUE
*     end of the jbc-loop
 2011       CONTINUE
*2011 bailed out of the jbc-loop
            DO 2012 p = igl, MIN( igl+KBL-1, N )
               SVA( p ) = DABS( SVA( p ) )
 2012       CONTINUE
*
 2000    CONTINUE
*2000 :: end of the ibr-loop
*
*     .. update SVA(N)
         IF( ( SVA( N ).LT.ROOTBIG ) .AND. ( SVA( N ).GT.ROOTSFMIN ) )
     $       THEN
            SVA( N ) = DNRM2( M, A( 1, N ), 1 )*D( N )
         ELSE
            T = ZERO
            AAPP = ONE
            CALL DLASSQ( M, A( 1, N ), 1, T, AAPP )
            SVA( N ) = T*DSQRT( AAPP )*D( N )
         END IF
*
*     Additional steering devices
*
         IF( ( i.LT.SWBAND ) .AND. ( ( MXAAPQ.LE.ROOTTOL ) .OR.
     $       ( ISWROT.LE.N ) ) )SWBAND = i
*
         IF( ( i.GT.SWBAND+1 ) .AND. ( MXAAPQ.LT.DBLE( N )*TOL ) .AND.
     $       ( DBLE( N )*MXAAPQ*MXSINJ.LT.TOL ) ) THEN
            GO TO 1994
         END IF
*
         IF( NOTROT.GE.EMPTSW )GO TO 1994

 1993 CONTINUE
*     end i=1:NSWEEP loop
* #:) Reaching this point means that the procedure has completed the given
*     number of iterations.
      INFO = NSWEEP - 1
      GO TO 1995
 1994 CONTINUE
* #:) Reaching this point means that during the i-th sweep all pivots were
*     below the given tolerance, causing early exit.
*
      INFO = 0
* #:) INFO = 0 confirms successful iterations.
 1995 CONTINUE
*
*     Sort the vector D.
      DO 5991 p = 1, N - 1
         q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
         IF( p.NE.q ) THEN
            TEMP1 = SVA( p )
            SVA( p ) = SVA( q )
            SVA( q ) = TEMP1
            TEMP1 = D( p )
            D( p ) = D( q )
            D( q ) = TEMP1
            CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
            IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 )
         END IF
 5991 CONTINUE
*
      RETURN
*     ..
*     .. END OF DGSVJ0
*     ..
      END
*> \brief \b DGSVJ1 pre-processor for the routine dgesvj, applies Jacobi rotations targeting only particular pivots.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGSVJ1 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgsvj1.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgsvj1.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgsvj1.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGSVJ1( JOBV, M, N, N1, A, LDA, D, SVA, MV, V, LDV,
*                          EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       DOUBLE PRECISION   EPS, SFMIN, TOL
*       INTEGER            INFO, LDA, LDV, LWORK, M, MV, N, N1, NSWEEP
*       CHARACTER*1        JOBV
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), D( N ), SVA( N ), V( LDV, * ),
*      $                   WORK( LWORK )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGSVJ1 is called from DGESVJ as a pre-processor and that is its main
*> purpose. It applies Jacobi rotations in the same way as DGESVJ does, but
*> it targets only particular pivots and it does not check convergence
*> (stopping criterion). Few tuning parameters (marked by [TP]) are
*> available for the implementer.
*>
*> Further Details
*> ~~~~~~~~~~~~~~~
*> DGSVJ1 applies few sweeps of Jacobi rotations in the column space of
*> the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
*> off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The
*> block-entries (tiles) of the (1,2) off-diagonal block are marked by the
*> [x]'s in the following scheme:
*>
*>    | *  *  * [x] [x] [x]|
*>    | *  *  * [x] [x] [x]|    Row-cycling in the nblr-by-nblc [x] blocks.
*>    | *  *  * [x] [x] [x]|    Row-cyclic pivoting inside each [x] block.
*>    |[x] [x] [x] *  *  * |
*>    |[x] [x] [x] *  *  * |
*>    |[x] [x] [x] *  *  * |
*>
*> In terms of the columns of A, the first N1 columns are rotated 'against'
*> the remaining N-N1 columns, trying to increase the angle between the
*> corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is
*> tiled using quadratic tiles of side KBL. Here, KBL is a tuning parameter.
*> The number of sweeps is given in NSWEEP and the orthogonality threshold
*> is given in TOL.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBV
*> \verbatim
*>          JOBV is CHARACTER*1
*>          Specifies whether the output from this procedure is used
*>          to compute the matrix V:
*>          = 'V': the product of the Jacobi rotations is accumulated
*>                 by postmultiplying the N-by-N array V.
*>                (See the description of V.)
*>          = 'A': the product of the Jacobi rotations is accumulated
*>                 by postmultiplying the MV-by-N array V.
*>                (See the descriptions of MV and V.)
*>          = 'N': the Jacobi rotations are not accumulated.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the input matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the input matrix A.
*>          M >= N >= 0.
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*>          N1 is INTEGER
*>          N1 specifies the 2 x 2 block partition, the first N1 columns are
*>          rotated 'against' the remaining N-N1 columns of A.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, M-by-N matrix A, such that A*diag(D) represents
*>          the input matrix.
*>          On exit,
*>          A_onexit * D_onexit represents the input matrix A*diag(D)
*>          post-multiplied by a sequence of Jacobi rotations, where the
*>          rotation threshold and the total number of sweeps are given in
*>          TOL and NSWEEP, respectively.
*>          (See the descriptions of N1, D, TOL and NSWEEP.)
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The array D accumulates the scaling factors from the fast scaled
*>          Jacobi rotations.
*>          On entry, A*diag(D) represents the input matrix.
*>          On exit, A_onexit*diag(D_onexit) represents the input matrix
*>          post-multiplied by a sequence of Jacobi rotations, where the
*>          rotation threshold and the total number of sweeps are given in
*>          TOL and NSWEEP, respectively.
*>          (See the descriptions of N1, A, TOL and NSWEEP.)
*> \endverbatim
*>
*> \param[in,out] SVA
*> \verbatim
*>          SVA is DOUBLE PRECISION array, dimension (N)
*>          On entry, SVA contains the Euclidean norms of the columns of
*>          the matrix A*diag(D).
*>          On exit, SVA contains the Euclidean norms of the columns of
*>          the matrix onexit*diag(D_onexit).
*> \endverbatim
*>
*> \param[in] MV
*> \verbatim
*>          MV is INTEGER
*>          If JOBV = 'A', then MV rows of V are post-multiplied by a
*>                         sequence of Jacobi rotations.
*>          If JOBV = 'N', then MV is not referenced.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension (LDV,N)
*>          If JOBV = 'V', then N rows of V are post-multiplied by a
*>                         sequence of Jacobi rotations.
*>          If JOBV = 'A', then MV rows of V are post-multiplied by a
*>                         sequence of Jacobi rotations.
*>          If JOBV = 'N', then V is not referenced.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of the array V,  LDV >= 1.
*>          If JOBV = 'V', LDV >= N.
*>          If JOBV = 'A', LDV >= MV.
*> \endverbatim
*>
*> \param[in] EPS
*> \verbatim
*>          EPS is DOUBLE PRECISION
*>          EPS = DLAMCH('Epsilon')
*> \endverbatim
*>
*> \param[in] SFMIN
*> \verbatim
*>          SFMIN is DOUBLE PRECISION
*>          SFMIN = DLAMCH('Safe Minimum')
*> \endverbatim
*>
*> \param[in] TOL
*> \verbatim
*>          TOL is DOUBLE PRECISION
*>          TOL is the threshold for Jacobi rotations. For a pair
*>          A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
*>          applied only if DABS(COS(angle(A(:,p),A(:,q)))) > TOL.
*> \endverbatim
*>
*> \param[in] NSWEEP
*> \verbatim
*>          NSWEEP is INTEGER
*>          NSWEEP is the number of sweeps of Jacobi rotations to be
*>          performed.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          LWORK is the dimension of WORK. LWORK >= M.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, then the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gsvj1
*
*> \par Contributors:
*  ==================
*>
*> Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
*
*  =====================================================================
      SUBROUTINE DGSVJ1( JOBV, M, N, N1, A, LDA, D, SVA, MV, V, LDV,
     $                   EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   EPS, SFMIN, TOL
      INTEGER            INFO, LDA, LDV, LWORK, M, MV, N, N1, NSWEEP
      CHARACTER*1        JOBV
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), D( N ), SVA( N ), V( LDV, * ),
     $                   WORK( LWORK )
*     ..
*
*  =====================================================================
*
*     .. Local Parameters ..
      DOUBLE PRECISION   ZERO, HALF, ONE
      PARAMETER          ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG,
     $                   BIGTHETA, CS, LARGE, MXAAPQ, MXSINJ, ROOTBIG,
     $                   ROOTEPS, ROOTSFMIN, ROOTTOL, SMALL, SN, T,
     $                   TEMP1, THETA, THSIGN
      INTEGER            BLSKIP, EMPTSW, i, ibr, igl, IERR, IJBLSK,
     $                   ISWROT, jbc, jgl, KBL, MVL, NOTROT, nblc, nblr,
     $                   p, PSKIPPED, q, ROWSKIP, SWBAND
      LOGICAL            APPLV, ROTOK, RSVEC
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   FASTR( 5 )
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DABS, MAX, DBLE, MIN, DSIGN, DSQRT
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DDOT, DNRM2
      INTEGER            IDAMAX
      LOGICAL            LSAME
      EXTERNAL           IDAMAX, LSAME, DDOT, DNRM2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DLASCL, DLASSQ, DROTM,
     $                   DSWAP,
     $                   XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      APPLV = LSAME( JOBV, 'A' )
      RSVEC = LSAME( JOBV, 'V' )
      IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( M.LT.0 ) THEN
         INFO = -2
      ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN
         INFO = -3
      ELSE IF( N1.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDA.LT.M ) THEN
         INFO = -6
      ELSE IF( ( RSVEC.OR.APPLV ) .AND. ( MV.LT.0 ) ) THEN
         INFO = -9
      ELSE IF( ( RSVEC.AND.( LDV.LT.N ) ).OR.
     $         ( APPLV.AND.( LDV.LT.MV ) )  ) THEN
         INFO = -11
      ELSE IF( TOL.LE.EPS ) THEN
         INFO = -14
      ELSE IF( NSWEEP.LT.0 ) THEN
         INFO = -15
      ELSE IF( LWORK.LT.M ) THEN
         INFO = -17
      ELSE
         INFO = 0
      END IF
*
*     #:(
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGSVJ1', -INFO )
         RETURN
      END IF
*
      IF( RSVEC ) THEN
         MVL = N
      ELSE IF( APPLV ) THEN
         MVL = MV
      END IF
      RSVEC = RSVEC .OR. APPLV

      ROOTEPS = DSQRT( EPS )
      ROOTSFMIN = DSQRT( SFMIN )
      SMALL = SFMIN / EPS
      BIG = ONE / SFMIN
      ROOTBIG = ONE / ROOTSFMIN
      LARGE = BIG / DSQRT( DBLE( M*N ) )
      BIGTHETA = ONE / ROOTEPS
      ROOTTOL = DSQRT( TOL )
*
*     .. Initialize the right singular vector matrix ..
*
*     RSVEC = LSAME( JOBV, 'Y' )
*
      EMPTSW = N1*( N-N1 )
      NOTROT = 0
      FASTR( 1 ) = ZERO
*
*     .. Row-cyclic pivot strategy with de Rijk's pivoting ..
*
      KBL = MIN( 8, N )
      NBLR = N1 / KBL
      IF( ( NBLR*KBL ).NE.N1 )NBLR = NBLR + 1

*     .. the tiling is nblr-by-nblc [tiles]

      NBLC = ( N-N1 ) / KBL
      IF( ( NBLC*KBL ).NE.( N-N1 ) )NBLC = NBLC + 1
      BLSKIP = ( KBL**2 ) + 1
*[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.

      ROWSKIP = MIN( 5, KBL )
*[TP] ROWSKIP is a tuning parameter.
      SWBAND = 0
*[TP] SWBAND is a tuning parameter. It is meaningful and effective
*     if SGESVJ is used as a computational routine in the preconditioned
*     Jacobi SVD algorithm SGESVJ.
*
*
*     | *   *   * [x] [x] [x]|
*     | *   *   * [x] [x] [x]|    Row-cycling in the nblr-by-nblc [x] blocks.
*     | *   *   * [x] [x] [x]|    Row-cyclic pivoting inside each [x] block.
*     |[x] [x] [x] *   *   * |
*     |[x] [x] [x] *   *   * |
*     |[x] [x] [x] *   *   * |
*
*
      DO 1993 i = 1, NSWEEP
*     .. go go go ...
*
         MXAAPQ = ZERO
         MXSINJ = ZERO
         ISWROT = 0
*
         NOTROT = 0
         PSKIPPED = 0
*
         DO 2000 ibr = 1, NBLR

            igl = ( ibr-1 )*KBL + 1
*
*
*........................................................
* ... go to the off diagonal blocks

            igl = ( ibr-1 )*KBL + 1

            DO 2010 jbc = 1, NBLC

               jgl = N1 + ( jbc-1 )*KBL + 1

*        doing the block at ( ibr, jbc )

               IJBLSK = 0
               DO 2100 p = igl, MIN( igl+KBL-1, N1 )

                  AAPP = SVA( p )

                  IF( AAPP.GT.ZERO ) THEN

                     PSKIPPED = 0

                     DO 2200 q = jgl, MIN( jgl+KBL-1, N )
*
                        AAQQ = SVA( q )

                        IF( AAQQ.GT.ZERO ) THEN
                           AAPP0 = AAPP
*
*     .. M x 2 Jacobi SVD ..
*
*        .. Safe Gram matrix computation ..
*
                           IF( AAQQ.GE.ONE ) THEN
                              IF( AAPP.GE.AAQQ ) THEN
                                 ROTOK = ( SMALL*AAPP ).LE.AAQQ
                              ELSE
                                 ROTOK = ( SMALL*AAQQ ).LE.AAPP
                              END IF
                              IF( AAPP.LT.( BIG / AAQQ ) ) THEN
                                 AAPQ = ( DDOT( M, A( 1, p ), 1,
     $                                    A( 1,
     $                                  q ), 1 )*D( p )*D( q ) / AAQQ )
     $                                  / AAPP
                              ELSE
                                 CALL DCOPY( M, A( 1, p ), 1, WORK,
     $                                       1 )
                                 CALL DLASCL( 'G', 0, 0, AAPP,
     $                                        D( p ),
     $                                        M, 1, WORK, LDA, IERR )
                                 AAPQ = DDOT( M, WORK, 1, A( 1, q ),
     $                                  1 )*D( q ) / AAQQ
                              END IF
                           ELSE
                              IF( AAPP.GE.AAQQ ) THEN
                                 ROTOK = AAPP.LE.( AAQQ / SMALL )
                              ELSE
                                 ROTOK = AAQQ.LE.( AAPP / SMALL )
                              END IF
                              IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
                                 AAPQ = ( DDOT( M, A( 1, p ), 1,
     $                                    A( 1,
     $                                  q ), 1 )*D( p )*D( q ) / AAQQ )
     $                                  / AAPP
                              ELSE
                                 CALL DCOPY( M, A( 1, q ), 1, WORK,
     $                                       1 )
                                 CALL DLASCL( 'G', 0, 0, AAQQ,
     $                                        D( q ),
     $                                        M, 1, WORK, LDA, IERR )
                                 AAPQ = DDOT( M, WORK, 1, A( 1, p ),
     $                                  1 )*D( p ) / AAPP
                              END IF
                           END IF

                           MXAAPQ = MAX( MXAAPQ, DABS( AAPQ ) )

*        TO rotate or NOT to rotate, THAT is the question ...
*
                           IF( DABS( AAPQ ).GT.TOL ) THEN
                              NOTROT = 0
*           ROTATED  = ROTATED + 1
                              PSKIPPED = 0
                              ISWROT = ISWROT + 1
*
                              IF( ROTOK ) THEN
*
                                 AQOAP = AAQQ / AAPP
                                 APOAQ = AAPP / AAQQ
                                 THETA = -HALF*DABS(AQOAP-APOAQ) / AAPQ
                                 IF( AAQQ.GT.AAPP0 )THETA = -THETA

                                 IF( DABS( THETA ).GT.BIGTHETA ) THEN
                                    T = HALF / THETA
                                    FASTR( 3 ) = T*D( p ) / D( q )
                                    FASTR( 4 ) = -T*D( q ) / D( p )
                                    CALL DROTM( M, A( 1, p ), 1,
     $                                          A( 1, q ), 1, FASTR )
                                    IF( RSVEC )CALL DROTM( MVL,
     $                                              V( 1, p ), 1,
     $                                              V( 1, q ), 1,
     $                                              FASTR )
                                    SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
     $                                         ONE+T*APOAQ*AAPQ ) )
                                    AAPP = AAPP*DSQRT( MAX( ZERO,
     $                                     ONE-T*AQOAP*AAPQ ) )
                                    MXSINJ = MAX( MXSINJ, DABS( T ) )
                                 ELSE
*
*                 .. choose correct signum for THETA and rotate
*
                                    THSIGN = -DSIGN( ONE, AAPQ )
                                    IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN
                                    T = ONE / ( THETA+THSIGN*
     $                                  DSQRT( ONE+THETA*THETA ) )
                                    CS = DSQRT( ONE / ( ONE+T*T ) )
                                    SN = T*CS
                                    MXSINJ = MAX( MXSINJ, DABS( SN ) )
                                    SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
     $                                         ONE+T*APOAQ*AAPQ ) )
                                    AAPP = AAPP*DSQRT( MAX( ZERO,
     $                                    ONE-T*AQOAP*AAPQ ) )

                                    APOAQ = D( p ) / D( q )
                                    AQOAP = D( q ) / D( p )
                                    IF( D( p ).GE.ONE ) THEN
*
                                       IF( D( q ).GE.ONE ) THEN
                                          FASTR( 3 ) = T*APOAQ
                                          FASTR( 4 ) = -T*AQOAP
                                          D( p ) = D( p )*CS
                                          D( q ) = D( q )*CS
                                          CALL DROTM( M, A( 1, p ),
     $                                                1,
     $                                                A( 1, q ), 1,
     $                                                FASTR )
                                          IF( RSVEC )CALL DROTM( MVL,
     $                                        V( 1, p ), 1, V( 1, q ),
     $                                        1, FASTR )
                                       ELSE
                                          CALL DAXPY( M, -T*AQOAP,
     $                                                A( 1, q ), 1,
     $                                                A( 1, p ), 1 )
                                          CALL DAXPY( M, CS*SN*APOAQ,
     $                                                A( 1, p ), 1,
     $                                                A( 1, q ), 1 )
                                          IF( RSVEC ) THEN
                                             CALL DAXPY( MVL,
     $                                                   -T*AQOAP,
     $                                                   V( 1, q ), 1,
     $                                                   V( 1, p ), 1 )
                                             CALL DAXPY( MVL,
     $                                                   CS*SN*APOAQ,
     $                                                   V( 1, p ), 1,
     $                                                   V( 1, q ), 1 )
                                          END IF
                                          D( p ) = D( p )*CS
                                          D( q ) = D( q ) / CS
                                       END IF
                                    ELSE
                                       IF( D( q ).GE.ONE ) THEN
                                          CALL DAXPY( M, T*APOAQ,
     $                                                A( 1, p ), 1,
     $                                                A( 1, q ), 1 )
                                          CALL DAXPY( M,
     $                                                -CS*SN*AQOAP,
     $                                                A( 1, q ), 1,
     $                                                A( 1, p ), 1 )
                                          IF( RSVEC ) THEN
                                             CALL DAXPY( MVL,
     $                                                   T*APOAQ,
     $                                                   V( 1, p ), 1,
     $                                                   V( 1, q ), 1 )
                                             CALL DAXPY( MVL,
     $                                                   -CS*SN*AQOAP,
     $                                                   V( 1, q ), 1,
     $                                                   V( 1, p ), 1 )
                                          END IF
                                          D( p ) = D( p ) / CS
                                          D( q ) = D( q )*CS
                                       ELSE
                                          IF( D( p ).GE.D( q ) ) THEN
                                             CALL DAXPY( M, -T*AQOAP,
     $                                                   A( 1, q ), 1,
     $                                                   A( 1, p ), 1 )
                                             CALL DAXPY( M,
     $                                                   CS*SN*APOAQ,
     $                                                   A( 1, p ), 1,
     $                                                   A( 1, q ), 1 )
                                             D( p ) = D( p )*CS
                                             D( q ) = D( q ) / CS
                                             IF( RSVEC ) THEN
                                                CALL DAXPY( MVL,
     $                                               -T*AQOAP,
     $                                               V( 1, q ), 1,
     $                                               V( 1, p ), 1 )
                                                CALL DAXPY( MVL,
     $                                               CS*SN*APOAQ,
     $                                               V( 1, p ), 1,
     $                                               V( 1, q ), 1 )
                                             END IF
                                          ELSE
                                             CALL DAXPY( M, T*APOAQ,
     $                                                   A( 1, p ), 1,
     $                                                   A( 1, q ), 1 )
                                             CALL DAXPY( M,
     $                                                   -CS*SN*AQOAP,
     $                                                   A( 1, q ), 1,
     $                                                   A( 1, p ), 1 )
                                             D( p ) = D( p ) / CS
                                             D( q ) = D( q )*CS
                                             IF( RSVEC ) THEN
                                                CALL DAXPY( MVL,
     $                                               T*APOAQ, V( 1, p ),
     $                                               1, V( 1, q ), 1 )
                                                CALL DAXPY( MVL,
     $                                               -CS*SN*AQOAP,
     $                                               V( 1, q ), 1,
     $                                               V( 1, p ), 1 )
                                             END IF
                                          END IF
                                       END IF
                                    END IF
                                 END IF

                              ELSE
                                 IF( AAPP.GT.AAQQ ) THEN
                                    CALL DCOPY( M, A( 1, p ), 1,
     $                                          WORK,
     $                                          1 )
                                    CALL DLASCL( 'G', 0, 0, AAPP,
     $                                           ONE,
     $                                           M, 1, WORK, LDA, IERR )
                                    CALL DLASCL( 'G', 0, 0, AAQQ,
     $                                           ONE,
     $                                           M, 1, A( 1, q ), LDA,
     $                                           IERR )
                                    TEMP1 = -AAPQ*D( p ) / D( q )
                                    CALL DAXPY( M, TEMP1, WORK, 1,
     $                                          A( 1, q ), 1 )
                                    CALL DLASCL( 'G', 0, 0, ONE,
     $                                           AAQQ,
     $                                           M, 1, A( 1, q ), LDA,
     $                                           IERR )
                                    SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
     $                                         ONE-AAPQ*AAPQ ) )
                                    MXSINJ = MAX( MXSINJ, SFMIN )
                                 ELSE
                                    CALL DCOPY( M, A( 1, q ), 1,
     $                                          WORK,
     $                                          1 )
                                    CALL DLASCL( 'G', 0, 0, AAQQ,
     $                                           ONE,
     $                                           M, 1, WORK, LDA, IERR )
                                    CALL DLASCL( 'G', 0, 0, AAPP,
     $                                           ONE,
     $                                           M, 1, A( 1, p ), LDA,
     $                                           IERR )
                                    TEMP1 = -AAPQ*D( q ) / D( p )
                                    CALL DAXPY( M, TEMP1, WORK, 1,
     $                                          A( 1, p ), 1 )
                                    CALL DLASCL( 'G', 0, 0, ONE,
     $                                           AAPP,
     $                                           M, 1, A( 1, p ), LDA,
     $                                           IERR )
                                    SVA( p ) = AAPP*DSQRT( MAX( ZERO,
     $                                         ONE-AAPQ*AAPQ ) )
                                    MXSINJ = MAX( MXSINJ, SFMIN )
                                 END IF
                              END IF
*           END IF ROTOK THEN ... ELSE
*
*           In the case of cancellation in updating SVA(q)
*           .. recompute SVA(q)
                              IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
     $                            THEN
                                 IF( ( AAQQ.LT.ROOTBIG ) .AND.
     $                               ( AAQQ.GT.ROOTSFMIN ) ) THEN
                                    SVA( q ) = DNRM2( M, A( 1, q ),
     $                                   1 )*
     $                                         D( q )
                                 ELSE
                                    T = ZERO
                                    AAQQ = ONE
                                    CALL DLASSQ( M, A( 1, q ), 1, T,
     $                                           AAQQ )
                                    SVA( q ) = T*DSQRT( AAQQ )*D( q )
                                 END IF
                              END IF
                              IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN
                                 IF( ( AAPP.LT.ROOTBIG ) .AND.
     $                               ( AAPP.GT.ROOTSFMIN ) ) THEN
                                    AAPP = DNRM2( M, A( 1, p ), 1 )*
     $                                     D( p )
                                 ELSE
                                    T = ZERO
                                    AAPP = ONE
                                    CALL DLASSQ( M, A( 1, p ), 1, T,
     $                                           AAPP )
                                    AAPP = T*DSQRT( AAPP )*D( p )
                                 END IF
                                 SVA( p ) = AAPP
                              END IF
*              end of OK rotation
                           ELSE
                              NOTROT = NOTROT + 1
*           SKIPPED  = SKIPPED  + 1
                              PSKIPPED = PSKIPPED + 1
                              IJBLSK = IJBLSK + 1
                           END IF
                        ELSE
                           NOTROT = NOTROT + 1
                           PSKIPPED = PSKIPPED + 1
                           IJBLSK = IJBLSK + 1
                        END IF

*      IF ( NOTROT .GE. EMPTSW )  GO TO 2011
                        IF( ( i.LE.SWBAND ) .AND. ( IJBLSK.GE.BLSKIP ) )
     $                      THEN
                           SVA( p ) = AAPP
                           NOTROT = 0
                           GO TO 2011
                        END IF
                        IF( ( i.LE.SWBAND ) .AND.
     $                      ( PSKIPPED.GT.ROWSKIP ) ) THEN
                           AAPP = -AAPP
                           NOTROT = 0
                           GO TO 2203
                        END IF

*
 2200                CONTINUE
*        end of the q-loop
 2203                CONTINUE

                     SVA( p ) = AAPP
*
                  ELSE
                     IF( AAPP.EQ.ZERO )NOTROT = NOTROT +
     $                   MIN( jgl+KBL-1, N ) - jgl + 1
                     IF( AAPP.LT.ZERO )NOTROT = 0
***      IF ( NOTROT .GE. EMPTSW )  GO TO 2011
                  END IF

 2100          CONTINUE
*     end of the p-loop
 2010       CONTINUE
*     end of the jbc-loop
 2011       CONTINUE
*2011 bailed out of the jbc-loop
            DO 2012 p = igl, MIN( igl+KBL-1, N )
               SVA( p ) = DABS( SVA( p ) )
 2012       CONTINUE
***   IF ( NOTROT .GE. EMPTSW ) GO TO 1994
 2000    CONTINUE
*2000 :: end of the ibr-loop
*
*     .. update SVA(N)
         IF( ( SVA( N ).LT.ROOTBIG ) .AND. ( SVA( N ).GT.ROOTSFMIN ) )
     $       THEN
            SVA( N ) = DNRM2( M, A( 1, N ), 1 )*D( N )
         ELSE
            T = ZERO
            AAPP = ONE
            CALL DLASSQ( M, A( 1, N ), 1, T, AAPP )
            SVA( N ) = T*DSQRT( AAPP )*D( N )
         END IF
*
*     Additional steering devices
*
         IF( ( i.LT.SWBAND ) .AND. ( ( MXAAPQ.LE.ROOTTOL ) .OR.
     $       ( ISWROT.LE.N ) ) )SWBAND = i

         IF( ( i.GT.SWBAND+1 ) .AND. ( MXAAPQ.LT.DBLE( N )*TOL ) .AND.
     $       ( DBLE( N )*MXAAPQ*MXSINJ.LT.TOL ) ) THEN
            GO TO 1994
         END IF

*
         IF( NOTROT.GE.EMPTSW )GO TO 1994

 1993 CONTINUE
*     end i=1:NSWEEP loop
* #:) Reaching this point means that the procedure has completed the given
*     number of sweeps.
      INFO = NSWEEP - 1
      GO TO 1995
 1994 CONTINUE
* #:) Reaching this point means that during the i-th sweep all pivots were
*     below the given threshold, causing early exit.

      INFO = 0
* #:) INFO = 0 confirms successful iterations.
 1995 CONTINUE
*
*     Sort the vector D
*
      DO 5991 p = 1, N - 1
         q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
         IF( p.NE.q ) THEN
            TEMP1 = SVA( p )
            SVA( p ) = SVA( q )
            SVA( q ) = TEMP1
            TEMP1 = D( p )
            D( p ) = D( q )
            D( q ) = TEMP1
            CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
            IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 )
         END IF
 5991 CONTINUE
*
      RETURN
*     ..
*     .. END OF DGSVJ1
*     ..
      END
*> \brief \b DGTCON
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGTCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgtcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgtcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgtcon.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGTCON( NORM, N, DL, D, DU, DU2, IPIV, ANORM, RCOND,
*                          WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          NORM
*       INTEGER            INFO, N
*       DOUBLE PRECISION   ANORM, RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), IWORK( * )
*       DOUBLE PRECISION   D( * ), DL( * ), DU( * ), DU2( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGTCON estimates the reciprocal of the condition number of a real
*> tridiagonal matrix A using the LU factorization as computed by
*> DGTTRF.
*>
*> An estimate is obtained for norm(inv(A)), and the reciprocal of the
*> condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NORM
*> \verbatim
*>          NORM is CHARACTER*1
*>          Specifies whether the 1-norm condition number or the
*>          infinity-norm condition number is required:
*>          = '1' or 'O':  1-norm;
*>          = 'I':         Infinity-norm.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] DL
*> \verbatim
*>          DL is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) multipliers that define the matrix L from the
*>          LU factorization of A as computed by DGTTRF.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The n diagonal elements of the upper triangular matrix U from
*>          the LU factorization of A.
*> \endverbatim
*>
*> \param[in] DU
*> \verbatim
*>          DU is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) elements of the first superdiagonal of U.
*> \endverbatim
*>
*> \param[in] DU2
*> \verbatim
*>          DU2 is DOUBLE PRECISION array, dimension (N-2)
*>          The (n-2) elements of the second superdiagonal of U.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          The pivot indices; for 1 <= i <= n, row i of the matrix was
*>          interchanged with row IPIV(i).  IPIV(i) will always be either
*>          i or i+1; IPIV(i) = i indicates a row interchange was not
*>          required.
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*>          ANORM is DOUBLE PRECISION
*>          If NORM = '1' or 'O', the 1-norm of the original matrix A.
*>          If NORM = 'I', the infinity-norm of the original matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The reciprocal of the condition number of the matrix A,
*>          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
*>          estimate of the 1-norm of inv(A) computed in this routine.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gtcon
*
*  =====================================================================
      SUBROUTINE DGTCON( NORM, N, DL, D, DU, DU2, IPIV, ANORM, RCOND,
     $                   WORK, IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          NORM
      INTEGER            INFO, N
      DOUBLE PRECISION   ANORM, RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), IWORK( * )
      DOUBLE PRECISION   D( * ), DL( * ), DU( * ), DU2( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ONENRM
      INTEGER            I, KASE, KASE1
      DOUBLE PRECISION   AINVNM
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGTTRS, DLACN2, XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments.
*
      INFO = 0
      ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' )
      IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( ANORM.LT.ZERO ) THEN
         INFO = -8
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGTCON', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      RCOND = ZERO
      IF( N.EQ.0 ) THEN
         RCOND = ONE
         RETURN
      ELSE IF( ANORM.EQ.ZERO ) THEN
         RETURN
      END IF
*
*     Check that D(1:N) is non-zero.
*
      DO 10 I = 1, N
         IF( D( I ).EQ.ZERO )
     $      RETURN
   10 CONTINUE
*
      AINVNM = ZERO
      IF( ONENRM ) THEN
         KASE1 = 1
      ELSE
         KASE1 = 2
      END IF
      KASE = 0
   20 CONTINUE
      CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
      IF( KASE.NE.0 ) THEN
         IF( KASE.EQ.KASE1 ) THEN
*
*           Multiply by inv(U)*inv(L).
*
            CALL DGTTRS( 'No transpose', N, 1, DL, D, DU, DU2, IPIV,
     $                   WORK, N, INFO )
         ELSE
*
*           Multiply by inv(L**T)*inv(U**T).
*
            CALL DGTTRS( 'Transpose', N, 1, DL, D, DU, DU2, IPIV,
     $                   WORK,
     $                   N, INFO )
         END IF
         GO TO 20
      END IF
*
*     Compute the estimate of the reciprocal condition number.
*
      IF( AINVNM.NE.ZERO )
     $   RCOND = ( ONE / AINVNM ) / ANORM
*
      RETURN
*
*     End of DGTCON
*
      END
*> \brief \b DGTRFS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGTRFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgtrfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgtrfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgtrfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2,
*                          IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK,
*                          INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANS
*       INTEGER            INFO, LDB, LDX, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), IWORK( * )
*       DOUBLE PRECISION   B( LDB, * ), BERR( * ), D( * ), DF( * ),
*      $                   DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
*      $                   FERR( * ), WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGTRFS improves the computed solution to a system of linear
*> equations when the coefficient matrix is tridiagonal, and provides
*> error bounds and backward error estimates for the solution.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the form of the system of equations:
*>          = 'N':  A * X = B     (No transpose)
*>          = 'T':  A**T * X = B  (Transpose)
*>          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] DL
*> \verbatim
*>          DL is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) subdiagonal elements of A.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The diagonal elements of A.
*> \endverbatim
*>
*> \param[in] DU
*> \verbatim
*>          DU is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) superdiagonal elements of A.
*> \endverbatim
*>
*> \param[in] DLF
*> \verbatim
*>          DLF is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) multipliers that define the matrix L from the
*>          LU factorization of A as computed by DGTTRF.
*> \endverbatim
*>
*> \param[in] DF
*> \verbatim
*>          DF is DOUBLE PRECISION array, dimension (N)
*>          The n diagonal elements of the upper triangular matrix U from
*>          the LU factorization of A.
*> \endverbatim
*>
*> \param[in] DUF
*> \verbatim
*>          DUF is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) elements of the first superdiagonal of U.
*> \endverbatim
*>
*> \param[in] DU2
*> \verbatim
*>          DU2 is DOUBLE PRECISION array, dimension (N-2)
*>          The (n-2) elements of the second superdiagonal of U.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          The pivot indices; for 1 <= i <= n, row i of the matrix was
*>          interchanged with row IPIV(i).  IPIV(i) will always be either
*>          i or i+1; IPIV(i) = i indicates a row interchange was not
*>          required.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          On entry, the solution matrix X, as computed by DGTTRS.
*>          On exit, the improved solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The estimated forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).  The estimate is as reliable as
*>          the estimate for RCOND, and is almost always a slight
*>          overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in
*>          any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  ITMAX is the maximum number of steps of iterative refinement.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gtrfs
*
*  =====================================================================
      SUBROUTINE DGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
     $                   DU2,
     $                   IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS
      INTEGER            INFO, LDB, LDX, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), IWORK( * )
      DOUBLE PRECISION   B( LDB, * ), BERR( * ), D( * ), DF( * ),
     $                   DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
     $                   FERR( * ), WORK( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            ITMAX
      PARAMETER          ( ITMAX = 5 )
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
      DOUBLE PRECISION   TWO
      PARAMETER          ( TWO = 2.0D+0 )
      DOUBLE PRECISION   THREE
      PARAMETER          ( THREE = 3.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOTRAN
      CHARACTER          TRANSN, TRANST
      INTEGER            COUNT, I, J, KASE, NZ
      DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DGTTRS, DLACN2, DLAGTM,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NOTRAN = LSAME( TRANS, 'N' )
      IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
     $    LSAME( TRANS, 'C' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -13
      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
         INFO = -15
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGTRFS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
         DO 10 J = 1, NRHS
            FERR( J ) = ZERO
            BERR( J ) = ZERO
   10    CONTINUE
         RETURN
      END IF
*
      IF( NOTRAN ) THEN
         TRANSN = 'N'
         TRANST = 'T'
      ELSE
         TRANSN = 'T'
         TRANST = 'N'
      END IF
*
*     NZ = maximum number of nonzero elements in each row of A, plus 1
*
      NZ = 4
      EPS = DLAMCH( 'Epsilon' )
      SAFMIN = DLAMCH( 'Safe minimum' )
      SAFE1 = NZ*SAFMIN
      SAFE2 = SAFE1 / EPS
*
*     Do for each right hand side
*
      DO 110 J = 1, NRHS
*
         COUNT = 1
         LSTRES = THREE
   20    CONTINUE
*
*        Loop until stopping criterion is satisfied.
*
*        Compute residual R = B - op(A) * X,
*        where op(A) = A, A**T, or A**H, depending on TRANS.
*
         CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
         CALL DLAGTM( TRANS, N, 1, -ONE, DL, D, DU, X( 1, J ), LDX,
     $                ONE,
     $                WORK( N+1 ), N )
*
*        Compute abs(op(A))*abs(x) + abs(b) for use in the backward
*        error bound.
*
         IF( NOTRAN ) THEN
            IF( N.EQ.1 ) THEN
               WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) )
            ELSE
               WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) +
     $                     ABS( DU( 1 )*X( 2, J ) )
               DO 30 I = 2, N - 1
                  WORK( I ) = ABS( B( I, J ) ) +
     $                        ABS( DL( I-1 )*X( I-1, J ) ) +
     $                        ABS( D( I )*X( I, J ) ) +
     $                        ABS( DU( I )*X( I+1, J ) )
   30          CONTINUE
               WORK( N ) = ABS( B( N, J ) ) +
     $                     ABS( DL( N-1 )*X( N-1, J ) ) +
     $                     ABS( D( N )*X( N, J ) )
            END IF
         ELSE
            IF( N.EQ.1 ) THEN
               WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) )
            ELSE
               WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) +
     $                     ABS( DL( 1 )*X( 2, J ) )
               DO 40 I = 2, N - 1
                  WORK( I ) = ABS( B( I, J ) ) +
     $                        ABS( DU( I-1 )*X( I-1, J ) ) +
     $                        ABS( D( I )*X( I, J ) ) +
     $                        ABS( DL( I )*X( I+1, J ) )
   40          CONTINUE
               WORK( N ) = ABS( B( N, J ) ) +
     $                     ABS( DU( N-1 )*X( N-1, J ) ) +
     $                     ABS( D( N )*X( N, J ) )
            END IF
         END IF
*
*        Compute componentwise relative backward error from formula
*
*        max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
*
*        where abs(Z) is the componentwise absolute value of the matrix
*        or vector Z.  If the i-th component of the denominator is less
*        than SAFE2, then SAFE1 is added to the i-th components of the
*        numerator and denominator before dividing.
*
         S = ZERO
         DO 50 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
            ELSE
               S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
     $             ( WORK( I )+SAFE1 ) )
            END IF
   50    CONTINUE
         BERR( J ) = S
*
*        Test stopping criterion. Continue iterating if
*           1) The residual BERR(J) is larger than machine epsilon, and
*           2) BERR(J) decreased by at least a factor of 2 during the
*              last iteration, and
*           3) At most ITMAX iterations tried.
*
         IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
     $       COUNT.LE.ITMAX ) THEN
*
*           Update solution and try again.
*
            CALL DGTTRS( TRANS, N, 1, DLF, DF, DUF, DU2, IPIV,
     $                   WORK( N+1 ), N, INFO )
            CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
            LSTRES = BERR( J )
            COUNT = COUNT + 1
            GO TO 20
         END IF
*
*        Bound error from formula
*
*        norm(X - XTRUE) / norm(X) .le. FERR =
*        norm( abs(inv(op(A)))*
*           ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
*
*        where
*          norm(Z) is the magnitude of the largest component of Z
*          inv(op(A)) is the inverse of op(A)
*          abs(Z) is the componentwise absolute value of the matrix or
*             vector Z
*          NZ is the maximum number of nonzeros in any row of A, plus 1
*          EPS is machine epsilon
*
*        The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
*        is incremented by SAFE1 if the i-th component of
*        abs(op(A))*abs(X) + abs(B) is less than SAFE2.
*
*        Use DLACN2 to estimate the infinity-norm of the matrix
*           inv(op(A)) * diag(W),
*        where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
*
         DO 60 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
            ELSE
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
            END IF
   60    CONTINUE
*
         KASE = 0
   70    CONTINUE
         CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
     $                FERR( J ),
     $                KASE, ISAVE )
         IF( KASE.NE.0 ) THEN
            IF( KASE.EQ.1 ) THEN
*
*              Multiply by diag(W)*inv(op(A)**T).
*
               CALL DGTTRS( TRANST, N, 1, DLF, DF, DUF, DU2, IPIV,
     $                      WORK( N+1 ), N, INFO )
               DO 80 I = 1, N
                  WORK( N+I ) = WORK( I )*WORK( N+I )
   80          CONTINUE
            ELSE
*
*              Multiply by inv(op(A))*diag(W).
*
               DO 90 I = 1, N
                  WORK( N+I ) = WORK( I )*WORK( N+I )
   90          CONTINUE
               CALL DGTTRS( TRANSN, N, 1, DLF, DF, DUF, DU2, IPIV,
     $                      WORK( N+1 ), N, INFO )
            END IF
            GO TO 70
         END IF
*
*        Normalize error.
*
         LSTRES = ZERO
         DO 100 I = 1, N
            LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
  100    CONTINUE
         IF( LSTRES.NE.ZERO )
     $      FERR( J ) = FERR( J ) / LSTRES
*
  110 CONTINUE
*
      RETURN
*
*     End of DGTRFS
*
      END
*> \brief <b> DGTSV computes the solution to system of linear equations A * X = B for GT matrices </b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGTSV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgtsv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgtsv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgtsv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGTSV( N, NRHS, DL, D, DU, B, LDB, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   B( LDB, * ), D( * ), DL( * ), DU( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGTSV  solves the equation
*>
*>    A*X = B,
*>
*> where A is an n by n tridiagonal matrix, by Gaussian elimination with
*> partial pivoting.
*>
*> Note that the equation  A**T*X = B  may be solved by interchanging the
*> order of the arguments DU and DL.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] DL
*> \verbatim
*>          DL is DOUBLE PRECISION array, dimension (N-1)
*>          On entry, DL must contain the (n-1) sub-diagonal elements of
*>          A.
*>
*>          On exit, DL is overwritten by the (n-2) elements of the
*>          second super-diagonal of the upper triangular matrix U from
*>          the LU factorization of A, in DL(1), ..., DL(n-2).
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, D must contain the diagonal elements of A.
*>
*>          On exit, D is overwritten by the n diagonal elements of U.
*> \endverbatim
*>
*> \param[in,out] DU
*> \verbatim
*>          DU is DOUBLE PRECISION array, dimension (N-1)
*>          On entry, DU must contain the (n-1) super-diagonal elements
*>          of A.
*>
*>          On exit, DU is overwritten by the (n-1) elements of the first
*>          super-diagonal of U.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the N by NRHS matrix of right hand side matrix B.
*>          On exit, if INFO = 0, the N by NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*>          > 0: if INFO = i, U(i,i) is exactly zero, and the solution
*>               has not been computed.  The factorization has not been
*>               completed unless i = N.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gtsv
*
*  =====================================================================
      SUBROUTINE DGTSV( N, NRHS, DL, D, DU, B, LDB, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   B( LDB, * ), D( * ), DL( * ), DU( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J
      DOUBLE PRECISION   FACT, TEMP
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      IF( N.LT.0 ) THEN
         INFO = -1
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGTSV ', -INFO )
         RETURN
      END IF
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( NRHS.EQ.1 ) THEN
         DO 10 I = 1, N - 2
            IF( ABS( D( I ) ).GE.ABS( DL( I ) ) ) THEN
*
*              No row interchange required
*
               IF( D( I ).NE.ZERO ) THEN
                  FACT = DL( I ) / D( I )
                  D( I+1 ) = D( I+1 ) - FACT*DU( I )
                  B( I+1, 1 ) = B( I+1, 1 ) - FACT*B( I, 1 )
               ELSE
                  INFO = I
                  RETURN
               END IF
               DL( I ) = ZERO
            ELSE
*
*              Interchange rows I and I+1
*
               FACT = D( I ) / DL( I )
               D( I ) = DL( I )
               TEMP = D( I+1 )
               D( I+1 ) = DU( I ) - FACT*TEMP
               DL( I ) = DU( I+1 )
               DU( I+1 ) = -FACT*DL( I )
               DU( I ) = TEMP
               TEMP = B( I, 1 )
               B( I, 1 ) = B( I+1, 1 )
               B( I+1, 1 ) = TEMP - FACT*B( I+1, 1 )
            END IF
   10    CONTINUE
         IF( N.GT.1 ) THEN
            I = N - 1
            IF( ABS( D( I ) ).GE.ABS( DL( I ) ) ) THEN
               IF( D( I ).NE.ZERO ) THEN
                  FACT = DL( I ) / D( I )
                  D( I+1 ) = D( I+1 ) - FACT*DU( I )
                  B( I+1, 1 ) = B( I+1, 1 ) - FACT*B( I, 1 )
               ELSE
                  INFO = I
                  RETURN
               END IF
            ELSE
               FACT = D( I ) / DL( I )
               D( I ) = DL( I )
               TEMP = D( I+1 )
               D( I+1 ) = DU( I ) - FACT*TEMP
               DU( I ) = TEMP
               TEMP = B( I, 1 )
               B( I, 1 ) = B( I+1, 1 )
               B( I+1, 1 ) = TEMP - FACT*B( I+1, 1 )
            END IF
         END IF
         IF( D( N ).EQ.ZERO ) THEN
            INFO = N
            RETURN
         END IF
      ELSE
         DO 40 I = 1, N - 2
            IF( ABS( D( I ) ).GE.ABS( DL( I ) ) ) THEN
*
*              No row interchange required
*
               IF( D( I ).NE.ZERO ) THEN
                  FACT = DL( I ) / D( I )
                  D( I+1 ) = D( I+1 ) - FACT*DU( I )
                  DO 20 J = 1, NRHS
                     B( I+1, J ) = B( I+1, J ) - FACT*B( I, J )
   20             CONTINUE
               ELSE
                  INFO = I
                  RETURN
               END IF
               DL( I ) = ZERO
            ELSE
*
*              Interchange rows I and I+1
*
               FACT = D( I ) / DL( I )
               D( I ) = DL( I )
               TEMP = D( I+1 )
               D( I+1 ) = DU( I ) - FACT*TEMP
               DL( I ) = DU( I+1 )
               DU( I+1 ) = -FACT*DL( I )
               DU( I ) = TEMP
               DO 30 J = 1, NRHS
                  TEMP = B( I, J )
                  B( I, J ) = B( I+1, J )
                  B( I+1, J ) = TEMP - FACT*B( I+1, J )
   30          CONTINUE
            END IF
   40    CONTINUE
         IF( N.GT.1 ) THEN
            I = N - 1
            IF( ABS( D( I ) ).GE.ABS( DL( I ) ) ) THEN
               IF( D( I ).NE.ZERO ) THEN
                  FACT = DL( I ) / D( I )
                  D( I+1 ) = D( I+1 ) - FACT*DU( I )
                  DO 50 J = 1, NRHS
                     B( I+1, J ) = B( I+1, J ) - FACT*B( I, J )
   50             CONTINUE
               ELSE
                  INFO = I
                  RETURN
               END IF
            ELSE
               FACT = D( I ) / DL( I )
               D( I ) = DL( I )
               TEMP = D( I+1 )
               D( I+1 ) = DU( I ) - FACT*TEMP
               DU( I ) = TEMP
               DO 60 J = 1, NRHS
                  TEMP = B( I, J )
                  B( I, J ) = B( I+1, J )
                  B( I+1, J ) = TEMP - FACT*B( I+1, J )
   60          CONTINUE
            END IF
         END IF
         IF( D( N ).EQ.ZERO ) THEN
            INFO = N
            RETURN
         END IF
      END IF
*
*     Back solve with the matrix U from the factorization.
*
      IF( NRHS.LE.2 ) THEN
         J = 1
   70    CONTINUE
         B( N, J ) = B( N, J ) / D( N )
         IF( N.GT.1 )
     $      B( N-1, J ) = ( B( N-1, J )-DU( N-1 )*B( N, J ) ) / D( N-1 )
         DO 80 I = N - 2, 1, -1
            B( I, J ) = ( B( I, J )-DU( I )*B( I+1, J )-DL( I )*
     $                  B( I+2, J ) ) / D( I )
   80    CONTINUE
         IF( J.LT.NRHS ) THEN
            J = J + 1
            GO TO 70
         END IF
      ELSE
         DO 100 J = 1, NRHS
            B( N, J ) = B( N, J ) / D( N )
            IF( N.GT.1 )
     $         B( N-1, J ) = ( B( N-1, J )-DU( N-1 )*B( N, J ) ) /
     $                       D( N-1 )
            DO 90 I = N - 2, 1, -1
               B( I, J ) = ( B( I, J )-DU( I )*B( I+1, J )-DL( I )*
     $                     B( I+2, J ) ) / D( I )
   90       CONTINUE
  100    CONTINUE
      END IF
*
      RETURN
*
*     End of DGTSV
*
      END
*> \brief <b> DGTSVX computes the solution to system of linear equations A * X = B for GT matrices </b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGTSVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgtsvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgtsvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgtsvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
*                          DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
*                          WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          FACT, TRANS
*       INTEGER            INFO, LDB, LDX, N, NRHS
*       DOUBLE PRECISION   RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), IWORK( * )
*       DOUBLE PRECISION   B( LDB, * ), BERR( * ), D( * ), DF( * ),
*      $                   DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
*      $                   FERR( * ), WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGTSVX uses the LU factorization to compute the solution to a real
*> system of linear equations A * X = B or A**T * X = B,
*> where A is a tridiagonal matrix of order N and X and B are N-by-NRHS
*> matrices.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
*  =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'N', the LU decomposition is used to factor the matrix A
*>    as A = L * U, where L is a product of permutation and unit lower
*>    bidiagonal matrices and U is upper triangular with nonzeros in
*>    only the main diagonal and first two superdiagonals.
*>
*> 2. If some U(i,i)=0, so that U is exactly singular, then the routine
*>    returns with INFO = i. Otherwise, the factored form of A is used
*>    to estimate the condition number of the matrix A.  If the
*>    reciprocal of the condition number is less than machine precision,
*>    INFO = N+1 is returned as a warning, but the routine still goes on
*>    to solve for X and compute error bounds as described below.
*>
*> 3. The system of equations is solved for X using the factored form
*>    of A.
*>
*> 4. Iterative refinement is applied to improve the computed solution
*>    matrix and calculate error bounds and backward error estimates
*>    for it.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] FACT
*> \verbatim
*>          FACT is CHARACTER*1
*>          Specifies whether or not the factored form of A has been
*>          supplied on entry.
*>          = 'F':  DLF, DF, DUF, DU2, and IPIV contain the factored
*>                  form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV
*>                  will not be modified.
*>          = 'N':  The matrix will be copied to DLF, DF, and DUF
*>                  and factored.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the form of the system of equations:
*>          = 'N':  A * X = B     (No transpose)
*>          = 'T':  A**T * X = B  (Transpose)
*>          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] DL
*> \verbatim
*>          DL is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) subdiagonal elements of A.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The n diagonal elements of A.
*> \endverbatim
*>
*> \param[in] DU
*> \verbatim
*>          DU is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) superdiagonal elements of A.
*> \endverbatim
*>
*> \param[in,out] DLF
*> \verbatim
*>          DLF is DOUBLE PRECISION array, dimension (N-1)
*>          If FACT = 'F', then DLF is an input argument and on entry
*>          contains the (n-1) multipliers that define the matrix L from
*>          the LU factorization of A as computed by DGTTRF.
*>
*>          If FACT = 'N', then DLF is an output argument and on exit
*>          contains the (n-1) multipliers that define the matrix L from
*>          the LU factorization of A.
*> \endverbatim
*>
*> \param[in,out] DF
*> \verbatim
*>          DF is DOUBLE PRECISION array, dimension (N)
*>          If FACT = 'F', then DF is an input argument and on entry
*>          contains the n diagonal elements of the upper triangular
*>          matrix U from the LU factorization of A.
*>
*>          If FACT = 'N', then DF is an output argument and on exit
*>          contains the n diagonal elements of the upper triangular
*>          matrix U from the LU factorization of A.
*> \endverbatim
*>
*> \param[in,out] DUF
*> \verbatim
*>          DUF is DOUBLE PRECISION array, dimension (N-1)
*>          If FACT = 'F', then DUF is an input argument and on entry
*>          contains the (n-1) elements of the first superdiagonal of U.
*>
*>          If FACT = 'N', then DUF is an output argument and on exit
*>          contains the (n-1) elements of the first superdiagonal of U.
*> \endverbatim
*>
*> \param[in,out] DU2
*> \verbatim
*>          DU2 is DOUBLE PRECISION array, dimension (N-2)
*>          If FACT = 'F', then DU2 is an input argument and on entry
*>          contains the (n-2) elements of the second superdiagonal of
*>          U.
*>
*>          If FACT = 'N', then DU2 is an output argument and on exit
*>          contains the (n-2) elements of the second superdiagonal of
*>          U.
*> \endverbatim
*>
*> \param[in,out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          If FACT = 'F', then IPIV is an input argument and on entry
*>          contains the pivot indices from the LU factorization of A as
*>          computed by DGTTRF.
*>
*>          If FACT = 'N', then IPIV is an output argument and on exit
*>          contains the pivot indices from the LU factorization of A;
*>          row i of the matrix was interchanged with row IPIV(i).
*>          IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
*>          a row interchange was not required.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          The N-by-NRHS right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The estimate of the reciprocal condition number of the matrix
*>          A.  If RCOND is less than the machine precision (in
*>          particular, if RCOND = 0), the matrix is singular to working
*>          precision.  This condition is indicated by a return code of
*>          INFO > 0.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The estimated forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).  The estimate is as reliable as
*>          the estimate for RCOND, and is almost always a slight
*>          overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in
*>          any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, and i is
*>                <= N:  U(i,i) is exactly zero.  The factorization
*>                       has not been completed unless i = N, but the
*>                       factor U is exactly singular, so the solution
*>                       and error bounds could not be computed.
*>                       RCOND = 0 is returned.
*>                = N+1: U is nonsingular, but RCOND is less than machine
*>                       precision, meaning that the matrix is singular
*>                       to working precision.  Nevertheless, the
*>                       solution and error bounds are computed because
*>                       there are a number of situations where the
*>                       computed solution can be more accurate than the
*>                       value of RCOND would suggest.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gtsvx
*
*  =====================================================================
      SUBROUTINE DGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF,
     $                   DUF,
     $                   DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
     $                   WORK, IWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          FACT, TRANS
      INTEGER            INFO, LDB, LDX, N, NRHS
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), IWORK( * )
      DOUBLE PRECISION   B( LDB, * ), BERR( * ), D( * ), DF( * ),
     $                   DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
     $                   FERR( * ), WORK( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOFACT, NOTRAN
      CHARACTER          NORM
      DOUBLE PRECISION   ANORM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANGT
      EXTERNAL           LSAME, DLAMCH, DLANGT
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGTCON, DGTRFS, DGTTRF, DGTTRS,
     $                   DLACPY,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      NOFACT = LSAME( FACT, 'N' )
      NOTRAN = LSAME( TRANS, 'N' )
      IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
     $         LSAME( TRANS, 'C' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -14
      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
         INFO = -16
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGTSVX', -INFO )
         RETURN
      END IF
*
      IF( NOFACT ) THEN
*
*        Compute the LU factorization of A.
*
         CALL DCOPY( N, D, 1, DF, 1 )
         IF( N.GT.1 ) THEN
            CALL DCOPY( N-1, DL, 1, DLF, 1 )
            CALL DCOPY( N-1, DU, 1, DUF, 1 )
         END IF
         CALL DGTTRF( N, DLF, DF, DUF, DU2, IPIV, INFO )
*
*        Return if INFO is non-zero.
*
         IF( INFO.GT.0 )THEN
            RCOND = ZERO
            RETURN
         END IF
      END IF
*
*     Compute the norm of the matrix A.
*
      IF( NOTRAN ) THEN
         NORM = '1'
      ELSE
         NORM = 'I'
      END IF
      ANORM = DLANGT( NORM, N, DL, D, DU )
*
*     Compute the reciprocal of the condition number of A.
*
      CALL DGTCON( NORM, N, DLF, DF, DUF, DU2, IPIV, ANORM, RCOND,
     $             WORK,
     $             IWORK, INFO )
*
*     Compute the solution vectors X.
*
      CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
      CALL DGTTRS( TRANS, N, NRHS, DLF, DF, DUF, DU2, IPIV, X, LDX,
     $             INFO )
*
*     Use iterative refinement to improve the computed solutions and
*     compute error bounds and backward error estimates for them.
*
      CALL DGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2,
     $             IPIV,
     $             B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
*     Set INFO = N+1 if the matrix is singular to working precision.
*
      IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
     $   INFO = N + 1
*
      RETURN
*
*     End of DGTSVX
*
      END
*> \brief \b DGTTRF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGTTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgttrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgttrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgttrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGTTRF( N, DL, D, DU, DU2, IPIV, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   D( * ), DL( * ), DU( * ), DU2( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGTTRF computes an LU factorization of a real tridiagonal matrix A
*> using elimination with partial pivoting and row interchanges.
*>
*> The factorization has the form
*>    A = L * U
*> where L is a product of permutation and unit lower bidiagonal
*> matrices and U is upper triangular with nonzeros in only the main
*> diagonal and first two superdiagonals.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.
*> \endverbatim
*>
*> \param[in,out] DL
*> \verbatim
*>          DL is DOUBLE PRECISION array, dimension (N-1)
*>          On entry, DL must contain the (n-1) sub-diagonal elements of
*>          A.
*>
*>          On exit, DL is overwritten by the (n-1) multipliers that
*>          define the matrix L from the LU factorization of A.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, D must contain the diagonal elements of A.
*>
*>          On exit, D is overwritten by the n diagonal elements of the
*>          upper triangular matrix U from the LU factorization of A.
*> \endverbatim
*>
*> \param[in,out] DU
*> \verbatim
*>          DU is DOUBLE PRECISION array, dimension (N-1)
*>          On entry, DU must contain the (n-1) super-diagonal elements
*>          of A.
*>
*>          On exit, DU is overwritten by the (n-1) elements of the first
*>          super-diagonal of U.
*> \endverbatim
*>
*> \param[out] DU2
*> \verbatim
*>          DU2 is DOUBLE PRECISION array, dimension (N-2)
*>          On exit, DU2 is overwritten by the (n-2) elements of the
*>          second super-diagonal of U.
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          The pivot indices; for 1 <= i <= n, row i of the matrix was
*>          interchanged with row IPIV(i).  IPIV(i) will always be either
*>          i or i+1; IPIV(i) = i indicates a row interchange was not
*>          required.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -k, the k-th argument had an illegal value
*>          > 0:  if INFO = k, U(k,k) is exactly zero. The factorization
*>                has been completed, but the factor U is exactly
*>                singular, and division by zero will occur if it is used
*>                to solve a system of equations.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gttrf
*
*  =====================================================================
      SUBROUTINE DGTTRF( N, DL, D, DU, DU2, IPIV, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   D( * ), DL( * ), DU( * ), DU2( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
      DOUBLE PRECISION   FACT, TEMP
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      IF( N.LT.0 ) THEN
         INFO = -1
         CALL XERBLA( 'DGTTRF', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Initialize IPIV(i) = i and DU2(I) = 0
*
      DO 10 I = 1, N
         IPIV( I ) = I
   10 CONTINUE
      DO 20 I = 1, N - 2
         DU2( I ) = ZERO
   20 CONTINUE
*
      DO 30 I = 1, N - 2
         IF( ABS( D( I ) ).GE.ABS( DL( I ) ) ) THEN
*
*           No row interchange required, eliminate DL(I)
*
            IF( D( I ).NE.ZERO ) THEN
               FACT = DL( I ) / D( I )
               DL( I ) = FACT
               D( I+1 ) = D( I+1 ) - FACT*DU( I )
            END IF
         ELSE
*
*           Interchange rows I and I+1, eliminate DL(I)
*
            FACT = D( I ) / DL( I )
            D( I ) = DL( I )
            DL( I ) = FACT
            TEMP = DU( I )
            DU( I ) = D( I+1 )
            D( I+1 ) = TEMP - FACT*D( I+1 )
            DU2( I ) = DU( I+1 )
            DU( I+1 ) = -FACT*DU( I+1 )
            IPIV( I ) = I + 1
         END IF
   30 CONTINUE
      IF( N.GT.1 ) THEN
         I = N - 1
         IF( ABS( D( I ) ).GE.ABS( DL( I ) ) ) THEN
            IF( D( I ).NE.ZERO ) THEN
               FACT = DL( I ) / D( I )
               DL( I ) = FACT
               D( I+1 ) = D( I+1 ) - FACT*DU( I )
            END IF
         ELSE
            FACT = D( I ) / DL( I )
            D( I ) = DL( I )
            DL( I ) = FACT
            TEMP = DU( I )
            DU( I ) = D( I+1 )
            D( I+1 ) = TEMP - FACT*D( I+1 )
            IPIV( I ) = I + 1
         END IF
      END IF
*
*     Check for a zero on the diagonal of U.
*
      DO 40 I = 1, N
         IF( D( I ).EQ.ZERO ) THEN
            INFO = I
            GO TO 50
         END IF
   40 CONTINUE
   50 CONTINUE
*
      RETURN
*
*     End of DGTTRF
*
      END
*> \brief \b DGTTRS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGTTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgttrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgttrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgttrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGTTRS( TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB,
*                          INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANS
*       INTEGER            INFO, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   B( LDB, * ), D( * ), DL( * ), DU( * ), DU2( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGTTRS solves one of the systems of equations
*>    A*X = B  or  A**T*X = B,
*> with a tridiagonal matrix A using the LU factorization computed
*> by DGTTRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the form of the system of equations.
*>          = 'N':  A * X = B  (No transpose)
*>          = 'T':  A**T* X = B  (Transpose)
*>          = 'C':  A**T* X = B  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] DL
*> \verbatim
*>          DL is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) multipliers that define the matrix L from the
*>          LU factorization of A.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The n diagonal elements of the upper triangular matrix U from
*>          the LU factorization of A.
*> \endverbatim
*>
*> \param[in] DU
*> \verbatim
*>          DU is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) elements of the first super-diagonal of U.
*> \endverbatim
*>
*> \param[in] DU2
*> \verbatim
*>          DU2 is DOUBLE PRECISION array, dimension (N-2)
*>          The (n-2) elements of the second super-diagonal of U.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          The pivot indices; for 1 <= i <= n, row i of the matrix was
*>          interchanged with row IPIV(i).  IPIV(i) will always be either
*>          i or i+1; IPIV(i) = i indicates a row interchange was not
*>          required.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the matrix of right hand side vectors B.
*>          On exit, B is overwritten by the solution vectors X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gttrs
*
*  =====================================================================
      SUBROUTINE DGTTRS( TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B,
     $                   LDB,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS
      INTEGER            INFO, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   B( LDB, * ), D( * ), DL( * ), DU( * ), DU2( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            NOTRAN
      INTEGER            ITRANS, J, JB, NB
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGTTS2, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      NOTRAN = ( TRANS.EQ.'N' .OR. TRANS.EQ.'n' )
      IF( .NOT.NOTRAN .AND. .NOT.( TRANS.EQ.'T' .OR. TRANS.EQ.
     $    't' ) .AND. .NOT.( TRANS.EQ.'C' .OR. TRANS.EQ.'c' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDB.LT.MAX( N, 1 ) ) THEN
         INFO = -10
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGTTRS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 )
     $   RETURN
*
*     Decode TRANS
*
      IF( NOTRAN ) THEN
         ITRANS = 0
      ELSE
         ITRANS = 1
      END IF
*
*     Determine the number of right-hand sides to solve at a time.
*
      IF( NRHS.EQ.1 ) THEN
         NB = 1
      ELSE
         NB = MAX( 1, ILAENV( 1, 'DGTTRS', TRANS, N, NRHS, -1, -1 ) )
      END IF
*
      IF( NB.GE.NRHS ) THEN
         CALL DGTTS2( ITRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB )
      ELSE
         DO 10 J = 1, NRHS, NB
            JB = MIN( NRHS-J+1, NB )
            CALL DGTTS2( ITRANS, N, JB, DL, D, DU, DU2, IPIV, B( 1,
     $                   J ),
     $                   LDB )
   10    CONTINUE
      END IF
*
*     End of DGTTRS
*
      END
*> \brief \b DGTTS2 solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by sgttrf.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGTTS2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgtts2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgtts2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgtts2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGTTS2( ITRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB )
*
*       .. Scalar Arguments ..
*       INTEGER            ITRANS, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   B( LDB, * ), D( * ), DL( * ), DU( * ), DU2( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGTTS2 solves one of the systems of equations
*>    A*X = B  or  A**T*X = B,
*> with a tridiagonal matrix A using the LU factorization computed
*> by DGTTRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ITRANS
*> \verbatim
*>          ITRANS is INTEGER
*>          Specifies the form of the system of equations.
*>          = 0:  A * X = B  (No transpose)
*>          = 1:  A**T* X = B  (Transpose)
*>          = 2:  A**T* X = B  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] DL
*> \verbatim
*>          DL is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) multipliers that define the matrix L from the
*>          LU factorization of A.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The n diagonal elements of the upper triangular matrix U from
*>          the LU factorization of A.
*> \endverbatim
*>
*> \param[in] DU
*> \verbatim
*>          DU is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) elements of the first super-diagonal of U.
*> \endverbatim
*>
*> \param[in] DU2
*> \verbatim
*>          DU2 is DOUBLE PRECISION array, dimension (N-2)
*>          The (n-2) elements of the second super-diagonal of U.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          The pivot indices; for 1 <= i <= n, row i of the matrix was
*>          interchanged with row IPIV(i).  IPIV(i) will always be either
*>          i or i+1; IPIV(i) = i indicates a row interchange was not
*>          required.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the matrix of right hand side vectors B.
*>          On exit, B is overwritten by the solution vectors X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gtts2
*
*  =====================================================================
      SUBROUTINE DGTTS2( ITRANS, N, NRHS, DL, D, DU, DU2, IPIV, B,
     $                   LDB )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            ITRANS, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   B( LDB, * ), D( * ), DL( * ), DU( * ), DU2( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            I, IP, J
      DOUBLE PRECISION   TEMP
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 )
     $   RETURN
*
      IF( ITRANS.EQ.0 ) THEN
*
*        Solve A*X = B using the LU factorization of A,
*        overwriting each right hand side vector with its solution.
*
         IF( NRHS.LE.1 ) THEN
            J = 1
   10       CONTINUE
*
*           Solve L*x = b.
*
            DO 20 I = 1, N - 1
               IP = IPIV( I )
               TEMP = B( I+1-IP+I, J ) - DL( I )*B( IP, J )
               B( I, J ) = B( IP, J )
               B( I+1, J ) = TEMP
   20       CONTINUE
*
*           Solve U*x = b.
*
            B( N, J ) = B( N, J ) / D( N )
            IF( N.GT.1 )
     $         B( N-1, J ) = ( B( N-1, J )-DU( N-1 )*B( N, J ) ) /
     $                       D( N-1 )
            DO 30 I = N - 2, 1, -1
               B( I, J ) = ( B( I, J )-DU( I )*B( I+1, J )-DU2( I )*
     $                     B( I+2, J ) ) / D( I )
   30       CONTINUE
            IF( J.LT.NRHS ) THEN
               J = J + 1
               GO TO 10
            END IF
         ELSE
            DO 60 J = 1, NRHS
*
*              Solve L*x = b.
*
               DO 40 I = 1, N - 1
                  IF( IPIV( I ).EQ.I ) THEN
                     B( I+1, J ) = B( I+1, J ) - DL( I )*B( I, J )
                  ELSE
                     TEMP = B( I, J )
                     B( I, J ) = B( I+1, J )
                     B( I+1, J ) = TEMP - DL( I )*B( I, J )
                  END IF
   40          CONTINUE
*
*              Solve U*x = b.
*
               B( N, J ) = B( N, J ) / D( N )
               IF( N.GT.1 )
     $            B( N-1, J ) = ( B( N-1, J )-DU( N-1 )*B( N, J ) ) /
     $                          D( N-1 )
               DO 50 I = N - 2, 1, -1
                  B( I, J ) = ( B( I, J )-DU( I )*B( I+1, J )-DU2( I )*
     $                        B( I+2, J ) ) / D( I )
   50          CONTINUE
   60       CONTINUE
         END IF
      ELSE
*
*        Solve A**T * X = B.
*
         IF( NRHS.LE.1 ) THEN
*
*           Solve U**T*x = b.
*
            J = 1
   70       CONTINUE
            B( 1, J ) = B( 1, J ) / D( 1 )
            IF( N.GT.1 )
     $         B( 2, J ) = ( B( 2, J )-DU( 1 )*B( 1, J ) ) / D( 2 )
            DO 80 I = 3, N
               B( I, J ) = ( B( I, J )-DU( I-1 )*B( I-1, J )-DU2( I-2 )*
     $                     B( I-2, J ) ) / D( I )
   80       CONTINUE
*
*           Solve L**T*x = b.
*
            DO 90 I = N - 1, 1, -1
               IP = IPIV( I )
               TEMP = B( I, J ) - DL( I )*B( I+1, J )
               B( I, J ) = B( IP, J )
               B( IP, J ) = TEMP
   90       CONTINUE
            IF( J.LT.NRHS ) THEN
               J = J + 1
               GO TO 70
            END IF
*
         ELSE
            DO 120 J = 1, NRHS
*
*              Solve U**T*x = b.
*
               B( 1, J ) = B( 1, J ) / D( 1 )
               IF( N.GT.1 )
     $            B( 2, J ) = ( B( 2, J )-DU( 1 )*B( 1, J ) ) / D( 2 )
               DO 100 I = 3, N
                  B( I, J ) = ( B( I, J )-DU( I-1 )*B( I-1, J )-
     $                        DU2( I-2 )*B( I-2, J ) ) / D( I )
  100          CONTINUE
               DO 110 I = N - 1, 1, -1
                  IF( IPIV( I ).EQ.I ) THEN
                     B( I, J ) = B( I, J ) - DL( I )*B( I+1, J )
                  ELSE
                     TEMP = B( I+1, J )
                     B( I+1, J ) = B( I, J ) - DL( I )*TEMP
                     B( I, J ) = TEMP
                  END IF
  110          CONTINUE
  120       CONTINUE
         END IF
      END IF
*
*     End of DGTTS2
*
      END
*> \brief \b DHGEQZ
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DHGEQZ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dhgeqz.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dhgeqz.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dhgeqz.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
*                          ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK,
*                          LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          COMPQ, COMPZ, JOB
*       INTEGER            IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   ALPHAI( * ), ALPHAR( * ), BETA( * ),
*      $                   H( LDH, * ), Q( LDQ, * ), T( LDT, * ),
*      $                   WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DHGEQZ computes the eigenvalues of a real matrix pair (H,T),
*> where H is an upper Hessenberg matrix and T is upper triangular,
*> using the double-shift QZ method.
*> Matrix pairs of this type are produced by the reduction to
*> generalized upper Hessenberg form of a real matrix pair (A,B):
*>
*>    A = Q1*H*Z1**T,  B = Q1*T*Z1**T,
*>
*> as computed by DGGHRD.
*>
*> If JOB='S', then the Hessenberg-triangular pair (H,T) is
*> also reduced to generalized Schur form,
*>
*>    H = Q*S*Z**T,  T = Q*P*Z**T,
*>
*> where Q and Z are orthogonal matrices, P is an upper triangular
*> matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
*> diagonal blocks.
*>
*> The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
*> (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
*> eigenvalues.
*>
*> Additionally, the 2-by-2 upper triangular diagonal blocks of P
*> corresponding to 2-by-2 blocks of S are reduced to positive diagonal
*> form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
*> P(j,j) > 0, and P(j+1,j+1) > 0.
*>
*> Optionally, the orthogonal matrix Q from the generalized Schur
*> factorization may be postmultiplied into an input matrix Q1, and the
*> orthogonal matrix Z may be postmultiplied into an input matrix Z1.
*> If Q1 and Z1 are the orthogonal matrices from DGGHRD that reduced
*> the matrix pair (A,B) to generalized upper Hessenberg form, then the
*> output matrices Q1*Q and Z1*Z are the orthogonal factors from the
*> generalized Schur factorization of (A,B):
*>
*>    A = (Q1*Q)*S*(Z1*Z)**T,  B = (Q1*Q)*P*(Z1*Z)**T.
*>
*> To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
*> of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
*> complex and beta real.
*> If beta is nonzero, lambda = alpha / beta is an eigenvalue of the
*> generalized nonsymmetric eigenvalue problem (GNEP)
*>    A*x = lambda*B*x
*> and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
*> alternate form of the GNEP
*>    mu*A*y = B*y.
*> Real eigenvalues can be read directly from the generalized Schur
*> form:
*>   alpha = S(i,i), beta = P(i,i).
*>
*> Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
*>      Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
*>      pp. 241--256.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOB
*> \verbatim
*>          JOB is CHARACTER*1
*>          = 'E': Compute eigenvalues only;
*>          = 'S': Compute eigenvalues and the Schur form.
*> \endverbatim
*>
*> \param[in] COMPQ
*> \verbatim
*>          COMPQ is CHARACTER*1
*>          = 'N': Left Schur vectors (Q) are not computed;
*>          = 'I': Q is initialized to the unit matrix and the matrix Q
*>                 of left Schur vectors of (H,T) is returned;
*>          = 'V': Q must contain an orthogonal matrix Q1 on entry and
*>                 the product Q1*Q is returned.
*> \endverbatim
*>
*> \param[in] COMPZ
*> \verbatim
*>          COMPZ is CHARACTER*1
*>          = 'N': Right Schur vectors (Z) are not computed;
*>          = 'I': Z is initialized to the unit matrix and the matrix Z
*>                 of right Schur vectors of (H,T) is returned;
*>          = 'V': Z must contain an orthogonal matrix Z1 on entry and
*>                 the product Z1*Z is returned.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices H, T, Q, and Z.  N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*>          ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*>          IHI is INTEGER
*>          ILO and IHI mark the rows and columns of H which are in
*>          Hessenberg form.  It is assumed that A is already upper
*>          triangular in rows and columns 1:ILO-1 and IHI+1:N.
*>          If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*>          H is DOUBLE PRECISION array, dimension (LDH, N)
*>          On entry, the N-by-N upper Hessenberg matrix H.
*>          On exit, if JOB = 'S', H contains the upper quasi-triangular
*>          matrix S from the generalized Schur factorization.
*>          If JOB = 'E', the diagonal blocks of H match those of S, but
*>          the rest of H is unspecified.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*>          LDH is INTEGER
*>          The leading dimension of the array H.  LDH >= max( 1, N ).
*> \endverbatim
*>
*> \param[in,out] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT, N)
*>          On entry, the N-by-N upper triangular matrix T.
*>          On exit, if JOB = 'S', T contains the upper triangular
*>          matrix P from the generalized Schur factorization;
*>          2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
*>          are reduced to positive diagonal form, i.e., if H(j+1,j) is
*>          non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
*>          T(j+1,j+1) > 0.
*>          If JOB = 'E', the diagonal blocks of T match those of P, but
*>          the rest of T is unspecified.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T.  LDT >= max( 1, N ).
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*>          ALPHAR is DOUBLE PRECISION array, dimension (N)
*>          The real parts of each scalar alpha defining an eigenvalue
*>          of GNEP.
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*>          ALPHAI is DOUBLE PRECISION array, dimension (N)
*>          The imaginary parts of each scalar alpha defining an
*>          eigenvalue of GNEP.
*>          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
*>          positive, then the j-th and (j+1)-st eigenvalues are a
*>          complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION array, dimension (N)
*>          The scalars beta that define the eigenvalues of GNEP.
*>          Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
*>          beta = BETA(j) represent the j-th eigenvalue of the matrix
*>          pair (A,B), in one of the forms lambda = alpha/beta or
*>          mu = beta/alpha.  Since either lambda or mu may overflow,
*>          they should not, in general, be computed.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ, N)
*>          On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in
*>          the reduction of (A,B) to generalized Hessenberg form.
*>          On exit, if COMPQ = 'I', the orthogonal matrix of left Schur
*>          vectors of (H,T), and if COMPQ = 'V', the orthogonal matrix
*>          of left Schur vectors of (A,B).
*>          Not referenced if COMPQ = 'N'.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q.  LDQ >= 1.
*>          If COMPQ='V' or 'I', then LDQ >= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
*>          On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
*>          the reduction of (A,B) to generalized Hessenberg form.
*>          On exit, if COMPZ = 'I', the orthogonal matrix of
*>          right Schur vectors of (H,T), and if COMPZ = 'V', the
*>          orthogonal matrix of right Schur vectors of (A,B).
*>          Not referenced if COMPZ = 'N'.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1.
*>          If COMPZ='V' or 'I', then LDZ >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.  LWORK >= max(1,N).
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*>          = 1,...,N: the QZ iteration did not converge.  (H,T) is not
*>                     in Schur form, but ALPHAR(i), ALPHAI(i), and
*>                     BETA(i), i=INFO+1,...,N should be correct.
*>          = N+1,...,2*N: the shift calculation failed.  (H,T) is not
*>                     in Schur form, but ALPHAR(i), ALPHAI(i), and
*>                     BETA(i), i=INFO-N+1,...,N should be correct.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hgeqz
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  Iteration counters:
*>
*>  JITER  -- counts iterations.
*>  IITER  -- counts iterations run since ILAST was last
*>            changed.  This is therefore reset only when a 1-by-1 or
*>            2-by-2 block deflates off the bottom.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T,
     $                   LDT,
     $                   ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK,
     $                   LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          COMPQ, COMPZ, JOB
      INTEGER            IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   ALPHAI( * ), ALPHAR( * ), BETA( * ),
     $                   H( LDH, * ), Q( LDQ, * ), T( LDT, * ),
     $                   WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
*    $                     SAFETY = 1.0E+0 )
      DOUBLE PRECISION   HALF, ZERO, ONE, SAFETY
      PARAMETER          ( HALF = 0.5D+0, ZERO = 0.0D+0, ONE = 1.0D+0,
     $                   SAFETY = 1.0D+2 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ILAZR2, ILAZRO, ILPIVT, ILQ, ILSCHR, ILZ,
     $                   LQUERY
      INTEGER            ICOMPQ, ICOMPZ, IFIRST, IFRSTM, IITER, ILAST,
     $                   ILASTM, IN, ISCHUR, ISTART, J, JC, JCH, JITER,
     $                   JR, MAXIT
      DOUBLE PRECISION   A11, A12, A1I, A1R, A21, A22, A2I, A2R, AD11,
     $                   AD11L, AD12, AD12L, AD21, AD21L, AD22, AD22L,
     $                   AD32L, AN, ANORM, ASCALE, ATOL, B11, B1A, B1I,
     $                   B1R, B22, B2A, B2I, B2R, BN, BNORM, BSCALE,
     $                   BTOL, C, C11I, C11R, C12, C21, C22I, C22R, CL,
     $                   CQ, CR, CZ, ESHIFT, S, S1, S1INV, S2, SAFMAX,
     $                   SAFMIN, SCALE, SL, SQI, SQR, SR, SZI, SZR, T1,
     $                   T2, T3, TAU, TEMP, TEMP2, TEMPI, TEMPR, U1,
     $                   U12, U12L, U2, ULP, VS, W11, W12, W21, W22,
     $                   WABS, WI, WR, WR2
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   V( 3 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANHS, DLAPY2, DLAPY3
      EXTERNAL           LSAME, DLAMCH, DLANHS, DLAPY2,
     $                   DLAPY3
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLAG2, DLARFG, DLARTG, DLASET, DLASV2,
     $                   DROT,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Decode JOB, COMPQ, COMPZ
*
      IF( LSAME( JOB, 'E' ) ) THEN
         ILSCHR = .FALSE.
         ISCHUR = 1
      ELSE IF( LSAME( JOB, 'S' ) ) THEN
         ILSCHR = .TRUE.
         ISCHUR = 2
      ELSE
         ISCHUR = 0
      END IF
*
      IF( LSAME( COMPQ, 'N' ) ) THEN
         ILQ = .FALSE.
         ICOMPQ = 1
      ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
         ILQ = .TRUE.
         ICOMPQ = 2
      ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
         ILQ = .TRUE.
         ICOMPQ = 3
      ELSE
         ICOMPQ = 0
      END IF
*
      IF( LSAME( COMPZ, 'N' ) ) THEN
         ILZ = .FALSE.
         ICOMPZ = 1
      ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
         ILZ = .TRUE.
         ICOMPZ = 2
      ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
         ILZ = .TRUE.
         ICOMPZ = 3
      ELSE
         ICOMPZ = 0
      END IF
*
*     Check Argument Values
*
      INFO = 0
      WORK( 1 ) = MAX( 1, N )
      LQUERY = ( LWORK.EQ.-1 )
      IF( ISCHUR.EQ.0 ) THEN
         INFO = -1
      ELSE IF( ICOMPQ.EQ.0 ) THEN
         INFO = -2
      ELSE IF( ICOMPZ.EQ.0 ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( ILO.LT.1 ) THEN
         INFO = -5
      ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
         INFO = -6
      ELSE IF( LDH.LT.N ) THEN
         INFO = -8
      ELSE IF( LDT.LT.N ) THEN
         INFO = -10
      ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN
         INFO = -15
      ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
         INFO = -17
      ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
         INFO = -19
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DHGEQZ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.LE.0 ) THEN
         WORK( 1 ) = DBLE( 1 )
         RETURN
      END IF
*
*     Initialize Q and Z
*
      IF( ICOMPQ.EQ.3 )
     $   CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
      IF( ICOMPZ.EQ.3 )
     $   CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
*
*     Machine Constants
*
      IN = IHI + 1 - ILO
      SAFMIN = DLAMCH( 'S' )
      SAFMAX = ONE / SAFMIN
      ULP = DLAMCH( 'E' )*DLAMCH( 'B' )
      ANORM = DLANHS( 'F', IN, H( ILO, ILO ), LDH, WORK )
      BNORM = DLANHS( 'F', IN, T( ILO, ILO ), LDT, WORK )
      ATOL = MAX( SAFMIN, ULP*ANORM )
      BTOL = MAX( SAFMIN, ULP*BNORM )
      ASCALE = ONE / MAX( SAFMIN, ANORM )
      BSCALE = ONE / MAX( SAFMIN, BNORM )
*
*     Set Eigenvalues IHI+1:N
*
      DO 30 J = IHI + 1, N
         IF( T( J, J ).LT.ZERO ) THEN
            IF( ILSCHR ) THEN
               DO 10 JR = 1, J
                  H( JR, J ) = -H( JR, J )
                  T( JR, J ) = -T( JR, J )
   10          CONTINUE
            ELSE
               H( J, J ) = -H( J, J )
               T( J, J ) = -T( J, J )
            END IF
            IF( ILZ ) THEN
               DO 20 JR = 1, N
                  Z( JR, J ) = -Z( JR, J )
   20          CONTINUE
            END IF
         END IF
         ALPHAR( J ) = H( J, J )
         ALPHAI( J ) = ZERO
         BETA( J ) = T( J, J )
   30 CONTINUE
*
*     If IHI < ILO, skip QZ steps
*
      IF( IHI.LT.ILO )
     $   GO TO 380
*
*     MAIN QZ ITERATION LOOP
*
*     Initialize dynamic indices
*
*     Eigenvalues ILAST+1:N have been found.
*        Column operations modify rows IFRSTM:whatever.
*        Row operations modify columns whatever:ILASTM.
*
*     If only eigenvalues are being computed, then
*        IFRSTM is the row of the last splitting row above row ILAST;
*        this is always at least ILO.
*     IITER counts iterations since the last eigenvalue was found,
*        to tell when to use an extraordinary shift.
*     MAXIT is the maximum number of QZ sweeps allowed.
*
      ILAST = IHI
      IF( ILSCHR ) THEN
         IFRSTM = 1
         ILASTM = N
      ELSE
         IFRSTM = ILO
         ILASTM = IHI
      END IF
      IITER = 0
      ESHIFT = ZERO
      MAXIT = 30*( IHI-ILO+1 )
*
      DO 360 JITER = 1, MAXIT
*
*        Split the matrix if possible.
*
*        Two tests:
*           1: H(j,j-1)=0  or  j=ILO
*           2: T(j,j)=0
*
         IF( ILAST.EQ.ILO ) THEN
*
*           Special case: j=ILAST
*
            GO TO 80
         ELSE
            IF( ABS( H( ILAST, ILAST-1 ) ).LE.MAX( SAFMIN, ULP*( 
     $         ABS( H( ILAST, ILAST ) ) + ABS( H( ILAST-1, ILAST-1 ) ) 
     $         ) ) ) THEN
               H( ILAST, ILAST-1 ) = ZERO
               GO TO 80
            END IF
         END IF
*
         IF( ABS( T( ILAST, ILAST ) ).LE.BTOL ) THEN
            T( ILAST, ILAST ) = ZERO
            GO TO 70
         END IF
*
*        General case: j<ILAST
*
         DO 60 J = ILAST - 1, ILO, -1
*
*           Test 1: for H(j,j-1)=0 or j=ILO
*
            IF( J.EQ.ILO ) THEN
               ILAZRO = .TRUE.
            ELSE
               IF( ABS( H( J, J-1 ) ).LE.MAX( SAFMIN, ULP*( 
     $         ABS( H( J, J ) ) + ABS( H( J-1, J-1 ) ) 
     $         ) ) ) THEN
                  H( J, J-1 ) = ZERO
                  ILAZRO = .TRUE.
               ELSE
                  ILAZRO = .FALSE.
               END IF
            END IF
*
*           Test 2: for T(j,j)=0
*
            IF( ABS( T( J, J ) ).LT.BTOL ) THEN
               T( J, J ) = ZERO
*
*              Test 1a: Check for 2 consecutive small subdiagonals in A
*
               ILAZR2 = .FALSE.
               IF( .NOT.ILAZRO ) THEN
                  TEMP = ABS( H( J, J-1 ) )
                  TEMP2 = ABS( H( J, J ) )
                  TEMPR = MAX( TEMP, TEMP2 )
                  IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
                     TEMP = TEMP / TEMPR
                     TEMP2 = TEMP2 / TEMPR
                  END IF
                  IF( TEMP*( ASCALE*ABS( H( J+1, J ) ) ).LE.TEMP2*
     $                ( ASCALE*ATOL ) )ILAZR2 = .TRUE.
               END IF
*
*              If both tests pass (1 & 2), i.e., the leading diagonal
*              element of B in the block is zero, split a 1x1 block off
*              at the top. (I.e., at the J-th row/column) The leading
*              diagonal element of the remainder can also be zero, so
*              this may have to be done repeatedly.
*
               IF( ILAZRO .OR. ILAZR2 ) THEN
                  DO 40 JCH = J, ILAST - 1
                     TEMP = H( JCH, JCH )
                     CALL DLARTG( TEMP, H( JCH+1, JCH ), C, S,
     $                            H( JCH, JCH ) )
                     H( JCH+1, JCH ) = ZERO
                     CALL DROT( ILASTM-JCH, H( JCH, JCH+1 ), LDH,
     $                          H( JCH+1, JCH+1 ), LDH, C, S )
                     CALL DROT( ILASTM-JCH, T( JCH, JCH+1 ), LDT,
     $                          T( JCH+1, JCH+1 ), LDT, C, S )
                     IF( ILQ )
     $                  CALL DROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ),
     $                             1,
     $                             C, S )
                     IF( ILAZR2 )
     $                  H( JCH, JCH-1 ) = H( JCH, JCH-1 )*C
                     ILAZR2 = .FALSE.
                     IF( ABS( T( JCH+1, JCH+1 ) ).GE.BTOL ) THEN
                        IF( JCH+1.GE.ILAST ) THEN
                           GO TO 80
                        ELSE
                           IFIRST = JCH + 1
                           GO TO 110
                        END IF
                     END IF
                     T( JCH+1, JCH+1 ) = ZERO
   40             CONTINUE
                  GO TO 70
               ELSE
*
*                 Only test 2 passed -- chase the zero to T(ILAST,ILAST)
*                 Then process as in the case T(ILAST,ILAST)=0
*
                  DO 50 JCH = J, ILAST - 1
                     TEMP = T( JCH, JCH+1 )
                     CALL DLARTG( TEMP, T( JCH+1, JCH+1 ), C, S,
     $                            T( JCH, JCH+1 ) )
                     T( JCH+1, JCH+1 ) = ZERO
                     IF( JCH.LT.ILASTM-1 )
     $                  CALL DROT( ILASTM-JCH-1, T( JCH, JCH+2 ),
     $                             LDT,
     $                             T( JCH+1, JCH+2 ), LDT, C, S )
                     CALL DROT( ILASTM-JCH+2, H( JCH, JCH-1 ), LDH,
     $                          H( JCH+1, JCH-1 ), LDH, C, S )
                     IF( ILQ )
     $                  CALL DROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ),
     $                             1,
     $                             C, S )
                     TEMP = H( JCH+1, JCH )
                     CALL DLARTG( TEMP, H( JCH+1, JCH-1 ), C, S,
     $                            H( JCH+1, JCH ) )
                     H( JCH+1, JCH-1 ) = ZERO
                     CALL DROT( JCH+1-IFRSTM, H( IFRSTM, JCH ), 1,
     $                          H( IFRSTM, JCH-1 ), 1, C, S )
                     CALL DROT( JCH-IFRSTM, T( IFRSTM, JCH ), 1,
     $                          T( IFRSTM, JCH-1 ), 1, C, S )
                     IF( ILZ )
     $                  CALL DROT( N, Z( 1, JCH ), 1, Z( 1, JCH-1 ),
     $                             1,
     $                             C, S )
   50             CONTINUE
                  GO TO 70
               END IF
            ELSE IF( ILAZRO ) THEN
*
*              Only test 1 passed -- work on J:ILAST
*
               IFIRST = J
               GO TO 110
            END IF
*
*           Neither test passed -- try next J
*
   60    CONTINUE
*
*        (Drop-through is "impossible")
*
         INFO = N + 1
         GO TO 420
*
*        T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a
*        1x1 block.
*
   70    CONTINUE
         TEMP = H( ILAST, ILAST )
         CALL DLARTG( TEMP, H( ILAST, ILAST-1 ), C, S,
     $                H( ILAST, ILAST ) )
         H( ILAST, ILAST-1 ) = ZERO
         CALL DROT( ILAST-IFRSTM, H( IFRSTM, ILAST ), 1,
     $              H( IFRSTM, ILAST-1 ), 1, C, S )
         CALL DROT( ILAST-IFRSTM, T( IFRSTM, ILAST ), 1,
     $              T( IFRSTM, ILAST-1 ), 1, C, S )
         IF( ILZ )
     $      CALL DROT( N, Z( 1, ILAST ), 1, Z( 1, ILAST-1 ), 1, C,
     $                 S )
*
*        H(ILAST,ILAST-1)=0 -- Standardize B, set ALPHAR, ALPHAI,
*                              and BETA
*
   80    CONTINUE
         IF( T( ILAST, ILAST ).LT.ZERO ) THEN
            IF( ILSCHR ) THEN
               DO 90 J = IFRSTM, ILAST
                  H( J, ILAST ) = -H( J, ILAST )
                  T( J, ILAST ) = -T( J, ILAST )
   90          CONTINUE
            ELSE
               H( ILAST, ILAST ) = -H( ILAST, ILAST )
               T( ILAST, ILAST ) = -T( ILAST, ILAST )
            END IF
            IF( ILZ ) THEN
               DO 100 J = 1, N
                  Z( J, ILAST ) = -Z( J, ILAST )
  100          CONTINUE
            END IF
         END IF
         ALPHAR( ILAST ) = H( ILAST, ILAST )
         ALPHAI( ILAST ) = ZERO
         BETA( ILAST ) = T( ILAST, ILAST )
*
*        Go to next block -- exit if finished.
*
         ILAST = ILAST - 1
         IF( ILAST.LT.ILO )
     $      GO TO 380
*
*        Reset counters
*
         IITER = 0
         ESHIFT = ZERO
         IF( .NOT.ILSCHR ) THEN
            ILASTM = ILAST
            IF( IFRSTM.GT.ILAST )
     $         IFRSTM = ILO
         END IF
         GO TO 350
*
*        QZ step
*
*        This iteration only involves rows/columns IFIRST:ILAST. We
*        assume IFIRST < ILAST, and that the diagonal of B is non-zero.
*
  110    CONTINUE
         IITER = IITER + 1
         IF( .NOT.ILSCHR ) THEN
            IFRSTM = IFIRST
         END IF
*
*        Compute single shifts.
*
*        At this point, IFIRST < ILAST, and the diagonal elements of
*        T(IFIRST:ILAST,IFIRST,ILAST) are larger than BTOL (in
*        magnitude)
*
         IF( ( IITER / 10 )*10.EQ.IITER ) THEN
*
*           Exceptional shift.  Chosen for no particularly good reason.
*           (Single shift only.)
*
            IF( ( DBLE( MAXIT )*SAFMIN )*ABS( H( ILAST, ILAST-1 ) ).LT.
     $          ABS( T( ILAST-1, ILAST-1 ) ) ) THEN
               ESHIFT = H( ILAST, ILAST-1 ) /
     $                  T( ILAST-1, ILAST-1 )
            ELSE
               ESHIFT = ESHIFT + ONE / ( SAFMIN*DBLE( MAXIT ) )
            END IF
            S1 = ONE
            WR = ESHIFT
*
         ELSE
*
*           Shifts based on the generalized eigenvalues of the
*           bottom-right 2x2 block of A and B. The first eigenvalue
*           returned by DLAG2 is the Wilkinson shift (AEP p.512),
*
            CALL DLAG2( H( ILAST-1, ILAST-1 ), LDH,
     $                  T( ILAST-1, ILAST-1 ), LDT, SAFMIN*SAFETY, S1,
     $                  S2, WR, WR2, WI )
*
            IF ( ABS( (WR/S1)*T( ILAST, ILAST ) - H( ILAST, ILAST ) )
     $         .GT. ABS( (WR2/S2)*T( ILAST, ILAST )
     $         - H( ILAST, ILAST ) ) ) THEN
               TEMP = WR
               WR = WR2
               WR2 = TEMP
               TEMP = S1
               S1 = S2
               S2 = TEMP
            END IF
            TEMP = MAX( S1, SAFMIN*MAX( ONE, ABS( WR ), ABS( WI ) ) )
            IF( WI.NE.ZERO )
     $         GO TO 200
         END IF
*
*        Fiddle with shift to avoid overflow
*
         TEMP = MIN( ASCALE, ONE )*( HALF*SAFMAX )
         IF( S1.GT.TEMP ) THEN
            SCALE = TEMP / S1
         ELSE
            SCALE = ONE
         END IF
*
         TEMP = MIN( BSCALE, ONE )*( HALF*SAFMAX )
         IF( ABS( WR ).GT.TEMP )
     $      SCALE = MIN( SCALE, TEMP / ABS( WR ) )
         S1 = SCALE*S1
         WR = SCALE*WR
*
*        Now check for two consecutive small subdiagonals.
*
         DO 120 J = ILAST - 1, IFIRST + 1, -1
            ISTART = J
            TEMP = ABS( S1*H( J, J-1 ) )
            TEMP2 = ABS( S1*H( J, J )-WR*T( J, J ) )
            TEMPR = MAX( TEMP, TEMP2 )
            IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
               TEMP = TEMP / TEMPR
               TEMP2 = TEMP2 / TEMPR
            END IF
            IF( ABS( ( ASCALE*H( J+1, J ) )*TEMP ).LE.( ASCALE*ATOL )*
     $          TEMP2 )GO TO 130
  120    CONTINUE
*
         ISTART = IFIRST
  130    CONTINUE
*
*        Do an implicit single-shift QZ sweep.
*
*        Initial Q
*
         TEMP = S1*H( ISTART, ISTART ) - WR*T( ISTART, ISTART )
         TEMP2 = S1*H( ISTART+1, ISTART )
         CALL DLARTG( TEMP, TEMP2, C, S, TEMPR )
*
*        Sweep
*
         DO 190 J = ISTART, ILAST - 1
            IF( J.GT.ISTART ) THEN
               TEMP = H( J, J-1 )
               CALL DLARTG( TEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
               H( J+1, J-1 ) = ZERO
            END IF
*
            DO 140 JC = J, ILASTM
               TEMP = C*H( J, JC ) + S*H( J+1, JC )
               H( J+1, JC ) = -S*H( J, JC ) + C*H( J+1, JC )
               H( J, JC ) = TEMP
               TEMP2 = C*T( J, JC ) + S*T( J+1, JC )
               T( J+1, JC ) = -S*T( J, JC ) + C*T( J+1, JC )
               T( J, JC ) = TEMP2
  140       CONTINUE
            IF( ILQ ) THEN
               DO 150 JR = 1, N
                  TEMP = C*Q( JR, J ) + S*Q( JR, J+1 )
                  Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
                  Q( JR, J ) = TEMP
  150          CONTINUE
            END IF
*
            TEMP = T( J+1, J+1 )
            CALL DLARTG( TEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
            T( J+1, J ) = ZERO
*
            DO 160 JR = IFRSTM, MIN( J+2, ILAST )
               TEMP = C*H( JR, J+1 ) + S*H( JR, J )
               H( JR, J ) = -S*H( JR, J+1 ) + C*H( JR, J )
               H( JR, J+1 ) = TEMP
  160       CONTINUE
            DO 170 JR = IFRSTM, J
               TEMP = C*T( JR, J+1 ) + S*T( JR, J )
               T( JR, J ) = -S*T( JR, J+1 ) + C*T( JR, J )
               T( JR, J+1 ) = TEMP
  170       CONTINUE
            IF( ILZ ) THEN
               DO 180 JR = 1, N
                  TEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
                  Z( JR, J ) = -S*Z( JR, J+1 ) + C*Z( JR, J )
                  Z( JR, J+1 ) = TEMP
  180          CONTINUE
            END IF
  190    CONTINUE
*
         GO TO 350
*
*        Use Francis double-shift
*
*        Note: the Francis double-shift should work with real shifts,
*              but only if the block is at least 3x3.
*              This code may break if this point is reached with
*              a 2x2 block with real eigenvalues.
*
  200    CONTINUE
         IF( IFIRST+1.EQ.ILAST ) THEN
*
*           Special case -- 2x2 block with complex eigenvectors
*
*           Step 1: Standardize, that is, rotate so that
*
*                       ( B11  0  )
*                   B = (         )  with B11 non-negative.
*                       (  0  B22 )
*
            CALL DLASV2( T( ILAST-1, ILAST-1 ), T( ILAST-1, ILAST ),
     $                   T( ILAST, ILAST ), B22, B11, SR, CR, SL, CL )
*
            IF( B11.LT.ZERO ) THEN
               CR = -CR
               SR = -SR
               B11 = -B11
               B22 = -B22
            END IF
*
            CALL DROT( ILASTM+1-IFIRST, H( ILAST-1, ILAST-1 ), LDH,
     $                 H( ILAST, ILAST-1 ), LDH, CL, SL )
            CALL DROT( ILAST+1-IFRSTM, H( IFRSTM, ILAST-1 ), 1,
     $                 H( IFRSTM, ILAST ), 1, CR, SR )
*
            IF( ILAST.LT.ILASTM )
     $         CALL DROT( ILASTM-ILAST, T( ILAST-1, ILAST+1 ), LDT,
     $                    T( ILAST, ILAST+1 ), LDT, CL, SL )
            IF( IFRSTM.LT.ILAST-1 )
     $         CALL DROT( IFIRST-IFRSTM, T( IFRSTM, ILAST-1 ), 1,
     $                    T( IFRSTM, ILAST ), 1, CR, SR )
*
            IF( ILQ )
     $         CALL DROT( N, Q( 1, ILAST-1 ), 1, Q( 1, ILAST ), 1,
     $                    CL,
     $                    SL )
            IF( ILZ )
     $         CALL DROT( N, Z( 1, ILAST-1 ), 1, Z( 1, ILAST ), 1,
     $                    CR,
     $                    SR )
*
            T( ILAST-1, ILAST-1 ) = B11
            T( ILAST-1, ILAST ) = ZERO
            T( ILAST, ILAST-1 ) = ZERO
            T( ILAST, ILAST ) = B22
*
*           If B22 is negative, negate column ILAST
*
            IF( B22.LT.ZERO ) THEN
               DO 210 J = IFRSTM, ILAST
                  H( J, ILAST ) = -H( J, ILAST )
                  T( J, ILAST ) = -T( J, ILAST )
  210          CONTINUE
*
               IF( ILZ ) THEN
                  DO 220 J = 1, N
                     Z( J, ILAST ) = -Z( J, ILAST )
  220             CONTINUE
               END IF
               B22 = -B22
            END IF
*
*           Step 2: Compute ALPHAR, ALPHAI, and BETA (see refs.)
*
*           Recompute shift
*
            CALL DLAG2( H( ILAST-1, ILAST-1 ), LDH,
     $                  T( ILAST-1, ILAST-1 ), LDT, SAFMIN*SAFETY, S1,
     $                  TEMP, WR, TEMP2, WI )
*
*           If standardization has perturbed the shift onto real line,
*           do another (real single-shift) QR step.
*
            IF( WI.EQ.ZERO )
     $         GO TO 350
            S1INV = ONE / S1
*
*           Do EISPACK (QZVAL) computation of alpha and beta
*
            A11 = H( ILAST-1, ILAST-1 )
            A21 = H( ILAST, ILAST-1 )
            A12 = H( ILAST-1, ILAST )
            A22 = H( ILAST, ILAST )
*
*           Compute complex Givens rotation on right
*           (Assume some element of C = (sA - wB) > unfl )
*                            __
*           (sA - wB) ( CZ   -SZ )
*                     ( SZ    CZ )
*
            C11R = S1*A11 - WR*B11
            C11I = -WI*B11
            C12 = S1*A12
            C21 = S1*A21
            C22R = S1*A22 - WR*B22
            C22I = -WI*B22
*
            IF( ABS( C11R )+ABS( C11I )+ABS( C12 ).GT.ABS( C21 )+
     $          ABS( C22R )+ABS( C22I ) ) THEN
               T1 = DLAPY3( C12, C11R, C11I )
               CZ = C12 / T1
               SZR = -C11R / T1
               SZI = -C11I / T1
            ELSE
               CZ = DLAPY2( C22R, C22I )
               IF( CZ.LE.SAFMIN ) THEN
                  CZ = ZERO
                  SZR = ONE
                  SZI = ZERO
               ELSE
                  TEMPR = C22R / CZ
                  TEMPI = C22I / CZ
                  T1 = DLAPY2( CZ, C21 )
                  CZ = CZ / T1
                  SZR = -C21*TEMPR / T1
                  SZI = C21*TEMPI / T1
               END IF
            END IF
*
*           Compute Givens rotation on left
*
*           (  CQ   SQ )
*           (  __      )  A or B
*           ( -SQ   CQ )
*
            AN = ABS( A11 ) + ABS( A12 ) + ABS( A21 ) + ABS( A22 )
            BN = ABS( B11 ) + ABS( B22 )
            WABS = ABS( WR ) + ABS( WI )
            IF( S1*AN.GT.WABS*BN ) THEN
               CQ = CZ*B11
               SQR = SZR*B22
               SQI = -SZI*B22
            ELSE
               A1R = CZ*A11 + SZR*A12
               A1I = SZI*A12
               A2R = CZ*A21 + SZR*A22
               A2I = SZI*A22
               CQ = DLAPY2( A1R, A1I )
               IF( CQ.LE.SAFMIN ) THEN
                  CQ = ZERO
                  SQR = ONE
                  SQI = ZERO
               ELSE
                  TEMPR = A1R / CQ
                  TEMPI = A1I / CQ
                  SQR = TEMPR*A2R + TEMPI*A2I
                  SQI = TEMPI*A2R - TEMPR*A2I
               END IF
            END IF
            T1 = DLAPY3( CQ, SQR, SQI )
            CQ = CQ / T1
            SQR = SQR / T1
            SQI = SQI / T1
*
*           Compute diagonal elements of QBZ
*
            TEMPR = SQR*SZR - SQI*SZI
            TEMPI = SQR*SZI + SQI*SZR
            B1R = CQ*CZ*B11 + TEMPR*B22
            B1I = TEMPI*B22
            B1A = DLAPY2( B1R, B1I )
            B2R = CQ*CZ*B22 + TEMPR*B11
            B2I = -TEMPI*B11
            B2A = DLAPY2( B2R, B2I )
*
*           Normalize so beta > 0, and Im( alpha1 ) > 0
*
            BETA( ILAST-1 ) = B1A
            BETA( ILAST ) = B2A
            ALPHAR( ILAST-1 ) = ( WR*B1A )*S1INV
            ALPHAI( ILAST-1 ) = ( WI*B1A )*S1INV
            ALPHAR( ILAST ) = ( WR*B2A )*S1INV
            ALPHAI( ILAST ) = -( WI*B2A )*S1INV
*
*           Step 3: Go to next block -- exit if finished.
*
            ILAST = IFIRST - 1
            IF( ILAST.LT.ILO )
     $         GO TO 380
*
*           Reset counters
*
            IITER = 0
            ESHIFT = ZERO
            IF( .NOT.ILSCHR ) THEN
               ILASTM = ILAST
               IF( IFRSTM.GT.ILAST )
     $            IFRSTM = ILO
            END IF
            GO TO 350
         ELSE
*
*           Usual case: 3x3 or larger block, using Francis implicit
*                       double-shift
*
*                                    2
*           Eigenvalue equation is  w  - c w + d = 0,
*
*                                         -1 2        -1
*           so compute 1st column of  (A B  )  - c A B   + d
*           using the formula in QZIT (from EISPACK)
*
*           We assume that the block is at least 3x3
*
            AD11 = ( ASCALE*H( ILAST-1, ILAST-1 ) ) /
     $             ( BSCALE*T( ILAST-1, ILAST-1 ) )
            AD21 = ( ASCALE*H( ILAST, ILAST-1 ) ) /
     $             ( BSCALE*T( ILAST-1, ILAST-1 ) )
            AD12 = ( ASCALE*H( ILAST-1, ILAST ) ) /
     $             ( BSCALE*T( ILAST, ILAST ) )
            AD22 = ( ASCALE*H( ILAST, ILAST ) ) /
     $             ( BSCALE*T( ILAST, ILAST ) )
            U12 = T( ILAST-1, ILAST ) / T( ILAST, ILAST )
            AD11L = ( ASCALE*H( IFIRST, IFIRST ) ) /
     $              ( BSCALE*T( IFIRST, IFIRST ) )
            AD21L = ( ASCALE*H( IFIRST+1, IFIRST ) ) /
     $              ( BSCALE*T( IFIRST, IFIRST ) )
            AD12L = ( ASCALE*H( IFIRST, IFIRST+1 ) ) /
     $              ( BSCALE*T( IFIRST+1, IFIRST+1 ) )
            AD22L = ( ASCALE*H( IFIRST+1, IFIRST+1 ) ) /
     $              ( BSCALE*T( IFIRST+1, IFIRST+1 ) )
            AD32L = ( ASCALE*H( IFIRST+2, IFIRST+1 ) ) /
     $              ( BSCALE*T( IFIRST+1, IFIRST+1 ) )
            U12L = T( IFIRST, IFIRST+1 ) / T( IFIRST+1, IFIRST+1 )
*
            V( 1 ) = ( AD11-AD11L )*( AD22-AD11L ) - AD12*AD21 +
     $               AD21*U12*AD11L + ( AD12L-AD11L*U12L )*AD21L
            V( 2 ) = ( ( AD22L-AD11L )-AD21L*U12L-( AD11-AD11L )-
     $               ( AD22-AD11L )+AD21*U12 )*AD21L
            V( 3 ) = AD32L*AD21L
*
            ISTART = IFIRST
*
            CALL DLARFG( 3, V( 1 ), V( 2 ), 1, TAU )
            V( 1 ) = ONE
*
*           Sweep
*
            DO 290 J = ISTART, ILAST - 2
*
*              All but last elements: use 3x3 Householder transforms.
*
*              Zero (j-1)st column of A
*
               IF( J.GT.ISTART ) THEN
                  V( 1 ) = H( J, J-1 )
                  V( 2 ) = H( J+1, J-1 )
                  V( 3 ) = H( J+2, J-1 )
*
                  CALL DLARFG( 3, H( J, J-1 ), V( 2 ), 1, TAU )
                  V( 1 ) = ONE
                  H( J+1, J-1 ) = ZERO
                  H( J+2, J-1 ) = ZERO
               END IF
*
               T2 = TAU*V( 2 )
               T3 = TAU*V( 3 )
               DO 230 JC = J, ILASTM
                  TEMP = H( J, JC )+V( 2 )*H( J+1, JC )+V( 3 )*
     $                   H( J+2, JC )
                  H( J, JC ) = H( J, JC ) - TEMP*TAU
                  H( J+1, JC ) = H( J+1, JC ) - TEMP*T2
                  H( J+2, JC ) = H( J+2, JC ) - TEMP*T3
                  TEMP2 = T( J, JC )+V( 2 )*T( J+1, JC )+V( 3 )*
     $                    T( J+2, JC )
                  T( J, JC ) = T( J, JC ) - TEMP2*TAU
                  T( J+1, JC ) = T( J+1, JC ) - TEMP2*T2
                  T( J+2, JC ) = T( J+2, JC ) - TEMP2*T3
  230          CONTINUE
               IF( ILQ ) THEN
                  DO 240 JR = 1, N
                     TEMP = Q( JR, J )+V( 2 )*Q( JR, J+1 )+V( 3 )*
     $                      Q( JR, J+2 )
                     Q( JR, J ) = Q( JR, J ) - TEMP*TAU
                     Q( JR, J+1 ) = Q( JR, J+1 ) - TEMP*T2
                     Q( JR, J+2 ) = Q( JR, J+2 ) - TEMP*T3
  240             CONTINUE
               END IF
*
*              Zero j-th column of B (see DLAGBC for details)
*
*              Swap rows to pivot
*
               ILPIVT = .FALSE.
               TEMP = MAX( ABS( T( J+1, J+1 ) ), ABS( T( J+1, J+2 ) ) )
               TEMP2 = MAX( ABS( T( J+2, J+1 ) ), ABS( T( J+2, J+2 ) ) )
               IF( MAX( TEMP, TEMP2 ).LT.SAFMIN ) THEN
                  SCALE = ZERO
                  U1 = ONE
                  U2 = ZERO
                  GO TO 250
               ELSE IF( TEMP.GE.TEMP2 ) THEN
                  W11 = T( J+1, J+1 )
                  W21 = T( J+2, J+1 )
                  W12 = T( J+1, J+2 )
                  W22 = T( J+2, J+2 )
                  U1 = T( J+1, J )
                  U2 = T( J+2, J )
               ELSE
                  W21 = T( J+1, J+1 )
                  W11 = T( J+2, J+1 )
                  W22 = T( J+1, J+2 )
                  W12 = T( J+2, J+2 )
                  U2 = T( J+1, J )
                  U1 = T( J+2, J )
               END IF
*
*              Swap columns if nec.
*
               IF( ABS( W12 ).GT.ABS( W11 ) ) THEN
                  ILPIVT = .TRUE.
                  TEMP = W12
                  TEMP2 = W22
                  W12 = W11
                  W22 = W21
                  W11 = TEMP
                  W21 = TEMP2
               END IF
*
*              LU-factor
*
               TEMP = W21 / W11
               U2 = U2 - TEMP*U1
               W22 = W22 - TEMP*W12
               W21 = ZERO
*
*              Compute SCALE
*
               SCALE = ONE
               IF( ABS( W22 ).LT.SAFMIN ) THEN
                  SCALE = ZERO
                  U2 = ONE
                  U1 = -W12 / W11
                  GO TO 250
               END IF
               IF( ABS( W22 ).LT.ABS( U2 ) )
     $            SCALE = ABS( W22 / U2 )
               IF( ABS( W11 ).LT.ABS( U1 ) )
     $            SCALE = MIN( SCALE, ABS( W11 / U1 ) )
*
*              Solve
*
               U2 = ( SCALE*U2 ) / W22
               U1 = ( SCALE*U1-W12*U2 ) / W11
*
  250          CONTINUE
               IF( ILPIVT ) THEN
                  TEMP = U2
                  U2 = U1
                  U1 = TEMP
               END IF
*
*              Compute Householder Vector
*
               T1 = SQRT( SCALE**2+U1**2+U2**2 )
               TAU = ONE + SCALE / T1
               VS = -ONE / ( SCALE+T1 )
               V( 1 ) = ONE
               V( 2 ) = VS*U1
               V( 3 ) = VS*U2
*
*              Apply transformations from the right.
*
               T2 = TAU*V(2)
               T3 = TAU*V(3)
               DO 260 JR = IFRSTM, MIN( J+3, ILAST )
                  TEMP = H( JR, J )+V( 2 )*H( JR, J+1 )+V( 3 )*
     $                   H( JR, J+2 )
                  H( JR, J ) = H( JR, J ) - TEMP*TAU
                  H( JR, J+1 ) = H( JR, J+1 ) - TEMP*T2
                  H( JR, J+2 ) = H( JR, J+2 ) - TEMP*T3
  260          CONTINUE
               DO 270 JR = IFRSTM, J + 2
                  TEMP = T( JR, J )+V( 2 )*T( JR, J+1 )+V( 3 )*
     $                   T( JR, J+2 )
                  T( JR, J ) = T( JR, J ) - TEMP*TAU
                  T( JR, J+1 ) = T( JR, J+1 ) - TEMP*T2
                  T( JR, J+2 ) = T( JR, J+2 ) - TEMP*T3
  270          CONTINUE
               IF( ILZ ) THEN
                  DO 280 JR = 1, N
                     TEMP = Z( JR, J )+V( 2 )*Z( JR, J+1 )+V( 3 )*
     $                      Z( JR, J+2 )
                     Z( JR, J ) = Z( JR, J ) - TEMP*TAU
                     Z( JR, J+1 ) = Z( JR, J+1 ) - TEMP*T2
                     Z( JR, J+2 ) = Z( JR, J+2 ) - TEMP*T3
  280             CONTINUE
               END IF
               T( J+1, J ) = ZERO
               T( J+2, J ) = ZERO
  290       CONTINUE
*
*           Last elements: Use Givens rotations
*
*           Rotations from the left
*
            J = ILAST - 1
            TEMP = H( J, J-1 )
            CALL DLARTG( TEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
            H( J+1, J-1 ) = ZERO
*
            DO 300 JC = J, ILASTM
               TEMP = C*H( J, JC ) + S*H( J+1, JC )
               H( J+1, JC ) = -S*H( J, JC ) + C*H( J+1, JC )
               H( J, JC ) = TEMP
               TEMP2 = C*T( J, JC ) + S*T( J+1, JC )
               T( J+1, JC ) = -S*T( J, JC ) + C*T( J+1, JC )
               T( J, JC ) = TEMP2
  300       CONTINUE
            IF( ILQ ) THEN
               DO 310 JR = 1, N
                  TEMP = C*Q( JR, J ) + S*Q( JR, J+1 )
                  Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
                  Q( JR, J ) = TEMP
  310          CONTINUE
            END IF
*
*           Rotations from the right.
*
            TEMP = T( J+1, J+1 )
            CALL DLARTG( TEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
            T( J+1, J ) = ZERO
*
            DO 320 JR = IFRSTM, ILAST
               TEMP = C*H( JR, J+1 ) + S*H( JR, J )
               H( JR, J ) = -S*H( JR, J+1 ) + C*H( JR, J )
               H( JR, J+1 ) = TEMP
  320       CONTINUE
            DO 330 JR = IFRSTM, ILAST - 1
               TEMP = C*T( JR, J+1 ) + S*T( JR, J )
               T( JR, J ) = -S*T( JR, J+1 ) + C*T( JR, J )
               T( JR, J+1 ) = TEMP
  330       CONTINUE
            IF( ILZ ) THEN
               DO 340 JR = 1, N
                  TEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
                  Z( JR, J ) = -S*Z( JR, J+1 ) + C*Z( JR, J )
                  Z( JR, J+1 ) = TEMP
  340          CONTINUE
            END IF
*
*           End of Double-Shift code
*
         END IF
*
         GO TO 350
*
*        End of iteration loop
*
  350    CONTINUE
  360 CONTINUE
*
*     Drop-through = non-convergence
*
      INFO = ILAST
      GO TO 420
*
*     Successful completion of all QZ steps
*
  380 CONTINUE
*
*     Set Eigenvalues 1:ILO-1
*
      DO 410 J = 1, ILO - 1
         IF( T( J, J ).LT.ZERO ) THEN
            IF( ILSCHR ) THEN
               DO 390 JR = 1, J
                  H( JR, J ) = -H( JR, J )
                  T( JR, J ) = -T( JR, J )
  390          CONTINUE
            ELSE
               H( J, J ) = -H( J, J )
               T( J, J ) = -T( J, J )
            END IF
            IF( ILZ ) THEN
               DO 400 JR = 1, N
                  Z( JR, J ) = -Z( JR, J )
  400          CONTINUE
            END IF
         END IF
         ALPHAR( J ) = H( J, J )
         ALPHAI( J ) = ZERO
         BETA( J ) = T( J, J )
  410 CONTINUE
*
*     Normal Termination
*
      INFO = 0
*
*     Exit (other than argument error) -- return optimal workspace size
*
  420 CONTINUE
      WORK( 1 ) = DBLE( N )
      RETURN
*
*     End of DHGEQZ
*
      END
*> \brief \b DHSEIN
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DHSEIN + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dhsein.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dhsein.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dhsein.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, WR, WI,
*                          VL, LDVL, VR, LDVR, MM, M, WORK, IFAILL,
*                          IFAILR, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          EIGSRC, INITV, SIDE
*       INTEGER            INFO, LDH, LDVL, LDVR, M, MM, N
*       ..
*       .. Array Arguments ..
*       LOGICAL            SELECT( * )
*       INTEGER            IFAILL( * ), IFAILR( * )
*       DOUBLE PRECISION   H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ),
*      $                   WI( * ), WORK( * ), WR( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DHSEIN uses inverse iteration to find specified right and/or left
*> eigenvectors of a real upper Hessenberg matrix H.
*>
*> The right eigenvector x and the left eigenvector y of the matrix H
*> corresponding to an eigenvalue w are defined by:
*>
*>              H * x = w * x,     y**h * H = w * y**h
*>
*> where y**h denotes the conjugate transpose of the vector y.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'R': compute right eigenvectors only;
*>          = 'L': compute left eigenvectors only;
*>          = 'B': compute both right and left eigenvectors.
*> \endverbatim
*>
*> \param[in] EIGSRC
*> \verbatim
*>          EIGSRC is CHARACTER*1
*>          Specifies the source of eigenvalues supplied in (WR,WI):
*>          = 'Q': the eigenvalues were found using DHSEQR; thus, if
*>                 H has zero subdiagonal elements, and so is
*>                 block-triangular, then the j-th eigenvalue can be
*>                 assumed to be an eigenvalue of the block containing
*>                 the j-th row/column.  This property allows DHSEIN to
*>                 perform inverse iteration on just one diagonal block.
*>          = 'N': no assumptions are made on the correspondence
*>                 between eigenvalues and diagonal blocks.  In this
*>                 case, DHSEIN must always perform inverse iteration
*>                 using the whole matrix H.
*> \endverbatim
*>
*> \param[in] INITV
*> \verbatim
*>          INITV is CHARACTER*1
*>          = 'N': no initial vectors are supplied;
*>          = 'U': user-supplied initial vectors are stored in the arrays
*>                 VL and/or VR.
*> \endverbatim
*>
*> \param[in,out] SELECT
*> \verbatim
*>          SELECT is LOGICAL array, dimension (N)
*>          Specifies the eigenvectors to be computed. To select the
*>          real eigenvector corresponding to a real eigenvalue WR(j),
*>          SELECT(j) must be set to .TRUE.. To select the complex
*>          eigenvector corresponding to a complex eigenvalue
*>          (WR(j),WI(j)), with complex conjugate (WR(j+1),WI(j+1)),
*>          either SELECT(j) or SELECT(j+1) or both must be set to
*>          .TRUE.; then on exit SELECT(j) is .TRUE. and SELECT(j+1) is
*>          .FALSE..
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix H.  N >= 0.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*>          H is DOUBLE PRECISION array, dimension (LDH,N)
*>          The upper Hessenberg matrix H.
*>          If a NaN is detected in H, the routine will return with INFO=-6.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*>          LDH is INTEGER
*>          The leading dimension of the array H.  LDH >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] WR
*> \verbatim
*>          WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[in] WI
*> \verbatim
*>          WI is DOUBLE PRECISION array, dimension (N)
*>
*>          On entry, the real and imaginary parts of the eigenvalues of
*>          H; a complex conjugate pair of eigenvalues must be stored in
*>          consecutive elements of WR and WI.
*>          On exit, WR may have been altered since close eigenvalues
*>          are perturbed slightly in searching for independent
*>          eigenvectors.
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*>          VL is DOUBLE PRECISION array, dimension (LDVL,MM)
*>          On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must
*>          contain starting vectors for the inverse iteration for the
*>          left eigenvectors; the starting vector for each eigenvector
*>          must be in the same column(s) in which the eigenvector will
*>          be stored.
*>          On exit, if SIDE = 'L' or 'B', the left eigenvectors
*>          specified by SELECT will be stored consecutively in the
*>          columns of VL, in the same order as their eigenvalues. A
*>          complex eigenvector corresponding to a complex eigenvalue is
*>          stored in two consecutive columns, the first holding the real
*>          part and the second the imaginary part.
*>          If SIDE = 'R', VL is not referenced.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*>          LDVL is INTEGER
*>          The leading dimension of the array VL.
*>          LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise.
*> \endverbatim
*>
*> \param[in,out] VR
*> \verbatim
*>          VR is DOUBLE PRECISION array, dimension (LDVR,MM)
*>          On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR must
*>          contain starting vectors for the inverse iteration for the
*>          right eigenvectors; the starting vector for each eigenvector
*>          must be in the same column(s) in which the eigenvector will
*>          be stored.
*>          On exit, if SIDE = 'R' or 'B', the right eigenvectors
*>          specified by SELECT will be stored consecutively in the
*>          columns of VR, in the same order as their eigenvalues. A
*>          complex eigenvector corresponding to a complex eigenvalue is
*>          stored in two consecutive columns, the first holding the real
*>          part and the second the imaginary part.
*>          If SIDE = 'L', VR is not referenced.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*>          LDVR is INTEGER
*>          The leading dimension of the array VR.
*>          LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise.
*> \endverbatim
*>
*> \param[in] MM
*> \verbatim
*>          MM is INTEGER
*>          The number of columns in the arrays VL and/or VR. MM >= M.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The number of columns in the arrays VL and/or VR required to
*>          store the eigenvectors; each selected real eigenvector
*>          occupies one column and each selected complex eigenvector
*>          occupies two columns.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension ((N+2)*N)
*> \endverbatim
*>
*> \param[out] IFAILL
*> \verbatim
*>          IFAILL is INTEGER array, dimension (MM)
*>          If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left
*>          eigenvector in the i-th column of VL (corresponding to the
*>          eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the
*>          eigenvector converged satisfactorily. If the i-th and (i+1)th
*>          columns of VL hold a complex eigenvector, then IFAILL(i) and
*>          IFAILL(i+1) are set to the same value.
*>          If SIDE = 'R', IFAILL is not referenced.
*> \endverbatim
*>
*> \param[out] IFAILR
*> \verbatim
*>          IFAILR is INTEGER array, dimension (MM)
*>          If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right
*>          eigenvector in the i-th column of VR (corresponding to the
*>          eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the
*>          eigenvector converged satisfactorily. If the i-th and (i+1)th
*>          columns of VR hold a complex eigenvector, then IFAILR(i) and
*>          IFAILR(i+1) are set to the same value.
*>          If SIDE = 'L', IFAILR is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, i is the number of eigenvectors which
*>                failed to converge; see IFAILL and IFAILR for further
*>                details.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hsein
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  Each eigenvector is normalized so that the element of largest
*>  magnitude has magnitude 1; here the magnitude of a complex number
*>  (x,y) is taken to be |x|+|y|.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, WR,
     $                   WI,
     $                   VL, LDVL, VR, LDVR, MM, M, WORK, IFAILL,
     $                   IFAILR, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          EIGSRC, INITV, SIDE
      INTEGER            INFO, LDH, LDVL, LDVR, M, MM, N
*     ..
*     .. Array Arguments ..
      LOGICAL            SELECT( * )
      INTEGER            IFAILL( * ), IFAILR( * )
      DOUBLE PRECISION   H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ),
     $                   WI( * ), WORK( * ), WR( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            BOTHV, FROMQR, LEFTV, NOINIT, PAIR, RIGHTV
      INTEGER            I, IINFO, K, KL, KLN, KR, KSI, KSR, LDWORK
      DOUBLE PRECISION   BIGNUM, EPS3, HNORM, SMLNUM, ULP, UNFL, WKI,
     $                   WKR
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      DOUBLE PRECISION   DLAMCH, DLANHS
      EXTERNAL           LSAME, DLAMCH, DLANHS, DISNAN
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLAEIN, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. Executable Statements ..
*
*     Decode and test the input parameters.
*
      BOTHV = LSAME( SIDE, 'B' )
      RIGHTV = LSAME( SIDE, 'R' ) .OR. BOTHV
      LEFTV = LSAME( SIDE, 'L' ) .OR. BOTHV
*
      FROMQR = LSAME( EIGSRC, 'Q' )
*
      NOINIT = LSAME( INITV, 'N' )
*
*     Set M to the number of columns required to store the selected
*     eigenvectors, and standardize the array SELECT.
*
      M = 0
      PAIR = .FALSE.
      DO 10 K = 1, N
         IF( PAIR ) THEN
            PAIR = .FALSE.
            SELECT( K ) = .FALSE.
         ELSE
            IF( WI( K ).EQ.ZERO ) THEN
               IF( SELECT( K ) )
     $            M = M + 1
            ELSE
               PAIR = .TRUE.
               IF( SELECT( K ) .OR. SELECT( K+1 ) ) THEN
                  SELECT( K ) = .TRUE.
                  M = M + 2
               END IF
            END IF
         END IF
   10 CONTINUE
*
      INFO = 0
      IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
         INFO = -1
      ELSE IF( .NOT.FROMQR .AND. .NOT.LSAME( EIGSRC, 'N' ) ) THEN
         INFO = -2
      ELSE IF( .NOT.NOINIT .AND. .NOT.LSAME( INITV, 'U' ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDVL.LT.1 .OR. ( LEFTV .AND. LDVL.LT.N ) ) THEN
         INFO = -11
      ELSE IF( LDVR.LT.1 .OR. ( RIGHTV .AND. LDVR.LT.N ) ) THEN
         INFO = -13
      ELSE IF( MM.LT.M ) THEN
         INFO = -14
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DHSEIN', -INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Set machine-dependent constants.
*
      UNFL = DLAMCH( 'Safe minimum' )
      ULP = DLAMCH( 'Precision' )
      SMLNUM = UNFL*( N / ULP )
      BIGNUM = ( ONE-ULP ) / SMLNUM
*
      LDWORK = N + 1
*
      KL = 1
      KLN = 0
      IF( FROMQR ) THEN
         KR = 0
      ELSE
         KR = N
      END IF
      KSR = 1
*
      DO 120 K = 1, N
         IF( SELECT( K ) ) THEN
*
*           Compute eigenvector(s) corresponding to W(K).
*
            IF( FROMQR ) THEN
*
*              If affiliation of eigenvalues is known, check whether
*              the matrix splits.
*
*              Determine KL and KR such that 1 <= KL <= K <= KR <= N
*              and H(KL,KL-1) and H(KR+1,KR) are zero (or KL = 1 or
*              KR = N).
*
*              Then inverse iteration can be performed with the
*              submatrix H(KL:N,KL:N) for a left eigenvector, and with
*              the submatrix H(1:KR,1:KR) for a right eigenvector.
*
               DO 20 I = K, KL + 1, -1
                  IF( H( I, I-1 ).EQ.ZERO )
     $               GO TO 30
   20          CONTINUE
   30          CONTINUE
               KL = I
               IF( K.GT.KR ) THEN
                  DO 40 I = K, N - 1
                     IF( H( I+1, I ).EQ.ZERO )
     $                  GO TO 50
   40             CONTINUE
   50             CONTINUE
                  KR = I
               END IF
            END IF
*
            IF( KL.NE.KLN ) THEN
               KLN = KL
*
*              Compute infinity-norm of submatrix H(KL:KR,KL:KR) if it
*              has not ben computed before.
*
               HNORM = DLANHS( 'I', KR-KL+1, H( KL, KL ), LDH, WORK )
               IF( DISNAN( HNORM ) ) THEN
                  INFO = -6
                  RETURN
               ELSE IF( HNORM.GT.ZERO ) THEN
                  EPS3 = HNORM*ULP
               ELSE
                  EPS3 = SMLNUM
               END IF
            END IF
*
*           Perturb eigenvalue if it is close to any previous
*           selected eigenvalues affiliated to the submatrix
*           H(KL:KR,KL:KR). Close roots are modified by EPS3.
*
            WKR = WR( K )
            WKI = WI( K )
   60       CONTINUE
            DO 70 I = K - 1, KL, -1
               IF( SELECT( I ) .AND. ABS( WR( I )-WKR )+
     $             ABS( WI( I )-WKI ).LT.EPS3 ) THEN
                  WKR = WKR + EPS3
                  GO TO 60
               END IF
   70       CONTINUE
            WR( K ) = WKR
*
            PAIR = WKI.NE.ZERO
            IF( PAIR ) THEN
               KSI = KSR + 1
            ELSE
               KSI = KSR
            END IF
            IF( LEFTV ) THEN
*
*              Compute left eigenvector.
*
               CALL DLAEIN( .FALSE., NOINIT, N-KL+1, H( KL, KL ),
     $                      LDH,
     $                      WKR, WKI, VL( KL, KSR ), VL( KL, KSI ),
     $                      WORK, LDWORK, WORK( N*N+N+1 ), EPS3, SMLNUM,
     $                      BIGNUM, IINFO )
               IF( IINFO.GT.0 ) THEN
                  IF( PAIR ) THEN
                     INFO = INFO + 2
                  ELSE
                     INFO = INFO + 1
                  END IF
                  IFAILL( KSR ) = K
                  IFAILL( KSI ) = K
               ELSE
                  IFAILL( KSR ) = 0
                  IFAILL( KSI ) = 0
               END IF
               DO 80 I = 1, KL - 1
                  VL( I, KSR ) = ZERO
   80          CONTINUE
               IF( PAIR ) THEN
                  DO 90 I = 1, KL - 1
                     VL( I, KSI ) = ZERO
   90             CONTINUE
               END IF
            END IF
            IF( RIGHTV ) THEN
*
*              Compute right eigenvector.
*
               CALL DLAEIN( .TRUE., NOINIT, KR, H, LDH, WKR, WKI,
     $                      VR( 1, KSR ), VR( 1, KSI ), WORK, LDWORK,
     $                      WORK( N*N+N+1 ), EPS3, SMLNUM, BIGNUM,
     $                      IINFO )
               IF( IINFO.GT.0 ) THEN
                  IF( PAIR ) THEN
                     INFO = INFO + 2
                  ELSE
                     INFO = INFO + 1
                  END IF
                  IFAILR( KSR ) = K
                  IFAILR( KSI ) = K
               ELSE
                  IFAILR( KSR ) = 0
                  IFAILR( KSI ) = 0
               END IF
               DO 100 I = KR + 1, N
                  VR( I, KSR ) = ZERO
  100          CONTINUE
               IF( PAIR ) THEN
                  DO 110 I = KR + 1, N
                     VR( I, KSI ) = ZERO
  110             CONTINUE
               END IF
            END IF
*
            IF( PAIR ) THEN
               KSR = KSR + 2
            ELSE
               KSR = KSR + 1
            END IF
         END IF
  120 CONTINUE
*
      RETURN
*
*     End of DHSEIN
*
      END
*> \brief \b DHSEQR
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DHSEQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dhseqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dhseqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dhseqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z,
*                          LDZ, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            IHI, ILO, INFO, LDH, LDZ, LWORK, N
*       CHARACTER          COMPZ, JOB
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   H( LDH, * ), WI( * ), WORK( * ), WR( * ),
*      $                   Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>    DHSEQR computes the eigenvalues of a Hessenberg matrix H
*>    and, optionally, the matrices T and Z from the Schur decomposition
*>    H = Z T Z**T, where T is an upper quasi-triangular matrix (the
*>    Schur form), and Z is the orthogonal matrix of Schur vectors.
*>
*>    Optionally Z may be postmultiplied into an input orthogonal
*>    matrix Q so that this routine can give the Schur factorization
*>    of a matrix A which has been reduced to the Hessenberg form H
*>    by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOB
*> \verbatim
*>          JOB is CHARACTER*1
*>           = 'E':  compute eigenvalues only;
*>           = 'S':  compute eigenvalues and the Schur form T.
*> \endverbatim
*>
*> \param[in] COMPZ
*> \verbatim
*>          COMPZ is CHARACTER*1
*>           = 'N':  no Schur vectors are computed;
*>           = 'I':  Z is initialized to the unit matrix and the matrix Z
*>                   of Schur vectors of H is returned;
*>           = 'V':  Z must contain an orthogonal matrix Q on entry, and
*>                   the product Q*Z is returned.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>           The order of the matrix H.  N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*>          ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*>          IHI is INTEGER
*>
*>           It is assumed that H is already upper triangular in rows
*>           and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
*>           set by a previous call to DGEBAL, and then passed to ZGEHRD
*>           when the matrix output by DGEBAL is reduced to Hessenberg
*>           form. Otherwise ILO and IHI should be set to 1 and N
*>           respectively.  If N > 0, then 1 <= ILO <= IHI <= N.
*>           If N = 0, then ILO = 1 and IHI = 0.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*>          H is DOUBLE PRECISION array, dimension (LDH,N)
*>           On entry, the upper Hessenberg matrix H.
*>           On exit, if INFO = 0 and JOB = 'S', then H contains the
*>           upper quasi-triangular matrix T from the Schur decomposition
*>           (the Schur form); 2-by-2 diagonal blocks (corresponding to
*>           complex conjugate pairs of eigenvalues) are returned in
*>           standard form, with H(i,i) = H(i+1,i+1) and
*>           H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and JOB = 'E', the
*>           contents of H are unspecified on exit.  (The output value of
*>           H when INFO > 0 is given under the description of INFO
*>           below.)
*>
*>           Unlike earlier versions of DHSEQR, this subroutine may
*>           explicitly H(i,j) = 0 for i > j and j = 1, 2, ... ILO-1
*>           or j = IHI+1, IHI+2, ... N.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*>          LDH is INTEGER
*>           The leading dimension of the array H. LDH >= max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*>          WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*>          WI is DOUBLE PRECISION array, dimension (N)
*>
*>           The real and imaginary parts, respectively, of the computed
*>           eigenvalues. If two eigenvalues are computed as a complex
*>           conjugate pair, they are stored in consecutive elements of
*>           WR and WI, say the i-th and (i+1)th, with WI(i) > 0 and
*>           WI(i+1) < 0. If JOB = 'S', the eigenvalues are stored in
*>           the same order as on the diagonal of the Schur form returned
*>           in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2
*>           diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
*>           WI(i+1) = -WI(i).
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ,N)
*>           If COMPZ = 'N', Z is not referenced.
*>           If COMPZ = 'I', on entry Z need not be set and on exit,
*>           if INFO = 0, Z contains the orthogonal matrix Z of the Schur
*>           vectors of H.  If COMPZ = 'V', on entry Z must contain an
*>           N-by-N matrix Q, which is assumed to be equal to the unit
*>           matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
*>           if INFO = 0, Z contains Q*Z.
*>           Normally Q is the orthogonal matrix generated by DORGHR
*>           after the call to DGEHRD which formed the Hessenberg matrix
*>           H. (The output value of Z when INFO > 0 is given under
*>           the description of INFO below.)
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>           The leading dimension of the array Z.  if COMPZ = 'I' or
*>           COMPZ = 'V', then LDZ >= MAX(1,N).  Otherwise, LDZ >= 1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (LWORK)
*>           On exit, if INFO = 0, WORK(1) returns an estimate of
*>           the optimal value for LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>           The dimension of the array WORK.  LWORK >= max(1,N)
*>           is sufficient and delivers very good and sometimes
*>           optimal performance.  However, LWORK as large as 11*N
*>           may be required for optimal performance.  A workspace
*>           query is recommended to determine the optimal workspace
*>           size.
*>
*>           If LWORK = -1, then DHSEQR does a workspace query.
*>           In this case, DHSEQR checks the input parameters and
*>           estimates the optimal workspace size for the given
*>           values of N, ILO and IHI.  The estimate is returned
*>           in WORK(1).  No error message related to LWORK is
*>           issued by XERBLA.  Neither H nor Z are accessed.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>             = 0:  successful exit
*>             < 0:  if INFO = -i, the i-th argument had an illegal
*>                    value
*>             > 0:  if INFO = i, DHSEQR failed to compute all of
*>                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
*>                and WI contain those eigenvalues which have been
*>                successfully computed.  (Failures are rare.)
*>
*>                If INFO > 0 and JOB = 'E', then on exit, the
*>                remaining unconverged eigenvalues are the eigen-
*>                values of the upper Hessenberg matrix rows and
*>                columns ILO through INFO of the final, output
*>                value of H.
*>
*>                If INFO > 0 and JOB   = 'S', then on exit
*>
*>           (*)  (initial value of H)*U  = U*(final value of H)
*>
*>                where U is an orthogonal matrix.  The final
*>                value of H is upper Hessenberg and quasi-triangular
*>                in rows and columns INFO+1 through IHI.
*>
*>                If INFO > 0 and COMPZ = 'V', then on exit
*>
*>                  (final value of Z)  =  (initial value of Z)*U
*>
*>                where U is the orthogonal matrix in (*) (regard-
*>                less of the value of JOB.)
*>
*>                If INFO > 0 and COMPZ = 'I', then on exit
*>                      (final value of Z)  = U
*>                where U is the orthogonal matrix in (*) (regard-
*>                less of the value of JOB.)
*>
*>                If INFO > 0 and COMPZ = 'N', then Z is not
*>                accessed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hseqr
*
*> \par Contributors:
*  ==================
*>
*>       Karen Braman and Ralph Byers, Department of Mathematics,
*>       University of Kansas, USA
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>             Default values supplied by
*>             ILAENV(ISPEC,'DHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
*>             It is suggested that these defaults be adjusted in order
*>             to attain best performance in each particular
*>             computational environment.
*>
*>            ISPEC=12: The DLAHQR vs DLAQR0 crossover point.
*>                      Default: 75. (Must be at least 11.)
*>
*>            ISPEC=13: Recommended deflation window size.
*>                      This depends on ILO, IHI and NS.  NS is the
*>                      number of simultaneous shifts returned
*>                      by ILAENV(ISPEC=15).  (See ISPEC=15 below.)
*>                      The default for (IHI-ILO+1) <= 500 is NS.
*>                      The default for (IHI-ILO+1) >  500 is 3*NS/2.
*>
*>            ISPEC=14: Nibble crossover point. (See IPARMQ for
*>                      details.)  Default: 14% of deflation window
*>                      size.
*>
*>            ISPEC=15: Number of simultaneous shifts in a multishift
*>                      QR iteration.
*>
*>                      If IHI-ILO+1 is ...
*>
*>                      greater than      ...but less    ... the
*>                      or equal to ...      than        default is
*>
*>                           1               30          NS =   2(+)
*>                          30               60          NS =   4(+)
*>                          60              150          NS =  10(+)
*>                         150              590          NS =  **
*>                         590             3000          NS =  64
*>                        3000             6000          NS = 128
*>                        6000             infinity      NS = 256
*>
*>                  (+)  By default some or all matrices of this order
*>                       are passed to the implicit double shift routine
*>                       DLAHQR and this parameter is ignored.  See
*>                       ISPEC=12 above and comments in IPARMQ for
*>                       details.
*>
*>                 (**)  The asterisks (**) indicate an ad-hoc
*>                       function of N increasing from 10 to 64.
*>
*>            ISPEC=16: Select structured matrix multiply.
*>                      If the number of simultaneous shifts (specified
*>                      by ISPEC=15) is less than 14, then the default
*>                      for ISPEC=16 is 0.  Otherwise the default for
*>                      ISPEC=16 is 2.
*> \endverbatim
*
*> \par References:
*  ================
*>
*>       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*>       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
*>       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
*>       929--947, 2002.
*> \n
*>       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*>       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
*>       of Matrix Analysis, volume 23, pages 948--973, 2002.
*
*  =====================================================================
      SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z,
     $                   LDZ, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            IHI, ILO, INFO, LDH, LDZ, LWORK, N
      CHARACTER          COMPZ, JOB
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   H( LDH, * ), WI( * ), WORK( * ), WR( * ),
     $                   Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
*
*     ==== Matrices of order NTINY or smaller must be processed by
*     .    DLAHQR because of insufficient subdiagonal scratch space.
*     .    (This is a hard limit.) ====
      INTEGER            NTINY
      PARAMETER          ( NTINY = 15 )
*
*     ==== NL allocates some local workspace to help small matrices
*     .    through a rare DLAHQR failure.  NL > NTINY = 15 is
*     .    required and NL <= NMIN = ILAENV(ISPEC=12,...) is recom-
*     .    mended.  (The default value of NMIN is 75.)  Using NL = 49
*     .    allows up to six simultaneous shifts and a 16-by-16
*     .    deflation window.  ====
      INTEGER            NL
      PARAMETER          ( NL = 49 )
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0d0, ONE = 1.0d0 )
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   HL( NL, NL ), WORKL( NL )
*     ..
*     .. Local Scalars ..
      INTEGER            I, KBOT, NMIN
      LOGICAL            INITZ, LQUERY, WANTT, WANTZ
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      LOGICAL            LSAME
      EXTERNAL           ILAENV, LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACPY, DLAHQR, DLAQR0, DLASET,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     ==== Decode and check the input parameters. ====
*
      WANTT = LSAME( JOB, 'S' )
      INITZ = LSAME( COMPZ, 'I' )
      WANTZ = INITZ .OR. LSAME( COMPZ, 'V' )
      WORK( 1 ) = DBLE( MAX( 1, N ) )
      LQUERY = LWORK.EQ.-1
*
      INFO = 0
      IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN
         INFO = -1
      ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
         INFO = -4
      ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
         INFO = -5
      ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.MAX( 1, N ) ) ) THEN
         INFO = -11
      ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
         INFO = -13
      END IF
*
      IF( INFO.NE.0 ) THEN
*
*        ==== Quick return in case of invalid argument. ====
*
         CALL XERBLA( 'DHSEQR', -INFO )
         RETURN
*
      ELSE IF( N.EQ.0 ) THEN
*
*        ==== Quick return in case N = 0; nothing to do. ====
*
         RETURN
*
      ELSE IF( LQUERY ) THEN
*
*        ==== Quick return in case of a workspace query ====
*
         CALL DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
     $                IHI, Z, LDZ, WORK, LWORK, INFO )
*        ==== Ensure reported workspace size is backward-compatible with
*        .    previous LAPACK versions. ====
         WORK( 1 ) = MAX( DBLE( MAX( 1, N ) ), WORK( 1 ) )
         RETURN
*
      ELSE
*
*        ==== copy eigenvalues isolated by DGEBAL ====
*
         DO 10 I = 1, ILO - 1
            WR( I ) = H( I, I )
            WI( I ) = ZERO
   10    CONTINUE
         DO 20 I = IHI + 1, N
            WR( I ) = H( I, I )
            WI( I ) = ZERO
   20    CONTINUE
*
*        ==== Initialize Z, if requested ====
*
         IF( INITZ )
     $      CALL DLASET( 'A', N, N, ZERO, ONE, Z, LDZ )
*
*        ==== Quick return if possible ====
*
         IF( ILO.EQ.IHI ) THEN
            WR( ILO ) = H( ILO, ILO )
            WI( ILO ) = ZERO
            RETURN
         END IF
*
*        ==== DLAHQR/DLAQR0 crossover point ====
*
         NMIN = ILAENV( 12, 'DHSEQR', JOB( : 1 ) // COMPZ( : 1 ), N,
     $          ILO, IHI, LWORK )
         NMIN = MAX( NTINY, NMIN )
*
*        ==== DLAQR0 for big matrices; DLAHQR for small ones ====
*
         IF( N.GT.NMIN ) THEN
            CALL DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
     $                   ILO,
     $                   IHI, Z, LDZ, WORK, LWORK, INFO )
         ELSE
*
*           ==== Small matrix ====
*
            CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
     $                   ILO,
     $                   IHI, Z, LDZ, INFO )
*
            IF( INFO.GT.0 ) THEN
*
*              ==== A rare DLAHQR failure!  DLAQR0 sometimes succeeds
*              .    when DLAHQR fails. ====
*
               KBOT = INFO
*
               IF( N.GE.NL ) THEN
*
*                 ==== Larger matrices have enough subdiagonal scratch
*                 .    space to call DLAQR0 directly. ====
*
                  CALL DLAQR0( WANTT, WANTZ, N, ILO, KBOT, H, LDH,
     $                         WR,
     $                         WI, ILO, IHI, Z, LDZ, WORK, LWORK, INFO )
*
               ELSE
*
*                 ==== Tiny matrices don't have enough subdiagonal
*                 .    scratch space to benefit from DLAQR0.  Hence,
*                 .    tiny matrices must be copied into a larger
*                 .    array before calling DLAQR0. ====
*
                  CALL DLACPY( 'A', N, N, H, LDH, HL, NL )
                  HL( N+1, N ) = ZERO
                  CALL DLASET( 'A', NL, NL-N, ZERO, ZERO, HL( 1,
     $                         N+1 ),
     $                         NL )
                  CALL DLAQR0( WANTT, WANTZ, NL, ILO, KBOT, HL, NL,
     $                         WR,
     $                         WI, ILO, IHI, Z, LDZ, WORKL, NL, INFO )
                  IF( WANTT .OR. INFO.NE.0 )
     $               CALL DLACPY( 'A', N, N, HL, NL, H, LDH )
               END IF
            END IF
         END IF
*
*        ==== Clear out the trash, if necessary. ====
*
         IF( ( WANTT .OR. INFO.NE.0 ) .AND. N.GT.2 )
     $      CALL DLASET( 'L', N-2, N-2, ZERO, ZERO, H( 3, 1 ), LDH )
*
*        ==== Ensure reported workspace size is backward-compatible with
*        .    previous LAPACK versions. ====
*
         WORK( 1 ) = MAX( DBLE( MAX( 1, N ) ), WORK( 1 ) )
      END IF
*
*     ==== End of DHSEQR ====
*
      END
*> \brief \b DISNAN tests input for NaN.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DISNAN + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/disnan.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/disnan.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/disnan.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       LOGICAL FUNCTION DISNAN( DIN )
*
*       .. Scalar Arguments ..
*       DOUBLE PRECISION, INTENT(IN) :: DIN
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DISNAN returns .TRUE. if its argument is NaN, and .FALSE.
*> otherwise.  To be replaced by the Fortran 2003 intrinsic in the
*> future.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] DIN
*> \verbatim
*>          DIN is DOUBLE PRECISION
*>          Input to test for NaN.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup isnan
*
*  =====================================================================
      LOGICAL FUNCTION DISNAN( DIN )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION, INTENT(IN) :: DIN
*     ..
*
*  =====================================================================
*
*  .. External Functions ..
      LOGICAL DLAISNAN
      EXTERNAL DLAISNAN
*  ..
*  .. Executable Statements ..
      DISNAN = DLAISNAN(DIN,DIN)
      RETURN
      END
*> \brief \b DLABAD
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLABAD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlabad.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlabad.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlabad.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLABAD( SMALL, LARGE )
*
*       .. Scalar Arguments ..
*       DOUBLE PRECISION   LARGE, SMALL
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLABAD is a no-op and kept for compatibility reasons. It used
*> to correct the overflow/underflow behavior of machines that
*> are not IEEE-754 compliant.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in,out] SMALL
*> \verbatim
*>          SMALL is DOUBLE PRECISION
*>          On entry, the underflow threshold as computed by DLAMCH.
*>          On exit, the unchanged value SMALL.
*> \endverbatim
*>
*> \param[in,out] LARGE
*> \verbatim
*>          LARGE is DOUBLE PRECISION
*>          On entry, the overflow threshold as computed by DLAMCH.
*>          On exit, the unchanged value LARGE.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup labad
*
*  =====================================================================
      SUBROUTINE DLABAD( SMALL, LARGE )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   LARGE, SMALL
*     ..
*
*  =====================================================================
*
*     .. Intrinsic Functions ..
      INTRINSIC          LOG10, SQRT
*     ..
*     .. Executable Statements ..
*
*     If it looks like we're on a Cray, take the square root of
*     SMALL and LARGE to avoid overflow and underflow problems.
*
*      IF( LOG10( LARGE ).GT.2000.D0 ) THEN
*         SMALL = SQRT( SMALL )
*         LARGE = SQRT( LARGE )
*      END IF
*
      RETURN
*
*     End of DLABAD
*
      END
*> \brief \b DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLABRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlabrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlabrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlabrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
*                          LDY )
*
*       .. Scalar Arguments ..
*       INTEGER            LDA, LDX, LDY, M, N, NB
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAUP( * ),
*      $                   TAUQ( * ), X( LDX, * ), Y( LDY, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLABRD reduces the first NB rows and columns of a real general
*> m by n matrix A to upper or lower bidiagonal form by an orthogonal
*> transformation Q**T * A * P, and returns the matrices X and Y which
*> are needed to apply the transformation to the unreduced part of A.
*>
*> If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
*> bidiagonal form.
*>
*> This is an auxiliary routine called by DGEBRD
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows in the matrix A.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns in the matrix A.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*>          NB is INTEGER
*>          The number of leading rows and columns of A to be reduced.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the m by n general matrix to be reduced.
*>          On exit, the first NB rows and columns of the matrix are
*>          overwritten; the rest of the array is unchanged.
*>          If m >= n, elements on and below the diagonal in the first NB
*>            columns, with the array TAUQ, represent the orthogonal
*>            matrix Q as a product of elementary reflectors; and
*>            elements above the diagonal in the first NB rows, with the
*>            array TAUP, represent the orthogonal matrix P as a product
*>            of elementary reflectors.
*>          If m < n, elements below the diagonal in the first NB
*>            columns, with the array TAUQ, represent the orthogonal
*>            matrix Q as a product of elementary reflectors, and
*>            elements on and above the diagonal in the first NB rows,
*>            with the array TAUP, represent the orthogonal matrix P as
*>            a product of elementary reflectors.
*>          See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (NB)
*>          The diagonal elements of the first NB rows and columns of
*>          the reduced matrix.  D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (NB)
*>          The off-diagonal elements of the first NB rows and columns of
*>          the reduced matrix.
*> \endverbatim
*>
*> \param[out] TAUQ
*> \verbatim
*>          TAUQ is DOUBLE PRECISION array, dimension (NB)
*>          The scalar factors of the elementary reflectors which
*>          represent the orthogonal matrix Q. See Further Details.
*> \endverbatim
*>
*> \param[out] TAUP
*> \verbatim
*>          TAUP is DOUBLE PRECISION array, dimension (NB)
*>          The scalar factors of the elementary reflectors which
*>          represent the orthogonal matrix P. See Further Details.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NB)
*>          The m-by-nb matrix X required to update the unreduced part
*>          of A.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X. LDX >= max(1,M).
*> \endverbatim
*>
*> \param[out] Y
*> \verbatim
*>          Y is DOUBLE PRECISION array, dimension (LDY,NB)
*>          The n-by-nb matrix Y required to update the unreduced part
*>          of A.
*> \endverbatim
*>
*> \param[in] LDY
*> \verbatim
*>          LDY is INTEGER
*>          The leading dimension of the array Y. LDY >= max(1,N).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup labrd
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrices Q and P are represented as products of elementary
*>  reflectors:
*>
*>     Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)
*>
*>  Each H(i) and G(i) has the form:
*>
*>     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T
*>
*>  where tauq and taup are real scalars, and v and u are real vectors.
*>
*>  If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
*>  A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
*>  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*>  If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
*>  A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
*>  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*>  The elements of the vectors v and u together form the m-by-nb matrix
*>  V and the nb-by-n matrix U**T which are needed, with X and Y, to apply
*>  the transformation to the unreduced part of the matrix, using a block
*>  update of the form:  A := A - V*Y**T - X*U**T.
*>
*>  The contents of A on exit are illustrated by the following examples
*>  with nb = 2:
*>
*>  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
*>
*>    (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
*>    (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
*>    (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
*>    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
*>    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
*>    (  v1  v2  a   a   a  )
*>
*>  where a denotes an element of the original matrix which is unchanged,
*>  vi denotes an element of the vector defining H(i), and ui an element
*>  of the vector defining G(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX,
     $                   Y,
     $                   LDY )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            LDA, LDX, LDY, M, N, NB
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAUP( * ),
     $                   TAUQ( * ), X( LDX, * ), Y( LDY, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMV, DLARFG, DSCAL
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( M.LE.0 .OR. N.LE.0 )
     $   RETURN
*
      IF( M.GE.N ) THEN
*
*        Reduce to upper bidiagonal form
*
         DO 10 I = 1, NB
*
*           Update A(i:m,i)
*
            CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, A( I, 1 ),
     $                  LDA, Y( I, 1 ), LDY, ONE, A( I, I ), 1 )
            CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, X( I, 1 ),
     $                  LDX, A( 1, I ), 1, ONE, A( I, I ), 1 )
*
*           Generate reflection Q(i) to annihilate A(i+1:m,i)
*
            CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
     $                   TAUQ( I ) )
            D( I ) = A( I, I )
            IF( I.LT.N ) THEN
               A( I, I ) = ONE
*
*              Compute Y(i+1:n,i)
*
               CALL DGEMV( 'Transpose', M-I+1, N-I, ONE, A( I, I+1 ),
     $                     LDA, A( I, I ), 1, ZERO, Y( I+1, I ), 1 )
               CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, A( I, 1 ),
     $                     LDA,
     $                     A( I, I ), 1, ZERO, Y( 1, I ), 1 )
               CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1,
     $                     1 ),
     $                     LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
               CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, X( I, 1 ),
     $                     LDX,
     $                     A( I, I ), 1, ZERO, Y( 1, I ), 1 )
               CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ),
     $                     LDA, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
               CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
*
*              Update A(i,i+1:n)
*
               CALL DGEMV( 'No transpose', N-I, I, -ONE, Y( I+1, 1 ),
     $                     LDY, A( I, 1 ), LDA, ONE, A( I, I+1 ), LDA )
               CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ),
     $                     LDA, X( I, 1 ), LDX, ONE, A( I, I+1 ), LDA )
*
*              Generate reflection P(i) to annihilate A(i,i+2:n)
*
               CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ),
     $                      LDA, TAUP( I ) )
               E( I ) = A( I, I+1 )
               A( I, I+1 ) = ONE
*
*              Compute X(i+1:m,i)
*
               CALL DGEMV( 'No transpose', M-I, N-I, ONE, A( I+1,
     $                     I+1 ),
     $                     LDA, A( I, I+1 ), LDA, ZERO, X( I+1, I ), 1 )
               CALL DGEMV( 'Transpose', N-I, I, ONE, Y( I+1, 1 ),
     $                     LDY,
     $                     A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
               CALL DGEMV( 'No transpose', M-I, I, -ONE, A( I+1, 1 ),
     $                     LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
               CALL DGEMV( 'No transpose', I-1, N-I, ONE, A( 1,
     $                     I+1 ),
     $                     LDA, A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
               CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1,
     $                     1 ),
     $                     LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
               CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
            END IF
   10    CONTINUE
      ELSE
*
*        Reduce to lower bidiagonal form
*
         DO 20 I = 1, NB
*
*           Update A(i,i:n)
*
            CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, Y( I, 1 ),
     $                  LDY, A( I, 1 ), LDA, ONE, A( I, I ), LDA )
            CALL DGEMV( 'Transpose', I-1, N-I+1, -ONE, A( 1, I ),
     $                  LDA,
     $                  X( I, 1 ), LDX, ONE, A( I, I ), LDA )
*
*           Generate reflection P(i) to annihilate A(i,i+1:n)
*
            CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ),
     $                   LDA,
     $                   TAUP( I ) )
            D( I ) = A( I, I )
            IF( I.LT.M ) THEN
               A( I, I ) = ONE
*
*              Compute X(i+1:m,i)
*
               CALL DGEMV( 'No transpose', M-I, N-I+1, ONE, A( I+1,
     $                     I ),
     $                     LDA, A( I, I ), LDA, ZERO, X( I+1, I ), 1 )
               CALL DGEMV( 'Transpose', N-I+1, I-1, ONE, Y( I, 1 ),
     $                     LDY,
     $                     A( I, I ), LDA, ZERO, X( 1, I ), 1 )
               CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1,
     $                     1 ),
     $                     LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
               CALL DGEMV( 'No transpose', I-1, N-I+1, ONE, A( 1,
     $                     I ),
     $                     LDA, A( I, I ), LDA, ZERO, X( 1, I ), 1 )
               CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1,
     $                     1 ),
     $                     LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
               CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
*
*              Update A(i+1:m,i)
*
               CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1,
     $                     1 ),
     $                     LDA, Y( I, 1 ), LDY, ONE, A( I+1, I ), 1 )
               CALL DGEMV( 'No transpose', M-I, I, -ONE, X( I+1, 1 ),
     $                     LDX, A( 1, I ), 1, ONE, A( I+1, I ), 1 )
*
*              Generate reflection Q(i) to annihilate A(i+2:m,i)
*
               CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ),
     $                      1,
     $                      TAUQ( I ) )
               E( I ) = A( I+1, I )
               A( I+1, I ) = ONE
*
*              Compute Y(i+1:n,i)
*
               CALL DGEMV( 'Transpose', M-I, N-I, ONE, A( I+1, I+1 ),
     $                     LDA, A( I+1, I ), 1, ZERO, Y( I+1, I ), 1 )
               CALL DGEMV( 'Transpose', M-I, I-1, ONE, A( I+1, 1 ),
     $                     LDA,
     $                     A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )
               CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1,
     $                     1 ),
     $                     LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
               CALL DGEMV( 'Transpose', M-I, I, ONE, X( I+1, 1 ),
     $                     LDX,
     $                     A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )
               CALL DGEMV( 'Transpose', I, N-I, -ONE, A( 1, I+1 ),
     $                     LDA,
     $                     Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
               CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
            END IF
   20    CONTINUE
      END IF
      RETURN
*
*     End of DLABRD
*
      END
*> \brief \b DLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLACN2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlacn2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlacn2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlacn2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLACN2( N, V, X, ISGN, EST, KASE, ISAVE )
*
*       .. Scalar Arguments ..
*       INTEGER            KASE, N
*       DOUBLE PRECISION   EST
*       ..
*       .. Array Arguments ..
*       INTEGER            ISGN( * ), ISAVE( 3 )
*       DOUBLE PRECISION   V( * ), X( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLACN2 estimates the 1-norm of a square, real matrix A.
*> Reverse communication is used for evaluating matrix-vector products.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>         The order of the matrix.  N >= 1.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension (N)
*>         On the final return, V = A*W,  where  EST = norm(V)/norm(W)
*>         (W is not returned).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (N)
*>         On an intermediate return, X should be overwritten by
*>               A * X,   if KASE=1,
*>               A**T * X,  if KASE=2,
*>         and DLACN2 must be re-called with all the other parameters
*>         unchanged.
*> \endverbatim
*>
*> \param[out] ISGN
*> \verbatim
*>          ISGN is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[in,out] EST
*> \verbatim
*>          EST is DOUBLE PRECISION
*>         On entry with KASE = 1 or 2 and ISAVE(1) = 3, EST should be
*>         unchanged from the previous call to DLACN2.
*>         On exit, EST is an estimate (a lower bound) for norm(A).
*> \endverbatim
*>
*> \param[in,out] KASE
*> \verbatim
*>          KASE is INTEGER
*>         On the initial call to DLACN2, KASE should be 0.
*>         On an intermediate return, KASE will be 1 or 2, indicating
*>         whether X should be overwritten by A * X  or A**T * X.
*>         On the final return from DLACN2, KASE will again be 0.
*> \endverbatim
*>
*> \param[in,out] ISAVE
*> \verbatim
*>          ISAVE is INTEGER array, dimension (3)
*>         ISAVE is used to save variables between calls to DLACN2
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lacn2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  Originally named SONEST, dated March 16, 1988.
*>
*>  This is a thread safe version of DLACON, which uses the array ISAVE
*>  in place of a SAVE statement, as follows:
*>
*>     DLACON     DLACN2
*>      JUMP     ISAVE(1)
*>      J        ISAVE(2)
*>      ITER     ISAVE(3)
*> \endverbatim
*
*> \par Contributors:
*  ==================
*>
*>     Nick Higham, University of Manchester
*
*> \par References:
*  ================
*>
*>  N.J. Higham, "FORTRAN codes for estimating the one-norm of
*>  a real or complex matrix, with applications to condition estimation",
*>  ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.
*>
*  =====================================================================
      SUBROUTINE DLACN2( N, V, X, ISGN, EST, KASE, ISAVE )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            KASE, N
      DOUBLE PRECISION   EST
*     ..
*     .. Array Arguments ..
      INTEGER            ISGN( * ), ISAVE( 3 )
      DOUBLE PRECISION   V( * ), X( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            ITMAX
      PARAMETER          ( ITMAX = 5 )
      DOUBLE PRECISION   ZERO, ONE, TWO
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, JLAST
      DOUBLE PRECISION   ALTSGN, ESTOLD, TEMP, XS
*     ..
*     .. External Functions ..
      INTEGER            IDAMAX
      DOUBLE PRECISION   DASUM
      EXTERNAL           IDAMAX, DASUM
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, NINT
*     ..
*     .. Executable Statements ..
*
      IF( KASE.EQ.0 ) THEN
         DO 10 I = 1, N
            X( I ) = ONE / DBLE( N )
   10    CONTINUE
         KASE = 1
         ISAVE( 1 ) = 1
         RETURN
      END IF
*
      GO TO ( 20, 40, 70, 110, 140 )ISAVE( 1 )
*
*     ................ ENTRY   (ISAVE( 1 ) = 1)
*     FIRST ITERATION.  X HAS BEEN OVERWRITTEN BY A*X.
*
   20 CONTINUE
      IF( N.EQ.1 ) THEN
         V( 1 ) = X( 1 )
         EST = ABS( V( 1 ) )
*        ... QUIT
         GO TO 150
      END IF
      EST = DASUM( N, X, 1 )
*
      DO 30 I = 1, N
         IF( X(I).GE.ZERO ) THEN
            X(I) = ONE
         ELSE
            X(I) = -ONE
         END IF
         ISGN( I ) = NINT( X( I ) )
   30 CONTINUE
      KASE = 2
      ISAVE( 1 ) = 2
      RETURN
*
*     ................ ENTRY   (ISAVE( 1 ) = 2)
*     FIRST ITERATION.  X HAS BEEN OVERWRITTEN BY TRANSPOSE(A)*X.
*
   40 CONTINUE
      ISAVE( 2 ) = IDAMAX( N, X, 1 )
      ISAVE( 3 ) = 2
*
*     MAIN LOOP - ITERATIONS 2,3,...,ITMAX.
*
   50 CONTINUE
      DO 60 I = 1, N
         X( I ) = ZERO
   60 CONTINUE
      X( ISAVE( 2 ) ) = ONE
      KASE = 1
      ISAVE( 1 ) = 3
      RETURN
*
*     ................ ENTRY   (ISAVE( 1 ) = 3)
*     X HAS BEEN OVERWRITTEN BY A*X.
*
   70 CONTINUE
      CALL DCOPY( N, X, 1, V, 1 )
      ESTOLD = EST
      EST = DASUM( N, V, 1 )
      DO 80 I = 1, N
         IF( X(I).GE.ZERO ) THEN
            XS = ONE
         ELSE
            XS = -ONE
         END IF
         IF( NINT( XS ).NE.ISGN( I ) )
     $      GO TO 90
   80 CONTINUE
*     REPEATED SIGN VECTOR DETECTED, HENCE ALGORITHM HAS CONVERGED.
      GO TO 120
*
   90 CONTINUE
*     TEST FOR CYCLING.
      IF( EST.LE.ESTOLD )
     $   GO TO 120
*
      DO 100 I = 1, N
         IF( X(I).GE.ZERO ) THEN
            X(I) = ONE
         ELSE
            X(I) = -ONE
         END IF
         ISGN( I ) = NINT( X( I ) )
  100 CONTINUE
      KASE = 2
      ISAVE( 1 ) = 4
      RETURN
*
*     ................ ENTRY   (ISAVE( 1 ) = 4)
*     X HAS BEEN OVERWRITTEN BY TRANSPOSE(A)*X.
*
  110 CONTINUE
      JLAST = ISAVE( 2 )
      ISAVE( 2 ) = IDAMAX( N, X, 1 )
      IF( ( X( JLAST ).NE.ABS( X( ISAVE( 2 ) ) ) ) .AND.
     $    ( ISAVE( 3 ).LT.ITMAX ) ) THEN
         ISAVE( 3 ) = ISAVE( 3 ) + 1
         GO TO 50
      END IF
*
*     ITERATION COMPLETE.  FINAL STAGE.
*
  120 CONTINUE
      ALTSGN = ONE
      DO 130 I = 1, N
         X( I ) = ALTSGN*( ONE+DBLE( I-1 ) / DBLE( N-1 ) )
         ALTSGN = -ALTSGN
  130 CONTINUE
      KASE = 1
      ISAVE( 1 ) = 5
      RETURN
*
*     ................ ENTRY   (ISAVE( 1 ) = 5)
*     X HAS BEEN OVERWRITTEN BY A*X.
*
  140 CONTINUE
      TEMP = TWO*( DASUM( N, X, 1 ) / DBLE( 3*N ) )
      IF( TEMP.GT.EST ) THEN
         CALL DCOPY( N, X, 1, V, 1 )
         EST = TEMP
      END IF
*
  150 CONTINUE
      KASE = 0
      RETURN
*
*     End of DLACN2
*
      END
*> \brief \b DLACON estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLACON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlacon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlacon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlacon.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLACON( N, V, X, ISGN, EST, KASE )
*
*       .. Scalar Arguments ..
*       INTEGER            KASE, N
*       DOUBLE PRECISION   EST
*       ..
*       .. Array Arguments ..
*       INTEGER            ISGN( * )
*       DOUBLE PRECISION   V( * ), X( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLACON estimates the 1-norm of a square, real matrix A.
*> Reverse communication is used for evaluating matrix-vector products.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>         The order of the matrix.  N >= 1.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension (N)
*>         On the final return, V = A*W,  where  EST = norm(V)/norm(W)
*>         (W is not returned).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (N)
*>         On an intermediate return, X should be overwritten by
*>               A * X,   if KASE=1,
*>               A**T * X,  if KASE=2,
*>         and DLACON must be re-called with all the other parameters
*>         unchanged.
*> \endverbatim
*>
*> \param[out] ISGN
*> \verbatim
*>          ISGN is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[in,out] EST
*> \verbatim
*>          EST is DOUBLE PRECISION
*>         On entry with KASE = 1 or 2 and JUMP = 3, EST should be
*>         unchanged from the previous call to DLACON.
*>         On exit, EST is an estimate (a lower bound) for norm(A).
*> \endverbatim
*>
*> \param[in,out] KASE
*> \verbatim
*>          KASE is INTEGER
*>         On the initial call to DLACON, KASE should be 0.
*>         On an intermediate return, KASE will be 1 or 2, indicating
*>         whether X should be overwritten by A * X  or A**T * X.
*>         On the final return from DLACON, KASE will again be 0.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lacon
*
*> \par Contributors:
*  ==================
*>
*>  Nick Higham, University of Manchester. \n
*>  Originally named SONEST, dated March 16, 1988.
*
*> \par References:
*  ================
*>
*>  N.J. Higham, "FORTRAN codes for estimating the one-norm of
*>  a real or complex matrix, with applications to condition estimation",
*>  ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.
*>
*  =====================================================================
      SUBROUTINE DLACON( N, V, X, ISGN, EST, KASE )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            KASE, N
      DOUBLE PRECISION   EST
*     ..
*     .. Array Arguments ..
      INTEGER            ISGN( * )
      DOUBLE PRECISION   V( * ), X( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            ITMAX
      PARAMETER          ( ITMAX = 5 )
      DOUBLE PRECISION   ZERO, ONE, TWO
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, ITER, J, JLAST, JUMP
      DOUBLE PRECISION   ALTSGN, ESTOLD, TEMP
*     ..
*     .. External Functions ..
      INTEGER            IDAMAX
      DOUBLE PRECISION   DASUM
      EXTERNAL           IDAMAX, DASUM
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, NINT, SIGN
*     ..
*     .. Save statement ..
      SAVE
*     ..
*     .. Executable Statements ..
*
      IF( KASE.EQ.0 ) THEN
         DO 10 I = 1, N
            X( I ) = ONE / DBLE( N )
   10    CONTINUE
         KASE = 1
         JUMP = 1
         RETURN
      END IF
*
      GO TO ( 20, 40, 70, 110, 140 )JUMP
*
*     ................ ENTRY   (JUMP = 1)
*     FIRST ITERATION.  X HAS BEEN OVERWRITTEN BY A*X.
*
   20 CONTINUE
      IF( N.EQ.1 ) THEN
         V( 1 ) = X( 1 )
         EST = ABS( V( 1 ) )
*        ... QUIT
         GO TO 150
      END IF
      EST = DASUM( N, X, 1 )
*
      DO 30 I = 1, N
         X( I ) = SIGN( ONE, X( I ) )
         ISGN( I ) = NINT( X( I ) )
   30 CONTINUE
      KASE = 2
      JUMP = 2
      RETURN
*
*     ................ ENTRY   (JUMP = 2)
*     FIRST ITERATION.  X HAS BEEN OVERWRITTEN BY TRANSPOSE(A)*X.
*
   40 CONTINUE
      J = IDAMAX( N, X, 1 )
      ITER = 2
*
*     MAIN LOOP - ITERATIONS 2,3,...,ITMAX.
*
   50 CONTINUE
      DO 60 I = 1, N
         X( I ) = ZERO
   60 CONTINUE
      X( J ) = ONE
      KASE = 1
      JUMP = 3
      RETURN
*
*     ................ ENTRY   (JUMP = 3)
*     X HAS BEEN OVERWRITTEN BY A*X.
*
   70 CONTINUE
      CALL DCOPY( N, X, 1, V, 1 )
      ESTOLD = EST
      EST = DASUM( N, V, 1 )
      DO 80 I = 1, N
         IF( NINT( SIGN( ONE, X( I ) ) ).NE.ISGN( I ) )
     $      GO TO 90
   80 CONTINUE
*     REPEATED SIGN VECTOR DETECTED, HENCE ALGORITHM HAS CONVERGED.
      GO TO 120
*
   90 CONTINUE
*     TEST FOR CYCLING.
      IF( EST.LE.ESTOLD )
     $   GO TO 120
*
      DO 100 I = 1, N
         X( I ) = SIGN( ONE, X( I ) )
         ISGN( I ) = NINT( X( I ) )
  100 CONTINUE
      KASE = 2
      JUMP = 4
      RETURN
*
*     ................ ENTRY   (JUMP = 4)
*     X HAS BEEN OVERWRITTEN BY TRANSPOSE(A)*X.
*
  110 CONTINUE
      JLAST = J
      J = IDAMAX( N, X, 1 )
      IF( ( X( JLAST ).NE.ABS( X( J ) ) ) .AND. ( ITER.LT.ITMAX ) ) THEN
         ITER = ITER + 1
         GO TO 50
      END IF
*
*     ITERATION COMPLETE.  FINAL STAGE.
*
  120 CONTINUE
      ALTSGN = ONE
      DO 130 I = 1, N
         X( I ) = ALTSGN*( ONE+DBLE( I-1 ) / DBLE( N-1 ) )
         ALTSGN = -ALTSGN
  130 CONTINUE
      KASE = 1
      JUMP = 5
      RETURN
*
*     ................ ENTRY   (JUMP = 5)
*     X HAS BEEN OVERWRITTEN BY A*X.
*
  140 CONTINUE
      TEMP = TWO*( DASUM( N, X, 1 ) / DBLE( 3*N ) )
      IF( TEMP.GT.EST ) THEN
         CALL DCOPY( N, X, 1, V, 1 )
         EST = TEMP
      END IF
*
  150 CONTINUE
      KASE = 0
      RETURN
*
*     End of DLACON
*
      END
*> \brief \b DLACPY copies all or part of one two-dimensional array to another.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLACPY + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlacpy.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlacpy.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlacpy.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLACPY( UPLO, M, N, A, LDA, B, LDB )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            LDA, LDB, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLACPY copies all or part of a two-dimensional matrix A to another
*> matrix B.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies the part of the matrix A to be copied to B.
*>          = 'U':      Upper triangular part
*>          = 'L':      Lower triangular part
*>          Otherwise:  All of the matrix A
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The m by n matrix A.  If UPLO = 'U', only the upper triangle
*>          or trapezoid is accessed; if UPLO = 'L', only the lower
*>          triangle or trapezoid is accessed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,N)
*>          On exit, B = A in the locations specified by UPLO.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,M).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lacpy
*
*  =====================================================================
      SUBROUTINE DLACPY( UPLO, M, N, A, LDA, B, LDB )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            LDA, LDB, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            I, J
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
      IF( LSAME( UPLO, 'U' ) ) THEN
         DO 20 J = 1, N
            DO 10 I = 1, MIN( J, M )
               B( I, J ) = A( I, J )
   10       CONTINUE
   20    CONTINUE
      ELSE IF( LSAME( UPLO, 'L' ) ) THEN
         DO 40 J = 1, N
            DO 30 I = J, M
               B( I, J ) = A( I, J )
   30       CONTINUE
   40    CONTINUE
      ELSE
         DO 60 J = 1, N
            DO 50 I = 1, M
               B( I, J ) = A( I, J )
   50       CONTINUE
   60    CONTINUE
      END IF
      RETURN
*
*     End of DLACPY
*
      END
*> \brief \b DLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLADIV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dladiv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dladiv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dladiv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLADIV( A, B, C, D, P, Q )
*
*       .. Scalar Arguments ..
*       DOUBLE PRECISION   A, B, C, D, P, Q
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLADIV performs complex division in  real arithmetic
*>
*>                       a + i*b
*>            p + i*q = ---------
*>                       c + i*d
*>
*> The algorithm is due to Michael Baudin and Robert L. Smith
*> and can be found in the paper
*> "A Robust Complex Division in Scilab"
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*>          C is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION
*>          The scalars a, b, c, and d in the above expression.
*> \endverbatim
*>
*> \param[out] P
*> \verbatim
*>          P is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION
*>          The scalars p and q in the above expression.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ladiv
*
*  =====================================================================
      SUBROUTINE DLADIV( A, B, C, D, P, Q )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   A, B, C, D, P, Q
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   BS
      PARAMETER          ( BS = 2.0D0 )
      DOUBLE PRECISION   HALF
      PARAMETER          ( HALF = 0.5D0 )
      DOUBLE PRECISION   TWO
      PARAMETER          ( TWO = 2.0D0 )
*
*     .. Local Scalars ..
      DOUBLE PRECISION   AA, BB, CC, DD, AB, CD, S, OV, UN, BE, EPS
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLADIV1
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. Executable Statements ..
*
      AA = A
      BB = B
      CC = C
      DD = D
      AB = MAX( ABS(A), ABS(B) )
      CD = MAX( ABS(C), ABS(D) )
      S = 1.0D0

      OV = DLAMCH( 'Overflow threshold' )
      UN = DLAMCH( 'Safe minimum' )
      EPS = DLAMCH( 'Epsilon' )
      BE = BS / (EPS*EPS)

      IF( AB >= HALF*OV ) THEN
         AA = HALF * AA
         BB = HALF * BB
         S  = TWO * S
      END IF
      IF( CD >= HALF*OV ) THEN
         CC = HALF * CC
         DD = HALF * DD
         S  = HALF * S
      END IF
      IF( AB <= UN*BS/EPS ) THEN
         AA = AA * BE
         BB = BB * BE
         S  = S / BE
      END IF
      IF( CD <= UN*BS/EPS ) THEN
         CC = CC * BE
         DD = DD * BE
         S  = S * BE
      END IF
      IF( ABS( D ).LE.ABS( C ) ) THEN
         CALL DLADIV1(AA, BB, CC, DD, P, Q)
      ELSE
         CALL DLADIV1(BB, AA, DD, CC, P, Q)
         Q = -Q
      END IF
      P = P * S
      Q = Q * S
*
      RETURN
*
*     End of DLADIV
*
      END

*> \ingroup ladiv


      SUBROUTINE DLADIV1( A, B, C, D, P, Q )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   A, B, C, D, P, Q
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D0 )
*
*     .. Local Scalars ..
      DOUBLE PRECISION   R, T
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLADIV2
      EXTERNAL           DLADIV2
*     ..
*     .. Executable Statements ..
*
      R = D / C
      T = ONE / (C + D * R)
      P = DLADIV2(A, B, C, D, R, T)
      A = -A
      Q = DLADIV2(B, A, C, D, R, T)
*
      RETURN
*
*     End of DLADIV1
*
      END

*> \ingroup ladiv

      DOUBLE PRECISION FUNCTION DLADIV2( A, B, C, D, R, T )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   A, B, C, D, R, T
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D0 )
*
*     .. Local Scalars ..
      DOUBLE PRECISION   BR
*     ..
*     .. Executable Statements ..
*
      IF( R.NE.ZERO ) THEN
         BR = B * R
         IF( BR.NE.ZERO ) THEN
            DLADIV2 = (A + BR) * T
         ELSE
            DLADIV2 = A * T + (B * T) * R
         END IF
      ELSE
         DLADIV2 = (A + D * (B / C)) * T
      END IF
*
      RETURN
*
*     End of DLADIV2
*
      END
*> \brief \b DLAE2 computes the eigenvalues of a 2-by-2 symmetric matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAE2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlae2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlae2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlae2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAE2( A, B, C, RT1, RT2 )
*
*       .. Scalar Arguments ..
*       DOUBLE PRECISION   A, B, C, RT1, RT2
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAE2  computes the eigenvalues of a 2-by-2 symmetric matrix
*>    [  A   B  ]
*>    [  B   C  ].
*> On return, RT1 is the eigenvalue of larger absolute value, and RT2
*> is the eigenvalue of smaller absolute value.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION
*>          The (1,1) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION
*>          The (1,2) and (2,1) elements of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*>          C is DOUBLE PRECISION
*>          The (2,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[out] RT1
*> \verbatim
*>          RT1 is DOUBLE PRECISION
*>          The eigenvalue of larger absolute value.
*> \endverbatim
*>
*> \param[out] RT2
*> \verbatim
*>          RT2 is DOUBLE PRECISION
*>          The eigenvalue of smaller absolute value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lae2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  RT1 is accurate to a few ulps barring over/underflow.
*>
*>  RT2 may be inaccurate if there is massive cancellation in the
*>  determinant A*C-B*B; higher precision or correctly rounded or
*>  correctly truncated arithmetic would be needed to compute RT2
*>  accurately in all cases.
*>
*>  Overflow is possible only if RT1 is within a factor of 5 of overflow.
*>  Underflow is harmless if the input data is 0 or exceeds
*>     underflow_threshold / macheps.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLAE2( A, B, C, RT1, RT2 )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   A, B, C, RT1, RT2
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D0 )
      DOUBLE PRECISION   TWO
      PARAMETER          ( TWO = 2.0D0 )
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D0 )
      DOUBLE PRECISION   HALF
      PARAMETER          ( HALF = 0.5D0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   AB, ACMN, ACMX, ADF, DF, RT, SM, TB
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, SQRT
*     ..
*     .. Executable Statements ..
*
*     Compute the eigenvalues
*
      SM = A + C
      DF = A - C
      ADF = ABS( DF )
      TB = B + B
      AB = ABS( TB )
      IF( ABS( A ).GT.ABS( C ) ) THEN
         ACMX = A
         ACMN = C
      ELSE
         ACMX = C
         ACMN = A
      END IF
      IF( ADF.GT.AB ) THEN
         RT = ADF*SQRT( ONE+( AB / ADF )**2 )
      ELSE IF( ADF.LT.AB ) THEN
         RT = AB*SQRT( ONE+( ADF / AB )**2 )
      ELSE
*
*        Includes case AB=ADF=0
*
         RT = AB*SQRT( TWO )
      END IF
      IF( SM.LT.ZERO ) THEN
         RT1 = HALF*( SM-RT )
*
*        Order of execution important.
*        To get fully accurate smaller eigenvalue,
*        next line needs to be executed in higher precision.
*
         RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
      ELSE IF( SM.GT.ZERO ) THEN
         RT1 = HALF*( SM+RT )
*
*        Order of execution important.
*        To get fully accurate smaller eigenvalue,
*        next line needs to be executed in higher precision.
*
         RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
      ELSE
*
*        Includes case RT1 = RT2 = 0
*
         RT1 = HALF*RT
         RT2 = -HALF*RT
      END IF
      RETURN
*
*     End of DLAE2
*
      END
*> \brief \b DLAEBZ computes the number of eigenvalues of a real symmetric tridiagonal matrix which are less than or equal to a given value, and performs other tasks required by the routine sstebz.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAEBZ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaebz.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaebz.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaebz.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAEBZ( IJOB, NITMAX, N, MMAX, MINP, NBMIN, ABSTOL,
*                          RELTOL, PIVMIN, D, E, E2, NVAL, AB, C, MOUT,
*                          NAB, WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            IJOB, INFO, MINP, MMAX, MOUT, N, NBMIN, NITMAX
*       DOUBLE PRECISION   ABSTOL, PIVMIN, RELTOL
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * ), NAB( MMAX, * ), NVAL( * )
*       DOUBLE PRECISION   AB( MMAX, * ), C( * ), D( * ), E( * ), E2( * ),
*      $                   WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAEBZ contains the iteration loops which compute and use the
*> function N(w), which is the count of eigenvalues of a symmetric
*> tridiagonal matrix T less than or equal to its argument  w.  It
*> performs a choice of two types of loops:
*>
*> IJOB=1, followed by
*> IJOB=2: It takes as input a list of intervals and returns a list of
*>         sufficiently small intervals whose union contains the same
*>         eigenvalues as the union of the original intervals.
*>         The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP.
*>         The output interval (AB(j,1),AB(j,2)] will contain
*>         eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT.
*>
*> IJOB=3: It performs a binary search in each input interval
*>         (AB(j,1),AB(j,2)] for a point  w(j)  such that
*>         N(w(j))=NVAL(j), and uses  C(j)  as the starting point of
*>         the search.  If such a w(j) is found, then on output
*>         AB(j,1)=AB(j,2)=w.  If no such w(j) is found, then on output
*>         (AB(j,1),AB(j,2)] will be a small interval containing the
*>         point where N(w) jumps through NVAL(j), unless that point
*>         lies outside the initial interval.
*>
*> Note that the intervals are in all cases half-open intervals,
*> i.e., of the form  (a,b] , which includes  b  but not  a .
*>
*> To avoid underflow, the matrix should be scaled so that its largest
*> element is no greater than  overflow**(1/2) * underflow**(1/4)
*> in absolute value.  To assure the most accurate computation
*> of small eigenvalues, the matrix should be scaled to be
*> not much smaller than that, either.
*>
*> See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
*> Matrix", Report CS41, Computer Science Dept., Stanford
*> University, July 21, 1966
*>
*> Note: the arguments are, in general, *not* checked for unreasonable
*> values.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] IJOB
*> \verbatim
*>          IJOB is INTEGER
*>          Specifies what is to be done:
*>          = 1:  Compute NAB for the initial intervals.
*>          = 2:  Perform bisection iteration to find eigenvalues of T.
*>          = 3:  Perform bisection iteration to invert N(w), i.e.,
*>                to find a point which has a specified number of
*>                eigenvalues of T to its left.
*>          Other values will cause DLAEBZ to return with INFO=-1.
*> \endverbatim
*>
*> \param[in] NITMAX
*> \verbatim
*>          NITMAX is INTEGER
*>          The maximum number of "levels" of bisection to be
*>          performed, i.e., an interval of width W will not be made
*>          smaller than 2^(-NITMAX) * W.  If not all intervals
*>          have converged after NITMAX iterations, then INFO is set
*>          to the number of non-converged intervals.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The dimension n of the tridiagonal matrix T.  It must be at
*>          least 1.
*> \endverbatim
*>
*> \param[in] MMAX
*> \verbatim
*>          MMAX is INTEGER
*>          The maximum number of intervals.  If more than MMAX intervals
*>          are generated, then DLAEBZ will quit with INFO=MMAX+1.
*> \endverbatim
*>
*> \param[in] MINP
*> \verbatim
*>          MINP is INTEGER
*>          The initial number of intervals.  It may not be greater than
*>          MMAX.
*> \endverbatim
*>
*> \param[in] NBMIN
*> \verbatim
*>          NBMIN is INTEGER
*>          The smallest number of intervals that should be processed
*>          using a vector loop.  If zero, then only the scalar loop
*>          will be used.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*>          ABSTOL is DOUBLE PRECISION
*>          The minimum (absolute) width of an interval.  When an
*>          interval is narrower than ABSTOL, or than RELTOL times the
*>          larger (in magnitude) endpoint, then it is considered to be
*>          sufficiently small, i.e., converged.  This must be at least
*>          zero.
*> \endverbatim
*>
*> \param[in] RELTOL
*> \verbatim
*>          RELTOL is DOUBLE PRECISION
*>          The minimum relative width of an interval.  When an interval
*>          is narrower than ABSTOL, or than RELTOL times the larger (in
*>          magnitude) endpoint, then it is considered to be
*>          sufficiently small, i.e., converged.  Note: this should
*>          always be at least radix*machine epsilon.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*>          PIVMIN is DOUBLE PRECISION
*>          The minimum absolute value of a "pivot" in the Sturm
*>          sequence loop.
*>          This must be at least  max |e(j)**2|*safe_min  and at
*>          least safe_min, where safe_min is at least
*>          the smallest number that can divide one without overflow.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N)
*>          The offdiagonal elements of the tridiagonal matrix T in
*>          positions 1 through N-1.  E(N) is arbitrary.
*> \endverbatim
*>
*> \param[in] E2
*> \verbatim
*>          E2 is DOUBLE PRECISION array, dimension (N)
*>          The squares of the offdiagonal elements of the tridiagonal
*>          matrix T.  E2(N) is ignored.
*> \endverbatim
*>
*> \param[in,out] NVAL
*> \verbatim
*>          NVAL is INTEGER array, dimension (MINP)
*>          If IJOB=1 or 2, not referenced.
*>          If IJOB=3, the desired values of N(w).  The elements of NVAL
*>          will be reordered to correspond with the intervals in AB.
*>          Thus, NVAL(j) on output will not, in general be the same as
*>          NVAL(j) on input, but it will correspond with the interval
*>          (AB(j,1),AB(j,2)] on output.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (MMAX,2)
*>          The endpoints of the intervals.  AB(j,1) is  a(j), the left
*>          endpoint of the j-th interval, and AB(j,2) is b(j), the
*>          right endpoint of the j-th interval.  The input intervals
*>          will, in general, be modified, split, and reordered by the
*>          calculation.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (MMAX)
*>          If IJOB=1, ignored.
*>          If IJOB=2, workspace.
*>          If IJOB=3, then on input C(j) should be initialized to the
*>          first search point in the binary search.
*> \endverbatim
*>
*> \param[out] MOUT
*> \verbatim
*>          MOUT is INTEGER
*>          If IJOB=1, the number of eigenvalues in the intervals.
*>          If IJOB=2 or 3, the number of intervals output.
*>          If IJOB=3, MOUT will equal MINP.
*> \endverbatim
*>
*> \param[in,out] NAB
*> \verbatim
*>          NAB is INTEGER array, dimension (MMAX,2)
*>          If IJOB=1, then on output NAB(i,j) will be set to N(AB(i,j)).
*>          If IJOB=2, then on input, NAB(i,j) should be set.  It must
*>             satisfy the condition:
*>             N(AB(i,1)) <= NAB(i,1) <= NAB(i,2) <= N(AB(i,2)),
*>             which means that in interval i only eigenvalues
*>             NAB(i,1)+1,...,NAB(i,2) will be considered.  Usually,
*>             NAB(i,j)=N(AB(i,j)), from a previous call to DLAEBZ with
*>             IJOB=1.
*>             On output, NAB(i,j) will contain
*>             max(na(k),min(nb(k),N(AB(i,j)))), where k is the index of
*>             the input interval that the output interval
*>             (AB(j,1),AB(j,2)] came from, and na(k) and nb(k) are the
*>             the input values of NAB(k,1) and NAB(k,2).
*>          If IJOB=3, then on output, NAB(i,j) contains N(AB(i,j)),
*>             unless N(w) > NVAL(i) for all search points  w , in which
*>             case NAB(i,1) will not be modified, i.e., the output
*>             value will be the same as the input value (modulo
*>             reorderings -- see NVAL and AB), or unless N(w) < NVAL(i)
*>             for all search points  w , in which case NAB(i,2) will
*>             not be modified.  Normally, NAB should be set to some
*>             distinctive value(s) before DLAEBZ is called.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MMAX)
*>          Workspace.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MMAX)
*>          Workspace.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:       All intervals converged.
*>          = 1--MMAX: The last INFO intervals did not converge.
*>          = MMAX+1:  More than MMAX intervals were generated.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laebz
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>      This routine is intended to be called only by other LAPACK
*>  routines, thus the interface is less user-friendly.  It is intended
*>  for two purposes:
*>
*>  (a) finding eigenvalues.  In this case, DLAEBZ should have one or
*>      more initial intervals set up in AB, and DLAEBZ should be called
*>      with IJOB=1.  This sets up NAB, and also counts the eigenvalues.
*>      Intervals with no eigenvalues would usually be thrown out at
*>      this point.  Also, if not all the eigenvalues in an interval i
*>      are desired, NAB(i,1) can be increased or NAB(i,2) decreased.
*>      For example, set NAB(i,1)=NAB(i,2)-1 to get the largest
*>      eigenvalue.  DLAEBZ is then called with IJOB=2 and MMAX
*>      no smaller than the value of MOUT returned by the call with
*>      IJOB=1.  After this (IJOB=2) call, eigenvalues NAB(i,1)+1
*>      through NAB(i,2) are approximately AB(i,1) (or AB(i,2)) to the
*>      tolerance specified by ABSTOL and RELTOL.
*>
*>  (b) finding an interval (a',b'] containing eigenvalues w(f),...,w(l).
*>      In this case, start with a Gershgorin interval  (a,b).  Set up
*>      AB to contain 2 search intervals, both initially (a,b).  One
*>      NVAL element should contain  f-1  and the other should contain  l
*>      , while C should contain a and b, resp.  NAB(i,1) should be -1
*>      and NAB(i,2) should be N+1, to flag an error if the desired
*>      interval does not lie in (a,b).  DLAEBZ is then called with
*>      IJOB=3.  On exit, if w(f-1) < w(f), then one of the intervals --
*>      j -- will have AB(j,1)=AB(j,2) and NAB(j,1)=NAB(j,2)=f-1, while
*>      if, to the specified tolerance, w(f-k)=...=w(f+r), k > 0 and r
*>      >= 0, then the interval will have  N(AB(j,1))=NAB(j,1)=f-k and
*>      N(AB(j,2))=NAB(j,2)=f+r.  The cases w(l) < w(l+1) and
*>      w(l-r)=...=w(l+k) are handled similarly.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLAEBZ( IJOB, NITMAX, N, MMAX, MINP, NBMIN, ABSTOL,
     $                   RELTOL, PIVMIN, D, E, E2, NVAL, AB, C, MOUT,
     $                   NAB, WORK, IWORK, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            IJOB, INFO, MINP, MMAX, MOUT, N, NBMIN, NITMAX
      DOUBLE PRECISION   ABSTOL, PIVMIN, RELTOL
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * ), NAB( MMAX, * ), NVAL( * )
      DOUBLE PRECISION   AB( MMAX, * ), C( * ), D( * ), E( * ), E2( * ),
     $                   WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, TWO, HALF
      PARAMETER          ( ZERO = 0.0D0, TWO = 2.0D0,
     $                   HALF = 1.0D0 / TWO )
*     ..
*     .. Local Scalars ..
      INTEGER            ITMP1, ITMP2, J, JI, JIT, JP, KF, KFNEW, KL,
     $                   KLNEW
      DOUBLE PRECISION   TMP1, TMP2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Check for Errors
*
      INFO = 0
      IF( IJOB.LT.1 .OR. IJOB.GT.3 ) THEN
         INFO = -1
         RETURN
      END IF
*
*     Initialize NAB
*
      IF( IJOB.EQ.1 ) THEN
*
*        Compute the number of eigenvalues in the initial intervals.
*
         MOUT = 0
         DO 30 JI = 1, MINP
            DO 20 JP = 1, 2
               TMP1 = D( 1 ) - AB( JI, JP )
               IF( ABS( TMP1 ).LT.PIVMIN )
     $            TMP1 = -PIVMIN
               NAB( JI, JP ) = 0
               IF( TMP1.LE.ZERO )
     $            NAB( JI, JP ) = 1
*
               DO 10 J = 2, N
                  TMP1 = D( J ) - E2( J-1 ) / TMP1 - AB( JI, JP )
                  IF( ABS( TMP1 ).LT.PIVMIN )
     $               TMP1 = -PIVMIN
                  IF( TMP1.LE.ZERO )
     $               NAB( JI, JP ) = NAB( JI, JP ) + 1
   10          CONTINUE
   20       CONTINUE
            MOUT = MOUT + NAB( JI, 2 ) - NAB( JI, 1 )
   30    CONTINUE
         RETURN
      END IF
*
*     Initialize for loop
*
*     KF and KL have the following meaning:
*        Intervals 1,...,KF-1 have converged.
*        Intervals KF,...,KL  still need to be refined.
*
      KF = 1
      KL = MINP
*
*     If IJOB=2, initialize C.
*     If IJOB=3, use the user-supplied starting point.
*
      IF( IJOB.EQ.2 ) THEN
         DO 40 JI = 1, MINP
            C( JI ) = HALF*( AB( JI, 1 )+AB( JI, 2 ) )
   40    CONTINUE
      END IF
*
*     Iteration loop
*
      DO 130 JIT = 1, NITMAX
*
*        Loop over intervals
*
         IF( KL-KF+1.GE.NBMIN .AND. NBMIN.GT.0 ) THEN
*
*           Begin of Parallel Version of the loop
*
            DO 60 JI = KF, KL
*
*              Compute N(c), the number of eigenvalues less than c
*
               WORK( JI ) = D( 1 ) - C( JI )
               IWORK( JI ) = 0
               IF( WORK( JI ).LE.PIVMIN ) THEN
                  IWORK( JI ) = 1
                  WORK( JI ) = MIN( WORK( JI ), -PIVMIN )
               END IF
*
               DO 50 J = 2, N
                  WORK( JI ) = D( J ) - E2( J-1 ) / WORK( JI ) - C( JI )
                  IF( WORK( JI ).LE.PIVMIN ) THEN
                     IWORK( JI ) = IWORK( JI ) + 1
                     WORK( JI ) = MIN( WORK( JI ), -PIVMIN )
                  END IF
   50          CONTINUE
   60       CONTINUE
*
            IF( IJOB.LE.2 ) THEN
*
*              IJOB=2: Choose all intervals containing eigenvalues.
*
               KLNEW = KL
               DO 70 JI = KF, KL
*
*                 Insure that N(w) is monotone
*
                  IWORK( JI ) = MIN( NAB( JI, 2 ),
     $                          MAX( NAB( JI, 1 ), IWORK( JI ) ) )
*
*                 Update the Queue -- add intervals if both halves
*                 contain eigenvalues.
*
                  IF( IWORK( JI ).EQ.NAB( JI, 2 ) ) THEN
*
*                    No eigenvalue in the upper interval:
*                    just use the lower interval.
*
                     AB( JI, 2 ) = C( JI )
*
                  ELSE IF( IWORK( JI ).EQ.NAB( JI, 1 ) ) THEN
*
*                    No eigenvalue in the lower interval:
*                    just use the upper interval.
*
                     AB( JI, 1 ) = C( JI )
                  ELSE
                     KLNEW = KLNEW + 1
                     IF( KLNEW.LE.MMAX ) THEN
*
*                       Eigenvalue in both intervals -- add upper to
*                       queue.
*
                        AB( KLNEW, 2 ) = AB( JI, 2 )
                        NAB( KLNEW, 2 ) = NAB( JI, 2 )
                        AB( KLNEW, 1 ) = C( JI )
                        NAB( KLNEW, 1 ) = IWORK( JI )
                        AB( JI, 2 ) = C( JI )
                        NAB( JI, 2 ) = IWORK( JI )
                     ELSE
                        INFO = MMAX + 1
                     END IF
                  END IF
   70          CONTINUE
               IF( INFO.NE.0 )
     $            RETURN
               KL = KLNEW
            ELSE
*
*              IJOB=3: Binary search.  Keep only the interval containing
*                      w   s.t. N(w) = NVAL
*
               DO 80 JI = KF, KL
                  IF( IWORK( JI ).LE.NVAL( JI ) ) THEN
                     AB( JI, 1 ) = C( JI )
                     NAB( JI, 1 ) = IWORK( JI )
                  END IF
                  IF( IWORK( JI ).GE.NVAL( JI ) ) THEN
                     AB( JI, 2 ) = C( JI )
                     NAB( JI, 2 ) = IWORK( JI )
                  END IF
   80          CONTINUE
            END IF
*
         ELSE
*
*           End of Parallel Version of the loop
*
*           Begin of Serial Version of the loop
*
            KLNEW = KL
            DO 100 JI = KF, KL
*
*              Compute N(w), the number of eigenvalues less than w
*
               TMP1 = C( JI )
               TMP2 = D( 1 ) - TMP1
               ITMP1 = 0
               IF( TMP2.LE.PIVMIN ) THEN
                  ITMP1 = 1
                  TMP2 = MIN( TMP2, -PIVMIN )
               END IF
*
               DO 90 J = 2, N
                  TMP2 = D( J ) - E2( J-1 ) / TMP2 - TMP1
                  IF( TMP2.LE.PIVMIN ) THEN
                     ITMP1 = ITMP1 + 1
                     TMP2 = MIN( TMP2, -PIVMIN )
                  END IF
   90          CONTINUE
*
               IF( IJOB.LE.2 ) THEN
*
*                 IJOB=2: Choose all intervals containing eigenvalues.
*
*                 Insure that N(w) is monotone
*
                  ITMP1 = MIN( NAB( JI, 2 ),
     $                    MAX( NAB( JI, 1 ), ITMP1 ) )
*
*                 Update the Queue -- add intervals if both halves
*                 contain eigenvalues.
*
                  IF( ITMP1.EQ.NAB( JI, 2 ) ) THEN
*
*                    No eigenvalue in the upper interval:
*                    just use the lower interval.
*
                     AB( JI, 2 ) = TMP1
*
                  ELSE IF( ITMP1.EQ.NAB( JI, 1 ) ) THEN
*
*                    No eigenvalue in the lower interval:
*                    just use the upper interval.
*
                     AB( JI, 1 ) = TMP1
                  ELSE IF( KLNEW.LT.MMAX ) THEN
*
*                    Eigenvalue in both intervals -- add upper to queue.
*
                     KLNEW = KLNEW + 1
                     AB( KLNEW, 2 ) = AB( JI, 2 )
                     NAB( KLNEW, 2 ) = NAB( JI, 2 )
                     AB( KLNEW, 1 ) = TMP1
                     NAB( KLNEW, 1 ) = ITMP1
                     AB( JI, 2 ) = TMP1
                     NAB( JI, 2 ) = ITMP1
                  ELSE
                     INFO = MMAX + 1
                     RETURN
                  END IF
               ELSE
*
*                 IJOB=3: Binary search.  Keep only the interval
*                         containing  w  s.t. N(w) = NVAL
*
                  IF( ITMP1.LE.NVAL( JI ) ) THEN
                     AB( JI, 1 ) = TMP1
                     NAB( JI, 1 ) = ITMP1
                  END IF
                  IF( ITMP1.GE.NVAL( JI ) ) THEN
                     AB( JI, 2 ) = TMP1
                     NAB( JI, 2 ) = ITMP1
                  END IF
               END IF
  100       CONTINUE
            KL = KLNEW
*
         END IF
*
*        Check for convergence
*
         KFNEW = KF
         DO 110 JI = KF, KL
            TMP1 = ABS( AB( JI, 2 )-AB( JI, 1 ) )
            TMP2 = MAX( ABS( AB( JI, 2 ) ), ABS( AB( JI, 1 ) ) )
            IF( TMP1.LT.MAX( ABSTOL, PIVMIN, RELTOL*TMP2 ) .OR.
     $          NAB( JI, 1 ).GE.NAB( JI, 2 ) ) THEN
*
*              Converged -- Swap with position KFNEW,
*                           then increment KFNEW
*
               IF( JI.GT.KFNEW ) THEN
                  TMP1 = AB( JI, 1 )
                  TMP2 = AB( JI, 2 )
                  ITMP1 = NAB( JI, 1 )
                  ITMP2 = NAB( JI, 2 )
                  AB( JI, 1 ) = AB( KFNEW, 1 )
                  AB( JI, 2 ) = AB( KFNEW, 2 )
                  NAB( JI, 1 ) = NAB( KFNEW, 1 )
                  NAB( JI, 2 ) = NAB( KFNEW, 2 )
                  AB( KFNEW, 1 ) = TMP1
                  AB( KFNEW, 2 ) = TMP2
                  NAB( KFNEW, 1 ) = ITMP1
                  NAB( KFNEW, 2 ) = ITMP2
                  IF( IJOB.EQ.3 ) THEN
                     ITMP1 = NVAL( JI )
                     NVAL( JI ) = NVAL( KFNEW )
                     NVAL( KFNEW ) = ITMP1
                  END IF
               END IF
               KFNEW = KFNEW + 1
            END IF
  110    CONTINUE
         KF = KFNEW
*
*        Choose Midpoints
*
         DO 120 JI = KF, KL
            C( JI ) = HALF*( AB( JI, 1 )+AB( JI, 2 ) )
  120    CONTINUE
*
*        If no more intervals to refine, quit.
*
         IF( KF.GT.KL )
     $      GO TO 140
  130 CONTINUE
*
*     Converged
*
  140 CONTINUE
      INFO = MAX( KL+1-KF, 0 )
      MOUT = KL
*
      RETURN
*
*     End of DLAEBZ
*
      END
*> \brief \b DLAED0 used by DSTEDC. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED0 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed0.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed0.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed0.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAED0( ICOMPQ, QSIZ, N, D, E, Q, LDQ, QSTORE, LDQS,
*                          WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            ICOMPQ, INFO, LDQ, LDQS, N, QSIZ
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   D( * ), E( * ), Q( LDQ, * ), QSTORE( LDQS, * ),
*      $                   WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAED0 computes all eigenvalues and corresponding eigenvectors of a
*> symmetric tridiagonal matrix using the divide and conquer method.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*>          ICOMPQ is INTEGER
*>          = 0:  Compute eigenvalues only.
*>          = 1:  Compute eigenvectors of original dense symmetric matrix
*>                also.  On entry, Q contains the orthogonal matrix used
*>                to reduce the original matrix to tridiagonal form.
*>          = 2:  Compute eigenvalues and eigenvectors of tridiagonal
*>                matrix.
*> \endverbatim
*>
*> \param[in] QSIZ
*> \verbatim
*>          QSIZ is INTEGER
*>         The dimension of the orthogonal matrix used to reduce
*>         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>         The dimension of the symmetric tridiagonal matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>         On entry, the main diagonal of the tridiagonal matrix.
*>         On exit, its eigenvalues.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>         The off-diagonal elements of the tridiagonal matrix.
*>         On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ, N)
*>         On entry, Q must contain an N-by-N orthogonal matrix.
*>         If ICOMPQ = 0    Q is not referenced.
*>         If ICOMPQ = 1    On entry, Q is a subset of the columns of the
*>                          orthogonal matrix used to reduce the full
*>                          matrix to tridiagonal form corresponding to
*>                          the subset of the full matrix which is being
*>                          decomposed at this time.
*>         If ICOMPQ = 2    On entry, Q will be the identity matrix.
*>                          On exit, Q contains the eigenvectors of the
*>                          tridiagonal matrix.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>         The leading dimension of the array Q.  If eigenvectors are
*>         desired, then  LDQ >= max(1,N).  In any case,  LDQ >= 1.
*> \endverbatim
*>
*> \param[out] QSTORE
*> \verbatim
*>          QSTORE is DOUBLE PRECISION array, dimension (LDQS, N)
*>         Referenced only when ICOMPQ = 1.  Used to store parts of
*>         the eigenvector matrix when the updating matrix multiplies
*>         take place.
*> \endverbatim
*>
*> \param[in] LDQS
*> \verbatim
*>          LDQS is INTEGER
*>         The leading dimension of the array QSTORE.  If ICOMPQ = 1,
*>         then  LDQS >= max(1,N).  In any case,  LDQS >= 1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array,
*>         If ICOMPQ = 0 or 1, the dimension of WORK must be at least
*>                     1 + 3*N + 2*N*lg N + 3*N**2
*>                     ( lg( N ) = smallest integer k
*>                                 such that 2^k >= N )
*>         If ICOMPQ = 2, the dimension of WORK must be at least
*>                     4*N + N**2.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array,
*>         If ICOMPQ = 0 or 1, the dimension of IWORK must be at least
*>                        6 + 6*N + 5*N*lg N.
*>                        ( lg( N ) = smallest integer k
*>                                    such that 2^k >= N )
*>         If ICOMPQ = 2, the dimension of IWORK must be at least
*>                        3 + 5*N.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  The algorithm failed to compute an eigenvalue while
*>                working on the submatrix lying in rows and columns
*>                INFO/(N+1) through mod(INFO,N+1).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed0
*
*> \par Contributors:
*  ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
*  =====================================================================
      SUBROUTINE DLAED0( ICOMPQ, QSIZ, N, D, E, Q, LDQ, QSTORE, LDQS,
     $                   WORK, IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            ICOMPQ, INFO, LDQ, LDQS, N, QSIZ
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   D( * ), E( * ), Q( LDQ, * ), QSTORE( LDQS, * ),
     $                   WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO
      PARAMETER          ( ZERO = 0.D0, ONE = 1.D0, TWO = 2.D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            CURLVL, CURPRB, CURR, I, IGIVCL, IGIVNM,
     $                   IGIVPT, INDXQ, IPERM, IPRMPT, IQ, IQPTR, IWREM,
     $                   J, K, LGN, MATSIZ, MSD2, SMLSIZ, SMM1, SPM1,
     $                   SPM2, SUBMAT, SUBPBS, TLVLS
      DOUBLE PRECISION   TEMP
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGEMM, DLACPY, DLAED1, DLAED7,
     $                   DSTEQR,
     $                   XERBLA
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, INT, LOG, MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.2 ) THEN
         INFO = -1
      ELSE IF( ( ICOMPQ.EQ.1 ) .AND. ( QSIZ.LT.MAX( 0, N ) ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDQS.LT.MAX( 1, N ) ) THEN
         INFO = -9
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLAED0', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      SMLSIZ = ILAENV( 9, 'DLAED0', ' ', 0, 0, 0, 0 )
*
*     Determine the size and placement of the submatrices, and save in
*     the leading elements of IWORK.
*
      IWORK( 1 ) = N
      SUBPBS = 1
      TLVLS = 0
   10 CONTINUE
      IF( IWORK( SUBPBS ).GT.SMLSIZ ) THEN
         DO 20 J = SUBPBS, 1, -1
            IWORK( 2*J ) = ( IWORK( J )+1 ) / 2
            IWORK( 2*J-1 ) = IWORK( J ) / 2
   20    CONTINUE
         TLVLS = TLVLS + 1
         SUBPBS = 2*SUBPBS
         GO TO 10
      END IF
      DO 30 J = 2, SUBPBS
         IWORK( J ) = IWORK( J ) + IWORK( J-1 )
   30 CONTINUE
*
*     Divide the matrix into SUBPBS submatrices of size at most SMLSIZ+1
*     using rank-1 modifications (cuts).
*
      SPM1 = SUBPBS - 1
      DO 40 I = 1, SPM1
         SUBMAT = IWORK( I ) + 1
         SMM1 = SUBMAT - 1
         D( SMM1 ) = D( SMM1 ) - ABS( E( SMM1 ) )
         D( SUBMAT ) = D( SUBMAT ) - ABS( E( SMM1 ) )
   40 CONTINUE
*
      INDXQ = 4*N + 3
      IF( ICOMPQ.NE.2 ) THEN
*
*        Set up workspaces for eigenvalues only/accumulate new vectors
*        routine
*
         TEMP = LOG( DBLE( N ) ) / LOG( TWO )
         LGN = INT( TEMP )
         IF( 2**LGN.LT.N )
     $      LGN = LGN + 1
         IF( 2**LGN.LT.N )
     $      LGN = LGN + 1
         IPRMPT = INDXQ + N + 1
         IPERM = IPRMPT + N*LGN
         IQPTR = IPERM + N*LGN
         IGIVPT = IQPTR + N + 2
         IGIVCL = IGIVPT + N*LGN
*
         IGIVNM = 1
         IQ = IGIVNM + 2*N*LGN
         IWREM = IQ + N**2 + 1
*
*        Initialize pointers
*
         DO 50 I = 0, SUBPBS
            IWORK( IPRMPT+I ) = 1
            IWORK( IGIVPT+I ) = 1
   50    CONTINUE
         IWORK( IQPTR ) = 1
      END IF
*
*     Solve each submatrix eigenproblem at the bottom of the divide and
*     conquer tree.
*
      CURR = 0
      DO 70 I = 0, SPM1
         IF( I.EQ.0 ) THEN
            SUBMAT = 1
            MATSIZ = IWORK( 1 )
         ELSE
            SUBMAT = IWORK( I ) + 1
            MATSIZ = IWORK( I+1 ) - IWORK( I )
         END IF
         IF( ICOMPQ.EQ.2 ) THEN
            CALL DSTEQR( 'I', MATSIZ, D( SUBMAT ), E( SUBMAT ),
     $                   Q( SUBMAT, SUBMAT ), LDQ, WORK, INFO )
            IF( INFO.NE.0 )
     $         GO TO 130
         ELSE
            CALL DSTEQR( 'I', MATSIZ, D( SUBMAT ), E( SUBMAT ),
     $                   WORK( IQ-1+IWORK( IQPTR+CURR ) ), MATSIZ, WORK,
     $                   INFO )
            IF( INFO.NE.0 )
     $         GO TO 130
            IF( ICOMPQ.EQ.1 ) THEN
               CALL DGEMM( 'N', 'N', QSIZ, MATSIZ, MATSIZ, ONE,
     $                     Q( 1, SUBMAT ), LDQ, WORK( IQ-1+IWORK( IQPTR+
     $                     CURR ) ), MATSIZ, ZERO, QSTORE( 1, SUBMAT ),
     $                     LDQS )
            END IF
            IWORK( IQPTR+CURR+1 ) = IWORK( IQPTR+CURR ) + MATSIZ**2
            CURR = CURR + 1
         END IF
         K = 1
         DO 60 J = SUBMAT, IWORK( I+1 )
            IWORK( INDXQ+J ) = K
            K = K + 1
   60    CONTINUE
   70 CONTINUE
*
*     Successively merge eigensystems of adjacent submatrices
*     into eigensystem for the corresponding larger matrix.
*
*     while ( SUBPBS > 1 )
*
      CURLVL = 1
   80 CONTINUE
      IF( SUBPBS.GT.1 ) THEN
         SPM2 = SUBPBS - 2
         DO 90 I = 0, SPM2, 2
            IF( I.EQ.0 ) THEN
               SUBMAT = 1
               MATSIZ = IWORK( 2 )
               MSD2 = IWORK( 1 )
               CURPRB = 0
            ELSE
               SUBMAT = IWORK( I ) + 1
               MATSIZ = IWORK( I+2 ) - IWORK( I )
               MSD2 = MATSIZ / 2
               CURPRB = CURPRB + 1
            END IF
*
*     Merge lower order eigensystems (of size MSD2 and MATSIZ - MSD2)
*     into an eigensystem of size MATSIZ.
*     DLAED1 is used only for the full eigensystem of a tridiagonal
*     matrix.
*     DLAED7 handles the cases in which eigenvalues only or eigenvalues
*     and eigenvectors of a full symmetric matrix (which was reduced to
*     tridiagonal form) are desired.
*
            IF( ICOMPQ.EQ.2 ) THEN
               CALL DLAED1( MATSIZ, D( SUBMAT ), Q( SUBMAT, SUBMAT ),
     $                      LDQ, IWORK( INDXQ+SUBMAT ),
     $                      E( SUBMAT+MSD2-1 ), MSD2, WORK,
     $                      IWORK( SUBPBS+1 ), INFO )
            ELSE
               CALL DLAED7( ICOMPQ, MATSIZ, QSIZ, TLVLS, CURLVL,
     $                      CURPRB,
     $                      D( SUBMAT ), QSTORE( 1, SUBMAT ), LDQS,
     $                      IWORK( INDXQ+SUBMAT ), E( SUBMAT+MSD2-1 ),
     $                      MSD2, WORK( IQ ), IWORK( IQPTR ),
     $                      IWORK( IPRMPT ), IWORK( IPERM ),
     $                      IWORK( IGIVPT ), IWORK( IGIVCL ),
     $                      WORK( IGIVNM ), WORK( IWREM ),
     $                      IWORK( SUBPBS+1 ), INFO )
            END IF
            IF( INFO.NE.0 )
     $         GO TO 130
            IWORK( I / 2+1 ) = IWORK( I+2 )
   90    CONTINUE
         SUBPBS = SUBPBS / 2
         CURLVL = CURLVL + 1
         GO TO 80
      END IF
*
*     end while
*
*     Re-merge the eigenvalues/vectors which were deflated at the final
*     merge step.
*
      IF( ICOMPQ.EQ.1 ) THEN
         DO 100 I = 1, N
            J = IWORK( INDXQ+I )
            WORK( I ) = D( J )
            CALL DCOPY( QSIZ, QSTORE( 1, J ), 1, Q( 1, I ), 1 )
  100    CONTINUE
         CALL DCOPY( N, WORK, 1, D, 1 )
      ELSE IF( ICOMPQ.EQ.2 ) THEN
         DO 110 I = 1, N
            J = IWORK( INDXQ+I )
            WORK( I ) = D( J )
            CALL DCOPY( N, Q( 1, J ), 1, WORK( N*I+1 ), 1 )
  110    CONTINUE
         CALL DCOPY( N, WORK, 1, D, 1 )
         CALL DLACPY( 'A', N, N, WORK( N+1 ), N, Q, LDQ )
      ELSE
         DO 120 I = 1, N
            J = IWORK( INDXQ+I )
            WORK( I ) = D( J )
  120    CONTINUE
         CALL DCOPY( N, WORK, 1, D, 1 )
      END IF
      GO TO 140
*
  130 CONTINUE
      INFO = SUBMAT*( N+1 ) + SUBMAT + MATSIZ - 1
*
  140 CONTINUE
      RETURN
*
*     End of DLAED0
*
      END
*> \brief \b DLAED1 used by DSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED1 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed1.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed1.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed1.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK,
*                          INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            CUTPNT, INFO, LDQ, N
*       DOUBLE PRECISION   RHO
*       ..
*       .. Array Arguments ..
*       INTEGER            INDXQ( * ), IWORK( * )
*       DOUBLE PRECISION   D( * ), Q( LDQ, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAED1 computes the updated eigensystem of a diagonal
*> matrix after modification by a rank-one symmetric matrix.  This
*> routine is used only for the eigenproblem which requires all
*> eigenvalues and eigenvectors of a tridiagonal matrix.  DLAED7 handles
*> the case in which eigenvalues only or eigenvalues and eigenvectors
*> of a full symmetric matrix (which was reduced to tridiagonal form)
*> are desired.
*>
*>   T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
*>
*>    where Z = Q**T*u, u is a vector of length N with ones in the
*>    CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
*>
*>    The eigenvectors of the original matrix are stored in Q, and the
*>    eigenvalues are in D.  The algorithm consists of three stages:
*>
*>       The first stage consists of deflating the size of the problem
*>       when there are multiple eigenvalues or if there is a zero in
*>       the Z vector.  For each such occurrence the dimension of the
*>       secular equation problem is reduced by one.  This stage is
*>       performed by the routine DLAED2.
*>
*>       The second stage consists of calculating the updated
*>       eigenvalues. This is done by finding the roots of the secular
*>       equation via the routine DLAED4 (as called by DLAED3).
*>       This routine also calculates the eigenvectors of the current
*>       problem.
*>
*>       The final stage consists of computing the updated eigenvectors
*>       directly using the updated eigenvalues.  The eigenvectors for
*>       the current problem are multiplied with the eigenvectors from
*>       the overall problem.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>         The dimension of the symmetric tridiagonal matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>         On entry, the eigenvalues of the rank-1-perturbed matrix.
*>         On exit, the eigenvalues of the repaired matrix.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ,N)
*>         On entry, the eigenvectors of the rank-1-perturbed matrix.
*>         On exit, the eigenvectors of the repaired tridiagonal matrix.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>         The leading dimension of the array Q.  LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] INDXQ
*> \verbatim
*>          INDXQ is INTEGER array, dimension (N)
*>         On entry, the permutation which separately sorts the two
*>         subproblems in D into ascending order.
*>         On exit, the permutation which will reintegrate the
*>         subproblems back into sorted order,
*>         i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*>          RHO is DOUBLE PRECISION
*>         The subdiagonal entry used to create the rank-1 modification.
*> \endverbatim
*>
*> \param[in] CUTPNT
*> \verbatim
*>          CUTPNT is INTEGER
*>         The location of the last eigenvalue in the leading sub-matrix.
*>         min(1,N) <= CUTPNT <= N/2.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (4*N + N**2)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = 1, an eigenvalue did not converge
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed1
*
*> \par Contributors:
*  ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA \n
*>  Modified by Francoise Tisseur, University of Tennessee
*>
*  =====================================================================
      SUBROUTINE DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK,
     $                   IWORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            CUTPNT, INFO, LDQ, N
      DOUBLE PRECISION   RHO
*     ..
*     .. Array Arguments ..
      INTEGER            INDXQ( * ), IWORK( * )
      DOUBLE PRECISION   D( * ), Q( LDQ, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            COLTYP, I, IDLMDA, INDX, INDXC, INDXP, IQ2, IS,
     $                   IW, IZ, K, N1, N2, ZPP1
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLAED2, DLAED3, DLAMRG,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IF( N.LT.0 ) THEN
         INFO = -1
      ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
         INFO = -4
      ELSE IF( MIN( 1, N / 2 ).GT.CUTPNT .OR. ( N / 2 ).LT.CUTPNT ) THEN
         INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLAED1', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     The following values are integer pointers which indicate
*     the portion of the workspace
*     used by a particular array in DLAED2 and DLAED3.
*
      IZ = 1
      IDLMDA = IZ + N
      IW = IDLMDA + N
      IQ2 = IW + N
*
      INDX = 1
      INDXC = INDX + N
      COLTYP = INDXC + N
      INDXP = COLTYP + N
*
*
*     Form the z-vector which consists of the last row of Q_1 and the
*     first row of Q_2.
*
      CALL DCOPY( CUTPNT, Q( CUTPNT, 1 ), LDQ, WORK( IZ ), 1 )
      ZPP1 = CUTPNT + 1
      CALL DCOPY( N-CUTPNT, Q( ZPP1, ZPP1 ), LDQ, WORK( IZ+CUTPNT ),
     $            1 )
*
*     Deflate eigenvalues.
*
      CALL DLAED2( K, N, CUTPNT, D, Q, LDQ, INDXQ, RHO, WORK( IZ ),
     $             WORK( IDLMDA ), WORK( IW ), WORK( IQ2 ),
     $             IWORK( INDX ), IWORK( INDXC ), IWORK( INDXP ),
     $             IWORK( COLTYP ), INFO )
*
      IF( INFO.NE.0 )
     $   GO TO 20
*
*     Solve Secular Equation.
*
      IF( K.NE.0 ) THEN
         IS = ( IWORK( COLTYP )+IWORK( COLTYP+1 ) )*CUTPNT +
     $        ( IWORK( COLTYP+1 )+IWORK( COLTYP+2 ) )*( N-CUTPNT ) + IQ2
         CALL DLAED3( K, N, CUTPNT, D, Q, LDQ, RHO, WORK( IDLMDA ),
     $                WORK( IQ2 ), IWORK( INDXC ), IWORK( COLTYP ),
     $                WORK( IW ), WORK( IS ), INFO )
         IF( INFO.NE.0 )
     $      GO TO 20
*
*     Prepare the INDXQ sorting permutation.
*
         N1 = K
         N2 = N - K
         CALL DLAMRG( N1, N2, D, 1, -1, INDXQ )
      ELSE
         DO 10 I = 1, N
            INDXQ( I ) = I
   10    CONTINUE
      END IF
*
   20 CONTINUE
      RETURN
*
*     End of DLAED1
*
      END
*> \brief \b DLAED2 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMBDA, W,
*                          Q2, INDX, INDXC, INDXP, COLTYP, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, K, LDQ, N, N1
*       DOUBLE PRECISION   RHO
*       ..
*       .. Array Arguments ..
*       INTEGER            COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
*      $                   INDXQ( * )
*       DOUBLE PRECISION   D( * ), DLAMBDA( * ), Q( LDQ, * ), Q2( * ),
*      $                   W( * ), Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAED2 merges the two sets of eigenvalues together into a single
*> sorted set.  Then it tries to deflate the size of the problem.
*> There are two ways in which deflation can occur:  when two or more
*> eigenvalues are close together or if there is a tiny entry in the
*> Z vector.  For each such occurrence the order of the related secular
*> equation problem is reduced by one.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[out] K
*> \verbatim
*>          K is INTEGER
*>         The number of non-deflated eigenvalues, and the order of the
*>         related secular equation. 0 <= K <=N.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>         The dimension of the symmetric tridiagonal matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*>          N1 is INTEGER
*>         The location of the last eigenvalue in the leading sub-matrix.
*>         min(1,N) <= N1 <= N/2.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>         On entry, D contains the eigenvalues of the two submatrices to
*>         be combined.
*>         On exit, D contains the trailing (N-K) updated eigenvalues
*>         (those which were deflated) sorted into increasing order.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ, N)
*>         On entry, Q contains the eigenvectors of two submatrices in
*>         the two square blocks with corners at (1,1), (N1,N1)
*>         and (N1+1, N1+1), (N,N).
*>         On exit, Q contains the trailing (N-K) updated eigenvectors
*>         (those which were deflated) in its last N-K columns.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>         The leading dimension of the array Q.  LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] INDXQ
*> \verbatim
*>          INDXQ is INTEGER array, dimension (N)
*>         The permutation which separately sorts the two sub-problems
*>         in D into ascending order.  Note that elements in the second
*>         half of this permutation must first have N1 added to their
*>         values. Destroyed on exit.
*> \endverbatim
*>
*> \param[in,out] RHO
*> \verbatim
*>          RHO is DOUBLE PRECISION
*>         On entry, the off-diagonal element associated with the rank-1
*>         cut which originally split the two submatrices which are now
*>         being recombined.
*>         On exit, RHO has been modified to the value required by
*>         DLAED3.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (N)
*>         On entry, Z contains the updating vector (the last
*>         row of the first sub-eigenvector matrix and the first row of
*>         the second sub-eigenvector matrix).
*>         On exit, the contents of Z have been destroyed by the updating
*>         process.
*> \endverbatim
*>
*> \param[out] DLAMBDA
*> \verbatim
*>          DLAMBDA is DOUBLE PRECISION array, dimension (N)
*>         A copy of the first K eigenvalues which will be used by
*>         DLAED3 to form the secular equation.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>         The first k values of the final deflation-altered z-vector
*>         which will be passed to DLAED3.
*> \endverbatim
*>
*> \param[out] Q2
*> \verbatim
*>          Q2 is DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2)
*>         A copy of the first K eigenvectors which will be used by
*>         DLAED3 in a matrix multiply (DGEMM) to solve for the new
*>         eigenvectors.
*> \endverbatim
*>
*> \param[out] INDX
*> \verbatim
*>          INDX is INTEGER array, dimension (N)
*>         The permutation used to sort the contents of DLAMBDA into
*>         ascending order.
*> \endverbatim
*>
*> \param[out] INDXC
*> \verbatim
*>          INDXC is INTEGER array, dimension (N)
*>         The permutation used to arrange the columns of the deflated
*>         Q matrix into three groups:  the first group contains non-zero
*>         elements only at and above N1, the second contains
*>         non-zero elements only below N1, and the third is dense.
*> \endverbatim
*>
*> \param[out] INDXP
*> \verbatim
*>          INDXP is INTEGER array, dimension (N)
*>         The permutation used to place deflated values of D at the end
*>         of the array.  INDXP(1:K) points to the nondeflated D-values
*>         and INDXP(K+1:N) points to the deflated eigenvalues.
*> \endverbatim
*>
*> \param[out] COLTYP
*> \verbatim
*>          COLTYP is INTEGER array, dimension (N)
*>         During execution, a label which will indicate which of the
*>         following types a column in the Q2 matrix is:
*>         1 : non-zero in the upper half only;
*>         2 : dense;
*>         3 : non-zero in the lower half only;
*>         4 : deflated.
*>         On exit, COLTYP(i) is the number of columns of type i,
*>         for i=1 to 4 only.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed2
*
*> \par Contributors:
*  ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA \n
*>  Modified by Francoise Tisseur, University of Tennessee
*>
*  =====================================================================
      SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMBDA,
     $                   W,
     $                   Q2, INDX, INDXC, INDXP, COLTYP, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, LDQ, N, N1
      DOUBLE PRECISION   RHO
*     ..
*     .. Array Arguments ..
      INTEGER            COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
     $                   INDXQ( * )
      DOUBLE PRECISION   D( * ), DLAMBDA( * ), Q( LDQ, * ), Q2( * ),
     $                   W( * ), Z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   MONE, ZERO, ONE, TWO, EIGHT
      PARAMETER          ( MONE = -1.0D0, ZERO = 0.0D0, ONE = 1.0D0,
     $                   TWO = 2.0D0, EIGHT = 8.0D0 )
*     ..
*     .. Local Arrays ..
      INTEGER            CTOT( 4 ), PSM( 4 )
*     ..
*     .. Local Scalars ..
      INTEGER            CT, I, IMAX, IQ1, IQ2, J, JMAX, JS, K2, N1P1,
     $                   N2, NJ, PJ
      DOUBLE PRECISION   C, EPS, S, T, TAU, TOL
*     ..
*     .. External Functions ..
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH, DLAPY2
      EXTERNAL           IDAMAX, DLAMCH, DLAPY2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLACPY, DLAMRG, DROT, DSCAL,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
         INFO = -6
      ELSE IF( MIN( 1, ( N / 2 ) ).GT.N1 .OR. ( N / 2 ).LT.N1 ) THEN
         INFO = -3
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLAED2', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      N2 = N - N1
      N1P1 = N1 + 1
*
      IF( RHO.LT.ZERO ) THEN
         CALL DSCAL( N2, MONE, Z( N1P1 ), 1 )
      END IF
*
*     Normalize z so that norm(z) = 1.  Since z is the concatenation of
*     two normalized vectors, norm2(z) = sqrt(2).
*
      T = ONE / SQRT( TWO )
      CALL DSCAL( N, T, Z, 1 )
*
*     RHO = ABS( norm(z)**2 * RHO )
*
      RHO = ABS( TWO*RHO )
*
*     Sort the eigenvalues into increasing order
*
      DO 10 I = N1P1, N
         INDXQ( I ) = INDXQ( I ) + N1
   10 CONTINUE
*
*     re-integrate the deflated parts from the last pass
*
      DO 20 I = 1, N
         DLAMBDA( I ) = D( INDXQ( I ) )
   20 CONTINUE
      CALL DLAMRG( N1, N2, DLAMBDA, 1, 1, INDXC )
      DO 30 I = 1, N
         INDX( I ) = INDXQ( INDXC( I ) )
   30 CONTINUE
*
*     Calculate the allowable deflation tolerance
*
      IMAX = IDAMAX( N, Z, 1 )
      JMAX = IDAMAX( N, D, 1 )
      EPS = DLAMCH( 'Epsilon' )
      TOL = EIGHT*EPS*MAX( ABS( D( JMAX ) ), ABS( Z( IMAX ) ) )
*
*     If the rank-1 modifier is small enough, no more needs to be done
*     except to reorganize Q so that its columns correspond with the
*     elements in D.
*
      IF( RHO*ABS( Z( IMAX ) ).LE.TOL ) THEN
         K = 0
         IQ2 = 1
         DO 40 J = 1, N
            I = INDX( J )
            CALL DCOPY( N, Q( 1, I ), 1, Q2( IQ2 ), 1 )
            DLAMBDA( J ) = D( I )
            IQ2 = IQ2 + N
   40    CONTINUE
         CALL DLACPY( 'A', N, N, Q2, N, Q, LDQ )
         CALL DCOPY( N, DLAMBDA, 1, D, 1 )
         GO TO 190
      END IF
*
*     If there are multiple eigenvalues then the problem deflates.  Here
*     the number of equal eigenvalues are found.  As each equal
*     eigenvalue is found, an elementary reflector is computed to rotate
*     the corresponding eigensubspace so that the corresponding
*     components of Z are zero in this new basis.
*
      DO 50 I = 1, N1
         COLTYP( I ) = 1
   50 CONTINUE
      DO 60 I = N1P1, N
         COLTYP( I ) = 3
   60 CONTINUE
*
*
      K = 0
      K2 = N + 1
      DO 70 J = 1, N
         NJ = INDX( J )
         IF( RHO*ABS( Z( NJ ) ).LE.TOL ) THEN
*
*           Deflate due to small z component.
*
            K2 = K2 - 1
            COLTYP( NJ ) = 4
            INDXP( K2 ) = NJ
            IF( J.EQ.N )
     $         GO TO 100
         ELSE
            PJ = NJ
            GO TO 80
         END IF
   70 CONTINUE
   80 CONTINUE
      J = J + 1
      NJ = INDX( J )
      IF( J.GT.N )
     $   GO TO 100
      IF( RHO*ABS( Z( NJ ) ).LE.TOL ) THEN
*
*        Deflate due to small z component.
*
         K2 = K2 - 1
         COLTYP( NJ ) = 4
         INDXP( K2 ) = NJ
      ELSE
*
*        Check if eigenvalues are close enough to allow deflation.
*
         S = Z( PJ )
         C = Z( NJ )
*
*        Find sqrt(a**2+b**2) without overflow or
*        destructive underflow.
*
         TAU = DLAPY2( C, S )
         T = D( NJ ) - D( PJ )
         C = C / TAU
         S = -S / TAU
         IF( ABS( T*C*S ).LE.TOL ) THEN
*
*           Deflation is possible.
*
            Z( NJ ) = TAU
            Z( PJ ) = ZERO
            IF( COLTYP( NJ ).NE.COLTYP( PJ ) )
     $         COLTYP( NJ ) = 2
            COLTYP( PJ ) = 4
            CALL DROT( N, Q( 1, PJ ), 1, Q( 1, NJ ), 1, C, S )
            T = D( PJ )*C**2 + D( NJ )*S**2
            D( NJ ) = D( PJ )*S**2 + D( NJ )*C**2
            D( PJ ) = T
            K2 = K2 - 1
            I = 1
   90       CONTINUE
            IF( K2+I.LE.N ) THEN
               IF( D( PJ ).LT.D( INDXP( K2+I ) ) ) THEN
                  INDXP( K2+I-1 ) = INDXP( K2+I )
                  INDXP( K2+I ) = PJ
                  I = I + 1
                  GO TO 90
               ELSE
                  INDXP( K2+I-1 ) = PJ
               END IF
            ELSE
               INDXP( K2+I-1 ) = PJ
            END IF
            PJ = NJ
         ELSE
            K = K + 1
            DLAMBDA( K ) = D( PJ )
            W( K ) = Z( PJ )
            INDXP( K ) = PJ
            PJ = NJ
         END IF
      END IF
      GO TO 80
  100 CONTINUE
*
*     Record the last eigenvalue.
*
      K = K + 1
      DLAMBDA( K ) = D( PJ )
      W( K ) = Z( PJ )
      INDXP( K ) = PJ
*
*     Count up the total number of the various types of columns, then
*     form a permutation which positions the four column types into
*     four uniform groups (although one or more of these groups may be
*     empty).
*
      DO 110 J = 1, 4
         CTOT( J ) = 0
  110 CONTINUE
      DO 120 J = 1, N
         CT = COLTYP( J )
         CTOT( CT ) = CTOT( CT ) + 1
  120 CONTINUE
*
*     PSM(*) = Position in SubMatrix (of types 1 through 4)
*
      PSM( 1 ) = 1
      PSM( 2 ) = 1 + CTOT( 1 )
      PSM( 3 ) = PSM( 2 ) + CTOT( 2 )
      PSM( 4 ) = PSM( 3 ) + CTOT( 3 )
      K = N - CTOT( 4 )
*
*     Fill out the INDXC array so that the permutation which it induces
*     will place all type-1 columns first, all type-2 columns next,
*     then all type-3's, and finally all type-4's.
*
      DO 130 J = 1, N
         JS = INDXP( J )
         CT = COLTYP( JS )
         INDX( PSM( CT ) ) = JS
         INDXC( PSM( CT ) ) = J
         PSM( CT ) = PSM( CT ) + 1
  130 CONTINUE
*
*     Sort the eigenvalues and corresponding eigenvectors into DLAMBDA
*     and Q2 respectively.  The eigenvalues/vectors which were not
*     deflated go into the first K slots of DLAMBDA and Q2 respectively,
*     while those which were deflated go into the last N - K slots.
*
      I = 1
      IQ1 = 1
      IQ2 = 1 + ( CTOT( 1 )+CTOT( 2 ) )*N1
      DO 140 J = 1, CTOT( 1 )
         JS = INDX( I )
         CALL DCOPY( N1, Q( 1, JS ), 1, Q2( IQ1 ), 1 )
         Z( I ) = D( JS )
         I = I + 1
         IQ1 = IQ1 + N1
  140 CONTINUE
*
      DO 150 J = 1, CTOT( 2 )
         JS = INDX( I )
         CALL DCOPY( N1, Q( 1, JS ), 1, Q2( IQ1 ), 1 )
         CALL DCOPY( N2, Q( N1+1, JS ), 1, Q2( IQ2 ), 1 )
         Z( I ) = D( JS )
         I = I + 1
         IQ1 = IQ1 + N1
         IQ2 = IQ2 + N2
  150 CONTINUE
*
      DO 160 J = 1, CTOT( 3 )
         JS = INDX( I )
         CALL DCOPY( N2, Q( N1+1, JS ), 1, Q2( IQ2 ), 1 )
         Z( I ) = D( JS )
         I = I + 1
         IQ2 = IQ2 + N2
  160 CONTINUE
*
      IQ1 = IQ2
      DO 170 J = 1, CTOT( 4 )
         JS = INDX( I )
         CALL DCOPY( N, Q( 1, JS ), 1, Q2( IQ2 ), 1 )
         IQ2 = IQ2 + N
         Z( I ) = D( JS )
         I = I + 1
  170 CONTINUE
*
*     The deflated eigenvalues and their corresponding vectors go back
*     into the last N - K slots of D and Q respectively.
*
      IF( K.LT.N ) THEN
         CALL DLACPY( 'A', N, CTOT( 4 ), Q2( IQ1 ), N,
     $                Q( 1, K+1 ), LDQ )
         CALL DCOPY( N-K, Z( K+1 ), 1, D( K+1 ), 1 )
      END IF
*
*     Copy CTOT into COLTYP for referencing in DLAED3.
*
      DO 180 J = 1, 4
         COLTYP( J ) = CTOT( J )
  180 CONTINUE
*
  190 CONTINUE
      RETURN
*
*     End of DLAED2
*
      END
*> \brief \b DLAED3 used by DSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed3.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMBDA, Q2, INDX,
*                          CTOT, W, S, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, K, LDQ, N, N1
*       DOUBLE PRECISION   RHO
*       ..
*       .. Array Arguments ..
*       INTEGER            CTOT( * ), INDX( * )
*       DOUBLE PRECISION   D( * ), DLAMBDA( * ), Q( LDQ, * ), Q2( * ),
*      $                   S( * ), W( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAED3 finds the roots of the secular equation, as defined by the
*> values in D, W, and RHO, between 1 and K.  It makes the
*> appropriate calls to DLAED4 and then updates the eigenvectors by
*> multiplying the matrix of eigenvectors of the pair of eigensystems
*> being combined by the matrix of eigenvectors of the K-by-K system
*> which is solved here.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of terms in the rational function to be solved by
*>          DLAED4.  K >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of rows and columns in the Q matrix.
*>          N >= K (deflation may result in N>K).
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*>          N1 is INTEGER
*>          The location of the last eigenvalue in the leading submatrix.
*>          min(1,N) <= N1 <= N/2.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          D(I) contains the updated eigenvalues for
*>          1 <= I <= K.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ,N)
*>          Initially the first K columns are used as workspace.
*>          On output the columns 1 to K contain
*>          the updated eigenvectors.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q.  LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*>          RHO is DOUBLE PRECISION
*>          The value of the parameter in the rank one update equation.
*>          RHO >= 0 required.
*> \endverbatim
*>
*> \param[in] DLAMBDA
*> \verbatim
*>          DLAMBDA is DOUBLE PRECISION array, dimension (K)
*>          The first K elements of this array contain the old roots
*>          of the deflated updating problem.  These are the poles
*>          of the secular equation.
*> \endverbatim
*>
*> \param[in] Q2
*> \verbatim
*>          Q2 is DOUBLE PRECISION array, dimension (LDQ2*N)
*>          The first K columns of this matrix contain the non-deflated
*>          eigenvectors for the split problem.
*> \endverbatim
*>
*> \param[in] INDX
*> \verbatim
*>          INDX is INTEGER array, dimension (N)
*>          The permutation used to arrange the columns of the deflated
*>          Q matrix into three groups (see DLAED2).
*>          The rows of the eigenvectors found by DLAED4 must be likewise
*>          permuted before the matrix multiply can take place.
*> \endverbatim
*>
*> \param[in] CTOT
*> \verbatim
*>          CTOT is INTEGER array, dimension (4)
*>          A count of the total number of the various types of columns
*>          in Q, as described in INDX.  The fourth column type is any
*>          column which has been deflated.
*> \endverbatim
*>
*> \param[in,out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (K)
*>          The first K elements of this array contain the components
*>          of the deflation-adjusted updating vector. Destroyed on
*>          output.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (N1 + 1)*K
*>          Will contain the eigenvectors of the repaired matrix which
*>          will be multiplied by the previously accumulated eigenvectors
*>          to update the system.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = 1, an eigenvalue did not converge
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed3
*
*> \par Contributors:
*  ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA \n
*>  Modified by Francoise Tisseur, University of Tennessee
*>
*  =====================================================================
      SUBROUTINE DLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMBDA, Q2, INDX,
     $                   CTOT, W, S, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, LDQ, N, N1
      DOUBLE PRECISION   RHO
*     ..
*     .. Array Arguments ..
      INTEGER            CTOT( * ), INDX( * )
      DOUBLE PRECISION   D( * ), DLAMBDA( * ), Q( LDQ, * ), Q2( * ),
     $                   S( * ), W( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D0, ZERO = 0.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, II, IQ2, J, N12, N2, N23
      DOUBLE PRECISION   TEMP
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DNRM2
      EXTERNAL           DNRM2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGEMM, DLACPY, DLAED4, DLASET,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, SIGN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IF( K.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.K ) THEN
         INFO = -2
      ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLAED3', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( K.EQ.0 )
     $   RETURN
*
*
      DO 20 J = 1, K
         CALL DLAED4( K, J, DLAMBDA, W, Q( 1, J ), RHO, D( J ),
     $                INFO )
*
*        If the zero finder fails, the computation is terminated.
*
         IF( INFO.NE.0 )
     $      GO TO 120
   20 CONTINUE
*
      IF( K.EQ.1 )
     $   GO TO 110
      IF( K.EQ.2 ) THEN
         DO 30 J = 1, K
            W( 1 ) = Q( 1, J )
            W( 2 ) = Q( 2, J )
            II = INDX( 1 )
            Q( 1, J ) = W( II )
            II = INDX( 2 )
            Q( 2, J ) = W( II )
   30    CONTINUE
         GO TO 110
      END IF
*
*     Compute updated W.
*
      CALL DCOPY( K, W, 1, S, 1 )
*
*     Initialize W(I) = Q(I,I)
*
      CALL DCOPY( K, Q, LDQ+1, W, 1 )
      DO 60 J = 1, K
         DO 40 I = 1, J - 1
            W( I ) = W( I )*( Q( I, J )/( DLAMBDA( I )-DLAMBDA( J ) ) )
   40    CONTINUE
         DO 50 I = J + 1, K
            W( I ) = W( I )*( Q( I, J )/( DLAMBDA( I )-DLAMBDA( J ) ) )
   50    CONTINUE
   60 CONTINUE
      DO 70 I = 1, K
         W( I ) = SIGN( SQRT( -W( I ) ), S( I ) )
   70 CONTINUE
*
*     Compute eigenvectors of the modified rank-1 modification.
*
      DO 100 J = 1, K
         DO 80 I = 1, K
            S( I ) = W( I ) / Q( I, J )
   80    CONTINUE
         TEMP = DNRM2( K, S, 1 )
         DO 90 I = 1, K
            II = INDX( I )
            Q( I, J ) = S( II ) / TEMP
   90    CONTINUE
  100 CONTINUE
*
*     Compute the updated eigenvectors.
*
  110 CONTINUE
*
      N2 = N - N1
      N12 = CTOT( 1 ) + CTOT( 2 )
      N23 = CTOT( 2 ) + CTOT( 3 )
*
      CALL DLACPY( 'A', N23, K, Q( CTOT( 1 )+1, 1 ), LDQ, S, N23 )
      IQ2 = N1*N12 + 1
      IF( N23.NE.0 ) THEN
         CALL DGEMM( 'N', 'N', N2, K, N23, ONE, Q2( IQ2 ), N2, S,
     $               N23,
     $               ZERO, Q( N1+1, 1 ), LDQ )
      ELSE
         CALL DLASET( 'A', N2, K, ZERO, ZERO, Q( N1+1, 1 ), LDQ )
      END IF
*
      CALL DLACPY( 'A', N12, K, Q, LDQ, S, N12 )
      IF( N12.NE.0 ) THEN
         CALL DGEMM( 'N', 'N', N1, K, N12, ONE, Q2, N1, S, N12, ZERO,
     $               Q,
     $               LDQ )
      ELSE
         CALL DLASET( 'A', N1, K, ZERO, ZERO, Q( 1, 1 ), LDQ )
      END IF
*
*
  120 CONTINUE
      RETURN
*
*     End of DLAED3
*
      END
*> \brief \b DLAED4 used by DSTEDC. Finds a single root of the secular equation.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED4 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed4.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed4.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed4.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAED4( N, I, D, Z, DELTA, RHO, DLAM, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            I, INFO, N
*       DOUBLE PRECISION   DLAM, RHO
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), DELTA( * ), Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> This subroutine computes the I-th updated eigenvalue of a symmetric
*> rank-one modification to a diagonal matrix whose elements are
*> given in the array d, and that
*>
*>            D(i) < D(j)  for  i < j
*>
*> and that RHO > 0.  This is arranged by the calling routine, and is
*> no loss in generality.  The rank-one modified system is thus
*>
*>            diag( D )  +  RHO * Z * Z_transpose.
*>
*> where we assume the Euclidean norm of Z is 1.
*>
*> The method consists of approximating the rational functions in the
*> secular equation by simpler interpolating rational functions.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>         The length of all arrays.
*> \endverbatim
*>
*> \param[in] I
*> \verbatim
*>          I is INTEGER
*>         The index of the eigenvalue to be computed.  1 <= I <= N.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>         The original eigenvalues.  It is assumed that they are in
*>         order, D(I) < D(J)  for I < J.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (N)
*>         The components of the updating vector.
*> \endverbatim
*>
*> \param[out] DELTA
*> \verbatim
*>          DELTA is DOUBLE PRECISION array, dimension (N)
*>         If N > 2, DELTA contains (D(j) - lambda_I) in its  j-th
*>         component.  If N = 1, then DELTA(1) = 1. If N = 2, see DLAED5
*>         for detail. The vector DELTA contains the information necessary
*>         to construct the eigenvectors by DLAED3 and DLAED9.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*>          RHO is DOUBLE PRECISION
*>         The scalar in the symmetric updating formula.
*> \endverbatim
*>
*> \param[out] DLAM
*> \verbatim
*>          DLAM is DOUBLE PRECISION
*>         The computed lambda_I, the I-th updated eigenvalue.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>         = 0:  successful exit
*>         > 0:  if INFO = 1, the updating process failed.
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  Logical variable ORGATI (origin-at-i?) is used for distinguishing
*>  whether D(i) or D(i+1) is treated as the origin.
*>
*>            ORGATI = .true.    origin at i
*>            ORGATI = .false.   origin at i+1
*>
*>   Logical variable SWTCH3 (switch-for-3-poles?) is for noting
*>   if we are working with THREE poles!
*>
*>   MAXIT is the maximum number of iterations allowed for each
*>   eigenvalue.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed4
*
*> \par Contributors:
*  ==================
*>
*>     Ren-Cang Li, Computer Science Division, University of California
*>     at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE DLAED4( N, I, D, Z, DELTA, RHO, DLAM, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            I, INFO, N
      DOUBLE PRECISION   DLAM, RHO
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), DELTA( * ), Z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            MAXIT
      PARAMETER          ( MAXIT = 30 )
      DOUBLE PRECISION   ZERO, ONE, TWO, THREE, FOUR, EIGHT, TEN
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
     $                   THREE = 3.0D0, FOUR = 4.0D0, EIGHT = 8.0D0,
     $                   TEN = 10.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ORGATI, SWTCH, SWTCH3
      INTEGER            II, IIM1, IIP1, IP1, ITER, J, NITER
      DOUBLE PRECISION   A, B, C, DEL, DLTLB, DLTUB, DPHI, DPSI, DW,
     $                   EPS, ERRETM, ETA, MIDPT, PHI, PREW, PSI,
     $                   RHOINV, TAU, TEMP, TEMP1, W
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   ZZ( 3 )
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLAED5, DLAED6
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Since this routine is called in an inner loop, we do no argument
*     checking.
*
*     Quick return for N=1 and 2.
*
      INFO = 0
      IF( N.EQ.1 ) THEN
*
*         Presumably, I=1 upon entry
*
         DLAM = D( 1 ) + RHO*Z( 1 )*Z( 1 )
         DELTA( 1 ) = ONE
         RETURN
      END IF
      IF( N.EQ.2 ) THEN
         CALL DLAED5( I, D, Z, DELTA, RHO, DLAM )
         RETURN
      END IF
*
*     Compute machine epsilon
*
      EPS = DLAMCH( 'Epsilon' )
      RHOINV = ONE / RHO
*
*     The case I = N
*
      IF( I.EQ.N ) THEN
*
*        Initialize some basic variables
*
         II = N - 1
         NITER = 1
*
*        Calculate initial guess
*
         MIDPT = RHO / TWO
*
*        If ||Z||_2 is not one, then TEMP should be set to
*        RHO * ||Z||_2^2 / TWO
*
         DO 10 J = 1, N
            DELTA( J ) = ( D( J )-D( I ) ) - MIDPT
   10    CONTINUE
*
         PSI = ZERO
         DO 20 J = 1, N - 2
            PSI = PSI + Z( J )*Z( J ) / DELTA( J )
   20    CONTINUE
*
         C = RHOINV + PSI
         W = C + Z( II )*Z( II ) / DELTA( II ) +
     $       Z( N )*Z( N ) / DELTA( N )
*
         IF( W.LE.ZERO ) THEN
            TEMP = Z( N-1 )*Z( N-1 ) / ( D( N )-D( N-1 )+RHO ) +
     $             Z( N )*Z( N ) / RHO
            IF( C.LE.TEMP ) THEN
               TAU = RHO
            ELSE
               DEL = D( N ) - D( N-1 )
               A = -C*DEL + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N )
               B = Z( N )*Z( N )*DEL
               IF( A.LT.ZERO ) THEN
                  TAU = TWO*B / ( SQRT( A*A+FOUR*B*C )-A )
               ELSE
                  TAU = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C )
               END IF
            END IF
*
*           It can be proved that
*               D(N)+RHO/2 <= LAMBDA(N) < D(N)+TAU <= D(N)+RHO
*
            DLTLB = MIDPT
            DLTUB = RHO
         ELSE
            DEL = D( N ) - D( N-1 )
            A = -C*DEL + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N )
            B = Z( N )*Z( N )*DEL
            IF( A.LT.ZERO ) THEN
               TAU = TWO*B / ( SQRT( A*A+FOUR*B*C )-A )
            ELSE
               TAU = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C )
            END IF
*
*           It can be proved that
*               D(N) < D(N)+TAU < LAMBDA(N) < D(N)+RHO/2
*
            DLTLB = ZERO
            DLTUB = MIDPT
         END IF
*
         DO 30 J = 1, N
            DELTA( J ) = ( D( J )-D( I ) ) - TAU
   30    CONTINUE
*
*        Evaluate PSI and the derivative DPSI
*
         DPSI = ZERO
         PSI = ZERO
         ERRETM = ZERO
         DO 40 J = 1, II
            TEMP = Z( J ) / DELTA( J )
            PSI = PSI + Z( J )*TEMP
            DPSI = DPSI + TEMP*TEMP
            ERRETM = ERRETM + PSI
   40    CONTINUE
         ERRETM = ABS( ERRETM )
*
*        Evaluate PHI and the derivative DPHI
*
         TEMP = Z( N ) / DELTA( N )
         PHI = Z( N )*TEMP
         DPHI = TEMP*TEMP
         ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV +
     $            ABS( TAU )*( DPSI+DPHI )
*
         W = RHOINV + PHI + PSI
*
*        Test for convergence
*
         IF( ABS( W ).LE.EPS*ERRETM ) THEN
            DLAM = D( I ) + TAU
            GO TO 250
         END IF
*
         IF( W.LE.ZERO ) THEN
            DLTLB = MAX( DLTLB, TAU )
         ELSE
            DLTUB = MIN( DLTUB, TAU )
         END IF
*
*        Calculate the new step
*
         NITER = NITER + 1
         C = W - DELTA( N-1 )*DPSI - DELTA( N )*DPHI
         A = ( DELTA( N-1 )+DELTA( N ) )*W -
     $       DELTA( N-1 )*DELTA( N )*( DPSI+DPHI )
         B = DELTA( N-1 )*DELTA( N )*W
         IF( C.LT.ZERO )
     $      C = ABS( C )
         IF( C.EQ.ZERO ) THEN
*           ETA = B/A
*           ETA = RHO - TAU
*           ETA = DLTUB - TAU
*
*           Update proposed by Li, Ren-Cang:
            ETA = -W / ( DPSI+DPHI )
         ELSE IF( A.GE.ZERO ) THEN
            ETA = ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
         ELSE
            ETA = TWO*B / ( A-SQRT( ABS( A*A-FOUR*B*C ) ) )
         END IF
*
*        Note, eta should be positive if w is negative, and
*        eta should be negative otherwise. However,
*        if for some reason caused by roundoff, eta*w > 0,
*        we simply use one Newton step instead. This way
*        will guarantee eta*w < 0.
*
         IF( W*ETA.GT.ZERO )
     $      ETA = -W / ( DPSI+DPHI )
         TEMP = TAU + ETA
         IF( TEMP.GT.DLTUB .OR. TEMP.LT.DLTLB ) THEN
            IF( W.LT.ZERO ) THEN
               ETA = ( DLTUB-TAU ) / TWO
            ELSE
               ETA = ( DLTLB-TAU ) / TWO
            END IF
         END IF
         DO 50 J = 1, N
            DELTA( J ) = DELTA( J ) - ETA
   50    CONTINUE
*
         TAU = TAU + ETA
*
*        Evaluate PSI and the derivative DPSI
*
         DPSI = ZERO
         PSI = ZERO
         ERRETM = ZERO
         DO 60 J = 1, II
            TEMP = Z( J ) / DELTA( J )
            PSI = PSI + Z( J )*TEMP
            DPSI = DPSI + TEMP*TEMP
            ERRETM = ERRETM + PSI
   60    CONTINUE
         ERRETM = ABS( ERRETM )
*
*        Evaluate PHI and the derivative DPHI
*
         TEMP = Z( N ) / DELTA( N )
         PHI = Z( N )*TEMP
         DPHI = TEMP*TEMP
         ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV +
     $            ABS( TAU )*( DPSI+DPHI )
*
         W = RHOINV + PHI + PSI
*
*        Main loop to update the values of the array   DELTA
*
         ITER = NITER + 1
*
         DO 90 NITER = ITER, MAXIT
*
*           Test for convergence
*
            IF( ABS( W ).LE.EPS*ERRETM ) THEN
               DLAM = D( I ) + TAU
               GO TO 250
            END IF
*
            IF( W.LE.ZERO ) THEN
               DLTLB = MAX( DLTLB, TAU )
            ELSE
               DLTUB = MIN( DLTUB, TAU )
            END IF
*
*           Calculate the new step
*
            C = W - DELTA( N-1 )*DPSI - DELTA( N )*DPHI
            A = ( DELTA( N-1 )+DELTA( N ) )*W -
     $          DELTA( N-1 )*DELTA( N )*( DPSI+DPHI )
            B = DELTA( N-1 )*DELTA( N )*W
            IF( A.GE.ZERO ) THEN
               ETA = ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
            ELSE
               ETA = TWO*B / ( A-SQRT( ABS( A*A-FOUR*B*C ) ) )
            END IF
*
*           Note, eta should be positive if w is negative, and
*           eta should be negative otherwise. However,
*           if for some reason caused by roundoff, eta*w > 0,
*           we simply use one Newton step instead. This way
*           will guarantee eta*w < 0.
*
            IF( W*ETA.GT.ZERO )
     $         ETA = -W / ( DPSI+DPHI )
            TEMP = TAU + ETA
            IF( TEMP.GT.DLTUB .OR. TEMP.LT.DLTLB ) THEN
               IF( W.LT.ZERO ) THEN
                  ETA = ( DLTUB-TAU ) / TWO
               ELSE
                  ETA = ( DLTLB-TAU ) / TWO
               END IF
            END IF
            DO 70 J = 1, N
               DELTA( J ) = DELTA( J ) - ETA
   70       CONTINUE
*
            TAU = TAU + ETA
*
*           Evaluate PSI and the derivative DPSI
*
            DPSI = ZERO
            PSI = ZERO
            ERRETM = ZERO
            DO 80 J = 1, II
               TEMP = Z( J ) / DELTA( J )
               PSI = PSI + Z( J )*TEMP
               DPSI = DPSI + TEMP*TEMP
               ERRETM = ERRETM + PSI
   80       CONTINUE
            ERRETM = ABS( ERRETM )
*
*           Evaluate PHI and the derivative DPHI
*
            TEMP = Z( N ) / DELTA( N )
            PHI = Z( N )*TEMP
            DPHI = TEMP*TEMP
            ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV +
     $               ABS( TAU )*( DPSI+DPHI )
*
            W = RHOINV + PHI + PSI
   90    CONTINUE
*
*        Return with INFO = 1, NITER = MAXIT and not converged
*
         INFO = 1
         DLAM = D( I ) + TAU
         GO TO 250
*
*        End for the case I = N
*
      ELSE
*
*        The case for I < N
*
         NITER = 1
         IP1 = I + 1
*
*        Calculate initial guess
*
         DEL = D( IP1 ) - D( I )
         MIDPT = DEL / TWO
         DO 100 J = 1, N
            DELTA( J ) = ( D( J )-D( I ) ) - MIDPT
  100    CONTINUE
*
         PSI = ZERO
         DO 110 J = 1, I - 1
            PSI = PSI + Z( J )*Z( J ) / DELTA( J )
  110    CONTINUE
*
         PHI = ZERO
         DO 120 J = N, I + 2, -1
            PHI = PHI + Z( J )*Z( J ) / DELTA( J )
  120    CONTINUE
         C = RHOINV + PSI + PHI
         W = C + Z( I )*Z( I ) / DELTA( I ) +
     $       Z( IP1 )*Z( IP1 ) / DELTA( IP1 )
*
         IF( W.GT.ZERO ) THEN
*
*           d(i)< the ith eigenvalue < (d(i)+d(i+1))/2
*
*           We choose d(i) as origin.
*
            ORGATI = .TRUE.
            A = C*DEL + Z( I )*Z( I ) + Z( IP1 )*Z( IP1 )
            B = Z( I )*Z( I )*DEL
            IF( A.GT.ZERO ) THEN
               TAU = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
            ELSE
               TAU = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
            END IF
            DLTLB = ZERO
            DLTUB = MIDPT
         ELSE
*
*           (d(i)+d(i+1))/2 <= the ith eigenvalue < d(i+1)
*
*           We choose d(i+1) as origin.
*
            ORGATI = .FALSE.
            A = C*DEL - Z( I )*Z( I ) - Z( IP1 )*Z( IP1 )
            B = Z( IP1 )*Z( IP1 )*DEL
            IF( A.LT.ZERO ) THEN
               TAU = TWO*B / ( A-SQRT( ABS( A*A+FOUR*B*C ) ) )
            ELSE
               TAU = -( A+SQRT( ABS( A*A+FOUR*B*C ) ) ) / ( TWO*C )
            END IF
            DLTLB = -MIDPT
            DLTUB = ZERO
         END IF
*
         IF( ORGATI ) THEN
            DO 130 J = 1, N
               DELTA( J ) = ( D( J )-D( I ) ) - TAU
  130       CONTINUE
         ELSE
            DO 140 J = 1, N
               DELTA( J ) = ( D( J )-D( IP1 ) ) - TAU
  140       CONTINUE
         END IF
         IF( ORGATI ) THEN
            II = I
         ELSE
            II = I + 1
         END IF
         IIM1 = II - 1
         IIP1 = II + 1
*
*        Evaluate PSI and the derivative DPSI
*
         DPSI = ZERO
         PSI = ZERO
         ERRETM = ZERO
         DO 150 J = 1, IIM1
            TEMP = Z( J ) / DELTA( J )
            PSI = PSI + Z( J )*TEMP
            DPSI = DPSI + TEMP*TEMP
            ERRETM = ERRETM + PSI
  150    CONTINUE
         ERRETM = ABS( ERRETM )
*
*        Evaluate PHI and the derivative DPHI
*
         DPHI = ZERO
         PHI = ZERO
         DO 160 J = N, IIP1, -1
            TEMP = Z( J ) / DELTA( J )
            PHI = PHI + Z( J )*TEMP
            DPHI = DPHI + TEMP*TEMP
            ERRETM = ERRETM + PHI
  160    CONTINUE
*
         W = RHOINV + PHI + PSI
*
*        W is the value of the secular function with
*        its ii-th element removed.
*
         SWTCH3 = .FALSE.
         IF( ORGATI ) THEN
            IF( W.LT.ZERO )
     $         SWTCH3 = .TRUE.
         ELSE
            IF( W.GT.ZERO )
     $         SWTCH3 = .TRUE.
         END IF
         IF( II.EQ.1 .OR. II.EQ.N )
     $      SWTCH3 = .FALSE.
*
         TEMP = Z( II ) / DELTA( II )
         DW = DPSI + DPHI + TEMP*TEMP
         TEMP = Z( II )*TEMP
         W = W + TEMP
         ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV +
     $            THREE*ABS( TEMP ) + ABS( TAU )*DW
*
*        Test for convergence
*
         IF( ABS( W ).LE.EPS*ERRETM ) THEN
            IF( ORGATI ) THEN
               DLAM = D( I ) + TAU
            ELSE
               DLAM = D( IP1 ) + TAU
            END IF
            GO TO 250
         END IF
*
         IF( W.LE.ZERO ) THEN
            DLTLB = MAX( DLTLB, TAU )
         ELSE
            DLTUB = MIN( DLTUB, TAU )
         END IF
*
*        Calculate the new step
*
         NITER = NITER + 1
         IF( .NOT.SWTCH3 ) THEN
            IF( ORGATI ) THEN
               C = W - DELTA( IP1 )*DW - ( D( I )-D( IP1 ) )*
     $             ( Z( I ) / DELTA( I ) )**2
            ELSE
               C = W - DELTA( I )*DW - ( D( IP1 )-D( I ) )*
     $             ( Z( IP1 ) / DELTA( IP1 ) )**2
            END IF
            A = ( DELTA( I )+DELTA( IP1 ) )*W -
     $          DELTA( I )*DELTA( IP1 )*DW
            B = DELTA( I )*DELTA( IP1 )*W
            IF( C.EQ.ZERO ) THEN
               IF( A.EQ.ZERO ) THEN
                  IF( ORGATI ) THEN
                     A = Z( I )*Z( I ) + DELTA( IP1 )*DELTA( IP1 )*
     $                   ( DPSI+DPHI )
                  ELSE
                     A = Z( IP1 )*Z( IP1 ) + DELTA( I )*DELTA( I )*
     $                   ( DPSI+DPHI )
                  END IF
               END IF
               ETA = B / A
            ELSE IF( A.LE.ZERO ) THEN
               ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
            ELSE
               ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
            END IF
         ELSE
*
*           Interpolation using THREE most relevant poles
*
            TEMP = RHOINV + PSI + PHI
            IF( ORGATI ) THEN
               TEMP1 = Z( IIM1 ) / DELTA( IIM1 )
               TEMP1 = TEMP1*TEMP1
               C = TEMP - DELTA( IIP1 )*( DPSI+DPHI ) -
     $             ( D( IIM1 )-D( IIP1 ) )*TEMP1
               ZZ( 1 ) = Z( IIM1 )*Z( IIM1 )
               ZZ( 3 ) = DELTA( IIP1 )*DELTA( IIP1 )*
     $                   ( ( DPSI-TEMP1 )+DPHI )
            ELSE
               TEMP1 = Z( IIP1 ) / DELTA( IIP1 )
               TEMP1 = TEMP1*TEMP1
               C = TEMP - DELTA( IIM1 )*( DPSI+DPHI ) -
     $             ( D( IIP1 )-D( IIM1 ) )*TEMP1
               ZZ( 1 ) = DELTA( IIM1 )*DELTA( IIM1 )*
     $                   ( DPSI+( DPHI-TEMP1 ) )
               ZZ( 3 ) = Z( IIP1 )*Z( IIP1 )
            END IF
            ZZ( 2 ) = Z( II )*Z( II )
            CALL DLAED6( NITER, ORGATI, C, DELTA( IIM1 ), ZZ, W, ETA,
     $                   INFO )
            IF( INFO.NE.0 )
     $         GO TO 250
         END IF
*
*        Note, eta should be positive if w is negative, and
*        eta should be negative otherwise. However,
*        if for some reason caused by roundoff, eta*w > 0,
*        we simply use one Newton step instead. This way
*        will guarantee eta*w < 0.
*
         IF( W*ETA.GE.ZERO )
     $      ETA = -W / DW
         TEMP = TAU + ETA
         IF( TEMP.GT.DLTUB .OR. TEMP.LT.DLTLB ) THEN
            IF( W.LT.ZERO ) THEN
               ETA = ( DLTUB-TAU ) / TWO
            ELSE
               ETA = ( DLTLB-TAU ) / TWO
            END IF
         END IF
*
         PREW = W
*
         DO 180 J = 1, N
            DELTA( J ) = DELTA( J ) - ETA
  180    CONTINUE
*
*        Evaluate PSI and the derivative DPSI
*
         DPSI = ZERO
         PSI = ZERO
         ERRETM = ZERO
         DO 190 J = 1, IIM1
            TEMP = Z( J ) / DELTA( J )
            PSI = PSI + Z( J )*TEMP
            DPSI = DPSI + TEMP*TEMP
            ERRETM = ERRETM + PSI
  190    CONTINUE
         ERRETM = ABS( ERRETM )
*
*        Evaluate PHI and the derivative DPHI
*
         DPHI = ZERO
         PHI = ZERO
         DO 200 J = N, IIP1, -1
            TEMP = Z( J ) / DELTA( J )
            PHI = PHI + Z( J )*TEMP
            DPHI = DPHI + TEMP*TEMP
            ERRETM = ERRETM + PHI
  200    CONTINUE
*
         TEMP = Z( II ) / DELTA( II )
         DW = DPSI + DPHI + TEMP*TEMP
         TEMP = Z( II )*TEMP
         W = RHOINV + PHI + PSI + TEMP
         ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV +
     $            THREE*ABS( TEMP ) + ABS( TAU+ETA )*DW
*
         SWTCH = .FALSE.
         IF( ORGATI ) THEN
            IF( -W.GT.ABS( PREW ) / TEN )
     $         SWTCH = .TRUE.
         ELSE
            IF( W.GT.ABS( PREW ) / TEN )
     $         SWTCH = .TRUE.
         END IF
*
         TAU = TAU + ETA
*
*        Main loop to update the values of the array   DELTA
*
         ITER = NITER + 1
*
         DO 240 NITER = ITER, MAXIT
*
*           Test for convergence
*
            IF( ABS( W ).LE.EPS*ERRETM ) THEN
               IF( ORGATI ) THEN
                  DLAM = D( I ) + TAU
               ELSE
                  DLAM = D( IP1 ) + TAU
               END IF
               GO TO 250
            END IF
*
            IF( W.LE.ZERO ) THEN
               DLTLB = MAX( DLTLB, TAU )
            ELSE
               DLTUB = MIN( DLTUB, TAU )
            END IF
*
*           Calculate the new step
*
            IF( .NOT.SWTCH3 ) THEN
               IF( .NOT.SWTCH ) THEN
                  IF( ORGATI ) THEN
                     C = W - DELTA( IP1 )*DW -
     $                   ( D( I )-D( IP1 ) )*( Z( I ) / DELTA( I ) )**2
                  ELSE
                     C = W - DELTA( I )*DW - ( D( IP1 )-D( I ) )*
     $                   ( Z( IP1 ) / DELTA( IP1 ) )**2
                  END IF
               ELSE
                  TEMP = Z( II ) / DELTA( II )
                  IF( ORGATI ) THEN
                     DPSI = DPSI + TEMP*TEMP
                  ELSE
                     DPHI = DPHI + TEMP*TEMP
                  END IF
                  C = W - DELTA( I )*DPSI - DELTA( IP1 )*DPHI
               END IF
               A = ( DELTA( I )+DELTA( IP1 ) )*W -
     $             DELTA( I )*DELTA( IP1 )*DW
               B = DELTA( I )*DELTA( IP1 )*W
               IF( C.EQ.ZERO ) THEN
                  IF( A.EQ.ZERO ) THEN
                     IF( .NOT.SWTCH ) THEN
                        IF( ORGATI ) THEN
                           A = Z( I )*Z( I ) + DELTA( IP1 )*
     $                         DELTA( IP1 )*( DPSI+DPHI )
                        ELSE
                           A = Z( IP1 )*Z( IP1 ) +
     $                         DELTA( I )*DELTA( I )*( DPSI+DPHI )
                        END IF
                     ELSE
                        A = DELTA( I )*DELTA( I )*DPSI +
     $                      DELTA( IP1 )*DELTA( IP1 )*DPHI
                     END IF
                  END IF
                  ETA = B / A
               ELSE IF( A.LE.ZERO ) THEN
                  ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
               ELSE
                  ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
               END IF
            ELSE
*
*              Interpolation using THREE most relevant poles
*
               TEMP = RHOINV + PSI + PHI
               IF( SWTCH ) THEN
                  C = TEMP - DELTA( IIM1 )*DPSI - DELTA( IIP1 )*DPHI
                  ZZ( 1 ) = DELTA( IIM1 )*DELTA( IIM1 )*DPSI
                  ZZ( 3 ) = DELTA( IIP1 )*DELTA( IIP1 )*DPHI
               ELSE
                  IF( ORGATI ) THEN
                     TEMP1 = Z( IIM1 ) / DELTA( IIM1 )
                     TEMP1 = TEMP1*TEMP1
                     C = TEMP - DELTA( IIP1 )*( DPSI+DPHI ) -
     $                   ( D( IIM1 )-D( IIP1 ) )*TEMP1
                     ZZ( 1 ) = Z( IIM1 )*Z( IIM1 )
                     ZZ( 3 ) = DELTA( IIP1 )*DELTA( IIP1 )*
     $                         ( ( DPSI-TEMP1 )+DPHI )
                  ELSE
                     TEMP1 = Z( IIP1 ) / DELTA( IIP1 )
                     TEMP1 = TEMP1*TEMP1
                     C = TEMP - DELTA( IIM1 )*( DPSI+DPHI ) -
     $                   ( D( IIP1 )-D( IIM1 ) )*TEMP1
                     ZZ( 1 ) = DELTA( IIM1 )*DELTA( IIM1 )*
     $                         ( DPSI+( DPHI-TEMP1 ) )
                     ZZ( 3 ) = Z( IIP1 )*Z( IIP1 )
                  END IF
               END IF
               CALL DLAED6( NITER, ORGATI, C, DELTA( IIM1 ), ZZ, W,
     $                      ETA,
     $                      INFO )
               IF( INFO.NE.0 )
     $            GO TO 250
            END IF
*
*           Note, eta should be positive if w is negative, and
*           eta should be negative otherwise. However,
*           if for some reason caused by roundoff, eta*w > 0,
*           we simply use one Newton step instead. This way
*           will guarantee eta*w < 0.
*
            IF( W*ETA.GE.ZERO )
     $         ETA = -W / DW
            TEMP = TAU + ETA
            IF( TEMP.GT.DLTUB .OR. TEMP.LT.DLTLB ) THEN
               IF( W.LT.ZERO ) THEN
                  ETA = ( DLTUB-TAU ) / TWO
               ELSE
                  ETA = ( DLTLB-TAU ) / TWO
               END IF
            END IF
*
            DO 210 J = 1, N
               DELTA( J ) = DELTA( J ) - ETA
  210       CONTINUE
*
            TAU = TAU + ETA
            PREW = W
*
*           Evaluate PSI and the derivative DPSI
*
            DPSI = ZERO
            PSI = ZERO
            ERRETM = ZERO
            DO 220 J = 1, IIM1
               TEMP = Z( J ) / DELTA( J )
               PSI = PSI + Z( J )*TEMP
               DPSI = DPSI + TEMP*TEMP
               ERRETM = ERRETM + PSI
  220       CONTINUE
            ERRETM = ABS( ERRETM )
*
*           Evaluate PHI and the derivative DPHI
*
            DPHI = ZERO
            PHI = ZERO
            DO 230 J = N, IIP1, -1
               TEMP = Z( J ) / DELTA( J )
               PHI = PHI + Z( J )*TEMP
               DPHI = DPHI + TEMP*TEMP
               ERRETM = ERRETM + PHI
  230       CONTINUE
*
            TEMP = Z( II ) / DELTA( II )
            DW = DPSI + DPHI + TEMP*TEMP
            TEMP = Z( II )*TEMP
            W = RHOINV + PHI + PSI + TEMP
            ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV +
     $               THREE*ABS( TEMP ) + ABS( TAU )*DW
            IF( W*PREW.GT.ZERO .AND. ABS( W ).GT.ABS( PREW ) / TEN )
     $         SWTCH = .NOT.SWTCH
*
  240    CONTINUE
*
*        Return with INFO = 1, NITER = MAXIT and not converged
*
         INFO = 1
         IF( ORGATI ) THEN
            DLAM = D( I ) + TAU
         ELSE
            DLAM = D( IP1 ) + TAU
         END IF
*
      END IF
*
  250 CONTINUE
*
      RETURN
*
*     End of DLAED4
*
      END
*> \brief \b DLAED5 used by DSTEDC. Solves the 2-by-2 secular equation.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED5 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed5.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed5.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed5.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAED5( I, D, Z, DELTA, RHO, DLAM )
*
*       .. Scalar Arguments ..
*       INTEGER            I
*       DOUBLE PRECISION   DLAM, RHO
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( 2 ), DELTA( 2 ), Z( 2 )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> This subroutine computes the I-th eigenvalue of a symmetric rank-one
*> modification of a 2-by-2 diagonal matrix
*>
*>            diag( D )  +  RHO * Z * transpose(Z) .
*>
*> The diagonal elements in the array D are assumed to satisfy
*>
*>            D(i) < D(j)  for  i < j .
*>
*> We also assume RHO > 0 and that the Euclidean norm of the vector
*> Z is one.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] I
*> \verbatim
*>          I is INTEGER
*>         The index of the eigenvalue to be computed.  I = 1 or I = 2.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (2)
*>         The original eigenvalues.  We assume D(1) < D(2).
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (2)
*>         The components of the updating vector.
*> \endverbatim
*>
*> \param[out] DELTA
*> \verbatim
*>          DELTA is DOUBLE PRECISION array, dimension (2)
*>         The vector DELTA contains the information necessary
*>         to construct the eigenvectors.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*>          RHO is DOUBLE PRECISION
*>         The scalar in the symmetric updating formula.
*> \endverbatim
*>
*> \param[out] DLAM
*> \verbatim
*>          DLAM is DOUBLE PRECISION
*>         The computed lambda_I, the I-th updated eigenvalue.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed5
*
*> \par Contributors:
*  ==================
*>
*>     Ren-Cang Li, Computer Science Division, University of California
*>     at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE DLAED5( I, D, Z, DELTA, RHO, DLAM )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            I
      DOUBLE PRECISION   DLAM, RHO
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( 2 ), DELTA( 2 ), Z( 2 )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO, FOUR
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
     $                   FOUR = 4.0D0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   B, C, DEL, TAU, TEMP, W
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, SQRT
*     ..
*     .. Executable Statements ..
*
      DEL = D( 2 ) - D( 1 )
      IF( I.EQ.1 ) THEN
         W = ONE + TWO*RHO*( Z( 2 )*Z( 2 )-Z( 1 )*Z( 1 ) ) / DEL
         IF( W.GT.ZERO ) THEN
            B = DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
            C = RHO*Z( 1 )*Z( 1 )*DEL
*
*           B > ZERO, always
*
            TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) )
            DLAM = D( 1 ) + TAU
            DELTA( 1 ) = -Z( 1 ) / TAU
            DELTA( 2 ) = Z( 2 ) / ( DEL-TAU )
         ELSE
            B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
            C = RHO*Z( 2 )*Z( 2 )*DEL
            IF( B.GT.ZERO ) THEN
               TAU = -TWO*C / ( B+SQRT( B*B+FOUR*C ) )
            ELSE
               TAU = ( B-SQRT( B*B+FOUR*C ) ) / TWO
            END IF
            DLAM = D( 2 ) + TAU
            DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
            DELTA( 2 ) = -Z( 2 ) / TAU
         END IF
         TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
         DELTA( 1 ) = DELTA( 1 ) / TEMP
         DELTA( 2 ) = DELTA( 2 ) / TEMP
      ELSE
*
*     Now I=2
*
         B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
         C = RHO*Z( 2 )*Z( 2 )*DEL
         IF( B.GT.ZERO ) THEN
            TAU = ( B+SQRT( B*B+FOUR*C ) ) / TWO
         ELSE
            TAU = TWO*C / ( -B+SQRT( B*B+FOUR*C ) )
         END IF
         DLAM = D( 2 ) + TAU
         DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
         DELTA( 2 ) = -Z( 2 ) / TAU
         TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
         DELTA( 1 ) = DELTA( 1 ) / TEMP
         DELTA( 2 ) = DELTA( 2 ) / TEMP
      END IF
      RETURN
*
*     End of DLAED5
*
      END
*> \brief \b DLAED6 used by DSTEDC. Computes one Newton step in solution of the secular equation.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED6 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed6.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed6.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed6.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAED6( KNITER, ORGATI, RHO, D, Z, FINIT, TAU, INFO )
*
*       .. Scalar Arguments ..
*       LOGICAL            ORGATI
*       INTEGER            INFO, KNITER
*       DOUBLE PRECISION   FINIT, RHO, TAU
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( 3 ), Z( 3 )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAED6 computes the positive or negative root (closest to the origin)
*> of
*>                  z(1)        z(2)        z(3)
*> f(x) =   rho + --------- + ---------- + ---------
*>                 d(1)-x      d(2)-x      d(3)-x
*>
*> It is assumed that
*>
*>       if ORGATI = .true. the root is between d(2) and d(3);
*>       otherwise it is between d(1) and d(2)
*>
*> This routine will be called by DLAED4 when necessary. In most cases,
*> the root sought is the smallest in magnitude, though it might not be
*> in some extremely rare situations.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] KNITER
*> \verbatim
*>          KNITER is INTEGER
*>               Refer to DLAED4 for its significance.
*> \endverbatim
*>
*> \param[in] ORGATI
*> \verbatim
*>          ORGATI is LOGICAL
*>               If ORGATI is true, the needed root is between d(2) and
*>               d(3); otherwise it is between d(1) and d(2).  See
*>               DLAED4 for further details.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*>          RHO is DOUBLE PRECISION
*>               Refer to the equation f(x) above.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (3)
*>               D satisfies d(1) < d(2) < d(3).
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (3)
*>               Each of the elements in z must be positive.
*> \endverbatim
*>
*> \param[in] FINIT
*> \verbatim
*>          FINIT is DOUBLE PRECISION
*>               The value of f at 0. It is more accurate than the one
*>               evaluated inside this routine (if someone wants to do
*>               so).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION
*>               The root of the equation f(x).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>               = 0: successful exit
*>               > 0: if INFO = 1, failure to converge
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed6
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  10/02/03: This version has a few statements commented out for thread
*>  safety (machine parameters are computed on each entry). SJH.
*>
*>  05/10/06: Modified from a new version of Ren-Cang Li, use
*>     Gragg-Thornton-Warner cubic convergent scheme for better stability.
*> \endverbatim
*
*> \par Contributors:
*  ==================
*>
*>     Ren-Cang Li, Computer Science Division, University of California
*>     at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE DLAED6( KNITER, ORGATI, RHO, D, Z, FINIT, TAU,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      LOGICAL            ORGATI
      INTEGER            INFO, KNITER
      DOUBLE PRECISION   FINIT, RHO, TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( 3 ), Z( 3 )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            MAXIT
      PARAMETER          ( MAXIT = 40 )
      DOUBLE PRECISION   ZERO, ONE, TWO, THREE, FOUR, EIGHT
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
     $                   THREE = 3.0D0, FOUR = 4.0D0, EIGHT = 8.0D0 )
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   DSCALE( 3 ), ZSCALE( 3 )
*     ..
*     .. Local Scalars ..
      LOGICAL            SCALE
      INTEGER            I, ITER, NITER
      DOUBLE PRECISION   A, B, BASE, C, DDF, DF, EPS, ERRETM, ETA, F,
     $                   FC, SCLFAC, SCLINV, SMALL1, SMALL2, SMINV1,
     $                   SMINV2, TEMP, TEMP1, TEMP2, TEMP3, TEMP4,
     $                   LBD, UBD
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, INT, LOG, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
      INFO = 0
*
      IF( ORGATI ) THEN
         LBD = D(2)
         UBD = D(3)
      ELSE
         LBD = D(1)
         UBD = D(2)
      END IF
      IF( FINIT .LT. ZERO )THEN
         LBD = ZERO
      ELSE
         UBD = ZERO
      END IF
*
      NITER = 1
      TAU = ZERO
      IF( KNITER.EQ.2 ) THEN
         IF( ORGATI ) THEN
            TEMP = ( D( 3 )-D( 2 ) ) / TWO
            C = RHO + Z( 1 ) / ( ( D( 1 )-D( 2 ) )-TEMP )
            A = C*( D( 2 )+D( 3 ) ) + Z( 2 ) + Z( 3 )
            B = C*D( 2 )*D( 3 ) + Z( 2 )*D( 3 ) + Z( 3 )*D( 2 )
         ELSE
            TEMP = ( D( 1 )-D( 2 ) ) / TWO
            C = RHO + Z( 3 ) / ( ( D( 3 )-D( 2 ) )-TEMP )
            A = C*( D( 1 )+D( 2 ) ) + Z( 1 ) + Z( 2 )
            B = C*D( 1 )*D( 2 ) + Z( 1 )*D( 2 ) + Z( 2 )*D( 1 )
         END IF
         TEMP = MAX( ABS( A ), ABS( B ), ABS( C ) )
         A = A / TEMP
         B = B / TEMP
         C = C / TEMP
         IF( C.EQ.ZERO ) THEN
            TAU = B / A
         ELSE IF( A.LE.ZERO ) THEN
            TAU = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
         ELSE
            TAU = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
         END IF
         IF( TAU .LT. LBD .OR. TAU .GT. UBD )
     $      TAU = ( LBD+UBD )/TWO
         IF( D(1).EQ.TAU .OR. D(2).EQ.TAU .OR. D(3).EQ.TAU ) THEN
            TAU = ZERO
         ELSE
            TEMP = FINIT + TAU*Z(1)/( D(1)*( D( 1 )-TAU ) ) +
     $                     TAU*Z(2)/( D(2)*( D( 2 )-TAU ) ) +
     $                     TAU*Z(3)/( D(3)*( D( 3 )-TAU ) )
            IF( TEMP .LE. ZERO )THEN
               LBD = TAU
            ELSE
               UBD = TAU
            END IF
            IF( ABS( FINIT ).LE.ABS( TEMP ) )
     $         TAU = ZERO
         END IF
      END IF
*
*     get machine parameters for possible scaling to avoid overflow
*
*     modified by Sven: parameters SMALL1, SMINV1, SMALL2,
*     SMINV2, EPS are not SAVEd anymore between one call to the
*     others but recomputed at each call
*
      EPS = DLAMCH( 'Epsilon' )
      BASE = DLAMCH( 'Base' )
      SMALL1 = BASE**( INT( LOG( DLAMCH( 'SafMin' ) ) / LOG( BASE ) /
     $         THREE ) )
      SMINV1 = ONE / SMALL1
      SMALL2 = SMALL1*SMALL1
      SMINV2 = SMINV1*SMINV1
*
*     Determine if scaling of inputs necessary to avoid overflow
*     when computing 1/TEMP**3
*
      IF( ORGATI ) THEN
         TEMP = MIN( ABS( D( 2 )-TAU ), ABS( D( 3 )-TAU ) )
      ELSE
         TEMP = MIN( ABS( D( 1 )-TAU ), ABS( D( 2 )-TAU ) )
      END IF
      SCALE = .FALSE.
      IF( TEMP.LE.SMALL1 ) THEN
         SCALE = .TRUE.
         IF( TEMP.LE.SMALL2 ) THEN
*
*        Scale up by power of radix nearest 1/SAFMIN**(2/3)
*
            SCLFAC = SMINV2
            SCLINV = SMALL2
         ELSE
*
*        Scale up by power of radix nearest 1/SAFMIN**(1/3)
*
            SCLFAC = SMINV1
            SCLINV = SMALL1
         END IF
*
*        Scaling up safe because D, Z, TAU scaled elsewhere to be O(1)
*
         DO 10 I = 1, 3
            DSCALE( I ) = D( I )*SCLFAC
            ZSCALE( I ) = Z( I )*SCLFAC
   10    CONTINUE
         TAU = TAU*SCLFAC
         LBD = LBD*SCLFAC
         UBD = UBD*SCLFAC
      ELSE
*
*        Copy D and Z to DSCALE and ZSCALE
*
         DO 20 I = 1, 3
            DSCALE( I ) = D( I )
            ZSCALE( I ) = Z( I )
   20    CONTINUE
      END IF
*
      FC = ZERO
      DF = ZERO
      DDF = ZERO
      DO 30 I = 1, 3
         TEMP = ONE / ( DSCALE( I )-TAU )
         TEMP1 = ZSCALE( I )*TEMP
         TEMP2 = TEMP1*TEMP
         TEMP3 = TEMP2*TEMP
         FC = FC + TEMP1 / DSCALE( I )
         DF = DF + TEMP2
         DDF = DDF + TEMP3
   30 CONTINUE
      F = FINIT + TAU*FC
*
      IF( ABS( F ).LE.ZERO )
     $   GO TO 60
      IF( F .LE. ZERO )THEN
         LBD = TAU
      ELSE
         UBD = TAU
      END IF
*
*        Iteration begins -- Use Gragg-Thornton-Warner cubic convergent
*                            scheme
*
*     It is not hard to see that
*
*           1) Iterations will go up monotonically
*              if FINIT < 0;
*
*           2) Iterations will go down monotonically
*              if FINIT > 0.
*
      ITER = NITER + 1
*
      DO 50 NITER = ITER, MAXIT
*
         IF( ORGATI ) THEN
            TEMP1 = DSCALE( 2 ) - TAU
            TEMP2 = DSCALE( 3 ) - TAU
         ELSE
            TEMP1 = DSCALE( 1 ) - TAU
            TEMP2 = DSCALE( 2 ) - TAU
         END IF
         A = ( TEMP1+TEMP2 )*F - TEMP1*TEMP2*DF
         B = TEMP1*TEMP2*F
         C = F - ( TEMP1+TEMP2 )*DF + TEMP1*TEMP2*DDF
         TEMP = MAX( ABS( A ), ABS( B ), ABS( C ) )
         A = A / TEMP
         B = B / TEMP
         C = C / TEMP
         IF( C.EQ.ZERO ) THEN
            ETA = B / A
         ELSE IF( A.LE.ZERO ) THEN
            ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
         ELSE
            ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
         END IF
         IF( F*ETA.GE.ZERO ) THEN
            ETA = -F / DF
         END IF
*
         TAU = TAU + ETA
         IF( TAU .LT. LBD .OR. TAU .GT. UBD )
     $      TAU = ( LBD + UBD )/TWO
*
         FC = ZERO
         ERRETM = ZERO
         DF = ZERO
         DDF = ZERO
         DO 40 I = 1, 3
            IF ( ( DSCALE( I )-TAU ).NE.ZERO ) THEN
               TEMP = ONE / ( DSCALE( I )-TAU )
               TEMP1 = ZSCALE( I )*TEMP
               TEMP2 = TEMP1*TEMP
               TEMP3 = TEMP2*TEMP
               TEMP4 = TEMP1 / DSCALE( I )
               FC = FC + TEMP4
               ERRETM = ERRETM + ABS( TEMP4 )
               DF = DF + TEMP2
               DDF = DDF + TEMP3
            ELSE
               GO TO 60
            END IF
   40    CONTINUE
         F = FINIT + TAU*FC
         ERRETM = EIGHT*( ABS( FINIT )+ABS( TAU )*ERRETM ) +
     $            ABS( TAU )*DF
         IF( ( ABS( F ).LE.FOUR*EPS*ERRETM ) .OR.
     $      ( (UBD-LBD).LE.FOUR*EPS*ABS(TAU) )  )
     $      GO TO 60
         IF( F .LE. ZERO )THEN
            LBD = TAU
         ELSE
            UBD = TAU
         END IF
   50 CONTINUE
      INFO = 1
   60 CONTINUE
*
*     Undo scaling
*
      IF( SCALE )
     $   TAU = TAU*SCLINV
      RETURN
*
*     End of DLAED6
*
      END
*> \brief \b DLAED7 used by DSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED7 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed7.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed7.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed7.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAED7( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q,
*                          LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR,
*                          PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK,
*                          INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N,
*      $                   QSIZ, TLVLS
*       DOUBLE PRECISION   RHO
*       ..
*       .. Array Arguments ..
*       INTEGER            GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
*      $                   IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
*       DOUBLE PRECISION   D( * ), GIVNUM( 2, * ), Q( LDQ, * ),
*      $                   QSTORE( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAED7 computes the updated eigensystem of a diagonal
*> matrix after modification by a rank-one symmetric matrix. This
*> routine is used only for the eigenproblem which requires all
*> eigenvalues and optionally eigenvectors of a dense symmetric matrix
*> that has been reduced to tridiagonal form.  DLAED1 handles
*> the case in which all eigenvalues and eigenvectors of a symmetric
*> tridiagonal matrix are desired.
*>
*>   T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
*>
*>    where Z = Q**Tu, u is a vector of length N with ones in the
*>    CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
*>
*>    The eigenvectors of the original matrix are stored in Q, and the
*>    eigenvalues are in D.  The algorithm consists of three stages:
*>
*>       The first stage consists of deflating the size of the problem
*>       when there are multiple eigenvalues or if there is a zero in
*>       the Z vector.  For each such occurrence the dimension of the
*>       secular equation problem is reduced by one.  This stage is
*>       performed by the routine DLAED8.
*>
*>       The second stage consists of calculating the updated
*>       eigenvalues. This is done by finding the roots of the secular
*>       equation via the routine DLAED4 (as called by DLAED9).
*>       This routine also calculates the eigenvectors of the current
*>       problem.
*>
*>       The final stage consists of computing the updated eigenvectors
*>       directly using the updated eigenvalues.  The eigenvectors for
*>       the current problem are multiplied with the eigenvectors from
*>       the overall problem.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*>          ICOMPQ is INTEGER
*>          = 0:  Compute eigenvalues only.
*>          = 1:  Compute eigenvectors of original dense symmetric matrix
*>                also.  On entry, Q contains the orthogonal matrix used
*>                to reduce the original matrix to tridiagonal form.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>         The dimension of the symmetric tridiagonal matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in] QSIZ
*> \verbatim
*>          QSIZ is INTEGER
*>         The dimension of the orthogonal matrix used to reduce
*>         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.
*> \endverbatim
*>
*> \param[in] TLVLS
*> \verbatim
*>          TLVLS is INTEGER
*>         The total number of merging levels in the overall divide and
*>         conquer tree.
*> \endverbatim
*>
*> \param[in] CURLVL
*> \verbatim
*>          CURLVL is INTEGER
*>         The current level in the overall merge routine,
*>         0 <= CURLVL <= TLVLS.
*> \endverbatim
*>
*> \param[in] CURPBM
*> \verbatim
*>          CURPBM is INTEGER
*>         The current problem in the current level in the overall
*>         merge routine (counting from upper left to lower right).
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>         On entry, the eigenvalues of the rank-1-perturbed matrix.
*>         On exit, the eigenvalues of the repaired matrix.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ, N)
*>         On entry, the eigenvectors of the rank-1-perturbed matrix.
*>         On exit, the eigenvectors of the repaired tridiagonal matrix.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>         The leading dimension of the array Q.  LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[out] INDXQ
*> \verbatim
*>          INDXQ is INTEGER array, dimension (N)
*>         The permutation which will reintegrate the subproblem just
*>         solved back into sorted order, i.e., D( INDXQ( I = 1, N ) )
*>         will be in ascending order.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*>          RHO is DOUBLE PRECISION
*>         The subdiagonal element used to create the rank-1
*>         modification.
*> \endverbatim
*>
*> \param[in] CUTPNT
*> \verbatim
*>          CUTPNT is INTEGER
*>         Contains the location of the last eigenvalue in the leading
*>         sub-matrix.  min(1,N) <= CUTPNT <= N.
*> \endverbatim
*>
*> \param[in,out] QSTORE
*> \verbatim
*>          QSTORE is DOUBLE PRECISION array, dimension (N**2+1)
*>         Stores eigenvectors of submatrices encountered during
*>         divide and conquer, packed together. QPTR points to
*>         beginning of the submatrices.
*> \endverbatim
*>
*> \param[in,out] QPTR
*> \verbatim
*>          QPTR is INTEGER array, dimension (N+2)
*>         List of indices pointing to beginning of submatrices stored
*>         in QSTORE. The submatrices are numbered starting at the
*>         bottom left of the divide and conquer tree, from left to
*>         right and bottom to top.
*> \endverbatim
*>
*> \param[in] PRMPTR
*> \verbatim
*>          PRMPTR is INTEGER array, dimension (N lg N)
*>         Contains a list of pointers which indicate where in PERM a
*>         level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
*>         indicates the size of the permutation and also the size of
*>         the full, non-deflated problem.
*> \endverbatim
*>
*> \param[in] PERM
*> \verbatim
*>          PERM is INTEGER array, dimension (N lg N)
*>         Contains the permutations (from deflation and sorting) to be
*>         applied to each eigenblock.
*> \endverbatim
*>
*> \param[in] GIVPTR
*> \verbatim
*>          GIVPTR is INTEGER array, dimension (N lg N)
*>         Contains a list of pointers which indicate where in GIVCOL a
*>         level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)
*>         indicates the number of Givens rotations.
*> \endverbatim
*>
*> \param[in] GIVCOL
*> \verbatim
*>          GIVCOL is INTEGER array, dimension (2, N lg N)
*>         Each pair of numbers indicates a pair of columns to take place
*>         in a Givens rotation.
*> \endverbatim
*>
*> \param[in] GIVNUM
*> \verbatim
*>          GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N)
*>         Each number indicates the S value to be used in the
*>         corresponding Givens rotation.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N+2*QSIZ*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = 1, an eigenvalue did not converge
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed7
*
*> \par Contributors:
*  ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
*  =====================================================================
      SUBROUTINE DLAED7( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D,
     $                   Q,
     $                   LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR,
     $                   PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N,
     $                   QSIZ, TLVLS
      DOUBLE PRECISION   RHO
*     ..
*     .. Array Arguments ..
      INTEGER            GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
     $                   IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
      DOUBLE PRECISION   D( * ), GIVNUM( 2, * ), Q( LDQ, * ),
     $                   QSTORE( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D0, ZERO = 0.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            COLTYP, CURR, I, IDLMDA, INDX, INDXC, INDXP,
     $                   IQ2, IS, IW, IZ, K, LDQ2, N1, N2, PTR
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMM, DLAED8, DLAED9, DLAEDA, DLAMRG,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( ICOMPQ.EQ.1 .AND. QSIZ.LT.N ) THEN
         INFO = -3
      ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
         INFO = -9
      ELSE IF( MIN( 1, N ).GT.CUTPNT .OR. N.LT.CUTPNT ) THEN
         INFO = -12
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLAED7', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     The following values are for bookkeeping purposes only.  They are
*     integer pointers which indicate the portion of the workspace
*     used by a particular array in DLAED8 and DLAED9.
*
      IF( ICOMPQ.EQ.1 ) THEN
         LDQ2 = QSIZ
      ELSE
         LDQ2 = N
      END IF
*
      IZ = 1
      IDLMDA = IZ + N
      IW = IDLMDA + N
      IQ2 = IW + N
      IS = IQ2 + N*LDQ2
*
      INDX = 1
      INDXC = INDX + N
      COLTYP = INDXC + N
      INDXP = COLTYP + N
*
*     Form the z-vector which consists of the last row of Q_1 and the
*     first row of Q_2.
*
      PTR = 1 + 2**TLVLS
      DO 10 I = 1, CURLVL - 1
         PTR = PTR + 2**( TLVLS-I )
   10 CONTINUE
      CURR = PTR + CURPBM
      CALL DLAEDA( N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR,
     $             GIVCOL, GIVNUM, QSTORE, QPTR, WORK( IZ ),
     $             WORK( IZ+N ), INFO )
*
*     When solving the final problem, we no longer need the stored data,
*     so we will overwrite the data from this level onto the previously
*     used storage space.
*
      IF( CURLVL.EQ.TLVLS ) THEN
         QPTR( CURR ) = 1
         PRMPTR( CURR ) = 1
         GIVPTR( CURR ) = 1
      END IF
*
*     Sort and Deflate eigenvalues.
*
      CALL DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, CUTPNT,
     $             WORK( IZ ), WORK( IDLMDA ), WORK( IQ2 ), LDQ2,
     $             WORK( IW ), PERM( PRMPTR( CURR ) ), GIVPTR( CURR+1 ),
     $             GIVCOL( 1, GIVPTR( CURR ) ),
     $             GIVNUM( 1, GIVPTR( CURR ) ), IWORK( INDXP ),
     $             IWORK( INDX ), INFO )
      PRMPTR( CURR+1 ) = PRMPTR( CURR ) + N
      GIVPTR( CURR+1 ) = GIVPTR( CURR+1 ) + GIVPTR( CURR )
*
*     Solve Secular Equation.
*
      IF( K.NE.0 ) THEN
         CALL DLAED9( K, 1, K, N, D, WORK( IS ), K, RHO,
     $                WORK( IDLMDA ),
     $                WORK( IW ), QSTORE( QPTR( CURR ) ), K, INFO )
         IF( INFO.NE.0 )
     $      GO TO 30
         IF( ICOMPQ.EQ.1 ) THEN
            CALL DGEMM( 'N', 'N', QSIZ, K, K, ONE, WORK( IQ2 ), LDQ2,
     $                  QSTORE( QPTR( CURR ) ), K, ZERO, Q, LDQ )
         END IF
         QPTR( CURR+1 ) = QPTR( CURR ) + K**2
*
*     Prepare the INDXQ sorting permutation.
*
         N1 = K
         N2 = N - K
         CALL DLAMRG( N1, N2, D, 1, -1, INDXQ )
      ELSE
         QPTR( CURR+1 ) = QPTR( CURR )
         DO 20 I = 1, N
            INDXQ( I ) = I
   20    CONTINUE
      END IF
*
   30 CONTINUE
      RETURN
*
*     End of DLAED7
*
      END
*> \brief \b DLAED8 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED8 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed8.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed8.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed8.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO,
*                          CUTPNT, Z, DLAMBDA, Q2, LDQ2, W, PERM, GIVPTR,
*                          GIVCOL, GIVNUM, INDXP, INDX, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N,
*      $                   QSIZ
*       DOUBLE PRECISION   RHO
*       ..
*       .. Array Arguments ..
*       INTEGER            GIVCOL( 2, * ), INDX( * ), INDXP( * ),
*      $                   INDXQ( * ), PERM( * )
*       DOUBLE PRECISION   D( * ), DLAMBDA( * ), GIVNUM( 2, * ),
*      $                   Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAED8 merges the two sets of eigenvalues together into a single
*> sorted set.  Then it tries to deflate the size of the problem.
*> There are two ways in which deflation can occur:  when two or more
*> eigenvalues are close together or if there is a tiny element in the
*> Z vector.  For each such occurrence the order of the related secular
*> equation problem is reduced by one.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*>          ICOMPQ is INTEGER
*>          = 0:  Compute eigenvalues only.
*>          = 1:  Compute eigenvectors of original dense symmetric matrix
*>                also.  On entry, Q contains the orthogonal matrix used
*>                to reduce the original matrix to tridiagonal form.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*>          K is INTEGER
*>         The number of non-deflated eigenvalues, and the order of the
*>         related secular equation.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>         The dimension of the symmetric tridiagonal matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in] QSIZ
*> \verbatim
*>          QSIZ is INTEGER
*>         The dimension of the orthogonal matrix used to reduce
*>         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>         On entry, the eigenvalues of the two submatrices to be
*>         combined.  On exit, the trailing (N-K) updated eigenvalues
*>         (those which were deflated) sorted into increasing order.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ,N)
*>         If ICOMPQ = 0, Q is not referenced.  Otherwise,
*>         on entry, Q contains the eigenvectors of the partially solved
*>         system which has been previously updated in matrix
*>         multiplies with other partially solved eigensystems.
*>         On exit, Q contains the trailing (N-K) updated eigenvectors
*>         (those which were deflated) in its last N-K columns.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>         The leading dimension of the array Q.  LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in] INDXQ
*> \verbatim
*>          INDXQ is INTEGER array, dimension (N)
*>         The permutation which separately sorts the two sub-problems
*>         in D into ascending order.  Note that elements in the second
*>         half of this permutation must first have CUTPNT added to
*>         their values in order to be accurate.
*> \endverbatim
*>
*> \param[in,out] RHO
*> \verbatim
*>          RHO is DOUBLE PRECISION
*>         On entry, the off-diagonal element associated with the rank-1
*>         cut which originally split the two submatrices which are now
*>         being recombined.
*>         On exit, RHO has been modified to the value required by
*>         DLAED3.
*> \endverbatim
*>
*> \param[in] CUTPNT
*> \verbatim
*>          CUTPNT is INTEGER
*>         The location of the last eigenvalue in the leading
*>         sub-matrix.  min(1,N) <= CUTPNT <= N.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (N)
*>         On entry, Z contains the updating vector (the last row of
*>         the first sub-eigenvector matrix and the first row of the
*>         second sub-eigenvector matrix).
*>         On exit, the contents of Z are destroyed by the updating
*>         process.
*> \endverbatim
*>
*> \param[out] DLAMBDA
*> \verbatim
*>          DLAMBDA is DOUBLE PRECISION array, dimension (N)
*>         A copy of the first K eigenvalues which will be used by
*>         DLAED3 to form the secular equation.
*> \endverbatim
*>
*> \param[out] Q2
*> \verbatim
*>          Q2 is DOUBLE PRECISION array, dimension (LDQ2,N)
*>         If ICOMPQ = 0, Q2 is not referenced.  Otherwise,
*>         a copy of the first K eigenvectors which will be used by
*>         DLAED7 in a matrix multiply (DGEMM) to update the new
*>         eigenvectors.
*> \endverbatim
*>
*> \param[in] LDQ2
*> \verbatim
*>          LDQ2 is INTEGER
*>         The leading dimension of the array Q2.  LDQ2 >= max(1,N).
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>         The first k values of the final deflation-altered z-vector and
*>         will be passed to DLAED3.
*> \endverbatim
*>
*> \param[out] PERM
*> \verbatim
*>          PERM is INTEGER array, dimension (N)
*>         The permutations (from deflation and sorting) to be applied
*>         to each eigenblock.
*> \endverbatim
*>
*> \param[out] GIVPTR
*> \verbatim
*>          GIVPTR is INTEGER
*>         The number of Givens rotations which took place in this
*>         subproblem.
*> \endverbatim
*>
*> \param[out] GIVCOL
*> \verbatim
*>          GIVCOL is INTEGER array, dimension (2, N)
*>         Each pair of numbers indicates a pair of columns to take place
*>         in a Givens rotation.
*> \endverbatim
*>
*> \param[out] GIVNUM
*> \verbatim
*>          GIVNUM is DOUBLE PRECISION array, dimension (2, N)
*>         Each number indicates the S value to be used in the
*>         corresponding Givens rotation.
*> \endverbatim
*>
*> \param[out] INDXP
*> \verbatim
*>          INDXP is INTEGER array, dimension (N)
*>         The permutation used to place deflated values of D at the end
*>         of the array.  INDXP(1:K) points to the nondeflated D-values
*>         and INDXP(K+1:N) points to the deflated eigenvalues.
*> \endverbatim
*>
*> \param[out] INDX
*> \verbatim
*>          INDX is INTEGER array, dimension (N)
*>         The permutation used to sort the contents of D into ascending
*>         order.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed8
*
*> \par Contributors:
*  ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
*  =====================================================================
      SUBROUTINE DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO,
     $                   CUTPNT, Z, DLAMBDA, Q2, LDQ2, W, PERM, GIVPTR,
     $                   GIVCOL, GIVNUM, INDXP, INDX, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N,
     $                   QSIZ
      DOUBLE PRECISION   RHO
*     ..
*     .. Array Arguments ..
      INTEGER            GIVCOL( 2, * ), INDX( * ), INDXP( * ),
     $                   INDXQ( * ), PERM( * )
      DOUBLE PRECISION   D( * ), DLAMBDA( * ), GIVNUM( 2, * ),
     $                   Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   MONE, ZERO, ONE, TWO, EIGHT
      PARAMETER          ( MONE = -1.0D0, ZERO = 0.0D0, ONE = 1.0D0,
     $                   TWO = 2.0D0, EIGHT = 8.0D0 )
*     ..
*     .. Local Scalars ..
*
      INTEGER            I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2
      DOUBLE PRECISION   C, EPS, S, T, TAU, TOL
*     ..
*     .. External Functions ..
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH, DLAPY2
      EXTERNAL           IDAMAX, DLAMCH, DLAPY2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLACPY, DLAMRG, DROT, DSCAL,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( ICOMPQ.EQ.1 .AND. QSIZ.LT.N ) THEN
         INFO = -4
      ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( CUTPNT.LT.MIN( 1, N ) .OR. CUTPNT.GT.N ) THEN
         INFO = -10
      ELSE IF( LDQ2.LT.MAX( 1, N ) ) THEN
         INFO = -14
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLAED8', -INFO )
         RETURN
      END IF
*
*     Need to initialize GIVPTR to O here in case of quick exit
*     to prevent an unspecified code behavior (usually sigfault)
*     when IWORK array on entry to *stedc is not zeroed
*     (or at least some IWORK entries which used in *laed7 for GIVPTR).
*
      GIVPTR = 0
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      N1 = CUTPNT
      N2 = N - N1
      N1P1 = N1 + 1
*
      IF( RHO.LT.ZERO ) THEN
         CALL DSCAL( N2, MONE, Z( N1P1 ), 1 )
      END IF
*
*     Normalize z so that norm(z) = 1
*
      T = ONE / SQRT( TWO )
      DO 10 J = 1, N
         INDX( J ) = J
   10 CONTINUE
      CALL DSCAL( N, T, Z, 1 )
      RHO = ABS( TWO*RHO )
*
*     Sort the eigenvalues into increasing order
*
      DO 20 I = CUTPNT + 1, N
         INDXQ( I ) = INDXQ( I ) + CUTPNT
   20 CONTINUE
      DO 30 I = 1, N
         DLAMBDA( I ) = D( INDXQ( I ) )
         W( I ) = Z( INDXQ( I ) )
   30 CONTINUE
      I = 1
      J = CUTPNT + 1
      CALL DLAMRG( N1, N2, DLAMBDA, 1, 1, INDX )
      DO 40 I = 1, N
         D( I ) = DLAMBDA( INDX( I ) )
         Z( I ) = W( INDX( I ) )
   40 CONTINUE
*
*     Calculate the allowable deflation tolerance
*
      IMAX = IDAMAX( N, Z, 1 )
      JMAX = IDAMAX( N, D, 1 )
      EPS = DLAMCH( 'Epsilon' )
      TOL = EIGHT*EPS*ABS( D( JMAX ) )
*
*     If the rank-1 modifier is small enough, no more needs to be done
*     except to reorganize Q so that its columns correspond with the
*     elements in D.
*
      IF( RHO*ABS( Z( IMAX ) ).LE.TOL ) THEN
         K = 0
         IF( ICOMPQ.EQ.0 ) THEN
            DO 50 J = 1, N
               PERM( J ) = INDXQ( INDX( J ) )
   50       CONTINUE
         ELSE
            DO 60 J = 1, N
               PERM( J ) = INDXQ( INDX( J ) )
               CALL DCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ),
     $                     1 )
   60       CONTINUE
            CALL DLACPY( 'A', QSIZ, N, Q2( 1, 1 ), LDQ2, Q( 1, 1 ),
     $                   LDQ )
         END IF
         RETURN
      END IF
*
*     If there are multiple eigenvalues then the problem deflates.  Here
*     the number of equal eigenvalues are found.  As each equal
*     eigenvalue is found, an elementary reflector is computed to rotate
*     the corresponding eigensubspace so that the corresponding
*     components of Z are zero in this new basis.
*
      K = 0
      K2 = N + 1
      DO 70 J = 1, N
         IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN
*
*           Deflate due to small z component.
*
            K2 = K2 - 1
            INDXP( K2 ) = J
            IF( J.EQ.N )
     $         GO TO 110
         ELSE
            JLAM = J
            GO TO 80
         END IF
   70 CONTINUE
   80 CONTINUE
      J = J + 1
      IF( J.GT.N )
     $   GO TO 100
      IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN
*
*        Deflate due to small z component.
*
         K2 = K2 - 1
         INDXP( K2 ) = J
      ELSE
*
*        Check if eigenvalues are close enough to allow deflation.
*
         S = Z( JLAM )
         C = Z( J )
*
*        Find sqrt(a**2+b**2) without overflow or
*        destructive underflow.
*
         TAU = DLAPY2( C, S )
         T = D( J ) - D( JLAM )
         C = C / TAU
         S = -S / TAU
         IF( ABS( T*C*S ).LE.TOL ) THEN
*
*           Deflation is possible.
*
            Z( J ) = TAU
            Z( JLAM ) = ZERO
*
*           Record the appropriate Givens rotation
*
            GIVPTR = GIVPTR + 1
            GIVCOL( 1, GIVPTR ) = INDXQ( INDX( JLAM ) )
            GIVCOL( 2, GIVPTR ) = INDXQ( INDX( J ) )
            GIVNUM( 1, GIVPTR ) = C
            GIVNUM( 2, GIVPTR ) = S
            IF( ICOMPQ.EQ.1 ) THEN
               CALL DROT( QSIZ, Q( 1, INDXQ( INDX( JLAM ) ) ), 1,
     $                    Q( 1, INDXQ( INDX( J ) ) ), 1, C, S )
            END IF
            T = D( JLAM )*C*C + D( J )*S*S
            D( J ) = D( JLAM )*S*S + D( J )*C*C
            D( JLAM ) = T
            K2 = K2 - 1
            I = 1
   90       CONTINUE
            IF( K2+I.LE.N ) THEN
               IF( D( JLAM ).LT.D( INDXP( K2+I ) ) ) THEN
                  INDXP( K2+I-1 ) = INDXP( K2+I )
                  INDXP( K2+I ) = JLAM
                  I = I + 1
                  GO TO 90
               ELSE
                  INDXP( K2+I-1 ) = JLAM
               END IF
            ELSE
               INDXP( K2+I-1 ) = JLAM
            END IF
            JLAM = J
         ELSE
            K = K + 1
            W( K ) = Z( JLAM )
            DLAMBDA( K ) = D( JLAM )
            INDXP( K ) = JLAM
            JLAM = J
         END IF
      END IF
      GO TO 80
  100 CONTINUE
*
*     Record the last eigenvalue.
*
      K = K + 1
      W( K ) = Z( JLAM )
      DLAMBDA( K ) = D( JLAM )
      INDXP( K ) = JLAM
*
  110 CONTINUE
*
*     Sort the eigenvalues and corresponding eigenvectors into DLAMBDA
*     and Q2 respectively.  The eigenvalues/vectors which were not
*     deflated go into the first K slots of DLAMBDA and Q2 respectively,
*     while those which were deflated go into the last N - K slots.
*
      IF( ICOMPQ.EQ.0 ) THEN
         DO 120 J = 1, N
            JP = INDXP( J )
            DLAMBDA( J ) = D( JP )
            PERM( J ) = INDXQ( INDX( JP ) )
  120    CONTINUE
      ELSE
         DO 130 J = 1, N
            JP = INDXP( J )
            DLAMBDA( J ) = D( JP )
            PERM( J ) = INDXQ( INDX( JP ) )
            CALL DCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 )
  130    CONTINUE
      END IF
*
*     The deflated eigenvalues and their corresponding vectors go back
*     into the last N - K slots of D and Q respectively.
*
      IF( K.LT.N ) THEN
         IF( ICOMPQ.EQ.0 ) THEN
            CALL DCOPY( N-K, DLAMBDA( K+1 ), 1, D( K+1 ), 1 )
         ELSE
            CALL DCOPY( N-K, DLAMBDA( K+1 ), 1, D( K+1 ), 1 )
            CALL DLACPY( 'A', QSIZ, N-K, Q2( 1, K+1 ), LDQ2,
     $                   Q( 1, K+1 ), LDQ )
         END IF
      END IF
*
      RETURN
*
*     End of DLAED8
*
      END
*> \brief \b DLAED9 used by DSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED9 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed9.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed9.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed9.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAED9( K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMBDA,
*                          W, S, LDS, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, K, KSTART, KSTOP, LDQ, LDS, N
*       DOUBLE PRECISION   RHO
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), DLAMBDA( * ), Q( LDQ, * ), S( LDS, * ),
*      $                   W( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAED9 finds the roots of the secular equation, as defined by the
*> values in D, Z, and RHO, between KSTART and KSTOP.  It makes the
*> appropriate calls to DLAED4 and then stores the new matrix of
*> eigenvectors for use in calculating the next level of Z vectors.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of terms in the rational function to be solved by
*>          DLAED4.  K >= 0.
*> \endverbatim
*>
*> \param[in] KSTART
*> \verbatim
*>          KSTART is INTEGER
*> \endverbatim
*>
*> \param[in] KSTOP
*> \verbatim
*>          KSTOP is INTEGER
*>          The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP
*>          are to be computed.  1 <= KSTART <= KSTOP <= K.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of rows and columns in the Q matrix.
*>          N >= K (delation may result in N > K).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          D(I) contains the updated eigenvalues
*>          for KSTART <= I <= KSTOP.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q.  LDQ >= max( 1, N ).
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*>          RHO is DOUBLE PRECISION
*>          The value of the parameter in the rank one update equation.
*>          RHO >= 0 required.
*> \endverbatim
*>
*> \param[in] DLAMBDA
*> \verbatim
*>          DLAMBDA is DOUBLE PRECISION array, dimension (K)
*>          The first K elements of this array contain the old roots
*>          of the deflated updating problem.  These are the poles
*>          of the secular equation.
*> \endverbatim
*>
*> \param[in] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (K)
*>          The first K elements of this array contain the components
*>          of the deflation-adjusted updating vector.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (LDS, K)
*>          Will contain the eigenvectors of the repaired matrix which
*>          will be stored for subsequent Z vector calculation and
*>          multiplied by the previously accumulated eigenvectors
*>          to update the system.
*> \endverbatim
*>
*> \param[in] LDS
*> \verbatim
*>          LDS is INTEGER
*>          The leading dimension of S.  LDS >= max( 1, K ).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = 1, an eigenvalue did not converge
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed9
*
*> \par Contributors:
*  ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
*  =====================================================================
      SUBROUTINE DLAED9( K, KSTART, KSTOP, N, D, Q, LDQ, RHO,
     $                   DLAMBDA,
     $                   W, S, LDS, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, KSTART, KSTOP, LDQ, LDS, N
      DOUBLE PRECISION   RHO
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), DLAMBDA( * ), Q( LDQ, * ), S( LDS, * ),
     $                   W( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            I, J
      DOUBLE PRECISION   TEMP
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DNRM2
      EXTERNAL           DNRM2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLAED4, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, SIGN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IF( K.LT.0 ) THEN
         INFO = -1
      ELSE IF( KSTART.LT.1 .OR. KSTART.GT.MAX( 1, K ) ) THEN
         INFO = -2
      ELSE IF( MAX( 1, KSTOP ).LT.KSTART .OR. KSTOP.GT.MAX( 1, K ) )
     $          THEN
         INFO = -3
      ELSE IF( N.LT.K ) THEN
         INFO = -4
      ELSE IF( LDQ.LT.MAX( 1, K ) ) THEN
         INFO = -7
      ELSE IF( LDS.LT.MAX( 1, K ) ) THEN
         INFO = -12
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLAED9', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( K.EQ.0 )
     $   RETURN
*
      DO 20 J = KSTART, KSTOP
         CALL DLAED4( K, J, DLAMBDA, W, Q( 1, J ), RHO, D( J ),
     $                INFO )
*
*        If the zero finder fails, the computation is terminated.
*
         IF( INFO.NE.0 )
     $      GO TO 120
   20 CONTINUE
*
      IF( K.EQ.1 .OR. K.EQ.2 ) THEN
         DO 40 I = 1, K
            DO 30 J = 1, K
               S( J, I ) = Q( J, I )
   30       CONTINUE
   40    CONTINUE
         GO TO 120
      END IF
*
*     Compute updated W.
*
      CALL DCOPY( K, W, 1, S, 1 )
*
*     Initialize W(I) = Q(I,I)
*
      CALL DCOPY( K, Q, LDQ+1, W, 1 )
      DO 70 J = 1, K
         DO 50 I = 1, J - 1
            W( I ) = W( I )*( Q( I, J )/( DLAMBDA( I )-DLAMBDA( J ) ) )
   50    CONTINUE
         DO 60 I = J + 1, K
            W( I ) = W( I )*( Q( I, J )/( DLAMBDA( I )-DLAMBDA( J ) ) )
   60    CONTINUE
   70 CONTINUE
      DO 80 I = 1, K
         W( I ) = SIGN( SQRT( -W( I ) ), S( I, 1 ) )
   80 CONTINUE
*
*     Compute eigenvectors of the modified rank-1 modification.
*
      DO 110 J = 1, K
         DO 90 I = 1, K
            Q( I, J ) = W( I ) / Q( I, J )
   90    CONTINUE
         TEMP = DNRM2( K, Q( 1, J ), 1 )
         DO 100 I = 1, K
            S( I, J ) = Q( I, J ) / TEMP
  100    CONTINUE
  110 CONTINUE
*
  120 CONTINUE
      RETURN
*
*     End of DLAED9
*
      END
*> \brief \b DLAEDA used by DSTEDC. Computes the Z vector determining the rank-one modification of the diagonal matrix. Used when the original matrix is dense.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAEDA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaeda.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaeda.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaeda.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAEDA( N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR,
*                          GIVCOL, GIVNUM, Q, QPTR, Z, ZTEMP, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            CURLVL, CURPBM, INFO, N, TLVLS
*       ..
*       .. Array Arguments ..
*       INTEGER            GIVCOL( 2, * ), GIVPTR( * ), PERM( * ),
*      $                   PRMPTR( * ), QPTR( * )
*       DOUBLE PRECISION   GIVNUM( 2, * ), Q( * ), Z( * ), ZTEMP( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAEDA computes the Z vector corresponding to the merge step in the
*> CURLVLth step of the merge process with TLVLS steps for the CURPBMth
*> problem.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>         The dimension of the symmetric tridiagonal matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in] TLVLS
*> \verbatim
*>          TLVLS is INTEGER
*>         The total number of merging levels in the overall divide and
*>         conquer tree.
*> \endverbatim
*>
*> \param[in] CURLVL
*> \verbatim
*>          CURLVL is INTEGER
*>         The current level in the overall merge routine,
*>         0 <= curlvl <= tlvls.
*> \endverbatim
*>
*> \param[in] CURPBM
*> \verbatim
*>          CURPBM is INTEGER
*>         The current problem in the current level in the overall
*>         merge routine (counting from upper left to lower right).
*> \endverbatim
*>
*> \param[in] PRMPTR
*> \verbatim
*>          PRMPTR is INTEGER array, dimension (N lg N)
*>         Contains a list of pointers which indicate where in PERM a
*>         level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
*>         indicates the size of the permutation and incidentally the
*>         size of the full, non-deflated problem.
*> \endverbatim
*>
*> \param[in] PERM
*> \verbatim
*>          PERM is INTEGER array, dimension (N lg N)
*>         Contains the permutations (from deflation and sorting) to be
*>         applied to each eigenblock.
*> \endverbatim
*>
*> \param[in] GIVPTR
*> \verbatim
*>          GIVPTR is INTEGER array, dimension (N lg N)
*>         Contains a list of pointers which indicate where in GIVCOL a
*>         level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)
*>         indicates the number of Givens rotations.
*> \endverbatim
*>
*> \param[in] GIVCOL
*> \verbatim
*>          GIVCOL is INTEGER array, dimension (2, N lg N)
*>         Each pair of numbers indicates a pair of columns to take place
*>         in a Givens rotation.
*> \endverbatim
*>
*> \param[in] GIVNUM
*> \verbatim
*>          GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N)
*>         Each number indicates the S value to be used in the
*>         corresponding Givens rotation.
*> \endverbatim
*>
*> \param[in] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (N**2)
*>         Contains the square eigenblocks from previous levels, the
*>         starting positions for blocks are given by QPTR.
*> \endverbatim
*>
*> \param[in] QPTR
*> \verbatim
*>          QPTR is INTEGER array, dimension (N+2)
*>         Contains a list of pointers which indicate where in Q an
*>         eigenblock is stored.  SQRT( QPTR(i+1) - QPTR(i) ) indicates
*>         the size of the block.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (N)
*>         On output this vector contains the updating vector (the last
*>         row of the first sub-eigenvector matrix and the first row of
*>         the second sub-eigenvector matrix).
*> \endverbatim
*>
*> \param[out] ZTEMP
*> \verbatim
*>          ZTEMP is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laeda
*
*> \par Contributors:
*  ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
*  =====================================================================
      SUBROUTINE DLAEDA( N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM,
     $                   GIVPTR,
     $                   GIVCOL, GIVNUM, Q, QPTR, Z, ZTEMP, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            CURLVL, CURPBM, INFO, N, TLVLS
*     ..
*     .. Array Arguments ..
      INTEGER            GIVCOL( 2, * ), GIVPTR( * ), PERM( * ),
     $                   PRMPTR( * ), QPTR( * )
      DOUBLE PRECISION   GIVNUM( 2, * ), Q( * ), Z( * ), ZTEMP( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, HALF, ONE
      PARAMETER          ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            BSIZ1, BSIZ2, CURR, I, K, MID, PSIZ1, PSIZ2,
     $                   PTR, ZPTR1
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGEMV, DROT, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE, INT, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IF( N.LT.0 ) THEN
         INFO = -1
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLAEDA', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Determine location of first number in second half.
*
      MID = N / 2 + 1
*
*     Gather last/first rows of appropriate eigenblocks into center of Z
*
      PTR = 1
*
*     Determine location of lowest level subproblem in the full storage
*     scheme
*
      CURR = PTR + CURPBM*2**CURLVL + 2**( CURLVL-1 ) - 1
*
*     Determine size of these matrices.  We add HALF to the value of
*     the SQRT in case the machine underestimates one of these square
*     roots.
*
      BSIZ1 = INT( HALF+SQRT( DBLE( QPTR( CURR+1 )-QPTR( CURR ) ) ) )
      BSIZ2 = INT( HALF+SQRT( DBLE( QPTR( CURR+2 )-QPTR( CURR+1 ) ) ) )
      DO 10 K = 1, MID - BSIZ1 - 1
         Z( K ) = ZERO
   10 CONTINUE
      CALL DCOPY( BSIZ1, Q( QPTR( CURR )+BSIZ1-1 ), BSIZ1,
     $            Z( MID-BSIZ1 ), 1 )
      CALL DCOPY( BSIZ2, Q( QPTR( CURR+1 ) ), BSIZ2, Z( MID ), 1 )
      DO 20 K = MID + BSIZ2, N
         Z( K ) = ZERO
   20 CONTINUE
*
*     Loop through remaining levels 1 -> CURLVL applying the Givens
*     rotations and permutation and then multiplying the center matrices
*     against the current Z.
*
      PTR = 2**TLVLS + 1
      DO 70 K = 1, CURLVL - 1
         CURR = PTR + CURPBM*2**( CURLVL-K ) + 2**( CURLVL-K-1 ) - 1
         PSIZ1 = PRMPTR( CURR+1 ) - PRMPTR( CURR )
         PSIZ2 = PRMPTR( CURR+2 ) - PRMPTR( CURR+1 )
         ZPTR1 = MID - PSIZ1
*
*       Apply Givens at CURR and CURR+1
*
         DO 30 I = GIVPTR( CURR ), GIVPTR( CURR+1 ) - 1
            CALL DROT( 1, Z( ZPTR1+GIVCOL( 1, I )-1 ), 1,
     $                 Z( ZPTR1+GIVCOL( 2, I )-1 ), 1, GIVNUM( 1, I ),
     $                 GIVNUM( 2, I ) )
   30    CONTINUE
         DO 40 I = GIVPTR( CURR+1 ), GIVPTR( CURR+2 ) - 1
            CALL DROT( 1, Z( MID-1+GIVCOL( 1, I ) ), 1,
     $                 Z( MID-1+GIVCOL( 2, I ) ), 1, GIVNUM( 1, I ),
     $                 GIVNUM( 2, I ) )
   40    CONTINUE
         PSIZ1 = PRMPTR( CURR+1 ) - PRMPTR( CURR )
         PSIZ2 = PRMPTR( CURR+2 ) - PRMPTR( CURR+1 )
         DO 50 I = 0, PSIZ1 - 1
            ZTEMP( I+1 ) = Z( ZPTR1+PERM( PRMPTR( CURR )+I )-1 )
   50    CONTINUE
         DO 60 I = 0, PSIZ2 - 1
            ZTEMP( PSIZ1+I+1 ) = Z( MID+PERM( PRMPTR( CURR+1 )+I )-1 )
   60    CONTINUE
*
*        Multiply Blocks at CURR and CURR+1
*
*        Determine size of these matrices.  We add HALF to the value of
*        the SQRT in case the machine underestimates one of these
*        square roots.
*
         BSIZ1 = INT( HALF+SQRT( DBLE( QPTR( CURR+1 )-QPTR( CURR ) ) ) )
         BSIZ2 = INT( HALF+SQRT( DBLE( QPTR( CURR+2 )-QPTR( CURR+
     $           1 ) ) ) )
         IF( BSIZ1.GT.0 ) THEN
            CALL DGEMV( 'T', BSIZ1, BSIZ1, ONE, Q( QPTR( CURR ) ),
     $                  BSIZ1, ZTEMP( 1 ), 1, ZERO, Z( ZPTR1 ), 1 )
         END IF
         CALL DCOPY( PSIZ1-BSIZ1, ZTEMP( BSIZ1+1 ), 1,
     $               Z( ZPTR1+BSIZ1 ),
     $               1 )
         IF( BSIZ2.GT.0 ) THEN
            CALL DGEMV( 'T', BSIZ2, BSIZ2, ONE, Q( QPTR( CURR+1 ) ),
     $                  BSIZ2, ZTEMP( PSIZ1+1 ), 1, ZERO, Z( MID ), 1 )
         END IF
         CALL DCOPY( PSIZ2-BSIZ2, ZTEMP( PSIZ1+BSIZ2+1 ), 1,
     $               Z( MID+BSIZ2 ), 1 )
*
         PTR = PTR + 2**( TLVLS-K )
   70 CONTINUE
*
      RETURN
*
*     End of DLAEDA
*
      END
*> \brief \b DLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAEIN + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaein.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaein.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaein.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAEIN( RIGHTV, NOINIT, N, H, LDH, WR, WI, VR, VI, B,
*                          LDB, WORK, EPS3, SMLNUM, BIGNUM, INFO )
*
*       .. Scalar Arguments ..
*       LOGICAL            NOINIT, RIGHTV
*       INTEGER            INFO, LDB, LDH, N
*       DOUBLE PRECISION   BIGNUM, EPS3, SMLNUM, WI, WR
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   B( LDB, * ), H( LDH, * ), VI( * ), VR( * ),
*      $                   WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAEIN uses inverse iteration to find a right or left eigenvector
*> corresponding to the eigenvalue (WR,WI) of a real upper Hessenberg
*> matrix H.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] RIGHTV
*> \verbatim
*>          RIGHTV is LOGICAL
*>          = .TRUE. : compute right eigenvector;
*>          = .FALSE.: compute left eigenvector.
*> \endverbatim
*>
*> \param[in] NOINIT
*> \verbatim
*>          NOINIT is LOGICAL
*>          = .TRUE. : no initial vector supplied in (VR,VI).
*>          = .FALSE.: initial vector supplied in (VR,VI).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix H.  N >= 0.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*>          H is DOUBLE PRECISION array, dimension (LDH,N)
*>          The upper Hessenberg matrix H.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*>          LDH is INTEGER
*>          The leading dimension of the array H.  LDH >= max(1,N).
*> \endverbatim
*>
*> \param[in] WR
*> \verbatim
*>          WR is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] WI
*> \verbatim
*>          WI is DOUBLE PRECISION
*>          The real and imaginary parts of the eigenvalue of H whose
*>          corresponding right or left eigenvector is to be computed.
*> \endverbatim
*>
*> \param[in,out] VR
*> \verbatim
*>          VR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[in,out] VI
*> \verbatim
*>          VI is DOUBLE PRECISION array, dimension (N)
*>          On entry, if NOINIT = .FALSE. and WI = 0.0, VR must contain
*>          a real starting vector for inverse iteration using the real
*>          eigenvalue WR; if NOINIT = .FALSE. and WI.ne.0.0, VR and VI
*>          must contain the real and imaginary parts of a complex
*>          starting vector for inverse iteration using the complex
*>          eigenvalue (WR,WI); otherwise VR and VI need not be set.
*>          On exit, if WI = 0.0 (real eigenvalue), VR contains the
*>          computed real eigenvector; if WI.ne.0.0 (complex eigenvalue),
*>          VR and VI contain the real and imaginary parts of the
*>          computed complex eigenvector. The eigenvector is normalized
*>          so that the component of largest magnitude has magnitude 1;
*>          here the magnitude of a complex number (x,y) is taken to be
*>          |x| + |y|.
*>          VI is not referenced if WI = 0.0.
*> \endverbatim
*>
*> \param[out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,N)
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= N+1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[in] EPS3
*> \verbatim
*>          EPS3 is DOUBLE PRECISION
*>          A small machine-dependent value which is used to perturb
*>          close eigenvalues, and to replace zero pivots.
*> \endverbatim
*>
*> \param[in] SMLNUM
*> \verbatim
*>          SMLNUM is DOUBLE PRECISION
*>          A machine-dependent value close to the underflow threshold.
*> \endverbatim
*>
*> \param[in] BIGNUM
*> \verbatim
*>          BIGNUM is DOUBLE PRECISION
*>          A machine-dependent value close to the overflow threshold.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          = 1:  inverse iteration did not converge; VR is set to the
*>                last iterate, and so is VI if WI.ne.0.0.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laein
*
*  =====================================================================
      SUBROUTINE DLAEIN( RIGHTV, NOINIT, N, H, LDH, WR, WI, VR, VI,
     $                   B,
     $                   LDB, WORK, EPS3, SMLNUM, BIGNUM, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      LOGICAL            NOINIT, RIGHTV
      INTEGER            INFO, LDB, LDH, N
      DOUBLE PRECISION   BIGNUM, EPS3, SMLNUM, WI, WR
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   B( LDB, * ), H( LDH, * ), VI( * ), VR( * ),
     $                   WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TENTH
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TENTH = 1.0D-1 )
*     ..
*     .. Local Scalars ..
      CHARACTER          NORMIN, TRANS
      INTEGER            I, I1, I2, I3, IERR, ITS, J
      DOUBLE PRECISION   ABSBII, ABSBJJ, EI, EJ, GROWTO, NORM, NRMSML,
     $                   REC, ROOTN, SCALE, TEMP, VCRIT, VMAX, VNORM, W,
     $                   W1, X, XI, XR, Y
*     ..
*     .. External Functions ..
      INTEGER            IDAMAX
      DOUBLE PRECISION   DASUM, DLAPY2, DNRM2
      EXTERNAL           IDAMAX, DASUM, DLAPY2, DNRM2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLADIV, DLATRS, DSCAL
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
      INFO = 0
*
*     GROWTO is the threshold used in the acceptance test for an
*     eigenvector.
*
      ROOTN = SQRT( DBLE( N ) )
      GROWTO = TENTH / ROOTN
      NRMSML = MAX( ONE, EPS3*ROOTN )*SMLNUM
*
*     Form B = H - (WR,WI)*I (except that the subdiagonal elements and
*     the imaginary parts of the diagonal elements are not stored).
*
      DO 20 J = 1, N
         DO 10 I = 1, J - 1
            B( I, J ) = H( I, J )
   10    CONTINUE
         B( J, J ) = H( J, J ) - WR
   20 CONTINUE
*
      IF( WI.EQ.ZERO ) THEN
*
*        Real eigenvalue.
*
         IF( NOINIT ) THEN
*
*           Set initial vector.
*
            DO 30 I = 1, N
               VR( I ) = EPS3
   30       CONTINUE
         ELSE
*
*           Scale supplied initial vector.
*
            VNORM = DNRM2( N, VR, 1 )
            CALL DSCAL( N, ( EPS3*ROOTN ) / MAX( VNORM, NRMSML ), VR,
     $                  1 )
         END IF
*
         IF( RIGHTV ) THEN
*
*           LU decomposition with partial pivoting of B, replacing zero
*           pivots by EPS3.
*
            DO 60 I = 1, N - 1
               EI = H( I+1, I )
               IF( ABS( B( I, I ) ).LT.ABS( EI ) ) THEN
*
*                 Interchange rows and eliminate.
*
                  X = B( I, I ) / EI
                  B( I, I ) = EI
                  DO 40 J = I + 1, N
                     TEMP = B( I+1, J )
                     B( I+1, J ) = B( I, J ) - X*TEMP
                     B( I, J ) = TEMP
   40             CONTINUE
               ELSE
*
*                 Eliminate without interchange.
*
                  IF( B( I, I ).EQ.ZERO )
     $               B( I, I ) = EPS3
                  X = EI / B( I, I )
                  IF( X.NE.ZERO ) THEN
                     DO 50 J = I + 1, N
                        B( I+1, J ) = B( I+1, J ) - X*B( I, J )
   50                CONTINUE
                  END IF
               END IF
   60       CONTINUE
            IF( B( N, N ).EQ.ZERO )
     $         B( N, N ) = EPS3
*
            TRANS = 'N'
*
         ELSE
*
*           UL decomposition with partial pivoting of B, replacing zero
*           pivots by EPS3.
*
            DO 90 J = N, 2, -1
               EJ = H( J, J-1 )
               IF( ABS( B( J, J ) ).LT.ABS( EJ ) ) THEN
*
*                 Interchange columns and eliminate.
*
                  X = B( J, J ) / EJ
                  B( J, J ) = EJ
                  DO 70 I = 1, J - 1
                     TEMP = B( I, J-1 )
                     B( I, J-1 ) = B( I, J ) - X*TEMP
                     B( I, J ) = TEMP
   70             CONTINUE
               ELSE
*
*                 Eliminate without interchange.
*
                  IF( B( J, J ).EQ.ZERO )
     $               B( J, J ) = EPS3
                  X = EJ / B( J, J )
                  IF( X.NE.ZERO ) THEN
                     DO 80 I = 1, J - 1
                        B( I, J-1 ) = B( I, J-1 ) - X*B( I, J )
   80                CONTINUE
                  END IF
               END IF
   90       CONTINUE
            IF( B( 1, 1 ).EQ.ZERO )
     $         B( 1, 1 ) = EPS3
*
            TRANS = 'T'
*
         END IF
*
         NORMIN = 'N'
         DO 110 ITS = 1, N
*
*           Solve U*x = scale*v for a right eigenvector
*             or U**T*x = scale*v for a left eigenvector,
*           overwriting x on v.
*
            CALL DLATRS( 'Upper', TRANS, 'Nonunit', NORMIN, N, B,
     $                   LDB,
     $                   VR, SCALE, WORK, IERR )
            NORMIN = 'Y'
*
*           Test for sufficient growth in the norm of v.
*
            VNORM = DASUM( N, VR, 1 )
            IF( VNORM.GE.GROWTO*SCALE )
     $         GO TO 120
*
*           Choose new orthogonal starting vector and try again.
*
            TEMP = EPS3 / ( ROOTN+ONE )
            VR( 1 ) = EPS3
            DO 100 I = 2, N
               VR( I ) = TEMP
  100       CONTINUE
            VR( N-ITS+1 ) = VR( N-ITS+1 ) - EPS3*ROOTN
  110    CONTINUE
*
*        Failure to find eigenvector in N iterations.
*
         INFO = 1
*
  120    CONTINUE
*
*        Normalize eigenvector.
*
         I = IDAMAX( N, VR, 1 )
         CALL DSCAL( N, ONE / ABS( VR( I ) ), VR, 1 )
      ELSE
*
*        Complex eigenvalue.
*
         IF( NOINIT ) THEN
*
*           Set initial vector.
*
            DO 130 I = 1, N
               VR( I ) = EPS3
               VI( I ) = ZERO
  130       CONTINUE
         ELSE
*
*           Scale supplied initial vector.
*
            NORM = DLAPY2( DNRM2( N, VR, 1 ),
     $                     DNRM2( N, VI, 1 ) )
            REC = ( EPS3*ROOTN ) / MAX( NORM, NRMSML )
            CALL DSCAL( N, REC, VR, 1 )
            CALL DSCAL( N, REC, VI, 1 )
         END IF
*
         IF( RIGHTV ) THEN
*
*           LU decomposition with partial pivoting of B, replacing zero
*           pivots by EPS3.
*
*           The imaginary part of the (i,j)-th element of U is stored in
*           B(j+1,i).
*
            B( 2, 1 ) = -WI
            DO 140 I = 2, N
               B( I+1, 1 ) = ZERO
  140       CONTINUE
*
            DO 170 I = 1, N - 1
               ABSBII = DLAPY2( B( I, I ), B( I+1, I ) )
               EI = H( I+1, I )
               IF( ABSBII.LT.ABS( EI ) ) THEN
*
*                 Interchange rows and eliminate.
*
                  XR = B( I, I ) / EI
                  XI = B( I+1, I ) / EI
                  B( I, I ) = EI
                  B( I+1, I ) = ZERO
                  DO 150 J = I + 1, N
                     TEMP = B( I+1, J )
                     B( I+1, J ) = B( I, J ) - XR*TEMP
                     B( J+1, I+1 ) = B( J+1, I ) - XI*TEMP
                     B( I, J ) = TEMP
                     B( J+1, I ) = ZERO
  150             CONTINUE
                  B( I+2, I ) = -WI
                  B( I+1, I+1 ) = B( I+1, I+1 ) - XI*WI
                  B( I+2, I+1 ) = B( I+2, I+1 ) + XR*WI
               ELSE
*
*                 Eliminate without interchanging rows.
*
                  IF( ABSBII.EQ.ZERO ) THEN
                     B( I, I ) = EPS3
                     B( I+1, I ) = ZERO
                     ABSBII = EPS3
                  END IF
                  EI = ( EI / ABSBII ) / ABSBII
                  XR = B( I, I )*EI
                  XI = -B( I+1, I )*EI
                  DO 160 J = I + 1, N
                     B( I+1, J ) = B( I+1, J ) - XR*B( I, J ) +
     $                             XI*B( J+1, I )
                     B( J+1, I+1 ) = -XR*B( J+1, I ) - XI*B( I, J )
  160             CONTINUE
                  B( I+2, I+1 ) = B( I+2, I+1 ) - WI
               END IF
*
*              Compute 1-norm of offdiagonal elements of i-th row.
*
               WORK( I ) = DASUM( N-I, B( I, I+1 ), LDB ) +
     $                     DASUM( N-I, B( I+2, I ), 1 )
  170       CONTINUE
            IF( B( N, N ).EQ.ZERO .AND. B( N+1, N ).EQ.ZERO )
     $         B( N, N ) = EPS3
            WORK( N ) = ZERO
*
            I1 = N
            I2 = 1
            I3 = -1
         ELSE
*
*           UL decomposition with partial pivoting of conjg(B),
*           replacing zero pivots by EPS3.
*
*           The imaginary part of the (i,j)-th element of U is stored in
*           B(j+1,i).
*
            B( N+1, N ) = WI
            DO 180 J = 1, N - 1
               B( N+1, J ) = ZERO
  180       CONTINUE
*
            DO 210 J = N, 2, -1
               EJ = H( J, J-1 )
               ABSBJJ = DLAPY2( B( J, J ), B( J+1, J ) )
               IF( ABSBJJ.LT.ABS( EJ ) ) THEN
*
*                 Interchange columns and eliminate
*
                  XR = B( J, J ) / EJ
                  XI = B( J+1, J ) / EJ
                  B( J, J ) = EJ
                  B( J+1, J ) = ZERO
                  DO 190 I = 1, J - 1
                     TEMP = B( I, J-1 )
                     B( I, J-1 ) = B( I, J ) - XR*TEMP
                     B( J, I ) = B( J+1, I ) - XI*TEMP
                     B( I, J ) = TEMP
                     B( J+1, I ) = ZERO
  190             CONTINUE
                  B( J+1, J-1 ) = WI
                  B( J-1, J-1 ) = B( J-1, J-1 ) + XI*WI
                  B( J, J-1 ) = B( J, J-1 ) - XR*WI
               ELSE
*
*                 Eliminate without interchange.
*
                  IF( ABSBJJ.EQ.ZERO ) THEN
                     B( J, J ) = EPS3
                     B( J+1, J ) = ZERO
                     ABSBJJ = EPS3
                  END IF
                  EJ = ( EJ / ABSBJJ ) / ABSBJJ
                  XR = B( J, J )*EJ
                  XI = -B( J+1, J )*EJ
                  DO 200 I = 1, J - 1
                     B( I, J-1 ) = B( I, J-1 ) - XR*B( I, J ) +
     $                             XI*B( J+1, I )
                     B( J, I ) = -XR*B( J+1, I ) - XI*B( I, J )
  200             CONTINUE
                  B( J, J-1 ) = B( J, J-1 ) + WI
               END IF
*
*              Compute 1-norm of offdiagonal elements of j-th column.
*
               WORK( J ) = DASUM( J-1, B( 1, J ), 1 ) +
     $                     DASUM( J-1, B( J+1, 1 ), LDB )
  210       CONTINUE
            IF( B( 1, 1 ).EQ.ZERO .AND. B( 2, 1 ).EQ.ZERO )
     $         B( 1, 1 ) = EPS3
            WORK( 1 ) = ZERO
*
            I1 = 1
            I2 = N
            I3 = 1
         END IF
*
         DO 270 ITS = 1, N
            SCALE = ONE
            VMAX = ONE
            VCRIT = BIGNUM
*
*           Solve U*(xr,xi) = scale*(vr,vi) for a right eigenvector,
*             or U**T*(xr,xi) = scale*(vr,vi) for a left eigenvector,
*           overwriting (xr,xi) on (vr,vi).
*
            DO 250 I = I1, I2, I3
*
               IF( WORK( I ).GT.VCRIT ) THEN
                  REC = ONE / VMAX
                  CALL DSCAL( N, REC, VR, 1 )
                  CALL DSCAL( N, REC, VI, 1 )
                  SCALE = SCALE*REC
                  VMAX = ONE
                  VCRIT = BIGNUM
               END IF
*
               XR = VR( I )
               XI = VI( I )
               IF( RIGHTV ) THEN
                  DO 220 J = I + 1, N
                     XR = XR - B( I, J )*VR( J ) + B( J+1, I )*VI( J )
                     XI = XI - B( I, J )*VI( J ) - B( J+1, I )*VR( J )
  220             CONTINUE
               ELSE
                  DO 230 J = 1, I - 1
                     XR = XR - B( J, I )*VR( J ) + B( I+1, J )*VI( J )
                     XI = XI - B( J, I )*VI( J ) - B( I+1, J )*VR( J )
  230             CONTINUE
               END IF
*
               W = ABS( B( I, I ) ) + ABS( B( I+1, I ) )
               IF( W.GT.SMLNUM ) THEN
                  IF( W.LT.ONE ) THEN
                     W1 = ABS( XR ) + ABS( XI )
                     IF( W1.GT.W*BIGNUM ) THEN
                        REC = ONE / W1
                        CALL DSCAL( N, REC, VR, 1 )
                        CALL DSCAL( N, REC, VI, 1 )
                        XR = VR( I )
                        XI = VI( I )
                        SCALE = SCALE*REC
                        VMAX = VMAX*REC
                     END IF
                  END IF
*
*                 Divide by diagonal element of B.
*
                  CALL DLADIV( XR, XI, B( I, I ), B( I+1, I ),
     $                         VR( I ),
     $                         VI( I ) )
                  VMAX = MAX( ABS( VR( I ) )+ABS( VI( I ) ), VMAX )
                  VCRIT = BIGNUM / VMAX
               ELSE
                  DO 240 J = 1, N
                     VR( J ) = ZERO
                     VI( J ) = ZERO
  240             CONTINUE
                  VR( I ) = ONE
                  VI( I ) = ONE
                  SCALE = ZERO
                  VMAX = ONE
                  VCRIT = BIGNUM
               END IF
  250       CONTINUE
*
*           Test for sufficient growth in the norm of (VR,VI).
*
            VNORM = DASUM( N, VR, 1 ) + DASUM( N, VI, 1 )
            IF( VNORM.GE.GROWTO*SCALE )
     $         GO TO 280
*
*           Choose a new orthogonal starting vector and try again.
*
            Y = EPS3 / ( ROOTN+ONE )
            VR( 1 ) = EPS3
            VI( 1 ) = ZERO
*
            DO 260 I = 2, N
               VR( I ) = Y
               VI( I ) = ZERO
  260       CONTINUE
            VR( N-ITS+1 ) = VR( N-ITS+1 ) - EPS3*ROOTN
  270    CONTINUE
*
*        Failure to find eigenvector in N iterations
*
         INFO = 1
*
  280    CONTINUE
*
*        Normalize eigenvector.
*
         VNORM = ZERO
         DO 290 I = 1, N
            VNORM = MAX( VNORM, ABS( VR( I ) )+ABS( VI( I ) ) )
  290    CONTINUE
         CALL DSCAL( N, ONE / VNORM, VR, 1 )
         CALL DSCAL( N, ONE / VNORM, VI, 1 )
*
      END IF
*
      RETURN
*
*     End of DLAEIN
*
      END
*> \brief \b DLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAEV2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaev2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaev2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaev2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAEV2( A, B, C, RT1, RT2, CS1, SN1 )
*
*       .. Scalar Arguments ..
*       DOUBLE PRECISION   A, B, C, CS1, RT1, RT2, SN1
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix
*>    [  A   B  ]
*>    [  B   C  ].
*> On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
*> eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
*> eigenvector for RT1, giving the decomposition
*>
*>    [ CS1  SN1 ] [  A   B  ] [ CS1 -SN1 ]  =  [ RT1  0  ]
*>    [-SN1  CS1 ] [  B   C  ] [ SN1  CS1 ]     [  0  RT2 ].
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION
*>          The (1,1) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION
*>          The (1,2) element and the conjugate of the (2,1) element of
*>          the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*>          C is DOUBLE PRECISION
*>          The (2,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[out] RT1
*> \verbatim
*>          RT1 is DOUBLE PRECISION
*>          The eigenvalue of larger absolute value.
*> \endverbatim
*>
*> \param[out] RT2
*> \verbatim
*>          RT2 is DOUBLE PRECISION
*>          The eigenvalue of smaller absolute value.
*> \endverbatim
*>
*> \param[out] CS1
*> \verbatim
*>          CS1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] SN1
*> \verbatim
*>          SN1 is DOUBLE PRECISION
*>          The vector (CS1, SN1) is a unit right eigenvector for RT1.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laev2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  RT1 is accurate to a few ulps barring over/underflow.
*>
*>  RT2 may be inaccurate if there is massive cancellation in the
*>  determinant A*C-B*B; higher precision or correctly rounded or
*>  correctly truncated arithmetic would be needed to compute RT2
*>  accurately in all cases.
*>
*>  CS1 and SN1 are accurate to a few ulps barring over/underflow.
*>
*>  Overflow is possible only if RT1 is within a factor of 5 of overflow.
*>  Underflow is harmless if the input data is 0 or exceeds
*>     underflow_threshold / macheps.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLAEV2( A, B, C, RT1, RT2, CS1, SN1 )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   A, B, C, CS1, RT1, RT2, SN1
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D0 )
      DOUBLE PRECISION   TWO
      PARAMETER          ( TWO = 2.0D0 )
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D0 )
      DOUBLE PRECISION   HALF
      PARAMETER          ( HALF = 0.5D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            SGN1, SGN2
      DOUBLE PRECISION   AB, ACMN, ACMX, ACS, ADF, CS, CT, DF, RT, SM,
     $                   TB, TN
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, SQRT
*     ..
*     .. Executable Statements ..
*
*     Compute the eigenvalues
*
      SM = A + C
      DF = A - C
      ADF = ABS( DF )
      TB = B + B
      AB = ABS( TB )
      IF( ABS( A ).GT.ABS( C ) ) THEN
         ACMX = A
         ACMN = C
      ELSE
         ACMX = C
         ACMN = A
      END IF
      IF( ADF.GT.AB ) THEN
         RT = ADF*SQRT( ONE+( AB / ADF )**2 )
      ELSE IF( ADF.LT.AB ) THEN
         RT = AB*SQRT( ONE+( ADF / AB )**2 )
      ELSE
*
*        Includes case AB=ADF=0
*
         RT = AB*SQRT( TWO )
      END IF
      IF( SM.LT.ZERO ) THEN
         RT1 = HALF*( SM-RT )
         SGN1 = -1
*
*        Order of execution important.
*        To get fully accurate smaller eigenvalue,
*        next line needs to be executed in higher precision.
*
         RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
      ELSE IF( SM.GT.ZERO ) THEN
         RT1 = HALF*( SM+RT )
         SGN1 = 1
*
*        Order of execution important.
*        To get fully accurate smaller eigenvalue,
*        next line needs to be executed in higher precision.
*
         RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
      ELSE
*
*        Includes case RT1 = RT2 = 0
*
         RT1 = HALF*RT
         RT2 = -HALF*RT
         SGN1 = 1
      END IF
*
*     Compute the eigenvector
*
      IF( DF.GE.ZERO ) THEN
         CS = DF + RT
         SGN2 = 1
      ELSE
         CS = DF - RT
         SGN2 = -1
      END IF
      ACS = ABS( CS )
      IF( ACS.GT.AB ) THEN
         CT = -TB / CS
         SN1 = ONE / SQRT( ONE+CT*CT )
         CS1 = CT*SN1
      ELSE
         IF( AB.EQ.ZERO ) THEN
            CS1 = ONE
            SN1 = ZERO
         ELSE
            TN = -CS / TB
            CS1 = ONE / SQRT( ONE+TN*TN )
            SN1 = TN*CS1
         END IF
      END IF
      IF( SGN1.EQ.SGN2 ) THEN
         TN = CS1
         CS1 = -SN1
         SN1 = TN
      END IF
      RETURN
*
*     End of DLAEV2
*
      END
*> \brief \b DLAEXC swaps adjacent diagonal blocks of a real upper quasi-triangular matrix in Schur canonical form, by an orthogonal similarity transformation.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAEXC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaexc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaexc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaexc.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAEXC( WANTQ, N, T, LDT, Q, LDQ, J1, N1, N2, WORK,
*                          INFO )
*
*       .. Scalar Arguments ..
*       LOGICAL            WANTQ
*       INTEGER            INFO, J1, LDQ, LDT, N, N1, N2
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   Q( LDQ, * ), T( LDT, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAEXC swaps adjacent diagonal blocks T11 and T22 of order 1 or 2 in
*> an upper quasi-triangular matrix T by an orthogonal similarity
*> transformation.
*>
*> T must be in Schur canonical form, that is, block upper triangular
*> with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block
*> has its diagonal elements equal and its off-diagonal elements of
*> opposite sign.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] WANTQ
*> \verbatim
*>          WANTQ is LOGICAL
*>          = .TRUE. : accumulate the transformation in the matrix Q;
*>          = .FALSE.: do not accumulate the transformation.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix T. N >= 0.
*> \endverbatim
*>
*> \param[in,out] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,N)
*>          On entry, the upper quasi-triangular matrix T, in Schur
*>          canonical form.
*>          On exit, the updated matrix T, again in Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ,N)
*>          On entry, if WANTQ is .TRUE., the orthogonal matrix Q.
*>          On exit, if WANTQ is .TRUE., the updated matrix Q.
*>          If WANTQ is .FALSE., Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q.
*>          LDQ >= 1; and if WANTQ is .TRUE., LDQ >= N.
*> \endverbatim
*>
*> \param[in] J1
*> \verbatim
*>          J1 is INTEGER
*>          The index of the first row of the first block T11.
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*>          N1 is INTEGER
*>          The order of the first block T11. N1 = 0, 1 or 2.
*> \endverbatim
*>
*> \param[in] N2
*> \verbatim
*>          N2 is INTEGER
*>          The order of the second block T22. N2 = 0, 1 or 2.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          = 1: the transformed matrix T would be too far from Schur
*>               form; the blocks are not swapped and T and Q are
*>               unchanged.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laexc
*
*  =====================================================================
      SUBROUTINE DLAEXC( WANTQ, N, T, LDT, Q, LDQ, J1, N1, N2, WORK,
     $                   INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      LOGICAL            WANTQ
      INTEGER            INFO, J1, LDQ, LDT, N, N1, N2
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   Q( LDQ, * ), T( LDT, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
      DOUBLE PRECISION   TEN
      PARAMETER          ( TEN = 1.0D+1 )
      INTEGER            LDD, LDX
      PARAMETER          ( LDD = 4, LDX = 2 )
*     ..
*     .. Local Scalars ..
      INTEGER            IERR, J2, J3, J4, K, ND
      DOUBLE PRECISION   CS, DNORM, EPS, SCALE, SMLNUM, SN, T11, T22,
     $                   T33, TAU, TAU1, TAU2, TEMP, THRESH, WI1, WI2,
     $                   WR1, WR2, XNORM
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   D( LDD, 4 ), U( 3 ), U1( 3 ), U2( 3 ),
     $                   X( LDX, 2 )
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           DLAMCH, DLANGE
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACPY, DLANV2, DLARFG, DLARFX, DLARTG,
     $                   DLASY2,
     $                   DROT
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. Executable Statements ..
*
      INFO = 0
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. N1.EQ.0 .OR. N2.EQ.0 )
     $   RETURN
      IF( J1+N1.GT.N )
     $   RETURN
*
      J2 = J1 + 1
      J3 = J1 + 2
      J4 = J1 + 3
*
      IF( N1.EQ.1 .AND. N2.EQ.1 ) THEN
*
*        Swap two 1-by-1 blocks.
*
         T11 = T( J1, J1 )
         T22 = T( J2, J2 )
*
*        Determine the transformation to perform the interchange.
*
         CALL DLARTG( T( J1, J2 ), T22-T11, CS, SN, TEMP )
*
*        Apply transformation to the matrix T.
*
         IF( J3.LE.N )
     $      CALL DROT( N-J1-1, T( J1, J3 ), LDT, T( J2, J3 ), LDT,
     $                 CS,
     $                 SN )
         CALL DROT( J1-1, T( 1, J1 ), 1, T( 1, J2 ), 1, CS, SN )
*
         T( J1, J1 ) = T22
         T( J2, J2 ) = T11
*
         IF( WANTQ ) THEN
*
*           Accumulate transformation in the matrix Q.
*
            CALL DROT( N, Q( 1, J1 ), 1, Q( 1, J2 ), 1, CS, SN )
         END IF
*
      ELSE
*
*        Swapping involves at least one 2-by-2 block.
*
*        Copy the diagonal block of order N1+N2 to the local array D
*        and compute its norm.
*
         ND = N1 + N2
         CALL DLACPY( 'Full', ND, ND, T( J1, J1 ), LDT, D, LDD )
         DNORM = DLANGE( 'Max', ND, ND, D, LDD, WORK )
*
*        Compute machine-dependent threshold for test for accepting
*        swap.
*
         EPS = DLAMCH( 'P' )
         SMLNUM = DLAMCH( 'S' ) / EPS
         THRESH = MAX( TEN*EPS*DNORM, SMLNUM )
*
*        Solve T11*X - X*T22 = scale*T12 for X.
*
         CALL DLASY2( .FALSE., .FALSE., -1, N1, N2, D, LDD,
     $                D( N1+1, N1+1 ), LDD, D( 1, N1+1 ), LDD, SCALE, X,
     $                LDX, XNORM, IERR )
*
*        Swap the adjacent diagonal blocks.
*
         K = N1 + N1 + N2 - 3
         GO TO ( 10, 20, 30 )K
*
   10    CONTINUE
*
*        N1 = 1, N2 = 2: generate elementary reflector H so that:
*
*        ( scale, X11, X12 ) H = ( 0, 0, * )
*
         U( 1 ) = SCALE
         U( 2 ) = X( 1, 1 )
         U( 3 ) = X( 1, 2 )
         CALL DLARFG( 3, U( 3 ), U, 1, TAU )
         U( 3 ) = ONE
         T11 = T( J1, J1 )
*
*        Perform swap provisionally on diagonal block in D.
*
         CALL DLARFX( 'L', 3, 3, U, TAU, D, LDD, WORK )
         CALL DLARFX( 'R', 3, 3, U, TAU, D, LDD, WORK )
*
*        Test whether to reject swap.
*
         IF( MAX( ABS( D( 3, 1 ) ), ABS( D( 3, 2 ) ), ABS( D( 3,
     $       3 )-T11 ) ).GT.THRESH )GO TO 50
*
*        Accept swap: apply transformation to the entire matrix T.
*
         CALL DLARFX( 'L', 3, N-J1+1, U, TAU, T( J1, J1 ), LDT,
     $                WORK )
         CALL DLARFX( 'R', J2, 3, U, TAU, T( 1, J1 ), LDT, WORK )
*
         T( J3, J1 ) = ZERO
         T( J3, J2 ) = ZERO
         T( J3, J3 ) = T11
*
         IF( WANTQ ) THEN
*
*           Accumulate transformation in the matrix Q.
*
            CALL DLARFX( 'R', N, 3, U, TAU, Q( 1, J1 ), LDQ, WORK )
         END IF
         GO TO 40
*
   20    CONTINUE
*
*        N1 = 2, N2 = 1: generate elementary reflector H so that:
*
*        H (  -X11 ) = ( * )
*          (  -X21 ) = ( 0 )
*          ( scale ) = ( 0 )
*
         U( 1 ) = -X( 1, 1 )
         U( 2 ) = -X( 2, 1 )
         U( 3 ) = SCALE
         CALL DLARFG( 3, U( 1 ), U( 2 ), 1, TAU )
         U( 1 ) = ONE
         T33 = T( J3, J3 )
*
*        Perform swap provisionally on diagonal block in D.
*
         CALL DLARFX( 'L', 3, 3, U, TAU, D, LDD, WORK )
         CALL DLARFX( 'R', 3, 3, U, TAU, D, LDD, WORK )
*
*        Test whether to reject swap.
*
         IF( MAX( ABS( D( 2, 1 ) ), ABS( D( 3, 1 ) ), ABS( D( 1,
     $       1 )-T33 ) ).GT.THRESH )GO TO 50
*
*        Accept swap: apply transformation to the entire matrix T.
*
         CALL DLARFX( 'R', J3, 3, U, TAU, T( 1, J1 ), LDT, WORK )
         CALL DLARFX( 'L', 3, N-J1, U, TAU, T( J1, J2 ), LDT, WORK )
*
         T( J1, J1 ) = T33
         T( J2, J1 ) = ZERO
         T( J3, J1 ) = ZERO
*
         IF( WANTQ ) THEN
*
*           Accumulate transformation in the matrix Q.
*
            CALL DLARFX( 'R', N, 3, U, TAU, Q( 1, J1 ), LDQ, WORK )
         END IF
         GO TO 40
*
   30    CONTINUE
*
*        N1 = 2, N2 = 2: generate elementary reflectors H(1) and H(2) so
*        that:
*
*        H(2) H(1) (  -X11  -X12 ) = (  *  * )
*                  (  -X21  -X22 )   (  0  * )
*                  ( scale    0  )   (  0  0 )
*                  (    0  scale )   (  0  0 )
*
         U1( 1 ) = -X( 1, 1 )
         U1( 2 ) = -X( 2, 1 )
         U1( 3 ) = SCALE
         CALL DLARFG( 3, U1( 1 ), U1( 2 ), 1, TAU1 )
         U1( 1 ) = ONE
*
         TEMP = -TAU1*( X( 1, 2 )+U1( 2 )*X( 2, 2 ) )
         U2( 1 ) = -TEMP*U1( 2 ) - X( 2, 2 )
         U2( 2 ) = -TEMP*U1( 3 )
         U2( 3 ) = SCALE
         CALL DLARFG( 3, U2( 1 ), U2( 2 ), 1, TAU2 )
         U2( 1 ) = ONE
*
*        Perform swap provisionally on diagonal block in D.
*
         CALL DLARFX( 'L', 3, 4, U1, TAU1, D, LDD, WORK )
         CALL DLARFX( 'R', 4, 3, U1, TAU1, D, LDD, WORK )
         CALL DLARFX( 'L', 3, 4, U2, TAU2, D( 2, 1 ), LDD, WORK )
         CALL DLARFX( 'R', 4, 3, U2, TAU2, D( 1, 2 ), LDD, WORK )
*
*        Test whether to reject swap.
*
         IF( MAX( ABS( D( 3, 1 ) ), ABS( D( 3, 2 ) ), ABS( D( 4, 1 ) ),
     $       ABS( D( 4, 2 ) ) ).GT.THRESH )GO TO 50
*
*        Accept swap: apply transformation to the entire matrix T.
*
         CALL DLARFX( 'L', 3, N-J1+1, U1, TAU1, T( J1, J1 ), LDT,
     $                WORK )
         CALL DLARFX( 'R', J4, 3, U1, TAU1, T( 1, J1 ), LDT, WORK )
         CALL DLARFX( 'L', 3, N-J1+1, U2, TAU2, T( J2, J1 ), LDT,
     $                WORK )
         CALL DLARFX( 'R', J4, 3, U2, TAU2, T( 1, J2 ), LDT, WORK )
*
         T( J3, J1 ) = ZERO
         T( J3, J2 ) = ZERO
         T( J4, J1 ) = ZERO
         T( J4, J2 ) = ZERO
*
         IF( WANTQ ) THEN
*
*           Accumulate transformation in the matrix Q.
*
            CALL DLARFX( 'R', N, 3, U1, TAU1, Q( 1, J1 ), LDQ, WORK )
            CALL DLARFX( 'R', N, 3, U2, TAU2, Q( 1, J2 ), LDQ, WORK )
         END IF
*
   40    CONTINUE
*
         IF( N2.EQ.2 ) THEN
*
*           Standardize new 2-by-2 block T11
*
            CALL DLANV2( T( J1, J1 ), T( J1, J2 ), T( J2, J1 ),
     $                   T( J2, J2 ), WR1, WI1, WR2, WI2, CS, SN )
            CALL DROT( N-J1-1, T( J1, J1+2 ), LDT, T( J2, J1+2 ),
     $                 LDT,
     $                 CS, SN )
            CALL DROT( J1-1, T( 1, J1 ), 1, T( 1, J2 ), 1, CS, SN )
            IF( WANTQ )
     $         CALL DROT( N, Q( 1, J1 ), 1, Q( 1, J2 ), 1, CS, SN )
         END IF
*
         IF( N1.EQ.2 ) THEN
*
*           Standardize new 2-by-2 block T22
*
            J3 = J1 + N2
            J4 = J3 + 1
            CALL DLANV2( T( J3, J3 ), T( J3, J4 ), T( J4, J3 ),
     $                   T( J4, J4 ), WR1, WI1, WR2, WI2, CS, SN )
            IF( J3+2.LE.N )
     $         CALL DROT( N-J3-1, T( J3, J3+2 ), LDT, T( J4, J3+2 ),
     $                    LDT, CS, SN )
            CALL DROT( J3-1, T( 1, J3 ), 1, T( 1, J4 ), 1, CS, SN )
            IF( WANTQ )
     $         CALL DROT( N, Q( 1, J3 ), 1, Q( 1, J4 ), 1, CS, SN )
         END IF
*
      END IF
      RETURN
*
*     Exit with INFO = 1 if swap was rejected.
*
   50 CONTINUE
      INFO = 1
      RETURN
*
*     End of DLAEXC
*
      END
*> \brief \b DLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAG2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlag2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlag2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlag2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
*                         WR2, WI )
*
*       .. Scalar Arguments ..
*       INTEGER            LDA, LDB
*       DOUBLE PRECISION   SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
*> problem  A - w B, with scaling as necessary to avoid over-/underflow.
*>
*> The scaling factor "s" results in a modified eigenvalue equation
*>
*>     s A - w B
*>
*> where  s  is a non-negative scaling factor chosen so that  w,  w B,
*> and  s A  do not overflow and, if possible, do not underflow, either.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, 2)
*>          On entry, the 2 x 2 matrix A.  It is assumed that its 1-norm
*>          is less than 1/SAFMIN.  Entries less than
*>          sqrt(SAFMIN)*norm(A) are subject to being treated as zero.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= 2.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB, 2)
*>          On entry, the 2 x 2 upper triangular matrix B.  It is
*>          assumed that the one-norm of B is less than 1/SAFMIN.  The
*>          diagonals should be at least sqrt(SAFMIN) times the largest
*>          element of B (in absolute value); if a diagonal is smaller
*>          than that, then  +/- sqrt(SAFMIN) will be used instead of
*>          that diagonal.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= 2.
*> \endverbatim
*>
*> \param[in] SAFMIN
*> \verbatim
*>          SAFMIN is DOUBLE PRECISION
*>          The smallest positive number s.t. 1/SAFMIN does not
*>          overflow.  (This should always be DLAMCH('S') -- it is an
*>          argument in order to avoid having to call DLAMCH frequently.)
*> \endverbatim
*>
*> \param[out] SCALE1
*> \verbatim
*>          SCALE1 is DOUBLE PRECISION
*>          A scaling factor used to avoid over-/underflow in the
*>          eigenvalue equation which defines the first eigenvalue.  If
*>          the eigenvalues are complex, then the eigenvalues are
*>          ( WR1  +/-  WI i ) / SCALE1  (which may lie outside the
*>          exponent range of the machine), SCALE1=SCALE2, and SCALE1
*>          will always be positive.  If the eigenvalues are real, then
*>          the first (real) eigenvalue is  WR1 / SCALE1 , but this may
*>          overflow or underflow, and in fact, SCALE1 may be zero or
*>          less than the underflow threshold if the exact eigenvalue
*>          is sufficiently large.
*> \endverbatim
*>
*> \param[out] SCALE2
*> \verbatim
*>          SCALE2 is DOUBLE PRECISION
*>          A scaling factor used to avoid over-/underflow in the
*>          eigenvalue equation which defines the second eigenvalue.  If
*>          the eigenvalues are complex, then SCALE2=SCALE1.  If the
*>          eigenvalues are real, then the second (real) eigenvalue is
*>          WR2 / SCALE2 , but this may overflow or underflow, and in
*>          fact, SCALE2 may be zero or less than the underflow
*>          threshold if the exact eigenvalue is sufficiently large.
*> \endverbatim
*>
*> \param[out] WR1
*> \verbatim
*>          WR1 is DOUBLE PRECISION
*>          If the eigenvalue is real, then WR1 is SCALE1 times the
*>          eigenvalue closest to the (2,2) element of A B**(-1).  If the
*>          eigenvalue is complex, then WR1=WR2 is SCALE1 times the real
*>          part of the eigenvalues.
*> \endverbatim
*>
*> \param[out] WR2
*> \verbatim
*>          WR2 is DOUBLE PRECISION
*>          If the eigenvalue is real, then WR2 is SCALE2 times the
*>          other eigenvalue.  If the eigenvalue is complex, then
*>          WR1=WR2 is SCALE1 times the real part of the eigenvalues.
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*>          WI is DOUBLE PRECISION
*>          If the eigenvalue is real, then WI is zero.  If the
*>          eigenvalue is complex, then WI is SCALE1 times the imaginary
*>          part of the eigenvalues.  WI will always be non-negative.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lag2
*
*  =====================================================================
      SUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
     $                  WR2, WI )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            LDA, LDB
      DOUBLE PRECISION   SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
      DOUBLE PRECISION   HALF
      PARAMETER          ( HALF = ONE / TWO )
      DOUBLE PRECISION   FUZZY1
      PARAMETER          ( FUZZY1 = ONE+1.0D-5 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   A11, A12, A21, A22, ABI22, ANORM, AS11, AS12,
     $                   AS22, ASCALE, B11, B12, B22, BINV11, BINV22,
     $                   BMIN, BNORM, BSCALE, BSIZE, C1, C2, C3, C4, C5,
     $                   DIFF, DISCR, PP, QQ, R, RTMAX, RTMIN, S1, S2,
     $                   SAFMAX, SHIFT, SS, SUM, WABS, WBIG, WDET,
     $                   WSCALE, WSIZE, WSMALL
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, SIGN, SQRT
*     ..
*     .. Executable Statements ..
*
      RTMIN = SQRT( SAFMIN )
      RTMAX = ONE / RTMIN
      SAFMAX = ONE / SAFMIN
*
*     Scale A
*
      ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ),
     $        ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN )
      ASCALE = ONE / ANORM
      A11 = ASCALE*A( 1, 1 )
      A21 = ASCALE*A( 2, 1 )
      A12 = ASCALE*A( 1, 2 )
      A22 = ASCALE*A( 2, 2 )
*
*     Perturb B if necessary to insure non-singularity
*
      B11 = B( 1, 1 )
      B12 = B( 1, 2 )
      B22 = B( 2, 2 )
      BMIN = RTMIN*MAX( ABS( B11 ), ABS( B12 ), ABS( B22 ), RTMIN )
      IF( ABS( B11 ).LT.BMIN )
     $   B11 = SIGN( BMIN, B11 )
      IF( ABS( B22 ).LT.BMIN )
     $   B22 = SIGN( BMIN, B22 )
*
*     Scale B
*
      BNORM = MAX( ABS( B11 ), ABS( B12 )+ABS( B22 ), SAFMIN )
      BSIZE = MAX( ABS( B11 ), ABS( B22 ) )
      BSCALE = ONE / BSIZE
      B11 = B11*BSCALE
      B12 = B12*BSCALE
      B22 = B22*BSCALE
*
*     Compute larger eigenvalue by method described by C. van Loan
*
*     ( AS is A shifted by -SHIFT*B )
*
      BINV11 = ONE / B11
      BINV22 = ONE / B22
      S1 = A11*BINV11
      S2 = A22*BINV22
      IF( ABS( S1 ).LE.ABS( S2 ) ) THEN
         AS12 = A12 - S1*B12
         AS22 = A22 - S1*B22
         SS = A21*( BINV11*BINV22 )
         ABI22 = AS22*BINV22 - SS*B12
         PP = HALF*ABI22
         SHIFT = S1
      ELSE
         AS12 = A12 - S2*B12
         AS11 = A11 - S2*B11
         SS = A21*( BINV11*BINV22 )
         ABI22 = -SS*B12
         PP = HALF*( AS11*BINV11+ABI22 )
         SHIFT = S2
      END IF
      QQ = SS*AS12
      IF( ABS( PP*RTMIN ).GE.ONE ) THEN
         DISCR = ( RTMIN*PP )**2 + QQ*SAFMIN
         R = SQRT( ABS( DISCR ) )*RTMAX
      ELSE
         IF( PP**2+ABS( QQ ).LE.SAFMIN ) THEN
            DISCR = ( RTMAX*PP )**2 + QQ*SAFMAX
            R = SQRT( ABS( DISCR ) )*RTMIN
         ELSE
            DISCR = PP**2 + QQ
            R = SQRT( ABS( DISCR ) )
         END IF
      END IF
*
*     Note: the test of R in the following IF is to cover the case when
*           DISCR is small and negative and is flushed to zero during
*           the calculation of R.  On machines which have a consistent
*           flush-to-zero threshold and handle numbers above that
*           threshold correctly, it would not be necessary.
*
      IF( DISCR.GE.ZERO .OR. R.EQ.ZERO ) THEN
         SUM = PP + SIGN( R, PP )
         DIFF = PP - SIGN( R, PP )
         WBIG = SHIFT + SUM
*
*        Compute smaller eigenvalue
*
         WSMALL = SHIFT + DIFF
         IF( HALF*ABS( WBIG ).GT.MAX( ABS( WSMALL ), SAFMIN ) ) THEN
            WDET = ( A11*A22-A12*A21 )*( BINV11*BINV22 )
            WSMALL = WDET / WBIG
         END IF
*
*        Choose (real) eigenvalue closest to 2,2 element of A*B**(-1)
*        for WR1.
*
         IF( PP.GT.ABI22 ) THEN
            WR1 = MIN( WBIG, WSMALL )
            WR2 = MAX( WBIG, WSMALL )
         ELSE
            WR1 = MAX( WBIG, WSMALL )
            WR2 = MIN( WBIG, WSMALL )
         END IF
         WI = ZERO
      ELSE
*
*        Complex eigenvalues
*
         WR1 = SHIFT + PP
         WR2 = WR1
         WI = R
      END IF
*
*     Further scaling to avoid underflow and overflow in computing
*     SCALE1 and overflow in computing w*B.
*
*     This scale factor (WSCALE) is bounded from above using C1 and C2,
*     and from below using C3 and C4.
*        C1 implements the condition  s A  must never overflow.
*        C2 implements the condition  w B  must never overflow.
*        C3, with C2,
*           implement the condition that s A - w B must never overflow.
*        C4 implements the condition  s    should not underflow.
*        C5 implements the condition  max(s,|w|) should be at least 2.
*
      C1 = BSIZE*( SAFMIN*MAX( ONE, ASCALE ) )
      C2 = SAFMIN*MAX( ONE, BNORM )
      C3 = BSIZE*SAFMIN
      IF( ASCALE.LE.ONE .AND. BSIZE.LE.ONE ) THEN
         C4 = MIN( ONE, ( ASCALE / SAFMIN )*BSIZE )
      ELSE
         C4 = ONE
      END IF
      IF( ASCALE.LE.ONE .OR. BSIZE.LE.ONE ) THEN
         C5 = MIN( ONE, ASCALE*BSIZE )
      ELSE
         C5 = ONE
      END IF
*
*     Scale first eigenvalue
*
      WABS = ABS( WR1 ) + ABS( WI )
      WSIZE = MAX( SAFMIN, C1, FUZZY1*( WABS*C2+C3 ),
     $        MIN( C4, HALF*MAX( WABS, C5 ) ) )
      IF( WSIZE.NE.ONE ) THEN
         WSCALE = ONE / WSIZE
         IF( WSIZE.GT.ONE ) THEN
            SCALE1 = ( MAX( ASCALE, BSIZE )*WSCALE )*
     $               MIN( ASCALE, BSIZE )
         ELSE
            SCALE1 = ( MIN( ASCALE, BSIZE )*WSCALE )*
     $               MAX( ASCALE, BSIZE )
         END IF
         WR1 = WR1*WSCALE
         IF( WI.NE.ZERO ) THEN
            WI = WI*WSCALE
            WR2 = WR1
            SCALE2 = SCALE1
         END IF
      ELSE
         SCALE1 = ASCALE*BSIZE
         SCALE2 = SCALE1
      END IF
*
*     Scale second eigenvalue (if real)
*
      IF( WI.EQ.ZERO ) THEN
         WSIZE = MAX( SAFMIN, C1, FUZZY1*( ABS( WR2 )*C2+C3 ),
     $           MIN( C4, HALF*MAX( ABS( WR2 ), C5 ) ) )
         IF( WSIZE.NE.ONE ) THEN
            WSCALE = ONE / WSIZE
            IF( WSIZE.GT.ONE ) THEN
               SCALE2 = ( MAX( ASCALE, BSIZE )*WSCALE )*
     $                  MIN( ASCALE, BSIZE )
            ELSE
               SCALE2 = ( MIN( ASCALE, BSIZE )*WSCALE )*
     $                  MAX( ASCALE, BSIZE )
            END IF
            WR2 = WR2*WSCALE
         ELSE
            SCALE2 = ASCALE*BSIZE
         END IF
      END IF
*
*     End of DLAG2
*
      RETURN
      END
*> \brief \b DLAGS2 computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A and B such that the rows of the transformed A and B are parallel.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAGS2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlags2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlags2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlags2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV,
*                          SNV, CSQ, SNQ )
*
*       .. Scalar Arguments ..
*       LOGICAL            UPPER
*       DOUBLE PRECISION   A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, SNQ,
*      $                   SNU, SNV
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such
*> that if ( UPPER ) then
*>
*>           U**T *A*Q = U**T *( A1 A2 )*Q = ( x  0  )
*>                             ( 0  A3 )     ( x  x  )
*> and
*>           V**T*B*Q = V**T *( B1 B2 )*Q = ( x  0  )
*>                            ( 0  B3 )     ( x  x  )
*>
*> or if ( .NOT.UPPER ) then
*>
*>           U**T *A*Q = U**T *( A1 0  )*Q = ( x  x  )
*>                             ( A2 A3 )     ( 0  x  )
*> and
*>           V**T*B*Q = V**T*( B1 0  )*Q = ( x  x  )
*>                           ( B2 B3 )     ( 0  x  )
*>
*> The rows of the transformed A and B are parallel, where
*>
*>   U = (  CSU  SNU ), V = (  CSV SNV ), Q = (  CSQ   SNQ )
*>       ( -SNU  CSU )      ( -SNV CSV )      ( -SNQ   CSQ )
*>
*> Z**T denotes the transpose of Z.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPPER
*> \verbatim
*>          UPPER is LOGICAL
*>          = .TRUE.: the input matrices A and B are upper triangular.
*>          = .FALSE.: the input matrices A and B are lower triangular.
*> \endverbatim
*>
*> \param[in] A1
*> \verbatim
*>          A1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] A2
*> \verbatim
*>          A2 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] A3
*> \verbatim
*>          A3 is DOUBLE PRECISION
*>          On entry, A1, A2 and A3 are elements of the input 2-by-2
*>          upper (lower) triangular matrix A.
*> \endverbatim
*>
*> \param[in] B1
*> \verbatim
*>          B1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] B2
*> \verbatim
*>          B2 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] B3
*> \verbatim
*>          B3 is DOUBLE PRECISION
*>          On entry, B1, B2 and B3 are elements of the input 2-by-2
*>          upper (lower) triangular matrix B.
*> \endverbatim
*>
*> \param[out] CSU
*> \verbatim
*>          CSU is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] SNU
*> \verbatim
*>          SNU is DOUBLE PRECISION
*>          The desired orthogonal matrix U.
*> \endverbatim
*>
*> \param[out] CSV
*> \verbatim
*>          CSV is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] SNV
*> \verbatim
*>          SNV is DOUBLE PRECISION
*>          The desired orthogonal matrix V.
*> \endverbatim
*>
*> \param[out] CSQ
*> \verbatim
*>          CSQ is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] SNQ
*> \verbatim
*>          SNQ is DOUBLE PRECISION
*>          The desired orthogonal matrix Q.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lags2
*
*  =====================================================================
      SUBROUTINE DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU,
     $                   CSV,
     $                   SNV, CSQ, SNQ )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      LOGICAL            UPPER
      DOUBLE PRECISION   A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, SNQ,
     $                   SNU, SNV
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   A, AUA11, AUA12, AUA21, AUA22, AVB11, AVB12,
     $                   AVB21, AVB22, B, C, CSL, CSR, D, R, S1, S2,
     $                   SNL, SNR, UA11, UA11R, UA12, UA21, UA22, UA22R,
     $                   VB11, VB11R, VB12, VB21, VB22, VB22R
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARTG, DLASV2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS
*     ..
*     .. Executable Statements ..
*
      IF( UPPER ) THEN
*
*        Input matrices A and B are upper triangular matrices
*
*        Form matrix C = A*adj(B) = ( a b )
*                                   ( 0 d )
*
         A = A1*B3
         D = A3*B1
         B = A2*B1 - A1*B2
*
*        The SVD of real 2-by-2 triangular C
*
*         ( CSL -SNL )*( A B )*(  CSR  SNR ) = ( R 0 )
*         ( SNL  CSL ) ( 0 D ) ( -SNR  CSR )   ( 0 T )
*
         CALL DLASV2( A, B, D, S1, S2, SNR, CSR, SNL, CSL )
*
         IF( ABS( CSL ).GE.ABS( SNL ) .OR. ABS( CSR ).GE.ABS( SNR ) )
     $        THEN
*
*           Compute the (1,1) and (1,2) elements of U**T *A and V**T *B,
*           and (1,2) element of |U|**T *|A| and |V|**T *|B|.
*
            UA11R = CSL*A1
            UA12 = CSL*A2 + SNL*A3
*
            VB11R = CSR*B1
            VB12 = CSR*B2 + SNR*B3
*
            AUA12 = ABS( CSL )*ABS( A2 ) + ABS( SNL )*ABS( A3 )
            AVB12 = ABS( CSR )*ABS( B2 ) + ABS( SNR )*ABS( B3 )
*
*           zero (1,2) elements of U**T *A and V**T *B
*
            IF( ( ABS( UA11R )+ABS( UA12 ) ).NE.ZERO ) THEN
               IF( AUA12 / ( ABS( UA11R )+ABS( UA12 ) ).LE.AVB12 /
     $             ( ABS( VB11R )+ABS( VB12 ) ) ) THEN
                  CALL DLARTG( -UA11R, UA12, CSQ, SNQ, R )
               ELSE
                  CALL DLARTG( -VB11R, VB12, CSQ, SNQ, R )
               END IF
            ELSE
               CALL DLARTG( -VB11R, VB12, CSQ, SNQ, R )
            END IF
*
            CSU = CSL
            SNU = -SNL
            CSV = CSR
            SNV = -SNR
*
         ELSE
*
*           Compute the (2,1) and (2,2) elements of U**T *A and V**T *B,
*           and (2,2) element of |U|**T *|A| and |V|**T *|B|.
*
            UA21 = -SNL*A1
            UA22 = -SNL*A2 + CSL*A3
*
            VB21 = -SNR*B1
            VB22 = -SNR*B2 + CSR*B3
*
            AUA22 = ABS( SNL )*ABS( A2 ) + ABS( CSL )*ABS( A3 )
            AVB22 = ABS( SNR )*ABS( B2 ) + ABS( CSR )*ABS( B3 )
*
*           zero (2,2) elements of U**T*A and V**T*B, and then swap.
*
            IF( ( ABS( UA21 )+ABS( UA22 ) ).NE.ZERO ) THEN
               IF( AUA22 / ( ABS( UA21 )+ABS( UA22 ) ).LE.AVB22 /
     $             ( ABS( VB21 )+ABS( VB22 ) ) ) THEN
                  CALL DLARTG( -UA21, UA22, CSQ, SNQ, R )
               ELSE
                  CALL DLARTG( -VB21, VB22, CSQ, SNQ, R )
               END IF
            ELSE
               CALL DLARTG( -VB21, VB22, CSQ, SNQ, R )
            END IF
*
            CSU = SNL
            SNU = CSL
            CSV = SNR
            SNV = CSR
*
         END IF
*
      ELSE
*
*        Input matrices A and B are lower triangular matrices
*
*        Form matrix C = A*adj(B) = ( a 0 )
*                                   ( c d )
*
         A = A1*B3
         D = A3*B1
         C = A2*B3 - A3*B2
*
*        The SVD of real 2-by-2 triangular C
*
*         ( CSL -SNL )*( A 0 )*(  CSR  SNR ) = ( R 0 )
*         ( SNL  CSL ) ( C D ) ( -SNR  CSR )   ( 0 T )
*
         CALL DLASV2( A, C, D, S1, S2, SNR, CSR, SNL, CSL )
*
         IF( ABS( CSR ).GE.ABS( SNR ) .OR. ABS( CSL ).GE.ABS( SNL ) )
     $        THEN
*
*           Compute the (2,1) and (2,2) elements of U**T *A and V**T *B,
*           and (2,1) element of |U|**T *|A| and |V|**T *|B|.
*
            UA21 = -SNR*A1 + CSR*A2
            UA22R = CSR*A3
*
            VB21 = -SNL*B1 + CSL*B2
            VB22R = CSL*B3
*
            AUA21 = ABS( SNR )*ABS( A1 ) + ABS( CSR )*ABS( A2 )
            AVB21 = ABS( SNL )*ABS( B1 ) + ABS( CSL )*ABS( B2 )
*
*           zero (2,1) elements of U**T *A and V**T *B.
*
            IF( ( ABS( UA21 )+ABS( UA22R ) ).NE.ZERO ) THEN
               IF( AUA21 / ( ABS( UA21 )+ABS( UA22R ) ).LE.AVB21 /
     $             ( ABS( VB21 )+ABS( VB22R ) ) ) THEN
                  CALL DLARTG( UA22R, UA21, CSQ, SNQ, R )
               ELSE
                  CALL DLARTG( VB22R, VB21, CSQ, SNQ, R )
               END IF
            ELSE
               CALL DLARTG( VB22R, VB21, CSQ, SNQ, R )
            END IF
*
            CSU = CSR
            SNU = -SNR
            CSV = CSL
            SNV = -SNL
*
         ELSE
*
*           Compute the (1,1) and (1,2) elements of U**T *A and V**T *B,
*           and (1,1) element of |U|**T *|A| and |V|**T *|B|.
*
            UA11 = CSR*A1 + SNR*A2
            UA12 = SNR*A3
*
            VB11 = CSL*B1 + SNL*B2
            VB12 = SNL*B3
*
            AUA11 = ABS( CSR )*ABS( A1 ) + ABS( SNR )*ABS( A2 )
            AVB11 = ABS( CSL )*ABS( B1 ) + ABS( SNL )*ABS( B2 )
*
*           zero (1,1) elements of U**T*A and V**T*B, and then swap.
*
            IF( ( ABS( UA11 )+ABS( UA12 ) ).NE.ZERO ) THEN
               IF( AUA11 / ( ABS( UA11 )+ABS( UA12 ) ).LE.AVB11 /
     $             ( ABS( VB11 )+ABS( VB12 ) ) ) THEN
                  CALL DLARTG( UA12, UA11, CSQ, SNQ, R )
               ELSE
                  CALL DLARTG( VB12, VB11, CSQ, SNQ, R )
               END IF
            ELSE
               CALL DLARTG( VB12, VB11, CSQ, SNQ, R )
            END IF
*
            CSU = SNR
            SNU = CSR
            CSV = SNL
            SNV = CSL
*
         END IF
*
      END IF
*
      RETURN
*
*     End of DLAGS2
*
      END
*> \brief \b DLAGTF computes an LU factorization of a matrix T-λI, where T is a general tridiagonal matrix, and λ a scalar, using partial pivoting with row interchanges.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAGTF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlagtf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlagtf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlagtf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAGTF( N, A, LAMBDA, B, C, TOL, D, IN, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, N
*       DOUBLE PRECISION   LAMBDA, TOL
*       ..
*       .. Array Arguments ..
*       INTEGER            IN( * )
*       DOUBLE PRECISION   A( * ), B( * ), C( * ), D( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAGTF factorizes the matrix (T - lambda*I), where T is an n by n
*> tridiagonal matrix and lambda is a scalar, as
*>
*>    T - lambda*I = PLU,
*>
*> where P is a permutation matrix, L is a unit lower tridiagonal matrix
*> with at most one non-zero sub-diagonal elements per column and U is
*> an upper triangular matrix with at most two non-zero super-diagonal
*> elements per column.
*>
*> The factorization is obtained by Gaussian elimination with partial
*> pivoting and implicit row scaling.
*>
*> The parameter LAMBDA is included in the routine so that DLAGTF may
*> be used, in conjunction with DLAGTS, to obtain eigenvectors of T by
*> inverse iteration.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix T.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (N)
*>          On entry, A must contain the diagonal elements of T.
*>
*>          On exit, A is overwritten by the n diagonal elements of the
*>          upper triangular matrix U of the factorization of T.
*> \endverbatim
*>
*> \param[in] LAMBDA
*> \verbatim
*>          LAMBDA is DOUBLE PRECISION
*>          On entry, the scalar lambda.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (N-1)
*>          On entry, B must contain the (n-1) super-diagonal elements of
*>          T.
*>
*>          On exit, B is overwritten by the (n-1) super-diagonal
*>          elements of the matrix U of the factorization of T.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (N-1)
*>          On entry, C must contain the (n-1) sub-diagonal elements of
*>          T.
*>
*>          On exit, C is overwritten by the (n-1) sub-diagonal elements
*>          of the matrix L of the factorization of T.
*> \endverbatim
*>
*> \param[in] TOL
*> \verbatim
*>          TOL is DOUBLE PRECISION
*>          On entry, a relative tolerance used to indicate whether or
*>          not the matrix (T - lambda*I) is nearly singular. TOL should
*>          normally be chose as approximately the largest relative error
*>          in the elements of T. For example, if the elements of T are
*>          correct to about 4 significant figures, then TOL should be
*>          set to about 5*10**(-4). If TOL is supplied as less than eps,
*>          where eps is the relative machine precision, then the value
*>          eps is used in place of TOL.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N-2)
*>          On exit, D is overwritten by the (n-2) second super-diagonal
*>          elements of the matrix U of the factorization of T.
*> \endverbatim
*>
*> \param[out] IN
*> \verbatim
*>          IN is INTEGER array, dimension (N)
*>          On exit, IN contains details of the permutation matrix P. If
*>          an interchange occurred at the kth step of the elimination,
*>          then IN(k) = 1, otherwise IN(k) = 0. The element IN(n)
*>          returns the smallest positive integer j such that
*>
*>             abs( u(j,j) ) <= norm( (T - lambda*I)(j) )*TOL,
*>
*>          where norm( A(j) ) denotes the sum of the absolute values of
*>          the jth row of the matrix A. If no such j exists then IN(n)
*>          is returned as zero. If IN(n) is returned as positive, then a
*>          diagonal element of U is small, indicating that
*>          (T - lambda*I) is singular or nearly singular,
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -k, the kth argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lagtf
*
*  =====================================================================
      SUBROUTINE DLAGTF( N, A, LAMBDA, B, C, TOL, D, IN, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, N
      DOUBLE PRECISION   LAMBDA, TOL
*     ..
*     .. Array Arguments ..
      INTEGER            IN( * )
      DOUBLE PRECISION   A( * ), B( * ), C( * ), D( * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            K
      DOUBLE PRECISION   EPS, MULT, PIV1, PIV2, SCALE1, SCALE2, TEMP, TL
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      IF( N.LT.0 ) THEN
         INFO = -1
         CALL XERBLA( 'DLAGTF', -INFO )
         RETURN
      END IF
*
      IF( N.EQ.0 )
     $   RETURN
*
      A( 1 ) = A( 1 ) - LAMBDA
      IN( N ) = 0
      IF( N.EQ.1 ) THEN
         IF( A( 1 ).EQ.ZERO )
     $      IN( 1 ) = 1
         RETURN
      END IF
*
      EPS = DLAMCH( 'Epsilon' )
*
      TL = MAX( TOL, EPS )
      SCALE1 = ABS( A( 1 ) ) + ABS( B( 1 ) )
      DO 10 K = 1, N - 1
         A( K+1 ) = A( K+1 ) - LAMBDA
         SCALE2 = ABS( C( K ) ) + ABS( A( K+1 ) )
         IF( K.LT.( N-1 ) )
     $      SCALE2 = SCALE2 + ABS( B( K+1 ) )
         IF( A( K ).EQ.ZERO ) THEN
            PIV1 = ZERO
         ELSE
            PIV1 = ABS( A( K ) ) / SCALE1
         END IF
         IF( C( K ).EQ.ZERO ) THEN
            IN( K ) = 0
            PIV2 = ZERO
            SCALE1 = SCALE2
            IF( K.LT.( N-1 ) )
     $         D( K ) = ZERO
         ELSE
            PIV2 = ABS( C( K ) ) / SCALE2
            IF( PIV2.LE.PIV1 ) THEN
               IN( K ) = 0
               SCALE1 = SCALE2
               C( K ) = C( K ) / A( K )
               A( K+1 ) = A( K+1 ) - C( K )*B( K )
               IF( K.LT.( N-1 ) )
     $            D( K ) = ZERO
            ELSE
               IN( K ) = 1
               MULT = A( K ) / C( K )
               A( K ) = C( K )
               TEMP = A( K+1 )
               A( K+1 ) = B( K ) - MULT*TEMP
               IF( K.LT.( N-1 ) ) THEN
                  D( K ) = B( K+1 )
                  B( K+1 ) = -MULT*D( K )
               END IF
               B( K ) = TEMP
               C( K ) = MULT
            END IF
         END IF
         IF( ( MAX( PIV1, PIV2 ).LE.TL ) .AND. ( IN( N ).EQ.0 ) )
     $      IN( N ) = K
   10 CONTINUE
      IF( ( ABS( A( N ) ).LE.SCALE1*TL ) .AND. ( IN( N ).EQ.0 ) )
     $   IN( N ) = N
*
      RETURN
*
*     End of DLAGTF
*
      END
*> \brief \b DLAGTM performs a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal matrix, B and C are rectangular matrices, and α and β are scalars, which may be 0, 1, or -1.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAGTM + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlagtm.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlagtm.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlagtm.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAGTM( TRANS, N, NRHS, ALPHA, DL, D, DU, X, LDX, BETA,
*                          B, LDB )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANS
*       INTEGER            LDB, LDX, N, NRHS
*       DOUBLE PRECISION   ALPHA, BETA
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   B( LDB, * ), D( * ), DL( * ), DU( * ),
*      $                   X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAGTM performs a matrix-matrix product of the form
*>
*>    B := alpha * A * X + beta * B
*>
*> where A is a tridiagonal matrix of order N, B and X are N by NRHS
*> matrices, and alpha and beta are real scalars, each of which may be
*> 0., 1., or -1.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the operation applied to A.
*>          = 'N':  No transpose, B := alpha * A * X + beta * B
*>          = 'T':  Transpose,    B := alpha * A'* X + beta * B
*>          = 'C':  Conjugate transpose = Transpose
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices X and B.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*>          ALPHA is DOUBLE PRECISION
*>          The scalar alpha.  ALPHA must be 0., 1., or -1.; otherwise,
*>          it is assumed to be 0.
*> \endverbatim
*>
*> \param[in] DL
*> \verbatim
*>          DL is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) sub-diagonal elements of T.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The diagonal elements of T.
*> \endverbatim
*>
*> \param[in] DU
*> \verbatim
*>          DU is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) super-diagonal elements of T.
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          The N by NRHS matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(N,1).
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION
*>          The scalar beta.  BETA must be 0., 1., or -1.; otherwise,
*>          it is assumed to be 1.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the N by NRHS matrix B.
*>          On exit, B is overwritten by the matrix expression
*>          B := alpha * A * X + beta * B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(N,1).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lagtm
*
*  =====================================================================
      SUBROUTINE DLAGTM( TRANS, N, NRHS, ALPHA, DL, D, DU, X, LDX,
     $                   BETA,
     $                   B, LDB )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS
      INTEGER            LDB, LDX, N, NRHS
      DOUBLE PRECISION   ALPHA, BETA
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   B( LDB, * ), D( * ), DL( * ), DU( * ),
     $                   X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. Executable Statements ..
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Multiply B by BETA if BETA.NE.1.
*
      IF( BETA.EQ.ZERO ) THEN
         DO 20 J = 1, NRHS
            DO 10 I = 1, N
               B( I, J ) = ZERO
   10       CONTINUE
   20    CONTINUE
      ELSE IF( BETA.EQ.-ONE ) THEN
         DO 40 J = 1, NRHS
            DO 30 I = 1, N
               B( I, J ) = -B( I, J )
   30       CONTINUE
   40    CONTINUE
      END IF
*
      IF( ALPHA.EQ.ONE ) THEN
         IF( LSAME( TRANS, 'N' ) ) THEN
*
*           Compute B := B + A*X
*
            DO 60 J = 1, NRHS
               IF( N.EQ.1 ) THEN
                  B( 1, J ) = B( 1, J ) + D( 1 )*X( 1, J )
               ELSE
                  B( 1, J ) = B( 1, J ) + D( 1 )*X( 1, J ) +
     $                        DU( 1 )*X( 2, J )
                  B( N, J ) = B( N, J ) + DL( N-1 )*X( N-1, J ) +
     $                        D( N )*X( N, J )
                  DO 50 I = 2, N - 1
                     B( I, J ) = B( I, J ) + DL( I-1 )*X( I-1, J ) +
     $                           D( I )*X( I, J ) + DU( I )*X( I+1, J )
   50             CONTINUE
               END IF
   60       CONTINUE
         ELSE
*
*           Compute B := B + A**T*X
*
            DO 80 J = 1, NRHS
               IF( N.EQ.1 ) THEN
                  B( 1, J ) = B( 1, J ) + D( 1 )*X( 1, J )
               ELSE
                  B( 1, J ) = B( 1, J ) + D( 1 )*X( 1, J ) +
     $                        DL( 1 )*X( 2, J )
                  B( N, J ) = B( N, J ) + DU( N-1 )*X( N-1, J ) +
     $                        D( N )*X( N, J )
                  DO 70 I = 2, N - 1
                     B( I, J ) = B( I, J ) + DU( I-1 )*X( I-1, J ) +
     $                           D( I )*X( I, J ) + DL( I )*X( I+1, J )
   70             CONTINUE
               END IF
   80       CONTINUE
         END IF
      ELSE IF( ALPHA.EQ.-ONE ) THEN
         IF( LSAME( TRANS, 'N' ) ) THEN
*
*           Compute B := B - A*X
*
            DO 100 J = 1, NRHS
               IF( N.EQ.1 ) THEN
                  B( 1, J ) = B( 1, J ) - D( 1 )*X( 1, J )
               ELSE
                  B( 1, J ) = B( 1, J ) - D( 1 )*X( 1, J ) -
     $                        DU( 1 )*X( 2, J )
                  B( N, J ) = B( N, J ) - DL( N-1 )*X( N-1, J ) -
     $                        D( N )*X( N, J )
                  DO 90 I = 2, N - 1
                     B( I, J ) = B( I, J ) - DL( I-1 )*X( I-1, J ) -
     $                           D( I )*X( I, J ) - DU( I )*X( I+1, J )
   90             CONTINUE
               END IF
  100       CONTINUE
         ELSE
*
*           Compute B := B - A**T*X
*
            DO 120 J = 1, NRHS
               IF( N.EQ.1 ) THEN
                  B( 1, J ) = B( 1, J ) - D( 1 )*X( 1, J )
               ELSE
                  B( 1, J ) = B( 1, J ) - D( 1 )*X( 1, J ) -
     $                        DL( 1 )*X( 2, J )
                  B( N, J ) = B( N, J ) - DU( N-1 )*X( N-1, J ) -
     $                        D( N )*X( N, J )
                  DO 110 I = 2, N - 1
                     B( I, J ) = B( I, J ) - DU( I-1 )*X( I-1, J ) -
     $                           D( I )*X( I, J ) - DL( I )*X( I+1, J )
  110             CONTINUE
               END IF
  120       CONTINUE
         END IF
      END IF
      RETURN
*
*     End of DLAGTM
*
      END
*> \brief \b DLAGTS solves the system of equations (T-λI)x = y
*> or (T-λI)^Tx = y, where T is a general tridiagonal matrix
*> and λ a scalar, using the LU factorization computed by slagtf.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAGTS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlagts.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlagts.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlagts.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAGTS( JOB, N, A, B, C, D, IN, Y, TOL, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, JOB, N
*       DOUBLE PRECISION   TOL
*       ..
*       .. Array Arguments ..
*       INTEGER            IN( * )
*       DOUBLE PRECISION   A( * ), B( * ), C( * ), D( * ), Y( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAGTS may be used to solve one of the systems of equations
*>
*>    (T - lambda*I)*x = y   or   (T - lambda*I)**T*x = y,
*>
*> where T is an n by n tridiagonal matrix, for x, following the
*> factorization of (T - lambda*I) as
*>
*>    (T - lambda*I) = P*L*U ,
*>
*> by routine DLAGTF. The choice of equation to be solved is
*> controlled by the argument JOB, and in each case there is an option
*> to perturb zero or very small diagonal elements of U, this option
*> being intended for use in applications such as inverse iteration.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOB
*> \verbatim
*>          JOB is INTEGER
*>          Specifies the job to be performed by DLAGTS as follows:
*>          =  1: The equations  (T - lambda*I)x = y  are to be solved,
*>                but diagonal elements of U are not to be perturbed.
*>          = -1: The equations  (T - lambda*I)x = y  are to be solved
*>                and, if overflow would otherwise occur, the diagonal
*>                elements of U are to be perturbed. See argument TOL
*>                below.
*>          =  2: The equations  (T - lambda*I)**Tx = y  are to be solved,
*>                but diagonal elements of U are not to be perturbed.
*>          = -2: The equations  (T - lambda*I)**Tx = y  are to be solved
*>                and, if overflow would otherwise occur, the diagonal
*>                elements of U are to be perturbed. See argument TOL
*>                below.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix T.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (N)
*>          On entry, A must contain the diagonal elements of U as
*>          returned from DLAGTF.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (N-1)
*>          On entry, B must contain the first super-diagonal elements of
*>          U as returned from DLAGTF.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (N-1)
*>          On entry, C must contain the sub-diagonal elements of L as
*>          returned from DLAGTF.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N-2)
*>          On entry, D must contain the second super-diagonal elements
*>          of U as returned from DLAGTF.
*> \endverbatim
*>
*> \param[in] IN
*> \verbatim
*>          IN is INTEGER array, dimension (N)
*>          On entry, IN must contain details of the matrix P as returned
*>          from DLAGTF.
*> \endverbatim
*>
*> \param[in,out] Y
*> \verbatim
*>          Y is DOUBLE PRECISION array, dimension (N)
*>          On entry, the right hand side vector y.
*>          On exit, Y is overwritten by the solution vector x.
*> \endverbatim
*>
*> \param[in,out] TOL
*> \verbatim
*>          TOL is DOUBLE PRECISION
*>          On entry, with  JOB < 0, TOL should be the minimum
*>          perturbation to be made to very small diagonal elements of U.
*>          TOL should normally be chosen as about eps*norm(U), where eps
*>          is the relative machine precision, but if TOL is supplied as
*>          non-positive, then it is reset to eps*max( abs( u(i,j) ) ).
*>          If  JOB > 0  then TOL is not referenced.
*>
*>          On exit, TOL is changed as described above, only if TOL is
*>          non-positive on entry. Otherwise TOL is unchanged.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  overflow would occur when computing the INFO(th)
*>                element of the solution vector x. This can only occur
*>                when JOB is supplied as positive and either means
*>                that a diagonal element of U is very small, or that
*>                the elements of the right-hand side vector y are very
*>                large.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lagts
*
*  =====================================================================
      SUBROUTINE DLAGTS( JOB, N, A, B, C, D, IN, Y, TOL, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, JOB, N
      DOUBLE PRECISION   TOL
*     ..
*     .. Array Arguments ..
      INTEGER            IN( * )
      DOUBLE PRECISION   A( * ), B( * ), C( * ), D( * ), Y( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            K
      DOUBLE PRECISION   ABSAK, AK, BIGNUM, EPS, PERT, SFMIN, TEMP
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SIGN
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      IF( ( ABS( JOB ).GT.2 ) .OR. ( JOB.EQ.0 ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLAGTS', -INFO )
         RETURN
      END IF
*
      IF( N.EQ.0 )
     $   RETURN
*
      EPS = DLAMCH( 'Epsilon' )
      SFMIN = DLAMCH( 'Safe minimum' )
      BIGNUM = ONE / SFMIN
*
      IF( JOB.LT.0 ) THEN
         IF( TOL.LE.ZERO ) THEN
            TOL = ABS( A( 1 ) )
            IF( N.GT.1 )
     $         TOL = MAX( TOL, ABS( A( 2 ) ), ABS( B( 1 ) ) )
            DO 10 K = 3, N
               TOL = MAX( TOL, ABS( A( K ) ), ABS( B( K-1 ) ),
     $               ABS( D( K-2 ) ) )
   10       CONTINUE
            TOL = TOL*EPS
            IF( TOL.EQ.ZERO )
     $         TOL = EPS
         END IF
      END IF
*
      IF( ABS( JOB ).EQ.1 ) THEN
         DO 20 K = 2, N
            IF( IN( K-1 ).EQ.0 ) THEN
               Y( K ) = Y( K ) - C( K-1 )*Y( K-1 )
            ELSE
               TEMP = Y( K-1 )
               Y( K-1 ) = Y( K )
               Y( K ) = TEMP - C( K-1 )*Y( K )
            END IF
   20    CONTINUE
         IF( JOB.EQ.1 ) THEN
            DO 30 K = N, 1, -1
               IF( K.LE.N-2 ) THEN
                  TEMP = Y( K ) - B( K )*Y( K+1 ) - D( K )*Y( K+2 )
               ELSE IF( K.EQ.N-1 ) THEN
                  TEMP = Y( K ) - B( K )*Y( K+1 )
               ELSE
                  TEMP = Y( K )
               END IF
               AK = A( K )
               ABSAK = ABS( AK )
               IF( ABSAK.LT.ONE ) THEN
                  IF( ABSAK.LT.SFMIN ) THEN
                     IF( ABSAK.EQ.ZERO .OR. ABS( TEMP )*SFMIN.GT.ABSAK )
     $                    THEN
                        INFO = K
                        RETURN
                     ELSE
                        TEMP = TEMP*BIGNUM
                        AK = AK*BIGNUM
                     END IF
                  ELSE IF( ABS( TEMP ).GT.ABSAK*BIGNUM ) THEN
                     INFO = K
                     RETURN
                  END IF
               END IF
               Y( K ) = TEMP / AK
   30       CONTINUE
         ELSE
            DO 50 K = N, 1, -1
               IF( K.LE.N-2 ) THEN
                  TEMP = Y( K ) - B( K )*Y( K+1 ) - D( K )*Y( K+2 )
               ELSE IF( K.EQ.N-1 ) THEN
                  TEMP = Y( K ) - B( K )*Y( K+1 )
               ELSE
                  TEMP = Y( K )
               END IF
               AK = A( K )
               PERT = SIGN( TOL, AK )
   40          CONTINUE
               ABSAK = ABS( AK )
               IF( ABSAK.LT.ONE ) THEN
                  IF( ABSAK.LT.SFMIN ) THEN
                     IF( ABSAK.EQ.ZERO .OR. ABS( TEMP )*SFMIN.GT.ABSAK )
     $                    THEN
                        AK = AK + PERT
                        PERT = 2*PERT
                        GO TO 40
                     ELSE
                        TEMP = TEMP*BIGNUM
                        AK = AK*BIGNUM
                     END IF
                  ELSE IF( ABS( TEMP ).GT.ABSAK*BIGNUM ) THEN
                     AK = AK + PERT
                     PERT = 2*PERT
                     GO TO 40
                  END IF
               END IF
               Y( K ) = TEMP / AK
   50       CONTINUE
         END IF
      ELSE
*
*        Come to here if  JOB = 2 or -2
*
         IF( JOB.EQ.2 ) THEN
            DO 60 K = 1, N
               IF( K.GE.3 ) THEN
                  TEMP = Y( K ) - B( K-1 )*Y( K-1 ) - D( K-2 )*Y( K-2 )
               ELSE IF( K.EQ.2 ) THEN
                  TEMP = Y( K ) - B( K-1 )*Y( K-1 )
               ELSE
                  TEMP = Y( K )
               END IF
               AK = A( K )
               ABSAK = ABS( AK )
               IF( ABSAK.LT.ONE ) THEN
                  IF( ABSAK.LT.SFMIN ) THEN
                     IF( ABSAK.EQ.ZERO .OR. ABS( TEMP )*SFMIN.GT.ABSAK )
     $                    THEN
                        INFO = K
                        RETURN
                     ELSE
                        TEMP = TEMP*BIGNUM
                        AK = AK*BIGNUM
                     END IF
                  ELSE IF( ABS( TEMP ).GT.ABSAK*BIGNUM ) THEN
                     INFO = K
                     RETURN
                  END IF
               END IF
               Y( K ) = TEMP / AK
   60       CONTINUE
         ELSE
            DO 80 K = 1, N
               IF( K.GE.3 ) THEN
                  TEMP = Y( K ) - B( K-1 )*Y( K-1 ) - D( K-2 )*Y( K-2 )
               ELSE IF( K.EQ.2 ) THEN
                  TEMP = Y( K ) - B( K-1 )*Y( K-1 )
               ELSE
                  TEMP = Y( K )
               END IF
               AK = A( K )
               PERT = SIGN( TOL, AK )
   70          CONTINUE
               ABSAK = ABS( AK )
               IF( ABSAK.LT.ONE ) THEN
                  IF( ABSAK.LT.SFMIN ) THEN
                     IF( ABSAK.EQ.ZERO .OR. ABS( TEMP )*SFMIN.GT.ABSAK )
     $                    THEN
                        AK = AK + PERT
                        PERT = 2*PERT
                        GO TO 70
                     ELSE
                        TEMP = TEMP*BIGNUM
                        AK = AK*BIGNUM
                     END IF
                  ELSE IF( ABS( TEMP ).GT.ABSAK*BIGNUM ) THEN
                     AK = AK + PERT
                     PERT = 2*PERT
                     GO TO 70
                  END IF
               END IF
               Y( K ) = TEMP / AK
   80       CONTINUE
         END IF
*
         DO 90 K = N, 2, -1
            IF( IN( K-1 ).EQ.0 ) THEN
               Y( K-1 ) = Y( K-1 ) - C( K-1 )*Y( K )
            ELSE
               TEMP = Y( K-1 )
               Y( K-1 ) = Y( K )
               Y( K ) = TEMP - C( K-1 )*Y( K )
            END IF
   90    CONTINUE
      END IF
*
*     End of DLAGTS
*
      END
*> \brief \b DLAGV2 computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAGV2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlagv2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlagv2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlagv2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAGV2( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL,
*                          CSR, SNR )
*
*       .. Scalar Arguments ..
*       INTEGER            LDA, LDB
*       DOUBLE PRECISION   CSL, CSR, SNL, SNR
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), ALPHAI( 2 ), ALPHAR( 2 ),
*      $                   B( LDB, * ), BETA( 2 )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAGV2 computes the Generalized Schur factorization of a real 2-by-2
*> matrix pencil (A,B) where B is upper triangular. This routine
*> computes orthogonal (rotation) matrices given by CSL, SNL and CSR,
*> SNR such that
*>
*> 1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0
*>    types), then
*>
*>    [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
*>    [  0  a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]
*>
*>    [ b11 b12 ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
*>    [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ],
*>
*> 2) if the pencil (A,B) has a pair of complex conjugate eigenvalues,
*>    then
*>
*>    [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
*>    [ a21 a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]
*>
*>    [ b11  0  ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
*>    [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ]
*>
*>    where b11 >= b22 > 0.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, 2)
*>          On entry, the 2 x 2 matrix A.
*>          On exit, A is overwritten by the ``A-part'' of the
*>          generalized Schur form.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          THe leading dimension of the array A.  LDA >= 2.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB, 2)
*>          On entry, the upper triangular 2 x 2 matrix B.
*>          On exit, B is overwritten by the ``B-part'' of the
*>          generalized Schur form.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          THe leading dimension of the array B.  LDB >= 2.
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*>          ALPHAR is DOUBLE PRECISION array, dimension (2)
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*>          ALPHAI is DOUBLE PRECISION array, dimension (2)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION array, dimension (2)
*>          (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the
*>          pencil (A,B), k=1,2, i = sqrt(-1).  Note that BETA(k) may
*>          be zero.
*> \endverbatim
*>
*> \param[out] CSL
*> \verbatim
*>          CSL is DOUBLE PRECISION
*>          The cosine of the left rotation matrix.
*> \endverbatim
*>
*> \param[out] SNL
*> \verbatim
*>          SNL is DOUBLE PRECISION
*>          The sine of the left rotation matrix.
*> \endverbatim
*>
*> \param[out] CSR
*> \verbatim
*>          CSR is DOUBLE PRECISION
*>          The cosine of the right rotation matrix.
*> \endverbatim
*>
*> \param[out] SNR
*> \verbatim
*>          SNR is DOUBLE PRECISION
*>          The sine of the right rotation matrix.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lagv2
*
*> \par Contributors:
*  ==================
*>
*>     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
*  =====================================================================
      SUBROUTINE DLAGV2( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL,
     $                   SNL,
     $                   CSR, SNR )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            LDA, LDB
      DOUBLE PRECISION   CSL, CSR, SNL, SNR
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), ALPHAI( 2 ), ALPHAR( 2 ),
     $                   B( LDB, * ), BETA( 2 )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   ANORM, ASCALE, BNORM, BSCALE, H1, H2, H3, QQ,
     $                   R, RR, SAFMIN, SCALE1, SCALE2, T, ULP, WI, WR1,
     $                   WR2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLAG2, DLARTG, DLASV2, DROT
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH, DLAPY2
      EXTERNAL           DLAMCH, DLAPY2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. Executable Statements ..
*
      SAFMIN = DLAMCH( 'S' )
      ULP = DLAMCH( 'P' )
*
*     Scale A
*
      ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ),
     $        ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN )
      ASCALE = ONE / ANORM
      A( 1, 1 ) = ASCALE*A( 1, 1 )
      A( 1, 2 ) = ASCALE*A( 1, 2 )
      A( 2, 1 ) = ASCALE*A( 2, 1 )
      A( 2, 2 ) = ASCALE*A( 2, 2 )
*
*     Scale B
*
      BNORM = MAX( ABS( B( 1, 1 ) ), ABS( B( 1, 2 ) )+ABS( B( 2, 2 ) ),
     $        SAFMIN )
      BSCALE = ONE / BNORM
      B( 1, 1 ) = BSCALE*B( 1, 1 )
      B( 1, 2 ) = BSCALE*B( 1, 2 )
      B( 2, 2 ) = BSCALE*B( 2, 2 )
*
*     Check if A can be deflated
*
      IF( ABS( A( 2, 1 ) ).LE.ULP ) THEN
         CSL = ONE
         SNL = ZERO
         CSR = ONE
         SNR = ZERO
         A( 2, 1 ) = ZERO
         B( 2, 1 ) = ZERO
         WI = ZERO
*
*     Check if B is singular
*
      ELSE IF( ABS( B( 1, 1 ) ).LE.ULP ) THEN
         CALL DLARTG( A( 1, 1 ), A( 2, 1 ), CSL, SNL, R )
         CSR = ONE
         SNR = ZERO
         CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL )
         CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL )
         A( 2, 1 ) = ZERO
         B( 1, 1 ) = ZERO
         B( 2, 1 ) = ZERO
         WI = ZERO
*
      ELSE IF( ABS( B( 2, 2 ) ).LE.ULP ) THEN
         CALL DLARTG( A( 2, 2 ), A( 2, 1 ), CSR, SNR, T )
         SNR = -SNR
         CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR )
         CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR )
         CSL = ONE
         SNL = ZERO
         A( 2, 1 ) = ZERO
         B( 2, 1 ) = ZERO
         B( 2, 2 ) = ZERO
         WI = ZERO
*
      ELSE
*
*        B is nonsingular, first compute the eigenvalues of (A,B)
*
         CALL DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
     $               WR2,
     $               WI )
*
         IF( WI.EQ.ZERO ) THEN
*
*           two real eigenvalues, compute s*A-w*B
*
            H1 = SCALE1*A( 1, 1 ) - WR1*B( 1, 1 )
            H2 = SCALE1*A( 1, 2 ) - WR1*B( 1, 2 )
            H3 = SCALE1*A( 2, 2 ) - WR1*B( 2, 2 )
*
            RR = DLAPY2( H1, H2 )
            QQ = DLAPY2( SCALE1*A( 2, 1 ), H3 )
*
            IF( RR.GT.QQ ) THEN
*
*              find right rotation matrix to zero 1,1 element of
*              (sA - wB)
*
               CALL DLARTG( H2, H1, CSR, SNR, T )
*
            ELSE
*
*              find right rotation matrix to zero 2,1 element of
*              (sA - wB)
*
               CALL DLARTG( H3, SCALE1*A( 2, 1 ), CSR, SNR, T )
*
            END IF
*
            SNR = -SNR
            CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR )
            CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR )
*
*           compute inf norms of A and B
*
            H1 = MAX( ABS( A( 1, 1 ) )+ABS( A( 1, 2 ) ),
     $           ABS( A( 2, 1 ) )+ABS( A( 2, 2 ) ) )
            H2 = MAX( ABS( B( 1, 1 ) )+ABS( B( 1, 2 ) ),
     $           ABS( B( 2, 1 ) )+ABS( B( 2, 2 ) ) )
*
            IF( ( SCALE1*H1 ).GE.ABS( WR1 )*H2 ) THEN
*
*              find left rotation matrix Q to zero out B(2,1)
*
               CALL DLARTG( B( 1, 1 ), B( 2, 1 ), CSL, SNL, R )
*
            ELSE
*
*              find left rotation matrix Q to zero out A(2,1)
*
               CALL DLARTG( A( 1, 1 ), A( 2, 1 ), CSL, SNL, R )
*
            END IF
*
            CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL )
            CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL )
*
            A( 2, 1 ) = ZERO
            B( 2, 1 ) = ZERO
*
         ELSE
*
*           a pair of complex conjugate eigenvalues
*           first compute the SVD of the matrix B
*
            CALL DLASV2( B( 1, 1 ), B( 1, 2 ), B( 2, 2 ), R, T, SNR,
     $                   CSR, SNL, CSL )
*
*           Form (A,B) := Q(A,B)Z**T where Q is left rotation matrix and
*           Z is right rotation matrix computed from DLASV2
*
            CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL )
            CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL )
            CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR )
            CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR )
*
            B( 2, 1 ) = ZERO
            B( 1, 2 ) = ZERO
*
         END IF
*
      END IF
*
*     Unscaling
*
      A( 1, 1 ) = ANORM*A( 1, 1 )
      A( 2, 1 ) = ANORM*A( 2, 1 )
      A( 1, 2 ) = ANORM*A( 1, 2 )
      A( 2, 2 ) = ANORM*A( 2, 2 )
      B( 1, 1 ) = BNORM*B( 1, 1 )
      B( 2, 1 ) = BNORM*B( 2, 1 )
      B( 1, 2 ) = BNORM*B( 1, 2 )
      B( 2, 2 ) = BNORM*B( 2, 2 )
*
      IF( WI.EQ.ZERO ) THEN
         ALPHAR( 1 ) = A( 1, 1 )
         ALPHAR( 2 ) = A( 2, 2 )
         ALPHAI( 1 ) = ZERO
         ALPHAI( 2 ) = ZERO
         BETA( 1 ) = B( 1, 1 )
         BETA( 2 ) = B( 2, 2 )
      ELSE
         ALPHAR( 1 ) = ANORM*WR1 / SCALE1 / BNORM
         ALPHAI( 1 ) = ANORM*WI / SCALE1 / BNORM
         ALPHAR( 2 ) = ALPHAR( 1 )
         ALPHAI( 2 ) = -ALPHAI( 1 )
         BETA( 1 ) = ONE
         BETA( 2 ) = ONE
      END IF
*
      RETURN
*
*     End of DLAGV2
*
      END
*> \brief \b DLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAHQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlahqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlahqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlahqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
*                          ILOZ, IHIZ, Z, LDZ, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N
*       LOGICAL            WANTT, WANTZ
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>    DLAHQR is an auxiliary routine called by DHSEQR to update the
*>    eigenvalues and Schur decomposition already computed by DHSEQR, by
*>    dealing with the Hessenberg submatrix in rows and columns ILO to
*>    IHI.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] WANTT
*> \verbatim
*>          WANTT is LOGICAL
*>          = .TRUE. : the full Schur form T is required;
*>          = .FALSE.: only eigenvalues are required.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*>          WANTZ is LOGICAL
*>          = .TRUE. : the matrix of Schur vectors Z is required;
*>          = .FALSE.: Schur vectors are not required.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix H.  N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*>          ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*>          IHI is INTEGER
*>          It is assumed that H is already upper quasi-triangular in
*>          rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
*>          ILO = 1). DLAHQR works primarily with the Hessenberg
*>          submatrix in rows and columns ILO to IHI, but applies
*>          transformations to all of H if WANTT is .TRUE..
*>          1 <= ILO <= max(1,IHI); IHI <= N.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*>          H is DOUBLE PRECISION array, dimension (LDH,N)
*>          On entry, the upper Hessenberg matrix H.
*>          On exit, if INFO is zero and if WANTT is .TRUE., H is upper
*>          quasi-triangular in rows and columns ILO:IHI, with any
*>          2-by-2 diagonal blocks in standard form. If INFO is zero
*>          and WANTT is .FALSE., the contents of H are unspecified on
*>          exit.  The output state of H if INFO is nonzero is given
*>          below under the description of INFO.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*>          LDH is INTEGER
*>          The leading dimension of the array H. LDH >= max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*>          WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*>          WI is DOUBLE PRECISION array, dimension (N)
*>          The real and imaginary parts, respectively, of the computed
*>          eigenvalues ILO to IHI are stored in the corresponding
*>          elements of WR and WI. If two eigenvalues are computed as a
*>          complex conjugate pair, they are stored in consecutive
*>          elements of WR and WI, say the i-th and (i+1)th, with
*>          WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the
*>          eigenvalues are stored in the same order as on the diagonal
*>          of the Schur form returned in H, with WR(i) = H(i,i), and, if
*>          H(i:i+1,i:i+1) is a 2-by-2 diagonal block,
*>          WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*>          ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*>          IHIZ is INTEGER
*>          Specify the rows of Z to which transformations must be
*>          applied if WANTZ is .TRUE..
*>          1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ,N)
*>          If WANTZ is .TRUE., on entry Z must contain the current
*>          matrix Z of transformations accumulated by DHSEQR, and on
*>          exit Z has been updated; transformations are applied only to
*>          the submatrix Z(ILOZ:IHIZ,ILO:IHI).
*>          If WANTZ is .FALSE., Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z. LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>           = 0:  successful exit
*>           > 0:  If INFO = i, DLAHQR failed to compute all the
*>                  eigenvalues ILO to IHI in a total of 30 iterations
*>                  per eigenvalue; elements i+1:ihi of WR and WI
*>                  contain those eigenvalues which have been
*>                  successfully computed.
*>
*>                  If INFO > 0 and WANTT is .FALSE., then on exit,
*>                  the remaining unconverged eigenvalues are the
*>                  eigenvalues of the upper Hessenberg matrix rows
*>                  and columns ILO through INFO of the final, output
*>                  value of H.
*>
*>                  If INFO > 0 and WANTT is .TRUE., then on exit
*>          (*)       (initial value of H)*U  = U*(final value of H)
*>                  where U is an orthogonal matrix.    The final
*>                  value of H is upper Hessenberg and triangular in
*>                  rows and columns INFO+1 through IHI.
*>
*>                  If INFO > 0 and WANTZ is .TRUE., then on exit
*>                      (final value of Z)  = (initial value of Z)*U
*>                  where U is the orthogonal matrix in (*)
*>                  (regardless of the value of WANTT.)
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lahqr
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>     02-96 Based on modifications by
*>     David Day, Sandia National Laboratory, USA
*>
*>     12-04 Further modifications by
*>     Ralph Byers, University of Kansas, USA
*>     This is a modified version of DLAHQR from LAPACK version 3.0.
*>     It is (1) more robust against overflow and underflow and
*>     (2) adopts the more conservative Ahues & Tisseur stopping
*>     criterion (LAWN 122, 1997).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
     $                   ILOZ, IHIZ, Z, LDZ, INFO )
      IMPLICIT NONE
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N
      LOGICAL            WANTT, WANTZ
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * )
*     ..
*
*  =========================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO
      PARAMETER          ( ZERO = 0.0d0, ONE = 1.0d0, TWO = 2.0d0 )
      DOUBLE PRECISION   DAT1, DAT2
      PARAMETER          ( DAT1 = 3.0d0 / 4.0d0, DAT2 = -0.4375d0 )
      INTEGER            KEXSH
      PARAMETER          ( KEXSH = 10 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   AA, AB, BA, BB, CS, DET, H11, H12, H21, H21S,
     $                   H22, RT1I, RT1R, RT2I, RT2R, RTDISC, S, SAFMAX,
     $                   SAFMIN, SMLNUM, SN, SUM, T1, T2, T3, TR, TST,
     $                   ULP, V2, V3
      INTEGER            I, I1, I2, ITS, ITMAX, J, K, L, M, NH, NR, NZ,
     $                   KDEFL 
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   V( 3 )
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLANV2, DLARFG, DROT
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
      INFO = 0
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
      IF( ILO.EQ.IHI ) THEN
         WR( ILO ) = H( ILO, ILO )
         WI( ILO ) = ZERO
         RETURN
      END IF
*
*     ==== clear out the trash ====
      DO 10 J = ILO, IHI - 3
         H( J+2, J ) = ZERO
         H( J+3, J ) = ZERO
   10 CONTINUE
      IF( ILO.LE.IHI-2 )
     $   H( IHI, IHI-2 ) = ZERO
*
      NH = IHI - ILO + 1
      NZ = IHIZ - ILOZ + 1
*
*     Set machine-dependent constants for the stopping criterion.
*
      SAFMIN = DLAMCH( 'SAFE MINIMUM' )
      SAFMAX = ONE / SAFMIN
      ULP = DLAMCH( 'PRECISION' )
      SMLNUM = SAFMIN*( DBLE( NH ) / ULP )
*
*     I1 and I2 are the indices of the first row and last column of H
*     to which transformations must be applied. If eigenvalues only are
*     being computed, I1 and I2 are set inside the main loop.
*
      IF( WANTT ) THEN
         I1 = 1
         I2 = N
      END IF
*
*     ITMAX is the total number of QR iterations allowed.
*
      ITMAX = 30 * MAX( 10, NH )
*
*     KDEFL counts the number of iterations since a deflation
*
      KDEFL = 0
*
*     The main loop begins here. I is the loop index and decreases from
*     IHI to ILO in steps of 1 or 2. Each iteration of the loop works
*     with the active submatrix in rows and columns L to I.
*     Eigenvalues I+1 to IHI have already converged. Either L = ILO or
*     H(L,L-1) is negligible so that the matrix splits.
*
      I = IHI
   20 CONTINUE
      L = ILO
      IF( I.LT.ILO )
     $   GO TO 160
*
*     Perform QR iterations on rows and columns ILO to I until a
*     submatrix of order 1 or 2 splits off at the bottom because a
*     subdiagonal element has become negligible.
*
      DO 140 ITS = 0, ITMAX
*
*        Look for a single small subdiagonal element.
*
         DO 30 K = I, L + 1, -1
            IF( ABS( H( K, K-1 ) ).LE.SMLNUM )
     $         GO TO 40
            TST = ABS( H( K-1, K-1 ) ) + ABS( H( K, K ) )
            IF( TST.EQ.ZERO ) THEN
               IF( K-2.GE.ILO )
     $            TST = TST + ABS( H( K-1, K-2 ) )
               IF( K+1.LE.IHI )
     $            TST = TST + ABS( H( K+1, K ) )
            END IF
*           ==== The following is a conservative small subdiagonal
*           .    deflation  criterion due to Ahues & Tisseur (LAWN 122,
*           .    1997). It has better mathematical foundation and
*           .    improves accuracy in some cases.  ====
            IF( ABS( H( K, K-1 ) ).LE.ULP*TST ) THEN
               AB = MAX( ABS( H( K, K-1 ) ), ABS( H( K-1, K ) ) )
               BA = MIN( ABS( H( K, K-1 ) ), ABS( H( K-1, K ) ) )
               AA = MAX( ABS( H( K, K ) ),
     $              ABS( H( K-1, K-1 )-H( K, K ) ) )
               BB = MIN( ABS( H( K, K ) ),
     $              ABS( H( K-1, K-1 )-H( K, K ) ) )
               S = AA + AB
               IF( BA*( AB / S ).LE.MAX( SMLNUM,
     $             ULP*( BB*( AA / S ) ) ) )GO TO 40
            END IF
   30    CONTINUE
   40    CONTINUE
         L = K
         IF( L.GT.ILO ) THEN
*
*           H(L,L-1) is negligible
*
            H( L, L-1 ) = ZERO
         END IF
*
*        Exit from loop if a submatrix of order 1 or 2 has split off.
*
         IF( L.GE.I-1 )
     $      GO TO 150
         KDEFL = KDEFL + 1
*
*        Now the active submatrix is in rows and columns L to I. If
*        eigenvalues only are being computed, only the active submatrix
*        need be transformed.
*
         IF( .NOT.WANTT ) THEN
            I1 = L
            I2 = I
         END IF
*
         IF( MOD(KDEFL,2*KEXSH).EQ.0 ) THEN
*
*           Exceptional shift.
*
            S = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
            H11 = DAT1*S + H( I, I )
            H12 = DAT2*S
            H21 = S
            H22 = H11
         ELSE IF( MOD(KDEFL,KEXSH).EQ.0 ) THEN
*
*           Exceptional shift.
*
            S = ABS( H( L+1, L ) ) + ABS( H( L+2, L+1 ) )
            H11 = DAT1*S + H( L, L )
            H12 = DAT2*S
            H21 = S
            H22 = H11
         ELSE
*
*           Prepare to use Francis' double shift
*           (i.e. 2nd degree generalized Rayleigh quotient)
*
            H11 = H( I-1, I-1 )
            H21 = H( I, I-1 )
            H12 = H( I-1, I )
            H22 = H( I, I )
         END IF
         S = ABS( H11 ) + ABS( H12 ) + ABS( H21 ) + ABS( H22 )
         IF( S.EQ.ZERO ) THEN
            RT1R = ZERO
            RT1I = ZERO
            RT2R = ZERO
            RT2I = ZERO
         ELSE
            H11 = H11 / S
            H21 = H21 / S
            H12 = H12 / S
            H22 = H22 / S
            TR = ( H11+H22 ) / TWO
            DET = ( H11-TR )*( H22-TR ) - H12*H21
            RTDISC = SQRT( ABS( DET ) )
            IF( DET.GE.ZERO ) THEN
*
*              ==== complex conjugate shifts ====
*
               RT1R = TR*S
               RT2R = RT1R
               RT1I = RTDISC*S
               RT2I = -RT1I
            ELSE
*
*              ==== real shifts (use only one of them)  ====
*
               RT1R = TR + RTDISC
               RT2R = TR - RTDISC
               IF( ABS( RT1R-H22 ).LE.ABS( RT2R-H22 ) ) THEN
                  RT1R = RT1R*S
                  RT2R = RT1R
               ELSE
                  RT2R = RT2R*S
                  RT1R = RT2R
               END IF
               RT1I = ZERO
               RT2I = ZERO
            END IF
         END IF
*
*        Look for two consecutive small subdiagonal elements.
*
         DO 50 M = I - 2, L, -1
*           Determine the effect of starting the double-shift QR
*           iteration at row M, and see if this would make H(M,M-1)
*           negligible.  (The following uses scaling to avoid
*           overflows and most underflows.)
*
            H21S = H( M+1, M )
            S = ABS( H( M, M )-RT2R ) + ABS( RT2I ) + ABS( H21S )
            H21S = H( M+1, M ) / S
            V( 1 ) = H21S*H( M, M+1 ) + ( H( M, M )-RT1R )*
     $               ( ( H( M, M )-RT2R ) / S ) - RT1I*( RT2I / S )
            V( 2 ) = H21S*( H( M, M )+H( M+1, M+1 )-RT1R-RT2R )
            V( 3 ) = H21S*H( M+2, M+1 )
            S = ABS( V( 1 ) ) + ABS( V( 2 ) ) + ABS( V( 3 ) )
            V( 1 ) = V( 1 ) / S
            V( 2 ) = V( 2 ) / S
            V( 3 ) = V( 3 ) / S
            IF( M.EQ.L )
     $         GO TO 60
            IF( ABS( H( M, M-1 ) )*( ABS( V( 2 ) )+ABS( V( 3 ) ) ).LE.
     $          ULP*ABS( V( 1 ) )*( ABS( H( M-1, M-1 ) )+ABS( H( M,
     $          M ) )+ABS( H( M+1, M+1 ) ) ) )GO TO 60
   50    CONTINUE
   60    CONTINUE
*
*        Double-shift QR step
*
         DO 130 K = M, I - 1
*
*           The first iteration of this loop determines a reflection G
*           from the vector V and applies it from left and right to H,
*           thus creating a nonzero bulge below the subdiagonal.
*
*           Each subsequent iteration determines a reflection G to
*           restore the Hessenberg form in the (K-1)th column, and thus
*           chases the bulge one step toward the bottom of the active
*           submatrix. NR is the order of G.
*
            NR = MIN( 3, I-K+1 )
            IF( K.GT.M )
     $         CALL DCOPY( NR, H( K, K-1 ), 1, V, 1 )
            CALL DLARFG( NR, V( 1 ), V( 2 ), 1, T1 )
            IF( K.GT.M ) THEN
               H( K, K-1 ) = V( 1 )
               H( K+1, K-1 ) = ZERO
               IF( K.LT.I-1 )
     $            H( K+2, K-1 ) = ZERO
            ELSE IF( M.GT.L ) THEN
*               ==== Use the following instead of
*               .    H( K, K-1 ) = -H( K, K-1 ) to
*               .    avoid a bug when v(2) and v(3)
*               .    underflow. ====
               H( K, K-1 ) = H( K, K-1 )*( ONE-T1 )
            END IF
            V2 = V( 2 )
            T2 = T1*V2
            IF( NR.EQ.3 ) THEN
               V3 = V( 3 )
               T3 = T1*V3
*
*              Apply G from the left to transform the rows of the matrix
*              in columns K to I2.
*
               DO 70 J = K, I2
                  SUM = H( K, J ) + V2*H( K+1, J ) + V3*H( K+2, J )
                  H( K, J ) = H( K, J ) - SUM*T1
                  H( K+1, J ) = H( K+1, J ) - SUM*T2
                  H( K+2, J ) = H( K+2, J ) - SUM*T3
   70          CONTINUE
*
*              Apply G from the right to transform the columns of the
*              matrix in rows I1 to min(K+3,I).
*
               DO 80 J = I1, MIN( K+3, I )
                  SUM = H( J, K ) + V2*H( J, K+1 ) + V3*H( J, K+2 )
                  H( J, K ) = H( J, K ) - SUM*T1
                  H( J, K+1 ) = H( J, K+1 ) - SUM*T2
                  H( J, K+2 ) = H( J, K+2 ) - SUM*T3
   80          CONTINUE
*
               IF( WANTZ ) THEN
*
*                 Accumulate transformations in the matrix Z
*
                  DO 90 J = ILOZ, IHIZ
                     SUM = Z( J, K ) + V2*Z( J, K+1 ) + V3*Z( J, K+2 )
                     Z( J, K ) = Z( J, K ) - SUM*T1
                     Z( J, K+1 ) = Z( J, K+1 ) - SUM*T2
                     Z( J, K+2 ) = Z( J, K+2 ) - SUM*T3
   90             CONTINUE
               END IF
            ELSE IF( NR.EQ.2 ) THEN
*
*              Apply G from the left to transform the rows of the matrix
*              in columns K to I2.
*
               DO 100 J = K, I2
                  SUM = H( K, J ) + V2*H( K+1, J )
                  H( K, J ) = H( K, J ) - SUM*T1
                  H( K+1, J ) = H( K+1, J ) - SUM*T2
  100          CONTINUE
*
*              Apply G from the right to transform the columns of the
*              matrix in rows I1 to min(K+3,I).
*
               DO 110 J = I1, I
                  SUM = H( J, K ) + V2*H( J, K+1 )
                  H( J, K ) = H( J, K ) - SUM*T1
                  H( J, K+1 ) = H( J, K+1 ) - SUM*T2
  110          CONTINUE
*
               IF( WANTZ ) THEN
*
*                 Accumulate transformations in the matrix Z
*
                  DO 120 J = ILOZ, IHIZ
                     SUM = Z( J, K ) + V2*Z( J, K+1 )
                     Z( J, K ) = Z( J, K ) - SUM*T1
                     Z( J, K+1 ) = Z( J, K+1 ) - SUM*T2
  120             CONTINUE
               END IF
            END IF
  130    CONTINUE
*
  140 CONTINUE
*
*     Failure to converge in remaining number of iterations
*
      INFO = I
      RETURN
*
  150 CONTINUE
*
      IF( L.EQ.I ) THEN
*
*        H(I,I-1) is negligible: one eigenvalue has converged.
*
         WR( I ) = H( I, I )
         WI( I ) = ZERO
      ELSE IF( L.EQ.I-1 ) THEN
*
*        H(I-1,I-2) is negligible: a pair of eigenvalues have converged.
*
*        Transform the 2-by-2 submatrix to standard Schur form,
*        and compute and store the eigenvalues.
*
         CALL DLANV2( H( I-1, I-1 ), H( I-1, I ), H( I, I-1 ),
     $                H( I, I ), WR( I-1 ), WI( I-1 ), WR( I ), WI( I ),
     $                CS, SN )
*
         IF( WANTT ) THEN
*
*           Apply the transformation to the rest of H.
*
            IF( I2.GT.I )
     $         CALL DROT( I2-I, H( I-1, I+1 ), LDH, H( I, I+1 ), LDH,
     $                    CS, SN )
            CALL DROT( I-I1-1, H( I1, I-1 ), 1, H( I1, I ), 1, CS,
     $                 SN )
         END IF
         IF( WANTZ ) THEN
*
*           Apply the transformation to Z.
*
            CALL DROT( NZ, Z( ILOZ, I-1 ), 1, Z( ILOZ, I ), 1, CS,
     $                 SN )
         END IF
      END IF
*     reset deflation counter
      KDEFL = 0
*
*     return to start of the main loop with new value of I.
*
      I = L - 1
      GO TO 20
*
  160 CONTINUE
      RETURN
*
*     End of DLAHQR
*
      END
*> \brief \b DLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAHR2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlahr2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlahr2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlahr2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
*
*       .. Scalar Arguments ..
*       INTEGER            K, LDA, LDT, LDY, N, NB
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION  A( LDA, * ), T( LDT, NB ), TAU( NB ),
*      $                   Y( LDY, NB )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1)
*> matrix A so that elements below the k-th subdiagonal are zero. The
*> reduction is performed by an orthogonal similarity transformation
*> Q**T * A * Q. The routine returns the matrices V and T which determine
*> Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.
*>
*> This is an auxiliary routine called by DGEHRD.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The offset for the reduction. Elements below the k-th
*>          subdiagonal in the first NB columns are reduced to zero.
*>          K < N.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*>          NB is INTEGER
*>          The number of columns to be reduced.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N-K+1)
*>          On entry, the n-by-(n-k+1) general matrix A.
*>          On exit, the elements on and above the k-th subdiagonal in
*>          the first NB columns are overwritten with the corresponding
*>          elements of the reduced matrix; the elements below the k-th
*>          subdiagonal, with the array TAU, represent the matrix Q as a
*>          product of elementary reflectors. The other columns of A are
*>          unchanged. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (NB)
*>          The scalar factors of the elementary reflectors. See Further
*>          Details.
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,NB)
*>          The upper triangular matrix T.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T.  LDT >= NB.
*> \endverbatim
*>
*> \param[out] Y
*> \verbatim
*>          Y is DOUBLE PRECISION array, dimension (LDY,NB)
*>          The n-by-nb matrix Y.
*> \endverbatim
*>
*> \param[in] LDY
*> \verbatim
*>          LDY is INTEGER
*>          The leading dimension of the array Y. LDY >= N.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lahr2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix Q is represented as a product of nb elementary reflectors
*>
*>     Q = H(1) H(2) . . . H(nb).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
*>  A(i+k+1:n,i), and tau in TAU(i).
*>
*>  The elements of the vectors v together form the (n-k+1)-by-nb matrix
*>  V which is needed, with T and Y, to apply the transformation to the
*>  unreduced part of the matrix, using an update of the form:
*>  A := (I - V*T*V**T) * (A - Y*V**T).
*>
*>  The contents of A on exit are illustrated by the following example
*>  with n = 7, k = 3 and nb = 2:
*>
*>     ( a   a   a   a   a )
*>     ( a   a   a   a   a )
*>     ( a   a   a   a   a )
*>     ( h   h   a   a   a )
*>     ( v1  h   a   a   a )
*>     ( v1  v2  a   a   a )
*>     ( v1  v2  a   a   a )
*>
*>  where a denotes an element of the original matrix A, h denotes a
*>  modified element of the upper Hessenberg matrix H, and vi denotes an
*>  element of the vector defining H(i).
*>
*>  This subroutine is a slight modification of LAPACK-3.0's DLAHRD
*>  incorporating improvements proposed by Quintana-Orti and Van de
*>  Gejin. Note that the entries of A(1:K,2:NB) differ from those
*>  returned by the original LAPACK-3.0's DLAHRD routine. (This
*>  subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)
*> \endverbatim
*
*> \par References:
*  ================
*>
*>  Gregorio Quintana-Orti and Robert van de Geijn, "Improving the
*>  performance of reduction to Hessenberg form," ACM Transactions on
*>  Mathematical Software, 32(2):180-194, June 2006.
*>
*  =====================================================================
      SUBROUTINE DLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            K, LDA, LDT, LDY, N, NB
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION  A( LDA, * ), T( LDT, NB ), TAU( NB ),
     $                   Y( LDY, NB )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION  ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0,
     $                     ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
      DOUBLE PRECISION  EI
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DGEMM, DGEMV, DLACPY,
     $                   DLARFG, DSCAL, DTRMM, DTRMV
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( N.LE.1 )
     $   RETURN
*
      DO 10 I = 1, NB
         IF( I.GT.1 ) THEN
*
*           Update A(K+1:N,I)
*
*           Update I-th column of A - Y * V**T
*
            CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1),
     $                  LDY,
     $                  A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 )
*
*           Apply I - V * T**T * V**T to this column (call it b) from the
*           left, using the last column of T as workspace
*
*           Let  V = ( V1 )   and   b = ( b1 )   (first I-1 rows)
*                    ( V2 )             ( b2 )
*
*           where V1 is unit lower triangular
*
*           w := V1**T * b1
*
            CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
            CALL DTRMV( 'Lower', 'Transpose', 'UNIT',
     $                  I-1, A( K+1, 1 ),
     $                  LDA, T( 1, NB ), 1 )
*
*           w := w + V2**T * b2
*
            CALL DGEMV( 'Transpose', N-K-I+1, I-1,
     $                  ONE, A( K+I, 1 ),
     $                  LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
*
*           w := T**T * w
*
            CALL DTRMV( 'Upper', 'Transpose', 'NON-UNIT',
     $                  I-1, T, LDT,
     $                  T( 1, NB ), 1 )
*
*           b2 := b2 - V2*w
*
            CALL DGEMV( 'NO TRANSPOSE', N-K-I+1, I-1, -ONE,
     $                  A( K+I, 1 ),
     $                  LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
*
*           b1 := b1 - V1*w
*
            CALL DTRMV( 'Lower', 'NO TRANSPOSE',
     $                  'UNIT', I-1,
     $                  A( K+1, 1 ), LDA, T( 1, NB ), 1 )
            CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
*
            A( K+I-1, I-1 ) = EI
         END IF
*
*        Generate the elementary reflector H(I) to annihilate
*        A(K+I+1:N,I)
*
         CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ),
     $                1,
     $                TAU( I ) )
         EI = A( K+I, I )
         A( K+I, I ) = ONE
*
*        Compute  Y(K+1:N,I)
*
         CALL DGEMV( 'NO TRANSPOSE', N-K, N-K-I+1,
     $               ONE, A( K+1, I+1 ),
     $               LDA, A( K+I, I ), 1, ZERO, Y( K+1, I ), 1 )
         CALL DGEMV( 'Transpose', N-K-I+1, I-1,
     $               ONE, A( K+I, 1 ), LDA,
     $               A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
         CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE,
     $               Y( K+1, 1 ), LDY,
     $               T( 1, I ), 1, ONE, Y( K+1, I ), 1 )
         CALL DSCAL( N-K, TAU( I ), Y( K+1, I ), 1 )
*
*        Compute T(1:I,I)
*
         CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
         CALL DTRMV( 'Upper', 'No Transpose', 'NON-UNIT',
     $               I-1, T, LDT,
     $               T( 1, I ), 1 )
         T( I, I ) = TAU( I )
*
   10 CONTINUE
      A( K+NB, NB ) = EI
*
*     Compute Y(1:K,1:NB)
*
      CALL DLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY )
      CALL DTRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE',
     $            'UNIT', K, NB,
     $            ONE, A( K+1, 1 ), LDA, Y, LDY )
      IF( N.GT.K+NB )
     $   CALL DGEMM( 'NO TRANSPOSE', 'NO TRANSPOSE', K,
     $               NB, N-K-NB, ONE,
     $               A( 1, 2+NB ), LDA, A( K+1+NB, 1 ), LDA, ONE, Y,
     $               LDY )
      CALL DTRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE',
     $            'NON-UNIT', K, NB,
     $            ONE, T, LDT, Y, LDY )
*
      RETURN
*
*     End of DLAHR2
*
      END
*> \brief \b DLAHRD reduces the first nb columns of a general rectangular matrix A so that elements below the k-th subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAHRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlahrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlahrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlahrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
*
*       .. Scalar Arguments ..
*       INTEGER            K, LDA, LDT, LDY, N, NB
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), T( LDT, NB ), TAU( NB ),
*      $                   Y( LDY, NB )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DLAHR2.
*>
*> DLAHRD reduces the first NB columns of a real general n-by-(n-k+1)
*> matrix A so that elements below the k-th subdiagonal are zero. The
*> reduction is performed by an orthogonal similarity transformation
*> Q**T * A * Q. The routine returns the matrices V and T which determine
*> Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The offset for the reduction. Elements below the k-th
*>          subdiagonal in the first NB columns are reduced to zero.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*>          NB is INTEGER
*>          The number of columns to be reduced.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N-K+1)
*>          On entry, the n-by-(n-k+1) general matrix A.
*>          On exit, the elements on and above the k-th subdiagonal in
*>          the first NB columns are overwritten with the corresponding
*>          elements of the reduced matrix; the elements below the k-th
*>          subdiagonal, with the array TAU, represent the matrix Q as a
*>          product of elementary reflectors. The other columns of A are
*>          unchanged. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (NB)
*>          The scalar factors of the elementary reflectors. See Further
*>          Details.
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,NB)
*>          The upper triangular matrix T.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T.  LDT >= NB.
*> \endverbatim
*>
*> \param[out] Y
*> \verbatim
*>          Y is DOUBLE PRECISION array, dimension (LDY,NB)
*>          The n-by-nb matrix Y.
*> \endverbatim
*>
*> \param[in] LDY
*> \verbatim
*>          LDY is INTEGER
*>          The leading dimension of the array Y. LDY >= N.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lahrd
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix Q is represented as a product of nb elementary reflectors
*>
*>     Q = H(1) H(2) . . . H(nb).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
*>  A(i+k+1:n,i), and tau in TAU(i).
*>
*>  The elements of the vectors v together form the (n-k+1)-by-nb matrix
*>  V which is needed, with T and Y, to apply the transformation to the
*>  unreduced part of the matrix, using an update of the form:
*>  A := (I - V*T*V**T) * (A - Y*V**T).
*>
*>  The contents of A on exit are illustrated by the following example
*>  with n = 7, k = 3 and nb = 2:
*>
*>     ( a   h   a   a   a )
*>     ( a   h   a   a   a )
*>     ( a   h   a   a   a )
*>     ( h   h   a   a   a )
*>     ( v1  h   a   a   a )
*>     ( v1  v2  a   a   a )
*>     ( v1  v2  a   a   a )
*>
*>  where a denotes an element of the original matrix A, h denotes a
*>  modified element of the upper Hessenberg matrix H, and vi denotes an
*>  element of the vector defining H(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            K, LDA, LDT, LDY, N, NB
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), T( LDT, NB ), TAU( NB ),
     $                   Y( LDY, NB )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
      DOUBLE PRECISION   EI
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DGEMV, DLARFG, DSCAL, DTRMV
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( N.LE.1 )
     $   RETURN
*
      DO 10 I = 1, NB
         IF( I.GT.1 ) THEN
*
*           Update A(1:n,i)
*
*           Compute i-th column of A - Y * V**T
*
            CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY,
     $                  A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 )
*
*           Apply I - V * T**T * V**T to this column (call it b) from the
*           left, using the last column of T as workspace
*
*           Let  V = ( V1 )   and   b = ( b1 )   (first I-1 rows)
*                    ( V2 )             ( b2 )
*
*           where V1 is unit lower triangular
*
*           w := V1**T * b1
*
            CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
            CALL DTRMV( 'Lower', 'Transpose', 'Unit', I-1,
     $                  A( K+1, 1 ), LDA, T( 1, NB ), 1 )
*
*           w := w + V2**T *b2
*
            CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ),
     $                  LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
*
*           w := T**T *w
*
            CALL DTRMV( 'Upper', 'Transpose', 'Non-unit', I-1, T, LDT,
     $                  T( 1, NB ), 1 )
*
*           b2 := b2 - V2*w
*
            CALL DGEMV( 'No transpose', N-K-I+1, I-1, -ONE,
     $                  A( K+I, 1 ), LDA, T( 1, NB ), 1, ONE,
     $                  A( K+I, I ), 1 )
*
*           b1 := b1 - V1*w
*
            CALL DTRMV( 'Lower', 'No transpose', 'Unit', I-1,
     $                  A( K+1, 1 ), LDA, T( 1, NB ), 1 )
            CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
*
            A( K+I-1, I-1 ) = EI
         END IF
*
*        Generate the elementary reflector H(i) to annihilate
*        A(k+i+1:n,i)
*
         CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
     $                TAU( I ) )
         EI = A( K+I, I )
         A( K+I, I ) = ONE
*
*        Compute  Y(1:n,i)
*
         CALL DGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ),
     $               LDA, A( K+I, I ), 1, ZERO, Y( 1, I ), 1 )
         CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ),
     $               LDA, A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
         CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ),
     $               1, ONE, Y( 1, I ), 1 )
         CALL DSCAL( N, TAU( I ), Y( 1, I ), 1 )
*
*        Compute T(1:i,i)
*
         CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
         CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T, LDT,
     $               T( 1, I ), 1 )
         T( I, I ) = TAU( I )
*
   10 CONTINUE
      A( K+NB, NB ) = EI
*
      RETURN
*
*     End of DLAHRD
*
      END
*> \brief \b DLAIC1 applies one step of incremental condition estimation.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAIC1 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaic1.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaic1.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaic1.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C )
*
*       .. Scalar Arguments ..
*       INTEGER            J, JOB
*       DOUBLE PRECISION   C, GAMMA, S, SEST, SESTPR
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   W( J ), X( J )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAIC1 applies one step of incremental condition estimation in
*> its simplest version:
*>
*> Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j
*> lower triangular matrix L, such that
*>          twonorm(L*x) = sest
*> Then DLAIC1 computes sestpr, s, c such that
*> the vector
*>                 [ s*x ]
*>          xhat = [  c  ]
*> is an approximate singular vector of
*>                 [ L       0  ]
*>          Lhat = [ w**T gamma ]
*> in the sense that
*>          twonorm(Lhat*xhat) = sestpr.
*>
*> Depending on JOB, an estimate for the largest or smallest singular
*> value is computed.
*>
*> Note that [s c]**T and sestpr**2 is an eigenpair of the system
*>
*>     diag(sest*sest, 0) + [alpha  gamma] * [ alpha ]
*>                                           [ gamma ]
*>
*> where  alpha =  x**T*w.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOB
*> \verbatim
*>          JOB is INTEGER
*>          = 1: an estimate for the largest singular value is computed.
*>          = 2: an estimate for the smallest singular value is computed.
*> \endverbatim
*>
*> \param[in] J
*> \verbatim
*>          J is INTEGER
*>          Length of X and W
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (J)
*>          The j-vector x.
*> \endverbatim
*>
*> \param[in] SEST
*> \verbatim
*>          SEST is DOUBLE PRECISION
*>          Estimated singular value of j by j matrix L
*> \endverbatim
*>
*> \param[in] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (J)
*>          The j-vector w.
*> \endverbatim
*>
*> \param[in] GAMMA
*> \verbatim
*>          GAMMA is DOUBLE PRECISION
*>          The diagonal element gamma.
*> \endverbatim
*>
*> \param[out] SESTPR
*> \verbatim
*>          SESTPR is DOUBLE PRECISION
*>          Estimated singular value of (j+1) by (j+1) matrix Lhat.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION
*>          Sine needed in forming xhat.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*>          C is DOUBLE PRECISION
*>          Cosine needed in forming xhat.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laic1
*
*  =====================================================================
      SUBROUTINE DLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            J, JOB
      DOUBLE PRECISION   C, GAMMA, S, SEST, SESTPR
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   W( J ), X( J )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
      DOUBLE PRECISION   HALF, FOUR
      PARAMETER          ( HALF = 0.5D0, FOUR = 4.0D0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   ABSALP, ABSEST, ABSGAM, ALPHA, B, COSINE, EPS,
     $                   NORMA, S1, S2, SINE, T, TEST, TMP, ZETA1, ZETA2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SIGN, SQRT
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DDOT, DLAMCH
      EXTERNAL           DDOT, DLAMCH
*     ..
*     .. Executable Statements ..
*
      EPS = DLAMCH( 'Epsilon' )
      ALPHA = DDOT( J, X, 1, W, 1 )
*
      ABSALP = ABS( ALPHA )
      ABSGAM = ABS( GAMMA )
      ABSEST = ABS( SEST )
*
      IF( JOB.EQ.1 ) THEN
*
*        Estimating largest singular value
*
*        special cases
*
         IF( SEST.EQ.ZERO ) THEN
            S1 = MAX( ABSGAM, ABSALP )
            IF( S1.EQ.ZERO ) THEN
               S = ZERO
               C = ONE
               SESTPR = ZERO
            ELSE
               S = ALPHA / S1
               C = GAMMA / S1
               TMP = SQRT( S*S+C*C )
               S = S / TMP
               C = C / TMP
               SESTPR = S1*TMP
            END IF
            RETURN
         ELSE IF( ABSGAM.LE.EPS*ABSEST ) THEN
            S = ONE
            C = ZERO
            TMP = MAX( ABSEST, ABSALP )
            S1 = ABSEST / TMP
            S2 = ABSALP / TMP
            SESTPR = TMP*SQRT( S1*S1+S2*S2 )
            RETURN
         ELSE IF( ABSALP.LE.EPS*ABSEST ) THEN
            S1 = ABSGAM
            S2 = ABSEST
            IF( S1.LE.S2 ) THEN
               S = ONE
               C = ZERO
               SESTPR = S2
            ELSE
               S = ZERO
               C = ONE
               SESTPR = S1
            END IF
            RETURN
         ELSE IF( ABSEST.LE.EPS*ABSALP .OR. ABSEST.LE.EPS*ABSGAM ) THEN
            S1 = ABSGAM
            S2 = ABSALP
            IF( S1.LE.S2 ) THEN
               TMP = S1 / S2
               S = SQRT( ONE+TMP*TMP )
               SESTPR = S2*S
               C = ( GAMMA / S2 ) / S
               S = SIGN( ONE, ALPHA ) / S
            ELSE
               TMP = S2 / S1
               C = SQRT( ONE+TMP*TMP )
               SESTPR = S1*C
               S = ( ALPHA / S1 ) / C
               C = SIGN( ONE, GAMMA ) / C
            END IF
            RETURN
         ELSE
*
*           normal case
*
            ZETA1 = ALPHA / ABSEST
            ZETA2 = GAMMA / ABSEST
*
            B = ( ONE-ZETA1*ZETA1-ZETA2*ZETA2 )*HALF
            C = ZETA1*ZETA1
            IF( B.GT.ZERO ) THEN
               T = C / ( B+SQRT( B*B+C ) )
            ELSE
               T = SQRT( B*B+C ) - B
            END IF
*
            SINE = -ZETA1 / T
            COSINE = -ZETA2 / ( ONE+T )
            TMP = SQRT( SINE*SINE+COSINE*COSINE )
            S = SINE / TMP
            C = COSINE / TMP
            SESTPR = SQRT( T+ONE )*ABSEST
            RETURN
         END IF
*
      ELSE IF( JOB.EQ.2 ) THEN
*
*        Estimating smallest singular value
*
*        special cases
*
         IF( SEST.EQ.ZERO ) THEN
            SESTPR = ZERO
            IF( MAX( ABSGAM, ABSALP ).EQ.ZERO ) THEN
               SINE = ONE
               COSINE = ZERO
            ELSE
               SINE = -GAMMA
               COSINE = ALPHA
            END IF
            S1 = MAX( ABS( SINE ), ABS( COSINE ) )
            S = SINE / S1
            C = COSINE / S1
            TMP = SQRT( S*S+C*C )
            S = S / TMP
            C = C / TMP
            RETURN
         ELSE IF( ABSGAM.LE.EPS*ABSEST ) THEN
            S = ZERO
            C = ONE
            SESTPR = ABSGAM
            RETURN
         ELSE IF( ABSALP.LE.EPS*ABSEST ) THEN
            S1 = ABSGAM
            S2 = ABSEST
            IF( S1.LE.S2 ) THEN
               S = ZERO
               C = ONE
               SESTPR = S1
            ELSE
               S = ONE
               C = ZERO
               SESTPR = S2
            END IF
            RETURN
         ELSE IF( ABSEST.LE.EPS*ABSALP .OR. ABSEST.LE.EPS*ABSGAM ) THEN
            S1 = ABSGAM
            S2 = ABSALP
            IF( S1.LE.S2 ) THEN
               TMP = S1 / S2
               C = SQRT( ONE+TMP*TMP )
               SESTPR = ABSEST*( TMP / C )
               S = -( GAMMA / S2 ) / C
               C = SIGN( ONE, ALPHA ) / C
            ELSE
               TMP = S2 / S1
               S = SQRT( ONE+TMP*TMP )
               SESTPR = ABSEST / S
               C = ( ALPHA / S1 ) / S
               S = -SIGN( ONE, GAMMA ) / S
            END IF
            RETURN
         ELSE
*
*           normal case
*
            ZETA1 = ALPHA / ABSEST
            ZETA2 = GAMMA / ABSEST
*
            NORMA = MAX( ONE+ZETA1*ZETA1+ABS( ZETA1*ZETA2 ),
     $              ABS( ZETA1*ZETA2 )+ZETA2*ZETA2 )
*
*           See if root is closer to zero or to ONE
*
            TEST = ONE + TWO*( ZETA1-ZETA2 )*( ZETA1+ZETA2 )
            IF( TEST.GE.ZERO ) THEN
*
*              root is close to zero, compute directly
*
               B = ( ZETA1*ZETA1+ZETA2*ZETA2+ONE )*HALF
               C = ZETA2*ZETA2
               T = C / ( B+SQRT( ABS( B*B-C ) ) )
               SINE = ZETA1 / ( ONE-T )
               COSINE = -ZETA2 / T
               SESTPR = SQRT( T+FOUR*EPS*EPS*NORMA )*ABSEST
            ELSE
*
*              root is closer to ONE, shift by that amount
*
               B = ( ZETA2*ZETA2+ZETA1*ZETA1-ONE )*HALF
               C = ZETA1*ZETA1
               IF( B.GE.ZERO ) THEN
                  T = -C / ( B+SQRT( B*B+C ) )
               ELSE
                  T = B - SQRT( B*B+C )
               END IF
               SINE = -ZETA1 / T
               COSINE = -ZETA2 / ( ONE+T )
               SESTPR = SQRT( ONE+T+FOUR*EPS*EPS*NORMA )*ABSEST
            END IF
            TMP = SQRT( SINE*SINE+COSINE*COSINE )
            S = SINE / TMP
            C = COSINE / TMP
            RETURN
*
         END IF
      END IF
      RETURN
*
*     End of DLAIC1
*
      END
*> \brief \b DLAISNAN tests input for NaN by comparing two arguments for inequality.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAISNAN + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaisnan.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaisnan.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaisnan.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       LOGICAL FUNCTION DLAISNAN( DIN1, DIN2 )
*
*       .. Scalar Arguments ..
*       DOUBLE PRECISION, INTENT(IN) :: DIN1, DIN2
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> This routine is not for general use.  It exists solely to avoid
*> over-optimization in DISNAN.
*>
*> DLAISNAN checks for NaNs by comparing its two arguments for
*> inequality.  NaN is the only floating-point value where NaN != NaN
*> returns .TRUE.  To check for NaNs, pass the same variable as both
*> arguments.
*>
*> A compiler must assume that the two arguments are
*> not the same variable, and the test will not be optimized away.
*> Interprocedural or whole-program optimization may delete this
*> test.  The ISNAN functions will be replaced by the correct
*> Fortran 03 intrinsic once the intrinsic is widely available.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] DIN1
*> \verbatim
*>          DIN1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] DIN2
*> \verbatim
*>          DIN2 is DOUBLE PRECISION
*>          Two numbers to compare for inequality.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laisnan
*
*  =====================================================================
      LOGICAL FUNCTION DLAISNAN( DIN1, DIN2 )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION, INTENT(IN) :: DIN1, DIN2
*     ..
*
*  =====================================================================
*
*  .. Executable Statements ..
      DLAISNAN = (DIN1.NE.DIN2)
      RETURN
      END
*> \brief \b DLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLALN2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaln2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaln2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaln2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B,
*                          LDB, WR, WI, X, LDX, SCALE, XNORM, INFO )
*
*       .. Scalar Arguments ..
*       LOGICAL            LTRANS
*       INTEGER            INFO, LDA, LDB, LDX, NA, NW
*       DOUBLE PRECISION   CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLALN2 solves a system of the form  (ca A - w D ) X = s B
*> or (ca A**T - w D) X = s B   with possible scaling ("s") and
*> perturbation of A.  (A**T means A-transpose.)
*>
*> A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA
*> real diagonal matrix, w is a real or complex value, and X and B are
*> NA x 1 matrices -- real if w is real, complex if w is complex.  NA
*> may be 1 or 2.
*>
*> If w is complex, X and B are represented as NA x 2 matrices,
*> the first column of each being the real part and the second
*> being the imaginary part.
*>
*> "s" is a scaling factor (<= 1), computed by DLALN2, which is
*> so chosen that X can be computed without overflow.  X is further
*> scaled if necessary to assure that norm(ca A - w D)*norm(X) is less
*> than overflow.
*>
*> If both singular values of (ca A - w D) are less than SMIN,
*> SMIN*identity will be used instead of (ca A - w D).  If only one
*> singular value is less than SMIN, one element of (ca A - w D) will be
*> perturbed enough to make the smallest singular value roughly SMIN.
*> If both singular values are at least SMIN, (ca A - w D) will not be
*> perturbed.  In any case, the perturbation will be at most some small
*> multiple of max( SMIN, ulp*norm(ca A - w D) ).  The singular values
*> are computed by infinity-norm approximations, and thus will only be
*> correct to a factor of 2 or so.
*>
*> Note: all input quantities are assumed to be smaller than overflow
*> by a reasonable factor.  (See BIGNUM.)
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] LTRANS
*> \verbatim
*>          LTRANS is LOGICAL
*>          =.TRUE.:  A-transpose will be used.
*>          =.FALSE.: A will be used (not transposed.)
*> \endverbatim
*>
*> \param[in] NA
*> \verbatim
*>          NA is INTEGER
*>          The size of the matrix A.  It may (only) be 1 or 2.
*> \endverbatim
*>
*> \param[in] NW
*> \verbatim
*>          NW is INTEGER
*>          1 if "w" is real, 2 if "w" is complex.  It may only be 1
*>          or 2.
*> \endverbatim
*>
*> \param[in] SMIN
*> \verbatim
*>          SMIN is DOUBLE PRECISION
*>          The desired lower bound on the singular values of A.  This
*>          should be a safe distance away from underflow or overflow,
*>          say, between (underflow/machine precision) and  (machine
*>          precision * overflow ).  (See BIGNUM and ULP.)
*> \endverbatim
*>
*> \param[in] CA
*> \verbatim
*>          CA is DOUBLE PRECISION
*>          The coefficient c, which A is multiplied by.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,NA)
*>          The NA x NA matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of A.  It must be at least NA.
*> \endverbatim
*>
*> \param[in] D1
*> \verbatim
*>          D1 is DOUBLE PRECISION
*>          The 1,1 element in the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] D2
*> \verbatim
*>          D2 is DOUBLE PRECISION
*>          The 2,2 element in the diagonal matrix D.  Not used if NA=1.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NW)
*>          The NA x NW matrix B (right-hand side).  If NW=2 ("w" is
*>          complex), column 1 contains the real part of B and column 2
*>          contains the imaginary part.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of B.  It must be at least NA.
*> \endverbatim
*>
*> \param[in] WR
*> \verbatim
*>          WR is DOUBLE PRECISION
*>          The real part of the scalar "w".
*> \endverbatim
*>
*> \param[in] WI
*> \verbatim
*>          WI is DOUBLE PRECISION
*>          The imaginary part of the scalar "w".  Not used if NW=1.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NW)
*>          The NA x NW matrix X (unknowns), as computed by DLALN2.
*>          If NW=2 ("w" is complex), on exit, column 1 will contain
*>          the real part of X and column 2 will contain the imaginary
*>          part.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of X.  It must be at least NA.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*>          SCALE is DOUBLE PRECISION
*>          The scale factor that B must be multiplied by to insure
*>          that overflow does not occur when computing X.  Thus,
*>          (ca A - w D) X  will be SCALE*B, not B (ignoring
*>          perturbations of A.)  It will be at most 1.
*> \endverbatim
*>
*> \param[out] XNORM
*> \verbatim
*>          XNORM is DOUBLE PRECISION
*>          The infinity-norm of X, when X is regarded as an NA x NW
*>          real matrix.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          An error flag.  It will be set to zero if no error occurs,
*>          a negative number if an argument is in error, or a positive
*>          number if  ca A - w D  had to be perturbed.
*>          The possible values are:
*>          = 0: No error occurred, and (ca A - w D) did not have to be
*>                 perturbed.
*>          = 1: (ca A - w D) had to be perturbed to make its smallest
*>               (or only) singular value greater than SMIN.
*>          NOTE: In the interests of speed, this routine does not
*>                check the inputs for errors.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laln2
*
*  =====================================================================
      SUBROUTINE DLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B,
     $                   LDB, WR, WI, X, LDX, SCALE, XNORM, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      LOGICAL            LTRANS
      INTEGER            INFO, LDA, LDB, LDX, NA, NW
      DOUBLE PRECISION   CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), X( LDX, * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
      DOUBLE PRECISION   TWO
      PARAMETER          ( TWO = 2.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            ICMAX, J
      DOUBLE PRECISION   BBND, BI1, BI2, BIGNUM, BNORM, BR1, BR2, CI21,
     $                   CI22, CMAX, CNORM, CR21, CR22, CSI, CSR, LI21,
     $                   LR21, SMINI, SMLNUM, TEMP, U22ABS, UI11, UI11R,
     $                   UI12, UI12S, UI22, UR11, UR11R, UR12, UR12S,
     $                   UR22, XI1, XI2, XR1, XR2
*     ..
*     .. Local Arrays ..
      LOGICAL            RSWAP( 4 ), ZSWAP( 4 )
      INTEGER            IPIVOT( 4, 4 )
      DOUBLE PRECISION   CI( 2, 2 ), CIV( 4 ), CR( 2, 2 ), CRV( 4 )
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLADIV
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. Equivalences ..
      EQUIVALENCE        ( CI( 1, 1 ), CIV( 1 ) ),
     $                   ( CR( 1, 1 ), CRV( 1 ) )
*     ..
*     .. Data statements ..
      DATA               ZSWAP / .FALSE., .FALSE., .TRUE., .TRUE. /
      DATA               RSWAP / .FALSE., .TRUE., .FALSE., .TRUE. /
      DATA               IPIVOT / 1, 2, 3, 4, 2, 1, 4, 3, 3, 4, 1, 2, 4,
     $                   3, 2, 1 /
*     ..
*     .. Executable Statements ..
*
*     Compute BIGNUM
*
      SMLNUM = TWO*DLAMCH( 'Safe minimum' )
      BIGNUM = ONE / SMLNUM
      SMINI = MAX( SMIN, SMLNUM )
*
*     Don't check for input errors
*
      INFO = 0
*
*     Standard Initializations
*
      SCALE = ONE
*
      IF( NA.EQ.1 ) THEN
*
*        1 x 1  (i.e., scalar) system   C X = B
*
         IF( NW.EQ.1 ) THEN
*
*           Real 1x1 system.
*
*           C = ca A - w D
*
            CSR = CA*A( 1, 1 ) - WR*D1
            CNORM = ABS( CSR )
*
*           If | C | < SMINI, use C = SMINI
*
            IF( CNORM.LT.SMINI ) THEN
               CSR = SMINI
               CNORM = SMINI
               INFO = 1
            END IF
*
*           Check scaling for  X = B / C
*
            BNORM = ABS( B( 1, 1 ) )
            IF( CNORM.LT.ONE .AND. BNORM.GT.ONE ) THEN
               IF( BNORM.GT.BIGNUM*CNORM )
     $            SCALE = ONE / BNORM
            END IF
*
*           Compute X
*
            X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / CSR
            XNORM = ABS( X( 1, 1 ) )
         ELSE
*
*           Complex 1x1 system (w is complex)
*
*           C = ca A - w D
*
            CSR = CA*A( 1, 1 ) - WR*D1
            CSI = -WI*D1
            CNORM = ABS( CSR ) + ABS( CSI )
*
*           If | C | < SMINI, use C = SMINI
*
            IF( CNORM.LT.SMINI ) THEN
               CSR = SMINI
               CSI = ZERO
               CNORM = SMINI
               INFO = 1
            END IF
*
*           Check scaling for  X = B / C
*
            BNORM = ABS( B( 1, 1 ) ) + ABS( B( 1, 2 ) )
            IF( CNORM.LT.ONE .AND. BNORM.GT.ONE ) THEN
               IF( BNORM.GT.BIGNUM*CNORM )
     $            SCALE = ONE / BNORM
            END IF
*
*           Compute X
*
            CALL DLADIV( SCALE*B( 1, 1 ), SCALE*B( 1, 2 ), CSR, CSI,
     $                   X( 1, 1 ), X( 1, 2 ) )
            XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) )
         END IF
*
      ELSE
*
*        2x2 System
*
*        Compute the real part of  C = ca A - w D  (or  ca A**T - w D )
*
         CR( 1, 1 ) = CA*A( 1, 1 ) - WR*D1
         CR( 2, 2 ) = CA*A( 2, 2 ) - WR*D2
         IF( LTRANS ) THEN
            CR( 1, 2 ) = CA*A( 2, 1 )
            CR( 2, 1 ) = CA*A( 1, 2 )
         ELSE
            CR( 2, 1 ) = CA*A( 2, 1 )
            CR( 1, 2 ) = CA*A( 1, 2 )
         END IF
*
         IF( NW.EQ.1 ) THEN
*
*           Real 2x2 system  (w is real)
*
*           Find the largest element in C
*
            CMAX = ZERO
            ICMAX = 0
*
            DO 10 J = 1, 4
               IF( ABS( CRV( J ) ).GT.CMAX ) THEN
                  CMAX = ABS( CRV( J ) )
                  ICMAX = J
               END IF
   10       CONTINUE
*
*           If norm(C) < SMINI, use SMINI*identity.
*
            IF( CMAX.LT.SMINI ) THEN
               BNORM = MAX( ABS( B( 1, 1 ) ), ABS( B( 2, 1 ) ) )
               IF( SMINI.LT.ONE .AND. BNORM.GT.ONE ) THEN
                  IF( BNORM.GT.BIGNUM*SMINI )
     $               SCALE = ONE / BNORM
               END IF
               TEMP = SCALE / SMINI
               X( 1, 1 ) = TEMP*B( 1, 1 )
               X( 2, 1 ) = TEMP*B( 2, 1 )
               XNORM = TEMP*BNORM
               INFO = 1
               RETURN
            END IF
*
*           Gaussian elimination with complete pivoting.
*
            UR11 = CRV( ICMAX )
            CR21 = CRV( IPIVOT( 2, ICMAX ) )
            UR12 = CRV( IPIVOT( 3, ICMAX ) )
            CR22 = CRV( IPIVOT( 4, ICMAX ) )
            UR11R = ONE / UR11
            LR21 = UR11R*CR21
            UR22 = CR22 - UR12*LR21
*
*           If smaller pivot < SMINI, use SMINI
*
            IF( ABS( UR22 ).LT.SMINI ) THEN
               UR22 = SMINI
               INFO = 1
            END IF
            IF( RSWAP( ICMAX ) ) THEN
               BR1 = B( 2, 1 )
               BR2 = B( 1, 1 )
            ELSE
               BR1 = B( 1, 1 )
               BR2 = B( 2, 1 )
            END IF
            BR2 = BR2 - LR21*BR1
            BBND = MAX( ABS( BR1*( UR22*UR11R ) ), ABS( BR2 ) )
            IF( BBND.GT.ONE .AND. ABS( UR22 ).LT.ONE ) THEN
               IF( BBND.GE.BIGNUM*ABS( UR22 ) )
     $            SCALE = ONE / BBND
            END IF
*
            XR2 = ( BR2*SCALE ) / UR22
            XR1 = ( SCALE*BR1 )*UR11R - XR2*( UR11R*UR12 )
            IF( ZSWAP( ICMAX ) ) THEN
               X( 1, 1 ) = XR2
               X( 2, 1 ) = XR1
            ELSE
               X( 1, 1 ) = XR1
               X( 2, 1 ) = XR2
            END IF
            XNORM = MAX( ABS( XR1 ), ABS( XR2 ) )
*
*           Further scaling if  norm(A) norm(X) > overflow
*
            IF( XNORM.GT.ONE .AND. CMAX.GT.ONE ) THEN
               IF( XNORM.GT.BIGNUM / CMAX ) THEN
                  TEMP = CMAX / BIGNUM
                  X( 1, 1 ) = TEMP*X( 1, 1 )
                  X( 2, 1 ) = TEMP*X( 2, 1 )
                  XNORM = TEMP*XNORM
                  SCALE = TEMP*SCALE
               END IF
            END IF
         ELSE
*
*           Complex 2x2 system  (w is complex)
*
*           Find the largest element in C
*
            CI( 1, 1 ) = -WI*D1
            CI( 2, 1 ) = ZERO
            CI( 1, 2 ) = ZERO
            CI( 2, 2 ) = -WI*D2
            CMAX = ZERO
            ICMAX = 0
*
            DO 20 J = 1, 4
               IF( ABS( CRV( J ) )+ABS( CIV( J ) ).GT.CMAX ) THEN
                  CMAX = ABS( CRV( J ) ) + ABS( CIV( J ) )
                  ICMAX = J
               END IF
   20       CONTINUE
*
*           If norm(C) < SMINI, use SMINI*identity.
*
            IF( CMAX.LT.SMINI ) THEN
               BNORM = MAX( ABS( B( 1, 1 ) )+ABS( B( 1, 2 ) ),
     $                 ABS( B( 2, 1 ) )+ABS( B( 2, 2 ) ) )
               IF( SMINI.LT.ONE .AND. BNORM.GT.ONE ) THEN
                  IF( BNORM.GT.BIGNUM*SMINI )
     $               SCALE = ONE / BNORM
               END IF
               TEMP = SCALE / SMINI
               X( 1, 1 ) = TEMP*B( 1, 1 )
               X( 2, 1 ) = TEMP*B( 2, 1 )
               X( 1, 2 ) = TEMP*B( 1, 2 )
               X( 2, 2 ) = TEMP*B( 2, 2 )
               XNORM = TEMP*BNORM
               INFO = 1
               RETURN
            END IF
*
*           Gaussian elimination with complete pivoting.
*
            UR11 = CRV( ICMAX )
            UI11 = CIV( ICMAX )
            CR21 = CRV( IPIVOT( 2, ICMAX ) )
            CI21 = CIV( IPIVOT( 2, ICMAX ) )
            UR12 = CRV( IPIVOT( 3, ICMAX ) )
            UI12 = CIV( IPIVOT( 3, ICMAX ) )
            CR22 = CRV( IPIVOT( 4, ICMAX ) )
            CI22 = CIV( IPIVOT( 4, ICMAX ) )
            IF( ICMAX.EQ.1 .OR. ICMAX.EQ.4 ) THEN
*
*              Code when off-diagonals of pivoted C are real
*
               IF( ABS( UR11 ).GT.ABS( UI11 ) ) THEN
                  TEMP = UI11 / UR11
                  UR11R = ONE / ( UR11*( ONE+TEMP**2 ) )
                  UI11R = -TEMP*UR11R
               ELSE
                  TEMP = UR11 / UI11
                  UI11R = -ONE / ( UI11*( ONE+TEMP**2 ) )
                  UR11R = -TEMP*UI11R
               END IF
               LR21 = CR21*UR11R
               LI21 = CR21*UI11R
               UR12S = UR12*UR11R
               UI12S = UR12*UI11R
               UR22 = CR22 - UR12*LR21
               UI22 = CI22 - UR12*LI21
            ELSE
*
*              Code when diagonals of pivoted C are real
*
               UR11R = ONE / UR11
               UI11R = ZERO
               LR21 = CR21*UR11R
               LI21 = CI21*UR11R
               UR12S = UR12*UR11R
               UI12S = UI12*UR11R
               UR22 = CR22 - UR12*LR21 + UI12*LI21
               UI22 = -UR12*LI21 - UI12*LR21
            END IF
            U22ABS = ABS( UR22 ) + ABS( UI22 )
*
*           If smaller pivot < SMINI, use SMINI
*
            IF( U22ABS.LT.SMINI ) THEN
               UR22 = SMINI
               UI22 = ZERO
               INFO = 1
            END IF
            IF( RSWAP( ICMAX ) ) THEN
               BR2 = B( 1, 1 )
               BR1 = B( 2, 1 )
               BI2 = B( 1, 2 )
               BI1 = B( 2, 2 )
            ELSE
               BR1 = B( 1, 1 )
               BR2 = B( 2, 1 )
               BI1 = B( 1, 2 )
               BI2 = B( 2, 2 )
            END IF
            BR2 = BR2 - LR21*BR1 + LI21*BI1
            BI2 = BI2 - LI21*BR1 - LR21*BI1
            BBND = MAX( ( ABS( BR1 )+ABS( BI1 ) )*
     $             ( U22ABS*( ABS( UR11R )+ABS( UI11R ) ) ),
     $             ABS( BR2 )+ABS( BI2 ) )
            IF( BBND.GT.ONE .AND. U22ABS.LT.ONE ) THEN
               IF( BBND.GE.BIGNUM*U22ABS ) THEN
                  SCALE = ONE / BBND
                  BR1 = SCALE*BR1
                  BI1 = SCALE*BI1
                  BR2 = SCALE*BR2
                  BI2 = SCALE*BI2
               END IF
            END IF
*
            CALL DLADIV( BR2, BI2, UR22, UI22, XR2, XI2 )
            XR1 = UR11R*BR1 - UI11R*BI1 - UR12S*XR2 + UI12S*XI2
            XI1 = UI11R*BR1 + UR11R*BI1 - UI12S*XR2 - UR12S*XI2
            IF( ZSWAP( ICMAX ) ) THEN
               X( 1, 1 ) = XR2
               X( 2, 1 ) = XR1
               X( 1, 2 ) = XI2
               X( 2, 2 ) = XI1
            ELSE
               X( 1, 1 ) = XR1
               X( 2, 1 ) = XR2
               X( 1, 2 ) = XI1
               X( 2, 2 ) = XI2
            END IF
            XNORM = MAX( ABS( XR1 )+ABS( XI1 ), ABS( XR2 )+ABS( XI2 ) )
*
*           Further scaling if  norm(A) norm(X) > overflow
*
            IF( XNORM.GT.ONE .AND. CMAX.GT.ONE ) THEN
               IF( XNORM.GT.BIGNUM / CMAX ) THEN
                  TEMP = CMAX / BIGNUM
                  X( 1, 1 ) = TEMP*X( 1, 1 )
                  X( 2, 1 ) = TEMP*X( 2, 1 )
                  X( 1, 2 ) = TEMP*X( 1, 2 )
                  X( 2, 2 ) = TEMP*X( 2, 2 )
                  XNORM = TEMP*XNORM
                  SCALE = TEMP*SCALE
               END IF
            END IF
         END IF
      END IF
*
      RETURN
*
*     End of DLALN2
*
      END
*> \brief \b DLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLALS0 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlals0.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlals0.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlals0.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX,
*                          PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM,
*                          POLES, DIFL, DIFR, Z, K, C, S, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            GIVPTR, ICOMPQ, INFO, K, LDB, LDBX, LDGCOL,
*      $                   LDGNUM, NL, NR, NRHS, SQRE
*       DOUBLE PRECISION   C, S
*       ..
*       .. Array Arguments ..
*       INTEGER            GIVCOL( LDGCOL, * ), PERM( * )
*       DOUBLE PRECISION   B( LDB, * ), BX( LDBX, * ), DIFL( * ),
*      $                   DIFR( LDGNUM, * ), GIVNUM( LDGNUM, * ),
*      $                   POLES( LDGNUM, * ), WORK( * ), Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLALS0 applies back the multiplying factors of either the left or the
*> right singular vector matrix of a diagonal matrix appended by a row
*> to the right hand side matrix B in solving the least squares problem
*> using the divide-and-conquer SVD approach.
*>
*> For the left singular vector matrix, three types of orthogonal
*> matrices are involved:
*>
*> (1L) Givens rotations: the number of such rotations is GIVPTR; the
*>      pairs of columns/rows they were applied to are stored in GIVCOL;
*>      and the C- and S-values of these rotations are stored in GIVNUM.
*>
*> (2L) Permutation. The (NL+1)-st row of B is to be moved to the first
*>      row, and for J=2:N, PERM(J)-th row of B is to be moved to the
*>      J-th row.
*>
*> (3L) The left singular vector matrix of the remaining matrix.
*>
*> For the right singular vector matrix, four types of orthogonal
*> matrices are involved:
*>
*> (1R) The right singular vector matrix of the remaining matrix.
*>
*> (2R) If SQRE = 1, one extra Givens rotation to generate the right
*>      null space.
*>
*> (3R) The inverse transformation of (2L).
*>
*> (4R) The inverse transformation of (1L).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*>          ICOMPQ is INTEGER
*>         Specifies whether singular vectors are to be computed in
*>         factored form:
*>         = 0: Left singular vector matrix.
*>         = 1: Right singular vector matrix.
*> \endverbatim
*>
*> \param[in] NL
*> \verbatim
*>          NL is INTEGER
*>         The row dimension of the upper block. NL >= 1.
*> \endverbatim
*>
*> \param[in] NR
*> \verbatim
*>          NR is INTEGER
*>         The row dimension of the lower block. NR >= 1.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*>          SQRE is INTEGER
*>         = 0: the lower block is an NR-by-NR square matrix.
*>         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
*>
*>         The bidiagonal matrix has row dimension N = NL + NR + 1,
*>         and column dimension M = N + SQRE.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>         The number of columns of B and BX. NRHS must be at least 1.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension ( LDB, NRHS )
*>         On input, B contains the right hand sides of the least
*>         squares problem in rows 1 through M. On output, B contains
*>         the solution X in rows 1 through N.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>         The leading dimension of B. LDB must be at least
*>         max(1,MAX( M, N ) ).
*> \endverbatim
*>
*> \param[out] BX
*> \verbatim
*>          BX is DOUBLE PRECISION array, dimension ( LDBX, NRHS )
*> \endverbatim
*>
*> \param[in] LDBX
*> \verbatim
*>          LDBX is INTEGER
*>         The leading dimension of BX.
*> \endverbatim
*>
*> \param[in] PERM
*> \verbatim
*>          PERM is INTEGER array, dimension ( N )
*>         The permutations (from deflation and sorting) applied
*>         to the two blocks.
*> \endverbatim
*>
*> \param[in] GIVPTR
*> \verbatim
*>          GIVPTR is INTEGER
*>         The number of Givens rotations which took place in this
*>         subproblem.
*> \endverbatim
*>
*> \param[in] GIVCOL
*> \verbatim
*>          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
*>         Each pair of numbers indicates a pair of rows/columns
*>         involved in a Givens rotation.
*> \endverbatim
*>
*> \param[in] LDGCOL
*> \verbatim
*>          LDGCOL is INTEGER
*>         The leading dimension of GIVCOL, must be at least N.
*> \endverbatim
*>
*> \param[in] GIVNUM
*> \verbatim
*>          GIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
*>         Each number indicates the C or S value used in the
*>         corresponding Givens rotation.
*> \endverbatim
*>
*> \param[in] LDGNUM
*> \verbatim
*>          LDGNUM is INTEGER
*>         The leading dimension of arrays DIFR, POLES and
*>         GIVNUM, must be at least K.
*> \endverbatim
*>
*> \param[in] POLES
*> \verbatim
*>          POLES is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
*>         On entry, POLES(1:K, 1) contains the new singular
*>         values obtained from solving the secular equation, and
*>         POLES(1:K, 2) is an array containing the poles in the secular
*>         equation.
*> \endverbatim
*>
*> \param[in] DIFL
*> \verbatim
*>          DIFL is DOUBLE PRECISION array, dimension ( K ).
*>         On entry, DIFL(I) is the distance between I-th updated
*>         (undeflated) singular value and the I-th (undeflated) old
*>         singular value.
*> \endverbatim
*>
*> \param[in] DIFR
*> \verbatim
*>          DIFR is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ).
*>         On entry, DIFR(I, 1) contains the distances between I-th
*>         updated (undeflated) singular value and the I+1-th
*>         (undeflated) old singular value. And DIFR(I, 2) is the
*>         normalizing factor for the I-th right singular vector.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension ( K )
*>         Contain the components of the deflation-adjusted updating row
*>         vector.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>         Contains the dimension of the non-deflated matrix,
*>         This is the order of the related secular equation. 1 <= K <=N.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*>          C is DOUBLE PRECISION
*>         C contains garbage if SQRE =0 and the C-value of a Givens
*>         rotation related to the right null space if SQRE = 1.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*>          S is DOUBLE PRECISION
*>         S contains garbage if SQRE =0 and the S-value of a Givens
*>         rotation related to the right null space if SQRE = 1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension ( K )
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lals0
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Ren-Cang Li, Computer Science Division, University of
*>       California at Berkeley, USA \n
*>     Osni Marques, LBNL/NERSC, USA \n
*
*  =====================================================================
      SUBROUTINE DLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX,
     $                   LDBX,
     $                   PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM,
     $                   POLES, DIFL, DIFR, Z, K, C, S, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            GIVPTR, ICOMPQ, INFO, K, LDB, LDBX, LDGCOL,
     $                   LDGNUM, NL, NR, NRHS, SQRE
      DOUBLE PRECISION   C, S
*     ..
*     .. Array Arguments ..
      INTEGER            GIVCOL( LDGCOL, * ), PERM( * )
      DOUBLE PRECISION   B( LDB, * ), BX( LDBX, * ), DIFL( * ),
     $                   DIFR( LDGNUM, * ), GIVNUM( LDGNUM, * ),
     $                   POLES( LDGNUM, * ), WORK( * ), Z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO, NEGONE
      PARAMETER          ( ONE = 1.0D0, ZERO = 0.0D0, NEGONE = -1.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J, M, N, NLP1
      DOUBLE PRECISION   DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, TEMP
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGEMV, DLACPY, DLASCL, DROT,
     $                   DSCAL,
     $                   XERBLA
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMC3, DNRM2
      EXTERNAL           DLAMC3, DNRM2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      N = NL + NR + 1
*
      IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
         INFO = -1
      ELSE IF( NL.LT.1 ) THEN
         INFO = -2
      ELSE IF( NR.LT.1 ) THEN
         INFO = -3
      ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
         INFO = -4
      ELSE IF( NRHS.LT.1 ) THEN
         INFO = -5
      ELSE IF( LDB.LT.N ) THEN
         INFO = -7
      ELSE IF( LDBX.LT.N ) THEN
         INFO = -9
      ELSE IF( GIVPTR.LT.0 ) THEN
         INFO = -11
      ELSE IF( LDGCOL.LT.N ) THEN
         INFO = -13
      ELSE IF( LDGNUM.LT.N ) THEN
         INFO = -15
      ELSE IF( K.LT.1 ) THEN
         INFO = -20
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLALS0', -INFO )
         RETURN
      END IF
*
      M = N + SQRE
      NLP1 = NL + 1
*
      IF( ICOMPQ.EQ.0 ) THEN
*
*        Apply back orthogonal transformations from the left.
*
*        Step (1L): apply back the Givens rotations performed.
*
         DO 10 I = 1, GIVPTR
            CALL DROT( NRHS, B( GIVCOL( I, 2 ), 1 ), LDB,
     $                 B( GIVCOL( I, 1 ), 1 ), LDB, GIVNUM( I, 2 ),
     $                 GIVNUM( I, 1 ) )
   10    CONTINUE
*
*        Step (2L): permute rows of B.
*
         CALL DCOPY( NRHS, B( NLP1, 1 ), LDB, BX( 1, 1 ), LDBX )
         DO 20 I = 2, N
            CALL DCOPY( NRHS, B( PERM( I ), 1 ), LDB, BX( I, 1 ),
     $                  LDBX )
   20    CONTINUE
*
*        Step (3L): apply the inverse of the left singular vector
*        matrix to BX.
*
         IF( K.EQ.1 ) THEN
            CALL DCOPY( NRHS, BX, LDBX, B, LDB )
            IF( Z( 1 ).LT.ZERO ) THEN
               CALL DSCAL( NRHS, NEGONE, B, LDB )
            END IF
         ELSE
            DO 50 J = 1, K
               DIFLJ = DIFL( J )
               DJ = POLES( J, 1 )
               DSIGJ = -POLES( J, 2 )
               IF( J.LT.K ) THEN
                  DIFRJ = -DIFR( J, 1 )
                  DSIGJP = -POLES( J+1, 2 )
               END IF
               IF( ( Z( J ).EQ.ZERO ) .OR. ( POLES( J, 2 ).EQ.ZERO ) )
     $              THEN
                  WORK( J ) = ZERO
               ELSE
                  WORK( J ) = -POLES( J, 2 )*Z( J ) / DIFLJ /
     $                        ( POLES( J, 2 )+DJ )
               END IF
               DO 30 I = 1, J - 1
                  IF( ( Z( I ).EQ.ZERO ) .OR.
     $                ( POLES( I, 2 ).EQ.ZERO ) ) THEN
                     WORK( I ) = ZERO
                  ELSE
*
*                    Use calls to the subroutine DLAMC3 to enforce the
*                    parentheses (x+y)+z. The goal is to prevent
*                    optimizing compilers from doing x+(y+z).
*
                     WORK( I ) = POLES( I, 2 )*Z( I ) /
     $                           ( DLAMC3( POLES( I, 2 ), DSIGJ )-
     $                           DIFLJ ) / ( POLES( I, 2 )+DJ )
                  END IF
   30          CONTINUE
               DO 40 I = J + 1, K
                  IF( ( Z( I ).EQ.ZERO ) .OR.
     $                ( POLES( I, 2 ).EQ.ZERO ) ) THEN
                     WORK( I ) = ZERO
                  ELSE
                     WORK( I ) = POLES( I, 2 )*Z( I ) /
     $                           ( DLAMC3( POLES( I, 2 ), DSIGJP )+
     $                           DIFRJ ) / ( POLES( I, 2 )+DJ )
                  END IF
   40          CONTINUE
               WORK( 1 ) = NEGONE
               TEMP = DNRM2( K, WORK, 1 )
               CALL DGEMV( 'T', K, NRHS, ONE, BX, LDBX, WORK, 1,
     $                     ZERO,
     $                     B( J, 1 ), LDB )
               CALL DLASCL( 'G', 0, 0, TEMP, ONE, 1, NRHS, B( J, 1 ),
     $                      LDB, INFO )
   50       CONTINUE
         END IF
*
*        Move the deflated rows of BX to B also.
*
         IF( K.LT.MAX( M, N ) )
     $      CALL DLACPY( 'A', N-K, NRHS, BX( K+1, 1 ), LDBX,
     $                   B( K+1, 1 ), LDB )
      ELSE
*
*        Apply back the right orthogonal transformations.
*
*        Step (1R): apply back the new right singular vector matrix
*        to B.
*
         IF( K.EQ.1 ) THEN
            CALL DCOPY( NRHS, B, LDB, BX, LDBX )
         ELSE
            DO 80 J = 1, K
               DSIGJ = POLES( J, 2 )
               IF( Z( J ).EQ.ZERO ) THEN
                  WORK( J ) = ZERO
               ELSE
                  WORK( J ) = -Z( J ) / DIFL( J ) /
     $                        ( DSIGJ+POLES( J, 1 ) ) / DIFR( J, 2 )
               END IF
               DO 60 I = 1, J - 1
                  IF( Z( J ).EQ.ZERO ) THEN
                     WORK( I ) = ZERO
                  ELSE
*
*                    Use calls to the subroutine DLAMC3 to enforce the
*                    parentheses (x+y)+z. The goal is to prevent
*                    optimizing compilers from doing x+(y+z).
*
                     WORK( I ) = Z( J ) / ( DLAMC3( DSIGJ,
     $                     -POLES( I+1,
     $                           2 ) )-DIFR( I, 1 ) ) /
     $                           ( DSIGJ+POLES( I, 1 ) ) / DIFR( I, 2 )
                  END IF
   60          CONTINUE
               DO 70 I = J + 1, K
                  IF( Z( J ).EQ.ZERO ) THEN
                     WORK( I ) = ZERO
                  ELSE
                     WORK( I ) = Z( J ) / ( DLAMC3( DSIGJ, -POLES( I,
     $                           2 ) )-DIFL( I ) ) /
     $                           ( DSIGJ+POLES( I, 1 ) ) / DIFR( I, 2 )
                  END IF
   70          CONTINUE
               CALL DGEMV( 'T', K, NRHS, ONE, B, LDB, WORK, 1, ZERO,
     $                     BX( J, 1 ), LDBX )
   80       CONTINUE
         END IF
*
*        Step (2R): if SQRE = 1, apply back the rotation that is
*        related to the right null space of the subproblem.
*
         IF( SQRE.EQ.1 ) THEN
            CALL DCOPY( NRHS, B( M, 1 ), LDB, BX( M, 1 ), LDBX )
            CALL DROT( NRHS, BX( 1, 1 ), LDBX, BX( M, 1 ), LDBX, C,
     $                 S )
         END IF
         IF( K.LT.MAX( M, N ) )
     $      CALL DLACPY( 'A', N-K, NRHS, B( K+1, 1 ), LDB, BX( K+1,
     $                   1 ),
     $                   LDBX )
*
*        Step (3R): permute rows of B.
*
         CALL DCOPY( NRHS, BX( 1, 1 ), LDBX, B( NLP1, 1 ), LDB )
         IF( SQRE.EQ.1 ) THEN
            CALL DCOPY( NRHS, BX( M, 1 ), LDBX, B( M, 1 ), LDB )
         END IF
         DO 90 I = 2, N
            CALL DCOPY( NRHS, BX( I, 1 ), LDBX, B( PERM( I ), 1 ),
     $                  LDB )
   90    CONTINUE
*
*        Step (4R): apply back the Givens rotations performed.
*
         DO 100 I = GIVPTR, 1, -1
            CALL DROT( NRHS, B( GIVCOL( I, 2 ), 1 ), LDB,
     $                 B( GIVCOL( I, 1 ), 1 ), LDB, GIVNUM( I, 2 ),
     $                 -GIVNUM( I, 1 ) )
  100    CONTINUE
      END IF
*
      RETURN
*
*     End of DLALS0
*
      END
*> \brief \b DLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLALSA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlalsa.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlalsa.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlalsa.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLALSA( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U,
*                          LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR,
*                          GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK,
*                          IWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS,
*      $                   SMLSIZ
*       ..
*       .. Array Arguments ..
*       INTEGER            GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
*      $                   K( * ), PERM( LDGCOL, * )
*       DOUBLE PRECISION   B( LDB, * ), BX( LDBX, * ), C( * ),
*      $                   DIFL( LDU, * ), DIFR( LDU, * ),
*      $                   GIVNUM( LDU, * ), POLES( LDU, * ), S( * ),
*      $                   U( LDU, * ), VT( LDU, * ), WORK( * ),
*      $                   Z( LDU, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLALSA is an intermediate step in solving the least squares problem
*> by computing the SVD of the coefficient matrix in compact form (The
*> singular vectors are computed as products of simple orthogonal
*> matrices.).
*>
*> If ICOMPQ = 0, DLALSA applies the inverse of the left singular vector
*> matrix of an upper bidiagonal matrix to the right hand side; and if
*> ICOMPQ = 1, DLALSA applies the right singular vector matrix to the
*> right hand side. The singular vector matrices were generated in
*> compact form by DLALSA.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*>          ICOMPQ is INTEGER
*>         Specifies whether the left or the right singular vector
*>         matrix is involved.
*>         = 0: Left singular vector matrix
*>         = 1: Right singular vector matrix
*> \endverbatim
*>
*> \param[in] SMLSIZ
*> \verbatim
*>          SMLSIZ is INTEGER
*>         The maximum size of the subproblems at the bottom of the
*>         computation tree.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>         The row and column dimensions of the upper bidiagonal matrix.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>         The number of columns of B and BX. NRHS must be at least 1.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension ( LDB, NRHS )
*>         On input, B contains the right hand sides of the least
*>         squares problem in rows 1 through M.
*>         On output, B contains the solution X in rows 1 through N.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>         The leading dimension of B in the calling subprogram.
*>         LDB must be at least max(1,MAX( M, N ) ).
*> \endverbatim
*>
*> \param[out] BX
*> \verbatim
*>          BX is DOUBLE PRECISION array, dimension ( LDBX, NRHS )
*>         On exit, the result of applying the left or right singular
*>         vector matrix to B.
*> \endverbatim
*>
*> \param[in] LDBX
*> \verbatim
*>          LDBX is INTEGER
*>         The leading dimension of BX.
*> \endverbatim
*>
*> \param[in] U
*> \verbatim
*>          U is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ ).
*>         On entry, U contains the left singular vector matrices of all
*>         subproblems at the bottom level.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER, LDU = > N.
*>         The leading dimension of arrays U, VT, DIFL, DIFR,
*>         POLES, GIVNUM, and Z.
*> \endverbatim
*>
*> \param[in] VT
*> \verbatim
*>          VT is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ+1 ).
*>         On entry, VT**T contains the right singular vector matrices of
*>         all subproblems at the bottom level.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER array, dimension ( N ).
*> \endverbatim
*>
*> \param[in] DIFL
*> \verbatim
*>          DIFL is DOUBLE PRECISION array, dimension ( LDU, NLVL ).
*>         where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.
*> \endverbatim
*>
*> \param[in] DIFR
*> \verbatim
*>          DIFR is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
*>         On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record
*>         distances between singular values on the I-th level and
*>         singular values on the (I -1)-th level, and DIFR(*, 2 * I)
*>         record the normalizing factors of the right singular vectors
*>         matrices of subproblems on I-th level.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension ( LDU, NLVL ).
*>         On entry, Z(1, I) contains the components of the deflation-
*>         adjusted updating row vector for subproblems on the I-th
*>         level.
*> \endverbatim
*>
*> \param[in] POLES
*> \verbatim
*>          POLES is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
*>         On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
*>         singular values involved in the secular equations on the I-th
*>         level.
*> \endverbatim
*>
*> \param[in] GIVPTR
*> \verbatim
*>          GIVPTR is INTEGER array, dimension ( N ).
*>         On entry, GIVPTR( I ) records the number of Givens
*>         rotations performed on the I-th problem on the computation
*>         tree.
*> \endverbatim
*>
*> \param[in] GIVCOL
*> \verbatim
*>          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 * NLVL ).
*>         On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the
*>         locations of Givens rotations performed on the I-th level on
*>         the computation tree.
*> \endverbatim
*>
*> \param[in] LDGCOL
*> \verbatim
*>          LDGCOL is INTEGER, LDGCOL = > N.
*>         The leading dimension of arrays GIVCOL and PERM.
*> \endverbatim
*>
*> \param[in] PERM
*> \verbatim
*>          PERM is INTEGER array, dimension ( LDGCOL, NLVL ).
*>         On entry, PERM(*, I) records permutations done on the I-th
*>         level of the computation tree.
*> \endverbatim
*>
*> \param[in] GIVNUM
*> \verbatim
*>          GIVNUM is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
*>         On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S-
*>         values of Givens rotations performed on the I-th level on the
*>         computation tree.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension ( N ).
*>         On entry, if the I-th subproblem is not square,
*>         C( I ) contains the C-value of a Givens rotation related to
*>         the right null space of the I-th subproblem.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension ( N ).
*>         On entry, if the I-th subproblem is not square,
*>         S( I ) contains the S-value of a Givens rotation related to
*>         the right null space of the I-th subproblem.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lalsa
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Ren-Cang Li, Computer Science Division, University of
*>       California at Berkeley, USA \n
*>     Osni Marques, LBNL/NERSC, USA \n
*
*  =====================================================================
      SUBROUTINE DLALSA( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX,
     $                   U,
     $                   LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR,
     $                   GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK,
     $                   IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS,
     $                   SMLSIZ
*     ..
*     .. Array Arguments ..
      INTEGER            GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
     $                   K( * ), PERM( LDGCOL, * )
      DOUBLE PRECISION   B( LDB, * ), BX( LDBX, * ), C( * ),
     $                   DIFL( LDU, * ), DIFR( LDU, * ),
     $                   GIVNUM( LDU, * ), POLES( LDU, * ), S( * ),
     $                   U( LDU, * ), VT( LDU, * ), WORK( * ),
     $                   Z( LDU, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, I1, IC, IM1, INODE, J, LF, LL, LVL, LVL2,
     $                   ND, NDB1, NDIML, NDIMR, NL, NLF, NLP1, NLVL,
     $                   NR, NRF, NRP1, SQRE
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGEMM, DLALS0, DLASDT,
     $                   XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
         INFO = -1
      ELSE IF( SMLSIZ.LT.3 ) THEN
         INFO = -2
      ELSE IF( N.LT.SMLSIZ ) THEN
         INFO = -3
      ELSE IF( NRHS.LT.1 ) THEN
         INFO = -4
      ELSE IF( LDB.LT.N ) THEN
         INFO = -6
      ELSE IF( LDBX.LT.N ) THEN
         INFO = -8
      ELSE IF( LDU.LT.N ) THEN
         INFO = -10
      ELSE IF( LDGCOL.LT.N ) THEN
         INFO = -19
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLALSA', -INFO )
         RETURN
      END IF
*
*     Book-keeping and  setting up the computation tree.
*
      INODE = 1
      NDIML = INODE + N
      NDIMR = NDIML + N
*
      CALL DLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ),
     $             IWORK( NDIMR ), SMLSIZ )
*
*     The following code applies back the left singular vector factors.
*     For applying back the right singular vector factors, go to 50.
*
      IF( ICOMPQ.EQ.1 ) THEN
         GO TO 50
      END IF
*
*     The nodes on the bottom level of the tree were solved
*     by DLASDQ. The corresponding left and right singular vector
*     matrices are in explicit form. First apply back the left
*     singular vector matrices.
*
      NDB1 = ( ND+1 ) / 2
      DO 10 I = NDB1, ND
*
*        IC : center row of each node
*        NL : number of rows of left  subproblem
*        NR : number of rows of right subproblem
*        NLF: starting row of the left   subproblem
*        NRF: starting row of the right  subproblem
*
         I1 = I - 1
         IC = IWORK( INODE+I1 )
         NL = IWORK( NDIML+I1 )
         NR = IWORK( NDIMR+I1 )
         NLF = IC - NL
         NRF = IC + 1
         CALL DGEMM( 'T', 'N', NL, NRHS, NL, ONE, U( NLF, 1 ), LDU,
     $               B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX )
         CALL DGEMM( 'T', 'N', NR, NRHS, NR, ONE, U( NRF, 1 ), LDU,
     $               B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX )
   10 CONTINUE
*
*     Next copy the rows of B that correspond to unchanged rows
*     in the bidiagonal matrix to BX.
*
      DO 20 I = 1, ND
         IC = IWORK( INODE+I-1 )
         CALL DCOPY( NRHS, B( IC, 1 ), LDB, BX( IC, 1 ), LDBX )
   20 CONTINUE
*
*     Finally go through the left singular vector matrices of all
*     the other subproblems bottom-up on the tree.
*
      J = 2**NLVL
      SQRE = 0
*
      DO 40 LVL = NLVL, 1, -1
         LVL2 = 2*LVL - 1
*
*        find the first node LF and last node LL on
*        the current level LVL
*
         IF( LVL.EQ.1 ) THEN
            LF = 1
            LL = 1
         ELSE
            LF = 2**( LVL-1 )
            LL = 2*LF - 1
         END IF
         DO 30 I = LF, LL
            IM1 = I - 1
            IC = IWORK( INODE+IM1 )
            NL = IWORK( NDIML+IM1 )
            NR = IWORK( NDIMR+IM1 )
            NLF = IC - NL
            NRF = IC + 1
            J = J - 1
            CALL DLALS0( ICOMPQ, NL, NR, SQRE, NRHS, BX( NLF, 1 ),
     $                   LDBX,
     $                   B( NLF, 1 ), LDB, PERM( NLF, LVL ),
     $                   GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL,
     $                   GIVNUM( NLF, LVL2 ), LDU, POLES( NLF, LVL2 ),
     $                   DIFL( NLF, LVL ), DIFR( NLF, LVL2 ),
     $                   Z( NLF, LVL ), K( J ), C( J ), S( J ), WORK,
     $                   INFO )
   30    CONTINUE
   40 CONTINUE
      GO TO 90
*
*     ICOMPQ = 1: applying back the right singular vector factors.
*
   50 CONTINUE
*
*     First now go through the right singular vector matrices of all
*     the tree nodes top-down.
*
      J = 0
      DO 70 LVL = 1, NLVL
         LVL2 = 2*LVL - 1
*
*        Find the first node LF and last node LL on
*        the current level LVL.
*
         IF( LVL.EQ.1 ) THEN
            LF = 1
            LL = 1
         ELSE
            LF = 2**( LVL-1 )
            LL = 2*LF - 1
         END IF
         DO 60 I = LL, LF, -1
            IM1 = I - 1
            IC = IWORK( INODE+IM1 )
            NL = IWORK( NDIML+IM1 )
            NR = IWORK( NDIMR+IM1 )
            NLF = IC - NL
            NRF = IC + 1
            IF( I.EQ.LL ) THEN
               SQRE = 0
            ELSE
               SQRE = 1
            END IF
            J = J + 1
            CALL DLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B( NLF, 1 ),
     $                   LDB,
     $                   BX( NLF, 1 ), LDBX, PERM( NLF, LVL ),
     $                   GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL,
     $                   GIVNUM( NLF, LVL2 ), LDU, POLES( NLF, LVL2 ),
     $                   DIFL( NLF, LVL ), DIFR( NLF, LVL2 ),
     $                   Z( NLF, LVL ), K( J ), C( J ), S( J ), WORK,
     $                   INFO )
   60    CONTINUE
   70 CONTINUE
*
*     The nodes on the bottom level of the tree were solved
*     by DLASDQ. The corresponding right singular vector
*     matrices are in explicit form. Apply them back.
*
      NDB1 = ( ND+1 ) / 2
      DO 80 I = NDB1, ND
         I1 = I - 1
         IC = IWORK( INODE+I1 )
         NL = IWORK( NDIML+I1 )
         NR = IWORK( NDIMR+I1 )
         NLP1 = NL + 1
         IF( I.EQ.ND ) THEN
            NRP1 = NR
         ELSE
            NRP1 = NR + 1
         END IF
         NLF = IC - NL
         NRF = IC + 1
         CALL DGEMM( 'T', 'N', NLP1, NRHS, NLP1, ONE, VT( NLF, 1 ),
     $               LDU,
     $               B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX )
         CALL DGEMM( 'T', 'N', NRP1, NRHS, NRP1, ONE, VT( NRF, 1 ),
     $               LDU,
     $               B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX )
   80 CONTINUE
*
   90 CONTINUE
*
      RETURN
*
*     End of DLALSA
*
      END
*> \brief \b DLALSD uses the singular value decomposition of A to solve the least squares problem.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLALSD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlalsd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlalsd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlalsd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND,
*                          RANK, WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDB, N, NRHS, RANK, SMLSIZ
*       DOUBLE PRECISION   RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   B( LDB, * ), D( * ), E( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLALSD uses the singular value decomposition of A to solve the least
*> squares problem of finding X to minimize the Euclidean norm of each
*> column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
*> are N-by-NRHS. The solution X overwrites B.
*>
*> The singular values of A smaller than RCOND times the largest
*> singular value are treated as zero in solving the least squares
*> problem; in this case a minimum norm solution is returned.
*> The actual singular values are returned in D in ascending order.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>         = 'U': D and E define an upper bidiagonal matrix.
*>         = 'L': D and E define a  lower bidiagonal matrix.
*> \endverbatim
*>
*> \param[in] SMLSIZ
*> \verbatim
*>          SMLSIZ is INTEGER
*>         The maximum size of the subproblems at the bottom of the
*>         computation tree.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>         The dimension of the  bidiagonal matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>         The number of columns of B. NRHS must be at least 1.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>         On entry D contains the main diagonal of the bidiagonal
*>         matrix. On exit, if INFO = 0, D contains its singular values.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>         Contains the super-diagonal entries of the bidiagonal matrix.
*>         On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>         On input, B contains the right hand sides of the least
*>         squares problem. On output, B contains the solution X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>         The leading dimension of B in the calling subprogram.
*>         LDB must be at least max(1,N).
*> \endverbatim
*>
*> \param[in] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>         The singular values of A less than or equal to RCOND times
*>         the largest singular value are treated as zero in solving
*>         the least squares problem. If RCOND is negative,
*>         machine precision is used instead.
*>         For example, if diag(S)*X=B were the least squares problem,
*>         where diag(S) is a diagonal matrix of singular values, the
*>         solution would be X(i) = B(i) / S(i) if S(i) is greater than
*>         RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
*>         RCOND*max(S).
*> \endverbatim
*>
*> \param[out] RANK
*> \verbatim
*>          RANK is INTEGER
*>         The number of singular values of A greater than RCOND times
*>         the largest singular value.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension at least
*>         (9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2),
*>         where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension at least
*>         (3*N*NLVL + 11*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>         = 0:  successful exit.
*>         < 0:  if INFO = -i, the i-th argument had an illegal value.
*>         > 0:  The algorithm failed to compute a singular value while
*>               working on the submatrix lying in rows and columns
*>               INFO/(N+1) through MOD(INFO,N+1).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lalsd
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Ren-Cang Li, Computer Science Division, University of
*>       California at Berkeley, USA \n
*>     Osni Marques, LBNL/NERSC, USA \n
*
*  =====================================================================
      SUBROUTINE DLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND,
     $                   RANK, WORK, IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDB, N, NRHS, RANK, SMLSIZ
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   B( LDB, * ), D( * ), E( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            BX, BXST, C, DIFL, DIFR, GIVCOL, GIVNUM,
     $                   GIVPTR, I, ICMPQ1, ICMPQ2, IWK, J, K, NLVL,
     $                   NM1, NSIZE, NSUB, NWORK, PERM, POLES, S, SIZEI,
     $                   SMLSZP, SQRE, ST, ST1, U, VT, Z
      DOUBLE PRECISION   CS, EPS, ORGNRM, R, RCND, SN, TOL
*     ..
*     .. External Functions ..
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH, DLANST
      EXTERNAL           IDAMAX, DLAMCH, DLANST
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGEMM, DLACPY, DLALSA, DLARTG,
     $                   DLASCL,
     $                   DLASDA, DLASDQ, DLASET, DLASRT, DROT, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, INT, LOG, SIGN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( NRHS.LT.1 ) THEN
         INFO = -4
      ELSE IF( ( LDB.LT.1 ) .OR. ( LDB.LT.N ) ) THEN
         INFO = -8
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLALSD', -INFO )
         RETURN
      END IF
*
      EPS = DLAMCH( 'Epsilon' )
*
*     Set up the tolerance.
*
      IF( ( RCOND.LE.ZERO ) .OR. ( RCOND.GE.ONE ) ) THEN
         RCND = EPS
      ELSE
         RCND = RCOND
      END IF
*
      RANK = 0
*
*     Quick return if possible.
*
      IF( N.EQ.0 ) THEN
         RETURN
      ELSE IF( N.EQ.1 ) THEN
         IF( D( 1 ).EQ.ZERO ) THEN
            CALL DLASET( 'A', 1, NRHS, ZERO, ZERO, B, LDB )
         ELSE
            RANK = 1
            CALL DLASCL( 'G', 0, 0, D( 1 ), ONE, 1, NRHS, B, LDB,
     $                   INFO )
            D( 1 ) = ABS( D( 1 ) )
         END IF
         RETURN
      END IF
*
*     Rotate the matrix if it is lower bidiagonal.
*
      IF( UPLO.EQ.'L' ) THEN
         DO 10 I = 1, N - 1
            CALL DLARTG( D( I ), E( I ), CS, SN, R )
            D( I ) = R
            E( I ) = SN*D( I+1 )
            D( I+1 ) = CS*D( I+1 )
            IF( NRHS.EQ.1 ) THEN
               CALL DROT( 1, B( I, 1 ), 1, B( I+1, 1 ), 1, CS, SN )
            ELSE
               WORK( I*2-1 ) = CS
               WORK( I*2 ) = SN
            END IF
   10    CONTINUE
         IF( NRHS.GT.1 ) THEN
            DO 30 I = 1, NRHS
               DO 20 J = 1, N - 1
                  CS = WORK( J*2-1 )
                  SN = WORK( J*2 )
                  CALL DROT( 1, B( J, I ), 1, B( J+1, I ), 1, CS,
     $                       SN )
   20          CONTINUE
   30       CONTINUE
         END IF
      END IF
*
*     Scale.
*
      NM1 = N - 1
      ORGNRM = DLANST( 'M', N, D, E )
      IF( ORGNRM.EQ.ZERO ) THEN
         CALL DLASET( 'A', N, NRHS, ZERO, ZERO, B, LDB )
         RETURN
      END IF
*
      CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO )
      CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, NM1, 1, E, NM1, INFO )
*
*     If N is smaller than the minimum divide size SMLSIZ, then solve
*     the problem with another solver.
*
      IF( N.LE.SMLSIZ ) THEN
         NWORK = 1 + N*N
         CALL DLASET( 'A', N, N, ZERO, ONE, WORK, N )
         CALL DLASDQ( 'U', 0, N, N, 0, NRHS, D, E, WORK, N, WORK, N,
     $                B,
     $                LDB, WORK( NWORK ), INFO )
         IF( INFO.NE.0 ) THEN
            RETURN
         END IF
         TOL = RCND*ABS( D( IDAMAX( N, D, 1 ) ) )
         DO 40 I = 1, N
            IF( D( I ).LE.TOL ) THEN
               CALL DLASET( 'A', 1, NRHS, ZERO, ZERO, B( I, 1 ),
     $                      LDB )
            ELSE
               CALL DLASCL( 'G', 0, 0, D( I ), ONE, 1, NRHS, B( I,
     $                      1 ),
     $                      LDB, INFO )
               RANK = RANK + 1
            END IF
   40    CONTINUE
         CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, WORK, N, B, LDB,
     $               ZERO,
     $               WORK( NWORK ), N )
         CALL DLACPY( 'A', N, NRHS, WORK( NWORK ), N, B, LDB )
*
*        Unscale.
*
         CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
         CALL DLASRT( 'D', N, D, INFO )
         CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, NRHS, B, LDB, INFO )
*
         RETURN
      END IF
*
*     Book-keeping and setting up some constants.
*
      NLVL = INT( LOG( DBLE( N ) / DBLE( SMLSIZ+1 ) ) / LOG( TWO ) ) + 1
*
      SMLSZP = SMLSIZ + 1
*
      U = 1
      VT = 1 + SMLSIZ*N
      DIFL = VT + SMLSZP*N
      DIFR = DIFL + NLVL*N
      Z = DIFR + NLVL*N*2
      C = Z + NLVL*N
      S = C + N
      POLES = S + N
      GIVNUM = POLES + 2*NLVL*N
      BX = GIVNUM + 2*NLVL*N
      NWORK = BX + N*NRHS
*
      SIZEI = 1 + N
      K = SIZEI + N
      GIVPTR = K + N
      PERM = GIVPTR + N
      GIVCOL = PERM + NLVL*N
      IWK = GIVCOL + NLVL*N*2
*
      ST = 1
      SQRE = 0
      ICMPQ1 = 1
      ICMPQ2 = 0
      NSUB = 0
*
      DO 50 I = 1, N
         IF( ABS( D( I ) ).LT.EPS ) THEN
            D( I ) = SIGN( EPS, D( I ) )
         END IF
   50 CONTINUE
*
      DO 60 I = 1, NM1
         IF( ( ABS( E( I ) ).LT.EPS ) .OR. ( I.EQ.NM1 ) ) THEN
            NSUB = NSUB + 1
            IWORK( NSUB ) = ST
*
*           Subproblem found. First determine its size and then
*           apply divide and conquer on it.
*
            IF( I.LT.NM1 ) THEN
*
*              A subproblem with E(I) small for I < NM1.
*
               NSIZE = I - ST + 1
               IWORK( SIZEI+NSUB-1 ) = NSIZE
            ELSE IF( ABS( E( I ) ).GE.EPS ) THEN
*
*              A subproblem with E(NM1) not too small but I = NM1.
*
               NSIZE = N - ST + 1
               IWORK( SIZEI+NSUB-1 ) = NSIZE
            ELSE
*
*              A subproblem with E(NM1) small. This implies an
*              1-by-1 subproblem at D(N), which is not solved
*              explicitly.
*
               NSIZE = I - ST + 1
               IWORK( SIZEI+NSUB-1 ) = NSIZE
               NSUB = NSUB + 1
               IWORK( NSUB ) = N
               IWORK( SIZEI+NSUB-1 ) = 1
               CALL DCOPY( NRHS, B( N, 1 ), LDB, WORK( BX+NM1 ), N )
            END IF
            ST1 = ST - 1
            IF( NSIZE.EQ.1 ) THEN
*
*              This is a 1-by-1 subproblem and is not solved
*              explicitly.
*
               CALL DCOPY( NRHS, B( ST, 1 ), LDB, WORK( BX+ST1 ), N )
            ELSE IF( NSIZE.LE.SMLSIZ ) THEN
*
*              This is a small subproblem and is solved by DLASDQ.
*
               CALL DLASET( 'A', NSIZE, NSIZE, ZERO, ONE,
     $                      WORK( VT+ST1 ), N )
               CALL DLASDQ( 'U', 0, NSIZE, NSIZE, 0, NRHS, D( ST ),
     $                      E( ST ), WORK( VT+ST1 ), N, WORK( NWORK ),
     $                      N, B( ST, 1 ), LDB, WORK( NWORK ), INFO )
               IF( INFO.NE.0 ) THEN
                  RETURN
               END IF
               CALL DLACPY( 'A', NSIZE, NRHS, B( ST, 1 ), LDB,
     $                      WORK( BX+ST1 ), N )
            ELSE
*
*              A large problem. Solve it using divide and conquer.
*
               CALL DLASDA( ICMPQ1, SMLSIZ, NSIZE, SQRE, D( ST ),
     $                      E( ST ), WORK( U+ST1 ), N, WORK( VT+ST1 ),
     $                      IWORK( K+ST1 ), WORK( DIFL+ST1 ),
     $                      WORK( DIFR+ST1 ), WORK( Z+ST1 ),
     $                      WORK( POLES+ST1 ), IWORK( GIVPTR+ST1 ),
     $                      IWORK( GIVCOL+ST1 ), N, IWORK( PERM+ST1 ),
     $                      WORK( GIVNUM+ST1 ), WORK( C+ST1 ),
     $                      WORK( S+ST1 ), WORK( NWORK ), IWORK( IWK ),
     $                      INFO )
               IF( INFO.NE.0 ) THEN
                  RETURN
               END IF
               BXST = BX + ST1
               CALL DLALSA( ICMPQ2, SMLSIZ, NSIZE, NRHS, B( ST, 1 ),
     $                      LDB, WORK( BXST ), N, WORK( U+ST1 ), N,
     $                      WORK( VT+ST1 ), IWORK( K+ST1 ),
     $                      WORK( DIFL+ST1 ), WORK( DIFR+ST1 ),
     $                      WORK( Z+ST1 ), WORK( POLES+ST1 ),
     $                      IWORK( GIVPTR+ST1 ), IWORK( GIVCOL+ST1 ), N,
     $                      IWORK( PERM+ST1 ), WORK( GIVNUM+ST1 ),
     $                      WORK( C+ST1 ), WORK( S+ST1 ), WORK( NWORK ),
     $                      IWORK( IWK ), INFO )
               IF( INFO.NE.0 ) THEN
                  RETURN
               END IF
            END IF
            ST = I + 1
         END IF
   60 CONTINUE
*
*     Apply the singular values and treat the tiny ones as zero.
*
      TOL = RCND*ABS( D( IDAMAX( N, D, 1 ) ) )
*
      DO 70 I = 1, N
*
*        Some of the elements in D can be negative because 1-by-1
*        subproblems were not solved explicitly.
*
         IF( ABS( D( I ) ).LE.TOL ) THEN
            CALL DLASET( 'A', 1, NRHS, ZERO, ZERO, WORK( BX+I-1 ),
     $                   N )
         ELSE
            RANK = RANK + 1
            CALL DLASCL( 'G', 0, 0, D( I ), ONE, 1, NRHS,
     $                   WORK( BX+I-1 ), N, INFO )
         END IF
         D( I ) = ABS( D( I ) )
   70 CONTINUE
*
*     Now apply back the right singular vectors.
*
      ICMPQ2 = 1
      DO 80 I = 1, NSUB
         ST = IWORK( I )
         ST1 = ST - 1
         NSIZE = IWORK( SIZEI+I-1 )
         BXST = BX + ST1
         IF( NSIZE.EQ.1 ) THEN
            CALL DCOPY( NRHS, WORK( BXST ), N, B( ST, 1 ), LDB )
         ELSE IF( NSIZE.LE.SMLSIZ ) THEN
            CALL DGEMM( 'T', 'N', NSIZE, NRHS, NSIZE, ONE,
     $                  WORK( VT+ST1 ), N, WORK( BXST ), N, ZERO,
     $                  B( ST, 1 ), LDB )
         ELSE
            CALL DLALSA( ICMPQ2, SMLSIZ, NSIZE, NRHS, WORK( BXST ),
     $                   N,
     $                   B( ST, 1 ), LDB, WORK( U+ST1 ), N,
     $                   WORK( VT+ST1 ), IWORK( K+ST1 ),
     $                   WORK( DIFL+ST1 ), WORK( DIFR+ST1 ),
     $                   WORK( Z+ST1 ), WORK( POLES+ST1 ),
     $                   IWORK( GIVPTR+ST1 ), IWORK( GIVCOL+ST1 ), N,
     $                   IWORK( PERM+ST1 ), WORK( GIVNUM+ST1 ),
     $                   WORK( C+ST1 ), WORK( S+ST1 ), WORK( NWORK ),
     $                   IWORK( IWK ), INFO )
            IF( INFO.NE.0 ) THEN
               RETURN
            END IF
         END IF
   80 CONTINUE
*
*     Unscale and sort the singular values.
*
      CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
      CALL DLASRT( 'D', N, D, INFO )
      CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, NRHS, B, LDB, INFO )
*
      RETURN
*
*     End of DLALSD
*
      END
*> \brief \b DLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single set sorted in ascending order.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAMRG + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlamrg.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlamrg.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlamrg.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAMRG( N1, N2, A, DTRD1, DTRD2, INDEX )
*
*       .. Scalar Arguments ..
*       INTEGER            DTRD1, DTRD2, N1, N2
*       ..
*       .. Array Arguments ..
*       INTEGER            INDEX( * )
*       DOUBLE PRECISION   A( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAMRG will create a permutation list which will merge the elements
*> of A (which is composed of two independently sorted sets) into a
*> single set which is sorted in ascending order.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N1
*> \verbatim
*>          N1 is INTEGER
*> \endverbatim
*>
*> \param[in] N2
*> \verbatim
*>          N2 is INTEGER
*>         These arguments contain the respective lengths of the two
*>         sorted lists to be merged.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (N1+N2)
*>         The first N1 elements of A contain a list of numbers which
*>         are sorted in either ascending or descending order.  Likewise
*>         for the final N2 elements.
*> \endverbatim
*>
*> \param[in] DTRD1
*> \verbatim
*>          DTRD1 is INTEGER
*> \endverbatim
*>
*> \param[in] DTRD2
*> \verbatim
*>          DTRD2 is INTEGER
*>         These are the strides to be taken through the array A.
*>         Allowable strides are 1 and -1.  They indicate whether a
*>         subset of A is sorted in ascending (DTRDx = 1) or descending
*>         (DTRDx = -1) order.
*> \endverbatim
*>
*> \param[out] INDEX
*> \verbatim
*>          INDEX is INTEGER array, dimension (N1+N2)
*>         On exit this array will contain a permutation such that
*>         if B( I ) = A( INDEX( I ) ) for I=1,N1+N2, then B will be
*>         sorted in ascending order.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lamrg
*
*  =====================================================================
      SUBROUTINE DLAMRG( N1, N2, A, DTRD1, DTRD2, INDEX )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            DTRD1, DTRD2, N1, N2
*     ..
*     .. Array Arguments ..
      INTEGER            INDEX( * )
      DOUBLE PRECISION   A( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            I, IND1, IND2, N1SV, N2SV
*     ..
*     .. Executable Statements ..
*
      N1SV = N1
      N2SV = N2
      IF( DTRD1.GT.0 ) THEN
         IND1 = 1
      ELSE
         IND1 = N1
      END IF
      IF( DTRD2.GT.0 ) THEN
         IND2 = 1 + N1
      ELSE
         IND2 = N1 + N2
      END IF
      I = 1
*     while ( (N1SV > 0) & (N2SV > 0) )
   10 CONTINUE
      IF( N1SV.GT.0 .AND. N2SV.GT.0 ) THEN
         IF( A( IND1 ).LE.A( IND2 ) ) THEN
            INDEX( I ) = IND1
            I = I + 1
            IND1 = IND1 + DTRD1
            N1SV = N1SV - 1
         ELSE
            INDEX( I ) = IND2
            I = I + 1
            IND2 = IND2 + DTRD2
            N2SV = N2SV - 1
         END IF
         GO TO 10
      END IF
*     end while
      IF( N1SV.EQ.0 ) THEN
         DO 20 N1SV = 1, N2SV
            INDEX( I ) = IND2
            I = I + 1
            IND2 = IND2 + DTRD2
   20    CONTINUE
      ELSE
*     N2SV .EQ. 0
         DO 30 N2SV = 1, N1SV
            INDEX( I ) = IND1
            I = I + 1
            IND1 = IND1 + DTRD1
   30    CONTINUE
      END IF
*
      RETURN
*
*     End of DLAMRG
*
      END
*> \brief \b DLANEG computes the Sturm count.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANEG + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaneg.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaneg.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaneg.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       INTEGER FUNCTION DLANEG( N, D, LLD, SIGMA, PIVMIN, R )
*
*       .. Scalar Arguments ..
*       INTEGER            N, R
*       DOUBLE PRECISION   PIVMIN, SIGMA
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), LLD( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLANEG computes the Sturm count, the number of negative pivots
*> encountered while factoring tridiagonal T - sigma I = L D L^T.
*> This implementation works directly on the factors without forming
*> the tridiagonal matrix T.  The Sturm count is also the number of
*> eigenvalues of T less than sigma.
*>
*> This routine is called from DLARRB.
*>
*> The current routine does not use the PIVMIN parameter but rather
*> requires IEEE-754 propagation of Infinities and NaNs.  This
*> routine also has no input range restrictions but does require
*> default exception handling such that x/0 produces Inf when x is
*> non-zero, and Inf/Inf produces NaN.  For more information, see:
*>
*>   Marques, Riedy, and Voemel, "Benefits of IEEE-754 Features in
*>   Modern Symmetric Tridiagonal Eigensolvers," SIAM Journal on
*>   Scientific Computing, v28, n5, 2006.  DOI 10.1137/050641624
*>   (Tech report version in LAWN 172 with the same title.)
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The N diagonal elements of the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] LLD
*> \verbatim
*>          LLD is DOUBLE PRECISION array, dimension (N-1)
*>          The (N-1) elements L(i)*L(i)*D(i).
*> \endverbatim
*>
*> \param[in] SIGMA
*> \verbatim
*>          SIGMA is DOUBLE PRECISION
*>          Shift amount in T - sigma I = L D L^T.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*>          PIVMIN is DOUBLE PRECISION
*>          The minimum pivot in the Sturm sequence.  May be used
*>          when zero pivots are encountered on non-IEEE-754
*>          architectures.
*> \endverbatim
*>
*> \param[in] R
*> \verbatim
*>          R is INTEGER
*>          The twist index for the twisted factorization that is used
*>          for the negcount.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laneg
*
*> \par Contributors:
*  ==================
*>
*>     Osni Marques, LBNL/NERSC, USA \n
*>     Christof Voemel, University of California, Berkeley, USA \n
*>     Jason Riedy, University of California, Berkeley, USA \n
*>
*  =====================================================================
      INTEGER FUNCTION DLANEG( N, D, LLD, SIGMA, PIVMIN, R )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            N, R
      DOUBLE PRECISION   PIVMIN, SIGMA
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), LLD( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER        ( ZERO = 0.0D0, ONE = 1.0D0 )
*     Some architectures propagate Infinities and NaNs very slowly, so
*     the code computes counts in BLKLEN chunks.  Then a NaN can
*     propagate at most BLKLEN columns before being detected.  This is
*     not a general tuning parameter; it needs only to be just large
*     enough that the overhead is tiny in common cases.
      INTEGER BLKLEN
      PARAMETER ( BLKLEN = 128 )
*     ..
*     .. Local Scalars ..
      INTEGER            BJ, J, NEG1, NEG2, NEGCNT
      DOUBLE PRECISION   BSAV, DMINUS, DPLUS, GAMMA, P, T, TMP
      LOGICAL SAWNAN
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC MIN, MAX
*     ..
*     .. External Functions ..
      LOGICAL DISNAN
      EXTERNAL DISNAN
*     ..
*     .. Executable Statements ..

      NEGCNT = 0

*     I) upper part: L D L^T - SIGMA I = L+ D+ L+^T
      T = -SIGMA
      DO 210 BJ = 1, R-1, BLKLEN
         NEG1 = 0
         BSAV = T
         DO 21 J = BJ, MIN(BJ+BLKLEN-1, R-1)
            DPLUS = D( J ) + T
            IF( DPLUS.LT.ZERO ) NEG1 = NEG1 + 1
            TMP = T / DPLUS
            T = TMP * LLD( J ) - SIGMA
 21      CONTINUE
         SAWNAN = DISNAN( T )
*     Run a slower version of the above loop if a NaN is detected.
*     A NaN should occur only with a zero pivot after an infinite
*     pivot.  In that case, substituting 1 for T/DPLUS is the
*     correct limit.
         IF( SAWNAN ) THEN
            NEG1 = 0
            T = BSAV
            DO 22 J = BJ, MIN(BJ+BLKLEN-1, R-1)
               DPLUS = D( J ) + T
               IF( DPLUS.LT.ZERO ) NEG1 = NEG1 + 1
               TMP = T / DPLUS
               IF (DISNAN(TMP)) TMP = ONE
               T = TMP * LLD(J) - SIGMA
 22         CONTINUE
         END IF
         NEGCNT = NEGCNT + NEG1
 210  CONTINUE
*
*     II) lower part: L D L^T - SIGMA I = U- D- U-^T
      P = D( N ) - SIGMA
      DO 230 BJ = N-1, R, -BLKLEN
         NEG2 = 0
         BSAV = P
         DO 23 J = BJ, MAX(BJ-BLKLEN+1, R), -1
            DMINUS = LLD( J ) + P
            IF( DMINUS.LT.ZERO ) NEG2 = NEG2 + 1
            TMP = P / DMINUS
            P = TMP * D( J ) - SIGMA
 23      CONTINUE
         SAWNAN = DISNAN( P )
*     As above, run a slower version that substitutes 1 for Inf/Inf.
*
         IF( SAWNAN ) THEN
            NEG2 = 0
            P = BSAV
            DO 24 J = BJ, MAX(BJ-BLKLEN+1, R), -1
               DMINUS = LLD( J ) + P
               IF( DMINUS.LT.ZERO ) NEG2 = NEG2 + 1
               TMP = P / DMINUS
               IF (DISNAN(TMP)) TMP = ONE
               P = TMP * D(J) - SIGMA
 24         CONTINUE
         END IF
         NEGCNT = NEGCNT + NEG2
 230  CONTINUE
*
*     III) Twist index
*       T was shifted by SIGMA initially.
      GAMMA = (T + SIGMA) + P
      IF( GAMMA.LT.ZERO ) NEGCNT = NEGCNT+1

      DLANEG = NEGCNT
      END
*> \brief \b DLANGB returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of general band matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANGB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlangb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlangb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlangb.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       DOUBLE PRECISION FUNCTION DLANGB( NORM, N, KL, KU, AB, LDAB,
*                        WORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          NORM
*       INTEGER            KL, KU, LDAB, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLANGB  returns the value of the one norm,  or the Frobenius norm, or
*> the  infinity norm,  or the element of  largest absolute value  of an
*> n by n band matrix  A,  with kl sub-diagonals and ku super-diagonals.
*> \endverbatim
*>
*> \return DLANGB
*> \verbatim
*>
*>    DLANGB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*>             (
*>             ( norm1(A),         NORM = '1', 'O' or 'o'
*>             (
*>             ( normI(A),         NORM = 'I' or 'i'
*>             (
*>             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
*>
*> where  norm1  denotes the  one norm of a matrix (maximum column sum),
*> normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
*> normF  denotes the  Frobenius norm of a matrix (square root of sum of
*> squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NORM
*> \verbatim
*>          NORM is CHARACTER*1
*>          Specifies the value to be returned in DLANGB as described
*>          above.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.  When N = 0, DLANGB is
*>          set to zero.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*>          KL is INTEGER
*>          The number of sub-diagonals of the matrix A.  KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*>          KU is INTEGER
*>          The number of super-diagonals of the matrix A.  KU >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          The band matrix A, stored in rows 1 to KL+KU+1.  The j-th
*>          column of A is stored in the j-th column of the array AB as
*>          follows:
*>          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KL+KU+1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*>          where LWORK >= N when NORM = 'I'; otherwise, WORK is not
*>          referenced.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup langb
*
*  =====================================================================
      DOUBLE PRECISION FUNCTION DLANGB( NORM, N, KL, KU, AB, LDAB,
     $                 WORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          NORM
      INTEGER            KL, KU, LDAB, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * ), WORK( * )
*     ..
*
* =====================================================================
*
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J, K, L
      DOUBLE PRECISION   SCALE, SUM, VALUE, TEMP
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASSQ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      EXTERNAL           LSAME, DISNAN
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
      IF( N.EQ.0 ) THEN
         VALUE = ZERO
      ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
*        Find max(abs(A(i,j))).
*
         VALUE = ZERO
         DO 20 J = 1, N
            DO 10 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 )
               TEMP = ABS( AB( I, J ) )
               IF( VALUE.LT.TEMP .OR. DISNAN( TEMP ) ) VALUE = TEMP
   10       CONTINUE
   20    CONTINUE
      ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN
*
*        Find norm1(A).
*
         VALUE = ZERO
         DO 40 J = 1, N
            SUM = ZERO
            DO 30 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 )
               SUM = SUM + ABS( AB( I, J ) )
   30       CONTINUE
            IF( VALUE.LT.SUM .OR. DISNAN( SUM ) ) VALUE = SUM
   40    CONTINUE
      ELSE IF( LSAME( NORM, 'I' ) ) THEN
*
*        Find normI(A).
*
         DO 50 I = 1, N
            WORK( I ) = ZERO
   50    CONTINUE
         DO 70 J = 1, N
            K = KU + 1 - J
            DO 60 I = MAX( 1, J-KU ), MIN( N, J+KL )
               WORK( I ) = WORK( I ) + ABS( AB( K+I, J ) )
   60       CONTINUE
   70    CONTINUE
         VALUE = ZERO
         DO 80 I = 1, N
            TEMP = WORK( I )
            IF( VALUE.LT.TEMP .OR. DISNAN( TEMP ) ) VALUE = TEMP
   80    CONTINUE
      ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
     $         ( LSAME( NORM, 'E' ) ) ) THEN
*
*        Find normF(A).
*
         SCALE = ZERO
         SUM = ONE
         DO 90 J = 1, N
            L = MAX( 1, J-KU )
            K = KU + 1 - J + L
            CALL DLASSQ( MIN( N, J+KL )-L+1, AB( K, J ), 1, SCALE,
     $                   SUM )
   90    CONTINUE
         VALUE = SCALE*SQRT( SUM )
      END IF
*
      DLANGB = VALUE
      RETURN
*
*     End of DLANGB
*
      END
*> \brief \b DLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general rectangular matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANGE + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlange.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlange.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlange.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       DOUBLE PRECISION FUNCTION DLANGE( NORM, M, N, A, LDA, WORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          NORM
*       INTEGER            LDA, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLANGE  returns the value of the one norm,  or the Frobenius norm, or
*> the  infinity norm,  or the  element of  largest absolute value  of a
*> real matrix A.
*> \endverbatim
*>
*> \return DLANGE
*> \verbatim
*>
*>    DLANGE = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*>             (
*>             ( norm1(A),         NORM = '1', 'O' or 'o'
*>             (
*>             ( normI(A),         NORM = 'I' or 'i'
*>             (
*>             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
*>
*> where  norm1  denotes the  one norm of a matrix (maximum column sum),
*> normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
*> normF  denotes the  Frobenius norm of a matrix (square root of sum of
*> squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NORM
*> \verbatim
*>          NORM is CHARACTER*1
*>          Specifies the value to be returned in DLANGE as described
*>          above.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.  When M = 0,
*>          DLANGE is set to zero.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.  When N = 0,
*>          DLANGE is set to zero.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The m by n matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(M,1).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*>          where LWORK >= M when NORM = 'I'; otherwise, WORK is not
*>          referenced.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lange
*
*  =====================================================================
      DOUBLE PRECISION FUNCTION DLANGE( NORM, M, N, A, LDA, WORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          NORM
      INTEGER            LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), WORK( * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J
      DOUBLE PRECISION   SCALE, SUM, VALUE, TEMP
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASSQ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      EXTERNAL           LSAME, DISNAN
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
      IF( MIN( M, N ).EQ.0 ) THEN
         VALUE = ZERO
      ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
*        Find max(abs(A(i,j))).
*
         VALUE = ZERO
         DO 20 J = 1, N
            DO 10 I = 1, M
               TEMP = ABS( A( I, J ) )
               IF( VALUE.LT.TEMP .OR. DISNAN( TEMP ) ) VALUE = TEMP
   10       CONTINUE
   20    CONTINUE
      ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN
*
*        Find norm1(A).
*
         VALUE = ZERO
         DO 40 J = 1, N
            SUM = ZERO
            DO 30 I = 1, M
               SUM = SUM + ABS( A( I, J ) )
   30       CONTINUE
            IF( VALUE.LT.SUM .OR. DISNAN( SUM ) ) VALUE = SUM
   40    CONTINUE
      ELSE IF( LSAME( NORM, 'I' ) ) THEN
*
*        Find normI(A).
*
         DO 50 I = 1, M
            WORK( I ) = ZERO
   50    CONTINUE
         DO 70 J = 1, N
            DO 60 I = 1, M
               WORK( I ) = WORK( I ) + ABS( A( I, J ) )
   60       CONTINUE
   70    CONTINUE
         VALUE = ZERO
         DO 80 I = 1, M
            TEMP = WORK( I )
            IF( VALUE.LT.TEMP .OR. DISNAN( TEMP ) ) VALUE = TEMP
   80    CONTINUE
      ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
     $         ( LSAME( NORM, 'E' ) ) ) THEN
*
*        Find normF(A).
*
         SCALE = ZERO
         SUM = ONE
         DO 90 J = 1, N
            CALL DLASSQ( M, A( 1, J ), 1, SCALE, SUM )
   90    CONTINUE
         VALUE = SCALE*SQRT( SUM )
      END IF
*
      DLANGE = VALUE
      RETURN
*
*     End of DLANGE
*
      END
*> \brief \b DLANGT returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general tridiagonal matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANGT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlangt.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlangt.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlangt.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       DOUBLE PRECISION FUNCTION DLANGT( NORM, N, DL, D, DU )
*
*       .. Scalar Arguments ..
*       CHARACTER          NORM
*       INTEGER            N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), DL( * ), DU( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLANGT  returns the value of the one norm,  or the Frobenius norm, or
*> the  infinity norm,  or the  element of  largest absolute value  of a
*> real tridiagonal matrix A.
*> \endverbatim
*>
*> \return DLANGT
*> \verbatim
*>
*>    DLANGT = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*>             (
*>             ( norm1(A),         NORM = '1', 'O' or 'o'
*>             (
*>             ( normI(A),         NORM = 'I' or 'i'
*>             (
*>             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
*>
*> where  norm1  denotes the  one norm of a matrix (maximum column sum),
*> normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
*> normF  denotes the  Frobenius norm of a matrix (square root of sum of
*> squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NORM
*> \verbatim
*>          NORM is CHARACTER*1
*>          Specifies the value to be returned in DLANGT as described
*>          above.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.  When N = 0, DLANGT is
*>          set to zero.
*> \endverbatim
*>
*> \param[in] DL
*> \verbatim
*>          DL is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) sub-diagonal elements of A.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The diagonal elements of A.
*> \endverbatim
*>
*> \param[in] DU
*> \verbatim
*>          DU is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) super-diagonal elements of A.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup langt
*
*  =====================================================================
      DOUBLE PRECISION FUNCTION DLANGT( NORM, N, DL, D, DU )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          NORM
      INTEGER            N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), DL( * ), DU( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
      DOUBLE PRECISION   ANORM, SCALE, SUM, TEMP
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      EXTERNAL           LSAME, DISNAN
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASSQ
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, SQRT
*     ..
*     .. Executable Statements ..
*
      IF( N.LE.0 ) THEN
         ANORM = ZERO
      ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
*        Find max(abs(A(i,j))).
*
         ANORM = ABS( D( N ) )
         DO 10 I = 1, N - 1
            IF( ANORM.LT.ABS( DL( I ) ) .OR.
     $          DISNAN( ABS( DL( I ) ) ) )
     $           ANORM = ABS(DL(I))
            IF( ANORM.LT.ABS( D( I ) ) .OR. DISNAN( ABS( D( I ) ) ) )
     $           ANORM = ABS(D(I))
            IF( ANORM.LT.ABS( DU( I ) ) .OR.
     $          DISNAN (ABS( DU( I ) ) ) )
     $           ANORM = ABS(DU(I))
   10    CONTINUE
      ELSE IF( LSAME( NORM, 'O' ) .OR. NORM.EQ.'1' ) THEN
*
*        Find norm1(A).
*
         IF( N.EQ.1 ) THEN
            ANORM = ABS( D( 1 ) )
         ELSE
            ANORM = ABS( D( 1 ) )+ABS( DL( 1 ) )
            TEMP = ABS( D( N ) )+ABS( DU( N-1 ) )
            IF( ANORM .LT. TEMP .OR. DISNAN( TEMP ) ) ANORM = TEMP
            DO 20 I = 2, N - 1
               TEMP = ABS( D( I ) )+ABS( DL( I ) )+ABS( DU( I-1 ) )
               IF( ANORM .LT. TEMP .OR. DISNAN( TEMP ) ) ANORM = TEMP
   20       CONTINUE
         END IF
      ELSE IF( LSAME( NORM, 'I' ) ) THEN
*
*        Find normI(A).
*
         IF( N.EQ.1 ) THEN
            ANORM = ABS( D( 1 ) )
         ELSE
            ANORM = ABS( D( 1 ) )+ABS( DU( 1 ) )
            TEMP = ABS( D( N ) )+ABS( DL( N-1 ) )
            IF( ANORM .LT. TEMP .OR. DISNAN( TEMP ) ) ANORM = TEMP
            DO 30 I = 2, N - 1
               TEMP = ABS( D( I ) )+ABS( DU( I ) )+ABS( DL( I-1 ) )
               IF( ANORM .LT. TEMP .OR. DISNAN( TEMP ) ) ANORM = TEMP
   30       CONTINUE
         END IF
      ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
     $         ( LSAME( NORM, 'E' ) ) ) THEN
*
*        Find normF(A).
*
         SCALE = ZERO
         SUM = ONE
         CALL DLASSQ( N, D, 1, SCALE, SUM )
         IF( N.GT.1 ) THEN
            CALL DLASSQ( N-1, DL, 1, SCALE, SUM )
            CALL DLASSQ( N-1, DU, 1, SCALE, SUM )
         END IF
         ANORM = SCALE*SQRT( SUM )
      END IF
*
      DLANGT = ANORM
      RETURN
*
*     End of DLANGT
*
      END
*> \brief \b DLANHS returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of an upper Hessenberg matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANHS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlanhs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlanhs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlanhs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       DOUBLE PRECISION FUNCTION DLANHS( NORM, N, A, LDA, WORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          NORM
*       INTEGER            LDA, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLANHS  returns the value of the one norm,  or the Frobenius norm, or
*> the  infinity norm,  or the  element of  largest absolute value  of a
*> Hessenberg matrix A.
*> \endverbatim
*>
*> \return DLANHS
*> \verbatim
*>
*>    DLANHS = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*>             (
*>             ( norm1(A),         NORM = '1', 'O' or 'o'
*>             (
*>             ( normI(A),         NORM = 'I' or 'i'
*>             (
*>             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
*>
*> where  norm1  denotes the  one norm of a matrix (maximum column sum),
*> normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
*> normF  denotes the  Frobenius norm of a matrix (square root of sum of
*> squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NORM
*> \verbatim
*>          NORM is CHARACTER*1
*>          Specifies the value to be returned in DLANHS as described
*>          above.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.  When N = 0, DLANHS is
*>          set to zero.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The n by n upper Hessenberg matrix A; the part of A below the
*>          first sub-diagonal is not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(N,1).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*>          where LWORK >= N when NORM = 'I'; otherwise, WORK is not
*>          referenced.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lanhs
*
*  =====================================================================
      DOUBLE PRECISION FUNCTION DLANHS( NORM, N, A, LDA, WORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          NORM
      INTEGER            LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), WORK( * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J
      DOUBLE PRECISION   SCALE, SUM, VALUE
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASSQ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      EXTERNAL           LSAME, DISNAN
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
      IF( N.EQ.0 ) THEN
         VALUE = ZERO
      ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
*        Find max(abs(A(i,j))).
*
         VALUE = ZERO
         DO 20 J = 1, N
            DO 10 I = 1, MIN( N, J+1 )
               SUM = ABS( A( I, J ) )
               IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
   10       CONTINUE
   20    CONTINUE
      ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN
*
*        Find norm1(A).
*
         VALUE = ZERO
         DO 40 J = 1, N
            SUM = ZERO
            DO 30 I = 1, MIN( N, J+1 )
               SUM = SUM + ABS( A( I, J ) )
   30       CONTINUE
            IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
   40    CONTINUE
      ELSE IF( LSAME( NORM, 'I' ) ) THEN
*
*        Find normI(A).
*
         DO 50 I = 1, N
            WORK( I ) = ZERO
   50    CONTINUE
         DO 70 J = 1, N
            DO 60 I = 1, MIN( N, J+1 )
               WORK( I ) = WORK( I ) + ABS( A( I, J ) )
   60       CONTINUE
   70    CONTINUE
         VALUE = ZERO
         DO 80 I = 1, N
            SUM = WORK( I )
            IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
   80    CONTINUE
      ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
     $         ( LSAME( NORM, 'E' ) ) ) THEN
*
*        Find normF(A).
*
         SCALE = ZERO
         SUM = ONE
         DO 90 J = 1, N
            CALL DLASSQ( MIN( N, J+1 ), A( 1, J ), 1, SCALE, SUM )
   90    CONTINUE
         VALUE = SCALE*SQRT( SUM )
      END IF
*
      DLANHS = VALUE
      RETURN
*
*     End of DLANHS
*
      END
*> \brief \b DLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANSB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlansb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlansb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlansb.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       DOUBLE PRECISION FUNCTION DLANSB( NORM, UPLO, N, K, AB, LDAB,
*                        WORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          NORM, UPLO
*       INTEGER            K, LDAB, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLANSB  returns the value of the one norm,  or the Frobenius norm, or
*> the  infinity norm,  or the element of  largest absolute value  of an
*> n by n symmetric band matrix A,  with k super-diagonals.
*> \endverbatim
*>
*> \return DLANSB
*> \verbatim
*>
*>    DLANSB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*>             (
*>             ( norm1(A),         NORM = '1', 'O' or 'o'
*>             (
*>             ( normI(A),         NORM = 'I' or 'i'
*>             (
*>             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
*>
*> where  norm1  denotes the  one norm of a matrix (maximum column sum),
*> normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
*> normF  denotes the  Frobenius norm of a matrix (square root of sum of
*> squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NORM
*> \verbatim
*>          NORM is CHARACTER*1
*>          Specifies the value to be returned in DLANSB as described
*>          above.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the upper or lower triangular part of the
*>          band matrix A is supplied.
*>          = 'U':  Upper triangular part is supplied
*>          = 'L':  Lower triangular part is supplied
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.  When N = 0, DLANSB is
*>          set to zero.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of super-diagonals or sub-diagonals of the
*>          band matrix A.  K >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          The upper or lower triangle of the symmetric band matrix A,
*>          stored in the first K+1 rows of AB.  The j-th column of A is
*>          stored in the j-th column of the array AB as follows:
*>          if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= K+1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*>          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
*>          WORK is not referenced.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lanhb
*
*  =====================================================================
      DOUBLE PRECISION FUNCTION DLANSB( NORM, UPLO, N, K, AB, LDAB,
     $                 WORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          NORM, UPLO
      INTEGER            K, LDAB, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * ), WORK( * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J, L
      DOUBLE PRECISION   ABSA, SCALE, SUM, VALUE
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASSQ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      EXTERNAL           LSAME, DISNAN
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
      IF( N.EQ.0 ) THEN
         VALUE = ZERO
      ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
*        Find max(abs(A(i,j))).
*
         VALUE = ZERO
         IF( LSAME( UPLO, 'U' ) ) THEN
            DO 20 J = 1, N
               DO 10 I = MAX( K+2-J, 1 ), K + 1
                  SUM = ABS( AB( I, J ) )
                  IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
   10          CONTINUE
   20       CONTINUE
         ELSE
            DO 40 J = 1, N
               DO 30 I = 1, MIN( N+1-J, K+1 )
                  SUM = ABS( AB( I, J ) )
                  IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
   30          CONTINUE
   40       CONTINUE
         END IF
      ELSE IF( ( LSAME( NORM, 'I' ) ) .OR.
     $         ( LSAME( NORM, 'O' ) ) .OR.
     $         ( NORM.EQ.'1' ) ) THEN
*
*        Find normI(A) ( = norm1(A), since A is symmetric).
*
         VALUE = ZERO
         IF( LSAME( UPLO, 'U' ) ) THEN
            DO 60 J = 1, N
               SUM = ZERO
               L = K + 1 - J
               DO 50 I = MAX( 1, J-K ), J - 1
                  ABSA = ABS( AB( L+I, J ) )
                  SUM = SUM + ABSA
                  WORK( I ) = WORK( I ) + ABSA
   50          CONTINUE
               WORK( J ) = SUM + ABS( AB( K+1, J ) )
   60       CONTINUE
            DO 70 I = 1, N
               SUM = WORK( I )
               IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
   70       CONTINUE
         ELSE
            DO 80 I = 1, N
               WORK( I ) = ZERO
   80       CONTINUE
            DO 100 J = 1, N
               SUM = WORK( J ) + ABS( AB( 1, J ) )
               L = 1 - J
               DO 90 I = J + 1, MIN( N, J+K )
                  ABSA = ABS( AB( L+I, J ) )
                  SUM = SUM + ABSA
                  WORK( I ) = WORK( I ) + ABSA
   90          CONTINUE
               IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
  100       CONTINUE
         END IF
      ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
     $         ( LSAME( NORM, 'E' ) ) ) THEN
*
*        Find normF(A).
*
         SCALE = ZERO
         SUM = ONE
         IF( K.GT.0 ) THEN
            IF( LSAME( UPLO, 'U' ) ) THEN
               DO 110 J = 2, N
                  CALL DLASSQ( MIN( J-1, K ), AB( MAX( K+2-J, 1 ),
     $                         J ),
     $                         1, SCALE, SUM )
  110          CONTINUE
               L = K + 1
            ELSE
               DO 120 J = 1, N - 1
                  CALL DLASSQ( MIN( N-J, K ), AB( 2, J ), 1, SCALE,
     $                         SUM )
  120          CONTINUE
               L = 1
            END IF
            SUM = 2*SUM
         ELSE
            L = 1
         END IF
         CALL DLASSQ( N, AB( L, 1 ), LDAB, SCALE, SUM )
         VALUE = SCALE*SQRT( SUM )
      END IF
*
      DLANSB = VALUE
      RETURN
*
*     End of DLANSB
*
      END
*> \brief \b DLANSF returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix in RFP format.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANSF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlansf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlansf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlansf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       DOUBLE PRECISION FUNCTION DLANSF( NORM, TRANSR, UPLO, N, A, WORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          NORM, TRANSR, UPLO
*       INTEGER            N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( 0: * ), WORK( 0: * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLANSF returns the value of the one norm, or the Frobenius norm, or
*> the infinity norm, or the element of largest absolute value of a
*> real symmetric matrix A in RFP format.
*> \endverbatim
*>
*> \return DLANSF
*> \verbatim
*>
*>    DLANSF = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*>             (
*>             ( norm1(A),         NORM = '1', 'O' or 'o'
*>             (
*>             ( normI(A),         NORM = 'I' or 'i'
*>             (
*>             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
*>
*> where  norm1  denotes the  one norm of a matrix (maximum column sum),
*> normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
*> normF  denotes the  Frobenius norm of a matrix (square root of sum of
*> squares).  Note that  max(abs(A(i,j)))  is not a  matrix norm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NORM
*> \verbatim
*>          NORM is CHARACTER*1
*>          Specifies the value to be returned in DLANSF as described
*>          above.
*> \endverbatim
*>
*> \param[in] TRANSR
*> \verbatim
*>          TRANSR is CHARACTER*1
*>          Specifies whether the RFP format of A is normal or
*>          transposed format.
*>          = 'N':  RFP format is Normal;
*>          = 'T':  RFP format is Transpose.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>           On entry, UPLO specifies whether the RFP matrix A came from
*>           an upper or lower triangular matrix as follows:
*>           = 'U': RFP A came from an upper triangular matrix;
*>           = 'L': RFP A came from a lower triangular matrix.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A. N >= 0. When N = 0, DLANSF is
*>          set to zero.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 );
*>          On entry, the upper (if UPLO = 'U') or lower (if UPLO = 'L')
*>          part of the symmetric matrix A stored in RFP format. See the
*>          "Notes" below for more details.
*>          Unchanged on exit.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*>          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
*>          WORK is not referenced.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lanhf
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  We first consider Rectangular Full Packed (RFP) Format when N is
*>  even. We give an example where N = 6.
*>
*>      AP is Upper             AP is Lower
*>
*>   00 01 02 03 04 05       00
*>      11 12 13 14 15       10 11
*>         22 23 24 25       20 21 22
*>            33 34 35       30 31 32 33
*>               44 45       40 41 42 43 44
*>                  55       50 51 52 53 54 55
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*>  the transpose of the first three columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*>  the transpose of the last three columns of AP lower.
*>  This covers the case N even and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>        03 04 05                33 43 53
*>        13 14 15                00 44 54
*>        23 24 25                10 11 55
*>        33 34 35                20 21 22
*>        00 44 45                30 31 32
*>        01 11 55                40 41 42
*>        02 12 22                50 51 52
*>
*>  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*>  transpose of RFP A above. One therefore gets:
*>
*>
*>           RFP A                   RFP A
*>
*>     03 13 23 33 00 01 02    33 00 10 20 30 40 50
*>     04 14 24 34 44 11 12    43 44 11 21 31 41 51
*>     05 15 25 35 45 55 22    53 54 55 22 32 42 52
*>
*>
*>  We then consider Rectangular Full Packed (RFP) Format when N is
*>  odd. We give an example where N = 5.
*>
*>     AP is Upper                 AP is Lower
*>
*>   00 01 02 03 04              00
*>      11 12 13 14              10 11
*>         22 23 24              20 21 22
*>            33 34              30 31 32 33
*>               44              40 41 42 43 44
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*>  the transpose of the first two columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*>  the transpose of the last two columns of AP lower.
*>  This covers the case N odd and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>        02 03 04                00 33 43
*>        12 13 14                10 11 44
*>        22 23 24                20 21 22
*>        00 33 34                30 31 32
*>        01 11 44                40 41 42
*>
*>  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*>  transpose of RFP A above. One therefore gets:
*>
*>           RFP A                   RFP A
*>
*>     02 12 22 00 01             00 10 20 30 40 50
*>     03 13 23 33 11             33 11 21 31 41 51
*>     04 14 24 34 44             43 44 22 32 42 52
*> \endverbatim
*
*  =====================================================================
      DOUBLE PRECISION FUNCTION DLANSF( NORM, TRANSR, UPLO, N, A,
     $                                  WORK )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          NORM, TRANSR, UPLO
      INTEGER            N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( 0: * ), WORK( 0: * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J, IFM, ILU, NOE, N1, K, L, LDA
      DOUBLE PRECISION   SCALE, S, VALUE, AA, TEMP
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      EXTERNAL           LSAME, DISNAN
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASSQ
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
      IF( N.EQ.0 ) THEN
         DLANSF = ZERO
         RETURN
      ELSE IF( N.EQ.1 ) THEN
         DLANSF = ABS( A(0) )
         RETURN
      END IF
*
*     set noe = 1 if n is odd. if n is even set noe=0
*
      NOE = 1
      IF( MOD( N, 2 ).EQ.0 )
     $   NOE = 0
*
*     set ifm = 0 when form='T or 't' and 1 otherwise
*
      IFM = 1
      IF( LSAME( TRANSR, 'T' ) )
     $   IFM = 0
*
*     set ilu = 0 when uplo='U or 'u' and 1 otherwise
*
      ILU = 1
      IF( LSAME( UPLO, 'U' ) )
     $   ILU = 0
*
*     set lda = (n+1)/2 when ifm = 0
*     set lda = n when ifm = 1 and noe = 1
*     set lda = n+1 when ifm = 1 and noe = 0
*
      IF( IFM.EQ.1 ) THEN
         IF( NOE.EQ.1 ) THEN
            LDA = N
         ELSE
*           noe=0
            LDA = N + 1
         END IF
      ELSE
*        ifm=0
         LDA = ( N+1 ) / 2
      END IF
*
      IF( LSAME( NORM, 'M' ) ) THEN
*
*       Find max(abs(A(i,j))).
*
         K = ( N+1 ) / 2
         VALUE = ZERO
         IF( NOE.EQ.1 ) THEN
*           n is odd
            IF( IFM.EQ.1 ) THEN
*           A is n by k
               DO J = 0, K - 1
                  DO I = 0, N - 1
                     TEMP = ABS( A( I+J*LDA ) )
                     IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
     $                    VALUE = TEMP
                  END DO
               END DO
            ELSE
*              xpose case; A is k by n
               DO J = 0, N - 1
                  DO I = 0, K - 1
                     TEMP = ABS( A( I+J*LDA ) )
                     IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
     $                    VALUE = TEMP
                  END DO
               END DO
            END IF
         ELSE
*           n is even
            IF( IFM.EQ.1 ) THEN
*              A is n+1 by k
               DO J = 0, K - 1
                  DO I = 0, N
                     TEMP = ABS( A( I+J*LDA ) )
                     IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
     $                    VALUE = TEMP
                  END DO
               END DO
            ELSE
*              xpose case; A is k by n+1
               DO J = 0, N
                  DO I = 0, K - 1
                     TEMP = ABS( A( I+J*LDA ) )
                     IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
     $                    VALUE = TEMP
                  END DO
               END DO
            END IF
         END IF
      ELSE IF( ( LSAME( NORM, 'I' ) ) .OR.
     $         ( LSAME( NORM, 'O' ) ) .OR.
     $         ( NORM.EQ.'1' ) ) THEN
*
*        Find normI(A) ( = norm1(A), since A is symmetric).
*
         IF( IFM.EQ.1 ) THEN
            K = N / 2
            IF( NOE.EQ.1 ) THEN
*              n is odd
               IF( ILU.EQ.0 ) THEN
                  DO I = 0, K - 1
                     WORK( I ) = ZERO
                  END DO
                  DO J = 0, K
                     S = ZERO
                     DO I = 0, K + J - 1
                        AA = ABS( A( I+J*LDA ) )
*                       -> A(i,j+k)
                        S = S + AA
                        WORK( I ) = WORK( I ) + AA
                     END DO
                     AA = ABS( A( I+J*LDA ) )
*                    -> A(j+k,j+k)
                     WORK( J+K ) = S + AA
                     IF( I.EQ.K+K )
     $                  GO TO 10
                     I = I + 1
                     AA = ABS( A( I+J*LDA ) )
*                    -> A(j,j)
                     WORK( J ) = WORK( J ) + AA
                     S = ZERO
                     DO L = J + 1, K - 1
                        I = I + 1
                        AA = ABS( A( I+J*LDA ) )
*                       -> A(l,j)
                        S = S + AA
                        WORK( L ) = WORK( L ) + AA
                     END DO
                     WORK( J ) = WORK( J ) + S
                  END DO
   10             CONTINUE
                  VALUE = WORK( 0 )
                  DO I = 1, N-1
                     TEMP = WORK( I )
                     IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
     $                    VALUE = TEMP
                  END DO
               ELSE
*                 ilu = 1
                  K = K + 1
*                 k=(n+1)/2 for n odd and ilu=1
                  DO I = K, N - 1
                     WORK( I ) = ZERO
                  END DO
                  DO J = K - 1, 0, -1
                     S = ZERO
                     DO I = 0, J - 2
                        AA = ABS( A( I+J*LDA ) )
*                       -> A(j+k,i+k)
                        S = S + AA
                        WORK( I+K ) = WORK( I+K ) + AA
                     END DO
                     IF( J.GT.0 ) THEN
                        AA = ABS( A( I+J*LDA ) )
*                       -> A(j+k,j+k)
                        S = S + AA
                        WORK( I+K ) = WORK( I+K ) + S
*                       i=j
                        I = I + 1
                     END IF
                     AA = ABS( A( I+J*LDA ) )
*                    -> A(j,j)
                     WORK( J ) = AA
                     S = ZERO
                     DO L = J + 1, N - 1
                        I = I + 1
                        AA = ABS( A( I+J*LDA ) )
*                       -> A(l,j)
                        S = S + AA
                        WORK( L ) = WORK( L ) + AA
                     END DO
                     WORK( J ) = WORK( J ) + S
                  END DO
                  VALUE = WORK( 0 )
                  DO I = 1, N-1
                     TEMP = WORK( I )
                     IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
     $                    VALUE = TEMP
                  END DO
               END IF
            ELSE
*              n is even
               IF( ILU.EQ.0 ) THEN
                  DO I = 0, K - 1
                     WORK( I ) = ZERO
                  END DO
                  DO J = 0, K - 1
                     S = ZERO
                     DO I = 0, K + J - 1
                        AA = ABS( A( I+J*LDA ) )
*                       -> A(i,j+k)
                        S = S + AA
                        WORK( I ) = WORK( I ) + AA
                     END DO
                     AA = ABS( A( I+J*LDA ) )
*                    -> A(j+k,j+k)
                     WORK( J+K ) = S + AA
                     I = I + 1
                     AA = ABS( A( I+J*LDA ) )
*                    -> A(j,j)
                     WORK( J ) = WORK( J ) + AA
                     S = ZERO
                     DO L = J + 1, K - 1
                        I = I + 1
                        AA = ABS( A( I+J*LDA ) )
*                       -> A(l,j)
                        S = S + AA
                        WORK( L ) = WORK( L ) + AA
                     END DO
                     WORK( J ) = WORK( J ) + S
                  END DO
                  VALUE = WORK( 0 )
                  DO I = 1, N-1
                     TEMP = WORK( I )
                     IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
     $                    VALUE = TEMP
                  END DO
               ELSE
*                 ilu = 1
                  DO I = K, N - 1
                     WORK( I ) = ZERO
                  END DO
                  DO J = K - 1, 0, -1
                     S = ZERO
                     DO I = 0, J - 1
                        AA = ABS( A( I+J*LDA ) )
*                       -> A(j+k,i+k)
                        S = S + AA
                        WORK( I+K ) = WORK( I+K ) + AA
                     END DO
                     AA = ABS( A( I+J*LDA ) )
*                    -> A(j+k,j+k)
                     S = S + AA
                     WORK( I+K ) = WORK( I+K ) + S
*                    i=j
                     I = I + 1
                     AA = ABS( A( I+J*LDA ) )
*                    -> A(j,j)
                     WORK( J ) = AA
                     S = ZERO
                     DO L = J + 1, N - 1
                        I = I + 1
                        AA = ABS( A( I+J*LDA ) )
*                       -> A(l,j)
                        S = S + AA
                        WORK( L ) = WORK( L ) + AA
                     END DO
                     WORK( J ) = WORK( J ) + S
                  END DO
                  VALUE = WORK( 0 )
                  DO I = 1, N-1
                     TEMP = WORK( I )
                     IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
     $                    VALUE = TEMP
                  END DO
               END IF
            END IF
         ELSE
*           ifm=0
            K = N / 2
            IF( NOE.EQ.1 ) THEN
*              n is odd
               IF( ILU.EQ.0 ) THEN
                  N1 = K
*                 n/2
                  K = K + 1
*                 k is the row size and lda
                  DO I = N1, N - 1
                     WORK( I ) = ZERO
                  END DO
                  DO J = 0, N1 - 1
                     S = ZERO
                     DO I = 0, K - 1
                        AA = ABS( A( I+J*LDA ) )
*                       A(j,n1+i)
                        WORK( I+N1 ) = WORK( I+N1 ) + AA
                        S = S + AA
                     END DO
                     WORK( J ) = S
                  END DO
*                 j=n1=k-1 is special
                  S = ABS( A( 0+J*LDA ) )
*                 A(k-1,k-1)
                  DO I = 1, K - 1
                     AA = ABS( A( I+J*LDA ) )
*                    A(k-1,i+n1)
                     WORK( I+N1 ) = WORK( I+N1 ) + AA
                     S = S + AA
                  END DO
                  WORK( J ) = WORK( J ) + S
                  DO J = K, N - 1
                     S = ZERO
                     DO I = 0, J - K - 1
                        AA = ABS( A( I+J*LDA ) )
*                       A(i,j-k)
                        WORK( I ) = WORK( I ) + AA
                        S = S + AA
                     END DO
*                    i=j-k
                     AA = ABS( A( I+J*LDA ) )
*                    A(j-k,j-k)
                     S = S + AA
                     WORK( J-K ) = WORK( J-K ) + S
                     I = I + 1
                     S = ABS( A( I+J*LDA ) )
*                    A(j,j)
                     DO L = J + 1, N - 1
                        I = I + 1
                        AA = ABS( A( I+J*LDA ) )
*                       A(j,l)
                        WORK( L ) = WORK( L ) + AA
                        S = S + AA
                     END DO
                     WORK( J ) = WORK( J ) + S
                  END DO
                  VALUE = WORK( 0 )
                  DO I = 1, N-1
                     TEMP = WORK( I )
                     IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
     $                    VALUE = TEMP
                  END DO
               ELSE
*                 ilu=1
                  K = K + 1
*                 k=(n+1)/2 for n odd and ilu=1
                  DO I = K, N - 1
                     WORK( I ) = ZERO
                  END DO
                  DO J = 0, K - 2
*                    process
                     S = ZERO
                     DO I = 0, J - 1
                        AA = ABS( A( I+J*LDA ) )
*                       A(j,i)
                        WORK( I ) = WORK( I ) + AA
                        S = S + AA
                     END DO
                     AA = ABS( A( I+J*LDA ) )
*                    i=j so process of A(j,j)
                     S = S + AA
                     WORK( J ) = S
*                    is initialised here
                     I = I + 1
*                    i=j process A(j+k,j+k)
                     AA = ABS( A( I+J*LDA ) )
                     S = AA
                     DO L = K + J + 1, N - 1
                        I = I + 1
                        AA = ABS( A( I+J*LDA ) )
*                       A(l,k+j)
                        S = S + AA
                        WORK( L ) = WORK( L ) + AA
                     END DO
                     WORK( K+J ) = WORK( K+J ) + S
                  END DO
*                 j=k-1 is special :process col A(k-1,0:k-1)
                  S = ZERO
                  DO I = 0, K - 2
                     AA = ABS( A( I+J*LDA ) )
*                    A(k,i)
                     WORK( I ) = WORK( I ) + AA
                     S = S + AA
                  END DO
*                 i=k-1
                  AA = ABS( A( I+J*LDA ) )
*                 A(k-1,k-1)
                  S = S + AA
                  WORK( I ) = S
*                 done with col j=k+1
                  DO J = K, N - 1
*                    process col j of A = A(j,0:k-1)
                     S = ZERO
                     DO I = 0, K - 1
                        AA = ABS( A( I+J*LDA ) )
*                       A(j,i)
                        WORK( I ) = WORK( I ) + AA
                        S = S + AA
                     END DO
                     WORK( J ) = WORK( J ) + S
                  END DO
                  VALUE = WORK( 0 )
                  DO I = 1, N-1
                     TEMP = WORK( I )
                     IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
     $                    VALUE = TEMP
                  END DO
               END IF
            ELSE
*              n is even
               IF( ILU.EQ.0 ) THEN
                  DO I = K, N - 1
                     WORK( I ) = ZERO
                  END DO
                  DO J = 0, K - 1
                     S = ZERO
                     DO I = 0, K - 1
                        AA = ABS( A( I+J*LDA ) )
*                       A(j,i+k)
                        WORK( I+K ) = WORK( I+K ) + AA
                        S = S + AA
                     END DO
                     WORK( J ) = S
                  END DO
*                 j=k
                  AA = ABS( A( 0+J*LDA ) )
*                 A(k,k)
                  S = AA
                  DO I = 1, K - 1
                     AA = ABS( A( I+J*LDA ) )
*                    A(k,k+i)
                     WORK( I+K ) = WORK( I+K ) + AA
                     S = S + AA
                  END DO
                  WORK( J ) = WORK( J ) + S
                  DO J = K + 1, N - 1
                     S = ZERO
                     DO I = 0, J - 2 - K
                        AA = ABS( A( I+J*LDA ) )
*                       A(i,j-k-1)
                        WORK( I ) = WORK( I ) + AA
                        S = S + AA
                     END DO
*                     i=j-1-k
                     AA = ABS( A( I+J*LDA ) )
*                    A(j-k-1,j-k-1)
                     S = S + AA
                     WORK( J-K-1 ) = WORK( J-K-1 ) + S
                     I = I + 1
                     AA = ABS( A( I+J*LDA ) )
*                    A(j,j)
                     S = AA
                     DO L = J + 1, N - 1
                        I = I + 1
                        AA = ABS( A( I+J*LDA ) )
*                       A(j,l)
                        WORK( L ) = WORK( L ) + AA
                        S = S + AA
                     END DO
                     WORK( J ) = WORK( J ) + S
                  END DO
*                 j=n
                  S = ZERO
                  DO I = 0, K - 2
                     AA = ABS( A( I+J*LDA ) )
*                    A(i,k-1)
                     WORK( I ) = WORK( I ) + AA
                     S = S + AA
                  END DO
*                 i=k-1
                  AA = ABS( A( I+J*LDA ) )
*                 A(k-1,k-1)
                  S = S + AA
                  WORK( I ) = WORK( I ) + S
                  VALUE = WORK( 0 )
                  DO I = 1, N-1
                     TEMP = WORK( I )
                     IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
     $                    VALUE = TEMP
                  END DO
               ELSE
*                 ilu=1
                  DO I = K, N - 1
                     WORK( I ) = ZERO
                  END DO
*                 j=0 is special :process col A(k:n-1,k)
                  S = ABS( A( 0 ) )
*                 A(k,k)
                  DO I = 1, K - 1
                     AA = ABS( A( I ) )
*                    A(k+i,k)
                     WORK( I+K ) = WORK( I+K ) + AA
                     S = S + AA
                  END DO
                  WORK( K ) = WORK( K ) + S
                  DO J = 1, K - 1
*                    process
                     S = ZERO
                     DO I = 0, J - 2
                        AA = ABS( A( I+J*LDA ) )
*                       A(j-1,i)
                        WORK( I ) = WORK( I ) + AA
                        S = S + AA
                     END DO
                     AA = ABS( A( I+J*LDA ) )
*                    i=j-1 so process of A(j-1,j-1)
                     S = S + AA
                     WORK( J-1 ) = S
*                    is initialised here
                     I = I + 1
*                    i=j process A(j+k,j+k)
                     AA = ABS( A( I+J*LDA ) )
                     S = AA
                     DO L = K + J + 1, N - 1
                        I = I + 1
                        AA = ABS( A( I+J*LDA ) )
*                       A(l,k+j)
                        S = S + AA
                        WORK( L ) = WORK( L ) + AA
                     END DO
                     WORK( K+J ) = WORK( K+J ) + S
                  END DO
*                 j=k is special :process col A(k,0:k-1)
                  S = ZERO
                  DO I = 0, K - 2
                     AA = ABS( A( I+J*LDA ) )
*                    A(k,i)
                     WORK( I ) = WORK( I ) + AA
                     S = S + AA
                  END DO
*                 i=k-1
                  AA = ABS( A( I+J*LDA ) )
*                 A(k-1,k-1)
                  S = S + AA
                  WORK( I ) = S
*                 done with col j=k+1
                  DO J = K + 1, N
*                    process col j-1 of A = A(j-1,0:k-1)
                     S = ZERO
                     DO I = 0, K - 1
                        AA = ABS( A( I+J*LDA ) )
*                       A(j-1,i)
                        WORK( I ) = WORK( I ) + AA
                        S = S + AA
                     END DO
                     WORK( J-1 ) = WORK( J-1 ) + S
                  END DO
                  VALUE = WORK( 0 )
                  DO I = 1, N-1
                     TEMP = WORK( I )
                     IF( VALUE .LT. TEMP .OR. DISNAN( TEMP ) )
     $                    VALUE = TEMP
                  END DO
               END IF
            END IF
         END IF
      ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
     $         ( LSAME( NORM, 'E' ) ) ) THEN
*
*       Find normF(A).
*
         K = ( N+1 ) / 2
         SCALE = ZERO
         S = ONE
         IF( NOE.EQ.1 ) THEN
*           n is odd
            IF( IFM.EQ.1 ) THEN
*              A is normal
               IF( ILU.EQ.0 ) THEN
*                 A is upper
                  DO J = 0, K - 3
                     CALL DLASSQ( K-J-2, A( K+J+1+J*LDA ), 1, SCALE,
     $                            S )
*                    L at A(k,0)
                  END DO
                  DO J = 0, K - 1
                     CALL DLASSQ( K+J-1, A( 0+J*LDA ), 1, SCALE, S )
*                    trap U at A(0,0)
                  END DO
                  S = S + S
*                 double s for the off diagonal elements
                  CALL DLASSQ( K-1, A( K ), LDA+1, SCALE, S )
*                 tri L at A(k,0)
                  CALL DLASSQ( K, A( K-1 ), LDA+1, SCALE, S )
*                 tri U at A(k-1,0)
               ELSE
*                 ilu=1 & A is lower
                  DO J = 0, K - 1
                     CALL DLASSQ( N-J-1, A( J+1+J*LDA ), 1, SCALE,
     $                            S )
*                    trap L at A(0,0)
                  END DO
                  DO J = 0, K - 2
                     CALL DLASSQ( J, A( 0+( 1+J )*LDA ), 1, SCALE,
     $                            S )
*                    U at A(0,1)
                  END DO
                  S = S + S
*                 double s for the off diagonal elements
                  CALL DLASSQ( K, A( 0 ), LDA+1, SCALE, S )
*                 tri L at A(0,0)
                  CALL DLASSQ( K-1, A( 0+LDA ), LDA+1, SCALE, S )
*                 tri U at A(0,1)
               END IF
            ELSE
*              A is xpose
               IF( ILU.EQ.0 ) THEN
*                 A**T is upper
                  DO J = 1, K - 2
                     CALL DLASSQ( J, A( 0+( K+J )*LDA ), 1, SCALE,
     $                            S )
*                    U at A(0,k)
                  END DO
                  DO J = 0, K - 2
                     CALL DLASSQ( K, A( 0+J*LDA ), 1, SCALE, S )
*                    k by k-1 rect. at A(0,0)
                  END DO
                  DO J = 0, K - 2
                     CALL DLASSQ( K-J-1, A( J+1+( J+K-1 )*LDA ), 1,
     $                            SCALE, S )
*                    L at A(0,k-1)
                  END DO
                  S = S + S
*                 double s for the off diagonal elements
                  CALL DLASSQ( K-1, A( 0+K*LDA ), LDA+1, SCALE, S )
*                 tri U at A(0,k)
                  CALL DLASSQ( K, A( 0+( K-1 )*LDA ), LDA+1, SCALE,
     $                         S )
*                 tri L at A(0,k-1)
               ELSE
*                 A**T is lower
                  DO J = 1, K - 1
                     CALL DLASSQ( J, A( 0+J*LDA ), 1, SCALE, S )
*                    U at A(0,0)
                  END DO
                  DO J = K, N - 1
                     CALL DLASSQ( K, A( 0+J*LDA ), 1, SCALE, S )
*                    k by k-1 rect. at A(0,k)
                  END DO
                  DO J = 0, K - 3
                     CALL DLASSQ( K-J-2, A( J+2+J*LDA ), 1, SCALE,
     $                            S )
*                    L at A(1,0)
                  END DO
                  S = S + S
*                 double s for the off diagonal elements
                  CALL DLASSQ( K, A( 0 ), LDA+1, SCALE, S )
*                 tri U at A(0,0)
                  CALL DLASSQ( K-1, A( 1 ), LDA+1, SCALE, S )
*                 tri L at A(1,0)
               END IF
            END IF
         ELSE
*           n is even
            IF( IFM.EQ.1 ) THEN
*              A is normal
               IF( ILU.EQ.0 ) THEN
*                 A is upper
                  DO J = 0, K - 2
                     CALL DLASSQ( K-J-1, A( K+J+2+J*LDA ), 1, SCALE,
     $                            S )
*                    L at A(k+1,0)
                  END DO
                  DO J = 0, K - 1
                     CALL DLASSQ( K+J, A( 0+J*LDA ), 1, SCALE, S )
*                    trap U at A(0,0)
                  END DO
                  S = S + S
*                 double s for the off diagonal elements
                  CALL DLASSQ( K, A( K+1 ), LDA+1, SCALE, S )
*                 tri L at A(k+1,0)
                  CALL DLASSQ( K, A( K ), LDA+1, SCALE, S )
*                 tri U at A(k,0)
               ELSE
*                 ilu=1 & A is lower
                  DO J = 0, K - 1
                     CALL DLASSQ( N-J-1, A( J+2+J*LDA ), 1, SCALE,
     $                            S )
*                    trap L at A(1,0)
                  END DO
                  DO J = 1, K - 1
                     CALL DLASSQ( J, A( 0+J*LDA ), 1, SCALE, S )
*                    U at A(0,0)
                  END DO
                  S = S + S
*                 double s for the off diagonal elements
                  CALL DLASSQ( K, A( 1 ), LDA+1, SCALE, S )
*                 tri L at A(1,0)
                  CALL DLASSQ( K, A( 0 ), LDA+1, SCALE, S )
*                 tri U at A(0,0)
               END IF
            ELSE
*              A is xpose
               IF( ILU.EQ.0 ) THEN
*                 A**T is upper
                  DO J = 1, K - 1
                     CALL DLASSQ( J, A( 0+( K+1+J )*LDA ), 1, SCALE,
     $                            S )
*                    U at A(0,k+1)
                  END DO
                  DO J = 0, K - 1
                     CALL DLASSQ( K, A( 0+J*LDA ), 1, SCALE, S )
*                    k by k rect. at A(0,0)
                  END DO
                  DO J = 0, K - 2
                     CALL DLASSQ( K-J-1, A( J+1+( J+K )*LDA ), 1,
     $                            SCALE,
     $                            S )
*                    L at A(0,k)
                  END DO
                  S = S + S
*                 double s for the off diagonal elements
                  CALL DLASSQ( K, A( 0+( K+1 )*LDA ), LDA+1, SCALE,
     $                         S )
*                 tri U at A(0,k+1)
                  CALL DLASSQ( K, A( 0+K*LDA ), LDA+1, SCALE, S )
*                 tri L at A(0,k)
               ELSE
*                 A**T is lower
                  DO J = 1, K - 1
                     CALL DLASSQ( J, A( 0+( J+1 )*LDA ), 1, SCALE,
     $                            S )
*                    U at A(0,1)
                  END DO
                  DO J = K + 1, N
                     CALL DLASSQ( K, A( 0+J*LDA ), 1, SCALE, S )
*                    k by k rect. at A(0,k+1)
                  END DO
                  DO J = 0, K - 2
                     CALL DLASSQ( K-J-1, A( J+1+J*LDA ), 1, SCALE,
     $                            S )
*                    L at A(0,0)
                  END DO
                  S = S + S
*                 double s for the off diagonal elements
                  CALL DLASSQ( K, A( LDA ), LDA+1, SCALE, S )
*                 tri L at A(0,1)
                  CALL DLASSQ( K, A( 0 ), LDA+1, SCALE, S )
*                 tri U at A(0,0)
               END IF
            END IF
         END IF
         VALUE = SCALE*SQRT( S )
      END IF
*
      DLANSF = VALUE
      RETURN
*
*     End of DLANSF
*
      END
*> \brief \b DLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix supplied in packed form.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANSP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlansp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlansp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlansp.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       DOUBLE PRECISION FUNCTION DLANSP( NORM, UPLO, N, AP, WORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          NORM, UPLO
*       INTEGER            N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AP( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLANSP  returns the value of the one norm,  or the Frobenius norm, or
*> the  infinity norm,  or the  element of  largest absolute value  of a
*> real symmetric matrix A,  supplied in packed form.
*> \endverbatim
*>
*> \return DLANSP
*> \verbatim
*>
*>    DLANSP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*>             (
*>             ( norm1(A),         NORM = '1', 'O' or 'o'
*>             (
*>             ( normI(A),         NORM = 'I' or 'i'
*>             (
*>             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
*>
*> where  norm1  denotes the  one norm of a matrix (maximum column sum),
*> normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
*> normF  denotes the  Frobenius norm of a matrix (square root of sum of
*> squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NORM
*> \verbatim
*>          NORM is CHARACTER*1
*>          Specifies the value to be returned in DLANSP as described
*>          above.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the upper or lower triangular part of the
*>          symmetric matrix A is supplied.
*>          = 'U':  Upper triangular part of A is supplied
*>          = 'L':  Lower triangular part of A is supplied
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.  When N = 0, DLANSP is
*>          set to zero.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          The upper or lower triangle of the symmetric matrix A, packed
*>          columnwise in a linear array.  The j-th column of A is stored
*>          in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*>          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
*>          WORK is not referenced.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lanhp
*
*  =====================================================================
      DOUBLE PRECISION FUNCTION DLANSP( NORM, UPLO, N, AP, WORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          NORM, UPLO
      INTEGER            N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( * ), WORK( * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J, K
      DOUBLE PRECISION   ABSA, SCALE, SUM, VALUE
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASSQ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      EXTERNAL           LSAME, DISNAN
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, SQRT
*     ..
*     .. Executable Statements ..
*
      IF( N.EQ.0 ) THEN
         VALUE = ZERO
      ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
*        Find max(abs(A(i,j))).
*
         VALUE = ZERO
         IF( LSAME( UPLO, 'U' ) ) THEN
            K = 1
            DO 20 J = 1, N
               DO 10 I = K, K + J - 1
                  SUM = ABS( AP( I ) )
                  IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
   10          CONTINUE
               K = K + J
   20       CONTINUE
         ELSE
            K = 1
            DO 40 J = 1, N
               DO 30 I = K, K + N - J
                  SUM = ABS( AP( I ) )
                  IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
   30          CONTINUE
               K = K + N - J + 1
   40       CONTINUE
         END IF
      ELSE IF( ( LSAME( NORM, 'I' ) ) .OR.
     $         ( LSAME( NORM, 'O' ) ) .OR.
     $         ( NORM.EQ.'1' ) ) THEN
*
*        Find normI(A) ( = norm1(A), since A is symmetric).
*
         VALUE = ZERO
         K = 1
         IF( LSAME( UPLO, 'U' ) ) THEN
            DO 60 J = 1, N
               SUM = ZERO
               DO 50 I = 1, J - 1
                  ABSA = ABS( AP( K ) )
                  SUM = SUM + ABSA
                  WORK( I ) = WORK( I ) + ABSA
                  K = K + 1
   50          CONTINUE
               WORK( J ) = SUM + ABS( AP( K ) )
               K = K + 1
   60       CONTINUE
            DO 70 I = 1, N
               SUM = WORK( I )
               IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
   70       CONTINUE
         ELSE
            DO 80 I = 1, N
               WORK( I ) = ZERO
   80       CONTINUE
            DO 100 J = 1, N
               SUM = WORK( J ) + ABS( AP( K ) )
               K = K + 1
               DO 90 I = J + 1, N
                  ABSA = ABS( AP( K ) )
                  SUM = SUM + ABSA
                  WORK( I ) = WORK( I ) + ABSA
                  K = K + 1
   90          CONTINUE
               IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
  100       CONTINUE
         END IF
      ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
     $         ( LSAME( NORM, 'E' ) ) ) THEN
*
*        Find normF(A).
*
         SCALE = ZERO
         SUM = ONE
         K = 2
         IF( LSAME( UPLO, 'U' ) ) THEN
            DO 110 J = 2, N
               CALL DLASSQ( J-1, AP( K ), 1, SCALE, SUM )
               K = K + J
  110       CONTINUE
         ELSE
            DO 120 J = 1, N - 1
               CALL DLASSQ( N-J, AP( K ), 1, SCALE, SUM )
               K = K + N - J + 1
  120       CONTINUE
         END IF
         SUM = 2*SUM
         K = 1
         DO 130 I = 1, N
            IF( AP( K ).NE.ZERO ) THEN
               ABSA = ABS( AP( K ) )
               IF( SCALE.LT.ABSA ) THEN
                  SUM = ONE + SUM*( SCALE / ABSA )**2
                  SCALE = ABSA
               ELSE
                  SUM = SUM + ( ABSA / SCALE )**2
               END IF
            END IF
            IF( LSAME( UPLO, 'U' ) ) THEN
               K = K + I + 1
            ELSE
               K = K + N - I + 1
            END IF
  130    CONTINUE
         VALUE = SCALE*SQRT( SUM )
      END IF
*
      DLANSP = VALUE
      RETURN
*
*     End of DLANSP
*
      END
*> \brief \b DLANST returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric tridiagonal matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANST + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlanst.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlanst.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlanst.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       DOUBLE PRECISION FUNCTION DLANST( NORM, N, D, E )
*
*       .. Scalar Arguments ..
*       CHARACTER          NORM
*       INTEGER            N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), E( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLANST  returns the value of the one norm,  or the Frobenius norm, or
*> the  infinity norm,  or the  element of  largest absolute value  of a
*> real symmetric tridiagonal matrix A.
*> \endverbatim
*>
*> \return DLANST
*> \verbatim
*>
*>    DLANST = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*>             (
*>             ( norm1(A),         NORM = '1', 'O' or 'o'
*>             (
*>             ( normI(A),         NORM = 'I' or 'i'
*>             (
*>             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
*>
*> where  norm1  denotes the  one norm of a matrix (maximum column sum),
*> normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
*> normF  denotes the  Frobenius norm of a matrix (square root of sum of
*> squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NORM
*> \verbatim
*>          NORM is CHARACTER*1
*>          Specifies the value to be returned in DLANST as described
*>          above.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.  When N = 0, DLANST is
*>          set to zero.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The diagonal elements of A.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) sub-diagonal or super-diagonal elements of A.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lanht
*
*  =====================================================================
      DOUBLE PRECISION FUNCTION DLANST( NORM, N, D, E )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          NORM
      INTEGER            N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), E( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
      DOUBLE PRECISION   ANORM, SCALE, SUM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      EXTERNAL           LSAME, DISNAN
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASSQ
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, SQRT
*     ..
*     .. Executable Statements ..
*
      IF( N.LE.0 ) THEN
         ANORM = ZERO
      ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
*        Find max(abs(A(i,j))).
*
         ANORM = ABS( D( N ) )
         DO 10 I = 1, N - 1
            SUM = ABS( D( I ) )
            IF( ANORM .LT. SUM .OR. DISNAN( SUM ) ) ANORM = SUM
            SUM = ABS( E( I ) )
            IF( ANORM .LT. SUM .OR. DISNAN( SUM ) ) ANORM = SUM
   10    CONTINUE
      ELSE IF( LSAME( NORM, 'O' ) .OR. NORM.EQ.'1' .OR.
     $         LSAME( NORM, 'I' ) ) THEN
*
*        Find norm1(A).
*
         IF( N.EQ.1 ) THEN
            ANORM = ABS( D( 1 ) )
         ELSE
            ANORM = ABS( D( 1 ) )+ABS( E( 1 ) )
            SUM = ABS( E( N-1 ) )+ABS( D( N ) )
            IF( ANORM .LT. SUM .OR. DISNAN( SUM ) ) ANORM = SUM
            DO 20 I = 2, N - 1
               SUM = ABS( D( I ) )+ABS( E( I ) )+ABS( E( I-1 ) )
               IF( ANORM .LT. SUM .OR. DISNAN( SUM ) ) ANORM = SUM
   20       CONTINUE
         END IF
      ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
     $         ( LSAME( NORM, 'E' ) ) ) THEN
*
*        Find normF(A).
*
         SCALE = ZERO
         SUM = ONE
         IF( N.GT.1 ) THEN
            CALL DLASSQ( N-1, E, 1, SCALE, SUM )
            SUM = 2*SUM
         END IF
         CALL DLASSQ( N, D, 1, SCALE, SUM )
         ANORM = SCALE*SQRT( SUM )
      END IF
*
      DLANST = ANORM
      RETURN
*
*     End of DLANST
*
      END
*> \brief \b DLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANSY + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlansy.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlansy.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlansy.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       DOUBLE PRECISION FUNCTION DLANSY( NORM, UPLO, N, A, LDA, WORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          NORM, UPLO
*       INTEGER            LDA, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLANSY  returns the value of the one norm,  or the Frobenius norm, or
*> the  infinity norm,  or the  element of  largest absolute value  of a
*> real symmetric matrix A.
*> \endverbatim
*>
*> \return DLANSY
*> \verbatim
*>
*>    DLANSY = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*>             (
*>             ( norm1(A),         NORM = '1', 'O' or 'o'
*>             (
*>             ( normI(A),         NORM = 'I' or 'i'
*>             (
*>             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
*>
*> where  norm1  denotes the  one norm of a matrix (maximum column sum),
*> normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
*> normF  denotes the  Frobenius norm of a matrix (square root of sum of
*> squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NORM
*> \verbatim
*>          NORM is CHARACTER*1
*>          Specifies the value to be returned in DLANSY as described
*>          above.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the upper or lower triangular part of the
*>          symmetric matrix A is to be referenced.
*>          = 'U':  Upper triangular part of A is referenced
*>          = 'L':  Lower triangular part of A is referenced
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.  When N = 0, DLANSY is
*>          set to zero.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The symmetric matrix A.  If UPLO = 'U', the leading n by n
*>          upper triangular part of A contains the upper triangular part
*>          of the matrix A, and the strictly lower triangular part of A
*>          is not referenced.  If UPLO = 'L', the leading n by n lower
*>          triangular part of A contains the lower triangular part of
*>          the matrix A, and the strictly upper triangular part of A is
*>          not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(N,1).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*>          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
*>          WORK is not referenced.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lanhe
*
*  =====================================================================
      DOUBLE PRECISION FUNCTION DLANSY( NORM, UPLO, N, A, LDA, WORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          NORM, UPLO
      INTEGER            LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), WORK( * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J
      DOUBLE PRECISION   ABSA, SCALE, SUM, VALUE
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASSQ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      EXTERNAL           LSAME, DISNAN
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, SQRT
*     ..
*     .. Executable Statements ..
*
      IF( N.EQ.0 ) THEN
         VALUE = ZERO
      ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
*        Find max(abs(A(i,j))).
*
         VALUE = ZERO
         IF( LSAME( UPLO, 'U' ) ) THEN
            DO 20 J = 1, N
               DO 10 I = 1, J
                  SUM = ABS( A( I, J ) )
                  IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
   10          CONTINUE
   20       CONTINUE
         ELSE
            DO 40 J = 1, N
               DO 30 I = J, N
                  SUM = ABS( A( I, J ) )
                  IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
   30          CONTINUE
   40       CONTINUE
         END IF
      ELSE IF( ( LSAME( NORM, 'I' ) ) .OR.
     $         ( LSAME( NORM, 'O' ) ) .OR.
     $         ( NORM.EQ.'1' ) ) THEN
*
*        Find normI(A) ( = norm1(A), since A is symmetric).
*
         VALUE = ZERO
         IF( LSAME( UPLO, 'U' ) ) THEN
            DO 60 J = 1, N
               SUM = ZERO
               DO 50 I = 1, J - 1
                  ABSA = ABS( A( I, J ) )
                  SUM = SUM + ABSA
                  WORK( I ) = WORK( I ) + ABSA
   50          CONTINUE
               WORK( J ) = SUM + ABS( A( J, J ) )
   60       CONTINUE
            DO 70 I = 1, N
               SUM = WORK( I )
               IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
   70       CONTINUE
         ELSE
            DO 80 I = 1, N
               WORK( I ) = ZERO
   80       CONTINUE
            DO 100 J = 1, N
               SUM = WORK( J ) + ABS( A( J, J ) )
               DO 90 I = J + 1, N
                  ABSA = ABS( A( I, J ) )
                  SUM = SUM + ABSA
                  WORK( I ) = WORK( I ) + ABSA
   90          CONTINUE
               IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
  100       CONTINUE
         END IF
      ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
     $         ( LSAME( NORM, 'E' ) ) ) THEN
*
*        Find normF(A).
*
         SCALE = ZERO
         SUM = ONE
         IF( LSAME( UPLO, 'U' ) ) THEN
            DO 110 J = 2, N
               CALL DLASSQ( J-1, A( 1, J ), 1, SCALE, SUM )
  110       CONTINUE
         ELSE
            DO 120 J = 1, N - 1
               CALL DLASSQ( N-J, A( J+1, J ), 1, SCALE, SUM )
  120       CONTINUE
         END IF
         SUM = 2*SUM
         CALL DLASSQ( N, A, LDA+1, SCALE, SUM )
         VALUE = SCALE*SQRT( SUM )
      END IF
*
      DLANSY = VALUE
      RETURN
*
*     End of DLANSY
*
      END
*> \brief \b DLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANTB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlantb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlantb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlantb.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       DOUBLE PRECISION FUNCTION DLANTB( NORM, UPLO, DIAG, N, K, AB,
*                        LDAB, WORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, NORM, UPLO
*       INTEGER            K, LDAB, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLANTB  returns the value of the one norm,  or the Frobenius norm, or
*> the  infinity norm,  or the element of  largest absolute value  of an
*> n by n triangular band matrix A,  with ( k + 1 ) diagonals.
*> \endverbatim
*>
*> \return DLANTB
*> \verbatim
*>
*>    DLANTB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*>             (
*>             ( norm1(A),         NORM = '1', 'O' or 'o'
*>             (
*>             ( normI(A),         NORM = 'I' or 'i'
*>             (
*>             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
*>
*> where  norm1  denotes the  one norm of a matrix (maximum column sum),
*> normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
*> normF  denotes the  Frobenius norm of a matrix (square root of sum of
*> squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NORM
*> \verbatim
*>          NORM is CHARACTER*1
*>          Specifies the value to be returned in DLANTB as described
*>          above.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the matrix A is upper or lower triangular.
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          Specifies whether or not the matrix A is unit triangular.
*>          = 'N':  Non-unit triangular
*>          = 'U':  Unit triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.  When N = 0, DLANTB is
*>          set to zero.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of super-diagonals of the matrix A if UPLO = 'U',
*>          or the number of sub-diagonals of the matrix A if UPLO = 'L'.
*>          K >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          The upper or lower triangular band matrix A, stored in the
*>          first k+1 rows of AB.  The j-th column of A is stored
*>          in the j-th column of the array AB as follows:
*>          if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).
*>          Note that when DIAG = 'U', the elements of the array AB
*>          corresponding to the diagonal elements of the matrix A are
*>          not referenced, but are assumed to be one.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= K+1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*>          where LWORK >= N when NORM = 'I'; otherwise, WORK is not
*>          referenced.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lantb
*
*  =====================================================================
      DOUBLE PRECISION FUNCTION DLANTB( NORM, UPLO, DIAG, N, K, AB,
     $                 LDAB, WORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, NORM, UPLO
      INTEGER            K, LDAB, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * ), WORK( * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UDIAG
      INTEGER            I, J, L
      DOUBLE PRECISION   SCALE, SUM, VALUE
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASSQ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      EXTERNAL           LSAME, DISNAN
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
      IF( N.EQ.0 ) THEN
         VALUE = ZERO
      ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
*        Find max(abs(A(i,j))).
*
         IF( LSAME( DIAG, 'U' ) ) THEN
            VALUE = ONE
            IF( LSAME( UPLO, 'U' ) ) THEN
               DO 20 J = 1, N
                  DO 10 I = MAX( K+2-J, 1 ), K
                     SUM = ABS( AB( I, J ) )
                     IF( VALUE .LT. SUM .OR.
     $                   DISNAN( SUM ) ) VALUE = SUM
   10             CONTINUE
   20          CONTINUE
            ELSE
               DO 40 J = 1, N
                  DO 30 I = 2, MIN( N+1-J, K+1 )
                     SUM = ABS( AB( I, J ) )
                     IF( VALUE .LT. SUM .OR.
     $                   DISNAN( SUM ) ) VALUE = SUM
   30             CONTINUE
   40          CONTINUE
            END IF
         ELSE
            VALUE = ZERO
            IF( LSAME( UPLO, 'U' ) ) THEN
               DO 60 J = 1, N
                  DO 50 I = MAX( K+2-J, 1 ), K + 1
                     SUM = ABS( AB( I, J ) )
                     IF( VALUE .LT. SUM .OR.
     $                   DISNAN( SUM ) ) VALUE = SUM
   50             CONTINUE
   60          CONTINUE
            ELSE
               DO 80 J = 1, N
                  DO 70 I = 1, MIN( N+1-J, K+1 )
                     SUM = ABS( AB( I, J ) )
                     IF( VALUE .LT. SUM .OR.
     $                   DISNAN( SUM ) ) VALUE = SUM
   70             CONTINUE
   80          CONTINUE
            END IF
         END IF
      ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN
*
*        Find norm1(A).
*
         VALUE = ZERO
         UDIAG = LSAME( DIAG, 'U' )
         IF( LSAME( UPLO, 'U' ) ) THEN
            DO 110 J = 1, N
               IF( UDIAG ) THEN
                  SUM = ONE
                  DO 90 I = MAX( K+2-J, 1 ), K
                     SUM = SUM + ABS( AB( I, J ) )
   90             CONTINUE
               ELSE
                  SUM = ZERO
                  DO 100 I = MAX( K+2-J, 1 ), K + 1
                     SUM = SUM + ABS( AB( I, J ) )
  100             CONTINUE
               END IF
               IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
  110       CONTINUE
         ELSE
            DO 140 J = 1, N
               IF( UDIAG ) THEN
                  SUM = ONE
                  DO 120 I = 2, MIN( N+1-J, K+1 )
                     SUM = SUM + ABS( AB( I, J ) )
  120             CONTINUE
               ELSE
                  SUM = ZERO
                  DO 130 I = 1, MIN( N+1-J, K+1 )
                     SUM = SUM + ABS( AB( I, J ) )
  130             CONTINUE
               END IF
               IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
  140       CONTINUE
         END IF
      ELSE IF( LSAME( NORM, 'I' ) ) THEN
*
*        Find normI(A).
*
         VALUE = ZERO
         IF( LSAME( UPLO, 'U' ) ) THEN
            IF( LSAME( DIAG, 'U' ) ) THEN
               DO 150 I = 1, N
                  WORK( I ) = ONE
  150          CONTINUE
               DO 170 J = 1, N
                  L = K + 1 - J
                  DO 160 I = MAX( 1, J-K ), J - 1
                     WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
  160             CONTINUE
  170          CONTINUE
            ELSE
               DO 180 I = 1, N
                  WORK( I ) = ZERO
  180          CONTINUE
               DO 200 J = 1, N
                  L = K + 1 - J
                  DO 190 I = MAX( 1, J-K ), J
                     WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
  190             CONTINUE
  200          CONTINUE
            END IF
         ELSE
            IF( LSAME( DIAG, 'U' ) ) THEN
               DO 210 I = 1, N
                  WORK( I ) = ONE
  210          CONTINUE
               DO 230 J = 1, N
                  L = 1 - J
                  DO 220 I = J + 1, MIN( N, J+K )
                     WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
  220             CONTINUE
  230          CONTINUE
            ELSE
               DO 240 I = 1, N
                  WORK( I ) = ZERO
  240          CONTINUE
               DO 260 J = 1, N
                  L = 1 - J
                  DO 250 I = J, MIN( N, J+K )
                     WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
  250             CONTINUE
  260          CONTINUE
            END IF
         END IF
         DO 270 I = 1, N
            SUM = WORK( I )
            IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
  270    CONTINUE
      ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
     $         ( LSAME( NORM, 'E' ) ) ) THEN
*
*        Find normF(A).
*
         IF( LSAME( UPLO, 'U' ) ) THEN
            IF( LSAME( DIAG, 'U' ) ) THEN
               SCALE = ONE
               SUM = N
               IF( K.GT.0 ) THEN
                  DO 280 J = 2, N
                     CALL DLASSQ( MIN( J-1, K ),
     $                            AB( MAX( K+2-J, 1 ), J ), 1, SCALE,
     $                            SUM )
  280             CONTINUE
               END IF
            ELSE
               SCALE = ZERO
               SUM = ONE
               DO 290 J = 1, N
                  CALL DLASSQ( MIN( J, K+1 ), AB( MAX( K+2-J, 1 ),
     $                         J ),
     $                         1, SCALE, SUM )
  290          CONTINUE
            END IF
         ELSE
            IF( LSAME( DIAG, 'U' ) ) THEN
               SCALE = ONE
               SUM = N
               IF( K.GT.0 ) THEN
                  DO 300 J = 1, N - 1
                     CALL DLASSQ( MIN( N-J, K ), AB( 2, J ), 1,
     $                            SCALE,
     $                            SUM )
  300             CONTINUE
               END IF
            ELSE
               SCALE = ZERO
               SUM = ONE
               DO 310 J = 1, N
                  CALL DLASSQ( MIN( N-J+1, K+1 ), AB( 1, J ), 1,
     $                         SCALE,
     $                         SUM )
  310          CONTINUE
            END IF
         END IF
         VALUE = SCALE*SQRT( SUM )
      END IF
*
      DLANTB = VALUE
      RETURN
*
*     End of DLANTB
*
      END
*> \brief \b DLANTP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix supplied in packed form.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANTP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlantp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlantp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlantp.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       DOUBLE PRECISION FUNCTION DLANTP( NORM, UPLO, DIAG, N, AP, WORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, NORM, UPLO
*       INTEGER            N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AP( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLANTP  returns the value of the one norm,  or the Frobenius norm, or
*> the  infinity norm,  or the  element of  largest absolute value  of a
*> triangular matrix A, supplied in packed form.
*> \endverbatim
*>
*> \return DLANTP
*> \verbatim
*>
*>    DLANTP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*>             (
*>             ( norm1(A),         NORM = '1', 'O' or 'o'
*>             (
*>             ( normI(A),         NORM = 'I' or 'i'
*>             (
*>             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
*>
*> where  norm1  denotes the  one norm of a matrix (maximum column sum),
*> normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
*> normF  denotes the  Frobenius norm of a matrix (square root of sum of
*> squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NORM
*> \verbatim
*>          NORM is CHARACTER*1
*>          Specifies the value to be returned in DLANTP as described
*>          above.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the matrix A is upper or lower triangular.
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          Specifies whether or not the matrix A is unit triangular.
*>          = 'N':  Non-unit triangular
*>          = 'U':  Unit triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.  When N = 0, DLANTP is
*>          set to zero.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          The upper or lower triangular matrix A, packed columnwise in
*>          a linear array.  The j-th column of A is stored in the array
*>          AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*>          Note that when DIAG = 'U', the elements of the array AP
*>          corresponding to the diagonal elements of the matrix A are
*>          not referenced, but are assumed to be one.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*>          where LWORK >= N when NORM = 'I'; otherwise, WORK is not
*>          referenced.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lantp
*
*  =====================================================================
      DOUBLE PRECISION FUNCTION DLANTP( NORM, UPLO, DIAG, N, AP,
     $                                  WORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, NORM, UPLO
      INTEGER            N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( * ), WORK( * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UDIAG
      INTEGER            I, J, K
      DOUBLE PRECISION   SCALE, SUM, VALUE
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASSQ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      EXTERNAL           LSAME, DISNAN
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, SQRT
*     ..
*     .. Executable Statements ..
*
      IF( N.EQ.0 ) THEN
         VALUE = ZERO
      ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
*        Find max(abs(A(i,j))).
*
         K = 1
         IF( LSAME( DIAG, 'U' ) ) THEN
            VALUE = ONE
            IF( LSAME( UPLO, 'U' ) ) THEN
               DO 20 J = 1, N
                  DO 10 I = K, K + J - 2
                     SUM = ABS( AP( I ) )
                     IF( VALUE .LT. SUM .OR.
     $                   DISNAN( SUM ) ) VALUE = SUM
   10             CONTINUE
                  K = K + J
   20          CONTINUE
            ELSE
               DO 40 J = 1, N
                  DO 30 I = K + 1, K + N - J
                     SUM = ABS( AP( I ) )
                     IF( VALUE .LT. SUM .OR.
     $                   DISNAN( SUM ) ) VALUE = SUM
   30             CONTINUE
                  K = K + N - J + 1
   40          CONTINUE
            END IF
         ELSE
            VALUE = ZERO
            IF( LSAME( UPLO, 'U' ) ) THEN
               DO 60 J = 1, N
                  DO 50 I = K, K + J - 1
                     SUM = ABS( AP( I ) )
                     IF( VALUE .LT. SUM .OR.
     $                   DISNAN( SUM ) ) VALUE = SUM
   50             CONTINUE
                  K = K + J
   60          CONTINUE
            ELSE
               DO 80 J = 1, N
                  DO 70 I = K, K + N - J
                     SUM = ABS( AP( I ) )
                     IF( VALUE .LT. SUM .OR.
     $                   DISNAN( SUM ) ) VALUE = SUM
   70             CONTINUE
                  K = K + N - J + 1
   80          CONTINUE
            END IF
         END IF
      ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN
*
*        Find norm1(A).
*
         VALUE = ZERO
         K = 1
         UDIAG = LSAME( DIAG, 'U' )
         IF( LSAME( UPLO, 'U' ) ) THEN
            DO 110 J = 1, N
               IF( UDIAG ) THEN
                  SUM = ONE
                  DO 90 I = K, K + J - 2
                     SUM = SUM + ABS( AP( I ) )
   90             CONTINUE
               ELSE
                  SUM = ZERO
                  DO 100 I = K, K + J - 1
                     SUM = SUM + ABS( AP( I ) )
  100             CONTINUE
               END IF
               K = K + J
               IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
  110       CONTINUE
         ELSE
            DO 140 J = 1, N
               IF( UDIAG ) THEN
                  SUM = ONE
                  DO 120 I = K + 1, K + N - J
                     SUM = SUM + ABS( AP( I ) )
  120             CONTINUE
               ELSE
                  SUM = ZERO
                  DO 130 I = K, K + N - J
                     SUM = SUM + ABS( AP( I ) )
  130             CONTINUE
               END IF
               K = K + N - J + 1
               IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
  140       CONTINUE
         END IF
      ELSE IF( LSAME( NORM, 'I' ) ) THEN
*
*        Find normI(A).
*
         K = 1
         IF( LSAME( UPLO, 'U' ) ) THEN
            IF( LSAME( DIAG, 'U' ) ) THEN
               DO 150 I = 1, N
                  WORK( I ) = ONE
  150          CONTINUE
               DO 170 J = 1, N
                  DO 160 I = 1, J - 1
                     WORK( I ) = WORK( I ) + ABS( AP( K ) )
                     K = K + 1
  160             CONTINUE
                  K = K + 1
  170          CONTINUE
            ELSE
               DO 180 I = 1, N
                  WORK( I ) = ZERO
  180          CONTINUE
               DO 200 J = 1, N
                  DO 190 I = 1, J
                     WORK( I ) = WORK( I ) + ABS( AP( K ) )
                     K = K + 1
  190             CONTINUE
  200          CONTINUE
            END IF
         ELSE
            IF( LSAME( DIAG, 'U' ) ) THEN
               DO 210 I = 1, N
                  WORK( I ) = ONE
  210          CONTINUE
               DO 230 J = 1, N
                  K = K + 1
                  DO 220 I = J + 1, N
                     WORK( I ) = WORK( I ) + ABS( AP( K ) )
                     K = K + 1
  220             CONTINUE
  230          CONTINUE
            ELSE
               DO 240 I = 1, N
                  WORK( I ) = ZERO
  240          CONTINUE
               DO 260 J = 1, N
                  DO 250 I = J, N
                     WORK( I ) = WORK( I ) + ABS( AP( K ) )
                     K = K + 1
  250             CONTINUE
  260          CONTINUE
            END IF
         END IF
         VALUE = ZERO
         DO 270 I = 1, N
            SUM = WORK( I )
            IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
  270    CONTINUE
      ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
     $         ( LSAME( NORM, 'E' ) ) ) THEN
*
*        Find normF(A).
*
         IF( LSAME( UPLO, 'U' ) ) THEN
            IF( LSAME( DIAG, 'U' ) ) THEN
               SCALE = ONE
               SUM = N
               K = 2
               DO 280 J = 2, N
                  CALL DLASSQ( J-1, AP( K ), 1, SCALE, SUM )
                  K = K + J
  280          CONTINUE
            ELSE
               SCALE = ZERO
               SUM = ONE
               K = 1
               DO 290 J = 1, N
                  CALL DLASSQ( J, AP( K ), 1, SCALE, SUM )
                  K = K + J
  290          CONTINUE
            END IF
         ELSE
            IF( LSAME( DIAG, 'U' ) ) THEN
               SCALE = ONE
               SUM = N
               K = 2
               DO 300 J = 1, N - 1
                  CALL DLASSQ( N-J, AP( K ), 1, SCALE, SUM )
                  K = K + N - J + 1
  300          CONTINUE
            ELSE
               SCALE = ZERO
               SUM = ONE
               K = 1
               DO 310 J = 1, N
                  CALL DLASSQ( N-J+1, AP( K ), 1, SCALE, SUM )
                  K = K + N - J + 1
  310          CONTINUE
            END IF
         END IF
         VALUE = SCALE*SQRT( SUM )
      END IF
*
      DLANTP = VALUE
      RETURN
*
*     End of DLANTP
*
      END
*> \brief \b DLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANTR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlantr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlantr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlantr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       DOUBLE PRECISION FUNCTION DLANTR( NORM, UPLO, DIAG, M, N, A, LDA,
*                        WORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, NORM, UPLO
*       INTEGER            LDA, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLANTR  returns the value of the one norm,  or the Frobenius norm, or
*> the  infinity norm,  or the  element of  largest absolute value  of a
*> trapezoidal or triangular matrix A.
*> \endverbatim
*>
*> \return DLANTR
*> \verbatim
*>
*>    DLANTR = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*>             (
*>             ( norm1(A),         NORM = '1', 'O' or 'o'
*>             (
*>             ( normI(A),         NORM = 'I' or 'i'
*>             (
*>             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
*>
*> where  norm1  denotes the  one norm of a matrix (maximum column sum),
*> normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
*> normF  denotes the  Frobenius norm of a matrix (square root of sum of
*> squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NORM
*> \verbatim
*>          NORM is CHARACTER*1
*>          Specifies the value to be returned in DLANTR as described
*>          above.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the matrix A is upper or lower trapezoidal.
*>          = 'U':  Upper trapezoidal
*>          = 'L':  Lower trapezoidal
*>          Note that A is triangular instead of trapezoidal if M = N.
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          Specifies whether or not the matrix A has unit diagonal.
*>          = 'N':  Non-unit diagonal
*>          = 'U':  Unit diagonal
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0, and if
*>          UPLO = 'U', M <= N.  When M = 0, DLANTR is set to zero.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0, and if
*>          UPLO = 'L', N <= M.  When N = 0, DLANTR is set to zero.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The trapezoidal matrix A (A is triangular if M = N).
*>          If UPLO = 'U', the leading m by n upper trapezoidal part of
*>          the array A contains the upper trapezoidal matrix, and the
*>          strictly lower triangular part of A is not referenced.
*>          If UPLO = 'L', the leading m by n lower trapezoidal part of
*>          the array A contains the lower trapezoidal matrix, and the
*>          strictly upper triangular part of A is not referenced.  Note
*>          that when DIAG = 'U', the diagonal elements of A are not
*>          referenced and are assumed to be one.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(M,1).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*>          where LWORK >= M when NORM = 'I'; otherwise, WORK is not
*>          referenced.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lantr
*
*  =====================================================================
      DOUBLE PRECISION FUNCTION DLANTR( NORM, UPLO, DIAG, M, N, A,
     $                                  LDA,
     $                 WORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, NORM, UPLO
      INTEGER            LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), WORK( * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UDIAG
      INTEGER            I, J
      DOUBLE PRECISION   SCALE, SUM, VALUE
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASSQ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      EXTERNAL           LSAME, DISNAN
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
      IF( MIN( M, N ).EQ.0 ) THEN
         VALUE = ZERO
      ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
*        Find max(abs(A(i,j))).
*
         IF( LSAME( DIAG, 'U' ) ) THEN
            VALUE = ONE
            IF( LSAME( UPLO, 'U' ) ) THEN
               DO 20 J = 1, N
                  DO 10 I = 1, MIN( M, J-1 )
                     SUM = ABS( A( I, J ) )
                     IF( VALUE .LT. SUM .OR.
     $                   DISNAN( SUM ) ) VALUE = SUM
   10             CONTINUE
   20          CONTINUE
            ELSE
               DO 40 J = 1, N
                  DO 30 I = J + 1, M
                     SUM = ABS( A( I, J ) )
                     IF( VALUE .LT. SUM .OR.
     $                   DISNAN( SUM ) ) VALUE = SUM
   30             CONTINUE
   40          CONTINUE
            END IF
         ELSE
            VALUE = ZERO
            IF( LSAME( UPLO, 'U' ) ) THEN
               DO 60 J = 1, N
                  DO 50 I = 1, MIN( M, J )
                     SUM = ABS( A( I, J ) )
                     IF( VALUE .LT. SUM .OR.
     $                   DISNAN( SUM ) ) VALUE = SUM
   50             CONTINUE
   60          CONTINUE
            ELSE
               DO 80 J = 1, N
                  DO 70 I = J, M
                     SUM = ABS( A( I, J ) )
                     IF( VALUE .LT. SUM .OR.
     $                   DISNAN( SUM ) ) VALUE = SUM
   70             CONTINUE
   80          CONTINUE
            END IF
         END IF
      ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN
*
*        Find norm1(A).
*
         VALUE = ZERO
         UDIAG = LSAME( DIAG, 'U' )
         IF( LSAME( UPLO, 'U' ) ) THEN
            DO 110 J = 1, N
               IF( ( UDIAG ) .AND. ( J.LE.M ) ) THEN
                  SUM = ONE
                  DO 90 I = 1, J - 1
                     SUM = SUM + ABS( A( I, J ) )
   90             CONTINUE
               ELSE
                  SUM = ZERO
                  DO 100 I = 1, MIN( M, J )
                     SUM = SUM + ABS( A( I, J ) )
  100             CONTINUE
               END IF
               IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
  110       CONTINUE
         ELSE
            DO 140 J = 1, N
               IF( UDIAG ) THEN
                  SUM = ONE
                  DO 120 I = J + 1, M
                     SUM = SUM + ABS( A( I, J ) )
  120             CONTINUE
               ELSE
                  SUM = ZERO
                  DO 130 I = J, M
                     SUM = SUM + ABS( A( I, J ) )
  130             CONTINUE
               END IF
               IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
  140       CONTINUE
         END IF
      ELSE IF( LSAME( NORM, 'I' ) ) THEN
*
*        Find normI(A).
*
         IF( LSAME( UPLO, 'U' ) ) THEN
            IF( LSAME( DIAG, 'U' ) ) THEN
               DO 150 I = 1, M
                  WORK( I ) = ONE
  150          CONTINUE
               DO 170 J = 1, N
                  DO 160 I = 1, MIN( M, J-1 )
                     WORK( I ) = WORK( I ) + ABS( A( I, J ) )
  160             CONTINUE
  170          CONTINUE
            ELSE
               DO 180 I = 1, M
                  WORK( I ) = ZERO
  180          CONTINUE
               DO 200 J = 1, N
                  DO 190 I = 1, MIN( M, J )
                     WORK( I ) = WORK( I ) + ABS( A( I, J ) )
  190             CONTINUE
  200          CONTINUE
            END IF
         ELSE
            IF( LSAME( DIAG, 'U' ) ) THEN
               DO 210 I = 1, MIN( M, N )
                  WORK( I ) = ONE
  210          CONTINUE
               DO 220 I = N + 1, M
                  WORK( I ) = ZERO
  220          CONTINUE
               DO 240 J = 1, N
                  DO 230 I = J + 1, M
                     WORK( I ) = WORK( I ) + ABS( A( I, J ) )
  230             CONTINUE
  240          CONTINUE
            ELSE
               DO 250 I = 1, M
                  WORK( I ) = ZERO
  250          CONTINUE
               DO 270 J = 1, N
                  DO 260 I = J, M
                     WORK( I ) = WORK( I ) + ABS( A( I, J ) )
  260             CONTINUE
  270          CONTINUE
            END IF
         END IF
         VALUE = ZERO
         DO 280 I = 1, M
            SUM = WORK( I )
            IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
  280    CONTINUE
      ELSE IF( ( LSAME( NORM, 'F' ) ) .OR.
     $         ( LSAME( NORM, 'E' ) ) ) THEN
*
*        Find normF(A).
*
         IF( LSAME( UPLO, 'U' ) ) THEN
            IF( LSAME( DIAG, 'U' ) ) THEN
               SCALE = ONE
               SUM = MIN( M, N )
               DO 290 J = 2, N
                  CALL DLASSQ( MIN( M, J-1 ), A( 1, J ), 1, SCALE,
     $                         SUM )
  290          CONTINUE
            ELSE
               SCALE = ZERO
               SUM = ONE
               DO 300 J = 1, N
                  CALL DLASSQ( MIN( M, J ), A( 1, J ), 1, SCALE,
     $                         SUM )
  300          CONTINUE
            END IF
         ELSE
            IF( LSAME( DIAG, 'U' ) ) THEN
               SCALE = ONE
               SUM = MIN( M, N )
               DO 310 J = 1, N
                  CALL DLASSQ( M-J, A( MIN( M, J+1 ), J ), 1, SCALE,
     $                         SUM )
  310          CONTINUE
            ELSE
               SCALE = ZERO
               SUM = ONE
               DO 320 J = 1, N
                  CALL DLASSQ( M-J+1, A( J, J ), 1, SCALE, SUM )
  320          CONTINUE
            END IF
         END IF
         VALUE = SCALE*SQRT( SUM )
      END IF
*
      DLANTR = VALUE
      RETURN
*
*     End of DLANTR
*
      END
*> \brief \b DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANV2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlanv2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlanv2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlanv2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLANV2( A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN )
*
*       .. Scalar Arguments ..
*       DOUBLE PRECISION   A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric
*> matrix in standard form:
*>
*>      [ A  B ] = [ CS -SN ] [ AA  BB ] [ CS  SN ]
*>      [ C  D ]   [ SN  CS ] [ CC  DD ] [-SN  CS ]
*>
*> where either
*> 1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or
*> 2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex
*> conjugate eigenvalues.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION
*>          On entry, the elements of the input matrix.
*>          On exit, they are overwritten by the elements of the
*>          standardised Schur form.
*> \endverbatim
*>
*> \param[out] RT1R
*> \verbatim
*>          RT1R is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] RT1I
*> \verbatim
*>          RT1I is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] RT2R
*> \verbatim
*>          RT2R is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] RT2I
*> \verbatim
*>          RT2I is DOUBLE PRECISION
*>          The real and imaginary parts of the eigenvalues. If the
*>          eigenvalues are a complex conjugate pair, RT1I > 0.
*> \endverbatim
*>
*> \param[out] CS
*> \verbatim
*>          CS is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] SN
*> \verbatim
*>          SN is DOUBLE PRECISION
*>          Parameters of the rotation matrix.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lanv2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  Modified by V. Sima, Research Institute for Informatics, Bucharest,
*>  Romania, to reduce the risk of cancellation errors,
*>  when computing real eigenvalues, and to ensure, if possible, that
*>  abs(RT1R) >= abs(RT2R).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLANV2( A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, HALF, ONE, TWO
      PARAMETER          ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0,
     $                     TWO = 2.0D0 )
      DOUBLE PRECISION   MULTPL
      PARAMETER          ( MULTPL = 4.0D+0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   AA, BB, BCMAX, BCMIS, CC, CS1, DD, EPS, P, SAB,
     $                   SAC, SCALE, SIGMA, SN1, TAU, TEMP, Z, SAFMIN,
     $                   SAFMN2, SAFMX2
      INTEGER            COUNT
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH, DLAPY2
      EXTERNAL           DLAMCH, DLAPY2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, SIGN, SQRT
*     ..
*     .. Executable Statements ..
*
      SAFMIN = DLAMCH( 'S' )
      EPS = DLAMCH( 'P' )
      SAFMN2 = DLAMCH( 'B' )**INT( LOG( SAFMIN / EPS ) /
     $            LOG( DLAMCH( 'B' ) ) / TWO )
      SAFMX2 = ONE / SAFMN2
      IF( C.EQ.ZERO ) THEN
         CS = ONE
         SN = ZERO
*
      ELSE IF( B.EQ.ZERO ) THEN
*
*        Swap rows and columns
*
         CS = ZERO
         SN = ONE
         TEMP = D
         D = A
         A = TEMP
         B = -C
         C = ZERO
*
      ELSE IF( ( A-D ).EQ.ZERO .AND. SIGN( ONE, B ).NE.SIGN( ONE, C ) )
     $          THEN
         CS = ONE
         SN = ZERO
*
      ELSE
*
         TEMP = A - D
         P = HALF*TEMP
         BCMAX = MAX( ABS( B ), ABS( C ) )
         BCMIS = MIN( ABS( B ), ABS( C ) )*SIGN( ONE, B )*SIGN( ONE, C )
         SCALE = MAX( ABS( P ), BCMAX )
         Z = ( P / SCALE )*P + ( BCMAX / SCALE )*BCMIS
*
*        If Z is of the order of the machine accuracy, postpone the
*        decision on the nature of eigenvalues
*
         IF( Z.GE.MULTPL*EPS ) THEN
*
*           Real eigenvalues. Compute A and D.
*
            Z = P + SIGN( SQRT( SCALE )*SQRT( Z ), P )
            A = D + Z
            D = D - ( BCMAX / Z )*BCMIS
*
*           Compute B and the rotation matrix
*
            TAU = DLAPY2( C, Z )
            CS = Z / TAU
            SN = C / TAU
            B = B - C
            C = ZERO
*
         ELSE
*
*           Complex eigenvalues, or real (almost) equal eigenvalues.
*           Make diagonal elements equal.
*
            COUNT = 0
            SIGMA = B + C
   10       CONTINUE
            COUNT = COUNT + 1
            SCALE = MAX( ABS(TEMP), ABS(SIGMA) )
            IF( SCALE.GE.SAFMX2 ) THEN
               SIGMA = SIGMA * SAFMN2
               TEMP = TEMP * SAFMN2
               IF (COUNT .LE. 20)
     $            GOTO 10
            END IF
            IF( SCALE.LE.SAFMN2 ) THEN
               SIGMA = SIGMA * SAFMX2
               TEMP = TEMP * SAFMX2
               IF (COUNT .LE. 20)
     $            GOTO 10
            END IF
            P = HALF*TEMP
            TAU = DLAPY2( SIGMA, TEMP )
            CS = SQRT( HALF*( ONE+ABS( SIGMA ) / TAU ) )
            SN = -( P / ( TAU*CS ) )*SIGN( ONE, SIGMA )
*
*           Compute [ AA  BB ] = [ A  B ] [ CS -SN ]
*                   [ CC  DD ]   [ C  D ] [ SN  CS ]
*
            AA = A*CS + B*SN
            BB = -A*SN + B*CS
            CC = C*CS + D*SN
            DD = -C*SN + D*CS
*
*           Compute [ A  B ] = [ CS  SN ] [ AA  BB ]
*                   [ C  D ]   [-SN  CS ] [ CC  DD ]
*
*           Note: Some of the multiplications are wrapped in parentheses to
*                 prevent compilers from using FMA instructions. See
*                 https://github.com/Reference-LAPACK/lapack/issues/1031.
*
            A = AA*CS + CC*SN
            B = ( BB*CS ) + ( DD*SN )
            C = -( AA*SN ) + ( CC*CS )
            D = -BB*SN + DD*CS
*
            TEMP = HALF*( A+D )
            A = TEMP
            D = TEMP
*
            IF( C.NE.ZERO ) THEN
               IF( B.NE.ZERO ) THEN
                  IF( SIGN( ONE, B ).EQ.SIGN( ONE, C ) ) THEN
*
*                    Real eigenvalues: reduce to upper triangular form
*
                     SAB = SQRT( ABS( B ) )
                     SAC = SQRT( ABS( C ) )
                     P = SIGN( SAB*SAC, C )
                     TAU = ONE / SQRT( ABS( B+C ) )
                     A = TEMP + P
                     D = TEMP - P
                     B = B - C
                     C = ZERO
                     CS1 = SAB*TAU
                     SN1 = SAC*TAU
                     TEMP = CS*CS1 - SN*SN1
                     SN = CS*SN1 + SN*CS1
                     CS = TEMP
                  END IF
               ELSE
                  B = -C
                  C = ZERO
                  TEMP = CS
                  CS = -SN
                  SN = TEMP
               END IF
            END IF
         END IF
*
      END IF
*
*     Store eigenvalues in (RT1R,RT1I) and (RT2R,RT2I).
*
      RT1R = A
      RT2R = D
      IF( C.EQ.ZERO ) THEN
         RT1I = ZERO
         RT2I = ZERO
      ELSE
         RT1I = SQRT( ABS( B ) )*SQRT( ABS( C ) )
         RT2I = -RT1I
      END IF
      RETURN
*
*     End of DLANV2
*
      END
*> \brief \b DLAPLL measures the linear dependence of two vectors.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAPLL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlapll.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlapll.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlapll.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAPLL( N, X, INCX, Y, INCY, SSMIN )
*
*       .. Scalar Arguments ..
*       INTEGER            INCX, INCY, N
*       DOUBLE PRECISION   SSMIN
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   X( * ), Y( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> Given two column vectors X and Y, let
*>
*>                      A = ( X Y ).
*>
*> The subroutine first computes the QR factorization of A = Q*R,
*> and then computes the SVD of the 2-by-2 upper triangular matrix R.
*> The smaller singular value of R is returned in SSMIN, which is used
*> as the measurement of the linear dependency of the vectors X and Y.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The length of the vectors X and Y.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array,
*>                         dimension (1+(N-1)*INCX)
*>          On entry, X contains the N-vector X.
*>          On exit, X is overwritten.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*>          INCX is INTEGER
*>          The increment between successive elements of X. INCX > 0.
*> \endverbatim
*>
*> \param[in,out] Y
*> \verbatim
*>          Y is DOUBLE PRECISION array,
*>                         dimension (1+(N-1)*INCY)
*>          On entry, Y contains the N-vector Y.
*>          On exit, Y is overwritten.
*> \endverbatim
*>
*> \param[in] INCY
*> \verbatim
*>          INCY is INTEGER
*>          The increment between successive elements of Y. INCY > 0.
*> \endverbatim
*>
*> \param[out] SSMIN
*> \verbatim
*>          SSMIN is DOUBLE PRECISION
*>          The smallest singular value of the N-by-2 matrix A = ( X Y ).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lapll
*
*  =====================================================================
      SUBROUTINE DLAPLL( N, X, INCX, Y, INCY, SSMIN )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INCX, INCY, N
      DOUBLE PRECISION   SSMIN
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   X( * ), Y( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   A11, A12, A22, C, SSMAX, TAU
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DDOT
      EXTERNAL           DDOT
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DLARFG, DLAS2
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( N.LE.1 ) THEN
         SSMIN = ZERO
         RETURN
      END IF
*
*     Compute the QR factorization of the N-by-2 matrix ( X Y )
*
      CALL DLARFG( N, X( 1 ), X( 1+INCX ), INCX, TAU )
      A11 = X( 1 )
      X( 1 ) = ONE
*
      C = -TAU*DDOT( N, X, INCX, Y, INCY )
      CALL DAXPY( N, C, X, INCX, Y, INCY )
*
      CALL DLARFG( N-1, Y( 1+INCY ), Y( 1+2*INCY ), INCY, TAU )
*
      A12 = Y( 1 )
      A22 = Y( 1+INCY )
*
*     Compute the SVD of 2-by-2 Upper triangular matrix.
*
      CALL DLAS2( A11, A12, A22, SSMIN, SSMAX )
*
      RETURN
*
*     End of DLAPLL
*
      END
*> \brief \b DLAPMR rearranges rows of a matrix as specified by a permutation vector.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAPMR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlapmr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlapmr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlapmr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAPMR( FORWRD, M, N, X, LDX, K )
*
*       .. Scalar Arguments ..
*       LOGICAL            FORWRD
*       INTEGER            LDX, M, N
*       ..
*       .. Array Arguments ..
*       INTEGER            K( * )
*       DOUBLE PRECISION   X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAPMR rearranges the rows of the M by N matrix X as specified
*> by the permutation K(1),K(2),...,K(M) of the integers 1,...,M.
*> If FORWRD = .TRUE.,  forward permutation:
*>
*>      X(K(I),*) is moved X(I,*) for I = 1,2,...,M.
*>
*> If FORWRD = .FALSE., backward permutation:
*>
*>      X(I,*) is moved to X(K(I),*) for I = 1,2,...,M.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] FORWRD
*> \verbatim
*>          FORWRD is LOGICAL
*>          = .TRUE., forward permutation
*>          = .FALSE., backward permutation
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix X. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix X. N >= 0.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,N)
*>          On entry, the M by N matrix X.
*>          On exit, X contains the permuted matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X, LDX >= MAX(1,M).
*> \endverbatim
*>
*> \param[in,out] K
*> \verbatim
*>          K is INTEGER array, dimension (M)
*>          On entry, K contains the permutation vector. K is used as
*>          internal workspace, but reset to its original value on
*>          output.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lapmr
*
*  =====================================================================
      SUBROUTINE DLAPMR( FORWRD, M, N, X, LDX, K )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      LOGICAL            FORWRD
      INTEGER            LDX, M, N
*     ..
*     .. Array Arguments ..
      INTEGER            K( * )
      DOUBLE PRECISION   X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            I, IN, J, JJ
      DOUBLE PRECISION   TEMP
*     ..
*     .. Executable Statements ..
*
      IF( M.LE.1 )
     $   RETURN
*
      DO 10 I = 1, M
         K( I ) = -K( I )
   10 CONTINUE
*
      IF( FORWRD ) THEN
*
*        Forward permutation
*
         DO 50 I = 1, M
*
            IF( K( I ).GT.0 )
     $         GO TO 40
*
            J = I
            K( J ) = -K( J )
            IN = K( J )
*
   20       CONTINUE
            IF( K( IN ).GT.0 )
     $         GO TO 40
*
            DO 30 JJ = 1, N
               TEMP = X( J, JJ )
               X( J, JJ ) = X( IN, JJ )
               X( IN, JJ ) = TEMP
   30       CONTINUE
*
            K( IN ) = -K( IN )
            J = IN
            IN = K( IN )
            GO TO 20
*
   40       CONTINUE
*
   50    CONTINUE
*
      ELSE
*
*        Backward permutation
*
         DO 90 I = 1, M
*
            IF( K( I ).GT.0 )
     $         GO TO 80
*
            K( I ) = -K( I )
            J = K( I )
   60       CONTINUE
            IF( J.EQ.I )
     $         GO TO 80
*
            DO 70 JJ = 1, N
               TEMP = X( I, JJ )
               X( I, JJ ) = X( J, JJ )
               X( J, JJ ) = TEMP
   70       CONTINUE
*
            K( J ) = -K( J )
            J = K( J )
            GO TO 60
*
   80       CONTINUE
*
   90    CONTINUE
*
      END IF
*
      RETURN
*
*     End of DLAPMR
*
      END

*> \brief \b DLAPMT performs a forward or backward permutation of the columns of a matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAPMT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlapmt.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlapmt.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlapmt.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAPMT( FORWRD, M, N, X, LDX, K )
*
*       .. Scalar Arguments ..
*       LOGICAL            FORWRD
*       INTEGER            LDX, M, N
*       ..
*       .. Array Arguments ..
*       INTEGER            K( * )
*       DOUBLE PRECISION   X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAPMT rearranges the columns of the M by N matrix X as specified
*> by the permutation K(1),K(2),...,K(N) of the integers 1,...,N.
*> If FORWRD = .TRUE.,  forward permutation:
*>
*>      X(*,K(J)) is moved X(*,J) for J = 1,2,...,N.
*>
*> If FORWRD = .FALSE., backward permutation:
*>
*>      X(*,J) is moved to X(*,K(J)) for J = 1,2,...,N.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] FORWRD
*> \verbatim
*>          FORWRD is LOGICAL
*>          = .TRUE., forward permutation
*>          = .FALSE., backward permutation
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix X. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix X. N >= 0.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,N)
*>          On entry, the M by N matrix X.
*>          On exit, X contains the permuted matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X, LDX >= MAX(1,M).
*> \endverbatim
*>
*> \param[in,out] K
*> \verbatim
*>          K is INTEGER array, dimension (N)
*>          On entry, K contains the permutation vector. K is used as
*>          internal workspace, but reset to its original value on
*>          output.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lapmt
*
*  =====================================================================
      SUBROUTINE DLAPMT( FORWRD, M, N, X, LDX, K )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      LOGICAL            FORWRD
      INTEGER            LDX, M, N
*     ..
*     .. Array Arguments ..
      INTEGER            K( * )
      DOUBLE PRECISION   X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            I, II, IN, J
      DOUBLE PRECISION   TEMP
*     ..
*     .. Executable Statements ..
*
      IF( N.LE.1 )
     $   RETURN
*
      DO 10 I = 1, N
         K( I ) = -K( I )
   10 CONTINUE
*
      IF( FORWRD ) THEN
*
*        Forward permutation
*
         DO 50 I = 1, N
*
            IF( K( I ).GT.0 )
     $         GO TO 40
*
            J = I
            K( J ) = -K( J )
            IN = K( J )
*
   20       CONTINUE
            IF( K( IN ).GT.0 )
     $         GO TO 40
*
            DO 30 II = 1, M
               TEMP = X( II, J )
               X( II, J ) = X( II, IN )
               X( II, IN ) = TEMP
   30       CONTINUE
*
            K( IN ) = -K( IN )
            J = IN
            IN = K( IN )
            GO TO 20
*
   40       CONTINUE
*
   50    CONTINUE
*
      ELSE
*
*        Backward permutation
*
         DO 90 I = 1, N
*
            IF( K( I ).GT.0 )
     $         GO TO 80
*
            K( I ) = -K( I )
            J = K( I )
   60       CONTINUE
            IF( J.EQ.I )
     $         GO TO 80
*
            DO 70 II = 1, M
               TEMP = X( II, I )
               X( II, I ) = X( II, J )
               X( II, J ) = TEMP
   70       CONTINUE
*
            K( J ) = -K( J )
            J = K( J )
            GO TO 60
*
   80       CONTINUE
*
   90    CONTINUE
*
      END IF
*
      RETURN
*
*     End of DLAPMT
*
      END
*> \brief \b DLAPY2 returns sqrt(x2+y2).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAPY2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlapy2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlapy2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlapy2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       DOUBLE PRECISION FUNCTION DLAPY2( X, Y )
*
*       .. Scalar Arguments ..
*       DOUBLE PRECISION   X, Y
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAPY2 returns sqrt(x**2+y**2), taking care not to cause unnecessary
*> overflow and unnecessary underflow.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] X
*> \verbatim
*>          X is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] Y
*> \verbatim
*>          Y is DOUBLE PRECISION
*>          X and Y specify the values x and y.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lapy2
*
*  =====================================================================
      DOUBLE PRECISION FUNCTION DLAPY2( X, Y )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   X, Y
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   W, XABS, YABS, Z, HUGEVAL
      LOGICAL            X_IS_NAN, Y_IS_NAN
*     ..
*     .. External Functions ..
      LOGICAL            DISNAN
      EXTERNAL           DISNAN
*     ..
*     .. External Subroutines ..
      DOUBLE PRECISION   DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
      X_IS_NAN = DISNAN( X )
      Y_IS_NAN = DISNAN( Y )
      IF ( X_IS_NAN ) DLAPY2 = X
      IF ( Y_IS_NAN ) DLAPY2 = Y
      HUGEVAL = DLAMCH( 'Overflow' )
*
      IF ( .NOT.( X_IS_NAN.OR.Y_IS_NAN ) ) THEN
         XABS = ABS( X )
         YABS = ABS( Y )
         W = MAX( XABS, YABS )
         Z = MIN( XABS, YABS )
         IF( Z.EQ.ZERO .OR. W.GT.HUGEVAL ) THEN
            DLAPY2 = W
         ELSE
            DLAPY2 = W*SQRT( ONE+( Z / W )**2 )
         END IF
      END IF
      RETURN
*
*     End of DLAPY2
*
      END
*> \brief \b DLAPY3 returns sqrt(x2+y2+z2).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAPY3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlapy3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlapy3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlapy3.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       DOUBLE PRECISION FUNCTION DLAPY3( X, Y, Z )
*
*       .. Scalar Arguments ..
*       DOUBLE PRECISION   X, Y, Z
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAPY3 returns sqrt(x**2+y**2+z**2), taking care not to cause
*> unnecessary overflow and unnecessary underflow.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] X
*> \verbatim
*>          X is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] Y
*> \verbatim
*>          Y is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*>          Z is DOUBLE PRECISION
*>          X, Y and Z specify the values x, y and z.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lapy3
*
*  =====================================================================
      DOUBLE PRECISION FUNCTION DLAPY3( X, Y, Z )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   X, Y, Z
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   W, XABS, YABS, ZABS, HUGEVAL
*     ..
*     .. External Subroutines ..
      DOUBLE PRECISION   DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
      HUGEVAL = DLAMCH( 'Overflow' )
      XABS = ABS( X )
      YABS = ABS( Y )
      ZABS = ABS( Z )
      W = MAX( XABS, YABS, ZABS )
      IF( W.EQ.ZERO .OR. W.GT.HUGEVAL ) THEN
*     W can be zero for max(0,nan,0)
*     adding all three entries together will make sure
*     NaN will not disappear.
         DLAPY3 =  XABS + YABS + ZABS
      ELSE
         DLAPY3 = W*SQRT( ( XABS / W )**2+( YABS / W )**2+
     $            ( ZABS / W )**2 )
      END IF
      RETURN
*
*     End of DLAPY3
*
      END
*> \brief \b DLAQGB scales a general band matrix, using row and column scaling factors computed by sgbequ.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQGB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqgb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqgb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqgb.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAQGB( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
*                          AMAX, EQUED )
*
*       .. Scalar Arguments ..
*       CHARACTER          EQUED
*       INTEGER            KL, KU, LDAB, M, N
*       DOUBLE PRECISION   AMAX, COLCND, ROWCND
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * ), C( * ), R( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAQGB equilibrates a general M by N band matrix A with KL
*> subdiagonals and KU superdiagonals using the row and scaling factors
*> in the vectors R and C.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*>          KL is INTEGER
*>          The number of subdiagonals within the band of A.  KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*>          KU is INTEGER
*>          The number of superdiagonals within the band of A.  KU >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
*>          The j-th column of A is stored in the j-th column of the
*>          array AB as follows:
*>          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)
*>
*>          On exit, the equilibrated matrix, in the same storage format
*>          as A.  See EQUED for the form of the equilibrated matrix.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDA >= KL+KU+1.
*> \endverbatim
*>
*> \param[in] R
*> \verbatim
*>          R is DOUBLE PRECISION array, dimension (M)
*>          The row scale factors for A.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (N)
*>          The column scale factors for A.
*> \endverbatim
*>
*> \param[in] ROWCND
*> \verbatim
*>          ROWCND is DOUBLE PRECISION
*>          Ratio of the smallest R(i) to the largest R(i).
*> \endverbatim
*>
*> \param[in] COLCND
*> \verbatim
*>          COLCND is DOUBLE PRECISION
*>          Ratio of the smallest C(i) to the largest C(i).
*> \endverbatim
*>
*> \param[in] AMAX
*> \verbatim
*>          AMAX is DOUBLE PRECISION
*>          Absolute value of largest matrix entry.
*> \endverbatim
*>
*> \param[out] EQUED
*> \verbatim
*>          EQUED is CHARACTER*1
*>          Specifies the form of equilibration that was done.
*>          = 'N':  No equilibration
*>          = 'R':  Row equilibration, i.e., A has been premultiplied by
*>                  diag(R).
*>          = 'C':  Column equilibration, i.e., A has been postmultiplied
*>                  by diag(C).
*>          = 'B':  Both row and column equilibration, i.e., A has been
*>                  replaced by diag(R) * A * diag(C).
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  THRESH is a threshold value used to decide if row or column scaling
*>  should be done based on the ratio of the row or column scaling
*>  factors.  If ROWCND < THRESH, row scaling is done, and if
*>  COLCND < THRESH, column scaling is done.
*>
*>  LARGE and SMALL are threshold values used to decide if row scaling
*>  should be done based on the absolute size of the largest matrix
*>  element.  If AMAX > LARGE or AMAX < SMALL, row scaling is done.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqgb
*
*  =====================================================================
      SUBROUTINE DLAQGB( M, N, KL, KU, AB, LDAB, R, C, ROWCND,
     $                   COLCND,
     $                   AMAX, EQUED )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          EQUED
      INTEGER            KL, KU, LDAB, M, N
      DOUBLE PRECISION   AMAX, COLCND, ROWCND
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * ), C( * ), R( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, THRESH
      PARAMETER          ( ONE = 1.0D+0, THRESH = 0.1D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J
      DOUBLE PRECISION   CJ, LARGE, SMALL
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( M.LE.0 .OR. N.LE.0 ) THEN
         EQUED = 'N'
         RETURN
      END IF
*
*     Initialize LARGE and SMALL.
*
      SMALL = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
      LARGE = ONE / SMALL
*
      IF( ROWCND.GE.THRESH .AND. AMAX.GE.SMALL .AND. AMAX.LE.LARGE )
     $     THEN
*
*        No row scaling
*
         IF( COLCND.GE.THRESH ) THEN
*
*           No column scaling
*
            EQUED = 'N'
         ELSE
*
*           Column scaling
*
            DO 20 J = 1, N
               CJ = C( J )
               DO 10 I = MAX( 1, J-KU ), MIN( M, J+KL )
                  AB( KU+1+I-J, J ) = CJ*AB( KU+1+I-J, J )
   10          CONTINUE
   20       CONTINUE
            EQUED = 'C'
         END IF
      ELSE IF( COLCND.GE.THRESH ) THEN
*
*        Row scaling, no column scaling
*
         DO 40 J = 1, N
            DO 30 I = MAX( 1, J-KU ), MIN( M, J+KL )
               AB( KU+1+I-J, J ) = R( I )*AB( KU+1+I-J, J )
   30       CONTINUE
   40    CONTINUE
         EQUED = 'R'
      ELSE
*
*        Row and column scaling
*
         DO 60 J = 1, N
            CJ = C( J )
            DO 50 I = MAX( 1, J-KU ), MIN( M, J+KL )
               AB( KU+1+I-J, J ) = CJ*R( I )*AB( KU+1+I-J, J )
   50       CONTINUE
   60    CONTINUE
         EQUED = 'B'
      END IF
*
      RETURN
*
*     End of DLAQGB
*
      END
*> \brief \b DLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQGE + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqge.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqge.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqge.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAQGE( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
*                          EQUED )
*
*       .. Scalar Arguments ..
*       CHARACTER          EQUED
*       INTEGER            LDA, M, N
*       DOUBLE PRECISION   AMAX, COLCND, ROWCND
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), C( * ), R( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAQGE equilibrates a general M by N matrix A using the row and
*> column scaling factors in the vectors R and C.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M by N matrix A.
*>          On exit, the equilibrated matrix.  See EQUED for the form of
*>          the equilibrated matrix.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(M,1).
*> \endverbatim
*>
*> \param[in] R
*> \verbatim
*>          R is DOUBLE PRECISION array, dimension (M)
*>          The row scale factors for A.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (N)
*>          The column scale factors for A.
*> \endverbatim
*>
*> \param[in] ROWCND
*> \verbatim
*>          ROWCND is DOUBLE PRECISION
*>          Ratio of the smallest R(i) to the largest R(i).
*> \endverbatim
*>
*> \param[in] COLCND
*> \verbatim
*>          COLCND is DOUBLE PRECISION
*>          Ratio of the smallest C(i) to the largest C(i).
*> \endverbatim
*>
*> \param[in] AMAX
*> \verbatim
*>          AMAX is DOUBLE PRECISION
*>          Absolute value of largest matrix entry.
*> \endverbatim
*>
*> \param[out] EQUED
*> \verbatim
*>          EQUED is CHARACTER*1
*>          Specifies the form of equilibration that was done.
*>          = 'N':  No equilibration
*>          = 'R':  Row equilibration, i.e., A has been premultiplied by
*>                  diag(R).
*>          = 'C':  Column equilibration, i.e., A has been postmultiplied
*>                  by diag(C).
*>          = 'B':  Both row and column equilibration, i.e., A has been
*>                  replaced by diag(R) * A * diag(C).
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  THRESH is a threshold value used to decide if row or column scaling
*>  should be done based on the ratio of the row or column scaling
*>  factors.  If ROWCND < THRESH, row scaling is done, and if
*>  COLCND < THRESH, column scaling is done.
*>
*>  LARGE and SMALL are threshold values used to decide if row scaling
*>  should be done based on the absolute size of the largest matrix
*>  element.  If AMAX > LARGE or AMAX < SMALL, row scaling is done.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqge
*
*  =====================================================================
      SUBROUTINE DLAQGE( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
     $                   EQUED )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          EQUED
      INTEGER            LDA, M, N
      DOUBLE PRECISION   AMAX, COLCND, ROWCND
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( * ), R( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, THRESH
      PARAMETER          ( ONE = 1.0D+0, THRESH = 0.1D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J
      DOUBLE PRECISION   CJ, LARGE, SMALL
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( M.LE.0 .OR. N.LE.0 ) THEN
         EQUED = 'N'
         RETURN
      END IF
*
*     Initialize LARGE and SMALL.
*
      SMALL = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
      LARGE = ONE / SMALL
*
      IF( ROWCND.GE.THRESH .AND. AMAX.GE.SMALL .AND. AMAX.LE.LARGE )
     $     THEN
*
*        No row scaling
*
         IF( COLCND.GE.THRESH ) THEN
*
*           No column scaling
*
            EQUED = 'N'
         ELSE
*
*           Column scaling
*
            DO 20 J = 1, N
               CJ = C( J )
               DO 10 I = 1, M
                  A( I, J ) = CJ*A( I, J )
   10          CONTINUE
   20       CONTINUE
            EQUED = 'C'
         END IF
      ELSE IF( COLCND.GE.THRESH ) THEN
*
*        Row scaling, no column scaling
*
         DO 40 J = 1, N
            DO 30 I = 1, M
               A( I, J ) = R( I )*A( I, J )
   30       CONTINUE
   40    CONTINUE
         EQUED = 'R'
      ELSE
*
*        Row and column scaling
*
         DO 60 J = 1, N
            CJ = C( J )
            DO 50 I = 1, M
               A( I, J ) = CJ*R( I )*A( I, J )
   50       CONTINUE
   60    CONTINUE
         EQUED = 'B'
      END IF
*
      RETURN
*
*     End of DLAQGE
*
      END
*> \brief \b DLAQP2 computes a QR factorization with column pivoting of the matrix block.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQP2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqp2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqp2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqp2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAQP2( M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2,
*                          WORK )
*
*       .. Scalar Arguments ..
*       INTEGER            LDA, M, N, OFFSET
*       ..
*       .. Array Arguments ..
*       INTEGER            JPVT( * )
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), VN1( * ), VN2( * ),
*      $                   WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAQP2 computes a QR factorization with column pivoting of
*> the block A(OFFSET+1:M,1:N).
*> The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] OFFSET
*> \verbatim
*>          OFFSET is INTEGER
*>          The number of rows of the matrix A that must be pivoted
*>          but no factorized. OFFSET >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit, the upper triangle of block A(OFFSET+1:M,1:N) is
*>          the triangular factor obtained; the elements in block
*>          A(OFFSET+1:M,1:N) below the diagonal, together with the
*>          array TAU, represent the orthogonal matrix Q as a product of
*>          elementary reflectors. Block A(1:OFFSET,1:N) has been
*>          accordingly pivoted, but no factorized.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] JPVT
*> \verbatim
*>          JPVT is INTEGER array, dimension (N)
*>          On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
*>          to the front of A*P (a leading column); if JPVT(i) = 0,
*>          the i-th column of A is a free column.
*>          On exit, if JPVT(i) = k, then the i-th column of A*P
*>          was the k-th column of A.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors.
*> \endverbatim
*>
*> \param[in,out] VN1
*> \verbatim
*>          VN1 is DOUBLE PRECISION array, dimension (N)
*>          The vector with the partial column norms.
*> \endverbatim
*>
*> \param[in,out] VN2
*> \verbatim
*>          VN2 is DOUBLE PRECISION array, dimension (N)
*>          The vector with the exact column norms.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqp2
*
*> \par Contributors:
*  ==================
*>
*>    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
*>    X. Sun, Computer Science Dept., Duke University, USA
*> \n
*>  Partial column norm updating strategy modified on April 2011
*>    Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
*>    University of Zagreb, Croatia.
*
*> \par References:
*  ================
*>
*> LAPACK Working Note 176
*
*> \htmlonly
*> <a href="http://www.netlib.org/lapack/lawnspdf/lawn176.pdf">[PDF]</a>
*> \endhtmlonly
*
*  =====================================================================
      SUBROUTINE DLAQP2( M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2,
     $                   WORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            LDA, M, N, OFFSET
*     ..
*     .. Array Arguments ..
      INTEGER            JPVT( * )
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), VN1( * ), VN2( * ),
     $                   WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, ITEMP, J, MN, OFFPI, PVT
      DOUBLE PRECISION   TEMP, TEMP2, TOL3Z
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF, DLARFG, DSWAP
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, SQRT
*     ..
*     .. External Functions ..
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH, DNRM2
      EXTERNAL           IDAMAX, DLAMCH, DNRM2
*     ..
*     .. Executable Statements ..
*
      MN = MIN( M-OFFSET, N )
      TOL3Z = SQRT(DLAMCH('Epsilon'))
*
*     Compute factorization.
*
      DO 20 I = 1, MN
*
         OFFPI = OFFSET + I
*
*        Determine ith pivot column and swap if necessary.
*
         PVT = ( I-1 ) + IDAMAX( N-I+1, VN1( I ), 1 )
*
         IF( PVT.NE.I ) THEN
            CALL DSWAP( M, A( 1, PVT ), 1, A( 1, I ), 1 )
            ITEMP = JPVT( PVT )
            JPVT( PVT ) = JPVT( I )
            JPVT( I ) = ITEMP
            VN1( PVT ) = VN1( I )
            VN2( PVT ) = VN2( I )
         END IF
*
*        Generate elementary reflector H(i).
*
         IF( OFFPI.LT.M ) THEN
            CALL DLARFG( M-OFFPI+1, A( OFFPI, I ), A( OFFPI+1, I ),
     $                   1,
     $                   TAU( I ) )
         ELSE
            CALL DLARFG( 1, A( M, I ), A( M, I ), 1, TAU( I ) )
         END IF
*
         IF( I.LT.N ) THEN
*
*           Apply H(i)**T to A(offset+i:m,i+1:n) from the left.
*
            CALL DLARF1F( 'Left', M-OFFPI+1, N-I, A( OFFPI, I ), 1,
     $                  TAU( I ), A( OFFPI, I+1 ), LDA, WORK( 1 ) )
         END IF
*
*        Update partial column norms.
*
         DO 10 J = I + 1, N
            IF( VN1( J ).NE.ZERO ) THEN
*
*              NOTE: The following 4 lines follow from the analysis in
*              Lapack Working Note 176.
*
               TEMP = ONE - ( ABS( A( OFFPI, J ) ) / VN1( J ) )**2
               TEMP = MAX( TEMP, ZERO )
               TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2
               IF( TEMP2 .LE. TOL3Z ) THEN
                  IF( OFFPI.LT.M ) THEN
                     VN1( J ) = DNRM2( M-OFFPI, A( OFFPI+1, J ), 1 )
                     VN2( J ) = VN1( J )
                  ELSE
                     VN1( J ) = ZERO
                     VN2( J ) = ZERO
                  END IF
               ELSE
                  VN1( J ) = VN1( J )*SQRT( TEMP )
               END IF
            END IF
   10    CONTINUE
*
   20 CONTINUE
*
      RETURN
*
*     End of DLAQP2
*
      END
*> \brief \b DLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQPS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqps.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqps.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqps.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
*                          VN2, AUXV, F, LDF )
*
*       .. Scalar Arguments ..
*       INTEGER            KB, LDA, LDF, M, N, NB, OFFSET
*       ..
*       .. Array Arguments ..
*       INTEGER            JPVT( * )
*       DOUBLE PRECISION   A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ),
*      $                   VN1( * ), VN2( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAQPS computes a step of QR factorization with column pivoting
*> of a real M-by-N matrix A by using Blas-3.  It tries to factorize
*> NB columns from A starting from the row OFFSET+1, and updates all
*> of the matrix with Blas-3 xGEMM.
*>
*> In some cases, due to catastrophic cancellations, it cannot
*> factorize NB columns.  Hence, the actual number of factorized
*> columns is returned in KB.
*>
*> Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A. N >= 0
*> \endverbatim
*>
*> \param[in] OFFSET
*> \verbatim
*>          OFFSET is INTEGER
*>          The number of rows of A that have been factorized in
*>          previous steps.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*>          NB is INTEGER
*>          The number of columns to factorize.
*> \endverbatim
*>
*> \param[out] KB
*> \verbatim
*>          KB is INTEGER
*>          The number of columns actually factorized.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit, block A(OFFSET+1:M,1:KB) is the triangular
*>          factor obtained and block A(1:OFFSET,1:N) has been
*>          accordingly pivoted, but no factorized.
*>          The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has
*>          been updated.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] JPVT
*> \verbatim
*>          JPVT is INTEGER array, dimension (N)
*>          JPVT(I) = K <==> Column K of the full matrix A has been
*>          permuted into position I in AP.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (KB)
*>          The scalar factors of the elementary reflectors.
*> \endverbatim
*>
*> \param[in,out] VN1
*> \verbatim
*>          VN1 is DOUBLE PRECISION array, dimension (N)
*>          The vector with the partial column norms.
*> \endverbatim
*>
*> \param[in,out] VN2
*> \verbatim
*>          VN2 is DOUBLE PRECISION array, dimension (N)
*>          The vector with the exact column norms.
*> \endverbatim
*>
*> \param[in,out] AUXV
*> \verbatim
*>          AUXV is DOUBLE PRECISION array, dimension (NB)
*>          Auxiliary vector.
*> \endverbatim
*>
*> \param[in,out] F
*> \verbatim
*>          F is DOUBLE PRECISION array, dimension (LDF,NB)
*>          Matrix F**T = L*Y**T*A.
*> \endverbatim
*>
*> \param[in] LDF
*> \verbatim
*>          LDF is INTEGER
*>          The leading dimension of the array F. LDF >= max(1,N).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqps
*
*> \par Contributors:
*  ==================
*>
*>    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
*>    X. Sun, Computer Science Dept., Duke University, USA
*> \n
*>  Partial column norm updating strategy modified on April 2011
*>    Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
*>    University of Zagreb, Croatia.
*
*> \par References:
*  ================
*>
*> LAPACK Working Note 176
*
*> \htmlonly
*> <a href="http://www.netlib.org/lapack/lawnspdf/lawn176.pdf">[PDF]</a>
*> \endhtmlonly
*
*  =====================================================================
      SUBROUTINE DLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU,
     $                   VN1,
     $                   VN2, AUXV, F, LDF )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            KB, LDA, LDF, M, N, NB, OFFSET
*     ..
*     .. Array Arguments ..
      INTEGER            JPVT( * )
      DOUBLE PRECISION   A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ),
     $                   VN1( * ), VN2( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            ITEMP, J, K, LASTRK, LSTICC, PVT, RK
      DOUBLE PRECISION   AKK, TEMP, TEMP2, TOL3Z
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMM, DGEMV, DLARFG, DSWAP
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, MAX, MIN, NINT, SQRT
*     ..
*     .. External Functions ..
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH, DNRM2
      EXTERNAL           IDAMAX, DLAMCH, DNRM2
*     ..
*     .. Executable Statements ..
*
      LASTRK = MIN( M, N+OFFSET )
      LSTICC = 0
      K = 0
      TOL3Z = SQRT(DLAMCH('Epsilon'))
*
*     Beginning of while loop.
*
   10 CONTINUE
      IF( ( K.LT.NB ) .AND. ( LSTICC.EQ.0 ) ) THEN
         K = K + 1
         RK = OFFSET + K
*
*        Determine ith pivot column and swap if necessary
*
         PVT = ( K-1 ) + IDAMAX( N-K+1, VN1( K ), 1 )
         IF( PVT.NE.K ) THEN
            CALL DSWAP( M, A( 1, PVT ), 1, A( 1, K ), 1 )
            CALL DSWAP( K-1, F( PVT, 1 ), LDF, F( K, 1 ), LDF )
            ITEMP = JPVT( PVT )
            JPVT( PVT ) = JPVT( K )
            JPVT( K ) = ITEMP
            VN1( PVT ) = VN1( K )
            VN2( PVT ) = VN2( K )
         END IF
*
*        Apply previous Householder reflectors to column K:
*        A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)**T.
*
         IF( K.GT.1 ) THEN
            CALL DGEMV( 'No transpose', M-RK+1, K-1, -ONE, A( RK,
     $                  1 ),
     $                  LDA, F( K, 1 ), LDF, ONE, A( RK, K ), 1 )
         END IF
*
*        Generate elementary reflector H(k).
*
         IF( RK.LT.M ) THEN
            CALL DLARFG( M-RK+1, A( RK, K ), A( RK+1, K ), 1,
     $                   TAU( K ) )
         ELSE
            CALL DLARFG( 1, A( RK, K ), A( RK, K ), 1, TAU( K ) )
         END IF
*
         AKK = A( RK, K )
         A( RK, K ) = ONE
*
*        Compute Kth column of F:
*
*        Compute  F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)**T*A(RK:M,K).
*
         IF( K.LT.N ) THEN
            CALL DGEMV( 'Transpose', M-RK+1, N-K, TAU( K ),
     $                  A( RK, K+1 ), LDA, A( RK, K ), 1, ZERO,
     $                  F( K+1, K ), 1 )
         END IF
*
*        Padding F(1:K,K) with zeros.
*
         DO 20 J = 1, K
            F( J, K ) = ZERO
   20    CONTINUE
*
*        Incremental updating of F:
*        F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)**T
*                    *A(RK:M,K).
*
         IF( K.GT.1 ) THEN
            CALL DGEMV( 'Transpose', M-RK+1, K-1, -TAU( K ), A( RK,
     $                  1 ),
     $                  LDA, A( RK, K ), 1, ZERO, AUXV( 1 ), 1 )
*
            CALL DGEMV( 'No transpose', N, K-1, ONE, F( 1, 1 ), LDF,
     $                  AUXV( 1 ), 1, ONE, F( 1, K ), 1 )
         END IF
*
*        Update the current row of A:
*        A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)**T.
*
         IF( K.LT.N ) THEN
            CALL DGEMV( 'No transpose', N-K, K, -ONE, F( K+1, 1 ),
     $                  LDF,
     $                  A( RK, 1 ), LDA, ONE, A( RK, K+1 ), LDA )
         END IF
*
*        Update partial column norms.
*
         IF( RK.LT.LASTRK ) THEN
            DO 30 J = K + 1, N
               IF( VN1( J ).NE.ZERO ) THEN
*
*                 NOTE: The following 4 lines follow from the analysis in
*                 Lapack Working Note 176.
*
                  TEMP = ABS( A( RK, J ) ) / VN1( J )
                  TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) )
                  TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2
                  IF( TEMP2 .LE. TOL3Z ) THEN
                     VN2( J ) = DBLE( LSTICC )
                     LSTICC = J
                  ELSE
                     VN1( J ) = VN1( J )*SQRT( TEMP )
                  END IF
               END IF
   30       CONTINUE
         END IF
*
         A( RK, K ) = AKK
*
*        End of while loop.
*
         GO TO 10
      END IF
      KB = K
      RK = OFFSET + KB
*
*     Apply the block reflector to the rest of the matrix:
*     A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) -
*                         A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)**T.
*
      IF( KB.LT.MIN( N, M-OFFSET ) ) THEN
         CALL DGEMM( 'No transpose', 'Transpose', M-RK, N-KB, KB,
     $               -ONE,
     $               A( RK+1, 1 ), LDA, F( KB+1, 1 ), LDF, ONE,
     $               A( RK+1, KB+1 ), LDA )
      END IF
*
*     Recomputation of difficult columns.
*
   40 CONTINUE
      IF( LSTICC.GT.0 ) THEN
         ITEMP = NINT( VN2( LSTICC ) )
         VN1( LSTICC ) = DNRM2( M-RK, A( RK+1, LSTICC ), 1 )
*
*        NOTE: The computation of VN1( LSTICC ) relies on the fact that
*        SNRM2 does not fail on vectors with norm below the value of
*        SQRT(DLAMCH('S'))
*
         VN2( LSTICC ) = VN1( LSTICC )
         LSTICC = ITEMP
         GO TO 40
      END IF
*
      RETURN
*
*     End of DLAQPS
*
      END
*> \brief \b DLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQR0 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr0.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqr0.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqr0.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
*                          ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
*       LOGICAL            WANTT, WANTZ
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   H( LDH, * ), WI( * ), WORK( * ), WR( * ),
*      $                   Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>    DLAQR0 computes the eigenvalues of a Hessenberg matrix H
*>    and, optionally, the matrices T and Z from the Schur decomposition
*>    H = Z T Z**T, where T is an upper quasi-triangular matrix (the
*>    Schur form), and Z is the orthogonal matrix of Schur vectors.
*>
*>    Optionally Z may be postmultiplied into an input orthogonal
*>    matrix Q so that this routine can give the Schur factorization
*>    of a matrix A which has been reduced to the Hessenberg form H
*>    by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] WANTT
*> \verbatim
*>          WANTT is LOGICAL
*>          = .TRUE. : the full Schur form T is required;
*>          = .FALSE.: only eigenvalues are required.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*>          WANTZ is LOGICAL
*>          = .TRUE. : the matrix of Schur vectors Z is required;
*>          = .FALSE.: Schur vectors are not required.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>           The order of the matrix H.  N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*>          ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*>          IHI is INTEGER
*>           It is assumed that H is already upper triangular in rows
*>           and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,
*>           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
*>           previous call to DGEBAL, and then passed to DGEHRD when the
*>           matrix output by DGEBAL is reduced to Hessenberg form.
*>           Otherwise, ILO and IHI should be set to 1 and N,
*>           respectively.  If N > 0, then 1 <= ILO <= IHI <= N.
*>           If N = 0, then ILO = 1 and IHI = 0.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*>          H is DOUBLE PRECISION array, dimension (LDH,N)
*>           On entry, the upper Hessenberg matrix H.
*>           On exit, if INFO = 0 and WANTT is .TRUE., then H contains
*>           the upper quasi-triangular matrix T from the Schur
*>           decomposition (the Schur form); 2-by-2 diagonal blocks
*>           (corresponding to complex conjugate pairs of eigenvalues)
*>           are returned in standard form, with H(i,i) = H(i+1,i+1)
*>           and H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and WANTT is
*>           .FALSE., then the contents of H are unspecified on exit.
*>           (The output value of H when INFO > 0 is given under the
*>           description of INFO below.)
*>
*>           This subroutine may explicitly set H(i,j) = 0 for i > j and
*>           j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*>          LDH is INTEGER
*>           The leading dimension of the array H. LDH >= max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*>          WR is DOUBLE PRECISION array, dimension (IHI)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*>          WI is DOUBLE PRECISION array, dimension (IHI)
*>           The real and imaginary parts, respectively, of the computed
*>           eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
*>           and WI(ILO:IHI). If two eigenvalues are computed as a
*>           complex conjugate pair, they are stored in consecutive
*>           elements of WR and WI, say the i-th and (i+1)th, with
*>           WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., then
*>           the eigenvalues are stored in the same order as on the
*>           diagonal of the Schur form returned in H, with
*>           WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
*>           block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
*>           WI(i+1) = -WI(i).
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*>          ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*>          IHIZ is INTEGER
*>           Specify the rows of Z to which transformations must be
*>           applied if WANTZ is .TRUE..
*>           1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ,IHI)
*>           If WANTZ is .FALSE., then Z is not referenced.
*>           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
*>           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
*>           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
*>           (The output value of Z when INFO > 0 is given under
*>           the description of INFO below.)
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>           The leading dimension of the array Z.  if WANTZ is .TRUE.
*>           then LDZ >= MAX(1,IHIZ).  Otherwise, LDZ >= 1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension LWORK
*>           On exit, if LWORK = -1, WORK(1) returns an estimate of
*>           the optimal value for LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>           The dimension of the array WORK.  LWORK >= max(1,N)
*>           is sufficient, but LWORK typically as large as 6*N may
*>           be required for optimal performance.  A workspace query
*>           to determine the optimal workspace size is recommended.
*>
*>           If LWORK = -1, then DLAQR0 does a workspace query.
*>           In this case, DLAQR0 checks the input parameters and
*>           estimates the optimal workspace size for the given
*>           values of N, ILO and IHI.  The estimate is returned
*>           in WORK(1).  No error message related to LWORK is
*>           issued by XERBLA.  Neither H nor Z are accessed.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>             = 0:  successful exit
*>             > 0:  if INFO = i, DLAQR0 failed to compute all of
*>                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
*>                and WI contain those eigenvalues which have been
*>                successfully computed.  (Failures are rare.)
*>
*>                If INFO > 0 and WANT is .FALSE., then on exit,
*>                the remaining unconverged eigenvalues are the eigen-
*>                values of the upper Hessenberg matrix rows and
*>                columns ILO through INFO of the final, output
*>                value of H.
*>
*>                If INFO > 0 and WANTT is .TRUE., then on exit
*>
*>           (*)  (initial value of H)*U  = U*(final value of H)
*>
*>                where U is an orthogonal matrix.  The final
*>                value of H is upper Hessenberg and quasi-triangular
*>                in rows and columns INFO+1 through IHI.
*>
*>                If INFO > 0 and WANTZ is .TRUE., then on exit
*>
*>                  (final value of Z(ILO:IHI,ILOZ:IHIZ)
*>                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
*>
*>                where U is the orthogonal matrix in (*) (regard-
*>                less of the value of WANTT.)
*>
*>                If INFO > 0 and WANTZ is .FALSE., then Z is not
*>                accessed.
*> \endverbatim
*
*> \par Contributors:
*  ==================
*>
*>       Karen Braman and Ralph Byers, Department of Mathematics,
*>       University of Kansas, USA
*
*> \par References:
*  ================
*>
*>       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*>       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
*>       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
*>       929--947, 2002.
*> \n
*>       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*>       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
*>       of Matrix Analysis, volume 23, pages 948--973, 2002.
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqr0
*
*  =====================================================================
      SUBROUTINE DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
     $                   ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
      LOGICAL            WANTT, WANTZ
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   H( LDH, * ), WI( * ), WORK( * ), WR( * ),
     $                   Z( LDZ, * )
*     ..
*
*  ================================================================
*
*     .. Parameters ..
*
*     ==== Matrices of order NTINY or smaller must be processed by
*     .    DLAHQR because of insufficient subdiagonal scratch space.
*     .    (This is a hard limit.) ====
      INTEGER            NTINY
      PARAMETER          ( NTINY = 15 )
*
*     ==== Exceptional deflation windows:  try to cure rare
*     .    slow convergence by varying the size of the
*     .    deflation window after KEXNW iterations. ====
      INTEGER            KEXNW
      PARAMETER          ( KEXNW = 5 )
*
*     ==== Exceptional shifts: try to cure rare slow convergence
*     .    with ad-hoc exceptional shifts every KEXSH iterations.
*     .    ====
      INTEGER            KEXSH
      PARAMETER          ( KEXSH = 6 )
*
*     ==== The constants WILK1 and WILK2 are used to form the
*     .    exceptional shifts. ====
      DOUBLE PRECISION   WILK1, WILK2
      PARAMETER          ( WILK1 = 0.75d0, WILK2 = -0.4375d0 )
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0d0, ONE = 1.0d0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   AA, BB, CC, CS, DD, SN, SS, SWAP
      INTEGER            I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
     $                   KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
     $                   LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
     $                   NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
      LOGICAL            SORTED
      CHARACTER          JBCMPZ*2
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   ZDUM( 1, 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACPY, DLAHQR, DLANV2, DLAQR3, DLAQR4,
     $                   DLAQR5
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, INT, MAX, MIN, MOD
*     ..
*     .. Executable Statements ..
      INFO = 0
*
*     ==== Quick return for N = 0: nothing to do. ====
*
      IF( N.EQ.0 ) THEN
         WORK( 1 ) = ONE
         RETURN
      END IF
*
      IF( N.LE.NTINY ) THEN
*
*        ==== Tiny matrices must use DLAHQR. ====
*
         LWKOPT = 1
         IF( LWORK.NE.-1 )
     $      CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
     $                   ILOZ, IHIZ, Z, LDZ, INFO )
      ELSE
*
*        ==== Use small bulge multi-shift QR with aggressive early
*        .    deflation on larger-than-tiny matrices. ====
*
*        ==== Hope for the best. ====
*
         INFO = 0
*
*        ==== Set up job flags for ILAENV. ====
*
         IF( WANTT ) THEN
            JBCMPZ( 1: 1 ) = 'S'
         ELSE
            JBCMPZ( 1: 1 ) = 'E'
         END IF
         IF( WANTZ ) THEN
            JBCMPZ( 2: 2 ) = 'V'
         ELSE
            JBCMPZ( 2: 2 ) = 'N'
         END IF
*
*        ==== NWR = recommended deflation window size.  At this
*        .    point,  N .GT. NTINY = 15, so there is enough
*        .    subdiagonal workspace for NWR.GE.2 as required.
*        .    (In fact, there is enough subdiagonal space for
*        .    NWR.GE.4.) ====
*
         NWR = ILAENV( 13, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
         NWR = MAX( 2, NWR )
         NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
*
*        ==== NSR = recommended number of simultaneous shifts.
*        .    At this point N .GT. NTINY = 15, so there is at
*        .    enough subdiagonal workspace for NSR to be even
*        .    and greater than or equal to two as required. ====
*
         NSR = ILAENV( 15, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
         NSR = MIN( NSR, ( N-3 ) / 6, IHI-ILO )
         NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
*
*        ==== Estimate optimal workspace ====
*
*        ==== Workspace query call to DLAQR3 ====
*
         CALL DLAQR3( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
     $                IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH,
     $                N, H, LDH, WORK, -1 )
*
*        ==== Optimal workspace = MAX(DLAQR5, DLAQR3) ====
*
         LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
*
*        ==== Quick return in case of workspace query. ====
*
         IF( LWORK.EQ.-1 ) THEN
            WORK( 1 ) = DBLE( LWKOPT )
            RETURN
         END IF
*
*        ==== DLAHQR/DLAQR0 crossover point ====
*
         NMIN = ILAENV( 12, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
         NMIN = MAX( NTINY, NMIN )
*
*        ==== Nibble crossover point ====
*
         NIBBLE = ILAENV( 14, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
         NIBBLE = MAX( 0, NIBBLE )
*
*        ==== Accumulate reflections during ttswp?  Use block
*        .    2-by-2 structure during matrix-matrix multiply? ====
*
         KACC22 = ILAENV( 16, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
         KACC22 = MAX( 0, KACC22 )
         KACC22 = MIN( 2, KACC22 )
*
*        ==== NWMAX = the largest possible deflation window for
*        .    which there is sufficient workspace. ====
*
         NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
         NW = NWMAX
*
*        ==== NSMAX = the Largest number of simultaneous shifts
*        .    for which there is sufficient workspace. ====
*
         NSMAX = MIN( ( N-3 ) / 6, 2*LWORK / 3 )
         NSMAX = NSMAX - MOD( NSMAX, 2 )
*
*        ==== NDFL: an iteration count restarted at deflation. ====
*
         NDFL = 1
*
*        ==== ITMAX = iteration limit ====
*
         ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
*
*        ==== Last row and column in the active block ====
*
         KBOT = IHI
*
*        ==== Main Loop ====
*
         DO 80 IT = 1, ITMAX
*
*           ==== Done when KBOT falls below ILO ====
*
            IF( KBOT.LT.ILO )
     $         GO TO 90
*
*           ==== Locate active block ====
*
            DO 10 K = KBOT, ILO + 1, -1
               IF( H( K, K-1 ).EQ.ZERO )
     $            GO TO 20
   10       CONTINUE
            K = ILO
   20       CONTINUE
            KTOP = K
*
*           ==== Select deflation window size:
*           .    Typical Case:
*           .      If possible and advisable, nibble the entire
*           .      active block.  If not, use size MIN(NWR,NWMAX)
*           .      or MIN(NWR+1,NWMAX) depending upon which has
*           .      the smaller corresponding subdiagonal entry
*           .      (a heuristic).
*           .
*           .    Exceptional Case:
*           .      If there have been no deflations in KEXNW or
*           .      more iterations, then vary the deflation window
*           .      size.   At first, because, larger windows are,
*           .      in general, more powerful than smaller ones,
*           .      rapidly increase the window to the maximum possible.
*           .      Then, gradually reduce the window size. ====
*
            NH = KBOT - KTOP + 1
            NWUPBD = MIN( NH, NWMAX )
            IF( NDFL.LT.KEXNW ) THEN
               NW = MIN( NWUPBD, NWR )
            ELSE
               NW = MIN( NWUPBD, 2*NW )
            END IF
            IF( NW.LT.NWMAX ) THEN
               IF( NW.GE.NH-1 ) THEN
                  NW = NH
               ELSE
                  KWTOP = KBOT - NW + 1
                  IF( ABS( H( KWTOP, KWTOP-1 ) ).GT.
     $                ABS( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
               END IF
            END IF
            IF( NDFL.LT.KEXNW ) THEN
               NDEC = -1
            ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
               NDEC = NDEC + 1
               IF( NW-NDEC.LT.2 )
     $            NDEC = 0
               NW = NW - NDEC
            END IF
*
*           ==== Aggressive early deflation:
*           .    split workspace under the subdiagonal into
*           .      - an nw-by-nw work array V in the lower
*           .        left-hand-corner,
*           .      - an NW-by-at-least-NW-but-more-is-better
*           .        (NW-by-NHO) horizontal work array along
*           .        the bottom edge,
*           .      - an at-least-NW-but-more-is-better (NHV-by-NW)
*           .        vertical work array along the left-hand-edge.
*           .        ====
*
            KV = N - NW + 1
            KT = NW + 1
            NHO = ( N-NW-1 ) - KT + 1
            KWV = NW + 2
            NVE = ( N-NW ) - KWV + 1
*
*           ==== Aggressive early deflation ====
*
            CALL DLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH,
     $                   ILOZ,
     $                   IHIZ, Z, LDZ, LS, LD, WR, WI, H( KV, 1 ), LDH,
     $                   NHO, H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH,
     $                   WORK, LWORK )
*
*           ==== Adjust KBOT accounting for new deflations. ====
*
            KBOT = KBOT - LD
*
*           ==== KS points to the shifts. ====
*
            KS = KBOT - LS + 1
*
*           ==== Skip an expensive QR sweep if there is a (partly
*           .    heuristic) reason to expect that many eigenvalues
*           .    will deflate without it.  Here, the QR sweep is
*           .    skipped if many eigenvalues have just been deflated
*           .    or if the remaining active block is small.
*
            IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
     $          KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
*
*              ==== NS = nominal number of simultaneous shifts.
*              .    This may be lowered (slightly) if DLAQR3
*              .    did not provide that many shifts. ====
*
               NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
               NS = NS - MOD( NS, 2 )
*
*              ==== If there have been no deflations
*              .    in a multiple of KEXSH iterations,
*              .    then try exceptional shifts.
*              .    Otherwise use shifts provided by
*              .    DLAQR3 above or from the eigenvalues
*              .    of a trailing principal submatrix. ====
*
               IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
                  KS = KBOT - NS + 1
                  DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2
                     SS = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
                     AA = WILK1*SS + H( I, I )
                     BB = SS
                     CC = WILK2*SS
                     DD = AA
                     CALL DLANV2( AA, BB, CC, DD, WR( I-1 ),
     $                            WI( I-1 ),
     $                            WR( I ), WI( I ), CS, SN )
   30             CONTINUE
                  IF( KS.EQ.KTOP ) THEN
                     WR( KS+1 ) = H( KS+1, KS+1 )
                     WI( KS+1 ) = ZERO
                     WR( KS ) = WR( KS+1 )
                     WI( KS ) = WI( KS+1 )
                  END IF
               ELSE
*
*                 ==== Got NS/2 or fewer shifts? Use DLAQR4 or
*                 .    DLAHQR on a trailing principal submatrix to
*                 .    get more. (Since NS.LE.NSMAX.LE.(N-3)/6,
*                 .    there is enough space below the subdiagonal
*                 .    to fit an NS-by-NS scratch array.) ====
*
                  IF( KBOT-KS+1.LE.NS / 2 ) THEN
                     KS = KBOT - NS + 1
                     KT = N - NS + 1
                     CALL DLACPY( 'A', NS, NS, H( KS, KS ), LDH,
     $                            H( KT, 1 ), LDH )
                     IF( NS.GT.NMIN ) THEN
                        CALL DLAQR4( .false., .false., NS, 1, NS,
     $                               H( KT, 1 ), LDH, WR( KS ),
     $                               WI( KS ), 1, 1, ZDUM, 1, WORK,
     $                               LWORK, INF )
                     ELSE
                        CALL DLAHQR( .false., .false., NS, 1, NS,
     $                               H( KT, 1 ), LDH, WR( KS ),
     $                               WI( KS ), 1, 1, ZDUM, 1, INF )
                     END IF
                     KS = KS + INF
*
*                    ==== In case of a rare QR failure use
*                    .    eigenvalues of the trailing 2-by-2
*                    .    principal submatrix.  ====
*
                     IF( KS.GE.KBOT ) THEN
                        AA = H( KBOT-1, KBOT-1 )
                        CC = H( KBOT, KBOT-1 )
                        BB = H( KBOT-1, KBOT )
                        DD = H( KBOT, KBOT )
                        CALL DLANV2( AA, BB, CC, DD, WR( KBOT-1 ),
     $                               WI( KBOT-1 ), WR( KBOT ),
     $                               WI( KBOT ), CS, SN )
                        KS = KBOT - 1
                     END IF
                  END IF
*
                  IF( KBOT-KS+1.GT.NS ) THEN
*
*                    ==== Sort the shifts (Helps a little)
*                    .    Bubble sort keeps complex conjugate
*                    .    pairs together. ====
*
                     SORTED = .false.
                     DO 50 K = KBOT, KS + 1, -1
                        IF( SORTED )
     $                     GO TO 60
                        SORTED = .true.
                        DO 40 I = KS, K - 1
                           IF( ABS( WR( I ) )+ABS( WI( I ) ).LT.
     $                         ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN
                              SORTED = .false.
*
                              SWAP = WR( I )
                              WR( I ) = WR( I+1 )
                              WR( I+1 ) = SWAP
*
                              SWAP = WI( I )
                              WI( I ) = WI( I+1 )
                              WI( I+1 ) = SWAP
                           END IF
   40                   CONTINUE
   50                CONTINUE
   60                CONTINUE
                  END IF
*
*                 ==== Shuffle shifts into pairs of real shifts
*                 .    and pairs of complex conjugate shifts
*                 .    assuming complex conjugate shifts are
*                 .    already adjacent to one another. (Yes,
*                 .    they are.)  ====
*
                  DO 70 I = KBOT, KS + 2, -2
                     IF( WI( I ).NE.-WI( I-1 ) ) THEN
*
                        SWAP = WR( I )
                        WR( I ) = WR( I-1 )
                        WR( I-1 ) = WR( I-2 )
                        WR( I-2 ) = SWAP
*
                        SWAP = WI( I )
                        WI( I ) = WI( I-1 )
                        WI( I-1 ) = WI( I-2 )
                        WI( I-2 ) = SWAP
                     END IF
   70             CONTINUE
               END IF
*
*              ==== If there are only two shifts and both are
*              .    real, then use only one.  ====
*
               IF( KBOT-KS+1.EQ.2 ) THEN
                  IF( WI( KBOT ).EQ.ZERO ) THEN
                     IF( ABS( WR( KBOT )-H( KBOT, KBOT ) ).LT.
     $                   ABS( WR( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
                        WR( KBOT-1 ) = WR( KBOT )
                     ELSE
                        WR( KBOT ) = WR( KBOT-1 )
                     END IF
                  END IF
               END IF
*
*              ==== Use up to NS of the the smallest magnitude
*              .    shifts.  If there aren't NS shifts available,
*              .    then use them all, possibly dropping one to
*              .    make the number of shifts even. ====
*
               NS = MIN( NS, KBOT-KS+1 )
               NS = NS - MOD( NS, 2 )
               KS = KBOT - NS + 1
*
*              ==== Small-bulge multi-shift QR sweep:
*              .    split workspace under the subdiagonal into
*              .    - a KDU-by-KDU work array U in the lower
*              .      left-hand-corner,
*              .    - a KDU-by-at-least-KDU-but-more-is-better
*              .      (KDU-by-NHo) horizontal work array WH along
*              .      the bottom edge,
*              .    - and an at-least-KDU-but-more-is-better-by-KDU
*              .      (NVE-by-KDU) vertical work WV arrow along
*              .      the left-hand-edge. ====
*
               KDU = 2*NS
               KU = N - KDU + 1
               KWH = KDU + 1
               NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
               KWV = KDU + 4
               NVE = N - KDU - KWV + 1
*
*              ==== Small-bulge multi-shift QR sweep ====
*
               CALL DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
     $                      WR( KS ), WI( KS ), H, LDH, ILOZ, IHIZ, Z,
     $                      LDZ, WORK, 3, H( KU, 1 ), LDH, NVE,
     $                      H( KWV, 1 ), LDH, NHO, H( KU, KWH ), LDH )
            END IF
*
*           ==== Note progress (or the lack of it). ====
*
            IF( LD.GT.0 ) THEN
               NDFL = 1
            ELSE
               NDFL = NDFL + 1
            END IF
*
*           ==== End of main loop ====
   80    CONTINUE
*
*        ==== Iteration limit exceeded.  Set INFO to show where
*        .    the problem occurred and exit. ====
*
         INFO = KBOT
   90    CONTINUE
      END IF
*
*     ==== Return the optimal value of LWORK. ====
*
      WORK( 1 ) = DBLE( LWKOPT )
*
*     ==== End of DLAQR0 ====
*
      END
*> \brief \b DLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQR1 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr1.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqr1.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqr1.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAQR1( N, H, LDH, SR1, SI1, SR2, SI2, V )
*
*       .. Scalar Arguments ..
*       DOUBLE PRECISION   SI1, SI2, SR1, SR2
*       INTEGER            LDH, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   H( LDH, * ), V( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>      Given a 2-by-2 or 3-by-3 matrix H, DLAQR1 sets v to a
*>      scalar multiple of the first column of the product
*>
*>      (*)  K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I)
*>
*>      scaling to avoid overflows and most underflows. It
*>      is assumed that either
*>
*>              1) sr1 = sr2 and si1 = -si2
*>          or
*>              2) si1 = si2 = 0.
*>
*>      This is useful for starting double implicit shift bulges
*>      in the QR algorithm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>              Order of the matrix H. N must be either 2 or 3.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*>          H is DOUBLE PRECISION array, dimension (LDH,N)
*>              The 2-by-2 or 3-by-3 matrix H in (*).
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*>          LDH is INTEGER
*>              The leading dimension of H as declared in
*>              the calling procedure.  LDH >= N
*> \endverbatim
*>
*> \param[in] SR1
*> \verbatim
*>          SR1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] SI1
*> \verbatim
*>          SI1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] SR2
*> \verbatim
*>          SR2 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] SI2
*> \verbatim
*>          SI2 is DOUBLE PRECISION
*>              The shifts in (*).
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension (N)
*>              A scalar multiple of the first column of the
*>              matrix K in (*).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqr1
*
*> \par Contributors:
*  ==================
*>
*>       Karen Braman and Ralph Byers, Department of Mathematics,
*>       University of Kansas, USA
*>
*  =====================================================================
      SUBROUTINE DLAQR1( N, H, LDH, SR1, SI1, SR2, SI2, V )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   SI1, SI2, SR1, SR2
      INTEGER            LDH, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   H( LDH, * ), V( * )
*     ..
*
*  ================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0d0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   H21S, H31S, S
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( N.NE.2 .AND. N.NE.3 ) THEN
         RETURN
      END IF
*
      IF( N.EQ.2 ) THEN
         S = ABS( H( 1, 1 )-SR2 ) + ABS( SI2 ) + ABS( H( 2, 1 ) )
         IF( S.EQ.ZERO ) THEN
            V( 1 ) = ZERO
            V( 2 ) = ZERO
         ELSE
            H21S = H( 2, 1 ) / S
            V( 1 ) = H21S*H( 1, 2 ) + ( H( 1, 1 )-SR1 )*
     $               ( ( H( 1, 1 )-SR2 ) / S ) - SI1*( SI2 / S )
            V( 2 ) = H21S*( H( 1, 1 )+H( 2, 2 )-SR1-SR2 )
         END IF
      ELSE
         S = ABS( H( 1, 1 )-SR2 ) + ABS( SI2 ) + ABS( H( 2, 1 ) ) +
     $       ABS( H( 3, 1 ) )
         IF( S.EQ.ZERO ) THEN
            V( 1 ) = ZERO
            V( 2 ) = ZERO
            V( 3 ) = ZERO
         ELSE
            H21S = H( 2, 1 ) / S
            H31S = H( 3, 1 ) / S
            V( 1 ) = ( H( 1, 1 )-SR1 )*( ( H( 1, 1 )-SR2 ) / S ) -
     $               SI1*( SI2 / S ) + H( 1, 2 )*H21S + H( 1, 3 )*H31S
            V( 2 ) = H21S*( H( 1, 1 )+H( 2, 2 )-SR1-SR2 ) +
     $               H( 2, 3 )*H31S
            V( 3 ) = H31S*( H( 1, 1 )+H( 3, 3 )-SR1-SR2 ) +
     $               H21S*H( 3, 2 )
         END IF
      END IF
      END
*> \brief \b DLAQR2 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQR2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqr2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqr2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
*                          IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T,
*                          LDT, NV, WV, LDWV, WORK, LWORK )
*
*       .. Scalar Arguments ..
*       INTEGER            IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
*      $                   LDZ, LWORK, N, ND, NH, NS, NV, NW
*       LOGICAL            WANTT, WANTZ
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   H( LDH, * ), SI( * ), SR( * ), T( LDT, * ),
*      $                   V( LDV, * ), WORK( * ), WV( LDWV, * ),
*      $                   Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>    DLAQR2 is identical to DLAQR3 except that it avoids
*>    recursion by calling DLAHQR instead of DLAQR4.
*>
*>    Aggressive early deflation:
*>
*>    This subroutine accepts as input an upper Hessenberg matrix
*>    H and performs an orthogonal similarity transformation
*>    designed to detect and deflate fully converged eigenvalues from
*>    a trailing principal submatrix.  On output H has been over-
*>    written by a new Hessenberg matrix that is a perturbation of
*>    an orthogonal similarity transformation of H.  It is to be
*>    hoped that the final version of H has many zero subdiagonal
*>    entries.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] WANTT
*> \verbatim
*>          WANTT is LOGICAL
*>          If .TRUE., then the Hessenberg matrix H is fully updated
*>          so that the quasi-triangular Schur factor may be
*>          computed (in cooperation with the calling subroutine).
*>          If .FALSE., then only enough of H is updated to preserve
*>          the eigenvalues.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*>          WANTZ is LOGICAL
*>          If .TRUE., then the orthogonal matrix Z is updated so
*>          so that the orthogonal Schur factor may be computed
*>          (in cooperation with the calling subroutine).
*>          If .FALSE., then Z is not referenced.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix H and (if WANTZ is .TRUE.) the
*>          order of the orthogonal matrix Z.
*> \endverbatim
*>
*> \param[in] KTOP
*> \verbatim
*>          KTOP is INTEGER
*>          It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
*>          KBOT and KTOP together determine an isolated block
*>          along the diagonal of the Hessenberg matrix.
*> \endverbatim
*>
*> \param[in] KBOT
*> \verbatim
*>          KBOT is INTEGER
*>          It is assumed without a check that either
*>          KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together
*>          determine an isolated block along the diagonal of the
*>          Hessenberg matrix.
*> \endverbatim
*>
*> \param[in] NW
*> \verbatim
*>          NW is INTEGER
*>          Deflation window size.  1 <= NW <= (KBOT-KTOP+1).
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*>          H is DOUBLE PRECISION array, dimension (LDH,N)
*>          On input the initial N-by-N section of H stores the
*>          Hessenberg matrix undergoing aggressive early deflation.
*>          On output H has been transformed by an orthogonal
*>          similarity transformation, perturbed, and the returned
*>          to Hessenberg form that (it is to be hoped) has some
*>          zero subdiagonal entries.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*>          LDH is INTEGER
*>          Leading dimension of H just as declared in the calling
*>          subroutine.  N <= LDH
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*>          ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*>          IHIZ is INTEGER
*>          Specify the rows of Z to which transformations must be
*>          applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ,N)
*>          IF WANTZ is .TRUE., then on output, the orthogonal
*>          similarity transformation mentioned above has been
*>          accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
*>          If WANTZ is .FALSE., then Z is unreferenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of Z just as declared in the
*>          calling subroutine.  1 <= LDZ.
*> \endverbatim
*>
*> \param[out] NS
*> \verbatim
*>          NS is INTEGER
*>          The number of unconverged (ie approximate) eigenvalues
*>          returned in SR and SI that may be used as shifts by the
*>          calling subroutine.
*> \endverbatim
*>
*> \param[out] ND
*> \verbatim
*>          ND is INTEGER
*>          The number of converged eigenvalues uncovered by this
*>          subroutine.
*> \endverbatim
*>
*> \param[out] SR
*> \verbatim
*>          SR is DOUBLE PRECISION array, dimension (KBOT)
*> \endverbatim
*>
*> \param[out] SI
*> \verbatim
*>          SI is DOUBLE PRECISION array, dimension (KBOT)
*>          On output, the real and imaginary parts of approximate
*>          eigenvalues that may be used for shifts are stored in
*>          SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
*>          SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
*>          The real and imaginary parts of converged eigenvalues
*>          are stored in SR(KBOT-ND+1) through SR(KBOT) and
*>          SI(KBOT-ND+1) through SI(KBOT), respectively.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension (LDV,NW)
*>          An NW-by-NW work array.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of V just as declared in the
*>          calling subroutine.  NW <= LDV
*> \endverbatim
*>
*> \param[in] NH
*> \verbatim
*>          NH is INTEGER
*>          The number of columns of T.  NH >= NW.
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,NW)
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of T just as declared in the
*>          calling subroutine.  NW <= LDT
*> \endverbatim
*>
*> \param[in] NV
*> \verbatim
*>          NV is INTEGER
*>          The number of rows of work array WV available for
*>          workspace.  NV >= NW.
*> \endverbatim
*>
*> \param[out] WV
*> \verbatim
*>          WV is DOUBLE PRECISION array, dimension (LDWV,NW)
*> \endverbatim
*>
*> \param[in] LDWV
*> \verbatim
*>          LDWV is INTEGER
*>          The leading dimension of W just as declared in the
*>          calling subroutine.  NW <= LDV
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (LWORK)
*>          On exit, WORK(1) is set to an estimate of the optimal value
*>          of LWORK for the given values of N, NW, KTOP and KBOT.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the work array WORK.  LWORK = 2*NW
*>          suffices, but greater efficiency may result from larger
*>          values of LWORK.
*>
*>          If LWORK = -1, then a workspace query is assumed; DLAQR2
*>          only estimates the optimal workspace size for the given
*>          values of N, NW, KTOP and KBOT.  The estimate is returned
*>          in WORK(1).  No error message related to LWORK is issued
*>          by XERBLA.  Neither H nor Z are accessed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqr2
*
*> \par Contributors:
*  ==================
*>
*>       Karen Braman and Ralph Byers, Department of Mathematics,
*>       University of Kansas, USA
*>
*  =====================================================================
      SUBROUTINE DLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH,
     $                   ILOZ,
     $                   IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T,
     $                   LDT, NV, WV, LDWV, WORK, LWORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
     $                   LDZ, LWORK, N, ND, NH, NS, NV, NW
      LOGICAL            WANTT, WANTZ
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   H( LDH, * ), SI( * ), SR( * ), T( LDT, * ),
     $                   V( LDV, * ), WORK( * ), WV( LDWV, * ),
     $                   Z( LDZ, * )
*     ..
*
*  ================================================================
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0d0, ONE = 1.0d0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   AA, BB, BETA, CC, CS, DD, EVI, EVK, FOO, S,
     $                   SAFMAX, SAFMIN, SMLNUM, SN, TAU, ULP
      INTEGER            I, IFST, ILST, INFO, INFQR, J, JW, K, KCOL,
     $                   KEND, KLN, KROW, KWTOP, LTOP, LWK1, LWK2,
     $                   LWKOPT
      LOGICAL            BULGE, SORTED
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGEHRD, DGEMM, DLACPY,
     $                   DLAHQR,
     $                   DLANV2, DLARF1F, DLARFG, DLASET, DORMHR, DTREXC
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, INT, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     ==== Estimate optimal workspace. ====
*
      JW = MIN( NW, KBOT-KTOP+1 )
      IF( JW.LE.2 ) THEN
         LWKOPT = 1
      ELSE
*
*        ==== Workspace query call to DGEHRD ====
*
         CALL DGEHRD( JW, 1, JW-1, T, LDT, WORK, WORK, -1, INFO )
         LWK1 = INT( WORK( 1 ) )
*
*        ==== Workspace query call to DORMHR ====
*
         CALL DORMHR( 'R', 'N', JW, JW, 1, JW-1, T, LDT, WORK, V,
     $                LDV,
     $                WORK, -1, INFO )
         LWK2 = INT( WORK( 1 ) )
*
*        ==== Optimal workspace ====
*
         LWKOPT = JW + MAX( LWK1, LWK2 )
      END IF
*
*     ==== Quick return in case of workspace query. ====
*
      IF( LWORK.EQ.-1 ) THEN
         WORK( 1 ) = DBLE( LWKOPT )
         RETURN
      END IF
*
*     ==== Nothing to do ...
*     ... for an empty active block ... ====
      NS = 0
      ND = 0
      WORK( 1 ) = ONE
      IF( KTOP.GT.KBOT )
     $   RETURN
*     ... nor for an empty deflation window. ====
      IF( NW.LT.1 )
     $   RETURN
*
*     ==== Machine constants ====
*
      SAFMIN = DLAMCH( 'SAFE MINIMUM' )
      SAFMAX = ONE / SAFMIN
      ULP = DLAMCH( 'PRECISION' )
      SMLNUM = SAFMIN*( DBLE( N ) / ULP )
*
*     ==== Setup deflation window ====
*
      JW = MIN( NW, KBOT-KTOP+1 )
      KWTOP = KBOT - JW + 1
      IF( KWTOP.EQ.KTOP ) THEN
         S = ZERO
      ELSE
         S = H( KWTOP, KWTOP-1 )
      END IF
*
      IF( KBOT.EQ.KWTOP ) THEN
*
*        ==== 1-by-1 deflation window: not much to do ====
*
         SR( KWTOP ) = H( KWTOP, KWTOP )
         SI( KWTOP ) = ZERO
         NS = 1
         ND = 0
         IF( ABS( S ).LE.MAX( SMLNUM, ULP*ABS( H( KWTOP, KWTOP ) ) ) )
     $        THEN
            NS = 0
            ND = 1
            IF( KWTOP.GT.KTOP )
     $         H( KWTOP, KWTOP-1 ) = ZERO
         END IF
         WORK( 1 ) = ONE
         RETURN
      END IF
*
*     ==== Convert to spike-triangular form.  (In case of a
*     .    rare QR failure, this routine continues to do
*     .    aggressive early deflation using that part of
*     .    the deflation window that converged using INFQR
*     .    here and there to keep track.) ====
*
      CALL DLACPY( 'U', JW, JW, H( KWTOP, KWTOP ), LDH, T, LDT )
      CALL DCOPY( JW-1, H( KWTOP+1, KWTOP ), LDH+1, T( 2, 1 ),
     $            LDT+1 )
*
      CALL DLASET( 'A', JW, JW, ZERO, ONE, V, LDV )
      CALL DLAHQR( .true., .true., JW, 1, JW, T, LDT, SR( KWTOP ),
     $             SI( KWTOP ), 1, JW, V, LDV, INFQR )
*
*     ==== DTREXC needs a clean margin near the diagonal ====
*
      DO 10 J = 1, JW - 3
         T( J+2, J ) = ZERO
         T( J+3, J ) = ZERO
   10 CONTINUE
      IF( JW.GT.2 )
     $   T( JW, JW-2 ) = ZERO
*
*     ==== Deflation detection loop ====
*
      NS = JW
      ILST = INFQR + 1
   20 CONTINUE
      IF( ILST.LE.NS ) THEN
         IF( NS.EQ.1 ) THEN
            BULGE = .FALSE.
         ELSE
            BULGE = T( NS, NS-1 ).NE.ZERO
         END IF
*
*        ==== Small spike tip test for deflation ====
*
         IF( .NOT.BULGE ) THEN
*
*           ==== Real eigenvalue ====
*
            FOO = ABS( T( NS, NS ) )
            IF( FOO.EQ.ZERO )
     $         FOO = ABS( S )
            IF( ABS( S*V( 1, NS ) ).LE.MAX( SMLNUM, ULP*FOO ) ) THEN
*
*              ==== Deflatable ====
*
               NS = NS - 1
            ELSE
*
*              ==== Undeflatable.   Move it up out of the way.
*              .    (DTREXC can not fail in this case.) ====
*
               IFST = NS
               CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST,
     $                      WORK,
     $                      INFO )
               ILST = ILST + 1
            END IF
         ELSE
*
*           ==== Complex conjugate pair ====
*
            FOO = ABS( T( NS, NS ) ) + SQRT( ABS( T( NS, NS-1 ) ) )*
     $            SQRT( ABS( T( NS-1, NS ) ) )
            IF( FOO.EQ.ZERO )
     $         FOO = ABS( S )
            IF( MAX( ABS( S*V( 1, NS ) ), ABS( S*V( 1, NS-1 ) ) ).LE.
     $          MAX( SMLNUM, ULP*FOO ) ) THEN
*
*              ==== Deflatable ====
*
               NS = NS - 2
            ELSE
*
*              ==== Undeflatable. Move them up out of the way.
*              .    Fortunately, DTREXC does the right thing with
*              .    ILST in case of a rare exchange failure. ====
*
               IFST = NS
               CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST,
     $                      WORK,
     $                      INFO )
               ILST = ILST + 2
            END IF
         END IF
*
*        ==== End deflation detection loop ====
*
         GO TO 20
      END IF
*
*        ==== Return to Hessenberg form ====
*
      IF( NS.EQ.0 )
     $   S = ZERO
*
      IF( NS.LT.JW ) THEN
*
*        ==== sorting diagonal blocks of T improves accuracy for
*        .    graded matrices.  Bubble sort deals well with
*        .    exchange failures. ====
*
         SORTED = .false.
         I = NS + 1
   30    CONTINUE
         IF( SORTED )
     $      GO TO 50
         SORTED = .true.
*
         KEND = I - 1
         I = INFQR + 1
         IF( I.EQ.NS ) THEN
            K = I + 1
         ELSE IF( T( I+1, I ).EQ.ZERO ) THEN
            K = I + 1
         ELSE
            K = I + 2
         END IF
   40    CONTINUE
         IF( K.LE.KEND ) THEN
            IF( K.EQ.I+1 ) THEN
               EVI = ABS( T( I, I ) )
            ELSE
               EVI = ABS( T( I, I ) ) + SQRT( ABS( T( I+1, I ) ) )*
     $               SQRT( ABS( T( I, I+1 ) ) )
            END IF
*
            IF( K.EQ.KEND ) THEN
               EVK = ABS( T( K, K ) )
            ELSE IF( T( K+1, K ).EQ.ZERO ) THEN
               EVK = ABS( T( K, K ) )
            ELSE
               EVK = ABS( T( K, K ) ) + SQRT( ABS( T( K+1, K ) ) )*
     $               SQRT( ABS( T( K, K+1 ) ) )
            END IF
*
            IF( EVI.GE.EVK ) THEN
               I = K
            ELSE
               SORTED = .false.
               IFST = I
               ILST = K
               CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST,
     $                      WORK,
     $                      INFO )
               IF( INFO.EQ.0 ) THEN
                  I = ILST
               ELSE
                  I = K
               END IF
            END IF
            IF( I.EQ.KEND ) THEN
               K = I + 1
            ELSE IF( T( I+1, I ).EQ.ZERO ) THEN
               K = I + 1
            ELSE
               K = I + 2
            END IF
            GO TO 40
         END IF
         GO TO 30
   50    CONTINUE
      END IF
*
*     ==== Restore shift/eigenvalue array from T ====
*
      I = JW
   60 CONTINUE
      IF( I.GE.INFQR+1 ) THEN
         IF( I.EQ.INFQR+1 ) THEN
            SR( KWTOP+I-1 ) = T( I, I )
            SI( KWTOP+I-1 ) = ZERO
            I = I - 1
         ELSE IF( T( I, I-1 ).EQ.ZERO ) THEN
            SR( KWTOP+I-1 ) = T( I, I )
            SI( KWTOP+I-1 ) = ZERO
            I = I - 1
         ELSE
            AA = T( I-1, I-1 )
            CC = T( I, I-1 )
            BB = T( I-1, I )
            DD = T( I, I )
            CALL DLANV2( AA, BB, CC, DD, SR( KWTOP+I-2 ),
     $                   SI( KWTOP+I-2 ), SR( KWTOP+I-1 ),
     $                   SI( KWTOP+I-1 ), CS, SN )
            I = I - 2
         END IF
         GO TO 60
      END IF
*
      IF( NS.LT.JW .OR. S.EQ.ZERO ) THEN
         IF( NS.GT.1 .AND. S.NE.ZERO ) THEN
*
*           ==== Reflect spike back into lower triangle ====
*
            CALL DCOPY( NS, V, LDV, WORK, 1 )
            BETA = WORK( 1 )
            CALL DLARFG( NS, BETA, WORK( 2 ), 1, TAU )
*
            CALL DLASET( 'L', JW-2, JW-2, ZERO, ZERO, T( 3, 1 ),
     $                   LDT )
*
            CALL DLARF1F( 'L', NS, JW, WORK, 1, TAU, T, LDT,
     $                  WORK( JW+1 ) )
            CALL DLARF1F( 'R', NS, NS, WORK, 1, TAU, T, LDT,
     $                  WORK( JW+1 ) )
            CALL DLARF1F( 'R', JW, NS, WORK, 1, TAU, V, LDV,
     $                  WORK( JW+1 ) )
*
            CALL DGEHRD( JW, 1, NS, T, LDT, WORK, WORK( JW+1 ),
     $                   LWORK-JW, INFO )
         END IF
*
*        ==== Copy updated reduced window into place ====
*
         IF( KWTOP.GT.1 )
     $      H( KWTOP, KWTOP-1 ) = S*V( 1, 1 )
         CALL DLACPY( 'U', JW, JW, T, LDT, H( KWTOP, KWTOP ), LDH )
         CALL DCOPY( JW-1, T( 2, 1 ), LDT+1, H( KWTOP+1, KWTOP ),
     $               LDH+1 )
*
*        ==== Accumulate orthogonal matrix in order update
*        .    H and Z, if requested.  ====
*
         IF( NS.GT.1 .AND. S.NE.ZERO )
     $      CALL DORMHR( 'R', 'N', JW, NS, 1, NS, T, LDT, WORK, V,
     $                   LDV,
     $                   WORK( JW+1 ), LWORK-JW, INFO )
*
*        ==== Update vertical slab in H ====
*
         IF( WANTT ) THEN
            LTOP = 1
         ELSE
            LTOP = KTOP
         END IF
         DO 70 KROW = LTOP, KWTOP - 1, NV
            KLN = MIN( NV, KWTOP-KROW )
            CALL DGEMM( 'N', 'N', KLN, JW, JW, ONE, H( KROW, KWTOP ),
     $                  LDH, V, LDV, ZERO, WV, LDWV )
            CALL DLACPY( 'A', KLN, JW, WV, LDWV, H( KROW, KWTOP ),
     $                   LDH )
   70    CONTINUE
*
*        ==== Update horizontal slab in H ====
*
         IF( WANTT ) THEN
            DO 80 KCOL = KBOT + 1, N, NH
               KLN = MIN( NH, N-KCOL+1 )
               CALL DGEMM( 'C', 'N', JW, KLN, JW, ONE, V, LDV,
     $                     H( KWTOP, KCOL ), LDH, ZERO, T, LDT )
               CALL DLACPY( 'A', JW, KLN, T, LDT, H( KWTOP, KCOL ),
     $                      LDH )
   80       CONTINUE
         END IF
*
*        ==== Update vertical slab in Z ====
*
         IF( WANTZ ) THEN
            DO 90 KROW = ILOZ, IHIZ, NV
               KLN = MIN( NV, IHIZ-KROW+1 )
               CALL DGEMM( 'N', 'N', KLN, JW, JW, ONE, Z( KROW,
     $                     KWTOP ),
     $                     LDZ, V, LDV, ZERO, WV, LDWV )
               CALL DLACPY( 'A', KLN, JW, WV, LDWV, Z( KROW, KWTOP ),
     $                      LDZ )
   90       CONTINUE
         END IF
      END IF
*
*     ==== Return the number of deflations ... ====
*
      ND = JW - NS
*
*     ==== ... and the number of shifts. (Subtracting
*     .    INFQR from the spike length takes care
*     .    of the case of a rare QR failure while
*     .    calculating eigenvalues of the deflation
*     .    window.)  ====
*
      NS = NS - INFQR
*
*      ==== Return optimal workspace. ====
*
      WORK( 1 ) = DBLE( LWKOPT )
*
*     ==== End of DLAQR2 ====
*
      END
*> \brief \b DLAQR3 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQR3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqr3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqr3.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
*                          IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T,
*                          LDT, NV, WV, LDWV, WORK, LWORK )
*
*       .. Scalar Arguments ..
*       INTEGER            IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
*      $                   LDZ, LWORK, N, ND, NH, NS, NV, NW
*       LOGICAL            WANTT, WANTZ
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   H( LDH, * ), SI( * ), SR( * ), T( LDT, * ),
*      $                   V( LDV, * ), WORK( * ), WV( LDWV, * ),
*      $                   Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>    Aggressive early deflation:
*>
*>    DLAQR3 accepts as input an upper Hessenberg matrix
*>    H and performs an orthogonal similarity transformation
*>    designed to detect and deflate fully converged eigenvalues from
*>    a trailing principal submatrix.  On output H has been over-
*>    written by a new Hessenberg matrix that is a perturbation of
*>    an orthogonal similarity transformation of H.  It is to be
*>    hoped that the final version of H has many zero subdiagonal
*>    entries.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] WANTT
*> \verbatim
*>          WANTT is LOGICAL
*>          If .TRUE., then the Hessenberg matrix H is fully updated
*>          so that the quasi-triangular Schur factor may be
*>          computed (in cooperation with the calling subroutine).
*>          If .FALSE., then only enough of H is updated to preserve
*>          the eigenvalues.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*>          WANTZ is LOGICAL
*>          If .TRUE., then the orthogonal matrix Z is updated so
*>          so that the orthogonal Schur factor may be computed
*>          (in cooperation with the calling subroutine).
*>          If .FALSE., then Z is not referenced.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix H and (if WANTZ is .TRUE.) the
*>          order of the orthogonal matrix Z.
*> \endverbatim
*>
*> \param[in] KTOP
*> \verbatim
*>          KTOP is INTEGER
*>          It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
*>          KBOT and KTOP together determine an isolated block
*>          along the diagonal of the Hessenberg matrix.
*> \endverbatim
*>
*> \param[in] KBOT
*> \verbatim
*>          KBOT is INTEGER
*>          It is assumed without a check that either
*>          KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together
*>          determine an isolated block along the diagonal of the
*>          Hessenberg matrix.
*> \endverbatim
*>
*> \param[in] NW
*> \verbatim
*>          NW is INTEGER
*>          Deflation window size.  1 <= NW <= (KBOT-KTOP+1).
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*>          H is DOUBLE PRECISION array, dimension (LDH,N)
*>          On input the initial N-by-N section of H stores the
*>          Hessenberg matrix undergoing aggressive early deflation.
*>          On output H has been transformed by an orthogonal
*>          similarity transformation, perturbed, and the returned
*>          to Hessenberg form that (it is to be hoped) has some
*>          zero subdiagonal entries.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*>          LDH is INTEGER
*>          Leading dimension of H just as declared in the calling
*>          subroutine.  N <= LDH
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*>          ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*>          IHIZ is INTEGER
*>          Specify the rows of Z to which transformations must be
*>          applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ,N)
*>          IF WANTZ is .TRUE., then on output, the orthogonal
*>          similarity transformation mentioned above has been
*>          accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
*>          If WANTZ is .FALSE., then Z is unreferenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of Z just as declared in the
*>          calling subroutine.  1 <= LDZ.
*> \endverbatim
*>
*> \param[out] NS
*> \verbatim
*>          NS is INTEGER
*>          The number of unconverged (ie approximate) eigenvalues
*>          returned in SR and SI that may be used as shifts by the
*>          calling subroutine.
*> \endverbatim
*>
*> \param[out] ND
*> \verbatim
*>          ND is INTEGER
*>          The number of converged eigenvalues uncovered by this
*>          subroutine.
*> \endverbatim
*>
*> \param[out] SR
*> \verbatim
*>          SR is DOUBLE PRECISION array, dimension (KBOT)
*> \endverbatim
*>
*> \param[out] SI
*> \verbatim
*>          SI is DOUBLE PRECISION array, dimension (KBOT)
*>          On output, the real and imaginary parts of approximate
*>          eigenvalues that may be used for shifts are stored in
*>          SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
*>          SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
*>          The real and imaginary parts of converged eigenvalues
*>          are stored in SR(KBOT-ND+1) through SR(KBOT) and
*>          SI(KBOT-ND+1) through SI(KBOT), respectively.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension (LDV,NW)
*>          An NW-by-NW work array.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of V just as declared in the
*>          calling subroutine.  NW <= LDV
*> \endverbatim
*>
*> \param[in] NH
*> \verbatim
*>          NH is INTEGER
*>          The number of columns of T.  NH >= NW.
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,NW)
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of T just as declared in the
*>          calling subroutine.  NW <= LDT
*> \endverbatim
*>
*> \param[in] NV
*> \verbatim
*>          NV is INTEGER
*>          The number of rows of work array WV available for
*>          workspace.  NV >= NW.
*> \endverbatim
*>
*> \param[out] WV
*> \verbatim
*>          WV is DOUBLE PRECISION array, dimension (LDWV,NW)
*> \endverbatim
*>
*> \param[in] LDWV
*> \verbatim
*>          LDWV is INTEGER
*>          The leading dimension of W just as declared in the
*>          calling subroutine.  NW <= LDV
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (LWORK)
*>          On exit, WORK(1) is set to an estimate of the optimal value
*>          of LWORK for the given values of N, NW, KTOP and KBOT.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the work array WORK.  LWORK = 2*NW
*>          suffices, but greater efficiency may result from larger
*>          values of LWORK.
*>
*>          If LWORK = -1, then a workspace query is assumed; DLAQR3
*>          only estimates the optimal workspace size for the given
*>          values of N, NW, KTOP and KBOT.  The estimate is returned
*>          in WORK(1).  No error message related to LWORK is issued
*>          by XERBLA.  Neither H nor Z are accessed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqr3
*
*> \par Contributors:
*  ==================
*>
*>       Karen Braman and Ralph Byers, Department of Mathematics,
*>       University of Kansas, USA
*>
*  =====================================================================
      SUBROUTINE DLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH,
     $                   ILOZ,
     $                   IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T,
     $                   LDT, NV, WV, LDWV, WORK, LWORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
     $                   LDZ, LWORK, N, ND, NH, NS, NV, NW
      LOGICAL            WANTT, WANTZ
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   H( LDH, * ), SI( * ), SR( * ), T( LDT, * ),
     $                   V( LDV, * ), WORK( * ), WV( LDWV, * ),
     $                   Z( LDZ, * )
*     ..
*
*  ================================================================
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0d0, ONE = 1.0d0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   AA, BB, BETA, CC, CS, DD, EVI, EVK, FOO, S,
     $                   SAFMAX, SAFMIN, SMLNUM, SN, TAU, ULP
      INTEGER            I, IFST, ILST, INFO, INFQR, J, JW, K, KCOL,
     $                   KEND, KLN, KROW, KWTOP, LTOP, LWK1, LWK2, LWK3,
     $                   LWKOPT, NMIN
      LOGICAL            BULGE, SORTED
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      INTEGER            ILAENV
      EXTERNAL           DLAMCH, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGEHRD, DGEMM, DLACPY, DLAHQR,
     $                   DLANV2,
     $                   DLAQR4, DLARF1F, DLARFG, DLASET, DORMHR, DTREXC
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, INT, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     ==== Estimate optimal workspace. ====
*
      JW = MIN( NW, KBOT-KTOP+1 )
      IF( JW.LE.2 ) THEN
         LWKOPT = 1
      ELSE
*
*        ==== Workspace query call to DGEHRD ====
*
         CALL DGEHRD( JW, 1, JW-1, T, LDT, WORK, WORK, -1, INFO )
         LWK1 = INT( WORK( 1 ) )
*
*        ==== Workspace query call to DORMHR ====
*
         CALL DORMHR( 'R', 'N', JW, JW, 1, JW-1, T, LDT, WORK, V,
     $                LDV,
     $                WORK, -1, INFO )
         LWK2 = INT( WORK( 1 ) )
*
*        ==== Workspace query call to DLAQR4 ====
*
         CALL DLAQR4( .true., .true., JW, 1, JW, T, LDT, SR, SI, 1,
     $                JW,
     $                V, LDV, WORK, -1, INFQR )
         LWK3 = INT( WORK( 1 ) )
*
*        ==== Optimal workspace ====
*
         LWKOPT = MAX( JW+MAX( LWK1, LWK2 ), LWK3 )
      END IF
*
*     ==== Quick return in case of workspace query. ====
*
      IF( LWORK.EQ.-1 ) THEN
         WORK( 1 ) = DBLE( LWKOPT )
         RETURN
      END IF
*
*     ==== Nothing to do ...
*     ... for an empty active block ... ====
      NS = 0
      ND = 0
      WORK( 1 ) = ONE
      IF( KTOP.GT.KBOT )
     $   RETURN
*     ... nor for an empty deflation window. ====
      IF( NW.LT.1 )
     $   RETURN
*
*     ==== Machine constants ====
*
      SAFMIN = DLAMCH( 'SAFE MINIMUM' )
      SAFMAX = ONE / SAFMIN
      ULP = DLAMCH( 'PRECISION' )
      SMLNUM = SAFMIN*( DBLE( N ) / ULP )
*
*     ==== Setup deflation window ====
*
      JW = MIN( NW, KBOT-KTOP+1 )
      KWTOP = KBOT - JW + 1
      IF( KWTOP.EQ.KTOP ) THEN
         S = ZERO
      ELSE
         S = H( KWTOP, KWTOP-1 )
      END IF
*
      IF( KBOT.EQ.KWTOP ) THEN
*
*        ==== 1-by-1 deflation window: not much to do ====
*
         SR( KWTOP ) = H( KWTOP, KWTOP )
         SI( KWTOP ) = ZERO
         NS = 1
         ND = 0
         IF( ABS( S ).LE.MAX( SMLNUM, ULP*ABS( H( KWTOP, KWTOP ) ) ) )
     $        THEN
            NS = 0
            ND = 1
            IF( KWTOP.GT.KTOP )
     $         H( KWTOP, KWTOP-1 ) = ZERO
         END IF
         WORK( 1 ) = ONE
         RETURN
      END IF
*
*     ==== Convert to spike-triangular form.  (In case of a
*     .    rare QR failure, this routine continues to do
*     .    aggressive early deflation using that part of
*     .    the deflation window that converged using INFQR
*     .    here and there to keep track.) ====
*
      CALL DLACPY( 'U', JW, JW, H( KWTOP, KWTOP ), LDH, T, LDT )
      CALL DCOPY( JW-1, H( KWTOP+1, KWTOP ), LDH+1, T( 2, 1 ),
     $            LDT+1 )
*
      CALL DLASET( 'A', JW, JW, ZERO, ONE, V, LDV )
      NMIN = ILAENV( 12, 'DLAQR3', 'SV', JW, 1, JW, LWORK )
      IF( JW.GT.NMIN ) THEN
         CALL DLAQR4( .true., .true., JW, 1, JW, T, LDT, SR( KWTOP ),
     $                SI( KWTOP ), 1, JW, V, LDV, WORK, LWORK, INFQR )
      ELSE
         CALL DLAHQR( .true., .true., JW, 1, JW, T, LDT, SR( KWTOP ),
     $                SI( KWTOP ), 1, JW, V, LDV, INFQR )
      END IF
*
*     ==== DTREXC needs a clean margin near the diagonal ====
*
      DO 10 J = 1, JW - 3
         T( J+2, J ) = ZERO
         T( J+3, J ) = ZERO
   10 CONTINUE
      IF( JW.GT.2 )
     $   T( JW, JW-2 ) = ZERO
*
*     ==== Deflation detection loop ====
*
      NS = JW
      ILST = INFQR + 1
   20 CONTINUE
      IF( ILST.LE.NS ) THEN
         IF( NS.EQ.1 ) THEN
            BULGE = .FALSE.
         ELSE
            BULGE = T( NS, NS-1 ).NE.ZERO
         END IF
*
*        ==== Small spike tip test for deflation ====
*
         IF( .NOT. BULGE ) THEN
*
*           ==== Real eigenvalue ====
*
            FOO = ABS( T( NS, NS ) )
            IF( FOO.EQ.ZERO )
     $         FOO = ABS( S )
            IF( ABS( S*V( 1, NS ) ).LE.MAX( SMLNUM, ULP*FOO ) ) THEN
*
*              ==== Deflatable ====
*
               NS = NS - 1
            ELSE
*
*              ==== Undeflatable.   Move it up out of the way.
*              .    (DTREXC can not fail in this case.) ====
*
               IFST = NS
               CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST,
     $                      WORK,
     $                      INFO )
               ILST = ILST + 1
            END IF
         ELSE
*
*           ==== Complex conjugate pair ====
*
            FOO = ABS( T( NS, NS ) ) + SQRT( ABS( T( NS, NS-1 ) ) )*
     $            SQRT( ABS( T( NS-1, NS ) ) )
            IF( FOO.EQ.ZERO )
     $         FOO = ABS( S )
            IF( MAX( ABS( S*V( 1, NS ) ), ABS( S*V( 1, NS-1 ) ) ).LE.
     $          MAX( SMLNUM, ULP*FOO ) ) THEN
*
*              ==== Deflatable ====
*
               NS = NS - 2
            ELSE
*
*              ==== Undeflatable. Move them up out of the way.
*              .    Fortunately, DTREXC does the right thing with
*              .    ILST in case of a rare exchange failure. ====
*
               IFST = NS
               CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST,
     $                      WORK,
     $                      INFO )
               ILST = ILST + 2
            END IF
         END IF
*
*        ==== End deflation detection loop ====
*
         GO TO 20
      END IF
*
*        ==== Return to Hessenberg form ====
*
      IF( NS.EQ.0 )
     $   S = ZERO
*
      IF( NS.LT.JW ) THEN
*
*        ==== sorting diagonal blocks of T improves accuracy for
*        .    graded matrices.  Bubble sort deals well with
*        .    exchange failures. ====
*
         SORTED = .false.
         I = NS + 1
   30    CONTINUE
         IF( SORTED )
     $      GO TO 50
         SORTED = .true.
*
         KEND = I - 1
         I = INFQR + 1
         IF( I.EQ.NS ) THEN
            K = I + 1
         ELSE IF( T( I+1, I ).EQ.ZERO ) THEN
            K = I + 1
         ELSE
            K = I + 2
         END IF
   40    CONTINUE
         IF( K.LE.KEND ) THEN
            IF( K.EQ.I+1 ) THEN
               EVI = ABS( T( I, I ) )
            ELSE
               EVI = ABS( T( I, I ) ) + SQRT( ABS( T( I+1, I ) ) )*
     $               SQRT( ABS( T( I, I+1 ) ) )
            END IF
*
            IF( K.EQ.KEND ) THEN
               EVK = ABS( T( K, K ) )
            ELSE IF( T( K+1, K ).EQ.ZERO ) THEN
               EVK = ABS( T( K, K ) )
            ELSE
               EVK = ABS( T( K, K ) ) + SQRT( ABS( T( K+1, K ) ) )*
     $               SQRT( ABS( T( K, K+1 ) ) )
            END IF
*
            IF( EVI.GE.EVK ) THEN
               I = K
            ELSE
               SORTED = .false.
               IFST = I
               ILST = K
               CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST,
     $                      WORK,
     $                      INFO )
               IF( INFO.EQ.0 ) THEN
                  I = ILST
               ELSE
                  I = K
               END IF
            END IF
            IF( I.EQ.KEND ) THEN
               K = I + 1
            ELSE IF( T( I+1, I ).EQ.ZERO ) THEN
               K = I + 1
            ELSE
               K = I + 2
            END IF
            GO TO 40
         END IF
         GO TO 30
   50    CONTINUE
      END IF
*
*     ==== Restore shift/eigenvalue array from T ====
*
      I = JW
   60 CONTINUE
      IF( I.GE.INFQR+1 ) THEN
         IF( I.EQ.INFQR+1 ) THEN
            SR( KWTOP+I-1 ) = T( I, I )
            SI( KWTOP+I-1 ) = ZERO
            I = I - 1
         ELSE IF( T( I, I-1 ).EQ.ZERO ) THEN
            SR( KWTOP+I-1 ) = T( I, I )
            SI( KWTOP+I-1 ) = ZERO
            I = I - 1
         ELSE
            AA = T( I-1, I-1 )
            CC = T( I, I-1 )
            BB = T( I-1, I )
            DD = T( I, I )
            CALL DLANV2( AA, BB, CC, DD, SR( KWTOP+I-2 ),
     $                   SI( KWTOP+I-2 ), SR( KWTOP+I-1 ),
     $                   SI( KWTOP+I-1 ), CS, SN )
            I = I - 2
         END IF
         GO TO 60
      END IF
*
      IF( NS.LT.JW .OR. S.EQ.ZERO ) THEN
         IF( NS.GT.1 .AND. S.NE.ZERO ) THEN
*
*           ==== Reflect spike back into lower triangle ====
*
            CALL DCOPY( NS, V, LDV, WORK, 1 )
            BETA = WORK( 1 )
            CALL DLARFG( NS, BETA, WORK( 2 ), 1, TAU )
*
            CALL DLASET( 'L', JW-2, JW-2, ZERO, ZERO, T( 3, 1 ),
     $                   LDT )
*
            CALL DLARF1F( 'L', NS, JW, WORK, 1, TAU, T, LDT,
     $                  WORK( JW+1 ) )
            CALL DLARF1F( 'R', NS, NS, WORK, 1, TAU, T, LDT,
     $                  WORK( JW+1 ) )
            CALL DLARF1F( 'R', JW, NS, WORK, 1, TAU, V, LDV,
     $                  WORK( JW+1 ) )
*
            CALL DGEHRD( JW, 1, NS, T, LDT, WORK, WORK( JW+1 ),
     $                   LWORK-JW, INFO )
         END IF
*
*        ==== Copy updated reduced window into place ====
*
         IF( KWTOP.GT.1 )
     $      H( KWTOP, KWTOP-1 ) = S*V( 1, 1 )
         CALL DLACPY( 'U', JW, JW, T, LDT, H( KWTOP, KWTOP ), LDH )
         CALL DCOPY( JW-1, T( 2, 1 ), LDT+1, H( KWTOP+1, KWTOP ),
     $               LDH+1 )
*
*        ==== Accumulate orthogonal matrix in order update
*        .    H and Z, if requested.  ====
*
         IF( NS.GT.1 .AND. S.NE.ZERO )
     $      CALL DORMHR( 'R', 'N', JW, NS, 1, NS, T, LDT, WORK, V,
     $                   LDV,
     $                   WORK( JW+1 ), LWORK-JW, INFO )
*
*        ==== Update vertical slab in H ====
*
         IF( WANTT ) THEN
            LTOP = 1
         ELSE
            LTOP = KTOP
         END IF
         DO 70 KROW = LTOP, KWTOP - 1, NV
            KLN = MIN( NV, KWTOP-KROW )
            CALL DGEMM( 'N', 'N', KLN, JW, JW, ONE, H( KROW, KWTOP ),
     $                  LDH, V, LDV, ZERO, WV, LDWV )
            CALL DLACPY( 'A', KLN, JW, WV, LDWV, H( KROW, KWTOP ),
     $                   LDH )
   70    CONTINUE
*
*        ==== Update horizontal slab in H ====
*
         IF( WANTT ) THEN
            DO 80 KCOL = KBOT + 1, N, NH
               KLN = MIN( NH, N-KCOL+1 )
               CALL DGEMM( 'C', 'N', JW, KLN, JW, ONE, V, LDV,
     $                     H( KWTOP, KCOL ), LDH, ZERO, T, LDT )
               CALL DLACPY( 'A', JW, KLN, T, LDT, H( KWTOP, KCOL ),
     $                      LDH )
   80       CONTINUE
         END IF
*
*        ==== Update vertical slab in Z ====
*
         IF( WANTZ ) THEN
            DO 90 KROW = ILOZ, IHIZ, NV
               KLN = MIN( NV, IHIZ-KROW+1 )
               CALL DGEMM( 'N', 'N', KLN, JW, JW, ONE, Z( KROW,
     $                     KWTOP ),
     $                     LDZ, V, LDV, ZERO, WV, LDWV )
               CALL DLACPY( 'A', KLN, JW, WV, LDWV, Z( KROW, KWTOP ),
     $                      LDZ )
   90       CONTINUE
         END IF
      END IF
*
*     ==== Return the number of deflations ... ====
*
      ND = JW - NS
*
*     ==== ... and the number of shifts. (Subtracting
*     .    INFQR from the spike length takes care
*     .    of the case of a rare QR failure while
*     .    calculating eigenvalues of the deflation
*     .    window.)  ====
*
      NS = NS - INFQR
*
*      ==== Return optimal workspace. ====
*
      WORK( 1 ) = DBLE( LWKOPT )
*
*     ==== End of DLAQR3 ====
*
      END
*> \brief \b DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQR4 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr4.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqr4.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqr4.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
*                          ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
*       LOGICAL            WANTT, WANTZ
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   H( LDH, * ), WI( * ), WORK( * ), WR( * ),
*      $                   Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>    DLAQR4 implements one level of recursion for DLAQR0.
*>    It is a complete implementation of the small bulge multi-shift
*>    QR algorithm.  It may be called by DLAQR0 and, for large enough
*>    deflation window size, it may be called by DLAQR3.  This
*>    subroutine is identical to DLAQR0 except that it calls DLAQR2
*>    instead of DLAQR3.
*>
*>    DLAQR4 computes the eigenvalues of a Hessenberg matrix H
*>    and, optionally, the matrices T and Z from the Schur decomposition
*>    H = Z T Z**T, where T is an upper quasi-triangular matrix (the
*>    Schur form), and Z is the orthogonal matrix of Schur vectors.
*>
*>    Optionally Z may be postmultiplied into an input orthogonal
*>    matrix Q so that this routine can give the Schur factorization
*>    of a matrix A which has been reduced to the Hessenberg form H
*>    by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] WANTT
*> \verbatim
*>          WANTT is LOGICAL
*>          = .TRUE. : the full Schur form T is required;
*>          = .FALSE.: only eigenvalues are required.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*>          WANTZ is LOGICAL
*>          = .TRUE. : the matrix of Schur vectors Z is required;
*>          = .FALSE.: Schur vectors are not required.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>           The order of the matrix H.  N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*>          ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*>          IHI is INTEGER
*>           It is assumed that H is already upper triangular in rows
*>           and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,
*>           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
*>           previous call to DGEBAL, and then passed to DGEHRD when the
*>           matrix output by DGEBAL is reduced to Hessenberg form.
*>           Otherwise, ILO and IHI should be set to 1 and N,
*>           respectively.  If N > 0, then 1 <= ILO <= IHI <= N.
*>           If N = 0, then ILO = 1 and IHI = 0.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*>          H is DOUBLE PRECISION array, dimension (LDH,N)
*>           On entry, the upper Hessenberg matrix H.
*>           On exit, if INFO = 0 and WANTT is .TRUE., then H contains
*>           the upper quasi-triangular matrix T from the Schur
*>           decomposition (the Schur form); 2-by-2 diagonal blocks
*>           (corresponding to complex conjugate pairs of eigenvalues)
*>           are returned in standard form, with H(i,i) = H(i+1,i+1)
*>           and H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and WANTT is
*>           .FALSE., then the contents of H are unspecified on exit.
*>           (The output value of H when INFO > 0 is given under the
*>           description of INFO below.)
*>
*>           This subroutine may explicitly set H(i,j) = 0 for i > j and
*>           j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*>          LDH is INTEGER
*>           The leading dimension of the array H. LDH >= max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*>          WR is DOUBLE PRECISION array, dimension (IHI)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*>          WI is DOUBLE PRECISION array, dimension (IHI)
*>           The real and imaginary parts, respectively, of the computed
*>           eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
*>           and WI(ILO:IHI). If two eigenvalues are computed as a
*>           complex conjugate pair, they are stored in consecutive
*>           elements of WR and WI, say the i-th and (i+1)th, with
*>           WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., then
*>           the eigenvalues are stored in the same order as on the
*>           diagonal of the Schur form returned in H, with
*>           WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
*>           block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
*>           WI(i+1) = -WI(i).
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*>          ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*>          IHIZ is INTEGER
*>           Specify the rows of Z to which transformations must be
*>           applied if WANTZ is .TRUE..
*>           1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ,IHI)
*>           If WANTZ is .FALSE., then Z is not referenced.
*>           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
*>           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
*>           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
*>           (The output value of Z when INFO > 0 is given under
*>           the description of INFO below.)
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>           The leading dimension of the array Z.  if WANTZ is .TRUE.
*>           then LDZ >= MAX(1,IHIZ).  Otherwise, LDZ >= 1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension LWORK
*>           On exit, if LWORK = -1, WORK(1) returns an estimate of
*>           the optimal value for LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>           The dimension of the array WORK.  LWORK >= max(1,N)
*>           is sufficient, but LWORK typically as large as 6*N may
*>           be required for optimal performance.  A workspace query
*>           to determine the optimal workspace size is recommended.
*>
*>           If LWORK = -1, then DLAQR4 does a workspace query.
*>           In this case, DLAQR4 checks the input parameters and
*>           estimates the optimal workspace size for the given
*>           values of N, ILO and IHI.  The estimate is returned
*>           in WORK(1).  No error message related to LWORK is
*>           issued by XERBLA.  Neither H nor Z are accessed.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>             = 0:  successful exit
*>             > 0:  if INFO = i, DLAQR4 failed to compute all of
*>                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
*>                and WI contain those eigenvalues which have been
*>                successfully computed.  (Failures are rare.)
*>
*>                If INFO > 0 and WANT is .FALSE., then on exit,
*>                the remaining unconverged eigenvalues are the eigen-
*>                values of the upper Hessenberg matrix rows and
*>                columns ILO through INFO of the final, output
*>                value of H.
*>
*>                If INFO > 0 and WANTT is .TRUE., then on exit
*>
*>           (*)  (initial value of H)*U  = U*(final value of H)
*>
*>                where U is a orthogonal matrix.  The final
*>                value of  H is upper Hessenberg and triangular in
*>                rows and columns INFO+1 through IHI.
*>
*>                If INFO > 0 and WANTZ is .TRUE., then on exit
*>
*>                  (final value of Z(ILO:IHI,ILOZ:IHIZ)
*>                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
*>
*>                where U is the orthogonal matrix in (*) (regard-
*>                less of the value of WANTT.)
*>
*>                If INFO > 0 and WANTZ is .FALSE., then Z is not
*>                accessed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqr4
*
*> \par Contributors:
*  ==================
*>
*>       Karen Braman and Ralph Byers, Department of Mathematics,
*>       University of Kansas, USA
*
*> \par References:
*  ================
*>
*>       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*>       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
*>       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
*>       929--947, 2002.
*> \n
*>       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*>       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
*>       of Matrix Analysis, volume 23, pages 948--973, 2002.
*>
*  =====================================================================
      SUBROUTINE DLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
     $                   ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
      LOGICAL            WANTT, WANTZ
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   H( LDH, * ), WI( * ), WORK( * ), WR( * ),
     $                   Z( LDZ, * )
*     ..
*
*  ================================================================
*     .. Parameters ..
*
*     ==== Matrices of order NTINY or smaller must be processed by
*     .    DLAHQR because of insufficient subdiagonal scratch space.
*     .    (This is a hard limit.) ====
      INTEGER            NTINY
      PARAMETER          ( NTINY = 15 )
*
*     ==== Exceptional deflation windows:  try to cure rare
*     .    slow convergence by varying the size of the
*     .    deflation window after KEXNW iterations. ====
      INTEGER            KEXNW
      PARAMETER          ( KEXNW = 5 )
*
*     ==== Exceptional shifts: try to cure rare slow convergence
*     .    with ad-hoc exceptional shifts every KEXSH iterations.
*     .    ====
      INTEGER            KEXSH
      PARAMETER          ( KEXSH = 6 )
*
*     ==== The constants WILK1 and WILK2 are used to form the
*     .    exceptional shifts. ====
      DOUBLE PRECISION   WILK1, WILK2
      PARAMETER          ( WILK1 = 0.75d0, WILK2 = -0.4375d0 )
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0d0, ONE = 1.0d0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   AA, BB, CC, CS, DD, SN, SS, SWAP
      INTEGER            I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
     $                   KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
     $                   LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
     $                   NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
      LOGICAL            SORTED
      CHARACTER          JBCMPZ*2
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   ZDUM( 1, 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACPY, DLAHQR, DLANV2, DLAQR2,
     $                   DLAQR5
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, INT, MAX, MIN, MOD
*     ..
*     .. Executable Statements ..
      INFO = 0
*
*     ==== Quick return for N = 0: nothing to do. ====
*
      IF( N.EQ.0 ) THEN
         WORK( 1 ) = ONE
         RETURN
      END IF
*
      IF( N.LE.NTINY ) THEN
*
*        ==== Tiny matrices must use DLAHQR. ====
*
         LWKOPT = 1
         IF( LWORK.NE.-1 )
     $      CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
     $                   ILOZ, IHIZ, Z, LDZ, INFO )
      ELSE
*
*        ==== Use small bulge multi-shift QR with aggressive early
*        .    deflation on larger-than-tiny matrices. ====
*
*        ==== Hope for the best. ====
*
         INFO = 0
*
*        ==== Set up job flags for ILAENV. ====
*
         IF( WANTT ) THEN
            JBCMPZ( 1: 1 ) = 'S'
         ELSE
            JBCMPZ( 1: 1 ) = 'E'
         END IF
         IF( WANTZ ) THEN
            JBCMPZ( 2: 2 ) = 'V'
         ELSE
            JBCMPZ( 2: 2 ) = 'N'
         END IF
*
*        ==== NWR = recommended deflation window size.  At this
*        .    point,  N .GT. NTINY = 15, so there is enough
*        .    subdiagonal workspace for NWR.GE.2 as required.
*        .    (In fact, there is enough subdiagonal space for
*        .    NWR.GE.4.) ====
*
         NWR = ILAENV( 13, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
         NWR = MAX( 2, NWR )
         NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
*
*        ==== NSR = recommended number of simultaneous shifts.
*        .    At this point N .GT. NTINY = 15, so there is at
*        .    enough subdiagonal workspace for NSR to be even
*        .    and greater than or equal to two as required. ====
*
         NSR = ILAENV( 15, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
         NSR = MIN( NSR, ( N-3 ) / 6, IHI-ILO )
         NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
*
*        ==== Estimate optimal workspace ====
*
*        ==== Workspace query call to DLAQR2 ====
*
         CALL DLAQR2( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
     $                IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH,
     $                N, H, LDH, WORK, -1 )
*
*        ==== Optimal workspace = MAX(DLAQR5, DLAQR2) ====
*
         LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
*
*        ==== Quick return in case of workspace query. ====
*
         IF( LWORK.EQ.-1 ) THEN
            WORK( 1 ) = DBLE( LWKOPT )
            RETURN
         END IF
*
*        ==== DLAHQR/DLAQR0 crossover point ====
*
         NMIN = ILAENV( 12, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
         NMIN = MAX( NTINY, NMIN )
*
*        ==== Nibble crossover point ====
*
         NIBBLE = ILAENV( 14, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
         NIBBLE = MAX( 0, NIBBLE )
*
*        ==== Accumulate reflections during ttswp?  Use block
*        .    2-by-2 structure during matrix-matrix multiply? ====
*
         KACC22 = ILAENV( 16, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
         KACC22 = MAX( 0, KACC22 )
         KACC22 = MIN( 2, KACC22 )
*
*        ==== NWMAX = the largest possible deflation window for
*        .    which there is sufficient workspace. ====
*
         NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
         NW = NWMAX
*
*        ==== NSMAX = the Largest number of simultaneous shifts
*        .    for which there is sufficient workspace. ====
*
         NSMAX = MIN( ( N-3 ) / 6, 2*LWORK / 3 )
         NSMAX = NSMAX - MOD( NSMAX, 2 )
*
*        ==== NDFL: an iteration count restarted at deflation. ====
*
         NDFL = 1
*
*        ==== ITMAX = iteration limit ====
*
         ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
*
*        ==== Last row and column in the active block ====
*
         KBOT = IHI
*
*        ==== Main Loop ====
*
         DO 80 IT = 1, ITMAX
*
*           ==== Done when KBOT falls below ILO ====
*
            IF( KBOT.LT.ILO )
     $         GO TO 90
*
*           ==== Locate active block ====
*
            DO 10 K = KBOT, ILO + 1, -1
               IF( H( K, K-1 ).EQ.ZERO )
     $            GO TO 20
   10       CONTINUE
            K = ILO
   20       CONTINUE
            KTOP = K
*
*           ==== Select deflation window size:
*           .    Typical Case:
*           .      If possible and advisable, nibble the entire
*           .      active block.  If not, use size MIN(NWR,NWMAX)
*           .      or MIN(NWR+1,NWMAX) depending upon which has
*           .      the smaller corresponding subdiagonal entry
*           .      (a heuristic).
*           .
*           .    Exceptional Case:
*           .      If there have been no deflations in KEXNW or
*           .      more iterations, then vary the deflation window
*           .      size.   At first, because, larger windows are,
*           .      in general, more powerful than smaller ones,
*           .      rapidly increase the window to the maximum possible.
*           .      Then, gradually reduce the window size. ====
*
            NH = KBOT - KTOP + 1
            NWUPBD = MIN( NH, NWMAX )
            IF( NDFL.LT.KEXNW ) THEN
               NW = MIN( NWUPBD, NWR )
            ELSE
               NW = MIN( NWUPBD, 2*NW )
            END IF
            IF( NW.LT.NWMAX ) THEN
               IF( NW.GE.NH-1 ) THEN
                  NW = NH
               ELSE
                  KWTOP = KBOT - NW + 1
                  IF( ABS( H( KWTOP, KWTOP-1 ) ).GT.
     $                ABS( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
               END IF
            END IF
            IF( NDFL.LT.KEXNW ) THEN
               NDEC = -1
            ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
               NDEC = NDEC + 1
               IF( NW-NDEC.LT.2 )
     $            NDEC = 0
               NW = NW - NDEC
            END IF
*
*           ==== Aggressive early deflation:
*           .    split workspace under the subdiagonal into
*           .      - an nw-by-nw work array V in the lower
*           .        left-hand-corner,
*           .      - an NW-by-at-least-NW-but-more-is-better
*           .        (NW-by-NHO) horizontal work array along
*           .        the bottom edge,
*           .      - an at-least-NW-but-more-is-better (NHV-by-NW)
*           .        vertical work array along the left-hand-edge.
*           .        ====
*
            KV = N - NW + 1
            KT = NW + 1
            NHO = ( N-NW-1 ) - KT + 1
            KWV = NW + 2
            NVE = ( N-NW ) - KWV + 1
*
*           ==== Aggressive early deflation ====
*
            CALL DLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH,
     $                   ILOZ,
     $                   IHIZ, Z, LDZ, LS, LD, WR, WI, H( KV, 1 ), LDH,
     $                   NHO, H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH,
     $                   WORK, LWORK )
*
*           ==== Adjust KBOT accounting for new deflations. ====
*
            KBOT = KBOT - LD
*
*           ==== KS points to the shifts. ====
*
            KS = KBOT - LS + 1
*
*           ==== Skip an expensive QR sweep if there is a (partly
*           .    heuristic) reason to expect that many eigenvalues
*           .    will deflate without it.  Here, the QR sweep is
*           .    skipped if many eigenvalues have just been deflated
*           .    or if the remaining active block is small.
*
            IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
     $          KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
*
*              ==== NS = nominal number of simultaneous shifts.
*              .    This may be lowered (slightly) if DLAQR2
*              .    did not provide that many shifts. ====
*
               NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
               NS = NS - MOD( NS, 2 )
*
*              ==== If there have been no deflations
*              .    in a multiple of KEXSH iterations,
*              .    then try exceptional shifts.
*              .    Otherwise use shifts provided by
*              .    DLAQR2 above or from the eigenvalues
*              .    of a trailing principal submatrix. ====
*
               IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
                  KS = KBOT - NS + 1
                  DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2
                     SS = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
                     AA = WILK1*SS + H( I, I )
                     BB = SS
                     CC = WILK2*SS
                     DD = AA
                     CALL DLANV2( AA, BB, CC, DD, WR( I-1 ),
     $                            WI( I-1 ),
     $                            WR( I ), WI( I ), CS, SN )
   30             CONTINUE
                  IF( KS.EQ.KTOP ) THEN
                     WR( KS+1 ) = H( KS+1, KS+1 )
                     WI( KS+1 ) = ZERO
                     WR( KS ) = WR( KS+1 )
                     WI( KS ) = WI( KS+1 )
                  END IF
               ELSE
*
*                 ==== Got NS/2 or fewer shifts? Use DLAHQR
*                 .    on a trailing principal submatrix to
*                 .    get more. (Since NS.LE.NSMAX.LE.(N-3)/6,
*                 .    there is enough space below the subdiagonal
*                 .    to fit an NS-by-NS scratch array.) ====
*
                  IF( KBOT-KS+1.LE.NS / 2 ) THEN
                     KS = KBOT - NS + 1
                     KT = N - NS + 1
                     CALL DLACPY( 'A', NS, NS, H( KS, KS ), LDH,
     $                            H( KT, 1 ), LDH )
                     CALL DLAHQR( .false., .false., NS, 1, NS,
     $                            H( KT, 1 ), LDH, WR( KS ), WI( KS ),
     $                            1, 1, ZDUM, 1, INF )
                     KS = KS + INF
*
*                    ==== In case of a rare QR failure use
*                    .    eigenvalues of the trailing 2-by-2
*                    .    principal submatrix.  ====
*
                     IF( KS.GE.KBOT ) THEN
                        AA = H( KBOT-1, KBOT-1 )
                        CC = H( KBOT, KBOT-1 )
                        BB = H( KBOT-1, KBOT )
                        DD = H( KBOT, KBOT )
                        CALL DLANV2( AA, BB, CC, DD, WR( KBOT-1 ),
     $                               WI( KBOT-1 ), WR( KBOT ),
     $                               WI( KBOT ), CS, SN )
                        KS = KBOT - 1
                     END IF
                  END IF
*
                  IF( KBOT-KS+1.GT.NS ) THEN
*
*                    ==== Sort the shifts (Helps a little)
*                    .    Bubble sort keeps complex conjugate
*                    .    pairs together. ====
*
                     SORTED = .false.
                     DO 50 K = KBOT, KS + 1, -1
                        IF( SORTED )
     $                     GO TO 60
                        SORTED = .true.
                        DO 40 I = KS, K - 1
                           IF( ABS( WR( I ) )+ABS( WI( I ) ).LT.
     $                         ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN
                              SORTED = .false.
*
                              SWAP = WR( I )
                              WR( I ) = WR( I+1 )
                              WR( I+1 ) = SWAP
*
                              SWAP = WI( I )
                              WI( I ) = WI( I+1 )
                              WI( I+1 ) = SWAP
                           END IF
   40                   CONTINUE
   50                CONTINUE
   60                CONTINUE
                  END IF
*
*                 ==== Shuffle shifts into pairs of real shifts
*                 .    and pairs of complex conjugate shifts
*                 .    assuming complex conjugate shifts are
*                 .    already adjacent to one another. (Yes,
*                 .    they are.)  ====
*
                  DO 70 I = KBOT, KS + 2, -2
                     IF( WI( I ).NE.-WI( I-1 ) ) THEN
*
                        SWAP = WR( I )
                        WR( I ) = WR( I-1 )
                        WR( I-1 ) = WR( I-2 )
                        WR( I-2 ) = SWAP
*
                        SWAP = WI( I )
                        WI( I ) = WI( I-1 )
                        WI( I-1 ) = WI( I-2 )
                        WI( I-2 ) = SWAP
                     END IF
   70             CONTINUE
               END IF
*
*              ==== If there are only two shifts and both are
*              .    real, then use only one.  ====
*
               IF( KBOT-KS+1.EQ.2 ) THEN
                  IF( WI( KBOT ).EQ.ZERO ) THEN
                     IF( ABS( WR( KBOT )-H( KBOT, KBOT ) ).LT.
     $                   ABS( WR( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
                        WR( KBOT-1 ) = WR( KBOT )
                     ELSE
                        WR( KBOT ) = WR( KBOT-1 )
                     END IF
                  END IF
               END IF
*
*              ==== Use up to NS of the the smallest magnitude
*              .    shifts.  If there aren't NS shifts available,
*              .    then use them all, possibly dropping one to
*              .    make the number of shifts even. ====
*
               NS = MIN( NS, KBOT-KS+1 )
               NS = NS - MOD( NS, 2 )
               KS = KBOT - NS + 1
*
*              ==== Small-bulge multi-shift QR sweep:
*              .    split workspace under the subdiagonal into
*              .    - a KDU-by-KDU work array U in the lower
*              .      left-hand-corner,
*              .    - a KDU-by-at-least-KDU-but-more-is-better
*              .      (KDU-by-NHo) horizontal work array WH along
*              .      the bottom edge,
*              .    - and an at-least-KDU-but-more-is-better-by-KDU
*              .      (NVE-by-KDU) vertical work WV arrow along
*              .      the left-hand-edge. ====
*
               KDU = 2*NS
               KU = N - KDU + 1
               KWH = KDU + 1
               NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
               KWV = KDU + 4
               NVE = N - KDU - KWV + 1
*
*              ==== Small-bulge multi-shift QR sweep ====
*
               CALL DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
     $                      WR( KS ), WI( KS ), H, LDH, ILOZ, IHIZ, Z,
     $                      LDZ, WORK, 3, H( KU, 1 ), LDH, NVE,
     $                      H( KWV, 1 ), LDH, NHO, H( KU, KWH ), LDH )
            END IF
*
*           ==== Note progress (or the lack of it). ====
*
            IF( LD.GT.0 ) THEN
               NDFL = 1
            ELSE
               NDFL = NDFL + 1
            END IF
*
*           ==== End of main loop ====
   80    CONTINUE
*
*        ==== Iteration limit exceeded.  Set INFO to show where
*        .    the problem occurred and exit. ====
*
         INFO = KBOT
   90    CONTINUE
      END IF
*
*     ==== Return the optimal value of LWORK. ====
*
      WORK( 1 ) = DBLE( LWKOPT )
*
*     ==== End of DLAQR4 ====
*
      END
*> \brief \b DLAQR5 performs a single small-bulge multi-shift QR sweep.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQR5 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr5.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqr5.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqr5.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS,
*                          SR, SI, H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U,
*                          LDU, NV, WV, LDWV, NH, WH, LDWH )
*
*       .. Scalar Arguments ..
*       INTEGER            IHIZ, ILOZ, KACC22, KBOT, KTOP, LDH, LDU, LDV,
*      $                   LDWH, LDWV, LDZ, N, NH, NSHFTS, NV
*       LOGICAL            WANTT, WANTZ
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   H( LDH, * ), SI( * ), SR( * ), U( LDU, * ),
*      $                   V( LDV, * ), WH( LDWH, * ), WV( LDWV, * ),
*      $                   Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>    DLAQR5, called by DLAQR0, performs a
*>    single small-bulge multi-shift QR sweep.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] WANTT
*> \verbatim
*>          WANTT is LOGICAL
*>             WANTT = .true. if the quasi-triangular Schur factor
*>             is being computed.  WANTT is set to .false. otherwise.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*>          WANTZ is LOGICAL
*>             WANTZ = .true. if the orthogonal Schur factor is being
*>             computed.  WANTZ is set to .false. otherwise.
*> \endverbatim
*>
*> \param[in] KACC22
*> \verbatim
*>          KACC22 is INTEGER with value 0, 1, or 2.
*>             Specifies the computation mode of far-from-diagonal
*>             orthogonal updates.
*>        = 0: DLAQR5 does not accumulate reflections and does not
*>             use matrix-matrix multiply to update far-from-diagonal
*>             matrix entries.
*>        = 1: DLAQR5 accumulates reflections and uses matrix-matrix
*>             multiply to update the far-from-diagonal matrix entries.
*>        = 2: Same as KACC22 = 1. This option used to enable exploiting
*>             the 2-by-2 structure during matrix multiplications, but
*>             this is no longer supported.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>             N is the order of the Hessenberg matrix H upon which this
*>             subroutine operates.
*> \endverbatim
*>
*> \param[in] KTOP
*> \verbatim
*>          KTOP is INTEGER
*> \endverbatim
*>
*> \param[in] KBOT
*> \verbatim
*>          KBOT is INTEGER
*>             These are the first and last rows and columns of an
*>             isolated diagonal block upon which the QR sweep is to be
*>             applied. It is assumed without a check that
*>                       either KTOP = 1  or   H(KTOP,KTOP-1) = 0
*>             and
*>                       either KBOT = N  or   H(KBOT+1,KBOT) = 0.
*> \endverbatim
*>
*> \param[in] NSHFTS
*> \verbatim
*>          NSHFTS is INTEGER
*>             NSHFTS gives the number of simultaneous shifts.  NSHFTS
*>             must be positive and even.
*> \endverbatim
*>
*> \param[in,out] SR
*> \verbatim
*>          SR is DOUBLE PRECISION array, dimension (NSHFTS)
*> \endverbatim
*>
*> \param[in,out] SI
*> \verbatim
*>          SI is DOUBLE PRECISION array, dimension (NSHFTS)
*>             SR contains the real parts and SI contains the imaginary
*>             parts of the NSHFTS shifts of origin that define the
*>             multi-shift QR sweep.  On output SR and SI may be
*>             reordered.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*>          H is DOUBLE PRECISION array, dimension (LDH,N)
*>             On input H contains a Hessenberg matrix.  On output a
*>             multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied
*>             to the isolated diagonal block in rows and columns KTOP
*>             through KBOT.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*>          LDH is INTEGER
*>             LDH is the leading dimension of H just as declared in the
*>             calling procedure.  LDH >= MAX(1,N).
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*>          ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*>          IHIZ is INTEGER
*>             Specify the rows of Z to which transformations must be
*>             applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ,IHIZ)
*>             If WANTZ = .TRUE., then the QR Sweep orthogonal
*>             similarity transformation is accumulated into
*>             Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
*>             If WANTZ = .FALSE., then Z is unreferenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>             LDA is the leading dimension of Z just as declared in
*>             the calling procedure. LDZ >= N.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension (LDV,NSHFTS/2)
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>             LDV is the leading dimension of V as declared in the
*>             calling procedure.  LDV >= 3.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*>          U is DOUBLE PRECISION array, dimension (LDU,2*NSHFTS)
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>             LDU is the leading dimension of U just as declared in the
*>             in the calling subroutine.  LDU >= 2*NSHFTS.
*> \endverbatim
*>
*> \param[in] NV
*> \verbatim
*>          NV is INTEGER
*>             NV is the number of rows in WV agailable for workspace.
*>             NV >= 1.
*> \endverbatim
*>
*> \param[out] WV
*> \verbatim
*>          WV is DOUBLE PRECISION array, dimension (LDWV,2*NSHFTS)
*> \endverbatim
*>
*> \param[in] LDWV
*> \verbatim
*>          LDWV is INTEGER
*>             LDWV is the leading dimension of WV as declared in the
*>             in the calling subroutine.  LDWV >= NV.
*> \endverbatim
*
*> \param[in] NH
*> \verbatim
*>          NH is INTEGER
*>             NH is the number of columns in array WH available for
*>             workspace. NH >= 1.
*> \endverbatim
*>
*> \param[out] WH
*> \verbatim
*>          WH is DOUBLE PRECISION array, dimension (LDWH,NH)
*> \endverbatim
*>
*> \param[in] LDWH
*> \verbatim
*>          LDWH is INTEGER
*>             Leading dimension of WH just as declared in the
*>             calling procedure.  LDWH >= 2*NSHFTS.
*> \endverbatim
*>
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqr5
*
*> \par Contributors:
*  ==================
*>
*>       Karen Braman and Ralph Byers, Department of Mathematics,
*>       University of Kansas, USA
*>
*>       Lars Karlsson, Daniel Kressner, and Bruno Lang
*>
*>       Thijs Steel, Department of Computer science,
*>       KU Leuven, Belgium
*
*> \par References:
*  ================
*>
*>       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*>       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
*>       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
*>       929--947, 2002.
*>
*>       Lars Karlsson, Daniel Kressner, and Bruno Lang, Optimally packed
*>       chains of bulges in multishift QR algorithms.
*>       ACM Trans. Math. Softw. 40, 2, Article 12 (February 2014).
*>
*  =====================================================================
      SUBROUTINE DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS,
     $                   SR, SI, H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U,
     $                   LDU, NV, WV, LDWV, NH, WH, LDWH )
      IMPLICIT NONE
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            IHIZ, ILOZ, KACC22, KBOT, KTOP, LDH, LDU, LDV,
     $                   LDWH, LDWV, LDZ, N, NH, NSHFTS, NV
      LOGICAL            WANTT, WANTZ
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   H( LDH, * ), SI( * ), SR( * ), U( LDU, * ),
     $                   V( LDV, * ), WH( LDWH, * ), WV( LDWV, * ),
     $                   Z( LDZ, * )
*     ..
*
*  ================================================================
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0d0, ONE = 1.0d0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   ALPHA, BETA, H11, H12, H21, H22, REFSUM,
     $                   SAFMAX, SAFMIN, SCL, SMLNUM, SWAP, T1, T2,
     $                   T3, TST1, TST2, ULP
      INTEGER            I, I2, I4, INCOL, J, JBOT, JCOL, JLEN,
     $                   JROW, JTOP, K, K1, KDU, KMS, KRCOL,
     $                   M, M22, MBOT, MTOP, NBMPS, NDCOL,
     $                   NS, NU
      LOGICAL            ACCUM, BMP22
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. Intrinsic Functions ..
*
      INTRINSIC          ABS, DBLE, MAX, MIN, MOD
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   VT( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMM, DLACPY, DLAQR1, DLARFG, DLASET,
     $                   DTRMM
*     ..
*     .. Executable Statements ..
*
*     ==== If there are no shifts, then there is nothing to do. ====
*
      IF( NSHFTS.LT.2 )
     $   RETURN
*
*     ==== If the active block is empty or 1-by-1, then there
*     .    is nothing to do. ====
*
      IF( KTOP.GE.KBOT )
     $   RETURN
*
*     ==== Shuffle shifts into pairs of real shifts and pairs
*     .    of complex conjugate shifts assuming complex
*     .    conjugate shifts are already adjacent to one
*     .    another. ====
*
      DO 10 I = 1, NSHFTS - 2, 2
         IF( SI( I ).NE.-SI( I+1 ) ) THEN
*
            SWAP = SR( I )
            SR( I ) = SR( I+1 )
            SR( I+1 ) = SR( I+2 )
            SR( I+2 ) = SWAP
*
            SWAP = SI( I )
            SI( I ) = SI( I+1 )
            SI( I+1 ) = SI( I+2 )
            SI( I+2 ) = SWAP
         END IF
   10 CONTINUE
*
*     ==== NSHFTS is supposed to be even, but if it is odd,
*     .    then simply reduce it by one.  The shuffle above
*     .    ensures that the dropped shift is real and that
*     .    the remaining shifts are paired. ====
*
      NS = NSHFTS - MOD( NSHFTS, 2 )
*
*     ==== Machine constants for deflation ====
*
      SAFMIN = DLAMCH( 'SAFE MINIMUM' )
      SAFMAX = ONE / SAFMIN
      ULP = DLAMCH( 'PRECISION' )
      SMLNUM = SAFMIN*( DBLE( N ) / ULP )
*
*     ==== Use accumulated reflections to update far-from-diagonal
*     .    entries ? ====
*
      ACCUM = ( KACC22.EQ.1 ) .OR. ( KACC22.EQ.2 )
*
*     ==== clear trash ====
*
      IF( KTOP+2.LE.KBOT )
     $   H( KTOP+2, KTOP ) = ZERO
*
*     ==== NBMPS = number of 2-shift bulges in the chain ====
*
      NBMPS = NS / 2
*
*     ==== KDU = width of slab ====
*
      KDU = 4*NBMPS
*
*     ==== Create and chase chains of NBMPS bulges ====
*
      DO 180 INCOL = KTOP - 2*NBMPS + 1, KBOT - 2, 2*NBMPS
*
*        JTOP = Index from which updates from the right start.
*
         IF( ACCUM ) THEN
            JTOP = MAX( KTOP, INCOL )
         ELSE IF( WANTT ) THEN
            JTOP = 1
         ELSE
            JTOP = KTOP
         END IF
*
         NDCOL = INCOL + KDU
         IF( ACCUM )
     $      CALL DLASET( 'ALL', KDU, KDU, ZERO, ONE, U, LDU )
*
*        ==== Near-the-diagonal bulge chase.  The following loop
*        .    performs the near-the-diagonal part of a small bulge
*        .    multi-shift QR sweep.  Each 4*NBMPS column diagonal
*        .    chunk extends from column INCOL to column NDCOL
*        .    (including both column INCOL and column NDCOL). The
*        .    following loop chases a 2*NBMPS+1 column long chain of
*        .    NBMPS bulges 2*NBMPS columns to the right.  (INCOL
*        .    may be less than KTOP and and NDCOL may be greater than
*        .    KBOT indicating phantom columns from which to chase
*        .    bulges before they are actually introduced or to which
*        .    to chase bulges beyond column KBOT.)  ====
*
         DO 145 KRCOL = INCOL, MIN( INCOL+2*NBMPS-1, KBOT-2 )
*
*           ==== Bulges number MTOP to MBOT are active double implicit
*           .    shift bulges.  There may or may not also be small
*           .    2-by-2 bulge, if there is room.  The inactive bulges
*           .    (if any) must wait until the active bulges have moved
*           .    down the diagonal to make room.  The phantom matrix
*           .    paradigm described above helps keep track.  ====
*
            MTOP = MAX( 1, ( KTOP-KRCOL ) / 2+1 )
            MBOT = MIN( NBMPS, ( KBOT-KRCOL-1 ) / 2 )
            M22 = MBOT + 1
            BMP22 = ( MBOT.LT.NBMPS ) .AND. ( KRCOL+2*( M22-1 ) ).EQ.
     $              ( KBOT-2 )
*
*           ==== Generate reflections to chase the chain right
*           .    one column.  (The minimum value of K is KTOP-1.) ====
*
            IF ( BMP22 ) THEN
*
*              ==== Special case: 2-by-2 reflection at bottom treated
*              .    separately ====
*
               K = KRCOL + 2*( M22-1 )
               IF( K.EQ.KTOP-1 ) THEN
                  CALL DLAQR1( 2, H( K+1, K+1 ), LDH, SR( 2*M22-1 ),
     $                         SI( 2*M22-1 ), SR( 2*M22 ), SI( 2*M22 ),
     $                         V( 1, M22 ) )
                  BETA = V( 1, M22 )
                  CALL DLARFG( 2, BETA, V( 2, M22 ), 1, V( 1, M22 ) )
               ELSE
                  BETA = H( K+1, K )
                  V( 2, M22 ) = H( K+2, K )
                  CALL DLARFG( 2, BETA, V( 2, M22 ), 1, V( 1, M22 ) )
                  H( K+1, K ) = BETA
                  H( K+2, K ) = ZERO
               END IF

*
*              ==== Perform update from right within 
*              .    computational window. ====
*
               T1 = V( 1, M22 )
               T2 = T1*V( 2, M22 )
               DO 30 J = JTOP, MIN( KBOT, K+3 )
                  REFSUM = H( J, K+1 ) + V( 2, M22 )*H( J, K+2 )
                  H( J, K+1 ) = H( J, K+1 ) - REFSUM*T1
                  H( J, K+2 ) = H( J, K+2 ) - REFSUM*T2
   30          CONTINUE
*
*              ==== Perform update from left within 
*              .    computational window. ====
*
               IF( ACCUM ) THEN
                  JBOT = MIN( NDCOL, KBOT )
               ELSE IF( WANTT ) THEN
                  JBOT = N
               ELSE
                  JBOT = KBOT
               END IF
               T1 = V( 1, M22 )
               T2 = T1*V( 2, M22 )
               DO 40 J = K+1, JBOT
                  REFSUM = H( K+1, J ) + V( 2, M22 )*H( K+2, J )
                  H( K+1, J ) = H( K+1, J ) - REFSUM*T1
                  H( K+2, J ) = H( K+2, J ) - REFSUM*T2
   40          CONTINUE
*
*              ==== The following convergence test requires that
*              .    the tradition small-compared-to-nearby-diagonals
*              .    criterion and the Ahues & Tisseur (LAWN 122, 1997)
*              .    criteria both be satisfied.  The latter improves
*              .    accuracy in some examples. Falling back on an
*              .    alternate convergence criterion when TST1 or TST2
*              .    is zero (as done here) is traditional but probably
*              .    unnecessary. ====
*
               IF( K.GE.KTOP ) THEN
                  IF( H( K+1, K ).NE.ZERO ) THEN
                     TST1 = ABS( H( K, K ) ) + ABS( H( K+1, K+1 ) )
                     IF( TST1.EQ.ZERO ) THEN
                        IF( K.GE.KTOP+1 )
     $                     TST1 = TST1 + ABS( H( K, K-1 ) )
                        IF( K.GE.KTOP+2 )
     $                     TST1 = TST1 + ABS( H( K, K-2 ) )
                        IF( K.GE.KTOP+3 )
     $                     TST1 = TST1 + ABS( H( K, K-3 ) )
                        IF( K.LE.KBOT-2 )
     $                     TST1 = TST1 + ABS( H( K+2, K+1 ) )
                        IF( K.LE.KBOT-3 )
     $                     TST1 = TST1 + ABS( H( K+3, K+1 ) )
                        IF( K.LE.KBOT-4 )
     $                     TST1 = TST1 + ABS( H( K+4, K+1 ) )
                     END IF
                     IF( ABS( H( K+1, K ) )
     $                   .LE.MAX( SMLNUM, ULP*TST1 ) ) THEN
                        H12 = MAX( ABS( H( K+1, K ) ),
     $                             ABS( H( K, K+1 ) ) )
                        H21 = MIN( ABS( H( K+1, K ) ),
     $                             ABS( H( K, K+1 ) ) )
                        H11 = MAX( ABS( H( K+1, K+1 ) ),
     $                             ABS( H( K, K )-H( K+1, K+1 ) ) )
                        H22 = MIN( ABS( H( K+1, K+1 ) ),
     $                        ABS( H( K, K )-H( K+1, K+1 ) ) )
                        SCL = H11 + H12
                        TST2 = H22*( H11 / SCL )
*
                        IF( TST2.EQ.ZERO .OR. H21*( H12 / SCL ).LE.
     $                      MAX( SMLNUM, ULP*TST2 ) ) THEN
                           H( K+1, K ) = ZERO
                        END IF
                     END IF
                  END IF
               END IF
*
*              ==== Accumulate orthogonal transformations. ====
*
               IF( ACCUM ) THEN
                  KMS = K - INCOL
                  T1 = V( 1, M22 )
                  T2 = T1*V( 2, M22 )
                  DO 50 J = MAX( 1, KTOP-INCOL ), KDU
                     REFSUM = U( J, KMS+1 ) + V( 2, M22 )*U( J, KMS+2 )
                     U( J, KMS+1 ) = U( J, KMS+1 ) - REFSUM*T1
                     U( J, KMS+2 ) = U( J, KMS+2 ) - REFSUM*T2
  50                 CONTINUE
               ELSE IF( WANTZ ) THEN
                  T1 = V( 1, M22 )
                  T2 = T1*V( 2, M22 )
                  DO 60 J = ILOZ, IHIZ
                     REFSUM = Z( J, K+1 )+V( 2, M22 )*Z( J, K+2 )
                     Z( J, K+1 ) = Z( J, K+1 ) - REFSUM*T1
                     Z( J, K+2 ) = Z( J, K+2 ) - REFSUM*T2
  60              CONTINUE
               END IF
            END IF
*
*           ==== Normal case: Chain of 3-by-3 reflections ====
*
            DO 80 M = MBOT, MTOP, -1
               K = KRCOL + 2*( M-1 )
               IF( K.EQ.KTOP-1 ) THEN
                  CALL DLAQR1( 3, H( KTOP, KTOP ), LDH, SR( 2*M-1 ),
     $                         SI( 2*M-1 ), SR( 2*M ), SI( 2*M ),
     $                         V( 1, M ) )
                  ALPHA = V( 1, M )
                  CALL DLARFG( 3, ALPHA, V( 2, M ), 1, V( 1, M ) )
               ELSE
*
*                 ==== Perform delayed transformation of row below
*                 .    Mth bulge. Exploit fact that first two elements
*                 .    of row are actually zero. ====
*
                  T1 = V( 1, M )
                  T2 = T1*V( 2, M )
                  T3 = T1*V( 3, M )
                  REFSUM = V( 3, M )*H( K+3, K+2 )
                  H( K+3, K   ) = -REFSUM*T1
                  H( K+3, K+1 ) = -REFSUM*T2
                  H( K+3, K+2 ) = H( K+3, K+2 ) - REFSUM*T3
*
*                 ==== Calculate reflection to move
*                 .    Mth bulge one step. ====
*
                  BETA      = H( K+1, K )
                  V( 2, M ) = H( K+2, K )
                  V( 3, M ) = H( K+3, K )
                  CALL DLARFG( 3, BETA, V( 2, M ), 1, V( 1, M ) )
*
*                 ==== A Bulge may collapse because of vigilant
*                 .    deflation or destructive underflow.  In the
*                 .    underflow case, try the two-small-subdiagonals
*                 .    trick to try to reinflate the bulge.  ====
*
                  IF( H( K+3, K ).NE.ZERO .OR. H( K+3, K+1 ).NE.
     $                ZERO .OR. H( K+3, K+2 ).EQ.ZERO ) THEN
*
*                    ==== Typical case: not collapsed (yet). ====
*
                     H( K+1, K ) = BETA
                     H( K+2, K ) = ZERO
                     H( K+3, K ) = ZERO
                  ELSE
*
*                    ==== Atypical case: collapsed.  Attempt to
*                    .    reintroduce ignoring H(K+1,K) and H(K+2,K).
*                    .    If the fill resulting from the new
*                    .    reflector is too large, then abandon it.
*                    .    Otherwise, use the new one. ====
*
                     CALL DLAQR1( 3, H( K+1, K+1 ), LDH, SR( 2*M-1 ),
     $                            SI( 2*M-1 ), SR( 2*M ), SI( 2*M ),
     $                            VT )
                     ALPHA = VT( 1 )
                     CALL DLARFG( 3, ALPHA, VT( 2 ), 1, VT( 1 ) )
                     T1 = VT( 1 )
                     T2 = T1*VT( 2 )
                     T3 = T1*VT( 3 )
                     REFSUM = H( K+1, K ) + VT( 2 )*H( K+2, K )
*
                     IF( ABS( H( K+2, K )-REFSUM*T2 )+
     $                   ABS( REFSUM*T3 ).GT.ULP*
     $                   ( ABS( H( K, K ) )+ABS( H( K+1,
     $                   K+1 ) )+ABS( H( K+2, K+2 ) ) ) ) THEN
*
*                       ==== Starting a new bulge here would
*                       .    create non-negligible fill.  Use
*                       .    the old one with trepidation. ====
*
                        H( K+1, K ) = BETA
                        H( K+2, K ) = ZERO
                        H( K+3, K ) = ZERO
                     ELSE
*
*                       ==== Starting a new bulge here would
*                       .    create only negligible fill.
*                       .    Replace the old reflector with
*                       .    the new one. ====
*
                        H( K+1, K ) = H( K+1, K ) - REFSUM*T1
                        H( K+2, K ) = ZERO
                        H( K+3, K ) = ZERO
                        V( 1, M ) = VT( 1 )
                        V( 2, M ) = VT( 2 )
                        V( 3, M ) = VT( 3 )
                     END IF
                  END IF
               END IF
*
*              ====  Apply reflection from the right and
*              .     the first column of update from the left.
*              .     These updates are required for the vigilant
*              .     deflation check. We still delay most of the
*              .     updates from the left for efficiency. ====      
*
               T1 = V( 1, M )
               T2 = T1*V( 2, M )
               T3 = T1*V( 3, M )
               DO 70 J = JTOP, MIN( KBOT, K+3 )
                  REFSUM = H( J, K+1 ) + V( 2, M )*H( J, K+2 )
     $                     + V( 3, M )*H( J, K+3 )
                  H( J, K+1 ) = H( J, K+1 ) - REFSUM*T1
                  H( J, K+2 ) = H( J, K+2 ) - REFSUM*T2
                  H( J, K+3 ) = H( J, K+3 ) - REFSUM*T3
   70          CONTINUE
*
*              ==== Perform update from left for subsequent
*              .    column. ====
*
               REFSUM = H( K+1, K+1 ) + V( 2, M )*H( K+2, K+1 )
     $                  + V( 3, M )*H( K+3, K+1 )
               H( K+1, K+1 ) = H( K+1, K+1 ) - REFSUM*T1
               H( K+2, K+1 ) = H( K+2, K+1 ) - REFSUM*T2
               H( K+3, K+1 ) = H( K+3, K+1 ) - REFSUM*T3
*
*              ==== The following convergence test requires that
*              .    the tradition small-compared-to-nearby-diagonals
*              .    criterion and the Ahues & Tisseur (LAWN 122, 1997)
*              .    criteria both be satisfied.  The latter improves
*              .    accuracy in some examples. Falling back on an
*              .    alternate convergence criterion when TST1 or TST2
*              .    is zero (as done here) is traditional but probably
*              .    unnecessary. ====
*
               IF( K.LT.KTOP)
     $              CYCLE
               IF( H( K+1, K ).NE.ZERO ) THEN
                  TST1 = ABS( H( K, K ) ) + ABS( H( K+1, K+1 ) )
                  IF( TST1.EQ.ZERO ) THEN
                     IF( K.GE.KTOP+1 )
     $                  TST1 = TST1 + ABS( H( K, K-1 ) )
                     IF( K.GE.KTOP+2 )
     $                  TST1 = TST1 + ABS( H( K, K-2 ) )
                     IF( K.GE.KTOP+3 )
     $                  TST1 = TST1 + ABS( H( K, K-3 ) )
                     IF( K.LE.KBOT-2 )
     $                  TST1 = TST1 + ABS( H( K+2, K+1 ) )
                     IF( K.LE.KBOT-3 )
     $                  TST1 = TST1 + ABS( H( K+3, K+1 ) )
                     IF( K.LE.KBOT-4 )
     $                  TST1 = TST1 + ABS( H( K+4, K+1 ) )
                  END IF
                  IF( ABS( H( K+1, K ) ).LE.MAX( SMLNUM, ULP*TST1 ) )
     $                 THEN
                     H12 = MAX( ABS( H( K+1, K ) ), ABS( H( K, K+1 ) ) )
                     H21 = MIN( ABS( H( K+1, K ) ), ABS( H( K, K+1 ) ) )
                     H11 = MAX( ABS( H( K+1, K+1 ) ),
     $                     ABS( H( K, K )-H( K+1, K+1 ) ) )
                     H22 = MIN( ABS( H( K+1, K+1 ) ),
     $                     ABS( H( K, K )-H( K+1, K+1 ) ) )
                     SCL = H11 + H12
                     TST2 = H22*( H11 / SCL )
*
                     IF( TST2.EQ.ZERO .OR. H21*( H12 / SCL ).LE.
     $                   MAX( SMLNUM, ULP*TST2 ) ) THEN
                        H( K+1, K ) = ZERO
                     END IF
                  END IF
               END IF
   80       CONTINUE
*
*           ==== Multiply H by reflections from the left ====
*
            IF( ACCUM ) THEN
               JBOT = MIN( NDCOL, KBOT )
            ELSE IF( WANTT ) THEN
               JBOT = N
            ELSE
               JBOT = KBOT
            END IF
*
            DO 100 M = MBOT, MTOP, -1
               K = KRCOL + 2*( M-1 )
               T1 = V( 1, M )
               T2 = T1*V( 2, M )
               T3 = T1*V( 3, M )
               DO 90 J = MAX( KTOP, KRCOL + 2*M ), JBOT
                  REFSUM = H( K+1, J ) + V( 2, M )*H( K+2, J )
     $                     + V( 3, M )*H( K+3, J )
                  H( K+1, J ) = H( K+1, J ) - REFSUM*T1
                  H( K+2, J ) = H( K+2, J ) - REFSUM*T2
                  H( K+3, J ) = H( K+3, J ) - REFSUM*T3
   90          CONTINUE
  100       CONTINUE
*
*           ==== Accumulate orthogonal transformations. ====
*
            IF( ACCUM ) THEN
*
*              ==== Accumulate U. (If needed, update Z later
*              .    with an efficient matrix-matrix
*              .    multiply.) ====
*
               DO 120 M = MBOT, MTOP, -1
                  K = KRCOL + 2*( M-1 )
                  KMS = K - INCOL
                  I2 = MAX( 1, KTOP-INCOL )
                  I2 = MAX( I2, KMS-(KRCOL-INCOL)+1 )
                  I4 = MIN( KDU, KRCOL + 2*( MBOT-1 ) - INCOL + 5 )
                  T1 = V( 1, M )
                  T2 = T1*V( 2, M )
                  T3 = T1*V( 3, M )
                  DO 110 J = I2, I4
                     REFSUM = U( J, KMS+1 ) + V( 2, M )*U( J, KMS+2 )
     $                        + V( 3, M )*U( J, KMS+3 )
                     U( J, KMS+1 ) = U( J, KMS+1 ) - REFSUM*T1
                     U( J, KMS+2 ) = U( J, KMS+2 ) - REFSUM*T2
                     U( J, KMS+3 ) = U( J, KMS+3 ) - REFSUM*T3
  110             CONTINUE
  120          CONTINUE
            ELSE IF( WANTZ ) THEN
*
*              ==== U is not accumulated, so update Z
*              .    now by multiplying by reflections
*              .    from the right. ====
*
               DO 140 M = MBOT, MTOP, -1
                  K = KRCOL + 2*( M-1 )
                  T1 = V( 1, M )
                  T2 = T1*V( 2, M )
                  T3 = T1*V( 3, M )
                  DO 130 J = ILOZ, IHIZ
                     REFSUM = Z( J, K+1 ) + V( 2, M )*Z( J, K+2 )
     $                        + V( 3, M )*Z( J, K+3 )
                     Z( J, K+1 ) = Z( J, K+1 ) - REFSUM*T1
                     Z( J, K+2 ) = Z( J, K+2 ) - REFSUM*T2
                     Z( J, K+3 ) = Z( J, K+3 ) - REFSUM*T3
  130             CONTINUE
  140          CONTINUE
            END IF
*
*           ==== End of near-the-diagonal bulge chase. ====
*
  145    CONTINUE
*
*        ==== Use U (if accumulated) to update far-from-diagonal
*        .    entries in H.  If required, use U to update Z as
*        .    well. ====
*
         IF( ACCUM ) THEN
            IF( WANTT ) THEN
               JTOP = 1
               JBOT = N
            ELSE
               JTOP = KTOP
               JBOT = KBOT
            END IF
            K1 = MAX( 1, KTOP-INCOL )
            NU = ( KDU-MAX( 0, NDCOL-KBOT ) ) - K1 + 1
*
*           ==== Horizontal Multiply ====
*
            DO 150 JCOL = MIN( NDCOL, KBOT ) + 1, JBOT, NH
               JLEN = MIN( NH, JBOT-JCOL+1 )
               CALL DGEMM( 'C', 'N', NU, JLEN, NU, ONE, U( K1, K1 ),
     $                        LDU, H( INCOL+K1, JCOL ), LDH, ZERO, WH,
     $                        LDWH )
               CALL DLACPY( 'ALL', NU, JLEN, WH, LDWH,
     $                         H( INCOL+K1, JCOL ), LDH )
  150       CONTINUE
*
*           ==== Vertical multiply ====
*
            DO 160 JROW = JTOP, MAX( KTOP, INCOL ) - 1, NV
               JLEN = MIN( NV, MAX( KTOP, INCOL )-JROW )
               CALL DGEMM( 'N', 'N', JLEN, NU, NU, ONE,
     $                     H( JROW, INCOL+K1 ), LDH, U( K1, K1 ),
     $                     LDU, ZERO, WV, LDWV )
               CALL DLACPY( 'ALL', JLEN, NU, WV, LDWV,
     $                      H( JROW, INCOL+K1 ), LDH )
  160       CONTINUE
*
*           ==== Z multiply (also vertical) ====
*
            IF( WANTZ ) THEN
               DO 170 JROW = ILOZ, IHIZ, NV
                  JLEN = MIN( NV, IHIZ-JROW+1 )
                  CALL DGEMM( 'N', 'N', JLEN, NU, NU, ONE,
     $                        Z( JROW, INCOL+K1 ), LDZ, U( K1, K1 ),
     $                        LDU, ZERO, WV, LDWV )
                  CALL DLACPY( 'ALL', JLEN, NU, WV, LDWV,
     $                         Z( JROW, INCOL+K1 ), LDZ )
  170          CONTINUE
            END IF
         END IF
  180 CONTINUE
*
*     ==== End of DLAQR5 ====
*
      END
*> \brief \b DLAQSB scales a symmetric/Hermitian band matrix, using scaling factors computed by spbequ.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQSB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqsb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqsb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqsb.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAQSB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED )
*
*       .. Scalar Arguments ..
*       CHARACTER          EQUED, UPLO
*       INTEGER            KD, LDAB, N
*       DOUBLE PRECISION   AMAX, SCOND
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * ), S( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAQSB equilibrates a symmetric band matrix A using the scaling
*> factors in the vector S.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the upper or lower triangular part of the
*>          symmetric matrix A is stored.
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of super-diagonals of the matrix A if UPLO = 'U',
*>          or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          On entry, the upper or lower triangle of the symmetric band
*>          matrix A, stored in the first KD+1 rows of the array.  The
*>          j-th column of A is stored in the j-th column of the array AB
*>          as follows:
*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*>
*>          On exit, if INFO = 0, the triangular factor U or L from the
*>          Cholesky factorization A = U**T*U or A = L*L**T of the band
*>          matrix A, in the same storage format as A.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (N)
*>          The scale factors for A.
*> \endverbatim
*>
*> \param[in] SCOND
*> \verbatim
*>          SCOND is DOUBLE PRECISION
*>          Ratio of the smallest S(i) to the largest S(i).
*> \endverbatim
*>
*> \param[in] AMAX
*> \verbatim
*>          AMAX is DOUBLE PRECISION
*>          Absolute value of largest matrix entry.
*> \endverbatim
*>
*> \param[out] EQUED
*> \verbatim
*>          EQUED is CHARACTER*1
*>          Specifies whether or not equilibration was done.
*>          = 'N':  No equilibration.
*>          = 'Y':  Equilibration was done, i.e., A has been replaced by
*>                  diag(S) * A * diag(S).
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  THRESH is a threshold value used to decide if scaling should be done
*>  based on the ratio of the scaling factors.  If SCOND < THRESH,
*>  scaling is done.
*>
*>  LARGE and SMALL are threshold values used to decide if scaling should
*>  be done based on the absolute size of the largest matrix element.
*>  If AMAX > LARGE or AMAX < SMALL, scaling is done.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqhb
*
*  =====================================================================
      SUBROUTINE DLAQSB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX,
     $                   EQUED )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          EQUED, UPLO
      INTEGER            KD, LDAB, N
      DOUBLE PRECISION   AMAX, SCOND
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * ), S( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, THRESH
      PARAMETER          ( ONE = 1.0D+0, THRESH = 0.1D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J
      DOUBLE PRECISION   CJ, LARGE, SMALL
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( N.LE.0 ) THEN
         EQUED = 'N'
         RETURN
      END IF
*
*     Initialize LARGE and SMALL.
*
      SMALL = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
      LARGE = ONE / SMALL
*
      IF( SCOND.GE.THRESH .AND. AMAX.GE.SMALL .AND. AMAX.LE.LARGE ) THEN
*
*        No equilibration
*
         EQUED = 'N'
      ELSE
*
*        Replace A by diag(S) * A * diag(S).
*
         IF( LSAME( UPLO, 'U' ) ) THEN
*
*           Upper triangle of A is stored in band format.
*
            DO 20 J = 1, N
               CJ = S( J )
               DO 10 I = MAX( 1, J-KD ), J
                  AB( KD+1+I-J, J ) = CJ*S( I )*AB( KD+1+I-J, J )
   10          CONTINUE
   20       CONTINUE
         ELSE
*
*           Lower triangle of A is stored.
*
            DO 40 J = 1, N
               CJ = S( J )
               DO 30 I = J, MIN( N, J+KD )
                  AB( 1+I-J, J ) = CJ*S( I )*AB( 1+I-J, J )
   30          CONTINUE
   40       CONTINUE
         END IF
         EQUED = 'Y'
      END IF
*
      RETURN
*
*     End of DLAQSB
*
      END
*> \brief \b DLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by sppequ.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQSP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqsp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqsp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqsp.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAQSP( UPLO, N, AP, S, SCOND, AMAX, EQUED )
*
*       .. Scalar Arguments ..
*       CHARACTER          EQUED, UPLO
*       INTEGER            N
*       DOUBLE PRECISION   AMAX, SCOND
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AP( * ), S( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAQSP equilibrates a symmetric matrix A using the scaling factors
*> in the vector S.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the upper or lower triangular part of the
*>          symmetric matrix A is stored.
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the symmetric matrix
*>          A, packed columnwise in a linear array.  The j-th column of A
*>          is stored in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*>
*>          On exit, the equilibrated matrix:  diag(S) * A * diag(S), in
*>          the same storage format as A.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (N)
*>          The scale factors for A.
*> \endverbatim
*>
*> \param[in] SCOND
*> \verbatim
*>          SCOND is DOUBLE PRECISION
*>          Ratio of the smallest S(i) to the largest S(i).
*> \endverbatim
*>
*> \param[in] AMAX
*> \verbatim
*>          AMAX is DOUBLE PRECISION
*>          Absolute value of largest matrix entry.
*> \endverbatim
*>
*> \param[out] EQUED
*> \verbatim
*>          EQUED is CHARACTER*1
*>          Specifies whether or not equilibration was done.
*>          = 'N':  No equilibration.
*>          = 'Y':  Equilibration was done, i.e., A has been replaced by
*>                  diag(S) * A * diag(S).
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  THRESH is a threshold value used to decide if scaling should be done
*>  based on the ratio of the scaling factors.  If SCOND < THRESH,
*>  scaling is done.
*>
*>  LARGE and SMALL are threshold values used to decide if scaling should
*>  be done based on the absolute size of the largest matrix element.
*>  If AMAX > LARGE or AMAX < SMALL, scaling is done.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqhp
*
*  =====================================================================
      SUBROUTINE DLAQSP( UPLO, N, AP, S, SCOND, AMAX, EQUED )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          EQUED, UPLO
      INTEGER            N
      DOUBLE PRECISION   AMAX, SCOND
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( * ), S( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, THRESH
      PARAMETER          ( ONE = 1.0D+0, THRESH = 0.1D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J, JC
      DOUBLE PRECISION   CJ, LARGE, SMALL
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( N.LE.0 ) THEN
         EQUED = 'N'
         RETURN
      END IF
*
*     Initialize LARGE and SMALL.
*
      SMALL = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
      LARGE = ONE / SMALL
*
      IF( SCOND.GE.THRESH .AND. AMAX.GE.SMALL .AND. AMAX.LE.LARGE ) THEN
*
*        No equilibration
*
         EQUED = 'N'
      ELSE
*
*        Replace A by diag(S) * A * diag(S).
*
         IF( LSAME( UPLO, 'U' ) ) THEN
*
*           Upper triangle of A is stored.
*
            JC = 1
            DO 20 J = 1, N
               CJ = S( J )
               DO 10 I = 1, J
                  AP( JC+I-1 ) = CJ*S( I )*AP( JC+I-1 )
   10          CONTINUE
               JC = JC + J
   20       CONTINUE
         ELSE
*
*           Lower triangle of A is stored.
*
            JC = 1
            DO 40 J = 1, N
               CJ = S( J )
               DO 30 I = J, N
                  AP( JC+I-J ) = CJ*S( I )*AP( JC+I-J )
   30          CONTINUE
               JC = JC + N - J + 1
   40       CONTINUE
         END IF
         EQUED = 'Y'
      END IF
*
      RETURN
*
*     End of DLAQSP
*
      END
*> \brief \b DLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQSY + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqsy.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqsy.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqsy.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
*
*       .. Scalar Arguments ..
*       CHARACTER          EQUED, UPLO
*       INTEGER            LDA, N
*       DOUBLE PRECISION   AMAX, SCOND
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), S( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAQSY equilibrates a symmetric matrix A using the scaling factors
*> in the vector S.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the upper or lower triangular part of the
*>          symmetric matrix A is stored.
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*>          n by n upper triangular part of A contains the upper
*>          triangular part of the matrix A, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading n by n lower triangular part of A contains the lower
*>          triangular part of the matrix A, and the strictly upper
*>          triangular part of A is not referenced.
*>
*>          On exit, if EQUED = 'Y', the equilibrated matrix:
*>          diag(S) * A * diag(S).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(N,1).
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (N)
*>          The scale factors for A.
*> \endverbatim
*>
*> \param[in] SCOND
*> \verbatim
*>          SCOND is DOUBLE PRECISION
*>          Ratio of the smallest S(i) to the largest S(i).
*> \endverbatim
*>
*> \param[in] AMAX
*> \verbatim
*>          AMAX is DOUBLE PRECISION
*>          Absolute value of largest matrix entry.
*> \endverbatim
*>
*> \param[out] EQUED
*> \verbatim
*>          EQUED is CHARACTER*1
*>          Specifies whether or not equilibration was done.
*>          = 'N':  No equilibration.
*>          = 'Y':  Equilibration was done, i.e., A has been replaced by
*>                  diag(S) * A * diag(S).
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  THRESH is a threshold value used to decide if scaling should be done
*>  based on the ratio of the scaling factors.  If SCOND < THRESH,
*>  scaling is done.
*>
*>  LARGE and SMALL are threshold values used to decide if scaling should
*>  be done based on the absolute size of the largest matrix element.
*>  If AMAX > LARGE or AMAX < SMALL, scaling is done.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqhe
*
*  =====================================================================
      SUBROUTINE DLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          EQUED, UPLO
      INTEGER            LDA, N
      DOUBLE PRECISION   AMAX, SCOND
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), S( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, THRESH
      PARAMETER          ( ONE = 1.0D+0, THRESH = 0.1D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J
      DOUBLE PRECISION   CJ, LARGE, SMALL
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( N.LE.0 ) THEN
         EQUED = 'N'
         RETURN
      END IF
*
*     Initialize LARGE and SMALL.
*
      SMALL = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
      LARGE = ONE / SMALL
*
      IF( SCOND.GE.THRESH .AND. AMAX.GE.SMALL .AND. AMAX.LE.LARGE ) THEN
*
*        No equilibration
*
         EQUED = 'N'
      ELSE
*
*        Replace A by diag(S) * A * diag(S).
*
         IF( LSAME( UPLO, 'U' ) ) THEN
*
*           Upper triangle of A is stored.
*
            DO 20 J = 1, N
               CJ = S( J )
               DO 10 I = 1, J
                  A( I, J ) = CJ*S( I )*A( I, J )
   10          CONTINUE
   20       CONTINUE
         ELSE
*
*           Lower triangle of A is stored.
*
            DO 40 J = 1, N
               CJ = S( J )
               DO 30 I = J, N
                  A( I, J ) = CJ*S( I )*A( I, J )
   30          CONTINUE
   40       CONTINUE
         END IF
         EQUED = 'Y'
      END IF
*
      RETURN
*
*     End of DLAQSY
*
      END
*> \brief \b DLAQTR solves a real quasi-triangular system of equations, or a complex quasi-triangular system of special form, in real arithmetic.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQTR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqtr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqtr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqtr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAQTR( LTRAN, LREAL, N, T, LDT, B, W, SCALE, X, WORK,
*                          INFO )
*
*       .. Scalar Arguments ..
*       LOGICAL            LREAL, LTRAN
*       INTEGER            INFO, LDT, N
*       DOUBLE PRECISION   SCALE, W
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   B( * ), T( LDT, * ), WORK( * ), X( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAQTR solves the real quasi-triangular system
*>
*>              op(T)*p = scale*c,               if LREAL = .TRUE.
*>
*> or the complex quasi-triangular systems
*>
*>            op(T + iB)*(p+iq) = scale*(c+id),  if LREAL = .FALSE.
*>
*> in real arithmetic, where T is upper quasi-triangular.
*> If LREAL = .FALSE., then the first diagonal block of T must be
*> 1 by 1, B is the specially structured matrix
*>
*>                B = [ b(1) b(2) ... b(n) ]
*>                    [       w            ]
*>                    [           w        ]
*>                    [              .     ]
*>                    [                 w  ]
*>
*> op(A) = A or A**T, A**T denotes the transpose of
*> matrix A.
*>
*> On input, X = [ c ].  On output, X = [ p ].
*>               [ d ]                  [ q ]
*>
*> This subroutine is designed for the condition number estimation
*> in routine DTRSNA.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] LTRAN
*> \verbatim
*>          LTRAN is LOGICAL
*>          On entry, LTRAN specifies the option of conjugate transpose:
*>             = .FALSE.,    op(T+i*B) = T+i*B,
*>             = .TRUE.,     op(T+i*B) = (T+i*B)**T.
*> \endverbatim
*>
*> \param[in] LREAL
*> \verbatim
*>          LREAL is LOGICAL
*>          On entry, LREAL specifies the input matrix structure:
*>             = .FALSE.,    the input is complex
*>             = .TRUE.,     the input is real
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          On entry, N specifies the order of T+i*B. N >= 0.
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,N)
*>          On entry, T contains a matrix in Schur canonical form.
*>          If LREAL = .FALSE., then the first diagonal block of T mu
*>          be 1 by 1.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the matrix T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (N)
*>          On entry, B contains the elements to form the matrix
*>          B as described above.
*>          If LREAL = .TRUE., B is not referenced.
*> \endverbatim
*>
*> \param[in] W
*> \verbatim
*>          W is DOUBLE PRECISION
*>          On entry, W is the diagonal element of the matrix B.
*>          If LREAL = .TRUE., W is not referenced.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*>          SCALE is DOUBLE PRECISION
*>          On exit, SCALE is the scale factor.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (2*N)
*>          On entry, X contains the right hand side of the system.
*>          On exit, X is overwritten by the solution.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          On exit, INFO is set to
*>             0: successful exit.
*>               1: the some diagonal 1 by 1 block has been perturbed by
*>                  a small number SMIN to keep nonsingularity.
*>               2: the some diagonal 2 by 2 block has been perturbed by
*>                  a small number in DLALN2 to keep nonsingularity.
*>          NOTE: In the interests of speed, this routine does not
*>                check the inputs for errors.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqtr
*
*  =====================================================================
      SUBROUTINE DLAQTR( LTRAN, LREAL, N, T, LDT, B, W, SCALE, X,
     $                   WORK,
     $                   INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      LOGICAL            LREAL, LTRAN
      INTEGER            INFO, LDT, N
      DOUBLE PRECISION   SCALE, W
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   B( * ), T( LDT, * ), WORK( * ), X( * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOTRAN
      INTEGER            I, IERR, J, J1, J2, JNEXT, K, N1, N2
      DOUBLE PRECISION   BIGNUM, EPS, REC, SCALOC, SI, SMIN, SMINW,
     $                   SMLNUM, SR, TJJ, TMP, XJ, XMAX, XNORM, Z
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   D( 2, 2 ), V( 2, 2 )
*     ..
*     .. External Functions ..
      INTEGER            IDAMAX
      DOUBLE PRECISION   DASUM, DDOT, DLAMCH, DLANGE
      EXTERNAL           IDAMAX, DASUM, DDOT, DLAMCH, DLANGE
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DLADIV, DLALN2, DSCAL
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. Executable Statements ..
*
*     Do not test the input parameters for errors
*
      NOTRAN = .NOT.LTRAN
      INFO = 0
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Set constants to control overflow
*
      EPS = DLAMCH( 'P' )
      SMLNUM = DLAMCH( 'S' ) / EPS
      BIGNUM = ONE / SMLNUM
*
      XNORM = DLANGE( 'M', N, N, T, LDT, D )
      IF( .NOT.LREAL )
     $   XNORM = MAX( XNORM, ABS( W ), DLANGE( 'M', N, 1, B, N, D ) )
      SMIN = MAX( SMLNUM, EPS*XNORM )
*
*     Compute 1-norm of each column of strictly upper triangular
*     part of T to control overflow in triangular solver.
*
      WORK( 1 ) = ZERO
      DO 10 J = 2, N
         WORK( J ) = DASUM( J-1, T( 1, J ), 1 )
   10 CONTINUE
*
      IF( .NOT.LREAL ) THEN
         DO 20 I = 2, N
            WORK( I ) = WORK( I ) + ABS( B( I ) )
   20    CONTINUE
      END IF
*
      N2 = 2*N
      N1 = N
      IF( .NOT.LREAL )
     $   N1 = N2
      K = IDAMAX( N1, X, 1 )
      XMAX = ABS( X( K ) )
      SCALE = ONE
*
      IF( XMAX.GT.BIGNUM ) THEN
         SCALE = BIGNUM / XMAX
         CALL DSCAL( N1, SCALE, X, 1 )
         XMAX = BIGNUM
      END IF
*
      IF( LREAL ) THEN
*
         IF( NOTRAN ) THEN
*
*           Solve T*p = scale*c
*
            JNEXT = N
            DO 30 J = N, 1, -1
               IF( J.GT.JNEXT )
     $            GO TO 30
               J1 = J
               J2 = J
               JNEXT = J - 1
               IF( J.GT.1 ) THEN
                  IF( T( J, J-1 ).NE.ZERO ) THEN
                     J1 = J - 1
                     JNEXT = J - 2
                  END IF
               END IF
*
               IF( J1.EQ.J2 ) THEN
*
*                 Meet 1 by 1 diagonal block
*
*                 Scale to avoid overflow when computing
*                     x(j) = b(j)/T(j,j)
*
                  XJ = ABS( X( J1 ) )
                  TJJ = ABS( T( J1, J1 ) )
                  TMP = T( J1, J1 )
                  IF( TJJ.LT.SMIN ) THEN
                     TMP = SMIN
                     TJJ = SMIN
                     INFO = 1
                  END IF
*
                  IF( XJ.EQ.ZERO )
     $               GO TO 30
*
                  IF( TJJ.LT.ONE ) THEN
                     IF( XJ.GT.BIGNUM*TJJ ) THEN
                        REC = ONE / XJ
                        CALL DSCAL( N, REC, X, 1 )
                        SCALE = SCALE*REC
                        XMAX = XMAX*REC
                     END IF
                  END IF
                  X( J1 ) = X( J1 ) / TMP
                  XJ = ABS( X( J1 ) )
*
*                 Scale x if necessary to avoid overflow when adding a
*                 multiple of column j1 of T.
*
                  IF( XJ.GT.ONE ) THEN
                     REC = ONE / XJ
                     IF( WORK( J1 ).GT.( BIGNUM-XMAX )*REC ) THEN
                        CALL DSCAL( N, REC, X, 1 )
                        SCALE = SCALE*REC
                     END IF
                  END IF
                  IF( J1.GT.1 ) THEN
                     CALL DAXPY( J1-1, -X( J1 ), T( 1, J1 ), 1, X,
     $                           1 )
                     K = IDAMAX( J1-1, X, 1 )
                     XMAX = ABS( X( K ) )
                  END IF
*
               ELSE
*
*                 Meet 2 by 2 diagonal block
*
*                 Call 2 by 2 linear system solve, to take
*                 care of possible overflow by scaling factor.
*
                  D( 1, 1 ) = X( J1 )
                  D( 2, 1 ) = X( J2 )
                  CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, T( J1, J1 ),
     $                         LDT, ONE, ONE, D, 2, ZERO, ZERO, V, 2,
     $                         SCALOC, XNORM, IERR )
                  IF( IERR.NE.0 )
     $               INFO = 2
*
                  IF( SCALOC.NE.ONE ) THEN
                     CALL DSCAL( N, SCALOC, X, 1 )
                     SCALE = SCALE*SCALOC
                  END IF
                  X( J1 ) = V( 1, 1 )
                  X( J2 ) = V( 2, 1 )
*
*                 Scale V(1,1) (= X(J1)) and/or V(2,1) (=X(J2))
*                 to avoid overflow in updating right-hand side.
*
                  XJ = MAX( ABS( V( 1, 1 ) ), ABS( V( 2, 1 ) ) )
                  IF( XJ.GT.ONE ) THEN
                     REC = ONE / XJ
                     IF( MAX( WORK( J1 ), WORK( J2 ) ).GT.
     $                   ( BIGNUM-XMAX )*REC ) THEN
                        CALL DSCAL( N, REC, X, 1 )
                        SCALE = SCALE*REC
                     END IF
                  END IF
*
*                 Update right-hand side
*
                  IF( J1.GT.1 ) THEN
                     CALL DAXPY( J1-1, -X( J1 ), T( 1, J1 ), 1, X,
     $                           1 )
                     CALL DAXPY( J1-1, -X( J2 ), T( 1, J2 ), 1, X,
     $                           1 )
                     K = IDAMAX( J1-1, X, 1 )
                     XMAX = ABS( X( K ) )
                  END IF
*
               END IF
*
   30       CONTINUE
*
         ELSE
*
*           Solve T**T*p = scale*c
*
            JNEXT = 1
            DO 40 J = 1, N
               IF( J.LT.JNEXT )
     $            GO TO 40
               J1 = J
               J2 = J
               JNEXT = J + 1
               IF( J.LT.N ) THEN
                  IF( T( J+1, J ).NE.ZERO ) THEN
                     J2 = J + 1
                     JNEXT = J + 2
                  END IF
               END IF
*
               IF( J1.EQ.J2 ) THEN
*
*                 1 by 1 diagonal block
*
*                 Scale if necessary to avoid overflow in forming the
*                 right-hand side element by inner product.
*
                  XJ = ABS( X( J1 ) )
                  IF( XMAX.GT.ONE ) THEN
                     REC = ONE / XMAX
                     IF( WORK( J1 ).GT.( BIGNUM-XJ )*REC ) THEN
                        CALL DSCAL( N, REC, X, 1 )
                        SCALE = SCALE*REC
                        XMAX = XMAX*REC
                     END IF
                  END IF
*
                  X( J1 ) = X( J1 ) - DDOT( J1-1, T( 1, J1 ), 1, X,
     $               1 )
*
                  XJ = ABS( X( J1 ) )
                  TJJ = ABS( T( J1, J1 ) )
                  TMP = T( J1, J1 )
                  IF( TJJ.LT.SMIN ) THEN
                     TMP = SMIN
                     TJJ = SMIN
                     INFO = 1
                  END IF
*
                  IF( TJJ.LT.ONE ) THEN
                     IF( XJ.GT.BIGNUM*TJJ ) THEN
                        REC = ONE / XJ
                        CALL DSCAL( N, REC, X, 1 )
                        SCALE = SCALE*REC
                        XMAX = XMAX*REC
                     END IF
                  END IF
                  X( J1 ) = X( J1 ) / TMP
                  XMAX = MAX( XMAX, ABS( X( J1 ) ) )
*
               ELSE
*
*                 2 by 2 diagonal block
*
*                 Scale if necessary to avoid overflow in forming the
*                 right-hand side elements by inner product.
*
                  XJ = MAX( ABS( X( J1 ) ), ABS( X( J2 ) ) )
                  IF( XMAX.GT.ONE ) THEN
                     REC = ONE / XMAX
                     IF( MAX( WORK( J2 ), WORK( J1 ) ).GT.( BIGNUM-XJ )*
     $                   REC ) THEN
                        CALL DSCAL( N, REC, X, 1 )
                        SCALE = SCALE*REC
                        XMAX = XMAX*REC
                     END IF
                  END IF
*
                  D( 1, 1 ) = X( J1 ) - DDOT( J1-1, T( 1, J1 ), 1, X,
     $                        1 )
                  D( 2, 1 ) = X( J2 ) - DDOT( J1-1, T( 1, J2 ), 1, X,
     $                        1 )
*
                  CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, T( J1, J1 ),
     $                         LDT, ONE, ONE, D, 2, ZERO, ZERO, V, 2,
     $                         SCALOC, XNORM, IERR )
                  IF( IERR.NE.0 )
     $               INFO = 2
*
                  IF( SCALOC.NE.ONE ) THEN
                     CALL DSCAL( N, SCALOC, X, 1 )
                     SCALE = SCALE*SCALOC
                  END IF
                  X( J1 ) = V( 1, 1 )
                  X( J2 ) = V( 2, 1 )
                  XMAX = MAX( ABS( X( J1 ) ), ABS( X( J2 ) ), XMAX )
*
               END IF
   40       CONTINUE
         END IF
*
      ELSE
*
         SMINW = MAX( EPS*ABS( W ), SMIN )
         IF( NOTRAN ) THEN
*
*           Solve (T + iB)*(p+iq) = c+id
*
            JNEXT = N
            DO 70 J = N, 1, -1
               IF( J.GT.JNEXT )
     $            GO TO 70
               J1 = J
               J2 = J
               JNEXT = J - 1
               IF( J.GT.1 ) THEN
                  IF( T( J, J-1 ).NE.ZERO ) THEN
                     J1 = J - 1
                     JNEXT = J - 2
                  END IF
               END IF
*
               IF( J1.EQ.J2 ) THEN
*
*                 1 by 1 diagonal block
*
*                 Scale if necessary to avoid overflow in division
*
                  Z = W
                  IF( J1.EQ.1 )
     $               Z = B( 1 )
                  XJ = ABS( X( J1 ) ) + ABS( X( N+J1 ) )
                  TJJ = ABS( T( J1, J1 ) ) + ABS( Z )
                  TMP = T( J1, J1 )
                  IF( TJJ.LT.SMINW ) THEN
                     TMP = SMINW
                     TJJ = SMINW
                     INFO = 1
                  END IF
*
                  IF( XJ.EQ.ZERO )
     $               GO TO 70
*
                  IF( TJJ.LT.ONE ) THEN
                     IF( XJ.GT.BIGNUM*TJJ ) THEN
                        REC = ONE / XJ
                        CALL DSCAL( N2, REC, X, 1 )
                        SCALE = SCALE*REC
                        XMAX = XMAX*REC
                     END IF
                  END IF
                  CALL DLADIV( X( J1 ), X( N+J1 ), TMP, Z, SR, SI )
                  X( J1 ) = SR
                  X( N+J1 ) = SI
                  XJ = ABS( X( J1 ) ) + ABS( X( N+J1 ) )
*
*                 Scale x if necessary to avoid overflow when adding a
*                 multiple of column j1 of T.
*
                  IF( XJ.GT.ONE ) THEN
                     REC = ONE / XJ
                     IF( WORK( J1 ).GT.( BIGNUM-XMAX )*REC ) THEN
                        CALL DSCAL( N2, REC, X, 1 )
                        SCALE = SCALE*REC
                     END IF
                  END IF
*
                  IF( J1.GT.1 ) THEN
                     CALL DAXPY( J1-1, -X( J1 ), T( 1, J1 ), 1, X,
     $                           1 )
                     CALL DAXPY( J1-1, -X( N+J1 ), T( 1, J1 ), 1,
     $                           X( N+1 ), 1 )
*
                     X( 1 ) = X( 1 ) + B( J1 )*X( N+J1 )
                     X( N+1 ) = X( N+1 ) - B( J1 )*X( J1 )
*
                     XMAX = ZERO
                     DO 50 K = 1, J1 - 1
                        XMAX = MAX( XMAX, ABS( X( K ) )+
     $                         ABS( X( K+N ) ) )
   50                CONTINUE
                  END IF
*
               ELSE
*
*                 Meet 2 by 2 diagonal block
*
                  D( 1, 1 ) = X( J1 )
                  D( 2, 1 ) = X( J2 )
                  D( 1, 2 ) = X( N+J1 )
                  D( 2, 2 ) = X( N+J2 )
                  CALL DLALN2( .FALSE., 2, 2, SMINW, ONE, T( J1,
     $                         J1 ),
     $                         LDT, ONE, ONE, D, 2, ZERO, -W, V, 2,
     $                         SCALOC, XNORM, IERR )
                  IF( IERR.NE.0 )
     $               INFO = 2
*
                  IF( SCALOC.NE.ONE ) THEN
                     CALL DSCAL( 2*N, SCALOC, X, 1 )
                     SCALE = SCALOC*SCALE
                  END IF
                  X( J1 ) = V( 1, 1 )
                  X( J2 ) = V( 2, 1 )
                  X( N+J1 ) = V( 1, 2 )
                  X( N+J2 ) = V( 2, 2 )
*
*                 Scale X(J1), .... to avoid overflow in
*                 updating right hand side.
*
                  XJ = MAX( ABS( V( 1, 1 ) )+ABS( V( 1, 2 ) ),
     $                 ABS( V( 2, 1 ) )+ABS( V( 2, 2 ) ) )
                  IF( XJ.GT.ONE ) THEN
                     REC = ONE / XJ
                     IF( MAX( WORK( J1 ), WORK( J2 ) ).GT.
     $                   ( BIGNUM-XMAX )*REC ) THEN
                        CALL DSCAL( N2, REC, X, 1 )
                        SCALE = SCALE*REC
                     END IF
                  END IF
*
*                 Update the right-hand side.
*
                  IF( J1.GT.1 ) THEN
                     CALL DAXPY( J1-1, -X( J1 ), T( 1, J1 ), 1, X,
     $                           1 )
                     CALL DAXPY( J1-1, -X( J2 ), T( 1, J2 ), 1, X,
     $                           1 )
*
                     CALL DAXPY( J1-1, -X( N+J1 ), T( 1, J1 ), 1,
     $                           X( N+1 ), 1 )
                     CALL DAXPY( J1-1, -X( N+J2 ), T( 1, J2 ), 1,
     $                           X( N+1 ), 1 )
*
                     X( 1 ) = X( 1 ) + B( J1 )*X( N+J1 ) +
     $                        B( J2 )*X( N+J2 )
                     X( N+1 ) = X( N+1 ) - B( J1 )*X( J1 ) -
     $                          B( J2 )*X( J2 )
*
                     XMAX = ZERO
                     DO 60 K = 1, J1 - 1
                        XMAX = MAX( ABS( X( K ) )+ABS( X( K+N ) ),
     $                         XMAX )
   60                CONTINUE
                  END IF
*
               END IF
   70       CONTINUE
*
         ELSE
*
*           Solve (T + iB)**T*(p+iq) = c+id
*
            JNEXT = 1
            DO 80 J = 1, N
               IF( J.LT.JNEXT )
     $            GO TO 80
               J1 = J
               J2 = J
               JNEXT = J + 1
               IF( J.LT.N ) THEN
                  IF( T( J+1, J ).NE.ZERO ) THEN
                     J2 = J + 1
                     JNEXT = J + 2
                  END IF
               END IF
*
               IF( J1.EQ.J2 ) THEN
*
*                 1 by 1 diagonal block
*
*                 Scale if necessary to avoid overflow in forming the
*                 right-hand side element by inner product.
*
                  XJ = ABS( X( J1 ) ) + ABS( X( J1+N ) )
                  IF( XMAX.GT.ONE ) THEN
                     REC = ONE / XMAX
                     IF( WORK( J1 ).GT.( BIGNUM-XJ )*REC ) THEN
                        CALL DSCAL( N2, REC, X, 1 )
                        SCALE = SCALE*REC
                        XMAX = XMAX*REC
                     END IF
                  END IF
*
                  X( J1 ) = X( J1 ) - DDOT( J1-1, T( 1, J1 ), 1, X,
     $               1 )
                  X( N+J1 ) = X( N+J1 ) - DDOT( J1-1, T( 1, J1 ), 1,
     $                        X( N+1 ), 1 )
                  IF( J1.GT.1 ) THEN
                     X( J1 ) = X( J1 ) - B( J1 )*X( N+1 )
                     X( N+J1 ) = X( N+J1 ) + B( J1 )*X( 1 )
                  END IF
                  XJ = ABS( X( J1 ) ) + ABS( X( J1+N ) )
*
                  Z = W
                  IF( J1.EQ.1 )
     $               Z = B( 1 )
*
*                 Scale if necessary to avoid overflow in
*                 complex division
*
                  TJJ = ABS( T( J1, J1 ) ) + ABS( Z )
                  TMP = T( J1, J1 )
                  IF( TJJ.LT.SMINW ) THEN
                     TMP = SMINW
                     TJJ = SMINW
                     INFO = 1
                  END IF
*
                  IF( TJJ.LT.ONE ) THEN
                     IF( XJ.GT.BIGNUM*TJJ ) THEN
                        REC = ONE / XJ
                        CALL DSCAL( N2, REC, X, 1 )
                        SCALE = SCALE*REC
                        XMAX = XMAX*REC
                     END IF
                  END IF
                  CALL DLADIV( X( J1 ), X( N+J1 ), TMP, -Z, SR, SI )
                  X( J1 ) = SR
                  X( J1+N ) = SI
                  XMAX = MAX( ABS( X( J1 ) )+ABS( X( J1+N ) ), XMAX )
*
               ELSE
*
*                 2 by 2 diagonal block
*
*                 Scale if necessary to avoid overflow in forming the
*                 right-hand side element by inner product.
*
                  XJ = MAX( ABS( X( J1 ) )+ABS( X( N+J1 ) ),
     $                 ABS( X( J2 ) )+ABS( X( N+J2 ) ) )
                  IF( XMAX.GT.ONE ) THEN
                     REC = ONE / XMAX
                     IF( MAX( WORK( J1 ), WORK( J2 ) ).GT.
     $                   ( BIGNUM-XJ ) / XMAX ) THEN
                        CALL DSCAL( N2, REC, X, 1 )
                        SCALE = SCALE*REC
                        XMAX = XMAX*REC
                     END IF
                  END IF
*
                  D( 1, 1 ) = X( J1 ) - DDOT( J1-1, T( 1, J1 ), 1, X,
     $                        1 )
                  D( 2, 1 ) = X( J2 ) - DDOT( J1-1, T( 1, J2 ), 1, X,
     $                        1 )
                  D( 1, 2 ) = X( N+J1 ) - DDOT( J1-1, T( 1, J1 ), 1,
     $                        X( N+1 ), 1 )
                  D( 2, 2 ) = X( N+J2 ) - DDOT( J1-1, T( 1, J2 ), 1,
     $                        X( N+1 ), 1 )
                  D( 1, 1 ) = D( 1, 1 ) - B( J1 )*X( N+1 )
                  D( 2, 1 ) = D( 2, 1 ) - B( J2 )*X( N+1 )
                  D( 1, 2 ) = D( 1, 2 ) + B( J1 )*X( 1 )
                  D( 2, 2 ) = D( 2, 2 ) + B( J2 )*X( 1 )
*
                  CALL DLALN2( .TRUE., 2, 2, SMINW, ONE, T( J1, J1 ),
     $                         LDT, ONE, ONE, D, 2, ZERO, W, V, 2,
     $                         SCALOC, XNORM, IERR )
                  IF( IERR.NE.0 )
     $               INFO = 2
*
                  IF( SCALOC.NE.ONE ) THEN
                     CALL DSCAL( N2, SCALOC, X, 1 )
                     SCALE = SCALOC*SCALE
                  END IF
                  X( J1 ) = V( 1, 1 )
                  X( J2 ) = V( 2, 1 )
                  X( N+J1 ) = V( 1, 2 )
                  X( N+J2 ) = V( 2, 2 )
                  XMAX = MAX( ABS( X( J1 ) )+ABS( X( N+J1 ) ),
     $                   ABS( X( J2 ) )+ABS( X( N+J2 ) ), XMAX )
*
               END IF
*
   80       CONTINUE
*
         END IF
*
      END IF
*
      RETURN
*
*     End of DLAQTR
*
      END
*> \brief \b DLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAR1V + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlar1v.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlar1v.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlar1v.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAR1V( N, B1, BN, LAMBDA, D, L, LD, LLD,
*                  PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA,
*                  R, ISUPPZ, NRMINV, RESID, RQCORR, WORK )
*
*       .. Scalar Arguments ..
*       LOGICAL            WANTNC
*       INTEGER   B1, BN, N, NEGCNT, R
*       DOUBLE PRECISION   GAPTOL, LAMBDA, MINGMA, NRMINV, PIVMIN, RESID,
*      $                   RQCORR, ZTZ
*       ..
*       .. Array Arguments ..
*       INTEGER            ISUPPZ( * )
*       DOUBLE PRECISION   D( * ), L( * ), LD( * ), LLD( * ),
*      $                  WORK( * )
*       DOUBLE PRECISION Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAR1V computes the (scaled) r-th column of the inverse of
*> the sumbmatrix in rows B1 through BN of the tridiagonal matrix
*> L D L**T - sigma I. When sigma is close to an eigenvalue, the
*> computed vector is an accurate eigenvector. Usually, r corresponds
*> to the index where the eigenvector is largest in magnitude.
*> The following steps accomplish this computation :
*> (a) Stationary qd transform,  L D L**T - sigma I = L(+) D(+) L(+)**T,
*> (b) Progressive qd transform, L D L**T - sigma I = U(-) D(-) U(-)**T,
*> (c) Computation of the diagonal elements of the inverse of
*>     L D L**T - sigma I by combining the above transforms, and choosing
*>     r as the index where the diagonal of the inverse is (one of the)
*>     largest in magnitude.
*> (d) Computation of the (scaled) r-th column of the inverse using the
*>     twisted factorization obtained by combining the top part of the
*>     the stationary and the bottom part of the progressive transform.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>           The order of the matrix L D L**T.
*> \endverbatim
*>
*> \param[in] B1
*> \verbatim
*>          B1 is INTEGER
*>           First index of the submatrix of L D L**T.
*> \endverbatim
*>
*> \param[in] BN
*> \verbatim
*>          BN is INTEGER
*>           Last index of the submatrix of L D L**T.
*> \endverbatim
*>
*> \param[in] LAMBDA
*> \verbatim
*>          LAMBDA is DOUBLE PRECISION
*>           The shift. In order to compute an accurate eigenvector,
*>           LAMBDA should be a good approximation to an eigenvalue
*>           of L D L**T.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*>          L is DOUBLE PRECISION array, dimension (N-1)
*>           The (n-1) subdiagonal elements of the unit bidiagonal matrix
*>           L, in elements 1 to N-1.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>           The n diagonal elements of the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] LD
*> \verbatim
*>          LD is DOUBLE PRECISION array, dimension (N-1)
*>           The n-1 elements L(i)*D(i).
*> \endverbatim
*>
*> \param[in] LLD
*> \verbatim
*>          LLD is DOUBLE PRECISION array, dimension (N-1)
*>           The n-1 elements L(i)*L(i)*D(i).
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*>          PIVMIN is DOUBLE PRECISION
*>           The minimum pivot in the Sturm sequence.
*> \endverbatim
*>
*> \param[in] GAPTOL
*> \verbatim
*>          GAPTOL is DOUBLE PRECISION
*>           Tolerance that indicates when eigenvector entries are negligible
*>           w.r.t. their contribution to the residual.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (N)
*>           On input, all entries of Z must be set to 0.
*>           On output, Z contains the (scaled) r-th column of the
*>           inverse. The scaling is such that Z(R) equals 1.
*> \endverbatim
*>
*> \param[in] WANTNC
*> \verbatim
*>          WANTNC is LOGICAL
*>           Specifies whether NEGCNT has to be computed.
*> \endverbatim
*>
*> \param[out] NEGCNT
*> \verbatim
*>          NEGCNT is INTEGER
*>           If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin
*>           in the  matrix factorization L D L**T, and NEGCNT = -1 otherwise.
*> \endverbatim
*>
*> \param[out] ZTZ
*> \verbatim
*>          ZTZ is DOUBLE PRECISION
*>           The square of the 2-norm of Z.
*> \endverbatim
*>
*> \param[out] MINGMA
*> \verbatim
*>          MINGMA is DOUBLE PRECISION
*>           The reciprocal of the largest (in magnitude) diagonal
*>           element of the inverse of L D L**T - sigma I.
*> \endverbatim
*>
*> \param[in,out] R
*> \verbatim
*>          R is INTEGER
*>           The twist index for the twisted factorization used to
*>           compute Z.
*>           On input, 0 <= R <= N. If R is input as 0, R is set to
*>           the index where (L D L**T - sigma I)^{-1} is largest
*>           in magnitude. If 1 <= R <= N, R is unchanged.
*>           On output, R contains the twist index used to compute Z.
*>           Ideally, R designates the position of the maximum entry in the
*>           eigenvector.
*> \endverbatim
*>
*> \param[out] ISUPPZ
*> \verbatim
*>          ISUPPZ is INTEGER array, dimension (2)
*>           The support of the vector in Z, i.e., the vector Z is
*>           nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).
*> \endverbatim
*>
*> \param[out] NRMINV
*> \verbatim
*>          NRMINV is DOUBLE PRECISION
*>           NRMINV = 1/SQRT( ZTZ )
*> \endverbatim
*>
*> \param[out] RESID
*> \verbatim
*>          RESID is DOUBLE PRECISION
*>           The residual of the FP vector.
*>           RESID = ABS( MINGMA )/SQRT( ZTZ )
*> \endverbatim
*>
*> \param[out] RQCORR
*> \verbatim
*>          RQCORR is DOUBLE PRECISION
*>           The Rayleigh Quotient correction to LAMBDA.
*>           RQCORR = MINGMA*TMP
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (4*N)
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lar1v
*
*> \par Contributors:
*  ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA
*
*  =====================================================================
      SUBROUTINE DLAR1V( N, B1, BN, LAMBDA, D, L, LD, LLD,
     $           PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA,
     $           R, ISUPPZ, NRMINV, RESID, RQCORR, WORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      LOGICAL            WANTNC
      INTEGER   B1, BN, N, NEGCNT, R
      DOUBLE PRECISION   GAPTOL, LAMBDA, MINGMA, NRMINV, PIVMIN, RESID,
     $                   RQCORR, ZTZ
*     ..
*     .. Array Arguments ..
      INTEGER            ISUPPZ( * )
      DOUBLE PRECISION   D( * ), L( * ), LD( * ), LLD( * ),
     $                  WORK( * )
      DOUBLE PRECISION Z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )

*     ..
*     .. Local Scalars ..
      LOGICAL            SAWNAN1, SAWNAN2
      INTEGER            I, INDLPL, INDP, INDS, INDUMN, NEG1, NEG2, R1,
     $                   R2
      DOUBLE PRECISION   DMINUS, DPLUS, EPS, S, TMP
*     ..
*     .. External Functions ..
      LOGICAL DISNAN
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DISNAN, DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS
*     ..
*     .. Executable Statements ..
*
      EPS = DLAMCH( 'Precision' )


      IF( R.EQ.0 ) THEN
         R1 = B1
         R2 = BN
      ELSE
         R1 = R
         R2 = R
      END IF

*     Storage for LPLUS
      INDLPL = 0
*     Storage for UMINUS
      INDUMN = N
      INDS = 2*N + 1
      INDP = 3*N + 1

      IF( B1.EQ.1 ) THEN
         WORK( INDS ) = ZERO
      ELSE
         WORK( INDS+B1-1 ) = LLD( B1-1 )
      END IF

*
*     Compute the stationary transform (using the differential form)
*     until the index R2.
*
      SAWNAN1 = .FALSE.
      NEG1 = 0
      S = WORK( INDS+B1-1 ) - LAMBDA
      DO 50 I = B1, R1 - 1
         DPLUS = D( I ) + S
         WORK( INDLPL+I ) = LD( I ) / DPLUS
         IF(DPLUS.LT.ZERO) NEG1 = NEG1 + 1
         WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I )
         S = WORK( INDS+I ) - LAMBDA
 50   CONTINUE
      SAWNAN1 = DISNAN( S )
      IF( SAWNAN1 ) GOTO 60
      DO 51 I = R1, R2 - 1
         DPLUS = D( I ) + S
         WORK( INDLPL+I ) = LD( I ) / DPLUS
         WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I )
         S = WORK( INDS+I ) - LAMBDA
 51   CONTINUE
      SAWNAN1 = DISNAN( S )
*
 60   CONTINUE
      IF( SAWNAN1 ) THEN
*        Runs a slower version of the above loop if a NaN is detected
         NEG1 = 0
         S = WORK( INDS+B1-1 ) - LAMBDA
         DO 70 I = B1, R1 - 1
            DPLUS = D( I ) + S
            IF(ABS(DPLUS).LT.PIVMIN) DPLUS = -PIVMIN
            WORK( INDLPL+I ) = LD( I ) / DPLUS
            IF(DPLUS.LT.ZERO) NEG1 = NEG1 + 1
            WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I )
            IF( WORK( INDLPL+I ).EQ.ZERO )
     $                      WORK( INDS+I ) = LLD( I )
            S = WORK( INDS+I ) - LAMBDA
 70      CONTINUE
         DO 71 I = R1, R2 - 1
            DPLUS = D( I ) + S
            IF(ABS(DPLUS).LT.PIVMIN) DPLUS = -PIVMIN
            WORK( INDLPL+I ) = LD( I ) / DPLUS
            WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I )
            IF( WORK( INDLPL+I ).EQ.ZERO )
     $                      WORK( INDS+I ) = LLD( I )
            S = WORK( INDS+I ) - LAMBDA
 71      CONTINUE
      END IF
*
*     Compute the progressive transform (using the differential form)
*     until the index R1
*
      SAWNAN2 = .FALSE.
      NEG2 = 0
      WORK( INDP+BN-1 ) = D( BN ) - LAMBDA
      DO 80 I = BN - 1, R1, -1
         DMINUS = LLD( I ) + WORK( INDP+I )
         TMP = D( I ) / DMINUS
         IF(DMINUS.LT.ZERO) NEG2 = NEG2 + 1
         WORK( INDUMN+I ) = L( I )*TMP
         WORK( INDP+I-1 ) = WORK( INDP+I )*TMP - LAMBDA
 80   CONTINUE
      TMP = WORK( INDP+R1-1 )
      SAWNAN2 = DISNAN( TMP )

      IF( SAWNAN2 ) THEN
*        Runs a slower version of the above loop if a NaN is detected
         NEG2 = 0
         DO 100 I = BN-1, R1, -1
            DMINUS = LLD( I ) + WORK( INDP+I )
            IF(ABS(DMINUS).LT.PIVMIN) DMINUS = -PIVMIN
            TMP = D( I ) / DMINUS
            IF(DMINUS.LT.ZERO) NEG2 = NEG2 + 1
            WORK( INDUMN+I ) = L( I )*TMP
            WORK( INDP+I-1 ) = WORK( INDP+I )*TMP - LAMBDA
            IF( TMP.EQ.ZERO )
     $          WORK( INDP+I-1 ) = D( I ) - LAMBDA
 100     CONTINUE
      END IF
*
*     Find the index (from R1 to R2) of the largest (in magnitude)
*     diagonal element of the inverse
*
      MINGMA = WORK( INDS+R1-1 ) + WORK( INDP+R1-1 )
      IF( MINGMA.LT.ZERO ) NEG1 = NEG1 + 1
      IF( WANTNC ) THEN
         NEGCNT = NEG1 + NEG2
      ELSE
         NEGCNT = -1
      ENDIF
      IF( ABS(MINGMA).EQ.ZERO )
     $   MINGMA = EPS*WORK( INDS+R1-1 )
      R = R1
      DO 110 I = R1, R2 - 1
         TMP = WORK( INDS+I ) + WORK( INDP+I )
         IF( TMP.EQ.ZERO )
     $      TMP = EPS*WORK( INDS+I )
         IF( ABS( TMP ).LE.ABS( MINGMA ) ) THEN
            MINGMA = TMP
            R = I + 1
         END IF
 110  CONTINUE
*
*     Compute the FP vector: solve N^T v = e_r
*
      ISUPPZ( 1 ) = B1
      ISUPPZ( 2 ) = BN
      Z( R ) = ONE
      ZTZ = ONE
*
*     Compute the FP vector upwards from R
*
      IF( .NOT.SAWNAN1 .AND. .NOT.SAWNAN2 ) THEN
         DO 210 I = R-1, B1, -1
            Z( I ) = -( WORK( INDLPL+I )*Z( I+1 ) )
            IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I)).LT.GAPTOL )
     $           THEN
               Z( I ) = ZERO
               ISUPPZ( 1 ) = I + 1
               GOTO 220
            ENDIF
            ZTZ = ZTZ + Z( I )*Z( I )
 210     CONTINUE
 220     CONTINUE
      ELSE
*        Run slower loop if NaN occurred.
         DO 230 I = R - 1, B1, -1
            IF( Z( I+1 ).EQ.ZERO ) THEN
               Z( I ) = -( LD( I+1 ) / LD( I ) )*Z( I+2 )
            ELSE
               Z( I ) = -( WORK( INDLPL+I )*Z( I+1 ) )
            END IF
            IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I)).LT.GAPTOL )
     $           THEN
               Z( I ) = ZERO
               ISUPPZ( 1 ) = I + 1
               GO TO 240
            END IF
            ZTZ = ZTZ + Z( I )*Z( I )
 230     CONTINUE
 240     CONTINUE
      ENDIF

*     Compute the FP vector downwards from R in blocks of size BLKSIZ
      IF( .NOT.SAWNAN1 .AND. .NOT.SAWNAN2 ) THEN
         DO 250 I = R, BN-1
            Z( I+1 ) = -( WORK( INDUMN+I )*Z( I ) )
            IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I)).LT.GAPTOL )
     $         THEN
               Z( I+1 ) = ZERO
               ISUPPZ( 2 ) = I
               GO TO 260
            END IF
            ZTZ = ZTZ + Z( I+1 )*Z( I+1 )
 250     CONTINUE
 260     CONTINUE
      ELSE
*        Run slower loop if NaN occurred.
         DO 270 I = R, BN - 1
            IF( Z( I ).EQ.ZERO ) THEN
               Z( I+1 ) = -( LD( I-1 ) / LD( I ) )*Z( I-1 )
            ELSE
               Z( I+1 ) = -( WORK( INDUMN+I )*Z( I ) )
            END IF
            IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I)).LT.GAPTOL )
     $           THEN
               Z( I+1 ) = ZERO
               ISUPPZ( 2 ) = I
               GO TO 280
            END IF
            ZTZ = ZTZ + Z( I+1 )*Z( I+1 )
 270     CONTINUE
 280     CONTINUE
      END IF
*
*     Compute quantities for convergence test
*
      TMP = ONE / ZTZ
      NRMINV = SQRT( TMP )
      RESID = ABS( MINGMA )*NRMINV
      RQCORR = MINGMA*TMP
*
*
      RETURN
*
*     End of DLAR1V
*
      END
*> \brief \b DLAR2V applies a vector of plane rotations with real cosines and real sines from both sides to a sequence of 2-by-2 symmetric/Hermitian matrices.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAR2V + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlar2v.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlar2v.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlar2v.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAR2V( N, X, Y, Z, INCX, C, S, INCC )
*
*       .. Scalar Arguments ..
*       INTEGER            INCC, INCX, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   C( * ), S( * ), X( * ), Y( * ), Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAR2V applies a vector of real plane rotations from both sides to
*> a sequence of 2-by-2 real symmetric matrices, defined by the elements
*> of the vectors x, y and z. For i = 1,2,...,n
*>
*>    ( x(i)  z(i) ) := (  c(i)  s(i) ) ( x(i)  z(i) ) ( c(i) -s(i) )
*>    ( z(i)  y(i) )    ( -s(i)  c(i) ) ( z(i)  y(i) ) ( s(i)  c(i) )
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of plane rotations to be applied.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array,
*>                         dimension (1+(N-1)*INCX)
*>          The vector x.
*> \endverbatim
*>
*> \param[in,out] Y
*> \verbatim
*>          Y is DOUBLE PRECISION array,
*>                         dimension (1+(N-1)*INCX)
*>          The vector y.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array,
*>                         dimension (1+(N-1)*INCX)
*>          The vector z.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*>          INCX is INTEGER
*>          The increment between elements of X, Y and Z. INCX > 0.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
*>          The cosines of the plane rotations.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
*>          The sines of the plane rotations.
*> \endverbatim
*>
*> \param[in] INCC
*> \verbatim
*>          INCC is INTEGER
*>          The increment between elements of C and S. INCC > 0.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lar2v
*
*  =====================================================================
      SUBROUTINE DLAR2V( N, X, Y, Z, INCX, C, S, INCC )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INCC, INCX, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   C( * ), S( * ), X( * ), Y( * ), Z( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            I, IC, IX
      DOUBLE PRECISION   CI, SI, T1, T2, T3, T4, T5, T6, XI, YI, ZI
*     ..
*     .. Executable Statements ..
*
      IX = 1
      IC = 1
      DO 10 I = 1, N
         XI = X( IX )
         YI = Y( IX )
         ZI = Z( IX )
         CI = C( IC )
         SI = S( IC )
         T1 = SI*ZI
         T2 = CI*ZI
         T3 = T2 - SI*XI
         T4 = T2 + SI*YI
         T5 = CI*XI + T1
         T6 = CI*YI - T1
         X( IX ) = CI*T5 + SI*T4
         Y( IX ) = CI*T6 - SI*T3
         Z( IX ) = CI*T4 - SI*T5
         IX = IX + INCX
         IC = IC + INCC
   10 CONTINUE
*
*     End of DLAR2V
*
      RETURN
      END
*> \brief \b DLARF applies an elementary reflector to a general rectangular matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARF( SIDE, M, N, V, INCV, TAU, C, LDC, WORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIDE
*       INTEGER            INCV, LDC, M, N
*       DOUBLE PRECISION   TAU
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   C( LDC, * ), V( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARF applies a real elementary reflector H to a real m by n matrix
*> C, from either the left or the right. H is represented in the form
*>
*>       H = I - tau * v * v**T
*>
*> where tau is a real scalar and v is a real vector.
*>
*> If tau = 0, then H is taken to be the unit matrix.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': form  H * C
*>          = 'R': form  C * H
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension
*>                     (1 + (M-1)*abs(INCV)) if SIDE = 'L'
*>                  or (1 + (N-1)*abs(INCV)) if SIDE = 'R'
*>          The vector v in the representation of H. V is not used if
*>          TAU = 0.
*> \endverbatim
*>
*> \param[in] INCV
*> \verbatim
*>          INCV is INTEGER
*>          The increment between elements of v. INCV <> 0.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION
*>          The value tau in the representation of H.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the m by n matrix C.
*>          On exit, C is overwritten by the matrix H * C if SIDE = 'L',
*>          or C * H if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension
*>                         (N) if SIDE = 'L'
*>                      or (M) if SIDE = 'R'
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larf
*
*  =====================================================================
      SUBROUTINE DLARF( SIDE, M, N, V, INCV, TAU, C, LDC, WORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE
      INTEGER            INCV, LDC, M, N
      DOUBLE PRECISION   TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   C( LDC, * ), V( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            APPLYLEFT
      INTEGER            I, LASTV, LASTC
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMV, DGER
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILADLR, ILADLC
      EXTERNAL           LSAME, ILADLR, ILADLC
*     ..
*     .. Executable Statements ..
*
      APPLYLEFT = LSAME( SIDE, 'L' )
      LASTV = 0
      LASTC = 0
      IF( TAU.NE.ZERO ) THEN
!     Set up variables for scanning V.  LASTV begins pointing to the end
!     of V.
         IF( APPLYLEFT ) THEN
            LASTV = M
         ELSE
            LASTV = N
         END IF
         IF( INCV.GT.0 ) THEN
            I = 1 + (LASTV-1) * INCV
         ELSE
            I = 1
         END IF
!     Look for the last non-zero row in V.
         DO WHILE( LASTV.GT.0 .AND. V( I ).EQ.ZERO )
            LASTV = LASTV - 1
            I = I - INCV
         END DO
         IF( APPLYLEFT ) THEN
!     Scan for the last non-zero column in C(1:lastv,:).
            LASTC = ILADLC(LASTV, N, C, LDC)
         ELSE
!     Scan for the last non-zero row in C(:,1:lastv).
            LASTC = ILADLR(M, LASTV, C, LDC)
         END IF
      END IF
!     Note that lastc.eq.0 renders the BLAS operations null; no special
!     case is needed at this level.
      IF( APPLYLEFT ) THEN
*
*        Form  H * C
*
         IF( LASTV.GT.0 ) THEN
*
*           w(1:lastc,1) := C(1:lastv,1:lastc)**T * v(1:lastv,1)
*
            CALL DGEMV( 'Transpose', LASTV, LASTC, ONE, C, LDC, V,
     $                  INCV,
     $           ZERO, WORK, 1 )
*
*           C(1:lastv,1:lastc) := C(...) - v(1:lastv,1) * w(1:lastc,1)**T
*
            CALL DGER( LASTV, LASTC, -TAU, V, INCV, WORK, 1, C, LDC )
         END IF
      ELSE
*
*        Form  C * H
*
         IF( LASTV.GT.0 ) THEN
*
*           w(1:lastc,1) := C(1:lastc,1:lastv) * v(1:lastv,1)
*
            CALL DGEMV( 'No transpose', LASTC, LASTV, ONE, C, LDC,
     $           V, INCV, ZERO, WORK, 1 )
*
*           C(1:lastc,1:lastv) := C(...) - w(1:lastc,1) * v(1:lastv,1)**T
*
            CALL DGER( LASTC, LASTV, -TAU, WORK, 1, V, INCV, C, LDC )
         END IF
      END IF
      RETURN
*
*     End of DLARF
*
      END
*> \brief \b DLARF1F applies an elementary reflector to a general rectangular
*              matrix assuming v(1) = 1.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARF1F + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarf1f.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarf1f.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarf1f.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARF1F( SIDE, M, N, V, INCV, TAU, C, LDC, WORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIDE
*       INTEGER            INCV, LDC, M, N
*       DOUBLE PRECISION   TAU
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   C( LDC, * ), V( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARF1F applies a real elementary reflector H to a real m by n matrix
*> C, from either the left or the right. H is represented in the form
*>
*>       H = I - tau * v * v**T
*>
*> where tau is a real scalar and v is a real vector.
*>
*> If tau = 0, then H is taken to be the unit matrix.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': form  H * C
*>          = 'R': form  C * H
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension
*>                     (1 + (M-1)*abs(INCV)) if SIDE = 'L'
*>                  or (1 + (N-1)*abs(INCV)) if SIDE = 'R'
*>          The vector v in the representation of H. V is not used if
*>          TAU = 0. V(1) is not referenced or modified.
*> \endverbatim
*>
*> \param[in] INCV
*> \verbatim
*>          INCV is INTEGER
*>          The increment between elements of v. INCV <> 0.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION
*>          The value tau in the representation of H.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the m by n matrix C.
*>          On exit, C is overwritten by the matrix H * C if SIDE = 'L',
*>          or C * H if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension
*>                         (N) if SIDE = 'L'
*>                      or (M) if SIDE = 'R'
*> \endverbatim
*
*  To take advantage of the fact that v(1) = 1, we do the following
*     v = [ 1 v_2 ]**T
*     If SIDE='L'
*           |-----|
*           | C_1 |
*        C =| C_2 |
*           |-----|
*        C_1\in\mathbb{R}^{1\times n}, C_2\in\mathbb{R}^{m-1\times n}
*        So we compute:
*        C = HC   = (I - \tau vv**T)C
*                 = C - \tau vv**T C
*        w = C**T v  = [ C_1**T C_2**T ] [ 1 v_2 ]**T
*                    = C_1**T + C_2**T v ( DGEMM then DAXPY )
*        C  = C - \tau vv**T C
*           = C - \tau vw**T
*        Giving us   C_1 = C_1 - \tau w**T ( DAXPY )
*                 and
*                    C_2 = C_2 - \tau v_2w**T ( DGER )
*     If SIDE='R'
*
*        C = [ C_1 C_2 ]
*        C_1\in\mathbb{R}^{m\times 1}, C_2\in\mathbb{R}^{m\times n-1}
*        So we compute: 
*        C = CH   = C(I - \tau vv**T)
*                 = C - \tau Cvv**T
*
*        w = Cv   = [ C_1 C_2 ] [ 1 v_2 ]**T
*                 = C_1 + C_2v_2 ( DGEMM then DAXPY )
*        C  = C - \tau Cvv**T
*           = C - \tau wv**T
*        Giving us   C_1 = C_1 - \tau w ( DAXPY )
*                 and
*                    C_2 = C_2 - \tau wv_2**T ( DGER )
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larf
*
*  =====================================================================
      SUBROUTINE DLARF1F( SIDE, M, N, V, INCV, TAU, C, LDC, WORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE
      INTEGER            INCV, LDC, M, N
      DOUBLE PRECISION   TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   C( LDC, * ), V( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            APPLYLEFT
      INTEGER            I, LASTV, LASTC
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMV, DGER, DAXPY, DSCAL
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILADLR, ILADLC
      EXTERNAL           LSAME, ILADLR, ILADLC
*     ..
*     .. Executable Statements ..
*
      APPLYLEFT = LSAME( SIDE, 'L' )
      LASTV = 1
      LASTC = 0
      IF( TAU.NE.ZERO ) THEN
!     Set up variables for scanning V.  LASTV begins pointing to the end
!     of V.
         IF( APPLYLEFT ) THEN
            LASTV = M
         ELSE
            LASTV = N
         END IF
         IF( INCV.GT.0 ) THEN
            I = 1 + (LASTV-1) * INCV
         ELSE
            I = 1
         END IF
!     Look for the last non-zero row in V.
!        Since we are assuming that V(1) = 1, and it is not stored, so we
!        shouldn't access it.
         DO WHILE( LASTV.GT.1 .AND. V( I ).EQ.ZERO )
            LASTV = LASTV - 1
            I = I - INCV
         END DO
         IF( APPLYLEFT ) THEN
!     Scan for the last non-zero column in C(1:lastv,:).
            LASTC = ILADLC(LASTV, N, C, LDC)
         ELSE
!     Scan for the last non-zero row in C(:,1:lastv).
            LASTC = ILADLR(M, LASTV, C, LDC)
         END IF
      END IF
      IF( LASTC.EQ.0 ) THEN
         RETURN
      END IF
      IF( APPLYLEFT ) THEN
*
*        Form  H * C
*
         ! Check if lastv = 1. This means v = 1, So we just need to compute
         ! C := HC = (1-\tau)C.
         IF( LASTV.EQ.1 ) THEN
*
*           C(1,1:lastc) := ( 1 - tau ) * C(1,1:lastc)
*
            CALL DSCAL(LASTC, ONE - TAU, C, LDC)
         ELSE
*
*           w(1:lastc,1) := C(1:lastv,1:lastc)**T * v(1:lastv,1)
*
            ! w(1:lastc,1) := C(2:lastv,1:lastc)**T * v(2:lastv,1)
            CALL DGEMV( 'Transpose', LASTV-1, LASTC, ONE, C(1+1,1),
     $                  LDC, V(1+INCV), INCV, ZERO, WORK, 1)
            ! w(1:lastc,1) += C(1,1:lastc)**T * v(1,1) = C(1,1:lastc)**T
            CALL DAXPY(LASTC, ONE, C, LDC, WORK, 1)
*
*           C(1:lastv,1:lastc) := C(...) - tau * v(1:lastv,1) * w(1:lastc,1)**T
*
            ! C(1, 1:lastc)   := C(...) - tau * v(1,1) * w(1:lastc,1)**T
            !                  = C(...) - tau * w(1:lastc,1)**T
            CALL DAXPY(LASTC, -TAU, WORK, 1, C, LDC)
            ! C(2:lastv,1:lastc) := C(...) - tau * v(2:lastv,1)*w(1:lastc,1)**T
            CALL DGER(LASTV-1, LASTC, -TAU, V(1+INCV), INCV, WORK, 1,
     $                  C(1+1,1), LDC)
         END IF
      ELSE
*
*        Form  C * H
*
         ! Check if n = 1. This means v = 1, so we just need to compute
         ! C := CH = C(1-\tau).
         IF( LASTV.EQ.1 ) THEN
*
*           C(1:lastc,1) := ( 1 - tau ) * C(1:lastc,1)
*
            CALL DSCAL(LASTC, ONE - TAU, C, 1)
         ELSE
*
*           w(1:lastc,1) := C(1:lastc,1:lastv) * v(1:lastv,1)
*
            ! w(1:lastc,1) := C(1:lastc,2:lastv) * v(2:lastv,1)
            CALL DGEMV( 'No transpose', LASTC, LASTV-1, ONE, 
     $         C(1,1+1), LDC, V(1+INCV), INCV, ZERO, WORK, 1 )
            ! w(1:lastc,1) += C(1:lastc,1) v(1,1) = C(1:lastc,1)
            CALL DAXPY(LASTC, ONE, C, 1, WORK, 1)
*
*           C(1:lastc,1:lastv) := C(...) - tau * w(1:lastc,1) * v(1:lastv,1)**T
*
            ! C(1:lastc,1)     := C(...) - tau * w(1:lastc,1) * v(1,1)**T
            !                   = C(...) - tau * w(1:lastc,1)
            CALL DAXPY(LASTC, -TAU, WORK, 1, C, 1)
            ! C(1:lastc,2:lastv) := C(...) - tau * w(1:lastc,1) * v(2:lastv)**T
            CALL DGER( LASTC, LASTV-1, -TAU, WORK, 1, V(1+INCV),
     $                  INCV, C(1,1+1), LDC )
         END IF
      END IF
      RETURN
*
*     End of DLARF1F
*
      END
*> \brief \b DLARF1L applies an elementary reflector to a general rectangular
*              matrix assuming v(lastv) = 1 where lastv is the last non-zero
*              element
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARF1L + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarf1l.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarf1l.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarf1l.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARF1L( SIDE, M, N, V, INCV, TAU, C, LDC, WORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIDE
*       INTEGER            INCV, LDC, M, N
*       DOUBLE PRECISION   TAU
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   C( LDC, * ), V( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARF1L applies a real elementary reflector H to a real m by n matrix
*> C, from either the left or the right. H is represented in the form
*>
*>       H = I - tau * v * v**T
*>
*> where tau is a real scalar and v is a real vector.
*>
*> If tau = 0, then H is taken to be the unit matrix.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': form  H * C
*>          = 'R': form  C * H
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension
*>                     (1 + (M-1)*abs(INCV)) if SIDE = 'L'
*>                  or (1 + (N-1)*abs(INCV)) if SIDE = 'R'
*>          The vector v in the representation of H. V is not used if
*>          TAU = 0.
*> \endverbatim
*>
*> \param[in] INCV
*> \verbatim
*>          INCV is INTEGER
*>          The increment between elements of v. INCV <> 0.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION
*>          The value tau in the representation of H.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the m by n matrix C.
*>          On exit, C is overwritten by the matrix H * C if SIDE = 'L',
*>          or C * H if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension
*>                         (N) if SIDE = 'L'
*>                      or (M) if SIDE = 'R'
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larf
*
*  =====================================================================
      SUBROUTINE DLARF1L( SIDE, M, N, V, INCV, TAU, C, LDC, WORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE
      INTEGER            INCV, LDC, M, N
      DOUBLE PRECISION   TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   C( LDC, * ), V( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            APPLYLEFT
      INTEGER            I, FIRSTV, LASTV, LASTC
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMV, DGER
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILADLR, ILADLC
      EXTERNAL           LSAME, ILADLR, ILADLC
*     ..
*     .. Executable Statements ..
*
      APPLYLEFT = LSAME( SIDE, 'L' )
      FIRSTV = 1
      LASTC = 0
      IF( TAU.NE.ZERO ) THEN
!     Set up variables for scanning V.  LASTV begins pointing to the end
!     of V.
         IF( APPLYLEFT ) THEN
            LASTV = M
         ELSE
            LASTV = N
         END IF
         I = 1
!     Look for the last non-zero row in V.
         DO WHILE( LASTV.GT.FIRSTV .AND. V( I ).EQ.ZERO )
            FIRSTV = FIRSTV + 1
            I = I + INCV
         END DO
         IF( APPLYLEFT ) THEN
!     Scan for the last non-zero column in C(1:lastv,:).
            LASTC = ILADLC(LASTV, N, C, LDC)
         ELSE
!     Scan for the last non-zero row in C(:,1:lastv).
            LASTC = ILADLR(M, LASTV, C, LDC)
         END IF
      END IF
      IF( LASTC.EQ.0 ) THEN
         RETURN
      END IF
      IF( APPLYLEFT ) THEN
*
*        Form  H * C
*
         IF( LASTV.GT.0 ) THEN
            ! Check if m = 1. This means v = 1, So we just need to compute
            ! C := HC = (1-\tau)C.
            IF( LASTV.EQ.FIRSTV ) THEN
               CALL DSCAL(LASTC, ONE - TAU, C( FIRSTV, 1), LDC)
            ELSE
*
*              w(1:lastc,1) := C(1:lastv,1:lastc)**T * v(1:lastv,1)
*
               ! w(1:lastc,1) := C(1:lastv-1,1:lastc)**T * v(1:lastv-1,1)
               CALL DGEMV( 'Transpose', LASTV-FIRSTV, LASTC, ONE,
     $                     C(FIRSTV,1), LDC, V(I), INCV, ZERO, 
     $                     WORK, 1)
               ! w(1:lastc,1) += C(lastv,1:lastc)**T * v(lastv,1) = C(lastv,1:lastc)**T
               CALL DAXPY(LASTC, ONE, C(LASTV,1), LDC, WORK, 1)
*
*              C(1:lastv,1:lastc) := C(...) - tau * v(1:lastv,1) * w(1:lastc,1)**T
*
               ! C(lastv, 1:lastc)   := C(...) - tau * v(lastv,1) * w(1:lastc,1)**T
               !                      = C(...) - tau * w(1:lastc,1)**T
               CALL DAXPY(LASTC, -TAU, WORK, 1, C(LASTV,1), LDC)
               ! C(1:lastv-1,1:lastc) := C(...) - tau * v(1:lastv-1,1)*w(1:lastc,1)**T
               CALL DGER(LASTV-FIRSTV, LASTC, -TAU, V(I), INCV,
     $                   WORK, 1, C(FIRSTV,1), LDC)
            END IF
         END IF
      ELSE
*
*        Form  C * H
*
         IF( LASTV.GT.0 ) THEN
            ! Check if n = 1. This means v = 1, so we just need to compute
            ! C := CH = C(1-\tau).
            IF( LASTV.EQ.FIRSTV ) THEN
               CALL DSCAL(LASTC, ONE - TAU, C, 1)
            ELSE
*
*              w(1:lastc,1) := C(1:lastc,1:lastv) * v(1:lastv,1)
*
               ! w(1:lastc,1) := C(1:lastc,1:lastv-1) * v(1:lastv-1,1)
               CALL DGEMV( 'No transpose', LASTC, LASTV-FIRSTV,
     $            ONE, C(1,FIRSTV), LDC, V(I), INCV, ZERO, WORK, 1 )
               ! w(1:lastc,1) += C(1:lastc,lastv) * v(lastv,1) = C(1:lastc,lastv)
               CALL DAXPY(LASTC, ONE, C(1,LASTV), 1, WORK, 1)
*
*              C(1:lastc,1:lastv) := C(...) - tau * w(1:lastc,1) * v(1:lastv,1)**T
*
               ! C(1:lastc,lastv)     := C(...) - tau * w(1:lastc,1) * v(lastv,1)**T
               !                       = C(...) - tau * w(1:lastc,1)
               CALL DAXPY(LASTC, -TAU, WORK, 1, C(1,LASTV), 1)
               ! C(1:lastc,1:lastv-1) := C(...) - tau * w(1:lastc,1) * v(1:lastv-1)**T
               CALL DGER( LASTC, LASTV-FIRSTV, -TAU, WORK, 1, V(I), 
     $                     INCV, C(1,FIRSTV), LDC )
            END IF
         END IF
      END IF
      RETURN
*
*     End of DLARF1L
*
      END
*> \brief \b DLARFB applies a block reflector or its transpose to a general rectangular matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARFB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarfb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarfb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarfb.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV,
*                          T, LDT, C, LDC, WORK, LDWORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIRECT, SIDE, STOREV, TRANS
*       INTEGER            K, LDC, LDT, LDV, LDWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   C( LDC, * ), T( LDT, * ), V( LDV, * ),
*      $                   WORK( LDWORK, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARFB applies a real block reflector H or its transpose H**T to a
*> real m by n matrix C, from either the left or the right.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': apply H or H**T from the Left
*>          = 'R': apply H or H**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N': apply H (No transpose)
*>          = 'T': apply H**T (Transpose)
*> \endverbatim
*>
*> \param[in] DIRECT
*> \verbatim
*>          DIRECT is CHARACTER*1
*>          Indicates how H is formed from a product of elementary
*>          reflectors
*>          = 'F': H = H(1) H(2) . . . H(k) (Forward)
*>          = 'B': H = H(k) . . . H(2) H(1) (Backward)
*> \endverbatim
*>
*> \param[in] STOREV
*> \verbatim
*>          STOREV is CHARACTER*1
*>          Indicates how the vectors which define the elementary
*>          reflectors are stored:
*>          = 'C': Columnwise
*>          = 'R': Rowwise
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The order of the matrix T (= the number of elementary
*>          reflectors whose product defines the block reflector).
*>          If SIDE = 'L', M >= K >= 0;
*>          if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension
*>                                (LDV,K) if STOREV = 'C'
*>                                (LDV,M) if STOREV = 'R' and SIDE = 'L'
*>                                (LDV,N) if STOREV = 'R' and SIDE = 'R'
*>          The matrix V. See Further Details.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of the array V.
*>          If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);
*>          if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);
*>          if STOREV = 'R', LDV >= K.
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,K)
*>          The triangular k by k matrix T in the representation of the
*>          block reflector.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T. LDT >= K.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the m by n matrix C.
*>          On exit, C is overwritten by H*C or H**T*C or C*H or C*H**T.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (LDWORK,K)
*> \endverbatim
*>
*> \param[in] LDWORK
*> \verbatim
*>          LDWORK is INTEGER
*>          The leading dimension of the array WORK.
*>          If SIDE = 'L', LDWORK >= max(1,N);
*>          if SIDE = 'R', LDWORK >= max(1,M).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larfb
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The shape of the matrix V and the storage of the vectors which define
*>  the H(i) is best illustrated by the following example with n = 5 and
*>  k = 3. The triangular part of V (including its diagonal) is not
*>  referenced.
*>
*>  DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
*>
*>               V = (  1       )                 V = (  1 v1 v1 v1 v1 )
*>                   ( v1  1    )                     (     1 v2 v2 v2 )
*>                   ( v1 v2  1 )                     (        1 v3 v3 )
*>                   ( v1 v2 v3 )
*>                   ( v1 v2 v3 )
*>
*>  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
*>
*>               V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
*>                   ( v1 v2 v3 )                     ( v2 v2 v2  1    )
*>                   (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
*>                   (     1 v3 )
*>                   (        1 )
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLARFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, V,
     $                   LDV,
     $                   T, LDT, C, LDC, WORK, LDWORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIRECT, SIDE, STOREV, TRANS
      INTEGER            K, LDC, LDT, LDV, LDWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   C( LDC, * ), T( LDT, * ), V( LDV, * ),
     $                   WORK( LDWORK, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      CHARACTER          TRANST
      INTEGER            I, J
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGEMM, DTRMM
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( M.LE.0 .OR. N.LE.0 )
     $   RETURN
*
      IF( LSAME( TRANS, 'N' ) ) THEN
         TRANST = 'T'
      ELSE
         TRANST = 'N'
      END IF
*
      IF( LSAME( STOREV, 'C' ) ) THEN
*
         IF( LSAME( DIRECT, 'F' ) ) THEN
*
*           Let  V =  ( V1 )    (first K rows)
*                     ( V2 )
*           where  V1  is unit lower triangular.
*
            IF( LSAME( SIDE, 'L' ) ) THEN
*
*              Form  H * C  or  H**T * C  where  C = ( C1 )
*                                                    ( C2 )
*
*              W := C**T * V  =  (C1**T * V1 + C2**T * V2)  (stored in WORK)
*
*              W := C1**T
*
               DO 10 J = 1, K
                  CALL DCOPY( N, C( J, 1 ), LDC, WORK( 1, J ), 1 )
   10          CONTINUE
*
*              W := W * V1
*
               CALL DTRMM( 'Right', 'Lower', 'No transpose', 'Unit',
     $                     N,
     $                     K, ONE, V, LDV, WORK, LDWORK )
               IF( M.GT.K ) THEN
*
*                 W := W + C2**T * V2
*
                  CALL DGEMM( 'Transpose', 'No transpose', N, K, M-K,
     $                        ONE, C( K+1, 1 ), LDC, V( K+1, 1 ), LDV,
     $                        ONE, WORK, LDWORK )
               END IF
*
*              W := W * T**T  or  W * T
*
               CALL DTRMM( 'Right', 'Upper', TRANST, 'Non-unit', N,
     $                     K,
     $                     ONE, T, LDT, WORK, LDWORK )
*
*              C := C - V * W**T
*
               IF( M.GT.K ) THEN
*
*                 C2 := C2 - V2 * W**T
*
                  CALL DGEMM( 'No transpose', 'Transpose', M-K, N, K,
     $                        -ONE, V( K+1, 1 ), LDV, WORK, LDWORK, ONE,
     $                        C( K+1, 1 ), LDC )
               END IF
*
*              W := W * V1**T
*
               CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit', N,
     $                     K,
     $                     ONE, V, LDV, WORK, LDWORK )
*
*              C1 := C1 - W**T
*
               DO 30 J = 1, K
                  DO 20 I = 1, N
                     C( J, I ) = C( J, I ) - WORK( I, J )
   20             CONTINUE
   30          CONTINUE
*
            ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
*              Form  C * H  or  C * H**T  where  C = ( C1  C2 )
*
*              W := C * V  =  (C1*V1 + C2*V2)  (stored in WORK)
*
*              W := C1
*
               DO 40 J = 1, K
                  CALL DCOPY( M, C( 1, J ), 1, WORK( 1, J ), 1 )
   40          CONTINUE
*
*              W := W * V1
*
               CALL DTRMM( 'Right', 'Lower', 'No transpose', 'Unit',
     $                     M,
     $                     K, ONE, V, LDV, WORK, LDWORK )
               IF( N.GT.K ) THEN
*
*                 W := W + C2 * V2
*
                  CALL DGEMM( 'No transpose', 'No transpose', M, K,
     $                        N-K,
     $                        ONE, C( 1, K+1 ), LDC, V( K+1, 1 ), LDV,
     $                        ONE, WORK, LDWORK )
               END IF
*
*              W := W * T  or  W * T**T
*
               CALL DTRMM( 'Right', 'Upper', TRANS, 'Non-unit', M, K,
     $                     ONE, T, LDT, WORK, LDWORK )
*
*              C := C - W * V**T
*
               IF( N.GT.K ) THEN
*
*                 C2 := C2 - W * V2**T
*
                  CALL DGEMM( 'No transpose', 'Transpose', M, N-K, K,
     $                        -ONE, WORK, LDWORK, V( K+1, 1 ), LDV, ONE,
     $                        C( 1, K+1 ), LDC )
               END IF
*
*              W := W * V1**T
*
               CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit', M,
     $                     K,
     $                     ONE, V, LDV, WORK, LDWORK )
*
*              C1 := C1 - W
*
               DO 60 J = 1, K
                  DO 50 I = 1, M
                     C( I, J ) = C( I, J ) - WORK( I, J )
   50             CONTINUE
   60          CONTINUE
            END IF
*
         ELSE
*
*           Let  V =  ( V1 )
*                     ( V2 )    (last K rows)
*           where  V2  is unit upper triangular.
*
            IF( LSAME( SIDE, 'L' ) ) THEN
*
*              Form  H * C  or  H**T * C  where  C = ( C1 )
*                                                    ( C2 )
*
*              W := C**T * V  =  (C1**T * V1 + C2**T * V2)  (stored in WORK)
*
*              W := C2**T
*
               DO 70 J = 1, K
                  CALL DCOPY( N, C( M-K+J, 1 ), LDC, WORK( 1, J ),
     $                        1 )
   70          CONTINUE
*
*              W := W * V2
*
               CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Unit',
     $                     N,
     $                     K, ONE, V( M-K+1, 1 ), LDV, WORK, LDWORK )
               IF( M.GT.K ) THEN
*
*                 W := W + C1**T * V1
*
                  CALL DGEMM( 'Transpose', 'No transpose', N, K, M-K,
     $                        ONE, C, LDC, V, LDV, ONE, WORK, LDWORK )
               END IF
*
*              W := W * T**T  or  W * T
*
               CALL DTRMM( 'Right', 'Lower', TRANST, 'Non-unit', N,
     $                     K,
     $                     ONE, T, LDT, WORK, LDWORK )
*
*              C := C - V * W**T
*
               IF( M.GT.K ) THEN
*
*                 C1 := C1 - V1 * W**T
*
                  CALL DGEMM( 'No transpose', 'Transpose', M-K, N, K,
     $                        -ONE, V, LDV, WORK, LDWORK, ONE, C, LDC )
               END IF
*
*              W := W * V2**T
*
               CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit', N,
     $                     K,
     $                     ONE, V( M-K+1, 1 ), LDV, WORK, LDWORK )
*
*              C2 := C2 - W**T
*
               DO 90 J = 1, K
                  DO 80 I = 1, N
                     C( M-K+J, I ) = C( M-K+J, I ) - WORK( I, J )
   80             CONTINUE
   90          CONTINUE
*
            ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
*              Form  C * H  or  C * H**T  where  C = ( C1  C2 )
*
*              W := C * V  =  (C1*V1 + C2*V2)  (stored in WORK)
*
*              W := C2
*
               DO 100 J = 1, K
                  CALL DCOPY( M, C( 1, N-K+J ), 1, WORK( 1, J ), 1 )
  100          CONTINUE
*
*              W := W * V2
*
               CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Unit',
     $                     M,
     $                     K, ONE, V( N-K+1, 1 ), LDV, WORK, LDWORK )
               IF( N.GT.K ) THEN
*
*                 W := W + C1 * V1
*
                  CALL DGEMM( 'No transpose', 'No transpose', M, K,
     $                        N-K,
     $                        ONE, C, LDC, V, LDV, ONE, WORK, LDWORK )
               END IF
*
*              W := W * T  or  W * T**T
*
               CALL DTRMM( 'Right', 'Lower', TRANS, 'Non-unit', M, K,
     $                     ONE, T, LDT, WORK, LDWORK )
*
*              C := C - W * V**T
*
               IF( N.GT.K ) THEN
*
*                 C1 := C1 - W * V1**T
*
                  CALL DGEMM( 'No transpose', 'Transpose', M, N-K, K,
     $                        -ONE, WORK, LDWORK, V, LDV, ONE, C, LDC )
               END IF
*
*              W := W * V2**T
*
               CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit', M,
     $                     K,
     $                     ONE, V( N-K+1, 1 ), LDV, WORK, LDWORK )
*
*              C2 := C2 - W
*
               DO 120 J = 1, K
                  DO 110 I = 1, M
                     C( I, N-K+J ) = C( I, N-K+J ) - WORK( I, J )
  110             CONTINUE
  120          CONTINUE
            END IF
         END IF
*
      ELSE IF( LSAME( STOREV, 'R' ) ) THEN
*
         IF( LSAME( DIRECT, 'F' ) ) THEN
*
*           Let  V =  ( V1  V2 )    (V1: first K columns)
*           where  V1  is unit upper triangular.
*
            IF( LSAME( SIDE, 'L' ) ) THEN
*
*              Form  H * C  or  H**T * C  where  C = ( C1 )
*                                                    ( C2 )
*
*              W := C**T * V**T  =  (C1**T * V1**T + C2**T * V2**T) (stored in WORK)
*
*              W := C1**T
*
               DO 130 J = 1, K
                  CALL DCOPY( N, C( J, 1 ), LDC, WORK( 1, J ), 1 )
  130          CONTINUE
*
*              W := W * V1**T
*
               CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit', N,
     $                     K,
     $                     ONE, V, LDV, WORK, LDWORK )
               IF( M.GT.K ) THEN
*
*                 W := W + C2**T * V2**T
*
                  CALL DGEMM( 'Transpose', 'Transpose', N, K, M-K,
     $                        ONE,
     $                        C( K+1, 1 ), LDC, V( 1, K+1 ), LDV, ONE,
     $                        WORK, LDWORK )
               END IF
*
*              W := W * T**T  or  W * T
*
               CALL DTRMM( 'Right', 'Upper', TRANST, 'Non-unit', N,
     $                     K,
     $                     ONE, T, LDT, WORK, LDWORK )
*
*              C := C - V**T * W**T
*
               IF( M.GT.K ) THEN
*
*                 C2 := C2 - V2**T * W**T
*
                  CALL DGEMM( 'Transpose', 'Transpose', M-K, N, K,
     $                        -ONE,
     $                        V( 1, K+1 ), LDV, WORK, LDWORK, ONE,
     $                        C( K+1, 1 ), LDC )
               END IF
*
*              W := W * V1
*
               CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Unit',
     $                     N,
     $                     K, ONE, V, LDV, WORK, LDWORK )
*
*              C1 := C1 - W**T
*
               DO 150 J = 1, K
                  DO 140 I = 1, N
                     C( J, I ) = C( J, I ) - WORK( I, J )
  140             CONTINUE
  150          CONTINUE
*
            ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
*              Form  C * H  or  C * H**T  where  C = ( C1  C2 )
*
*              W := C * V**T  =  (C1*V1**T + C2*V2**T)  (stored in WORK)
*
*              W := C1
*
               DO 160 J = 1, K
                  CALL DCOPY( M, C( 1, J ), 1, WORK( 1, J ), 1 )
  160          CONTINUE
*
*              W := W * V1**T
*
               CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit', M,
     $                     K,
     $                     ONE, V, LDV, WORK, LDWORK )
               IF( N.GT.K ) THEN
*
*                 W := W + C2 * V2**T
*
                  CALL DGEMM( 'No transpose', 'Transpose', M, K, N-K,
     $                        ONE, C( 1, K+1 ), LDC, V( 1, K+1 ), LDV,
     $                        ONE, WORK, LDWORK )
               END IF
*
*              W := W * T  or  W * T**T
*
               CALL DTRMM( 'Right', 'Upper', TRANS, 'Non-unit', M, K,
     $                     ONE, T, LDT, WORK, LDWORK )
*
*              C := C - W * V
*
               IF( N.GT.K ) THEN
*
*                 C2 := C2 - W * V2
*
                  CALL DGEMM( 'No transpose', 'No transpose', M, N-K,
     $                        K,
     $                        -ONE, WORK, LDWORK, V( 1, K+1 ), LDV, ONE,
     $                        C( 1, K+1 ), LDC )
               END IF
*
*              W := W * V1
*
               CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Unit',
     $                     M,
     $                     K, ONE, V, LDV, WORK, LDWORK )
*
*              C1 := C1 - W
*
               DO 180 J = 1, K
                  DO 170 I = 1, M
                     C( I, J ) = C( I, J ) - WORK( I, J )
  170             CONTINUE
  180          CONTINUE
*
            END IF
*
         ELSE
*
*           Let  V =  ( V1  V2 )    (V2: last K columns)
*           where  V2  is unit lower triangular.
*
            IF( LSAME( SIDE, 'L' ) ) THEN
*
*              Form  H * C  or  H**T * C  where  C = ( C1 )
*                                                    ( C2 )
*
*              W := C**T * V**T  =  (C1**T * V1**T + C2**T * V2**T) (stored in WORK)
*
*              W := C2**T
*
               DO 190 J = 1, K
                  CALL DCOPY( N, C( M-K+J, 1 ), LDC, WORK( 1, J ),
     $                        1 )
  190          CONTINUE
*
*              W := W * V2**T
*
               CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit', N,
     $                     K,
     $                     ONE, V( 1, M-K+1 ), LDV, WORK, LDWORK )
               IF( M.GT.K ) THEN
*
*                 W := W + C1**T * V1**T
*
                  CALL DGEMM( 'Transpose', 'Transpose', N, K, M-K,
     $                        ONE,
     $                        C, LDC, V, LDV, ONE, WORK, LDWORK )
               END IF
*
*              W := W * T**T  or  W * T
*
               CALL DTRMM( 'Right', 'Lower', TRANST, 'Non-unit', N,
     $                     K,
     $                     ONE, T, LDT, WORK, LDWORK )
*
*              C := C - V**T * W**T
*
               IF( M.GT.K ) THEN
*
*                 C1 := C1 - V1**T * W**T
*
                  CALL DGEMM( 'Transpose', 'Transpose', M-K, N, K,
     $                        -ONE,
     $                        V, LDV, WORK, LDWORK, ONE, C, LDC )
               END IF
*
*              W := W * V2
*
               CALL DTRMM( 'Right', 'Lower', 'No transpose', 'Unit',
     $                     N,
     $                     K, ONE, V( 1, M-K+1 ), LDV, WORK, LDWORK )
*
*              C2 := C2 - W**T
*
               DO 210 J = 1, K
                  DO 200 I = 1, N
                     C( M-K+J, I ) = C( M-K+J, I ) - WORK( I, J )
  200             CONTINUE
  210          CONTINUE
*
            ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
*              Form  C * H  or  C * H'  where  C = ( C1  C2 )
*
*              W := C * V**T  =  (C1*V1**T + C2*V2**T)  (stored in WORK)
*
*              W := C2
*
               DO 220 J = 1, K
                  CALL DCOPY( M, C( 1, N-K+J ), 1, WORK( 1, J ), 1 )
  220          CONTINUE
*
*              W := W * V2**T
*
               CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit', M,
     $                     K,
     $                     ONE, V( 1, N-K+1 ), LDV, WORK, LDWORK )
               IF( N.GT.K ) THEN
*
*                 W := W + C1 * V1**T
*
                  CALL DGEMM( 'No transpose', 'Transpose', M, K, N-K,
     $                        ONE, C, LDC, V, LDV, ONE, WORK, LDWORK )
               END IF
*
*              W := W * T  or  W * T**T
*
               CALL DTRMM( 'Right', 'Lower', TRANS, 'Non-unit', M, K,
     $                     ONE, T, LDT, WORK, LDWORK )
*
*              C := C - W * V
*
               IF( N.GT.K ) THEN
*
*                 C1 := C1 - W * V1
*
                  CALL DGEMM( 'No transpose', 'No transpose', M, N-K,
     $                        K,
     $                        -ONE, WORK, LDWORK, V, LDV, ONE, C, LDC )
               END IF
*
*              W := W * V2
*
               CALL DTRMM( 'Right', 'Lower', 'No transpose', 'Unit',
     $                     M,
     $                     K, ONE, V( 1, N-K+1 ), LDV, WORK, LDWORK )
*
*              C1 := C1 - W
*
               DO 240 J = 1, K
                  DO 230 I = 1, M
                     C( I, N-K+J ) = C( I, N-K+J ) - WORK( I, J )
  230             CONTINUE
  240          CONTINUE
*
            END IF
*
         END IF
      END IF
*
      RETURN
*
*     End of DLARFB
*
      END
*> \brief \b DLARFG generates an elementary reflector (Householder matrix).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARFG + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarfg.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarfg.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarfg.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARFG( N, ALPHA, X, INCX, TAU )
*
*       .. Scalar Arguments ..
*       INTEGER            INCX, N
*       DOUBLE PRECISION   ALPHA, TAU
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   X( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARFG generates a real elementary reflector H of order n, such
*> that
*>
*>       H * ( alpha ) = ( beta ),   H**T * H = I.
*>           (   x   )   (   0  )
*>
*> where alpha and beta are scalars, and x is an (n-1)-element real
*> vector. H is represented in the form
*>
*>       H = I - tau * ( 1 ) * ( 1 v**T ) ,
*>                     ( v )
*>
*> where tau is a real scalar and v is a real (n-1)-element
*> vector.
*>
*> If the elements of x are all zero, then tau = 0 and H is taken to be
*> the unit matrix.
*>
*> Otherwise  1 <= tau <= 2.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the elementary reflector.
*> \endverbatim
*>
*> \param[in,out] ALPHA
*> \verbatim
*>          ALPHA is DOUBLE PRECISION
*>          On entry, the value alpha.
*>          On exit, it is overwritten with the value beta.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension
*>                         (1+(N-2)*abs(INCX))
*>          On entry, the vector x.
*>          On exit, it is overwritten with the vector v.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*>          INCX is INTEGER
*>          The increment between elements of X. INCX > 0.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION
*>          The value tau.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larfg
*
*  =====================================================================
      SUBROUTINE DLARFG( N, ALPHA, X, INCX, TAU )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INCX, N
      DOUBLE PRECISION   ALPHA, TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   X( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            J, KNT
      DOUBLE PRECISION   BETA, RSAFMN, SAFMIN, XNORM
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH, DLAPY2, DNRM2
      EXTERNAL           DLAMCH, DLAPY2, DNRM2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, SIGN
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL
*     ..
*     .. Executable Statements ..
*
      IF( N.LE.1 ) THEN
         TAU = ZERO
         RETURN
      END IF
*
      XNORM = DNRM2( N-1, X, INCX )
*
      IF( XNORM.EQ.ZERO ) THEN
*
*        H  =  I
*
         TAU = ZERO
      ELSE
*
*        general case
*
         BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
         SAFMIN = DLAMCH( 'S' ) / DLAMCH( 'E' )
         KNT = 0
         IF( ABS( BETA ).LT.SAFMIN ) THEN
*
*           XNORM, BETA may be inaccurate; scale X and recompute them
*
            RSAFMN = ONE / SAFMIN
   10       CONTINUE
            KNT = KNT + 1
            CALL DSCAL( N-1, RSAFMN, X, INCX )
            BETA = BETA*RSAFMN
            ALPHA = ALPHA*RSAFMN
            IF( (ABS( BETA ).LT.SAFMIN) .AND. (KNT .LT. 20) )
     $         GO TO 10
*
*           New BETA is at most 1, at least SAFMIN
*
            XNORM = DNRM2( N-1, X, INCX )
            BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
         END IF
         TAU = ( BETA-ALPHA ) / BETA
         CALL DSCAL( N-1, ONE / ( ALPHA-BETA ), X, INCX )
*
*        If ALPHA is subnormal, it may lose relative accuracy
*
         DO 20 J = 1, KNT
            BETA = BETA*SAFMIN
 20      CONTINUE
         ALPHA = BETA
      END IF
*
      RETURN
*
*     End of DLARFG
*
      END
*> \brief \b DLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARFGP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarfgp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarfgp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarfgp.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARFGP( N, ALPHA, X, INCX, TAU )
*
*       .. Scalar Arguments ..
*       INTEGER            INCX, N
*       DOUBLE PRECISION   ALPHA, TAU
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   X( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARFGP generates a real elementary reflector H of order n, such
*> that
*>
*>       H * ( alpha ) = ( beta ),   H**T * H = I.
*>           (   x   )   (   0  )
*>
*> where alpha and beta are scalars, beta is non-negative, and x is
*> an (n-1)-element real vector.  H is represented in the form
*>
*>       H = I - tau * ( 1 ) * ( 1 v**T ) ,
*>                     ( v )
*>
*> where tau is a real scalar and v is a real (n-1)-element
*> vector.
*>
*> If the elements of x are all zero, then tau = 0 and H is taken to be
*> the unit matrix.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the elementary reflector.
*> \endverbatim
*>
*> \param[in,out] ALPHA
*> \verbatim
*>          ALPHA is DOUBLE PRECISION
*>          On entry, the value alpha.
*>          On exit, it is overwritten with the value beta.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension
*>                         (1+(N-2)*abs(INCX))
*>          On entry, the vector x.
*>          On exit, it is overwritten with the vector v.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*>          INCX is INTEGER
*>          The increment between elements of X. INCX > 0.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION
*>          The value tau.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larfgp
*
*  =====================================================================
      SUBROUTINE DLARFGP( N, ALPHA, X, INCX, TAU )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INCX, N
      DOUBLE PRECISION   ALPHA, TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   X( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   TWO, ONE, ZERO
      PARAMETER          ( TWO = 2.0D+0, ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            J, KNT
      DOUBLE PRECISION   BETA, BIGNUM, EPS, SAVEALPHA, SMLNUM, XNORM
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH, DLAPY2, DNRM2
      EXTERNAL           DLAMCH, DLAPY2, DNRM2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, SIGN
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL
*     ..
*     .. Executable Statements ..
*
      IF( N.LE.0 ) THEN
         TAU = ZERO
         RETURN
      END IF
*
      EPS = DLAMCH( 'Precision' )
      XNORM = DNRM2( N-1, X, INCX )
*
      IF( XNORM.LE.EPS*ABS(ALPHA) ) THEN
*
*        H  =  [+/-1, 0; I], sign chosen so ALPHA >= 0.
*
         IF( ALPHA.GE.ZERO ) THEN
*           When TAU.eq.ZERO, the vector is special-cased to be
*           all zeros in the application routines.  We do not need
*           to clear it.
            TAU = ZERO
         ELSE
*           However, the application routines rely on explicit
*           zero checks when TAU.ne.ZERO, and we must clear X.
            TAU = TWO
            DO J = 1, N-1
               X( 1 + (J-1)*INCX ) = 0
            END DO
            ALPHA = -ALPHA
         END IF
      ELSE
*
*        general case
*
         BETA = SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
         SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'E' )
         KNT = 0
         IF( ABS( BETA ).LT.SMLNUM ) THEN
*
*           XNORM, BETA may be inaccurate; scale X and recompute them
*
            BIGNUM = ONE / SMLNUM
   10       CONTINUE
            KNT = KNT + 1
            CALL DSCAL( N-1, BIGNUM, X, INCX )
            BETA = BETA*BIGNUM
            ALPHA = ALPHA*BIGNUM
            IF( (ABS( BETA ).LT.SMLNUM) .AND. (KNT .LT. 20) )
     $         GO TO 10
*
*           New BETA is at most 1, at least SMLNUM
*
            XNORM = DNRM2( N-1, X, INCX )
            BETA = SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
         END IF
         SAVEALPHA = ALPHA
         ALPHA = ALPHA + BETA
         IF( BETA.LT.ZERO ) THEN
            BETA = -BETA
            TAU = -ALPHA / BETA
         ELSE
            ALPHA = XNORM * (XNORM/ALPHA)
            TAU = ALPHA / BETA
            ALPHA = -ALPHA
         END IF
*
         IF ( ABS(TAU).LE.SMLNUM ) THEN
*
*           In the case where the computed TAU ends up being a denormalized number,
*           it loses relative accuracy. This is a BIG problem. Solution: flush TAU
*           to ZERO. This explains the next IF statement.
*
*           (Bug report provided by Pat Quillen from MathWorks on Jul 29, 2009.)
*           (Thanks Pat. Thanks MathWorks.)
*
            IF( SAVEALPHA.GE.ZERO ) THEN
               TAU = ZERO
            ELSE
               TAU = TWO
               DO J = 1, N-1
                  X( 1 + (J-1)*INCX ) = 0
               END DO
               BETA = -SAVEALPHA
            END IF
*
         ELSE
*
*           This is the general case.
*
            CALL DSCAL( N-1, ONE / ALPHA, X, INCX )
*
         END IF
*
*        If BETA is subnormal, it may lose relative accuracy
*
         DO 20 J = 1, KNT
            BETA = BETA*SMLNUM
 20      CONTINUE
         ALPHA = BETA
      END IF
*
      RETURN
*
*     End of DLARFGP
*
      END
*> \brief \b DLARFT forms the triangular factor T of a block reflector H = I - vtvH
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARFT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarft.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarft.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarft.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       RECURSIVE SUBROUTINE DLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIRECT, STOREV
*       INTEGER            K, LDT, LDV, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   T( LDT, * ), TAU( * ), V( LDV, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARFT forms the triangular factor T of a real block reflector H
*> of order n, which is defined as a product of k elementary reflectors.
*>
*> If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
*>
*> If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
*>
*> If STOREV = 'C', the vector which defines the elementary reflector
*> H(i) is stored in the i-th column of the array V, and
*>
*>    H  =  I - V * T * V**T
*>
*> If STOREV = 'R', the vector which defines the elementary reflector
*> H(i) is stored in the i-th row of the array V, and
*>
*>    H  =  I - V**T * T * V
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] DIRECT
*> \verbatim
*>          DIRECT is CHARACTER*1
*>          Specifies the order in which the elementary reflectors are
*>          multiplied to form the block reflector:
*>          = 'F': H = H(1) H(2) . . . H(k) (Forward)
*>          = 'B': H = H(k) . . . H(2) H(1) (Backward)
*> \endverbatim
*>
*> \param[in] STOREV
*> \verbatim
*>          STOREV is CHARACTER*1
*>          Specifies how the vectors which define the elementary
*>          reflectors are stored (see also Further Details):
*>          = 'C': columnwise
*>          = 'R': rowwise
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the block reflector H. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The order of the triangular factor T (= the number of
*>          elementary reflectors). K >= 1.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension
*>                               (LDV,K) if STOREV = 'C'
*>                               (LDV,N) if STOREV = 'R'
*>          The matrix V. See further details.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of the array V.
*>          If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,K)
*>          The k by k triangular factor T of the block reflector.
*>          If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
*>          lower triangular. The rest of the array is not used.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T. LDT >= K.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larft
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The shape of the matrix V and the storage of the vectors which define
*>  the H(i) is best illustrated by the following example with n = 5 and
*>  k = 3. The elements equal to 1 are not stored.
*>
*>  DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
*>
*>               V = (  1       )                 V = (  1 v1 v1 v1 v1 )
*>                   ( v1  1    )                     (     1 v2 v2 v2 )
*>                   ( v1 v2  1 )                     (        1 v3 v3 )
*>                   ( v1 v2 v3 )
*>                   ( v1 v2 v3 )
*>
*>  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
*>
*>               V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
*>                   ( v1 v2 v3 )                     ( v2 v2 v2  1    )
*>                   (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
*>                   (     1 v3 )
*>                   (        1 )
*> \endverbatim
*>
*  =====================================================================
      RECURSIVE SUBROUTINE DLARFT( DIRECT, STOREV, N, K, V, LDV,
     $                             TAU, T, LDT )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*        .. Scalar Arguments
*
      CHARACTER          DIRECT, STOREV
      INTEGER            K, LDT, LDV, N
*     ..
*     .. Array Arguments ..
*
      DOUBLE PRECISION   T( LDT, * ), TAU( * ), V( LDV, * )
*     ..
*
*     .. Parameters ..
*
      DOUBLE PRECISION ONE, NEG_ONE, ZERO
      PARAMETER(ONE=1.0D+0, ZERO = 0.0D+0, NEG_ONE=-1.0D+0)
*
*     .. Local Scalars ..
*
      INTEGER           I,J,L
      LOGICAL           QR,LQ,QL,DIRF,COLV
*
*     .. External Subroutines ..
*
      EXTERNAL          DTRMM,DGEMM,DLACPY
*
*     .. External Functions..
*
      LOGICAL           LSAME
      EXTERNAL          LSAME
*     
*     The general scheme used is inspired by the approach inside DGEQRT3
*     which was (at the time of writing this code):
*     Based on the algorithm of Elmroth and Gustavson,
*     IBM J. Res. Develop. Vol 44 No. 4 July 2000.
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF(N.EQ.0.OR.K.EQ.0) THEN
         RETURN
      END IF
*
*     Base case
*
      IF(N.EQ.1.OR.K.EQ.1) THEN
         T(1,1) = TAU(1)
         RETURN
      END IF
*
*     Beginning of executable statements
*
      L = K / 2
*
*     Determine what kind of Q we need to compute
*     We assume that if the user doesn't provide 'F' for DIRECT,
*     then they meant to provide 'B' and if they don't provide
*     'C' for STOREV, then they meant to provide 'R'
*
      DIRF = LSAME(DIRECT,'F')
      COLV = LSAME(STOREV,'C')
*
*     QR happens when we have forward direction in column storage
*
      QR = DIRF.AND.COLV
*
*     LQ happens when we have forward direction in row storage
*
      LQ = DIRF.AND.(.NOT.COLV)
*
*     QL happens when we have backward direction in column storage
*
      QL = (.NOT.DIRF).AND.COLV
*
*     The last case is RQ. Due to how we structured this, if the
*     above 3 are false, then RQ must be true, so we never store 
*     this
*     RQ happens when we have backward direction in row storage
*     RQ = (.NOT.DIRF).AND.(.NOT.COLV)
*
      IF(QR) THEN
*
*        Break V apart into 6 components
*
*        V = |---------------|
*            |V_{1,1} 0      |
*            |V_{2,1} V_{2,2}|
*            |V_{3,1} V_{3,2}|
*            |---------------|
*
*        V_{1,1}\in\R^{l,l}      unit lower triangular
*        V_{2,1}\in\R^{k-l,l}    rectangular
*        V_{3,1}\in\R^{n-k,l}    rectangular
*        
*        V_{2,2}\in\R^{k-l,k-l}  unit lower triangular
*        V_{3,2}\in\R^{n-k,k-l}  rectangular
*
*        We will construct the T matrix 
*        T = |---------------|
*            |T_{1,1} T_{1,2}|
*            |0       T_{2,2}|
*            |---------------|
*
*        T is the triangular factor obtained from block reflectors. 
*        To motivate the structure, assume we have already computed T_{1,1}
*        and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
*        T_{1,1}\in\R^{l, l}     upper triangular
*        T_{2,2}\in\R^{k-l, k-l} upper triangular
*        T_{1,2}\in\R^{l, k-l}   rectangular
*
*        Where l = floor(k/2)
*
*        Then, consider the product:
*        
*        (I - V_1*T_{1,1}*V_1')*(I - V_2*T_{2,2}*V_2')
*        = I - V_1*T_{1,1}*V_1' - V_2*T_{2,2}*V_2' + V_1*T_{1,1}*V_1'*V_2*T_{2,2}*V_2'
*        
*        Define T_{1,2} = -T_{1,1}*V_1'*V_2*T_{2,2}
*        
*        Then, we can define the matrix V as 
*        V = |-------|
*            |V_1 V_2|
*            |-------|
*        
*        So, our product is equivalent to the matrix product
*        I - V*T*V'
*        This means, we can compute T_{1,1} and T_{2,2}, then use this information
*        to compute T_{1,2}
*
*        Compute T_{1,1} recursively
*
         CALL DLARFT(DIRECT, STOREV, N, L, V, LDV, TAU, T, LDT)
*
*        Compute T_{2,2} recursively
*
         CALL DLARFT(DIRECT, STOREV, N-L, K-L, V(L+1, L+1), LDV, 
     $               TAU(L+1), T(L+1, L+1), LDT)
*
*        Compute T_{1,2} 
*        T_{1,2} = V_{2,1}'
*
         DO J = 1, L
            DO I = 1, K-L
               T(J, L+I) = V(L+I, J)
            END DO
         END DO
*
*        T_{1,2} = T_{1,2}*V_{2,2}
*
         CALL DTRMM('Right', 'Lower', 'No transpose', 'Unit', L,
     $               K-L, ONE, V(L+1, L+1), LDV, T(1, L+1), LDT)

*
*        T_{1,2} = V_{3,1}'*V_{3,2} + T_{1,2}
*        Note: We assume K <= N, and GEMM will do nothing if N=K
*
         CALL DGEMM('Transpose', 'No transpose', L, K-L, N-K, ONE, 
     $               V(K+1, 1), LDV, V(K+1, L+1), LDV, ONE, 
     $               T(1, L+1), LDT)
*
*        At this point, we have that T_{1,2} = V_1'*V_2
*        All that is left is to pre and post multiply by -T_{1,1} and T_{2,2}
*        respectively.
*
*        T_{1,2} = -T_{1,1}*T_{1,2}
*
         CALL DTRMM('Left', 'Upper', 'No transpose', 'Non-unit', L,
     $               K-L, NEG_ONE, T, LDT, T(1, L+1), LDT)
*
*        T_{1,2} = T_{1,2}*T_{2,2}
*
         CALL DTRMM('Right', 'Upper', 'No transpose', 'Non-unit', L, 
     $               K-L, ONE, T(L+1, L+1), LDT, T(1, L+1), LDT)

      ELSE IF(LQ) THEN
*
*        Break V apart into 6 components
*
*        V = |----------------------|
*            |V_{1,1} V_{1,2} V{1,3}|
*            |0       V_{2,2} V{2,3}|
*            |----------------------|
*
*        V_{1,1}\in\R^{l,l}      unit upper triangular
*        V_{1,2}\in\R^{l,k-l}    rectangular
*        V_{1,3}\in\R^{l,n-k}    rectangular
*        
*        V_{2,2}\in\R^{k-l,k-l}  unit upper triangular
*        V_{2,3}\in\R^{k-l,n-k}  rectangular
*
*        Where l = floor(k/2)
*
*        We will construct the T matrix 
*        T = |---------------|
*            |T_{1,1} T_{1,2}|
*            |0       T_{2,2}|
*            |---------------|
*
*        T is the triangular factor obtained from block reflectors. 
*        To motivate the structure, assume we have already computed T_{1,1}
*        and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
*        T_{1,1}\in\R^{l, l}     upper triangular
*        T_{2,2}\in\R^{k-l, k-l} upper triangular
*        T_{1,2}\in\R^{l, k-l}   rectangular
*
*        Then, consider the product:
*        
*        (I - V_1'*T_{1,1}*V_1)*(I - V_2'*T_{2,2}*V_2)
*        = I - V_1'*T_{1,1}*V_1 - V_2'*T_{2,2}*V_2 + V_1'*T_{1,1}*V_1*V_2'*T_{2,2}*V_2
*        
*        Define T_{1,2} = -T_{1,1}*V_1*V_2'*T_{2,2}
*        
*        Then, we can define the matrix V as 
*        V = |---|
*            |V_1|
*            |V_2|
*            |---|
*        
*        So, our product is equivalent to the matrix product
*        I - V'*T*V
*        This means, we can compute T_{1,1} and T_{2,2}, then use this information
*        to compute T_{1,2}
*
*        Compute T_{1,1} recursively
*
         CALL DLARFT(DIRECT, STOREV, N, L, V, LDV, TAU, T, LDT)
*
*        Compute T_{2,2} recursively
*
         CALL DLARFT(DIRECT, STOREV, N-L, K-L, V(L+1, L+1), LDV, 
     $      TAU(L+1), T(L+1, L+1), LDT)

*
*        Compute T_{1,2}
*        T_{1,2} = V_{1,2}
*
         CALL DLACPY('All', L, K-L, V(1, L+1), LDV, T(1, L+1), LDT)
*
*        T_{1,2} = T_{1,2}*V_{2,2}'
*
         CALL DTRMM('Right', 'Upper', 'Transpose', 'Unit', L, K-L,
     $               ONE, V(L+1, L+1), LDV, T(1, L+1), LDT)

*
*        T_{1,2} = V_{1,3}*V_{2,3}' + T_{1,2}
*        Note: We assume K <= N, and GEMM will do nothing if N=K
*
         CALL DGEMM('No transpose', 'Transpose', L, K-L, N-K, ONE,
     $               V(1, K+1), LDV, V(L+1, K+1), LDV, ONE, 
     $               T(1, L+1), LDT)
*
*        At this point, we have that T_{1,2} = V_1*V_2'
*        All that is left is to pre and post multiply by -T_{1,1} and T_{2,2}
*        respectively.
*
*        T_{1,2} = -T_{1,1}*T_{1,2}
*
         CALL DTRMM('Left', 'Upper', 'No transpose', 'Non-unit', L,
     $               K-L, NEG_ONE, T, LDT, T(1, L+1), LDT)

*
*        T_{1,2} = T_{1,2}*T_{2,2}
*
         CALL DTRMM('Right', 'Upper', 'No transpose', 'Non-unit', L,
     $               K-L, ONE, T(L+1, L+1), LDT, T(1, L+1), LDT)
      ELSE IF(QL) THEN
*
*        Break V apart into 6 components
*
*        V = |---------------|
*            |V_{1,1} V_{1,2}|
*            |V_{2,1} V_{2,2}|
*            |0       V_{3,2}|
*            |---------------|
*
*        V_{1,1}\in\R^{n-k,k-l}  rectangular
*        V_{2,1}\in\R^{k-l,k-l}  unit upper triangular
*        
*        V_{1,2}\in\R^{n-k,l}    rectangular
*        V_{2,2}\in\R^{k-l,l}    rectangular
*        V_{3,2}\in\R^{l,l}      unit upper triangular
*
*        We will construct the T matrix 
*        T = |---------------|
*            |T_{1,1} 0      |
*            |T_{2,1} T_{2,2}|
*            |---------------|
*
*        T is the triangular factor obtained from block reflectors. 
*        To motivate the structure, assume we have already computed T_{1,1}
*        and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
*        T_{1,1}\in\R^{k-l, k-l} non-unit lower triangular
*        T_{2,2}\in\R^{l, l}     non-unit lower triangular
*        T_{2,1}\in\R^{k-l, l}   rectangular
*
*        Where l = floor(k/2)
*
*        Then, consider the product:
*        
*        (I - V_2*T_{2,2}*V_2')*(I - V_1*T_{1,1}*V_1')
*        = I - V_2*T_{2,2}*V_2' - V_1*T_{1,1}*V_1' + V_2*T_{2,2}*V_2'*V_1*T_{1,1}*V_1'
*        
*        Define T_{2,1} = -T_{2,2}*V_2'*V_1*T_{1,1}
*        
*        Then, we can define the matrix V as 
*        V = |-------|
*            |V_1 V_2|
*            |-------|
*        
*        So, our product is equivalent to the matrix product
*        I - V*T*V'
*        This means, we can compute T_{1,1} and T_{2,2}, then use this information
*        to compute T_{2,1}
*
*        Compute T_{1,1} recursively
*
         CALL DLARFT(DIRECT, STOREV, N-L, K-L, V, LDV, TAU, T, LDT)
*
*        Compute T_{2,2} recursively
*
         CALL DLARFT(DIRECT, STOREV, N, L, V(1, K-L+1), LDV,
     $               TAU(K-L+1), T(K-L+1, K-L+1), LDT)
*
*        Compute T_{2,1}
*        T_{2,1} = V_{2,2}'
*
         DO J = 1, K-L
            DO I = 1, L
               T(K-L+I, J) = V(N-K+J, K-L+I)
            END DO
         END DO
*
*        T_{2,1} = T_{2,1}*V_{2,1}
*
         CALL DTRMM('Right', 'Upper', 'No transpose', 'Unit', L,
     $               K-L, ONE, V(N-K+1, 1), LDV, T(K-L+1, 1), LDT)

*
*        T_{2,1} = V_{2,2}'*V_{2,1} + T_{2,1}
*        Note: We assume K <= N, and GEMM will do nothing if N=K
*
         CALL DGEMM('Transpose', 'No transpose', L, K-L, N-K, ONE,
     $               V(1, K-L+1), LDV, V, LDV, ONE, T(K-L+1, 1),
     $               LDT)
*
*        At this point, we have that T_{2,1} = V_2'*V_1
*        All that is left is to pre and post multiply by -T_{2,2} and T_{1,1}
*        respectively.
*
*        T_{2,1} = -T_{2,2}*T_{2,1}
*
         CALL DTRMM('Left', 'Lower', 'No transpose', 'Non-unit', L,
     $               K-L, NEG_ONE, T(K-L+1, K-L+1), LDT,
     $               T(K-L+1, 1), LDT)
*
*        T_{2,1} = T_{2,1}*T_{1,1}
*
         CALL DTRMM('Right', 'Lower', 'No transpose', 'Non-unit', L,
     $               K-L, ONE, T, LDT, T(K-L+1, 1), LDT)
      ELSE
*
*        Else means RQ case
*
*        Break V apart into 6 components
*
*        V = |-----------------------|
*            |V_{1,1} V_{1,2} 0      |
*            |V_{2,1} V_{2,2} V_{2,3}|
*            |-----------------------|
*
*        V_{1,1}\in\R^{k-l,n-k}  rectangular
*        V_{1,2}\in\R^{k-l,k-l}  unit lower triangular
*
*        V_{2,1}\in\R^{l,n-k}    rectangular
*        V_{2,2}\in\R^{l,k-l}    rectangular
*        V_{2,3}\in\R^{l,l}      unit lower triangular
*
*        We will construct the T matrix 
*        T = |---------------|
*            |T_{1,1} 0      |
*            |T_{2,1} T_{2,2}|
*            |---------------|
*
*        T is the triangular factor obtained from block reflectors. 
*        To motivate the structure, assume we have already computed T_{1,1}
*        and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
*        T_{1,1}\in\R^{k-l, k-l} non-unit lower triangular
*        T_{2,2}\in\R^{l, l}     non-unit lower triangular
*        T_{2,1}\in\R^{k-l, l}   rectangular
*
*        Where l = floor(k/2)
*
*        Then, consider the product:
*        
*        (I - V_2'*T_{2,2}*V_2)*(I - V_1'*T_{1,1}*V_1)
*        = I - V_2'*T_{2,2}*V_2 - V_1'*T_{1,1}*V_1 + V_2'*T_{2,2}*V_2*V_1'*T_{1,1}*V_1
*        
*        Define T_{2,1} = -T_{2,2}*V_2*V_1'*T_{1,1}
*        
*        Then, we can define the matrix V as 
*        V = |---|
*            |V_1|
*            |V_2|
*            |---|
*        
*        So, our product is equivalent to the matrix product
*        I - V'*T*V
*        This means, we can compute T_{1,1} and T_{2,2}, then use this information
*        to compute T_{2,1}
*
*        Compute T_{1,1} recursively
*
         CALL DLARFT(DIRECT, STOREV, N-L, K-L, V, LDV, TAU, T, LDT)
*
*        Compute T_{2,2} recursively
*
         CALL DLARFT(DIRECT, STOREV, N, L, V(K-L+1, 1), LDV,
     $               TAU(K-L+1), T(K-L+1, K-L+1), LDT)
*
*        Compute T_{2,1}
*        T_{2,1} = V_{2,2}
*
         CALL DLACPY('All', L, K-L, V(K-L+1, N-K+1), LDV,
     $               T(K-L+1, 1), LDT)

*
*        T_{2,1} = T_{2,1}*V_{1,2}'
*
         CALL DTRMM('Right', 'Lower', 'Transpose', 'Unit', L, K-L,
     $               ONE, V(1, N-K+1), LDV, T(K-L+1, 1), LDT)

*
*        T_{2,1} = V_{2,1}*V_{1,1}' + T_{2,1} 
*        Note: We assume K <= N, and GEMM will do nothing if N=K
*
         CALL DGEMM('No transpose', 'Transpose', L, K-L, N-K, ONE, 
     $               V(K-L+1, 1), LDV, V, LDV, ONE, T(K-L+1, 1),
     $               LDT)

*
*        At this point, we have that T_{2,1} = V_2*V_1'
*        All that is left is to pre and post multiply by -T_{2,2} and T_{1,1}
*        respectively.
*
*        T_{2,1} = -T_{2,2}*T_{2,1}
*
         CALL DTRMM('Left', 'Lower', 'No tranpose', 'Non-unit', L,
     $               K-L, NEG_ONE, T(K-L+1, K-L+1), LDT,
     $               T(K-L+1, 1), LDT)

*
*        T_{2,1} = T_{2,1}*T_{1,1}
*
         CALL DTRMM('Right', 'Lower', 'No tranpose', 'Non-unit', L,
     $               K-L, ONE, T, LDT, T(K-L+1, 1), LDT)
      END IF
      END SUBROUTINE
*> \brief \b DLARFX applies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order ≤ 10.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARFX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarfx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarfx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarfx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARFX( SIDE, M, N, V, TAU, C, LDC, WORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIDE
*       INTEGER            LDC, M, N
*       DOUBLE PRECISION   TAU
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   C( LDC, * ), V( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARFX applies a real elementary reflector H to a real m by n
*> matrix C, from either the left or the right. H is represented in the
*> form
*>
*>       H = I - tau * v * v**T
*>
*> where tau is a real scalar and v is a real vector.
*>
*> If tau = 0, then H is taken to be the unit matrix
*>
*> This version uses inline code if H has order < 11.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': form  H * C
*>          = 'R': form  C * H
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension (M) if SIDE = 'L'
*>                                     or (N) if SIDE = 'R'
*>          The vector v in the representation of H.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION
*>          The value tau in the representation of H.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the m by n matrix C.
*>          On exit, C is overwritten by the matrix H * C if SIDE = 'L',
*>          or C * H if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= (1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension
*>                      (N) if SIDE = 'L'
*>                      or (M) if SIDE = 'R'
*>          WORK is not referenced if H has order < 11.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larfx
*
*  =====================================================================
      SUBROUTINE DLARFX( SIDE, M, N, V, TAU, C, LDC, WORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE
      INTEGER            LDC, M, N
      DOUBLE PRECISION   TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   C( LDC, * ), V( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            J
      DOUBLE PRECISION   SUM, T1, T10, T2, T3, T4, T5, T6, T7, T8, T9,
     $                   V1, V10, V2, V3, V4, V5, V6, V7, V8, V9
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF
*     ..
*     .. Executable Statements ..
*
      IF( TAU.EQ.ZERO )
     $   RETURN
      IF( LSAME( SIDE, 'L' ) ) THEN
*
*        Form  H * C, where H has order m.
*
         GO TO ( 10, 30, 50, 70, 90, 110, 130, 150,
     $           170, 190 )M
*
*        Code for general M
*
         CALL DLARF( SIDE, M, N, V, 1, TAU, C, LDC, WORK )
         GO TO 410
   10    CONTINUE
*
*        Special code for 1 x 1 Householder
*
         T1 = ONE - TAU*V( 1 )*V( 1 )
         DO 20 J = 1, N
            C( 1, J ) = T1*C( 1, J )
   20    CONTINUE
         GO TO 410
   30    CONTINUE
*
*        Special code for 2 x 2 Householder
*
         V1 = V( 1 )
         T1 = TAU*V1
         V2 = V( 2 )
         T2 = TAU*V2
         DO 40 J = 1, N
            SUM = V1*C( 1, J ) + V2*C( 2, J )
            C( 1, J ) = C( 1, J ) - SUM*T1
            C( 2, J ) = C( 2, J ) - SUM*T2
   40    CONTINUE
         GO TO 410
   50    CONTINUE
*
*        Special code for 3 x 3 Householder
*
         V1 = V( 1 )
         T1 = TAU*V1
         V2 = V( 2 )
         T2 = TAU*V2
         V3 = V( 3 )
         T3 = TAU*V3
         DO 60 J = 1, N
            SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J )
            C( 1, J ) = C( 1, J ) - SUM*T1
            C( 2, J ) = C( 2, J ) - SUM*T2
            C( 3, J ) = C( 3, J ) - SUM*T3
   60    CONTINUE
         GO TO 410
   70    CONTINUE
*
*        Special code for 4 x 4 Householder
*
         V1 = V( 1 )
         T1 = TAU*V1
         V2 = V( 2 )
         T2 = TAU*V2
         V3 = V( 3 )
         T3 = TAU*V3
         V4 = V( 4 )
         T4 = TAU*V4
         DO 80 J = 1, N
            SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
     $            V4*C( 4, J )
            C( 1, J ) = C( 1, J ) - SUM*T1
            C( 2, J ) = C( 2, J ) - SUM*T2
            C( 3, J ) = C( 3, J ) - SUM*T3
            C( 4, J ) = C( 4, J ) - SUM*T4
   80    CONTINUE
         GO TO 410
   90    CONTINUE
*
*        Special code for 5 x 5 Householder
*
         V1 = V( 1 )
         T1 = TAU*V1
         V2 = V( 2 )
         T2 = TAU*V2
         V3 = V( 3 )
         T3 = TAU*V3
         V4 = V( 4 )
         T4 = TAU*V4
         V5 = V( 5 )
         T5 = TAU*V5
         DO 100 J = 1, N
            SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
     $            V4*C( 4, J ) + V5*C( 5, J )
            C( 1, J ) = C( 1, J ) - SUM*T1
            C( 2, J ) = C( 2, J ) - SUM*T2
            C( 3, J ) = C( 3, J ) - SUM*T3
            C( 4, J ) = C( 4, J ) - SUM*T4
            C( 5, J ) = C( 5, J ) - SUM*T5
  100    CONTINUE
         GO TO 410
  110    CONTINUE
*
*        Special code for 6 x 6 Householder
*
         V1 = V( 1 )
         T1 = TAU*V1
         V2 = V( 2 )
         T2 = TAU*V2
         V3 = V( 3 )
         T3 = TAU*V3
         V4 = V( 4 )
         T4 = TAU*V4
         V5 = V( 5 )
         T5 = TAU*V5
         V6 = V( 6 )
         T6 = TAU*V6
         DO 120 J = 1, N
            SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
     $            V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J )
            C( 1, J ) = C( 1, J ) - SUM*T1
            C( 2, J ) = C( 2, J ) - SUM*T2
            C( 3, J ) = C( 3, J ) - SUM*T3
            C( 4, J ) = C( 4, J ) - SUM*T4
            C( 5, J ) = C( 5, J ) - SUM*T5
            C( 6, J ) = C( 6, J ) - SUM*T6
  120    CONTINUE
         GO TO 410
  130    CONTINUE
*
*        Special code for 7 x 7 Householder
*
         V1 = V( 1 )
         T1 = TAU*V1
         V2 = V( 2 )
         T2 = TAU*V2
         V3 = V( 3 )
         T3 = TAU*V3
         V4 = V( 4 )
         T4 = TAU*V4
         V5 = V( 5 )
         T5 = TAU*V5
         V6 = V( 6 )
         T6 = TAU*V6
         V7 = V( 7 )
         T7 = TAU*V7
         DO 140 J = 1, N
            SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
     $            V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) +
     $            V7*C( 7, J )
            C( 1, J ) = C( 1, J ) - SUM*T1
            C( 2, J ) = C( 2, J ) - SUM*T2
            C( 3, J ) = C( 3, J ) - SUM*T3
            C( 4, J ) = C( 4, J ) - SUM*T4
            C( 5, J ) = C( 5, J ) - SUM*T5
            C( 6, J ) = C( 6, J ) - SUM*T6
            C( 7, J ) = C( 7, J ) - SUM*T7
  140    CONTINUE
         GO TO 410
  150    CONTINUE
*
*        Special code for 8 x 8 Householder
*
         V1 = V( 1 )
         T1 = TAU*V1
         V2 = V( 2 )
         T2 = TAU*V2
         V3 = V( 3 )
         T3 = TAU*V3
         V4 = V( 4 )
         T4 = TAU*V4
         V5 = V( 5 )
         T5 = TAU*V5
         V6 = V( 6 )
         T6 = TAU*V6
         V7 = V( 7 )
         T7 = TAU*V7
         V8 = V( 8 )
         T8 = TAU*V8
         DO 160 J = 1, N
            SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
     $            V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) +
     $            V7*C( 7, J ) + V8*C( 8, J )
            C( 1, J ) = C( 1, J ) - SUM*T1
            C( 2, J ) = C( 2, J ) - SUM*T2
            C( 3, J ) = C( 3, J ) - SUM*T3
            C( 4, J ) = C( 4, J ) - SUM*T4
            C( 5, J ) = C( 5, J ) - SUM*T5
            C( 6, J ) = C( 6, J ) - SUM*T6
            C( 7, J ) = C( 7, J ) - SUM*T7
            C( 8, J ) = C( 8, J ) - SUM*T8
  160    CONTINUE
         GO TO 410
  170    CONTINUE
*
*        Special code for 9 x 9 Householder
*
         V1 = V( 1 )
         T1 = TAU*V1
         V2 = V( 2 )
         T2 = TAU*V2
         V3 = V( 3 )
         T3 = TAU*V3
         V4 = V( 4 )
         T4 = TAU*V4
         V5 = V( 5 )
         T5 = TAU*V5
         V6 = V( 6 )
         T6 = TAU*V6
         V7 = V( 7 )
         T7 = TAU*V7
         V8 = V( 8 )
         T8 = TAU*V8
         V9 = V( 9 )
         T9 = TAU*V9
         DO 180 J = 1, N
            SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
     $            V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) +
     $            V7*C( 7, J ) + V8*C( 8, J ) + V9*C( 9, J )
            C( 1, J ) = C( 1, J ) - SUM*T1
            C( 2, J ) = C( 2, J ) - SUM*T2
            C( 3, J ) = C( 3, J ) - SUM*T3
            C( 4, J ) = C( 4, J ) - SUM*T4
            C( 5, J ) = C( 5, J ) - SUM*T5
            C( 6, J ) = C( 6, J ) - SUM*T6
            C( 7, J ) = C( 7, J ) - SUM*T7
            C( 8, J ) = C( 8, J ) - SUM*T8
            C( 9, J ) = C( 9, J ) - SUM*T9
  180    CONTINUE
         GO TO 410
  190    CONTINUE
*
*        Special code for 10 x 10 Householder
*
         V1 = V( 1 )
         T1 = TAU*V1
         V2 = V( 2 )
         T2 = TAU*V2
         V3 = V( 3 )
         T3 = TAU*V3
         V4 = V( 4 )
         T4 = TAU*V4
         V5 = V( 5 )
         T5 = TAU*V5
         V6 = V( 6 )
         T6 = TAU*V6
         V7 = V( 7 )
         T7 = TAU*V7
         V8 = V( 8 )
         T8 = TAU*V8
         V9 = V( 9 )
         T9 = TAU*V9
         V10 = V( 10 )
         T10 = TAU*V10
         DO 200 J = 1, N
            SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) +
     $            V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) +
     $            V7*C( 7, J ) + V8*C( 8, J ) + V9*C( 9, J ) +
     $            V10*C( 10, J )
            C( 1, J ) = C( 1, J ) - SUM*T1
            C( 2, J ) = C( 2, J ) - SUM*T2
            C( 3, J ) = C( 3, J ) - SUM*T3
            C( 4, J ) = C( 4, J ) - SUM*T4
            C( 5, J ) = C( 5, J ) - SUM*T5
            C( 6, J ) = C( 6, J ) - SUM*T6
            C( 7, J ) = C( 7, J ) - SUM*T7
            C( 8, J ) = C( 8, J ) - SUM*T8
            C( 9, J ) = C( 9, J ) - SUM*T9
            C( 10, J ) = C( 10, J ) - SUM*T10
  200    CONTINUE
         GO TO 410
      ELSE
*
*        Form  C * H, where H has order n.
*
         GO TO ( 210, 230, 250, 270, 290, 310, 330, 350,
     $           370, 390 )N
*
*        Code for general N
*
         CALL DLARF( SIDE, M, N, V, 1, TAU, C, LDC, WORK )
         GO TO 410
  210    CONTINUE
*
*        Special code for 1 x 1 Householder
*
         T1 = ONE - TAU*V( 1 )*V( 1 )
         DO 220 J = 1, M
            C( J, 1 ) = T1*C( J, 1 )
  220    CONTINUE
         GO TO 410
  230    CONTINUE
*
*        Special code for 2 x 2 Householder
*
         V1 = V( 1 )
         T1 = TAU*V1
         V2 = V( 2 )
         T2 = TAU*V2
         DO 240 J = 1, M
            SUM = V1*C( J, 1 ) + V2*C( J, 2 )
            C( J, 1 ) = C( J, 1 ) - SUM*T1
            C( J, 2 ) = C( J, 2 ) - SUM*T2
  240    CONTINUE
         GO TO 410
  250    CONTINUE
*
*        Special code for 3 x 3 Householder
*
         V1 = V( 1 )
         T1 = TAU*V1
         V2 = V( 2 )
         T2 = TAU*V2
         V3 = V( 3 )
         T3 = TAU*V3
         DO 260 J = 1, M
            SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 )
            C( J, 1 ) = C( J, 1 ) - SUM*T1
            C( J, 2 ) = C( J, 2 ) - SUM*T2
            C( J, 3 ) = C( J, 3 ) - SUM*T3
  260    CONTINUE
         GO TO 410
  270    CONTINUE
*
*        Special code for 4 x 4 Householder
*
         V1 = V( 1 )
         T1 = TAU*V1
         V2 = V( 2 )
         T2 = TAU*V2
         V3 = V( 3 )
         T3 = TAU*V3
         V4 = V( 4 )
         T4 = TAU*V4
         DO 280 J = 1, M
            SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
     $            V4*C( J, 4 )
            C( J, 1 ) = C( J, 1 ) - SUM*T1
            C( J, 2 ) = C( J, 2 ) - SUM*T2
            C( J, 3 ) = C( J, 3 ) - SUM*T3
            C( J, 4 ) = C( J, 4 ) - SUM*T4
  280    CONTINUE
         GO TO 410
  290    CONTINUE
*
*        Special code for 5 x 5 Householder
*
         V1 = V( 1 )
         T1 = TAU*V1
         V2 = V( 2 )
         T2 = TAU*V2
         V3 = V( 3 )
         T3 = TAU*V3
         V4 = V( 4 )
         T4 = TAU*V4
         V5 = V( 5 )
         T5 = TAU*V5
         DO 300 J = 1, M
            SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
     $            V4*C( J, 4 ) + V5*C( J, 5 )
            C( J, 1 ) = C( J, 1 ) - SUM*T1
            C( J, 2 ) = C( J, 2 ) - SUM*T2
            C( J, 3 ) = C( J, 3 ) - SUM*T3
            C( J, 4 ) = C( J, 4 ) - SUM*T4
            C( J, 5 ) = C( J, 5 ) - SUM*T5
  300    CONTINUE
         GO TO 410
  310    CONTINUE
*
*        Special code for 6 x 6 Householder
*
         V1 = V( 1 )
         T1 = TAU*V1
         V2 = V( 2 )
         T2 = TAU*V2
         V3 = V( 3 )
         T3 = TAU*V3
         V4 = V( 4 )
         T4 = TAU*V4
         V5 = V( 5 )
         T5 = TAU*V5
         V6 = V( 6 )
         T6 = TAU*V6
         DO 320 J = 1, M
            SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
     $            V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 )
            C( J, 1 ) = C( J, 1 ) - SUM*T1
            C( J, 2 ) = C( J, 2 ) - SUM*T2
            C( J, 3 ) = C( J, 3 ) - SUM*T3
            C( J, 4 ) = C( J, 4 ) - SUM*T4
            C( J, 5 ) = C( J, 5 ) - SUM*T5
            C( J, 6 ) = C( J, 6 ) - SUM*T6
  320    CONTINUE
         GO TO 410
  330    CONTINUE
*
*        Special code for 7 x 7 Householder
*
         V1 = V( 1 )
         T1 = TAU*V1
         V2 = V( 2 )
         T2 = TAU*V2
         V3 = V( 3 )
         T3 = TAU*V3
         V4 = V( 4 )
         T4 = TAU*V4
         V5 = V( 5 )
         T5 = TAU*V5
         V6 = V( 6 )
         T6 = TAU*V6
         V7 = V( 7 )
         T7 = TAU*V7
         DO 340 J = 1, M
            SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
     $            V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) +
     $            V7*C( J, 7 )
            C( J, 1 ) = C( J, 1 ) - SUM*T1
            C( J, 2 ) = C( J, 2 ) - SUM*T2
            C( J, 3 ) = C( J, 3 ) - SUM*T3
            C( J, 4 ) = C( J, 4 ) - SUM*T4
            C( J, 5 ) = C( J, 5 ) - SUM*T5
            C( J, 6 ) = C( J, 6 ) - SUM*T6
            C( J, 7 ) = C( J, 7 ) - SUM*T7
  340    CONTINUE
         GO TO 410
  350    CONTINUE
*
*        Special code for 8 x 8 Householder
*
         V1 = V( 1 )
         T1 = TAU*V1
         V2 = V( 2 )
         T2 = TAU*V2
         V3 = V( 3 )
         T3 = TAU*V3
         V4 = V( 4 )
         T4 = TAU*V4
         V5 = V( 5 )
         T5 = TAU*V5
         V6 = V( 6 )
         T6 = TAU*V6
         V7 = V( 7 )
         T7 = TAU*V7
         V8 = V( 8 )
         T8 = TAU*V8
         DO 360 J = 1, M
            SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
     $            V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) +
     $            V7*C( J, 7 ) + V8*C( J, 8 )
            C( J, 1 ) = C( J, 1 ) - SUM*T1
            C( J, 2 ) = C( J, 2 ) - SUM*T2
            C( J, 3 ) = C( J, 3 ) - SUM*T3
            C( J, 4 ) = C( J, 4 ) - SUM*T4
            C( J, 5 ) = C( J, 5 ) - SUM*T5
            C( J, 6 ) = C( J, 6 ) - SUM*T6
            C( J, 7 ) = C( J, 7 ) - SUM*T7
            C( J, 8 ) = C( J, 8 ) - SUM*T8
  360    CONTINUE
         GO TO 410
  370    CONTINUE
*
*        Special code for 9 x 9 Householder
*
         V1 = V( 1 )
         T1 = TAU*V1
         V2 = V( 2 )
         T2 = TAU*V2
         V3 = V( 3 )
         T3 = TAU*V3
         V4 = V( 4 )
         T4 = TAU*V4
         V5 = V( 5 )
         T5 = TAU*V5
         V6 = V( 6 )
         T6 = TAU*V6
         V7 = V( 7 )
         T7 = TAU*V7
         V8 = V( 8 )
         T8 = TAU*V8
         V9 = V( 9 )
         T9 = TAU*V9
         DO 380 J = 1, M
            SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
     $            V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) +
     $            V7*C( J, 7 ) + V8*C( J, 8 ) + V9*C( J, 9 )
            C( J, 1 ) = C( J, 1 ) - SUM*T1
            C( J, 2 ) = C( J, 2 ) - SUM*T2
            C( J, 3 ) = C( J, 3 ) - SUM*T3
            C( J, 4 ) = C( J, 4 ) - SUM*T4
            C( J, 5 ) = C( J, 5 ) - SUM*T5
            C( J, 6 ) = C( J, 6 ) - SUM*T6
            C( J, 7 ) = C( J, 7 ) - SUM*T7
            C( J, 8 ) = C( J, 8 ) - SUM*T8
            C( J, 9 ) = C( J, 9 ) - SUM*T9
  380    CONTINUE
         GO TO 410
  390    CONTINUE
*
*        Special code for 10 x 10 Householder
*
         V1 = V( 1 )
         T1 = TAU*V1
         V2 = V( 2 )
         T2 = TAU*V2
         V3 = V( 3 )
         T3 = TAU*V3
         V4 = V( 4 )
         T4 = TAU*V4
         V5 = V( 5 )
         T5 = TAU*V5
         V6 = V( 6 )
         T6 = TAU*V6
         V7 = V( 7 )
         T7 = TAU*V7
         V8 = V( 8 )
         T8 = TAU*V8
         V9 = V( 9 )
         T9 = TAU*V9
         V10 = V( 10 )
         T10 = TAU*V10
         DO 400 J = 1, M
            SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) +
     $            V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) +
     $            V7*C( J, 7 ) + V8*C( J, 8 ) + V9*C( J, 9 ) +
     $            V10*C( J, 10 )
            C( J, 1 ) = C( J, 1 ) - SUM*T1
            C( J, 2 ) = C( J, 2 ) - SUM*T2
            C( J, 3 ) = C( J, 3 ) - SUM*T3
            C( J, 4 ) = C( J, 4 ) - SUM*T4
            C( J, 5 ) = C( J, 5 ) - SUM*T5
            C( J, 6 ) = C( J, 6 ) - SUM*T6
            C( J, 7 ) = C( J, 7 ) - SUM*T7
            C( J, 8 ) = C( J, 8 ) - SUM*T8
            C( J, 9 ) = C( J, 9 ) - SUM*T9
            C( J, 10 ) = C( J, 10 ) - SUM*T10
  400    CONTINUE
         GO TO 410
      END IF
  410 CONTINUE
      RETURN
*
*     End of DLARFX
*
      END
*> \brief \b DLARGV generates a vector of plane rotations with real cosines and real sines.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARGV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlargv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlargv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlargv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARGV( N, X, INCX, Y, INCY, C, INCC )
*
*       .. Scalar Arguments ..
*       INTEGER            INCC, INCX, INCY, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   C( * ), X( * ), Y( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARGV generates a vector of real plane rotations, determined by
*> elements of the real vectors x and y. For i = 1,2,...,n
*>
*>    (  c(i)  s(i) ) ( x(i) ) = ( a(i) )
*>    ( -s(i)  c(i) ) ( y(i) ) = (   0  )
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of plane rotations to be generated.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array,
*>                         dimension (1+(N-1)*INCX)
*>          On entry, the vector x.
*>          On exit, x(i) is overwritten by a(i), for i = 1,...,n.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*>          INCX is INTEGER
*>          The increment between elements of X. INCX > 0.
*> \endverbatim
*>
*> \param[in,out] Y
*> \verbatim
*>          Y is DOUBLE PRECISION array,
*>                         dimension (1+(N-1)*INCY)
*>          On entry, the vector y.
*>          On exit, the sines of the plane rotations.
*> \endverbatim
*>
*> \param[in] INCY
*> \verbatim
*>          INCY is INTEGER
*>          The increment between elements of Y. INCY > 0.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
*>          The cosines of the plane rotations.
*> \endverbatim
*>
*> \param[in] INCC
*> \verbatim
*>          INCC is INTEGER
*>          The increment between elements of C. INCC > 0.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup largv
*
*  =====================================================================
      SUBROUTINE DLARGV( N, X, INCX, Y, INCY, C, INCC )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INCC, INCX, INCY, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   C( * ), X( * ), Y( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, IC, IX, IY
      DOUBLE PRECISION   F, G, T, TT
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, SQRT
*     ..
*     .. Executable Statements ..
*
      IX = 1
      IY = 1
      IC = 1
      DO 10 I = 1, N
         F = X( IX )
         G = Y( IY )
         IF( G.EQ.ZERO ) THEN
            C( IC ) = ONE
         ELSE IF( F.EQ.ZERO ) THEN
            C( IC ) = ZERO
            Y( IY ) = ONE
            X( IX ) = G
         ELSE IF( ABS( F ).GT.ABS( G ) ) THEN
            T = G / F
            TT = SQRT( ONE+T*T )
            C( IC ) = ONE / TT
            Y( IY ) = T*C( IC )
            X( IX ) = F*TT
         ELSE
            T = F / G
            TT = SQRT( ONE+T*T )
            Y( IY ) = ONE / TT
            C( IC ) = T*Y( IY )
            X( IX ) = G*TT
         END IF
         IC = IC + INCC
         IY = IY + INCY
         IX = IX + INCX
   10 CONTINUE
      RETURN
*
*     End of DLARGV
*
      END
*> \brief \b DLARNV returns a vector of random numbers from a uniform or normal distribution.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARNV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarnv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarnv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarnv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARNV( IDIST, ISEED, N, X )
*
*       .. Scalar Arguments ..
*       INTEGER            IDIST, N
*       ..
*       .. Array Arguments ..
*       INTEGER            ISEED( 4 )
*       DOUBLE PRECISION   X( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARNV returns a vector of n random real numbers from a uniform or
*> normal distribution.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] IDIST
*> \verbatim
*>          IDIST is INTEGER
*>          Specifies the distribution of the random numbers:
*>          = 1:  uniform (0,1)
*>          = 2:  uniform (-1,1)
*>          = 3:  normal (0,1)
*> \endverbatim
*>
*> \param[in,out] ISEED
*> \verbatim
*>          ISEED is INTEGER array, dimension (4)
*>          On entry, the seed of the random number generator; the array
*>          elements must be between 0 and 4095, and ISEED(4) must be
*>          odd.
*>          On exit, the seed is updated.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of random numbers to be generated.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (N)
*>          The generated random numbers.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larnv
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  This routine calls the auxiliary routine DLARUV to generate random
*>  real numbers from a uniform (0,1) distribution, in batches of up to
*>  128 using vectorisable code. The Box-Muller method is used to
*>  transform numbers from a uniform to a normal distribution.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLARNV( IDIST, ISEED, N, X )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            IDIST, N
*     ..
*     .. Array Arguments ..
      INTEGER            ISEED( 4 )
      DOUBLE PRECISION   X( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, TWO
      PARAMETER          ( ONE = 1.0D+0, TWO = 2.0D+0 )
      INTEGER            LV
      PARAMETER          ( LV = 128 )
      DOUBLE PRECISION   TWOPI
      PARAMETER  ( TWOPI = 6.28318530717958647692528676655900576839D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, IL, IL2, IV
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   U( LV )
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          COS, LOG, MIN, SQRT
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARUV
*     ..
*     .. Executable Statements ..
*
      DO 40 IV = 1, N, LV / 2
         IL = MIN( LV / 2, N-IV+1 )
         IF( IDIST.EQ.3 ) THEN
            IL2 = 2*IL
         ELSE
            IL2 = IL
         END IF
*
*        Call DLARUV to generate IL2 numbers from a uniform (0,1)
*        distribution (IL2 <= LV)
*
         CALL DLARUV( ISEED, IL2, U )
*
         IF( IDIST.EQ.1 ) THEN
*
*           Copy generated numbers
*
            DO 10 I = 1, IL
               X( IV+I-1 ) = U( I )
   10       CONTINUE
         ELSE IF( IDIST.EQ.2 ) THEN
*
*           Convert generated numbers to uniform (-1,1) distribution
*
            DO 20 I = 1, IL
               X( IV+I-1 ) = TWO*U( I ) - ONE
   20       CONTINUE
         ELSE IF( IDIST.EQ.3 ) THEN
*
*           Convert generated numbers to normal (0,1) distribution
*
            DO 30 I = 1, IL
               X( IV+I-1 ) = SQRT( -TWO*LOG( U( 2*I-1 ) ) )*
     $                       COS( TWOPI*U( 2*I ) )
   30       CONTINUE
         END IF
   40 CONTINUE
      RETURN
*
*     End of DLARNV
*
      END
*> \brief \b DLARRA computes the splitting points with the specified threshold.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarra.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarra.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarra.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARRA( N, D, E, E2, SPLTOL, TNRM,
*                           NSPLIT, ISPLIT, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, N, NSPLIT
*       DOUBLE PRECISION    SPLTOL, TNRM
*       ..
*       .. Array Arguments ..
*       INTEGER            ISPLIT( * )
*       DOUBLE PRECISION   D( * ), E( * ), E2( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> Compute the splitting points with threshold SPLTOL.
*> DLARRA sets any "small" off-diagonal elements to zero.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix. N > 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, the N diagonal elements of the tridiagonal
*>          matrix T.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N)
*>          On entry, the first (N-1) entries contain the subdiagonal
*>          elements of the tridiagonal matrix T; E(N) need not be set.
*>          On exit, the entries E( ISPLIT( I ) ), 1 <= I <= NSPLIT,
*>          are set to zero, the other entries of E are untouched.
*> \endverbatim
*>
*> \param[in,out] E2
*> \verbatim
*>          E2 is DOUBLE PRECISION array, dimension (N)
*>          On entry, the first (N-1) entries contain the SQUARES of the
*>          subdiagonal elements of the tridiagonal matrix T;
*>          E2(N) need not be set.
*>          On exit, the entries E2( ISPLIT( I ) ),
*>          1 <= I <= NSPLIT, have been set to zero
*> \endverbatim
*>
*> \param[in] SPLTOL
*> \verbatim
*>          SPLTOL is DOUBLE PRECISION
*>          The threshold for splitting. Two criteria can be used:
*>          SPLTOL<0 : criterion based on absolute off-diagonal value
*>          SPLTOL>0 : criterion that preserves relative accuracy
*> \endverbatim
*>
*> \param[in] TNRM
*> \verbatim
*>          TNRM is DOUBLE PRECISION
*>          The norm of the matrix.
*> \endverbatim
*>
*> \param[out] NSPLIT
*> \verbatim
*>          NSPLIT is INTEGER
*>          The number of blocks T splits into. 1 <= NSPLIT <= N.
*> \endverbatim
*>
*> \param[out] ISPLIT
*> \verbatim
*>          ISPLIT is INTEGER array, dimension (N)
*>          The splitting points, at which T breaks up into blocks.
*>          The first block consists of rows/columns 1 to ISPLIT(1),
*>          the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
*>          etc., and the NSPLIT-th consists of rows/columns
*>          ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larra
*
*> \par Contributors:
*  ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA
*
*  =====================================================================
      SUBROUTINE DLARRA( N, D, E, E2, SPLTOL, TNRM,
     $                    NSPLIT, ISPLIT, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, N, NSPLIT
      DOUBLE PRECISION    SPLTOL, TNRM
*     ..
*     .. Array Arguments ..
      INTEGER            ISPLIT( * )
      DOUBLE PRECISION   D( * ), E( * ), E2( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
      DOUBLE PRECISION   EABS, TMP1

*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      NSPLIT = 1
*
*     Quick return if possible
*
      IF( N.LE.0 ) THEN
         RETURN
      END IF
*
*     Compute splitting points
      IF(SPLTOL.LT.ZERO) THEN
*        Criterion based on absolute off-diagonal value
         TMP1 = ABS(SPLTOL)* TNRM
         DO 9 I = 1, N-1
            EABS = ABS( E(I) )
            IF( EABS .LE. TMP1) THEN
               E(I) = ZERO
               E2(I) = ZERO
               ISPLIT( NSPLIT ) = I
               NSPLIT = NSPLIT + 1
            END IF
 9       CONTINUE
      ELSE
*        Criterion that guarantees relative accuracy
         DO 10 I = 1, N-1
            EABS = ABS( E(I) )
            IF( EABS .LE. SPLTOL * SQRT(ABS(D(I)))*SQRT(ABS(D(I+1))) )
     $      THEN
               E(I) = ZERO
               E2(I) = ZERO
               ISPLIT( NSPLIT ) = I
               NSPLIT = NSPLIT + 1
            END IF
 10      CONTINUE
      ENDIF
      ISPLIT( NSPLIT ) = N

      RETURN
*
*     End of DLARRA
*
      END
*> \brief \b DLARRB provides limited bisection to locate eigenvalues for more accuracy.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrb.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARRB( N, D, LLD, IFIRST, ILAST, RTOL1,
*                          RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK,
*                          PIVMIN, SPDIAM, TWIST, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            IFIRST, ILAST, INFO, N, OFFSET, TWIST
*       DOUBLE PRECISION   PIVMIN, RTOL1, RTOL2, SPDIAM
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   D( * ), LLD( * ), W( * ),
*      $                   WERR( * ), WGAP( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> Given the relatively robust representation(RRR) L D L^T, DLARRB
*> does "limited" bisection to refine the eigenvalues of L D L^T,
*> W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial
*> guesses for these eigenvalues are input in W, the corresponding estimate
*> of the error in these guesses and their gaps are input in WERR
*> and WGAP, respectively. During bisection, intervals
*> [left, right] are maintained by storing their mid-points and
*> semi-widths in the arrays W and WERR respectively.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The N diagonal elements of the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] LLD
*> \verbatim
*>          LLD is DOUBLE PRECISION array, dimension (N-1)
*>          The (N-1) elements L(i)*L(i)*D(i).
*> \endverbatim
*>
*> \param[in] IFIRST
*> \verbatim
*>          IFIRST is INTEGER
*>          The index of the first eigenvalue to be computed.
*> \endverbatim
*>
*> \param[in] ILAST
*> \verbatim
*>          ILAST is INTEGER
*>          The index of the last eigenvalue to be computed.
*> \endverbatim
*>
*> \param[in] RTOL1
*> \verbatim
*>          RTOL1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] RTOL2
*> \verbatim
*>          RTOL2 is DOUBLE PRECISION
*>          Tolerance for the convergence of the bisection intervals.
*>          An interval [LEFT,RIGHT] has converged if
*>          RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
*>          where GAP is the (estimated) distance to the nearest
*>          eigenvalue.
*> \endverbatim
*>
*> \param[in] OFFSET
*> \verbatim
*>          OFFSET is INTEGER
*>          Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFSET
*>          through ILAST-OFFSET elements of these arrays are to be used.
*> \endverbatim
*>
*> \param[in,out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are
*>          estimates of the eigenvalues of L D L^T indexed IFIRST through
*>          ILAST.
*>          On output, these estimates are refined.
*> \endverbatim
*>
*> \param[in,out] WGAP
*> \verbatim
*>          WGAP is DOUBLE PRECISION array, dimension (N-1)
*>          On input, the (estimated) gaps between consecutive
*>          eigenvalues of L D L^T, i.e., WGAP(I-OFFSET) is the gap between
*>          eigenvalues I and I+1. Note that if IFIRST = ILAST
*>          then WGAP(IFIRST-OFFSET) must be set to ZERO.
*>          On output, these gaps are refined.
*> \endverbatim
*>
*> \param[in,out] WERR
*> \verbatim
*>          WERR is DOUBLE PRECISION array, dimension (N)
*>          On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are
*>          the errors in the estimates of the corresponding elements in W.
*>          On output, these errors are refined.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (2*N)
*>          Workspace.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (2*N)
*>          Workspace.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*>          PIVMIN is DOUBLE PRECISION
*>          The minimum pivot in the Sturm sequence.
*> \endverbatim
*>
*> \param[in] SPDIAM
*> \verbatim
*>          SPDIAM is DOUBLE PRECISION
*>          The spectral diameter of the matrix.
*> \endverbatim
*>
*> \param[in] TWIST
*> \verbatim
*>          TWIST is INTEGER
*>          The twist index for the twisted factorization that is used
*>          for the negcount.
*>          TWIST = N: Compute negcount from L D L^T - LAMBDA I = L+ D+ L+^T
*>          TWIST = 1: Compute negcount from L D L^T - LAMBDA I = U- D- U-^T
*>          TWIST = R: Compute negcount from L D L^T - LAMBDA I = N(r) D(r) N(r)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          Error flag.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larrb
*
*> \par Contributors:
*  ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA
*
*  =====================================================================
      SUBROUTINE DLARRB( N, D, LLD, IFIRST, ILAST, RTOL1,
     $                   RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK,
     $                   PIVMIN, SPDIAM, TWIST, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            IFIRST, ILAST, INFO, N, OFFSET, TWIST
      DOUBLE PRECISION   PIVMIN, RTOL1, RTOL2, SPDIAM
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   D( * ), LLD( * ), W( * ),
     $                   WERR( * ), WGAP( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, TWO, HALF
      PARAMETER        ( ZERO = 0.0D0, TWO = 2.0D0,
     $                   HALF = 0.5D0 )
      INTEGER   MAXITR
*     ..
*     .. Local Scalars ..
      INTEGER            I, I1, II, IP, ITER, K, NEGCNT, NEXT, NINT,
     $                   OLNINT, PREV, R
      DOUBLE PRECISION   BACK, CVRGD, GAP, LEFT, LGAP, MID, MNWDTH,
     $                   RGAP, RIGHT, TMP, WIDTH
*     ..
*     .. External Functions ..
      INTEGER            DLANEG
      EXTERNAL           DLANEG
*
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. Executable Statements ..
*
      INFO = 0
*
*     Quick return if possible
*
      IF( N.LE.0 ) THEN
         RETURN
      END IF
*
      MAXITR = INT( ( LOG( SPDIAM+PIVMIN )-LOG( PIVMIN ) ) /
     $           LOG( TWO ) ) + 2
      MNWDTH = TWO * PIVMIN
*
      R = TWIST
      IF((R.LT.1).OR.(R.GT.N)) R = N
*
*     Initialize unconverged intervals in [ WORK(2*I-1), WORK(2*I) ].
*     The Sturm Count, Count( WORK(2*I-1) ) is arranged to be I-1, while
*     Count( WORK(2*I) ) is stored in IWORK( 2*I ). The integer IWORK( 2*I-1 )
*     for an unconverged interval is set to the index of the next unconverged
*     interval, and is -1 or 0 for a converged interval. Thus a linked
*     list of unconverged intervals is set up.
*
      I1 = IFIRST
*     The number of unconverged intervals
      NINT = 0
*     The last unconverged interval found
      PREV = 0

      RGAP = WGAP( I1-OFFSET )
      DO 75 I = I1, ILAST
         K = 2*I
         II = I - OFFSET
         LEFT = W( II ) - WERR( II )
         RIGHT = W( II ) + WERR( II )
         LGAP = RGAP
         RGAP = WGAP( II )
         GAP = MIN( LGAP, RGAP )

*        Make sure that [LEFT,RIGHT] contains the desired eigenvalue
*        Compute negcount from dstqds facto L+D+L+^T = L D L^T - LEFT
*
*        Do while( NEGCNT(LEFT).GT.I-1 )
*
         BACK = WERR( II )
 20      CONTINUE
         NEGCNT = DLANEG( N, D, LLD, LEFT, PIVMIN, R )
         IF( NEGCNT.GT.I-1 ) THEN
            LEFT = LEFT - BACK
            BACK = TWO*BACK
            GO TO 20
         END IF
*
*        Do while( NEGCNT(RIGHT).LT.I )
*        Compute negcount from dstqds facto L+D+L+^T = L D L^T - RIGHT
*
         BACK = WERR( II )
 50      CONTINUE

         NEGCNT = DLANEG( N, D, LLD, RIGHT, PIVMIN, R )
          IF( NEGCNT.LT.I ) THEN
             RIGHT = RIGHT + BACK
             BACK = TWO*BACK
             GO TO 50
          END IF
         WIDTH = HALF*ABS( LEFT - RIGHT )
         TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
         CVRGD = MAX(RTOL1*GAP,RTOL2*TMP)
         IF( WIDTH.LE.CVRGD .OR. WIDTH.LE.MNWDTH ) THEN
*           This interval has already converged and does not need refinement.
*           (Note that the gaps might change through refining the
*            eigenvalues, however, they can only get bigger.)
*           Remove it from the list.
            IWORK( K-1 ) = -1
*           Make sure that I1 always points to the first unconverged interval
            IF((I.EQ.I1).AND.(I.LT.ILAST)) I1 = I + 1
            IF((PREV.GE.I1).AND.(I.LE.ILAST)) IWORK( 2*PREV-1 ) = I + 1
         ELSE
*           unconverged interval found
            PREV = I
            NINT = NINT + 1
            IWORK( K-1 ) = I + 1
            IWORK( K ) = NEGCNT
         END IF
         WORK( K-1 ) = LEFT
         WORK( K ) = RIGHT
 75   CONTINUE

*
*     Do while( NINT.GT.0 ), i.e. there are still unconverged intervals
*     and while (ITER.LT.MAXITR)
*
      ITER = 0
 80   CONTINUE
      PREV = I1 - 1
      I = I1
      OLNINT = NINT

      DO 100 IP = 1, OLNINT
         K = 2*I
         II = I - OFFSET
         RGAP = WGAP( II )
         LGAP = RGAP
         IF(II.GT.1) LGAP = WGAP( II-1 )
         GAP = MIN( LGAP, RGAP )
         NEXT = IWORK( K-1 )
         LEFT = WORK( K-1 )
         RIGHT = WORK( K )
         MID = HALF*( LEFT + RIGHT )

*        semiwidth of interval
         WIDTH = RIGHT - MID
         TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
         CVRGD = MAX(RTOL1*GAP,RTOL2*TMP)
         IF( ( WIDTH.LE.CVRGD ) .OR. ( WIDTH.LE.MNWDTH ).OR.
     $       ( ITER.EQ.MAXITR ) )THEN
*           reduce number of unconverged intervals
            NINT = NINT - 1
*           Mark interval as converged.
            IWORK( K-1 ) = 0
            IF( I1.EQ.I ) THEN
               I1 = NEXT
            ELSE
*              Prev holds the last unconverged interval previously examined
               IF(PREV.GE.I1) IWORK( 2*PREV-1 ) = NEXT
            END IF
            I = NEXT
            GO TO 100
         END IF
         PREV = I
*
*        Perform one bisection step
*
         NEGCNT = DLANEG( N, D, LLD, MID, PIVMIN, R )
         IF( NEGCNT.LE.I-1 ) THEN
            WORK( K-1 ) = MID
         ELSE
            WORK( K ) = MID
         END IF
         I = NEXT
 100  CONTINUE
      ITER = ITER + 1
*     do another loop if there are still unconverged intervals
*     However, in the last iteration, all intervals are accepted
*     since this is the best we can do.
      IF( ( NINT.GT.0 ).AND.(ITER.LE.MAXITR) ) GO TO 80
*
*
*     At this point, all the intervals have converged
      DO 110 I = IFIRST, ILAST
         K = 2*I
         II = I - OFFSET
*        All intervals marked by '0' have been refined.
         IF( IWORK( K-1 ).EQ.0 ) THEN
            W( II ) = HALF*( WORK( K-1 )+WORK( K ) )
            WERR( II ) = WORK( K ) - W( II )
         END IF
 110  CONTINUE
*
      DO 111 I = IFIRST+1, ILAST
         K = 2*I
         II = I - OFFSET
         WGAP( II-1 ) = MAX( ZERO,
     $                     W(II) - WERR (II) - W( II-1 ) - WERR( II-1 ))
 111  CONTINUE

      RETURN
*
*     End of DLARRB
*
      END
*> \brief \b DLARRC computes the number of eigenvalues of the symmetric tridiagonal matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrc.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARRC( JOBT, N, VL, VU, D, E, PIVMIN,
*                                   EIGCNT, LCNT, RCNT, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBT
*       INTEGER            EIGCNT, INFO, LCNT, N, RCNT
*       DOUBLE PRECISION   PIVMIN, VL, VU
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), E( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> Find the number of eigenvalues of the symmetric tridiagonal matrix T
*> that are in the interval (VL,VU] if JOBT = 'T', and of L D L^T
*> if JOBT = 'L'.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBT
*> \verbatim
*>          JOBT is CHARACTER*1
*>          = 'T':  Compute Sturm count for matrix T.
*>          = 'L':  Compute Sturm count for matrix L D L^T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix. N > 0.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is DOUBLE PRECISION
*>          The lower bound for the eigenvalues.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*>          VU is DOUBLE PRECISION
*>          The upper bound for the eigenvalues.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          JOBT = 'T': The N diagonal elements of the tridiagonal matrix T.
*>          JOBT = 'L': The N diagonal elements of the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N)
*>          JOBT = 'T': The N-1 offdiagonal elements of the matrix T.
*>          JOBT = 'L': The N-1 offdiagonal elements of the matrix L.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*>          PIVMIN is DOUBLE PRECISION
*>          The minimum pivot in the Sturm sequence for T.
*> \endverbatim
*>
*> \param[out] EIGCNT
*> \verbatim
*>          EIGCNT is INTEGER
*>          The number of eigenvalues of the symmetric tridiagonal matrix T
*>          that are in the interval (VL,VU]
*> \endverbatim
*>
*> \param[out] LCNT
*> \verbatim
*>          LCNT is INTEGER
*> \endverbatim
*>
*> \param[out] RCNT
*> \verbatim
*>          RCNT is INTEGER
*>          The left and right negcounts of the interval.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larrc
*
*> \par Contributors:
*  ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA
*
*  =====================================================================
      SUBROUTINE DLARRC( JOBT, N, VL, VU, D, E, PIVMIN,
     $                            EIGCNT, LCNT, RCNT, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBT
      INTEGER            EIGCNT, INFO, LCNT, N, RCNT
      DOUBLE PRECISION   PIVMIN, VL, VU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), E( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
      LOGICAL            MATT
      DOUBLE PRECISION   LPIVOT, RPIVOT, SL, SU, TMP, TMP2

*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      LCNT = 0
      RCNT = 0
      EIGCNT = 0
*
*     Quick return if possible
*
      IF( N.LE.0 ) THEN
         RETURN
      END IF
*
      MATT = LSAME( JOBT, 'T' )


      IF (MATT) THEN
*        Sturm sequence count on T
         LPIVOT = D( 1 ) - VL
         RPIVOT = D( 1 ) - VU
         IF( LPIVOT.LE.ZERO ) THEN
            LCNT = LCNT + 1
         ENDIF
         IF( RPIVOT.LE.ZERO ) THEN
            RCNT = RCNT + 1
         ENDIF
         DO 10 I = 1, N-1
            TMP = E(I)**2
            LPIVOT = ( D( I+1 )-VL ) - TMP/LPIVOT
            RPIVOT = ( D( I+1 )-VU ) - TMP/RPIVOT
            IF( LPIVOT.LE.ZERO ) THEN
               LCNT = LCNT + 1
            ENDIF
            IF( RPIVOT.LE.ZERO ) THEN
               RCNT = RCNT + 1
            ENDIF
 10      CONTINUE
      ELSE
*        Sturm sequence count on L D L^T
         SL = -VL
         SU = -VU
         DO 20 I = 1, N - 1
            LPIVOT = D( I ) + SL
            RPIVOT = D( I ) + SU
            IF( LPIVOT.LE.ZERO ) THEN
               LCNT = LCNT + 1
            ENDIF
            IF( RPIVOT.LE.ZERO ) THEN
               RCNT = RCNT + 1
            ENDIF
            TMP = E(I) * D(I) * E(I)
*
            TMP2 = TMP / LPIVOT
            IF( TMP2.EQ.ZERO ) THEN
               SL =  TMP - VL
            ELSE
               SL = SL*TMP2 - VL
            END IF
*
            TMP2 = TMP / RPIVOT
            IF( TMP2.EQ.ZERO ) THEN
               SU =  TMP - VU
            ELSE
               SU = SU*TMP2 - VU
            END IF
 20      CONTINUE
         LPIVOT = D( N ) + SL
         RPIVOT = D( N ) + SU
         IF( LPIVOT.LE.ZERO ) THEN
            LCNT = LCNT + 1
         ENDIF
         IF( RPIVOT.LE.ZERO ) THEN
            RCNT = RCNT + 1
         ENDIF
      ENDIF
      EIGCNT = RCNT - LCNT

      RETURN
*
*     End of DLARRC
*
      END
*> \brief \b DLARRD computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARRD( RANGE, ORDER, N, VL, VU, IL, IU, GERS,
*                           RELTOL, D, E, E2, PIVMIN, NSPLIT, ISPLIT,
*                           M, W, WERR, WL, WU, IBLOCK, INDEXW,
*                           WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          ORDER, RANGE
*       INTEGER            IL, INFO, IU, M, N, NSPLIT
*       DOUBLE PRECISION    PIVMIN, RELTOL, VL, VU, WL, WU
*       ..
*       .. Array Arguments ..
*       INTEGER            IBLOCK( * ), INDEXW( * ),
*      $                   ISPLIT( * ), IWORK( * )
*       DOUBLE PRECISION   D( * ), E( * ), E2( * ),
*      $                   GERS( * ), W( * ), WERR( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARRD computes the eigenvalues of a symmetric tridiagonal
*> matrix T to suitable accuracy. This is an auxiliary code to be
*> called from DSTEMR.
*> The user may ask for all eigenvalues, all eigenvalues
*> in the half-open interval (VL, VU], or the IL-th through IU-th
*> eigenvalues.
*>
*> To avoid overflow, the matrix must be scaled so that its
*> largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
*> accuracy, it should not be much smaller than that.
*>
*> See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
*> Matrix", Report CS41, Computer Science Dept., Stanford
*> University, July 21, 1966.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] RANGE
*> \verbatim
*>          RANGE is CHARACTER*1
*>          = 'A': ("All")   all eigenvalues will be found.
*>          = 'V': ("Value") all eigenvalues in the half-open interval
*>                           (VL, VU] will be found.
*>          = 'I': ("Index") the IL-th through IU-th eigenvalues (of the
*>                           entire matrix) will be found.
*> \endverbatim
*>
*> \param[in] ORDER
*> \verbatim
*>          ORDER is CHARACTER*1
*>          = 'B': ("By Block") the eigenvalues will be grouped by
*>                              split-off block (see IBLOCK, ISPLIT) and
*>                              ordered from smallest to largest within
*>                              the block.
*>          = 'E': ("Entire matrix")
*>                              the eigenvalues for the entire matrix
*>                              will be ordered from smallest to
*>                              largest.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the tridiagonal matrix T.  N >= 0.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is DOUBLE PRECISION
*>          If RANGE='V', the lower bound of the interval to
*>          be searched for eigenvalues.  Eigenvalues less than or equal
*>          to VL, or greater than VU, will not be returned.  VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*>          VU is DOUBLE PRECISION
*>          If RANGE='V', the upper bound of the interval to
*>          be searched for eigenvalues.  Eigenvalues less than or equal
*>          to VL, or greater than VU, will not be returned.  VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*>          IL is INTEGER
*>          If RANGE='I', the index of the
*>          smallest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*>          IU is INTEGER
*>          If RANGE='I', the index of the
*>          largest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] GERS
*> \verbatim
*>          GERS is DOUBLE PRECISION array, dimension (2*N)
*>          The N Gerschgorin intervals (the i-th Gerschgorin interval
*>          is (GERS(2*i-1), GERS(2*i)).
*> \endverbatim
*>
*> \param[in] RELTOL
*> \verbatim
*>          RELTOL is DOUBLE PRECISION
*>          The minimum relative width of an interval.  When an interval
*>          is narrower than RELTOL times the larger (in
*>          magnitude) endpoint, then it is considered to be
*>          sufficiently small, i.e., converged.  Note: this should
*>          always be at least radix*machine epsilon.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The n diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) off-diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] E2
*> \verbatim
*>          E2 is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) squared off-diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*>          PIVMIN is DOUBLE PRECISION
*>          The minimum pivot allowed in the Sturm sequence for T.
*> \endverbatim
*>
*> \param[in] NSPLIT
*> \verbatim
*>          NSPLIT is INTEGER
*>          The number of diagonal blocks in the matrix T.
*>          1 <= NSPLIT <= N.
*> \endverbatim
*>
*> \param[in] ISPLIT
*> \verbatim
*>          ISPLIT is INTEGER array, dimension (N)
*>          The splitting points, at which T breaks up into submatrices.
*>          The first submatrix consists of rows/columns 1 to ISPLIT(1),
*>          the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
*>          etc., and the NSPLIT-th consists of rows/columns
*>          ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
*>          (Only the first NSPLIT elements will actually be used, but
*>          since the user cannot know a priori what value NSPLIT will
*>          have, N words must be reserved for ISPLIT.)
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The actual number of eigenvalues found. 0 <= M <= N.
*>          (See also the description of INFO=2,3.)
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          On exit, the first M elements of W will contain the
*>          eigenvalue approximations. DLARRD computes an interval
*>          I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue
*>          approximation is given as the interval midpoint
*>          W(j)= ( a_j + b_j)/2. The corresponding error is bounded by
*>          WERR(j) = abs( a_j - b_j)/2
*> \endverbatim
*>
*> \param[out] WERR
*> \verbatim
*>          WERR is DOUBLE PRECISION array, dimension (N)
*>          The error bound on the corresponding eigenvalue approximation
*>          in W.
*> \endverbatim
*>
*> \param[out] WL
*> \verbatim
*>          WL is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] WU
*> \verbatim
*>          WU is DOUBLE PRECISION
*>          The interval (WL, WU] contains all the wanted eigenvalues.
*>          If RANGE='V', then WL=VL and WU=VU.
*>          If RANGE='A', then WL and WU are the global Gerschgorin bounds
*>                        on the spectrum.
*>          If RANGE='I', then WL and WU are computed by DLAEBZ from the
*>                        index range specified.
*> \endverbatim
*>
*> \param[out] IBLOCK
*> \verbatim
*>          IBLOCK is INTEGER array, dimension (N)
*>          At each row/column j where E(j) is zero or small, the
*>          matrix T is considered to split into a block diagonal
*>          matrix.  On exit, if INFO = 0, IBLOCK(i) specifies to which
*>          block (from 1 to the number of blocks) the eigenvalue W(i)
*>          belongs.  (DLARRD may use the remaining N-M elements as
*>          workspace.)
*> \endverbatim
*>
*> \param[out] INDEXW
*> \verbatim
*>          INDEXW is INTEGER array, dimension (N)
*>          The indices of the eigenvalues within each block (submatrix);
*>          for example, INDEXW(i)= j and IBLOCK(i)=k imply that the
*>          i-th eigenvalue W(i) is the j-th eigenvalue in block k.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  some or all of the eigenvalues failed to converge or
*>                were not computed:
*>                =1 or 3: Bisection failed to converge for some
*>                        eigenvalues; these eigenvalues are flagged by a
*>                        negative block number.  The effect is that the
*>                        eigenvalues may not be as accurate as the
*>                        absolute and relative tolerances.  This is
*>                        generally caused by unexpectedly inaccurate
*>                        arithmetic.
*>                =2 or 3: RANGE='I' only: Not all of the eigenvalues
*>                        IL:IU were found.
*>                        Effect: M < IU+1-IL
*>                        Cause:  non-monotonic arithmetic, causing the
*>                                Sturm sequence to be non-monotonic.
*>                        Cure:   recalculate, using RANGE='A', and pick
*>                                out eigenvalues IL:IU.  In some cases,
*>                                increasing the PARAMETER "FUDGE" may
*>                                make things work.
*>                = 4:    RANGE='I', and the Gershgorin interval
*>                        initially used was too small.  No eigenvalues
*>                        were computed.
*>                        Probable cause: your machine has sloppy
*>                                        floating-point arithmetic.
*>                        Cure: Increase the PARAMETER "FUDGE",
*>                              recompile, and try again.
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  FUDGE   DOUBLE PRECISION, default = 2
*>          A "fudge factor" to widen the Gershgorin intervals.  Ideally,
*>          a value of 1 should work, but on machines with sloppy
*>          arithmetic, this needs to be larger.  The default for
*>          publicly released versions should be large enough to handle
*>          the worst machine around.  Note that this has no effect
*>          on accuracy of the solution.
*> \endverbatim
*>
*> \par Contributors:
*  ==================
*>
*>     W. Kahan, University of California, Berkeley, USA \n
*>     Beresford Parlett, University of California, Berkeley, USA \n
*>     Jim Demmel, University of California, Berkeley, USA \n
*>     Inderjit Dhillon, University of Texas, Austin, USA \n
*>     Osni Marques, LBNL/NERSC, USA \n
*>     Christof Voemel, University of California, Berkeley, USA \n
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larrd
*
*  =====================================================================
      SUBROUTINE DLARRD( RANGE, ORDER, N, VL, VU, IL, IU, GERS,
     $                    RELTOL, D, E, E2, PIVMIN, NSPLIT, ISPLIT,
     $                    M, W, WERR, WL, WU, IBLOCK, INDEXW,
     $                    WORK, IWORK, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          ORDER, RANGE
      INTEGER            IL, INFO, IU, M, N, NSPLIT
      DOUBLE PRECISION    PIVMIN, RELTOL, VL, VU, WL, WU
*     ..
*     .. Array Arguments ..
      INTEGER            IBLOCK( * ), INDEXW( * ),
     $                   ISPLIT( * ), IWORK( * )
      DOUBLE PRECISION   D( * ), E( * ), E2( * ),
     $                   GERS( * ), W( * ), WERR( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO, HALF, FUDGE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0,
     $                     TWO = 2.0D0, HALF = ONE/TWO,
     $                     FUDGE = TWO )
      INTEGER   ALLRNG, VALRNG, INDRNG
      PARAMETER ( ALLRNG = 1, VALRNG = 2, INDRNG = 3 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NCNVRG, TOOFEW
      INTEGER            I, IB, IBEGIN, IDISCL, IDISCU, IE, IEND, IINFO,
     $                   IM, IN, IOFF, IOUT, IRANGE, ITMAX, ITMP1,
     $                   ITMP2, IW, IWOFF, J, JBLK, JDISC, JE, JEE, NB,
     $                   NWL, NWU
      DOUBLE PRECISION   ATOLI, EPS, GL, GU, RTOLI, TMP1, TMP2,
     $                   TNORM, UFLOW, WKILL, WLU, WUL

*     ..
*     .. Local Arrays ..
      INTEGER            IDUMMA( 1 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, ILAENV, DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLAEBZ
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, INT, LOG, MAX, MIN
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      M = 0
*
*     Quick return if possible
*
      IF( N.LE.0 ) THEN
         RETURN
      END IF
*
*     Decode RANGE
*
      IF( LSAME( RANGE, 'A' ) ) THEN
         IRANGE = ALLRNG
      ELSE IF( LSAME( RANGE, 'V' ) ) THEN
         IRANGE = VALRNG
      ELSE IF( LSAME( RANGE, 'I' ) ) THEN
         IRANGE = INDRNG
      ELSE
         IRANGE = 0
      END IF
*
*     Check for Errors
*
      IF( IRANGE.LE.0 ) THEN
         INFO = -1
      ELSE IF( .NOT.(LSAME(ORDER,'B').OR.LSAME(ORDER,'E')) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( IRANGE.EQ.VALRNG ) THEN
         IF( VL.GE.VU )
     $      INFO = -5
      ELSE IF( IRANGE.EQ.INDRNG .AND.
     $        ( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) ) THEN
         INFO = -6
      ELSE IF( IRANGE.EQ.INDRNG .AND.
     $        ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) THEN
         INFO = -7
      END IF
*
      IF( INFO.NE.0 ) THEN
         RETURN
      END IF

*     Initialize error flags
      NCNVRG = .FALSE.
      TOOFEW = .FALSE.

*     Simplification:
      IF( IRANGE.EQ.INDRNG .AND. IL.EQ.1 .AND. IU.EQ.N ) IRANGE = 1

*     Get machine constants
      EPS = DLAMCH( 'P' )
      UFLOW = DLAMCH( 'U' )


*     Special Case when N=1
*     Treat case of 1x1 matrix for quick return
      IF( N.EQ.1 ) THEN
         IF( (IRANGE.EQ.ALLRNG).OR.
     $       ((IRANGE.EQ.VALRNG).AND.(D(1).GT.VL).AND.(D(1).LE.VU)).OR.
     $       ((IRANGE.EQ.INDRNG).AND.(IL.EQ.1).AND.(IU.EQ.1)) ) THEN
            M = 1
            W(1) = D(1)
*           The computation error of the eigenvalue is zero
            WERR(1) = ZERO
            IBLOCK( 1 ) = 1
            INDEXW( 1 ) = 1
         ENDIF
         RETURN
      END IF

*     NB is the minimum vector length for vector bisection, or 0
*     if only scalar is to be done.
      NB = ILAENV( 1, 'DSTEBZ', ' ', N, -1, -1, -1 )
      IF( NB.LE.1 ) NB = 0

*     Find global spectral radius
      GL = D(1)
      GU = D(1)
      DO 5 I = 1,N
         GL =  MIN( GL, GERS( 2*I - 1))
         GU = MAX( GU, GERS(2*I) )
 5    CONTINUE
*     Compute global Gerschgorin bounds and spectral diameter
      TNORM = MAX( ABS( GL ), ABS( GU ) )
      GL = GL - FUDGE*TNORM*EPS*N - FUDGE*TWO*PIVMIN
      GU = GU + FUDGE*TNORM*EPS*N + FUDGE*TWO*PIVMIN
*     [JAN/28/2009] remove the line below since SPDIAM variable not use
*     SPDIAM = GU - GL
*     Input arguments for DLAEBZ:
*     The relative tolerance.  An interval (a,b] lies within
*     "relative tolerance" if  b-a < RELTOL*max(|a|,|b|),
      RTOLI = RELTOL
*     Set the absolute tolerance for interval convergence to zero to force
*     interval convergence based on relative size of the interval.
*     This is dangerous because intervals might not converge when RELTOL is
*     small. But at least a very small number should be selected so that for
*     strongly graded matrices, the code can get relatively accurate
*     eigenvalues.
      ATOLI = FUDGE*TWO*UFLOW + FUDGE*TWO*PIVMIN

      IF( IRANGE.EQ.INDRNG ) THEN

*        RANGE='I': Compute an interval containing eigenvalues
*        IL through IU. The initial interval [GL,GU] from the global
*        Gerschgorin bounds GL and GU is refined by DLAEBZ.
         ITMAX = INT( ( LOG( TNORM+PIVMIN )-LOG( PIVMIN ) ) /
     $           LOG( TWO ) ) + 2
         WORK( N+1 ) = GL
         WORK( N+2 ) = GL
         WORK( N+3 ) = GU
         WORK( N+4 ) = GU
         WORK( N+5 ) = GL
         WORK( N+6 ) = GU
         IWORK( 1 ) = -1
         IWORK( 2 ) = -1
         IWORK( 3 ) = N + 1
         IWORK( 4 ) = N + 1
         IWORK( 5 ) = IL - 1
         IWORK( 6 ) = IU
*
         CALL DLAEBZ( 3, ITMAX, N, 2, 2, NB, ATOLI, RTOLI, PIVMIN,
     $         D, E, E2, IWORK( 5 ), WORK( N+1 ), WORK( N+5 ), IOUT,
     $                IWORK, W, IBLOCK, IINFO )
         IF( IINFO .NE. 0 ) THEN
            INFO = IINFO
            RETURN
         END IF
*        On exit, output intervals may not be ordered by ascending negcount
         IF( IWORK( 6 ).EQ.IU ) THEN
            WL = WORK( N+1 )
            WLU = WORK( N+3 )
            NWL = IWORK( 1 )
            WU = WORK( N+4 )
            WUL = WORK( N+2 )
            NWU = IWORK( 4 )
         ELSE
            WL = WORK( N+2 )
            WLU = WORK( N+4 )
            NWL = IWORK( 2 )
            WU = WORK( N+3 )
            WUL = WORK( N+1 )
            NWU = IWORK( 3 )
         END IF
*        On exit, the interval [WL, WLU] contains a value with negcount NWL,
*        and [WUL, WU] contains a value with negcount NWU.
         IF( NWL.LT.0 .OR. NWL.GE.N .OR. NWU.LT.1 .OR. NWU.GT.N ) THEN
            INFO = 4
            RETURN
         END IF

      ELSEIF( IRANGE.EQ.VALRNG ) THEN
         WL = VL
         WU = VU

      ELSEIF( IRANGE.EQ.ALLRNG ) THEN
         WL = GL
         WU = GU
      ENDIF



*     Find Eigenvalues -- Loop Over blocks and recompute NWL and NWU.
*     NWL accumulates the number of eigenvalues .le. WL,
*     NWU accumulates the number of eigenvalues .le. WU
      M = 0
      IEND = 0
      INFO = 0
      NWL = 0
      NWU = 0
*
      DO 70 JBLK = 1, NSPLIT
         IOFF = IEND
         IBEGIN = IOFF + 1
         IEND = ISPLIT( JBLK )
         IN = IEND - IOFF
*
         IF( IN.EQ.1 ) THEN
*           1x1 block
            IF( WL.GE.D( IBEGIN )-PIVMIN )
     $         NWL = NWL + 1
            IF( WU.GE.D( IBEGIN )-PIVMIN )
     $         NWU = NWU + 1
            IF( IRANGE.EQ.ALLRNG .OR.
     $           ( WL.LT.D( IBEGIN )-PIVMIN
     $             .AND. WU.GE. D( IBEGIN )-PIVMIN ) ) THEN
               M = M + 1
               W( M ) = D( IBEGIN )
               WERR(M) = ZERO
*              The gap for a single block doesn't matter for the later
*              algorithm and is assigned an arbitrary large value
               IBLOCK( M ) = JBLK
               INDEXW( M ) = 1
            END IF

*        Disabled 2x2 case because of a failure on the following matrix
*        RANGE = 'I', IL = IU = 4
*          Original Tridiagonal, d = [
*           -0.150102010615740E+00
*           -0.849897989384260E+00
*           -0.128208148052635E-15
*            0.128257718286320E-15
*          ];
*          e = [
*           -0.357171383266986E+00
*           -0.180411241501588E-15
*           -0.175152352710251E-15
*          ];
*
*         ELSE IF( IN.EQ.2 ) THEN
**           2x2 block
*            DISC = SQRT( (HALF*(D(IBEGIN)-D(IEND)))**2 + E(IBEGIN)**2 )
*            TMP1 = HALF*(D(IBEGIN)+D(IEND))
*            L1 = TMP1 - DISC
*            IF( WL.GE. L1-PIVMIN )
*     $         NWL = NWL + 1
*            IF( WU.GE. L1-PIVMIN )
*     $         NWU = NWU + 1
*            IF( IRANGE.EQ.ALLRNG .OR. ( WL.LT.L1-PIVMIN .AND. WU.GE.
*     $          L1-PIVMIN ) ) THEN
*               M = M + 1
*               W( M ) = L1
**              The uncertainty of eigenvalues of a 2x2 matrix is very small
*               WERR( M ) = EPS * ABS( W( M ) ) * TWO
*               IBLOCK( M ) = JBLK
*               INDEXW( M ) = 1
*            ENDIF
*            L2 = TMP1 + DISC
*            IF( WL.GE. L2-PIVMIN )
*     $         NWL = NWL + 1
*            IF( WU.GE. L2-PIVMIN )
*     $         NWU = NWU + 1
*            IF( IRANGE.EQ.ALLRNG .OR. ( WL.LT.L2-PIVMIN .AND. WU.GE.
*     $          L2-PIVMIN ) ) THEN
*               M = M + 1
*               W( M ) = L2
**              The uncertainty of eigenvalues of a 2x2 matrix is very small
*               WERR( M ) = EPS * ABS( W( M ) ) * TWO
*               IBLOCK( M ) = JBLK
*               INDEXW( M ) = 2
*            ENDIF
         ELSE
*           General Case - block of size IN >= 2
*           Compute local Gerschgorin interval and use it as the initial
*           interval for DLAEBZ
            GU = D( IBEGIN )
            GL = D( IBEGIN )
            TMP1 = ZERO

            DO 40 J = IBEGIN, IEND
               GL =  MIN( GL, GERS( 2*J - 1))
               GU = MAX( GU, GERS(2*J) )
   40       CONTINUE
*           [JAN/28/2009]
*           change SPDIAM by TNORM in lines 2 and 3 thereafter
*           line 1: remove computation of SPDIAM (not useful anymore)
*           SPDIAM = GU - GL
*           GL = GL - FUDGE*SPDIAM*EPS*IN - FUDGE*PIVMIN
*           GU = GU + FUDGE*SPDIAM*EPS*IN + FUDGE*PIVMIN
            GL = GL - FUDGE*TNORM*EPS*IN - FUDGE*PIVMIN
            GU = GU + FUDGE*TNORM*EPS*IN + FUDGE*PIVMIN
*
            IF( IRANGE.GT.1 ) THEN
               IF( GU.LT.WL ) THEN
*                 the local block contains none of the wanted eigenvalues
                  NWL = NWL + IN
                  NWU = NWU + IN
                  GO TO 70
               END IF
*              refine search interval if possible, only range (WL,WU] matters
               GL = MAX( GL, WL )
               GU = MIN( GU, WU )
               IF( GL.GE.GU )
     $            GO TO 70
            END IF

*           Find negcount of initial interval boundaries GL and GU
            WORK( N+1 ) = GL
            WORK( N+IN+1 ) = GU
            CALL DLAEBZ( 1, 0, IN, IN, 1, NB, ATOLI, RTOLI, PIVMIN,
     $                   D( IBEGIN ), E( IBEGIN ), E2( IBEGIN ),
     $                   IDUMMA, WORK( N+1 ), WORK( N+2*IN+1 ), IM,
     $                   IWORK, W( M+1 ), IBLOCK( M+1 ), IINFO )
            IF( IINFO .NE. 0 ) THEN
               INFO = IINFO
               RETURN
            END IF
*
            NWL = NWL + IWORK( 1 )
            NWU = NWU + IWORK( IN+1 )
            IWOFF = M - IWORK( 1 )

*           Compute Eigenvalues
            ITMAX = INT( ( LOG( GU-GL+PIVMIN )-LOG( PIVMIN ) ) /
     $              LOG( TWO ) ) + 2
            CALL DLAEBZ( 2, ITMAX, IN, IN, 1, NB, ATOLI, RTOLI,
     $                   PIVMIN,
     $                   D( IBEGIN ), E( IBEGIN ), E2( IBEGIN ),
     $                   IDUMMA, WORK( N+1 ), WORK( N+2*IN+1 ), IOUT,
     $                   IWORK, W( M+1 ), IBLOCK( M+1 ), IINFO )
            IF( IINFO .NE. 0 ) THEN
               INFO = IINFO
               RETURN
            END IF
*
*           Copy eigenvalues into W and IBLOCK
*           Use -JBLK for block number for unconverged eigenvalues.
*           Loop over the number of output intervals from DLAEBZ
            DO 60 J = 1, IOUT
*              eigenvalue approximation is middle point of interval
               TMP1 = HALF*( WORK( J+N )+WORK( J+IN+N ) )
*              semi length of error interval
               TMP2 = HALF*ABS( WORK( J+N )-WORK( J+IN+N ) )
               IF( J.GT.IOUT-IINFO ) THEN
*                 Flag non-convergence.
                  NCNVRG = .TRUE.
                  IB = -JBLK
               ELSE
                  IB = JBLK
               END IF
               DO 50 JE = IWORK( J ) + 1 + IWOFF,
     $                 IWORK( J+IN ) + IWOFF
                  W( JE ) = TMP1
                  WERR( JE ) = TMP2
                  INDEXW( JE ) = JE - IWOFF
                  IBLOCK( JE ) = IB
   50          CONTINUE
   60       CONTINUE
*
            M = M + IM
         END IF
   70 CONTINUE

*     If RANGE='I', then (WL,WU) contains eigenvalues NWL+1,...,NWU
*     If NWL+1 < IL or NWU > IU, discard extra eigenvalues.
      IF( IRANGE.EQ.INDRNG ) THEN
         IDISCL = IL - 1 - NWL
         IDISCU = NWU - IU
*
         IF( IDISCL.GT.0 ) THEN
            IM = 0
            DO 80 JE = 1, M
*              Remove some of the smallest eigenvalues from the left so that
*              at the end IDISCL =0. Move all eigenvalues up to the left.
               IF( W( JE ).LE.WLU .AND. IDISCL.GT.0 ) THEN
                  IDISCL = IDISCL - 1
               ELSE
                  IM = IM + 1
                  W( IM ) = W( JE )
                  WERR( IM ) = WERR( JE )
                  INDEXW( IM ) = INDEXW( JE )
                  IBLOCK( IM ) = IBLOCK( JE )
               END IF
 80         CONTINUE
            M = IM
         END IF
         IF( IDISCU.GT.0 ) THEN
*           Remove some of the largest eigenvalues from the right so that
*           at the end IDISCU =0. Move all eigenvalues up to the left.
            IM=M+1
            DO 81 JE = M, 1, -1
               IF( W( JE ).GE.WUL .AND. IDISCU.GT.0 ) THEN
                  IDISCU = IDISCU - 1
               ELSE
                  IM = IM - 1
                  W( IM ) = W( JE )
                  WERR( IM ) = WERR( JE )
                  INDEXW( IM ) = INDEXW( JE )
                  IBLOCK( IM ) = IBLOCK( JE )
               END IF
 81         CONTINUE
            JEE = 0
            DO 82 JE = IM, M
               JEE = JEE + 1
               W( JEE ) = W( JE )
               WERR( JEE ) = WERR( JE )
               INDEXW( JEE ) = INDEXW( JE )
               IBLOCK( JEE ) = IBLOCK( JE )
 82         CONTINUE
            M = M-IM+1
         END IF

         IF( IDISCL.GT.0 .OR. IDISCU.GT.0 ) THEN
*           Code to deal with effects of bad arithmetic. (If N(w) is
*           monotone non-decreasing, this should never happen.)
*           Some low eigenvalues to be discarded are not in (WL,WLU],
*           or high eigenvalues to be discarded are not in (WUL,WU]
*           so just kill off the smallest IDISCL/largest IDISCU
*           eigenvalues, by marking the corresponding IBLOCK = 0
            IF( IDISCL.GT.0 ) THEN
               WKILL = WU
               DO 100 JDISC = 1, IDISCL
                  IW = 0
                  DO 90 JE = 1, M
                     IF( IBLOCK( JE ).NE.0 .AND.
     $                    ( W( JE ).LT.WKILL .OR. IW.EQ.0 ) ) THEN
                        IW = JE
                        WKILL = W( JE )
                     END IF
 90               CONTINUE
                  IBLOCK( IW ) = 0
 100           CONTINUE
            END IF
            IF( IDISCU.GT.0 ) THEN
               WKILL = WL
               DO 120 JDISC = 1, IDISCU
                  IW = 0
                  DO 110 JE = 1, M
                     IF( IBLOCK( JE ).NE.0 .AND.
     $                    ( W( JE ).GE.WKILL .OR. IW.EQ.0 ) ) THEN
                        IW = JE
                        WKILL = W( JE )
                     END IF
 110              CONTINUE
                  IBLOCK( IW ) = 0
 120           CONTINUE
            END IF
*           Now erase all eigenvalues with IBLOCK set to zero
            IM = 0
            DO 130 JE = 1, M
               IF( IBLOCK( JE ).NE.0 ) THEN
                  IM = IM + 1
                  W( IM ) = W( JE )
                  WERR( IM ) = WERR( JE )
                  INDEXW( IM ) = INDEXW( JE )
                  IBLOCK( IM ) = IBLOCK( JE )
               END IF
 130        CONTINUE
            M = IM
         END IF
         IF( IDISCL.LT.0 .OR. IDISCU.LT.0 ) THEN
            TOOFEW = .TRUE.
         END IF
      END IF
*
      IF(( IRANGE.EQ.ALLRNG .AND. M.NE.N ).OR.
     $   ( IRANGE.EQ.INDRNG .AND. M.NE.IU-IL+1 ) ) THEN
         TOOFEW = .TRUE.
      END IF

*     If ORDER='B', do nothing the eigenvalues are already sorted by
*        block.
*     If ORDER='E', sort the eigenvalues from smallest to largest

      IF( LSAME(ORDER,'E') .AND. NSPLIT.GT.1 ) THEN
         DO 150 JE = 1, M - 1
            IE = 0
            TMP1 = W( JE )
            DO 140 J = JE + 1, M
               IF( W( J ).LT.TMP1 ) THEN
                  IE = J
                  TMP1 = W( J )
               END IF
  140       CONTINUE
            IF( IE.NE.0 ) THEN
               TMP2 = WERR( IE )
               ITMP1 = IBLOCK( IE )
               ITMP2 = INDEXW( IE )
               W( IE ) = W( JE )
               WERR( IE ) = WERR( JE )
               IBLOCK( IE ) = IBLOCK( JE )
               INDEXW( IE ) = INDEXW( JE )
               W( JE ) = TMP1
               WERR( JE ) = TMP2
               IBLOCK( JE ) = ITMP1
               INDEXW( JE ) = ITMP2
            END IF
  150    CONTINUE
      END IF
*
      INFO = 0
      IF( NCNVRG )
     $   INFO = INFO + 1
      IF( TOOFEW )
     $   INFO = INFO + 2
      RETURN
*
*     End of DLARRD
*
      END
*> \brief \b DLARRE given the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRE + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarre.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarre.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarre.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARRE( RANGE, N, VL, VU, IL, IU, D, E, E2,
*                           RTOL1, RTOL2, SPLTOL, NSPLIT, ISPLIT, M,
*                           W, WERR, WGAP, IBLOCK, INDEXW, GERS, PIVMIN,
*                           WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          RANGE
*       INTEGER            IL, INFO, IU, M, N, NSPLIT
*       DOUBLE PRECISION  PIVMIN, RTOL1, RTOL2, SPLTOL, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            IBLOCK( * ), ISPLIT( * ), IWORK( * ),
*      $                   INDEXW( * )
*       DOUBLE PRECISION   D( * ), E( * ), E2( * ), GERS( * ),
*      $                   W( * ),WERR( * ), WGAP( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> To find the desired eigenvalues of a given real symmetric
*> tridiagonal matrix T, DLARRE sets any "small" off-diagonal
*> elements to zero, and for each unreduced block T_i, it finds
*> (a) a suitable shift at one end of the block's spectrum,
*> (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and
*> (c) eigenvalues of each L_i D_i L_i^T.
*> The representations and eigenvalues found are then used by
*> DSTEMR to compute the eigenvectors of T.
*> The accuracy varies depending on whether bisection is used to
*> find a few eigenvalues or the dqds algorithm (subroutine DLASQ2) to
*> compute all and then discard any unwanted one.
*> As an added benefit, DLARRE also outputs the n
*> Gerschgorin intervals for the matrices L_i D_i L_i^T.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] RANGE
*> \verbatim
*>          RANGE is CHARACTER*1
*>          = 'A': ("All")   all eigenvalues will be found.
*>          = 'V': ("Value") all eigenvalues in the half-open interval
*>                           (VL, VU] will be found.
*>          = 'I': ("Index") the IL-th through IU-th eigenvalues (of the
*>                           entire matrix) will be found.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix. N > 0.
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*>          VL is DOUBLE PRECISION
*>          If RANGE='V', the lower bound for the eigenvalues.
*>          Eigenvalues less than or equal to VL, or greater than VU,
*>          will not be returned.  VL < VU.
*>          If RANGE='I' or ='A', DLARRE computes bounds on the desired
*>          part of the spectrum.
*> \endverbatim
*>
*> \param[in,out] VU
*> \verbatim
*>          VU is DOUBLE PRECISION
*>          If RANGE='V', the upper bound for the eigenvalues.
*>          Eigenvalues less than or equal to VL, or greater than VU,
*>          will not be returned.  VL < VU.
*>          If RANGE='I' or ='A', DLARRE computes bounds on the desired
*>          part of the spectrum.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*>          IL is INTEGER
*>          If RANGE='I', the index of the
*>          smallest eigenvalue to be returned.
*>          1 <= IL <= IU <= N.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*>          IU is INTEGER
*>          If RANGE='I', the index of the
*>          largest eigenvalue to be returned.
*>          1 <= IL <= IU <= N.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, the N diagonal elements of the tridiagonal
*>          matrix T.
*>          On exit, the N diagonal elements of the diagonal
*>          matrices D_i.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N)
*>          On entry, the first (N-1) entries contain the subdiagonal
*>          elements of the tridiagonal matrix T; E(N) need not be set.
*>          On exit, E contains the subdiagonal elements of the unit
*>          bidiagonal matrices L_i. The entries E( ISPLIT( I ) ),
*>          1 <= I <= NSPLIT, contain the base points sigma_i on output.
*> \endverbatim
*>
*> \param[in,out] E2
*> \verbatim
*>          E2 is DOUBLE PRECISION array, dimension (N)
*>          On entry, the first (N-1) entries contain the SQUARES of the
*>          subdiagonal elements of the tridiagonal matrix T;
*>          E2(N) need not be set.
*>          On exit, the entries E2( ISPLIT( I ) ),
*>          1 <= I <= NSPLIT, have been set to zero
*> \endverbatim
*>
*> \param[in] RTOL1
*> \verbatim
*>          RTOL1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] RTOL2
*> \verbatim
*>          RTOL2 is DOUBLE PRECISION
*>           Parameters for bisection.
*>           An interval [LEFT,RIGHT] has converged if
*>           RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
*> \endverbatim
*>
*> \param[in] SPLTOL
*> \verbatim
*>          SPLTOL is DOUBLE PRECISION
*>          The threshold for splitting.
*> \endverbatim
*>
*> \param[out] NSPLIT
*> \verbatim
*>          NSPLIT is INTEGER
*>          The number of blocks T splits into. 1 <= NSPLIT <= N.
*> \endverbatim
*>
*> \param[out] ISPLIT
*> \verbatim
*>          ISPLIT is INTEGER array, dimension (N)
*>          The splitting points, at which T breaks up into blocks.
*>          The first block consists of rows/columns 1 to ISPLIT(1),
*>          the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
*>          etc., and the NSPLIT-th consists of rows/columns
*>          ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The total number of eigenvalues (of all L_i D_i L_i^T)
*>          found.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          The first M elements contain the eigenvalues. The
*>          eigenvalues of each of the blocks, L_i D_i L_i^T, are
*>          sorted in ascending order ( DLARRE may use the
*>          remaining N-M elements as workspace).
*> \endverbatim
*>
*> \param[out] WERR
*> \verbatim
*>          WERR is DOUBLE PRECISION array, dimension (N)
*>          The error bound on the corresponding eigenvalue in W.
*> \endverbatim
*>
*> \param[out] WGAP
*> \verbatim
*>          WGAP is DOUBLE PRECISION array, dimension (N)
*>          The separation from the right neighbor eigenvalue in W.
*>          The gap is only with respect to the eigenvalues of the same block
*>          as each block has its own representation tree.
*>          Exception: at the right end of a block we store the left gap
*> \endverbatim
*>
*> \param[out] IBLOCK
*> \verbatim
*>          IBLOCK is INTEGER array, dimension (N)
*>          The indices of the blocks (submatrices) associated with the
*>          corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
*>          W(i) belongs to the first block from the top, =2 if W(i)
*>          belongs to the second block, etc.
*> \endverbatim
*>
*> \param[out] INDEXW
*> \verbatim
*>          INDEXW is INTEGER array, dimension (N)
*>          The indices of the eigenvalues within each block (submatrix);
*>          for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
*>          i-th eigenvalue W(i) is the 10-th eigenvalue in block 2
*> \endverbatim
*>
*> \param[out] GERS
*> \verbatim
*>          GERS is DOUBLE PRECISION array, dimension (2*N)
*>          The N Gerschgorin intervals (the i-th Gerschgorin interval
*>          is (GERS(2*i-1), GERS(2*i)).
*> \endverbatim
*>
*> \param[out] PIVMIN
*> \verbatim
*>          PIVMIN is DOUBLE PRECISION
*>          The minimum pivot in the Sturm sequence for T.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (6*N)
*>          Workspace.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (5*N)
*>          Workspace.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          > 0:  A problem occurred in DLARRE.
*>          < 0:  One of the called subroutines signaled an internal problem.
*>                Needs inspection of the corresponding parameter IINFO
*>                for further information.
*>
*>          =-1:  Problem in DLARRD.
*>          = 2:  No base representation could be found in MAXTRY iterations.
*>                Increasing MAXTRY and recompilation might be a remedy.
*>          =-3:  Problem in DLARRB when computing the refined root
*>                representation for DLASQ2.
*>          =-4:  Problem in DLARRB when preforming bisection on the
*>                desired part of the spectrum.
*>          =-5:  Problem in DLASQ2.
*>          =-6:  Problem in DLASQ2.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larre
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The base representations are required to suffer very little
*>  element growth and consequently define all their eigenvalues to
*>  high relative accuracy.
*> \endverbatim
*
*> \par Contributors:
*  ==================
*>
*>     Beresford Parlett, University of California, Berkeley, USA \n
*>     Jim Demmel, University of California, Berkeley, USA \n
*>     Inderjit Dhillon, University of Texas, Austin, USA \n
*>     Osni Marques, LBNL/NERSC, USA \n
*>     Christof Voemel, University of California, Berkeley, USA \n
*>
*  =====================================================================
      SUBROUTINE DLARRE( RANGE, N, VL, VU, IL, IU, D, E, E2,
     $                    RTOL1, RTOL2, SPLTOL, NSPLIT, ISPLIT, M,
     $                    W, WERR, WGAP, IBLOCK, INDEXW, GERS, PIVMIN,
     $                    WORK, IWORK, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          RANGE
      INTEGER            IL, INFO, IU, M, N, NSPLIT
      DOUBLE PRECISION  PIVMIN, RTOL1, RTOL2, SPLTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            IBLOCK( * ), ISPLIT( * ), IWORK( * ),
     $                   INDEXW( * )
      DOUBLE PRECISION   D( * ), E( * ), E2( * ), GERS( * ),
     $                   W( * ),WERR( * ), WGAP( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   FAC, FOUR, FOURTH, FUDGE, HALF, HNDRD,
     $                   MAXGROWTH, ONE, PERT, TWO, ZERO
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0,
     $                     TWO = 2.0D0, FOUR=4.0D0,
     $                     HNDRD = 100.0D0,
     $                     PERT = 8.0D0,
     $                     HALF = ONE/TWO, FOURTH = ONE/FOUR, FAC= HALF,
     $                     MAXGROWTH = 64.0D0, FUDGE = 2.0D0 )
      INTEGER            MAXTRY, ALLRNG, INDRNG, VALRNG
      PARAMETER          ( MAXTRY = 6, ALLRNG = 1, INDRNG = 2,
     $                     VALRNG = 3 )
*     ..
*     .. Local Scalars ..
      LOGICAL            FORCEB, NOREP, USEDQD
      INTEGER            CNT, CNT1, CNT2, I, IBEGIN, IDUM, IEND, IINFO,
     $                   IN, INDL, INDU, IRANGE, J, JBLK, MB, MM,
     $                   WBEGIN, WEND
      DOUBLE PRECISION   AVGAP, BSRTOL, CLWDTH, DMAX, DPIVOT, EABS,
     $                   EMAX, EOLD, EPS, GL, GU, ISLEFT, ISRGHT, RTL,
     $                   RTOL, S1, S2, SAFMIN, SGNDEF, SIGMA, SPDIAM,
     $                   TAU, TMP, TMP1


*     ..
*     .. Local Arrays ..
      INTEGER            ISEED( 4 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION            DLAMCH
      EXTERNAL           DLAMCH, LSAME

*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLARNV, DLARRA, DLARRB, DLARRC,
     $                   DLARRD,
     $                   DLASQ2, DLARRK
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN

*     ..
*     .. Executable Statements ..
*

      INFO = 0
      NSPLIT = 0
      M = 0
*
*     Quick return if possible
*
      IF( N.LE.0 ) THEN
         RETURN
      END IF
*
*     Decode RANGE
*
      IF( LSAME( RANGE, 'A' ) ) THEN
         IRANGE = ALLRNG
      ELSE IF( LSAME( RANGE, 'V' ) ) THEN
         IRANGE = VALRNG
      ELSE IF( LSAME( RANGE, 'I' ) ) THEN
         IRANGE = INDRNG
      END IF

*     Get machine constants
      SAFMIN = DLAMCH( 'S' )
      EPS = DLAMCH( 'P' )

*     Set parameters
      RTL = SQRT(EPS)
      BSRTOL = SQRT(EPS)

*     Treat case of 1x1 matrix for quick return
      IF( N.EQ.1 ) THEN
         IF( (IRANGE.EQ.ALLRNG).OR.
     $       ((IRANGE.EQ.VALRNG).AND.(D(1).GT.VL).AND.(D(1).LE.VU)).OR.
     $       ((IRANGE.EQ.INDRNG).AND.(IL.EQ.1).AND.(IU.EQ.1)) ) THEN
            M = 1
            W(1) = D(1)
*           The computation error of the eigenvalue is zero
            WERR(1) = ZERO
            WGAP(1) = ZERO
            IBLOCK( 1 ) = 1
            INDEXW( 1 ) = 1
            GERS(1) = D( 1 )
            GERS(2) = D( 1 )
         ENDIF
*        store the shift for the initial RRR, which is zero in this case
         E(1) = ZERO
         RETURN
      END IF

*     General case: tridiagonal matrix of order > 1
*
*     Init WERR, WGAP. Compute Gerschgorin intervals and spectral diameter.
*     Compute maximum off-diagonal entry and pivmin.
      GL = D(1)
      GU = D(1)
      EOLD = ZERO
      EMAX = ZERO
      E(N) = ZERO
      DO 5 I = 1,N
         WERR(I) = ZERO
         WGAP(I) = ZERO
         EABS = ABS( E(I) )
         IF( EABS .GE. EMAX ) THEN
            EMAX = EABS
         END IF
         TMP1 = EABS + EOLD
         GERS( 2*I-1) = D(I) - TMP1
         GL =  MIN( GL, GERS( 2*I - 1))
         GERS( 2*I ) = D(I) + TMP1
         GU = MAX( GU, GERS(2*I) )
         EOLD  = EABS
 5    CONTINUE
*     The minimum pivot allowed in the Sturm sequence for T
      PIVMIN = SAFMIN * MAX( ONE, EMAX**2 )
*     Compute spectral diameter. The Gerschgorin bounds give an
*     estimate that is wrong by at most a factor of SQRT(2)
      SPDIAM = GU - GL

*     Compute splitting points
      CALL DLARRA( N, D, E, E2, SPLTOL, SPDIAM,
     $                    NSPLIT, ISPLIT, IINFO )

*     Can force use of bisection instead of faster DQDS.
*     Option left in the code for future multisection work.
      FORCEB = .FALSE.

*     Initialize USEDQD, DQDS should be used for ALLRNG unless someone
*     explicitly wants bisection.
      USEDQD = (( IRANGE.EQ.ALLRNG ) .AND. (.NOT.FORCEB))

      IF( (IRANGE.EQ.ALLRNG) .AND. (.NOT. FORCEB) ) THEN
*        Set interval [VL,VU] that contains all eigenvalues
         VL = GL
         VU = GU
      ELSE
*        We call DLARRD to find crude approximations to the eigenvalues
*        in the desired range. In case IRANGE = INDRNG, we also obtain the
*        interval (VL,VU] that contains all the wanted eigenvalues.
*        An interval [LEFT,RIGHT] has converged if
*        RIGHT-LEFT.LT.RTOL*MAX(ABS(LEFT),ABS(RIGHT))
*        DLARRD needs a WORK of size 4*N, IWORK of size 3*N
         CALL DLARRD( RANGE, 'B', N, VL, VU, IL, IU, GERS,
     $                    BSRTOL, D, E, E2, PIVMIN, NSPLIT, ISPLIT,
     $                    MM, W, WERR, VL, VU, IBLOCK, INDEXW,
     $                    WORK, IWORK, IINFO )
         IF( IINFO.NE.0 ) THEN
            INFO = -1
            RETURN
         ENDIF
*        Make sure that the entries M+1 to N in W, WERR, IBLOCK, INDEXW are 0
         DO 14 I = MM+1,N
            W( I ) = ZERO
            WERR( I ) = ZERO
            IBLOCK( I ) = 0
            INDEXW( I ) = 0
 14      CONTINUE
      END IF


***
*     Loop over unreduced blocks
      IBEGIN = 1
      WBEGIN = 1
      DO 170 JBLK = 1, NSPLIT
         IEND = ISPLIT( JBLK )
         IN = IEND - IBEGIN + 1

*        1 X 1 block
         IF( IN.EQ.1 ) THEN
            IF( (IRANGE.EQ.ALLRNG).OR.( (IRANGE.EQ.VALRNG).AND.
     $         ( D( IBEGIN ).GT.VL ).AND.( D( IBEGIN ).LE.VU ) )
     $        .OR. ( (IRANGE.EQ.INDRNG).AND.(IBLOCK(WBEGIN).EQ.JBLK))
     $        ) THEN
               M = M + 1
               W( M ) = D( IBEGIN )
               WERR(M) = ZERO
*              The gap for a single block doesn't matter for the later
*              algorithm and is assigned an arbitrary large value
               WGAP(M) = ZERO
               IBLOCK( M ) = JBLK
               INDEXW( M ) = 1
               WBEGIN = WBEGIN + 1
            ENDIF
*           E( IEND ) holds the shift for the initial RRR
            E( IEND ) = ZERO
            IBEGIN = IEND + 1
            GO TO 170
         END IF
*
*        Blocks of size larger than 1x1
*
*        E( IEND ) will hold the shift for the initial RRR, for now set it =0
         E( IEND ) = ZERO
*
*        Find local outer bounds GL,GU for the block
         GL = D(IBEGIN)
         GU = D(IBEGIN)
         DO 15 I = IBEGIN , IEND
            GL = MIN( GERS( 2*I-1 ), GL )
            GU = MAX( GERS( 2*I ), GU )
 15      CONTINUE
         SPDIAM = GU - GL

         IF(.NOT. ((IRANGE.EQ.ALLRNG).AND.(.NOT.FORCEB)) ) THEN
*           Count the number of eigenvalues in the current block.
            MB = 0
            DO 20 I = WBEGIN,MM
               IF( IBLOCK(I).EQ.JBLK ) THEN
                  MB = MB+1
               ELSE
                  GOTO 21
               ENDIF
 20         CONTINUE
 21         CONTINUE

            IF( MB.EQ.0) THEN
*              No eigenvalue in the current block lies in the desired range
*              E( IEND ) holds the shift for the initial RRR
               E( IEND ) = ZERO
               IBEGIN = IEND + 1
               GO TO 170
            ELSE

*              Decide whether dqds or bisection is more efficient
               USEDQD = ( (MB .GT. FAC*IN) .AND. (.NOT.FORCEB) )
               WEND = WBEGIN + MB - 1
*              Calculate gaps for the current block
*              In later stages, when representations for individual
*              eigenvalues are different, we use SIGMA = E( IEND ).
               SIGMA = ZERO
               DO 30 I = WBEGIN, WEND - 1
                  WGAP( I ) = MAX( ZERO,
     $                        W(I+1)-WERR(I+1) - (W(I)+WERR(I)) )
 30            CONTINUE
               WGAP( WEND ) = MAX( ZERO,
     $                     VU - SIGMA - (W( WEND )+WERR( WEND )))
*              Find local index of the first and last desired evalue.
               INDL = INDEXW(WBEGIN)
               INDU = INDEXW( WEND )
            ENDIF
         ENDIF
         IF(( (IRANGE.EQ.ALLRNG) .AND. (.NOT. FORCEB) ).OR.USEDQD) THEN
*           Case of DQDS
*           Find approximations to the extremal eigenvalues of the block
            CALL DLARRK( IN, 1, GL, GU, D(IBEGIN),
     $               E2(IBEGIN), PIVMIN, RTL, TMP, TMP1, IINFO )
            IF( IINFO.NE.0 ) THEN
               INFO = -1
               RETURN
            ENDIF
            ISLEFT = MAX(GL, TMP - TMP1
     $               - HNDRD * EPS* ABS(TMP - TMP1))

            CALL DLARRK( IN, IN, GL, GU, D(IBEGIN),
     $               E2(IBEGIN), PIVMIN, RTL, TMP, TMP1, IINFO )
            IF( IINFO.NE.0 ) THEN
               INFO = -1
               RETURN
            ENDIF
            ISRGHT = MIN(GU, TMP + TMP1
     $                 + HNDRD * EPS * ABS(TMP + TMP1))
*           Improve the estimate of the spectral diameter
            SPDIAM = ISRGHT - ISLEFT
         ELSE
*           Case of bisection
*           Find approximations to the wanted extremal eigenvalues
            ISLEFT = MAX(GL, W(WBEGIN) - WERR(WBEGIN)
     $                  - HNDRD * EPS*ABS(W(WBEGIN)- WERR(WBEGIN) ))
            ISRGHT = MIN(GU,W(WEND) + WERR(WEND)
     $                  + HNDRD * EPS * ABS(W(WEND)+ WERR(WEND)))
         ENDIF


*        Decide whether the base representation for the current block
*        L_JBLK D_JBLK L_JBLK^T = T_JBLK - sigma_JBLK I
*        should be on the left or the right end of the current block.
*        The strategy is to shift to the end which is "more populated"
*        Furthermore, decide whether to use DQDS for the computation of
*        the eigenvalue approximations at the end of DLARRE or bisection.
*        dqds is chosen if all eigenvalues are desired or the number of
*        eigenvalues to be computed is large compared to the blocksize.
         IF( ( IRANGE.EQ.ALLRNG ) .AND. (.NOT.FORCEB) ) THEN
*           If all the eigenvalues have to be computed, we use dqd
            USEDQD = .TRUE.
*           INDL is the local index of the first eigenvalue to compute
            INDL = 1
            INDU = IN
*           MB =  number of eigenvalues to compute
            MB = IN
            WEND = WBEGIN + MB - 1
*           Define 1/4 and 3/4 points of the spectrum
            S1 = ISLEFT + FOURTH * SPDIAM
            S2 = ISRGHT - FOURTH * SPDIAM
         ELSE
*           DLARRD has computed IBLOCK and INDEXW for each eigenvalue
*           approximation.
*           choose sigma
            IF( USEDQD ) THEN
               S1 = ISLEFT + FOURTH * SPDIAM
               S2 = ISRGHT - FOURTH * SPDIAM
            ELSE
               TMP = MIN(ISRGHT,VU) -  MAX(ISLEFT,VL)
               S1 =  MAX(ISLEFT,VL) + FOURTH * TMP
               S2 =  MIN(ISRGHT,VU) - FOURTH * TMP
            ENDIF
         ENDIF

*        Compute the negcount at the 1/4 and 3/4 points
         IF(MB.GT.1) THEN
            CALL DLARRC( 'T', IN, S1, S2, D(IBEGIN),
     $                    E(IBEGIN), PIVMIN, CNT, CNT1, CNT2, IINFO)
         ENDIF

         IF(MB.EQ.1) THEN
            SIGMA = GL
            SGNDEF = ONE
         ELSEIF( CNT1 - INDL .GE. INDU - CNT2 ) THEN
            IF( ( IRANGE.EQ.ALLRNG ) .AND. (.NOT.FORCEB) ) THEN
               SIGMA = MAX(ISLEFT,GL)
            ELSEIF( USEDQD ) THEN
*              use Gerschgorin bound as shift to get pos def matrix
*              for dqds
               SIGMA = ISLEFT
            ELSE
*              use approximation of the first desired eigenvalue of the
*              block as shift
               SIGMA = MAX(ISLEFT,VL)
            ENDIF
            SGNDEF = ONE
         ELSE
            IF( ( IRANGE.EQ.ALLRNG ) .AND. (.NOT.FORCEB) ) THEN
               SIGMA = MIN(ISRGHT,GU)
            ELSEIF( USEDQD ) THEN
*              use Gerschgorin bound as shift to get neg def matrix
*              for dqds
               SIGMA = ISRGHT
            ELSE
*              use approximation of the first desired eigenvalue of the
*              block as shift
               SIGMA = MIN(ISRGHT,VU)
            ENDIF
            SGNDEF = -ONE
         ENDIF


*        An initial SIGMA has been chosen that will be used for computing
*        T - SIGMA I = L D L^T
*        Define the increment TAU of the shift in case the initial shift
*        needs to be refined to obtain a factorization with not too much
*        element growth.
         IF( USEDQD ) THEN
*           The initial SIGMA was to the outer end of the spectrum
*           the matrix is definite and we need not retreat.
            TAU = SPDIAM*EPS*N + TWO*PIVMIN
            TAU = MAX( TAU,TWO*EPS*ABS(SIGMA) )
         ELSE
            IF(MB.GT.1) THEN
               CLWDTH = W(WEND) + WERR(WEND) - W(WBEGIN) - WERR(WBEGIN)
               AVGAP = ABS(CLWDTH / DBLE(WEND-WBEGIN))
               IF( SGNDEF.EQ.ONE ) THEN
                  TAU = HALF*MAX(WGAP(WBEGIN),AVGAP)
                  TAU = MAX(TAU,WERR(WBEGIN))
               ELSE
                  TAU = HALF*MAX(WGAP(WEND-1),AVGAP)
                  TAU = MAX(TAU,WERR(WEND))
               ENDIF
            ELSE
               TAU = WERR(WBEGIN)
            ENDIF
         ENDIF
*
         DO 80 IDUM = 1, MAXTRY
*           Compute L D L^T factorization of tridiagonal matrix T - sigma I.
*           Store D in WORK(1:IN), L in WORK(IN+1:2*IN), and reciprocals of
*           pivots in WORK(2*IN+1:3*IN)
            DPIVOT = D( IBEGIN ) - SIGMA
            WORK( 1 ) = DPIVOT
            DMAX = ABS( WORK(1) )
            J = IBEGIN
            DO 70 I = 1, IN - 1
               WORK( 2*IN+I ) = ONE / WORK( I )
               TMP = E( J )*WORK( 2*IN+I )
               WORK( IN+I ) = TMP
               DPIVOT = ( D( J+1 )-SIGMA ) - TMP*E( J )
               WORK( I+1 ) = DPIVOT
               DMAX = MAX( DMAX, ABS(DPIVOT) )
               J = J + 1
 70         CONTINUE
*           check for element growth
            IF( DMAX .GT. MAXGROWTH*SPDIAM ) THEN
               NOREP = .TRUE.
            ELSE
               NOREP = .FALSE.
            ENDIF
            IF( USEDQD .AND. .NOT.NOREP ) THEN
*              Ensure the definiteness of the representation
*              All entries of D (of L D L^T) must have the same sign
               DO 71 I = 1, IN
                  TMP = SGNDEF*WORK( I )
                  IF( TMP.LT.ZERO ) NOREP = .TRUE.
 71            CONTINUE
            ENDIF
            IF(NOREP) THEN
*              Note that in the case of IRANGE=ALLRNG, we use the Gerschgorin
*              shift which makes the matrix definite. So we should end up
*              here really only in the case of IRANGE = VALRNG or INDRNG.
               IF( IDUM.EQ.MAXTRY-1 ) THEN
                  IF( SGNDEF.EQ.ONE ) THEN
*                    The fudged Gerschgorin shift should succeed
                     SIGMA =
     $                    GL - FUDGE*SPDIAM*EPS*N - FUDGE*TWO*PIVMIN
                  ELSE
                     SIGMA =
     $                    GU + FUDGE*SPDIAM*EPS*N + FUDGE*TWO*PIVMIN
                  END IF
               ELSE
                  SIGMA = SIGMA - SGNDEF * TAU
                  TAU = TWO * TAU
               END IF
            ELSE
*              an initial RRR is found
               GO TO 83
            END IF
 80      CONTINUE
*        if the program reaches this point, no base representation could be
*        found in MAXTRY iterations.
         INFO = 2
         RETURN

 83      CONTINUE
*        At this point, we have found an initial base representation
*        T - SIGMA I = L D L^T with not too much element growth.
*        Store the shift.
         E( IEND ) = SIGMA
*        Store D and L.
         CALL DCOPY( IN, WORK, 1, D( IBEGIN ), 1 )
         CALL DCOPY( IN-1, WORK( IN+1 ), 1, E( IBEGIN ), 1 )


         IF(MB.GT.1 ) THEN
*
*           Perturb each entry of the base representation by a small
*           (but random) relative amount to overcome difficulties with
*           glued matrices.
*
            DO 122 I = 1, 4
               ISEED( I ) = 1
 122        CONTINUE

            CALL DLARNV(2, ISEED, 2*IN-1, WORK(1))
            DO 125 I = 1,IN-1
               D(IBEGIN+I-1) = D(IBEGIN+I-1)*(ONE+EPS*PERT*WORK(I))
               E(IBEGIN+I-1) = E(IBEGIN+I-1)*(ONE+EPS*PERT*WORK(IN+I))
 125        CONTINUE
            D(IEND) = D(IEND)*(ONE+EPS*FOUR*WORK(IN))
*
         ENDIF
*
*        Don't update the Gerschgorin intervals because keeping track
*        of the updates would be too much work in DLARRV.
*        We update W instead and use it to locate the proper Gerschgorin
*        intervals.

*        Compute the required eigenvalues of L D L' by bisection or dqds
         IF ( .NOT.USEDQD ) THEN
*           If DLARRD has been used, shift the eigenvalue approximations
*           according to their representation. This is necessary for
*           a uniform DLARRV since dqds computes eigenvalues of the
*           shifted representation. In DLARRV, W will always hold the
*           UNshifted eigenvalue approximation.
            DO 134 J=WBEGIN,WEND
               W(J) = W(J) - SIGMA
               WERR(J) = WERR(J) + ABS(W(J)) * EPS
 134        CONTINUE
*           call DLARRB to reduce eigenvalue error of the approximations
*           from DLARRD
            DO 135 I = IBEGIN, IEND-1
               WORK( I ) = D( I ) * E( I )**2
 135        CONTINUE
*           use bisection to find EV from INDL to INDU
            CALL DLARRB(IN, D(IBEGIN), WORK(IBEGIN),
     $                  INDL, INDU, RTOL1, RTOL2, INDL-1,
     $                  W(WBEGIN), WGAP(WBEGIN), WERR(WBEGIN),
     $                  WORK( 2*N+1 ), IWORK, PIVMIN, SPDIAM,
     $                  IN, IINFO )
            IF( IINFO .NE. 0 ) THEN
               INFO = -4
               RETURN
            END IF
*           DLARRB computes all gaps correctly except for the last one
*           Record distance to VU/GU
            WGAP( WEND ) = MAX( ZERO,
     $           ( VU-SIGMA ) - ( W( WEND ) + WERR( WEND ) ) )
            DO 138 I = INDL, INDU
               M = M + 1
               IBLOCK(M) = JBLK
               INDEXW(M) = I
 138        CONTINUE
         ELSE
*           Call dqds to get all eigs (and then possibly delete unwanted
*           eigenvalues).
*           Note that dqds finds the eigenvalues of the L D L^T representation
*           of T to high relative accuracy. High relative accuracy
*           might be lost when the shift of the RRR is subtracted to obtain
*           the eigenvalues of T. However, T is not guaranteed to define its
*           eigenvalues to high relative accuracy anyway.
*           Set RTOL to the order of the tolerance used in DLASQ2
*           This is an ESTIMATED error, the worst case bound is 4*N*EPS
*           which is usually too large and requires unnecessary work to be
*           done by bisection when computing the eigenvectors
            RTOL = LOG(DBLE(IN)) * FOUR * EPS
            J = IBEGIN
            DO 140 I = 1, IN - 1
               WORK( 2*I-1 ) = ABS( D( J ) )
               WORK( 2*I ) = E( J )*E( J )*WORK( 2*I-1 )
               J = J + 1
  140       CONTINUE
            WORK( 2*IN-1 ) = ABS( D( IEND ) )
            WORK( 2*IN ) = ZERO
            CALL DLASQ2( IN, WORK, IINFO )
            IF( IINFO .NE. 0 ) THEN
*              If IINFO = -5 then an index is part of a tight cluster
*              and should be changed. The index is in IWORK(1) and the
*              gap is in WORK(N+1)
               INFO = -5
               RETURN
            ELSE
*              Test that all eigenvalues are positive as expected
               DO 149 I = 1, IN
                  IF( WORK( I ).LT.ZERO ) THEN
                     INFO = -6
                     RETURN
                  ENDIF
 149           CONTINUE
            END IF
            IF( SGNDEF.GT.ZERO ) THEN
               DO 150 I = INDL, INDU
                  M = M + 1
                  W( M ) = WORK( IN-I+1 )
                  IBLOCK( M ) = JBLK
                  INDEXW( M ) = I
 150           CONTINUE
            ELSE
               DO 160 I = INDL, INDU
                  M = M + 1
                  W( M ) = -WORK( I )
                  IBLOCK( M ) = JBLK
                  INDEXW( M ) = I
 160           CONTINUE
            END IF

            DO 165 I = M - MB + 1, M
*              the value of RTOL below should be the tolerance in DLASQ2
               WERR( I ) = RTOL * ABS( W(I) )
 165        CONTINUE
            DO 166 I = M - MB + 1, M - 1
*              compute the right gap between the intervals
               WGAP( I ) = MAX( ZERO,
     $                          W(I+1)-WERR(I+1) - (W(I)+WERR(I)) )
 166        CONTINUE
            WGAP( M ) = MAX( ZERO,
     $           ( VU-SIGMA ) - ( W( M ) + WERR( M ) ) )
         END IF
*        proceed with next block
         IBEGIN = IEND + 1
         WBEGIN = WEND + 1
 170  CONTINUE
*

      RETURN
*
*     End of DLARRE
*
      END
*> \brief \b DLARRF finds a new relatively robust representation such that at least one of the eigenvalues is relatively isolated.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARRF( N, D, L, LD, CLSTRT, CLEND,
*                          W, WGAP, WERR,
*                          SPDIAM, CLGAPL, CLGAPR, PIVMIN, SIGMA,
*                          DPLUS, LPLUS, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            CLSTRT, CLEND, INFO, N
*       DOUBLE PRECISION   CLGAPL, CLGAPR, PIVMIN, SIGMA, SPDIAM
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), DPLUS( * ), L( * ), LD( * ),
*      $          LPLUS( * ), W( * ), WGAP( * ), WERR( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> Given the initial representation L D L^T and its cluster of close
*> eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ...
*> W( CLEND ), DLARRF finds a new relatively robust representation
*> L D L^T - SIGMA I = L(+) D(+) L(+)^T such that at least one of the
*> eigenvalues of L(+) D(+) L(+)^T is relatively isolated.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix (subblock, if the matrix split).
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The N diagonal elements of the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*>          L is DOUBLE PRECISION array, dimension (N-1)
*>          The (N-1) subdiagonal elements of the unit bidiagonal
*>          matrix L.
*> \endverbatim
*>
*> \param[in] LD
*> \verbatim
*>          LD is DOUBLE PRECISION array, dimension (N-1)
*>          The (N-1) elements L(i)*D(i).
*> \endverbatim
*>
*> \param[in] CLSTRT
*> \verbatim
*>          CLSTRT is INTEGER
*>          The index of the first eigenvalue in the cluster.
*> \endverbatim
*>
*> \param[in] CLEND
*> \verbatim
*>          CLEND is INTEGER
*>          The index of the last eigenvalue in the cluster.
*> \endverbatim
*>
*> \param[in] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension
*>          dimension is >=  (CLEND-CLSTRT+1)
*>          The eigenvalue APPROXIMATIONS of L D L^T in ascending order.
*>          W( CLSTRT ) through W( CLEND ) form the cluster of relatively
*>          close eigenalues.
*> \endverbatim
*>
*> \param[in,out] WGAP
*> \verbatim
*>          WGAP is DOUBLE PRECISION array, dimension
*>          dimension is >=  (CLEND-CLSTRT+1)
*>          The separation from the right neighbor eigenvalue in W.
*> \endverbatim
*>
*> \param[in] WERR
*> \verbatim
*>          WERR is DOUBLE PRECISION array, dimension
*>          dimension is  >=  (CLEND-CLSTRT+1)
*>          WERR contain the semiwidth of the uncertainty
*>          interval of the corresponding eigenvalue APPROXIMATION in W
*> \endverbatim
*>
*> \param[in] SPDIAM
*> \verbatim
*>          SPDIAM is DOUBLE PRECISION
*>          estimate of the spectral diameter obtained from the
*>          Gerschgorin intervals
*> \endverbatim
*>
*> \param[in] CLGAPL
*> \verbatim
*>          CLGAPL is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] CLGAPR
*> \verbatim
*>          CLGAPR is DOUBLE PRECISION
*>          absolute gap on each end of the cluster.
*>          Set by the calling routine to protect against shifts too close
*>          to eigenvalues outside the cluster.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*>          PIVMIN is DOUBLE PRECISION
*>          The minimum pivot allowed in the Sturm sequence.
*> \endverbatim
*>
*> \param[out] SIGMA
*> \verbatim
*>          SIGMA is DOUBLE PRECISION
*>          The shift used to form L(+) D(+) L(+)^T.
*> \endverbatim
*>
*> \param[out] DPLUS
*> \verbatim
*>          DPLUS is DOUBLE PRECISION array, dimension (N)
*>          The N diagonal elements of the diagonal matrix D(+).
*> \endverbatim
*>
*> \param[out] LPLUS
*> \verbatim
*>          LPLUS is DOUBLE PRECISION array, dimension (N-1)
*>          The first (N-1) elements of LPLUS contain the subdiagonal
*>          elements of the unit bidiagonal matrix L(+).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (2*N)
*>          Workspace.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          Signals processing OK (=0) or failure (=1)
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larrf
*
*> \par Contributors:
*  ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA
*
*  =====================================================================
      SUBROUTINE DLARRF( N, D, L, LD, CLSTRT, CLEND,
     $                   W, WGAP, WERR,
     $                   SPDIAM, CLGAPL, CLGAPR, PIVMIN, SIGMA,
     $                   DPLUS, LPLUS, WORK, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            CLSTRT, CLEND, INFO, N
      DOUBLE PRECISION   CLGAPL, CLGAPR, PIVMIN, SIGMA, SPDIAM
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), DPLUS( * ), L( * ), LD( * ),
     $          LPLUS( * ), W( * ), WGAP( * ), WERR( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   FOUR, MAXGROWTH1, MAXGROWTH2, ONE, QUART, TWO
      PARAMETER          ( ONE = 1.0D0, TWO = 2.0D0, FOUR = 4.0D0,
     $                     QUART = 0.25D0,
     $                     MAXGROWTH1 = 8.D0,
     $                     MAXGROWTH2 = 8.D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL   DORRR1, FORCER, NOFAIL, SAWNAN1, SAWNAN2, TRYRRR1
      INTEGER            I, INDX, KTRY, KTRYMAX, SLEFT, SRIGHT, SHIFT
      PARAMETER          ( KTRYMAX = 1, SLEFT = 1, SRIGHT = 2 )
      DOUBLE PRECISION   AVGAP, BESTSHIFT, CLWDTH, EPS, FACT, FAIL,
     $                   FAIL2, GROWTHBOUND, LDELTA, LDMAX, LSIGMA,
     $                   MAX1, MAX2, MINGAP, OLDP, PROD, RDELTA, RDMAX,
     $                   RRR1, RRR2, RSIGMA, S, SMLGROWTH, TMP, ZNM2
*     ..
*     .. External Functions ..
      LOGICAL DISNAN
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DISNAN, DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS
*     ..
*     .. Executable Statements ..
*
      INFO = 0
*
*     Quick return if possible
*
      IF( N.LE.0 ) THEN
         RETURN
      END IF
*
      FACT = DBLE(2**KTRYMAX)
      EPS = DLAMCH( 'Precision' )
      SHIFT = 0
      FORCER = .FALSE.


*     Note that we cannot guarantee that for any of the shifts tried,
*     the factorization has a small or even moderate element growth.
*     There could be Ritz values at both ends of the cluster and despite
*     backing off, there are examples where all factorizations tried
*     (in IEEE mode, allowing zero pivots & infinities) have INFINITE
*     element growth.
*     For this reason, we should use PIVMIN in this subroutine so that at
*     least the L D L^T factorization exists. It can be checked afterwards
*     whether the element growth caused bad residuals/orthogonality.

*     Decide whether the code should accept the best among all
*     representations despite large element growth or signal INFO=1
*     Setting NOFAIL to .FALSE. for quick fix for bug 113
      NOFAIL = .FALSE.
*

*     Compute the average gap length of the cluster
      CLWDTH = ABS(W(CLEND)-W(CLSTRT)) + WERR(CLEND) + WERR(CLSTRT)
      AVGAP = CLWDTH / DBLE(CLEND-CLSTRT)
      MINGAP = MIN(CLGAPL, CLGAPR)
*     Initial values for shifts to both ends of cluster
      LSIGMA = MIN(W( CLSTRT ),W( CLEND )) - WERR( CLSTRT )
      RSIGMA = MAX(W( CLSTRT ),W( CLEND )) + WERR( CLEND )

*     Use a small fudge to make sure that we really shift to the outside
      LSIGMA = LSIGMA - ABS(LSIGMA)* FOUR * EPS
      RSIGMA = RSIGMA + ABS(RSIGMA)* FOUR * EPS

*     Compute upper bounds for how much to back off the initial shifts
      LDMAX = QUART * MINGAP + TWO * PIVMIN
      RDMAX = QUART * MINGAP + TWO * PIVMIN

      LDELTA = MAX(AVGAP,WGAP( CLSTRT ))/FACT
      RDELTA = MAX(AVGAP,WGAP( CLEND-1 ))/FACT
*
*     Initialize the record of the best representation found
*
      S = DLAMCH( 'S' )
      SMLGROWTH = ONE / S
      FAIL = DBLE(N-1)*MINGAP/(SPDIAM*EPS)
      FAIL2 = DBLE(N-1)*MINGAP/(SPDIAM*SQRT(EPS))
      BESTSHIFT = LSIGMA
*
*     while (KTRY <= KTRYMAX)
      KTRY = 0
      GROWTHBOUND = MAXGROWTH1*SPDIAM

 5    CONTINUE
      SAWNAN1 = .FALSE.
      SAWNAN2 = .FALSE.
*     Ensure that we do not back off too much of the initial shifts
      LDELTA = MIN(LDMAX,LDELTA)
      RDELTA = MIN(RDMAX,RDELTA)

*     Compute the element growth when shifting to both ends of the cluster
*     accept the shift if there is no element growth at one of the two ends

*     Left end
      S = -LSIGMA
      DPLUS( 1 ) = D( 1 ) + S
      IF(ABS(DPLUS(1)).LT.PIVMIN) THEN
         DPLUS(1) = -PIVMIN
*        Need to set SAWNAN1 because refined RRR test should not be used
*        in this case
         SAWNAN1 = .TRUE.
      ENDIF
      MAX1 = ABS( DPLUS( 1 ) )
      DO 6 I = 1, N - 1
         LPLUS( I ) = LD( I ) / DPLUS( I )
         S = S*LPLUS( I )*L( I ) - LSIGMA
         DPLUS( I+1 ) = D( I+1 ) + S
         IF(ABS(DPLUS(I+1)).LT.PIVMIN) THEN
            DPLUS(I+1) = -PIVMIN
*           Need to set SAWNAN1 because refined RRR test should not be used
*           in this case
            SAWNAN1 = .TRUE.
         ENDIF
         MAX1 = MAX( MAX1,ABS(DPLUS(I+1)) )
 6    CONTINUE
      SAWNAN1 = SAWNAN1 .OR.  DISNAN( MAX1 )

      IF( FORCER .OR.
     $   (MAX1.LE.GROWTHBOUND .AND. .NOT.SAWNAN1 ) ) THEN
         SIGMA = LSIGMA
         SHIFT = SLEFT
         GOTO 100
      ENDIF

*     Right end
      S = -RSIGMA
      WORK( 1 ) = D( 1 ) + S
      IF(ABS(WORK(1)).LT.PIVMIN) THEN
         WORK(1) = -PIVMIN
*        Need to set SAWNAN2 because refined RRR test should not be used
*        in this case
         SAWNAN2 = .TRUE.
      ENDIF
      MAX2 = ABS( WORK( 1 ) )
      DO 7 I = 1, N - 1
         WORK( N+I ) = LD( I ) / WORK( I )
         S = S*WORK( N+I )*L( I ) - RSIGMA
         WORK( I+1 ) = D( I+1 ) + S
         IF(ABS(WORK(I+1)).LT.PIVMIN) THEN
            WORK(I+1) = -PIVMIN
*           Need to set SAWNAN2 because refined RRR test should not be used
*           in this case
            SAWNAN2 = .TRUE.
         ENDIF
         MAX2 = MAX( MAX2,ABS(WORK(I+1)) )
 7    CONTINUE
      SAWNAN2 = SAWNAN2 .OR.  DISNAN( MAX2 )

      IF( FORCER .OR.
     $   (MAX2.LE.GROWTHBOUND .AND. .NOT.SAWNAN2 ) ) THEN
         SIGMA = RSIGMA
         SHIFT = SRIGHT
         GOTO 100
      ENDIF
*     If we are at this point, both shifts led to too much element growth

*     Record the better of the two shifts (provided it didn't lead to NaN)
      IF(SAWNAN1.AND.SAWNAN2) THEN
*        both MAX1 and MAX2 are NaN
         GOTO 50
      ELSE
         IF( .NOT.SAWNAN1 ) THEN
            INDX = 1
            IF(MAX1.LE.SMLGROWTH) THEN
               SMLGROWTH = MAX1
               BESTSHIFT = LSIGMA
            ENDIF
         ENDIF
         IF( .NOT.SAWNAN2 ) THEN
            IF(SAWNAN1 .OR. MAX2.LE.MAX1) INDX = 2
            IF(MAX2.LE.SMLGROWTH) THEN
               SMLGROWTH = MAX2
               BESTSHIFT = RSIGMA
            ENDIF
         ENDIF
      ENDIF

*     If we are here, both the left and the right shift led to
*     element growth. If the element growth is moderate, then
*     we may still accept the representation, if it passes a
*     refined test for RRR. This test supposes that no NaN occurred.
*     Moreover, we use the refined RRR test only for isolated clusters.
      IF((CLWDTH.LT.MINGAP/DBLE(128)) .AND.
     $   (MIN(MAX1,MAX2).LT.FAIL2)
     $  .AND.(.NOT.SAWNAN1).AND.(.NOT.SAWNAN2)) THEN
         DORRR1 = .TRUE.
      ELSE
         DORRR1 = .FALSE.
      ENDIF
      TRYRRR1 = .TRUE.
      IF( TRYRRR1 .AND. DORRR1 ) THEN
      IF(INDX.EQ.1) THEN
         TMP = ABS( DPLUS( N ) )
         ZNM2 = ONE
         PROD = ONE
         OLDP = ONE
         DO 15 I = N-1, 1, -1
            IF( PROD .LE. EPS ) THEN
               PROD =
     $         ((DPLUS(I+1)*WORK(N+I+1))/(DPLUS(I)*WORK(N+I)))*OLDP
            ELSE
               PROD = PROD*ABS(WORK(N+I))
            END IF
            OLDP = PROD
            ZNM2 = ZNM2 + PROD**2
            TMP = MAX( TMP, ABS( DPLUS( I ) * PROD ))
 15      CONTINUE
         RRR1 = TMP/( SPDIAM * SQRT( ZNM2 ) )
         IF (RRR1.LE.MAXGROWTH2) THEN
            SIGMA = LSIGMA
            SHIFT = SLEFT
            GOTO 100
         ENDIF
      ELSE IF(INDX.EQ.2) THEN
         TMP = ABS( WORK( N ) )
         ZNM2 = ONE
         PROD = ONE
         OLDP = ONE
         DO 16 I = N-1, 1, -1
            IF( PROD .LE. EPS ) THEN
               PROD = ((WORK(I+1)*LPLUS(I+1))/(WORK(I)*LPLUS(I)))*OLDP
            ELSE
               PROD = PROD*ABS(LPLUS(I))
            END IF
            OLDP = PROD
            ZNM2 = ZNM2 + PROD**2
            TMP = MAX( TMP, ABS( WORK( I ) * PROD ))
 16      CONTINUE
         RRR2 = TMP/( SPDIAM * SQRT( ZNM2 ) )
         IF (RRR2.LE.MAXGROWTH2) THEN
            SIGMA = RSIGMA
            SHIFT = SRIGHT
            GOTO 100
         ENDIF
      END IF
      ENDIF

 50   CONTINUE

      IF (KTRY.LT.KTRYMAX) THEN
*        If we are here, both shifts failed also the RRR test.
*        Back off to the outside
         LSIGMA = MAX( LSIGMA - LDELTA,
     $     LSIGMA - LDMAX)
         RSIGMA = MIN( RSIGMA + RDELTA,
     $     RSIGMA + RDMAX )
         LDELTA = TWO * LDELTA
         RDELTA = TWO * RDELTA
         KTRY = KTRY + 1
         GOTO 5
      ELSE
*        None of the representations investigated satisfied our
*        criteria. Take the best one we found.
         IF((SMLGROWTH.LT.FAIL).OR.NOFAIL) THEN
            LSIGMA = BESTSHIFT
            RSIGMA = BESTSHIFT
            FORCER = .TRUE.
            GOTO 5
         ELSE
            INFO = 1
            RETURN
         ENDIF
      END IF

 100  CONTINUE
      IF (SHIFT.EQ.SLEFT) THEN
      ELSEIF (SHIFT.EQ.SRIGHT) THEN
*        store new L and D back into DPLUS, LPLUS
         CALL DCOPY( N, WORK, 1, DPLUS, 1 )
         CALL DCOPY( N-1, WORK(N+1), 1, LPLUS, 1 )
      ENDIF

      RETURN
*
*     End of DLARRF
*
      END
*> \brief \b DLARRJ performs refinement of the initial estimates of the eigenvalues of the matrix T.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRJ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrj.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrj.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrj.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARRJ( N, D, E2, IFIRST, ILAST,
*                          RTOL, OFFSET, W, WERR, WORK, IWORK,
*                          PIVMIN, SPDIAM, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            IFIRST, ILAST, INFO, N, OFFSET
*       DOUBLE PRECISION   PIVMIN, RTOL, SPDIAM
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   D( * ), E2( * ), W( * ),
*      $                   WERR( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> Given the initial eigenvalue approximations of T, DLARRJ
*> does  bisection to refine the eigenvalues of T,
*> W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial
*> guesses for these eigenvalues are input in W, the corresponding estimate
*> of the error in these guesses in WERR. During bisection, intervals
*> [left, right] are maintained by storing their mid-points and
*> semi-widths in the arrays W and WERR respectively.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The N diagonal elements of T.
*> \endverbatim
*>
*> \param[in] E2
*> \verbatim
*>          E2 is DOUBLE PRECISION array, dimension (N-1)
*>          The Squares of the (N-1) subdiagonal elements of T.
*> \endverbatim
*>
*> \param[in] IFIRST
*> \verbatim
*>          IFIRST is INTEGER
*>          The index of the first eigenvalue to be computed.
*> \endverbatim
*>
*> \param[in] ILAST
*> \verbatim
*>          ILAST is INTEGER
*>          The index of the last eigenvalue to be computed.
*> \endverbatim
*>
*> \param[in] RTOL
*> \verbatim
*>          RTOL is DOUBLE PRECISION
*>          Tolerance for the convergence of the bisection intervals.
*>          An interval [LEFT,RIGHT] has converged if
*>          RIGHT-LEFT < RTOL*MAX(|LEFT|,|RIGHT|).
*> \endverbatim
*>
*> \param[in] OFFSET
*> \verbatim
*>          OFFSET is INTEGER
*>          Offset for the arrays W and WERR, i.e., the IFIRST-OFFSET
*>          through ILAST-OFFSET elements of these arrays are to be used.
*> \endverbatim
*>
*> \param[in,out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are
*>          estimates of the eigenvalues of L D L^T indexed IFIRST through
*>          ILAST.
*>          On output, these estimates are refined.
*> \endverbatim
*>
*> \param[in,out] WERR
*> \verbatim
*>          WERR is DOUBLE PRECISION array, dimension (N)
*>          On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are
*>          the errors in the estimates of the corresponding elements in W.
*>          On output, these errors are refined.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (2*N)
*>          Workspace.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (2*N)
*>          Workspace.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*>          PIVMIN is DOUBLE PRECISION
*>          The minimum pivot in the Sturm sequence for T.
*> \endverbatim
*>
*> \param[in] SPDIAM
*> \verbatim
*>          SPDIAM is DOUBLE PRECISION
*>          The spectral diameter of T.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          Error flag.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larrj
*
*> \par Contributors:
*  ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA
*
*  =====================================================================
      SUBROUTINE DLARRJ( N, D, E2, IFIRST, ILAST,
     $                   RTOL, OFFSET, W, WERR, WORK, IWORK,
     $                   PIVMIN, SPDIAM, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            IFIRST, ILAST, INFO, N, OFFSET
      DOUBLE PRECISION   PIVMIN, RTOL, SPDIAM
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   D( * ), E2( * ), W( * ),
     $                   WERR( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO, HALF
      PARAMETER        ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
     $                   HALF = 0.5D0 )
      INTEGER   MAXITR
*     ..
*     .. Local Scalars ..
      INTEGER            CNT, I, I1, I2, II, ITER, J, K, NEXT, NINT,
     $                   OLNINT, P, PREV, SAVI1
      DOUBLE PRECISION   DPLUS, FAC, LEFT, MID, RIGHT, S, TMP, WIDTH
*
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. Executable Statements ..
*
      INFO = 0
*
*     Quick return if possible
*
      IF( N.LE.0 ) THEN
         RETURN
      END IF
*
      MAXITR = INT( ( LOG( SPDIAM+PIVMIN )-LOG( PIVMIN ) ) /
     $           LOG( TWO ) ) + 2
*
*     Initialize unconverged intervals in [ WORK(2*I-1), WORK(2*I) ].
*     The Sturm Count, Count( WORK(2*I-1) ) is arranged to be I-1, while
*     Count( WORK(2*I) ) is stored in IWORK( 2*I ). The integer IWORK( 2*I-1 )
*     for an unconverged interval is set to the index of the next unconverged
*     interval, and is -1 or 0 for a converged interval. Thus a linked
*     list of unconverged intervals is set up.
*

      I1 = IFIRST
      I2 = ILAST
*     The number of unconverged intervals
      NINT = 0
*     The last unconverged interval found
      PREV = 0
      DO 75 I = I1, I2
         K = 2*I
         II = I - OFFSET
         LEFT = W( II ) - WERR( II )
         MID = W(II)
         RIGHT = W( II ) + WERR( II )
         WIDTH = RIGHT - MID
         TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )

*        The following test prevents the test of converged intervals
         IF( WIDTH.LT.RTOL*TMP ) THEN
*           This interval has already converged and does not need refinement.
*           (Note that the gaps might change through refining the
*            eigenvalues, however, they can only get bigger.)
*           Remove it from the list.
            IWORK( K-1 ) = -1
*           Make sure that I1 always points to the first unconverged interval
            IF((I.EQ.I1).AND.(I.LT.I2)) I1 = I + 1
            IF((PREV.GE.I1).AND.(I.LE.I2)) IWORK( 2*PREV-1 ) = I + 1
         ELSE
*           unconverged interval found
            PREV = I
*           Make sure that [LEFT,RIGHT] contains the desired eigenvalue
*
*           Do while( CNT(LEFT).GT.I-1 )
*
            FAC = ONE
 20         CONTINUE
            CNT = 0
            S = LEFT
            DPLUS = D( 1 ) - S
            IF( DPLUS.LT.ZERO ) CNT = CNT + 1
            DO 30 J = 2, N
               DPLUS = D( J ) - S - E2( J-1 )/DPLUS
               IF( DPLUS.LT.ZERO ) CNT = CNT + 1
 30         CONTINUE
            IF( CNT.GT.I-1 ) THEN
               LEFT = LEFT - WERR( II )*FAC
               FAC = TWO*FAC
               GO TO 20
            END IF
*
*           Do while( CNT(RIGHT).LT.I )
*
            FAC = ONE
 50         CONTINUE
            CNT = 0
            S = RIGHT
            DPLUS = D( 1 ) - S
            IF( DPLUS.LT.ZERO ) CNT = CNT + 1
            DO 60 J = 2, N
               DPLUS = D( J ) - S - E2( J-1 )/DPLUS
               IF( DPLUS.LT.ZERO ) CNT = CNT + 1
 60         CONTINUE
            IF( CNT.LT.I ) THEN
               RIGHT = RIGHT + WERR( II )*FAC
               FAC = TWO*FAC
               GO TO 50
            END IF
            NINT = NINT + 1
            IWORK( K-1 ) = I + 1
            IWORK( K ) = CNT
         END IF
         WORK( K-1 ) = LEFT
         WORK( K ) = RIGHT
 75   CONTINUE


      SAVI1 = I1
*
*     Do while( NINT.GT.0 ), i.e. there are still unconverged intervals
*     and while (ITER.LT.MAXITR)
*
      ITER = 0
 80   CONTINUE
      PREV = I1 - 1
      I = I1
      OLNINT = NINT

      DO 100 P = 1, OLNINT
         K = 2*I
         II = I - OFFSET
         NEXT = IWORK( K-1 )
         LEFT = WORK( K-1 )
         RIGHT = WORK( K )
         MID = HALF*( LEFT + RIGHT )

*        semiwidth of interval
         WIDTH = RIGHT - MID
         TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )

         IF( ( WIDTH.LT.RTOL*TMP ) .OR.
     $      (ITER.EQ.MAXITR) )THEN
*           reduce number of unconverged intervals
            NINT = NINT - 1
*           Mark interval as converged.
            IWORK( K-1 ) = 0
            IF( I1.EQ.I ) THEN
               I1 = NEXT
            ELSE
*              Prev holds the last unconverged interval previously examined
               IF(PREV.GE.I1) IWORK( 2*PREV-1 ) = NEXT
            END IF
            I = NEXT
            GO TO 100
         END IF
         PREV = I
*
*        Perform one bisection step
*
         CNT = 0
         S = MID
         DPLUS = D( 1 ) - S
         IF( DPLUS.LT.ZERO ) CNT = CNT + 1
         DO 90 J = 2, N
            DPLUS = D( J ) - S - E2( J-1 )/DPLUS
            IF( DPLUS.LT.ZERO ) CNT = CNT + 1
 90      CONTINUE
         IF( CNT.LE.I-1 ) THEN
            WORK( K-1 ) = MID
         ELSE
            WORK( K ) = MID
         END IF
         I = NEXT

 100  CONTINUE
      ITER = ITER + 1
*     do another loop if there are still unconverged intervals
*     However, in the last iteration, all intervals are accepted
*     since this is the best we can do.
      IF( ( NINT.GT.0 ).AND.(ITER.LE.MAXITR) ) GO TO 80
*
*
*     At this point, all the intervals have converged
      DO 110 I = SAVI1, ILAST
         K = 2*I
         II = I - OFFSET
*        All intervals marked by '0' have been refined.
         IF( IWORK( K-1 ).EQ.0 ) THEN
            W( II ) = HALF*( WORK( K-1 )+WORK( K ) )
            WERR( II ) = WORK( K ) - W( II )
         END IF
 110  CONTINUE
*

      RETURN
*
*     End of DLARRJ
*
      END
*> \brief \b DLARRK computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRK + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrk.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrk.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrk.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARRK( N, IW, GL, GU,
*                           D, E2, PIVMIN, RELTOL, W, WERR, INFO)
*
*       .. Scalar Arguments ..
*       INTEGER   INFO, IW, N
*       DOUBLE PRECISION    PIVMIN, RELTOL, GL, GU, W, WERR
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), E2( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARRK computes one eigenvalue of a symmetric tridiagonal
*> matrix T to suitable accuracy. This is an auxiliary code to be
*> called from DSTEMR.
*>
*> To avoid overflow, the matrix must be scaled so that its
*> largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
*> accuracy, it should not be much smaller than that.
*>
*> See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
*> Matrix", Report CS41, Computer Science Dept., Stanford
*> University, July 21, 1966.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the tridiagonal matrix T.  N >= 0.
*> \endverbatim
*>
*> \param[in] IW
*> \verbatim
*>          IW is INTEGER
*>          The index of the eigenvalues to be returned.
*> \endverbatim
*>
*> \param[in] GL
*> \verbatim
*>          GL is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] GU
*> \verbatim
*>          GU is DOUBLE PRECISION
*>          An upper and a lower bound on the eigenvalue.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The n diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] E2
*> \verbatim
*>          E2 is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) squared off-diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*>          PIVMIN is DOUBLE PRECISION
*>          The minimum pivot allowed in the Sturm sequence for T.
*> \endverbatim
*>
*> \param[in] RELTOL
*> \verbatim
*>          RELTOL is DOUBLE PRECISION
*>          The minimum relative width of an interval.  When an interval
*>          is narrower than RELTOL times the larger (in
*>          magnitude) endpoint, then it is considered to be
*>          sufficiently small, i.e., converged.  Note: this should
*>          always be at least radix*machine epsilon.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] WERR
*> \verbatim
*>          WERR is DOUBLE PRECISION
*>          The error bound on the corresponding eigenvalue approximation
*>          in W.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:       Eigenvalue converged
*>          = -1:      Eigenvalue did NOT converge
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  FUDGE   DOUBLE PRECISION, default = 2
*>          A "fudge factor" to widen the Gershgorin intervals.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larrk
*
*  =====================================================================
      SUBROUTINE DLARRK( N, IW, GL, GU,
     $                    D, E2, PIVMIN, RELTOL, W, WERR, INFO)
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER   INFO, IW, N
      DOUBLE PRECISION    PIVMIN, RELTOL, GL, GU, W, WERR
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), E2( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   FUDGE, HALF, TWO, ZERO
      PARAMETER          ( HALF = 0.5D0, TWO = 2.0D0,
     $                     FUDGE = TWO, ZERO = 0.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER   I, IT, ITMAX, NEGCNT
      DOUBLE PRECISION   ATOLI, EPS, LEFT, MID, RIGHT, RTOLI, TMP1,
     $                   TMP2, TNORM
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL   DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, INT, LOG, MAX
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( N.LE.0 ) THEN
         INFO = 0
         RETURN
      END IF
*
*     Get machine constants
      EPS = DLAMCH( 'P' )

      TNORM = MAX( ABS( GL ), ABS( GU ) )
      RTOLI = RELTOL
      ATOLI = FUDGE*TWO*PIVMIN

      ITMAX = INT( ( LOG( TNORM+PIVMIN )-LOG( PIVMIN ) ) /
     $           LOG( TWO ) ) + 2

      INFO = -1

      LEFT = GL - FUDGE*TNORM*EPS*N - FUDGE*TWO*PIVMIN
      RIGHT = GU + FUDGE*TNORM*EPS*N + FUDGE*TWO*PIVMIN
      IT = 0

 10   CONTINUE
*
*     Check if interval converged or maximum number of iterations reached
*
      TMP1 = ABS( RIGHT - LEFT )
      TMP2 = MAX( ABS(RIGHT), ABS(LEFT) )
      IF( TMP1.LT.MAX( ATOLI, PIVMIN, RTOLI*TMP2 ) ) THEN
         INFO = 0
         GOTO 30
      ENDIF
      IF(IT.GT.ITMAX)
     $   GOTO 30

*
*     Count number of negative pivots for mid-point
*
      IT = IT + 1
      MID = HALF * (LEFT + RIGHT)
      NEGCNT = 0
      TMP1 = D( 1 ) - MID
      IF( ABS( TMP1 ).LT.PIVMIN )
     $   TMP1 = -PIVMIN
      IF( TMP1.LE.ZERO )
     $   NEGCNT = NEGCNT + 1
*
      DO 20 I = 2, N
         TMP1 = D( I ) - E2( I-1 ) / TMP1 - MID
         IF( ABS( TMP1 ).LT.PIVMIN )
     $      TMP1 = -PIVMIN
         IF( TMP1.LE.ZERO )
     $      NEGCNT = NEGCNT + 1
 20   CONTINUE

      IF(NEGCNT.GE.IW) THEN
         RIGHT = MID
      ELSE
         LEFT = MID
      ENDIF
      GOTO 10

 30   CONTINUE
*
*     Converged or maximum number of iterations reached
*
      W = HALF * (LEFT + RIGHT)
      WERR = HALF * ABS( RIGHT - LEFT )

      RETURN
*
*     End of DLARRK
*
      END
*> \brief \b DLARRR performs tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARRR( N, D, E, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            N, INFO
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), E( * )
*       ..
*
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> Perform tests to decide whether the symmetric tridiagonal matrix T
*> warrants expensive computations which guarantee high relative accuracy
*> in the eigenvalues.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix. N > 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The N diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N)
*>          On entry, the first (N-1) entries contain the subdiagonal
*>          elements of the tridiagonal matrix T; E(N) is set to ZERO.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          INFO = 0(default) : the matrix warrants computations preserving
*>                              relative accuracy.
*>          INFO = 1          : the matrix warrants computations guaranteeing
*>                              only absolute accuracy.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larrr
*
*> \par Contributors:
*  ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA
*
*  =====================================================================
      SUBROUTINE DLARRR( N, D, E, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            N, INFO
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), E( * )
*     ..
*
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, RELCOND
      PARAMETER          ( ZERO = 0.0D0,
     $                     RELCOND = 0.999D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
      LOGICAL            YESREL
      DOUBLE PRECISION   EPS, SAFMIN, SMLNUM, RMIN, TMP, TMP2,
     $          OFFDIG, OFFDIG2

*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( N.LE.0 ) THEN
         INFO = 0
         RETURN
      END IF
*
*     As a default, do NOT go for relative-accuracy preserving computations.
      INFO = 1

      SAFMIN = DLAMCH( 'Safe minimum' )
      EPS = DLAMCH( 'Precision' )
      SMLNUM = SAFMIN / EPS
      RMIN = SQRT( SMLNUM )

*     Tests for relative accuracy
*
*     Test for scaled diagonal dominance
*     Scale the diagonal entries to one and check whether the sum of the
*     off-diagonals is less than one
*
*     The sdd relative error bounds have a 1/(1- 2*x) factor in them,
*     x = max(OFFDIG + OFFDIG2), so when x is close to 1/2, no relative
*     accuracy is promised.  In the notation of the code fragment below,
*     1/(1 - (OFFDIG + OFFDIG2)) is the condition number.
*     We don't think it is worth going into "sdd mode" unless the relative
*     condition number is reasonable, not 1/macheps.
*     The threshold should be compatible with other thresholds used in the
*     code. We set  OFFDIG + OFFDIG2 <= .999 =: RELCOND, it corresponds
*     to losing at most 3 decimal digits: 1 / (1 - (OFFDIG + OFFDIG2)) <= 1000
*     instead of the current OFFDIG + OFFDIG2 < 1
*
      YESREL = .TRUE.
      OFFDIG = ZERO
      TMP = SQRT(ABS(D(1)))
      IF (TMP.LT.RMIN) YESREL = .FALSE.
      IF(.NOT.YESREL) GOTO 11
      DO 10 I = 2, N
         TMP2 = SQRT(ABS(D(I)))
         IF (TMP2.LT.RMIN) YESREL = .FALSE.
         IF(.NOT.YESREL) GOTO 11
         OFFDIG2 = ABS(E(I-1))/(TMP*TMP2)
         IF(OFFDIG+OFFDIG2.GE.RELCOND) YESREL = .FALSE.
         IF(.NOT.YESREL) GOTO 11
         TMP = TMP2
         OFFDIG = OFFDIG2
 10   CONTINUE
 11   CONTINUE

      IF( YESREL ) THEN
         INFO = 0
         RETURN
      ELSE
      ENDIF
*

*
*     *** MORE TO BE IMPLEMENTED ***
*

*
*     Test if the lower bidiagonal matrix L from T = L D L^T
*     (zero shift facto) is well conditioned
*

*
*     Test if the upper bidiagonal matrix U from T = U D U^T
*     (zero shift facto) is well conditioned.
*     In this case, the matrix needs to be flipped and, at the end
*     of the eigenvector computation, the flip needs to be applied
*     to the computed eigenvectors (and the support)
*

*
      RETURN
*
*     End of DLARRR
*
      END
*> \brief \b DLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARRV( N, VL, VU, D, L, PIVMIN,
*                          ISPLIT, M, DOL, DOU, MINRGP,
*                          RTOL1, RTOL2, W, WERR, WGAP,
*                          IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ,
*                          WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            DOL, DOU, INFO, LDZ, M, N
*       DOUBLE PRECISION   MINRGP, PIVMIN, RTOL1, RTOL2, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            IBLOCK( * ), INDEXW( * ), ISPLIT( * ),
*      $                   ISUPPZ( * ), IWORK( * )
*       DOUBLE PRECISION   D( * ), GERS( * ), L( * ), W( * ), WERR( * ),
*      $                   WGAP( * ), WORK( * )
*       DOUBLE PRECISION  Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARRV computes the eigenvectors of the tridiagonal matrix
*> T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T.
*> The input eigenvalues should have been computed by DLARRE.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is DOUBLE PRECISION
*>          Lower bound of the interval that contains the desired
*>          eigenvalues. VL < VU. Needed to compute gaps on the left or right
*>          end of the extremal eigenvalues in the desired RANGE.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*>          VU is DOUBLE PRECISION
*>          Upper bound of the interval that contains the desired
*>          eigenvalues. VL < VU. 
*>          Note: VU is currently not used by this implementation of DLARRV, VU is
*>          passed to DLARRV because it could be used compute gaps on the right end
*>          of the extremal eigenvalues. However, with not much initial accuracy in
*>          LAMBDA and VU, the formula can lead to an overestimation of the right gap
*>          and thus to inadequately early RQI 'convergence'. This is currently
*>          prevented this by forcing a small right gap. And so it turns out that VU
*>          is currently not used by this implementation of DLARRV.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, the N diagonal elements of the diagonal matrix D.
*>          On exit, D may be overwritten.
*> \endverbatim
*>
*> \param[in,out] L
*> \verbatim
*>          L is DOUBLE PRECISION array, dimension (N)
*>          On entry, the (N-1) subdiagonal elements of the unit
*>          bidiagonal matrix L are in elements 1 to N-1 of L
*>          (if the matrix is not split.) At the end of each block
*>          is stored the corresponding shift as given by DLARRE.
*>          On exit, L is overwritten.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*>          PIVMIN is DOUBLE PRECISION
*>          The minimum pivot allowed in the Sturm sequence.
*> \endverbatim
*>
*> \param[in] ISPLIT
*> \verbatim
*>          ISPLIT is INTEGER array, dimension (N)
*>          The splitting points, at which T breaks up into blocks.
*>          The first block consists of rows/columns 1 to
*>          ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
*>          through ISPLIT( 2 ), etc.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The total number of input eigenvalues.  0 <= M <= N.
*> \endverbatim
*>
*> \param[in] DOL
*> \verbatim
*>          DOL is INTEGER
*> \endverbatim
*>
*> \param[in] DOU
*> \verbatim
*>          DOU is INTEGER
*>          If the user wants to compute only selected eigenvectors from all
*>          the eigenvalues supplied, he can specify an index range DOL:DOU.
*>          Or else the setting DOL=1, DOU=M should be applied.
*>          Note that DOL and DOU refer to the order in which the eigenvalues
*>          are stored in W.
*>          If the user wants to compute only selected eigenpairs, then
*>          the columns DOL-1 to DOU+1 of the eigenvector space Z contain the
*>          computed eigenvectors. All other columns of Z are set to zero.
*> \endverbatim
*>
*> \param[in] MINRGP
*> \verbatim
*>          MINRGP is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] RTOL1
*> \verbatim
*>          RTOL1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] RTOL2
*> \verbatim
*>          RTOL2 is DOUBLE PRECISION
*>           Parameters for bisection.
*>           An interval [LEFT,RIGHT] has converged if
*>           RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
*> \endverbatim
*>
*> \param[in,out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          The first M elements of W contain the APPROXIMATE eigenvalues for
*>          which eigenvectors are to be computed.  The eigenvalues
*>          should be grouped by split-off block and ordered from
*>          smallest to largest within the block ( The output array
*>          W from DLARRE is expected here ). Furthermore, they are with
*>          respect to the shift of the corresponding root representation
*>          for their block. On exit, W holds the eigenvalues of the
*>          UNshifted matrix.
*> \endverbatim
*>
*> \param[in,out] WERR
*> \verbatim
*>          WERR is DOUBLE PRECISION array, dimension (N)
*>          The first M elements contain the semiwidth of the uncertainty
*>          interval of the corresponding eigenvalue in W
*> \endverbatim
*>
*> \param[in,out] WGAP
*> \verbatim
*>          WGAP is DOUBLE PRECISION array, dimension (N)
*>          The separation from the right neighbor eigenvalue in W.
*> \endverbatim
*>
*> \param[in] IBLOCK
*> \verbatim
*>          IBLOCK is INTEGER array, dimension (N)
*>          The indices of the blocks (submatrices) associated with the
*>          corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
*>          W(i) belongs to the first block from the top, =2 if W(i)
*>          belongs to the second block, etc.
*> \endverbatim
*>
*> \param[in] INDEXW
*> \verbatim
*>          INDEXW is INTEGER array, dimension (N)
*>          The indices of the eigenvalues within each block (submatrix);
*>          for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
*>          i-th eigenvalue W(i) is the 10-th eigenvalue in the second block.
*> \endverbatim
*>
*> \param[in] GERS
*> \verbatim
*>          GERS is DOUBLE PRECISION array, dimension (2*N)
*>          The N Gerschgorin intervals (the i-th Gerschgorin interval
*>          is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should
*>          be computed from the original UNshifted matrix.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
*>          If INFO = 0, the first M columns of Z contain the
*>          orthonormal eigenvectors of the matrix T
*>          corresponding to the input eigenvalues, with the i-th
*>          column of Z holding the eigenvector associated with W(i).
*>          Note: the user must ensure that at least max(1,M) columns are
*>          supplied in the array Z.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] ISUPPZ
*> \verbatim
*>          ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
*>          The support of the eigenvectors in Z, i.e., the indices
*>          indicating the nonzero elements in Z. The I-th eigenvector
*>          is nonzero only in elements ISUPPZ( 2*I-1 ) through
*>          ISUPPZ( 2*I ).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (12*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (7*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>
*>          > 0:  A problem occurred in DLARRV.
*>          < 0:  One of the called subroutines signaled an internal problem.
*>                Needs inspection of the corresponding parameter IINFO
*>                for further information.
*>
*>          =-1:  Problem in DLARRB when refining a child's eigenvalues.
*>          =-2:  Problem in DLARRF when computing the RRR of a child.
*>                When a child is inside a tight cluster, it can be difficult
*>                to find an RRR. A partial remedy from the user's point of
*>                view is to make the parameter MINRGP smaller and recompile.
*>                However, as the orthogonality of the computed vectors is
*>                proportional to 1/MINRGP, the user should be aware that
*>                he might be trading in precision when he decreases MINRGP.
*>          =-3:  Problem in DLARRB when refining a single eigenvalue
*>                after the Rayleigh correction was rejected.
*>          = 5:  The Rayleigh Quotient Iteration failed to converge to
*>                full accuracy in MAXITR steps.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larrv
*
*> \par Contributors:
*  ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA
*
*  =====================================================================
      SUBROUTINE DLARRV( N, VL, VU, D, L, PIVMIN,
     $                   ISPLIT, M, DOL, DOU, MINRGP,
     $                   RTOL1, RTOL2, W, WERR, WGAP,
     $                   IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ,
     $                   WORK, IWORK, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            DOL, DOU, INFO, LDZ, M, N
      DOUBLE PRECISION   MINRGP, PIVMIN, RTOL1, RTOL2, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            IBLOCK( * ), INDEXW( * ), ISPLIT( * ),
     $                   ISUPPZ( * ), IWORK( * )
      DOUBLE PRECISION   D( * ), GERS( * ), L( * ), W( * ), WERR( * ),
     $                   WGAP( * ), WORK( * )
      DOUBLE PRECISION  Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            MAXITR
      PARAMETER          ( MAXITR = 10 )
      DOUBLE PRECISION   ZERO, ONE, TWO, THREE, FOUR, HALF
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0,
     $                     TWO = 2.0D0, THREE = 3.0D0,
     $                     FOUR = 4.0D0, HALF = 0.5D0)
*     ..
*     .. Local Scalars ..
      LOGICAL            ESKIP, NEEDBS, STP2II, TRYRQC, USEDBS, USEDRQ
      INTEGER            DONE, I, IBEGIN, IDONE, IEND, II, IINDC1,
     $                   IINDC2, IINDR, IINDWK, IINFO, IM, IN, INDEIG,
     $                   INDLD, INDLLD, INDWRK, ISUPMN, ISUPMX, ITER,
     $                   ITMP1, J, JBLK, K, MINIWSIZE, MINWSIZE, NCLUS,
     $                   NDEPTH, NEGCNT, NEWCLS, NEWFST, NEWFTT, NEWLST,
     $                   NEWSIZ, OFFSET, OLDCLS, OLDFST, OLDIEN, OLDLST,
     $                   OLDNCL, P, PARITY, Q, WBEGIN, WEND, WINDEX,
     $                   WINDMN, WINDPL, ZFROM, ZTO, ZUSEDL, ZUSEDU,
     $                   ZUSEDW
      DOUBLE PRECISION   BSTRES, BSTW, EPS, FUDGE, GAP, GAPTOL, GL, GU,
     $                   LAMBDA, LEFT, LGAP, MINGMA, NRMINV, RESID,
     $                   RGAP, RIGHT, RQCORR, RQTOL, SAVGAP, SGNDEF,
     $                   SIGMA, SPDIAM, SSIGMA, TAU, TMP, TOL, ZTZ
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLAR1V, DLARRB, DLARRF,
     $                   DLASET,
     $                   DSCAL
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC ABS, DBLE, MAX, MIN
*     ..
*     .. Executable Statements ..
*     ..

      INFO = 0
*
*     Quick return if possible
*
      IF( (N.LE.0).OR.(M.LE.0) ) THEN
         RETURN
      END IF
*
*     The first N entries of WORK are reserved for the eigenvalues
      INDLD = N+1
      INDLLD= 2*N+1
      INDWRK= 3*N+1
      MINWSIZE = 12 * N

      DO 5 I= 1,MINWSIZE
         WORK( I ) = ZERO
 5    CONTINUE

*     IWORK(IINDR+1:IINDR+N) hold the twist indices R for the
*     factorization used to compute the FP vector
      IINDR = 0
*     IWORK(IINDC1+1:IINC2+N) are used to store the clusters of the current
*     layer and the one above.
      IINDC1 = N
      IINDC2 = 2*N
      IINDWK = 3*N + 1

      MINIWSIZE = 7 * N
      DO 10 I= 1,MINIWSIZE
         IWORK( I ) = 0
 10   CONTINUE

      ZUSEDL = 1
      IF(DOL.GT.1) THEN
*        Set lower bound for use of Z
         ZUSEDL = DOL-1
      ENDIF
      ZUSEDU = M
      IF(DOU.LT.M) THEN
*        Set lower bound for use of Z
         ZUSEDU = DOU+1
      ENDIF
*     The width of the part of Z that is used
      ZUSEDW = ZUSEDU - ZUSEDL + 1


      CALL DLASET( 'Full', N, ZUSEDW, ZERO, ZERO,
     $                    Z(1,ZUSEDL), LDZ )

      EPS = DLAMCH( 'Precision' )
      RQTOL = TWO * EPS
*
*     Set expert flags for standard code.
      TRYRQC = .TRUE.

      IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN
      ELSE
*        Only selected eigenpairs are computed. Since the other evalues
*        are not refined by RQ iteration, bisection has to compute to full
*        accuracy.
         RTOL1 = FOUR * EPS
         RTOL2 = FOUR * EPS
      ENDIF

*     The entries WBEGIN:WEND in W, WERR, WGAP correspond to the
*     desired eigenvalues. The support of the nonzero eigenvector
*     entries is contained in the interval IBEGIN:IEND.
*     Remark that if k eigenpairs are desired, then the eigenvectors
*     are stored in k contiguous columns of Z.

*     DONE is the number of eigenvectors already computed
      DONE = 0
      IBEGIN = 1
      WBEGIN = 1
      DO 170 JBLK = 1, IBLOCK( M )
         IEND = ISPLIT( JBLK )
         SIGMA = L( IEND )
*        Find the eigenvectors of the submatrix indexed IBEGIN
*        through IEND.
         WEND = WBEGIN - 1
 15      CONTINUE
         IF( WEND.LT.M ) THEN
            IF( IBLOCK( WEND+1 ).EQ.JBLK ) THEN
               WEND = WEND + 1
               GO TO 15
            END IF
         END IF
         IF( WEND.LT.WBEGIN ) THEN
            IBEGIN = IEND + 1
            GO TO 170
         ELSEIF( (WEND.LT.DOL).OR.(WBEGIN.GT.DOU) ) THEN
            IBEGIN = IEND + 1
            WBEGIN = WEND + 1
            GO TO 170
         END IF

*        Find local spectral diameter of the block
         GL = GERS( 2*IBEGIN-1 )
         GU = GERS( 2*IBEGIN )
         DO 20 I = IBEGIN+1 , IEND
            GL = MIN( GERS( 2*I-1 ), GL )
            GU = MAX( GERS( 2*I ), GU )
 20      CONTINUE
         SPDIAM = GU - GL

*        OLDIEN is the last index of the previous block
         OLDIEN = IBEGIN - 1
*        Calculate the size of the current block
         IN = IEND - IBEGIN + 1
*        The number of eigenvalues in the current block
         IM = WEND - WBEGIN + 1

*        This is for a 1x1 block
         IF( IBEGIN.EQ.IEND ) THEN
            DONE = DONE+1
            Z( IBEGIN, WBEGIN ) = ONE
            ISUPPZ( 2*WBEGIN-1 ) = IBEGIN
            ISUPPZ( 2*WBEGIN ) = IBEGIN
            W( WBEGIN ) = W( WBEGIN ) + SIGMA
            WORK( WBEGIN ) = W( WBEGIN )
            IBEGIN = IEND + 1
            WBEGIN = WBEGIN + 1
            GO TO 170
         END IF

*        The desired (shifted) eigenvalues are stored in W(WBEGIN:WEND)
*        Note that these can be approximations, in this case, the corresp.
*        entries of WERR give the size of the uncertainty interval.
*        The eigenvalue approximations will be refined when necessary as
*        high relative accuracy is required for the computation of the
*        corresponding eigenvectors.
         CALL DCOPY( IM, W( WBEGIN ), 1,
     $                   WORK( WBEGIN ), 1 )

*        We store in W the eigenvalue approximations w.r.t. the original
*        matrix T.
         DO 30 I=1,IM
            W(WBEGIN+I-1) = W(WBEGIN+I-1)+SIGMA
 30      CONTINUE


*        NDEPTH is the current depth of the representation tree
         NDEPTH = 0
*        PARITY is either 1 or 0
         PARITY = 1
*        NCLUS is the number of clusters for the next level of the
*        representation tree, we start with NCLUS = 1 for the root
         NCLUS = 1
         IWORK( IINDC1+1 ) = 1
         IWORK( IINDC1+2 ) = IM

*        IDONE is the number of eigenvectors already computed in the current
*        block
         IDONE = 0
*        loop while( IDONE.LT.IM )
*        generate the representation tree for the current block and
*        compute the eigenvectors
   40    CONTINUE
         IF( IDONE.LT.IM ) THEN
*           This is a crude protection against infinitely deep trees
            IF( NDEPTH.GT.M ) THEN
               INFO = -2
               RETURN
            ENDIF
*           breadth first processing of the current level of the representation
*           tree: OLDNCL = number of clusters on current level
            OLDNCL = NCLUS
*           reset NCLUS to count the number of child clusters
            NCLUS = 0
*
            PARITY = 1 - PARITY
            IF( PARITY.EQ.0 ) THEN
               OLDCLS = IINDC1
               NEWCLS = IINDC2
            ELSE
               OLDCLS = IINDC2
               NEWCLS = IINDC1
            END IF
*           Process the clusters on the current level
            DO 150 I = 1, OLDNCL
               J = OLDCLS + 2*I
*              OLDFST, OLDLST = first, last index of current cluster.
*                               cluster indices start with 1 and are relative
*                               to WBEGIN when accessing W, WGAP, WERR, Z
               OLDFST = IWORK( J-1 )
               OLDLST = IWORK( J )
               IF( NDEPTH.GT.0 ) THEN
*                 Retrieve relatively robust representation (RRR) of cluster
*                 that has been computed at the previous level
*                 The RRR is stored in Z and overwritten once the eigenvectors
*                 have been computed or when the cluster is refined

                  IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN
*                    Get representation from location of the leftmost evalue
*                    of the cluster
                     J = WBEGIN + OLDFST - 1
                  ELSE
                     IF(WBEGIN+OLDFST-1.LT.DOL) THEN
*                       Get representation from the left end of Z array
                        J = DOL - 1
                     ELSEIF(WBEGIN+OLDFST-1.GT.DOU) THEN
*                       Get representation from the right end of Z array
                        J = DOU
                     ELSE
                        J = WBEGIN + OLDFST - 1
                     ENDIF
                  ENDIF
                  CALL DCOPY( IN, Z( IBEGIN, J ), 1, D( IBEGIN ), 1 )
                  CALL DCOPY( IN-1, Z( IBEGIN, J+1 ), 1, L( IBEGIN ),
     $               1 )
                  SIGMA = Z( IEND, J+1 )

*                 Set the corresponding entries in Z to zero
                  CALL DLASET( 'Full', IN, 2, ZERO, ZERO,
     $                         Z( IBEGIN, J), LDZ )
               END IF

*              Compute DL and DLL of current RRR
               DO 50 J = IBEGIN, IEND-1
                  TMP = D( J )*L( J )
                  WORK( INDLD-1+J ) = TMP
                  WORK( INDLLD-1+J ) = TMP*L( J )
   50          CONTINUE

               IF( NDEPTH.GT.0 ) THEN
*                 P and Q are index of the first and last eigenvalue to compute
*                 within the current block
                  P = INDEXW( WBEGIN-1+OLDFST )
                  Q = INDEXW( WBEGIN-1+OLDLST )
*                 Offset for the arrays WORK, WGAP and WERR, i.e., the P-OFFSET
*                 through the Q-OFFSET elements of these arrays are to be used.
*                  OFFSET = P-OLDFST
                  OFFSET = INDEXW( WBEGIN ) - 1
*                 perform limited bisection (if necessary) to get approximate
*                 eigenvalues to the precision needed.
                  CALL DLARRB( IN, D( IBEGIN ),
     $                         WORK(INDLLD+IBEGIN-1),
     $                         P, Q, RTOL1, RTOL2, OFFSET,
     $                         WORK(WBEGIN),WGAP(WBEGIN),WERR(WBEGIN),
     $                         WORK( INDWRK ), IWORK( IINDWK ),
     $                         PIVMIN, SPDIAM, IN, IINFO )
                  IF( IINFO.NE.0 ) THEN
                     INFO = -1
                     RETURN
                  ENDIF
*                 We also recompute the extremal gaps. W holds all eigenvalues
*                 of the unshifted matrix and must be used for computation
*                 of WGAP, the entries of WORK might stem from RRRs with
*                 different shifts. The gaps from WBEGIN-1+OLDFST to
*                 WBEGIN-1+OLDLST are correctly computed in DLARRB.
*                 However, we only allow the gaps to become greater since
*                 this is what should happen when we decrease WERR
                  IF( OLDFST.GT.1) THEN
                     WGAP( WBEGIN+OLDFST-2 ) =
     $             MAX(WGAP(WBEGIN+OLDFST-2),
     $                 W(WBEGIN+OLDFST-1)-WERR(WBEGIN+OLDFST-1)
     $                 - W(WBEGIN+OLDFST-2)-WERR(WBEGIN+OLDFST-2) )
                  ENDIF
                  IF( WBEGIN + OLDLST -1 .LT. WEND ) THEN
                     WGAP( WBEGIN+OLDLST-1 ) =
     $               MAX(WGAP(WBEGIN+OLDLST-1),
     $                   W(WBEGIN+OLDLST)-WERR(WBEGIN+OLDLST)
     $                   - W(WBEGIN+OLDLST-1)-WERR(WBEGIN+OLDLST-1) )
                  ENDIF
*                 Each time the eigenvalues in WORK get refined, we store
*                 the newly found approximation with all shifts applied in W
                  DO 53 J=OLDFST,OLDLST
                     W(WBEGIN+J-1) = WORK(WBEGIN+J-1)+SIGMA
 53               CONTINUE
               END IF

*              Process the current node.
               NEWFST = OLDFST
               DO 140 J = OLDFST, OLDLST
                  IF( J.EQ.OLDLST ) THEN
*                    we are at the right end of the cluster, this is also the
*                    boundary of the child cluster
                     NEWLST = J
                  ELSE IF ( WGAP( WBEGIN + J -1).GE.
     $                    MINRGP* ABS( WORK(WBEGIN + J -1) ) ) THEN
*                    the right relative gap is big enough, the child cluster
*                    (NEWFST,..,NEWLST) is well separated from the following
                     NEWLST = J
                   ELSE
*                    inside a child cluster, the relative gap is not
*                    big enough.
                     GOTO 140
                  END IF

*                 Compute size of child cluster found
                  NEWSIZ = NEWLST - NEWFST + 1

*                 NEWFTT is the place in Z where the new RRR or the computed
*                 eigenvector is to be stored
                  IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN
*                    Store representation at location of the leftmost evalue
*                    of the cluster
                     NEWFTT = WBEGIN + NEWFST - 1
                  ELSE
                     IF(WBEGIN+NEWFST-1.LT.DOL) THEN
*                       Store representation at the left end of Z array
                        NEWFTT = DOL - 1
                     ELSEIF(WBEGIN+NEWFST-1.GT.DOU) THEN
*                       Store representation at the right end of Z array
                        NEWFTT = DOU
                     ELSE
                        NEWFTT = WBEGIN + NEWFST - 1
                     ENDIF
                  ENDIF

                  IF( NEWSIZ.GT.1) THEN
*
*                    Current child is not a singleton but a cluster.
*                    Compute and store new representation of child.
*
*
*                    Compute left and right cluster gap.
*
*                    LGAP and RGAP are not computed from WORK because
*                    the eigenvalue approximations may stem from RRRs
*                    different shifts. However, W hold all eigenvalues
*                    of the unshifted matrix. Still, the entries in WGAP
*                    have to be computed from WORK since the entries
*                    in W might be of the same order so that gaps are not
*                    exhibited correctly for very close eigenvalues.
                     IF( NEWFST.EQ.1 ) THEN
                        LGAP = MAX( ZERO,
     $                       W(WBEGIN)-WERR(WBEGIN) - VL )
                    ELSE
                        LGAP = WGAP( WBEGIN+NEWFST-2 )
                     ENDIF
                     RGAP = WGAP( WBEGIN+NEWLST-1 )
*
*                    Compute left- and rightmost eigenvalue of child
*                    to high precision in order to shift as close
*                    as possible and obtain as large relative gaps
*                    as possible
*
                     DO 55 K =1,2
                        IF(K.EQ.1) THEN
                           P = INDEXW( WBEGIN-1+NEWFST )
                        ELSE
                           P = INDEXW( WBEGIN-1+NEWLST )
                        ENDIF
                        OFFSET = INDEXW( WBEGIN ) - 1
                        CALL DLARRB( IN, D(IBEGIN),
     $                       WORK( INDLLD+IBEGIN-1 ),P,P,
     $                       RQTOL, RQTOL, OFFSET,
     $                       WORK(WBEGIN),WGAP(WBEGIN),
     $                       WERR(WBEGIN),WORK( INDWRK ),
     $                       IWORK( IINDWK ), PIVMIN, SPDIAM,
     $                       IN, IINFO )
 55                  CONTINUE
*
                     IF((WBEGIN+NEWLST-1.LT.DOL).OR.
     $                  (WBEGIN+NEWFST-1.GT.DOU)) THEN
*                       if the cluster contains no desired eigenvalues
*                       skip the computation of that branch of the rep. tree
*
*                       We could skip before the refinement of the extremal
*                       eigenvalues of the child, but then the representation
*                       tree could be different from the one when nothing is
*                       skipped. For this reason we skip at this place.
                        IDONE = IDONE + NEWLST - NEWFST + 1
                        GOTO 139
                     ENDIF
*
*                    Compute RRR of child cluster.
*                    Note that the new RRR is stored in Z
*
*                    DLARRF needs LWORK = 2*N
                     CALL DLARRF( IN, D( IBEGIN ), L( IBEGIN ),
     $                         WORK(INDLD+IBEGIN-1),
     $                         NEWFST, NEWLST, WORK(WBEGIN),
     $                         WGAP(WBEGIN), WERR(WBEGIN),
     $                         SPDIAM, LGAP, RGAP, PIVMIN, TAU,
     $                         Z(IBEGIN, NEWFTT),Z(IBEGIN, NEWFTT+1),
     $                         WORK( INDWRK ), IINFO )
                     IF( IINFO.EQ.0 ) THEN
*                       a new RRR for the cluster was found by DLARRF
*                       update shift and store it
                        SSIGMA = SIGMA + TAU
                        Z( IEND, NEWFTT+1 ) = SSIGMA
*                       WORK() are the midpoints and WERR() the semi-width
*                       Note that the entries in W are unchanged.
                        DO 116 K = NEWFST, NEWLST
                           FUDGE =
     $                          THREE*EPS*ABS(WORK(WBEGIN+K-1))
                           WORK( WBEGIN + K - 1 ) =
     $                          WORK( WBEGIN + K - 1) - TAU
                           FUDGE = FUDGE +
     $                          FOUR*EPS*ABS(WORK(WBEGIN+K-1))
*                          Fudge errors
                           WERR( WBEGIN + K - 1 ) =
     $                          WERR( WBEGIN + K - 1 ) + FUDGE
*                          Gaps are not fudged. Provided that WERR is small
*                          when eigenvalues are close, a zero gap indicates
*                          that a new representation is needed for resolving
*                          the cluster. A fudge could lead to a wrong decision
*                          of judging eigenvalues 'separated' which in
*                          reality are not. This could have a negative impact
*                          on the orthogonality of the computed eigenvectors.
 116                    CONTINUE

                        NCLUS = NCLUS + 1
                        K = NEWCLS + 2*NCLUS
                        IWORK( K-1 ) = NEWFST
                        IWORK( K ) = NEWLST
                     ELSE
                        INFO = -2
                        RETURN
                     ENDIF
                  ELSE
*
*                    Compute eigenvector of singleton
*
                     ITER = 0
*
                     TOL = FOUR * LOG(DBLE(IN)) * EPS
*
                     K = NEWFST
                     WINDEX = WBEGIN + K - 1
                     WINDMN = MAX(WINDEX - 1,1)
                     WINDPL = MIN(WINDEX + 1,M)
                     LAMBDA = WORK( WINDEX )
                     DONE = DONE + 1
*                    Check if eigenvector computation is to be skipped
                     IF((WINDEX.LT.DOL).OR.
     $                  (WINDEX.GT.DOU)) THEN
                        ESKIP = .TRUE.
                        GOTO 125
                     ELSE
                        ESKIP = .FALSE.
                     ENDIF
                     LEFT = WORK( WINDEX ) - WERR( WINDEX )
                     RIGHT = WORK( WINDEX ) + WERR( WINDEX )
                     INDEIG = INDEXW( WINDEX )
*                    Note that since we compute the eigenpairs for a child,
*                    all eigenvalue approximations are w.r.t the same shift.
*                    In this case, the entries in WORK should be used for
*                    computing the gaps since they exhibit even very small
*                    differences in the eigenvalues, as opposed to the
*                    entries in W which might "look" the same.

                     IF( K .EQ. 1) THEN
*                       In the case RANGE='I' and with not much initial
*                       accuracy in LAMBDA and VL, the formula
*                       LGAP = MAX( ZERO, (SIGMA - VL) + LAMBDA )
*                       can lead to an overestimation of the left gap and
*                       thus to inadequately early RQI 'convergence'.
*                       Prevent this by forcing a small left gap.
                        LGAP = EPS*MAX(ABS(LEFT),ABS(RIGHT))
                     ELSE
                        LGAP = WGAP(WINDMN)
                     ENDIF
                     IF( K .EQ. IM) THEN
*                       In the case RANGE='I' and with not much initial
*                       accuracy in LAMBDA and VU, the formula
*                       can lead to an overestimation of the right gap and
*                       thus to inadequately early RQI 'convergence'.
*                       Prevent this by forcing a small right gap.
                        RGAP = EPS*MAX(ABS(LEFT),ABS(RIGHT))
                     ELSE
                        RGAP = WGAP(WINDEX)
                     ENDIF
                     GAP = MIN( LGAP, RGAP )
                     IF(( K .EQ. 1).OR.(K .EQ. IM)) THEN
*                       The eigenvector support can become wrong
*                       because significant entries could be cut off due to a
*                       large GAPTOL parameter in LAR1V. Prevent this.
                        GAPTOL = ZERO
                     ELSE
                        GAPTOL = GAP * EPS
                     ENDIF
                     ISUPMN = IN
                     ISUPMX = 1
*                    Update WGAP so that it holds the minimum gap
*                    to the left or the right. This is crucial in the
*                    case where bisection is used to ensure that the
*                    eigenvalue is refined up to the required precision.
*                    The correct value is restored afterwards.
                     SAVGAP = WGAP(WINDEX)
                     WGAP(WINDEX) = GAP
*                    We want to use the Rayleigh Quotient Correction
*                    as often as possible since it converges quadratically
*                    when we are close enough to the desired eigenvalue.
*                    However, the Rayleigh Quotient can have the wrong sign
*                    and lead us away from the desired eigenvalue. In this
*                    case, the best we can do is to use bisection.
                     USEDBS = .FALSE.
                     USEDRQ = .FALSE.
*                    Bisection is initially turned off unless it is forced
                     NEEDBS =  .NOT.TRYRQC
 120                 CONTINUE
*                    Check if bisection should be used to refine eigenvalue
                     IF(NEEDBS) THEN
*                       Take the bisection as new iterate
                        USEDBS = .TRUE.
                        ITMP1 = IWORK( IINDR+WINDEX )
                        OFFSET = INDEXW( WBEGIN ) - 1
                        CALL DLARRB( IN, D(IBEGIN),
     $                       WORK(INDLLD+IBEGIN-1),INDEIG,INDEIG,
     $                       ZERO, TWO*EPS, OFFSET,
     $                       WORK(WBEGIN),WGAP(WBEGIN),
     $                       WERR(WBEGIN),WORK( INDWRK ),
     $                       IWORK( IINDWK ), PIVMIN, SPDIAM,
     $                       ITMP1, IINFO )
                        IF( IINFO.NE.0 ) THEN
                           INFO = -3
                           RETURN
                        ENDIF
                        LAMBDA = WORK( WINDEX )
*                       Reset twist index from inaccurate LAMBDA to
*                       force computation of true MINGMA
                        IWORK( IINDR+WINDEX ) = 0
                     ENDIF
*                    Given LAMBDA, compute the eigenvector.
                     CALL DLAR1V( IN, 1, IN, LAMBDA, D( IBEGIN ),
     $                    L( IBEGIN ), WORK(INDLD+IBEGIN-1),
     $                    WORK(INDLLD+IBEGIN-1),
     $                    PIVMIN, GAPTOL, Z( IBEGIN, WINDEX ),
     $                    .NOT.USEDBS, NEGCNT, ZTZ, MINGMA,
     $                    IWORK( IINDR+WINDEX ), ISUPPZ( 2*WINDEX-1 ),
     $                    NRMINV, RESID, RQCORR, WORK( INDWRK ) )
                     IF(ITER .EQ. 0) THEN
                        BSTRES = RESID
                        BSTW = LAMBDA
                     ELSEIF(RESID.LT.BSTRES) THEN
                        BSTRES = RESID
                        BSTW = LAMBDA
                     ENDIF
                     ISUPMN = MIN(ISUPMN,ISUPPZ( 2*WINDEX-1 ))
                     ISUPMX = MAX(ISUPMX,ISUPPZ( 2*WINDEX ))
                     ITER = ITER + 1

*                    sin alpha <= |resid|/gap
*                    Note that both the residual and the gap are
*                    proportional to the matrix, so ||T|| doesn't play
*                    a role in the quotient

*
*                    Convergence test for Rayleigh-Quotient iteration
*                    (omitted when Bisection has been used)
*
                     IF( RESID.GT.TOL*GAP .AND. ABS( RQCORR ).GT.
     $                    RQTOL*ABS( LAMBDA ) .AND. .NOT. USEDBS)
     $                    THEN
*                       We need to check that the RQCORR update doesn't
*                       move the eigenvalue away from the desired one and
*                       towards a neighbor. -> protection with bisection
                        IF(INDEIG.LE.NEGCNT) THEN
*                          The wanted eigenvalue lies to the left
                           SGNDEF = -ONE
                        ELSE
*                          The wanted eigenvalue lies to the right
                           SGNDEF = ONE
                        ENDIF
*                       We only use the RQCORR if it improves the
*                       the iterate reasonably.
                        IF( ( RQCORR*SGNDEF.GE.ZERO )
     $                       .AND.( LAMBDA + RQCORR.LE. RIGHT)
     $                       .AND.( LAMBDA + RQCORR.GE. LEFT)
     $                       ) THEN
                           USEDRQ = .TRUE.
*                          Store new midpoint of bisection interval in WORK
                           IF(SGNDEF.EQ.ONE) THEN
*                             The current LAMBDA is on the left of the true
*                             eigenvalue
                              LEFT = LAMBDA
*                             We prefer to assume that the error estimate
*                             is correct. We could make the interval not
*                             as a bracket but to be modified if the RQCORR
*                             chooses to. In this case, the RIGHT side should
*                             be modified as follows:
*                              RIGHT = MAX(RIGHT, LAMBDA + RQCORR)
                           ELSE
*                             The current LAMBDA is on the right of the true
*                             eigenvalue
                              RIGHT = LAMBDA
*                             See comment about assuming the error estimate is
*                             correct above.
*                              LEFT = MIN(LEFT, LAMBDA + RQCORR)
                           ENDIF
                           WORK( WINDEX ) =
     $                       HALF * (RIGHT + LEFT)
*                          Take RQCORR since it has the correct sign and
*                          improves the iterate reasonably
                           LAMBDA = LAMBDA + RQCORR
*                          Update width of error interval
                           WERR( WINDEX ) =
     $                             HALF * (RIGHT-LEFT)
                        ELSE
                           NEEDBS = .TRUE.
                        ENDIF
                        IF(RIGHT-LEFT.LT.RQTOL*ABS(LAMBDA)) THEN
*                             The eigenvalue is computed to bisection accuracy
*                             compute eigenvector and stop
                           USEDBS = .TRUE.
                           GOTO 120
                        ELSEIF( ITER.LT.MAXITR ) THEN
                           GOTO 120
                        ELSEIF( ITER.EQ.MAXITR ) THEN
                           NEEDBS = .TRUE.
                           GOTO 120
                        ELSE
                           INFO = 5
                           RETURN
                        END IF
                     ELSE
                        STP2II = .FALSE.
        IF(USEDRQ .AND. USEDBS .AND.
     $                     BSTRES.LE.RESID) THEN
                           LAMBDA = BSTW
                           STP2II = .TRUE.
                        ENDIF
                        IF (STP2II) THEN
*                          improve error angle by second step
                           CALL DLAR1V( IN, 1, IN, LAMBDA,
     $                          D( IBEGIN ), L( IBEGIN ),
     $                          WORK(INDLD+IBEGIN-1),
     $                          WORK(INDLLD+IBEGIN-1),
     $                          PIVMIN, GAPTOL, Z( IBEGIN, WINDEX ),
     $                          .NOT.USEDBS, NEGCNT, ZTZ, MINGMA,
     $                          IWORK( IINDR+WINDEX ),
     $                          ISUPPZ( 2*WINDEX-1 ),
     $                          NRMINV, RESID, RQCORR, WORK( INDWRK ) )
                        ENDIF
                        WORK( WINDEX ) = LAMBDA
                     END IF
*
*                    Compute FP-vector support w.r.t. whole matrix
*
                     ISUPPZ( 2*WINDEX-1 ) = ISUPPZ( 2*WINDEX-1 )+OLDIEN
                     ISUPPZ( 2*WINDEX ) = ISUPPZ( 2*WINDEX )+OLDIEN
                     ZFROM = ISUPPZ( 2*WINDEX-1 )
                     ZTO = ISUPPZ( 2*WINDEX )
                     ISUPMN = ISUPMN + OLDIEN
                     ISUPMX = ISUPMX + OLDIEN
*                    Ensure vector is ok if support in the RQI has changed
                     IF(ISUPMN.LT.ZFROM) THEN
                        DO 122 II = ISUPMN,ZFROM-1
                           Z( II, WINDEX ) = ZERO
 122                    CONTINUE
                     ENDIF
                     IF(ISUPMX.GT.ZTO) THEN
                        DO 123 II = ZTO+1,ISUPMX
                           Z( II, WINDEX ) = ZERO
 123                    CONTINUE
                     ENDIF
                     CALL DSCAL( ZTO-ZFROM+1, NRMINV,
     $                       Z( ZFROM, WINDEX ), 1 )
 125                 CONTINUE
*                    Update W
                     W( WINDEX ) = LAMBDA+SIGMA
*                    Recompute the gaps on the left and right
*                    But only allow them to become larger and not
*                    smaller (which can only happen through "bad"
*                    cancellation and doesn't reflect the theory
*                    where the initial gaps are underestimated due
*                    to WERR being too crude.)
                     IF(.NOT.ESKIP) THEN
                        IF( K.GT.1) THEN
                           WGAP( WINDMN ) = MAX( WGAP(WINDMN),
     $                          W(WINDEX)-WERR(WINDEX)
     $                          - W(WINDMN)-WERR(WINDMN) )
                        ENDIF
                        IF( WINDEX.LT.WEND ) THEN
                           WGAP( WINDEX ) = MAX( SAVGAP,
     $                          W( WINDPL )-WERR( WINDPL )
     $                          - W( WINDEX )-WERR( WINDEX) )
                        ENDIF
                     ENDIF
                     IDONE = IDONE + 1
                  ENDIF
*                 here ends the code for the current child
*
 139              CONTINUE
*                 Proceed to any remaining child nodes
                  NEWFST = J + 1
 140           CONTINUE
 150        CONTINUE
            NDEPTH = NDEPTH + 1
            GO TO 40
         END IF
         IBEGIN = IEND + 1
         WBEGIN = WEND + 1
 170  CONTINUE
*

      RETURN
*
*     End of DLARRV
*
      END
*> \brief \b DLARSCL2 performs reciprocal diagonal scaling on a matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARSCL2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarscl2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarscl2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarscl2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARSCL2 ( M, N, D, X, LDX )
*
*       .. Scalar Arguments ..
*       INTEGER            M, N, LDX
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARSCL2 performs a reciprocal diagonal scaling on a matrix:
*>   x <-- inv(D) * x
*> where the diagonal matrix D is stored as a vector.
*>
*> Eventually to be replaced by BLAS_dge_diag_scale in the new BLAS
*> standard.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>     The number of rows of D and X. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>     The number of columns of X. N >= 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (M)
*>     Diagonal matrix D, stored as a vector of length M.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,N)
*>     On entry, the matrix X to be scaled by D.
*>     On exit, the scaled matrix.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>     The leading dimension of the matrix X. LDX >= M.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larscl2
*
*  =====================================================================
      SUBROUTINE DLARSCL2 ( M, N, D, X, LDX )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            M, N, LDX
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            I, J
*     ..
*     .. Executable Statements ..
*
      DO J = 1, N
         DO I = 1, M
            X( I, J ) = X( I, J ) / D( I )
         END DO
      END DO

      RETURN
      END
*> \brief \b DLARTGP generates a plane rotation so that the diagonal is nonnegative.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARTGP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlartgp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlartgp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlartgp.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARTGP( F, G, CS, SN, R )
*
*       .. Scalar Arguments ..
*       DOUBLE PRECISION   CS, F, G, R, SN
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARTGP generates a plane rotation so that
*>
*>    [  CS  SN  ]  .  [ F ]  =  [ R ]   where CS**2 + SN**2 = 1.
*>    [ -SN  CS  ]     [ G ]     [ 0 ]
*>
*> This is a slower, more accurate version of the Level 1 BLAS routine DROTG,
*> with the following other differences:
*>    F and G are unchanged on return.
*>    If G=0, then CS=(+/-)1 and SN=0.
*>    If F=0 and (G .ne. 0), then CS=0 and SN=(+/-)1.
*>
*> The sign is chosen so that R >= 0.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] F
*> \verbatim
*>          F is DOUBLE PRECISION
*>          The first component of vector to be rotated.
*> \endverbatim
*>
*> \param[in] G
*> \verbatim
*>          G is DOUBLE PRECISION
*>          The second component of vector to be rotated.
*> \endverbatim
*>
*> \param[out] CS
*> \verbatim
*>          CS is DOUBLE PRECISION
*>          The cosine of the rotation.
*> \endverbatim
*>
*> \param[out] SN
*> \verbatim
*>          SN is DOUBLE PRECISION
*>          The sine of the rotation.
*> \endverbatim
*>
*> \param[out] R
*> \verbatim
*>          R is DOUBLE PRECISION
*>          The nonzero component of the rotated vector.
*>
*>  This version has a few statements commented out for thread safety
*>  (machine parameters are computed on each entry). 10 feb 03, SJH.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lartgp
*
*  =====================================================================
      SUBROUTINE DLARTGP( F, G, CS, SN, R )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   CS, F, G, R, SN
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D0 )
      DOUBLE PRECISION   TWO
      PARAMETER          ( TWO = 2.0D0 )
*     ..
*     .. Local Scalars ..
*     LOGICAL            FIRST
      INTEGER            COUNT, I
      DOUBLE PRECISION   EPS, F1, G1, SAFMIN, SAFMN2, SAFMX2, SCALE
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, INT, LOG, MAX, SIGN, SQRT
*     ..
*     .. Save statement ..
*     SAVE               FIRST, SAFMX2, SAFMIN, SAFMN2
*     ..
*     .. Data statements ..
*     DATA               FIRST / .TRUE. /
*     ..
*     .. Executable Statements ..
*
*     IF( FIRST ) THEN
         SAFMIN = DLAMCH( 'S' )
         EPS = DLAMCH( 'E' )
         SAFMN2 = DLAMCH( 'B' )**INT( LOG( SAFMIN / EPS ) /
     $            LOG( DLAMCH( 'B' ) ) / TWO )
         SAFMX2 = ONE / SAFMN2
*        FIRST = .FALSE.
*     END IF
      IF( G.EQ.ZERO ) THEN
         CS = SIGN( ONE, F )
         SN = ZERO
         R = ABS( F )
      ELSE IF( F.EQ.ZERO ) THEN
         CS = ZERO
         SN = SIGN( ONE, G )
         R = ABS( G )
      ELSE
         F1 = F
         G1 = G
         SCALE = MAX( ABS( F1 ), ABS( G1 ) )
         IF( SCALE.GE.SAFMX2 ) THEN
            COUNT = 0
   10       CONTINUE
            COUNT = COUNT + 1
            F1 = F1*SAFMN2
            G1 = G1*SAFMN2
            SCALE = MAX( ABS( F1 ), ABS( G1 ) )
            IF( SCALE.GE.SAFMX2 .AND. COUNT .LT. 20 )
     $         GO TO 10
            R = SQRT( F1**2+G1**2 )
            CS = F1 / R
            SN = G1 / R
            DO 20 I = 1, COUNT
               R = R*SAFMX2
   20       CONTINUE
         ELSE IF( SCALE.LE.SAFMN2 ) THEN
            COUNT = 0
   30       CONTINUE
            COUNT = COUNT + 1
            F1 = F1*SAFMX2
            G1 = G1*SAFMX2
            SCALE = MAX( ABS( F1 ), ABS( G1 ) )
            IF( SCALE.LE.SAFMN2 )
     $         GO TO 30
            R = SQRT( F1**2+G1**2 )
            CS = F1 / R
            SN = G1 / R
            DO 40 I = 1, COUNT
               R = R*SAFMN2
   40       CONTINUE
         ELSE
            R = SQRT( F1**2+G1**2 )
            CS = F1 / R
            SN = G1 / R
         END IF
         IF( R.LT.ZERO ) THEN
            CS = -CS
            SN = -SN
            R = -R
         END IF
      END IF
      RETURN
*
*     End of DLARTGP
*
      END
*> \brief \b DLARTGS generates a plane rotation designed to introduce a bulge in implicit QR iteration for the bidiagonal SVD problem.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARTGS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlartgs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlartgs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlartgs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARTGS( X, Y, SIGMA, CS, SN )
*
*       .. Scalar Arguments ..
*       DOUBLE PRECISION        CS, SIGMA, SN, X, Y
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARTGS generates a plane rotation designed to introduce a bulge in
*> Golub-Reinsch-style implicit QR iteration for the bidiagonal SVD
*> problem. X and Y are the top-row entries, and SIGMA is the shift.
*> The computed CS and SN define a plane rotation satisfying
*>
*>    [  CS  SN  ]  .  [ X^2 - SIGMA ]  =  [ R ],
*>    [ -SN  CS  ]     [    X * Y    ]     [ 0 ]
*>
*> with R nonnegative.  If X^2 - SIGMA and X * Y are 0, then the
*> rotation is by PI/2.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] X
*> \verbatim
*>          X is DOUBLE PRECISION
*>          The (1,1) entry of an upper bidiagonal matrix.
*> \endverbatim
*>
*> \param[in] Y
*> \verbatim
*>          Y is DOUBLE PRECISION
*>          The (1,2) entry of an upper bidiagonal matrix.
*> \endverbatim
*>
*> \param[in] SIGMA
*> \verbatim
*>          SIGMA is DOUBLE PRECISION
*>          The shift.
*> \endverbatim
*>
*> \param[out] CS
*> \verbatim
*>          CS is DOUBLE PRECISION
*>          The cosine of the rotation.
*> \endverbatim
*>
*> \param[out] SN
*> \verbatim
*>          SN is DOUBLE PRECISION
*>          The sine of the rotation.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lartgs
*
*  =====================================================================
      SUBROUTINE DLARTGS( X, Y, SIGMA, CS, SN )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION        CS, SIGMA, SN, X, Y
*     ..
*
*  ===================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION        NEGONE, ONE, ZERO
      PARAMETER          ( NEGONE = -1.0D0, ONE = 1.0D0, ZERO = 0.0D0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION        R, S, THRESH, W, Z
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARTGP
*     ..
*     .. External Functions ..
      DOUBLE PRECISION        DLAMCH
      EXTERNAL           DLAMCH
*     .. Executable Statements ..
*
      THRESH = DLAMCH('E')
*
*     Compute the first column of B**T*B - SIGMA^2*I, up to a scale
*     factor.
*
      IF( (SIGMA .EQ. ZERO .AND. ABS(X) .LT. THRESH) .OR.
     $          (ABS(X) .EQ. SIGMA .AND. Y .EQ. ZERO) ) THEN
         Z = ZERO
         W = ZERO
      ELSE IF( SIGMA .EQ. ZERO ) THEN
         IF( X .GE. ZERO ) THEN
            Z = X
            W = Y
         ELSE
            Z = -X
            W = -Y
         END IF
      ELSE IF( ABS(X) .LT. THRESH ) THEN
         Z = -SIGMA*SIGMA
         W = ZERO
      ELSE
         IF( X .GE. ZERO ) THEN
            S = ONE
         ELSE
            S = NEGONE
         END IF
         Z = S * (ABS(X)-SIGMA) * (S+SIGMA/X)
         W = S * Y
      END IF
*
*     Generate the rotation.
*     CALL DLARTGP( Z, W, CS, SN, R ) might seem more natural;
*     reordering the arguments ensures that if Z = 0 then the rotation
*     is by PI/2.
*
      CALL DLARTGP( W, Z, SN, CS, R )
*
      RETURN
*
*     End DLARTGS
*
      END

*> \brief \b DLARTV applies a vector of plane rotations with real cosines and real sines to the elements of a pair of vectors.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARTV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlartv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlartv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlartv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARTV( N, X, INCX, Y, INCY, C, S, INCC )
*
*       .. Scalar Arguments ..
*       INTEGER            INCC, INCX, INCY, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   C( * ), S( * ), X( * ), Y( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARTV applies a vector of real plane rotations to elements of the
*> real vectors x and y. For i = 1,2,...,n
*>
*>    ( x(i) ) := (  c(i)  s(i) ) ( x(i) )
*>    ( y(i) )    ( -s(i)  c(i) ) ( y(i) )
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of plane rotations to be applied.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array,
*>                         dimension (1+(N-1)*INCX)
*>          The vector x.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*>          INCX is INTEGER
*>          The increment between elements of X. INCX > 0.
*> \endverbatim
*>
*> \param[in,out] Y
*> \verbatim
*>          Y is DOUBLE PRECISION array,
*>                         dimension (1+(N-1)*INCY)
*>          The vector y.
*> \endverbatim
*>
*> \param[in] INCY
*> \verbatim
*>          INCY is INTEGER
*>          The increment between elements of Y. INCY > 0.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
*>          The cosines of the plane rotations.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
*>          The sines of the plane rotations.
*> \endverbatim
*>
*> \param[in] INCC
*> \verbatim
*>          INCC is INTEGER
*>          The increment between elements of C and S. INCC > 0.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lartv
*
*  =====================================================================
      SUBROUTINE DLARTV( N, X, INCX, Y, INCY, C, S, INCC )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INCC, INCX, INCY, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   C( * ), S( * ), X( * ), Y( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            I, IC, IX, IY
      DOUBLE PRECISION   XI, YI
*     ..
*     .. Executable Statements ..
*
      IX = 1
      IY = 1
      IC = 1
      DO 10 I = 1, N
         XI = X( IX )
         YI = Y( IY )
         X( IX ) = C( IC )*XI + S( IC )*YI
         Y( IY ) = C( IC )*YI - S( IC )*XI
         IX = IX + INCX
         IY = IY + INCY
         IC = IC + INCC
   10 CONTINUE
      RETURN
*
*     End of DLARTV
*
      END
*> \brief \b DLARUV returns a vector of n random real numbers from a uniform distribution.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARUV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaruv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaruv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaruv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARUV( ISEED, N, X )
*
*       .. Scalar Arguments ..
*       INTEGER            N
*       ..
*       .. Array Arguments ..
*       INTEGER            ISEED( 4 )
*       DOUBLE PRECISION   X( N )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARUV returns a vector of n random real numbers from a uniform (0,1)
*> distribution (n <= 128).
*>
*> This is an auxiliary routine called by DLARNV and ZLARNV.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in,out] ISEED
*> \verbatim
*>          ISEED is INTEGER array, dimension (4)
*>          On entry, the seed of the random number generator; the array
*>          elements must be between 0 and 4095, and ISEED(4) must be
*>          odd.
*>          On exit, the seed is updated.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of random numbers to be generated. N <= 128.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (N)
*>          The generated random numbers.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laruv
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  This routine uses a multiplicative congruential method with modulus
*>  2**48 and multiplier 33952834046453 (see G.S.Fishman,
*>  'Multiplicative congruential random number generators with modulus
*>  2**b: an exhaustive analysis for b = 32 and a partial analysis for
*>  b = 48', Math. Comp. 189, pp 331-344, 1990).
*>
*>  48-bit integers are stored in 4 integer array elements with 12 bits
*>  per element. Hence the routine is portable across machines with
*>  integers of 32 bits or more.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLARUV( ISEED, N, X )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            N
*     ..
*     .. Array Arguments ..
      INTEGER            ISEED( 4 )
      DOUBLE PRECISION   X( N )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D0 )
      INTEGER            LV, IPW2
      DOUBLE PRECISION   R
      PARAMETER          ( LV = 128, IPW2 = 4096, R = ONE / IPW2 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, I1, I2, I3, I4, IT1, IT2, IT3, IT4, J
*     ..
*     .. Local Arrays ..
      INTEGER            MM( LV, 4 )
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE, MIN, MOD
*     ..
*     .. Data statements ..
      DATA               ( MM( 1, J ), J = 1, 4 ) / 494, 322, 2508,
     $                   2549 /
      DATA               ( MM( 2, J ), J = 1, 4 ) / 2637, 789, 3754,
     $                   1145 /
      DATA               ( MM( 3, J ), J = 1, 4 ) / 255, 1440, 1766,
     $                   2253 /
      DATA               ( MM( 4, J ), J = 1, 4 ) / 2008, 752, 3572,
     $                   305 /
      DATA               ( MM( 5, J ), J = 1, 4 ) / 1253, 2859, 2893,
     $                   3301 /
      DATA               ( MM( 6, J ), J = 1, 4 ) / 3344, 123, 307,
     $                   1065 /
      DATA               ( MM( 7, J ), J = 1, 4 ) / 4084, 1848, 1297,
     $                   3133 /
      DATA               ( MM( 8, J ), J = 1, 4 ) / 1739, 643, 3966,
     $                   2913 /
      DATA               ( MM( 9, J ), J = 1, 4 ) / 3143, 2405, 758,
     $                   3285 /
      DATA               ( MM( 10, J ), J = 1, 4 ) / 3468, 2638, 2598,
     $                   1241 /
      DATA               ( MM( 11, J ), J = 1, 4 ) / 688, 2344, 3406,
     $                   1197 /
      DATA               ( MM( 12, J ), J = 1, 4 ) / 1657, 46, 2922,
     $                   3729 /
      DATA               ( MM( 13, J ), J = 1, 4 ) / 1238, 3814, 1038,
     $                   2501 /
      DATA               ( MM( 14, J ), J = 1, 4 ) / 3166, 913, 2934,
     $                   1673 /
      DATA               ( MM( 15, J ), J = 1, 4 ) / 1292, 3649, 2091,
     $                   541 /
      DATA               ( MM( 16, J ), J = 1, 4 ) / 3422, 339, 2451,
     $                   2753 /
      DATA               ( MM( 17, J ), J = 1, 4 ) / 1270, 3808, 1580,
     $                   949 /
      DATA               ( MM( 18, J ), J = 1, 4 ) / 2016, 822, 1958,
     $                   2361 /
      DATA               ( MM( 19, J ), J = 1, 4 ) / 154, 2832, 2055,
     $                   1165 /
      DATA               ( MM( 20, J ), J = 1, 4 ) / 2862, 3078, 1507,
     $                   4081 /
      DATA               ( MM( 21, J ), J = 1, 4 ) / 697, 3633, 1078,
     $                   2725 /
      DATA               ( MM( 22, J ), J = 1, 4 ) / 1706, 2970, 3273,
     $                   3305 /
      DATA               ( MM( 23, J ), J = 1, 4 ) / 491, 637, 17,
     $                   3069 /
      DATA               ( MM( 24, J ), J = 1, 4 ) / 931, 2249, 854,
     $                   3617 /
      DATA               ( MM( 25, J ), J = 1, 4 ) / 1444, 2081, 2916,
     $                   3733 /
      DATA               ( MM( 26, J ), J = 1, 4 ) / 444, 4019, 3971,
     $                   409 /
      DATA               ( MM( 27, J ), J = 1, 4 ) / 3577, 1478, 2889,
     $                   2157 /
      DATA               ( MM( 28, J ), J = 1, 4 ) / 3944, 242, 3831,
     $                   1361 /
      DATA               ( MM( 29, J ), J = 1, 4 ) / 2184, 481, 2621,
     $                   3973 /
      DATA               ( MM( 30, J ), J = 1, 4 ) / 1661, 2075, 1541,
     $                   1865 /
      DATA               ( MM( 31, J ), J = 1, 4 ) / 3482, 4058, 893,
     $                   2525 /
      DATA               ( MM( 32, J ), J = 1, 4 ) / 657, 622, 736,
     $                   1409 /
      DATA               ( MM( 33, J ), J = 1, 4 ) / 3023, 3376, 3992,
     $                   3445 /
      DATA               ( MM( 34, J ), J = 1, 4 ) / 3618, 812, 787,
     $                   3577 /
      DATA               ( MM( 35, J ), J = 1, 4 ) / 1267, 234, 2125,
     $                   77 /
      DATA               ( MM( 36, J ), J = 1, 4 ) / 1828, 641, 2364,
     $                   3761 /
      DATA               ( MM( 37, J ), J = 1, 4 ) / 164, 4005, 2460,
     $                   2149 /
      DATA               ( MM( 38, J ), J = 1, 4 ) / 3798, 1122, 257,
     $                   1449 /
      DATA               ( MM( 39, J ), J = 1, 4 ) / 3087, 3135, 1574,
     $                   3005 /
      DATA               ( MM( 40, J ), J = 1, 4 ) / 2400, 2640, 3912,
     $                   225 /
      DATA               ( MM( 41, J ), J = 1, 4 ) / 2870, 2302, 1216,
     $                   85 /
      DATA               ( MM( 42, J ), J = 1, 4 ) / 3876, 40, 3248,
     $                   3673 /
      DATA               ( MM( 43, J ), J = 1, 4 ) / 1905, 1832, 3401,
     $                   3117 /
      DATA               ( MM( 44, J ), J = 1, 4 ) / 1593, 2247, 2124,
     $                   3089 /
      DATA               ( MM( 45, J ), J = 1, 4 ) / 1797, 2034, 2762,
     $                   1349 /
      DATA               ( MM( 46, J ), J = 1, 4 ) / 1234, 2637, 149,
     $                   2057 /
      DATA               ( MM( 47, J ), J = 1, 4 ) / 3460, 1287, 2245,
     $                   413 /
      DATA               ( MM( 48, J ), J = 1, 4 ) / 328, 1691, 166,
     $                   65 /
      DATA               ( MM( 49, J ), J = 1, 4 ) / 2861, 496, 466,
     $                   1845 /
      DATA               ( MM( 50, J ), J = 1, 4 ) / 1950, 1597, 4018,
     $                   697 /
      DATA               ( MM( 51, J ), J = 1, 4 ) / 617, 2394, 1399,
     $                   3085 /
      DATA               ( MM( 52, J ), J = 1, 4 ) / 2070, 2584, 190,
     $                   3441 /
      DATA               ( MM( 53, J ), J = 1, 4 ) / 3331, 1843, 2879,
     $                   1573 /
      DATA               ( MM( 54, J ), J = 1, 4 ) / 769, 336, 153,
     $                   3689 /
      DATA               ( MM( 55, J ), J = 1, 4 ) / 1558, 1472, 2320,
     $                   2941 /
      DATA               ( MM( 56, J ), J = 1, 4 ) / 2412, 2407, 18,
     $                   929 /
      DATA               ( MM( 57, J ), J = 1, 4 ) / 2800, 433, 712,
     $                   533 /
      DATA               ( MM( 58, J ), J = 1, 4 ) / 189, 2096, 2159,
     $                   2841 /
      DATA               ( MM( 59, J ), J = 1, 4 ) / 287, 1761, 2318,
     $                   4077 /
      DATA               ( MM( 60, J ), J = 1, 4 ) / 2045, 2810, 2091,
     $                   721 /
      DATA               ( MM( 61, J ), J = 1, 4 ) / 1227, 566, 3443,
     $                   2821 /
      DATA               ( MM( 62, J ), J = 1, 4 ) / 2838, 442, 1510,
     $                   2249 /
      DATA               ( MM( 63, J ), J = 1, 4 ) / 209, 41, 449,
     $                   2397 /
      DATA               ( MM( 64, J ), J = 1, 4 ) / 2770, 1238, 1956,
     $                   2817 /
      DATA               ( MM( 65, J ), J = 1, 4 ) / 3654, 1086, 2201,
     $                   245 /
      DATA               ( MM( 66, J ), J = 1, 4 ) / 3993, 603, 3137,
     $                   1913 /
      DATA               ( MM( 67, J ), J = 1, 4 ) / 192, 840, 3399,
     $                   1997 /
      DATA               ( MM( 68, J ), J = 1, 4 ) / 2253, 3168, 1321,
     $                   3121 /
      DATA               ( MM( 69, J ), J = 1, 4 ) / 3491, 1499, 2271,
     $                   997 /
      DATA               ( MM( 70, J ), J = 1, 4 ) / 2889, 1084, 3667,
     $                   1833 /
      DATA               ( MM( 71, J ), J = 1, 4 ) / 2857, 3438, 2703,
     $                   2877 /
      DATA               ( MM( 72, J ), J = 1, 4 ) / 2094, 2408, 629,
     $                   1633 /
      DATA               ( MM( 73, J ), J = 1, 4 ) / 1818, 1589, 2365,
     $                   981 /
      DATA               ( MM( 74, J ), J = 1, 4 ) / 688, 2391, 2431,
     $                   2009 /
      DATA               ( MM( 75, J ), J = 1, 4 ) / 1407, 288, 1113,
     $                   941 /
      DATA               ( MM( 76, J ), J = 1, 4 ) / 634, 26, 3922,
     $                   2449 /
      DATA               ( MM( 77, J ), J = 1, 4 ) / 3231, 512, 2554,
     $                   197 /
      DATA               ( MM( 78, J ), J = 1, 4 ) / 815, 1456, 184,
     $                   2441 /
      DATA               ( MM( 79, J ), J = 1, 4 ) / 3524, 171, 2099,
     $                   285 /
      DATA               ( MM( 80, J ), J = 1, 4 ) / 1914, 1677, 3228,
     $                   1473 /
      DATA               ( MM( 81, J ), J = 1, 4 ) / 516, 2657, 4012,
     $                   2741 /
      DATA               ( MM( 82, J ), J = 1, 4 ) / 164, 2270, 1921,
     $                   3129 /
      DATA               ( MM( 83, J ), J = 1, 4 ) / 303, 2587, 3452,
     $                   909 /
      DATA               ( MM( 84, J ), J = 1, 4 ) / 2144, 2961, 3901,
     $                   2801 /
      DATA               ( MM( 85, J ), J = 1, 4 ) / 3480, 1970, 572,
     $                   421 /
      DATA               ( MM( 86, J ), J = 1, 4 ) / 119, 1817, 3309,
     $                   4073 /
      DATA               ( MM( 87, J ), J = 1, 4 ) / 3357, 676, 3171,
     $                   2813 /
      DATA               ( MM( 88, J ), J = 1, 4 ) / 837, 1410, 817,
     $                   2337 /
      DATA               ( MM( 89, J ), J = 1, 4 ) / 2826, 3723, 3039,
     $                   1429 /
      DATA               ( MM( 90, J ), J = 1, 4 ) / 2332, 2803, 1696,
     $                   1177 /
      DATA               ( MM( 91, J ), J = 1, 4 ) / 2089, 3185, 1256,
     $                   1901 /
      DATA               ( MM( 92, J ), J = 1, 4 ) / 3780, 184, 3715,
     $                   81 /
      DATA               ( MM( 93, J ), J = 1, 4 ) / 1700, 663, 2077,
     $                   1669 /
      DATA               ( MM( 94, J ), J = 1, 4 ) / 3712, 499, 3019,
     $                   2633 /
      DATA               ( MM( 95, J ), J = 1, 4 ) / 150, 3784, 1497,
     $                   2269 /
      DATA               ( MM( 96, J ), J = 1, 4 ) / 2000, 1631, 1101,
     $                   129 /
      DATA               ( MM( 97, J ), J = 1, 4 ) / 3375, 1925, 717,
     $                   1141 /
      DATA               ( MM( 98, J ), J = 1, 4 ) / 1621, 3912, 51,
     $                   249 /
      DATA               ( MM( 99, J ), J = 1, 4 ) / 3090, 1398, 981,
     $                   3917 /
      DATA               ( MM( 100, J ), J = 1, 4 ) / 3765, 1349, 1978,
     $                   2481 /
      DATA               ( MM( 101, J ), J = 1, 4 ) / 1149, 1441, 1813,
     $                   3941 /
      DATA               ( MM( 102, J ), J = 1, 4 ) / 3146, 2224, 3881,
     $                   2217 /
      DATA               ( MM( 103, J ), J = 1, 4 ) / 33, 2411, 76,
     $                   2749 /
      DATA               ( MM( 104, J ), J = 1, 4 ) / 3082, 1907, 3846,
     $                   3041 /
      DATA               ( MM( 105, J ), J = 1, 4 ) / 2741, 3192, 3694,
     $                   1877 /
      DATA               ( MM( 106, J ), J = 1, 4 ) / 359, 2786, 1682,
     $                   345 /
      DATA               ( MM( 107, J ), J = 1, 4 ) / 3316, 382, 124,
     $                   2861 /
      DATA               ( MM( 108, J ), J = 1, 4 ) / 1749, 37, 1660,
     $                   1809 /
      DATA               ( MM( 109, J ), J = 1, 4 ) / 185, 759, 3997,
     $                   3141 /
      DATA               ( MM( 110, J ), J = 1, 4 ) / 2784, 2948, 479,
     $                   2825 /
      DATA               ( MM( 111, J ), J = 1, 4 ) / 2202, 1862, 1141,
     $                   157 /
      DATA               ( MM( 112, J ), J = 1, 4 ) / 2199, 3802, 886,
     $                   2881 /
      DATA               ( MM( 113, J ), J = 1, 4 ) / 1364, 2423, 3514,
     $                   3637 /
      DATA               ( MM( 114, J ), J = 1, 4 ) / 1244, 2051, 1301,
     $                   1465 /
      DATA               ( MM( 115, J ), J = 1, 4 ) / 2020, 2295, 3604,
     $                   2829 /
      DATA               ( MM( 116, J ), J = 1, 4 ) / 3160, 1332, 1888,
     $                   2161 /
      DATA               ( MM( 117, J ), J = 1, 4 ) / 2785, 1832, 1836,
     $                   3365 /
      DATA               ( MM( 118, J ), J = 1, 4 ) / 2772, 2405, 1990,
     $                   361 /
      DATA               ( MM( 119, J ), J = 1, 4 ) / 1217, 3638, 2058,
     $                   2685 /
      DATA               ( MM( 120, J ), J = 1, 4 ) / 1822, 3661, 692,
     $                   3745 /
      DATA               ( MM( 121, J ), J = 1, 4 ) / 1245, 327, 1194,
     $                   2325 /
      DATA               ( MM( 122, J ), J = 1, 4 ) / 2252, 3660, 20,
     $                   3609 /
      DATA               ( MM( 123, J ), J = 1, 4 ) / 3904, 716, 3285,
     $                   3821 /
      DATA               ( MM( 124, J ), J = 1, 4 ) / 2774, 1842, 2046,
     $                   3537 /
      DATA               ( MM( 125, J ), J = 1, 4 ) / 997, 3987, 2107,
     $                   517 /
      DATA               ( MM( 126, J ), J = 1, 4 ) / 2573, 1368, 3508,
     $                   3017 /
      DATA               ( MM( 127, J ), J = 1, 4 ) / 1148, 1848, 3525,
     $                   2141 /
      DATA               ( MM( 128, J ), J = 1, 4 ) / 545, 2366, 3801,
     $                   1537 /
*     ..
*     .. Executable Statements ..
*
*     Quick return for N < 1
      IF ( N < 1 ) THEN
         RETURN
      END IF
*
      I1 = ISEED( 1 )
      I2 = ISEED( 2 )
      I3 = ISEED( 3 )
      I4 = ISEED( 4 )
*
      DO 10 I = 1, MIN( N, LV )
*
  20     CONTINUE
*
*        Multiply the seed by i-th power of the multiplier modulo 2**48
*
         IT4 = I4*MM( I, 4 )
         IT3 = IT4 / IPW2
         IT4 = IT4 - IPW2*IT3
         IT3 = IT3 + I3*MM( I, 4 ) + I4*MM( I, 3 )
         IT2 = IT3 / IPW2
         IT3 = IT3 - IPW2*IT2
         IT2 = IT2 + I2*MM( I, 4 ) + I3*MM( I, 3 ) + I4*MM( I, 2 )
         IT1 = IT2 / IPW2
         IT2 = IT2 - IPW2*IT1
         IT1 = IT1 + I1*MM( I, 4 ) + I2*MM( I, 3 ) + I3*MM( I, 2 ) +
     $         I4*MM( I, 1 )
         IT1 = MOD( IT1, IPW2 )
*
*        Convert 48-bit integer to a real number in the interval (0,1)
*
         X( I ) = R*( DBLE( IT1 )+R*( DBLE( IT2 )+R*( DBLE( IT3 )+R*
     $            DBLE( IT4 ) ) ) )
*
         IF (X( I ).EQ.1.0D0) THEN
*           If a real number has n bits of precision, and the first
*           n bits of the 48-bit integer above happen to be all 1 (which
*           will occur about once every 2**n calls), then X( I ) will
*           be rounded to exactly 1.0.
*           Since X( I ) is not supposed to return exactly 0.0 or 1.0,
*           the statistically correct thing to do in this situation is
*           simply to iterate again.
*           N.B. the case X( I ) = 0.0 should not be possible.
            I1 = I1 + 2
            I2 = I2 + 2
            I3 = I3 + 2
            I4 = I4 + 2
            GOTO 20
         END IF
*
   10 CONTINUE
*
*     Return final value of seed
*
      ISEED( 1 ) = IT1
      ISEED( 2 ) = IT2
      ISEED( 3 ) = IT3
      ISEED( 4 ) = IT4
      RETURN
*
*     End of DLARUV
*
      END
*> \brief \b DLARZ applies an elementary reflector (as returned by stzrzf) to a general matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARZ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarz.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarz.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarz.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARZ( SIDE, M, N, L, V, INCV, TAU, C, LDC, WORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIDE
*       INTEGER            INCV, L, LDC, M, N
*       DOUBLE PRECISION   TAU
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   C( LDC, * ), V( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARZ applies a real elementary reflector H to a real M-by-N
*> matrix C, from either the left or the right. H is represented in the
*> form
*>
*>       H = I - tau * v * v**T
*>
*> where tau is a real scalar and v is a real vector.
*>
*> If tau = 0, then H is taken to be the unit matrix.
*>
*>
*> H is a product of k elementary reflectors as returned by DTZRZF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': form  H * C
*>          = 'R': form  C * H
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*>          L is INTEGER
*>          The number of entries of the vector V containing
*>          the meaningful part of the Householder vectors.
*>          If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension (1+(L-1)*abs(INCV))
*>          The vector v in the representation of H as returned by
*>          DTZRZF. V is not used if TAU = 0.
*> \endverbatim
*>
*> \param[in] INCV
*> \verbatim
*>          INCV is INTEGER
*>          The increment between elements of v. INCV <> 0.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION
*>          The value tau in the representation of H.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the M-by-N matrix C.
*>          On exit, C is overwritten by the matrix H * C if SIDE = 'L',
*>          or C * H if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension
*>                         (N) if SIDE = 'L'
*>                      or (M) if SIDE = 'R'
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larz
*
*> \par Contributors:
*  ==================
*>
*>    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLARZ( SIDE, M, N, L, V, INCV, TAU, C, LDC, WORK )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE
      INTEGER            INCV, L, LDC, M, N
      DOUBLE PRECISION   TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   C( LDC, * ), V( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DGEMV, DGER
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. Executable Statements ..
*
      IF( LSAME( SIDE, 'L' ) ) THEN
*
*        Form  H * C
*
         IF( TAU.NE.ZERO ) THEN
*
*           w( 1:n ) = C( 1, 1:n )
*
            CALL DCOPY( N, C, LDC, WORK, 1 )
*
*           w( 1:n ) = w( 1:n ) + C( m-l+1:m, 1:n )**T * v( 1:l )
*
            CALL DGEMV( 'Transpose', L, N, ONE, C( M-L+1, 1 ), LDC,
     $                  V,
     $                  INCV, ONE, WORK, 1 )
*
*           C( 1, 1:n ) = C( 1, 1:n ) - tau * w( 1:n )
*
            CALL DAXPY( N, -TAU, WORK, 1, C, LDC )
*
*           C( m-l+1:m, 1:n ) = C( m-l+1:m, 1:n ) - ...
*                               tau * v( 1:l ) * w( 1:n )**T
*
            CALL DGER( L, N, -TAU, V, INCV, WORK, 1, C( M-L+1, 1 ),
     $                 LDC )
         END IF
*
      ELSE
*
*        Form  C * H
*
         IF( TAU.NE.ZERO ) THEN
*
*           w( 1:m ) = C( 1:m, 1 )
*
            CALL DCOPY( M, C, 1, WORK, 1 )
*
*           w( 1:m ) = w( 1:m ) + C( 1:m, n-l+1:n, 1:n ) * v( 1:l )
*
            CALL DGEMV( 'No transpose', M, L, ONE, C( 1, N-L+1 ),
     $                  LDC,
     $                  V, INCV, ONE, WORK, 1 )
*
*           C( 1:m, 1 ) = C( 1:m, 1 ) - tau * w( 1:m )
*
            CALL DAXPY( M, -TAU, WORK, 1, C, 1 )
*
*           C( 1:m, n-l+1:n ) = C( 1:m, n-l+1:n ) - ...
*                               tau * w( 1:m ) * v( 1:l )**T
*
            CALL DGER( M, L, -TAU, WORK, 1, V, INCV, C( 1, N-L+1 ),
     $                 LDC )
*
         END IF
*
      END IF
*
      RETURN
*
*     End of DLARZ
*
      END
*> \brief \b DLARZB applies a block reflector or its transpose to a general matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARZB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarzb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarzb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarzb.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARZB( SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V,
*                          LDV, T, LDT, C, LDC, WORK, LDWORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIRECT, SIDE, STOREV, TRANS
*       INTEGER            K, L, LDC, LDT, LDV, LDWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   C( LDC, * ), T( LDT, * ), V( LDV, * ),
*      $                   WORK( LDWORK, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARZB applies a real block reflector H or its transpose H**T to
*> a real distributed M-by-N  C from the left or the right.
*>
*> Currently, only STOREV = 'R' and DIRECT = 'B' are supported.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': apply H or H**T from the Left
*>          = 'R': apply H or H**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N': apply H (No transpose)
*>          = 'C': apply H**T (Transpose)
*> \endverbatim
*>
*> \param[in] DIRECT
*> \verbatim
*>          DIRECT is CHARACTER*1
*>          Indicates how H is formed from a product of elementary
*>          reflectors
*>          = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
*>          = 'B': H = H(k) . . . H(2) H(1) (Backward)
*> \endverbatim
*>
*> \param[in] STOREV
*> \verbatim
*>          STOREV is CHARACTER*1
*>          Indicates how the vectors which define the elementary
*>          reflectors are stored:
*>          = 'C': Columnwise                        (not supported yet)
*>          = 'R': Rowwise
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The order of the matrix T (= the number of elementary
*>          reflectors whose product defines the block reflector).
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*>          L is INTEGER
*>          The number of columns of the matrix V containing the
*>          meaningful part of the Householder reflectors.
*>          If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension (LDV,NV).
*>          If STOREV = 'C', NV = K; if STOREV = 'R', NV = L.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of the array V.
*>          If STOREV = 'C', LDV >= L; if STOREV = 'R', LDV >= K.
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,K)
*>          The triangular K-by-K matrix T in the representation of the
*>          block reflector.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T. LDT >= K.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the M-by-N matrix C.
*>          On exit, C is overwritten by H*C or H**T*C or C*H or C*H**T.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (LDWORK,K)
*> \endverbatim
*>
*> \param[in] LDWORK
*> \verbatim
*>          LDWORK is INTEGER
*>          The leading dimension of the array WORK.
*>          If SIDE = 'L', LDWORK >= max(1,N);
*>          if SIDE = 'R', LDWORK >= max(1,M).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larzb
*
*> \par Contributors:
*  ==================
*>
*>    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLARZB( SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V,
     $                   LDV, T, LDT, C, LDC, WORK, LDWORK )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIRECT, SIDE, STOREV, TRANS
      INTEGER            K, L, LDC, LDT, LDV, LDWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   C( LDC, * ), T( LDT, * ), V( LDV, * ),
     $                   WORK( LDWORK, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      CHARACTER          TRANST
      INTEGER            I, INFO, J
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGEMM, DTRMM, XERBLA
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( M.LE.0 .OR. N.LE.0 )
     $   RETURN
*
*     Check for currently supported options
*
      INFO = 0
      IF( .NOT.LSAME( DIRECT, 'B' ) ) THEN
         INFO = -3
      ELSE IF( .NOT.LSAME( STOREV, 'R' ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLARZB', -INFO )
         RETURN
      END IF
*
      IF( LSAME( TRANS, 'N' ) ) THEN
         TRANST = 'T'
      ELSE
         TRANST = 'N'
      END IF
*
      IF( LSAME( SIDE, 'L' ) ) THEN
*
*        Form  H * C  or  H**T * C
*
*        W( 1:n, 1:k ) = C( 1:k, 1:n )**T
*
         DO 10 J = 1, K
            CALL DCOPY( N, C( J, 1 ), LDC, WORK( 1, J ), 1 )
   10    CONTINUE
*
*        W( 1:n, 1:k ) = W( 1:n, 1:k ) + ...
*                        C( m-l+1:m, 1:n )**T * V( 1:k, 1:l )**T
*
         IF( L.GT.0 )
     $      CALL DGEMM( 'Transpose', 'Transpose', N, K, L, ONE,
     $                  C( M-L+1, 1 ), LDC, V, LDV, ONE, WORK, LDWORK )
*
*        W( 1:n, 1:k ) = W( 1:n, 1:k ) * T**T  or  W( 1:m, 1:k ) * T
*
         CALL DTRMM( 'Right', 'Lower', TRANST, 'Non-unit', N, K, ONE,
     $               T,
     $               LDT, WORK, LDWORK )
*
*        C( 1:k, 1:n ) = C( 1:k, 1:n ) - W( 1:n, 1:k )**T
*
         DO 30 J = 1, N
            DO 20 I = 1, K
               C( I, J ) = C( I, J ) - WORK( J, I )
   20       CONTINUE
   30    CONTINUE
*
*        C( m-l+1:m, 1:n ) = C( m-l+1:m, 1:n ) - ...
*                            V( 1:k, 1:l )**T * W( 1:n, 1:k )**T
*
         IF( L.GT.0 )
     $      CALL DGEMM( 'Transpose', 'Transpose', L, N, K, -ONE, V,
     $                  LDV,
     $                  WORK, LDWORK, ONE, C( M-L+1, 1 ), LDC )
*
      ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
*        Form  C * H  or  C * H**T
*
*        W( 1:m, 1:k ) = C( 1:m, 1:k )
*
         DO 40 J = 1, K
            CALL DCOPY( M, C( 1, J ), 1, WORK( 1, J ), 1 )
   40    CONTINUE
*
*        W( 1:m, 1:k ) = W( 1:m, 1:k ) + ...
*                        C( 1:m, n-l+1:n ) * V( 1:k, 1:l )**T
*
         IF( L.GT.0 )
     $      CALL DGEMM( 'No transpose', 'Transpose', M, K, L, ONE,
     $                  C( 1, N-L+1 ), LDC, V, LDV, ONE, WORK, LDWORK )
*
*        W( 1:m, 1:k ) = W( 1:m, 1:k ) * T  or  W( 1:m, 1:k ) * T**T
*
         CALL DTRMM( 'Right', 'Lower', TRANS, 'Non-unit', M, K, ONE,
     $               T,
     $               LDT, WORK, LDWORK )
*
*        C( 1:m, 1:k ) = C( 1:m, 1:k ) - W( 1:m, 1:k )
*
         DO 60 J = 1, K
            DO 50 I = 1, M
               C( I, J ) = C( I, J ) - WORK( I, J )
   50       CONTINUE
   60    CONTINUE
*
*        C( 1:m, n-l+1:n ) = C( 1:m, n-l+1:n ) - ...
*                            W( 1:m, 1:k ) * V( 1:k, 1:l )
*
         IF( L.GT.0 )
     $      CALL DGEMM( 'No transpose', 'No transpose', M, L, K,
     $                  -ONE,
     $                  WORK, LDWORK, V, LDV, ONE, C( 1, N-L+1 ), LDC )
*
      END IF
*
      RETURN
*
*     End of DLARZB
*
      END
*> \brief \b DLARZT forms the triangular factor T of a block reflector H = I - vtvH.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARZT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarzt.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarzt.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarzt.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARZT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIRECT, STOREV
*       INTEGER            K, LDT, LDV, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   T( LDT, * ), TAU( * ), V( LDV, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARZT forms the triangular factor T of a real block reflector
*> H of order > n, which is defined as a product of k elementary
*> reflectors.
*>
*> If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
*>
*> If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
*>
*> If STOREV = 'C', the vector which defines the elementary reflector
*> H(i) is stored in the i-th column of the array V, and
*>
*>    H  =  I - V * T * V**T
*>
*> If STOREV = 'R', the vector which defines the elementary reflector
*> H(i) is stored in the i-th row of the array V, and
*>
*>    H  =  I - V**T * T * V
*>
*> Currently, only STOREV = 'R' and DIRECT = 'B' are supported.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] DIRECT
*> \verbatim
*>          DIRECT is CHARACTER*1
*>          Specifies the order in which the elementary reflectors are
*>          multiplied to form the block reflector:
*>          = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
*>          = 'B': H = H(k) . . . H(2) H(1) (Backward)
*> \endverbatim
*>
*> \param[in] STOREV
*> \verbatim
*>          STOREV is CHARACTER*1
*>          Specifies how the vectors which define the elementary
*>          reflectors are stored (see also Further Details):
*>          = 'C': columnwise                        (not supported yet)
*>          = 'R': rowwise
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the block reflector H. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The order of the triangular factor T (= the number of
*>          elementary reflectors). K >= 1.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension
*>                               (LDV,K) if STOREV = 'C'
*>                               (LDV,N) if STOREV = 'R'
*>          The matrix V. See further details.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of the array V.
*>          If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,K)
*>          The k by k triangular factor T of the block reflector.
*>          If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
*>          lower triangular. The rest of the array is not used.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T. LDT >= K.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larzt
*
*> \par Contributors:
*  ==================
*>
*>    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The shape of the matrix V and the storage of the vectors which define
*>  the H(i) is best illustrated by the following example with n = 5 and
*>  k = 3. The elements equal to 1 are not stored; the corresponding
*>  array elements are modified but restored on exit. The rest of the
*>  array is not used.
*>
*>  DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
*>
*>                                              ______V_____
*>         ( v1 v2 v3 )                        /            \
*>         ( v1 v2 v3 )                      ( v1 v1 v1 v1 v1 . . . . 1 )
*>     V = ( v1 v2 v3 )                      ( v2 v2 v2 v2 v2 . . . 1   )
*>         ( v1 v2 v3 )                      ( v3 v3 v3 v3 v3 . . 1     )
*>         ( v1 v2 v3 )
*>            .  .  .
*>            .  .  .
*>            1  .  .
*>               1  .
*>                  1
*>
*>  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
*>
*>                                                        ______V_____
*>            1                                          /            \
*>            .  1                           ( 1 . . . . v1 v1 v1 v1 v1 )
*>            .  .  1                        ( . 1 . . . v2 v2 v2 v2 v2 )
*>            .  .  .                        ( . . 1 . . v3 v3 v3 v3 v3 )
*>            .  .  .
*>         ( v1 v2 v3 )
*>         ( v1 v2 v3 )
*>     V = ( v1 v2 v3 )
*>         ( v1 v2 v3 )
*>         ( v1 v2 v3 )
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLARZT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIRECT, STOREV
      INTEGER            K, LDT, LDV, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   T( LDT, * ), TAU( * ), V( LDV, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, INFO, J
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMV, DTRMV, XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. Executable Statements ..
*
*     Check for currently supported options
*
      INFO = 0
      IF( .NOT.LSAME( DIRECT, 'B' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.LSAME( STOREV, 'R' ) ) THEN
         INFO = -2
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLARZT', -INFO )
         RETURN
      END IF
*
      DO 20 I = K, 1, -1
         IF( TAU( I ).EQ.ZERO ) THEN
*
*           H(i)  =  I
*
            DO 10 J = I, K
               T( J, I ) = ZERO
   10       CONTINUE
         ELSE
*
*           general case
*
            IF( I.LT.K ) THEN
*
*              T(i+1:k,i) = - tau(i) * V(i+1:k,1:n) * V(i,1:n)**T
*
               CALL DGEMV( 'No transpose', K-I, N, -TAU( I ),
     $                     V( I+1, 1 ), LDV, V( I, 1 ), LDV, ZERO,
     $                     T( I+1, I ), 1 )
*
*              T(i+1:k,i) = T(i+1:k,i+1:k) * T(i+1:k,i)
*
               CALL DTRMV( 'Lower', 'No transpose', 'Non-unit', K-I,
     $                     T( I+1, I+1 ), LDT, T( I+1, I ), 1 )
            END IF
            T( I, I ) = TAU( I )
         END IF
   20 CONTINUE
      RETURN
*
*     End of DLARZT
*
      END
*> \brief \b DLAS2 computes singular values of a 2-by-2 triangular matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAS2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlas2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlas2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlas2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAS2( F, G, H, SSMIN, SSMAX )
*
*       .. Scalar Arguments ..
*       DOUBLE PRECISION   F, G, H, SSMAX, SSMIN
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAS2  computes the singular values of the 2-by-2 matrix
*>    [  F   G  ]
*>    [  0   H  ].
*> On return, SSMIN is the smaller singular value and SSMAX is the
*> larger singular value.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] F
*> \verbatim
*>          F is DOUBLE PRECISION
*>          The (1,1) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] G
*> \verbatim
*>          G is DOUBLE PRECISION
*>          The (1,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*>          H is DOUBLE PRECISION
*>          The (2,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[out] SSMIN
*> \verbatim
*>          SSMIN is DOUBLE PRECISION
*>          The smaller singular value.
*> \endverbatim
*>
*> \param[out] SSMAX
*> \verbatim
*>          SSMAX is DOUBLE PRECISION
*>          The larger singular value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup las2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  Barring over/underflow, all output quantities are correct to within
*>  a few units in the last place (ulps), even in the absence of a guard
*>  digit in addition/subtraction.
*>
*>  In IEEE arithmetic, the code works correctly if one matrix element is
*>  infinite.
*>
*>  Overflow will not occur unless the largest singular value itself
*>  overflows, or is within a few ulps of overflow.
*>
*>  Underflow is harmless if underflow is gradual. Otherwise, results
*>  may correspond to a matrix modified by perturbations of size near
*>  the underflow threshold.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLAS2( F, G, H, SSMIN, SSMAX )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   F, G, H, SSMAX, SSMIN
*     ..
*
*  ====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D0 )
      DOUBLE PRECISION   TWO
      PARAMETER          ( TWO = 2.0D0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   AS, AT, AU, C, FA, FHMN, FHMX, GA, HA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
      FA = ABS( F )
      GA = ABS( G )
      HA = ABS( H )
      FHMN = MIN( FA, HA )
      FHMX = MAX( FA, HA )
      IF( FHMN.EQ.ZERO ) THEN
         SSMIN = ZERO
         IF( FHMX.EQ.ZERO ) THEN
            SSMAX = GA
         ELSE
            SSMAX = MAX( FHMX, GA )*SQRT( ONE+
     $              ( MIN( FHMX, GA ) / MAX( FHMX, GA ) )**2 )
         END IF
      ELSE
         IF( GA.LT.FHMX ) THEN
            AS = ONE + FHMN / FHMX
            AT = ( FHMX-FHMN ) / FHMX
            AU = ( GA / FHMX )**2
            C = TWO / ( SQRT( AS*AS+AU )+SQRT( AT*AT+AU ) )
            SSMIN = FHMN*C
            SSMAX = FHMX / C
         ELSE
            AU = FHMX / GA
            IF( AU.EQ.ZERO ) THEN
*
*              Avoid possible harmful underflow if exponent range
*              asymmetric (true SSMIN may not underflow even if
*              AU underflows)
*
               SSMIN = ( FHMN*FHMX ) / GA
               SSMAX = GA
            ELSE
               AS = ONE + FHMN / FHMX
               AT = ( FHMX-FHMN ) / FHMX
               C = ONE / ( SQRT( ONE+( AS*AU )**2 )+
     $             SQRT( ONE+( AT*AU )**2 ) )
               SSMIN = ( FHMN*C )*AU
               SSMIN = SSMIN + SSMIN
               SSMAX = GA / ( C+C )
            END IF
         END IF
      END IF
      RETURN
*
*     End of DLAS2
*
      END
*> \brief \b DLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASCL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlascl.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlascl.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlascl.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASCL( TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TYPE
*       INTEGER            INFO, KL, KU, LDA, M, N
*       DOUBLE PRECISION   CFROM, CTO
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASCL multiplies the M by N real matrix A by the real scalar
*> CTO/CFROM.  This is done without over/underflow as long as the final
*> result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that
*> A may be full, upper triangular, lower triangular, upper Hessenberg,
*> or banded.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TYPE
*> \verbatim
*>          TYPE is CHARACTER*1
*>          TYPE indices the storage type of the input matrix.
*>          = 'G':  A is a full matrix.
*>          = 'L':  A is a lower triangular matrix.
*>          = 'U':  A is an upper triangular matrix.
*>          = 'H':  A is an upper Hessenberg matrix.
*>          = 'B':  A is a symmetric band matrix with lower bandwidth KL
*>                  and upper bandwidth KU and with the only the lower
*>                  half stored.
*>          = 'Q':  A is a symmetric band matrix with lower bandwidth KL
*>                  and upper bandwidth KU and with the only the upper
*>                  half stored.
*>          = 'Z':  A is a band matrix with lower bandwidth KL and upper
*>                  bandwidth KU. See DGBTRF for storage details.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*>          KL is INTEGER
*>          The lower bandwidth of A.  Referenced only if TYPE = 'B',
*>          'Q' or 'Z'.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*>          KU is INTEGER
*>          The upper bandwidth of A.  Referenced only if TYPE = 'B',
*>          'Q' or 'Z'.
*> \endverbatim
*>
*> \param[in] CFROM
*> \verbatim
*>          CFROM is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] CTO
*> \verbatim
*>          CTO is DOUBLE PRECISION
*>
*>          The matrix A is multiplied by CTO/CFROM. A(I,J) is computed
*>          without over/underflow if the final result CTO*A(I,J)/CFROM
*>          can be represented without over/underflow.  CFROM must be
*>          nonzero.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The matrix to be multiplied by CTO/CFROM.  See TYPE for the
*>          storage type.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.
*>          If TYPE = 'G', 'L', 'U', 'H', LDA >= max(1,M);
*>             TYPE = 'B', LDA >= KL+1;
*>             TYPE = 'Q', LDA >= KU+1;
*>             TYPE = 'Z', LDA >= 2*KL+KU+1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          0  - successful exit
*>          <0 - if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lascl
*
*  =====================================================================
      SUBROUTINE DLASCL( TYPE, KL, KU, CFROM, CTO, M, N, A, LDA,
     $                   INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TYPE
      INTEGER            INFO, KL, KU, LDA, M, N
      DOUBLE PRECISION   CFROM, CTO
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            DONE
      INTEGER            I, ITYPE, J, K1, K2, K3, K4
      DOUBLE PRECISION   BIGNUM, CFROM1, CFROMC, CTO1, CTOC, MUL, SMLNUM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH, DISNAN
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
*
      IF( LSAME( TYPE, 'G' ) ) THEN
         ITYPE = 0
      ELSE IF( LSAME( TYPE, 'L' ) ) THEN
         ITYPE = 1
      ELSE IF( LSAME( TYPE, 'U' ) ) THEN
         ITYPE = 2
      ELSE IF( LSAME( TYPE, 'H' ) ) THEN
         ITYPE = 3
      ELSE IF( LSAME( TYPE, 'B' ) ) THEN
         ITYPE = 4
      ELSE IF( LSAME( TYPE, 'Q' ) ) THEN
         ITYPE = 5
      ELSE IF( LSAME( TYPE, 'Z' ) ) THEN
         ITYPE = 6
      ELSE
         ITYPE = -1
      END IF
*
      IF( ITYPE.EQ.-1 ) THEN
         INFO = -1
      ELSE IF( CFROM.EQ.ZERO .OR. DISNAN(CFROM) ) THEN
         INFO = -4
      ELSE IF( DISNAN(CTO) ) THEN
         INFO = -5
      ELSE IF( M.LT.0 ) THEN
         INFO = -6
      ELSE IF( N.LT.0 .OR. ( ITYPE.EQ.4 .AND. N.NE.M ) .OR.
     $         ( ITYPE.EQ.5 .AND. N.NE.M ) ) THEN
         INFO = -7
      ELSE IF( ITYPE.LE.3 .AND. LDA.LT.MAX( 1, M ) ) THEN
         INFO = -9
      ELSE IF( ITYPE.GE.4 ) THEN
         IF( KL.LT.0 .OR. KL.GT.MAX( M-1, 0 ) ) THEN
            INFO = -2
         ELSE IF( KU.LT.0 .OR. KU.GT.MAX( N-1, 0 ) .OR.
     $            ( ( ITYPE.EQ.4 .OR. ITYPE.EQ.5 ) .AND. KL.NE.KU ) )
     $             THEN
            INFO = -3
         ELSE IF( ( ITYPE.EQ.4 .AND. LDA.LT.KL+1 ) .OR.
     $            ( ITYPE.EQ.5 .AND. LDA.LT.KU+1 ) .OR.
     $            ( ITYPE.EQ.6 .AND. LDA.LT.2*KL+KU+1 ) ) THEN
            INFO = -9
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLASCL', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. M.EQ.0 )
     $   RETURN
*
*     Get machine parameters
*
      SMLNUM = DLAMCH( 'S' )
      BIGNUM = ONE / SMLNUM
*
      CFROMC = CFROM
      CTOC = CTO
*
   10 CONTINUE
      CFROM1 = CFROMC*SMLNUM
      IF( CFROM1.EQ.CFROMC ) THEN
!        CFROMC is an inf.  Multiply by a correctly signed zero for
!        finite CTOC, or a NaN if CTOC is infinite.
         MUL = CTOC / CFROMC
         DONE = .TRUE.
         CTO1 = CTOC
      ELSE
         CTO1 = CTOC / BIGNUM
         IF( CTO1.EQ.CTOC ) THEN
!           CTOC is either 0 or an inf.  In both cases, CTOC itself
!           serves as the correct multiplication factor.
            MUL = CTOC
            DONE = .TRUE.
            CFROMC = ONE
         ELSE IF( ABS( CFROM1 ).GT.ABS( CTOC ) .AND. CTOC.NE.ZERO ) THEN
            MUL = SMLNUM
            DONE = .FALSE.
            CFROMC = CFROM1
         ELSE IF( ABS( CTO1 ).GT.ABS( CFROMC ) ) THEN
            MUL = BIGNUM
            DONE = .FALSE.
            CTOC = CTO1
         ELSE
            MUL = CTOC / CFROMC
            DONE = .TRUE.
            IF (MUL .EQ. ONE)
     $         RETURN
         END IF
      END IF
*
      IF( ITYPE.EQ.0 ) THEN
*
*        Full matrix
*
         DO 30 J = 1, N
            DO 20 I = 1, M
               A( I, J ) = A( I, J )*MUL
   20       CONTINUE
   30    CONTINUE
*
      ELSE IF( ITYPE.EQ.1 ) THEN
*
*        Lower triangular matrix
*
         DO 50 J = 1, N
            DO 40 I = J, M
               A( I, J ) = A( I, J )*MUL
   40       CONTINUE
   50    CONTINUE
*
      ELSE IF( ITYPE.EQ.2 ) THEN
*
*        Upper triangular matrix
*
         DO 70 J = 1, N
            DO 60 I = 1, MIN( J, M )
               A( I, J ) = A( I, J )*MUL
   60       CONTINUE
   70    CONTINUE
*
      ELSE IF( ITYPE.EQ.3 ) THEN
*
*        Upper Hessenberg matrix
*
         DO 90 J = 1, N
            DO 80 I = 1, MIN( J+1, M )
               A( I, J ) = A( I, J )*MUL
   80       CONTINUE
   90    CONTINUE
*
      ELSE IF( ITYPE.EQ.4 ) THEN
*
*        Lower half of a symmetric band matrix
*
         K3 = KL + 1
         K4 = N + 1
         DO 110 J = 1, N
            DO 100 I = 1, MIN( K3, K4-J )
               A( I, J ) = A( I, J )*MUL
  100       CONTINUE
  110    CONTINUE
*
      ELSE IF( ITYPE.EQ.5 ) THEN
*
*        Upper half of a symmetric band matrix
*
         K1 = KU + 2
         K3 = KU + 1
         DO 130 J = 1, N
            DO 120 I = MAX( K1-J, 1 ), K3
               A( I, J ) = A( I, J )*MUL
  120       CONTINUE
  130    CONTINUE
*
      ELSE IF( ITYPE.EQ.6 ) THEN
*
*        Band matrix
*
         K1 = KL + KU + 2
         K2 = KL + 1
         K3 = 2*KL + KU + 1
         K4 = KL + KU + 1 + M
         DO 150 J = 1, N
            DO 140 I = MAX( K1-J, K2 ), MIN( K3, K4-J )
               A( I, J ) = A( I, J )*MUL
  140       CONTINUE
  150    CONTINUE
*
      END IF
*
      IF( .NOT.DONE )
     $   GO TO 10
*
      RETURN
*
*     End of DLASCL
*
      END
*> \brief \b DLASCL2 performs diagonal scaling on a matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASCL2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlascl2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlascl2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlascl2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASCL2 ( M, N, D, X, LDX )
*
*       .. Scalar Arguments ..
*       INTEGER            M, N, LDX
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASCL2 performs a diagonal scaling on a matrix:
*>   x <-- D * x
*> where the diagonal matrix D is stored as a vector.
*>
*> Eventually to be replaced by BLAS_dge_diag_scale in the new BLAS
*> standard.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>     The number of rows of D and X. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>     The number of columns of X. N >= 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, length M
*>     Diagonal matrix D, stored as a vector of length M.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,N)
*>     On entry, the matrix X to be scaled by D.
*>     On exit, the scaled matrix.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>     The leading dimension of the matrix X. LDX >= M.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lascl2
*
*  =====================================================================
      SUBROUTINE DLASCL2 ( M, N, D, X, LDX )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            M, N, LDX
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            I, J
*     ..
*     .. Executable Statements ..
*
      DO J = 1, N
         DO I = 1, M
            X( I, J ) = X( I, J ) * D( I )
         END DO
      END DO

      RETURN
      END
*> \brief \b DLASD0 computes the singular values of a real upper bidiagonal n-by-m matrix B with diagonal d and off-diagonal e. Used by sbdsdc.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASD0 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd0.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd0.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd0.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASD0( N, SQRE, D, E, U, LDU, VT, LDVT, SMLSIZ, IWORK,
*                          WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDU, LDVT, N, SMLSIZ, SQRE
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   D( * ), E( * ), U( LDU, * ), VT( LDVT, * ),
*      $                   WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> Using a divide and conquer approach, DLASD0 computes the singular
*> value decomposition (SVD) of a real upper bidiagonal N-by-M
*> matrix B with diagonal D and offdiagonal E, where M = N + SQRE.
*> The algorithm computes orthogonal matrices U and VT such that
*> B = U * S * VT. The singular values S are overwritten on D.
*>
*> A related subroutine, DLASDA, computes only the singular values,
*> and optionally, the singular vectors in compact form.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>         On entry, the row dimension of the upper bidiagonal matrix.
*>         This is also the dimension of the main diagonal array D.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*>          SQRE is INTEGER
*>         Specifies the column dimension of the bidiagonal matrix.
*>         = 0: The bidiagonal matrix has column dimension M = N;
*>         = 1: The bidiagonal matrix has column dimension M = N+1;
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>         On entry D contains the main diagonal of the bidiagonal
*>         matrix.
*>         On exit D, if INFO = 0, contains its singular values.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (M-1)
*>         Contains the subdiagonal entries of the bidiagonal matrix.
*>         On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] U
*> \verbatim
*>          U is DOUBLE PRECISION array, dimension (LDU, N)
*>         On exit, U contains the left singular vectors, 
*>          if U passed in as (N, N) Identity.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>         On entry, leading dimension of U.
*> \endverbatim
*>
*> \param[in,out] VT
*> \verbatim
*>          VT is DOUBLE PRECISION array, dimension (LDVT, M)
*>         On exit, VT**T contains the right singular vectors,
*>          if VT passed in as (M, M) Identity.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*>          LDVT is INTEGER
*>         On entry, leading dimension of VT.
*> \endverbatim
*>
*> \param[in] SMLSIZ
*> \verbatim
*>          SMLSIZ is INTEGER
*>         On entry, maximum size of the subproblems at the
*>         bottom of the computation tree.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (8*N)
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*M**2+2*M)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = 1, a singular value did not converge
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasd0
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Huan Ren, Computer Science Division, University of
*>     California at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE DLASD0( N, SQRE, D, E, U, LDU, VT, LDVT, SMLSIZ,
     $                   IWORK,
     $                   WORK, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDU, LDVT, N, SMLSIZ, SQRE
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   D( * ), E( * ), U( LDU, * ), VT( LDVT, * ),
     $                   WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            I, I1, IC, IDXQ, IDXQC, IM1, INODE, ITEMP, IWK,
     $                   J, LF, LL, LVL, M, NCC, ND, NDB1, NDIML, NDIMR,
     $                   NL, NLF, NLP1, NLVL, NR, NRF, NRP1, SQREI
      DOUBLE PRECISION   ALPHA, BETA
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASD1, DLASDQ, DLASDT, XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IF( N.LT.0 ) THEN
         INFO = -1
      ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
         INFO = -2
      END IF
*
      M = N + SQRE
*
      IF( LDU.LT.N ) THEN
         INFO = -6
      ELSE IF( LDVT.LT.M ) THEN
         INFO = -8
      ELSE IF( SMLSIZ.LT.3 ) THEN
         INFO = -9
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLASD0', -INFO )
         RETURN
      END IF
*
*     If the input matrix is too small, call DLASDQ to find the SVD.
*
      IF( N.LE.SMLSIZ ) THEN
         CALL DLASDQ( 'U', SQRE, N, M, N, 0, D, E, VT, LDVT, U, LDU,
     $                U,
     $                LDU, WORK, INFO )
         RETURN
      END IF
*
*     Set up the computation tree.
*
      INODE = 1
      NDIML = INODE + N
      NDIMR = NDIML + N
      IDXQ = NDIMR + N
      IWK = IDXQ + N
      CALL DLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ),
     $             IWORK( NDIMR ), SMLSIZ )
*
*     For the nodes on bottom level of the tree, solve
*     their subproblems by DLASDQ.
*
      NDB1 = ( ND+1 ) / 2
      NCC = 0
      DO 30 I = NDB1, ND
*
*     IC : center row of each node
*     NL : number of rows of left  subproblem
*     NR : number of rows of right subproblem
*     NLF: starting row of the left   subproblem
*     NRF: starting row of the right  subproblem
*
         I1 = I - 1
         IC = IWORK( INODE+I1 )
         NL = IWORK( NDIML+I1 )
         NLP1 = NL + 1
         NR = IWORK( NDIMR+I1 )
         NRP1 = NR + 1
         NLF = IC - NL
         NRF = IC + 1
         SQREI = 1
         CALL DLASDQ( 'U', SQREI, NL, NLP1, NL, NCC, D( NLF ),
     $                E( NLF ),
     $                VT( NLF, NLF ), LDVT, U( NLF, NLF ), LDU,
     $                U( NLF, NLF ), LDU, WORK, INFO )
         IF( INFO.NE.0 ) THEN
            RETURN
         END IF
         ITEMP = IDXQ + NLF - 2
         DO 10 J = 1, NL
            IWORK( ITEMP+J ) = J
   10    CONTINUE
         IF( I.EQ.ND ) THEN
            SQREI = SQRE
         ELSE
            SQREI = 1
         END IF
         NRP1 = NR + SQREI
         CALL DLASDQ( 'U', SQREI, NR, NRP1, NR, NCC, D( NRF ),
     $                E( NRF ),
     $                VT( NRF, NRF ), LDVT, U( NRF, NRF ), LDU,
     $                U( NRF, NRF ), LDU, WORK, INFO )
         IF( INFO.NE.0 ) THEN
            RETURN
         END IF
         ITEMP = IDXQ + IC
         DO 20 J = 1, NR
            IWORK( ITEMP+J-1 ) = J
   20    CONTINUE
   30 CONTINUE
*
*     Now conquer each subproblem bottom-up.
*
      DO 50 LVL = NLVL, 1, -1
*
*        Find the first node LF and last node LL on the
*        current level LVL.
*
         IF( LVL.EQ.1 ) THEN
            LF = 1
            LL = 1
         ELSE
            LF = 2**( LVL-1 )
            LL = 2*LF - 1
         END IF
         DO 40 I = LF, LL
            IM1 = I - 1
            IC = IWORK( INODE+IM1 )
            NL = IWORK( NDIML+IM1 )
            NR = IWORK( NDIMR+IM1 )
            NLF = IC - NL
            IF( ( SQRE.EQ.0 ) .AND. ( I.EQ.LL ) ) THEN
               SQREI = SQRE
            ELSE
               SQREI = 1
            END IF
            IDXQC = IDXQ + NLF - 1
            ALPHA = D( IC )
            BETA = E( IC )
            CALL DLASD1( NL, NR, SQREI, D( NLF ), ALPHA, BETA,
     $                   U( NLF, NLF ), LDU, VT( NLF, NLF ), LDVT,
     $                   IWORK( IDXQC ), IWORK( IWK ), WORK, INFO )
*
*        Report the possible convergence failure.
*
            IF( INFO.NE.0 ) THEN
               RETURN
            END IF
   40    CONTINUE
   50 CONTINUE
*
      RETURN
*
*     End of DLASD0
*
      END
*> \brief \b DLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASD1 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd1.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd1.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd1.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASD1( NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT,
*                          IDXQ, IWORK, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDU, LDVT, NL, NR, SQRE
*       DOUBLE PRECISION   ALPHA, BETA
*       ..
*       .. Array Arguments ..
*       INTEGER            IDXQ( * ), IWORK( * )
*       DOUBLE PRECISION   D( * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B,
*> where N = NL + NR + 1 and M = N + SQRE. DLASD1 is called from DLASD0.
*>
*> A related subroutine DLASD7 handles the case in which the singular
*> values (and the singular vectors in factored form) are desired.
*>
*> DLASD1 computes the SVD as follows:
*>
*>               ( D1(in)    0    0       0 )
*>   B = U(in) * (   Z1**T   a   Z2**T    b ) * VT(in)
*>               (   0       0   D2(in)   0 )
*>
*>     = U(out) * ( D(out) 0) * VT(out)
*>
*> where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M
*> with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
*> elsewhere; and the entry b is empty if SQRE = 0.
*>
*> The left singular vectors of the original matrix are stored in U, and
*> the transpose of the right singular vectors are stored in VT, and the
*> singular values are in D.  The algorithm consists of three stages:
*>
*>    The first stage consists of deflating the size of the problem
*>    when there are multiple singular values or when there are zeros in
*>    the Z vector.  For each such occurrence the dimension of the
*>    secular equation problem is reduced by one.  This stage is
*>    performed by the routine DLASD2.
*>
*>    The second stage consists of calculating the updated
*>    singular values. This is done by finding the square roots of the
*>    roots of the secular equation via the routine DLASD4 (as called
*>    by DLASD3). This routine also calculates the singular vectors of
*>    the current problem.
*>
*>    The final stage consists of computing the updated singular vectors
*>    directly using the updated singular values.  The singular vectors
*>    for the current problem are multiplied with the singular vectors
*>    from the overall problem.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NL
*> \verbatim
*>          NL is INTEGER
*>         The row dimension of the upper block.  NL >= 1.
*> \endverbatim
*>
*> \param[in] NR
*> \verbatim
*>          NR is INTEGER
*>         The row dimension of the lower block.  NR >= 1.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*>          SQRE is INTEGER
*>         = 0: the lower block is an NR-by-NR square matrix.
*>         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
*>
*>         The bidiagonal matrix has row dimension N = NL + NR + 1,
*>         and column dimension M = N + SQRE.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array,
*>                        dimension (N = NL+NR+1).
*>         On entry D(1:NL,1:NL) contains the singular values of the
*>         upper block; and D(NL+2:N) contains the singular values of
*>         the lower block. On exit D(1:N) contains the singular values
*>         of the modified matrix.
*> \endverbatim
*>
*> \param[in,out] ALPHA
*> \verbatim
*>          ALPHA is DOUBLE PRECISION
*>         Contains the diagonal element associated with the added row.
*> \endverbatim
*>
*> \param[in,out] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION
*>         Contains the off-diagonal element associated with the added
*>         row.
*> \endverbatim
*>
*> \param[in,out] U
*> \verbatim
*>          U is DOUBLE PRECISION array, dimension(LDU,N)
*>         On entry U(1:NL, 1:NL) contains the left singular vectors of
*>         the upper block; U(NL+2:N, NL+2:N) contains the left singular
*>         vectors of the lower block. On exit U contains the left
*>         singular vectors of the bidiagonal matrix.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>         The leading dimension of the array U.  LDU >= max( 1, N ).
*> \endverbatim
*>
*> \param[in,out] VT
*> \verbatim
*>          VT is DOUBLE PRECISION array, dimension(LDVT,M)
*>         where M = N + SQRE.
*>         On entry VT(1:NL+1, 1:NL+1)**T contains the right singular
*>         vectors of the upper block; VT(NL+2:M, NL+2:M)**T contains
*>         the right singular vectors of the lower block. On exit
*>         VT**T contains the right singular vectors of the
*>         bidiagonal matrix.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*>          LDVT is INTEGER
*>         The leading dimension of the array VT.  LDVT >= max( 1, M ).
*> \endverbatim
*>
*> \param[in,out] IDXQ
*> \verbatim
*>          IDXQ is INTEGER array, dimension(N)
*>         This contains the permutation which will reintegrate the
*>         subproblem just solved back into sorted order, i.e.
*>         D( IDXQ( I = 1, N ) ) will be in ascending order.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension( 4 * N )
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension( 3*M**2 + 2*M )
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = 1, a singular value did not converge
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasd1
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Huan Ren, Computer Science Division, University of
*>     California at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE DLASD1( NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT,
     $                   LDVT,
     $                   IDXQ, IWORK, WORK, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDU, LDVT, NL, NR, SQRE
      DOUBLE PRECISION   ALPHA, BETA
*     ..
*     .. Array Arguments ..
      INTEGER            IDXQ( * ), IWORK( * )
      DOUBLE PRECISION   D( * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
*
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            COLTYP, I, IDX, IDXC, IDXP, IQ, ISIGMA, IU2,
     $                   IVT2, IZ, K, LDQ, LDU2, LDVT2, M, N, N1, N2
      DOUBLE PRECISION   ORGNRM
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLAMRG, DLASCL, DLASD2, DLASD3,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IF( NL.LT.1 ) THEN
         INFO = -1
      ELSE IF( NR.LT.1 ) THEN
         INFO = -2
      ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
         INFO = -3
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLASD1', -INFO )
         RETURN
      END IF
*
      N = NL + NR + 1
      M = N + SQRE
*
*     The following values are for bookkeeping purposes only.  They are
*     integer pointers which indicate the portion of the workspace
*     used by a particular array in DLASD2 and DLASD3.
*
      LDU2 = N
      LDVT2 = M
*
      IZ = 1
      ISIGMA = IZ + M
      IU2 = ISIGMA + N
      IVT2 = IU2 + LDU2*N
      IQ = IVT2 + LDVT2*M
*
      IDX = 1
      IDXC = IDX + N
      COLTYP = IDXC + N
      IDXP = COLTYP + N
*
*     Scale.
*
      ORGNRM = MAX( ABS( ALPHA ), ABS( BETA ) )
      D( NL+1 ) = ZERO
      DO 10 I = 1, N
         IF( ABS( D( I ) ).GT.ORGNRM ) THEN
            ORGNRM = ABS( D( I ) )
         END IF
   10 CONTINUE
      CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO )
      ALPHA = ALPHA / ORGNRM
      BETA = BETA / ORGNRM
*
*     Deflate singular values.
*
      CALL DLASD2( NL, NR, SQRE, K, D, WORK( IZ ), ALPHA, BETA, U,
     $             LDU,
     $             VT, LDVT, WORK( ISIGMA ), WORK( IU2 ), LDU2,
     $             WORK( IVT2 ), LDVT2, IWORK( IDXP ), IWORK( IDX ),
     $             IWORK( IDXC ), IDXQ, IWORK( COLTYP ), INFO )
*
*     Solve Secular Equation and update singular vectors.
*
      LDQ = K
      CALL DLASD3( NL, NR, SQRE, K, D, WORK( IQ ), LDQ,
     $             WORK( ISIGMA ),
     $             U, LDU, WORK( IU2 ), LDU2, VT, LDVT, WORK( IVT2 ),
     $             LDVT2, IWORK( IDXC ), IWORK( COLTYP ), WORK( IZ ),
     $             INFO )
*
*     Report the convergence failure.
*
      IF( INFO.NE.0 ) THEN
         RETURN
      END IF
*
*     Unscale.
*
      CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
*
*     Prepare the IDXQ sorting permutation.
*
      N1 = K
      N2 = N - K
      CALL DLAMRG( N1, N2, D, 1, -1, IDXQ )
*
      RETURN
*
*     End of DLASD1
*
      END
*> \brief \b DLASD2 merges the two sets of singular values together into a single sorted set. Used by sbdsdc.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASD2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASD2( NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT,
*                          LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX,
*                          IDXC, IDXQ, COLTYP, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, K, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE
*       DOUBLE PRECISION   ALPHA, BETA
*       ..
*       .. Array Arguments ..
*       INTEGER            COLTYP( * ), IDX( * ), IDXC( * ), IDXP( * ),
*      $                   IDXQ( * )
*       DOUBLE PRECISION   D( * ), DSIGMA( * ), U( LDU, * ),
*      $                   U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
*      $                   Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASD2 merges the two sets of singular values together into a single
*> sorted set.  Then it tries to deflate the size of the problem.
*> There are two ways in which deflation can occur:  when two or more
*> singular values are close together or if there is a tiny entry in the
*> Z vector.  For each such occurrence the order of the related secular
*> equation problem is reduced by one.
*>
*> DLASD2 is called from DLASD1.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NL
*> \verbatim
*>          NL is INTEGER
*>         The row dimension of the upper block.  NL >= 1.
*> \endverbatim
*>
*> \param[in] NR
*> \verbatim
*>          NR is INTEGER
*>         The row dimension of the lower block.  NR >= 1.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*>          SQRE is INTEGER
*>         = 0: the lower block is an NR-by-NR square matrix.
*>         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
*>
*>         The bidiagonal matrix has N = NL + NR + 1 rows and
*>         M = N + SQRE >= N columns.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*>          K is INTEGER
*>         Contains the dimension of the non-deflated matrix,
*>         This is the order of the related secular equation. 1 <= K <=N.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension(N)
*>         On entry D contains the singular values of the two submatrices
*>         to be combined.  On exit D contains the trailing (N-K) updated
*>         singular values (those which were deflated) sorted into
*>         increasing order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension(N)
*>         On exit Z contains the updating row vector in the secular
*>         equation.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*>          ALPHA is DOUBLE PRECISION
*>         Contains the diagonal element associated with the added row.
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION
*>         Contains the off-diagonal element associated with the added
*>         row.
*> \endverbatim
*>
*> \param[in,out] U
*> \verbatim
*>          U is DOUBLE PRECISION array, dimension(LDU,N)
*>         On entry U contains the left singular vectors of two
*>         submatrices in the two square blocks with corners at (1,1),
*>         (NL, NL), and (NL+2, NL+2), (N,N).
*>         On exit U contains the trailing (N-K) updated left singular
*>         vectors (those which were deflated) in its last N-K columns.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>         The leading dimension of the array U.  LDU >= N.
*> \endverbatim
*>
*> \param[in,out] VT
*> \verbatim
*>          VT is DOUBLE PRECISION array, dimension(LDVT,M)
*>         On entry VT**T contains the right singular vectors of two
*>         submatrices in the two square blocks with corners at (1,1),
*>         (NL+1, NL+1), and (NL+2, NL+2), (M,M).
*>         On exit VT**T contains the trailing (N-K) updated right singular
*>         vectors (those which were deflated) in its last N-K columns.
*>         In case SQRE =1, the last row of VT spans the right null
*>         space.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*>          LDVT is INTEGER
*>         The leading dimension of the array VT.  LDVT >= M.
*> \endverbatim
*>
*> \param[out] DSIGMA
*> \verbatim
*>          DSIGMA is DOUBLE PRECISION array, dimension (N)
*>         Contains a copy of the diagonal elements (K-1 singular values
*>         and one zero) in the secular equation.
*> \endverbatim
*>
*> \param[out] U2
*> \verbatim
*>          U2 is DOUBLE PRECISION array, dimension(LDU2,N)
*>         Contains a copy of the first K-1 left singular vectors which
*>         will be used by DLASD3 in a matrix multiply (DGEMM) to solve
*>         for the new left singular vectors. U2 is arranged into four
*>         blocks. The first block contains a column with 1 at NL+1 and
*>         zero everywhere else; the second block contains non-zero
*>         entries only at and above NL; the third contains non-zero
*>         entries only below NL+1; and the fourth is dense.
*> \endverbatim
*>
*> \param[in] LDU2
*> \verbatim
*>          LDU2 is INTEGER
*>         The leading dimension of the array U2.  LDU2 >= N.
*> \endverbatim
*>
*> \param[out] VT2
*> \verbatim
*>          VT2 is DOUBLE PRECISION array, dimension(LDVT2,N)
*>         VT2**T contains a copy of the first K right singular vectors
*>         which will be used by DLASD3 in a matrix multiply (DGEMM) to
*>         solve for the new right singular vectors. VT2 is arranged into
*>         three blocks. The first block contains a row that corresponds
*>         to the special 0 diagonal element in SIGMA; the second block
*>         contains non-zeros only at and before NL +1; the third block
*>         contains non-zeros only at and after  NL +2.
*> \endverbatim
*>
*> \param[in] LDVT2
*> \verbatim
*>          LDVT2 is INTEGER
*>         The leading dimension of the array VT2.  LDVT2 >= M.
*> \endverbatim
*>
*> \param[out] IDXP
*> \verbatim
*>          IDXP is INTEGER array, dimension(N)
*>         This will contain the permutation used to place deflated
*>         values of D at the end of the array. On output IDXP(2:K)
*>         points to the nondeflated D-values and IDXP(K+1:N)
*>         points to the deflated singular values.
*> \endverbatim
*>
*> \param[out] IDX
*> \verbatim
*>          IDX is INTEGER array, dimension(N)
*>         This will contain the permutation used to sort the contents of
*>         D into ascending order.
*> \endverbatim
*>
*> \param[out] IDXC
*> \verbatim
*>          IDXC is INTEGER array, dimension(N)
*>         This will contain the permutation used to arrange the columns
*>         of the deflated U matrix into three groups:  the first group
*>         contains non-zero entries only at and above NL, the second
*>         contains non-zero entries only below NL+2, and the third is
*>         dense.
*> \endverbatim
*>
*> \param[in,out] IDXQ
*> \verbatim
*>          IDXQ is INTEGER array, dimension(N)
*>         This contains the permutation which separately sorts the two
*>         sub-problems in D into ascending order.  Note that entries in
*>         the first hlaf of this permutation must first be moved one
*>         position backward; and entries in the second half
*>         must first have NL+1 added to their values.
*> \endverbatim
*>
*> \param[out] COLTYP
*> \verbatim
*>          COLTYP is INTEGER array, dimension(N)
*>         As workspace, this will contain a label which will indicate
*>         which of the following types a column in the U2 matrix or a
*>         row in the VT2 matrix is:
*>         1 : non-zero in the upper half only
*>         2 : non-zero in the lower half only
*>         3 : dense
*>         4 : deflated
*>
*>         On exit, it is an array of dimension 4, with COLTYP(I) being
*>         the dimension of the I-th type columns.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasd2
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Huan Ren, Computer Science Division, University of
*>     California at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE DLASD2( NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU,
     $                   VT,
     $                   LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX,
     $                   IDXC, IDXQ, COLTYP, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE
      DOUBLE PRECISION   ALPHA, BETA
*     ..
*     .. Array Arguments ..
      INTEGER            COLTYP( * ), IDX( * ), IDXC( * ), IDXP( * ),
     $                   IDXQ( * )
      DOUBLE PRECISION   D( * ), DSIGMA( * ), U( LDU, * ),
     $                   U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
     $                   Z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO, EIGHT
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0,
     $                   EIGHT = 8.0D+0 )
*     ..
*     .. Local Arrays ..
      INTEGER            CTOT( 4 ), PSM( 4 )
*     ..
*     .. Local Scalars ..
      INTEGER            CT, I, IDXI, IDXJ, IDXJP, J, JP, JPREV, K2, M,
     $                   N, NLP1, NLP2
      DOUBLE PRECISION   C, EPS, HLFTOL, S, TAU, TOL, Z1
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH, DLAPY2
      EXTERNAL           DLAMCH, DLAPY2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLACPY, DLAMRG, DLASET, DROT,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IF( NL.LT.1 ) THEN
         INFO = -1
      ELSE IF( NR.LT.1 ) THEN
         INFO = -2
      ELSE IF( ( SQRE.NE.1 ) .AND. ( SQRE.NE.0 ) ) THEN
         INFO = -3
      END IF
*
      N = NL + NR + 1
      M = N + SQRE
*
      IF( LDU.LT.N ) THEN
         INFO = -10
      ELSE IF( LDVT.LT.M ) THEN
         INFO = -12
      ELSE IF( LDU2.LT.N ) THEN
         INFO = -15
      ELSE IF( LDVT2.LT.M ) THEN
         INFO = -17
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLASD2', -INFO )
         RETURN
      END IF
*
      NLP1 = NL + 1
      NLP2 = NL + 2
*
*     Generate the first part of the vector Z; and move the singular
*     values in the first part of D one position backward.
*
      Z1 = ALPHA*VT( NLP1, NLP1 )
      Z( 1 ) = Z1
      DO 10 I = NL, 1, -1
         Z( I+1 ) = ALPHA*VT( I, NLP1 )
         D( I+1 ) = D( I )
         IDXQ( I+1 ) = IDXQ( I ) + 1
   10 CONTINUE
*
*     Generate the second part of the vector Z.
*
      DO 20 I = NLP2, M
         Z( I ) = BETA*VT( I, NLP2 )
   20 CONTINUE
*
*     Initialize some reference arrays.
*
      DO 30 I = 2, NLP1
         COLTYP( I ) = 1
   30 CONTINUE
      DO 40 I = NLP2, N
         COLTYP( I ) = 2
   40 CONTINUE
*
*     Sort the singular values into increasing order
*
      DO 50 I = NLP2, N
         IDXQ( I ) = IDXQ( I ) + NLP1
   50 CONTINUE
*
*     DSIGMA, IDXC, IDXC, and the first column of U2
*     are used as storage space.
*
      DO 60 I = 2, N
         DSIGMA( I ) = D( IDXQ( I ) )
         U2( I, 1 ) = Z( IDXQ( I ) )
         IDXC( I ) = COLTYP( IDXQ( I ) )
   60 CONTINUE
*
      CALL DLAMRG( NL, NR, DSIGMA( 2 ), 1, 1, IDX( 2 ) )
*
      DO 70 I = 2, N
         IDXI = 1 + IDX( I )
         D( I ) = DSIGMA( IDXI )
         Z( I ) = U2( IDXI, 1 )
         COLTYP( I ) = IDXC( IDXI )
   70 CONTINUE
*
*     Calculate the allowable deflation tolerance
*
      EPS = DLAMCH( 'Epsilon' )
      TOL = MAX( ABS( ALPHA ), ABS( BETA ) )
      TOL = EIGHT*EPS*MAX( ABS( D( N ) ), TOL )
*
*     There are 2 kinds of deflation -- first a value in the z-vector
*     is small, second two (or more) singular values are very close
*     together (their difference is small).
*
*     If the value in the z-vector is small, we simply permute the
*     array so that the corresponding singular value is moved to the
*     end.
*
*     If two values in the D-vector are close, we perform a two-sided
*     rotation designed to make one of the corresponding z-vector
*     entries zero, and then permute the array so that the deflated
*     singular value is moved to the end.
*
*     If there are multiple singular values then the problem deflates.
*     Here the number of equal singular values are found.  As each equal
*     singular value is found, an elementary reflector is computed to
*     rotate the corresponding singular subspace so that the
*     corresponding components of Z are zero in this new basis.
*
      K = 1
      K2 = N + 1
      DO 80 J = 2, N
         IF( ABS( Z( J ) ).LE.TOL ) THEN
*
*           Deflate due to small z component.
*
            K2 = K2 - 1
            IDXP( K2 ) = J
            COLTYP( J ) = 4
            IF( J.EQ.N )
     $         GO TO 120
         ELSE
            JPREV = J
            GO TO 90
         END IF
   80 CONTINUE
   90 CONTINUE
      J = JPREV
  100 CONTINUE
      J = J + 1
      IF( J.GT.N )
     $   GO TO 110
      IF( ABS( Z( J ) ).LE.TOL ) THEN
*
*        Deflate due to small z component.
*
         K2 = K2 - 1
         IDXP( K2 ) = J
         COLTYP( J ) = 4
      ELSE
*
*        Check if singular values are close enough to allow deflation.
*
         IF( ABS( D( J )-D( JPREV ) ).LE.TOL ) THEN
*
*           Deflation is possible.
*
            S = Z( JPREV )
            C = Z( J )
*
*           Find sqrt(a**2+b**2) without overflow or
*           destructive underflow.
*
            TAU = DLAPY2( C, S )
            C = C / TAU
            S = -S / TAU
            Z( J ) = TAU
            Z( JPREV ) = ZERO
*
*           Apply back the Givens rotation to the left and right
*           singular vector matrices.
*
            IDXJP = IDXQ( IDX( JPREV )+1 )
            IDXJ = IDXQ( IDX( J )+1 )
            IF( IDXJP.LE.NLP1 ) THEN
               IDXJP = IDXJP - 1
            END IF
            IF( IDXJ.LE.NLP1 ) THEN
               IDXJ = IDXJ - 1
            END IF
            CALL DROT( N, U( 1, IDXJP ), 1, U( 1, IDXJ ), 1, C, S )
            CALL DROT( M, VT( IDXJP, 1 ), LDVT, VT( IDXJ, 1 ), LDVT,
     $                 C,
     $                 S )
            IF( COLTYP( J ).NE.COLTYP( JPREV ) ) THEN
               COLTYP( J ) = 3
            END IF
            COLTYP( JPREV ) = 4
            K2 = K2 - 1
            IDXP( K2 ) = JPREV
            JPREV = J
         ELSE
            K = K + 1
            U2( K, 1 ) = Z( JPREV )
            DSIGMA( K ) = D( JPREV )
            IDXP( K ) = JPREV
            JPREV = J
         END IF
      END IF
      GO TO 100
  110 CONTINUE
*
*     Record the last singular value.
*
      K = K + 1
      U2( K, 1 ) = Z( JPREV )
      DSIGMA( K ) = D( JPREV )
      IDXP( K ) = JPREV
*
  120 CONTINUE
*
*     Count up the total number of the various types of columns, then
*     form a permutation which positions the four column types into
*     four groups of uniform structure (although one or more of these
*     groups may be empty).
*
      DO 130 J = 1, 4
         CTOT( J ) = 0
  130 CONTINUE
      DO 140 J = 2, N
         CT = COLTYP( J )
         CTOT( CT ) = CTOT( CT ) + 1
  140 CONTINUE
*
*     PSM(*) = Position in SubMatrix (of types 1 through 4)
*
      PSM( 1 ) = 2
      PSM( 2 ) = 2 + CTOT( 1 )
      PSM( 3 ) = PSM( 2 ) + CTOT( 2 )
      PSM( 4 ) = PSM( 3 ) + CTOT( 3 )
*
*     Fill out the IDXC array so that the permutation which it induces
*     will place all type-1 columns first, all type-2 columns next,
*     then all type-3's, and finally all type-4's, starting from the
*     second column. This applies similarly to the rows of VT.
*
      DO 150 J = 2, N
         JP = IDXP( J )
         CT = COLTYP( JP )
         IDXC( PSM( CT ) ) = J
         PSM( CT ) = PSM( CT ) + 1
  150 CONTINUE
*
*     Sort the singular values and corresponding singular vectors into
*     DSIGMA, U2, and VT2 respectively.  The singular values/vectors
*     which were not deflated go into the first K slots of DSIGMA, U2,
*     and VT2 respectively, while those which were deflated go into the
*     last N - K slots, except that the first column/row will be treated
*     separately.
*
      DO 160 J = 2, N
         JP = IDXP( J )
         DSIGMA( J ) = D( JP )
         IDXJ = IDXQ( IDX( IDXP( IDXC( J ) ) )+1 )
         IF( IDXJ.LE.NLP1 ) THEN
            IDXJ = IDXJ - 1
         END IF
         CALL DCOPY( N, U( 1, IDXJ ), 1, U2( 1, J ), 1 )
         CALL DCOPY( M, VT( IDXJ, 1 ), LDVT, VT2( J, 1 ), LDVT2 )
  160 CONTINUE
*
*     Determine DSIGMA(1), DSIGMA(2) and Z(1)
*
      DSIGMA( 1 ) = ZERO
      HLFTOL = TOL / TWO
      IF( ABS( DSIGMA( 2 ) ).LE.HLFTOL )
     $   DSIGMA( 2 ) = HLFTOL
      IF( M.GT.N ) THEN
         Z( 1 ) = DLAPY2( Z1, Z( M ) )
         IF( Z( 1 ).LE.TOL ) THEN
            C = ONE
            S = ZERO
            Z( 1 ) = TOL
         ELSE
            C = Z1 / Z( 1 )
            S = Z( M ) / Z( 1 )
         END IF
      ELSE
         IF( ABS( Z1 ).LE.TOL ) THEN
            Z( 1 ) = TOL
         ELSE
            Z( 1 ) = Z1
         END IF
      END IF
*
*     Move the rest of the updating row to Z.
*
      CALL DCOPY( K-1, U2( 2, 1 ), 1, Z( 2 ), 1 )
*
*     Determine the first column of U2, the first row of VT2 and the
*     last row of VT.
*
      CALL DLASET( 'A', N, 1, ZERO, ZERO, U2, LDU2 )
      U2( NLP1, 1 ) = ONE
      IF( M.GT.N ) THEN
         DO 170 I = 1, NLP1
            VT( M, I ) = -S*VT( NLP1, I )
            VT2( 1, I ) = C*VT( NLP1, I )
  170    CONTINUE
         DO 180 I = NLP2, M
            VT2( 1, I ) = S*VT( M, I )
            VT( M, I ) = C*VT( M, I )
  180    CONTINUE
      ELSE
         CALL DCOPY( M, VT( NLP1, 1 ), LDVT, VT2( 1, 1 ), LDVT2 )
      END IF
      IF( M.GT.N ) THEN
         CALL DCOPY( M, VT( M, 1 ), LDVT, VT2( M, 1 ), LDVT2 )
      END IF
*
*     The deflated singular values and their corresponding vectors go
*     into the back of D, U, and V respectively.
*
      IF( N.GT.K ) THEN
         CALL DCOPY( N-K, DSIGMA( K+1 ), 1, D( K+1 ), 1 )
         CALL DLACPY( 'A', N, N-K, U2( 1, K+1 ), LDU2, U( 1, K+1 ),
     $                LDU )
         CALL DLACPY( 'A', N-K, M, VT2( K+1, 1 ), LDVT2, VT( K+1,
     $                1 ),
     $                LDVT )
      END IF
*
*     Copy CTOT into COLTYP for referencing in DLASD3.
*
      DO 190 J = 1, 4
         COLTYP( J ) = CTOT( J )
  190 CONTINUE
*
      RETURN
*
*     End of DLASD2
*
      END
*> \brief \b DLASD3 finds all square roots of the roots of the secular equation, as defined by the values in D and Z, and then updates the singular vectors by matrix multiplication. Used by sbdsdc.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASD3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd3.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASD3( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2,
*                          LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z,
*                          INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, K, LDQ, LDU, LDU2, LDVT, LDVT2, NL, NR,
*      $                   SQRE
*       ..
*       .. Array Arguments ..
*       INTEGER            CTOT( * ), IDXC( * )
*       DOUBLE PRECISION   D( * ), DSIGMA( * ), Q( LDQ, * ), U( LDU, * ),
*      $                   U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
*      $                   Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASD3 finds all the square roots of the roots of the secular
*> equation, as defined by the values in D and Z.  It makes the
*> appropriate calls to DLASD4 and then updates the singular
*> vectors by matrix multiplication.
*>
*> DLASD3 is called from DLASD1.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NL
*> \verbatim
*>          NL is INTEGER
*>         The row dimension of the upper block.  NL >= 1.
*> \endverbatim
*>
*> \param[in] NR
*> \verbatim
*>          NR is INTEGER
*>         The row dimension of the lower block.  NR >= 1.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*>          SQRE is INTEGER
*>         = 0: the lower block is an NR-by-NR square matrix.
*>         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
*>
*>         The bidiagonal matrix has N = NL + NR + 1 rows and
*>         M = N + SQRE >= N columns.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>         The size of the secular equation, 1 =< K = < N.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension(K)
*>         On exit the square roots of the roots of the secular equation,
*>         in ascending order.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ,K)
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>         The leading dimension of the array Q.  LDQ >= K.
*> \endverbatim
*>
*> \param[in] DSIGMA
*> \verbatim
*>          DSIGMA is DOUBLE PRECISION array, dimension(K)
*>         The first K elements of this array contain the old roots
*>         of the deflated updating problem.  These are the poles
*>         of the secular equation.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*>          U is DOUBLE PRECISION array, dimension (LDU, N)
*>         The last N - K columns of this matrix contain the deflated
*>         left singular vectors.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>         The leading dimension of the array U.  LDU >= N.
*> \endverbatim
*>
*> \param[in] U2
*> \verbatim
*>          U2 is DOUBLE PRECISION array, dimension (LDU2, N)
*>         The first K columns of this matrix contain the non-deflated
*>         left singular vectors for the split problem.
*> \endverbatim
*>
*> \param[in] LDU2
*> \verbatim
*>          LDU2 is INTEGER
*>         The leading dimension of the array U2.  LDU2 >= N.
*> \endverbatim
*>
*> \param[out] VT
*> \verbatim
*>          VT is DOUBLE PRECISION array, dimension (LDVT, M)
*>         The last M - K columns of VT**T contain the deflated
*>         right singular vectors.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*>          LDVT is INTEGER
*>         The leading dimension of the array VT.  LDVT >= N.
*> \endverbatim
*>
*> \param[in,out] VT2
*> \verbatim
*>          VT2 is DOUBLE PRECISION array, dimension (LDVT2, N)
*>         The first K columns of VT2**T contain the non-deflated
*>         right singular vectors for the split problem.
*> \endverbatim
*>
*> \param[in] LDVT2
*> \verbatim
*>          LDVT2 is INTEGER
*>         The leading dimension of the array VT2.  LDVT2 >= N.
*> \endverbatim
*>
*> \param[in] IDXC
*> \verbatim
*>          IDXC is INTEGER array, dimension ( N )
*>         The permutation used to arrange the columns of U (and rows of
*>         VT) into three groups:  the first group contains non-zero
*>         entries only at and above (or before) NL +1; the second
*>         contains non-zero entries only at and below (or after) NL+2;
*>         and the third is dense. The first column of U and the row of
*>         VT are treated separately, however.
*>
*>         The rows of the singular vectors found by DLASD4
*>         must be likewise permuted before the matrix multiplies can
*>         take place.
*> \endverbatim
*>
*> \param[in] CTOT
*> \verbatim
*>          CTOT is INTEGER array, dimension ( 4 )
*>         A count of the total number of the various types of columns
*>         in U (or rows in VT), as described in IDXC. The fourth column
*>         type is any column which has been deflated.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (K)
*>         The first K elements of this array contain the components
*>         of the deflation-adjusted updating row vector.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>         = 0:  successful exit.
*>         < 0:  if INFO = -i, the i-th argument had an illegal value.
*>         > 0:  if INFO = 1, a singular value did not converge
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasd3
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Huan Ren, Computer Science Division, University of
*>     California at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE DLASD3( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU,
     $                   U2,
     $                   LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z,
     $                   INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, LDQ, LDU, LDU2, LDVT, LDVT2, NL, NR,
     $                   SQRE
*     ..
*     .. Array Arguments ..
      INTEGER            CTOT( * ), IDXC( * )
      DOUBLE PRECISION   D( * ), DSIGMA( * ), Q( LDQ, * ), U( LDU, * ),
     $                   U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
     $                   Z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO, NEGONE
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0,
     $                   NEGONE = -1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            CTEMP, I, J, JC, KTEMP, M, N, NLP1, NLP2, NRP1
      DOUBLE PRECISION   RHO, TEMP
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DNRM2
      EXTERNAL           DNRM2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGEMM, DLACPY, DLASCL, DLASD4,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, SIGN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IF( NL.LT.1 ) THEN
         INFO = -1
      ELSE IF( NR.LT.1 ) THEN
         INFO = -2
      ELSE IF( ( SQRE.NE.1 ) .AND. ( SQRE.NE.0 ) ) THEN
         INFO = -3
      END IF
*
      N = NL + NR + 1
      M = N + SQRE
      NLP1 = NL + 1
      NLP2 = NL + 2
*
      IF( ( K.LT.1 ) .OR. ( K.GT.N ) ) THEN
         INFO = -4
      ELSE IF( LDQ.LT.K ) THEN
         INFO = -7
      ELSE IF( LDU.LT.N ) THEN
         INFO = -10
      ELSE IF( LDU2.LT.N ) THEN
         INFO = -12
      ELSE IF( LDVT.LT.M ) THEN
         INFO = -14
      ELSE IF( LDVT2.LT.M ) THEN
         INFO = -16
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLASD3', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( K.EQ.1 ) THEN
         D( 1 ) = ABS( Z( 1 ) )
         CALL DCOPY( M, VT2( 1, 1 ), LDVT2, VT( 1, 1 ), LDVT )
         IF( Z( 1 ).GT.ZERO ) THEN
            CALL DCOPY( N, U2( 1, 1 ), 1, U( 1, 1 ), 1 )
         ELSE
            DO 10 I = 1, N
               U( I, 1 ) = -U2( I, 1 )
   10       CONTINUE
         END IF
         RETURN
      END IF
*
*     Keep a copy of Z.
*
      CALL DCOPY( K, Z, 1, Q, 1 )
*
*     Normalize Z.
*
      RHO = DNRM2( K, Z, 1 )
      CALL DLASCL( 'G', 0, 0, RHO, ONE, K, 1, Z, K, INFO )
      RHO = RHO*RHO
*
*     Find the new singular values.
*
      DO 30 J = 1, K
         CALL DLASD4( K, J, DSIGMA, Z, U( 1, J ), RHO, D( J ),
     $                VT( 1, J ), INFO )
*
*        If the zero finder fails, report the convergence failure.
*
         IF( INFO.NE.0 ) THEN
            RETURN
         END IF
   30 CONTINUE
*
*     Compute updated Z.
*
      DO 60 I = 1, K
         Z( I ) = U( I, K )*VT( I, K )
         DO 40 J = 1, I - 1
            Z( I ) = Z( I )*( U( I, J )*VT( I, J ) /
     $               ( DSIGMA( I )-DSIGMA( J ) ) /
     $               ( DSIGMA( I )+DSIGMA( J ) ) )
   40    CONTINUE
         DO 50 J = I, K - 1
            Z( I ) = Z( I )*( U( I, J )*VT( I, J ) /
     $               ( DSIGMA( I )-DSIGMA( J+1 ) ) /
     $               ( DSIGMA( I )+DSIGMA( J+1 ) ) )
   50    CONTINUE
         Z( I ) = SIGN( SQRT( ABS( Z( I ) ) ), Q( I, 1 ) )
   60 CONTINUE
*
*     Compute left singular vectors of the modified diagonal matrix,
*     and store related information for the right singular vectors.
*
      DO 90 I = 1, K
         VT( 1, I ) = Z( 1 ) / U( 1, I ) / VT( 1, I )
         U( 1, I ) = NEGONE
         DO 70 J = 2, K
            VT( J, I ) = Z( J ) / U( J, I ) / VT( J, I )
            U( J, I ) = DSIGMA( J )*VT( J, I )
   70    CONTINUE
         TEMP = DNRM2( K, U( 1, I ), 1 )
         Q( 1, I ) = U( 1, I ) / TEMP
         DO 80 J = 2, K
            JC = IDXC( J )
            Q( J, I ) = U( JC, I ) / TEMP
   80    CONTINUE
   90 CONTINUE
*
*     Update the left singular vector matrix.
*
      IF( K.EQ.2 ) THEN
         CALL DGEMM( 'N', 'N', N, K, K, ONE, U2, LDU2, Q, LDQ, ZERO,
     $               U,
     $               LDU )
         GO TO 100
      END IF
      IF( CTOT( 1 ).GT.0 ) THEN
         CALL DGEMM( 'N', 'N', NL, K, CTOT( 1 ), ONE, U2( 1, 2 ),
     $               LDU2,
     $               Q( 2, 1 ), LDQ, ZERO, U( 1, 1 ), LDU )
         IF( CTOT( 3 ).GT.0 ) THEN
            KTEMP = 2 + CTOT( 1 ) + CTOT( 2 )
            CALL DGEMM( 'N', 'N', NL, K, CTOT( 3 ), ONE, U2( 1,
     $                  KTEMP ),
     $                  LDU2, Q( KTEMP, 1 ), LDQ, ONE, U( 1, 1 ), LDU )
         END IF
      ELSE IF( CTOT( 3 ).GT.0 ) THEN
         KTEMP = 2 + CTOT( 1 ) + CTOT( 2 )
         CALL DGEMM( 'N', 'N', NL, K, CTOT( 3 ), ONE, U2( 1, KTEMP ),
     $               LDU2, Q( KTEMP, 1 ), LDQ, ZERO, U( 1, 1 ), LDU )
      ELSE
         CALL DLACPY( 'F', NL, K, U2, LDU2, U, LDU )
      END IF
      CALL DCOPY( K, Q( 1, 1 ), LDQ, U( NLP1, 1 ), LDU )
      KTEMP = 2 + CTOT( 1 )
      CTEMP = CTOT( 2 ) + CTOT( 3 )
      CALL DGEMM( 'N', 'N', NR, K, CTEMP, ONE, U2( NLP2, KTEMP ),
     $            LDU2,
     $            Q( KTEMP, 1 ), LDQ, ZERO, U( NLP2, 1 ), LDU )
*
*     Generate the right singular vectors.
*
  100 CONTINUE
      DO 120 I = 1, K
         TEMP = DNRM2( K, VT( 1, I ), 1 )
         Q( I, 1 ) = VT( 1, I ) / TEMP
         DO 110 J = 2, K
            JC = IDXC( J )
            Q( I, J ) = VT( JC, I ) / TEMP
  110    CONTINUE
  120 CONTINUE
*
*     Update the right singular vector matrix.
*
      IF( K.EQ.2 ) THEN
         CALL DGEMM( 'N', 'N', K, M, K, ONE, Q, LDQ, VT2, LDVT2,
     $               ZERO,
     $               VT, LDVT )
         RETURN
      END IF
      KTEMP = 1 + CTOT( 1 )
      CALL DGEMM( 'N', 'N', K, NLP1, KTEMP, ONE, Q( 1, 1 ), LDQ,
     $            VT2( 1, 1 ), LDVT2, ZERO, VT( 1, 1 ), LDVT )
      KTEMP = 2 + CTOT( 1 ) + CTOT( 2 )
      IF( KTEMP.LE.LDVT2 )
     $   CALL DGEMM( 'N', 'N', K, NLP1, CTOT( 3 ), ONE, Q( 1,
     $               KTEMP ),
     $               LDQ, VT2( KTEMP, 1 ), LDVT2, ONE, VT( 1, 1 ),
     $               LDVT )
*
      KTEMP = CTOT( 1 ) + 1
      NRP1 = NR + SQRE
      IF( KTEMP.GT.1 ) THEN
         DO 130 I = 1, K
            Q( I, KTEMP ) = Q( I, 1 )
  130    CONTINUE
         DO 140 I = NLP2, M
            VT2( KTEMP, I ) = VT2( 1, I )
  140    CONTINUE
      END IF
      CTEMP = 1 + CTOT( 2 ) + CTOT( 3 )
      CALL DGEMM( 'N', 'N', K, NRP1, CTEMP, ONE, Q( 1, KTEMP ), LDQ,
     $            VT2( KTEMP, NLP2 ), LDVT2, ZERO, VT( 1, NLP2 ), LDVT )
*
      RETURN
*
*     End of DLASD3
*
      END
*> \brief \b DLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by dbdsdc.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASD4 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd4.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd4.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd4.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            I, INFO, N
*       DOUBLE PRECISION   RHO, SIGMA
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), DELTA( * ), WORK( * ), Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> This subroutine computes the square root of the I-th updated
*> eigenvalue of a positive symmetric rank-one modification to
*> a positive diagonal matrix whose entries are given as the squares
*> of the corresponding entries in the array d, and that
*>
*>        0 <= D(i) < D(j)  for  i < j
*>
*> and that RHO > 0. This is arranged by the calling routine, and is
*> no loss in generality.  The rank-one modified system is thus
*>
*>        diag( D ) * diag( D ) +  RHO * Z * Z_transpose.
*>
*> where we assume the Euclidean norm of Z is 1.
*>
*> The method consists of approximating the rational functions in the
*> secular equation by simpler interpolating rational functions.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>         The length of all arrays.
*> \endverbatim
*>
*> \param[in] I
*> \verbatim
*>          I is INTEGER
*>         The index of the eigenvalue to be computed.  1 <= I <= N.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension ( N )
*>         The original eigenvalues.  It is assumed that they are in
*>         order, 0 <= D(I) < D(J)  for I < J.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension ( N )
*>         The components of the updating vector.
*> \endverbatim
*>
*> \param[out] DELTA
*> \verbatim
*>          DELTA is DOUBLE PRECISION array, dimension ( N )
*>         If N .ne. 1, DELTA contains (D(j) - sigma_I) in its  j-th
*>         component.  If N = 1, then DELTA(1) = 1.  The vector DELTA
*>         contains the information necessary to construct the
*>         (singular) eigenvectors.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*>          RHO is DOUBLE PRECISION
*>         The scalar in the symmetric updating formula.
*> \endverbatim
*>
*> \param[out] SIGMA
*> \verbatim
*>          SIGMA is DOUBLE PRECISION
*>         The computed sigma_I, the I-th updated eigenvalue.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension ( N )
*>         If N .ne. 1, WORK contains (D(j) + sigma_I) in its  j-th
*>         component.  If N = 1, then WORK( 1 ) = 1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>         = 0:  successful exit
*>         > 0:  if INFO = 1, the updating process failed.
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  Logical variable ORGATI (origin-at-i?) is used for distinguishing
*>  whether D(i) or D(i+1) is treated as the origin.
*>
*>            ORGATI = .true.    origin at i
*>            ORGATI = .false.   origin at i+1
*>
*>  Logical variable SWTCH3 (switch-for-3-poles?) is for noting
*>  if we are working with THREE poles!
*>
*>  MAXIT is the maximum number of iterations allowed for each
*>  eigenvalue.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasd4
*
*> \par Contributors:
*  ==================
*>
*>     Ren-Cang Li, Computer Science Division, University of California
*>     at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE DLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            I, INFO, N
      DOUBLE PRECISION   RHO, SIGMA
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), DELTA( * ), WORK( * ), Z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            MAXIT
      PARAMETER          ( MAXIT = 400 )
      DOUBLE PRECISION   ZERO, ONE, TWO, THREE, FOUR, EIGHT, TEN
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0,
     $                   THREE = 3.0D+0, FOUR = 4.0D+0, EIGHT = 8.0D+0,
     $                   TEN = 10.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ORGATI, SWTCH, SWTCH3, GEOMAVG
      INTEGER            II, IIM1, IIP1, IP1, ITER, J, NITER
      DOUBLE PRECISION   A, B, C, DELSQ, DELSQ2, SQ2, DPHI, DPSI, DTIIM,
     $                   DTIIP, DTIPSQ, DTISQ, DTNSQ, DTNSQ1, DW, EPS,
     $                   ERRETM, ETA, PHI, PREW, PSI, RHOINV, SGLB,
     $                   SGUB, TAU, TAU2, TEMP, TEMP1, TEMP2, W
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   DD( 3 ), ZZ( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLAED6, DLASD5
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Since this routine is called in an inner loop, we do no argument
*     checking.
*
*     Quick return for N=1 and 2.
*
      INFO = 0
      IF( N.EQ.1 ) THEN
*
*        Presumably, I=1 upon entry
*
         SIGMA = SQRT( D( 1 )*D( 1 )+RHO*Z( 1 )*Z( 1 ) )
         DELTA( 1 ) = ONE
         WORK( 1 ) = ONE
         RETURN
      END IF
      IF( N.EQ.2 ) THEN
         CALL DLASD5( I, D, Z, DELTA, RHO, SIGMA, WORK )
         RETURN
      END IF
*
*     Compute machine epsilon
*
      EPS = DLAMCH( 'Epsilon' )
      RHOINV = ONE / RHO
      TAU2= ZERO
*
*     The case I = N
*
      IF( I.EQ.N ) THEN
*
*        Initialize some basic variables
*
         II = N - 1
         NITER = 1
*
*        Calculate initial guess
*
         TEMP = RHO / TWO
*
*        If ||Z||_2 is not one, then TEMP should be set to
*        RHO * ||Z||_2^2 / TWO
*
         TEMP1 = TEMP / ( D( N )+SQRT( D( N )*D( N )+TEMP ) )
         DO 10 J = 1, N
            WORK( J ) = D( J ) + D( N ) + TEMP1
            DELTA( J ) = ( D( J )-D( N ) ) - TEMP1
   10    CONTINUE
*
         PSI = ZERO
         DO 20 J = 1, N - 2
            PSI = PSI + Z( J )*Z( J ) / ( DELTA( J )*WORK( J ) )
   20    CONTINUE
*
         C = RHOINV + PSI
         W = C + Z( II )*Z( II ) / ( DELTA( II )*WORK( II ) ) +
     $       Z( N )*Z( N ) / ( DELTA( N )*WORK( N ) )
*
         IF( W.LE.ZERO ) THEN
            TEMP1 = SQRT( D( N )*D( N )+RHO )
            TEMP = Z( N-1 )*Z( N-1 ) / ( ( D( N-1 )+TEMP1 )*
     $             ( D( N )-D( N-1 )+RHO / ( D( N )+TEMP1 ) ) ) +
     $             Z( N )*Z( N ) / RHO
*
*           The following TAU2 is to approximate
*           SIGMA_n^2 - D( N )*D( N )
*
            IF( C.LE.TEMP ) THEN
               TAU = RHO
            ELSE
               DELSQ = ( D( N )-D( N-1 ) )*( D( N )+D( N-1 ) )
               A = -C*DELSQ + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N )
               B = Z( N )*Z( N )*DELSQ
               IF( A.LT.ZERO ) THEN
                  TAU2 = TWO*B / ( SQRT( A*A+FOUR*B*C )-A )
               ELSE
                  TAU2 = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C )
               END IF
               TAU = TAU2 / ( D( N )+SQRT( D( N )*D( N )+TAU2 ) )
            END IF
*
*           It can be proved that
*               D(N)^2+RHO/2 <= SIGMA_n^2 < D(N)^2+TAU2 <= D(N)^2+RHO
*
         ELSE
            DELSQ = ( D( N )-D( N-1 ) )*( D( N )+D( N-1 ) )
            A = -C*DELSQ + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N )
            B = Z( N )*Z( N )*DELSQ
*
*           The following TAU2 is to approximate
*           SIGMA_n^2 - D( N )*D( N )
*
            IF( A.LT.ZERO ) THEN
               TAU2 = TWO*B / ( SQRT( A*A+FOUR*B*C )-A )
            ELSE
               TAU2 = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C )
            END IF
            TAU = TAU2 / ( D( N )+SQRT( D( N )*D( N )+TAU2 ) )

*
*           It can be proved that
*           D(N)^2 < D(N)^2+TAU2 < SIGMA(N)^2 < D(N)^2+RHO/2
*
         END IF
*
*        The following TAU is to approximate SIGMA_n - D( N )
*
*         TAU = TAU2 / ( D( N )+SQRT( D( N )*D( N )+TAU2 ) )
*
         SIGMA = D( N ) + TAU
         DO 30 J = 1, N
            DELTA( J ) = ( D( J )-D( N ) ) - TAU
            WORK( J ) = D( J ) + D( N ) + TAU
   30    CONTINUE
*
*        Evaluate PSI and the derivative DPSI
*
         DPSI = ZERO
         PSI = ZERO
         ERRETM = ZERO
         DO 40 J = 1, II
            TEMP = Z( J ) / ( DELTA( J )*WORK( J ) )
            PSI = PSI + Z( J )*TEMP
            DPSI = DPSI + TEMP*TEMP
            ERRETM = ERRETM + PSI
   40    CONTINUE
         ERRETM = ABS( ERRETM )
*
*        Evaluate PHI and the derivative DPHI
*
         TEMP = Z( N ) / ( DELTA( N )*WORK( N ) )
         PHI = Z( N )*TEMP
         DPHI = TEMP*TEMP
         ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV
*    $          + ABS( TAU2 )*( DPSI+DPHI )
*
         W = RHOINV + PHI + PSI
*
*        Test for convergence
*
         IF( ABS( W ).LE.EPS*ERRETM ) THEN
            GO TO 240
         END IF
*
*        Calculate the new step
*
         NITER = NITER + 1
         DTNSQ1 = WORK( N-1 )*DELTA( N-1 )
         DTNSQ = WORK( N )*DELTA( N )
         C = W - DTNSQ1*DPSI - DTNSQ*DPHI
         A = ( DTNSQ+DTNSQ1 )*W - DTNSQ*DTNSQ1*( DPSI+DPHI )
         B = DTNSQ*DTNSQ1*W
         IF( C.LT.ZERO )
     $      C = ABS( C )
         IF( C.EQ.ZERO ) THEN
            ETA = RHO - SIGMA*SIGMA
         ELSE IF( A.GE.ZERO ) THEN
            ETA = ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
         ELSE
            ETA = TWO*B / ( A-SQRT( ABS( A*A-FOUR*B*C ) ) )
         END IF
*
*        Note, eta should be positive if w is negative, and
*        eta should be negative otherwise. However,
*        if for some reason caused by roundoff, eta*w > 0,
*        we simply use one Newton step instead. This way
*        will guarantee eta*w < 0.
*
         IF( W*ETA.GT.ZERO )
     $      ETA = -W / ( DPSI+DPHI )
         TEMP = ETA - DTNSQ
         IF( TEMP.GT.RHO )
     $      ETA = RHO + DTNSQ
*
         ETA = ETA / ( SIGMA+SQRT( ETA+SIGMA*SIGMA ) )
         TAU = TAU + ETA
         SIGMA = SIGMA + ETA
*
         DO 50 J = 1, N
            DELTA( J ) = DELTA( J ) - ETA
            WORK( J ) = WORK( J ) + ETA
   50    CONTINUE
*
*        Evaluate PSI and the derivative DPSI
*
         DPSI = ZERO
         PSI = ZERO
         ERRETM = ZERO
         DO 60 J = 1, II
            TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
            PSI = PSI + Z( J )*TEMP
            DPSI = DPSI + TEMP*TEMP
            ERRETM = ERRETM + PSI
   60    CONTINUE
         ERRETM = ABS( ERRETM )
*
*        Evaluate PHI and the derivative DPHI
*
         TAU2 = WORK( N )*DELTA( N )
         TEMP = Z( N ) / TAU2
         PHI = Z( N )*TEMP
         DPHI = TEMP*TEMP
         ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV
*    $          + ABS( TAU2 )*( DPSI+DPHI )
*
         W = RHOINV + PHI + PSI
*
*        Main loop to update the values of the array   DELTA
*
         ITER = NITER + 1
*
         DO 90 NITER = ITER, MAXIT
*
*           Test for convergence
*
            IF( ABS( W ).LE.EPS*ERRETM ) THEN
               GO TO 240
            END IF
*
*           Calculate the new step
*
            DTNSQ1 = WORK( N-1 )*DELTA( N-1 )
            DTNSQ = WORK( N )*DELTA( N )
            C = W - DTNSQ1*DPSI - DTNSQ*DPHI
            A = ( DTNSQ+DTNSQ1 )*W - DTNSQ1*DTNSQ*( DPSI+DPHI )
            B = DTNSQ1*DTNSQ*W
            IF( A.GE.ZERO ) THEN
               ETA = ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
            ELSE
               ETA = TWO*B / ( A-SQRT( ABS( A*A-FOUR*B*C ) ) )
            END IF
*
*           Note, eta should be positive if w is negative, and
*           eta should be negative otherwise. However,
*           if for some reason caused by roundoff, eta*w > 0,
*           we simply use one Newton step instead. This way
*           will guarantee eta*w < 0.
*
            IF( W*ETA.GT.ZERO )
     $         ETA = -W / ( DPSI+DPHI )
            TEMP = ETA - DTNSQ
            IF( TEMP.LE.ZERO )
     $         ETA = ETA / TWO
*
            ETA = ETA / ( SIGMA+SQRT( ETA+SIGMA*SIGMA ) )
            TAU = TAU + ETA
            SIGMA = SIGMA + ETA
*
            DO 70 J = 1, N
               DELTA( J ) = DELTA( J ) - ETA
               WORK( J ) = WORK( J ) + ETA
   70       CONTINUE
*
*           Evaluate PSI and the derivative DPSI
*
            DPSI = ZERO
            PSI = ZERO
            ERRETM = ZERO
            DO 80 J = 1, II
               TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
               PSI = PSI + Z( J )*TEMP
               DPSI = DPSI + TEMP*TEMP
               ERRETM = ERRETM + PSI
   80       CONTINUE
            ERRETM = ABS( ERRETM )
*
*           Evaluate PHI and the derivative DPHI
*
            TAU2 = WORK( N )*DELTA( N )
            TEMP = Z( N ) / TAU2
            PHI = Z( N )*TEMP
            DPHI = TEMP*TEMP
            ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV
*    $             + ABS( TAU2 )*( DPSI+DPHI )
*
            W = RHOINV + PHI + PSI
   90    CONTINUE
*
*        Return with INFO = 1, NITER = MAXIT and not converged
*
         INFO = 1
         GO TO 240
*
*        End for the case I = N
*
      ELSE
*
*        The case for I < N
*
         NITER = 1
         IP1 = I + 1
*
*        Calculate initial guess
*
         DELSQ = ( D( IP1 )-D( I ) )*( D( IP1 )+D( I ) )
         DELSQ2 = DELSQ / TWO
         SQ2=SQRT( ( D( I )*D( I )+D( IP1 )*D( IP1 ) ) / TWO )
         TEMP = DELSQ2 / ( D( I )+SQ2 )
         DO 100 J = 1, N
            WORK( J ) = D( J ) + D( I ) + TEMP
            DELTA( J ) = ( D( J )-D( I ) ) - TEMP
  100    CONTINUE
*
         PSI = ZERO
         DO 110 J = 1, I - 1
            PSI = PSI + Z( J )*Z( J ) / ( WORK( J )*DELTA( J ) )
  110    CONTINUE
*
         PHI = ZERO
         DO 120 J = N, I + 2, -1
            PHI = PHI + Z( J )*Z( J ) / ( WORK( J )*DELTA( J ) )
  120    CONTINUE
         C = RHOINV + PSI + PHI
         W = C + Z( I )*Z( I ) / ( WORK( I )*DELTA( I ) ) +
     $       Z( IP1 )*Z( IP1 ) / ( WORK( IP1 )*DELTA( IP1 ) )
*
         GEOMAVG = .FALSE.
         IF( W.GT.ZERO ) THEN
*
*           d(i)^2 < the ith sigma^2 < (d(i)^2+d(i+1)^2)/2
*
*           We choose d(i) as origin.
*
            ORGATI = .TRUE.
            II = I
            SGLB = ZERO
            SGUB = DELSQ2  / ( D( I )+SQ2 )
            A = C*DELSQ + Z( I )*Z( I ) + Z( IP1 )*Z( IP1 )
            B = Z( I )*Z( I )*DELSQ
            IF( A.GT.ZERO ) THEN
               TAU2 = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
            ELSE
               TAU2 = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
            END IF
*
*           TAU2 now is an estimation of SIGMA^2 - D( I )^2. The
*           following, however, is the corresponding estimation of
*           SIGMA - D( I ).
*
            TAU = TAU2 / ( D( I )+SQRT( D( I )*D( I )+TAU2 ) )
            TEMP = SQRT(EPS)
            IF( (D(I).LE.TEMP*D(IP1)).AND.(ABS(Z(I)).LE.TEMP)
     $                               .AND.(D(I).GT.ZERO) ) THEN
               TAU = MIN( TEN*D(I), SGUB )
               GEOMAVG = .TRUE.
            END IF
         ELSE
*
*           (d(i)^2+d(i+1)^2)/2 <= the ith sigma^2 < d(i+1)^2/2
*
*           We choose d(i+1) as origin.
*
            ORGATI = .FALSE.
            II = IP1
            SGLB = -DELSQ2  / ( D( II )+SQ2 )
            SGUB = ZERO
            A = C*DELSQ - Z( I )*Z( I ) - Z( IP1 )*Z( IP1 )
            B = Z( IP1 )*Z( IP1 )*DELSQ
            IF( A.LT.ZERO ) THEN
               TAU2 = TWO*B / ( A-SQRT( ABS( A*A+FOUR*B*C ) ) )
            ELSE
               TAU2 = -( A+SQRT( ABS( A*A+FOUR*B*C ) ) ) / ( TWO*C )
            END IF
*
*           TAU2 now is an estimation of SIGMA^2 - D( IP1 )^2. The
*           following, however, is the corresponding estimation of
*           SIGMA - D( IP1 ).
*
            TAU = TAU2 / ( D( IP1 )+SQRT( ABS( D( IP1 )*D( IP1 )+
     $            TAU2 ) ) )
         END IF
*
         SIGMA = D( II ) + TAU
         DO 130 J = 1, N
            WORK( J ) = D( J ) + D( II ) + TAU
            DELTA( J ) = ( D( J )-D( II ) ) - TAU
  130    CONTINUE
         IIM1 = II - 1
         IIP1 = II + 1
*
*        Evaluate PSI and the derivative DPSI
*
         DPSI = ZERO
         PSI = ZERO
         ERRETM = ZERO
         DO 150 J = 1, IIM1
            TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
            PSI = PSI + Z( J )*TEMP
            DPSI = DPSI + TEMP*TEMP
            ERRETM = ERRETM + PSI
  150    CONTINUE
         ERRETM = ABS( ERRETM )
*
*        Evaluate PHI and the derivative DPHI
*
         DPHI = ZERO
         PHI = ZERO
         DO 160 J = N, IIP1, -1
            TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
            PHI = PHI + Z( J )*TEMP
            DPHI = DPHI + TEMP*TEMP
            ERRETM = ERRETM + PHI
  160    CONTINUE
*
         W = RHOINV + PHI + PSI
*
*        W is the value of the secular function with
*        its ii-th element removed.
*
         SWTCH3 = .FALSE.
         IF( ORGATI ) THEN
            IF( W.LT.ZERO )
     $         SWTCH3 = .TRUE.
         ELSE
            IF( W.GT.ZERO )
     $         SWTCH3 = .TRUE.
         END IF
         IF( II.EQ.1 .OR. II.EQ.N )
     $      SWTCH3 = .FALSE.
*
         TEMP = Z( II ) / ( WORK( II )*DELTA( II ) )
         DW = DPSI + DPHI + TEMP*TEMP
         TEMP = Z( II )*TEMP
         W = W + TEMP
         ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV
     $          + THREE*ABS( TEMP )
*    $          + ABS( TAU2 )*DW
*
*        Test for convergence
*
         IF( ABS( W ).LE.EPS*ERRETM ) THEN
            GO TO 240
         END IF
*
         IF( W.LE.ZERO ) THEN
            SGLB = MAX( SGLB, TAU )
         ELSE
            SGUB = MIN( SGUB, TAU )
         END IF
*
*        Calculate the new step
*
         NITER = NITER + 1
         IF( .NOT.SWTCH3 ) THEN
            DTIPSQ = WORK( IP1 )*DELTA( IP1 )
            DTISQ = WORK( I )*DELTA( I )
            IF( ORGATI ) THEN
               C = W - DTIPSQ*DW + DELSQ*( Z( I ) / DTISQ )**2
            ELSE
               C = W - DTISQ*DW - DELSQ*( Z( IP1 ) / DTIPSQ )**2
            END IF
            A = ( DTIPSQ+DTISQ )*W - DTIPSQ*DTISQ*DW
            B = DTIPSQ*DTISQ*W
            IF( C.EQ.ZERO ) THEN
               IF( A.EQ.ZERO ) THEN
                  IF( ORGATI ) THEN
                     A = Z( I )*Z( I ) + DTIPSQ*DTIPSQ*( DPSI+DPHI )
                  ELSE
                     A = Z( IP1 )*Z( IP1 ) + DTISQ*DTISQ*( DPSI+DPHI )
                  END IF
               END IF
               ETA = B / A
            ELSE IF( A.LE.ZERO ) THEN
               ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
            ELSE
               ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
            END IF
         ELSE
*
*           Interpolation using THREE most relevant poles
*
            DTIIM = WORK( IIM1 )*DELTA( IIM1 )
            DTIIP = WORK( IIP1 )*DELTA( IIP1 )
            TEMP = RHOINV + PSI + PHI
            IF( ORGATI ) THEN
               TEMP1 = Z( IIM1 ) / DTIIM
               TEMP1 = TEMP1*TEMP1
               C = ( TEMP - DTIIP*( DPSI+DPHI ) ) -
     $             ( D( IIM1 )-D( IIP1 ) )*( D( IIM1 )+D( IIP1 ) )*TEMP1
               ZZ( 1 ) = Z( IIM1 )*Z( IIM1 )
               IF( DPSI.LT.TEMP1 ) THEN
                  ZZ( 3 ) = DTIIP*DTIIP*DPHI
               ELSE
                  ZZ( 3 ) = DTIIP*DTIIP*( ( DPSI-TEMP1 )+DPHI )
               END IF
            ELSE
               TEMP1 = Z( IIP1 ) / DTIIP
               TEMP1 = TEMP1*TEMP1
               C = ( TEMP - DTIIM*( DPSI+DPHI ) ) -
     $             ( D( IIP1 )-D( IIM1 ) )*( D( IIM1 )+D( IIP1 ) )*TEMP1
               IF( DPHI.LT.TEMP1 ) THEN
                  ZZ( 1 ) = DTIIM*DTIIM*DPSI
               ELSE
                  ZZ( 1 ) = DTIIM*DTIIM*( DPSI+( DPHI-TEMP1 ) )
               END IF
               ZZ( 3 ) = Z( IIP1 )*Z( IIP1 )
            END IF
            ZZ( 2 ) = Z( II )*Z( II )
            DD( 1 ) = DTIIM
            DD( 2 ) = DELTA( II )*WORK( II )
            DD( 3 ) = DTIIP
            CALL DLAED6( NITER, ORGATI, C, DD, ZZ, W, ETA, INFO )
*
            IF( INFO.NE.0 ) THEN
*
*              If INFO is not 0, i.e., DLAED6 failed, switch back
*              to 2 pole interpolation.
*
               SWTCH3 = .FALSE.
               INFO = 0
               DTIPSQ = WORK( IP1 )*DELTA( IP1 )
               DTISQ = WORK( I )*DELTA( I )
               IF( ORGATI ) THEN
                  C = W - DTIPSQ*DW + DELSQ*( Z( I ) / DTISQ )**2
               ELSE
                  C = W - DTISQ*DW - DELSQ*( Z( IP1 ) / DTIPSQ )**2
               END IF
               A = ( DTIPSQ+DTISQ )*W - DTIPSQ*DTISQ*DW
               B = DTIPSQ*DTISQ*W
               IF( C.EQ.ZERO ) THEN
                  IF( A.EQ.ZERO ) THEN
                     IF( ORGATI ) THEN
                        A = Z( I )*Z( I ) + DTIPSQ*DTIPSQ*( DPSI+DPHI )
                     ELSE
                        A = Z( IP1 )*Z( IP1 ) + DTISQ*DTISQ*( DPSI+DPHI)
                     END IF
                  END IF
                  ETA = B / A
               ELSE IF( A.LE.ZERO ) THEN
                  ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
               ELSE
                  ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
               END IF
            END IF
         END IF
*
*        Note, eta should be positive if w is negative, and
*        eta should be negative otherwise. However,
*        if for some reason caused by roundoff, eta*w > 0,
*        we simply use one Newton step instead. This way
*        will guarantee eta*w < 0.
*
         IF( W*ETA.GE.ZERO )
     $      ETA = -W / DW
*
         ETA = ETA / ( SIGMA+SQRT( SIGMA*SIGMA+ETA ) )
         TEMP = TAU + ETA
         IF( TEMP.GT.SGUB .OR. TEMP.LT.SGLB ) THEN
            IF( W.LT.ZERO ) THEN
               ETA = ( SGUB-TAU ) / TWO
            ELSE
               ETA = ( SGLB-TAU ) / TWO
            END IF
            IF( GEOMAVG ) THEN
               IF( W .LT. ZERO ) THEN
                  IF( TAU .GT. ZERO ) THEN
                     ETA = SQRT(SGUB*TAU)-TAU
                  END IF
               ELSE
                  IF( SGLB .GT. ZERO ) THEN
                     ETA = SQRT(SGLB*TAU)-TAU
                  END IF
               END IF
            END IF
         END IF
*
         PREW = W
*
         TAU = TAU + ETA
         SIGMA = SIGMA + ETA
*
         DO 170 J = 1, N
            WORK( J ) = WORK( J ) + ETA
            DELTA( J ) = DELTA( J ) - ETA
  170    CONTINUE
*
*        Evaluate PSI and the derivative DPSI
*
         DPSI = ZERO
         PSI = ZERO
         ERRETM = ZERO
         DO 180 J = 1, IIM1
            TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
            PSI = PSI + Z( J )*TEMP
            DPSI = DPSI + TEMP*TEMP
            ERRETM = ERRETM + PSI
  180    CONTINUE
         ERRETM = ABS( ERRETM )
*
*        Evaluate PHI and the derivative DPHI
*
         DPHI = ZERO
         PHI = ZERO
         DO 190 J = N, IIP1, -1
            TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
            PHI = PHI + Z( J )*TEMP
            DPHI = DPHI + TEMP*TEMP
            ERRETM = ERRETM + PHI
  190    CONTINUE
*
         TAU2 = WORK( II )*DELTA( II )
         TEMP = Z( II ) / TAU2
         DW = DPSI + DPHI + TEMP*TEMP
         TEMP = Z( II )*TEMP
         W = RHOINV + PHI + PSI + TEMP
         ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV
     $          + THREE*ABS( TEMP )
*    $          + ABS( TAU2 )*DW
*
         SWTCH = .FALSE.
         IF( ORGATI ) THEN
            IF( -W.GT.ABS( PREW ) / TEN )
     $         SWTCH = .TRUE.
         ELSE
            IF( W.GT.ABS( PREW ) / TEN )
     $         SWTCH = .TRUE.
         END IF
*
*        Main loop to update the values of the array   DELTA and WORK
*
         ITER = NITER + 1
*
         DO 230 NITER = ITER, MAXIT
*
*           Test for convergence
*
            IF( ABS( W ).LE.EPS*ERRETM ) THEN
*     $          .OR. (SGUB-SGLB).LE.EIGHT*ABS(SGUB+SGLB) ) THEN
               GO TO 240
            END IF
*
            IF( W.LE.ZERO ) THEN
               SGLB = MAX( SGLB, TAU )
            ELSE
               SGUB = MIN( SGUB, TAU )
            END IF
*
*           Calculate the new step
*
            IF( .NOT.SWTCH3 ) THEN
               DTIPSQ = WORK( IP1 )*DELTA( IP1 )
               DTISQ = WORK( I )*DELTA( I )
               IF( .NOT.SWTCH ) THEN
                  IF( ORGATI ) THEN
                     C = W - DTIPSQ*DW + DELSQ*( Z( I ) / DTISQ )**2
                  ELSE
                     C = W - DTISQ*DW - DELSQ*( Z( IP1 ) / DTIPSQ )**2
                  END IF
               ELSE
                  TEMP = Z( II ) / ( WORK( II )*DELTA( II ) )
                  IF( ORGATI ) THEN
                     DPSI = DPSI + TEMP*TEMP
                  ELSE
                     DPHI = DPHI + TEMP*TEMP
                  END IF
                  C = W - DTISQ*DPSI - DTIPSQ*DPHI
               END IF
               A = ( DTIPSQ+DTISQ )*W - DTIPSQ*DTISQ*DW
               B = DTIPSQ*DTISQ*W
               IF( C.EQ.ZERO ) THEN
                  IF( A.EQ.ZERO ) THEN
                     IF( .NOT.SWTCH ) THEN
                        IF( ORGATI ) THEN
                           A = Z( I )*Z( I ) + DTIPSQ*DTIPSQ*
     $                         ( DPSI+DPHI )
                        ELSE
                           A = Z( IP1 )*Z( IP1 ) +
     $                         DTISQ*DTISQ*( DPSI+DPHI )
                        END IF
                     ELSE
                        A = DTISQ*DTISQ*DPSI + DTIPSQ*DTIPSQ*DPHI
                     END IF
                  END IF
                  ETA = B / A
               ELSE IF( A.LE.ZERO ) THEN
                  ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
               ELSE
                  ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
               END IF
            ELSE
*
*              Interpolation using THREE most relevant poles
*
               DTIIM = WORK( IIM1 )*DELTA( IIM1 )
               DTIIP = WORK( IIP1 )*DELTA( IIP1 )
               TEMP = RHOINV + PSI + PHI
               IF( SWTCH ) THEN
                  C = TEMP - DTIIM*DPSI - DTIIP*DPHI
                  ZZ( 1 ) = DTIIM*DTIIM*DPSI
                  ZZ( 3 ) = DTIIP*DTIIP*DPHI
               ELSE
                  IF( ORGATI ) THEN
                     TEMP1 = Z( IIM1 ) / DTIIM
                     TEMP1 = TEMP1*TEMP1
                     TEMP2 = ( D( IIM1 )-D( IIP1 ) )*
     $                       ( D( IIM1 )+D( IIP1 ) )*TEMP1
                     C = TEMP - DTIIP*( DPSI+DPHI ) - TEMP2
                     ZZ( 1 ) = Z( IIM1 )*Z( IIM1 )
                     IF( DPSI.LT.TEMP1 ) THEN
                        ZZ( 3 ) = DTIIP*DTIIP*DPHI
                     ELSE
                        ZZ( 3 ) = DTIIP*DTIIP*( ( DPSI-TEMP1 )+DPHI )
                     END IF
                  ELSE
                     TEMP1 = Z( IIP1 ) / DTIIP
                     TEMP1 = TEMP1*TEMP1
                     TEMP2 = ( D( IIP1 )-D( IIM1 ) )*
     $                       ( D( IIM1 )+D( IIP1 ) )*TEMP1
                     C = TEMP - DTIIM*( DPSI+DPHI ) - TEMP2
                     IF( DPHI.LT.TEMP1 ) THEN
                        ZZ( 1 ) = DTIIM*DTIIM*DPSI
                     ELSE
                        ZZ( 1 ) = DTIIM*DTIIM*( DPSI+( DPHI-TEMP1 ) )
                     END IF
                     ZZ( 3 ) = Z( IIP1 )*Z( IIP1 )
                  END IF
               END IF
               DD( 1 ) = DTIIM
               DD( 2 ) = DELTA( II )*WORK( II )
               DD( 3 ) = DTIIP
               CALL DLAED6( NITER, ORGATI, C, DD, ZZ, W, ETA, INFO )
*
               IF( INFO.NE.0 ) THEN
*
*                 If INFO is not 0, i.e., DLAED6 failed, switch
*                 back to two pole interpolation
*
                  SWTCH3 = .FALSE.
                  INFO = 0
                  DTIPSQ = WORK( IP1 )*DELTA( IP1 )
                  DTISQ = WORK( I )*DELTA( I )
                  IF( .NOT.SWTCH ) THEN
                     IF( ORGATI ) THEN
                        C = W - DTIPSQ*DW + DELSQ*( Z( I )/DTISQ )**2
                     ELSE
                        C = W - DTISQ*DW - DELSQ*( Z( IP1 )/DTIPSQ )**2
                     END IF
                  ELSE
                     TEMP = Z( II ) / ( WORK( II )*DELTA( II ) )
                     IF( ORGATI ) THEN
                        DPSI = DPSI + TEMP*TEMP
                     ELSE
                        DPHI = DPHI + TEMP*TEMP
                     END IF
                     C = W - DTISQ*DPSI - DTIPSQ*DPHI
                  END IF
                  A = ( DTIPSQ+DTISQ )*W - DTIPSQ*DTISQ*DW
                  B = DTIPSQ*DTISQ*W
                  IF( C.EQ.ZERO ) THEN
                     IF( A.EQ.ZERO ) THEN
                        IF( .NOT.SWTCH ) THEN
                           IF( ORGATI ) THEN
                              A = Z( I )*Z( I ) + DTIPSQ*DTIPSQ*
     $                            ( DPSI+DPHI )
                           ELSE
                              A = Z( IP1 )*Z( IP1 ) +
     $                            DTISQ*DTISQ*( DPSI+DPHI )
                           END IF
                        ELSE
                           A = DTISQ*DTISQ*DPSI + DTIPSQ*DTIPSQ*DPHI
                        END IF
                     END IF
                     ETA = B / A
                  ELSE IF( A.LE.ZERO ) THEN
                     ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
                  ELSE
                     ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
                  END IF
               END IF
            END IF
*
*           Note, eta should be positive if w is negative, and
*           eta should be negative otherwise. However,
*           if for some reason caused by roundoff, eta*w > 0,
*           we simply use one Newton step instead. This way
*           will guarantee eta*w < 0.
*
            IF( W*ETA.GE.ZERO )
     $         ETA = -W / DW
*
            ETA = ETA / ( SIGMA+SQRT( SIGMA*SIGMA+ETA ) )
            TEMP=TAU+ETA
            IF( TEMP.GT.SGUB .OR. TEMP.LT.SGLB ) THEN
               IF( W.LT.ZERO ) THEN
                  ETA = ( SGUB-TAU ) / TWO
               ELSE
                  ETA = ( SGLB-TAU ) / TWO
               END IF
               IF( GEOMAVG ) THEN
                  IF( W .LT. ZERO ) THEN
                     IF( TAU .GT. ZERO ) THEN
                        ETA = SQRT(SGUB*TAU)-TAU
                     END IF
                  ELSE
                     IF( SGLB .GT. ZERO ) THEN
                        ETA = SQRT(SGLB*TAU)-TAU
                     END IF
                  END IF
               END IF
            END IF
*
            PREW = W
*
            TAU = TAU + ETA
            SIGMA = SIGMA + ETA
*
            DO 200 J = 1, N
               WORK( J ) = WORK( J ) + ETA
               DELTA( J ) = DELTA( J ) - ETA
  200       CONTINUE
*
*           Evaluate PSI and the derivative DPSI
*
            DPSI = ZERO
            PSI = ZERO
            ERRETM = ZERO
            DO 210 J = 1, IIM1
               TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
               PSI = PSI + Z( J )*TEMP
               DPSI = DPSI + TEMP*TEMP
               ERRETM = ERRETM + PSI
  210       CONTINUE
            ERRETM = ABS( ERRETM )
*
*           Evaluate PHI and the derivative DPHI
*
            DPHI = ZERO
            PHI = ZERO
            DO 220 J = N, IIP1, -1
               TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
               PHI = PHI + Z( J )*TEMP
               DPHI = DPHI + TEMP*TEMP
               ERRETM = ERRETM + PHI
  220       CONTINUE
*
            TAU2 = WORK( II )*DELTA( II )
            TEMP = Z( II ) / TAU2
            DW = DPSI + DPHI + TEMP*TEMP
            TEMP = Z( II )*TEMP
            W = RHOINV + PHI + PSI + TEMP
            ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV
     $             + THREE*ABS( TEMP )
*    $             + ABS( TAU2 )*DW
*
            IF( W*PREW.GT.ZERO .AND. ABS( W ).GT.ABS( PREW ) / TEN )
     $         SWTCH = .NOT.SWTCH
*
  230    CONTINUE
*
*        Return with INFO = 1, NITER = MAXIT and not converged
*
         INFO = 1
*
      END IF
*
  240 CONTINUE
      RETURN
*
*     End of DLASD4
*
      END
*> \brief \b DLASD5 computes the square root of the i-th eigenvalue of a positive symmetric rank-one modification of a 2-by-2 diagonal matrix. Used by sbdsdc.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASD5 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd5.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd5.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd5.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASD5( I, D, Z, DELTA, RHO, DSIGMA, WORK )
*
*       .. Scalar Arguments ..
*       INTEGER            I
*       DOUBLE PRECISION   DSIGMA, RHO
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( 2 ), DELTA( 2 ), WORK( 2 ), Z( 2 )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> This subroutine computes the square root of the I-th eigenvalue
*> of a positive symmetric rank-one modification of a 2-by-2 diagonal
*> matrix
*>
*>            diag( D ) * diag( D ) +  RHO * Z * transpose(Z) .
*>
*> The diagonal entries in the array D are assumed to satisfy
*>
*>            0 <= D(i) < D(j)  for  i < j .
*>
*> We also assume RHO > 0 and that the Euclidean norm of the vector
*> Z is one.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] I
*> \verbatim
*>          I is INTEGER
*>         The index of the eigenvalue to be computed.  I = 1 or I = 2.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension ( 2 )
*>         The original eigenvalues.  We assume 0 <= D(1) < D(2).
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension ( 2 )
*>         The components of the updating vector.
*> \endverbatim
*>
*> \param[out] DELTA
*> \verbatim
*>          DELTA is DOUBLE PRECISION array, dimension ( 2 )
*>         Contains (D(j) - sigma_I) in its  j-th component.
*>         The vector DELTA contains the information necessary
*>         to construct the eigenvectors.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*>          RHO is DOUBLE PRECISION
*>         The scalar in the symmetric updating formula.
*> \endverbatim
*>
*> \param[out] DSIGMA
*> \verbatim
*>          DSIGMA is DOUBLE PRECISION
*>         The computed sigma_I, the I-th updated eigenvalue.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension ( 2 )
*>         WORK contains (D(j) + sigma_I) in its  j-th component.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasd5
*
*> \par Contributors:
*  ==================
*>
*>     Ren-Cang Li, Computer Science Division, University of California
*>     at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE DLASD5( I, D, Z, DELTA, RHO, DSIGMA, WORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            I
      DOUBLE PRECISION   DSIGMA, RHO
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( 2 ), DELTA( 2 ), WORK( 2 ), Z( 2 )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO, THREE, FOUR
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0,
     $                   THREE = 3.0D+0, FOUR = 4.0D+0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   B, C, DEL, DELSQ, TAU, W
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, SQRT
*     ..
*     .. Executable Statements ..
*
      DEL = D( 2 ) - D( 1 )
      DELSQ = DEL*( D( 2 )+D( 1 ) )
      IF( I.EQ.1 ) THEN
         W = ONE + FOUR*RHO*( Z( 2 )*Z( 2 ) / ( D( 1 )+THREE*D( 2 ) )-
     $       Z( 1 )*Z( 1 ) / ( THREE*D( 1 )+D( 2 ) ) ) / DEL
         IF( W.GT.ZERO ) THEN
            B = DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
            C = RHO*Z( 1 )*Z( 1 )*DELSQ
*
*           B > ZERO, always
*
*           The following TAU is DSIGMA * DSIGMA - D( 1 ) * D( 1 )
*
            TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) )
*
*           The following TAU is DSIGMA - D( 1 )
*
            TAU = TAU / ( D( 1 )+SQRT( D( 1 )*D( 1 )+TAU ) )
            DSIGMA = D( 1 ) + TAU
            DELTA( 1 ) = -TAU
            DELTA( 2 ) = DEL - TAU
            WORK( 1 ) = TWO*D( 1 ) + TAU
            WORK( 2 ) = ( D( 1 )+TAU ) + D( 2 )
*           DELTA( 1 ) = -Z( 1 ) / TAU
*           DELTA( 2 ) = Z( 2 ) / ( DEL-TAU )
         ELSE
            B = -DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
            C = RHO*Z( 2 )*Z( 2 )*DELSQ
*
*           The following TAU is DSIGMA * DSIGMA - D( 2 ) * D( 2 )
*
            IF( B.GT.ZERO ) THEN
               TAU = -TWO*C / ( B+SQRT( B*B+FOUR*C ) )
            ELSE
               TAU = ( B-SQRT( B*B+FOUR*C ) ) / TWO
            END IF
*
*           The following TAU is DSIGMA - D( 2 )
*
            TAU = TAU / ( D( 2 )+SQRT( ABS( D( 2 )*D( 2 )+TAU ) ) )
            DSIGMA = D( 2 ) + TAU
            DELTA( 1 ) = -( DEL+TAU )
            DELTA( 2 ) = -TAU
            WORK( 1 ) = D( 1 ) + TAU + D( 2 )
            WORK( 2 ) = TWO*D( 2 ) + TAU
*           DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
*           DELTA( 2 ) = -Z( 2 ) / TAU
         END IF
*        TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
*        DELTA( 1 ) = DELTA( 1 ) / TEMP
*        DELTA( 2 ) = DELTA( 2 ) / TEMP
      ELSE
*
*        Now I=2
*
         B = -DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
         C = RHO*Z( 2 )*Z( 2 )*DELSQ
*
*        The following TAU is DSIGMA * DSIGMA - D( 2 ) * D( 2 )
*
         IF( B.GT.ZERO ) THEN
            TAU = ( B+SQRT( B*B+FOUR*C ) ) / TWO
         ELSE
            TAU = TWO*C / ( -B+SQRT( B*B+FOUR*C ) )
         END IF
*
*        The following TAU is DSIGMA - D( 2 )
*
         TAU = TAU / ( D( 2 )+SQRT( D( 2 )*D( 2 )+TAU ) )
         DSIGMA = D( 2 ) + TAU
         DELTA( 1 ) = -( DEL+TAU )
         DELTA( 2 ) = -TAU
         WORK( 1 ) = D( 1 ) + TAU + D( 2 )
         WORK( 2 ) = TWO*D( 2 ) + TAU
*        DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
*        DELTA( 2 ) = -Z( 2 ) / TAU
*        TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
*        DELTA( 1 ) = DELTA( 1 ) / TEMP
*        DELTA( 2 ) = DELTA( 2 ) / TEMP
      END IF
      RETURN
*
*     End of DLASD5
*
      END
*> \brief \b DLASD6 computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASD6 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd6.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd6.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd6.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA,
*                          IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM,
*                          LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK,
*                          IWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
*      $                   NR, SQRE
*       DOUBLE PRECISION   ALPHA, BETA, C, S
*       ..
*       .. Array Arguments ..
*       INTEGER            GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ),
*      $                   PERM( * )
*       DOUBLE PRECISION   D( * ), DIFL( * ), DIFR( * ),
*      $                   GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ),
*      $                   VF( * ), VL( * ), WORK( * ), Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASD6 computes the SVD of an updated upper bidiagonal matrix B
*> obtained by merging two smaller ones by appending a row. This
*> routine is used only for the problem which requires all singular
*> values and optionally singular vector matrices in factored form.
*> B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE.
*> A related subroutine, DLASD1, handles the case in which all singular
*> values and singular vectors of the bidiagonal matrix are desired.
*>
*> DLASD6 computes the SVD as follows:
*>
*>               ( D1(in)    0    0       0 )
*>   B = U(in) * (   Z1**T   a   Z2**T    b ) * VT(in)
*>               (   0       0   D2(in)   0 )
*>
*>     = U(out) * ( D(out) 0) * VT(out)
*>
*> where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M
*> with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
*> elsewhere; and the entry b is empty if SQRE = 0.
*>
*> The singular values of B can be computed using D1, D2, the first
*> components of all the right singular vectors of the lower block, and
*> the last components of all the right singular vectors of the upper
*> block. These components are stored and updated in VF and VL,
*> respectively, in DLASD6. Hence U and VT are not explicitly
*> referenced.
*>
*> The singular values are stored in D. The algorithm consists of two
*> stages:
*>
*>       The first stage consists of deflating the size of the problem
*>       when there are multiple singular values or if there is a zero
*>       in the Z vector. For each such occurrence the dimension of the
*>       secular equation problem is reduced by one. This stage is
*>       performed by the routine DLASD7.
*>
*>       The second stage consists of calculating the updated
*>       singular values. This is done by finding the roots of the
*>       secular equation via the routine DLASD4 (as called by DLASD8).
*>       This routine also updates VF and VL and computes the distances
*>       between the updated singular values and the old singular
*>       values.
*>
*> DLASD6 is called from DLASDA.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*>          ICOMPQ is INTEGER
*>         Specifies whether singular vectors are to be computed in
*>         factored form:
*>         = 0: Compute singular values only.
*>         = 1: Compute singular vectors in factored form as well.
*> \endverbatim
*>
*> \param[in] NL
*> \verbatim
*>          NL is INTEGER
*>         The row dimension of the upper block.  NL >= 1.
*> \endverbatim
*>
*> \param[in] NR
*> \verbatim
*>          NR is INTEGER
*>         The row dimension of the lower block.  NR >= 1.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*>          SQRE is INTEGER
*>         = 0: the lower block is an NR-by-NR square matrix.
*>         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
*>
*>         The bidiagonal matrix has row dimension N = NL + NR + 1,
*>         and column dimension M = N + SQRE.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension ( NL+NR+1 ).
*>         On entry D(1:NL,1:NL) contains the singular values of the
*>         upper block, and D(NL+2:N) contains the singular values
*>         of the lower block. On exit D(1:N) contains the singular
*>         values of the modified matrix.
*> \endverbatim
*>
*> \param[in,out] VF
*> \verbatim
*>          VF is DOUBLE PRECISION array, dimension ( M )
*>         On entry, VF(1:NL+1) contains the first components of all
*>         right singular vectors of the upper block; and VF(NL+2:M)
*>         contains the first components of all right singular vectors
*>         of the lower block. On exit, VF contains the first components
*>         of all right singular vectors of the bidiagonal matrix.
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*>          VL is DOUBLE PRECISION array, dimension ( M )
*>         On entry, VL(1:NL+1) contains the  last components of all
*>         right singular vectors of the upper block; and VL(NL+2:M)
*>         contains the last components of all right singular vectors of
*>         the lower block. On exit, VL contains the last components of
*>         all right singular vectors of the bidiagonal matrix.
*> \endverbatim
*>
*> \param[in,out] ALPHA
*> \verbatim
*>          ALPHA is DOUBLE PRECISION
*>         Contains the diagonal element associated with the added row.
*> \endverbatim
*>
*> \param[in,out] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION
*>         Contains the off-diagonal element associated with the added
*>         row.
*> \endverbatim
*>
*> \param[in,out] IDXQ
*> \verbatim
*>          IDXQ is INTEGER array, dimension ( N )
*>         This contains the permutation which will reintegrate the
*>         subproblem just solved back into sorted order, i.e.
*>         D( IDXQ( I = 1, N ) ) will be in ascending order.
*> \endverbatim
*>
*> \param[out] PERM
*> \verbatim
*>          PERM is INTEGER array, dimension ( N )
*>         The permutations (from deflation and sorting) to be applied
*>         to each block. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[out] GIVPTR
*> \verbatim
*>          GIVPTR is INTEGER
*>         The number of Givens rotations which took place in this
*>         subproblem. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[out] GIVCOL
*> \verbatim
*>          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
*>         Each pair of numbers indicates a pair of columns to take place
*>         in a Givens rotation. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[in] LDGCOL
*> \verbatim
*>          LDGCOL is INTEGER
*>         leading dimension of GIVCOL, must be at least N.
*> \endverbatim
*>
*> \param[out] GIVNUM
*> \verbatim
*>          GIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
*>         Each number indicates the C or S value to be used in the
*>         corresponding Givens rotation. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[in] LDGNUM
*> \verbatim
*>          LDGNUM is INTEGER
*>         The leading dimension of GIVNUM and POLES, must be at least N.
*> \endverbatim
*>
*> \param[out] POLES
*> \verbatim
*>          POLES is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
*>         On exit, POLES(1,*) is an array containing the new singular
*>         values obtained from solving the secular equation, and
*>         POLES(2,*) is an array containing the poles in the secular
*>         equation. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[out] DIFL
*> \verbatim
*>          DIFL is DOUBLE PRECISION array, dimension ( N )
*>         On exit, DIFL(I) is the distance between I-th updated
*>         (undeflated) singular value and the I-th (undeflated) old
*>         singular value.
*> \endverbatim
*>
*> \param[out] DIFR
*> \verbatim
*>          DIFR is DOUBLE PRECISION array,
*>                   dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and
*>                   dimension ( K ) if ICOMPQ = 0.
*>          On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not
*>          defined and will not be referenced.
*>
*>          If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
*>          normalizing factors for the right singular vector matrix.
*>
*>         See DLASD8 for details on DIFL and DIFR.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension ( M )
*>         The first elements of this array contain the components
*>         of the deflation-adjusted updating row vector.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*>          K is INTEGER
*>         Contains the dimension of the non-deflated matrix,
*>         This is the order of the related secular equation. 1 <= K <=N.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*>          C is DOUBLE PRECISION
*>         C contains garbage if SQRE =0 and the C-value of a Givens
*>         rotation related to the right null space if SQRE = 1.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION
*>         S contains garbage if SQRE =0 and the S-value of a Givens
*>         rotation related to the right null space if SQRE = 1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension ( 4 * M )
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension ( 3 * N )
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = 1, a singular value did not converge
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasd6
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Huan Ren, Computer Science Division, University of
*>     California at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE DLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA,
     $                   BETA,
     $                   IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM,
     $                   LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK,
     $                   IWORK, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
     $                   NR, SQRE
      DOUBLE PRECISION   ALPHA, BETA, C, S
*     ..
*     .. Array Arguments ..
      INTEGER            GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ),
     $                   PERM( * )
      DOUBLE PRECISION   D( * ), DIFL( * ), DIFR( * ),
     $                   GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ),
     $                   VF( * ), VL( * ), WORK( * ), Z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, IDX, IDXC, IDXP, ISIGMA, IVFW, IVLW, IW, M,
     $                   N, N1, N2
      DOUBLE PRECISION   ORGNRM
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLAMRG, DLASCL, DLASD7, DLASD8,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      N = NL + NR + 1
      M = N + SQRE
*
      IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
         INFO = -1
      ELSE IF( NL.LT.1 ) THEN
         INFO = -2
      ELSE IF( NR.LT.1 ) THEN
         INFO = -3
      ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
         INFO = -4
      ELSE IF( LDGCOL.LT.N ) THEN
         INFO = -14
      ELSE IF( LDGNUM.LT.N ) THEN
         INFO = -16
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLASD6', -INFO )
         RETURN
      END IF
*
*     The following values are for bookkeeping purposes only.  They are
*     integer pointers which indicate the portion of the workspace
*     used by a particular array in DLASD7 and DLASD8.
*
      ISIGMA = 1
      IW = ISIGMA + N
      IVFW = IW + M
      IVLW = IVFW + M
*
      IDX = 1
      IDXC = IDX + N
      IDXP = IDXC + N
*
*     Scale.
*
      ORGNRM = MAX( ABS( ALPHA ), ABS( BETA ) )
      D( NL+1 ) = ZERO
      DO 10 I = 1, N
         IF( ABS( D( I ) ).GT.ORGNRM ) THEN
            ORGNRM = ABS( D( I ) )
         END IF
   10 CONTINUE
      CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO )
      ALPHA = ALPHA / ORGNRM
      BETA = BETA / ORGNRM
*
*     Sort and Deflate singular values.
*
      CALL DLASD7( ICOMPQ, NL, NR, SQRE, K, D, Z, WORK( IW ), VF,
     $             WORK( IVFW ), VL, WORK( IVLW ), ALPHA, BETA,
     $             WORK( ISIGMA ), IWORK( IDX ), IWORK( IDXP ), IDXQ,
     $             PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, C, S,
     $             INFO )
*
*     Solve Secular Equation, compute DIFL, DIFR, and update VF, VL.
*
      CALL DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDGNUM,
     $             WORK( ISIGMA ), WORK( IW ), INFO )
*
*     Report the possible convergence failure.
*
      IF( INFO.NE.0 ) THEN
         RETURN
      END IF
*
*     Save the poles if ICOMPQ = 1.
*
      IF( ICOMPQ.EQ.1 ) THEN
         CALL DCOPY( K, D, 1, POLES( 1, 1 ), 1 )
         CALL DCOPY( K, WORK( ISIGMA ), 1, POLES( 1, 2 ), 1 )
      END IF
*
*     Unscale.
*
      CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
*
*     Prepare the IDXQ sorting permutation.
*
      N1 = K
      N2 = N - K
      CALL DLAMRG( N1, N2, D, 1, -1, IDXQ )
*
      RETURN
*
*     End of DLASD6
*
      END
*> \brief \b DLASD7 merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. Used by sbdsdc.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASD7 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd7.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd7.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd7.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASD7( ICOMPQ, NL, NR, SQRE, K, D, Z, ZW, VF, VFW, VL,
*                          VLW, ALPHA, BETA, DSIGMA, IDX, IDXP, IDXQ,
*                          PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM,
*                          C, S, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
*      $                   NR, SQRE
*       DOUBLE PRECISION   ALPHA, BETA, C, S
*       ..
*       .. Array Arguments ..
*       INTEGER            GIVCOL( LDGCOL, * ), IDX( * ), IDXP( * ),
*      $                   IDXQ( * ), PERM( * )
*       DOUBLE PRECISION   D( * ), DSIGMA( * ), GIVNUM( LDGNUM, * ),
*      $                   VF( * ), VFW( * ), VL( * ), VLW( * ), Z( * ),
*      $                   ZW( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASD7 merges the two sets of singular values together into a single
*> sorted set. Then it tries to deflate the size of the problem. There
*> are two ways in which deflation can occur:  when two or more singular
*> values are close together or if there is a tiny entry in the Z
*> vector. For each such occurrence the order of the related
*> secular equation problem is reduced by one.
*>
*> DLASD7 is called from DLASD6.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*>          ICOMPQ is INTEGER
*>          Specifies whether singular vectors are to be computed
*>          in compact form, as follows:
*>          = 0: Compute singular values only.
*>          = 1: Compute singular vectors of upper
*>               bidiagonal matrix in compact form.
*> \endverbatim
*>
*> \param[in] NL
*> \verbatim
*>          NL is INTEGER
*>         The row dimension of the upper block. NL >= 1.
*> \endverbatim
*>
*> \param[in] NR
*> \verbatim
*>          NR is INTEGER
*>         The row dimension of the lower block. NR >= 1.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*>          SQRE is INTEGER
*>         = 0: the lower block is an NR-by-NR square matrix.
*>         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
*>
*>         The bidiagonal matrix has
*>         N = NL + NR + 1 rows and
*>         M = N + SQRE >= N columns.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*>          K is INTEGER
*>         Contains the dimension of the non-deflated matrix, this is
*>         the order of the related secular equation. 1 <= K <=N.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension ( N )
*>         On entry D contains the singular values of the two submatrices
*>         to be combined. On exit D contains the trailing (N-K) updated
*>         singular values (those which were deflated) sorted into
*>         increasing order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension ( M )
*>         On exit Z contains the updating row vector in the secular
*>         equation.
*> \endverbatim
*>
*> \param[out] ZW
*> \verbatim
*>          ZW is DOUBLE PRECISION array, dimension ( M )
*>         Workspace for Z.
*> \endverbatim
*>
*> \param[in,out] VF
*> \verbatim
*>          VF is DOUBLE PRECISION array, dimension ( M )
*>         On entry, VF(1:NL+1) contains the first components of all
*>         right singular vectors of the upper block; and VF(NL+2:M)
*>         contains the first components of all right singular vectors
*>         of the lower block. On exit, VF contains the first components
*>         of all right singular vectors of the bidiagonal matrix.
*> \endverbatim
*>
*> \param[out] VFW
*> \verbatim
*>          VFW is DOUBLE PRECISION array, dimension ( M )
*>         Workspace for VF.
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*>          VL is DOUBLE PRECISION array, dimension ( M )
*>         On entry, VL(1:NL+1) contains the  last components of all
*>         right singular vectors of the upper block; and VL(NL+2:M)
*>         contains the last components of all right singular vectors
*>         of the lower block. On exit, VL contains the last components
*>         of all right singular vectors of the bidiagonal matrix.
*> \endverbatim
*>
*> \param[out] VLW
*> \verbatim
*>          VLW is DOUBLE PRECISION array, dimension ( M )
*>         Workspace for VL.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*>          ALPHA is DOUBLE PRECISION
*>         Contains the diagonal element associated with the added row.
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION
*>         Contains the off-diagonal element associated with the added
*>         row.
*> \endverbatim
*>
*> \param[out] DSIGMA
*> \verbatim
*>          DSIGMA is DOUBLE PRECISION array, dimension ( N )
*>         Contains a copy of the diagonal elements (K-1 singular values
*>         and one zero) in the secular equation.
*> \endverbatim
*>
*> \param[out] IDX
*> \verbatim
*>          IDX is INTEGER array, dimension ( N )
*>         This will contain the permutation used to sort the contents of
*>         D into ascending order.
*> \endverbatim
*>
*> \param[out] IDXP
*> \verbatim
*>          IDXP is INTEGER array, dimension ( N )
*>         This will contain the permutation used to place deflated
*>         values of D at the end of the array. On output IDXP(2:K)
*>         points to the nondeflated D-values and IDXP(K+1:N)
*>         points to the deflated singular values.
*> \endverbatim
*>
*> \param[in] IDXQ
*> \verbatim
*>          IDXQ is INTEGER array, dimension ( N )
*>         This contains the permutation which separately sorts the two
*>         sub-problems in D into ascending order.  Note that entries in
*>         the first half of this permutation must first be moved one
*>         position backward; and entries in the second half
*>         must first have NL+1 added to their values.
*> \endverbatim
*>
*> \param[out] PERM
*> \verbatim
*>          PERM is INTEGER array, dimension ( N )
*>         The permutations (from deflation and sorting) to be applied
*>         to each singular block. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[out] GIVPTR
*> \verbatim
*>          GIVPTR is INTEGER
*>         The number of Givens rotations which took place in this
*>         subproblem. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[out] GIVCOL
*> \verbatim
*>          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
*>         Each pair of numbers indicates a pair of columns to take place
*>         in a Givens rotation. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[in] LDGCOL
*> \verbatim
*>          LDGCOL is INTEGER
*>         The leading dimension of GIVCOL, must be at least N.
*> \endverbatim
*>
*> \param[out] GIVNUM
*> \verbatim
*>          GIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
*>         Each number indicates the C or S value to be used in the
*>         corresponding Givens rotation. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[in] LDGNUM
*> \verbatim
*>          LDGNUM is INTEGER
*>         The leading dimension of GIVNUM, must be at least N.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*>          C is DOUBLE PRECISION
*>         C contains garbage if SQRE =0 and the C-value of a Givens
*>         rotation related to the right null space if SQRE = 1.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION
*>         S contains garbage if SQRE =0 and the S-value of a Givens
*>         rotation related to the right null space if SQRE = 1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>         = 0:  successful exit.
*>         < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasd7
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Huan Ren, Computer Science Division, University of
*>     California at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE DLASD7( ICOMPQ, NL, NR, SQRE, K, D, Z, ZW, VF, VFW,
     $                   VL,
     $                   VLW, ALPHA, BETA, DSIGMA, IDX, IDXP, IDXQ,
     $                   PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM,
     $                   C, S, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
     $                   NR, SQRE
      DOUBLE PRECISION   ALPHA, BETA, C, S
*     ..
*     .. Array Arguments ..
      INTEGER            GIVCOL( LDGCOL, * ), IDX( * ), IDXP( * ),
     $                   IDXQ( * ), PERM( * )
      DOUBLE PRECISION   D( * ), DSIGMA( * ), GIVNUM( LDGNUM, * ),
     $                   VF( * ), VFW( * ), VL( * ), VLW( * ), Z( * ),
     $                   ZW( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO, EIGHT
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0,
     $                   EIGHT = 8.0D+0 )
*     ..
*     .. Local Scalars ..
*
      INTEGER            I, IDXI, IDXJ, IDXJP, J, JP, JPREV, K2, M, N,
     $                   NLP1, NLP2
      DOUBLE PRECISION   EPS, HLFTOL, TAU, TOL, Z1
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLAMRG, DROT, XERBLA
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH, DLAPY2
      EXTERNAL           DLAMCH, DLAPY2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      N = NL + NR + 1
      M = N + SQRE
*
      IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
         INFO = -1
      ELSE IF( NL.LT.1 ) THEN
         INFO = -2
      ELSE IF( NR.LT.1 ) THEN
         INFO = -3
      ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
         INFO = -4
      ELSE IF( LDGCOL.LT.N ) THEN
         INFO = -22
      ELSE IF( LDGNUM.LT.N ) THEN
         INFO = -24
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLASD7', -INFO )
         RETURN
      END IF
*
      NLP1 = NL + 1
      NLP2 = NL + 2
      IF( ICOMPQ.EQ.1 ) THEN
         GIVPTR = 0
      END IF
*
*     Generate the first part of the vector Z and move the singular
*     values in the first part of D one position backward.
*
      Z1 = ALPHA*VL( NLP1 )
      VL( NLP1 ) = ZERO
      TAU = VF( NLP1 )
      DO 10 I = NL, 1, -1
         Z( I+1 ) = ALPHA*VL( I )
         VL( I ) = ZERO
         VF( I+1 ) = VF( I )
         D( I+1 ) = D( I )
         IDXQ( I+1 ) = IDXQ( I ) + 1
   10 CONTINUE
      VF( 1 ) = TAU
*
*     Generate the second part of the vector Z.
*
      DO 20 I = NLP2, M
         Z( I ) = BETA*VF( I )
         VF( I ) = ZERO
   20 CONTINUE
*
*     Sort the singular values into increasing order
*
      DO 30 I = NLP2, N
         IDXQ( I ) = IDXQ( I ) + NLP1
   30 CONTINUE
*
*     DSIGMA, IDXC, IDXC, and ZW are used as storage space.
*
      DO 40 I = 2, N
         DSIGMA( I ) = D( IDXQ( I ) )
         ZW( I ) = Z( IDXQ( I ) )
         VFW( I ) = VF( IDXQ( I ) )
         VLW( I ) = VL( IDXQ( I ) )
   40 CONTINUE
*
      CALL DLAMRG( NL, NR, DSIGMA( 2 ), 1, 1, IDX( 2 ) )
*
      DO 50 I = 2, N
         IDXI = 1 + IDX( I )
         D( I ) = DSIGMA( IDXI )
         Z( I ) = ZW( IDXI )
         VF( I ) = VFW( IDXI )
         VL( I ) = VLW( IDXI )
   50 CONTINUE
*
*     Calculate the allowable deflation tolerance
*
      EPS = DLAMCH( 'Epsilon' )
      TOL = MAX( ABS( ALPHA ), ABS( BETA ) )
      TOL = EIGHT*EIGHT*EPS*MAX( ABS( D( N ) ), TOL )
*
*     There are 2 kinds of deflation -- first a value in the z-vector
*     is small, second two (or more) singular values are very close
*     together (their difference is small).
*
*     If the value in the z-vector is small, we simply permute the
*     array so that the corresponding singular value is moved to the
*     end.
*
*     If two values in the D-vector are close, we perform a two-sided
*     rotation designed to make one of the corresponding z-vector
*     entries zero, and then permute the array so that the deflated
*     singular value is moved to the end.
*
*     If there are multiple singular values then the problem deflates.
*     Here the number of equal singular values are found.  As each equal
*     singular value is found, an elementary reflector is computed to
*     rotate the corresponding singular subspace so that the
*     corresponding components of Z are zero in this new basis.
*
      K = 1
      K2 = N + 1
      DO 60 J = 2, N
         IF( ABS( Z( J ) ).LE.TOL ) THEN
*
*           Deflate due to small z component.
*
            K2 = K2 - 1
            IDXP( K2 ) = J
            IF( J.EQ.N )
     $         GO TO 100
         ELSE
            JPREV = J
            GO TO 70
         END IF
   60 CONTINUE
   70 CONTINUE
      J = JPREV
   80 CONTINUE
      J = J + 1
      IF( J.GT.N )
     $   GO TO 90
      IF( ABS( Z( J ) ).LE.TOL ) THEN
*
*        Deflate due to small z component.
*
         K2 = K2 - 1
         IDXP( K2 ) = J
      ELSE
*
*        Check if singular values are close enough to allow deflation.
*
         IF( ABS( D( J )-D( JPREV ) ).LE.TOL ) THEN
*
*           Deflation is possible.
*
            S = Z( JPREV )
            C = Z( J )
*
*           Find sqrt(a**2+b**2) without overflow or
*           destructive underflow.
*
            TAU = DLAPY2( C, S )
            Z( J ) = TAU
            Z( JPREV ) = ZERO
            C = C / TAU
            S = -S / TAU
*
*           Record the appropriate Givens rotation
*
            IF( ICOMPQ.EQ.1 ) THEN
               GIVPTR = GIVPTR + 1
               IDXJP = IDXQ( IDX( JPREV )+1 )
               IDXJ = IDXQ( IDX( J )+1 )
               IF( IDXJP.LE.NLP1 ) THEN
                  IDXJP = IDXJP - 1
               END IF
               IF( IDXJ.LE.NLP1 ) THEN
                  IDXJ = IDXJ - 1
               END IF
               GIVCOL( GIVPTR, 2 ) = IDXJP
               GIVCOL( GIVPTR, 1 ) = IDXJ
               GIVNUM( GIVPTR, 2 ) = C
               GIVNUM( GIVPTR, 1 ) = S
            END IF
            CALL DROT( 1, VF( JPREV ), 1, VF( J ), 1, C, S )
            CALL DROT( 1, VL( JPREV ), 1, VL( J ), 1, C, S )
            K2 = K2 - 1
            IDXP( K2 ) = JPREV
            JPREV = J
         ELSE
            K = K + 1
            ZW( K ) = Z( JPREV )
            DSIGMA( K ) = D( JPREV )
            IDXP( K ) = JPREV
            JPREV = J
         END IF
      END IF
      GO TO 80
   90 CONTINUE
*
*     Record the last singular value.
*
      K = K + 1
      ZW( K ) = Z( JPREV )
      DSIGMA( K ) = D( JPREV )
      IDXP( K ) = JPREV
*
  100 CONTINUE
*
*     Sort the singular values into DSIGMA. The singular values which
*     were not deflated go into the first K slots of DSIGMA, except
*     that DSIGMA(1) is treated separately.
*
      DO 110 J = 2, N
         JP = IDXP( J )
         DSIGMA( J ) = D( JP )
         VFW( J ) = VF( JP )
         VLW( J ) = VL( JP )
  110 CONTINUE
      IF( ICOMPQ.EQ.1 ) THEN
         DO 120 J = 2, N
            JP = IDXP( J )
            PERM( J ) = IDXQ( IDX( JP )+1 )
            IF( PERM( J ).LE.NLP1 ) THEN
               PERM( J ) = PERM( J ) - 1
            END IF
  120    CONTINUE
      END IF
*
*     The deflated singular values go back into the last N - K slots of
*     D.
*
      CALL DCOPY( N-K, DSIGMA( K+1 ), 1, D( K+1 ), 1 )
*
*     Determine DSIGMA(1), DSIGMA(2), Z(1), VF(1), VL(1), VF(M), and
*     VL(M).
*
      DSIGMA( 1 ) = ZERO
      HLFTOL = TOL / TWO
      IF( ABS( DSIGMA( 2 ) ).LE.HLFTOL )
     $   DSIGMA( 2 ) = HLFTOL
      IF( M.GT.N ) THEN
         Z( 1 ) = DLAPY2( Z1, Z( M ) )
         IF( Z( 1 ).LE.TOL ) THEN
            C = ONE
            S = ZERO
            Z( 1 ) = TOL
         ELSE
            C = Z1 / Z( 1 )
            S = -Z( M ) / Z( 1 )
         END IF
         CALL DROT( 1, VF( M ), 1, VF( 1 ), 1, C, S )
         CALL DROT( 1, VL( M ), 1, VL( 1 ), 1, C, S )
      ELSE
         IF( ABS( Z1 ).LE.TOL ) THEN
            Z( 1 ) = TOL
         ELSE
            Z( 1 ) = Z1
         END IF
      END IF
*
*     Restore Z, VF, and VL.
*
      CALL DCOPY( K-1, ZW( 2 ), 1, Z( 2 ), 1 )
      CALL DCOPY( N-1, VFW( 2 ), 1, VF( 2 ), 1 )
      CALL DCOPY( N-1, VLW( 2 ), 1, VL( 2 ), 1 )
*
      RETURN
*
*     End of DLASD7
*
      END
*> \brief \b DLASD8 finds the square roots of the roots of the secular equation, and stores, for each element in D, the distance to its two nearest poles. Used by sbdsdc.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASD8 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd8.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd8.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd8.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR,
*                          DSIGMA, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            ICOMPQ, INFO, K, LDDIFR
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), DIFL( * ), DIFR( LDDIFR, * ),
*      $                   DSIGMA( * ), VF( * ), VL( * ), WORK( * ),
*      $                   Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASD8 finds the square roots of the roots of the secular equation,
*> as defined by the values in DSIGMA and Z. It makes the appropriate
*> calls to DLASD4, and stores, for each  element in D, the distance
*> to its two nearest poles (elements in DSIGMA). It also updates
*> the arrays VF and VL, the first and last components of all the
*> right singular vectors of the original bidiagonal matrix.
*>
*> DLASD8 is called from DLASD6.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*>          ICOMPQ is INTEGER
*>          Specifies whether singular vectors are to be computed in
*>          factored form in the calling routine:
*>          = 0: Compute singular values only.
*>          = 1: Compute singular vectors in factored form as well.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of terms in the rational function to be solved
*>          by DLASD4.  K >= 1.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension ( K )
*>          On output, D contains the updated singular values.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension ( K )
*>          On entry, the first K elements of this array contain the
*>          components of the deflation-adjusted updating row vector.
*>          On exit, Z is updated.
*> \endverbatim
*>
*> \param[in,out] VF
*> \verbatim
*>          VF is DOUBLE PRECISION array, dimension ( K )
*>          On entry, VF contains  information passed through DBEDE8.
*>          On exit, VF contains the first K components of the first
*>          components of all right singular vectors of the bidiagonal
*>          matrix.
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*>          VL is DOUBLE PRECISION array, dimension ( K )
*>          On entry, VL contains  information passed through DBEDE8.
*>          On exit, VL contains the first K components of the last
*>          components of all right singular vectors of the bidiagonal
*>          matrix.
*> \endverbatim
*>
*> \param[out] DIFL
*> \verbatim
*>          DIFL is DOUBLE PRECISION array, dimension ( K )
*>          On exit, DIFL(I) = D(I) - DSIGMA(I).
*> \endverbatim
*>
*> \param[out] DIFR
*> \verbatim
*>          DIFR is DOUBLE PRECISION array,
*>                   dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and
*>                   dimension ( K ) if ICOMPQ = 0.
*>          On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not
*>          defined and will not be referenced.
*>
*>          If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
*>          normalizing factors for the right singular vector matrix.
*> \endverbatim
*>
*> \param[in] LDDIFR
*> \verbatim
*>          LDDIFR is INTEGER
*>          The leading dimension of DIFR, must be at least K.
*> \endverbatim
*>
*> \param[in] DSIGMA
*> \verbatim
*>          DSIGMA is DOUBLE PRECISION array, dimension ( K )
*>          On entry, the first K elements of this array contain the old
*>          roots of the deflated updating problem.  These are the poles
*>          of the secular equation.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*K)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = 1, a singular value did not converge
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasd8
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Huan Ren, Computer Science Division, University of
*>     California at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR,
     $                   DSIGMA, WORK, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            ICOMPQ, INFO, K, LDDIFR
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), DIFL( * ), DIFR( LDDIFR, * ),
     $                   DSIGMA( * ), VF( * ), VL( * ), WORK( * ),
     $                   Z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, IWK1, IWK2, IWK2I, IWK3, IWK3I, J
      DOUBLE PRECISION   DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, RHO, TEMP
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLASCL, DLASD4, DLASET,
     $                   XERBLA
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DDOT, DLAMC3, DNRM2
      EXTERNAL           DDOT, DLAMC3, DNRM2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, SIGN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
         INFO = -1
      ELSE IF( K.LT.1 ) THEN
         INFO = -2
      ELSE IF( LDDIFR.LT.K ) THEN
         INFO = -9
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLASD8', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( K.EQ.1 ) THEN
         D( 1 ) = ABS( Z( 1 ) )
         DIFL( 1 ) = D( 1 )
         IF( ICOMPQ.EQ.1 ) THEN
            DIFL( 2 ) = ONE
            DIFR( 1, 2 ) = ONE
         END IF
         RETURN
      END IF
*
*     Book keeping.
*
      IWK1 = 1
      IWK2 = IWK1 + K
      IWK3 = IWK2 + K
      IWK2I = IWK2 - 1
      IWK3I = IWK3 - 1
*
*     Normalize Z.
*
      RHO = DNRM2( K, Z, 1 )
      CALL DLASCL( 'G', 0, 0, RHO, ONE, K, 1, Z, K, INFO )
      RHO = RHO*RHO
*
*     Initialize WORK(IWK3).
*
      CALL DLASET( 'A', K, 1, ONE, ONE, WORK( IWK3 ), K )
*
*     Compute the updated singular values, the arrays DIFL, DIFR,
*     and the updated Z.
*
      DO 40 J = 1, K
         CALL DLASD4( K, J, DSIGMA, Z, WORK( IWK1 ), RHO, D( J ),
     $                WORK( IWK2 ), INFO )
*
*        If the root finder fails, report the convergence failure.
*
         IF( INFO.NE.0 ) THEN
            RETURN
         END IF
         WORK( IWK3I+J ) = WORK( IWK3I+J )*WORK( J )*WORK( IWK2I+J )
         DIFL( J ) = -WORK( J )
         DIFR( J, 1 ) = -WORK( J+1 )
         DO 20 I = 1, J - 1
            WORK( IWK3I+I ) = WORK( IWK3I+I )*WORK( I )*
     $                        WORK( IWK2I+I ) / ( DSIGMA( I )-
     $                        DSIGMA( J ) ) / ( DSIGMA( I )+
     $                        DSIGMA( J ) )
   20    CONTINUE
         DO 30 I = J + 1, K
            WORK( IWK3I+I ) = WORK( IWK3I+I )*WORK( I )*
     $                        WORK( IWK2I+I ) / ( DSIGMA( I )-
     $                        DSIGMA( J ) ) / ( DSIGMA( I )+
     $                        DSIGMA( J ) )
   30    CONTINUE
   40 CONTINUE
*
*     Compute updated Z.
*
      DO 50 I = 1, K
         Z( I ) = SIGN( SQRT( ABS( WORK( IWK3I+I ) ) ), Z( I ) )
   50 CONTINUE
*
*     Update VF and VL.
*
      DO 80 J = 1, K
         DIFLJ = DIFL( J )
         DJ = D( J )
         DSIGJ = -DSIGMA( J )
         IF( J.LT.K ) THEN
            DIFRJ = -DIFR( J, 1 )
            DSIGJP = -DSIGMA( J+1 )
         END IF
         WORK( J ) = -Z( J ) / DIFLJ / ( DSIGMA( J )+DJ )
*
*        Use calls to the subroutine DLAMC3 to enforce the parentheses
*        (x+y)+z. The goal is to prevent optimizing compilers
*        from doing x+(y+z).
*
         DO 60 I = 1, J - 1
            WORK( I ) = Z( I ) / ( DLAMC3( DSIGMA( I ),
     $            DSIGJ )-DIFLJ )
     $                   / ( DSIGMA( I )+DJ )
   60    CONTINUE
         DO 70 I = J + 1, K
            WORK( I ) = Z( I ) / ( DLAMC3( DSIGMA( I ),
     $            DSIGJP )+DIFRJ )
     $                   / ( DSIGMA( I )+DJ )
   70    CONTINUE
         TEMP = DNRM2( K, WORK, 1 )
         WORK( IWK2I+J ) = DDOT( K, WORK, 1, VF, 1 ) / TEMP
         WORK( IWK3I+J ) = DDOT( K, WORK, 1, VL, 1 ) / TEMP
         IF( ICOMPQ.EQ.1 ) THEN
            DIFR( J, 2 ) = TEMP
         END IF
   80 CONTINUE
*
      CALL DCOPY( K, WORK( IWK2 ), 1, VF, 1 )
      CALL DCOPY( K, WORK( IWK3 ), 1, VL, 1 )
*
      RETURN
*
*     End of DLASD8
*
      END

*> \brief \b DLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASDA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasda.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasda.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasda.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K,
*                          DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL,
*                          PERM, GIVNUM, C, S, WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE
*       ..
*       .. Array Arguments ..
*       INTEGER            GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
*      $                   K( * ), PERM( LDGCOL, * )
*       DOUBLE PRECISION   C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU, * ),
*      $                   E( * ), GIVNUM( LDU, * ), POLES( LDU, * ),
*      $                   S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ),
*      $                   Z( LDU, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> Using a divide and conquer approach, DLASDA computes the singular
*> value decomposition (SVD) of a real upper bidiagonal N-by-M matrix
*> B with diagonal D and offdiagonal E, where M = N + SQRE. The
*> algorithm computes the singular values in the SVD B = U * S * VT.
*> The orthogonal matrices U and VT are optionally computed in
*> compact form.
*>
*> A related subroutine, DLASD0, computes the singular values and
*> the singular vectors in explicit form.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*>          ICOMPQ is INTEGER
*>         Specifies whether singular vectors are to be computed
*>         in compact form, as follows
*>         = 0: Compute singular values only.
*>         = 1: Compute singular vectors of upper bidiagonal
*>              matrix in compact form.
*> \endverbatim
*>
*> \param[in] SMLSIZ
*> \verbatim
*>          SMLSIZ is INTEGER
*>         The maximum size of the subproblems at the bottom of the
*>         computation tree.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>         The row dimension of the upper bidiagonal matrix. This is
*>         also the dimension of the main diagonal array D.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*>          SQRE is INTEGER
*>         Specifies the column dimension of the bidiagonal matrix.
*>         = 0: The bidiagonal matrix has column dimension M = N;
*>         = 1: The bidiagonal matrix has column dimension M = N + 1.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension ( N )
*>         On entry D contains the main diagonal of the bidiagonal
*>         matrix. On exit D, if INFO = 0, contains its singular values.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension ( M-1 )
*>         Contains the subdiagonal entries of the bidiagonal matrix.
*>         On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*>          U is DOUBLE PRECISION array,
*>         dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced
*>         if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left
*>         singular vector matrices of all subproblems at the bottom
*>         level.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER, LDU = > N.
*>         The leading dimension of arrays U, VT, DIFL, DIFR, POLES,
*>         GIVNUM, and Z.
*> \endverbatim
*>
*> \param[out] VT
*> \verbatim
*>          VT is DOUBLE PRECISION array,
*>         dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced
*>         if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT**T contains the right
*>         singular vector matrices of all subproblems at the bottom
*>         level.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*>          K is INTEGER array,
*>         dimension ( N ) if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0.
*>         If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th
*>         secular equation on the computation tree.
*> \endverbatim
*>
*> \param[out] DIFL
*> \verbatim
*>          DIFL is DOUBLE PRECISION array, dimension ( LDU, NLVL ),
*>         where NLVL = floor(log_2 (N/SMLSIZ))).
*> \endverbatim
*>
*> \param[out] DIFR
*> \verbatim
*>          DIFR is DOUBLE PRECISION array,
*>                  dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and
*>                  dimension ( N ) if ICOMPQ = 0.
*>         If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1)
*>         record distances between singular values on the I-th
*>         level and singular values on the (I -1)-th level, and
*>         DIFR(1:N, 2 * I ) contains the normalizing factors for
*>         the right singular vector matrix. See DLASD8 for details.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array,
*>                  dimension ( LDU, NLVL ) if ICOMPQ = 1 and
*>                  dimension ( N ) if ICOMPQ = 0.
*>         The first K elements of Z(1, I) contain the components of
*>         the deflation-adjusted updating row vector for subproblems
*>         on the I-th level.
*> \endverbatim
*>
*> \param[out] POLES
*> \verbatim
*>          POLES is DOUBLE PRECISION array,
*>         dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced
*>         if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and
*>         POLES(1, 2*I) contain  the new and old singular values
*>         involved in the secular equations on the I-th level.
*> \endverbatim
*>
*> \param[out] GIVPTR
*> \verbatim
*>          GIVPTR is INTEGER array,
*>         dimension ( N ) if ICOMPQ = 1, and not referenced if
*>         ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records
*>         the number of Givens rotations performed on the I-th
*>         problem on the computation tree.
*> \endverbatim
*>
*> \param[out] GIVCOL
*> \verbatim
*>          GIVCOL is INTEGER array,
*>         dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not
*>         referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
*>         GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations
*>         of Givens rotations performed on the I-th level on the
*>         computation tree.
*> \endverbatim
*>
*> \param[in] LDGCOL
*> \verbatim
*>          LDGCOL is INTEGER, LDGCOL = > N.
*>         The leading dimension of arrays GIVCOL and PERM.
*> \endverbatim
*>
*> \param[out] PERM
*> \verbatim
*>          PERM is INTEGER array,
*>         dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not referenced
*>         if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records
*>         permutations done on the I-th level of the computation tree.
*> \endverbatim
*>
*> \param[out] GIVNUM
*> \verbatim
*>          GIVNUM is DOUBLE PRECISION array,
*>         dimension ( LDU,  2 * NLVL ) if ICOMPQ = 1, and not
*>         referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
*>         GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S-
*>         values of Givens rotations performed on the I-th level on
*>         the computation tree.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*>          C is DOUBLE PRECISION array,
*>         dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0.
*>         If ICOMPQ = 1 and the I-th subproblem is not square, on exit,
*>         C( I ) contains the C-value of a Givens rotation related to
*>         the right null space of the I-th subproblem.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension ( N ) if
*>         ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1
*>         and the I-th subproblem is not square, on exit, S( I )
*>         contains the S-value of a Givens rotation related to
*>         the right null space of the I-th subproblem.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension
*>         (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (7*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = 1, a singular value did not converge
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasda
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Huan Ren, Computer Science Division, University of
*>     California at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE DLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT,
     $                   K,
     $                   DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL,
     $                   PERM, GIVNUM, C, S, WORK, IWORK, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE
*     ..
*     .. Array Arguments ..
      INTEGER            GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
     $                   K( * ), PERM( LDGCOL, * )
      DOUBLE PRECISION   C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU, * ),
     $                   E( * ), GIVNUM( LDU, * ), POLES( LDU, * ),
     $                   S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ),
     $                   Z( LDU, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, I1, IC, IDXQ, IDXQI, IM1, INODE, ITEMP, IWK,
     $                   J, LF, LL, LVL, LVL2, M, NCC, ND, NDB1, NDIML,
     $                   NDIMR, NL, NLF, NLP1, NLVL, NR, NRF, NRP1, NRU,
     $                   NWORK1, NWORK2, SMLSZP, SQREI, VF, VFI, VL, VLI
      DOUBLE PRECISION   ALPHA, BETA
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLASD6, DLASDQ, DLASDT, DLASET,
     $                   XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
         INFO = -1
      ELSE IF( SMLSIZ.LT.3 ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
         INFO = -4
      ELSE IF( LDU.LT.( N+SQRE ) ) THEN
         INFO = -8
      ELSE IF( LDGCOL.LT.N ) THEN
         INFO = -17
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLASDA', -INFO )
         RETURN
      END IF
*
      M = N + SQRE
*
*     If the input matrix is too small, call DLASDQ to find the SVD.
*
      IF( N.LE.SMLSIZ ) THEN
         IF( ICOMPQ.EQ.0 ) THEN
            CALL DLASDQ( 'U', SQRE, N, 0, 0, 0, D, E, VT, LDU, U,
     $                   LDU,
     $                   U, LDU, WORK, INFO )
         ELSE
            CALL DLASDQ( 'U', SQRE, N, M, N, 0, D, E, VT, LDU, U,
     $                   LDU,
     $                   U, LDU, WORK, INFO )
         END IF
         RETURN
      END IF
*
*     Book-keeping and  set up the computation tree.
*
      INODE = 1
      NDIML = INODE + N
      NDIMR = NDIML + N
      IDXQ = NDIMR + N
      IWK = IDXQ + N
*
      NCC = 0
      NRU = 0
*
      SMLSZP = SMLSIZ + 1
      VF = 1
      VL = VF + M
      NWORK1 = VL + M
      NWORK2 = NWORK1 + SMLSZP*SMLSZP
*
      CALL DLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ),
     $             IWORK( NDIMR ), SMLSIZ )
*
*     for the nodes on bottom level of the tree, solve
*     their subproblems by DLASDQ.
*
      NDB1 = ( ND+1 ) / 2
      DO 30 I = NDB1, ND
*
*        IC : center row of each node
*        NL : number of rows of left  subproblem
*        NR : number of rows of right subproblem
*        NLF: starting row of the left   subproblem
*        NRF: starting row of the right  subproblem
*
         I1 = I - 1
         IC = IWORK( INODE+I1 )
         NL = IWORK( NDIML+I1 )
         NLP1 = NL + 1
         NR = IWORK( NDIMR+I1 )
         NLF = IC - NL
         NRF = IC + 1
         IDXQI = IDXQ + NLF - 2
         VFI = VF + NLF - 1
         VLI = VL + NLF - 1
         SQREI = 1
         IF( ICOMPQ.EQ.0 ) THEN
            CALL DLASET( 'A', NLP1, NLP1, ZERO, ONE, WORK( NWORK1 ),
     $                   SMLSZP )
            CALL DLASDQ( 'U', SQREI, NL, NLP1, NRU, NCC, D( NLF ),
     $                   E( NLF ), WORK( NWORK1 ), SMLSZP,
     $                   WORK( NWORK2 ), NL, WORK( NWORK2 ), NL,
     $                   WORK( NWORK2 ), INFO )
            ITEMP = NWORK1 + NL*SMLSZP
            CALL DCOPY( NLP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 )
            CALL DCOPY( NLP1, WORK( ITEMP ), 1, WORK( VLI ), 1 )
         ELSE
            CALL DLASET( 'A', NL, NL, ZERO, ONE, U( NLF, 1 ), LDU )
            CALL DLASET( 'A', NLP1, NLP1, ZERO, ONE, VT( NLF, 1 ),
     $                   LDU )
            CALL DLASDQ( 'U', SQREI, NL, NLP1, NL, NCC, D( NLF ),
     $                   E( NLF ), VT( NLF, 1 ), LDU, U( NLF, 1 ), LDU,
     $                   U( NLF, 1 ), LDU, WORK( NWORK1 ), INFO )
            CALL DCOPY( NLP1, VT( NLF, 1 ), 1, WORK( VFI ), 1 )
            CALL DCOPY( NLP1, VT( NLF, NLP1 ), 1, WORK( VLI ), 1 )
         END IF
         IF( INFO.NE.0 ) THEN
            RETURN
         END IF
         DO 10 J = 1, NL
            IWORK( IDXQI+J ) = J
   10    CONTINUE
         IF( ( I.EQ.ND ) .AND. ( SQRE.EQ.0 ) ) THEN
            SQREI = 0
         ELSE
            SQREI = 1
         END IF
         IDXQI = IDXQI + NLP1
         VFI = VFI + NLP1
         VLI = VLI + NLP1
         NRP1 = NR + SQREI
         IF( ICOMPQ.EQ.0 ) THEN
            CALL DLASET( 'A', NRP1, NRP1, ZERO, ONE, WORK( NWORK1 ),
     $                   SMLSZP )
            CALL DLASDQ( 'U', SQREI, NR, NRP1, NRU, NCC, D( NRF ),
     $                   E( NRF ), WORK( NWORK1 ), SMLSZP,
     $                   WORK( NWORK2 ), NR, WORK( NWORK2 ), NR,
     $                   WORK( NWORK2 ), INFO )
            ITEMP = NWORK1 + ( NRP1-1 )*SMLSZP
            CALL DCOPY( NRP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 )
            CALL DCOPY( NRP1, WORK( ITEMP ), 1, WORK( VLI ), 1 )
         ELSE
            CALL DLASET( 'A', NR, NR, ZERO, ONE, U( NRF, 1 ), LDU )
            CALL DLASET( 'A', NRP1, NRP1, ZERO, ONE, VT( NRF, 1 ),
     $                   LDU )
            CALL DLASDQ( 'U', SQREI, NR, NRP1, NR, NCC, D( NRF ),
     $                   E( NRF ), VT( NRF, 1 ), LDU, U( NRF, 1 ), LDU,
     $                   U( NRF, 1 ), LDU, WORK( NWORK1 ), INFO )
            CALL DCOPY( NRP1, VT( NRF, 1 ), 1, WORK( VFI ), 1 )
            CALL DCOPY( NRP1, VT( NRF, NRP1 ), 1, WORK( VLI ), 1 )
         END IF
         IF( INFO.NE.0 ) THEN
            RETURN
         END IF
         DO 20 J = 1, NR
            IWORK( IDXQI+J ) = J
   20    CONTINUE
   30 CONTINUE
*
*     Now conquer each subproblem bottom-up.
*
      J = 2**NLVL
      DO 50 LVL = NLVL, 1, -1
         LVL2 = LVL*2 - 1
*
*        Find the first node LF and last node LL on
*        the current level LVL.
*
         IF( LVL.EQ.1 ) THEN
            LF = 1
            LL = 1
         ELSE
            LF = 2**( LVL-1 )
            LL = 2*LF - 1
         END IF
         DO 40 I = LF, LL
            IM1 = I - 1
            IC = IWORK( INODE+IM1 )
            NL = IWORK( NDIML+IM1 )
            NR = IWORK( NDIMR+IM1 )
            NLF = IC - NL
            NRF = IC + 1
            IF( I.EQ.LL ) THEN
               SQREI = SQRE
            ELSE
               SQREI = 1
            END IF
            VFI = VF + NLF - 1
            VLI = VL + NLF - 1
            IDXQI = IDXQ + NLF - 1
            ALPHA = D( IC )
            BETA = E( IC )
            IF( ICOMPQ.EQ.0 ) THEN
               CALL DLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ),
     $                      WORK( VFI ), WORK( VLI ), ALPHA, BETA,
     $                      IWORK( IDXQI ), PERM, GIVPTR( 1 ), GIVCOL,
     $                      LDGCOL, GIVNUM, LDU, POLES, DIFL, DIFR, Z,
     $                      K( 1 ), C( 1 ), S( 1 ), WORK( NWORK1 ),
     $                      IWORK( IWK ), INFO )
            ELSE
               J = J - 1
               CALL DLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ),
     $                      WORK( VFI ), WORK( VLI ), ALPHA, BETA,
     $                      IWORK( IDXQI ), PERM( NLF, LVL ),
     $                      GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL,
     $                      GIVNUM( NLF, LVL2 ), LDU,
     $                      POLES( NLF, LVL2 ), DIFL( NLF, LVL ),
     $                      DIFR( NLF, LVL2 ), Z( NLF, LVL ), K( J ),
     $                      C( J ), S( J ), WORK( NWORK1 ),
     $                      IWORK( IWK ), INFO )
            END IF
            IF( INFO.NE.0 ) THEN
               RETURN
            END IF
   40    CONTINUE
   50 CONTINUE
*
      RETURN
*
*     End of DLASDA
*
      END
*> \brief \b DLASDQ computes the SVD of a real bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASDQ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasdq.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasdq.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasdq.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASDQ( UPLO, SQRE, N, NCVT, NRU, NCC, D, E, VT, LDVT,
*                          U, LDU, C, LDC, WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU, SQRE
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   C( LDC, * ), D( * ), E( * ), U( LDU, * ),
*      $                   VT( LDVT, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASDQ computes the singular value decomposition (SVD) of a real
*> (upper or lower) bidiagonal matrix with diagonal D and offdiagonal
*> E, accumulating the transformations if desired. Letting B denote
*> the input bidiagonal matrix, the algorithm computes orthogonal
*> matrices Q and P such that B = Q * S * P**T (P**T denotes the transpose
*> of P). The singular values S are overwritten on D.
*>
*> The input matrix U  is changed to U  * Q  if desired.
*> The input matrix VT is changed to P**T * VT if desired.
*> The input matrix C  is changed to Q**T * C  if desired.
*>
*> See "Computing  Small Singular Values of Bidiagonal Matrices With
*> Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan,
*> LAPACK Working Note #3, for a detailed description of the algorithm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>        On entry, UPLO specifies whether the input bidiagonal matrix
*>        is upper or lower bidiagonal, and whether it is square are
*>        not.
*>           UPLO = 'U' or 'u'   B is upper bidiagonal.
*>           UPLO = 'L' or 'l'   B is lower bidiagonal.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*>          SQRE is INTEGER
*>        = 0: then the input matrix is N-by-N.
*>        = 1: then the input matrix is N-by-(N+1) if UPLU = 'U' and
*>             (N+1)-by-N if UPLU = 'L'.
*>
*>        The bidiagonal matrix has
*>        N = NL + NR + 1 rows and
*>        M = N + SQRE >= N columns.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>        On entry, N specifies the number of rows and columns
*>        in the matrix. N must be at least 0.
*> \endverbatim
*>
*> \param[in] NCVT
*> \verbatim
*>          NCVT is INTEGER
*>        On entry, NCVT specifies the number of columns of
*>        the matrix VT. NCVT must be at least 0.
*> \endverbatim
*>
*> \param[in] NRU
*> \verbatim
*>          NRU is INTEGER
*>        On entry, NRU specifies the number of rows of
*>        the matrix U. NRU must be at least 0.
*> \endverbatim
*>
*> \param[in] NCC
*> \verbatim
*>          NCC is INTEGER
*>        On entry, NCC specifies the number of columns of
*>        the matrix C. NCC must be at least 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>        On entry, D contains the diagonal entries of the
*>        bidiagonal matrix whose SVD is desired. On normal exit,
*>        D contains the singular values in ascending order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array.
*>        dimension is (N-1) if SQRE = 0 and N if SQRE = 1.
*>        On entry, the entries of E contain the offdiagonal entries
*>        of the bidiagonal matrix whose SVD is desired. On normal
*>        exit, E will contain 0. If the algorithm does not converge,
*>        D and E will contain the diagonal and superdiagonal entries
*>        of a bidiagonal matrix orthogonally equivalent to the one
*>        given as input.
*> \endverbatim
*>
*> \param[in,out] VT
*> \verbatim
*>          VT is DOUBLE PRECISION array, dimension (LDVT, NCVT)
*>        On entry, contains a matrix which on exit has been
*>        premultiplied by P**T, dimension N-by-NCVT if SQRE = 0
*>        and (N+1)-by-NCVT if SQRE = 1 (not referenced if NCVT=0).
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*>          LDVT is INTEGER
*>        On entry, LDVT specifies the leading dimension of VT as
*>        declared in the calling (sub) program. LDVT must be at
*>        least 1. If NCVT is nonzero LDVT must also be at least N.
*> \endverbatim
*>
*> \param[in,out] U
*> \verbatim
*>          U is DOUBLE PRECISION array, dimension (LDU, N)
*>        On entry, contains a  matrix which on exit has been
*>        postmultiplied by Q, dimension NRU-by-N if SQRE = 0
*>        and NRU-by-(N+1) if SQRE = 1 (not referenced if NRU=0).
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>        On entry, LDU  specifies the leading dimension of U as
*>        declared in the calling (sub) program. LDU must be at
*>        least max( 1, NRU ) .
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC, NCC)
*>        On entry, contains an N-by-NCC matrix which on exit
*>        has been premultiplied by Q**T  dimension N-by-NCC if SQRE = 0
*>        and (N+1)-by-NCC if SQRE = 1 (not referenced if NCC=0).
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>        On entry, LDC  specifies the leading dimension of C as
*>        declared in the calling (sub) program. LDC must be at
*>        least 1. If NCC is nonzero, LDC must also be at least N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (4*N)
*>        Workspace. Only referenced if one of NCVT, NRU, or NCC is
*>        nonzero, and if N is at least 2.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>        On exit, a value of 0 indicates a successful exit.
*>        If INFO < 0, argument number -INFO is illegal.
*>        If INFO > 0, the algorithm did not converge, and INFO
*>        specifies how many superdiagonals did not converge.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasdq
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Huan Ren, Computer Science Division, University of
*>     California at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE DLASDQ( UPLO, SQRE, N, NCVT, NRU, NCC, D, E, VT,
     $                   LDVT,
     $                   U, LDU, C, LDC, WORK, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU, SQRE
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   C( LDC, * ), D( * ), E( * ), U( LDU, * ),
     $                   VT( LDVT, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ROTATE
      INTEGER            I, ISUB, IUPLO, J, NP1, SQRE1
      DOUBLE PRECISION   CS, R, SMIN, SN
*     ..
*     .. External Subroutines ..
      EXTERNAL           DBDSQR, DLARTG, DLASR, DSWAP,
     $                   XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IUPLO = 0
      IF( LSAME( UPLO, 'U' ) )
     $   IUPLO = 1
      IF( LSAME( UPLO, 'L' ) )
     $   IUPLO = 2
      IF( IUPLO.EQ.0 ) THEN
         INFO = -1
      ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( NCVT.LT.0 ) THEN
         INFO = -4
      ELSE IF( NRU.LT.0 ) THEN
         INFO = -5
      ELSE IF( NCC.LT.0 ) THEN
         INFO = -6
      ELSE IF( ( NCVT.EQ.0 .AND. LDVT.LT.1 ) .OR.
     $         ( NCVT.GT.0 .AND. LDVT.LT.MAX( 1, N ) ) ) THEN
         INFO = -10
      ELSE IF( LDU.LT.MAX( 1, NRU ) ) THEN
         INFO = -12
      ELSE IF( ( NCC.EQ.0 .AND. LDC.LT.1 ) .OR.
     $         ( NCC.GT.0 .AND. LDC.LT.MAX( 1, N ) ) ) THEN
         INFO = -14
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLASDQ', -INFO )
         RETURN
      END IF
      IF( N.EQ.0 )
     $   RETURN
*
*     ROTATE is true if any singular vectors desired, false otherwise
*
      ROTATE = ( NCVT.GT.0 ) .OR. ( NRU.GT.0 ) .OR. ( NCC.GT.0 )
      NP1 = N + 1
      SQRE1 = SQRE
*
*     If matrix non-square upper bidiagonal, rotate to be lower
*     bidiagonal.  The rotations are on the right.
*
      IF( ( IUPLO.EQ.1 ) .AND. ( SQRE1.EQ.1 ) ) THEN
         DO 10 I = 1, N - 1
            CALL DLARTG( D( I ), E( I ), CS, SN, R )
            D( I ) = R
            E( I ) = SN*D( I+1 )
            D( I+1 ) = CS*D( I+1 )
            IF( ROTATE ) THEN
               WORK( I ) = CS
               WORK( N+I ) = SN
            END IF
   10    CONTINUE
         CALL DLARTG( D( N ), E( N ), CS, SN, R )
         D( N ) = R
         E( N ) = ZERO
         IF( ROTATE ) THEN
            WORK( N ) = CS
            WORK( N+N ) = SN
         END IF
         IUPLO = 2
         SQRE1 = 0
*
*        Update singular vectors if desired.
*
         IF( NCVT.GT.0 )
     $      CALL DLASR( 'L', 'V', 'F', NP1, NCVT, WORK( 1 ),
     $                  WORK( NP1 ), VT, LDVT )
      END IF
*
*     If matrix lower bidiagonal, rotate to be upper bidiagonal
*     by applying Givens rotations on the left.
*
      IF( IUPLO.EQ.2 ) THEN
         DO 20 I = 1, N - 1
            CALL DLARTG( D( I ), E( I ), CS, SN, R )
            D( I ) = R
            E( I ) = SN*D( I+1 )
            D( I+1 ) = CS*D( I+1 )
            IF( ROTATE ) THEN
               WORK( I ) = CS
               WORK( N+I ) = SN
            END IF
   20    CONTINUE
*
*        If matrix (N+1)-by-N lower bidiagonal, one additional
*        rotation is needed.
*
         IF( SQRE1.EQ.1 ) THEN
            CALL DLARTG( D( N ), E( N ), CS, SN, R )
            D( N ) = R
            IF( ROTATE ) THEN
               WORK( N ) = CS
               WORK( N+N ) = SN
            END IF
         END IF
*
*        Update singular vectors if desired.
*
         IF( NRU.GT.0 ) THEN
            IF( SQRE1.EQ.0 ) THEN
               CALL DLASR( 'R', 'V', 'F', NRU, N, WORK( 1 ),
     $                     WORK( NP1 ), U, LDU )
            ELSE
               CALL DLASR( 'R', 'V', 'F', NRU, NP1, WORK( 1 ),
     $                     WORK( NP1 ), U, LDU )
            END IF
         END IF
         IF( NCC.GT.0 ) THEN
            IF( SQRE1.EQ.0 ) THEN
               CALL DLASR( 'L', 'V', 'F', N, NCC, WORK( 1 ),
     $                     WORK( NP1 ), C, LDC )
            ELSE
               CALL DLASR( 'L', 'V', 'F', NP1, NCC, WORK( 1 ),
     $                     WORK( NP1 ), C, LDC )
            END IF
         END IF
      END IF
*
*     Call DBDSQR to compute the SVD of the reduced real
*     N-by-N upper bidiagonal matrix.
*
      CALL DBDSQR( 'U', N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C,
     $             LDC, WORK, INFO )
*
*     Sort the singular values into ascending order (insertion sort on
*     singular values, but only one transposition per singular vector)
*
      DO 40 I = 1, N
*
*        Scan for smallest D(I).
*
         ISUB = I
         SMIN = D( I )
         DO 30 J = I + 1, N
            IF( D( J ).LT.SMIN ) THEN
               ISUB = J
               SMIN = D( J )
            END IF
   30    CONTINUE
         IF( ISUB.NE.I ) THEN
*
*           Swap singular values and vectors.
*
            D( ISUB ) = D( I )
            D( I ) = SMIN
            IF( NCVT.GT.0 )
     $         CALL DSWAP( NCVT, VT( ISUB, 1 ), LDVT, VT( I, 1 ),
     $                     LDVT )
            IF( NRU.GT.0 )
     $         CALL DSWAP( NRU, U( 1, ISUB ), 1, U( 1, I ), 1 )
            IF( NCC.GT.0 )
     $         CALL DSWAP( NCC, C( ISUB, 1 ), LDC, C( I, 1 ), LDC )
         END IF
   40 CONTINUE
*
      RETURN
*
*     End of DLASDQ
*
      END
*> \brief \b DLASDT creates a tree of subproblems for bidiagonal divide and conquer. Used by sbdsdc.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASDT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasdt.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasdt.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasdt.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASDT( N, LVL, ND, INODE, NDIML, NDIMR, MSUB )
*
*       .. Scalar Arguments ..
*       INTEGER            LVL, MSUB, N, ND
*       ..
*       .. Array Arguments ..
*       INTEGER            INODE( * ), NDIML( * ), NDIMR( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASDT creates a tree of subproblems for bidiagonal divide and
*> conquer.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          On entry, the number of diagonal elements of the
*>          bidiagonal matrix.
*> \endverbatim
*>
*> \param[out] LVL
*> \verbatim
*>          LVL is INTEGER
*>          On exit, the number of levels on the computation tree.
*> \endverbatim
*>
*> \param[out] ND
*> \verbatim
*>          ND is INTEGER
*>          On exit, the number of nodes on the tree.
*> \endverbatim
*>
*> \param[out] INODE
*> \verbatim
*>          INODE is INTEGER array, dimension ( N )
*>          On exit, centers of subproblems.
*> \endverbatim
*>
*> \param[out] NDIML
*> \verbatim
*>          NDIML is INTEGER array, dimension ( N )
*>          On exit, row dimensions of left children.
*> \endverbatim
*>
*> \param[out] NDIMR
*> \verbatim
*>          NDIMR is INTEGER array, dimension ( N )
*>          On exit, row dimensions of right children.
*> \endverbatim
*>
*> \param[in] MSUB
*> \verbatim
*>          MSUB is INTEGER
*>          On entry, the maximum row dimension each subproblem at the
*>          bottom of the tree can be of.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasdt
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Huan Ren, Computer Science Division, University of
*>     California at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE DLASDT( N, LVL, ND, INODE, NDIML, NDIMR, MSUB )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            LVL, MSUB, N, ND
*     ..
*     .. Array Arguments ..
      INTEGER            INODE( * ), NDIML( * ), NDIMR( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   TWO
      PARAMETER          ( TWO = 2.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, IL, IR, LLST, MAXN, NCRNT, NLVL
      DOUBLE PRECISION   TEMP
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE, INT, LOG, MAX
*     ..
*     .. Executable Statements ..
*
*     Find the number of levels on the tree.
*
      MAXN = MAX( 1, N )
      TEMP = LOG( DBLE( MAXN ) / DBLE( MSUB+1 ) ) / LOG( TWO )
      LVL = INT( TEMP ) + 1
*
      I = N / 2
      INODE( 1 ) = I + 1
      NDIML( 1 ) = I
      NDIMR( 1 ) = N - I - 1
      IL = 0
      IR = 1
      LLST = 1
      DO 20 NLVL = 1, LVL - 1
*
*        Constructing the tree at (NLVL+1)-st level. The number of
*        nodes created on this level is LLST * 2.
*
         DO 10 I = 0, LLST - 1
            IL = IL + 2
            IR = IR + 2
            NCRNT = LLST + I
            NDIML( IL ) = NDIML( NCRNT ) / 2
            NDIMR( IL ) = NDIML( NCRNT ) - NDIML( IL ) - 1
            INODE( IL ) = INODE( NCRNT ) - NDIMR( IL ) - 1
            NDIML( IR ) = NDIMR( NCRNT ) / 2
            NDIMR( IR ) = NDIMR( NCRNT ) - NDIML( IR ) - 1
            INODE( IR ) = INODE( NCRNT ) + NDIML( IR ) + 1
   10    CONTINUE
         LLST = LLST*2
   20 CONTINUE
      ND = LLST*2 - 1
*
      RETURN
*
*     End of DLASDT
*
      END
*> \brief \b DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASET + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaset.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaset.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaset.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASET( UPLO, M, N, ALPHA, BETA, A, LDA )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            LDA, M, N
*       DOUBLE PRECISION   ALPHA, BETA
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASET initializes an m-by-n matrix A to BETA on the diagonal and
*> ALPHA on the offdiagonals.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies the part of the matrix A to be set.
*>          = 'U':      Upper triangular part is set; the strictly lower
*>                      triangular part of A is not changed.
*>          = 'L':      Lower triangular part is set; the strictly upper
*>                      triangular part of A is not changed.
*>          Otherwise:  All of the matrix A is set.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*>          ALPHA is DOUBLE PRECISION
*>          The constant to which the offdiagonal elements are to be set.
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION
*>          The constant to which the diagonal elements are to be set.
*> \endverbatim
*>
*> \param[out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On exit, the leading m-by-n submatrix of A is set as follows:
*>
*>          if UPLO = 'U', A(i,j) = ALPHA, 1<=i<=j-1, 1<=j<=n,
*>          if UPLO = 'L', A(i,j) = ALPHA, j+1<=i<=m, 1<=j<=n,
*>          otherwise,     A(i,j) = ALPHA, 1<=i<=m, 1<=j<=n, i.ne.j,
*>
*>          and, for all UPLO, A(i,i) = BETA, 1<=i<=min(m,n).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laset
*
*  =====================================================================
      SUBROUTINE DLASET( UPLO, M, N, ALPHA, BETA, A, LDA )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            LDA, M, N
      DOUBLE PRECISION   ALPHA, BETA
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
* =====================================================================
*
*     .. Local Scalars ..
      INTEGER            I, J
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
      IF( LSAME( UPLO, 'U' ) ) THEN
*
*        Set the strictly upper triangular or trapezoidal part of the
*        array to ALPHA.
*
         DO 20 J = 2, N
            DO 10 I = 1, MIN( J-1, M )
               A( I, J ) = ALPHA
   10       CONTINUE
   20    CONTINUE
*
      ELSE IF( LSAME( UPLO, 'L' ) ) THEN
*
*        Set the strictly lower triangular or trapezoidal part of the
*        array to ALPHA.
*
         DO 40 J = 1, MIN( M, N )
            DO 30 I = J + 1, M
               A( I, J ) = ALPHA
   30       CONTINUE
   40    CONTINUE
*
      ELSE
*
*        Set the leading m-by-n submatrix to ALPHA.
*
         DO 60 J = 1, N
            DO 50 I = 1, M
               A( I, J ) = ALPHA
   50       CONTINUE
   60    CONTINUE
      END IF
*
*     Set the first min(M,N) diagonal elements to BETA.
*
      DO 70 I = 1, MIN( M, N )
         A( I, I ) = BETA
   70 CONTINUE
*
      RETURN
*
*     End of DLASET
*
      END
*> \brief \b DLASQ1 computes the singular values of a real square bidiagonal matrix. Used by sbdsqr.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ1 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq1.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq1.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq1.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASQ1( N, D, E, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), E( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASQ1 computes the singular values of a real N-by-N bidiagonal
*> matrix with diagonal D and off-diagonal E. The singular values
*> are computed to high relative accuracy, in the absence of
*> denormalization, underflow and overflow. The algorithm was first
*> presented in
*>
*> "Accurate singular values and differential qd algorithms" by K. V.
*> Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230,
*> 1994,
*>
*> and the present implementation is described in "An implementation of
*> the dqds Algorithm (Positive Case)", LAPACK Working Note.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>        The number of rows and columns in the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>        On entry, D contains the diagonal elements of the
*>        bidiagonal matrix whose SVD is desired. On normal exit,
*>        D contains the singular values in decreasing order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N)
*>        On entry, elements E(1:N-1) contain the off-diagonal elements
*>        of the bidiagonal matrix whose SVD is desired.
*>        On exit, E is overwritten.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>        = 0: successful exit
*>        < 0: if INFO = -i, the i-th argument had an illegal value
*>        > 0: the algorithm failed
*>             = 1, a split was marked by a positive value in E
*>             = 2, current block of Z not diagonalized after 100*N
*>                  iterations (in inner while loop)  On exit D and E
*>                  represent a matrix with the same singular values
*>                  which the calling subroutine could use to finish the
*>                  computation, or even feed back into DLASQ1
*>             = 3, termination criterion of outer while loop not met
*>                  (program created more than N unreduced blocks)
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasq1
*
*  =====================================================================
      SUBROUTINE DLASQ1( N, D, E, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), E( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, IINFO
      DOUBLE PRECISION   EPS, SCALE, SAFMIN, SIGMN, SIGMX
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLAS2, DLASCL, DLASQ2, DLASRT,
     $                   XERBLA
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      IF( N.LT.0 ) THEN
         INFO = -1
         CALL XERBLA( 'DLASQ1', -INFO )
         RETURN
      ELSE IF( N.EQ.0 ) THEN
         RETURN
      ELSE IF( N.EQ.1 ) THEN
         D( 1 ) = ABS( D( 1 ) )
         RETURN
      ELSE IF( N.EQ.2 ) THEN
         CALL DLAS2( D( 1 ), E( 1 ), D( 2 ), SIGMN, SIGMX )
         D( 1 ) = SIGMX
         D( 2 ) = SIGMN
         RETURN
      END IF
*
*     Estimate the largest singular value.
*
      SIGMX = ZERO
      DO 10 I = 1, N - 1
         D( I ) = ABS( D( I ) )
         SIGMX = MAX( SIGMX, ABS( E( I ) ) )
   10 CONTINUE
      D( N ) = ABS( D( N ) )
*
*     Early return if SIGMX is zero (matrix is already diagonal).
*
      IF( SIGMX.EQ.ZERO ) THEN
         CALL DLASRT( 'D', N, D, IINFO )
         RETURN
      END IF
*
      DO 20 I = 1, N
         SIGMX = MAX( SIGMX, D( I ) )
   20 CONTINUE
*
*     Copy D and E into WORK (in the Z format) and scale (squaring the
*     input data makes scaling by a power of the radix pointless).
*
      EPS = DLAMCH( 'Precision' )
      SAFMIN = DLAMCH( 'Safe minimum' )
      SCALE = SQRT( EPS / SAFMIN )
      CALL DCOPY( N, D, 1, WORK( 1 ), 2 )
      CALL DCOPY( N-1, E, 1, WORK( 2 ), 2 )
      CALL DLASCL( 'G', 0, 0, SIGMX, SCALE, 2*N-1, 1, WORK, 2*N-1,
     $             IINFO )
*
*     Compute the q's and e's.
*
      DO 30 I = 1, 2*N - 1
         WORK( I ) = WORK( I )**2
   30 CONTINUE
      WORK( 2*N ) = ZERO
*
      CALL DLASQ2( N, WORK, INFO )
*
      IF( INFO.EQ.0 ) THEN
         DO 40 I = 1, N
            D( I ) = SQRT( WORK( I ) )
   40    CONTINUE
         CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, D, N, IINFO )
      ELSE IF( INFO.EQ.2 ) THEN
*
*     Maximum number of iterations exceeded.  Move data from WORK
*     into D and E so the calling subroutine can try to finish
*
         DO I = 1, N
            D( I ) = SQRT( WORK( 2*I-1 ) )
            E( I ) = SQRT( WORK( 2*I ) )
         END DO
         CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, D, N, IINFO )
         CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, E, N, IINFO )
      END IF
*
      RETURN
*
*     End of DLASQ1
*
      END
*> \brief \b DLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASQ2( N, Z, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASQ2 computes all the eigenvalues of the symmetric positive
*> definite tridiagonal matrix associated with the qd array Z to high
*> relative accuracy are computed to high relative accuracy, in the
*> absence of denormalization, underflow and overflow.
*>
*> To see the relation of Z to the tridiagonal matrix, let L be a
*> unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and
*> let U be an upper bidiagonal matrix with 1's above and diagonal
*> Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the
*> symmetric tridiagonal to which it is similar.
*>
*> Note : DLASQ2 defines a logical variable, IEEE, which is true
*> on machines which follow ieee-754 floating-point standard in their
*> handling of infinities and NaNs, and false otherwise. This variable
*> is passed to DLASQ3.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>        The number of rows and columns in the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension ( 4*N )
*>        On entry Z holds the qd array. On exit, entries 1 to N hold
*>        the eigenvalues in decreasing order, Z( 2*N+1 ) holds the
*>        trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If
*>        N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 )
*>        holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of
*>        shifts that failed.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>        = 0: successful exit
*>        < 0: if the i-th argument is a scalar and had an illegal
*>             value, then INFO = -i, if the i-th argument is an
*>             array and the j-entry had an illegal value, then
*>             INFO = -(i*100+j)
*>        > 0: the algorithm failed
*>              = 1, a split was marked by a positive value in E
*>              = 2, current block of Z not diagonalized after 100*N
*>                   iterations (in inner while loop).  On exit Z holds
*>                   a qd array with the same eigenvalues as the given Z.
*>              = 3, termination criterion of outer while loop not met
*>                   (program created more than N unreduced blocks)
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasq2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  Local Variables: I0:N0 defines a current unreduced segment of Z.
*>  The shifts are accumulated in SIGMA. Iteration count is in ITER.
*>  Ping-pong is controlled by PP (alternates between 0 and 1).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLASQ2( N, Z, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   Z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   CBIAS
      PARAMETER          ( CBIAS = 1.50D0 )
      DOUBLE PRECISION   ZERO, HALF, ONE, TWO, FOUR, HUNDRD
      PARAMETER          ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0,
     $                     TWO = 2.0D0, FOUR = 4.0D0, HUNDRD = 100.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            IEEE
      INTEGER            I0, I1, I4, IINFO, IPN4, ITER, IWHILA, IWHILB,
     $                   K, KMIN, N0, N1, NBIG, NDIV, NFAIL, PP, SPLT,
     $                   TTYPE
      DOUBLE PRECISION   D, DEE, DEEMIN, DESIG, DMIN, DMIN1, DMIN2, DN,
     $                   DN1, DN2, E, EMAX, EMIN, EPS, G, OLDEMN, QMAX,
     $                   QMIN, S, SAFMIN, SIGMA, T, TAU, TEMP, TOL,
     $                   TOL2, TRACE, ZMAX, TEMPE, TEMPQ
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASQ3, DLASRT, XERBLA
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH, ILAENV
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments.
*     (in case DLASQ2 is not called by DLASQ1)
*
      INFO = 0
      EPS = DLAMCH( 'Precision' )
      SAFMIN = DLAMCH( 'Safe minimum' )
      TOL = EPS*HUNDRD
      TOL2 = TOL**2
*
      IF( N.LT.0 ) THEN
         INFO = -1
         CALL XERBLA( 'DLASQ2', 1 )
         RETURN
      ELSE IF( N.EQ.0 ) THEN
         RETURN
      ELSE IF( N.EQ.1 ) THEN
*
*        1-by-1 case.
*
         IF( Z( 1 ).LT.ZERO ) THEN
            INFO = -201
            CALL XERBLA( 'DLASQ2', 2 )
         END IF
         RETURN
      ELSE IF( N.EQ.2 ) THEN
*
*        2-by-2 case.
*
         IF( Z( 1 ).LT.ZERO ) THEN
            INFO = -201
            CALL XERBLA( 'DLASQ2', 2 )
            RETURN
         ELSE IF( Z( 2 ).LT.ZERO ) THEN
            INFO = -202
            CALL XERBLA( 'DLASQ2', 2 )
            RETURN
         ELSE IF( Z( 3 ).LT.ZERO ) THEN
           INFO = -203
           CALL XERBLA( 'DLASQ2', 2 )
           RETURN
         ELSE IF( Z( 3 ).GT.Z( 1 ) ) THEN
            D = Z( 3 )
            Z( 3 ) = Z( 1 )
            Z( 1 ) = D
         END IF
         Z( 5 ) = Z( 1 ) + Z( 2 ) + Z( 3 )
         IF( Z( 2 ).GT.Z( 3 )*TOL2 ) THEN
            T = HALF*( ( Z( 1 )-Z( 3 ) )+Z( 2 ) )
            S = Z( 3 )*( Z( 2 ) / T )
            IF( S.LE.T ) THEN
               S = Z( 3 )*( Z( 2 ) / ( T*( ONE+SQRT( ONE+S / T ) ) ) )
            ELSE
               S = Z( 3 )*( Z( 2 ) / ( T+SQRT( T )*SQRT( T+S ) ) )
            END IF
            T = Z( 1 ) + ( S+Z( 2 ) )
            Z( 3 ) = Z( 3 )*( Z( 1 ) / T )
            Z( 1 ) = T
         END IF
         Z( 2 ) = Z( 3 )
         Z( 6 ) = Z( 2 ) + Z( 1 )
         RETURN
      END IF
*
*     Check for negative data and compute sums of q's and e's.
*
      Z( 2*N ) = ZERO
      EMIN = Z( 2 )
      QMAX = ZERO
      ZMAX = ZERO
      D = ZERO
      E = ZERO
*
      DO 10 K = 1, 2*( N-1 ), 2
         IF( Z( K ).LT.ZERO ) THEN
            INFO = -( 200+K )
            CALL XERBLA( 'DLASQ2', 2 )
            RETURN
         ELSE IF( Z( K+1 ).LT.ZERO ) THEN
            INFO = -( 200+K+1 )
            CALL XERBLA( 'DLASQ2', 2 )
            RETURN
         END IF
         D = D + Z( K )
         E = E + Z( K+1 )
         QMAX = MAX( QMAX, Z( K ) )
         EMIN = MIN( EMIN, Z( K+1 ) )
         ZMAX = MAX( QMAX, ZMAX, Z( K+1 ) )
   10 CONTINUE
      IF( Z( 2*N-1 ).LT.ZERO ) THEN
         INFO = -( 200+2*N-1 )
         CALL XERBLA( 'DLASQ2', 2 )
         RETURN
      END IF
      D = D + Z( 2*N-1 )
      QMAX = MAX( QMAX, Z( 2*N-1 ) )
      ZMAX = MAX( QMAX, ZMAX )
*
*     Check for diagonality.
*
      IF( E.EQ.ZERO ) THEN
         DO 20 K = 2, N
            Z( K ) = Z( 2*K-1 )
   20    CONTINUE
         CALL DLASRT( 'D', N, Z, IINFO )
         Z( 2*N-1 ) = D
         RETURN
      END IF
*
      TRACE = D + E
*
*     Check for zero data.
*
      IF( TRACE.EQ.ZERO ) THEN
         Z( 2*N-1 ) = ZERO
         RETURN
      END IF
*
*     Check whether the machine is IEEE conformable.
*
      IEEE = ( ILAENV( 10, 'DLASQ2', 'N', 1, 2, 3, 4 ).EQ.1 )
*
*     Rearrange data for locality: Z=(q1,qq1,e1,ee1,q2,qq2,e2,ee2,...).
*
      DO 30 K = 2*N, 2, -2
         Z( 2*K ) = ZERO
         Z( 2*K-1 ) = Z( K )
         Z( 2*K-2 ) = ZERO
         Z( 2*K-3 ) = Z( K-1 )
   30 CONTINUE
*
      I0 = 1
      N0 = N
*
*     Reverse the qd-array, if warranted.
*
      IF( CBIAS*Z( 4*I0-3 ).LT.Z( 4*N0-3 ) ) THEN
         IPN4 = 4*( I0+N0 )
         DO 40 I4 = 4*I0, 2*( I0+N0-1 ), 4
            TEMP = Z( I4-3 )
            Z( I4-3 ) = Z( IPN4-I4-3 )
            Z( IPN4-I4-3 ) = TEMP
            TEMP = Z( I4-1 )
            Z( I4-1 ) = Z( IPN4-I4-5 )
            Z( IPN4-I4-5 ) = TEMP
   40    CONTINUE
      END IF
*
*     Initial split checking via dqd and Li's test.
*
      PP = 0
*
      DO 80 K = 1, 2
*
         D = Z( 4*N0+PP-3 )
         DO 50 I4 = 4*( N0-1 ) + PP, 4*I0 + PP, -4
            IF( Z( I4-1 ).LE.TOL2*D ) THEN
               Z( I4-1 ) = -ZERO
               D = Z( I4-3 )
            ELSE
               D = Z( I4-3 )*( D / ( D+Z( I4-1 ) ) )
            END IF
   50    CONTINUE
*
*        dqd maps Z to ZZ plus Li's test.
*
         EMIN = Z( 4*I0+PP+1 )
         D = Z( 4*I0+PP-3 )
         DO 60 I4 = 4*I0 + PP, 4*( N0-1 ) + PP, 4
            Z( I4-2*PP-2 ) = D + Z( I4-1 )
            IF( Z( I4-1 ).LE.TOL2*D ) THEN
               Z( I4-1 ) = -ZERO
               Z( I4-2*PP-2 ) = D
               Z( I4-2*PP ) = ZERO
               D = Z( I4+1 )
            ELSE IF( SAFMIN*Z( I4+1 ).LT.Z( I4-2*PP-2 ) .AND.
     $               SAFMIN*Z( I4-2*PP-2 ).LT.Z( I4+1 ) ) THEN
               TEMP = Z( I4+1 ) / Z( I4-2*PP-2 )
               Z( I4-2*PP ) = Z( I4-1 )*TEMP
               D = D*TEMP
            ELSE
               Z( I4-2*PP ) = Z( I4+1 )*( Z( I4-1 ) / Z( I4-2*PP-2 ) )
               D = Z( I4+1 )*( D / Z( I4-2*PP-2 ) )
            END IF
            EMIN = MIN( EMIN, Z( I4-2*PP ) )
   60    CONTINUE
         Z( 4*N0-PP-2 ) = D
*
*        Now find qmax.
*
         QMAX = Z( 4*I0-PP-2 )
         DO 70 I4 = 4*I0 - PP + 2, 4*N0 - PP - 2, 4
            QMAX = MAX( QMAX, Z( I4 ) )
   70    CONTINUE
*
*        Prepare for the next iteration on K.
*
         PP = 1 - PP
   80 CONTINUE
*
*     Initialise variables to pass to DLASQ3.
*
      TTYPE = 0
      DMIN1 = ZERO
      DMIN2 = ZERO
      DN    = ZERO
      DN1   = ZERO
      DN2   = ZERO
      G     = ZERO
      TAU   = ZERO
*
      ITER = 2
      NFAIL = 0
      NDIV = 2*( N0-I0 )
*
      DO 160 IWHILA = 1, N + 1
         IF( N0.LT.1 )
     $      GO TO 170
*
*        While array unfinished do
*
*        E(N0) holds the value of SIGMA when submatrix in I0:N0
*        splits from the rest of the array, but is negated.
*
         DESIG = ZERO
         IF( N0.EQ.N ) THEN
            SIGMA = ZERO
         ELSE
            SIGMA = -Z( 4*N0-1 )
         END IF
         IF( SIGMA.LT.ZERO ) THEN
            INFO = 1
            RETURN
         END IF
*
*        Find last unreduced submatrix's top index I0, find QMAX and
*        EMIN. Find Gershgorin-type bound if Q's much greater than E's.
*
         EMAX = ZERO
         IF( N0.GT.I0 ) THEN
            EMIN = ABS( Z( 4*N0-5 ) )
         ELSE
            EMIN = ZERO
         END IF
         QMIN = Z( 4*N0-3 )
         QMAX = QMIN
         DO 90 I4 = 4*N0, 8, -4
            IF( Z( I4-5 ).LE.ZERO )
     $         GO TO 100
            IF( QMIN.GE.FOUR*EMAX ) THEN
               QMIN = MIN( QMIN, Z( I4-3 ) )
               EMAX = MAX( EMAX, Z( I4-5 ) )
            END IF
            QMAX = MAX( QMAX, Z( I4-7 )+Z( I4-5 ) )
            EMIN = MIN( EMIN, Z( I4-5 ) )
   90    CONTINUE
         I4 = 4
*
  100    CONTINUE
         I0 = I4 / 4
         PP = 0
*
         IF( N0-I0.GT.1 ) THEN
            DEE = Z( 4*I0-3 )
            DEEMIN = DEE
            KMIN = I0
            DO 110 I4 = 4*I0+1, 4*N0-3, 4
               DEE = Z( I4 )*( DEE /( DEE+Z( I4-2 ) ) )
               IF( DEE.LE.DEEMIN ) THEN
                  DEEMIN = DEE
                  KMIN = ( I4+3 )/4
               END IF
  110       CONTINUE
            IF( (KMIN-I0)*2.LT.N0-KMIN .AND.
     $         DEEMIN.LE.HALF*Z(4*N0-3) ) THEN
               IPN4 = 4*( I0+N0 )
               PP = 2
               DO 120 I4 = 4*I0, 2*( I0+N0-1 ), 4
                  TEMP = Z( I4-3 )
                  Z( I4-3 ) = Z( IPN4-I4-3 )
                  Z( IPN4-I4-3 ) = TEMP
                  TEMP = Z( I4-2 )
                  Z( I4-2 ) = Z( IPN4-I4-2 )
                  Z( IPN4-I4-2 ) = TEMP
                  TEMP = Z( I4-1 )
                  Z( I4-1 ) = Z( IPN4-I4-5 )
                  Z( IPN4-I4-5 ) = TEMP
                  TEMP = Z( I4 )
                  Z( I4 ) = Z( IPN4-I4-4 )
                  Z( IPN4-I4-4 ) = TEMP
  120          CONTINUE
            END IF
         END IF
*
*        Put -(initial shift) into DMIN.
*
         DMIN = -MAX( ZERO, QMIN-TWO*SQRT( QMIN )*SQRT( EMAX ) )
*
*        Now I0:N0 is unreduced.
*        PP = 0 for ping, PP = 1 for pong.
*        PP = 2 indicates that flipping was applied to the Z array and
*               and that the tests for deflation upon entry in DLASQ3
*               should not be performed.
*
         NBIG = 100*( N0-I0+1 )
         DO 140 IWHILB = 1, NBIG
            IF( I0.GT.N0 )
     $         GO TO 150
*
*           While submatrix unfinished take a good dqds step.
*
            CALL DLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX,
     $                   NFAIL,
     $                   ITER, NDIV, IEEE, TTYPE, DMIN1, DMIN2, DN, DN1,
     $                   DN2, G, TAU )
*
            PP = 1 - PP
*
*           When EMIN is very small check for splits.
*
            IF( PP.EQ.0 .AND. N0-I0.GE.3 ) THEN
               IF( Z( 4*N0 ).LE.TOL2*QMAX .OR.
     $             Z( 4*N0-1 ).LE.TOL2*SIGMA ) THEN
                  SPLT = I0 - 1
                  QMAX = Z( 4*I0-3 )
                  EMIN = Z( 4*I0-1 )
                  OLDEMN = Z( 4*I0 )
                  DO 130 I4 = 4*I0, 4*( N0-3 ), 4
                     IF( Z( I4 ).LE.TOL2*Z( I4-3 ) .OR.
     $                   Z( I4-1 ).LE.TOL2*SIGMA ) THEN
                        Z( I4-1 ) = -SIGMA
                        SPLT = I4 / 4
                        QMAX = ZERO
                        EMIN = Z( I4+3 )
                        OLDEMN = Z( I4+4 )
                     ELSE
                        QMAX = MAX( QMAX, Z( I4+1 ) )
                        EMIN = MIN( EMIN, Z( I4-1 ) )
                        OLDEMN = MIN( OLDEMN, Z( I4 ) )
                     END IF
  130             CONTINUE
                  Z( 4*N0-1 ) = EMIN
                  Z( 4*N0 ) = OLDEMN
                  I0 = SPLT + 1
               END IF
            END IF
*
  140    CONTINUE
*
         INFO = 2
*
*        Maximum number of iterations exceeded, restore the shift
*        SIGMA and place the new d's and e's in a qd array.
*        This might need to be done for several blocks
*
         I1 = I0
         N1 = N0
 145     CONTINUE
         TEMPQ = Z( 4*I0-3 )
         Z( 4*I0-3 ) = Z( 4*I0-3 ) + SIGMA
         DO K = I0+1, N0
            TEMPE = Z( 4*K-5 )
            Z( 4*K-5 ) = Z( 4*K-5 ) * (TEMPQ / Z( 4*K-7 ))
            TEMPQ = Z( 4*K-3 )
            Z( 4*K-3 ) = Z( 4*K-3 ) + SIGMA + TEMPE - Z( 4*K-5 )
         END DO
*
*        Prepare to do this on the previous block if there is one
*
         IF( I1.GT.1 ) THEN
            N1 = I1-1
            DO WHILE( ( I1.GE.2 ) .AND. ( Z(4*I1-5).GE.ZERO ) )
               I1 = I1 - 1
            END DO
            SIGMA = -Z(4*N1-1)
            GO TO 145
         END IF

         DO K = 1, N
            Z( 2*K-1 ) = Z( 4*K-3 )
*
*        Only the block 1..N0 is unfinished.  The rest of the e's
*        must be essentially zero, although sometimes other data
*        has been stored in them.
*
            IF( K.LT.N0 ) THEN
               Z( 2*K ) = Z( 4*K-1 )
            ELSE
               Z( 2*K ) = 0
            END IF
         END DO
         RETURN
*
*        end IWHILB
*
  150    CONTINUE
*
  160 CONTINUE
*
      INFO = 3
      RETURN
*
*     end IWHILA
*
  170 CONTINUE
*
*     Move q's to the front.
*
      DO 180 K = 2, N
         Z( K ) = Z( 4*K-3 )
  180 CONTINUE
*
*     Sort and compute sum of eigenvalues.
*
      CALL DLASRT( 'D', N, Z, IINFO )
*
      E = ZERO
      DO 190 K = N, 1, -1
         E = E + Z( K )
  190 CONTINUE
*
*     Store trace, sum(eigenvalues) and information on performance.
*
      Z( 2*N+1 ) = TRACE
      Z( 2*N+2 ) = E
      Z( 2*N+3 ) = DBLE( ITER )
      Z( 2*N+4 ) = DBLE( NDIV ) / DBLE( N**2 )
      Z( 2*N+5 ) = HUNDRD*NFAIL / DBLE( ITER )
      RETURN
*
*     End of DLASQ2
*
      END
*> \brief \b DLASQ3 checks for deflation, computes a shift and calls dqds. Used by sbdsqr.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq3.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX, NFAIL,
*                          ITER, NDIV, IEEE, TTYPE, DMIN1, DMIN2, DN, DN1,
*                          DN2, G, TAU )
*
*       .. Scalar Arguments ..
*       LOGICAL            IEEE
*       INTEGER            I0, ITER, N0, NDIV, NFAIL, PP
*       DOUBLE PRECISION   DESIG, DMIN, DMIN1, DMIN2, DN, DN1, DN2, G,
*      $                   QMAX, SIGMA, TAU
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASQ3 checks for deflation, computes a shift (TAU) and calls dqds.
*> In case of failure it changes shifts, and tries again until output
*> is positive.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] I0
*> \verbatim
*>          I0 is INTEGER
*>         First index.
*> \endverbatim
*>
*> \param[in,out] N0
*> \verbatim
*>          N0 is INTEGER
*>         Last index.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension ( 4*N0 )
*>         Z holds the qd array.
*> \endverbatim
*>
*> \param[in,out] PP
*> \verbatim
*>          PP is INTEGER
*>         PP=0 for ping, PP=1 for pong.
*>         PP=2 indicates that flipping was applied to the Z array
*>         and that the initial tests for deflation should not be
*>         performed.
*> \endverbatim
*>
*> \param[out] DMIN
*> \verbatim
*>          DMIN is DOUBLE PRECISION
*>         Minimum value of d.
*> \endverbatim
*>
*> \param[out] SIGMA
*> \verbatim
*>          SIGMA is DOUBLE PRECISION
*>         Sum of shifts used in current segment.
*> \endverbatim
*>
*> \param[in,out] DESIG
*> \verbatim
*>          DESIG is DOUBLE PRECISION
*>         Lower order part of SIGMA
*> \endverbatim
*>
*> \param[in] QMAX
*> \verbatim
*>          QMAX is DOUBLE PRECISION
*>         Maximum value of q.
*> \endverbatim
*>
*> \param[in,out] NFAIL
*> \verbatim
*>          NFAIL is INTEGER
*>         Increment NFAIL by 1 each time the shift was too big.
*> \endverbatim
*>
*> \param[in,out] ITER
*> \verbatim
*>          ITER is INTEGER
*>         Increment ITER by 1 for each iteration.
*> \endverbatim
*>
*> \param[in,out] NDIV
*> \verbatim
*>          NDIV is INTEGER
*>         Increment NDIV by 1 for each division.
*> \endverbatim
*>
*> \param[in] IEEE
*> \verbatim
*>          IEEE is LOGICAL
*>         Flag for IEEE or non IEEE arithmetic (passed to DLASQ5).
*> \endverbatim
*>
*> \param[in,out] TTYPE
*> \verbatim
*>          TTYPE is INTEGER
*>         Shift type.
*> \endverbatim
*>
*> \param[in,out] DMIN1
*> \verbatim
*>          DMIN1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] DMIN2
*> \verbatim
*>          DMIN2 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] DN
*> \verbatim
*>          DN is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] DN1
*> \verbatim
*>          DN1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] DN2
*> \verbatim
*>          DN2 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] G
*> \verbatim
*>          G is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION
*>
*>         These are passed as arguments in order to save their values
*>         between calls to DLASQ3.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasq3
*
*  =====================================================================
      SUBROUTINE DLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX,
     $                   NFAIL,
     $                   ITER, NDIV, IEEE, TTYPE, DMIN1, DMIN2, DN, DN1,
     $                   DN2, G, TAU )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      LOGICAL            IEEE
      INTEGER            I0, ITER, N0, NDIV, NFAIL, PP
      DOUBLE PRECISION   DESIG, DMIN, DMIN1, DMIN2, DN, DN1, DN2, G,
     $                   QMAX, SIGMA, TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   Z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   CBIAS
      PARAMETER          ( CBIAS = 1.50D0 )
      DOUBLE PRECISION   ZERO, QURTR, HALF, ONE, TWO, HUNDRD
      PARAMETER          ( ZERO = 0.0D0, QURTR = 0.250D0, HALF = 0.5D0,
     $                     ONE = 1.0D0, TWO = 2.0D0, HUNDRD = 100.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            IPN4, J4, N0IN, NN, TTYPE
      DOUBLE PRECISION   EPS, S, T, TEMP, TOL, TOL2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASQ4, DLASQ5, DLASQ6
*     ..
*     .. External Function ..
      DOUBLE PRECISION   DLAMCH
      LOGICAL            DISNAN
      EXTERNAL           DISNAN, DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
      N0IN = N0
      EPS = DLAMCH( 'Precision' )
      TOL = EPS*HUNDRD
      TOL2 = TOL**2
*
*     Check for deflation.
*
   10 CONTINUE
*
      IF( N0.LT.I0 )
     $   RETURN
      IF( N0.EQ.I0 )
     $   GO TO 20
      NN = 4*N0 + PP
      IF( N0.EQ.( I0+1 ) )
     $   GO TO 40
*
*     Check whether E(N0-1) is negligible, 1 eigenvalue.
*
      IF( Z( NN-5 ).GT.TOL2*( SIGMA+Z( NN-3 ) ) .AND.
     $    Z( NN-2*PP-4 ).GT.TOL2*Z( NN-7 ) )
     $   GO TO 30
*
   20 CONTINUE
*
      Z( 4*N0-3 ) = Z( 4*N0+PP-3 ) + SIGMA
      N0 = N0 - 1
      GO TO 10
*
*     Check  whether E(N0-2) is negligible, 2 eigenvalues.
*
   30 CONTINUE
*
      IF( Z( NN-9 ).GT.TOL2*SIGMA .AND.
     $    Z( NN-2*PP-8 ).GT.TOL2*Z( NN-11 ) )
     $   GO TO 50
*
   40 CONTINUE
*
      IF( Z( NN-3 ).GT.Z( NN-7 ) ) THEN
         S = Z( NN-3 )
         Z( NN-3 ) = Z( NN-7 )
         Z( NN-7 ) = S
      END IF
      T = HALF*( ( Z( NN-7 )-Z( NN-3 ) )+Z( NN-5 ) )
      IF( Z( NN-5 ).GT.Z( NN-3 )*TOL2.AND.T.NE.ZERO ) THEN
         S = Z( NN-3 )*( Z( NN-5 ) / T )
         IF( S.LE.T ) THEN
            S = Z( NN-3 )*( Z( NN-5 ) /
     $          ( T*( ONE+SQRT( ONE+S / T ) ) ) )
         ELSE
            S = Z( NN-3 )*( Z( NN-5 ) / ( T+SQRT( T )*SQRT( T+S ) ) )
         END IF
         T = Z( NN-7 ) + ( S+Z( NN-5 ) )
         Z( NN-3 ) = Z( NN-3 )*( Z( NN-7 ) / T )
         Z( NN-7 ) = T
      END IF
      Z( 4*N0-7 ) = Z( NN-7 ) + SIGMA
      Z( 4*N0-3 ) = Z( NN-3 ) + SIGMA
      N0 = N0 - 2
      GO TO 10
*
   50 CONTINUE
      IF( PP.EQ.2 )
     $   PP = 0
*
*     Reverse the qd-array, if warranted.
*
      IF( DMIN.LE.ZERO .OR. N0.LT.N0IN ) THEN
         IF( CBIAS*Z( 4*I0+PP-3 ).LT.Z( 4*N0+PP-3 ) ) THEN
            IPN4 = 4*( I0+N0 )
            DO 60 J4 = 4*I0, 2*( I0+N0-1 ), 4
               TEMP = Z( J4-3 )
               Z( J4-3 ) = Z( IPN4-J4-3 )
               Z( IPN4-J4-3 ) = TEMP
               TEMP = Z( J4-2 )
               Z( J4-2 ) = Z( IPN4-J4-2 )
               Z( IPN4-J4-2 ) = TEMP
               TEMP = Z( J4-1 )
               Z( J4-1 ) = Z( IPN4-J4-5 )
               Z( IPN4-J4-5 ) = TEMP
               TEMP = Z( J4 )
               Z( J4 ) = Z( IPN4-J4-4 )
               Z( IPN4-J4-4 ) = TEMP
   60       CONTINUE
            IF( N0-I0.LE.4 ) THEN
               Z( 4*N0+PP-1 ) = Z( 4*I0+PP-1 )
               Z( 4*N0-PP ) = Z( 4*I0-PP )
            END IF
            DMIN2 = MIN( DMIN2, Z( 4*N0+PP-1 ) )
            Z( 4*N0+PP-1 ) = MIN( Z( 4*N0+PP-1 ), Z( 4*I0+PP-1 ),
     $                            Z( 4*I0+PP+3 ) )
            Z( 4*N0-PP ) = MIN( Z( 4*N0-PP ), Z( 4*I0-PP ),
     $                          Z( 4*I0-PP+4 ) )
            QMAX = MAX( QMAX, Z( 4*I0+PP-3 ), Z( 4*I0+PP+1 ) )
            DMIN = -ZERO
         END IF
      END IF
*
*     Choose a shift.
*
      CALL DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN, DN1,
     $             DN2, TAU, TTYPE, G )
*
*     Call dqds until DMIN > 0.
*
   70 CONTINUE
*
      CALL DLASQ5( I0, N0, Z, PP, TAU, SIGMA, DMIN, DMIN1, DMIN2, DN,
     $             DN1, DN2, IEEE, EPS )
*
      NDIV = NDIV + ( N0-I0+2 )
      ITER = ITER + 1
*
*     Check status.
*
      IF( DMIN.GE.ZERO .AND. DMIN1.GE.ZERO ) THEN
*
*        Success.
*
         GO TO 90
*
      ELSE IF( DMIN.LT.ZERO .AND. DMIN1.GT.ZERO .AND.
     $         Z( 4*( N0-1 )-PP ).LT.TOL*( SIGMA+DN1 ) .AND.
     $         ABS( DN ).LT.TOL*SIGMA ) THEN
*
*        Convergence hidden by negative DN.
*
         Z( 4*( N0-1 )-PP+2 ) = ZERO
         DMIN = ZERO
         GO TO 90
      ELSE IF( DMIN.LT.ZERO ) THEN
*
*        TAU too big. Select new TAU and try again.
*
         NFAIL = NFAIL + 1
         IF( TTYPE.LT.-22 ) THEN
*
*           Failed twice. Play it safe.
*
            TAU = ZERO
         ELSE IF( DMIN1.GT.ZERO ) THEN
*
*           Late failure. Gives excellent shift.
*
            TAU = ( TAU+DMIN )*( ONE-TWO*EPS )
            TTYPE = TTYPE - 11
         ELSE
*
*           Early failure. Divide by 4.
*
            TAU = QURTR*TAU
            TTYPE = TTYPE - 12
         END IF
         GO TO 70
      ELSE IF( DISNAN( DMIN ) ) THEN
*
*        NaN.
*
         IF( TAU.EQ.ZERO ) THEN
            GO TO 80
         ELSE
            TAU = ZERO
            GO TO 70
         END IF
      ELSE
*
*        Possible underflow. Play it safe.
*
         GO TO 80
      END IF
*
*     Risk of underflow.
*
   80 CONTINUE
      CALL DLASQ6( I0, N0, Z, PP, DMIN, DMIN1, DMIN2, DN, DN1, DN2 )
      NDIV = NDIV + ( N0-I0+2 )
      ITER = ITER + 1
      TAU = ZERO
*
   90 CONTINUE
      IF( TAU.LT.SIGMA ) THEN
         DESIG = DESIG + TAU
         T = SIGMA + DESIG
         DESIG = DESIG - ( T-SIGMA )
      ELSE
         T = SIGMA + TAU
         DESIG = SIGMA - ( T-TAU ) + DESIG
      END IF
      SIGMA = T
*
      RETURN
*
*     End of DLASQ3
*
      END
*> \brief \b DLASQ4 computes an approximation to the smallest eigenvalue using values of d from the previous transform. Used by sbdsqr.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ4 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq4.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq4.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq4.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN,
*                          DN1, DN2, TAU, TTYPE, G )
*
*       .. Scalar Arguments ..
*       INTEGER            I0, N0, N0IN, PP, TTYPE
*       DOUBLE PRECISION   DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASQ4 computes an approximation TAU to the smallest eigenvalue
*> using values of d from the previous transform.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] I0
*> \verbatim
*>          I0 is INTEGER
*>        First index.
*> \endverbatim
*>
*> \param[in] N0
*> \verbatim
*>          N0 is INTEGER
*>        Last index.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension ( 4*N0 )
*>        Z holds the qd array.
*> \endverbatim
*>
*> \param[in] PP
*> \verbatim
*>          PP is INTEGER
*>        PP=0 for ping, PP=1 for pong.
*> \endverbatim
*>
*> \param[in] N0IN
*> \verbatim
*>          N0IN is INTEGER
*>        The value of N0 at start of EIGTEST.
*> \endverbatim
*>
*> \param[in] DMIN
*> \verbatim
*>          DMIN is DOUBLE PRECISION
*>        Minimum value of d.
*> \endverbatim
*>
*> \param[in] DMIN1
*> \verbatim
*>          DMIN1 is DOUBLE PRECISION
*>        Minimum value of d, excluding D( N0 ).
*> \endverbatim
*>
*> \param[in] DMIN2
*> \verbatim
*>          DMIN2 is DOUBLE PRECISION
*>        Minimum value of d, excluding D( N0 ) and D( N0-1 ).
*> \endverbatim
*>
*> \param[in] DN
*> \verbatim
*>          DN is DOUBLE PRECISION
*>        d(N)
*> \endverbatim
*>
*> \param[in] DN1
*> \verbatim
*>          DN1 is DOUBLE PRECISION
*>        d(N-1)
*> \endverbatim
*>
*> \param[in] DN2
*> \verbatim
*>          DN2 is DOUBLE PRECISION
*>        d(N-2)
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION
*>        This is the shift.
*> \endverbatim
*>
*> \param[out] TTYPE
*> \verbatim
*>          TTYPE is INTEGER
*>        Shift type.
*> \endverbatim
*>
*> \param[in,out] G
*> \verbatim
*>          G is DOUBLE PRECISION
*>        G is passed as an argument in order to save its value between
*>        calls to DLASQ4.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasq4
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  CNST1 = 9/16
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN,
     $                   DN1, DN2, TAU, TTYPE, G )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            I0, N0, N0IN, PP, TTYPE
      DOUBLE PRECISION   DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   Z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   CNST1, CNST2, CNST3
      PARAMETER          ( CNST1 = 0.5630D0, CNST2 = 1.010D0,
     $                   CNST3 = 1.050D0 )
      DOUBLE PRECISION   QURTR, THIRD, HALF, ZERO, ONE, TWO, HUNDRD
      PARAMETER          ( QURTR = 0.250D0, THIRD = 0.3330D0,
     $                   HALF = 0.50D0, ZERO = 0.0D0, ONE = 1.0D0,
     $                   TWO = 2.0D0, HUNDRD = 100.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I4, NN, NP
      DOUBLE PRECISION   A2, B1, B2, GAM, GAP1, GAP2, S
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     A negative DMIN forces the shift to take that absolute value
*     TTYPE records the type of shift.
*
      IF( DMIN.LE.ZERO ) THEN
         TAU = -DMIN
         TTYPE = -1
         RETURN
      END IF
*
      NN = 4*N0 + PP
      IF( N0IN.EQ.N0 ) THEN
*
*        No eigenvalues deflated.
*
         IF( DMIN.EQ.DN .OR. DMIN.EQ.DN1 ) THEN
*
            B1 = SQRT( Z( NN-3 ) )*SQRT( Z( NN-5 ) )
            B2 = SQRT( Z( NN-7 ) )*SQRT( Z( NN-9 ) )
            A2 = Z( NN-7 ) + Z( NN-5 )
*
*           Cases 2 and 3.
*
            IF( DMIN.EQ.DN .AND. DMIN1.EQ.DN1 ) THEN
               GAP2 = DMIN2 - A2 - DMIN2*QURTR
               IF( GAP2.GT.ZERO .AND. GAP2.GT.B2 ) THEN
                  GAP1 = A2 - DN - ( B2 / GAP2 )*B2
               ELSE
                  GAP1 = A2 - DN - ( B1+B2 )
               END IF
               IF( GAP1.GT.ZERO .AND. GAP1.GT.B1 ) THEN
                  S = MAX( DN-( B1 / GAP1 )*B1, HALF*DMIN )
                  TTYPE = -2
               ELSE
                  S = ZERO
                  IF( DN.GT.B1 )
     $               S = DN - B1
                  IF( A2.GT.( B1+B2 ) )
     $               S = MIN( S, A2-( B1+B2 ) )
                  S = MAX( S, THIRD*DMIN )
                  TTYPE = -3
               END IF
            ELSE
*
*              Case 4.
*
               TTYPE = -4
               S = QURTR*DMIN
               IF( DMIN.EQ.DN ) THEN
                  GAM = DN
                  A2 = ZERO
                  IF( Z( NN-5 ) .GT. Z( NN-7 ) )
     $               RETURN
                  B2 = Z( NN-5 ) / Z( NN-7 )
                  NP = NN - 9
               ELSE
                  NP = NN - 2*PP
                  GAM = DN1
                  IF( Z( NP-4 ) .GT. Z( NP-2 ) )
     $               RETURN
                  A2 = Z( NP-4 ) / Z( NP-2 )
                  IF( Z( NN-9 ) .GT. Z( NN-11 ) )
     $               RETURN
                  B2 = Z( NN-9 ) / Z( NN-11 )
                  NP = NN - 13
               END IF
*
*              Approximate contribution to norm squared from I < NN-1.
*
               A2 = A2 + B2
               DO 10 I4 = NP, 4*I0 - 1 + PP, -4
                  IF( B2.EQ.ZERO )
     $               GO TO 20
                  B1 = B2
                  IF( Z( I4 ) .GT. Z( I4-2 ) )
     $               RETURN
                  B2 = B2*( Z( I4 ) / Z( I4-2 ) )
                  A2 = A2 + B2
                  IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 )
     $               GO TO 20
   10          CONTINUE
   20          CONTINUE
               A2 = CNST3*A2
*
*              Rayleigh quotient residual bound.
*
               IF( A2.LT.CNST1 )
     $            S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
            END IF
         ELSE IF( DMIN.EQ.DN2 ) THEN
*
*           Case 5.
*
            TTYPE = -5
            S = QURTR*DMIN
*
*           Compute contribution to norm squared from I > NN-2.
*
            NP = NN - 2*PP
            B1 = Z( NP-2 )
            B2 = Z( NP-6 )
            GAM = DN2
            IF( Z( NP-8 ).GT.B2 .OR. Z( NP-4 ).GT.B1 )
     $         RETURN
            A2 = ( Z( NP-8 ) / B2 )*( ONE+Z( NP-4 ) / B1 )
*
*           Approximate contribution to norm squared from I < NN-2.
*
            IF( N0-I0.GT.2 ) THEN
               B2 = Z( NN-13 ) / Z( NN-15 )
               A2 = A2 + B2
               DO 30 I4 = NN - 17, 4*I0 - 1 + PP, -4
                  IF( B2.EQ.ZERO )
     $               GO TO 40
                  B1 = B2
                  IF( Z( I4 ) .GT. Z( I4-2 ) )
     $               RETURN
                  B2 = B2*( Z( I4 ) / Z( I4-2 ) )
                  A2 = A2 + B2
                  IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 )
     $               GO TO 40
   30          CONTINUE
   40          CONTINUE
               A2 = CNST3*A2
            END IF
*
            IF( A2.LT.CNST1 )
     $         S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
         ELSE
*
*           Case 6, no information to guide us.
*
            IF( TTYPE.EQ.-6 ) THEN
               G = G + THIRD*( ONE-G )
            ELSE IF( TTYPE.EQ.-18 ) THEN
               G = QURTR*THIRD
            ELSE
               G = QURTR
            END IF
            S = G*DMIN
            TTYPE = -6
         END IF
*
      ELSE IF( N0IN.EQ.( N0+1 ) ) THEN
*
*        One eigenvalue just deflated. Use DMIN1, DN1 for DMIN and DN.
*
         IF( DMIN1.EQ.DN1 .AND. DMIN2.EQ.DN2 ) THEN
*
*           Cases 7 and 8.
*
            TTYPE = -7
            S = THIRD*DMIN1
            IF( Z( NN-5 ).GT.Z( NN-7 ) )
     $         RETURN
            B1 = Z( NN-5 ) / Z( NN-7 )
            B2 = B1
            IF( B2.EQ.ZERO )
     $         GO TO 60
            DO 50 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4
               A2 = B1
               IF( Z( I4 ).GT.Z( I4-2 ) )
     $            RETURN
               B1 = B1*( Z( I4 ) / Z( I4-2 ) )
               B2 = B2 + B1
               IF( HUNDRD*MAX( B1, A2 ).LT.B2 )
     $            GO TO 60
   50       CONTINUE
   60       CONTINUE
            B2 = SQRT( CNST3*B2 )
            A2 = DMIN1 / ( ONE+B2**2 )
            GAP2 = HALF*DMIN2 - A2
            IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN
               S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) )
            ELSE
               S = MAX( S, A2*( ONE-CNST2*B2 ) )
               TTYPE = -8
            END IF
         ELSE
*
*           Case 9.
*
            S = QURTR*DMIN1
            IF( DMIN1.EQ.DN1 )
     $         S = HALF*DMIN1
            TTYPE = -9
         END IF
*
      ELSE IF( N0IN.EQ.( N0+2 ) ) THEN
*
*        Two eigenvalues deflated. Use DMIN2, DN2 for DMIN and DN.
*
*        Cases 10 and 11.
*
         IF( DMIN2.EQ.DN2 .AND. TWO*Z( NN-5 ).LT.Z( NN-7 ) ) THEN
            TTYPE = -10
            S = THIRD*DMIN2
            IF( Z( NN-5 ).GT.Z( NN-7 ) )
     $         RETURN
            B1 = Z( NN-5 ) / Z( NN-7 )
            B2 = B1
            IF( B2.EQ.ZERO )
     $         GO TO 80
            DO 70 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4
               IF( Z( I4 ).GT.Z( I4-2 ) )
     $            RETURN
               B1 = B1*( Z( I4 ) / Z( I4-2 ) )
               B2 = B2 + B1
               IF( HUNDRD*B1.LT.B2 )
     $            GO TO 80
   70       CONTINUE
   80       CONTINUE
            B2 = SQRT( CNST3*B2 )
            A2 = DMIN2 / ( ONE+B2**2 )
            GAP2 = Z( NN-7 ) + Z( NN-9 ) -
     $             SQRT( Z( NN-11 ) )*SQRT( Z( NN-9 ) ) - A2
            IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN
               S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) )
            ELSE
               S = MAX( S, A2*( ONE-CNST2*B2 ) )
            END IF
         ELSE
            S = QURTR*DMIN2
            TTYPE = -11
         END IF
      ELSE IF( N0IN.GT.( N0+2 ) ) THEN
*
*        Case 12, more than two eigenvalues deflated. No information.
*
         S = ZERO
         TTYPE = -12
      END IF
*
      TAU = S
      RETURN
*
*     End of DLASQ4
*
      END
*> \brief \b DLASQ5 computes one dqds transform in ping-pong form. Used by sbdsqr and sstegr.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ5 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq5.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq5.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq5.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASQ5( I0, N0, Z, PP, TAU, SIGMA, DMIN, DMIN1, DMIN2, DN,
*                          DNM1, DNM2, IEEE, EPS )
*
*       .. Scalar Arguments ..
*       LOGICAL            IEEE
*       INTEGER            I0, N0, PP
*       DOUBLE PRECISION   DMIN, DMIN1, DMIN2, DN, DNM1, DNM2, TAU, SIGMA, EPS
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASQ5 computes one dqds transform in ping-pong form, one
*> version for IEEE machines another for non IEEE machines.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] I0
*> \verbatim
*>          I0 is INTEGER
*>        First index.
*> \endverbatim
*>
*> \param[in] N0
*> \verbatim
*>          N0 is INTEGER
*>        Last index.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension ( 4*N )
*>        Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
*>        an extra argument.
*> \endverbatim
*>
*> \param[in] PP
*> \verbatim
*>          PP is INTEGER
*>        PP=0 for ping, PP=1 for pong.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION
*>        This is the shift.
*> \endverbatim
*>
*> \param[in] SIGMA
*> \verbatim
*>          SIGMA is DOUBLE PRECISION
*>        This is the accumulated shift up to this step.
*> \endverbatim
*>
*> \param[out] DMIN
*> \verbatim
*>          DMIN is DOUBLE PRECISION
*>        Minimum value of d.
*> \endverbatim
*>
*> \param[out] DMIN1
*> \verbatim
*>          DMIN1 is DOUBLE PRECISION
*>        Minimum value of d, excluding D( N0 ).
*> \endverbatim
*>
*> \param[out] DMIN2
*> \verbatim
*>          DMIN2 is DOUBLE PRECISION
*>        Minimum value of d, excluding D( N0 ) and D( N0-1 ).
*> \endverbatim
*>
*> \param[out] DN
*> \verbatim
*>          DN is DOUBLE PRECISION
*>        d(N0), the last value of d.
*> \endverbatim
*>
*> \param[out] DNM1
*> \verbatim
*>          DNM1 is DOUBLE PRECISION
*>        d(N0-1).
*> \endverbatim
*>
*> \param[out] DNM2
*> \verbatim
*>          DNM2 is DOUBLE PRECISION
*>        d(N0-2).
*> \endverbatim
*>
*> \param[in] IEEE
*> \verbatim
*>          IEEE is LOGICAL
*>        Flag for IEEE or non IEEE arithmetic.
*> \endverbatim
*>
*> \param[in] EPS
*> \verbatim
*>          EPS is DOUBLE PRECISION
*>        This is the value of epsilon used.
*> \endverbatim
*>
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasq5
*
*  =====================================================================
      SUBROUTINE DLASQ5( I0, N0, Z, PP, TAU, SIGMA, DMIN, DMIN1,
     $                   DMIN2,
     $                   DN, DNM1, DNM2, IEEE, EPS )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      LOGICAL            IEEE
      INTEGER            I0, N0, PP
      DOUBLE PRECISION   DMIN, DMIN1, DMIN2, DN, DNM1, DNM2, TAU,
     $                   SIGMA, EPS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   Z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameter ..
      DOUBLE PRECISION   ZERO, HALF
      PARAMETER          ( ZERO = 0.0D0, HALF = 0.5 )
*     ..
*     .. Local Scalars ..
      INTEGER            J4, J4P2
      DOUBLE PRECISION   D, EMIN, TEMP, DTHRESH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
      IF( ( N0-I0-1 ).LE.0 )
     $   RETURN
*
      DTHRESH = EPS*(SIGMA+TAU)
      IF( TAU.LT.DTHRESH*HALF ) TAU = ZERO
      IF( TAU.NE.ZERO ) THEN
      J4 = 4*I0 + PP - 3
      EMIN = Z( J4+4 )
      D = Z( J4 ) - TAU
      DMIN = D
      DMIN1 = -Z( J4 )
*
      IF( IEEE ) THEN
*
*        Code for IEEE arithmetic.
*
         IF( PP.EQ.0 ) THEN
            DO 10 J4 = 4*I0, 4*( N0-3 ), 4
               Z( J4-2 ) = D + Z( J4-1 )
               TEMP = Z( J4+1 ) / Z( J4-2 )
               D = D*TEMP - TAU
               DMIN = MIN( DMIN, D )
               Z( J4 ) = Z( J4-1 )*TEMP
               EMIN = MIN( Z( J4 ), EMIN )
   10       CONTINUE
         ELSE
            DO 20 J4 = 4*I0, 4*( N0-3 ), 4
               Z( J4-3 ) = D + Z( J4 )
               TEMP = Z( J4+2 ) / Z( J4-3 )
               D = D*TEMP - TAU
               DMIN = MIN( DMIN, D )
               Z( J4-1 ) = Z( J4 )*TEMP
               EMIN = MIN( Z( J4-1 ), EMIN )
   20       CONTINUE
         END IF
*
*        Unroll last two steps.
*
         DNM2 = D
         DMIN2 = DMIN
         J4 = 4*( N0-2 ) - PP
         J4P2 = J4 + 2*PP - 1
         Z( J4-2 ) = DNM2 + Z( J4P2 )
         Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
         DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) ) - TAU
         DMIN = MIN( DMIN, DNM1 )
*
         DMIN1 = DMIN
         J4 = J4 + 4
         J4P2 = J4 + 2*PP - 1
         Z( J4-2 ) = DNM1 + Z( J4P2 )
         Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
         DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) ) - TAU
         DMIN = MIN( DMIN, DN )
*
      ELSE
*
*        Code for non IEEE arithmetic.
*
         IF( PP.EQ.0 ) THEN
            DO 30 J4 = 4*I0, 4*( N0-3 ), 4
               Z( J4-2 ) = D + Z( J4-1 )
               IF( D.LT.ZERO ) THEN
                  RETURN
               ELSE
                  Z( J4 ) = Z( J4+1 )*( Z( J4-1 ) / Z( J4-2 ) )
                  D = Z( J4+1 )*( D / Z( J4-2 ) ) - TAU
               END IF
               DMIN = MIN( DMIN, D )
               EMIN = MIN( EMIN, Z( J4 ) )
   30       CONTINUE
         ELSE
            DO 40 J4 = 4*I0, 4*( N0-3 ), 4
               Z( J4-3 ) = D + Z( J4 )
               IF( D.LT.ZERO ) THEN
                  RETURN
               ELSE
                  Z( J4-1 ) = Z( J4+2 )*( Z( J4 ) / Z( J4-3 ) )
                  D = Z( J4+2 )*( D / Z( J4-3 ) ) - TAU
               END IF
               DMIN = MIN( DMIN, D )
               EMIN = MIN( EMIN, Z( J4-1 ) )
   40       CONTINUE
         END IF
*
*        Unroll last two steps.
*
         DNM2 = D
         DMIN2 = DMIN
         J4 = 4*( N0-2 ) - PP
         J4P2 = J4 + 2*PP - 1
         Z( J4-2 ) = DNM2 + Z( J4P2 )
         IF( DNM2.LT.ZERO ) THEN
            RETURN
         ELSE
            Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
            DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) ) - TAU
         END IF
         DMIN = MIN( DMIN, DNM1 )
*
         DMIN1 = DMIN
         J4 = J4 + 4
         J4P2 = J4 + 2*PP - 1
         Z( J4-2 ) = DNM1 + Z( J4P2 )
         IF( DNM1.LT.ZERO ) THEN
            RETURN
         ELSE
            Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
            DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) ) - TAU
         END IF
         DMIN = MIN( DMIN, DN )
*
      END IF
      ELSE
*     This is the version that sets d's to zero if they are small enough
         J4 = 4*I0 + PP - 3
         EMIN = Z( J4+4 )
         D = Z( J4 ) - TAU
         DMIN = D
         DMIN1 = -Z( J4 )
         IF( IEEE ) THEN
*
*     Code for IEEE arithmetic.
*
            IF( PP.EQ.0 ) THEN
               DO 50 J4 = 4*I0, 4*( N0-3 ), 4
                  Z( J4-2 ) = D + Z( J4-1 )
                  TEMP = Z( J4+1 ) / Z( J4-2 )
                  D = D*TEMP - TAU
                  IF( D.LT.DTHRESH ) D = ZERO
                  DMIN = MIN( DMIN, D )
                  Z( J4 ) = Z( J4-1 )*TEMP
                  EMIN = MIN( Z( J4 ), EMIN )
 50            CONTINUE
            ELSE
               DO 60 J4 = 4*I0, 4*( N0-3 ), 4
                  Z( J4-3 ) = D + Z( J4 )
                  TEMP = Z( J4+2 ) / Z( J4-3 )
                  D = D*TEMP - TAU
                  IF( D.LT.DTHRESH ) D = ZERO
                  DMIN = MIN( DMIN, D )
                  Z( J4-1 ) = Z( J4 )*TEMP
                  EMIN = MIN( Z( J4-1 ), EMIN )
 60            CONTINUE
            END IF
*
*     Unroll last two steps.
*
            DNM2 = D
            DMIN2 = DMIN
            J4 = 4*( N0-2 ) - PP
            J4P2 = J4 + 2*PP - 1
            Z( J4-2 ) = DNM2 + Z( J4P2 )
            Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
            DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) ) - TAU
            DMIN = MIN( DMIN, DNM1 )
*
            DMIN1 = DMIN
            J4 = J4 + 4
            J4P2 = J4 + 2*PP - 1
            Z( J4-2 ) = DNM1 + Z( J4P2 )
            Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
            DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) ) - TAU
            DMIN = MIN( DMIN, DN )
*
         ELSE
*
*     Code for non IEEE arithmetic.
*
            IF( PP.EQ.0 ) THEN
               DO 70 J4 = 4*I0, 4*( N0-3 ), 4
                  Z( J4-2 ) = D + Z( J4-1 )
                  IF( D.LT.ZERO ) THEN
                     RETURN
                  ELSE
                     Z( J4 ) = Z( J4+1 )*( Z( J4-1 ) / Z( J4-2 ) )
                     D = Z( J4+1 )*( D / Z( J4-2 ) ) - TAU
                  END IF
                  IF( D.LT.DTHRESH) D = ZERO
                  DMIN = MIN( DMIN, D )
                  EMIN = MIN( EMIN, Z( J4 ) )
 70            CONTINUE
            ELSE
               DO 80 J4 = 4*I0, 4*( N0-3 ), 4
                  Z( J4-3 ) = D + Z( J4 )
                  IF( D.LT.ZERO ) THEN
                     RETURN
                  ELSE
                     Z( J4-1 ) = Z( J4+2 )*( Z( J4 ) / Z( J4-3 ) )
                     D = Z( J4+2 )*( D / Z( J4-3 ) ) - TAU
                  END IF
                  IF( D.LT.DTHRESH) D = ZERO
                  DMIN = MIN( DMIN, D )
                  EMIN = MIN( EMIN, Z( J4-1 ) )
 80            CONTINUE
            END IF
*
*     Unroll last two steps.
*
            DNM2 = D
            DMIN2 = DMIN
            J4 = 4*( N0-2 ) - PP
            J4P2 = J4 + 2*PP - 1
            Z( J4-2 ) = DNM2 + Z( J4P2 )
            IF( DNM2.LT.ZERO ) THEN
               RETURN
            ELSE
               Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
               DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) ) - TAU
            END IF
            DMIN = MIN( DMIN, DNM1 )
*
            DMIN1 = DMIN
            J4 = J4 + 4
            J4P2 = J4 + 2*PP - 1
            Z( J4-2 ) = DNM1 + Z( J4P2 )
            IF( DNM1.LT.ZERO ) THEN
               RETURN
            ELSE
               Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
               DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) ) - TAU
            END IF
            DMIN = MIN( DMIN, DN )
*
         END IF
      END IF
*
      Z( J4+2 ) = DN
      Z( 4*N0-PP ) = EMIN
      RETURN
*
*     End of DLASQ5
*
      END
*> \brief \b DLASQ6 computes one dqd transform in ping-pong form. Used by sbdsqr and sstegr.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ6 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq6.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq6.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq6.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASQ6( I0, N0, Z, PP, DMIN, DMIN1, DMIN2, DN,
*                          DNM1, DNM2 )
*
*       .. Scalar Arguments ..
*       INTEGER            I0, N0, PP
*       DOUBLE PRECISION   DMIN, DMIN1, DMIN2, DN, DNM1, DNM2
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASQ6 computes one dqd (shift equal to zero) transform in
*> ping-pong form, with protection against underflow and overflow.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] I0
*> \verbatim
*>          I0 is INTEGER
*>        First index.
*> \endverbatim
*>
*> \param[in] N0
*> \verbatim
*>          N0 is INTEGER
*>        Last index.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension ( 4*N )
*>        Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
*>        an extra argument.
*> \endverbatim
*>
*> \param[in] PP
*> \verbatim
*>          PP is INTEGER
*>        PP=0 for ping, PP=1 for pong.
*> \endverbatim
*>
*> \param[out] DMIN
*> \verbatim
*>          DMIN is DOUBLE PRECISION
*>        Minimum value of d.
*> \endverbatim
*>
*> \param[out] DMIN1
*> \verbatim
*>          DMIN1 is DOUBLE PRECISION
*>        Minimum value of d, excluding D( N0 ).
*> \endverbatim
*>
*> \param[out] DMIN2
*> \verbatim
*>          DMIN2 is DOUBLE PRECISION
*>        Minimum value of d, excluding D( N0 ) and D( N0-1 ).
*> \endverbatim
*>
*> \param[out] DN
*> \verbatim
*>          DN is DOUBLE PRECISION
*>        d(N0), the last value of d.
*> \endverbatim
*>
*> \param[out] DNM1
*> \verbatim
*>          DNM1 is DOUBLE PRECISION
*>        d(N0-1).
*> \endverbatim
*>
*> \param[out] DNM2
*> \verbatim
*>          DNM2 is DOUBLE PRECISION
*>        d(N0-2).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasq6
*
*  =====================================================================
      SUBROUTINE DLASQ6( I0, N0, Z, PP, DMIN, DMIN1, DMIN2, DN,
     $                   DNM1, DNM2 )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            I0, N0, PP
      DOUBLE PRECISION   DMIN, DMIN1, DMIN2, DN, DNM1, DNM2
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   Z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameter ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            J4, J4P2
      DOUBLE PRECISION   D, EMIN, SAFMIN, TEMP
*     ..
*     .. External Function ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
      IF( ( N0-I0-1 ).LE.0 )
     $   RETURN
*
      SAFMIN = DLAMCH( 'Safe minimum' )
      J4 = 4*I0 + PP - 3
      EMIN = Z( J4+4 )
      D = Z( J4 )
      DMIN = D
*
      IF( PP.EQ.0 ) THEN
         DO 10 J4 = 4*I0, 4*( N0-3 ), 4
            Z( J4-2 ) = D + Z( J4-1 )
            IF( Z( J4-2 ).EQ.ZERO ) THEN
               Z( J4 ) = ZERO
               D = Z( J4+1 )
               DMIN = D
               EMIN = ZERO
            ELSE IF( SAFMIN*Z( J4+1 ).LT.Z( J4-2 ) .AND.
     $               SAFMIN*Z( J4-2 ).LT.Z( J4+1 ) ) THEN
               TEMP = Z( J4+1 ) / Z( J4-2 )
               Z( J4 ) = Z( J4-1 )*TEMP
               D = D*TEMP
            ELSE
               Z( J4 ) = Z( J4+1 )*( Z( J4-1 ) / Z( J4-2 ) )
               D = Z( J4+1 )*( D / Z( J4-2 ) )
            END IF
            DMIN = MIN( DMIN, D )
            EMIN = MIN( EMIN, Z( J4 ) )
   10    CONTINUE
      ELSE
         DO 20 J4 = 4*I0, 4*( N0-3 ), 4
            Z( J4-3 ) = D + Z( J4 )
            IF( Z( J4-3 ).EQ.ZERO ) THEN
               Z( J4-1 ) = ZERO
               D = Z( J4+2 )
               DMIN = D
               EMIN = ZERO
            ELSE IF( SAFMIN*Z( J4+2 ).LT.Z( J4-3 ) .AND.
     $               SAFMIN*Z( J4-3 ).LT.Z( J4+2 ) ) THEN
               TEMP = Z( J4+2 ) / Z( J4-3 )
               Z( J4-1 ) = Z( J4 )*TEMP
               D = D*TEMP
            ELSE
               Z( J4-1 ) = Z( J4+2 )*( Z( J4 ) / Z( J4-3 ) )
               D = Z( J4+2 )*( D / Z( J4-3 ) )
            END IF
            DMIN = MIN( DMIN, D )
            EMIN = MIN( EMIN, Z( J4-1 ) )
   20    CONTINUE
      END IF
*
*     Unroll last two steps.
*
      DNM2 = D
      DMIN2 = DMIN
      J4 = 4*( N0-2 ) - PP
      J4P2 = J4 + 2*PP - 1
      Z( J4-2 ) = DNM2 + Z( J4P2 )
      IF( Z( J4-2 ).EQ.ZERO ) THEN
         Z( J4 ) = ZERO
         DNM1 = Z( J4P2+2 )
         DMIN = DNM1
         EMIN = ZERO
      ELSE IF( SAFMIN*Z( J4P2+2 ).LT.Z( J4-2 ) .AND.
     $         SAFMIN*Z( J4-2 ).LT.Z( J4P2+2 ) ) THEN
         TEMP = Z( J4P2+2 ) / Z( J4-2 )
         Z( J4 ) = Z( J4P2 )*TEMP
         DNM1 = DNM2*TEMP
      ELSE
         Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
         DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) )
      END IF
      DMIN = MIN( DMIN, DNM1 )
*
      DMIN1 = DMIN
      J4 = J4 + 4
      J4P2 = J4 + 2*PP - 1
      Z( J4-2 ) = DNM1 + Z( J4P2 )
      IF( Z( J4-2 ).EQ.ZERO ) THEN
         Z( J4 ) = ZERO
         DN = Z( J4P2+2 )
         DMIN = DN
         EMIN = ZERO
      ELSE IF( SAFMIN*Z( J4P2+2 ).LT.Z( J4-2 ) .AND.
     $         SAFMIN*Z( J4-2 ).LT.Z( J4P2+2 ) ) THEN
         TEMP = Z( J4P2+2 ) / Z( J4-2 )
         Z( J4 ) = Z( J4P2 )*TEMP
         DN = DNM1*TEMP
      ELSE
         Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
         DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) )
      END IF
      DMIN = MIN( DMIN, DN )
*
      Z( J4+2 ) = DN
      Z( 4*N0-PP ) = EMIN
      RETURN
*
*     End of DLASQ6
*
      END
*> \brief \b DLASR applies a sequence of plane rotations to a general rectangular matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASR( SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIRECT, PIVOT, SIDE
*       INTEGER            LDA, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), C( * ), S( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASR applies a sequence of plane rotations to a real matrix A,
*> from either the left or the right.
*>
*> When SIDE = 'L', the transformation takes the form
*>
*>    A := P*A
*>
*> and when SIDE = 'R', the transformation takes the form
*>
*>    A := A*P**T
*>
*> where P is an orthogonal matrix consisting of a sequence of z plane
*> rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R',
*> and P**T is the transpose of P.
*>
*> When DIRECT = 'F' (Forward sequence), then
*>
*>    P = P(z-1) * ... * P(2) * P(1)
*>
*> and when DIRECT = 'B' (Backward sequence), then
*>
*>    P = P(1) * P(2) * ... * P(z-1)
*>
*> where P(k) is a plane rotation matrix defined by the 2-by-2 rotation
*>
*>    R(k) = (  c(k)  s(k) )
*>         = ( -s(k)  c(k) ).
*>
*> When PIVOT = 'V' (Variable pivot), the rotation is performed
*> for the plane (k,k+1), i.e., P(k) has the form
*>
*>    P(k) = (  1                                            )
*>           (       ...                                     )
*>           (              1                                )
*>           (                   c(k)  s(k)                  )
*>           (                  -s(k)  c(k)                  )
*>           (                                1              )
*>           (                                     ...       )
*>           (                                            1  )
*>
*> where R(k) appears as a rank-2 modification to the identity matrix in
*> rows and columns k and k+1.
*>
*> When PIVOT = 'T' (Top pivot), the rotation is performed for the
*> plane (1,k+1), so P(k) has the form
*>
*>    P(k) = (  c(k)                    s(k)                 )
*>           (         1                                     )
*>           (              ...                              )
*>           (                     1                         )
*>           ( -s(k)                    c(k)                 )
*>           (                                 1             )
*>           (                                      ...      )
*>           (                                             1 )
*>
*> where R(k) appears in rows and columns 1 and k+1.
*>
*> Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is
*> performed for the plane (k,z), giving P(k) the form
*>
*>    P(k) = ( 1                                             )
*>           (      ...                                      )
*>           (             1                                 )
*>           (                  c(k)                    s(k) )
*>           (                         1                     )
*>           (                              ...              )
*>           (                                     1         )
*>           (                 -s(k)                    c(k) )
*>
*> where R(k) appears in rows and columns k and z.  The rotations are
*> performed without ever forming P(k) explicitly.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          Specifies whether the plane rotation matrix P is applied to
*>          A on the left or the right.
*>          = 'L':  Left, compute A := P*A
*>          = 'R':  Right, compute A:= A*P**T
*> \endverbatim
*>
*> \param[in] PIVOT
*> \verbatim
*>          PIVOT is CHARACTER*1
*>          Specifies the plane for which P(k) is a plane rotation
*>          matrix.
*>          = 'V':  Variable pivot, the plane (k,k+1)
*>          = 'T':  Top pivot, the plane (1,k+1)
*>          = 'B':  Bottom pivot, the plane (k,z)
*> \endverbatim
*>
*> \param[in] DIRECT
*> \verbatim
*>          DIRECT is CHARACTER*1
*>          Specifies whether P is a forward or backward sequence of
*>          plane rotations.
*>          = 'F':  Forward, P = P(z-1)*...*P(2)*P(1)
*>          = 'B':  Backward, P = P(1)*P(2)*...*P(z-1)
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  If m <= 1, an immediate
*>          return is effected.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  If n <= 1, an
*>          immediate return is effected.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension
*>                  (M-1) if SIDE = 'L'
*>                  (N-1) if SIDE = 'R'
*>          The cosines c(k) of the plane rotations.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension
*>                  (M-1) if SIDE = 'L'
*>                  (N-1) if SIDE = 'R'
*>          The sines s(k) of the plane rotations.  The 2-by-2 plane
*>          rotation part of the matrix P(k), R(k), has the form
*>          R(k) = (  c(k)  s(k) )
*>                 ( -s(k)  c(k) ).
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The M-by-N matrix A.  On exit, A is overwritten by P*A if
*>          SIDE = 'L' or by A*P**T if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasr
*
*  =====================================================================
      SUBROUTINE DLASR( SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIRECT, PIVOT, SIDE
      INTEGER            LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( * ), S( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, INFO, J
      DOUBLE PRECISION   CTEMP, STEMP, TEMP
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INFO = 0
      IF( .NOT.( LSAME( SIDE, 'L' ) .OR.
     $    LSAME( SIDE, 'R' ) ) ) THEN
         INFO = 1
      ELSE IF( .NOT.( LSAME( PIVOT, 'V' ) .OR. LSAME( PIVOT,
     $         'T' ) .OR. LSAME( PIVOT, 'B' ) ) ) THEN
         INFO = 2
      ELSE IF( .NOT.( LSAME( DIRECT, 'F' ) .OR.
     $         LSAME( DIRECT, 'B' ) ) )
     $          THEN
         INFO = 3
      ELSE IF( M.LT.0 ) THEN
         INFO = 4
      ELSE IF( N.LT.0 ) THEN
         INFO = 5
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = 9
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLASR ', INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( ( M.EQ.0 ) .OR. ( N.EQ.0 ) )
     $   RETURN
      IF( LSAME( SIDE, 'L' ) ) THEN
*
*        Form  P * A
*
         IF( LSAME( PIVOT, 'V' ) ) THEN
            IF( LSAME( DIRECT, 'F' ) ) THEN
               DO 20 J = 1, M - 1
                  CTEMP = C( J )
                  STEMP = S( J )
                  IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
                     DO 10 I = 1, N
                        TEMP = A( J+1, I )
                        A( J+1, I ) = CTEMP*TEMP - STEMP*A( J, I )
                        A( J, I ) = STEMP*TEMP + CTEMP*A( J, I )
   10                CONTINUE
                  END IF
   20          CONTINUE
            ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
               DO 40 J = M - 1, 1, -1
                  CTEMP = C( J )
                  STEMP = S( J )
                  IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
                     DO 30 I = 1, N
                        TEMP = A( J+1, I )
                        A( J+1, I ) = CTEMP*TEMP - STEMP*A( J, I )
                        A( J, I ) = STEMP*TEMP + CTEMP*A( J, I )
   30                CONTINUE
                  END IF
   40          CONTINUE
            END IF
         ELSE IF( LSAME( PIVOT, 'T' ) ) THEN
            IF( LSAME( DIRECT, 'F' ) ) THEN
               DO 60 J = 2, M
                  CTEMP = C( J-1 )
                  STEMP = S( J-1 )
                  IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
                     DO 50 I = 1, N
                        TEMP = A( J, I )
                        A( J, I ) = CTEMP*TEMP - STEMP*A( 1, I )
                        A( 1, I ) = STEMP*TEMP + CTEMP*A( 1, I )
   50                CONTINUE
                  END IF
   60          CONTINUE
            ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
               DO 80 J = M, 2, -1
                  CTEMP = C( J-1 )
                  STEMP = S( J-1 )
                  IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
                     DO 70 I = 1, N
                        TEMP = A( J, I )
                        A( J, I ) = CTEMP*TEMP - STEMP*A( 1, I )
                        A( 1, I ) = STEMP*TEMP + CTEMP*A( 1, I )
   70                CONTINUE
                  END IF
   80          CONTINUE
            END IF
         ELSE IF( LSAME( PIVOT, 'B' ) ) THEN
            IF( LSAME( DIRECT, 'F' ) ) THEN
               DO 100 J = 1, M - 1
                  CTEMP = C( J )
                  STEMP = S( J )
                  IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
                     DO 90 I = 1, N
                        TEMP = A( J, I )
                        A( J, I ) = STEMP*A( M, I ) + CTEMP*TEMP
                        A( M, I ) = CTEMP*A( M, I ) - STEMP*TEMP
   90                CONTINUE
                  END IF
  100          CONTINUE
            ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
               DO 120 J = M - 1, 1, -1
                  CTEMP = C( J )
                  STEMP = S( J )
                  IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
                     DO 110 I = 1, N
                        TEMP = A( J, I )
                        A( J, I ) = STEMP*A( M, I ) + CTEMP*TEMP
                        A( M, I ) = CTEMP*A( M, I ) - STEMP*TEMP
  110                CONTINUE
                  END IF
  120          CONTINUE
            END IF
         END IF
      ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
*        Form A * P**T
*
         IF( LSAME( PIVOT, 'V' ) ) THEN
            IF( LSAME( DIRECT, 'F' ) ) THEN
               DO 140 J = 1, N - 1
                  CTEMP = C( J )
                  STEMP = S( J )
                  IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
                     DO 130 I = 1, M
                        TEMP = A( I, J+1 )
                        A( I, J+1 ) = CTEMP*TEMP - STEMP*A( I, J )
                        A( I, J ) = STEMP*TEMP + CTEMP*A( I, J )
  130                CONTINUE
                  END IF
  140          CONTINUE
            ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
               DO 160 J = N - 1, 1, -1
                  CTEMP = C( J )
                  STEMP = S( J )
                  IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
                     DO 150 I = 1, M
                        TEMP = A( I, J+1 )
                        A( I, J+1 ) = CTEMP*TEMP - STEMP*A( I, J )
                        A( I, J ) = STEMP*TEMP + CTEMP*A( I, J )
  150                CONTINUE
                  END IF
  160          CONTINUE
            END IF
         ELSE IF( LSAME( PIVOT, 'T' ) ) THEN
            IF( LSAME( DIRECT, 'F' ) ) THEN
               DO 180 J = 2, N
                  CTEMP = C( J-1 )
                  STEMP = S( J-1 )
                  IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
                     DO 170 I = 1, M
                        TEMP = A( I, J )
                        A( I, J ) = CTEMP*TEMP - STEMP*A( I, 1 )
                        A( I, 1 ) = STEMP*TEMP + CTEMP*A( I, 1 )
  170                CONTINUE
                  END IF
  180          CONTINUE
            ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
               DO 200 J = N, 2, -1
                  CTEMP = C( J-1 )
                  STEMP = S( J-1 )
                  IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
                     DO 190 I = 1, M
                        TEMP = A( I, J )
                        A( I, J ) = CTEMP*TEMP - STEMP*A( I, 1 )
                        A( I, 1 ) = STEMP*TEMP + CTEMP*A( I, 1 )
  190                CONTINUE
                  END IF
  200          CONTINUE
            END IF
         ELSE IF( LSAME( PIVOT, 'B' ) ) THEN
            IF( LSAME( DIRECT, 'F' ) ) THEN
               DO 220 J = 1, N - 1
                  CTEMP = C( J )
                  STEMP = S( J )
                  IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
                     DO 210 I = 1, M
                        TEMP = A( I, J )
                        A( I, J ) = STEMP*A( I, N ) + CTEMP*TEMP
                        A( I, N ) = CTEMP*A( I, N ) - STEMP*TEMP
  210                CONTINUE
                  END IF
  220          CONTINUE
            ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
               DO 240 J = N - 1, 1, -1
                  CTEMP = C( J )
                  STEMP = S( J )
                  IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
                     DO 230 I = 1, M
                        TEMP = A( I, J )
                        A( I, J ) = STEMP*A( I, N ) + CTEMP*TEMP
                        A( I, N ) = CTEMP*A( I, N ) - STEMP*TEMP
  230                CONTINUE
                  END IF
  240          CONTINUE
            END IF
         END IF
      END IF
*
      RETURN
*
*     End of DLASR
*
      END
*> \brief \b DLASRT sorts numbers in increasing or decreasing order.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASRT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasrt.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasrt.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasrt.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASRT( ID, N, D, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          ID
*       INTEGER            INFO, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> Sort the numbers in D in increasing order (if ID = 'I') or
*> in decreasing order (if ID = 'D' ).
*>
*> Use Quick Sort, reverting to Insertion sort on arrays of
*> size <= 20. Dimension of STACK limits N to about 2**32.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ID
*> \verbatim
*>          ID is CHARACTER*1
*>          = 'I': sort D in increasing order;
*>          = 'D': sort D in decreasing order.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The length of the array D.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, the array to be sorted.
*>          On exit, D has been sorted into increasing order
*>          (D(1) <= ... <= D(N) ) or into decreasing order
*>          (D(1) >= ... >= D(N) ), depending on ID.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasrt
*
*  =====================================================================
      SUBROUTINE DLASRT( ID, N, D, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          ID
      INTEGER            INFO, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            SELECT
      PARAMETER          ( SELECT = 20 )
*     ..
*     .. Local Scalars ..
      INTEGER            DIR, ENDD, I, J, START, STKPNT
      DOUBLE PRECISION   D1, D2, D3, DMNMX, TMP
*     ..
*     .. Local Arrays ..
      INTEGER            STACK( 2, 32 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      DIR = -1
      IF( LSAME( ID, 'D' ) ) THEN
         DIR = 0
      ELSE IF( LSAME( ID, 'I' ) ) THEN
         DIR = 1
      END IF
      IF( DIR.EQ.-1 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLASRT', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.LE.1 )
     $   RETURN
*
      STKPNT = 1
      STACK( 1, 1 ) = 1
      STACK( 2, 1 ) = N
   10 CONTINUE
      START = STACK( 1, STKPNT )
      ENDD = STACK( 2, STKPNT )
      STKPNT = STKPNT - 1
      IF( ENDD-START.LE.SELECT .AND. ENDD-START.GT.0 ) THEN
*
*        Do Insertion sort on D( START:ENDD )
*
         IF( DIR.EQ.0 ) THEN
*
*           Sort into decreasing order
*
            DO 30 I = START + 1, ENDD
               DO 20 J = I, START + 1, -1
                  IF( D( J ).GT.D( J-1 ) ) THEN
                     DMNMX = D( J )
                     D( J ) = D( J-1 )
                     D( J-1 ) = DMNMX
                  ELSE
                     GO TO 30
                  END IF
   20          CONTINUE
   30       CONTINUE
*
         ELSE
*
*           Sort into increasing order
*
            DO 50 I = START + 1, ENDD
               DO 40 J = I, START + 1, -1
                  IF( D( J ).LT.D( J-1 ) ) THEN
                     DMNMX = D( J )
                     D( J ) = D( J-1 )
                     D( J-1 ) = DMNMX
                  ELSE
                     GO TO 50
                  END IF
   40          CONTINUE
   50       CONTINUE
*
         END IF
*
      ELSE IF( ENDD-START.GT.SELECT ) THEN
*
*        Partition D( START:ENDD ) and stack parts, largest one first
*
*        Choose partition entry as median of 3
*
         D1 = D( START )
         D2 = D( ENDD )
         I = ( START+ENDD ) / 2
         D3 = D( I )
         IF( D1.LT.D2 ) THEN
            IF( D3.LT.D1 ) THEN
               DMNMX = D1
            ELSE IF( D3.LT.D2 ) THEN
               DMNMX = D3
            ELSE
               DMNMX = D2
            END IF
         ELSE
            IF( D3.LT.D2 ) THEN
               DMNMX = D2
            ELSE IF( D3.LT.D1 ) THEN
               DMNMX = D3
            ELSE
               DMNMX = D1
            END IF
         END IF
*
         IF( DIR.EQ.0 ) THEN
*
*           Sort into decreasing order
*
            I = START - 1
            J = ENDD + 1
   60       CONTINUE
   70       CONTINUE
            J = J - 1
            IF( D( J ).LT.DMNMX )
     $         GO TO 70
   80       CONTINUE
            I = I + 1
            IF( D( I ).GT.DMNMX )
     $         GO TO 80
            IF( I.LT.J ) THEN
               TMP = D( I )
               D( I ) = D( J )
               D( J ) = TMP
               GO TO 60
            END IF
            IF( J-START.GT.ENDD-J-1 ) THEN
               STKPNT = STKPNT + 1
               STACK( 1, STKPNT ) = START
               STACK( 2, STKPNT ) = J
               STKPNT = STKPNT + 1
               STACK( 1, STKPNT ) = J + 1
               STACK( 2, STKPNT ) = ENDD
            ELSE
               STKPNT = STKPNT + 1
               STACK( 1, STKPNT ) = J + 1
               STACK( 2, STKPNT ) = ENDD
               STKPNT = STKPNT + 1
               STACK( 1, STKPNT ) = START
               STACK( 2, STKPNT ) = J
            END IF
         ELSE
*
*           Sort into increasing order
*
            I = START - 1
            J = ENDD + 1
   90       CONTINUE
  100       CONTINUE
            J = J - 1
            IF( D( J ).GT.DMNMX )
     $         GO TO 100
  110       CONTINUE
            I = I + 1
            IF( D( I ).LT.DMNMX )
     $         GO TO 110
            IF( I.LT.J ) THEN
               TMP = D( I )
               D( I ) = D( J )
               D( J ) = TMP
               GO TO 90
            END IF
            IF( J-START.GT.ENDD-J-1 ) THEN
               STKPNT = STKPNT + 1
               STACK( 1, STKPNT ) = START
               STACK( 2, STKPNT ) = J
               STKPNT = STKPNT + 1
               STACK( 1, STKPNT ) = J + 1
               STACK( 2, STKPNT ) = ENDD
            ELSE
               STKPNT = STKPNT + 1
               STACK( 1, STKPNT ) = J + 1
               STACK( 2, STKPNT ) = ENDD
               STKPNT = STKPNT + 1
               STACK( 1, STKPNT ) = START
               STACK( 2, STKPNT ) = J
            END IF
         END IF
      END IF
      IF( STKPNT.GT.0 )
     $   GO TO 10
      RETURN
*
*     End of DLASRT
*
      END
*> \brief \b DLASV2 computes the singular value decomposition of a 2-by-2 triangular matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASV2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasv2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasv2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasv2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASV2( F, G, H, SSMIN, SSMAX, SNR, CSR, SNL, CSL )
*
*       .. Scalar Arguments ..
*       DOUBLE PRECISION   CSL, CSR, F, G, H, SNL, SNR, SSMAX, SSMIN
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASV2 computes the singular value decomposition of a 2-by-2
*> triangular matrix
*>    [  F   G  ]
*>    [  0   H  ].
*> On return, abs(SSMAX) is the larger singular value, abs(SSMIN) is the
*> smaller singular value, and (CSL,SNL) and (CSR,SNR) are the left and
*> right singular vectors for abs(SSMAX), giving the decomposition
*>
*>    [ CSL  SNL ] [  F   G  ] [ CSR -SNR ]  =  [ SSMAX   0   ]
*>    [-SNL  CSL ] [  0   H  ] [ SNR  CSR ]     [  0    SSMIN ].
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] F
*> \verbatim
*>          F is DOUBLE PRECISION
*>          The (1,1) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] G
*> \verbatim
*>          G is DOUBLE PRECISION
*>          The (1,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*>          H is DOUBLE PRECISION
*>          The (2,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[out] SSMIN
*> \verbatim
*>          SSMIN is DOUBLE PRECISION
*>          abs(SSMIN) is the smaller singular value.
*> \endverbatim
*>
*> \param[out] SSMAX
*> \verbatim
*>          SSMAX is DOUBLE PRECISION
*>          abs(SSMAX) is the larger singular value.
*> \endverbatim
*>
*> \param[out] SNL
*> \verbatim
*>          SNL is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] CSL
*> \verbatim
*>          CSL is DOUBLE PRECISION
*>          The vector (CSL, SNL) is a unit left singular vector for the
*>          singular value abs(SSMAX).
*> \endverbatim
*>
*> \param[out] SNR
*> \verbatim
*>          SNR is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] CSR
*> \verbatim
*>          CSR is DOUBLE PRECISION
*>          The vector (CSR, SNR) is a unit right singular vector for the
*>          singular value abs(SSMAX).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasv2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  Any input parameter may be aliased with any output parameter.
*>
*>  Barring over/underflow and assuming a guard digit in subtraction, all
*>  output quantities are correct to within a few units in the last
*>  place (ulps).
*>
*>  In IEEE arithmetic, the code works correctly if one matrix element is
*>  infinite.
*>
*>  Overflow will not occur unless the largest singular value itself
*>  overflows or is within a few ulps of overflow.
*>
*>  Underflow is harmless if underflow is gradual. Otherwise, results
*>  may correspond to a matrix modified by perturbations of size near
*>  the underflow threshold.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLASV2( F, G, H, SSMIN, SSMAX, SNR, CSR, SNL, CSL )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   CSL, CSR, F, G, H, SNL, SNR, SSMAX, SSMIN
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D0 )
      DOUBLE PRECISION   HALF
      PARAMETER          ( HALF = 0.5D0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D0 )
      DOUBLE PRECISION   TWO
      PARAMETER          ( TWO = 2.0D0 )
      DOUBLE PRECISION   FOUR
      PARAMETER          ( FOUR = 4.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            GASMAL, SWAP
      INTEGER            PMAX
      DOUBLE PRECISION   A, CLT, CRT, D, FA, FT, GA, GT, HA, HT, L, M,
     $                   MM, R, S, SLT, SRT, T, TEMP, TSIGN, TT
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, SIGN, SQRT
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. Executable Statements ..
*
      FT = F
      FA = ABS( FT )
      HT = H
      HA = ABS( H )
*
*     PMAX points to the maximum absolute element of matrix
*       PMAX = 1 if F largest in absolute values
*       PMAX = 2 if G largest in absolute values
*       PMAX = 3 if H largest in absolute values
*
      PMAX = 1
      SWAP = ( HA.GT.FA )
      IF( SWAP ) THEN
         PMAX = 3
         TEMP = FT
         FT = HT
         HT = TEMP
         TEMP = FA
         FA = HA
         HA = TEMP
*
*        Now FA .ge. HA
*
      END IF
      GT = G
      GA = ABS( GT )
      IF( GA.EQ.ZERO ) THEN
*
*        Diagonal matrix
*
         SSMIN = HA
         SSMAX = FA
         CLT = ONE
         CRT = ONE
         SLT = ZERO
         SRT = ZERO
      ELSE
         GASMAL = .TRUE.
         IF( GA.GT.FA ) THEN
            PMAX = 2
            IF( ( FA / GA ).LT.DLAMCH( 'EPS' ) ) THEN
*
*              Case of very large GA
*
               GASMAL = .FALSE.
               SSMAX = GA
               IF( HA.GT.ONE ) THEN
                  SSMIN = FA / ( GA / HA )
               ELSE
                  SSMIN = ( FA / GA )*HA
               END IF
               CLT = ONE
               SLT = HT / GT
               SRT = ONE
               CRT = FT / GT
            END IF
         END IF
         IF( GASMAL ) THEN
*
*           Normal case
*
            D = FA - HA
            IF( D.EQ.FA ) THEN
*
*              Copes with infinite F or H
*
               L = ONE
            ELSE
               L = D / FA
            END IF
*
*           Note that 0 .le. L .le. 1
*
            M = GT / FT
*
*           Note that abs(M) .le. 1/macheps
*
            T = TWO - L
*
*           Note that T .ge. 1
*
            MM = M*M
            TT = T*T
            S = SQRT( TT+MM )
*
*           Note that 1 .le. S .le. 1 + 1/macheps
*
            IF( L.EQ.ZERO ) THEN
               R = ABS( M )
            ELSE
               R = SQRT( L*L+MM )
            END IF
*
*           Note that 0 .le. R .le. 1 + 1/macheps
*
            A = HALF*( S+R )
*
*           Note that 1 .le. A .le. 1 + abs(M)
*
            SSMIN = HA / A
            SSMAX = FA*A
            IF( MM.EQ.ZERO ) THEN
*
*              Note that M is very tiny
*
               IF( L.EQ.ZERO ) THEN
                  T = SIGN( TWO, FT )*SIGN( ONE, GT )
               ELSE
                  T = GT / SIGN( D, FT ) + M / T
               END IF
            ELSE
               T = ( M / ( S+T )+M / ( R+L ) )*( ONE+A )
            END IF
            L = SQRT( T*T+FOUR )
            CRT = TWO / L
            SRT = T / L
            CLT = ( CRT+SRT*M ) / A
            SLT = ( HT / FT )*SRT / A
         END IF
      END IF
      IF( SWAP ) THEN
         CSL = SRT
         SNL = CRT
         CSR = SLT
         SNR = CLT
      ELSE
         CSL = CLT
         SNL = SLT
         CSR = CRT
         SNR = SRT
      END IF
*
*     Correct signs of SSMAX and SSMIN
*
      IF( PMAX.EQ.1 )
     $   TSIGN = SIGN( ONE, CSR )*SIGN( ONE, CSL )*SIGN( ONE, F )
      IF( PMAX.EQ.2 )
     $   TSIGN = SIGN( ONE, SNR )*SIGN( ONE, CSL )*SIGN( ONE, G )
      IF( PMAX.EQ.3 )
     $   TSIGN = SIGN( ONE, SNR )*SIGN( ONE, SNL )*SIGN( ONE, H )
      SSMAX = SIGN( SSMAX, TSIGN )
      SSMIN = SIGN( SSMIN, TSIGN*SIGN( ONE, F )*SIGN( ONE, H ) )
      RETURN
*
*     End of DLASV2
*
      END
*> \brief \b DLASWP performs a series of row interchanges on a general rectangular matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASWP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaswp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaswp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaswp.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASWP( N, A, LDA, K1, K2, IPIV, INCX )
*
*       .. Scalar Arguments ..
*       INTEGER            INCX, K1, K2, LDA, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   A( LDA, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASWP performs a series of row interchanges on the matrix A.
*> One row interchange is initiated for each of rows K1 through K2 of A.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the matrix of column dimension N to which the row
*>          interchanges will be applied.
*>          On exit, the permuted matrix.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.
*> \endverbatim
*>
*> \param[in] K1
*> \verbatim
*>          K1 is INTEGER
*>          The first element of IPIV for which a row interchange will
*>          be done.
*> \endverbatim
*>
*> \param[in] K2
*> \verbatim
*>          K2 is INTEGER
*>          (K2-K1+1) is the number of elements of IPIV for which a row
*>          interchange will be done.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (K1+(K2-K1)*abs(INCX))
*>          The vector of pivot indices. Only the elements in positions
*>          K1 through K1+(K2-K1)*abs(INCX) of IPIV are accessed.
*>          IPIV(K1+(K-K1)*abs(INCX)) = L implies rows K and L are to be
*>          interchanged.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*>          INCX is INTEGER
*>          The increment between successive values of IPIV. If INCX
*>          is negative, the pivots are applied in reverse order.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laswp
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  Modified by
*>   R. C. Whaley, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLASWP( N, A, LDA, K1, K2, IPIV, INCX )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INCX, K1, K2, LDA, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
* =====================================================================
*
*     .. Local Scalars ..
      INTEGER            I, I1, I2, INC, IP, IX, IX0, J, K, N32
      DOUBLE PRECISION   TEMP
*     ..
*     .. Executable Statements ..
*
*     Interchange row I with row IPIV(K1+(I-K1)*abs(INCX)) for each of rows
*     K1 through K2.
*
      IF( INCX.GT.0 ) THEN
         IX0 = K1
         I1 = K1
         I2 = K2
         INC = 1
      ELSE IF( INCX.LT.0 ) THEN
         IX0 = K1 + ( K1-K2 )*INCX
         I1 = K2
         I2 = K1
         INC = -1
      ELSE
         RETURN
      END IF
*
      N32 = ( N / 32 )*32
      IF( N32.NE.0 ) THEN
         DO 30 J = 1, N32, 32
            IX = IX0
            DO 20 I = I1, I2, INC
               IP = IPIV( IX )
               IF( IP.NE.I ) THEN
                  DO 10 K = J, J + 31
                     TEMP = A( I, K )
                     A( I, K ) = A( IP, K )
                     A( IP, K ) = TEMP
   10             CONTINUE
               END IF
               IX = IX + INCX
   20       CONTINUE
   30    CONTINUE
      END IF
      IF( N32.NE.N ) THEN
         N32 = N32 + 1
         IX = IX0
         DO 50 I = I1, I2, INC
            IP = IPIV( IX )
            IF( IP.NE.I ) THEN
               DO 40 K = N32, N
                  TEMP = A( I, K )
                  A( I, K ) = A( IP, K )
                  A( IP, K ) = TEMP
   40          CONTINUE
            END IF
            IX = IX + INCX
   50    CONTINUE
      END IF
*
      RETURN
*
*     End of DLASWP
*
      END
*> \brief \b DLASY2 solves the Sylvester matrix equation where the matrices are of order 1 or 2.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASY2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasy2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasy2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasy2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
*                          LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
*
*       .. Scalar Arguments ..
*       LOGICAL            LTRANL, LTRANR
*       INTEGER            INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
*       DOUBLE PRECISION   SCALE, XNORM
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
*      $                   X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in
*>
*>        op(TL)*X + ISGN*X*op(TR) = SCALE*B,
*>
*> where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or
*> -1.  op(T) = T or T**T, where T**T denotes the transpose of T.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] LTRANL
*> \verbatim
*>          LTRANL is LOGICAL
*>          On entry, LTRANL specifies the op(TL):
*>             = .FALSE., op(TL) = TL,
*>             = .TRUE., op(TL) = TL**T.
*> \endverbatim
*>
*> \param[in] LTRANR
*> \verbatim
*>          LTRANR is LOGICAL
*>          On entry, LTRANR specifies the op(TR):
*>            = .FALSE., op(TR) = TR,
*>            = .TRUE., op(TR) = TR**T.
*> \endverbatim
*>
*> \param[in] ISGN
*> \verbatim
*>          ISGN is INTEGER
*>          On entry, ISGN specifies the sign of the equation
*>          as described before. ISGN may only be 1 or -1.
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*>          N1 is INTEGER
*>          On entry, N1 specifies the order of matrix TL.
*>          N1 may only be 0, 1 or 2.
*> \endverbatim
*>
*> \param[in] N2
*> \verbatim
*>          N2 is INTEGER
*>          On entry, N2 specifies the order of matrix TR.
*>          N2 may only be 0, 1 or 2.
*> \endverbatim
*>
*> \param[in] TL
*> \verbatim
*>          TL is DOUBLE PRECISION array, dimension (LDTL,2)
*>          On entry, TL contains an N1 by N1 matrix.
*> \endverbatim
*>
*> \param[in] LDTL
*> \verbatim
*>          LDTL is INTEGER
*>          The leading dimension of the matrix TL. LDTL >= max(1,N1).
*> \endverbatim
*>
*> \param[in] TR
*> \verbatim
*>          TR is DOUBLE PRECISION array, dimension (LDTR,2)
*>          On entry, TR contains an N2 by N2 matrix.
*> \endverbatim
*>
*> \param[in] LDTR
*> \verbatim
*>          LDTR is INTEGER
*>          The leading dimension of the matrix TR. LDTR >= max(1,N2).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,2)
*>          On entry, the N1 by N2 matrix B contains the right-hand
*>          side of the equation.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the matrix B. LDB >= max(1,N1).
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*>          SCALE is DOUBLE PRECISION
*>          On exit, SCALE contains the scale factor. SCALE is chosen
*>          less than or equal to 1 to prevent the solution overflowing.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,2)
*>          On exit, X contains the N1 by N2 solution.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the matrix X. LDX >= max(1,N1).
*> \endverbatim
*>
*> \param[out] XNORM
*> \verbatim
*>          XNORM is DOUBLE PRECISION
*>          On exit, XNORM is the infinity-norm of the solution.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          On exit, INFO is set to
*>             0: successful exit.
*>             1: TL and TR have too close eigenvalues, so TL or
*>                TR is perturbed to get a nonsingular equation.
*>          NOTE: In the interests of speed, this routine does not
*>                check the inputs for errors.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasy2
*
*  =====================================================================
      SUBROUTINE DLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
     $                   LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      LOGICAL            LTRANL, LTRANR
      INTEGER            INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
      DOUBLE PRECISION   SCALE, XNORM
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
     $                   X( LDX, * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
      DOUBLE PRECISION   TWO, HALF, EIGHT
      PARAMETER          ( TWO = 2.0D+0, HALF = 0.5D+0, EIGHT = 8.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            BSWAP, XSWAP
      INTEGER            I, IP, IPIV, IPSV, J, JP, JPSV, K
      DOUBLE PRECISION   BET, EPS, GAM, L21, SGN, SMIN, SMLNUM, TAU1,
     $                   TEMP, U11, U12, U22, XMAX
*     ..
*     .. Local Arrays ..
      LOGICAL            BSWPIV( 4 ), XSWPIV( 4 )
      INTEGER            JPIV( 4 ), LOCL21( 4 ), LOCU12( 4 ),
     $                   LOCU22( 4 )
      DOUBLE PRECISION   BTMP( 4 ), T16( 4, 4 ), TMP( 4 ), X2( 2 )
*     ..
*     .. External Functions ..
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           IDAMAX, DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DSWAP
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. Data statements ..
      DATA               LOCU12 / 3, 4, 1, 2 / , LOCL21 / 2, 1, 4, 3 / ,
     $                   LOCU22 / 4, 3, 2, 1 /
      DATA               XSWPIV / .FALSE., .FALSE., .TRUE., .TRUE. /
      DATA               BSWPIV / .FALSE., .TRUE., .FALSE., .TRUE. /
*     ..
*     .. Executable Statements ..
*
*     Do not check the input parameters for errors
*
      INFO = 0
*
*     Quick return if possible
*
      IF( N1.EQ.0 .OR. N2.EQ.0 )
     $   RETURN
*
*     Set constants to control overflow
*
      EPS = DLAMCH( 'P' )
      SMLNUM = DLAMCH( 'S' ) / EPS
      SGN = ISGN
*
      K = N1 + N1 + N2 - 2
      GO TO ( 10, 20, 30, 50 )K
*
*     1 by 1: TL11*X + SGN*X*TR11 = B11
*
   10 CONTINUE
      TAU1 = TL( 1, 1 ) + SGN*TR( 1, 1 )
      BET = ABS( TAU1 )
      IF( BET.LE.SMLNUM ) THEN
         TAU1 = SMLNUM
         BET = SMLNUM
         INFO = 1
      END IF
*
      SCALE = ONE
      GAM = ABS( B( 1, 1 ) )
      IF( SMLNUM*GAM.GT.BET )
     $   SCALE = ONE / GAM
*
      X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / TAU1
      XNORM = ABS( X( 1, 1 ) )
      RETURN
*
*     1 by 2:
*     TL11*[X11 X12] + ISGN*[X11 X12]*op[TR11 TR12]  = [B11 B12]
*                                       [TR21 TR22]
*
   20 CONTINUE
*
      SMIN = MAX( EPS*MAX( ABS( TL( 1, 1 ) ), ABS( TR( 1, 1 ) ),
     $       ABS( TR( 1, 2 ) ), ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) ),
     $       SMLNUM )
      TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
      TMP( 4 ) = TL( 1, 1 ) + SGN*TR( 2, 2 )
      IF( LTRANR ) THEN
         TMP( 2 ) = SGN*TR( 2, 1 )
         TMP( 3 ) = SGN*TR( 1, 2 )
      ELSE
         TMP( 2 ) = SGN*TR( 1, 2 )
         TMP( 3 ) = SGN*TR( 2, 1 )
      END IF
      BTMP( 1 ) = B( 1, 1 )
      BTMP( 2 ) = B( 1, 2 )
      GO TO 40
*
*     2 by 1:
*          op[TL11 TL12]*[X11] + ISGN* [X11]*TR11  = [B11]
*            [TL21 TL22] [X21]         [X21]         [B21]
*
   30 CONTINUE
      SMIN = MAX( EPS*MAX( ABS( TR( 1, 1 ) ), ABS( TL( 1, 1 ) ),
     $       ABS( TL( 1, 2 ) ), ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) ),
     $       SMLNUM )
      TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
      TMP( 4 ) = TL( 2, 2 ) + SGN*TR( 1, 1 )
      IF( LTRANL ) THEN
         TMP( 2 ) = TL( 1, 2 )
         TMP( 3 ) = TL( 2, 1 )
      ELSE
         TMP( 2 ) = TL( 2, 1 )
         TMP( 3 ) = TL( 1, 2 )
      END IF
      BTMP( 1 ) = B( 1, 1 )
      BTMP( 2 ) = B( 2, 1 )
   40 CONTINUE
*
*     Solve 2 by 2 system using complete pivoting.
*     Set pivots less than SMIN to SMIN.
*
      IPIV = IDAMAX( 4, TMP, 1 )
      U11 = TMP( IPIV )
      IF( ABS( U11 ).LE.SMIN ) THEN
         INFO = 1
         U11 = SMIN
      END IF
      U12 = TMP( LOCU12( IPIV ) )
      L21 = TMP( LOCL21( IPIV ) ) / U11
      U22 = TMP( LOCU22( IPIV ) ) - U12*L21
      XSWAP = XSWPIV( IPIV )
      BSWAP = BSWPIV( IPIV )
      IF( ABS( U22 ).LE.SMIN ) THEN
         INFO = 1
         U22 = SMIN
      END IF
      IF( BSWAP ) THEN
         TEMP = BTMP( 2 )
         BTMP( 2 ) = BTMP( 1 ) - L21*TEMP
         BTMP( 1 ) = TEMP
      ELSE
         BTMP( 2 ) = BTMP( 2 ) - L21*BTMP( 1 )
      END IF
      SCALE = ONE
      IF( ( TWO*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( U22 ) .OR.
     $    ( TWO*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( U11 ) ) THEN
         SCALE = HALF / MAX( ABS( BTMP( 1 ) ), ABS( BTMP( 2 ) ) )
         BTMP( 1 ) = BTMP( 1 )*SCALE
         BTMP( 2 ) = BTMP( 2 )*SCALE
      END IF
      X2( 2 ) = BTMP( 2 ) / U22
      X2( 1 ) = BTMP( 1 ) / U11 - ( U12 / U11 )*X2( 2 )
      IF( XSWAP ) THEN
         TEMP = X2( 2 )
         X2( 2 ) = X2( 1 )
         X2( 1 ) = TEMP
      END IF
      X( 1, 1 ) = X2( 1 )
      IF( N1.EQ.1 ) THEN
         X( 1, 2 ) = X2( 2 )
         XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) )
      ELSE
         X( 2, 1 ) = X2( 2 )
         XNORM = MAX( ABS( X( 1, 1 ) ), ABS( X( 2, 1 ) ) )
      END IF
      RETURN
*
*     2 by 2:
*     op[TL11 TL12]*[X11 X12] +ISGN* [X11 X12]*op[TR11 TR12] = [B11 B12]
*       [TL21 TL22] [X21 X22]        [X21 X22]   [TR21 TR22]   [B21 B22]
*
*     Solve equivalent 4 by 4 system using complete pivoting.
*     Set pivots less than SMIN to SMIN.
*
   50 CONTINUE
      SMIN = MAX( ABS( TR( 1, 1 ) ), ABS( TR( 1, 2 ) ),
     $       ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) )
      SMIN = MAX( SMIN, ABS( TL( 1, 1 ) ), ABS( TL( 1, 2 ) ),
     $       ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) )
      SMIN = MAX( EPS*SMIN, SMLNUM )
      BTMP( 1 ) = ZERO
      CALL DCOPY( 16, BTMP, 0, T16, 1 )
      T16( 1, 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
      T16( 2, 2 ) = TL( 2, 2 ) + SGN*TR( 1, 1 )
      T16( 3, 3 ) = TL( 1, 1 ) + SGN*TR( 2, 2 )
      T16( 4, 4 ) = TL( 2, 2 ) + SGN*TR( 2, 2 )
      IF( LTRANL ) THEN
         T16( 1, 2 ) = TL( 2, 1 )
         T16( 2, 1 ) = TL( 1, 2 )
         T16( 3, 4 ) = TL( 2, 1 )
         T16( 4, 3 ) = TL( 1, 2 )
      ELSE
         T16( 1, 2 ) = TL( 1, 2 )
         T16( 2, 1 ) = TL( 2, 1 )
         T16( 3, 4 ) = TL( 1, 2 )
         T16( 4, 3 ) = TL( 2, 1 )
      END IF
      IF( LTRANR ) THEN
         T16( 1, 3 ) = SGN*TR( 1, 2 )
         T16( 2, 4 ) = SGN*TR( 1, 2 )
         T16( 3, 1 ) = SGN*TR( 2, 1 )
         T16( 4, 2 ) = SGN*TR( 2, 1 )
      ELSE
         T16( 1, 3 ) = SGN*TR( 2, 1 )
         T16( 2, 4 ) = SGN*TR( 2, 1 )
         T16( 3, 1 ) = SGN*TR( 1, 2 )
         T16( 4, 2 ) = SGN*TR( 1, 2 )
      END IF
      BTMP( 1 ) = B( 1, 1 )
      BTMP( 2 ) = B( 2, 1 )
      BTMP( 3 ) = B( 1, 2 )
      BTMP( 4 ) = B( 2, 2 )
*
*     Perform elimination
*
      DO 100 I = 1, 3
         XMAX = ZERO
         DO 70 IP = I, 4
            DO 60 JP = I, 4
               IF( ABS( T16( IP, JP ) ).GE.XMAX ) THEN
                  XMAX = ABS( T16( IP, JP ) )
                  IPSV = IP
                  JPSV = JP
               END IF
   60       CONTINUE
   70    CONTINUE
         IF( IPSV.NE.I ) THEN
            CALL DSWAP( 4, T16( IPSV, 1 ), 4, T16( I, 1 ), 4 )
            TEMP = BTMP( I )
            BTMP( I ) = BTMP( IPSV )
            BTMP( IPSV ) = TEMP
         END IF
         IF( JPSV.NE.I )
     $      CALL DSWAP( 4, T16( 1, JPSV ), 1, T16( 1, I ), 1 )
         JPIV( I ) = JPSV
         IF( ABS( T16( I, I ) ).LT.SMIN ) THEN
            INFO = 1
            T16( I, I ) = SMIN
         END IF
         DO 90 J = I + 1, 4
            T16( J, I ) = T16( J, I ) / T16( I, I )
            BTMP( J ) = BTMP( J ) - T16( J, I )*BTMP( I )
            DO 80 K = I + 1, 4
               T16( J, K ) = T16( J, K ) - T16( J, I )*T16( I, K )
   80       CONTINUE
   90    CONTINUE
  100 CONTINUE
      IF( ABS( T16( 4, 4 ) ).LT.SMIN ) THEN
         INFO = 1
         T16( 4, 4 ) = SMIN
      END IF
      SCALE = ONE
      IF( ( EIGHT*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( T16( 1, 1 ) ) .OR.
     $    ( EIGHT*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( T16( 2, 2 ) ) .OR.
     $    ( EIGHT*SMLNUM )*ABS( BTMP( 3 ) ).GT.ABS( T16( 3, 3 ) ) .OR.
     $    ( EIGHT*SMLNUM )*ABS( BTMP( 4 ) ).GT.ABS( T16( 4, 4 ) ) ) THEN
         SCALE = ( ONE / EIGHT ) / MAX( ABS( BTMP( 1 ) ),
     $           ABS( BTMP( 2 ) ), ABS( BTMP( 3 ) ), ABS( BTMP( 4 ) ) )
         BTMP( 1 ) = BTMP( 1 )*SCALE
         BTMP( 2 ) = BTMP( 2 )*SCALE
         BTMP( 3 ) = BTMP( 3 )*SCALE
         BTMP( 4 ) = BTMP( 4 )*SCALE
      END IF
      DO 120 I = 1, 4
         K = 5 - I
         TEMP = ONE / T16( K, K )
         TMP( K ) = BTMP( K )*TEMP
         DO 110 J = K + 1, 4
            TMP( K ) = TMP( K ) - ( TEMP*T16( K, J ) )*TMP( J )
  110    CONTINUE
  120 CONTINUE
      DO 130 I = 1, 3
         IF( JPIV( 4-I ).NE.4-I ) THEN
            TEMP = TMP( 4-I )
            TMP( 4-I ) = TMP( JPIV( 4-I ) )
            TMP( JPIV( 4-I ) ) = TEMP
         END IF
  130 CONTINUE
      X( 1, 1 ) = TMP( 1 )
      X( 2, 1 ) = TMP( 2 )
      X( 1, 2 ) = TMP( 3 )
      X( 2, 2 ) = TMP( 4 )
      XNORM = MAX( ABS( TMP( 1 ) )+ABS( TMP( 3 ) ),
     $        ABS( TMP( 2 ) )+ABS( TMP( 4 ) ) )
      RETURN
*
*     End of DLASY2
*
      END
*> \brief \b DLASYF computes a partial factorization of a real symmetric matrix using the Bunch-Kaufman diagonal pivoting method.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASYF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasyf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasyf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasyf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASYF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, KB, LDA, LDW, N, NB
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   A( LDA, * ), W( LDW, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASYF computes a partial factorization of a real symmetric matrix A
*> using the Bunch-Kaufman diagonal pivoting method. The partial
*> factorization has the form:
*>
*> A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
*>       ( 0  U22 ) (  0   D  ) ( U12**T U22**T )
*>
*> A  =  ( L11  0 ) (  D   0  ) ( L11**T L21**T )  if UPLO = 'L'
*>       ( L21  I ) (  0  A22 ) (  0       I    )
*>
*> where the order of D is at most NB. The actual order is returned in
*> the argument KB, and is either NB or NB-1, or N if N <= NB.
*>
*> DLASYF is an auxiliary routine called by DSYTRF. It uses blocked code
*> (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
*> A22 (if UPLO = 'L').
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the upper or lower triangular part of the
*>          symmetric matrix A is stored:
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*>          NB is INTEGER
*>          The maximum number of columns of the matrix A that should be
*>          factored.  NB should be at least 2 to allow for 2-by-2 pivot
*>          blocks.
*> \endverbatim
*>
*> \param[out] KB
*> \verbatim
*>          KB is INTEGER
*>          The number of columns of A that were actually factored.
*>          KB is either NB-1 or NB, or N if N <= NB.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*>          n-by-n upper triangular part of A contains the upper
*>          triangular part of the matrix A, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading n-by-n lower triangular part of A contains the lower
*>          triangular part of the matrix A, and the strictly upper
*>          triangular part of A is not referenced.
*>          On exit, A contains details of the partial factorization.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the block structure of D.
*>
*>          If UPLO = 'U':
*>             Only the last KB elements of IPIV are set.
*>
*>             If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*>             interchanged and D(k,k) is a 1-by-1 diagonal block.
*>
*>             If IPIV(k) = IPIV(k-1) < 0, then rows and columns
*>             k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
*>             is a 2-by-2 diagonal block.
*>
*>          If UPLO = 'L':
*>             Only the first KB elements of IPIV are set.
*>
*>             If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*>             interchanged and D(k,k) is a 1-by-1 diagonal block.
*>
*>             If IPIV(k) = IPIV(k+1) < 0, then rows and columns
*>             k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
*>             is a 2-by-2 diagonal block.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (LDW,NB)
*> \endverbatim
*>
*> \param[in] LDW
*> \verbatim
*>          LDW is INTEGER
*>          The leading dimension of the array W.  LDW >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
*>               has been completed, but the block diagonal matrix D is
*>               exactly singular.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lahef
*
*> \par Contributors:
*  ==================
*>
*> \verbatim
*>
*>  November 2013,  Igor Kozachenko,
*>                  Computer Science Division,
*>                  University of California, Berkeley
*> \endverbatim
*
*  =====================================================================
      SUBROUTINE DLASYF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, KB, LDA, LDW, N, NB
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * ), W( LDW, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
      DOUBLE PRECISION   EIGHT, SEVTEN
      PARAMETER          ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            IMAX, J, JJ, JMAX, JP, K, KK, KKW, KP,
     $                   KSTEP, KW
      DOUBLE PRECISION   ABSAKK, ALPHA, COLMAX, D11, D21, D22, R1,
     $                   ROWMAX, T
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX
      EXTERNAL           LSAME, IDAMAX
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGEMMTR, DGEMV, DSCAL, DSWAP
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
      INFO = 0
*
*     Initialize ALPHA for use in choosing pivot block size.
*
      ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
*
      IF( LSAME( UPLO, 'U' ) ) THEN
*
*        Factorize the trailing columns of A using the upper triangle
*        of A and working backwards, and compute the matrix W = U12*D
*        for use in updating A11
*
*        K is the main loop index, decreasing from N in steps of 1 or 2
*
*        KW is the column of W which corresponds to column K of A
*
         K = N
   10    CONTINUE
         KW = NB + K - N
*
*        Exit from loop
*
         IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
     $      GO TO 30
*
*        Copy column K of A to column KW of W and update it
*
         CALL DCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 )
         IF( K.LT.N )
     $      CALL DGEMV( 'No transpose', K, N-K, -ONE, A( 1, K+1 ),
     $                  LDA,
     $                  W( K, KW+1 ), LDW, ONE, W( 1, KW ), 1 )
*
         KSTEP = 1
*
*        Determine rows and columns to be interchanged and whether
*        a 1-by-1 or 2-by-2 pivot block will be used
*
         ABSAKK = ABS( W( K, KW ) )
*
*        IMAX is the row-index of the largest off-diagonal element in
*        column K, and COLMAX is its absolute value.
*        Determine both COLMAX and IMAX.
*
         IF( K.GT.1 ) THEN
            IMAX = IDAMAX( K-1, W( 1, KW ), 1 )
            COLMAX = ABS( W( IMAX, KW ) )
         ELSE
            COLMAX = ZERO
         END IF
*
         IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
*
*           Column K is zero or underflow: set INFO and continue
*
            IF( INFO.EQ.0 )
     $         INFO = K
            KP = K
         ELSE
            IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
*
*              no interchange, use 1-by-1 pivot block
*
               KP = K
            ELSE
*
*              Copy column IMAX to column KW-1 of W and update it
*
               CALL DCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
               CALL DCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
     $                     W( IMAX+1, KW-1 ), 1 )
               IF( K.LT.N )
     $            CALL DGEMV( 'No transpose', K, N-K, -ONE, A( 1,
     $                        K+1 ),
     $                        LDA, W( IMAX, KW+1 ), LDW, ONE,
     $                        W( 1, KW-1 ), 1 )
*
*              JMAX is the column-index of the largest off-diagonal
*              element in row IMAX, and ROWMAX is its absolute value
*
               JMAX = IMAX + IDAMAX( K-IMAX, W( IMAX+1, KW-1 ), 1 )
               ROWMAX = ABS( W( JMAX, KW-1 ) )
               IF( IMAX.GT.1 ) THEN
                  JMAX = IDAMAX( IMAX-1, W( 1, KW-1 ), 1 )
                  ROWMAX = MAX( ROWMAX, ABS( W( JMAX, KW-1 ) ) )
               END IF
*
               IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
*
*                 no interchange, use 1-by-1 pivot block
*
                  KP = K
               ELSE IF( ABS( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX ) THEN
*
*                 interchange rows and columns K and IMAX, use 1-by-1
*                 pivot block
*
                  KP = IMAX
*
*                 copy column KW-1 of W to column KW of W
*
                  CALL DCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
               ELSE
*
*                 interchange rows and columns K-1 and IMAX, use 2-by-2
*                 pivot block
*
                  KP = IMAX
                  KSTEP = 2
               END IF
            END IF
*
*           ============================================================
*
*           KK is the column of A where pivoting step stopped
*
            KK = K - KSTEP + 1
*
*           KKW is the column of W which corresponds to column KK of A
*
            KKW = NB + KK - N
*
*           Interchange rows and columns KP and KK.
*           Updated column KP is already stored in column KKW of W.
*
            IF( KP.NE.KK ) THEN
*
*              Copy non-updated column KK to column KP of submatrix A
*              at step K. No need to copy element into column K
*              (or K and K-1 for 2-by-2 pivot) of A, since these columns
*              will be later overwritten.
*
               A( KP, KP ) = A( KK, KK )
               CALL DCOPY( KK-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
     $                     LDA )
               IF( KP.GT.1 )
     $            CALL DCOPY( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
*
*              Interchange rows KK and KP in last K+1 to N columns of A
*              (columns K (or K and K-1 for 2-by-2 pivot) of A will be
*              later overwritten). Interchange rows KK and KP
*              in last KKW to NB columns of W.
*
               IF( K.LT.N )
     $            CALL DSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ),
     $                        LDA )
               CALL DSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
     $                     LDW )
            END IF
*
            IF( KSTEP.EQ.1 ) THEN
*
*              1-by-1 pivot block D(k): column kw of W now holds
*
*              W(kw) = U(k)*D(k),
*
*              where U(k) is the k-th column of U
*
*              Store subdiag. elements of column U(k)
*              and 1-by-1 block D(k) in column k of A.
*              NOTE: Diagonal element U(k,k) is a UNIT element
*              and not stored.
*                 A(k,k) := D(k,k) = W(k,kw)
*                 A(1:k-1,k) := U(1:k-1,k) = W(1:k-1,kw)/D(k,k)
*
               CALL DCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
               R1 = ONE / A( K, K )
               CALL DSCAL( K-1, R1, A( 1, K ), 1 )
*
            ELSE
*
*              2-by-2 pivot block D(k): columns kw and kw-1 of W now hold
*
*              ( W(kw-1) W(kw) ) = ( U(k-1) U(k) )*D(k)
*
*              where U(k) and U(k-1) are the k-th and (k-1)-th columns
*              of U
*
*              Store U(1:k-2,k-1) and U(1:k-2,k) and 2-by-2
*              block D(k-1:k,k-1:k) in columns k-1 and k of A.
*              NOTE: 2-by-2 diagonal block U(k-1:k,k-1:k) is a UNIT
*              block and not stored.
*                 A(k-1:k,k-1:k) := D(k-1:k,k-1:k) = W(k-1:k,kw-1:kw)
*                 A(1:k-2,k-1:k) := U(1:k-2,k:k-1:k) =
*                 = W(1:k-2,kw-1:kw) * ( D(k-1:k,k-1:k)**(-1) )
*
               IF( K.GT.2 ) THEN
*
*                 Compose the columns of the inverse of 2-by-2 pivot
*                 block D in the following way to reduce the number
*                 of FLOPS when we myltiply panel ( W(kw-1) W(kw) ) by
*                 this inverse
*
*                 D**(-1) = ( d11 d21 )**(-1) =
*                           ( d21 d22 )
*
*                 = 1/(d11*d22-d21**2) * ( ( d22 ) (-d21 ) ) =
*                                        ( (-d21 ) ( d11 ) )
*
*                 = 1/d21 * 1/((d11/d21)*(d22/d21)-1) *
*
*                   * ( ( d22/d21 ) (      -1 ) ) =
*                     ( (      -1 ) ( d11/d21 ) )
*
*                 = 1/d21 * 1/(D22*D11-1) * ( ( D11 ) (  -1 ) ) =
*                                           ( ( -1  ) ( D22 ) )
*
*                 = 1/d21 * T * ( ( D11 ) (  -1 ) )
*                               ( (  -1 ) ( D22 ) )
*
*                 = D21 * ( ( D11 ) (  -1 ) )
*                         ( (  -1 ) ( D22 ) )
*
                  D21 = W( K-1, KW )
                  D11 = W( K, KW ) / D21
                  D22 = W( K-1, KW-1 ) / D21
                  T = ONE / ( D11*D22-ONE )
                  D21 = T / D21
*
*                 Update elements in columns A(k-1) and A(k) as
*                 dot products of rows of ( W(kw-1) W(kw) ) and columns
*                 of D**(-1)
*
                  DO 20 J = 1, K - 2
                     A( J, K-1 ) = D21*( D11*W( J, KW-1 )-W( J, KW ) )
                     A( J, K ) = D21*( D22*W( J, KW )-W( J, KW-1 ) )
   20             CONTINUE
               END IF
*
*              Copy D(k) to A
*
               A( K-1, K-1 ) = W( K-1, KW-1 )
               A( K-1, K ) = W( K-1, KW )
               A( K, K ) = W( K, KW )
*
            END IF
*
         END IF
*
*        Store details of the interchanges in IPIV
*
         IF( KSTEP.EQ.1 ) THEN
            IPIV( K ) = KP
         ELSE
            IPIV( K ) = -KP
            IPIV( K-1 ) = -KP
         END IF
*
*        Decrease K and return to the start of the main loop
*
         K = K - KSTEP
         GO TO 10
*
   30    CONTINUE
*
*        Update the upper triangle of A11 (= A(1:k,1:k)) as
*
*        A11 := A11 - U12*D*U12**T = A11 - U12*W**T
*
         CALL DGEMMTR( 'Upper', 'No transpose', 'Transpose', K, N-K,
     $                 -ONE, A( 1, K+1 ), LDA, W( 1, KW+1 ), LDW,
     $                 ONE, A( 1, 1 ), LDA )
*
*        Put U12 in standard form by partially undoing the interchanges
*        in columns k+1:n looping backwards from k+1 to n
*
         J = K + 1
   60    CONTINUE
*
*           Undo the interchanges (if any) of rows JJ and JP at each
*           step J
*
*           (Here, J is a diagonal index)
            JJ = J
            JP = IPIV( J )
            IF( JP.LT.0 ) THEN
               JP = -JP
*              (Here, J is a diagonal index)
               J = J + 1
            END IF
*           (NOTE: Here, J is used to determine row length. Length N-J+1
*           of the rows to swap back doesn't include diagonal element)
            J = J + 1
            IF( JP.NE.JJ .AND. J.LE.N )
     $         CALL DSWAP( N-J+1, A( JP, J ), LDA, A( JJ, J ), LDA )
         IF( J.LT.N )
     $      GO TO 60
*
*        Set KB to the number of columns factorized
*
         KB = N - K
*
      ELSE
*
*        Factorize the leading columns of A using the lower triangle
*        of A and working forwards, and compute the matrix W = L21*D
*        for use in updating A22
*
*        K is the main loop index, increasing from 1 in steps of 1 or 2
*
         K = 1
   70    CONTINUE
*
*        Exit from loop
*
         IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
     $      GO TO 90
*
*        Copy column K of A to column K of W and update it
*
         CALL DCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 )
         CALL DGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K, 1 ),
     $               LDA,
     $               W( K, 1 ), LDW, ONE, W( K, K ), 1 )
*
         KSTEP = 1
*
*        Determine rows and columns to be interchanged and whether
*        a 1-by-1 or 2-by-2 pivot block will be used
*
         ABSAKK = ABS( W( K, K ) )
*
*        IMAX is the row-index of the largest off-diagonal element in
*        column K, and COLMAX is its absolute value.
*        Determine both COLMAX and IMAX.
*
         IF( K.LT.N ) THEN
            IMAX = K + IDAMAX( N-K, W( K+1, K ), 1 )
            COLMAX = ABS( W( IMAX, K ) )
         ELSE
            COLMAX = ZERO
         END IF
*
         IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
*
*           Column K is zero or underflow: set INFO and continue
*
            IF( INFO.EQ.0 )
     $         INFO = K
            KP = K
         ELSE
            IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
*
*              no interchange, use 1-by-1 pivot block
*
               KP = K
            ELSE
*
*              Copy column IMAX to column K+1 of W and update it
*
               CALL DCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ),
     $                     1 )
               CALL DCOPY( N-IMAX+1, A( IMAX, IMAX ), 1, W( IMAX,
     $                     K+1 ),
     $                     1 )
               CALL DGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K,
     $                     1 ),
     $                     LDA, W( IMAX, 1 ), LDW, ONE, W( K, K+1 ), 1 )
*
*              JMAX is the column-index of the largest off-diagonal
*              element in row IMAX, and ROWMAX is its absolute value
*
               JMAX = K - 1 + IDAMAX( IMAX-K, W( K, K+1 ), 1 )
               ROWMAX = ABS( W( JMAX, K+1 ) )
               IF( IMAX.LT.N ) THEN
                  JMAX = IMAX + IDAMAX( N-IMAX, W( IMAX+1, K+1 ), 1 )
                  ROWMAX = MAX( ROWMAX, ABS( W( JMAX, K+1 ) ) )
               END IF
*
               IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
*
*                 no interchange, use 1-by-1 pivot block
*
                  KP = K
               ELSE IF( ABS( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX ) THEN
*
*                 interchange rows and columns K and IMAX, use 1-by-1
*                 pivot block
*
                  KP = IMAX
*
*                 copy column K+1 of W to column K of W
*
                  CALL DCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
               ELSE
*
*                 interchange rows and columns K+1 and IMAX, use 2-by-2
*                 pivot block
*
                  KP = IMAX
                  KSTEP = 2
               END IF
            END IF
*
*           ============================================================
*
*           KK is the column of A where pivoting step stopped
*
            KK = K + KSTEP - 1
*
*           Interchange rows and columns KP and KK.
*           Updated column KP is already stored in column KK of W.
*
            IF( KP.NE.KK ) THEN
*
*              Copy non-updated column KK to column KP of submatrix A
*              at step K. No need to copy element into column K
*              (or K and K+1 for 2-by-2 pivot) of A, since these columns
*              will be later overwritten.
*
               A( KP, KP ) = A( KK, KK )
               CALL DCOPY( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
     $                     LDA )
               IF( KP.LT.N )
     $            CALL DCOPY( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ),
     $                        1 )
*
*              Interchange rows KK and KP in first K-1 columns of A
*              (columns K (or K and K+1 for 2-by-2 pivot) of A will be
*              later overwritten). Interchange rows KK and KP
*              in first KK columns of W.
*
               IF( K.GT.1 )
     $            CALL DSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
               CALL DSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
            END IF
*
            IF( KSTEP.EQ.1 ) THEN
*
*              1-by-1 pivot block D(k): column k of W now holds
*
*              W(k) = L(k)*D(k),
*
*              where L(k) is the k-th column of L
*
*              Store subdiag. elements of column L(k)
*              and 1-by-1 block D(k) in column k of A.
*              (NOTE: Diagonal element L(k,k) is a UNIT element
*              and not stored)
*                 A(k,k) := D(k,k) = W(k,k)
*                 A(k+1:N,k) := L(k+1:N,k) = W(k+1:N,k)/D(k,k)
*
               CALL DCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
               IF( K.LT.N ) THEN
                  R1 = ONE / A( K, K )
                  CALL DSCAL( N-K, R1, A( K+1, K ), 1 )
               END IF
*
            ELSE
*
*              2-by-2 pivot block D(k): columns k and k+1 of W now hold
*
*              ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
*
*              where L(k) and L(k+1) are the k-th and (k+1)-th columns
*              of L
*
*              Store L(k+2:N,k) and L(k+2:N,k+1) and 2-by-2
*              block D(k:k+1,k:k+1) in columns k and k+1 of A.
*              (NOTE: 2-by-2 diagonal block L(k:k+1,k:k+1) is a UNIT
*              block and not stored)
*                 A(k:k+1,k:k+1) := D(k:k+1,k:k+1) = W(k:k+1,k:k+1)
*                 A(k+2:N,k:k+1) := L(k+2:N,k:k+1) =
*                 = W(k+2:N,k:k+1) * ( D(k:k+1,k:k+1)**(-1) )
*
               IF( K.LT.N-1 ) THEN
*
*                 Compose the columns of the inverse of 2-by-2 pivot
*                 block D in the following way to reduce the number
*                 of FLOPS when we myltiply panel ( W(k) W(k+1) ) by
*                 this inverse
*
*                 D**(-1) = ( d11 d21 )**(-1) =
*                           ( d21 d22 )
*
*                 = 1/(d11*d22-d21**2) * ( ( d22 ) (-d21 ) ) =
*                                        ( (-d21 ) ( d11 ) )
*
*                 = 1/d21 * 1/((d11/d21)*(d22/d21)-1) *
*
*                   * ( ( d22/d21 ) (      -1 ) ) =
*                     ( (      -1 ) ( d11/d21 ) )
*
*                 = 1/d21 * 1/(D22*D11-1) * ( ( D11 ) (  -1 ) ) =
*                                           ( ( -1  ) ( D22 ) )
*
*                 = 1/d21 * T * ( ( D11 ) (  -1 ) )
*                               ( (  -1 ) ( D22 ) )
*
*                 = D21 * ( ( D11 ) (  -1 ) )
*                         ( (  -1 ) ( D22 ) )
*
                  D21 = W( K+1, K )
                  D11 = W( K+1, K+1 ) / D21
                  D22 = W( K, K ) / D21
                  T = ONE / ( D11*D22-ONE )
                  D21 = T / D21
*
*                 Update elements in columns A(k) and A(k+1) as
*                 dot products of rows of ( W(k) W(k+1) ) and columns
*                 of D**(-1)
*
                  DO 80 J = K + 2, N
                     A( J, K ) = D21*( D11*W( J, K )-W( J, K+1 ) )
                     A( J, K+1 ) = D21*( D22*W( J, K+1 )-W( J, K ) )
   80             CONTINUE
               END IF
*
*              Copy D(k) to A
*
               A( K, K ) = W( K, K )
               A( K+1, K ) = W( K+1, K )
               A( K+1, K+1 ) = W( K+1, K+1 )
*
            END IF
*
         END IF
*
*        Store details of the interchanges in IPIV
*
         IF( KSTEP.EQ.1 ) THEN
            IPIV( K ) = KP
         ELSE
            IPIV( K ) = -KP
            IPIV( K+1 ) = -KP
         END IF
*
*        Increase K and return to the start of the main loop
*
         K = K + KSTEP
         GO TO 70
*
   90    CONTINUE
*
*        Update the lower triangle of A22 (= A(k:n,k:n)) as
*
*        A22 := A22 - L21*D*L21**T = A22 - L21*W**T
*
         CALL DGEMMTR( 'Lower', 'No transpose', 'Transpose', N-K+1,
     $                 K-1, -ONE, A( K, 1 ), LDA, W( K, 1 ), LDW,
     $                 ONE, A( K, K ), LDA )
*
*        Put L21 in standard form by partially undoing the interchanges
*        of rows in columns 1:k-1 looping backwards from k-1 to 1
*
         J = K - 1
  120    CONTINUE
*
*           Undo the interchanges (if any) of rows JJ and JP at each
*           step J
*
*           (Here, J is a diagonal index)
            JJ = J
            JP = IPIV( J )
            IF( JP.LT.0 ) THEN
               JP = -JP
*              (Here, J is a diagonal index)
               J = J - 1
            END IF
*           (NOTE: Here, J is used to determine row length. Length J
*           of the rows to swap back doesn't include diagonal element)
            J = J - 1
            IF( JP.NE.JJ .AND. J.GE.1 )
     $         CALL DSWAP( J, A( JP, 1 ), LDA, A( JJ, 1 ), LDA )
         IF( J.GT.1 )
     $      GO TO 120
*
*        Set KB to the number of columns factorized
*
         KB = K - 1
*
      END IF
      RETURN
*
*     End of DLASYF
*
      END
*> \brief \b DLATBS solves a triangular banded system of equations.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLATBS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatbs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatbs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatbs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLATBS( UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X,
*                          SCALE, CNORM, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, NORMIN, TRANS, UPLO
*       INTEGER            INFO, KD, LDAB, N
*       DOUBLE PRECISION   SCALE
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * ), CNORM( * ), X( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLATBS solves one of the triangular systems
*>
*>    A *x = s*b  or  A**T*x = s*b
*>
*> with scaling to prevent overflow, where A is an upper or lower
*> triangular band matrix.  Here A**T denotes the transpose of A, x and b
*> are n-element vectors, and s is a scaling factor, usually less than
*> or equal to 1, chosen so that the components of x will be less than
*> the overflow threshold.  If the unscaled problem will not cause
*> overflow, the Level 2 BLAS routine DTBSV is called.  If the matrix A
*> is singular (A(j,j) = 0 for some j), then s is set to 0 and a
*> non-trivial solution to A*x = 0 is returned.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the matrix A is upper or lower triangular.
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the operation applied to A.
*>          = 'N':  Solve A * x = s*b  (No transpose)
*>          = 'T':  Solve A**T* x = s*b  (Transpose)
*>          = 'C':  Solve A**T* x = s*b  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          Specifies whether or not the matrix A is unit triangular.
*>          = 'N':  Non-unit triangular
*>          = 'U':  Unit triangular
*> \endverbatim
*>
*> \param[in] NORMIN
*> \verbatim
*>          NORMIN is CHARACTER*1
*>          Specifies whether CNORM has been set or not.
*>          = 'Y':  CNORM contains the column norms on entry
*>          = 'N':  CNORM is not set on entry.  On exit, the norms will
*>                  be computed and stored in CNORM.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of subdiagonals or superdiagonals in the
*>          triangular matrix A.  KD >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          The upper or lower triangular band matrix A, stored in the
*>          first KD+1 rows of the array. The j-th column of A is stored
*>          in the j-th column of the array AB as follows:
*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (N)
*>          On entry, the right hand side b of the triangular system.
*>          On exit, X is overwritten by the solution vector x.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*>          SCALE is DOUBLE PRECISION
*>          The scaling factor s for the triangular system
*>             A * x = s*b  or  A**T* x = s*b.
*>          If SCALE = 0, the matrix A is singular or badly scaled, and
*>          the vector x is an exact or approximate solution to A*x = 0.
*> \endverbatim
*>
*> \param[in,out] CNORM
*> \verbatim
*>          CNORM is DOUBLE PRECISION array, dimension (N)
*>
*>          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
*>          contains the norm of the off-diagonal part of the j-th column
*>          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
*>          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
*>          must be greater than or equal to the 1-norm.
*>
*>          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
*>          returns the 1-norm of the offdiagonal part of the j-th column
*>          of A.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -k, the k-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup latbs
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  A rough bound on x is computed; if that is less than overflow, DTBSV
*>  is called, otherwise, specific code is used which checks for possible
*>  overflow or divide-by-zero at every operation.
*>
*>  A columnwise scheme is used for solving A*x = b.  The basic algorithm
*>  if A is lower triangular is
*>
*>       x[1:n] := b[1:n]
*>       for j = 1, ..., n
*>            x(j) := x(j) / A(j,j)
*>            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
*>       end
*>
*>  Define bounds on the components of x after j iterations of the loop:
*>     M(j) = bound on x[1:j]
*>     G(j) = bound on x[j+1:n]
*>  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
*>
*>  Then for iteration j+1 we have
*>     M(j+1) <= G(j) / | A(j+1,j+1) |
*>     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
*>            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
*>
*>  where CNORM(j+1) is greater than or equal to the infinity-norm of
*>  column j+1 of A, not counting the diagonal.  Hence
*>
*>     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
*>                  1<=i<=j
*>  and
*>
*>     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
*>                                   1<=i< j
*>
*>  Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTBSV if the
*>  reciprocal of the largest M(j), j=1,..,n, is larger than
*>  max(underflow, 1/overflow).
*>
*>  The bound on x(j) is also used to determine when a step in the
*>  columnwise method can be performed without fear of overflow.  If
*>  the computed bound is greater than a large constant, x is scaled to
*>  prevent overflow, but if the bound overflows, x is set to 0, x(j) to
*>  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
*>
*>  Similarly, a row-wise scheme is used to solve A**T*x = b.  The basic
*>  algorithm for A upper triangular is
*>
*>       for j = 1, ..., n
*>            x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
*>       end
*>
*>  We simultaneously compute two bounds
*>       G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j
*>       M(j) = bound on x(i), 1<=i<=j
*>
*>  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
*>  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
*>  Then the bound on x(j) is
*>
*>       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
*>
*>            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
*>                      1<=i<=j
*>
*>  and we can safely call DTBSV if 1/M(n) and 1/G(n) are both greater
*>  than max(underflow, 1/overflow).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLATBS( UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB,
     $                   X,
     $                   SCALE, CNORM, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, NORMIN, TRANS, UPLO
      INTEGER            INFO, KD, LDAB, N
      DOUBLE PRECISION   SCALE
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * ), CNORM( * ), X( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, HALF, ONE
      PARAMETER          ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOTRAN, NOUNIT, UPPER
      INTEGER            I, IMAX, J, JFIRST, JINC, JLAST, JLEN, MAIND
      DOUBLE PRECISION   BIGNUM, GROW, REC, SMLNUM, SUMJ, TJJ, TJJS,
     $                   TMAX, TSCAL, USCAL, XBND, XJ, XMAX
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX
      DOUBLE PRECISION   DASUM, DDOT, DLAMCH
      EXTERNAL           LSAME, IDAMAX, DASUM, DDOT, DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DSCAL, DTBSV, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      NOTRAN = LSAME( TRANS, 'N' )
      NOUNIT = LSAME( DIAG, 'N' )
*
*     Test the input parameters.
*
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
     $         LSAME( TRANS, 'C' ) ) THEN
         INFO = -2
      ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
         INFO = -3
      ELSE IF( .NOT.LSAME( NORMIN, 'Y' ) .AND. .NOT.
     $         LSAME( NORMIN, 'N' ) ) THEN
         INFO = -4
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      ELSE IF( KD.LT.0 ) THEN
         INFO = -6
      ELSE IF( LDAB.LT.KD+1 ) THEN
         INFO = -8
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLATBS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      SCALE = ONE
      IF( N.EQ.0 )
     $   RETURN
*
*     Determine machine dependent parameters to control overflow.
*
      SMLNUM = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
      BIGNUM = ONE / SMLNUM
*
      IF( LSAME( NORMIN, 'N' ) ) THEN
*
*        Compute the 1-norm of each column, not including the diagonal.
*
         IF( UPPER ) THEN
*
*           A is upper triangular.
*
            DO 10 J = 1, N
               JLEN = MIN( KD, J-1 )
               CNORM( J ) = DASUM( JLEN, AB( KD+1-JLEN, J ), 1 )
   10       CONTINUE
         ELSE
*
*           A is lower triangular.
*
            DO 20 J = 1, N
               JLEN = MIN( KD, N-J )
               IF( JLEN.GT.0 ) THEN
                  CNORM( J ) = DASUM( JLEN, AB( 2, J ), 1 )
               ELSE
                  CNORM( J ) = ZERO
               END IF
   20       CONTINUE
         END IF
      END IF
*
*     Scale the column norms by TSCAL if the maximum element in CNORM is
*     greater than BIGNUM.
*
      IMAX = IDAMAX( N, CNORM, 1 )
      TMAX = CNORM( IMAX )
      IF( TMAX.LE.BIGNUM ) THEN
         TSCAL = ONE
      ELSE
         TSCAL = ONE / ( SMLNUM*TMAX )
         CALL DSCAL( N, TSCAL, CNORM, 1 )
      END IF
*
*     Compute a bound on the computed solution vector to see if the
*     Level 2 BLAS routine DTBSV can be used.
*
      J = IDAMAX( N, X, 1 )
      XMAX = ABS( X( J ) )
      XBND = XMAX
      IF( NOTRAN ) THEN
*
*        Compute the growth in A * x = b.
*
         IF( UPPER ) THEN
            JFIRST = N
            JLAST = 1
            JINC = -1
            MAIND = KD + 1
         ELSE
            JFIRST = 1
            JLAST = N
            JINC = 1
            MAIND = 1
         END IF
*
         IF( TSCAL.NE.ONE ) THEN
            GROW = ZERO
            GO TO 50
         END IF
*
         IF( NOUNIT ) THEN
*
*           A is non-unit triangular.
*
*           Compute GROW = 1/G(j) and XBND = 1/M(j).
*           Initially, G(0) = max{x(i), i=1,...,n}.
*
            GROW = ONE / MAX( XBND, SMLNUM )
            XBND = GROW
            DO 30 J = JFIRST, JLAST, JINC
*
*              Exit the loop if the growth factor is too small.
*
               IF( GROW.LE.SMLNUM )
     $            GO TO 50
*
*              M(j) = G(j-1) / abs(A(j,j))
*
               TJJ = ABS( AB( MAIND, J ) )
               XBND = MIN( XBND, MIN( ONE, TJJ )*GROW )
               IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN
*
*                 G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
*
                  GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) )
               ELSE
*
*                 G(j) could overflow, set GROW to 0.
*
                  GROW = ZERO
               END IF
   30       CONTINUE
            GROW = XBND
         ELSE
*
*           A is unit triangular.
*
*           Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
*
            GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
            DO 40 J = JFIRST, JLAST, JINC
*
*              Exit the loop if the growth factor is too small.
*
               IF( GROW.LE.SMLNUM )
     $            GO TO 50
*
*              G(j) = G(j-1)*( 1 + CNORM(j) )
*
               GROW = GROW*( ONE / ( ONE+CNORM( J ) ) )
   40       CONTINUE
         END IF
   50    CONTINUE
*
      ELSE
*
*        Compute the growth in A**T * x = b.
*
         IF( UPPER ) THEN
            JFIRST = 1
            JLAST = N
            JINC = 1
            MAIND = KD + 1
         ELSE
            JFIRST = N
            JLAST = 1
            JINC = -1
            MAIND = 1
         END IF
*
         IF( TSCAL.NE.ONE ) THEN
            GROW = ZERO
            GO TO 80
         END IF
*
         IF( NOUNIT ) THEN
*
*           A is non-unit triangular.
*
*           Compute GROW = 1/G(j) and XBND = 1/M(j).
*           Initially, M(0) = max{x(i), i=1,...,n}.
*
            GROW = ONE / MAX( XBND, SMLNUM )
            XBND = GROW
            DO 60 J = JFIRST, JLAST, JINC
*
*              Exit the loop if the growth factor is too small.
*
               IF( GROW.LE.SMLNUM )
     $            GO TO 80
*
*              G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
*
               XJ = ONE + CNORM( J )
               GROW = MIN( GROW, XBND / XJ )
*
*              M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
*
               TJJ = ABS( AB( MAIND, J ) )
               IF( XJ.GT.TJJ )
     $            XBND = XBND*( TJJ / XJ )
   60       CONTINUE
            GROW = MIN( GROW, XBND )
         ELSE
*
*           A is unit triangular.
*
*           Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
*
            GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
            DO 70 J = JFIRST, JLAST, JINC
*
*              Exit the loop if the growth factor is too small.
*
               IF( GROW.LE.SMLNUM )
     $            GO TO 80
*
*              G(j) = ( 1 + CNORM(j) )*G(j-1)
*
               XJ = ONE + CNORM( J )
               GROW = GROW / XJ
   70       CONTINUE
         END IF
   80    CONTINUE
      END IF
*
      IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN
*
*        Use the Level 2 BLAS solve if the reciprocal of the bound on
*        elements of X is not too small.
*
         CALL DTBSV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, X, 1 )
      ELSE
*
*        Use a Level 1 BLAS solve, scaling intermediate results.
*
         IF( XMAX.GT.BIGNUM ) THEN
*
*           Scale X so that its components are less than or equal to
*           BIGNUM in absolute value.
*
            SCALE = BIGNUM / XMAX
            CALL DSCAL( N, SCALE, X, 1 )
            XMAX = BIGNUM
         END IF
*
         IF( NOTRAN ) THEN
*
*           Solve A * x = b
*
            DO 110 J = JFIRST, JLAST, JINC
*
*              Compute x(j) = b(j) / A(j,j), scaling x if necessary.
*
               XJ = ABS( X( J ) )
               IF( NOUNIT ) THEN
                  TJJS = AB( MAIND, J )*TSCAL
               ELSE
                  TJJS = TSCAL
                  IF( TSCAL.EQ.ONE )
     $               GO TO 100
               END IF
               TJJ = ABS( TJJS )
               IF( TJJ.GT.SMLNUM ) THEN
*
*                    abs(A(j,j)) > SMLNUM:
*
                  IF( TJJ.LT.ONE ) THEN
                     IF( XJ.GT.TJJ*BIGNUM ) THEN
*
*                          Scale x by 1/b(j).
*
                        REC = ONE / XJ
                        CALL DSCAL( N, REC, X, 1 )
                        SCALE = SCALE*REC
                        XMAX = XMAX*REC
                     END IF
                  END IF
                  X( J ) = X( J ) / TJJS
                  XJ = ABS( X( J ) )
               ELSE IF( TJJ.GT.ZERO ) THEN
*
*                    0 < abs(A(j,j)) <= SMLNUM:
*
                  IF( XJ.GT.TJJ*BIGNUM ) THEN
*
*                       Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
*                       to avoid overflow when dividing by A(j,j).
*
                     REC = ( TJJ*BIGNUM ) / XJ
                     IF( CNORM( J ).GT.ONE ) THEN
*
*                          Scale by 1/CNORM(j) to avoid overflow when
*                          multiplying x(j) times column j.
*
                        REC = REC / CNORM( J )
                     END IF
                     CALL DSCAL( N, REC, X, 1 )
                     SCALE = SCALE*REC
                     XMAX = XMAX*REC
                  END IF
                  X( J ) = X( J ) / TJJS
                  XJ = ABS( X( J ) )
               ELSE
*
*                    A(j,j) = 0:  Set x(1:n) = 0, x(j) = 1, and
*                    scale = 0, and compute a solution to A*x = 0.
*
                  DO 90 I = 1, N
                     X( I ) = ZERO
   90             CONTINUE
                  X( J ) = ONE
                  XJ = ONE
                  SCALE = ZERO
                  XMAX = ZERO
               END IF
  100          CONTINUE
*
*              Scale x if necessary to avoid overflow when adding a
*              multiple of column j of A.
*
               IF( XJ.GT.ONE ) THEN
                  REC = ONE / XJ
                  IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN
*
*                    Scale x by 1/(2*abs(x(j))).
*
                     REC = REC*HALF
                     CALL DSCAL( N, REC, X, 1 )
                     SCALE = SCALE*REC
                  END IF
               ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN
*
*                 Scale x by 1/2.
*
                  CALL DSCAL( N, HALF, X, 1 )
                  SCALE = SCALE*HALF
               END IF
*
               IF( UPPER ) THEN
                  IF( J.GT.1 ) THEN
*
*                    Compute the update
*                       x(max(1,j-kd):j-1) := x(max(1,j-kd):j-1) -
*                                             x(j)* A(max(1,j-kd):j-1,j)
*
                     JLEN = MIN( KD, J-1 )
                     CALL DAXPY( JLEN, -X( J )*TSCAL,
     $                           AB( KD+1-JLEN, J ), 1, X( J-JLEN ), 1 )
                     I = IDAMAX( J-1, X, 1 )
                     XMAX = ABS( X( I ) )
                  END IF
               ELSE IF( J.LT.N ) THEN
*
*                 Compute the update
*                    x(j+1:min(j+kd,n)) := x(j+1:min(j+kd,n)) -
*                                          x(j) * A(j+1:min(j+kd,n),j)
*
                  JLEN = MIN( KD, N-J )
                  IF( JLEN.GT.0 )
     $               CALL DAXPY( JLEN, -X( J )*TSCAL, AB( 2, J ), 1,
     $                           X( J+1 ), 1 )
                  I = J + IDAMAX( N-J, X( J+1 ), 1 )
                  XMAX = ABS( X( I ) )
               END IF
  110       CONTINUE
*
         ELSE
*
*           Solve A**T * x = b
*
            DO 160 J = JFIRST, JLAST, JINC
*
*              Compute x(j) = b(j) - sum A(k,j)*x(k).
*                                    k<>j
*
               XJ = ABS( X( J ) )
               USCAL = TSCAL
               REC = ONE / MAX( XMAX, ONE )
               IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN
*
*                 If x(j) could overflow, scale x by 1/(2*XMAX).
*
                  REC = REC*HALF
                  IF( NOUNIT ) THEN
                     TJJS = AB( MAIND, J )*TSCAL
                  ELSE
                     TJJS = TSCAL
                  END IF
                  TJJ = ABS( TJJS )
                  IF( TJJ.GT.ONE ) THEN
*
*                       Divide by A(j,j) when scaling x if A(j,j) > 1.
*
                     REC = MIN( ONE, REC*TJJ )
                     USCAL = USCAL / TJJS
                  END IF
                  IF( REC.LT.ONE ) THEN
                     CALL DSCAL( N, REC, X, 1 )
                     SCALE = SCALE*REC
                     XMAX = XMAX*REC
                  END IF
               END IF
*
               SUMJ = ZERO
               IF( USCAL.EQ.ONE ) THEN
*
*                 If the scaling needed for A in the dot product is 1,
*                 call DDOT to perform the dot product.
*
                  IF( UPPER ) THEN
                     JLEN = MIN( KD, J-1 )
                     SUMJ = DDOT( JLEN, AB( KD+1-JLEN, J ), 1,
     $                      X( J-JLEN ), 1 )
                  ELSE
                     JLEN = MIN( KD, N-J )
                     IF( JLEN.GT.0 )
     $                  SUMJ = DDOT( JLEN, AB( 2, J ), 1, X( J+1 ),
     $                               1 )
                  END IF
               ELSE
*
*                 Otherwise, use in-line code for the dot product.
*
                  IF( UPPER ) THEN
                     JLEN = MIN( KD, J-1 )
                     DO 120 I = 1, JLEN
                        SUMJ = SUMJ + ( AB( KD+I-JLEN, J )*USCAL )*
     $                         X( J-JLEN-1+I )
  120                CONTINUE
                  ELSE
                     JLEN = MIN( KD, N-J )
                     DO 130 I = 1, JLEN
                        SUMJ = SUMJ + ( AB( I+1, J )*USCAL )*X( J+I )
  130                CONTINUE
                  END IF
               END IF
*
               IF( USCAL.EQ.TSCAL ) THEN
*
*                 Compute x(j) := ( x(j) - sumj ) / A(j,j) if 1/A(j,j)
*                 was not used to scale the dotproduct.
*
                  X( J ) = X( J ) - SUMJ
                  XJ = ABS( X( J ) )
                  IF( NOUNIT ) THEN
*
*                    Compute x(j) = x(j) / A(j,j), scaling if necessary.
*
                     TJJS = AB( MAIND, J )*TSCAL
                  ELSE
                     TJJS = TSCAL
                     IF( TSCAL.EQ.ONE )
     $                  GO TO 150
                  END IF
                  TJJ = ABS( TJJS )
                  IF( TJJ.GT.SMLNUM ) THEN
*
*                       abs(A(j,j)) > SMLNUM:
*
                     IF( TJJ.LT.ONE ) THEN
                        IF( XJ.GT.TJJ*BIGNUM ) THEN
*
*                             Scale X by 1/abs(x(j)).
*
                           REC = ONE / XJ
                           CALL DSCAL( N, REC, X, 1 )
                           SCALE = SCALE*REC
                           XMAX = XMAX*REC
                        END IF
                     END IF
                     X( J ) = X( J ) / TJJS
                  ELSE IF( TJJ.GT.ZERO ) THEN
*
*                       0 < abs(A(j,j)) <= SMLNUM:
*
                     IF( XJ.GT.TJJ*BIGNUM ) THEN
*
*                          Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
*
                        REC = ( TJJ*BIGNUM ) / XJ
                        CALL DSCAL( N, REC, X, 1 )
                        SCALE = SCALE*REC
                        XMAX = XMAX*REC
                     END IF
                     X( J ) = X( J ) / TJJS
                  ELSE
*
*                       A(j,j) = 0:  Set x(1:n) = 0, x(j) = 1, and
*                       scale = 0, and compute a solution to A**T*x = 0.
*
                     DO 140 I = 1, N
                        X( I ) = ZERO
  140                CONTINUE
                     X( J ) = ONE
                     SCALE = ZERO
                     XMAX = ZERO
                  END IF
  150             CONTINUE
               ELSE
*
*                 Compute x(j) := x(j) / A(j,j) - sumj if the dot
*                 product has already been divided by 1/A(j,j).
*
                  X( J ) = X( J ) / TJJS - SUMJ
               END IF
               XMAX = MAX( XMAX, ABS( X( J ) ) )
  160       CONTINUE
         END IF
         SCALE = SCALE / TSCAL
      END IF
*
*     Scale the column norms by 1/TSCAL for return.
*
      IF( TSCAL.NE.ONE ) THEN
         CALL DSCAL( N, ONE / TSCAL, CNORM, 1 )
      END IF
*
      RETURN
*
*     End of DLATBS
*
      END
*> \brief \b DLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLATDF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatdf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatdf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatdf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
*                          JPIV )
*
*       .. Scalar Arguments ..
*       INTEGER            IJOB, LDZ, N
*       DOUBLE PRECISION   RDSCAL, RDSUM
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), JPIV( * )
*       DOUBLE PRECISION   RHS( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLATDF uses the LU factorization of the n-by-n matrix Z computed by
*> DGETC2 and computes a contribution to the reciprocal Dif-estimate
*> by solving Z * x = b for x, and choosing the r.h.s. b such that
*> the norm of x is as large as possible. On entry RHS = b holds the
*> contribution from earlier solved sub-systems, and on return RHS = x.
*>
*> The factorization of Z returned by DGETC2 has the form Z = P*L*U*Q,
*> where P and Q are permutation matrices. L is lower triangular with
*> unit diagonal elements and U is upper triangular.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] IJOB
*> \verbatim
*>          IJOB is INTEGER
*>          IJOB = 2: First compute an approximative null-vector e
*>              of Z using DGECON, e is normalized and solve for
*>              Zx = +-e - f with the sign giving the greater value
*>              of 2-norm(x). About 5 times as expensive as Default.
*>          IJOB .ne. 2: Local look ahead strategy where all entries of
*>              the r.h.s. b is chosen as either +1 or -1 (Default).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix Z.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
*>          On entry, the LU part of the factorization of the n-by-n
*>          matrix Z computed by DGETC2:  Z = P * L * U * Q
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDA >= max(1, N).
*> \endverbatim
*>
*> \param[in,out] RHS
*> \verbatim
*>          RHS is DOUBLE PRECISION array, dimension (N)
*>          On entry, RHS contains contributions from other subsystems.
*>          On exit, RHS contains the solution of the subsystem with
*>          entries according to the value of IJOB (see above).
*> \endverbatim
*>
*> \param[in,out] RDSUM
*> \verbatim
*>          RDSUM is DOUBLE PRECISION
*>          On entry, the sum of squares of computed contributions to
*>          the Dif-estimate under computation by DTGSYL, where the
*>          scaling factor RDSCAL (see below) has been factored out.
*>          On exit, the corresponding sum of squares updated with the
*>          contributions from the current sub-system.
*>          If TRANS = 'T' RDSUM is not touched.
*>          NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL.
*> \endverbatim
*>
*> \param[in,out] RDSCAL
*> \verbatim
*>          RDSCAL is DOUBLE PRECISION
*>          On entry, scaling factor used to prevent overflow in RDSUM.
*>          On exit, RDSCAL is updated w.r.t. the current contributions
*>          in RDSUM.
*>          If TRANS = 'T', RDSCAL is not touched.
*>          NOTE: RDSCAL only makes sense when DTGSY2 is called by
*>                DTGSYL.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N).
*>          The pivot indices; for 1 <= i <= N, row i of the
*>          matrix has been interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in] JPIV
*> \verbatim
*>          JPIV is INTEGER array, dimension (N).
*>          The pivot indices; for 1 <= j <= N, column j of the
*>          matrix has been interchanged with column JPIV(j).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup latdf
*
*> \par Further Details:
*  =====================
*>
*>  This routine is a further developed implementation of algorithm
*>  BSOLVE in [1] using complete pivoting in the LU factorization.
*
*> \par Contributors:
*  ==================
*>
*>     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*>     Umea University, S-901 87 Umea, Sweden.
*
*> \par References:
*  ================
*>
*> \verbatim
*>
*>
*>  [1] Bo Kagstrom and Lars Westin,
*>      Generalized Schur Methods with Condition Estimators for
*>      Solving the Generalized Sylvester Equation, IEEE Transactions
*>      on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
*>
*>  [2] Peter Poromaa,
*>      On Efficient and Robust Estimators for the Separation
*>      between two Regular Matrix Pairs with Applications in
*>      Condition Estimation. Report IMINF-95.05, Departement of
*>      Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
     $                   JPIV )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            IJOB, LDZ, N
      DOUBLE PRECISION   RDSCAL, RDSUM
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), JPIV( * )
      DOUBLE PRECISION   RHS( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            MAXDIM
      PARAMETER          ( MAXDIM = 8 )
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, INFO, J, K
      DOUBLE PRECISION   BM, BP, PMONE, SMINU, SPLUS, TEMP
*     ..
*     .. Local Arrays ..
      INTEGER            IWORK( MAXDIM )
      DOUBLE PRECISION   WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DGECON, DGESC2, DLASSQ,
     $                   DLASWP,
     $                   DSCAL
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DASUM, DDOT
      EXTERNAL           DASUM, DDOT
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, SQRT
*     ..
*     .. Executable Statements ..
*
      IF( IJOB.NE.2 ) THEN
*
*        Apply permutations IPIV to RHS
*
         CALL DLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 )
*
*        Solve for L-part choosing RHS either to +1 or -1.
*
         PMONE = -ONE
*
         DO 10 J = 1, N - 1
            BP = RHS( J ) + ONE
            BM = RHS( J ) - ONE
            SPLUS = ONE
*
*           Look-ahead for L-part RHS(1:N-1) = + or -1, SPLUS and
*           SMIN computed more efficiently than in BSOLVE [1].
*
            SPLUS = SPLUS + DDOT( N-J, Z( J+1, J ), 1, Z( J+1, J ),
     $                            1 )
            SMINU = DDOT( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 )
            SPLUS = SPLUS*RHS( J )
            IF( SPLUS.GT.SMINU ) THEN
               RHS( J ) = BP
            ELSE IF( SMINU.GT.SPLUS ) THEN
               RHS( J ) = BM
            ELSE
*
*              In this case the updating sums are equal and we can
*              choose RHS(J) +1 or -1. The first time this happens
*              we choose -1, thereafter +1. This is a simple way to
*              get good estimates of matrices like Byers well-known
*              example (see [1]). (Not done in BSOLVE.)
*
               RHS( J ) = RHS( J ) + PMONE
               PMONE = ONE
            END IF
*
*           Compute the remaining r.h.s.
*
            TEMP = -RHS( J )
            CALL DAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 )
*
   10    CONTINUE
*
*        Solve for U-part, look-ahead for RHS(N) = +-1. This is not done
*        in BSOLVE and will hopefully give us a better estimate because
*        any ill-conditioning of the original matrix is transferred to U
*        and not to L. U(N, N) is an approximation to sigma_min(LU).
*
         CALL DCOPY( N-1, RHS, 1, XP, 1 )
         XP( N ) = RHS( N ) + ONE
         RHS( N ) = RHS( N ) - ONE
         SPLUS = ZERO
         SMINU = ZERO
         DO 30 I = N, 1, -1
            TEMP = ONE / Z( I, I )
            XP( I ) = XP( I )*TEMP
            RHS( I ) = RHS( I )*TEMP
            DO 20 K = I + 1, N
               XP( I ) = XP( I ) - XP( K )*( Z( I, K )*TEMP )
               RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP )
   20       CONTINUE
            SPLUS = SPLUS + ABS( XP( I ) )
            SMINU = SMINU + ABS( RHS( I ) )
   30    CONTINUE
         IF( SPLUS.GT.SMINU )
     $      CALL DCOPY( N, XP, 1, RHS, 1 )
*
*        Apply the permutations JPIV to the computed solution (RHS)
*
         CALL DLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 )
*
*        Compute the sum of squares
*
         CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM )
*
      ELSE
*
*        IJOB = 2, Compute approximate nullvector XM of Z
*
         CALL DGECON( 'I', N, Z, LDZ, ONE, TEMP, WORK, IWORK, INFO )
         CALL DCOPY( N, WORK( N+1 ), 1, XM, 1 )
*
*        Compute RHS
*
         CALL DLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 )
         TEMP = ONE / SQRT( DDOT( N, XM, 1, XM, 1 ) )
         CALL DSCAL( N, TEMP, XM, 1 )
         CALL DCOPY( N, XM, 1, XP, 1 )
         CALL DAXPY( N, ONE, RHS, 1, XP, 1 )
         CALL DAXPY( N, -ONE, XM, 1, RHS, 1 )
         CALL DGESC2( N, Z, LDZ, RHS, IPIV, JPIV, TEMP )
         CALL DGESC2( N, Z, LDZ, XP, IPIV, JPIV, TEMP )
         IF( DASUM( N, XP, 1 ).GT.DASUM( N, RHS, 1 ) )
     $      CALL DCOPY( N, XP, 1, RHS, 1 )
*
*        Compute the sum of squares
*
         CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM )
*
      END IF
*
      RETURN
*
*     End of DLATDF
*
      END
*> \brief \b DLATPS solves a triangular system of equations with the matrix held in packed storage.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLATPS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatps.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatps.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatps.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLATPS( UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE,
*                          CNORM, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, NORMIN, TRANS, UPLO
*       INTEGER            INFO, N
*       DOUBLE PRECISION   SCALE
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AP( * ), CNORM( * ), X( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLATPS solves one of the triangular systems
*>
*>    A *x = s*b  or  A**T*x = s*b
*>
*> with scaling to prevent overflow, where A is an upper or lower
*> triangular matrix stored in packed form.  Here A**T denotes the
*> transpose of A, x and b are n-element vectors, and s is a scaling
*> factor, usually less than or equal to 1, chosen so that the
*> components of x will be less than the overflow threshold.  If the
*> unscaled problem will not cause overflow, the Level 2 BLAS routine
*> DTPSV is called. If the matrix A is singular (A(j,j) = 0 for some j),
*> then s is set to 0 and a non-trivial solution to A*x = 0 is returned.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the matrix A is upper or lower triangular.
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the operation applied to A.
*>          = 'N':  Solve A * x = s*b  (No transpose)
*>          = 'T':  Solve A**T* x = s*b  (Transpose)
*>          = 'C':  Solve A**T* x = s*b  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          Specifies whether or not the matrix A is unit triangular.
*>          = 'N':  Non-unit triangular
*>          = 'U':  Unit triangular
*> \endverbatim
*>
*> \param[in] NORMIN
*> \verbatim
*>          NORMIN is CHARACTER*1
*>          Specifies whether CNORM has been set or not.
*>          = 'Y':  CNORM contains the column norms on entry
*>          = 'N':  CNORM is not set on entry.  On exit, the norms will
*>                  be computed and stored in CNORM.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          The upper or lower triangular matrix A, packed columnwise in
*>          a linear array.  The j-th column of A is stored in the array
*>          AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (N)
*>          On entry, the right hand side b of the triangular system.
*>          On exit, X is overwritten by the solution vector x.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*>          SCALE is DOUBLE PRECISION
*>          The scaling factor s for the triangular system
*>             A * x = s*b  or  A**T* x = s*b.
*>          If SCALE = 0, the matrix A is singular or badly scaled, and
*>          the vector x is an exact or approximate solution to A*x = 0.
*> \endverbatim
*>
*> \param[in,out] CNORM
*> \verbatim
*>          CNORM is DOUBLE PRECISION array, dimension (N)
*>
*>          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
*>          contains the norm of the off-diagonal part of the j-th column
*>          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
*>          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
*>          must be greater than or equal to the 1-norm.
*>
*>          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
*>          returns the 1-norm of the offdiagonal part of the j-th column
*>          of A.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -k, the k-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup latps
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  A rough bound on x is computed; if that is less than overflow, DTPSV
*>  is called, otherwise, specific code is used which checks for possible
*>  overflow or divide-by-zero at every operation.
*>
*>  A columnwise scheme is used for solving A*x = b.  The basic algorithm
*>  if A is lower triangular is
*>
*>       x[1:n] := b[1:n]
*>       for j = 1, ..., n
*>            x(j) := x(j) / A(j,j)
*>            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
*>       end
*>
*>  Define bounds on the components of x after j iterations of the loop:
*>     M(j) = bound on x[1:j]
*>     G(j) = bound on x[j+1:n]
*>  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
*>
*>  Then for iteration j+1 we have
*>     M(j+1) <= G(j) / | A(j+1,j+1) |
*>     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
*>            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
*>
*>  where CNORM(j+1) is greater than or equal to the infinity-norm of
*>  column j+1 of A, not counting the diagonal.  Hence
*>
*>     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
*>                  1<=i<=j
*>  and
*>
*>     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
*>                                   1<=i< j
*>
*>  Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTPSV if the
*>  reciprocal of the largest M(j), j=1,..,n, is larger than
*>  max(underflow, 1/overflow).
*>
*>  The bound on x(j) is also used to determine when a step in the
*>  columnwise method can be performed without fear of overflow.  If
*>  the computed bound is greater than a large constant, x is scaled to
*>  prevent overflow, but if the bound overflows, x is set to 0, x(j) to
*>  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
*>
*>  Similarly, a row-wise scheme is used to solve A**T*x = b.  The basic
*>  algorithm for A upper triangular is
*>
*>       for j = 1, ..., n
*>            x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
*>       end
*>
*>  We simultaneously compute two bounds
*>       G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j
*>       M(j) = bound on x(i), 1<=i<=j
*>
*>  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
*>  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
*>  Then the bound on x(j) is
*>
*>       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
*>
*>            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
*>                      1<=i<=j
*>
*>  and we can safely call DTPSV if 1/M(n) and 1/G(n) are both greater
*>  than max(underflow, 1/overflow).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLATPS( UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE,
     $                   CNORM, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, NORMIN, TRANS, UPLO
      INTEGER            INFO, N
      DOUBLE PRECISION   SCALE
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( * ), CNORM( * ), X( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, HALF, ONE
      PARAMETER          ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOTRAN, NOUNIT, UPPER
      INTEGER            I, IMAX, IP, J, JFIRST, JINC, JLAST, JLEN
      DOUBLE PRECISION   BIGNUM, GROW, REC, SMLNUM, SUMJ, TJJ, TJJS,
     $                   TMAX, TSCAL, USCAL, XBND, XJ, XMAX
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX
      DOUBLE PRECISION   DASUM, DDOT, DLAMCH
      EXTERNAL           LSAME, IDAMAX, DASUM, DDOT, DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DSCAL, DTPSV, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      NOTRAN = LSAME( TRANS, 'N' )
      NOUNIT = LSAME( DIAG, 'N' )
*
*     Test the input parameters.
*
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
     $         LSAME( TRANS, 'C' ) ) THEN
         INFO = -2
      ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
         INFO = -3
      ELSE IF( .NOT.LSAME( NORMIN, 'Y' ) .AND. .NOT.
     $         LSAME( NORMIN, 'N' ) ) THEN
         INFO = -4
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLATPS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Determine machine dependent parameters to control overflow.
*
      SMLNUM = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
      BIGNUM = ONE / SMLNUM
      SCALE = ONE
*
      IF( LSAME( NORMIN, 'N' ) ) THEN
*
*        Compute the 1-norm of each column, not including the diagonal.
*
         IF( UPPER ) THEN
*
*           A is upper triangular.
*
            IP = 1
            DO 10 J = 1, N
               CNORM( J ) = DASUM( J-1, AP( IP ), 1 )
               IP = IP + J
   10       CONTINUE
         ELSE
*
*           A is lower triangular.
*
            IP = 1
            DO 20 J = 1, N - 1
               CNORM( J ) = DASUM( N-J, AP( IP+1 ), 1 )
               IP = IP + N - J + 1
   20       CONTINUE
            CNORM( N ) = ZERO
         END IF
      END IF
*
*     Scale the column norms by TSCAL if the maximum element in CNORM is
*     greater than BIGNUM.
*
      IMAX = IDAMAX( N, CNORM, 1 )
      TMAX = CNORM( IMAX )
      IF( TMAX.LE.BIGNUM ) THEN
         TSCAL = ONE
      ELSE
         TSCAL = ONE / ( SMLNUM*TMAX )
         CALL DSCAL( N, TSCAL, CNORM, 1 )
      END IF
*
*     Compute a bound on the computed solution vector to see if the
*     Level 2 BLAS routine DTPSV can be used.
*
      J = IDAMAX( N, X, 1 )
      XMAX = ABS( X( J ) )
      XBND = XMAX
      IF( NOTRAN ) THEN
*
*        Compute the growth in A * x = b.
*
         IF( UPPER ) THEN
            JFIRST = N
            JLAST = 1
            JINC = -1
         ELSE
            JFIRST = 1
            JLAST = N
            JINC = 1
         END IF
*
         IF( TSCAL.NE.ONE ) THEN
            GROW = ZERO
            GO TO 50
         END IF
*
         IF( NOUNIT ) THEN
*
*           A is non-unit triangular.
*
*           Compute GROW = 1/G(j) and XBND = 1/M(j).
*           Initially, G(0) = max{x(i), i=1,...,n}.
*
            GROW = ONE / MAX( XBND, SMLNUM )
            XBND = GROW
            IP = JFIRST*( JFIRST+1 ) / 2
            JLEN = N
            DO 30 J = JFIRST, JLAST, JINC
*
*              Exit the loop if the growth factor is too small.
*
               IF( GROW.LE.SMLNUM )
     $            GO TO 50
*
*              M(j) = G(j-1) / abs(A(j,j))
*
               TJJ = ABS( AP( IP ) )
               XBND = MIN( XBND, MIN( ONE, TJJ )*GROW )
               IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN
*
*                 G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
*
                  GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) )
               ELSE
*
*                 G(j) could overflow, set GROW to 0.
*
                  GROW = ZERO
               END IF
               IP = IP + JINC*JLEN
               JLEN = JLEN - 1
   30       CONTINUE
            GROW = XBND
         ELSE
*
*           A is unit triangular.
*
*           Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
*
            GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
            DO 40 J = JFIRST, JLAST, JINC
*
*              Exit the loop if the growth factor is too small.
*
               IF( GROW.LE.SMLNUM )
     $            GO TO 50
*
*              G(j) = G(j-1)*( 1 + CNORM(j) )
*
               GROW = GROW*( ONE / ( ONE+CNORM( J ) ) )
   40       CONTINUE
         END IF
   50    CONTINUE
*
      ELSE
*
*        Compute the growth in A**T * x = b.
*
         IF( UPPER ) THEN
            JFIRST = 1
            JLAST = N
            JINC = 1
         ELSE
            JFIRST = N
            JLAST = 1
            JINC = -1
         END IF
*
         IF( TSCAL.NE.ONE ) THEN
            GROW = ZERO
            GO TO 80
         END IF
*
         IF( NOUNIT ) THEN
*
*           A is non-unit triangular.
*
*           Compute GROW = 1/G(j) and XBND = 1/M(j).
*           Initially, M(0) = max{x(i), i=1,...,n}.
*
            GROW = ONE / MAX( XBND, SMLNUM )
            XBND = GROW
            IP = JFIRST*( JFIRST+1 ) / 2
            JLEN = 1
            DO 60 J = JFIRST, JLAST, JINC
*
*              Exit the loop if the growth factor is too small.
*
               IF( GROW.LE.SMLNUM )
     $            GO TO 80
*
*              G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
*
               XJ = ONE + CNORM( J )
               GROW = MIN( GROW, XBND / XJ )
*
*              M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
*
               TJJ = ABS( AP( IP ) )
               IF( XJ.GT.TJJ )
     $            XBND = XBND*( TJJ / XJ )
               JLEN = JLEN + 1
               IP = IP + JINC*JLEN
   60       CONTINUE
            GROW = MIN( GROW, XBND )
         ELSE
*
*           A is unit triangular.
*
*           Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
*
            GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
            DO 70 J = JFIRST, JLAST, JINC
*
*              Exit the loop if the growth factor is too small.
*
               IF( GROW.LE.SMLNUM )
     $            GO TO 80
*
*              G(j) = ( 1 + CNORM(j) )*G(j-1)
*
               XJ = ONE + CNORM( J )
               GROW = GROW / XJ
   70       CONTINUE
         END IF
   80    CONTINUE
      END IF
*
      IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN
*
*        Use the Level 2 BLAS solve if the reciprocal of the bound on
*        elements of X is not too small.
*
         CALL DTPSV( UPLO, TRANS, DIAG, N, AP, X, 1 )
      ELSE
*
*        Use a Level 1 BLAS solve, scaling intermediate results.
*
         IF( XMAX.GT.BIGNUM ) THEN
*
*           Scale X so that its components are less than or equal to
*           BIGNUM in absolute value.
*
            SCALE = BIGNUM / XMAX
            CALL DSCAL( N, SCALE, X, 1 )
            XMAX = BIGNUM
         END IF
*
         IF( NOTRAN ) THEN
*
*           Solve A * x = b
*
            IP = JFIRST*( JFIRST+1 ) / 2
            DO 110 J = JFIRST, JLAST, JINC
*
*              Compute x(j) = b(j) / A(j,j), scaling x if necessary.
*
               XJ = ABS( X( J ) )
               IF( NOUNIT ) THEN
                  TJJS = AP( IP )*TSCAL
               ELSE
                  TJJS = TSCAL
                  IF( TSCAL.EQ.ONE )
     $               GO TO 100
               END IF
               TJJ = ABS( TJJS )
               IF( TJJ.GT.SMLNUM ) THEN
*
*                    abs(A(j,j)) > SMLNUM:
*
                  IF( TJJ.LT.ONE ) THEN
                     IF( XJ.GT.TJJ*BIGNUM ) THEN
*
*                          Scale x by 1/b(j).
*
                        REC = ONE / XJ
                        CALL DSCAL( N, REC, X, 1 )
                        SCALE = SCALE*REC
                        XMAX = XMAX*REC
                     END IF
                  END IF
                  X( J ) = X( J ) / TJJS
                  XJ = ABS( X( J ) )
               ELSE IF( TJJ.GT.ZERO ) THEN
*
*                    0 < abs(A(j,j)) <= SMLNUM:
*
                  IF( XJ.GT.TJJ*BIGNUM ) THEN
*
*                       Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
*                       to avoid overflow when dividing by A(j,j).
*
                     REC = ( TJJ*BIGNUM ) / XJ
                     IF( CNORM( J ).GT.ONE ) THEN
*
*                          Scale by 1/CNORM(j) to avoid overflow when
*                          multiplying x(j) times column j.
*
                        REC = REC / CNORM( J )
                     END IF
                     CALL DSCAL( N, REC, X, 1 )
                     SCALE = SCALE*REC
                     XMAX = XMAX*REC
                  END IF
                  X( J ) = X( J ) / TJJS
                  XJ = ABS( X( J ) )
               ELSE
*
*                    A(j,j) = 0:  Set x(1:n) = 0, x(j) = 1, and
*                    scale = 0, and compute a solution to A*x = 0.
*
                  DO 90 I = 1, N
                     X( I ) = ZERO
   90             CONTINUE
                  X( J ) = ONE
                  XJ = ONE
                  SCALE = ZERO
                  XMAX = ZERO
               END IF
  100          CONTINUE
*
*              Scale x if necessary to avoid overflow when adding a
*              multiple of column j of A.
*
               IF( XJ.GT.ONE ) THEN
                  REC = ONE / XJ
                  IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN
*
*                    Scale x by 1/(2*abs(x(j))).
*
                     REC = REC*HALF
                     CALL DSCAL( N, REC, X, 1 )
                     SCALE = SCALE*REC
                  END IF
               ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN
*
*                 Scale x by 1/2.
*
                  CALL DSCAL( N, HALF, X, 1 )
                  SCALE = SCALE*HALF
               END IF
*
               IF( UPPER ) THEN
                  IF( J.GT.1 ) THEN
*
*                    Compute the update
*                       x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j)
*
                     CALL DAXPY( J-1, -X( J )*TSCAL, AP( IP-J+1 ), 1,
     $                           X,
     $                           1 )
                     I = IDAMAX( J-1, X, 1 )
                     XMAX = ABS( X( I ) )
                  END IF
                  IP = IP - J
               ELSE
                  IF( J.LT.N ) THEN
*
*                    Compute the update
*                       x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j)
*
                     CALL DAXPY( N-J, -X( J )*TSCAL, AP( IP+1 ), 1,
     $                           X( J+1 ), 1 )
                     I = J + IDAMAX( N-J, X( J+1 ), 1 )
                     XMAX = ABS( X( I ) )
                  END IF
                  IP = IP + N - J + 1
               END IF
  110       CONTINUE
*
         ELSE
*
*           Solve A**T * x = b
*
            IP = JFIRST*( JFIRST+1 ) / 2
            JLEN = 1
            DO 160 J = JFIRST, JLAST, JINC
*
*              Compute x(j) = b(j) - sum A(k,j)*x(k).
*                                    k<>j
*
               XJ = ABS( X( J ) )
               USCAL = TSCAL
               REC = ONE / MAX( XMAX, ONE )
               IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN
*
*                 If x(j) could overflow, scale x by 1/(2*XMAX).
*
                  REC = REC*HALF
                  IF( NOUNIT ) THEN
                     TJJS = AP( IP )*TSCAL
                  ELSE
                     TJJS = TSCAL
                  END IF
                  TJJ = ABS( TJJS )
                  IF( TJJ.GT.ONE ) THEN
*
*                       Divide by A(j,j) when scaling x if A(j,j) > 1.
*
                     REC = MIN( ONE, REC*TJJ )
                     USCAL = USCAL / TJJS
                  END IF
                  IF( REC.LT.ONE ) THEN
                     CALL DSCAL( N, REC, X, 1 )
                     SCALE = SCALE*REC
                     XMAX = XMAX*REC
                  END IF
               END IF
*
               SUMJ = ZERO
               IF( USCAL.EQ.ONE ) THEN
*
*                 If the scaling needed for A in the dot product is 1,
*                 call DDOT to perform the dot product.
*
                  IF( UPPER ) THEN
                     SUMJ = DDOT( J-1, AP( IP-J+1 ), 1, X, 1 )
                  ELSE IF( J.LT.N ) THEN
                     SUMJ = DDOT( N-J, AP( IP+1 ), 1, X( J+1 ), 1 )
                  END IF
               ELSE
*
*                 Otherwise, use in-line code for the dot product.
*
                  IF( UPPER ) THEN
                     DO 120 I = 1, J - 1
                        SUMJ = SUMJ + ( AP( IP-J+I )*USCAL )*X( I )
  120                CONTINUE
                  ELSE IF( J.LT.N ) THEN
                     DO 130 I = 1, N - J
                        SUMJ = SUMJ + ( AP( IP+I )*USCAL )*X( J+I )
  130                CONTINUE
                  END IF
               END IF
*
               IF( USCAL.EQ.TSCAL ) THEN
*
*                 Compute x(j) := ( x(j) - sumj ) / A(j,j) if 1/A(j,j)
*                 was not used to scale the dotproduct.
*
                  X( J ) = X( J ) - SUMJ
                  XJ = ABS( X( J ) )
                  IF( NOUNIT ) THEN
*
*                    Compute x(j) = x(j) / A(j,j), scaling if necessary.
*
                     TJJS = AP( IP )*TSCAL
                  ELSE
                     TJJS = TSCAL
                     IF( TSCAL.EQ.ONE )
     $                  GO TO 150
                  END IF
                  TJJ = ABS( TJJS )
                  IF( TJJ.GT.SMLNUM ) THEN
*
*                       abs(A(j,j)) > SMLNUM:
*
                     IF( TJJ.LT.ONE ) THEN
                        IF( XJ.GT.TJJ*BIGNUM ) THEN
*
*                             Scale X by 1/abs(x(j)).
*
                           REC = ONE / XJ
                           CALL DSCAL( N, REC, X, 1 )
                           SCALE = SCALE*REC
                           XMAX = XMAX*REC
                        END IF
                     END IF
                     X( J ) = X( J ) / TJJS
                  ELSE IF( TJJ.GT.ZERO ) THEN
*
*                       0 < abs(A(j,j)) <= SMLNUM:
*
                     IF( XJ.GT.TJJ*BIGNUM ) THEN
*
*                          Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
*
                        REC = ( TJJ*BIGNUM ) / XJ
                        CALL DSCAL( N, REC, X, 1 )
                        SCALE = SCALE*REC
                        XMAX = XMAX*REC
                     END IF
                     X( J ) = X( J ) / TJJS
                  ELSE
*
*                       A(j,j) = 0:  Set x(1:n) = 0, x(j) = 1, and
*                       scale = 0, and compute a solution to A**T*x = 0.
*
                     DO 140 I = 1, N
                        X( I ) = ZERO
  140                CONTINUE
                     X( J ) = ONE
                     SCALE = ZERO
                     XMAX = ZERO
                  END IF
  150             CONTINUE
               ELSE
*
*                 Compute x(j) := x(j) / A(j,j)  - sumj if the dot
*                 product has already been divided by 1/A(j,j).
*
                  X( J ) = X( J ) / TJJS - SUMJ
               END IF
               XMAX = MAX( XMAX, ABS( X( J ) ) )
               JLEN = JLEN + 1
               IP = IP + JINC*JLEN
  160       CONTINUE
         END IF
         SCALE = SCALE / TSCAL
      END IF
*
*     Scale the column norms by 1/TSCAL for return.
*
      IF( TSCAL.NE.ONE ) THEN
         CALL DSCAL( N, ONE / TSCAL, CNORM, 1 )
      END IF
*
      RETURN
*
*     End of DLATPS
*
      END
*> \brief \b DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLATRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            LDA, LDW, N, NB
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLATRD reduces NB rows and columns of a real symmetric matrix A to
*> symmetric tridiagonal form by an orthogonal similarity
*> transformation Q**T * A * Q, and returns the matrices V and W which are
*> needed to apply the transformation to the unreduced part of A.
*>
*> If UPLO = 'U', DLATRD reduces the last NB rows and columns of a
*> matrix, of which the upper triangle is supplied;
*> if UPLO = 'L', DLATRD reduces the first NB rows and columns of a
*> matrix, of which the lower triangle is supplied.
*>
*> This is an auxiliary routine called by DSYTRD.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the upper or lower triangular part of the
*>          symmetric matrix A is stored:
*>          = 'U': Upper triangular
*>          = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*>          NB is INTEGER
*>          The number of rows and columns to be reduced.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*>          n-by-n upper triangular part of A contains the upper
*>          triangular part of the matrix A, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading n-by-n lower triangular part of A contains the lower
*>          triangular part of the matrix A, and the strictly upper
*>          triangular part of A is not referenced.
*>          On exit:
*>          if UPLO = 'U', the last NB columns have been reduced to
*>            tridiagonal form, with the diagonal elements overwriting
*>            the diagonal elements of A; the elements above the diagonal
*>            with the array TAU, represent the orthogonal matrix Q as a
*>            product of elementary reflectors;
*>          if UPLO = 'L', the first NB columns have been reduced to
*>            tridiagonal form, with the diagonal elements overwriting
*>            the diagonal elements of A; the elements below the diagonal
*>            with the array TAU, represent the  orthogonal matrix Q as a
*>            product of elementary reflectors.
*>          See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= (1,N).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
*>          elements of the last NB columns of the reduced matrix;
*>          if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
*>          the first NB columns of the reduced matrix.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (N-1)
*>          The scalar factors of the elementary reflectors, stored in
*>          TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
*>          See Further Details.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (LDW,NB)
*>          The n-by-nb matrix W required to update the unreduced part
*>          of A.
*> \endverbatim
*>
*> \param[in] LDW
*> \verbatim
*>          LDW is INTEGER
*>          The leading dimension of the array W. LDW >= max(1,N).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup latrd
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  If UPLO = 'U', the matrix Q is represented as a product of elementary
*>  reflectors
*>
*>     Q = H(n) H(n-1) . . . H(n-nb+1).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
*>  and tau in TAU(i-1).
*>
*>  If UPLO = 'L', the matrix Q is represented as a product of elementary
*>  reflectors
*>
*>     Q = H(1) H(2) . . . H(nb).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
*>  and tau in TAU(i).
*>
*>  The elements of the vectors v together form the n-by-nb matrix V
*>  which is needed, with W, to apply the transformation to the unreduced
*>  part of the matrix, using a symmetric rank-2k update of the form:
*>  A := A - V*W**T - W*V**T.
*>
*>  The contents of A on exit are illustrated by the following examples
*>  with n = 5 and nb = 2:
*>
*>  if UPLO = 'U':                       if UPLO = 'L':
*>
*>    (  a   a   a   v4  v5 )              (  d                  )
*>    (      a   a   v4  v5 )              (  1   d              )
*>    (          a   1   v5 )              (  v1  1   a          )
*>    (              d   1  )              (  v1  v2  a   a      )
*>    (                  d  )              (  v1  v2  a   a   a  )
*>
*>  where d denotes a diagonal element of the reduced matrix, a denotes
*>  an element of the original matrix that is unchanged, and vi denotes
*>  an element of the vector defining H(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            LDA, LDW, N, NB
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, HALF
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, HALF = 0.5D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, IW
      DOUBLE PRECISION   ALPHA
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DGEMV, DLARFG, DSCAL, DSYMV
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DDOT
      EXTERNAL           LSAME, DDOT
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( N.LE.0 )
     $   RETURN
*
      IF( LSAME( UPLO, 'U' ) ) THEN
*
*        Reduce last NB columns of upper triangle
*
         DO 10 I = N, N - NB + 1, -1
            IW = I - N + NB
            IF( I.LT.N ) THEN
*
*              Update A(1:i,i)
*
               CALL DGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ),
     $                     LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 )
               CALL DGEMV( 'No transpose', I, N-I, -ONE, W( 1,
     $                     IW+1 ),
     $                     LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 )
            END IF
            IF( I.GT.1 ) THEN
*
*              Generate elementary reflector H(i) to annihilate
*              A(1:i-2,i)
*
               CALL DLARFG( I-1, A( I-1, I ), A( 1, I ), 1,
     $                      TAU( I-1 ) )
               E( I-1 ) = A( I-1, I )
               A( I-1, I ) = ONE
*
*              Compute W(1:i-1,i)
*
               CALL DSYMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1,
     $                     ZERO, W( 1, IW ), 1 )
               IF( I.LT.N ) THEN
                  CALL DGEMV( 'Transpose', I-1, N-I, ONE, W( 1,
     $                        IW+1 ),
     $                        LDW, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 )
                  CALL DGEMV( 'No transpose', I-1, N-I, -ONE,
     $                        A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE,
     $                        W( 1, IW ), 1 )
                  CALL DGEMV( 'Transpose', I-1, N-I, ONE, A( 1,
     $                        I+1 ),
     $                        LDA, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 )
                  CALL DGEMV( 'No transpose', I-1, N-I, -ONE,
     $                        W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE,
     $                        W( 1, IW ), 1 )
               END IF
               CALL DSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 )
               ALPHA = -HALF*TAU( I-1 )*DDOT( I-1, W( 1, IW ), 1,
     $                 A( 1, I ), 1 )
               CALL DAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 )
            END IF
*
   10    CONTINUE
      ELSE
*
*        Reduce first NB columns of lower triangle
*
         DO 20 I = 1, NB
*
*           Update A(i:n,i)
*
            CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ),
     $                  LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 )
            CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ),
     $                  LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 )
            IF( I.LT.N ) THEN
*
*              Generate elementary reflector H(i) to annihilate
*              A(i+2:n,i)
*
               CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ),
     $                      1,
     $                      TAU( I ) )
               E( I ) = A( I+1, I )
               A( I+1, I ) = ONE
*
*              Compute W(i+1:n,i)
*
               CALL DSYMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA,
     $                     A( I+1, I ), 1, ZERO, W( I+1, I ), 1 )
               CALL DGEMV( 'Transpose', N-I, I-1, ONE, W( I+1, 1 ),
     $                     LDW,
     $                     A( I+1, I ), 1, ZERO, W( 1, I ), 1 )
               CALL DGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1,
     $                     1 ),
     $                     LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
               CALL DGEMV( 'Transpose', N-I, I-1, ONE, A( I+1, 1 ),
     $                     LDA,
     $                     A( I+1, I ), 1, ZERO, W( 1, I ), 1 )
               CALL DGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1,
     $                     1 ),
     $                     LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
               CALL DSCAL( N-I, TAU( I ), W( I+1, I ), 1 )
               ALPHA = -HALF*TAU( I )*DDOT( N-I, W( I+1, I ), 1,
     $                 A( I+1, I ), 1 )
               CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ),
     $                     1 )
            END IF
*
   20    CONTINUE
      END IF
*
      RETURN
*
*     End of DLATRD
*
      END
*> \brief \b DLATRS solves a triangular system of equations with the scale factor set to prevent overflow.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLATRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE,
*                          CNORM, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, NORMIN, TRANS, UPLO
*       INTEGER            INFO, LDA, N
*       DOUBLE PRECISION   SCALE
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), CNORM( * ), X( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLATRS solves one of the triangular systems
*>
*>    A *x = s*b  or  A**T *x = s*b
*>
*> with scaling to prevent overflow.  Here A is an upper or lower
*> triangular matrix, A**T denotes the transpose of A, x and b are
*> n-element vectors, and s is a scaling factor, usually less than
*> or equal to 1, chosen so that the components of x will be less than
*> the overflow threshold.  If the unscaled problem will not cause
*> overflow, the Level 2 BLAS routine DTRSV is called.  If the matrix A
*> is singular (A(j,j) = 0 for some j), then s is set to 0 and a
*> non-trivial solution to A*x = 0 is returned.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the matrix A is upper or lower triangular.
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the operation applied to A.
*>          = 'N':  Solve A * x = s*b  (No transpose)
*>          = 'T':  Solve A**T* x = s*b  (Transpose)
*>          = 'C':  Solve A**T* x = s*b  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          Specifies whether or not the matrix A is unit triangular.
*>          = 'N':  Non-unit triangular
*>          = 'U':  Unit triangular
*> \endverbatim
*>
*> \param[in] NORMIN
*> \verbatim
*>          NORMIN is CHARACTER*1
*>          Specifies whether CNORM has been set or not.
*>          = 'Y':  CNORM contains the column norms on entry
*>          = 'N':  CNORM is not set on entry.  On exit, the norms will
*>                  be computed and stored in CNORM.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The triangular matrix A.  If UPLO = 'U', the leading n by n
*>          upper triangular part of the array A contains the upper
*>          triangular matrix, and the strictly lower triangular part of
*>          A is not referenced.  If UPLO = 'L', the leading n by n lower
*>          triangular part of the array A contains the lower triangular
*>          matrix, and the strictly upper triangular part of A is not
*>          referenced.  If DIAG = 'U', the diagonal elements of A are
*>          also not referenced and are assumed to be 1.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max (1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (N)
*>          On entry, the right hand side b of the triangular system.
*>          On exit, X is overwritten by the solution vector x.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*>          SCALE is DOUBLE PRECISION
*>          The scaling factor s for the triangular system
*>             A * x = s*b  or  A**T* x = s*b.
*>          If SCALE = 0, the matrix A is singular or badly scaled, and
*>          the vector x is an exact or approximate solution to A*x = 0.
*> \endverbatim
*>
*> \param[in,out] CNORM
*> \verbatim
*>          CNORM is DOUBLE PRECISION array, dimension (N)
*>
*>          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
*>          contains the norm of the off-diagonal part of the j-th column
*>          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
*>          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
*>          must be greater than or equal to the 1-norm.
*>
*>          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
*>          returns the 1-norm of the offdiagonal part of the j-th column
*>          of A.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -k, the k-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup latrs
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  A rough bound on x is computed; if that is less than overflow, DTRSV
*>  is called, otherwise, specific code is used which checks for possible
*>  overflow or divide-by-zero at every operation.
*>
*>  A columnwise scheme is used for solving A*x = b.  The basic algorithm
*>  if A is lower triangular is
*>
*>       x[1:n] := b[1:n]
*>       for j = 1, ..., n
*>            x(j) := x(j) / A(j,j)
*>            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
*>       end
*>
*>  Define bounds on the components of x after j iterations of the loop:
*>     M(j) = bound on x[1:j]
*>     G(j) = bound on x[j+1:n]
*>  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
*>
*>  Then for iteration j+1 we have
*>     M(j+1) <= G(j) / | A(j+1,j+1) |
*>     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
*>            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
*>
*>  where CNORM(j+1) is greater than or equal to the infinity-norm of
*>  column j+1 of A, not counting the diagonal.  Hence
*>
*>     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
*>                  1<=i<=j
*>  and
*>
*>     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
*>                                   1<=i< j
*>
*>  Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTRSV if the
*>  reciprocal of the largest M(j), j=1,..,n, is larger than
*>  max(underflow, 1/overflow).
*>
*>  The bound on x(j) is also used to determine when a step in the
*>  columnwise method can be performed without fear of overflow.  If
*>  the computed bound is greater than a large constant, x is scaled to
*>  prevent overflow, but if the bound overflows, x is set to 0, x(j) to
*>  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
*>
*>  Similarly, a row-wise scheme is used to solve A**T*x = b.  The basic
*>  algorithm for A upper triangular is
*>
*>       for j = 1, ..., n
*>            x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
*>       end
*>
*>  We simultaneously compute two bounds
*>       G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j
*>       M(j) = bound on x(i), 1<=i<=j
*>
*>  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
*>  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
*>  Then the bound on x(j) is
*>
*>       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
*>
*>            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
*>                      1<=i<=j
*>
*>  and we can safely call DTRSV if 1/M(n) and 1/G(n) are both greater
*>  than max(underflow, 1/overflow).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X,
     $                   SCALE,
     $                   CNORM, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, NORMIN, TRANS, UPLO
      INTEGER            INFO, LDA, N
      DOUBLE PRECISION   SCALE
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), CNORM( * ), X( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, HALF, ONE
      PARAMETER          ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOTRAN, NOUNIT, UPPER
      INTEGER            I, IMAX, J, JFIRST, JINC, JLAST
      DOUBLE PRECISION   BIGNUM, GROW, REC, SMLNUM, SUMJ, TJJ, TJJS,
     $                   TMAX, TSCAL, USCAL, XBND, XJ, XMAX
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   WORK(1)
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX
      DOUBLE PRECISION   DASUM, DDOT, DLAMCH, DLANGE
      EXTERNAL           LSAME, IDAMAX, DASUM, DDOT, DLAMCH,
     $                   DLANGE
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DSCAL, DTRSV, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      NOTRAN = LSAME( TRANS, 'N' )
      NOUNIT = LSAME( DIAG, 'N' )
*
*     Test the input parameters.
*
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
     $         LSAME( TRANS, 'C' ) ) THEN
         INFO = -2
      ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
         INFO = -3
      ELSE IF( .NOT.LSAME( NORMIN, 'Y' ) .AND. .NOT.
     $         LSAME( NORMIN, 'N' ) ) THEN
         INFO = -4
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLATRS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      SCALE = ONE
      IF( N.EQ.0 )
     $   RETURN
*
*     Determine machine dependent parameters to control overflow.
*
      SMLNUM = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
      BIGNUM = ONE / SMLNUM
*
      IF( LSAME( NORMIN, 'N' ) ) THEN
*
*        Compute the 1-norm of each column, not including the diagonal.
*
         IF( UPPER ) THEN
*
*           A is upper triangular.
*
            DO 10 J = 1, N
               CNORM( J ) = DASUM( J-1, A( 1, J ), 1 )
   10       CONTINUE
         ELSE
*
*           A is lower triangular.
*
            DO 20 J = 1, N - 1
               CNORM( J ) = DASUM( N-J, A( J+1, J ), 1 )
   20       CONTINUE
            CNORM( N ) = ZERO
         END IF
      END IF
*
*     Scale the column norms by TSCAL if the maximum element in CNORM is
*     greater than BIGNUM.
*
      IMAX = IDAMAX( N, CNORM, 1 )
      TMAX = CNORM( IMAX )
      IF( TMAX.LE.BIGNUM ) THEN
         TSCAL = ONE
      ELSE
*
*        Avoid NaN generation if entries in CNORM exceed the
*        overflow threshold
*
         IF( TMAX.LE.DLAMCH('Overflow') ) THEN
*           Case 1: All entries in CNORM are valid floating-point numbers
            TSCAL = ONE / ( SMLNUM*TMAX )
            CALL DSCAL( N, TSCAL, CNORM, 1 )
         ELSE
*           Case 2: At least one column norm of A cannot be represented
*           as floating-point number. Find the offdiagonal entry A( I, J )
*           with the largest absolute value. If this entry is not +/- Infinity,
*           use this value as TSCAL.
            TMAX = ZERO
            IF( UPPER ) THEN
*
*              A is upper triangular.
*
               DO J = 2, N
                  TMAX = MAX( DLANGE( 'M', J-1, 1, A( 1, J ), 1,
     $                        WORK ), TMAX )
               END DO
            ELSE
*
*              A is lower triangular.
*
               DO J = 1, N - 1
                  TMAX = MAX( DLANGE( 'M', N-J, 1, A( J+1, J ), 1,
     $                        WORK ), TMAX )
               END DO
            END IF
*
            IF( TMAX.LE.DLAMCH('Overflow') ) THEN
               TSCAL = ONE / ( SMLNUM*TMAX )
               DO J = 1, N
                  IF( CNORM( J ).LE.DLAMCH('Overflow') ) THEN
                     CNORM( J ) = CNORM( J )*TSCAL
                  ELSE
*                    Recompute the 1-norm without introducing Infinity
*                    in the summation
                     CNORM( J ) = ZERO
                     IF( UPPER ) THEN
                        DO I = 1, J - 1
                           CNORM( J ) = CNORM( J ) +
     $                                  TSCAL * ABS( A( I, J ) )
                        END DO
                     ELSE
                        DO I = J + 1, N
                           CNORM( J ) = CNORM( J ) +
     $                                  TSCAL * ABS( A( I, J ) )
                        END DO
                     END IF
                  END IF
               END DO
            ELSE
*              At least one entry of A is not a valid floating-point entry.
*              Rely on TRSV to propagate Inf and NaN.
               CALL DTRSV( UPLO, TRANS, DIAG, N, A, LDA, X, 1 )
               RETURN
            END IF
         END IF
      END IF
*
*     Compute a bound on the computed solution vector to see if the
*     Level 2 BLAS routine DTRSV can be used.
*
      J = IDAMAX( N, X, 1 )
      XMAX = ABS( X( J ) )
      XBND = XMAX
      IF( NOTRAN ) THEN
*
*        Compute the growth in A * x = b.
*
         IF( UPPER ) THEN
            JFIRST = N
            JLAST = 1
            JINC = -1
         ELSE
            JFIRST = 1
            JLAST = N
            JINC = 1
         END IF
*
         IF( TSCAL.NE.ONE ) THEN
            GROW = ZERO
            GO TO 50
         END IF
*
         IF( NOUNIT ) THEN
*
*           A is non-unit triangular.
*
*           Compute GROW = 1/G(j) and XBND = 1/M(j).
*           Initially, G(0) = max{x(i), i=1,...,n}.
*
            GROW = ONE / MAX( XBND, SMLNUM )
            XBND = GROW
            DO 30 J = JFIRST, JLAST, JINC
*
*              Exit the loop if the growth factor is too small.
*
               IF( GROW.LE.SMLNUM )
     $            GO TO 50
*
*              M(j) = G(j-1) / abs(A(j,j))
*
               TJJ = ABS( A( J, J ) )
               XBND = MIN( XBND, MIN( ONE, TJJ )*GROW )
               IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN
*
*                 G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
*
                  GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) )
               ELSE
*
*                 G(j) could overflow, set GROW to 0.
*
                  GROW = ZERO
               END IF
   30       CONTINUE
            GROW = XBND
         ELSE
*
*           A is unit triangular.
*
*           Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
*
            GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
            DO 40 J = JFIRST, JLAST, JINC
*
*              Exit the loop if the growth factor is too small.
*
               IF( GROW.LE.SMLNUM )
     $            GO TO 50
*
*              G(j) = G(j-1)*( 1 + CNORM(j) )
*
               GROW = GROW*( ONE / ( ONE+CNORM( J ) ) )
   40       CONTINUE
         END IF
   50    CONTINUE
*
      ELSE
*
*        Compute the growth in A**T * x = b.
*
         IF( UPPER ) THEN
            JFIRST = 1
            JLAST = N
            JINC = 1
         ELSE
            JFIRST = N
            JLAST = 1
            JINC = -1
         END IF
*
         IF( TSCAL.NE.ONE ) THEN
            GROW = ZERO
            GO TO 80
         END IF
*
         IF( NOUNIT ) THEN
*
*           A is non-unit triangular.
*
*           Compute GROW = 1/G(j) and XBND = 1/M(j).
*           Initially, M(0) = max{x(i), i=1,...,n}.
*
            GROW = ONE / MAX( XBND, SMLNUM )
            XBND = GROW
            DO 60 J = JFIRST, JLAST, JINC
*
*              Exit the loop if the growth factor is too small.
*
               IF( GROW.LE.SMLNUM )
     $            GO TO 80
*
*              G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
*
               XJ = ONE + CNORM( J )
               GROW = MIN( GROW, XBND / XJ )
*
*              M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
*
               TJJ = ABS( A( J, J ) )
               IF( XJ.GT.TJJ )
     $            XBND = XBND*( TJJ / XJ )
   60       CONTINUE
            GROW = MIN( GROW, XBND )
         ELSE
*
*           A is unit triangular.
*
*           Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
*
            GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
            DO 70 J = JFIRST, JLAST, JINC
*
*              Exit the loop if the growth factor is too small.
*
               IF( GROW.LE.SMLNUM )
     $            GO TO 80
*
*              G(j) = ( 1 + CNORM(j) )*G(j-1)
*
               XJ = ONE + CNORM( J )
               GROW = GROW / XJ
   70       CONTINUE
         END IF
   80    CONTINUE
      END IF
*
      IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN
*
*        Use the Level 2 BLAS solve if the reciprocal of the bound on
*        elements of X is not too small.
*
         CALL DTRSV( UPLO, TRANS, DIAG, N, A, LDA, X, 1 )
      ELSE
*
*        Use a Level 1 BLAS solve, scaling intermediate results.
*
         IF( XMAX.GT.BIGNUM ) THEN
*
*           Scale X so that its components are less than or equal to
*           BIGNUM in absolute value.
*
            SCALE = BIGNUM / XMAX
            CALL DSCAL( N, SCALE, X, 1 )
            XMAX = BIGNUM
         END IF
*
         IF( NOTRAN ) THEN
*
*           Solve A * x = b
*
            DO 110 J = JFIRST, JLAST, JINC
*
*              Compute x(j) = b(j) / A(j,j), scaling x if necessary.
*
               XJ = ABS( X( J ) )
               IF( NOUNIT ) THEN
                  TJJS = A( J, J )*TSCAL
               ELSE
                  TJJS = TSCAL
                  IF( TSCAL.EQ.ONE )
     $               GO TO 100
               END IF
               TJJ = ABS( TJJS )
               IF( TJJ.GT.SMLNUM ) THEN
*
*                    abs(A(j,j)) > SMLNUM:
*
                  IF( TJJ.LT.ONE ) THEN
                     IF( XJ.GT.TJJ*BIGNUM ) THEN
*
*                          Scale x by 1/b(j).
*
                        REC = ONE / XJ
                        CALL DSCAL( N, REC, X, 1 )
                        SCALE = SCALE*REC
                        XMAX = XMAX*REC
                     END IF
                  END IF
                  X( J ) = X( J ) / TJJS
                  XJ = ABS( X( J ) )
               ELSE IF( TJJ.GT.ZERO ) THEN
*
*                    0 < abs(A(j,j)) <= SMLNUM:
*
                  IF( XJ.GT.TJJ*BIGNUM ) THEN
*
*                       Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
*                       to avoid overflow when dividing by A(j,j).
*
                     REC = ( TJJ*BIGNUM ) / XJ
                     IF( CNORM( J ).GT.ONE ) THEN
*
*                          Scale by 1/CNORM(j) to avoid overflow when
*                          multiplying x(j) times column j.
*
                        REC = REC / CNORM( J )
                     END IF
                     CALL DSCAL( N, REC, X, 1 )
                     SCALE = SCALE*REC
                     XMAX = XMAX*REC
                  END IF
                  X( J ) = X( J ) / TJJS
                  XJ = ABS( X( J ) )
               ELSE
*
*                    A(j,j) = 0:  Set x(1:n) = 0, x(j) = 1, and
*                    scale = 0, and compute a solution to A*x = 0.
*
                  DO 90 I = 1, N
                     X( I ) = ZERO
   90             CONTINUE
                  X( J ) = ONE
                  XJ = ONE
                  SCALE = ZERO
                  XMAX = ZERO
               END IF
  100          CONTINUE
*
*              Scale x if necessary to avoid overflow when adding a
*              multiple of column j of A.
*
               IF( XJ.GT.ONE ) THEN
                  REC = ONE / XJ
                  IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN
*
*                    Scale x by 1/(2*abs(x(j))).
*
                     REC = REC*HALF
                     CALL DSCAL( N, REC, X, 1 )
                     SCALE = SCALE*REC
                  END IF
               ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN
*
*                 Scale x by 1/2.
*
                  CALL DSCAL( N, HALF, X, 1 )
                  SCALE = SCALE*HALF
               END IF
*
               IF( UPPER ) THEN
                  IF( J.GT.1 ) THEN
*
*                    Compute the update
*                       x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j)
*
                     CALL DAXPY( J-1, -X( J )*TSCAL, A( 1, J ), 1, X,
     $                           1 )
                     I = IDAMAX( J-1, X, 1 )
                     XMAX = ABS( X( I ) )
                  END IF
               ELSE
                  IF( J.LT.N ) THEN
*
*                    Compute the update
*                       x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j)
*
                     CALL DAXPY( N-J, -X( J )*TSCAL, A( J+1, J ), 1,
     $                           X( J+1 ), 1 )
                     I = J + IDAMAX( N-J, X( J+1 ), 1 )
                     XMAX = ABS( X( I ) )
                  END IF
               END IF
  110       CONTINUE
*
         ELSE
*
*           Solve A**T * x = b
*
            DO 160 J = JFIRST, JLAST, JINC
*
*              Compute x(j) = b(j) - sum A(k,j)*x(k).
*                                    k<>j
*
               XJ = ABS( X( J ) )
               USCAL = TSCAL
               REC = ONE / MAX( XMAX, ONE )
               IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN
*
*                 If x(j) could overflow, scale x by 1/(2*XMAX).
*
                  REC = REC*HALF
                  IF( NOUNIT ) THEN
                     TJJS = A( J, J )*TSCAL
                  ELSE
                     TJJS = TSCAL
                  END IF
                  TJJ = ABS( TJJS )
                  IF( TJJ.GT.ONE ) THEN
*
*                       Divide by A(j,j) when scaling x if A(j,j) > 1.
*
                     REC = MIN( ONE, REC*TJJ )
                     USCAL = USCAL / TJJS
                  END IF
                  IF( REC.LT.ONE ) THEN
                     CALL DSCAL( N, REC, X, 1 )
                     SCALE = SCALE*REC
                     XMAX = XMAX*REC
                  END IF
               END IF
*
               SUMJ = ZERO
               IF( USCAL.EQ.ONE ) THEN
*
*                 If the scaling needed for A in the dot product is 1,
*                 call DDOT to perform the dot product.
*
                  IF( UPPER ) THEN
                     SUMJ = DDOT( J-1, A( 1, J ), 1, X, 1 )
                  ELSE IF( J.LT.N ) THEN
                     SUMJ = DDOT( N-J, A( J+1, J ), 1, X( J+1 ), 1 )
                  END IF
               ELSE
*
*                 Otherwise, use in-line code for the dot product.
*
                  IF( UPPER ) THEN
                     DO 120 I = 1, J - 1
                        SUMJ = SUMJ + ( A( I, J )*USCAL )*X( I )
  120                CONTINUE
                  ELSE IF( J.LT.N ) THEN
                     DO 130 I = J + 1, N
                        SUMJ = SUMJ + ( A( I, J )*USCAL )*X( I )
  130                CONTINUE
                  END IF
               END IF
*
               IF( USCAL.EQ.TSCAL ) THEN
*
*                 Compute x(j) := ( x(j) - sumj ) / A(j,j) if 1/A(j,j)
*                 was not used to scale the dotproduct.
*
                  X( J ) = X( J ) - SUMJ
                  XJ = ABS( X( J ) )
                  IF( NOUNIT ) THEN
                     TJJS = A( J, J )*TSCAL
                  ELSE
                     TJJS = TSCAL
                     IF( TSCAL.EQ.ONE )
     $                  GO TO 150
                  END IF
*
*                    Compute x(j) = x(j) / A(j,j), scaling if necessary.
*
                  TJJ = ABS( TJJS )
                  IF( TJJ.GT.SMLNUM ) THEN
*
*                       abs(A(j,j)) > SMLNUM:
*
                     IF( TJJ.LT.ONE ) THEN
                        IF( XJ.GT.TJJ*BIGNUM ) THEN
*
*                             Scale X by 1/abs(x(j)).
*
                           REC = ONE / XJ
                           CALL DSCAL( N, REC, X, 1 )
                           SCALE = SCALE*REC
                           XMAX = XMAX*REC
                        END IF
                     END IF
                     X( J ) = X( J ) / TJJS
                  ELSE IF( TJJ.GT.ZERO ) THEN
*
*                       0 < abs(A(j,j)) <= SMLNUM:
*
                     IF( XJ.GT.TJJ*BIGNUM ) THEN
*
*                          Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
*
                        REC = ( TJJ*BIGNUM ) / XJ
                        CALL DSCAL( N, REC, X, 1 )
                        SCALE = SCALE*REC
                        XMAX = XMAX*REC
                     END IF
                     X( J ) = X( J ) / TJJS
                  ELSE
*
*                       A(j,j) = 0:  Set x(1:n) = 0, x(j) = 1, and
*                       scale = 0, and compute a solution to A**T*x = 0.
*
                     DO 140 I = 1, N
                        X( I ) = ZERO
  140                CONTINUE
                     X( J ) = ONE
                     SCALE = ZERO
                     XMAX = ZERO
                  END IF
  150             CONTINUE
               ELSE
*
*                 Compute x(j) := x(j) / A(j,j)  - sumj if the dot
*                 product has already been divided by 1/A(j,j).
*
                  X( J ) = X( J ) / TJJS - SUMJ
               END IF
               XMAX = MAX( XMAX, ABS( X( J ) ) )
  160       CONTINUE
         END IF
         SCALE = SCALE / TSCAL
      END IF
*
*     Scale the column norms by 1/TSCAL for return.
*
      IF( TSCAL.NE.ONE ) THEN
         CALL DSCAL( N, ONE / TSCAL, CNORM, 1 )
      END IF
*
      RETURN
*
*     End of DLATRS
*
      END
*> \brief \b DLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLATRZ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatrz.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatrz.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatrz.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLATRZ( M, N, L, A, LDA, TAU, WORK )
*
*       .. Scalar Arguments ..
*       INTEGER            L, LDA, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
*> [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0 ) * Z, by means
*> of orthogonal transformations.  Z is an (M+L)-by-(M+L) orthogonal
*> matrix and, R and A1 are M-by-M upper triangular matrices.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*>          L is INTEGER
*>          The number of columns of the matrix A containing the
*>          meaningful part of the Householder vectors. N-M >= L >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the leading M-by-N upper trapezoidal part of the
*>          array A must contain the matrix to be factorized.
*>          On exit, the leading M-by-M upper triangular part of A
*>          contains the upper triangular matrix R, and elements N-L+1 to
*>          N of the first M rows of A, with the array TAU, represent the
*>          orthogonal matrix Z as a product of M elementary reflectors.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (M)
*>          The scalar factors of the elementary reflectors.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (M)
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup latrz
*
*> \par Contributors:
*  ==================
*>
*>    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The factorization is obtained by Householder's method.  The kth
*>  transformation matrix, Z( k ), which is used to introduce zeros into
*>  the ( m - k + 1 )th row of A, is given in the form
*>
*>     Z( k ) = ( I     0   ),
*>              ( 0  T( k ) )
*>
*>  where
*>
*>     T( k ) = I - tau*u( k )*u( k )**T,   u( k ) = (   1    ),
*>                                                 (   0    )
*>                                                 ( z( k ) )
*>
*>  tau is a scalar and z( k ) is an l element vector. tau and z( k )
*>  are chosen to annihilate the elements of the kth row of A2.
*>
*>  The scalar tau is returned in the kth element of TAU and the vector
*>  u( k ) in the kth row of A2, such that the elements of z( k ) are
*>  in  a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
*>  the upper triangular part of A1.
*>
*>  Z is given by
*>
*>     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLATRZ( M, N, L, A, LDA, TAU, WORK )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            L, LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARFG, DLARZ
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
*     Quick return if possible
*
      IF( M.EQ.0 ) THEN
         RETURN
      ELSE IF( M.EQ.N ) THEN
         DO 10 I = 1, N
            TAU( I ) = ZERO
   10    CONTINUE
         RETURN
      END IF
*
      DO 20 I = M, 1, -1
*
*        Generate elementary reflector H(i) to annihilate
*        [ A(i,i) A(i,n-l+1:n) ]
*
         CALL DLARFG( L+1, A( I, I ), A( I, N-L+1 ), LDA, TAU( I ) )
*
*        Apply H(i) to A(1:i-1,i:n) from the right
*
         CALL DLARZ( 'Right', I-1, N-I+1, L, A( I, N-L+1 ), LDA,
     $               TAU( I ), A( 1, I ), LDA, WORK )
*
   20 CONTINUE
*
      RETURN
*
*     End of DLATRZ
*
      END
*> \brief \b DLATZM
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLATZM + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatzm.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatzm.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatzm.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLATZM( SIDE, M, N, V, INCV, TAU, C1, C2, LDC, WORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIDE
*       INTEGER            INCV, LDC, M, N
*       DOUBLE PRECISION   TAU
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   C1( LDC, * ), C2( LDC, * ), V( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DORMRZ.
*>
*> DLATZM applies a Householder matrix generated by DTZRQF to a matrix.
*>
*> Let P = I - tau*u*u**T,   u = ( 1 ),
*>                               ( v )
*> where v is an (m-1) vector if SIDE = 'L', or a (n-1) vector if
*> SIDE = 'R'.
*>
*> If SIDE equals 'L', let
*>        C = [ C1 ] 1
*>            [ C2 ] m-1
*>              n
*> Then C is overwritten by P*C.
*>
*> If SIDE equals 'R', let
*>        C = [ C1, C2 ] m
*>               1  n-1
*> Then C is overwritten by C*P.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': form P * C
*>          = 'R': form C * P
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension
*>                  (1 + (M-1)*abs(INCV)) if SIDE = 'L'
*>                  (1 + (N-1)*abs(INCV)) if SIDE = 'R'
*>          The vector v in the representation of P. V is not used
*>          if TAU = 0.
*> \endverbatim
*>
*> \param[in] INCV
*> \verbatim
*>          INCV is INTEGER
*>          The increment between elements of v. INCV <> 0
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION
*>          The value tau in the representation of P.
*> \endverbatim
*>
*> \param[in,out] C1
*> \verbatim
*>          C1 is DOUBLE PRECISION array, dimension
*>                         (LDC,N) if SIDE = 'L'
*>                         (M,1)   if SIDE = 'R'
*>          On entry, the n-vector C1 if SIDE = 'L', or the m-vector C1
*>          if SIDE = 'R'.
*>
*>          On exit, the first row of P*C if SIDE = 'L', or the first
*>          column of C*P if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in,out] C2
*> \verbatim
*>          C2 is DOUBLE PRECISION array, dimension
*>                         (LDC, N)   if SIDE = 'L'
*>                         (LDC, N-1) if SIDE = 'R'
*>          On entry, the (m - 1) x n matrix C2 if SIDE = 'L', or the
*>          m x (n - 1) matrix C2 if SIDE = 'R'.
*>
*>          On exit, rows 2:m of P*C if SIDE = 'L', or columns 2:m of C*P
*>          if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the arrays C1 and C2. LDC >= (1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension
*>                      (N) if SIDE = 'L'
*>                      (M) if SIDE = 'R'
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleOTHERcomputational
*
*  =====================================================================
      SUBROUTINE DLATZM( SIDE, M, N, V, INCV, TAU, C1, C2, LDC,
     $                   WORK )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE
      INTEGER            INCV, LDC, M, N
      DOUBLE PRECISION   TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   C1( LDC, * ), C2( LDC, * ), V( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DGEMV, DGER
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
      IF( ( MIN( M, N ).EQ.0 ) .OR. ( TAU.EQ.ZERO ) )
     $   RETURN
*
      IF( LSAME( SIDE, 'L' ) ) THEN
*
*        w :=  (C1 + v**T * C2)**T
*
         CALL DCOPY( N, C1, LDC, WORK, 1 )
         CALL DGEMV( 'Transpose', M-1, N, ONE, C2, LDC, V, INCV, ONE,
     $               WORK, 1 )
*
*        [ C1 ] := [ C1 ] - tau* [ 1 ] * w**T
*        [ C2 ]    [ C2 ]        [ v ]
*
         CALL DAXPY( N, -TAU, WORK, 1, C1, LDC )
         CALL DGER( M-1, N, -TAU, V, INCV, WORK, 1, C2, LDC )
*
      ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
*        w := C1 + C2 * v
*
         CALL DCOPY( M, C1, 1, WORK, 1 )
         CALL DGEMV( 'No transpose', M, N-1, ONE, C2, LDC, V, INCV,
     $               ONE, WORK, 1 )
*
*        [ C1, C2 ] := [ C1, C2 ] - tau* w * [ 1 , v**T]
*
         CALL DAXPY( M, -TAU, WORK, 1, C1, 1 )
         CALL DGER( M, N-1, -TAU, WORK, 1, V, INCV, C2, LDC )
      END IF
*
      RETURN
*
*     End of DLATZM
*
      END
*> \brief \b DLAUU2 computes the product UUH or LHL, where U and L are upper or lower triangular matrices (unblocked algorithm).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAUU2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlauu2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlauu2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlauu2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAUU2( UPLO, N, A, LDA, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAUU2 computes the product U * U**T or L**T * L, where the triangular
*> factor U or L is stored in the upper or lower triangular part of
*> the array A.
*>
*> If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
*> overwriting the factor U in A.
*> If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
*> overwriting the factor L in A.
*>
*> This is the unblocked form of the algorithm, calling Level 2 BLAS.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the triangular factor stored in the array A
*>          is upper or lower triangular:
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the triangular factor U or L.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the triangular factor U or L.
*>          On exit, if UPLO = 'U', the upper triangle of A is
*>          overwritten with the upper triangle of the product U * U**T;
*>          if UPLO = 'L', the lower triangle of A is overwritten with
*>          the lower triangle of the product L**T * L.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -k, the k-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lauu2
*
*  =====================================================================
      SUBROUTINE DLAUU2( UPLO, N, A, LDA, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I
      DOUBLE PRECISION   AII
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DDOT
      EXTERNAL           LSAME, DDOT
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMV, DSCAL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLAUU2', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( UPPER ) THEN
*
*        Compute the product U * U**T.
*
         DO 10 I = 1, N
            AII = A( I, I )
            IF( I.LT.N ) THEN
               A( I, I ) = DDOT( N-I+1, A( I, I ), LDA, A( I, I ),
     $            LDA )
               CALL DGEMV( 'No transpose', I-1, N-I, ONE, A( 1,
     $                     I+1 ),
     $                     LDA, A( I, I+1 ), LDA, AII, A( 1, I ), 1 )
            ELSE
               CALL DSCAL( I, AII, A( 1, I ), 1 )
            END IF
   10    CONTINUE
*
      ELSE
*
*        Compute the product L**T * L.
*
         DO 20 I = 1, N
            AII = A( I, I )
            IF( I.LT.N ) THEN
               A( I, I ) = DDOT( N-I+1, A( I, I ), 1, A( I, I ), 1 )
               CALL DGEMV( 'Transpose', N-I, I-1, ONE, A( I+1, 1 ),
     $                     LDA,
     $                     A( I+1, I ), 1, AII, A( I, 1 ), LDA )
            ELSE
               CALL DSCAL( I, AII, A( I, 1 ), LDA )
            END IF
   20    CONTINUE
      END IF
*
      RETURN
*
*     End of DLAUU2
*
      END
*> \brief \b DLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular matrices (blocked algorithm).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAUUM + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlauum.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlauum.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlauum.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAUUM( UPLO, N, A, LDA, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAUUM computes the product U * U**T or L**T * L, where the triangular
*> factor U or L is stored in the upper or lower triangular part of
*> the array A.
*>
*> If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
*> overwriting the factor U in A.
*> If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
*> overwriting the factor L in A.
*>
*> This is the blocked form of the algorithm, calling Level 3 BLAS.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the triangular factor stored in the array A
*>          is upper or lower triangular:
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the triangular factor U or L.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the triangular factor U or L.
*>          On exit, if UPLO = 'U', the upper triangle of A is
*>          overwritten with the upper triangle of the product U * U**T;
*>          if UPLO = 'L', the lower triangle of A is overwritten with
*>          the lower triangle of the product L**T * L.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -k, the k-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lauum
*
*  =====================================================================
      SUBROUTINE DLAUUM( UPLO, N, A, LDA, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I, IB, NB
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMM, DLAUU2, DSYRK, DTRMM, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLAUUM', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Determine the block size for this environment.
*
      NB = ILAENV( 1, 'DLAUUM', UPLO, N, -1, -1, -1 )
*
      IF( NB.LE.1 .OR. NB.GE.N ) THEN
*
*        Use unblocked code
*
         CALL DLAUU2( UPLO, N, A, LDA, INFO )
      ELSE
*
*        Use blocked code
*
         IF( UPPER ) THEN
*
*           Compute the product U * U**T.
*
            DO 10 I = 1, N, NB
               IB = MIN( NB, N-I+1 )
               CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Non-unit',
     $                     I-1, IB, ONE, A( I, I ), LDA, A( 1, I ),
     $                     LDA )
               CALL DLAUU2( 'Upper', IB, A( I, I ), LDA, INFO )
               IF( I+IB.LE.N ) THEN
                  CALL DGEMM( 'No transpose', 'Transpose', I-1, IB,
     $                        N-I-IB+1, ONE, A( 1, I+IB ), LDA,
     $                        A( I, I+IB ), LDA, ONE, A( 1, I ), LDA )
                  CALL DSYRK( 'Upper', 'No transpose', IB, N-I-IB+1,
     $                        ONE, A( I, I+IB ), LDA, ONE, A( I, I ),
     $                        LDA )
               END IF
   10       CONTINUE
         ELSE
*
*           Compute the product L**T * L.
*
            DO 20 I = 1, N, NB
               IB = MIN( NB, N-I+1 )
               CALL DTRMM( 'Left', 'Lower', 'Transpose', 'Non-unit',
     $                     IB,
     $                     I-1, ONE, A( I, I ), LDA, A( I, 1 ), LDA )
               CALL DLAUU2( 'Lower', IB, A( I, I ), LDA, INFO )
               IF( I+IB.LE.N ) THEN
                  CALL DGEMM( 'Transpose', 'No transpose', IB, I-1,
     $                        N-I-IB+1, ONE, A( I+IB, I ), LDA,
     $                        A( I+IB, 1 ), LDA, ONE, A( I, 1 ), LDA )
                  CALL DSYRK( 'Lower', 'Transpose', IB, N-I-IB+1,
     $                        ONE,
     $                        A( I+IB, I ), LDA, ONE, A( I, I ), LDA )
               END IF
   20       CONTINUE
         END IF
      END IF
*
      RETURN
*
*     End of DLAUUM
*
      END
*> \brief \b DOPGTR
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DOPGTR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dopgtr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dopgtr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dopgtr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DOPGTR( UPLO, N, AP, TAU, Q, LDQ, WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDQ, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AP( * ), Q( LDQ, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DOPGTR generates a real orthogonal matrix Q which is defined as the
*> product of n-1 elementary reflectors H(i) of order n, as returned by
*> DSPTRD using packed storage:
*>
*> if UPLO = 'U', Q = H(n-1) . . . H(2) H(1),
*>
*> if UPLO = 'L', Q = H(1) H(2) . . . H(n-1).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U': Upper triangular packed storage used in previous
*>                 call to DSPTRD;
*>          = 'L': Lower triangular packed storage used in previous
*>                 call to DSPTRD.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix Q. N >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          The vectors which define the elementary reflectors, as
*>          returned by DSPTRD.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (N-1)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DSPTRD.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ,N)
*>          The N-by-N orthogonal matrix Q.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q. LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N-1)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup upgtr
*
*  =====================================================================
      SUBROUTINE DOPGTR( UPLO, N, AP, TAU, Q, LDQ, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDQ, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( * ), Q( LDQ, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I, IINFO, IJ, J
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DORG2L, DORG2R, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DOPGTR', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( UPPER ) THEN
*
*        Q was determined by a call to DSPTRD with UPLO = 'U'
*
*        Unpack the vectors which define the elementary reflectors and
*        set the last row and column of Q equal to those of the unit
*        matrix
*
         IJ = 2
         DO 20 J = 1, N - 1
            DO 10 I = 1, J - 1
               Q( I, J ) = AP( IJ )
               IJ = IJ + 1
   10       CONTINUE
            IJ = IJ + 2
            Q( N, J ) = ZERO
   20    CONTINUE
         DO 30 I = 1, N - 1
            Q( I, N ) = ZERO
   30    CONTINUE
         Q( N, N ) = ONE
*
*        Generate Q(1:n-1,1:n-1)
*
         CALL DORG2L( N-1, N-1, N-1, Q, LDQ, TAU, WORK, IINFO )
*
      ELSE
*
*        Q was determined by a call to DSPTRD with UPLO = 'L'.
*
*        Unpack the vectors which define the elementary reflectors and
*        set the first row and column of Q equal to those of the unit
*        matrix
*
         Q( 1, 1 ) = ONE
         DO 40 I = 2, N
            Q( I, 1 ) = ZERO
   40    CONTINUE
         IJ = 3
         DO 60 J = 2, N
            Q( 1, J ) = ZERO
            DO 50 I = J + 1, N
               Q( I, J ) = AP( IJ )
               IJ = IJ + 1
   50       CONTINUE
            IJ = IJ + 2
   60    CONTINUE
         IF( N.GT.1 ) THEN
*
*           Generate Q(2:n,2:n)
*
            CALL DORG2R( N-1, N-1, N-1, Q( 2, 2 ), LDQ, TAU, WORK,
     $                   IINFO )
         END IF
      END IF
      RETURN
*
*     End of DOPGTR
*
      END
*> \brief \b DOPMTR
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DOPMTR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dopmtr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dopmtr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dopmtr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DOPMTR( SIDE, UPLO, TRANS, M, N, AP, TAU, C, LDC, WORK,
*                          INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIDE, TRANS, UPLO
*       INTEGER            INFO, LDC, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AP( * ), C( LDC, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DOPMTR overwrites the general real M-by-N matrix C with
*>
*>                 SIDE = 'L'     SIDE = 'R'
*> TRANS = 'N':      Q * C          C * Q
*> TRANS = 'T':      Q**T * C       C * Q**T
*>
*> where Q is a real orthogonal matrix of order nq, with nq = m if
*> SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
*> nq-1 elementary reflectors, as returned by DSPTRD using packed
*> storage:
*>
*> if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1);
*>
*> if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': apply Q or Q**T from the Left;
*>          = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U': Upper triangular packed storage used in previous
*>                 call to DSPTRD;
*>          = 'L': Lower triangular packed storage used in previous
*>                 call to DSPTRD.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N':  No transpose, apply Q;
*>          = 'T':  Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension
*>                               (M*(M+1)/2) if SIDE = 'L'
*>                               (N*(N+1)/2) if SIDE = 'R'
*>          The vectors which define the elementary reflectors, as
*>          returned by DSPTRD.  AP is modified by the routine but
*>          restored on exit.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (M-1) if SIDE = 'L'
*>                                     or (N-1) if SIDE = 'R'
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DSPTRD.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the M-by-N matrix C.
*>          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension
*>                                   (N) if SIDE = 'L'
*>                                   (M) if SIDE = 'R'
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup upmtr
*
*  =====================================================================
      SUBROUTINE DOPMTR( SIDE, UPLO, TRANS, M, N, AP, TAU, C, LDC,
     $                   WORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS, UPLO
      INTEGER            INFO, LDC, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            FORWRD, LEFT, NOTRAN, UPPER
      INTEGER            I, I1, I2, I3, IC, II, JC, MI, NI, NQ
      DOUBLE PRECISION   AII
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LEFT = LSAME( SIDE, 'L' )
      NOTRAN = LSAME( TRANS, 'N' )
      UPPER = LSAME( UPLO, 'U' )
*
*     NQ is the order of Q
*
      IF( LEFT ) THEN
         NQ = M
      ELSE
         NQ = N
      END IF
      IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -2
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
         INFO = -3
      ELSE IF( M.LT.0 ) THEN
         INFO = -4
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
         INFO = -9
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DOPMTR', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 )
     $   RETURN
*
      IF( UPPER ) THEN
*
*        Q was determined by a call to DSPTRD with UPLO = 'U'
*
         FORWRD = ( LEFT .AND. NOTRAN ) .OR.
     $            ( .NOT.LEFT .AND. .NOT.NOTRAN )
*
         IF( FORWRD ) THEN
            I1 = 1
            I2 = NQ - 1
            I3 = 1
            II = 2
         ELSE
            I1 = NQ - 1
            I2 = 1
            I3 = -1
            II = NQ*( NQ+1 ) / 2 - 1
         END IF
*
         IF( LEFT ) THEN
            NI = N
         ELSE
            MI = M
         END IF
*
         DO 10 I = I1, I2, I3
            IF( LEFT ) THEN
*
*              H(i) is applied to C(1:i,1:n)
*
               MI = I
            ELSE
*
*              H(i) is applied to C(1:m,1:i)
*
               NI = I
            END IF
*
*           Apply H(i)
*
            CALL DLARF1L( SIDE, MI, NI, AP( II-I+1 ), 1, TAU( I ), C,
     $                  LDC,
     $                  WORK )
*
            IF( FORWRD ) THEN
               II = II + I + 2
            ELSE
               II = II - I - 1
            END IF
   10    CONTINUE
      ELSE
*
*        Q was determined by a call to DSPTRD with UPLO = 'L'.
*
         FORWRD = ( LEFT .AND. .NOT.NOTRAN ) .OR.
     $            ( .NOT.LEFT .AND. NOTRAN )
*
         IF( FORWRD ) THEN
            I1 = 1
            I2 = NQ - 1
            I3 = 1
            II = 2
         ELSE
            I1 = NQ - 1
            I2 = 1
            I3 = -1
            II = NQ*( NQ+1 ) / 2 - 1
         END IF
*
         IF( LEFT ) THEN
            NI = N
            JC = 1
         ELSE
            MI = M
            IC = 1
         END IF
*
         DO 20 I = I1, I2, I3
            AII = AP( II )
            AP( II ) = ONE
            IF( LEFT ) THEN
*
*              H(i) is applied to C(i+1:m,1:n)
*
               MI = M - I
               IC = I + 1
            ELSE
*
*              H(i) is applied to C(1:m,i+1:n)
*
               NI = N - I
               JC = I + 1
            END IF
*
*           Apply H(i)
*
            CALL DLARF( SIDE, MI, NI, AP( II ), 1, TAU( I ),
     $                  C( IC, JC ), LDC, WORK )
            AP( II ) = AII
*
            IF( FORWRD ) THEN
               II = II + NQ - I + 1
            ELSE
               II = II - NQ + I - 2
            END IF
   20    CONTINUE
      END IF
      RETURN
*
*     End of DOPMTR
*
      END
*> \brief \b DORBDB
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORBDB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorbdb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorbdb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorbdb.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12,
*                          X21, LDX21, X22, LDX22, THETA, PHI, TAUP1,
*                          TAUP2, TAUQ1, TAUQ2, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIGNS, TRANS
*       INTEGER            INFO, LDX11, LDX12, LDX21, LDX22, LWORK, M, P,
*      $                   Q
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   PHI( * ), THETA( * )
*       DOUBLE PRECISION   TAUP1( * ), TAUP2( * ), TAUQ1( * ), TAUQ2( * ),
*      $                   WORK( * ), X11( LDX11, * ), X12( LDX12, * ),
*      $                   X21( LDX21, * ), X22( LDX22, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORBDB simultaneously bidiagonalizes the blocks of an M-by-M
*> partitioned orthogonal matrix X:
*>
*>                                 [ B11 | B12 0  0 ]
*>     [ X11 | X12 ]   [ P1 |    ] [  0  |  0 -I  0 ] [ Q1 |    ]**T
*> X = [-----------] = [---------] [----------------] [---------]   .
*>     [ X21 | X22 ]   [    | P2 ] [ B21 | B22 0  0 ] [    | Q2 ]
*>                                 [  0  |  0  0  I ]
*>
*> X11 is P-by-Q. Q must be no larger than P, M-P, or M-Q. (If this is
*> not the case, then X must be transposed and/or permuted. This can be
*> done in constant time using the TRANS and SIGNS options. See DORCSD
*> for details.)
*>
*> The orthogonal matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
*> (M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are
*> represented implicitly by Householder vectors.
*>
*> B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
*> implicitly by angles THETA, PHI.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER
*>          = 'T':      X, U1, U2, V1T, and V2T are stored in row-major
*>                      order;
*>          otherwise:  X, U1, U2, V1T, and V2T are stored in column-
*>                      major order.
*> \endverbatim
*>
*> \param[in] SIGNS
*> \verbatim
*>          SIGNS is CHARACTER
*>          = 'O':      The lower-left block is made nonpositive (the
*>                      "other" convention);
*>          otherwise:  The upper-right block is made nonpositive (the
*>                      "default" convention).
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows and columns in X.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*>          P is INTEGER
*>          The number of rows in X11 and X12. 0 <= P <= M.
*> \endverbatim
*>
*> \param[in] Q
*> \verbatim
*>          Q is INTEGER
*>          The number of columns in X11 and X21. 0 <= Q <=
*>          MIN(P,M-P,M-Q).
*> \endverbatim
*>
*> \param[in,out] X11
*> \verbatim
*>          X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
*>          On entry, the top-left block of the orthogonal matrix to be
*>          reduced. On exit, the form depends on TRANS:
*>          If TRANS = 'N', then
*>             the columns of tril(X11) specify reflectors for P1,
*>             the rows of triu(X11,1) specify reflectors for Q1;
*>          else TRANS = 'T', and
*>             the rows of triu(X11) specify reflectors for P1,
*>             the columns of tril(X11,-1) specify reflectors for Q1.
*> \endverbatim
*>
*> \param[in] LDX11
*> \verbatim
*>          LDX11 is INTEGER
*>          The leading dimension of X11. If TRANS = 'N', then LDX11 >=
*>          P; else LDX11 >= Q.
*> \endverbatim
*>
*> \param[in,out] X12
*> \verbatim
*>          X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q)
*>          On entry, the top-right block of the orthogonal matrix to
*>          be reduced. On exit, the form depends on TRANS:
*>          If TRANS = 'N', then
*>             the rows of triu(X12) specify the first P reflectors for
*>             Q2;
*>          else TRANS = 'T', and
*>             the columns of tril(X12) specify the first P reflectors
*>             for Q2.
*> \endverbatim
*>
*> \param[in] LDX12
*> \verbatim
*>          LDX12 is INTEGER
*>          The leading dimension of X12. If TRANS = 'N', then LDX12 >=
*>          P; else LDX11 >= M-Q.
*> \endverbatim
*>
*> \param[in,out] X21
*> \verbatim
*>          X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
*>          On entry, the bottom-left block of the orthogonal matrix to
*>          be reduced. On exit, the form depends on TRANS:
*>          If TRANS = 'N', then
*>             the columns of tril(X21) specify reflectors for P2;
*>          else TRANS = 'T', and
*>             the rows of triu(X21) specify reflectors for P2.
*> \endverbatim
*>
*> \param[in] LDX21
*> \verbatim
*>          LDX21 is INTEGER
*>          The leading dimension of X21. If TRANS = 'N', then LDX21 >=
*>          M-P; else LDX21 >= Q.
*> \endverbatim
*>
*> \param[in,out] X22
*> \verbatim
*>          X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q)
*>          On entry, the bottom-right block of the orthogonal matrix to
*>          be reduced. On exit, the form depends on TRANS:
*>          If TRANS = 'N', then
*>             the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
*>             M-P-Q reflectors for Q2,
*>          else TRANS = 'T', and
*>             the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
*>             M-P-Q reflectors for P2.
*> \endverbatim
*>
*> \param[in] LDX22
*> \verbatim
*>          LDX22 is INTEGER
*>          The leading dimension of X22. If TRANS = 'N', then LDX22 >=
*>          M-P; else LDX22 >= M-Q.
*> \endverbatim
*>
*> \param[out] THETA
*> \verbatim
*>          THETA is DOUBLE PRECISION array, dimension (Q)
*>          The entries of the bidiagonal blocks B11, B12, B21, B22 can
*>          be computed from the angles THETA and PHI. See Further
*>          Details.
*> \endverbatim
*>
*> \param[out] PHI
*> \verbatim
*>          PHI is DOUBLE PRECISION array, dimension (Q-1)
*>          The entries of the bidiagonal blocks B11, B12, B21, B22 can
*>          be computed from the angles THETA and PHI. See Further
*>          Details.
*> \endverbatim
*>
*> \param[out] TAUP1
*> \verbatim
*>          TAUP1 is DOUBLE PRECISION array, dimension (P)
*>          The scalar factors of the elementary reflectors that define
*>          P1.
*> \endverbatim
*>
*> \param[out] TAUP2
*> \verbatim
*>          TAUP2 is DOUBLE PRECISION array, dimension (M-P)
*>          The scalar factors of the elementary reflectors that define
*>          P2.
*> \endverbatim
*>
*> \param[out] TAUQ1
*> \verbatim
*>          TAUQ1 is DOUBLE PRECISION array, dimension (Q)
*>          The scalar factors of the elementary reflectors that define
*>          Q1.
*> \endverbatim
*>
*> \param[out] TAUQ2
*> \verbatim
*>          TAUQ2 is DOUBLE PRECISION array, dimension (M-Q)
*>          The scalar factors of the elementary reflectors that define
*>          Q2.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= M-Q.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unbdb
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The bidiagonal blocks B11, B12, B21, and B22 are represented
*>  implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ...,
*>  PHI(Q-1). B11 and B21 are upper bidiagonal, while B21 and B22 are
*>  lower bidiagonal. Every entry in each bidiagonal band is a product
*>  of a sine or cosine of a THETA with a sine or cosine of a PHI. See
*>  [1] or DORCSD for details.
*>
*>  P1, P2, Q1, and Q2 are represented as products of elementary
*>  reflectors. See DORCSD for details on generating P1, P2, Q1, and Q2
*>  using DORGQR and DORGLQ.
*> \endverbatim
*
*> \par References:
*  ================
*>
*>  [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
*>      Algorithms, 50(1):33-65, 2009.
*>
*  =====================================================================
      SUBROUTINE DORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12,
     $                   LDX12,
     $                   X21, LDX21, X22, LDX22, THETA, PHI, TAUP1,
     $                   TAUP2, TAUQ1, TAUQ2, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIGNS, TRANS
      INTEGER            INFO, LDX11, LDX12, LDX21, LDX22, LWORK, M, P,
     $                   Q
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   PHI( * ), THETA( * )
      DOUBLE PRECISION   TAUP1( * ), TAUP2( * ), TAUQ1( * ), TAUQ2( * ),
     $                   WORK( * ), X11( LDX11, * ), X12( LDX12, * ),
     $                   X21( LDX21, * ), X22( LDX22, * )
*     ..
*
*  ====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   REALONE
      PARAMETER          ( REALONE = 1.0D0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            COLMAJOR, LQUERY
      INTEGER            I, LWORKMIN, LWORKOPT
      DOUBLE PRECISION   Z1, Z2, Z3, Z4
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DLARF1F, DLARFGP, DSCAL,
     $                   XERBLA
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DNRM2
      LOGICAL            LSAME
      EXTERNAL           DNRM2, LSAME
*     ..
*     .. Intrinsic Functions
      INTRINSIC          ATAN2, COS, MAX, SIN
*     ..
*     .. Executable Statements ..
*
*     Test input arguments
*
      INFO = 0
      COLMAJOR = .NOT. LSAME( TRANS, 'T' )
      IF( .NOT. LSAME( SIGNS, 'O' ) ) THEN
         Z1 = REALONE
         Z2 = REALONE
         Z3 = REALONE
         Z4 = REALONE
      ELSE
         Z1 = REALONE
         Z2 = -REALONE
         Z3 = REALONE
         Z4 = -REALONE
      END IF
      LQUERY = LWORK .EQ. -1
*
      IF( M .LT. 0 ) THEN
         INFO = -3
      ELSE IF( P .LT. 0 .OR. P .GT. M ) THEN
         INFO = -4
      ELSE IF( Q .LT. 0 .OR. Q .GT. P .OR. Q .GT. M-P .OR.
     $         Q .GT. M-Q ) THEN
         INFO = -5
      ELSE IF( COLMAJOR .AND. LDX11 .LT. MAX( 1, P ) ) THEN
         INFO = -7
      ELSE IF( .NOT.COLMAJOR .AND. LDX11 .LT. MAX( 1, Q ) ) THEN
         INFO = -7
      ELSE IF( COLMAJOR .AND. LDX12 .LT. MAX( 1, P ) ) THEN
         INFO = -9
      ELSE IF( .NOT.COLMAJOR .AND. LDX12 .LT. MAX( 1, M-Q ) ) THEN
         INFO = -9
      ELSE IF( COLMAJOR .AND. LDX21 .LT. MAX( 1, M-P ) ) THEN
         INFO = -11
      ELSE IF( .NOT.COLMAJOR .AND. LDX21 .LT. MAX( 1, Q ) ) THEN
         INFO = -11
      ELSE IF( COLMAJOR .AND. LDX22 .LT. MAX( 1, M-P ) ) THEN
         INFO = -13
      ELSE IF( .NOT.COLMAJOR .AND. LDX22 .LT. MAX( 1, M-Q ) ) THEN
         INFO = -13
      END IF
*
*     Compute workspace
*
      IF( INFO .EQ. 0 ) THEN
         LWORKOPT = M - Q
         LWORKMIN = M - Q
         WORK(1) = LWORKOPT
         IF( LWORK .LT. LWORKMIN .AND. .NOT. LQUERY ) THEN
            INFO = -21
         END IF
      END IF
      IF( INFO .NE. 0 ) THEN
         CALL XERBLA( 'xORBDB', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Handle column-major and row-major separately
*
      IF( COLMAJOR ) THEN
*
*        Reduce columns 1, ..., Q of X11, X12, X21, and X22
*
         DO I = 1, Q
*
            IF( I .EQ. 1 ) THEN
               CALL DSCAL( P-I+1, Z1, X11(I,I), 1 )
            ELSE
               CALL DSCAL( P-I+1, Z1*COS(PHI(I-1)), X11(I,I), 1 )
               CALL DAXPY( P-I+1, -Z1*Z3*Z4*SIN(PHI(I-1)), X12(I,
     $                     I-1),
     $                     1, X11(I,I), 1 )
            END IF
            IF( I .EQ. 1 ) THEN
               CALL DSCAL( M-P-I+1, Z2, X21(I,I), 1 )
            ELSE
               CALL DSCAL( M-P-I+1, Z2*COS(PHI(I-1)), X21(I,I), 1 )
               CALL DAXPY( M-P-I+1, -Z2*Z3*Z4*SIN(PHI(I-1)), X22(I,
     $                     I-1),
     $                     1, X21(I,I), 1 )
            END IF
*
            THETA(I) = ATAN2( DNRM2( M-P-I+1, X21(I,I), 1 ),
     $                 DNRM2( P-I+1, X11(I,I), 1 ) )
*
            IF( P .GT. I ) THEN
               CALL DLARFGP( P-I+1, X11(I,I), X11(I+1,I), 1,
     $                       TAUP1(I) )
            ELSE IF( P .EQ. I ) THEN
               CALL DLARFGP( P-I+1, X11(I,I), X11(I,I), 1, TAUP1(I) )
            END IF
            IF ( M-P .GT. I ) THEN
               CALL DLARFGP( M-P-I+1, X21(I,I), X21(I+1,I), 1,
     $                       TAUP2(I) )
            ELSE IF ( M-P .EQ. I ) THEN
               CALL DLARFGP( M-P-I+1, X21(I,I), X21(I,I), 1,
     $                       TAUP2(I) )
            END IF
*
            IF ( Q .GT. I ) THEN
               CALL DLARF1F( 'L', P-I+1, Q-I, X11(I,I), 1, TAUP1(I),
     $                     X11(I,I+1), LDX11, WORK )
            END IF
            IF ( M-Q+1 .GT. I ) THEN
               CALL DLARF1F( 'L', P-I+1, M-Q-I+1, X11(I,I), 1,
     $                     TAUP1(I),
     $                     X12(I,I), LDX12, WORK )
            END IF
            IF ( Q .GT. I ) THEN
               CALL DLARF1F( 'L', M-P-I+1, Q-I, X21(I,I), 1,
     $                       TAUP2(I), X21(I,I+1), LDX21, WORK )
            END IF
            IF ( M-Q+1 .GT. I ) THEN
               CALL DLARF1F( 'L', M-P-I+1, M-Q-I+1, X21(I,I), 1,
     $                       TAUP2(I), X22(I,I), LDX22, WORK )
            END IF
*
            IF( I .LT. Q ) THEN
               CALL DSCAL( Q-I, -Z1*Z3*SIN(THETA(I)), X11(I,I+1),
     $                     LDX11 )
               CALL DAXPY( Q-I, Z2*Z3*COS(THETA(I)), X21(I,I+1),
     $                     LDX21,
     $                     X11(I,I+1), LDX11 )
            END IF
            CALL DSCAL( M-Q-I+1, -Z1*Z4*SIN(THETA(I)), X12(I,I),
     $                  LDX12 )
            CALL DAXPY( M-Q-I+1, Z2*Z4*COS(THETA(I)), X22(I,I),
     $                  LDX22,
     $                  X12(I,I), LDX12 )
*
            IF( I .LT. Q )
     $         PHI(I) = ATAN2( DNRM2( Q-I, X11(I,I+1), LDX11 ),
     $                  DNRM2( M-Q-I+1, X12(I,I), LDX12 ) )
*
            IF( I .LT. Q ) THEN
               IF ( Q-I .EQ. 1 ) THEN
                  CALL DLARFGP( Q-I, X11(I,I+1), X11(I,I+1), LDX11,
     $                          TAUQ1(I) )
               ELSE
                  CALL DLARFGP( Q-I, X11(I,I+1), X11(I,I+2), LDX11,
     $                          TAUQ1(I) )
               END IF
            END IF
            IF ( Q+I-1 .LT. M ) THEN
               IF ( M-Q .EQ. I ) THEN
                  CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I,I), LDX12,
     $                          TAUQ2(I) )
               ELSE
                  CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I,I+1), LDX12,
     $                          TAUQ2(I) )
               END IF
            END IF
*
            IF( I .LT. Q ) THEN
               CALL DLARF1F( 'R', P-I, Q-I, X11(I,I+1), LDX11,
     $                     TAUQ1(I),
     $                     X11(I+1,I+1), LDX11, WORK )
               CALL DLARF1F( 'R', M-P-I, Q-I, X11(I,I+1), LDX11,
     $                     TAUQ1(I),
     $                     X21(I+1,I+1), LDX21, WORK )
            END IF
            IF ( P .GT. I ) THEN
               CALL DLARF1F( 'R', P-I, M-Q-I+1, X12(I,I), LDX12,
     $                     TAUQ2(I),
     $                     X12(I+1,I), LDX12, WORK )
            END IF
            IF ( M-P .GT. I ) THEN
               CALL DLARF1F( 'R', M-P-I, M-Q-I+1, X12(I,I), LDX12,
     $                     TAUQ2(I), X22(I+1,I), LDX22, WORK )
            END IF
*
         END DO
*
*        Reduce columns Q + 1, ..., P of X12, X22
*
         DO I = Q + 1, P
*
            CALL DSCAL( M-Q-I+1, -Z1*Z4, X12(I,I), LDX12 )
            IF ( I .GE. M-Q ) THEN
               CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I,I), LDX12,
     $                       TAUQ2(I) )
            ELSE
               CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I,I+1), LDX12,
     $                       TAUQ2(I) )
            END IF
*
            IF ( P .GT. I ) THEN
               CALL DLARF1F( 'R', P-I, M-Q-I+1, X12(I,I), LDX12,
     $                     TAUQ2(I),
     $                     X12(I+1,I), LDX12, WORK )
            END IF
            IF( M-P-Q .GE. 1 )
     $         CALL DLARF1F( 'R', M-P-Q, M-Q-I+1, X12(I,I), LDX12,
     $                     TAUQ2(I), X22(Q+1,I), LDX22, WORK )
*
         END DO
*
*        Reduce columns P + 1, ..., M - Q of X12, X22
*
         DO I = 1, M - P - Q
*
            CALL DSCAL( M-P-Q-I+1, Z2*Z4, X22(Q+I,P+I), LDX22 )
            IF ( I .EQ. M-P-Q ) THEN
               CALL DLARFGP( M-P-Q-I+1, X22(Q+I,P+I), X22(Q+I,P+I),
     $                       LDX22, TAUQ2(P+I) )
            ELSE
               CALL DLARFGP( M-P-Q-I+1, X22(Q+I,P+I), X22(Q+I,P+I+1),
     $                       LDX22, TAUQ2(P+I) )
            END IF
            IF ( I .LT. M-P-Q ) THEN
               CALL DLARF1F( 'R', M-P-Q-I, M-P-Q-I+1, X22(Q+I,P+I),
     $                     LDX22,
     $                     TAUQ2(P+I), X22(Q+I+1,P+I), LDX22, WORK )
            END IF
*
         END DO
*
      ELSE
*
*        Reduce columns 1, ..., Q of X11, X12, X21, X22
*
         DO I = 1, Q
*
            IF( I .EQ. 1 ) THEN
               CALL DSCAL( P-I+1, Z1, X11(I,I), LDX11 )
            ELSE
               CALL DSCAL( P-I+1, Z1*COS(PHI(I-1)), X11(I,I), LDX11 )
               CALL DAXPY( P-I+1, -Z1*Z3*Z4*SIN(PHI(I-1)), X12(I-1,
     $                     I),
     $                     LDX12, X11(I,I), LDX11 )
            END IF
            IF( I .EQ. 1 ) THEN
               CALL DSCAL( M-P-I+1, Z2, X21(I,I), LDX21 )
            ELSE
               CALL DSCAL( M-P-I+1, Z2*COS(PHI(I-1)), X21(I,I),
     $                     LDX21 )
               CALL DAXPY( M-P-I+1, -Z2*Z3*Z4*SIN(PHI(I-1)), X22(I-1,
     $                     I),
     $                     LDX22, X21(I,I), LDX21 )
            END IF
*
            THETA(I) = ATAN2( DNRM2( M-P-I+1, X21(I,I), LDX21 ),
     $                 DNRM2( P-I+1, X11(I,I), LDX11 ) )
*
            CALL DLARFGP( P-I+1, X11(I,I), X11(I,I+1), LDX11,
     $                    TAUP1(I) )
            IF ( I .EQ. M-P ) THEN
               CALL DLARFGP( M-P-I+1, X21(I,I), X21(I,I), LDX21,
     $                    TAUP2(I) )
            ELSE
               CALL DLARFGP( M-P-I+1, X21(I,I), X21(I,I+1), LDX21,
     $                    TAUP2(I) )
            END IF
*
            IF ( Q .GT. I ) THEN
               CALL DLARF1F( 'R', Q-I, P-I+1, X11(I,I), LDX11,
     $                     TAUP1(I),
     $                     X11(I+1,I), LDX11, WORK )
            END IF
            IF ( M-Q+1 .GT. I ) THEN
               CALL DLARF1F( 'R', M-Q-I+1, P-I+1, X11(I,I), LDX11,
     $                     TAUP1(I), X12(I,I), LDX12, WORK )
            END IF
            IF ( Q .GT. I ) THEN
               CALL DLARF1F( 'R', Q-I, M-P-I+1, X21(I,I), LDX21,
     $                     TAUP2(I),
     $                     X21(I+1,I), LDX21, WORK )
            END IF
            IF ( M-Q+1 .GT. I ) THEN
               CALL DLARF1F( 'R', M-Q-I+1, M-P-I+1, X21(I,I), LDX21,
     $                     TAUP2(I), X22(I,I), LDX22, WORK )
            END IF
*
            IF( I .LT. Q ) THEN
               CALL DSCAL( Q-I, -Z1*Z3*SIN(THETA(I)), X11(I+1,I), 1 )
               CALL DAXPY( Q-I, Z2*Z3*COS(THETA(I)), X21(I+1,I), 1,
     $                     X11(I+1,I), 1 )
            END IF
            CALL DSCAL( M-Q-I+1, -Z1*Z4*SIN(THETA(I)), X12(I,I), 1 )
            CALL DAXPY( M-Q-I+1, Z2*Z4*COS(THETA(I)), X22(I,I), 1,
     $                  X12(I,I), 1 )
*
            IF( I .LT. Q )
     $         PHI(I) = ATAN2( DNRM2( Q-I, X11(I+1,I), 1 ),
     $                  DNRM2( M-Q-I+1, X12(I,I), 1 ) )
*
            IF( I .LT. Q ) THEN
               IF ( Q-I .EQ. 1) THEN
                  CALL DLARFGP( Q-I, X11(I+1,I), X11(I+1,I), 1,
     $                          TAUQ1(I) )
               ELSE
                  CALL DLARFGP( Q-I, X11(I+1,I), X11(I+2,I), 1,
     $                          TAUQ1(I) )
               END IF
            END IF
            IF ( M-Q .GT. I ) THEN
               CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I+1,I), 1,
     $                       TAUQ2(I) )
            ELSE
               CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I,I), 1,
     $                       TAUQ2(I) )
            END IF
*
            IF( I .LT. Q ) THEN
               CALL DLARF1F( 'L', Q-I, P-I, X11(I+1,I), 1, TAUQ1(I),
     $                     X11(I+1,I+1), LDX11, WORK )
               CALL DLARF1F( 'L', Q-I, M-P-I, X11(I+1,I), 1,
     $                       TAUQ1(I), X21(I+1,I+1), LDX21, WORK )
            END IF
            CALL DLARF1F( 'L', M-Q-I+1, P-I, X12(I,I), 1, TAUQ2(I),
     $                    X12(I,I+1), LDX12, WORK )
            IF ( M-P-I .GT. 0 ) THEN
               CALL DLARF1F( 'L', M-Q-I+1, M-P-I, X12(I,I), 1,
     $                       TAUQ2(I), X22(I,I+1), LDX22, WORK )
            END IF
*
         END DO
*
*        Reduce columns Q + 1, ..., P of X12, X22
*
         DO I = Q + 1, P
*
            CALL DSCAL( M-Q-I+1, -Z1*Z4, X12(I,I), 1 )
            CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I+1,I), 1,
     $                    TAUQ2(I) )
*
            IF ( P .GT. I ) THEN
               CALL DLARF1F( 'L', M-Q-I+1, P-I, X12(I,I), 1,
     $                       TAUQ2(I), X12(I,I+1), LDX12, WORK )
            END IF
            IF( M-P-Q .GE. 1 )
     $         CALL DLARF1F( 'L', M-Q-I+1, M-P-Q, X12(I,I), 1,
     $                       TAUQ2(I), X22(I,Q+1), LDX22, WORK )
*
         END DO
*
*        Reduce columns P + 1, ..., M - Q of X12, X22
*
         DO I = 1, M - P - Q
*
            CALL DSCAL( M-P-Q-I+1, Z2*Z4, X22(P+I,Q+I), 1 )
            IF ( M-P-Q .EQ. I ) THEN
               CALL DLARFGP( M-P-Q-I+1, X22(P+I,Q+I), X22(P+I,Q+I),
     $                       1,
     $                       TAUQ2(P+I) )
            ELSE
               CALL DLARFGP( M-P-Q-I+1, X22(P+I,Q+I), X22(P+I+1,Q+I),
     $                       1,
     $                       TAUQ2(P+I) )
               CALL DLARF1F( 'L', M-P-Q-I+1, M-P-Q-I, X22(P+I,Q+I),
     $                       1, TAUQ2(P+I), X22(P+I,Q+I+1), LDX22,
     $                       WORK )
            END IF
*
         END DO
*
      END IF
*
      RETURN
*
*     End of DORBDB
*
      END

*> \brief \b DORCSD
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORCSD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorcsd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorcsd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorcsd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       RECURSIVE SUBROUTINE DORCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS,
*                                    SIGNS, M, P, Q, X11, LDX11, X12,
*                                    LDX12, X21, LDX21, X22, LDX22, THETA,
*                                    U1, LDU1, U2, LDU2, V1T, LDV1T, V2T,
*                                    LDV2T, WORK, LWORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBU1, JOBU2, JOBV1T, JOBV2T, SIGNS, TRANS
*       INTEGER            INFO, LDU1, LDU2, LDV1T, LDV2T, LDX11, LDX12,
*      $                   LDX21, LDX22, LWORK, M, P, Q
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   THETA( * )
*       DOUBLE PRECISION   U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ),
*      $                   V2T( LDV2T, * ), WORK( * ), X11( LDX11, * ),
*      $                   X12( LDX12, * ), X21( LDX21, * ), X22( LDX22,
*      $                   * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORCSD computes the CS decomposition of an M-by-M partitioned
*> orthogonal matrix X:
*>
*>                                 [  I  0  0 |  0  0  0 ]
*>                                 [  0  C  0 |  0 -S  0 ]
*>     [ X11 | X12 ]   [ U1 |    ] [  0  0  0 |  0  0 -I ] [ V1 |    ]**T
*> X = [-----------] = [---------] [---------------------] [---------]   .
*>     [ X21 | X22 ]   [    | U2 ] [  0  0  0 |  I  0  0 ] [    | V2 ]
*>                                 [  0  S  0 |  0  C  0 ]
*>                                 [  0  0  I |  0  0  0 ]
*>
*> X11 is P-by-Q. The orthogonal matrices U1, U2, V1, and V2 are P-by-P,
*> (M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are
*> R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in
*> which R = MIN(P,M-P,Q,M-Q).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBU1
*> \verbatim
*>          JOBU1 is CHARACTER
*>          = 'Y':      U1 is computed;
*>          otherwise:  U1 is not computed.
*> \endverbatim
*>
*> \param[in] JOBU2
*> \verbatim
*>          JOBU2 is CHARACTER
*>          = 'Y':      U2 is computed;
*>          otherwise:  U2 is not computed.
*> \endverbatim
*>
*> \param[in] JOBV1T
*> \verbatim
*>          JOBV1T is CHARACTER
*>          = 'Y':      V1T is computed;
*>          otherwise:  V1T is not computed.
*> \endverbatim
*>
*> \param[in] JOBV2T
*> \verbatim
*>          JOBV2T is CHARACTER
*>          = 'Y':      V2T is computed;
*>          otherwise:  V2T is not computed.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER
*>          = 'T':      X, U1, U2, V1T, and V2T are stored in row-major
*>                      order;
*>          otherwise:  X, U1, U2, V1T, and V2T are stored in column-
*>                      major order.
*> \endverbatim
*>
*> \param[in] SIGNS
*> \verbatim
*>          SIGNS is CHARACTER
*>          = 'O':      The lower-left block is made nonpositive (the
*>                      "other" convention);
*>          otherwise:  The upper-right block is made nonpositive (the
*>                      "default" convention).
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows and columns in X.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*>          P is INTEGER
*>          The number of rows in X11 and X12. 0 <= P <= M.
*> \endverbatim
*>
*> \param[in] Q
*> \verbatim
*>          Q is INTEGER
*>          The number of columns in X11 and X21. 0 <= Q <= M.
*> \endverbatim
*>
*> \param[in,out] X11
*> \verbatim
*>          X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
*>          On entry, part of the orthogonal matrix whose CSD is desired.
*> \endverbatim
*>
*> \param[in] LDX11
*> \verbatim
*>          LDX11 is INTEGER
*>          The leading dimension of X11. LDX11 >= MAX(1,P).
*> \endverbatim
*>
*> \param[in,out] X12
*> \verbatim
*>          X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q)
*>          On entry, part of the orthogonal matrix whose CSD is desired.
*> \endverbatim
*>
*> \param[in] LDX12
*> \verbatim
*>          LDX12 is INTEGER
*>          The leading dimension of X12. LDX12 >= MAX(1,P).
*> \endverbatim
*>
*> \param[in,out] X21
*> \verbatim
*>          X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
*>          On entry, part of the orthogonal matrix whose CSD is desired.
*> \endverbatim
*>
*> \param[in] LDX21
*> \verbatim
*>          LDX21 is INTEGER
*>          The leading dimension of X11. LDX21 >= MAX(1,M-P).
*> \endverbatim
*>
*> \param[in,out] X22
*> \verbatim
*>          X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q)
*>          On entry, part of the orthogonal matrix whose CSD is desired.
*> \endverbatim
*>
*> \param[in] LDX22
*> \verbatim
*>          LDX22 is INTEGER
*>          The leading dimension of X11. LDX22 >= MAX(1,M-P).
*> \endverbatim
*>
*> \param[out] THETA
*> \verbatim
*>          THETA is DOUBLE PRECISION array, dimension (R), in which R =
*>          MIN(P,M-P,Q,M-Q).
*>          C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
*>          S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).
*> \endverbatim
*>
*> \param[out] U1
*> \verbatim
*>          U1 is DOUBLE PRECISION array, dimension (LDU1,P)
*>          If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1.
*> \endverbatim
*>
*> \param[in] LDU1
*> \verbatim
*>          LDU1 is INTEGER
*>          The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=
*>          MAX(1,P).
*> \endverbatim
*>
*> \param[out] U2
*> \verbatim
*>          U2 is DOUBLE PRECISION array, dimension (LDU2,M-P)
*>          If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal
*>          matrix U2.
*> \endverbatim
*>
*> \param[in] LDU2
*> \verbatim
*>          LDU2 is INTEGER
*>          The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=
*>          MAX(1,M-P).
*> \endverbatim
*>
*> \param[out] V1T
*> \verbatim
*>          V1T is DOUBLE PRECISION array, dimension (LDV1T,Q)
*>          If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal
*>          matrix V1**T.
*> \endverbatim
*>
*> \param[in] LDV1T
*> \verbatim
*>          LDV1T is INTEGER
*>          The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=
*>          MAX(1,Q).
*> \endverbatim
*>
*> \param[out] V2T
*> \verbatim
*>          V2T is DOUBLE PRECISION array, dimension (LDV2T,M-Q)
*>          If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) orthogonal
*>          matrix V2**T.
*> \endverbatim
*>
*> \param[in] LDV2T
*> \verbatim
*>          LDV2T is INTEGER
*>          The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >=
*>          MAX(1,M-Q).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*>          If INFO > 0 on exit, WORK(2:R) contains the values PHI(1),
*>          ..., PHI(R-1) that, together with THETA(1), ..., THETA(R),
*>          define the matrix in intermediate bidiagonal-block form
*>          remaining after nonconvergence. INFO specifies the number
*>          of nonzero PHI's.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the work array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (M-MIN(P, M-P, Q, M-Q))
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  DBBCSD did not converge. See the description of WORK
*>                above for details.
*> \endverbatim
*
*> \par References:
*  ================
*>
*>  [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
*>      Algorithms, 50(1):33-65, 2009.
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup uncsd
*
*  =====================================================================
      RECURSIVE SUBROUTINE DORCSD( JOBU1, JOBU2, JOBV1T, JOBV2T,
     $                             TRANS,
     $                             SIGNS, M, P, Q, X11, LDX11, X12,
     $                             LDX12, X21, LDX21, X22, LDX22, THETA,
     $                             U1, LDU1, U2, LDU2, V1T, LDV1T, V2T,
     $                             LDV2T, WORK, LWORK, IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBU1, JOBU2, JOBV1T, JOBV2T, SIGNS, TRANS
      INTEGER            INFO, LDU1, LDU2, LDV1T, LDV2T, LDX11, LDX12,
     $                   LDX21, LDX22, LWORK, M, P, Q
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   THETA( * )
      DOUBLE PRECISION   U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ),
     $                   V2T( LDV2T, * ), WORK( * ), X11( LDX11, * ),
     $                   X12( LDX12, * ), X21( LDX21, * ), X22( LDX22,
     $                   * )
*     ..
*
*  ===================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D0,
     $                     ZERO = 0.0D0 )
*     ..
*     .. Local Scalars ..
      CHARACTER          TRANST, SIGNST
      INTEGER            CHILDINFO, I, IB11D, IB11E, IB12D, IB12E,
     $                   IB21D, IB21E, IB22D, IB22E, IBBCSD, IORBDB,
     $                   IORGLQ, IORGQR, IPHI, ITAUP1, ITAUP2, ITAUQ1,
     $                   ITAUQ2, J, LBBCSDWORK, LBBCSDWORKMIN,
     $                   LBBCSDWORKOPT, LORBDBWORK, LORBDBWORKMIN,
     $                   LORBDBWORKOPT, LORGLQWORK, LORGLQWORKMIN,
     $                   LORGLQWORKOPT, LORGQRWORK, LORGQRWORKMIN,
     $                   LORGQRWORKOPT, LWORKMIN, LWORKOPT
      LOGICAL            COLMAJOR, DEFAULTSIGNS, LQUERY, WANTU1, WANTU2,
     $                   WANTV1T, WANTV2T
*     ..
*     .. External Subroutines ..
      EXTERNAL           DBBCSD, DLACPY, DLAPMR, DLAPMT,
     $                   DORBDB, DORGLQ, DORGQR, XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. Intrinsic Functions
      INTRINSIC          INT, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test input arguments
*
      INFO = 0
      WANTU1 = LSAME( JOBU1, 'Y' )
      WANTU2 = LSAME( JOBU2, 'Y' )
      WANTV1T = LSAME( JOBV1T, 'Y' )
      WANTV2T = LSAME( JOBV2T, 'Y' )
      COLMAJOR = .NOT. LSAME( TRANS, 'T' )
      DEFAULTSIGNS = .NOT. LSAME( SIGNS, 'O' )
      LQUERY = LWORK .EQ. -1
      IF( M .LT. 0 ) THEN
         INFO = -7
      ELSE IF( P .LT. 0 .OR. P .GT. M ) THEN
         INFO = -8
      ELSE IF( Q .LT. 0 .OR. Q .GT. M ) THEN
         INFO = -9
      ELSE IF ( COLMAJOR .AND.  LDX11 .LT. MAX( 1, P ) ) THEN
        INFO = -11
      ELSE IF (.NOT. COLMAJOR .AND. LDX11 .LT. MAX( 1, Q ) ) THEN
        INFO = -11
      ELSE IF (COLMAJOR .AND. LDX12 .LT. MAX( 1, P ) ) THEN
        INFO = -13
      ELSE IF (.NOT. COLMAJOR .AND. LDX12 .LT. MAX( 1, M-Q ) ) THEN
        INFO = -13
      ELSE IF (COLMAJOR .AND. LDX21 .LT. MAX( 1, M-P ) ) THEN
        INFO = -15
      ELSE IF (.NOT. COLMAJOR .AND. LDX21 .LT. MAX( 1, Q ) ) THEN
        INFO = -15
      ELSE IF (COLMAJOR .AND. LDX22 .LT. MAX( 1, M-P ) ) THEN
        INFO = -17
      ELSE IF (.NOT. COLMAJOR .AND. LDX22 .LT. MAX( 1, M-Q ) ) THEN
        INFO = -17
      ELSE IF( WANTU1 .AND. LDU1 .LT. P ) THEN
         INFO = -20
      ELSE IF( WANTU2 .AND. LDU2 .LT. M-P ) THEN
         INFO = -22
      ELSE IF( WANTV1T .AND. LDV1T .LT. Q ) THEN
         INFO = -24
      ELSE IF( WANTV2T .AND. LDV2T .LT. M-Q ) THEN
         INFO = -26
      END IF
*
*     Work with transpose if convenient
*
      IF( INFO .EQ. 0 .AND. MIN( P, M-P ) .LT. MIN( Q, M-Q ) ) THEN
         IF( COLMAJOR ) THEN
            TRANST = 'T'
         ELSE
            TRANST = 'N'
         END IF
         IF( DEFAULTSIGNS ) THEN
            SIGNST = 'O'
         ELSE
            SIGNST = 'D'
         END IF
         CALL DORCSD( JOBV1T, JOBV2T, JOBU1, JOBU2, TRANST, SIGNST,
     $                M,
     $                Q, P, X11, LDX11, X21, LDX21, X12, LDX12, X22,
     $                LDX22, THETA, V1T, LDV1T, V2T, LDV2T, U1, LDU1,
     $                U2, LDU2, WORK, LWORK, IWORK, INFO )
         RETURN
      END IF
*
*     Work with permutation [ 0 I; I 0 ] * X * [ 0 I; I 0 ] if
*     convenient
*
      IF( INFO .EQ. 0 .AND. M-Q .LT. Q ) THEN
         IF( DEFAULTSIGNS ) THEN
            SIGNST = 'O'
         ELSE
            SIGNST = 'D'
         END IF
         CALL DORCSD( JOBU2, JOBU1, JOBV2T, JOBV1T, TRANS, SIGNST, M,
     $                M-P, M-Q, X22, LDX22, X21, LDX21, X12, LDX12, X11,
     $                LDX11, THETA, U2, LDU2, U1, LDU1, V2T, LDV2T, V1T,
     $                LDV1T, WORK, LWORK, IWORK, INFO )
         RETURN
      END IF
*
*     Compute workspace
*
      IF( INFO .EQ. 0 ) THEN
*
         IPHI = 2
         ITAUP1 = IPHI + MAX( 1, Q - 1 )
         ITAUP2 = ITAUP1 + MAX( 1, P )
         ITAUQ1 = ITAUP2 + MAX( 1, M - P )
         ITAUQ2 = ITAUQ1 + MAX( 1, Q )
         IORGQR = ITAUQ2 + MAX( 1, M - Q )
         CALL DORGQR( M-Q, M-Q, M-Q, U1, MAX(1,M-Q), U1, WORK, -1,
     $                CHILDINFO )
         LORGQRWORKOPT = INT( WORK(1) )
         LORGQRWORKMIN = MAX( 1, M - Q )
         IORGLQ = ITAUQ2 + MAX( 1, M - Q )
         CALL DORGLQ( M-Q, M-Q, M-Q, U1, MAX(1,M-Q), U1, WORK, -1,
     $                CHILDINFO )
         LORGLQWORKOPT = INT( WORK(1) )
         LORGLQWORKMIN = MAX( 1, M - Q )
         IORBDB = ITAUQ2 + MAX( 1, M - Q )
         CALL DORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12,
     $                X21, LDX21, X22, LDX22, THETA, V1T, U1, U2, V1T,
     $                V2T, WORK, -1, CHILDINFO )
         LORBDBWORKOPT = INT( WORK(1) )
         LORBDBWORKMIN = LORBDBWORKOPT
         IB11D = ITAUQ2 + MAX( 1, M - Q )
         IB11E = IB11D + MAX( 1, Q )
         IB12D = IB11E + MAX( 1, Q - 1 )
         IB12E = IB12D + MAX( 1, Q )
         IB21D = IB12E + MAX( 1, Q - 1 )
         IB21E = IB21D + MAX( 1, Q )
         IB22D = IB21E + MAX( 1, Q - 1 )
         IB22E = IB22D + MAX( 1, Q )
         IBBCSD = IB22E + MAX( 1, Q - 1 )
         CALL DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q,
     $                THETA, THETA, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T,
     $                LDV2T, U1, U1, U1, U1, U1, U1, U1, U1, WORK, -1,
     $                CHILDINFO )
         LBBCSDWORKOPT = INT( WORK(1) )
         LBBCSDWORKMIN = LBBCSDWORKOPT
         LWORKOPT = MAX( IORGQR + LORGQRWORKOPT, IORGLQ + LORGLQWORKOPT,
     $              IORBDB + LORBDBWORKOPT, IBBCSD + LBBCSDWORKOPT ) - 1
         LWORKMIN = MAX( IORGQR + LORGQRWORKMIN, IORGLQ + LORGLQWORKMIN,
     $              IORBDB + LORBDBWORKOPT, IBBCSD + LBBCSDWORKMIN ) - 1
         WORK(1) = MAX(LWORKOPT,LWORKMIN)
*
         IF( LWORK .LT. LWORKMIN .AND. .NOT. LQUERY ) THEN
            INFO = -22
         ELSE
            LORGQRWORK = LWORK - IORGQR + 1
            LORGLQWORK = LWORK - IORGLQ + 1
            LORBDBWORK = LWORK - IORBDB + 1
            LBBCSDWORK = LWORK - IBBCSD + 1
         END IF
      END IF
*
*     Abort if any illegal arguments
*
      IF( INFO .NE. 0 ) THEN
         CALL XERBLA( 'DORCSD', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Transform to bidiagonal block form
*
      CALL DORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12,
     $             X21,
     $             LDX21, X22, LDX22, THETA, WORK(IPHI), WORK(ITAUP1),
     $             WORK(ITAUP2), WORK(ITAUQ1), WORK(ITAUQ2),
     $             WORK(IORBDB), LORBDBWORK, CHILDINFO )
*
*     Accumulate Householder reflectors
*
      IF( COLMAJOR ) THEN
         IF( WANTU1 .AND. P .GT. 0 ) THEN
            CALL DLACPY( 'L', P, Q, X11, LDX11, U1, LDU1 )
            CALL DORGQR( P, P, Q, U1, LDU1, WORK(ITAUP1),
     $                   WORK(IORGQR),
     $                   LORGQRWORK, INFO)
         END IF
         IF( WANTU2 .AND. M-P .GT. 0 ) THEN
            CALL DLACPY( 'L', M-P, Q, X21, LDX21, U2, LDU2 )
            CALL DORGQR( M-P, M-P, Q, U2, LDU2, WORK(ITAUP2),
     $                   WORK(IORGQR), LORGQRWORK, INFO )
         END IF
         IF( WANTV1T .AND. Q .GT. 0 ) THEN
            CALL DLACPY( 'U', Q-1, Q-1, X11(1,2), LDX11, V1T(2,2),
     $                   LDV1T )
            V1T(1, 1) = ONE
            DO J = 2, Q
               V1T(1,J) = ZERO
               V1T(J,1) = ZERO
            END DO
            CALL DORGLQ( Q-1, Q-1, Q-1, V1T(2,2), LDV1T,
     $                   WORK(ITAUQ1),
     $                   WORK(IORGLQ), LORGLQWORK, INFO )
         END IF
         IF( WANTV2T .AND. M-Q .GT. 0 ) THEN
            CALL DLACPY( 'U', P, M-Q, X12, LDX12, V2T, LDV2T )
            IF (M-P .GT. Q) Then
               CALL DLACPY( 'U', M-P-Q, M-P-Q, X22(Q+1,P+1), LDX22,
     $                      V2T(P+1,P+1), LDV2T )
            END IF
            IF (M .GT. Q) THEN
               CALL DORGLQ( M-Q, M-Q, M-Q, V2T, LDV2T, WORK(ITAUQ2),
     $                      WORK(IORGLQ), LORGLQWORK, INFO )
            END IF
         END IF
      ELSE
         IF( WANTU1 .AND. P .GT. 0 ) THEN
            CALL DLACPY( 'U', Q, P, X11, LDX11, U1, LDU1 )
            CALL DORGLQ( P, P, Q, U1, LDU1, WORK(ITAUP1),
     $                   WORK(IORGLQ),
     $                   LORGLQWORK, INFO)
         END IF
         IF( WANTU2 .AND. M-P .GT. 0 ) THEN
            CALL DLACPY( 'U', Q, M-P, X21, LDX21, U2, LDU2 )
            CALL DORGLQ( M-P, M-P, Q, U2, LDU2, WORK(ITAUP2),
     $                   WORK(IORGLQ), LORGLQWORK, INFO )
         END IF
         IF( WANTV1T .AND. Q .GT. 0 ) THEN
            CALL DLACPY( 'L', Q-1, Q-1, X11(2,1), LDX11, V1T(2,2),
     $                   LDV1T )
            V1T(1, 1) = ONE
            DO J = 2, Q
               V1T(1,J) = ZERO
               V1T(J,1) = ZERO
            END DO
            CALL DORGQR( Q-1, Q-1, Q-1, V1T(2,2), LDV1T,
     $                   WORK(ITAUQ1),
     $                   WORK(IORGQR), LORGQRWORK, INFO )
         END IF
         IF( WANTV2T .AND. M-Q .GT. 0 ) THEN
            CALL DLACPY( 'L', M-Q, P, X12, LDX12, V2T, LDV2T )
            CALL DLACPY( 'L', M-P-Q, M-P-Q, X22(P+1,Q+1), LDX22,
     $                   V2T(P+1,P+1), LDV2T )
            CALL DORGQR( M-Q, M-Q, M-Q, V2T, LDV2T, WORK(ITAUQ2),
     $                   WORK(IORGQR), LORGQRWORK, INFO )
         END IF
      END IF
*
*     Compute the CSD of the matrix in bidiagonal-block form
*
      CALL DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q,
     $             THETA,
     $             WORK(IPHI), U1, LDU1, U2, LDU2, V1T, LDV1T, V2T,
     $             LDV2T, WORK(IB11D), WORK(IB11E), WORK(IB12D),
     $             WORK(IB12E), WORK(IB21D), WORK(IB21E), WORK(IB22D),
     $             WORK(IB22E), WORK(IBBCSD), LBBCSDWORK, INFO )
*
*     Permute rows and columns to place identity submatrices in top-
*     left corner of (1,1)-block and/or bottom-right corner of (1,2)-
*     block and/or bottom-right corner of (2,1)-block and/or top-left
*     corner of (2,2)-block
*
      IF( Q .GT. 0 .AND. WANTU2 ) THEN
         DO I = 1, Q
            IWORK(I) = M - P - Q + I
         END DO
         DO I = Q + 1, M - P
            IWORK(I) = I - Q
         END DO
         IF( COLMAJOR ) THEN
            CALL DLAPMT( .FALSE., M-P, M-P, U2, LDU2, IWORK )
         ELSE
            CALL DLAPMR( .FALSE., M-P, M-P, U2, LDU2, IWORK )
         END IF
      END IF
      IF( M .GT. 0 .AND. WANTV2T ) THEN
         DO I = 1, P
            IWORK(I) = M - P - Q + I
         END DO
         DO I = P + 1, M - Q
            IWORK(I) = I - P
         END DO
         IF( .NOT. COLMAJOR ) THEN
            CALL DLAPMT( .FALSE., M-Q, M-Q, V2T, LDV2T, IWORK )
         ELSE
            CALL DLAPMR( .FALSE., M-Q, M-Q, V2T, LDV2T, IWORK )
         END IF
      END IF
*
      RETURN
*
*     End DORCSD
*
      END

*> \brief \b DORG2L generates all or part of the orthogonal matrix Q from a QL factorization determined by sgeqlf (unblocked algorithm).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORG2L + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorg2l.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorg2l.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorg2l.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORG2L( M, N, K, A, LDA, TAU, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, K, LDA, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORG2L generates an m by n real matrix Q with orthonormal columns,
*> which is defined as the last n columns of a product of k elementary
*> reflectors of order m
*>
*>       Q  =  H(k) . . . H(2) H(1)
*>
*> as returned by DGEQLF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix Q. M >= N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines the
*>          matrix Q. N >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the (n-k+i)-th column must contain the vector which
*>          defines the elementary reflector H(i), for i = 1,2,...,k, as
*>          returned by DGEQLF in the last k columns of its array
*>          argument A.
*>          On exit, the m by n matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DGEQLF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ung2l
*
*  =====================================================================
      SUBROUTINE DORG2L( M, N, K, A, LDA, TAU, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, II, J, L
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF1L, DSCAL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 .OR. N.GT.M ) THEN
         INFO = -2
      ELSE IF( K.LT.0 .OR. K.GT.N ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -5
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORG2L', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.LE.0 )
     $   RETURN
*
*     Initialise columns 1:n-k to columns of the unit matrix
*
      DO 20 J = 1, N - K
         DO 10 L = 1, M
            A( L, J ) = ZERO
   10    CONTINUE
         A( M-N+J, J ) = ONE
   20 CONTINUE
*
      DO 40 I = 1, K
         II = N - K + I
*
*        Apply H(i) to A(1:m-k+i,1:n-k+i) from the left
*
         !A(M-N+II, II) = ONE
         CALL DLARF1L( 'Left', M-N+II, II-1, A( 1, II ), 1, TAU( I ),
     $               A,
     $               LDA, WORK )
         CALL DSCAL( M-N+II-1, -TAU( I ), A( 1, II ), 1 )
         A( M-N+II, II ) = ONE - TAU( I )
*
*        Set A(m-k+i+1:m,n-k+i) to zero
*
         DO 30 L = M - N + II + 1, M
            A( L, II ) = ZERO
   30    CONTINUE
   40 CONTINUE
      RETURN
*
*     End of DORG2L
*
      END
*> \brief \b DORG2R generates all or part of the orthogonal matrix Q from a QR factorization determined by sgeqrf (unblocked algorithm).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORG2R + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorg2r.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorg2r.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorg2r.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORG2R( M, N, K, A, LDA, TAU, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, K, LDA, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORG2R generates an m by n real matrix Q with orthonormal columns,
*> which is defined as the first n columns of a product of k elementary
*> reflectors of order m
*>
*>       Q  =  H(1) H(2) . . . H(k)
*>
*> as returned by DGEQRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix Q. M >= N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines the
*>          matrix Q. N >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the i-th column must contain the vector which
*>          defines the elementary reflector H(i), for i = 1,2,...,k, as
*>          returned by DGEQRF in the first k columns of its array
*>          argument A.
*>          On exit, the m-by-n matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DGEQRF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ung2r
*
*  =====================================================================
      SUBROUTINE DORG2R( M, N, K, A, LDA, TAU, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J, L
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF1F, DSCAL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 .OR. N.GT.M ) THEN
         INFO = -2
      ELSE IF( K.LT.0 .OR. K.GT.N ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -5
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORG2R', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.LE.0 )
     $   RETURN
*
*     Initialise columns k+1:n to columns of the unit matrix
*
      DO 20 J = K + 1, N
         DO 10 L = 1, M
            A( L, J ) = ZERO
   10    CONTINUE
         A( J, J ) = ONE
   20 CONTINUE
*
      DO 40 I = K, 1, -1
*
*        Apply H(i) to A(i:m,i:n) from the left
*
         IF( I.LT.N ) THEN
            CALL DLARF1F( 'Left', M-I+1, N-I, A( I, I ), 1, TAU( I ),
     $                  A( I, I+1 ), LDA, WORK )
         END IF
         IF( I.LT.M )
     $      CALL DSCAL( M-I, -TAU( I ), A( I+1, I ), 1 )
         A( I, I ) = ONE - TAU( I )
*
*        Set A(1:i-1,i) to zero
*
         DO 30 L = 1, I - 1
            A( L, I ) = ZERO
   30    CONTINUE
   40 CONTINUE
      RETURN
*
*     End of DORG2R
*
      END
*> \brief \b DORGBR
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGBR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorgbr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorgbr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorgbr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORGBR( VECT, M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          VECT
*       INTEGER            INFO, K, LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORGBR generates one of the real orthogonal matrices Q or P**T
*> determined by DGEBRD when reducing a real matrix A to bidiagonal
*> form: A = Q * B * P**T.  Q and P**T are defined as products of
*> elementary reflectors H(i) or G(i) respectively.
*>
*> If VECT = 'Q', A is assumed to have been an M-by-K matrix, and Q
*> is of order M:
*> if m >= k, Q = H(1) H(2) . . . H(k) and DORGBR returns the first n
*> columns of Q, where m >= n >= k;
*> if m < k, Q = H(1) H(2) . . . H(m-1) and DORGBR returns Q as an
*> M-by-M matrix.
*>
*> If VECT = 'P', A is assumed to have been a K-by-N matrix, and P**T
*> is of order N:
*> if k < n, P**T = G(k) . . . G(2) G(1) and DORGBR returns the first m
*> rows of P**T, where n >= m >= k;
*> if k >= n, P**T = G(n-1) . . . G(2) G(1) and DORGBR returns P**T as
*> an N-by-N matrix.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] VECT
*> \verbatim
*>          VECT is CHARACTER*1
*>          Specifies whether the matrix Q or the matrix P**T is
*>          required, as defined in the transformation applied by DGEBRD:
*>          = 'Q':  generate Q;
*>          = 'P':  generate P**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix Q or P**T to be returned.
*>          M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix Q or P**T to be returned.
*>          N >= 0.
*>          If VECT = 'Q', M >= N >= min(M,K);
*>          if VECT = 'P', N >= M >= min(N,K).
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          If VECT = 'Q', the number of columns in the original M-by-K
*>          matrix reduced by DGEBRD.
*>          If VECT = 'P', the number of rows in the original K-by-N
*>          matrix reduced by DGEBRD.
*>          K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the vectors which define the elementary reflectors,
*>          as returned by DGEBRD.
*>          On exit, the M-by-N matrix Q or P**T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension
*>                                (min(M,K)) if VECT = 'Q'
*>                                (min(N,K)) if VECT = 'P'
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i) or G(i), which determines Q or P**T, as
*>          returned by DGEBRD in its array argument TAUQ or TAUP.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= max(1,min(M,N)).
*>          For optimum performance LWORK >= min(M,N)*NB, where NB
*>          is the optimal blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ungbr
*
*  =====================================================================
      SUBROUTINE DORGBR( VECT, M, N, K, A, LDA, TAU, WORK, LWORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          VECT
      INTEGER            INFO, K, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, WANTQ
      INTEGER            I, IINFO, J, LWKOPT, MN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DORGLQ, DORGQR, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      WANTQ = LSAME( VECT, 'Q' )
      MN = MIN( M, N )
      LQUERY = ( LWORK.EQ.-1 )
      IF( .NOT.WANTQ .AND. .NOT.LSAME( VECT, 'P' ) ) THEN
         INFO = -1
      ELSE IF( M.LT.0 ) THEN
         INFO = -2
      ELSE IF( N.LT.0 .OR. ( WANTQ .AND. ( N.GT.M .OR. N.LT.MIN( M,
     $         K ) ) ) .OR. ( .NOT.WANTQ .AND. ( M.GT.N .OR. M.LT.
     $         MIN( N, K ) ) ) ) THEN
         INFO = -3
      ELSE IF( K.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -6
      ELSE IF( LWORK.LT.MAX( 1, MN ) .AND. .NOT.LQUERY ) THEN
         INFO = -9
      END IF
*
      IF( INFO.EQ.0 ) THEN
         WORK( 1 ) = 1
         IF( WANTQ ) THEN
            IF( M.GE.K ) THEN
               CALL DORGQR( M, N, K, A, LDA, TAU, WORK, -1, IINFO )
            ELSE
               IF( M.GT.1 ) THEN
                  CALL DORGQR( M-1, M-1, M-1, A, LDA, TAU, WORK, -1,
     $                         IINFO )
               END IF
            END IF
         ELSE
            IF( K.LT.N ) THEN
               CALL DORGLQ( M, N, K, A, LDA, TAU, WORK, -1, IINFO )
            ELSE
               IF( N.GT.1 ) THEN
                  CALL DORGLQ( N-1, N-1, N-1, A, LDA, TAU, WORK, -1,
     $                         IINFO )
               END IF
            END IF
         END IF
         LWKOPT = INT( WORK( 1 ) )
         LWKOPT = MAX (LWKOPT, MN)
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORGBR', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         WORK( 1 ) = LWKOPT
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 ) THEN
         WORK( 1 ) = 1
         RETURN
      END IF
*
      IF( WANTQ ) THEN
*
*        Form Q, determined by a call to DGEBRD to reduce an m-by-k
*        matrix
*
         IF( M.GE.K ) THEN
*
*           If m >= k, assume m >= n >= k
*
            CALL DORGQR( M, N, K, A, LDA, TAU, WORK, LWORK, IINFO )
*
         ELSE
*
*           If m < k, assume m = n
*
*           Shift the vectors which define the elementary reflectors one
*           column to the right, and set the first row and column of Q
*           to those of the unit matrix
*
            DO 20 J = M, 2, -1
               A( 1, J ) = ZERO
               DO 10 I = J + 1, M
                  A( I, J ) = A( I, J-1 )
   10          CONTINUE
   20       CONTINUE
            A( 1, 1 ) = ONE
            DO 30 I = 2, M
               A( I, 1 ) = ZERO
   30       CONTINUE
            IF( M.GT.1 ) THEN
*
*              Form Q(2:m,2:m)
*
               CALL DORGQR( M-1, M-1, M-1, A( 2, 2 ), LDA, TAU, WORK,
     $                      LWORK, IINFO )
            END IF
         END IF
      ELSE
*
*        Form P**T, determined by a call to DGEBRD to reduce a k-by-n
*        matrix
*
         IF( K.LT.N ) THEN
*
*           If k < n, assume k <= m <= n
*
            CALL DORGLQ( M, N, K, A, LDA, TAU, WORK, LWORK, IINFO )
*
         ELSE
*
*           If k >= n, assume m = n
*
*           Shift the vectors which define the elementary reflectors one
*           row downward, and set the first row and column of P**T to
*           those of the unit matrix
*
            A( 1, 1 ) = ONE
            DO 40 I = 2, N
               A( I, 1 ) = ZERO
   40       CONTINUE
            DO 60 J = 2, N
               DO 50 I = J - 1, 2, -1
                  A( I, J ) = A( I-1, J )
   50          CONTINUE
               A( 1, J ) = ZERO
   60       CONTINUE
            IF( N.GT.1 ) THEN
*
*              Form P**T(2:n,2:n)
*
               CALL DORGLQ( N-1, N-1, N-1, A( 2, 2 ), LDA, TAU, WORK,
     $                      LWORK, IINFO )
            END IF
         END IF
      END IF
      WORK( 1 ) = LWKOPT
      RETURN
*
*     End of DORGBR
*
      END
*> \brief \b DORGHR
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGHR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorghr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorghr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorghr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORGHR( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            IHI, ILO, INFO, LDA, LWORK, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORGHR generates a real orthogonal matrix Q which is defined as the
*> product of IHI-ILO elementary reflectors of order N, as returned by
*> DGEHRD:
*>
*> Q = H(ilo) H(ilo+1) . . . H(ihi-1).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix Q. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*>          ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*>          IHI is INTEGER
*>
*>          ILO and IHI must have the same values as in the previous call
*>          of DGEHRD. Q is equal to the unit matrix except in the
*>          submatrix Q(ilo+1:ihi,ilo+1:ihi).
*>          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the vectors which define the elementary reflectors,
*>          as returned by DGEHRD.
*>          On exit, the N-by-N orthogonal matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (N-1)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DGEHRD.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= IHI-ILO.
*>          For optimum performance LWORK >= (IHI-ILO)*NB, where NB is
*>          the optimal blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unghr
*
*  =====================================================================
      SUBROUTINE DORGHR( N, ILO, IHI, A, LDA, TAU, WORK, LWORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            IHI, ILO, INFO, LDA, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IINFO, J, LWKOPT, NB, NH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DORGQR, XERBLA
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      NH = IHI - ILO
      LQUERY = ( LWORK.EQ.-1 )
      IF( N.LT.0 ) THEN
         INFO = -1
      ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
         INFO = -2
      ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LWORK.LT.MAX( 1, NH ) .AND. .NOT.LQUERY ) THEN
         INFO = -8
      END IF
*
      IF( INFO.EQ.0 ) THEN
         NB = ILAENV( 1, 'DORGQR', ' ', NH, NH, NH, -1 )
         LWKOPT = MAX( 1, NH )*NB
         WORK( 1 ) = LWKOPT
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORGHR', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 ) THEN
         WORK( 1 ) = 1
         RETURN
      END IF
*
*     Shift the vectors which define the elementary reflectors one
*     column to the right, and set the first ilo and the last n-ihi
*     rows and columns to those of the unit matrix
*
      DO 40 J = IHI, ILO + 1, -1
         DO 10 I = 1, J - 1
            A( I, J ) = ZERO
   10    CONTINUE
         DO 20 I = J + 1, IHI
            A( I, J ) = A( I, J-1 )
   20    CONTINUE
         DO 30 I = IHI + 1, N
            A( I, J ) = ZERO
   30    CONTINUE
   40 CONTINUE
      DO 60 J = 1, ILO
         DO 50 I = 1, N
            A( I, J ) = ZERO
   50    CONTINUE
         A( J, J ) = ONE
   60 CONTINUE
      DO 80 J = IHI + 1, N
         DO 70 I = 1, N
            A( I, J ) = ZERO
   70    CONTINUE
         A( J, J ) = ONE
   80 CONTINUE
*
      IF( NH.GT.0 ) THEN
*
*        Generate Q(ilo+1:ihi,ilo+1:ihi)
*
         CALL DORGQR( NH, NH, NH, A( ILO+1, ILO+1 ), LDA, TAU( ILO ),
     $                WORK, LWORK, IINFO )
      END IF
      WORK( 1 ) = LWKOPT
      RETURN
*
*     End of DORGHR
*
      END
*> \brief \b DORGL2
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGL2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorgl2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorgl2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorgl2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORGL2( M, N, K, A, LDA, TAU, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, K, LDA, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORGL2 generates an m by n real matrix Q with orthonormal rows,
*> which is defined as the first m rows of a product of k elementary
*> reflectors of order n
*>
*>       Q  =  H(k) . . . H(2) H(1)
*>
*> as returned by DGELQF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix Q. N >= M.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines the
*>          matrix Q. M >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the i-th row must contain the vector which defines
*>          the elementary reflector H(i), for i = 1,2,...,k, as returned
*>          by DGELQF in the first k rows of its array argument A.
*>          On exit, the m-by-n matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DGELQF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (M)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ungl2
*
*  =====================================================================
      SUBROUTINE DORGL2( M, N, K, A, LDA, TAU, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J, L
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF1F, DSCAL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.M ) THEN
         INFO = -2
      ELSE IF( K.LT.0 .OR. K.GT.M ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -5
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORGL2', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.LE.0 )
     $   RETURN
*
      IF( K.LT.M ) THEN
*
*        Initialise rows k+1:m to rows of the unit matrix
*
         DO 20 J = 1, N
            DO 10 L = K + 1, M
               A( L, J ) = ZERO
   10       CONTINUE
            IF( J.GT.K .AND. J.LE.M )
     $         A( J, J ) = ONE
   20    CONTINUE
      END IF
*
      DO 40 I = K, 1, -1
*
*        Apply H(i) to A(i:m,i:n) from the right
*
         IF( I.LT.N ) THEN
            IF( I.LT.M ) THEN
               CALL DLARF1F( 'Right', M-I, N-I+1, A( I, I ), LDA,
     $                     TAU( I ), A( I+1, I ), LDA, WORK )
            END IF
            CALL DSCAL( N-I, -TAU( I ), A( I, I+1 ), LDA )
         END IF
         A( I, I ) = ONE - TAU( I )
*
*        Set A(i,1:i-1) to zero
*
         DO 30 L = 1, I - 1
            A( I, L ) = ZERO
   30    CONTINUE
   40 CONTINUE
      RETURN
*
*     End of DORGL2
*
      END
*> \brief \b DORGLQ
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGLQ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorglq.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorglq.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorglq.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORGLQ( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, K, LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORGLQ generates an M-by-N real matrix Q with orthonormal rows,
*> which is defined as the first M rows of a product of K elementary
*> reflectors of order N
*>
*>       Q  =  H(k) . . . H(2) H(1)
*>
*> as returned by DGELQF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix Q. N >= M.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines the
*>          matrix Q. M >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the i-th row must contain the vector which defines
*>          the elementary reflector H(i), for i = 1,2,...,k, as returned
*>          by DGELQF in the first k rows of its array argument A.
*>          On exit, the M-by-N matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DGELQF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= max(1,M).
*>          For optimum performance LWORK >= M*NB, where NB is
*>          the optimal blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unglq
*
*  =====================================================================
      SUBROUTINE DORGLQ( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IB, IINFO, IWS, J, KI, KK, L, LDWORK,
     $                   LWKOPT, NB, NBMIN, NX
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARFB, DLARFT, DORGL2, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      NB = ILAENV( 1, 'DORGLQ', ' ', M, N, K, -1 )
      LWKOPT = MAX( 1, M )*NB
      WORK( 1 ) = LWKOPT
      LQUERY = ( LWORK.EQ.-1 )
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.M ) THEN
         INFO = -2
      ELSE IF( K.LT.0 .OR. K.GT.M ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -5
      ELSE IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN
         INFO = -8
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORGLQ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.LE.0 ) THEN
         WORK( 1 ) = 1
         RETURN
      END IF
*
      NBMIN = 2
      NX = 0
      IWS = M
      IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
*        Determine when to cross over from blocked to unblocked code.
*
         NX = MAX( 0, ILAENV( 3, 'DORGLQ', ' ', M, N, K, -1 ) )
         IF( NX.LT.K ) THEN
*
*           Determine if workspace is large enough for blocked code.
*
            LDWORK = M
            IWS = LDWORK*NB
            IF( LWORK.LT.IWS ) THEN
*
*              Not enough workspace to use optimal NB:  reduce NB and
*              determine the minimum value of NB.
*
               NB = LWORK / LDWORK
               NBMIN = MAX( 2, ILAENV( 2, 'DORGLQ', ' ', M, N, K,
     $                      -1 ) )
            END IF
         END IF
      END IF
*
      IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
*        Use blocked code after the last block.
*        The first kk rows are handled by the block method.
*
         KI = ( ( K-NX-1 ) / NB )*NB
         KK = MIN( K, KI+NB )
*
*        Set A(kk+1:m,1:kk) to zero.
*
         DO 20 J = 1, KK
            DO 10 I = KK + 1, M
               A( I, J ) = ZERO
   10       CONTINUE
   20    CONTINUE
      ELSE
         KK = 0
      END IF
*
*     Use unblocked code for the last or only block.
*
      IF( KK.LT.M )
     $   CALL DORGL2( M-KK, N-KK, K-KK, A( KK+1, KK+1 ), LDA,
     $                TAU( KK+1 ), WORK, IINFO )
*
      IF( KK.GT.0 ) THEN
*
*        Use blocked code
*
         DO 50 I = KI + 1, 1, -NB
            IB = MIN( NB, K-I+1 )
            IF( I+IB.LE.M ) THEN
*
*              Form the triangular factor of the block reflector
*              H = H(i) H(i+1) . . . H(i+ib-1)
*
               CALL DLARFT( 'Forward', 'Rowwise', N-I+1, IB, A( I,
     $                      I ),
     $                      LDA, TAU( I ), WORK, LDWORK )
*
*              Apply H**T to A(i+ib:m,i:n) from the right
*
               CALL DLARFB( 'Right', 'Transpose', 'Forward',
     $                      'Rowwise',
     $                      M-I-IB+1, N-I+1, IB, A( I, I ), LDA, WORK,
     $                      LDWORK, A( I+IB, I ), LDA, WORK( IB+1 ),
     $                      LDWORK )
            END IF
*
*           Apply H**T to columns i:n of current block
*
            CALL DORGL2( IB, N-I+1, IB, A( I, I ), LDA, TAU( I ),
     $                   WORK,
     $                   IINFO )
*
*           Set columns 1:i-1 of current block to zero
*
            DO 40 J = 1, I - 1
               DO 30 L = I, I + IB - 1
                  A( L, J ) = ZERO
   30          CONTINUE
   40       CONTINUE
   50    CONTINUE
      END IF
*
      WORK( 1 ) = IWS
      RETURN
*
*     End of DORGLQ
*
      END
*> \brief \b DORGQL
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGQL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorgql.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorgql.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorgql.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORGQL( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, K, LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORGQL generates an M-by-N real matrix Q with orthonormal columns,
*> which is defined as the last N columns of a product of K elementary
*> reflectors of order M
*>
*>       Q  =  H(k) . . . H(2) H(1)
*>
*> as returned by DGEQLF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix Q. M >= N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines the
*>          matrix Q. N >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the (n-k+i)-th column must contain the vector which
*>          defines the elementary reflector H(i), for i = 1,2,...,k, as
*>          returned by DGEQLF in the last k columns of its array
*>          argument A.
*>          On exit, the M-by-N matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DGEQLF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= max(1,N).
*>          For optimum performance LWORK >= N*NB, where NB is the
*>          optimal blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ungql
*
*  =====================================================================
      SUBROUTINE DORGQL( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IB, IINFO, IWS, J, KK, L, LDWORK, LWKOPT,
     $                   NB, NBMIN, NX
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARFB, DLARFT, DORG2L, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 .OR. N.GT.M ) THEN
         INFO = -2
      ELSE IF( K.LT.0 .OR. K.GT.N ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -5
      END IF
*
      IF( INFO.EQ.0 ) THEN
         IF( N.EQ.0 ) THEN
            LWKOPT = 1
         ELSE
            NB = ILAENV( 1, 'DORGQL', ' ', M, N, K, -1 )
            LWKOPT = N*NB
         END IF
         WORK( 1 ) = LWKOPT
*
         IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
            INFO = -8
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORGQL', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.LE.0 ) THEN
         RETURN
      END IF
*
      NBMIN = 2
      NX = 0
      IWS = N
      IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
*        Determine when to cross over from blocked to unblocked code.
*
         NX = MAX( 0, ILAENV( 3, 'DORGQL', ' ', M, N, K, -1 ) )
         IF( NX.LT.K ) THEN
*
*           Determine if workspace is large enough for blocked code.
*
            LDWORK = N
            IWS = LDWORK*NB
            IF( LWORK.LT.IWS ) THEN
*
*              Not enough workspace to use optimal NB:  reduce NB and
*              determine the minimum value of NB.
*
               NB = LWORK / LDWORK
               NBMIN = MAX( 2, ILAENV( 2, 'DORGQL', ' ', M, N, K,
     $                      -1 ) )
            END IF
         END IF
      END IF
*
      IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
*        Use blocked code after the first block.
*        The last kk columns are handled by the block method.
*
         KK = MIN( K, ( ( K-NX+NB-1 ) / NB )*NB )
*
*        Set A(m-kk+1:m,1:n-kk) to zero.
*
         DO 20 J = 1, N - KK
            DO 10 I = M - KK + 1, M
               A( I, J ) = ZERO
   10       CONTINUE
   20    CONTINUE
      ELSE
         KK = 0
      END IF
*
*     Use unblocked code for the first or only block.
*
      CALL DORG2L( M-KK, N-KK, K-KK, A, LDA, TAU, WORK, IINFO )
*
      IF( KK.GT.0 ) THEN
*
*        Use blocked code
*
         DO 50 I = K - KK + 1, K, NB
            IB = MIN( NB, K-I+1 )
            IF( N-K+I.GT.1 ) THEN
*
*              Form the triangular factor of the block reflector
*              H = H(i+ib-1) . . . H(i+1) H(i)
*
               CALL DLARFT( 'Backward', 'Columnwise', M-K+I+IB-1, IB,
     $                      A( 1, N-K+I ), LDA, TAU( I ), WORK, LDWORK )
*
*              Apply H to A(1:m-k+i+ib-1,1:n-k+i-1) from the left
*
               CALL DLARFB( 'Left', 'No transpose', 'Backward',
     $                      'Columnwise', M-K+I+IB-1, N-K+I-1, IB,
     $                      A( 1, N-K+I ), LDA, WORK, LDWORK, A, LDA,
     $                      WORK( IB+1 ), LDWORK )
            END IF
*
*           Apply H to rows 1:m-k+i+ib-1 of current block
*
            CALL DORG2L( M-K+I+IB-1, IB, IB, A( 1, N-K+I ), LDA,
     $                   TAU( I ), WORK, IINFO )
*
*           Set rows m-k+i+ib:m of current block to zero
*
            DO 40 J = N - K + I, N - K + I + IB - 1
               DO 30 L = M - K + I + IB, M
                  A( L, J ) = ZERO
   30          CONTINUE
   40       CONTINUE
   50    CONTINUE
      END IF
*
      WORK( 1 ) = IWS
      RETURN
*
*     End of DORGQL
*
      END
*> \brief \b DORGQR
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorgqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorgqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorgqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORGQR( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, K, LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORGQR generates an M-by-N real matrix Q with orthonormal columns,
*> which is defined as the first N columns of a product of K elementary
*> reflectors of order M
*>
*>       Q  =  H(1) H(2) . . . H(k)
*>
*> as returned by DGEQRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix Q. M >= N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines the
*>          matrix Q. N >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the i-th column must contain the vector which
*>          defines the elementary reflector H(i), for i = 1,2,...,k, as
*>          returned by DGEQRF in the first k columns of its array
*>          argument A.
*>          On exit, the M-by-N matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DGEQRF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= max(1,N).
*>          For optimum performance LWORK >= N*NB, where NB is the
*>          optimal blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ungqr
*
*  =====================================================================
      SUBROUTINE DORGQR( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IB, IINFO, IWS, J, KI, KK, L, LDWORK,
     $                   LWKOPT, NB, NBMIN, NX
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARFB, DLARFT, DORG2R, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      NB = ILAENV( 1, 'DORGQR', ' ', M, N, K, -1 )
      LWKOPT = MAX( 1, N )*NB
      WORK( 1 ) = LWKOPT
      LQUERY = ( LWORK.EQ.-1 )
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 .OR. N.GT.M ) THEN
         INFO = -2
      ELSE IF( K.LT.0 .OR. K.GT.N ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -5
      ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
         INFO = -8
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORGQR', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.LE.0 ) THEN
         WORK( 1 ) = 1
         RETURN
      END IF
*
      NBMIN = 2
      NX = 0
      IWS = N
      IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
*        Determine when to cross over from blocked to unblocked code.
*
         NX = MAX( 0, ILAENV( 3, 'DORGQR', ' ', M, N, K, -1 ) )
         IF( NX.LT.K ) THEN
*
*           Determine if workspace is large enough for blocked code.
*
            LDWORK = N
            IWS = LDWORK*NB
            IF( LWORK.LT.IWS ) THEN
*
*              Not enough workspace to use optimal NB:  reduce NB and
*              determine the minimum value of NB.
*
               NB = LWORK / LDWORK
               NBMIN = MAX( 2, ILAENV( 2, 'DORGQR', ' ', M, N, K,
     $                      -1 ) )
            END IF
         END IF
      END IF
*
      IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
*        Use blocked code after the last block.
*        The first kk columns are handled by the block method.
*
         KI = ( ( K-NX-1 ) / NB )*NB
         KK = MIN( K, KI+NB )
*
*        Set A(1:kk,kk+1:n) to zero.
*
         DO 20 J = KK + 1, N
            DO 10 I = 1, KK
               A( I, J ) = ZERO
   10       CONTINUE
   20    CONTINUE
      ELSE
         KK = 0
      END IF
*
*     Use unblocked code for the last or only block.
*
      IF( KK.LT.N )
     $   CALL DORG2R( M-KK, N-KK, K-KK, A( KK+1, KK+1 ), LDA,
     $                TAU( KK+1 ), WORK, IINFO )
*
      IF( KK.GT.0 ) THEN
*
*        Use blocked code
*
         DO 50 I = KI + 1, 1, -NB
            IB = MIN( NB, K-I+1 )
            IF( I+IB.LE.N ) THEN
*
*              Form the triangular factor of the block reflector
*              H = H(i) H(i+1) . . . H(i+ib-1)
*
               CALL DLARFT( 'Forward', 'Columnwise', M-I+1, IB,
     $                      A( I, I ), LDA, TAU( I ), WORK, LDWORK )
*
*              Apply H to A(i:m,i+ib:n) from the left
*
               CALL DLARFB( 'Left', 'No transpose', 'Forward',
     $                      'Columnwise', M-I+1, N-I-IB+1, IB,
     $                      A( I, I ), LDA, WORK, LDWORK, A( I, I+IB ),
     $                      LDA, WORK( IB+1 ), LDWORK )
            END IF
*
*           Apply H to rows i:m of current block
*
            CALL DORG2R( M-I+1, IB, IB, A( I, I ), LDA, TAU( I ),
     $                   WORK,
     $                   IINFO )
*
*           Set rows 1:i-1 of current block to zero
*
            DO 40 J = I, I + IB - 1
               DO 30 L = 1, I - 1
                  A( L, J ) = ZERO
   30          CONTINUE
   40       CONTINUE
   50    CONTINUE
      END IF
*
      WORK( 1 ) = IWS
      RETURN
*
*     End of DORGQR
*
      END
*> \brief \b DORGR2 generates all or part of the orthogonal matrix Q from an RQ factorization determined by sgerqf (unblocked algorithm).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGR2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorgr2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorgr2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorgr2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORGR2( M, N, K, A, LDA, TAU, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, K, LDA, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORGR2 generates an m by n real matrix Q with orthonormal rows,
*> which is defined as the last m rows of a product of k elementary
*> reflectors of order n
*>
*>       Q  =  H(1) H(2) . . . H(k)
*>
*> as returned by DGERQF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix Q. N >= M.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines the
*>          matrix Q. M >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the (m-k+i)-th row must contain the vector which
*>          defines the elementary reflector H(i), for i = 1,2,...,k, as
*>          returned by DGERQF in the last k rows of its array argument
*>          A.
*>          On exit, the m by n matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DGERQF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (M)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ungr2
*
*  =====================================================================
      SUBROUTINE DORGR2( M, N, K, A, LDA, TAU, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, II, J, L
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF, DSCAL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.M ) THEN
         INFO = -2
      ELSE IF( K.LT.0 .OR. K.GT.M ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -5
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORGR2', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.LE.0 )
     $   RETURN
*
      IF( K.LT.M ) THEN
*
*        Initialise rows 1:m-k to rows of the unit matrix
*
         DO 20 J = 1, N
            DO 10 L = 1, M - K
               A( L, J ) = ZERO
   10       CONTINUE
            IF( J.GT.N-M .AND. J.LE.N-K )
     $         A( M-N+J, J ) = ONE
   20    CONTINUE
      END IF
*
      DO 40 I = 1, K
         II = M - K + I
*
*        Apply H(i) to A(1:m-k+i,1:n-k+i) from the right
*
         !A( II, N-M+II ) = ONE
         CALL DLARF1L( 'Right', II-1, N-M+II, A( II, 1 ), LDA,
     $               TAU( I ),
     $               A, LDA, WORK )
         CALL DSCAL( N-M+II-1, -TAU( I ), A( II, 1 ), LDA )
         A( II, N-M+II ) = ONE - TAU( I )
*
*        Set A(m-k+i,n-k+i+1:n) to zero
*
         DO 30 L = N - M + II + 1, N
            A( II, L ) = ZERO
   30    CONTINUE
   40 CONTINUE
      RETURN
*
*     End of DORGR2
*
      END
*> \brief \b DORGRQ
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGRQ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorgrq.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorgrq.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorgrq.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORGRQ( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, K, LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORGRQ generates an M-by-N real matrix Q with orthonormal rows,
*> which is defined as the last M rows of a product of K elementary
*> reflectors of order N
*>
*>       Q  =  H(1) H(2) . . . H(k)
*>
*> as returned by DGERQF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix Q. N >= M.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines the
*>          matrix Q. M >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the (m-k+i)-th row must contain the vector which
*>          defines the elementary reflector H(i), for i = 1,2,...,k, as
*>          returned by DGERQF in the last k rows of its array argument
*>          A.
*>          On exit, the M-by-N matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DGERQF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= max(1,M).
*>          For optimum performance LWORK >= M*NB, where NB is the
*>          optimal blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ungrq
*
*  =====================================================================
      SUBROUTINE DORGRQ( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IB, II, IINFO, IWS, J, KK, L, LDWORK,
     $                   LWKOPT, NB, NBMIN, NX
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARFB, DLARFT, DORGR2, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.M ) THEN
         INFO = -2
      ELSE IF( K.LT.0 .OR. K.GT.M ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -5
      END IF
*
      IF( INFO.EQ.0 ) THEN
         IF( M.LE.0 ) THEN
            LWKOPT = 1
         ELSE
            NB = ILAENV( 1, 'DORGRQ', ' ', M, N, K, -1 )
            LWKOPT = M*NB
         END IF
         WORK( 1 ) = LWKOPT
*
         IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN
            INFO = -8
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORGRQ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.LE.0 ) THEN
         RETURN
      END IF
*
      NBMIN = 2
      NX = 0
      IWS = M
      IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
*        Determine when to cross over from blocked to unblocked code.
*
         NX = MAX( 0, ILAENV( 3, 'DORGRQ', ' ', M, N, K, -1 ) )
         IF( NX.LT.K ) THEN
*
*           Determine if workspace is large enough for blocked code.
*
            LDWORK = M
            IWS = LDWORK*NB
            IF( LWORK.LT.IWS ) THEN
*
*              Not enough workspace to use optimal NB:  reduce NB and
*              determine the minimum value of NB.
*
               NB = LWORK / LDWORK
               NBMIN = MAX( 2, ILAENV( 2, 'DORGRQ', ' ', M, N, K,
     $                      -1 ) )
            END IF
         END IF
      END IF
*
      IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
*        Use blocked code after the first block.
*        The last kk rows are handled by the block method.
*
         KK = MIN( K, ( ( K-NX+NB-1 ) / NB )*NB )
*
*        Set A(1:m-kk,n-kk+1:n) to zero.
*
         DO 20 J = N - KK + 1, N
            DO 10 I = 1, M - KK
               A( I, J ) = ZERO
   10       CONTINUE
   20    CONTINUE
      ELSE
         KK = 0
      END IF
*
*     Use unblocked code for the first or only block.
*
      CALL DORGR2( M-KK, N-KK, K-KK, A, LDA, TAU, WORK, IINFO )
*
      IF( KK.GT.0 ) THEN
*
*        Use blocked code
*
         DO 50 I = K - KK + 1, K, NB
            IB = MIN( NB, K-I+1 )
            II = M - K + I
            IF( II.GT.1 ) THEN
*
*              Form the triangular factor of the block reflector
*              H = H(i+ib-1) . . . H(i+1) H(i)
*
               CALL DLARFT( 'Backward', 'Rowwise', N-K+I+IB-1, IB,
     $                      A( II, 1 ), LDA, TAU( I ), WORK, LDWORK )
*
*              Apply H**T to A(1:m-k+i-1,1:n-k+i+ib-1) from the right
*
               CALL DLARFB( 'Right', 'Transpose', 'Backward',
     $                      'Rowwise',
     $                      II-1, N-K+I+IB-1, IB, A( II, 1 ), LDA, WORK,
     $                      LDWORK, A, LDA, WORK( IB+1 ), LDWORK )
            END IF
*
*           Apply H**T to columns 1:n-k+i+ib-1 of current block
*
            CALL DORGR2( IB, N-K+I+IB-1, IB, A( II, 1 ), LDA,
     $                   TAU( I ),
     $                   WORK, IINFO )
*
*           Set columns n-k+i+ib:n of current block to zero
*
            DO 40 L = N - K + I + IB, N
               DO 30 J = II, II + IB - 1
                  A( J, L ) = ZERO
   30          CONTINUE
   40       CONTINUE
   50    CONTINUE
      END IF
*
      WORK( 1 ) = IWS
      RETURN
*
*     End of DORGRQ
*
      END
*> \brief \b DORGTR
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGTR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorgtr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorgtr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorgtr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORGTR( UPLO, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, LWORK, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORGTR generates a real orthogonal matrix Q which is defined as the
*> product of n-1 elementary reflectors of order N, as returned by
*> DSYTRD:
*>
*> if UPLO = 'U', Q = H(n-1) . . . H(2) H(1),
*>
*> if UPLO = 'L', Q = H(1) H(2) . . . H(n-1).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U': Upper triangle of A contains elementary reflectors
*>                 from DSYTRD;
*>          = 'L': Lower triangle of A contains elementary reflectors
*>                 from DSYTRD.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix Q. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the vectors which define the elementary reflectors,
*>          as returned by DSYTRD.
*>          On exit, the N-by-N orthogonal matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (N-1)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DSYTRD.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= max(1,N-1).
*>          For optimum performance LWORK >= (N-1)*NB, where NB is
*>          the optimal blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ungtr
*
*  =====================================================================
      SUBROUTINE DORGTR( UPLO, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, UPPER
      INTEGER            I, IINFO, J, LWKOPT, NB
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DORGQL, DORGQR, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      ELSE IF( LWORK.LT.MAX( 1, N-1 ) .AND. .NOT.LQUERY ) THEN
         INFO = -7
      END IF
*
      IF( INFO.EQ.0 ) THEN
         IF( UPPER ) THEN
            NB = ILAENV( 1, 'DORGQL', ' ', N-1, N-1, N-1, -1 )
         ELSE
            NB = ILAENV( 1, 'DORGQR', ' ', N-1, N-1, N-1, -1 )
         END IF
         LWKOPT = MAX( 1, N-1 )*NB
         WORK( 1 ) = LWKOPT
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORGTR', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 ) THEN
         WORK( 1 ) = 1
         RETURN
      END IF
*
      IF( UPPER ) THEN
*
*        Q was determined by a call to DSYTRD with UPLO = 'U'
*
*        Shift the vectors which define the elementary reflectors one
*        column to the left, and set the last row and column of Q to
*        those of the unit matrix
*
         DO 20 J = 1, N - 1
            DO 10 I = 1, J - 1
               A( I, J ) = A( I, J+1 )
   10       CONTINUE
            A( N, J ) = ZERO
   20    CONTINUE
         DO 30 I = 1, N - 1
            A( I, N ) = ZERO
   30    CONTINUE
         A( N, N ) = ONE
*
*        Generate Q(1:n-1,1:n-1)
*
         CALL DORGQL( N-1, N-1, N-1, A, LDA, TAU, WORK, LWORK,
     $                IINFO )
*
      ELSE
*
*        Q was determined by a call to DSYTRD with UPLO = 'L'.
*
*        Shift the vectors which define the elementary reflectors one
*        column to the right, and set the first row and column of Q to
*        those of the unit matrix
*
         DO 50 J = N, 2, -1
            A( 1, J ) = ZERO
            DO 40 I = J + 1, N
               A( I, J ) = A( I, J-1 )
   40       CONTINUE
   50    CONTINUE
         A( 1, 1 ) = ONE
         DO 60 I = 2, N
            A( I, 1 ) = ZERO
   60    CONTINUE
         IF( N.GT.1 ) THEN
*
*           Generate Q(2:n,2:n)
*
            CALL DORGQR( N-1, N-1, N-1, A( 2, 2 ), LDA, TAU, WORK,
     $                   LWORK, IINFO )
         END IF
      END IF
      WORK( 1 ) = LWKOPT
      RETURN
*
*     End of DORGTR
*
      END
*> \brief \b DORM2L multiplies a general matrix by the orthogonal matrix from a QL factorization determined by sgeqlf (unblocked algorithm).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORM2L + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorm2l.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorm2l.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorm2l.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORM2L( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
*                          WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIDE, TRANS
*       INTEGER            INFO, K, LDA, LDC, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORM2L overwrites the general real m by n matrix C with
*>
*>       Q * C  if SIDE = 'L' and TRANS = 'N', or
*>
*>       Q**T * C  if SIDE = 'L' and TRANS = 'T', or
*>
*>       C * Q  if SIDE = 'R' and TRANS = 'N', or
*>
*>       C * Q**T if SIDE = 'R' and TRANS = 'T',
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*>       Q = H(k) . . . H(2) H(1)
*>
*> as returned by DGEQLF. Q is of order m if SIDE = 'L' and of order n
*> if SIDE = 'R'.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': apply Q or Q**T from the Left
*>          = 'R': apply Q or Q**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N': apply Q  (No transpose)
*>          = 'T': apply Q**T (Transpose)
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines
*>          the matrix Q.
*>          If SIDE = 'L', M >= K >= 0;
*>          if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,K)
*>          The i-th column must contain the vector which defines the
*>          elementary reflector H(i), for i = 1,2,...,k, as returned by
*>          DGEQLF in the last k columns of its array argument A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.
*>          If SIDE = 'L', LDA >= max(1,M);
*>          if SIDE = 'R', LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DGEQLF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the m by n matrix C.
*>          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension
*>                                   (N) if SIDE = 'L',
*>                                   (M) if SIDE = 'R'
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unm2l
*
*  =====================================================================
      SUBROUTINE DORM2L( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
     $                   WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS
      INTEGER            INFO, K, LDA, LDC, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LEFT, NOTRAN
      INTEGER            I, I1, I2, I3, MI, NI, NQ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF1L, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LEFT = LSAME( SIDE, 'L' )
      NOTRAN = LSAME( TRANS, 'N' )
*
*     NQ is the order of Q
*
      IF( LEFT ) THEN
         NQ = M
      ELSE
         NQ = N
      END IF
      IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
         INFO = -2
      ELSE IF( M.LT.0 ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
         INFO = -7
      ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
         INFO = -10
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORM2L', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 )
     $   RETURN
*
      IF( ( LEFT .AND. NOTRAN ) .OR. ( .NOT.LEFT .AND. .NOT.NOTRAN ) )
     $     THEN
         I1 = 1
         I2 = K
         I3 = 1
      ELSE
         I1 = K
         I2 = 1
         I3 = -1
      END IF
*
      IF( LEFT ) THEN
         NI = N
      ELSE
         MI = M
      END IF
*
      DO 10 I = I1, I2, I3
         IF( LEFT ) THEN
*
*           H(i) is applied to C(1:m-k+i,1:n)
*
            MI = M - K + I
         ELSE
*
*           H(i) is applied to C(1:m,1:n-k+i)
*
            NI = N - K + I
         END IF
*
*        Apply H(i)
*
         CALL DLARF1L( SIDE, MI, NI, A( 1, I ), 1, TAU( I ), C, LDC,
     $               WORK )
   10 CONTINUE
      RETURN
*
*     End of DORM2L
*
      END
*> \brief \b DORM2R multiplies a general matrix by the orthogonal matrix from a QR factorization determined by sgeqrf (unblocked algorithm).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORM2R + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorm2r.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorm2r.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorm2r.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORM2R( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
*                          WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIDE, TRANS
*       INTEGER            INFO, K, LDA, LDC, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORM2R overwrites the general real m by n matrix C with
*>
*>       Q * C  if SIDE = 'L' and TRANS = 'N', or
*>
*>       Q**T* C  if SIDE = 'L' and TRANS = 'T', or
*>
*>       C * Q  if SIDE = 'R' and TRANS = 'N', or
*>
*>       C * Q**T if SIDE = 'R' and TRANS = 'T',
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*>       Q = H(1) H(2) . . . H(k)
*>
*> as returned by DGEQRF. Q is of order m if SIDE = 'L' and of order n
*> if SIDE = 'R'.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': apply Q or Q**T from the Left
*>          = 'R': apply Q or Q**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N': apply Q  (No transpose)
*>          = 'T': apply Q**T (Transpose)
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines
*>          the matrix Q.
*>          If SIDE = 'L', M >= K >= 0;
*>          if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,K)
*>          The i-th column must contain the vector which defines the
*>          elementary reflector H(i), for i = 1,2,...,k, as returned by
*>          DGEQRF in the first k columns of its array argument A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.
*>          If SIDE = 'L', LDA >= max(1,M);
*>          if SIDE = 'R', LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DGEQRF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the m by n matrix C.
*>          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension
*>                                   (N) if SIDE = 'L',
*>                                   (M) if SIDE = 'R'
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unm2r
*
*  =====================================================================
      SUBROUTINE DORM2R( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
     $                   WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS
      INTEGER            INFO, K, LDA, LDC, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LEFT, NOTRAN
      INTEGER            I, I1, I2, I3, IC, JC, MI, NI, NQ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL            XERBLA, DLARF1F
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LEFT = LSAME( SIDE, 'L' )
      NOTRAN = LSAME( TRANS, 'N' )
*
*     NQ is the order of Q
*
      IF( LEFT ) THEN
         NQ = M
      ELSE
         NQ = N
      END IF
      IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
         INFO = -2
      ELSE IF( M.LT.0 ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
         INFO = -7
      ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
         INFO = -10
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORM2R', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 )
     $   RETURN
*
      IF( ( LEFT .AND. .NOT.NOTRAN ) .OR. ( .NOT.LEFT .AND. NOTRAN ) )
     $     THEN
         I1 = 1
         I2 = K
         I3 = 1
      ELSE
         I1 = K
         I2 = 1
         I3 = -1
      END IF
*
      IF( LEFT ) THEN
         NI = N
         JC = 1
      ELSE
         MI = M
         IC = 1
      END IF
*
      DO 10 I = I1, I2, I3
         IF( LEFT ) THEN
*
*           H(i) is applied to C(i:m,1:n)
*
            MI = M - I + 1
            IC = I
         ELSE
*
*           H(i) is applied to C(1:m,i:n)
*
            NI = N - I + 1
            JC = I
         END IF
*
*        Apply H(i)
*
         CALL DLARF1F( SIDE, MI, NI, A( I, I ), 1, TAU( I ), C( IC,
     $               JC ),
     $               LDC, WORK )
   10 CONTINUE
      RETURN
*
*     End of DORM2R
*
      END
*> \brief \b DORMBR
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMBR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormbr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormbr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormbr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORMBR( VECT, SIDE, TRANS, M, N, K, A, LDA, TAU, C,
*                          LDC, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIDE, TRANS, VECT
*       INTEGER            INFO, K, LDA, LDC, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> If VECT = 'Q', DORMBR overwrites the general real M-by-N matrix C
*> with
*>                 SIDE = 'L'     SIDE = 'R'
*> TRANS = 'N':      Q * C          C * Q
*> TRANS = 'T':      Q**T * C       C * Q**T
*>
*> If VECT = 'P', DORMBR overwrites the general real M-by-N matrix C
*> with
*>                 SIDE = 'L'     SIDE = 'R'
*> TRANS = 'N':      P * C          C * P
*> TRANS = 'T':      P**T * C       C * P**T
*>
*> Here Q and P**T are the orthogonal matrices determined by DGEBRD when
*> reducing a real matrix A to bidiagonal form: A = Q * B * P**T. Q and
*> P**T are defined as products of elementary reflectors H(i) and G(i)
*> respectively.
*>
*> Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the
*> order of the orthogonal matrix Q or P**T that is applied.
*>
*> If VECT = 'Q', A is assumed to have been an NQ-by-K matrix:
*> if nq >= k, Q = H(1) H(2) . . . H(k);
*> if nq < k, Q = H(1) H(2) . . . H(nq-1).
*>
*> If VECT = 'P', A is assumed to have been a K-by-NQ matrix:
*> if k < nq, P = G(1) G(2) . . . G(k);
*> if k >= nq, P = G(1) G(2) . . . G(nq-1).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] VECT
*> \verbatim
*>          VECT is CHARACTER*1
*>          = 'Q': apply Q or Q**T;
*>          = 'P': apply P or P**T.
*> \endverbatim
*>
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': apply Q, Q**T, P or P**T from the Left;
*>          = 'R': apply Q, Q**T, P or P**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N':  No transpose, apply Q  or P;
*>          = 'T':  Transpose, apply Q**T or P**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          If VECT = 'Q', the number of columns in the original
*>          matrix reduced by DGEBRD.
*>          If VECT = 'P', the number of rows in the original
*>          matrix reduced by DGEBRD.
*>          K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension
*>                                (LDA,min(nq,K)) if VECT = 'Q'
*>                                (LDA,nq)        if VECT = 'P'
*>          The vectors which define the elementary reflectors H(i) and
*>          G(i), whose products determine the matrices Q and P, as
*>          returned by DGEBRD.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.
*>          If VECT = 'Q', LDA >= max(1,nq);
*>          if VECT = 'P', LDA >= max(1,min(nq,K)).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (min(nq,K))
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i) or G(i) which determines Q or P, as returned
*>          by DGEBRD in the array argument TAUQ or TAUP.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the M-by-N matrix C.
*>          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q
*>          or P*C or P**T*C or C*P or C*P**T.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If SIDE = 'L', LWORK >= max(1,N);
*>          if SIDE = 'R', LWORK >= max(1,M).
*>          For optimum performance LWORK >= N*NB if SIDE = 'L', and
*>          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
*>          blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unmbr
*
*  =====================================================================
      SUBROUTINE DORMBR( VECT, SIDE, TRANS, M, N, K, A, LDA, TAU, C,
     $                   LDC, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS, VECT
      INTEGER            INFO, K, LDA, LDC, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            APPLYQ, LEFT, LQUERY, NOTRAN
      CHARACTER          TRANST
      INTEGER            I1, I2, IINFO, LWKOPT, MI, NB, NI, NQ, NW
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DORMLQ, DORMQR, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      APPLYQ = LSAME( VECT, 'Q' )
      LEFT = LSAME( SIDE, 'L' )
      NOTRAN = LSAME( TRANS, 'N' )
      LQUERY = ( LWORK.EQ.-1 )
*
*     NQ is the order of Q or P and NW is the minimum dimension of WORK
*
      IF( LEFT ) THEN
         NQ = M
         NW = MAX( 1, N )
      ELSE
         NQ = N
         NW = MAX( 1, M )
      END IF
      IF( .NOT.APPLYQ .AND. .NOT.LSAME( VECT, 'P' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
         INFO = -2
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
         INFO = -3
      ELSE IF( M.LT.0 ) THEN
         INFO = -4
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      ELSE IF( K.LT.0 ) THEN
         INFO = -6
      ELSE IF( ( APPLYQ .AND. LDA.LT.MAX( 1, NQ ) ) .OR.
     $         ( .NOT.APPLYQ .AND. LDA.LT.MAX( 1, MIN( NQ, K ) ) ) )
     $          THEN
         INFO = -8
      ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
         INFO = -11
      ELSE IF( LWORK.LT.NW .AND. .NOT.LQUERY ) THEN
         INFO = -13
      END IF
*
      IF( INFO.EQ.0 ) THEN
         IF( APPLYQ ) THEN
            IF( LEFT ) THEN
               NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, M-1, N, M-1,
     $              -1 )
            ELSE
               NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, M, N-1, N-1,
     $              -1 )
            END IF
         ELSE
            IF( LEFT ) THEN
               NB = ILAENV( 1, 'DORMLQ', SIDE // TRANS, M-1, N, M-1,
     $              -1 )
            ELSE
               NB = ILAENV( 1, 'DORMLQ', SIDE // TRANS, M, N-1, N-1,
     $              -1 )
            END IF
         END IF
         LWKOPT = NW*NB
         WORK( 1 ) = LWKOPT
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORMBR', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      WORK( 1 ) = 1
      IF( M.EQ.0 .OR. N.EQ.0 )
     $   RETURN
*
      IF( APPLYQ ) THEN
*
*        Apply Q
*
         IF( NQ.GE.K ) THEN
*
*           Q was determined by a call to DGEBRD with nq >= k
*
            CALL DORMQR( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
     $                   WORK, LWORK, IINFO )
         ELSE IF( NQ.GT.1 ) THEN
*
*           Q was determined by a call to DGEBRD with nq < k
*
            IF( LEFT ) THEN
               MI = M - 1
               NI = N
               I1 = 2
               I2 = 1
            ELSE
               MI = M
               NI = N - 1
               I1 = 1
               I2 = 2
            END IF
            CALL DORMQR( SIDE, TRANS, MI, NI, NQ-1, A( 2, 1 ), LDA,
     $                   TAU,
     $                   C( I1, I2 ), LDC, WORK, LWORK, IINFO )
         END IF
      ELSE
*
*        Apply P
*
         IF( NOTRAN ) THEN
            TRANST = 'T'
         ELSE
            TRANST = 'N'
         END IF
         IF( NQ.GT.K ) THEN
*
*           P was determined by a call to DGEBRD with nq > k
*
            CALL DORMLQ( SIDE, TRANST, M, N, K, A, LDA, TAU, C, LDC,
     $                   WORK, LWORK, IINFO )
         ELSE IF( NQ.GT.1 ) THEN
*
*           P was determined by a call to DGEBRD with nq <= k
*
            IF( LEFT ) THEN
               MI = M - 1
               NI = N
               I1 = 2
               I2 = 1
            ELSE
               MI = M
               NI = N - 1
               I1 = 1
               I2 = 2
            END IF
            CALL DORMLQ( SIDE, TRANST, MI, NI, NQ-1, A( 1, 2 ), LDA,
     $                   TAU, C( I1, I2 ), LDC, WORK, LWORK, IINFO )
         END IF
      END IF
      WORK( 1 ) = LWKOPT
      RETURN
*
*     End of DORMBR
*
      END
*> \brief \b DORMHR
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMHR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormhr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormhr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormhr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORMHR( SIDE, TRANS, M, N, ILO, IHI, A, LDA, TAU, C,
*                          LDC, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIDE, TRANS
*       INTEGER            IHI, ILO, INFO, LDA, LDC, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORMHR overwrites the general real M-by-N matrix C with
*>
*>                 SIDE = 'L'     SIDE = 'R'
*> TRANS = 'N':      Q * C          C * Q
*> TRANS = 'T':      Q**T * C       C * Q**T
*>
*> where Q is a real orthogonal matrix of order nq, with nq = m if
*> SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
*> IHI-ILO elementary reflectors, as returned by DGEHRD:
*>
*> Q = H(ilo) H(ilo+1) . . . H(ihi-1).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': apply Q or Q**T from the Left;
*>          = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N':  No transpose, apply Q;
*>          = 'T':  Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*>          ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*>          IHI is INTEGER
*>
*>          ILO and IHI must have the same values as in the previous call
*>          of DGEHRD. Q is equal to the unit matrix except in the
*>          submatrix Q(ilo+1:ihi,ilo+1:ihi).
*>          If SIDE = 'L', then 1 <= ILO <= IHI <= M, if M > 0, and
*>          ILO = 1 and IHI = 0, if M = 0;
*>          if SIDE = 'R', then 1 <= ILO <= IHI <= N, if N > 0, and
*>          ILO = 1 and IHI = 0, if N = 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension
*>                               (LDA,M) if SIDE = 'L'
*>                               (LDA,N) if SIDE = 'R'
*>          The vectors which define the elementary reflectors, as
*>          returned by DGEHRD.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.
*>          LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension
*>                               (M-1) if SIDE = 'L'
*>                               (N-1) if SIDE = 'R'
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DGEHRD.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the M-by-N matrix C.
*>          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If SIDE = 'L', LWORK >= max(1,N);
*>          if SIDE = 'R', LWORK >= max(1,M).
*>          For optimum performance LWORK >= N*NB if SIDE = 'L', and
*>          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
*>          blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unmhr
*
*  =====================================================================
      SUBROUTINE DORMHR( SIDE, TRANS, M, N, ILO, IHI, A, LDA, TAU, C,
     $                   LDC, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS
      INTEGER            IHI, ILO, INFO, LDA, LDC, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            LEFT, LQUERY
      INTEGER            I1, I2, IINFO, LWKOPT, MI, NB, NH, NI, NQ, NW
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DORMQR, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      NH = IHI - ILO
      LEFT = LSAME( SIDE, 'L' )
      LQUERY = ( LWORK.EQ.-1 )
*
*     NQ is the order of Q and NW is the minimum dimension of WORK
*
      IF( LEFT ) THEN
         NQ = M
         NW = MAX( 1, N )
      ELSE
         NQ = N
         NW = MAX( 1, M )
      END IF
      IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.LSAME( TRANS, 'N' ) .AND.
     $         .NOT.LSAME( TRANS, 'T' ) )
     $          THEN
         INFO = -2
      ELSE IF( M.LT.0 ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, NQ ) ) THEN
         INFO = -5
      ELSE IF( IHI.LT.MIN( ILO, NQ ) .OR. IHI.GT.NQ ) THEN
         INFO = -6
      ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
         INFO = -8
      ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
         INFO = -11
      ELSE IF( LWORK.LT.NW .AND. .NOT.LQUERY ) THEN
         INFO = -13
      END IF
*
      IF( INFO.EQ.0 ) THEN
         IF( LEFT ) THEN
            NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, NH, N, NH, -1 )
         ELSE
            NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, M, NH, NH, -1 )
         END IF
         LWKOPT = NW*NB
         WORK( 1 ) = LWKOPT
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORMHR', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 .OR. NH.EQ.0 ) THEN
         WORK( 1 ) = 1
         RETURN
      END IF
*
      IF( LEFT ) THEN
         MI = NH
         NI = N
         I1 = ILO + 1
         I2 = 1
      ELSE
         MI = M
         NI = NH
         I1 = 1
         I2 = ILO + 1
      END IF
*
      CALL DORMQR( SIDE, TRANS, MI, NI, NH, A( ILO+1, ILO ), LDA,
     $             TAU( ILO ), C( I1, I2 ), LDC, WORK, LWORK, IINFO )
*
      WORK( 1 ) = LWKOPT
      RETURN
*
*     End of DORMHR
*
      END
*> \brief \b DORML2 multiplies a general matrix by the orthogonal matrix from a LQ factorization determined by sgelqf (unblocked algorithm).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORML2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorml2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorml2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorml2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORML2( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
*                          WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIDE, TRANS
*       INTEGER            INFO, K, LDA, LDC, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORML2 overwrites the general real m by n matrix C with
*>
*>       Q * C  if SIDE = 'L' and TRANS = 'N', or
*>
*>       Q**T* C  if SIDE = 'L' and TRANS = 'T', or
*>
*>       C * Q  if SIDE = 'R' and TRANS = 'N', or
*>
*>       C * Q**T if SIDE = 'R' and TRANS = 'T',
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*>       Q = H(k) . . . H(2) H(1)
*>
*> as returned by DGELQF. Q is of order m if SIDE = 'L' and of order n
*> if SIDE = 'R'.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': apply Q or Q**T from the Left
*>          = 'R': apply Q or Q**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N': apply Q  (No transpose)
*>          = 'T': apply Q**T (Transpose)
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines
*>          the matrix Q.
*>          If SIDE = 'L', M >= K >= 0;
*>          if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension
*>                               (LDA,M) if SIDE = 'L',
*>                               (LDA,N) if SIDE = 'R'
*>          The i-th row must contain the vector which defines the
*>          elementary reflector H(i), for i = 1,2,...,k, as returned by
*>          DGELQF in the first k rows of its array argument A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,K).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DGELQF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the m by n matrix C.
*>          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension
*>                                   (N) if SIDE = 'L',
*>                                   (M) if SIDE = 'R'
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unml2
*
*  =====================================================================
      SUBROUTINE DORML2( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
     $                   WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS
      INTEGER            INFO, K, LDA, LDC, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LEFT, NOTRAN
      INTEGER            I, I1, I2, I3, IC, JC, MI, NI, NQ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF1F, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LEFT = LSAME( SIDE, 'L' )
      NOTRAN = LSAME( TRANS, 'N' )
*
*     NQ is the order of Q
*
      IF( LEFT ) THEN
         NQ = M
      ELSE
         NQ = N
      END IF
      IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
         INFO = -2
      ELSE IF( M.LT.0 ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, K ) ) THEN
         INFO = -7
      ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
         INFO = -10
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORML2', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 )
     $   RETURN
*
      IF( ( LEFT .AND. NOTRAN ) .OR. ( .NOT.LEFT .AND. .NOT.NOTRAN ) )
     $     THEN
         I1 = 1
         I2 = K
         I3 = 1
      ELSE
         I1 = K
         I2 = 1
         I3 = -1
      END IF
*
      IF( LEFT ) THEN
         NI = N
         JC = 1
      ELSE
         MI = M
         IC = 1
      END IF
*
      DO 10 I = I1, I2, I3
         IF( LEFT ) THEN
*
*           H(i) is applied to C(i:m,1:n)
*
            MI = M - I + 1
            IC = I
         ELSE
*
*           H(i) is applied to C(1:m,i:n)
*
            NI = N - I + 1
            JC = I
         END IF
*
*        Apply H(i)
*
         CALL DLARF1F( SIDE, MI, NI, A( I, I ), LDA, TAU( I ),
     $               C( IC, JC ), LDC, WORK )
   10 CONTINUE
      RETURN
*
*     End of DORML2
*
      END
*> \brief \b DORMLQ
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMLQ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormlq.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormlq.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormlq.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORMLQ( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
*                          WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIDE, TRANS
*       INTEGER            INFO, K, LDA, LDC, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORMLQ overwrites the general real M-by-N matrix C with
*>
*>                 SIDE = 'L'     SIDE = 'R'
*> TRANS = 'N':      Q * C          C * Q
*> TRANS = 'T':      Q**T * C       C * Q**T
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*>       Q = H(k) . . . H(2) H(1)
*>
*> as returned by DGELQF. Q is of order M if SIDE = 'L' and of order N
*> if SIDE = 'R'.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': apply Q or Q**T from the Left;
*>          = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N':  No transpose, apply Q;
*>          = 'T':  Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines
*>          the matrix Q.
*>          If SIDE = 'L', M >= K >= 0;
*>          if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension
*>                               (LDA,M) if SIDE = 'L',
*>                               (LDA,N) if SIDE = 'R'
*>          The i-th row must contain the vector which defines the
*>          elementary reflector H(i), for i = 1,2,...,k, as returned by
*>          DGELQF in the first k rows of its array argument A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,K).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DGELQF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the M-by-N matrix C.
*>          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If SIDE = 'L', LWORK >= max(1,N);
*>          if SIDE = 'R', LWORK >= max(1,M).
*>          For good performance, LWORK should generally be larger.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unmlq
*
*  =====================================================================
      SUBROUTINE DORMLQ( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
     $                   WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS
      INTEGER            INFO, K, LDA, LDC, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            NBMAX, LDT, TSIZE
      PARAMETER          ( NBMAX = 64, LDT = NBMAX+1,
     $                     TSIZE = LDT*NBMAX )
*     ..
*     .. Local Scalars ..
      LOGICAL            LEFT, LQUERY, NOTRAN
      CHARACTER          TRANST
      INTEGER            I, I1, I2, I3, IB, IC, IINFO, IWT, JC, LDWORK,
     $                   LWKOPT, MI, NB, NBMIN, NI, NQ, NW
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARFB, DLARFT, DORML2, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LEFT = LSAME( SIDE, 'L' )
      NOTRAN = LSAME( TRANS, 'N' )
      LQUERY = ( LWORK.EQ.-1 )
*
*     NQ is the order of Q and NW is the minimum dimension of WORK
*
      IF( LEFT ) THEN
         NQ = M
         NW = MAX( 1, N )
      ELSE
         NQ = N
         NW = MAX( 1, M )
      END IF
      IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
         INFO = -2
      ELSE IF( M.LT.0 ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, K ) ) THEN
         INFO = -7
      ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
         INFO = -10
      ELSE IF( LWORK.LT.NW .AND. .NOT.LQUERY ) THEN
         INFO = -12
      END IF
*
      IF( INFO.EQ.0 ) THEN
*
*        Compute the workspace requirements
*
         NB = MIN( NBMAX, ILAENV( 1, 'DORMLQ', SIDE // TRANS, M, N,
     $             K,
     $        -1 ) )
         LWKOPT = NW*NB + TSIZE
         WORK( 1 ) = LWKOPT
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORMLQ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) THEN
         WORK( 1 ) = 1
         RETURN
      END IF
*
      NBMIN = 2
      LDWORK = NW
      IF( NB.GT.1 .AND. NB.LT.K ) THEN
         IF( LWORK.LT.LWKOPT ) THEN
            NB = (LWORK-TSIZE) / LDWORK
            NBMIN = MAX( 2, ILAENV( 2, 'DORMLQ', SIDE // TRANS, M, N,
     $                   K,
     $              -1 ) )
         END IF
      END IF
*
      IF( NB.LT.NBMIN .OR. NB.GE.K ) THEN
*
*        Use unblocked code
*
         CALL DORML2( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
     $                WORK,
     $                IINFO )
      ELSE
*
*        Use blocked code
*
         IWT = 1 + NW*NB
         IF( ( LEFT .AND. NOTRAN ) .OR.
     $       ( .NOT.LEFT .AND. .NOT.NOTRAN ) ) THEN
            I1 = 1
            I2 = K
            I3 = NB
         ELSE
            I1 = ( ( K-1 ) / NB )*NB + 1
            I2 = 1
            I3 = -NB
         END IF
*
         IF( LEFT ) THEN
            NI = N
            JC = 1
         ELSE
            MI = M
            IC = 1
         END IF
*
         IF( NOTRAN ) THEN
            TRANST = 'T'
         ELSE
            TRANST = 'N'
         END IF
*
         DO 10 I = I1, I2, I3
            IB = MIN( NB, K-I+1 )
*
*           Form the triangular factor of the block reflector
*           H = H(i) H(i+1) . . . H(i+ib-1)
*
            CALL DLARFT( 'Forward', 'Rowwise', NQ-I+1, IB, A( I, I ),
     $                   LDA, TAU( I ), WORK( IWT ), LDT )
            IF( LEFT ) THEN
*
*              H or H**T is applied to C(i:m,1:n)
*
               MI = M - I + 1
               IC = I
            ELSE
*
*              H or H**T is applied to C(1:m,i:n)
*
               NI = N - I + 1
               JC = I
            END IF
*
*           Apply H or H**T
*
            CALL DLARFB( SIDE, TRANST, 'Forward', 'Rowwise', MI, NI,
     $                   IB,
     $                   A( I, I ), LDA, WORK( IWT ), LDT,
     $                   C( IC, JC ), LDC, WORK, LDWORK )
   10    CONTINUE
      END IF
      WORK( 1 ) = LWKOPT
      RETURN
*
*     End of DORMLQ
*
      END
*> \brief \b DORMQL
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMQL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormql.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormql.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormql.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORMQL( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
*                          WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIDE, TRANS
*       INTEGER            INFO, K, LDA, LDC, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORMQL overwrites the general real M-by-N matrix C with
*>
*>                 SIDE = 'L'     SIDE = 'R'
*> TRANS = 'N':      Q * C          C * Q
*> TRANS = 'T':      Q**T * C       C * Q**T
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*>       Q = H(k) . . . H(2) H(1)
*>
*> as returned by DGEQLF. Q is of order M if SIDE = 'L' and of order N
*> if SIDE = 'R'.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': apply Q or Q**T from the Left;
*>          = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N':  No transpose, apply Q;
*>          = 'T':  Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines
*>          the matrix Q.
*>          If SIDE = 'L', M >= K >= 0;
*>          if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,K)
*>          The i-th column must contain the vector which defines the
*>          elementary reflector H(i), for i = 1,2,...,k, as returned by
*>          DGEQLF in the last k columns of its array argument A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.
*>          If SIDE = 'L', LDA >= max(1,M);
*>          if SIDE = 'R', LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DGEQLF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the M-by-N matrix C.
*>          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If SIDE = 'L', LWORK >= max(1,N);
*>          if SIDE = 'R', LWORK >= max(1,M).
*>          For good performance, LWORK should generally be larger.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unmql
*
*  =====================================================================
      SUBROUTINE DORMQL( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
     $                   WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS
      INTEGER            INFO, K, LDA, LDC, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            NBMAX, LDT, TSIZE
      PARAMETER          ( NBMAX = 64, LDT = NBMAX+1,
     $                     TSIZE = LDT*NBMAX )
*     ..
*     .. Local Scalars ..
      LOGICAL            LEFT, LQUERY, NOTRAN
      INTEGER            I, I1, I2, I3, IB, IINFO, IWT, LDWORK, LWKOPT,
     $                   MI, NB, NBMIN, NI, NQ, NW
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARFB, DLARFT, DORM2L, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LEFT = LSAME( SIDE, 'L' )
      NOTRAN = LSAME( TRANS, 'N' )
      LQUERY = ( LWORK.EQ.-1 )
*
*     NQ is the order of Q and NW is the minimum dimension of WORK
*
      IF( LEFT ) THEN
         NQ = M
         NW = MAX( 1, N )
      ELSE
         NQ = N
         NW = MAX( 1, M )
      END IF
      IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
         INFO = -2
      ELSE IF( M.LT.0 ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
         INFO = -7
      ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
         INFO = -10
      ELSE IF( LWORK.LT.NW .AND. .NOT.LQUERY ) THEN
         INFO = -12
      END IF
*
      IF( INFO.EQ.0 ) THEN
*
*        Compute the workspace requirements
*
         IF( M.EQ.0 .OR. N.EQ.0 ) THEN
            LWKOPT = 1
         ELSE
            NB = MIN( NBMAX, ILAENV( 1, 'DORMQL', SIDE // TRANS, M,
     $                N,
     $                               K, -1 ) )
            LWKOPT = NW*NB + TSIZE
         END IF
         WORK( 1 ) = LWKOPT
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORMQL', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 ) THEN
         RETURN
      END IF
*
      NBMIN = 2
      LDWORK = NW
      IF( NB.GT.1 .AND. NB.LT.K ) THEN
         IF( LWORK.LT.LWKOPT ) THEN
            NB = (LWORK-TSIZE) / LDWORK
            NBMIN = MAX( 2, ILAENV( 2, 'DORMQL', SIDE // TRANS, M, N,
     $                   K,
     $              -1 ) )
         END IF
      END IF
*
      IF( NB.LT.NBMIN .OR. NB.GE.K ) THEN
*
*        Use unblocked code
*
         CALL DORM2L( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
     $                WORK,
     $                IINFO )
      ELSE
*
*        Use blocked code
*
         IWT = 1 + NW*NB
         IF( ( LEFT .AND. NOTRAN ) .OR.
     $       ( .NOT.LEFT .AND. .NOT.NOTRAN ) ) THEN
            I1 = 1
            I2 = K
            I3 = NB
         ELSE
            I1 = ( ( K-1 ) / NB )*NB + 1
            I2 = 1
            I3 = -NB
         END IF
*
         IF( LEFT ) THEN
            NI = N
         ELSE
            MI = M
         END IF
*
         DO 10 I = I1, I2, I3
            IB = MIN( NB, K-I+1 )
*
*           Form the triangular factor of the block reflector
*           H = H(i+ib-1) . . . H(i+1) H(i)
*
            CALL DLARFT( 'Backward', 'Columnwise', NQ-K+I+IB-1, IB,
     $                   A( 1, I ), LDA, TAU( I ), WORK( IWT ), LDT )
            IF( LEFT ) THEN
*
*              H or H**T is applied to C(1:m-k+i+ib-1,1:n)
*
               MI = M - K + I + IB - 1
            ELSE
*
*              H or H**T is applied to C(1:m,1:n-k+i+ib-1)
*
               NI = N - K + I + IB - 1
            END IF
*
*           Apply H or H**T
*
            CALL DLARFB( SIDE, TRANS, 'Backward', 'Columnwise', MI,
     $                   NI,
     $                   IB, A( 1, I ), LDA, WORK( IWT ), LDT, C, LDC,
     $                   WORK, LDWORK )
   10    CONTINUE
      END IF
      WORK( 1 ) = LWKOPT
      RETURN
*
*     End of DORMQL
*
      END
*> \brief \b DORMQR
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORMQR( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
*                          WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIDE, TRANS
*       INTEGER            INFO, K, LDA, LDC, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORMQR overwrites the general real M-by-N matrix C with
*>
*>                 SIDE = 'L'     SIDE = 'R'
*> TRANS = 'N':      Q * C          C * Q
*> TRANS = 'T':      Q**T * C       C * Q**T
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*>       Q = H(1) H(2) . . . H(k)
*>
*> as returned by DGEQRF. Q is of order M if SIDE = 'L' and of order N
*> if SIDE = 'R'.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': apply Q or Q**T from the Left;
*>          = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N':  No transpose, apply Q;
*>          = 'T':  Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines
*>          the matrix Q.
*>          If SIDE = 'L', M >= K >= 0;
*>          if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,K)
*>          The i-th column must contain the vector which defines the
*>          elementary reflector H(i), for i = 1,2,...,k, as returned by
*>          DGEQRF in the first k columns of its array argument A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.
*>          If SIDE = 'L', LDA >= max(1,M);
*>          if SIDE = 'R', LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DGEQRF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the M-by-N matrix C.
*>          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If SIDE = 'L', LWORK >= max(1,N);
*>          if SIDE = 'R', LWORK >= max(1,M).
*>          For good performance, LWORK should generally be larger.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unmqr
*
*  =====================================================================
      SUBROUTINE DORMQR( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
     $                   WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS
      INTEGER            INFO, K, LDA, LDC, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            NBMAX, LDT, TSIZE
      PARAMETER          ( NBMAX = 64, LDT = NBMAX+1,
     $                     TSIZE = LDT*NBMAX )
*     ..
*     .. Local Scalars ..
      LOGICAL            LEFT, LQUERY, NOTRAN
      INTEGER            I, I1, I2, I3, IB, IC, IINFO, IWT, JC, LDWORK,
     $                   LWKOPT, MI, NB, NBMIN, NI, NQ, NW
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARFB, DLARFT, DORM2R, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LEFT = LSAME( SIDE, 'L' )
      NOTRAN = LSAME( TRANS, 'N' )
      LQUERY = ( LWORK.EQ.-1 )
*
*     NQ is the order of Q and NW is the minimum dimension of WORK
*
      IF( LEFT ) THEN
         NQ = M
         NW = MAX( 1, N )
      ELSE
         NQ = N
         NW = MAX( 1, M )
      END IF
      IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
         INFO = -2
      ELSE IF( M.LT.0 ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
         INFO = -7
      ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
         INFO = -10
      ELSE IF( LWORK.LT.NW .AND. .NOT.LQUERY ) THEN
         INFO = -12
      END IF
*
      IF( INFO.EQ.0 ) THEN
*
*        Compute the workspace requirements
*
         NB = MIN( NBMAX, ILAENV( 1, 'DORMQR', SIDE // TRANS, M, N,
     $             K,
     $        -1 ) )
         LWKOPT = NW*NB + TSIZE
         WORK( 1 ) = LWKOPT
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORMQR', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) THEN
         WORK( 1 ) = 1
         RETURN
      END IF
*
      NBMIN = 2
      LDWORK = NW
      IF( NB.GT.1 .AND. NB.LT.K ) THEN
         IF( LWORK.LT.LWKOPT ) THEN
            NB = (LWORK-TSIZE) / LDWORK
            NBMIN = MAX( 2, ILAENV( 2, 'DORMQR', SIDE // TRANS, M, N,
     $                   K,
     $              -1 ) )
         END IF
      END IF
*
      IF( NB.LT.NBMIN .OR. NB.GE.K ) THEN
*
*        Use unblocked code
*
         CALL DORM2R( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
     $                WORK,
     $                IINFO )
      ELSE
*
*        Use blocked code
*
         IWT = 1 + NW*NB
         IF( ( LEFT .AND. .NOT.NOTRAN ) .OR.
     $       ( .NOT.LEFT .AND. NOTRAN ) ) THEN
            I1 = 1
            I2 = K
            I3 = NB
         ELSE
            I1 = ( ( K-1 ) / NB )*NB + 1
            I2 = 1
            I3 = -NB
         END IF
*
         IF( LEFT ) THEN
            NI = N
            JC = 1
         ELSE
            MI = M
            IC = 1
         END IF
*
         DO 10 I = I1, I2, I3
            IB = MIN( NB, K-I+1 )
*
*           Form the triangular factor of the block reflector
*           H = H(i) H(i+1) . . . H(i+ib-1)
*
            CALL DLARFT( 'Forward', 'Columnwise', NQ-I+1, IB, A( I,
     $                   I ),
     $                   LDA, TAU( I ), WORK( IWT ), LDT )
            IF( LEFT ) THEN
*
*              H or H**T is applied to C(i:m,1:n)
*
               MI = M - I + 1
               IC = I
            ELSE
*
*              H or H**T is applied to C(1:m,i:n)
*
               NI = N - I + 1
               JC = I
            END IF
*
*           Apply H or H**T
*
            CALL DLARFB( SIDE, TRANS, 'Forward', 'Columnwise', MI,
     $                   NI,
     $                   IB, A( I, I ), LDA, WORK( IWT ), LDT,
     $                   C( IC, JC ), LDC, WORK, LDWORK )
   10    CONTINUE
      END IF
      WORK( 1 ) = LWKOPT
      RETURN
*
*     End of DORMQR
*
      END
*> \brief \b DORMR2 multiplies a general matrix by the orthogonal matrix from a RQ factorization determined by sgerqf (unblocked algorithm).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMR2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormr2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormr2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormr2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORMR2( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
*                          WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIDE, TRANS
*       INTEGER            INFO, K, LDA, LDC, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORMR2 overwrites the general real m by n matrix C with
*>
*>       Q * C  if SIDE = 'L' and TRANS = 'N', or
*>
*>       Q**T* C  if SIDE = 'L' and TRANS = 'T', or
*>
*>       C * Q  if SIDE = 'R' and TRANS = 'N', or
*>
*>       C * Q**T if SIDE = 'R' and TRANS = 'T',
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*>       Q = H(1) H(2) . . . H(k)
*>
*> as returned by DGERQF. Q is of order m if SIDE = 'L' and of order n
*> if SIDE = 'R'.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': apply Q or Q**T from the Left
*>          = 'R': apply Q or Q**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N': apply Q  (No transpose)
*>          = 'T': apply Q' (Transpose)
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines
*>          the matrix Q.
*>          If SIDE = 'L', M >= K >= 0;
*>          if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension
*>                               (LDA,M) if SIDE = 'L',
*>                               (LDA,N) if SIDE = 'R'
*>          The i-th row must contain the vector which defines the
*>          elementary reflector H(i), for i = 1,2,...,k, as returned by
*>          DGERQF in the last k rows of its array argument A.
*>          A is modified by the routine but restored on exit.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,K).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DGERQF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the m by n matrix C.
*>          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension
*>                                   (N) if SIDE = 'L',
*>                                   (M) if SIDE = 'R'
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unmr2
*
*  =====================================================================
      SUBROUTINE DORMR2( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
     $                   WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS
      INTEGER            INFO, K, LDA, LDC, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LEFT, NOTRAN
      INTEGER            I, I1, I2, I3, MI, NI, NQ
      DOUBLE PRECISION   AII
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARF, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LEFT = LSAME( SIDE, 'L' )
      NOTRAN = LSAME( TRANS, 'N' )
*
*     NQ is the order of Q
*
      IF( LEFT ) THEN
         NQ = M
      ELSE
         NQ = N
      END IF
      IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
         INFO = -2
      ELSE IF( M.LT.0 ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, K ) ) THEN
         INFO = -7
      ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
         INFO = -10
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORMR2', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 )
     $   RETURN
*
      IF( ( LEFT .AND. .NOT.NOTRAN ) .OR. ( .NOT.LEFT .AND. NOTRAN ) )
     $     THEN
         I1 = 1
         I2 = K
         I3 = 1
      ELSE
         I1 = K
         I2 = 1
         I3 = -1
      END IF
*
      IF( LEFT ) THEN
         NI = N
      ELSE
         MI = M
      END IF
*
      DO 10 I = I1, I2, I3
         IF( LEFT ) THEN
*
*           H(i) is applied to C(1:m-k+i,1:n)
*
            MI = M - K + I
         ELSE
*
*           H(i) is applied to C(1:m,1:n-k+i)
*
            NI = N - K + I
         END IF
*
*        Apply H(i)
*
         AII = A( I, NQ-K+I )
         A( I, NQ-K+I ) = ONE
         CALL DLARF( SIDE, MI, NI, A( I, 1 ), LDA, TAU( I ), C, LDC,
     $               WORK )
         A( I, NQ-K+I ) = AII
   10 CONTINUE
      RETURN
*
*     End of DORMR2
*
      END
*> \brief \b DORMR3 multiplies a general matrix by the orthogonal matrix from a RZ factorization determined by stzrzf (unblocked algorithm).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMR3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormr3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormr3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormr3.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORMR3( SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC,
*                          WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIDE, TRANS
*       INTEGER            INFO, K, L, LDA, LDC, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORMR3 overwrites the general real m by n matrix C with
*>
*>       Q * C  if SIDE = 'L' and TRANS = 'N', or
*>
*>       Q**T* C  if SIDE = 'L' and TRANS = 'C', or
*>
*>       C * Q  if SIDE = 'R' and TRANS = 'N', or
*>
*>       C * Q**T if SIDE = 'R' and TRANS = 'C',
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*>       Q = H(1) H(2) . . . H(k)
*>
*> as returned by DTZRZF. Q is of order m if SIDE = 'L' and of order n
*> if SIDE = 'R'.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': apply Q or Q**T from the Left
*>          = 'R': apply Q or Q**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N': apply Q  (No transpose)
*>          = 'T': apply Q**T (Transpose)
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines
*>          the matrix Q.
*>          If SIDE = 'L', M >= K >= 0;
*>          if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*>          L is INTEGER
*>          The number of columns of the matrix A containing
*>          the meaningful part of the Householder reflectors.
*>          If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension
*>                               (LDA,M) if SIDE = 'L',
*>                               (LDA,N) if SIDE = 'R'
*>          The i-th row must contain the vector which defines the
*>          elementary reflector H(i), for i = 1,2,...,k, as returned by
*>          DTZRZF in the last k rows of its array argument A.
*>          A is modified by the routine but restored on exit.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,K).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DTZRZF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the m-by-n matrix C.
*>          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension
*>                                   (N) if SIDE = 'L',
*>                                   (M) if SIDE = 'R'
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unmr3
*
*> \par Contributors:
*  ==================
*>
*>    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DORMR3( SIDE, TRANS, M, N, K, L, A, LDA, TAU, C,
     $                   LDC,
     $                   WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS
      INTEGER            INFO, K, L, LDA, LDC, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            LEFT, NOTRAN
      INTEGER            I, I1, I2, I3, IC, JA, JC, MI, NI, NQ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARZ, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LEFT = LSAME( SIDE, 'L' )
      NOTRAN = LSAME( TRANS, 'N' )
*
*     NQ is the order of Q
*
      IF( LEFT ) THEN
         NQ = M
      ELSE
         NQ = N
      END IF
      IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
         INFO = -2
      ELSE IF( M.LT.0 ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
         INFO = -5
      ELSE IF( L.LT.0 .OR. ( LEFT .AND. ( L.GT.M ) ) .OR.
     $         ( .NOT.LEFT .AND. ( L.GT.N ) ) ) THEN
         INFO = -6
      ELSE IF( LDA.LT.MAX( 1, K ) ) THEN
         INFO = -8
      ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
         INFO = -11
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORMR3', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 )
     $   RETURN
*
      IF( ( LEFT .AND. .NOT.NOTRAN .OR. .NOT.LEFT .AND. NOTRAN ) ) THEN
         I1 = 1
         I2 = K
         I3 = 1
      ELSE
         I1 = K
         I2 = 1
         I3 = -1
      END IF
*
      IF( LEFT ) THEN
         NI = N
         JA = M - L + 1
         JC = 1
      ELSE
         MI = M
         JA = N - L + 1
         IC = 1
      END IF
*
      DO 10 I = I1, I2, I3
         IF( LEFT ) THEN
*
*           H(i) or H(i)**T is applied to C(i:m,1:n)
*
            MI = M - I + 1
            IC = I
         ELSE
*
*           H(i) or H(i)**T is applied to C(1:m,i:n)
*
            NI = N - I + 1
            JC = I
         END IF
*
*        Apply H(i) or H(i)**T
*
         CALL DLARZ( SIDE, MI, NI, L, A( I, JA ), LDA, TAU( I ),
     $               C( IC, JC ), LDC, WORK )
*
   10 CONTINUE
*
      RETURN
*
*     End of DORMR3
*
      END
*> \brief \b DORMRQ
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMRQ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormrq.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormrq.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormrq.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORMRQ( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
*                          WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIDE, TRANS
*       INTEGER            INFO, K, LDA, LDC, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORMRQ overwrites the general real M-by-N matrix C with
*>
*>                 SIDE = 'L'     SIDE = 'R'
*> TRANS = 'N':      Q * C          C * Q
*> TRANS = 'T':      Q**T * C       C * Q**T
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*>       Q = H(1) H(2) . . . H(k)
*>
*> as returned by DGERQF. Q is of order M if SIDE = 'L' and of order N
*> if SIDE = 'R'.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': apply Q or Q**T from the Left;
*>          = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N':  No transpose, apply Q;
*>          = 'T':  Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines
*>          the matrix Q.
*>          If SIDE = 'L', M >= K >= 0;
*>          if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension
*>                               (LDA,M) if SIDE = 'L',
*>                               (LDA,N) if SIDE = 'R'
*>          The i-th row must contain the vector which defines the
*>          elementary reflector H(i), for i = 1,2,...,k, as returned by
*>          DGERQF in the last k rows of its array argument A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,K).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DGERQF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the M-by-N matrix C.
*>          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If SIDE = 'L', LWORK >= max(1,N);
*>          if SIDE = 'R', LWORK >= max(1,M).
*>          For good performance, LWORK should generally be larger.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unmrq
*
*  =====================================================================
      SUBROUTINE DORMRQ( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
     $                   WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS
      INTEGER            INFO, K, LDA, LDC, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            NBMAX, LDT, TSIZE
      PARAMETER          ( NBMAX = 64, LDT = NBMAX+1,
     $                     TSIZE = LDT*NBMAX )
*     ..
*     .. Local Scalars ..
      LOGICAL            LEFT, LQUERY, NOTRAN
      CHARACTER          TRANST
      INTEGER            I, I1, I2, I3, IB, IINFO, IWT, LDWORK, LWKOPT,
     $                   MI, NB, NBMIN, NI, NQ, NW
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARFB, DLARFT, DORMR2, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LEFT = LSAME( SIDE, 'L' )
      NOTRAN = LSAME( TRANS, 'N' )
      LQUERY = ( LWORK.EQ.-1 )
*
*     NQ is the order of Q and NW is the minimum dimension of WORK
*
      IF( LEFT ) THEN
         NQ = M
         NW = MAX( 1, N )
      ELSE
         NQ = N
         NW = MAX( 1, M )
      END IF
      IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
         INFO = -2
      ELSE IF( M.LT.0 ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, K ) ) THEN
         INFO = -7
      ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
         INFO = -10
      ELSE IF( LWORK.LT.NW .AND. .NOT.LQUERY ) THEN
         INFO = -12
      END IF
*
      IF( INFO.EQ.0 ) THEN
*
*        Compute the workspace requirements
*
         IF( M.EQ.0 .OR. N.EQ.0 ) THEN
            LWKOPT = 1
         ELSE
            NB = MIN( NBMAX, ILAENV( 1, 'DORMRQ', SIDE // TRANS, M,
     $                N,
     $                               K, -1 ) )
            LWKOPT = NW*NB + TSIZE
         END IF
         WORK( 1 ) = LWKOPT
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORMRQ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 ) THEN
         RETURN
      END IF
*
      NBMIN = 2
      LDWORK = NW
      IF( NB.GT.1 .AND. NB.LT.K ) THEN
         IF( LWORK.LT.LWKOPT ) THEN
            NB = (LWORK-TSIZE) / LDWORK
            NBMIN = MAX( 2, ILAENV( 2, 'DORMRQ', SIDE // TRANS, M, N,
     $                   K,
     $              -1 ) )
         END IF
      END IF
*
      IF( NB.LT.NBMIN .OR. NB.GE.K ) THEN
*
*        Use unblocked code
*
         CALL DORMR2( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
     $                WORK,
     $                IINFO )
      ELSE
*
*        Use blocked code
*
         IWT = 1 + NW*NB
         IF( ( LEFT .AND. .NOT.NOTRAN ) .OR.
     $       ( .NOT.LEFT .AND. NOTRAN ) ) THEN
            I1 = 1
            I2 = K
            I3 = NB
         ELSE
            I1 = ( ( K-1 ) / NB )*NB + 1
            I2 = 1
            I3 = -NB
         END IF
*
         IF( LEFT ) THEN
            NI = N
         ELSE
            MI = M
         END IF
*
         IF( NOTRAN ) THEN
            TRANST = 'T'
         ELSE
            TRANST = 'N'
         END IF
*
         DO 10 I = I1, I2, I3
            IB = MIN( NB, K-I+1 )
*
*           Form the triangular factor of the block reflector
*           H = H(i+ib-1) . . . H(i+1) H(i)
*
            CALL DLARFT( 'Backward', 'Rowwise', NQ-K+I+IB-1, IB,
     $                   A( I, 1 ), LDA, TAU( I ), WORK( IWT ), LDT )
            IF( LEFT ) THEN
*
*              H or H**T is applied to C(1:m-k+i+ib-1,1:n)
*
               MI = M - K + I + IB - 1
            ELSE
*
*              H or H**T is applied to C(1:m,1:n-k+i+ib-1)
*
               NI = N - K + I + IB - 1
            END IF
*
*           Apply H or H**T
*
            CALL DLARFB( SIDE, TRANST, 'Backward', 'Rowwise', MI, NI,
     $                   IB, A( I, 1 ), LDA, WORK( IWT ), LDT, C, LDC,
     $                   WORK, LDWORK )
   10    CONTINUE
      END IF
      WORK( 1 ) = LWKOPT
      RETURN
*
*     End of DORMRQ
*
      END
*> \brief \b DORMRZ
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMRZ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormrz.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormrz.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormrz.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORMRZ( SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC,
*                          WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIDE, TRANS
*       INTEGER            INFO, K, L, LDA, LDC, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORMRZ overwrites the general real M-by-N matrix C with
*>
*>                 SIDE = 'L'     SIDE = 'R'
*> TRANS = 'N':      Q * C          C * Q
*> TRANS = 'T':      Q**T * C       C * Q**T
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*>       Q = H(1) H(2) . . . H(k)
*>
*> as returned by DTZRZF. Q is of order M if SIDE = 'L' and of order N
*> if SIDE = 'R'.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': apply Q or Q**T from the Left;
*>          = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N':  No transpose, apply Q;
*>          = 'T':  Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines
*>          the matrix Q.
*>          If SIDE = 'L', M >= K >= 0;
*>          if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*>          L is INTEGER
*>          The number of columns of the matrix A containing
*>          the meaningful part of the Householder reflectors.
*>          If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension
*>                               (LDA,M) if SIDE = 'L',
*>                               (LDA,N) if SIDE = 'R'
*>          The i-th row must contain the vector which defines the
*>          elementary reflector H(i), for i = 1,2,...,k, as returned by
*>          DTZRZF in the last k rows of its array argument A.
*>          A is modified by the routine but restored on exit.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,K).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DTZRZF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the M-by-N matrix C.
*>          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If SIDE = 'L', LWORK >= max(1,N);
*>          if SIDE = 'R', LWORK >= max(1,M).
*>          For good performance, LWORK should generally be larger.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unmrz
*
*> \par Contributors:
*  ==================
*>
*>    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DORMRZ( SIDE, TRANS, M, N, K, L, A, LDA, TAU, C,
     $                   LDC,
     $                   WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS
      INTEGER            INFO, K, L, LDA, LDC, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            NBMAX, LDT, TSIZE
      PARAMETER          ( NBMAX = 64, LDT = NBMAX+1,
     $                     TSIZE = LDT*NBMAX )
*     ..
*     .. Local Scalars ..
      LOGICAL            LEFT, LQUERY, NOTRAN
      CHARACTER          TRANST
      INTEGER            I, I1, I2, I3, IB, IC, IINFO, IWT, JA, JC,
     $                   LDWORK, LWKOPT, MI, NB, NBMIN, NI, NQ, NW
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARZB, DLARZT, DORMR3, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LEFT = LSAME( SIDE, 'L' )
      NOTRAN = LSAME( TRANS, 'N' )
      LQUERY = ( LWORK.EQ.-1 )
*
*     NQ is the order of Q and NW is the minimum dimension of WORK
*
      IF( LEFT ) THEN
         NQ = M
         NW = MAX( 1, N )
      ELSE
         NQ = N
         NW = MAX( 1, M )
      END IF
      IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
         INFO = -2
      ELSE IF( M.LT.0 ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
         INFO = -5
      ELSE IF( L.LT.0 .OR. ( LEFT .AND. ( L.GT.M ) ) .OR.
     $         ( .NOT.LEFT .AND. ( L.GT.N ) ) ) THEN
         INFO = -6
      ELSE IF( LDA.LT.MAX( 1, K ) ) THEN
         INFO = -8
      ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
         INFO = -11
      ELSE IF( LWORK.LT.NW .AND. .NOT.LQUERY ) THEN
         INFO = -13
      END IF
*
      IF( INFO.EQ.0 ) THEN
*
*        Compute the workspace requirements
*
         IF( M.EQ.0 .OR. N.EQ.0 ) THEN
            LWKOPT = 1
         ELSE
            NB = MIN( NBMAX, ILAENV( 1, 'DORMRQ', SIDE // TRANS, M,
     $                N,
     $                               K, -1 ) )
            LWKOPT = NW*NB + TSIZE
         END IF
         WORK( 1 ) = LWKOPT
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORMRZ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 ) THEN
         WORK( 1 ) = 1
         RETURN
      END IF
*
      NBMIN = 2
      LDWORK = NW
      IF( NB.GT.1 .AND. NB.LT.K ) THEN
         IF( LWORK.LT.LWKOPT ) THEN
            NB = (LWORK-TSIZE) / LDWORK
            NBMIN = MAX( 2, ILAENV( 2, 'DORMRQ', SIDE // TRANS, M, N,
     $                   K,
     $              -1 ) )
         END IF
      END IF
*
      IF( NB.LT.NBMIN .OR. NB.GE.K ) THEN
*
*        Use unblocked code
*
         CALL DORMR3( SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC,
     $                WORK, IINFO )
      ELSE
*
*        Use blocked code
*
         IWT = 1 + NW*NB
         IF( ( LEFT .AND. .NOT.NOTRAN ) .OR.
     $       ( .NOT.LEFT .AND. NOTRAN ) ) THEN
            I1 = 1
            I2 = K
            I3 = NB
         ELSE
            I1 = ( ( K-1 ) / NB )*NB + 1
            I2 = 1
            I3 = -NB
         END IF
*
         IF( LEFT ) THEN
            NI = N
            JC = 1
            JA = M - L + 1
         ELSE
            MI = M
            IC = 1
            JA = N - L + 1
         END IF
*
         IF( NOTRAN ) THEN
            TRANST = 'T'
         ELSE
            TRANST = 'N'
         END IF
*
         DO 10 I = I1, I2, I3
            IB = MIN( NB, K-I+1 )
*
*           Form the triangular factor of the block reflector
*           H = H(i+ib-1) . . . H(i+1) H(i)
*
            CALL DLARZT( 'Backward', 'Rowwise', L, IB, A( I, JA ),
     $                   LDA,
     $                   TAU( I ), WORK( IWT ), LDT )
*
            IF( LEFT ) THEN
*
*              H or H**T is applied to C(i:m,1:n)
*
               MI = M - I + 1
               IC = I
            ELSE
*
*              H or H**T is applied to C(1:m,i:n)
*
               NI = N - I + 1
               JC = I
            END IF
*
*           Apply H or H**T
*
            CALL DLARZB( SIDE, TRANST, 'Backward', 'Rowwise', MI, NI,
     $                   IB, L, A( I, JA ), LDA, WORK( IWT ), LDT,
     $                   C( IC, JC ), LDC, WORK, LDWORK )
   10    CONTINUE
*
      END IF
*
      WORK( 1 ) = LWKOPT
*
      RETURN
*
*     End of DORMRZ
*
      END
*> \brief \b DORMTR
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMTR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormtr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormtr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormtr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DORMTR( SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC,
*                          WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIDE, TRANS, UPLO
*       INTEGER            INFO, LDA, LDC, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORMTR overwrites the general real M-by-N matrix C with
*>
*>                 SIDE = 'L'     SIDE = 'R'
*> TRANS = 'N':      Q * C          C * Q
*> TRANS = 'T':      Q**T * C       C * Q**T
*>
*> where Q is a real orthogonal matrix of order nq, with nq = m if
*> SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
*> nq-1 elementary reflectors, as returned by DSYTRD:
*>
*> if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1);
*>
*> if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': apply Q or Q**T from the Left;
*>          = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U': Upper triangle of A contains elementary reflectors
*>                 from DSYTRD;
*>          = 'L': Lower triangle of A contains elementary reflectors
*>                 from DSYTRD.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N':  No transpose, apply Q;
*>          = 'T':  Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension
*>                               (LDA,M) if SIDE = 'L'
*>                               (LDA,N) if SIDE = 'R'
*>          The vectors which define the elementary reflectors, as
*>          returned by DSYTRD.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.
*>          LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension
*>                               (M-1) if SIDE = 'L'
*>                               (N-1) if SIDE = 'R'
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DSYTRD.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the M-by-N matrix C.
*>          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If SIDE = 'L', LWORK >= max(1,N);
*>          if SIDE = 'R', LWORK >= max(1,M).
*>          For optimum performance LWORK >= N*NB if SIDE = 'L', and
*>          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
*>          blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup unmtr
*
*  =====================================================================
      SUBROUTINE DORMTR( SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C,
     $                   LDC,
     $                   WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          SIDE, TRANS, UPLO
      INTEGER            INFO, LDA, LDC, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            LEFT, LQUERY, UPPER
      INTEGER            I1, I2, IINFO, LWKOPT, MI, NB, NI, NQ, NW
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DORMQL, DORMQR, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LEFT = LSAME( SIDE, 'L' )
      UPPER = LSAME( UPLO, 'U' )
      LQUERY = ( LWORK.EQ.-1 )
*
*     NQ is the order of Q and NW is the minimum dimension of WORK
*
      IF( LEFT ) THEN
         NQ = M
         NW = MAX( 1, N )
      ELSE
         NQ = N
         NW = MAX( 1, M )
      END IF
      IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -2
      ELSE IF( .NOT.LSAME( TRANS, 'N' ) .AND.
     $         .NOT.LSAME( TRANS, 'T' ) )
     $          THEN
         INFO = -3
      ELSE IF( M.LT.0 ) THEN
         INFO = -4
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
         INFO = -7
      ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
         INFO = -10
      ELSE IF( LWORK.LT.NW .AND. .NOT.LQUERY ) THEN
         INFO = -12
      END IF
*
      IF( INFO.EQ.0 ) THEN
         IF( UPPER ) THEN
            IF( LEFT ) THEN
               NB = ILAENV( 1, 'DORMQL', SIDE // TRANS, M-1, N, M-1,
     $              -1 )
            ELSE
               NB = ILAENV( 1, 'DORMQL', SIDE // TRANS, M, N-1, N-1,
     $              -1 )
            END IF
         ELSE
            IF( LEFT ) THEN
               NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, M-1, N, M-1,
     $              -1 )
            ELSE
               NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, M, N-1, N-1,
     $              -1 )
            END IF
         END IF
         LWKOPT = NW*NB
         WORK( 1 ) = LWKOPT
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DORMTR', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 .OR. NQ.EQ.1 ) THEN
         WORK( 1 ) = 1
         RETURN
      END IF
*
      IF( LEFT ) THEN
         MI = M - 1
         NI = N
      ELSE
         MI = M
         NI = N - 1
      END IF
*
      IF( UPPER ) THEN
*
*        Q was determined by a call to DSYTRD with UPLO = 'U'
*
         CALL DORMQL( SIDE, TRANS, MI, NI, NQ-1, A( 1, 2 ), LDA, TAU,
     $                C,
     $                LDC, WORK, LWORK, IINFO )
      ELSE
*
*        Q was determined by a call to DSYTRD with UPLO = 'L'
*
         IF( LEFT ) THEN
            I1 = 2
            I2 = 1
         ELSE
            I1 = 1
            I2 = 2
         END IF
         CALL DORMQR( SIDE, TRANS, MI, NI, NQ-1, A( 2, 1 ), LDA, TAU,
     $                C( I1, I2 ), LDC, WORK, LWORK, IINFO )
      END IF
      WORK( 1 ) = LWKOPT
      RETURN
*
*     End of DORMTR
*
      END
*> \brief \b DPBCON
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPBCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbcon.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPBCON( UPLO, N, KD, AB, LDAB, ANORM, RCOND, WORK,
*                          IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, KD, LDAB, N
*       DOUBLE PRECISION   ANORM, RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   AB( LDAB, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPBCON estimates the reciprocal of the condition number (in the
*> 1-norm) of a real symmetric positive definite band matrix using the
*> Cholesky factorization A = U**T*U or A = L*L**T computed by DPBTRF.
*>
*> An estimate is obtained for norm(inv(A)), and the reciprocal of the
*> condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangular factor stored in AB;
*>          = 'L':  Lower triangular factor stored in AB.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          The triangular factor U or L from the Cholesky factorization
*>          A = U**T*U or A = L*L**T of the band matrix A, stored in the
*>          first KD+1 rows of the array.  The j-th column of U or L is
*>          stored in the j-th column of the array AB as follows:
*>          if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO ='L', AB(1+i-j,j)    = L(i,j) for j<=i<=min(n,j+kd).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*>          ANORM is DOUBLE PRECISION
*>          The 1-norm (or infinity-norm) of the symmetric band matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The reciprocal of the condition number of the matrix A,
*>          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
*>          estimate of the 1-norm of inv(A) computed in this routine.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pbcon
*
*  =====================================================================
      SUBROUTINE DPBCON( UPLO, N, KD, AB, LDAB, ANORM, RCOND, WORK,
     $                   IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, KD, LDAB, N
      DOUBLE PRECISION   ANORM, RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   AB( LDAB, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      CHARACTER          NORMIN
      INTEGER            IX, KASE
      DOUBLE PRECISION   AINVNM, SCALE, SCALEL, SCALEU, SMLNUM
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, IDAMAX, DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACN2, DLATBS, DRSCL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( KD.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDAB.LT.KD+1 ) THEN
         INFO = -5
      ELSE IF( ANORM.LT.ZERO ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPBCON', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      RCOND = ZERO
      IF( N.EQ.0 ) THEN
         RCOND = ONE
         RETURN
      ELSE IF( ANORM.EQ.ZERO ) THEN
         RETURN
      END IF
*
      SMLNUM = DLAMCH( 'Safe minimum' )
*
*     Estimate the 1-norm of the inverse.
*
      KASE = 0
      NORMIN = 'N'
   10 CONTINUE
      CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
      IF( KASE.NE.0 ) THEN
         IF( UPPER ) THEN
*
*           Multiply by inv(U**T).
*
            CALL DLATBS( 'Upper', 'Transpose', 'Non-unit', NORMIN, N,
     $                   KD, AB, LDAB, WORK, SCALEL, WORK( 2*N+1 ),
     $                   INFO )
            NORMIN = 'Y'
*
*           Multiply by inv(U).
*
            CALL DLATBS( 'Upper', 'No transpose', 'Non-unit', NORMIN,
     $                   N,
     $                   KD, AB, LDAB, WORK, SCALEU, WORK( 2*N+1 ),
     $                   INFO )
         ELSE
*
*           Multiply by inv(L).
*
            CALL DLATBS( 'Lower', 'No transpose', 'Non-unit', NORMIN,
     $                   N,
     $                   KD, AB, LDAB, WORK, SCALEL, WORK( 2*N+1 ),
     $                   INFO )
            NORMIN = 'Y'
*
*           Multiply by inv(L**T).
*
            CALL DLATBS( 'Lower', 'Transpose', 'Non-unit', NORMIN, N,
     $                   KD, AB, LDAB, WORK, SCALEU, WORK( 2*N+1 ),
     $                   INFO )
         END IF
*
*        Multiply by 1/SCALE if doing so will not cause overflow.
*
         SCALE = SCALEL*SCALEU
         IF( SCALE.NE.ONE ) THEN
            IX = IDAMAX( N, WORK, 1 )
            IF( SCALE.LT.ABS( WORK( IX ) )*SMLNUM .OR. SCALE.EQ.ZERO )
     $         GO TO 20
            CALL DRSCL( N, SCALE, WORK, 1 )
         END IF
         GO TO 10
      END IF
*
*     Compute the estimate of the reciprocal condition number.
*
      IF( AINVNM.NE.ZERO )
     $   RCOND = ( ONE / AINVNM ) / ANORM
*
   20 CONTINUE
*
      RETURN
*
*     End of DPBCON
*
      END
*> \brief \b DPBEQU
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPBEQU + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbequ.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbequ.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbequ.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, KD, LDAB, N
*       DOUBLE PRECISION   AMAX, SCOND
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * ), S( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPBEQU computes row and column scalings intended to equilibrate a
*> symmetric positive definite band matrix A and reduce its condition
*> number (with respect to the two-norm).  S contains the scale factors,
*> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
*> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
*> choice of S puts the condition number of B within a factor N of the
*> smallest possible condition number over all possible diagonal
*> scalings.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangular of A is stored;
*>          = 'L':  Lower triangular of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          The upper or lower triangle of the symmetric band matrix A,
*>          stored in the first KD+1 rows of the array.  The j-th column
*>          of A is stored in the j-th column of the array AB as follows:
*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array A.  LDAB >= KD+1.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, S contains the scale factors for A.
*> \endverbatim
*>
*> \param[out] SCOND
*> \verbatim
*>          SCOND is DOUBLE PRECISION
*>          If INFO = 0, S contains the ratio of the smallest S(i) to
*>          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
*>          large nor too small, it is not worth scaling by S.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*>          AMAX is DOUBLE PRECISION
*>          Absolute value of largest matrix element.  If AMAX is very
*>          close to overflow or very close to underflow, the matrix
*>          should be scaled.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pbequ
*
*  =====================================================================
      SUBROUTINE DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, KD, LDAB, N
      DOUBLE PRECISION   AMAX, SCOND
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * ), S( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I, J
      DOUBLE PRECISION   SMIN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( KD.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDAB.LT.KD+1 ) THEN
         INFO = -5
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPBEQU', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 ) THEN
         SCOND = ONE
         AMAX = ZERO
         RETURN
      END IF
*
      IF( UPPER ) THEN
         J = KD + 1
      ELSE
         J = 1
      END IF
*
*     Initialize SMIN and AMAX.
*
      S( 1 ) = AB( J, 1 )
      SMIN = S( 1 )
      AMAX = S( 1 )
*
*     Find the minimum and maximum diagonal elements.
*
      DO 10 I = 2, N
         S( I ) = AB( J, I )
         SMIN = MIN( SMIN, S( I ) )
         AMAX = MAX( AMAX, S( I ) )
   10 CONTINUE
*
      IF( SMIN.LE.ZERO ) THEN
*
*        Find the first non-positive diagonal element and return.
*
         DO 20 I = 1, N
            IF( S( I ).LE.ZERO ) THEN
               INFO = I
               RETURN
            END IF
   20    CONTINUE
      ELSE
*
*        Set the scale factors to the reciprocals
*        of the diagonal elements.
*
         DO 30 I = 1, N
            S( I ) = ONE / SQRT( S( I ) )
   30    CONTINUE
*
*        Compute SCOND = min(S(I)) / max(S(I))
*
         SCOND = SQRT( SMIN ) / SQRT( AMAX )
      END IF
      RETURN
*
*     End of DPBEQU
*
      END
*> \brief \b DPBRFS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPBRFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbrfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbrfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbrfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
*                          LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
*      $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPBRFS improves the computed solution to a system of linear
*> equations when the coefficient matrix is symmetric positive definite
*> and banded, and provides error bounds and backward error estimates
*> for the solution.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices B and X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          The upper or lower triangle of the symmetric band matrix A,
*>          stored in the first KD+1 rows of the array.  The j-th column
*>          of A is stored in the j-th column of the array AB as follows:
*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in] AFB
*> \verbatim
*>          AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
*>          The triangular factor U or L from the Cholesky factorization
*>          A = U**T*U or A = L*L**T of the band matrix A as computed by
*>          DPBTRF, in the same storage format as A (see AB).
*> \endverbatim
*>
*> \param[in] LDAFB
*> \verbatim
*>          LDAFB is INTEGER
*>          The leading dimension of the array AFB.  LDAFB >= KD+1.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          On entry, the solution matrix X, as computed by DPBTRS.
*>          On exit, the improved solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The estimated forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).  The estimate is as reliable as
*>          the estimate for RCOND, and is almost always a slight
*>          overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in
*>          any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  ITMAX is the maximum number of steps of iterative refinement.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pbrfs
*
*  =====================================================================
      SUBROUTINE DPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
     $                   LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
     $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            ITMAX
      PARAMETER          ( ITMAX = 5 )
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
      DOUBLE PRECISION   TWO
      PARAMETER          ( TWO = 2.0D+0 )
      DOUBLE PRECISION   THREE
      PARAMETER          ( THREE = 3.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            COUNT, I, J, K, KASE, L, NZ
      DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DLACN2, DPBTRS, DSBMV,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( KD.LT.0 ) THEN
         INFO = -3
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDAB.LT.KD+1 ) THEN
         INFO = -6
      ELSE IF( LDAFB.LT.KD+1 ) THEN
         INFO = -8
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -10
      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
         INFO = -12
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPBRFS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
         DO 10 J = 1, NRHS
            FERR( J ) = ZERO
            BERR( J ) = ZERO
   10    CONTINUE
         RETURN
      END IF
*
*     NZ = maximum number of nonzero elements in each row of A, plus 1
*
      NZ = MIN( N+1, 2*KD+2 )
      EPS = DLAMCH( 'Epsilon' )
      SAFMIN = DLAMCH( 'Safe minimum' )
      SAFE1 = NZ*SAFMIN
      SAFE2 = SAFE1 / EPS
*
*     Do for each right hand side
*
      DO 140 J = 1, NRHS
*
         COUNT = 1
         LSTRES = THREE
   20    CONTINUE
*
*        Loop until stopping criterion is satisfied.
*
*        Compute residual R = B - A * X
*
         CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
         CALL DSBMV( UPLO, N, KD, -ONE, AB, LDAB, X( 1, J ), 1, ONE,
     $               WORK( N+1 ), 1 )
*
*        Compute componentwise relative backward error from formula
*
*        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
*
*        where abs(Z) is the componentwise absolute value of the matrix
*        or vector Z.  If the i-th component of the denominator is less
*        than SAFE2, then SAFE1 is added to the i-th components of the
*        numerator and denominator before dividing.
*
         DO 30 I = 1, N
            WORK( I ) = ABS( B( I, J ) )
   30    CONTINUE
*
*        Compute abs(A)*abs(X) + abs(B).
*
         IF( UPPER ) THEN
            DO 50 K = 1, N
               S = ZERO
               XK = ABS( X( K, J ) )
               L = KD + 1 - K
               DO 40 I = MAX( 1, K-KD ), K - 1
                  WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK
                  S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) )
   40          CONTINUE
               WORK( K ) = WORK( K ) + ABS( AB( KD+1, K ) )*XK + S
   50       CONTINUE
         ELSE
            DO 70 K = 1, N
               S = ZERO
               XK = ABS( X( K, J ) )
               WORK( K ) = WORK( K ) + ABS( AB( 1, K ) )*XK
               L = 1 - K
               DO 60 I = K + 1, MIN( N, K+KD )
                  WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK
                  S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) )
   60          CONTINUE
               WORK( K ) = WORK( K ) + S
   70       CONTINUE
         END IF
         S = ZERO
         DO 80 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
            ELSE
               S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
     $             ( WORK( I )+SAFE1 ) )
            END IF
   80    CONTINUE
         BERR( J ) = S
*
*        Test stopping criterion. Continue iterating if
*           1) The residual BERR(J) is larger than machine epsilon, and
*           2) BERR(J) decreased by at least a factor of 2 during the
*              last iteration, and
*           3) At most ITMAX iterations tried.
*
         IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
     $       COUNT.LE.ITMAX ) THEN
*
*           Update solution and try again.
*
            CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N,
     $                   INFO )
            CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
            LSTRES = BERR( J )
            COUNT = COUNT + 1
            GO TO 20
         END IF
*
*        Bound error from formula
*
*        norm(X - XTRUE) / norm(X) .le. FERR =
*        norm( abs(inv(A))*
*           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
*
*        where
*          norm(Z) is the magnitude of the largest component of Z
*          inv(A) is the inverse of A
*          abs(Z) is the componentwise absolute value of the matrix or
*             vector Z
*          NZ is the maximum number of nonzeros in any row of A, plus 1
*          EPS is machine epsilon
*
*        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
*        is incremented by SAFE1 if the i-th component of
*        abs(A)*abs(X) + abs(B) is less than SAFE2.
*
*        Use DLACN2 to estimate the infinity-norm of the matrix
*           inv(A) * diag(W),
*        where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
*
         DO 90 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
            ELSE
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
            END IF
   90    CONTINUE
*
         KASE = 0
  100    CONTINUE
         CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
     $                FERR( J ),
     $                KASE, ISAVE )
         IF( KASE.NE.0 ) THEN
            IF( KASE.EQ.1 ) THEN
*
*              Multiply by diag(W)*inv(A**T).
*
               CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ),
     $                      N,
     $                      INFO )
               DO 110 I = 1, N
                  WORK( N+I ) = WORK( N+I )*WORK( I )
  110          CONTINUE
            ELSE IF( KASE.EQ.2 ) THEN
*
*              Multiply by inv(A)*diag(W).
*
               DO 120 I = 1, N
                  WORK( N+I ) = WORK( N+I )*WORK( I )
  120          CONTINUE
               CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ),
     $                      N,
     $                      INFO )
            END IF
            GO TO 100
         END IF
*
*        Normalize error.
*
         LSTRES = ZERO
         DO 130 I = 1, N
            LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
  130    CONTINUE
         IF( LSTRES.NE.ZERO )
     $      FERR( J ) = FERR( J ) / LSTRES
*
  140 CONTINUE
*
      RETURN
*
*     End of DPBRFS
*
      END
*> \brief \b DPBSTF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPBSTF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbstf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbstf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbstf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPBSTF( UPLO, N, KD, AB, LDAB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, KD, LDAB, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPBSTF computes a split Cholesky factorization of a real
*> symmetric positive definite band matrix A.
*>
*> This routine is designed to be used in conjunction with DSBGST.
*>
*> The factorization has the form  A = S**T*S  where S is a band matrix
*> of the same bandwidth as A and the following structure:
*>
*>   S = ( U    )
*>       ( M  L )
*>
*> where U is upper triangular of order m = (n+kd)/2, and L is lower
*> triangular of order n-m.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          On entry, the upper or lower triangle of the symmetric band
*>          matrix A, stored in the first kd+1 rows of the array.  The
*>          j-th column of A is stored in the j-th column of the array AB
*>          as follows:
*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*>
*>          On exit, if INFO = 0, the factor S from the split Cholesky
*>          factorization A = S**T*S. See Further Details.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD+1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*>          > 0: if INFO = i, the factorization could not be completed,
*>               because the updated element a(i,i) was negative; the
*>               matrix A is not positive definite.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pbstf
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The band storage scheme is illustrated by the following example, when
*>  N = 7, KD = 2:
*>
*>  S = ( s11  s12  s13                     )
*>      (      s22  s23  s24                )
*>      (           s33  s34                )
*>      (                s44                )
*>      (           s53  s54  s55           )
*>      (                s64  s65  s66      )
*>      (                     s75  s76  s77 )
*>
*>  If UPLO = 'U', the array AB holds:
*>
*>  on entry:                          on exit:
*>
*>   *    *   a13  a24  a35  a46  a57   *    *   s13  s24  s53  s64  s75
*>   *   a12  a23  a34  a45  a56  a67   *   s12  s23  s34  s54  s65  s76
*>  a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77
*>
*>  If UPLO = 'L', the array AB holds:
*>
*>  on entry:                          on exit:
*>
*>  a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77
*>  a21  a32  a43  a54  a65  a76   *   s12  s23  s34  s54  s65  s76   *
*>  a31  a42  a53  a64  a64   *    *   s13  s24  s53  s64  s75   *    *
*>
*>  Array elements marked * are not used by the routine.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DPBSTF( UPLO, N, KD, AB, LDAB, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, KD, LDAB, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            J, KLD, KM, M
      DOUBLE PRECISION   AJJ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL, DSYR, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( KD.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDAB.LT.KD+1 ) THEN
         INFO = -5
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPBSTF', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      KLD = MAX( 1, LDAB-1 )
*
*     Set the splitting point m.
*
      M = ( N+KD ) / 2
*
      IF( UPPER ) THEN
*
*        Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m).
*
         DO 10 J = N, M + 1, -1
*
*           Compute s(j,j) and test for non-positive-definiteness.
*
            AJJ = AB( KD+1, J )
            IF( AJJ.LE.ZERO )
     $         GO TO 50
            AJJ = SQRT( AJJ )
            AB( KD+1, J ) = AJJ
            KM = MIN( J-1, KD )
*
*           Compute elements j-km:j-1 of the j-th column and update the
*           the leading submatrix within the band.
*
            CALL DSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 )
            CALL DSYR( 'Upper', KM, -ONE, AB( KD+1-KM, J ), 1,
     $                 AB( KD+1, J-KM ), KLD )
   10    CONTINUE
*
*        Factorize the updated submatrix A(1:m,1:m) as U**T*U.
*
         DO 20 J = 1, M
*
*           Compute s(j,j) and test for non-positive-definiteness.
*
            AJJ = AB( KD+1, J )
            IF( AJJ.LE.ZERO )
     $         GO TO 50
            AJJ = SQRT( AJJ )
            AB( KD+1, J ) = AJJ
            KM = MIN( KD, M-J )
*
*           Compute elements j+1:j+km of the j-th row and update the
*           trailing submatrix within the band.
*
            IF( KM.GT.0 ) THEN
               CALL DSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD )
               CALL DSYR( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD,
     $                    AB( KD+1, J+1 ), KLD )
            END IF
   20    CONTINUE
      ELSE
*
*        Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m).
*
         DO 30 J = N, M + 1, -1
*
*           Compute s(j,j) and test for non-positive-definiteness.
*
            AJJ = AB( 1, J )
            IF( AJJ.LE.ZERO )
     $         GO TO 50
            AJJ = SQRT( AJJ )
            AB( 1, J ) = AJJ
            KM = MIN( J-1, KD )
*
*           Compute elements j-km:j-1 of the j-th row and update the
*           trailing submatrix within the band.
*
            CALL DSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD )
            CALL DSYR( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD,
     $                 AB( 1, J-KM ), KLD )
   30    CONTINUE
*
*        Factorize the updated submatrix A(1:m,1:m) as U**T*U.
*
         DO 40 J = 1, M
*
*           Compute s(j,j) and test for non-positive-definiteness.
*
            AJJ = AB( 1, J )
            IF( AJJ.LE.ZERO )
     $         GO TO 50
            AJJ = SQRT( AJJ )
            AB( 1, J ) = AJJ
            KM = MIN( KD, M-J )
*
*           Compute elements j+1:j+km of the j-th column and update the
*           trailing submatrix within the band.
*
            IF( KM.GT.0 ) THEN
               CALL DSCAL( KM, ONE / AJJ, AB( 2, J ), 1 )
               CALL DSYR( 'Lower', KM, -ONE, AB( 2, J ), 1,
     $                    AB( 1, J+1 ), KLD )
            END IF
   40    CONTINUE
      END IF
      RETURN
*
   50 CONTINUE
      INFO = J
      RETURN
*
*     End of DPBSTF
*
      END
*> \brief <b> DPBSV computes the solution to system of linear equations A * X = B for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPBSV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbsv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbsv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbsv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPBSV( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, KD, LDAB, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPBSV computes the solution to a real system of linear equations
*>    A * X = B,
*> where A is an N-by-N symmetric positive definite band matrix and X
*> and B are N-by-NRHS matrices.
*>
*> The Cholesky decomposition is used to factor A as
*>    A = U**T * U,  if UPLO = 'U', or
*>    A = L * L**T,  if UPLO = 'L',
*> where U is an upper triangular band matrix, and L is a lower
*> triangular band matrix, with the same number of superdiagonals or
*> subdiagonals as A.  The factored form of A is then used to solve the
*> system of equations A * X = B.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of linear equations, i.e., the order of the
*>          matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          On entry, the upper or lower triangle of the symmetric band
*>          matrix A, stored in the first KD+1 rows of the array.  The
*>          j-th column of A is stored in the j-th column of the array AB
*>          as follows:
*>          if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(N,j+KD).
*>          See below for further details.
*>
*>          On exit, if INFO = 0, the triangular factor U or L from the
*>          Cholesky factorization A = U**T*U or A = L*L**T of the band
*>          matrix A, in the same storage format as A.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the N-by-NRHS right hand side matrix B.
*>          On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the leading principal minor of order i
*>                of A is not positive, so the factorization could not
*>                be completed, and the solution has not been computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pbsv
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The band storage scheme is illustrated by the following example, when
*>  N = 6, KD = 2, and UPLO = 'U':
*>
*>  On entry:                       On exit:
*>
*>      *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
*>      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
*>     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
*>
*>  Similarly, if UPLO = 'L' the format of A is as follows:
*>
*>  On entry:                       On exit:
*>
*>     a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
*>     a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
*>     a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *
*>
*>  Array elements marked * are not used by the routine.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DPBSV( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, KD, LDAB, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DPBTRF, DPBTRS, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF( .NOT.LSAME( UPLO, 'U' ) .AND.
     $    .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( KD.LT.0 ) THEN
         INFO = -3
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDAB.LT.KD+1 ) THEN
         INFO = -6
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -8
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPBSV ', -INFO )
         RETURN
      END IF
*
*     Compute the Cholesky factorization A = U**T*U or A = L*L**T.
*
      CALL DPBTRF( UPLO, N, KD, AB, LDAB, INFO )
      IF( INFO.EQ.0 ) THEN
*
*        Solve the system A*X = B, overwriting B with X.
*
         CALL DPBTRS( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )
*
      END IF
      RETURN
*
*     End of DPBSV
*
      END
*> \brief <b> DPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPBSVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbsvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbsvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbsvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
*                          EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
*                          WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          EQUED, FACT, UPLO
*       INTEGER            INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
*       DOUBLE PRECISION   RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
*      $                   BERR( * ), FERR( * ), S( * ), WORK( * ),
*      $                   X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
*> compute the solution to a real system of linear equations
*>    A * X = B,
*> where A is an N-by-N symmetric positive definite band matrix and X
*> and B are N-by-NRHS matrices.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
*  =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
*>    the system:
*>       diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
*>    Whether or not the system will be equilibrated depends on the
*>    scaling of the matrix A, but if equilibration is used, A is
*>    overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
*>
*> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
*>    factor the matrix A (after equilibration if FACT = 'E') as
*>       A = U**T * U,  if UPLO = 'U', or
*>       A = L * L**T,  if UPLO = 'L',
*>    where U is an upper triangular band matrix, and L is a lower
*>    triangular band matrix.
*>
*> 3. If the leading principal minor of order i is not positive,
*>    then the routine returns with INFO = i. Otherwise, the factored
*>    form of A is used to estimate the condition number of the matrix
*>    A.  If the reciprocal of the condition number is less than machine
*>    precision, INFO = N+1 is returned as a warning, but the routine
*>    still goes on to solve for X and compute error bounds as
*>    described below.
*>
*> 4. The system of equations is solved for X using the factored form
*>    of A.
*>
*> 5. Iterative refinement is applied to improve the computed solution
*>    matrix and calculate error bounds and backward error estimates
*>    for it.
*>
*> 6. If equilibration was used, the matrix X is premultiplied by
*>    diag(S) so that it solves the original system before
*>    equilibration.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] FACT
*> \verbatim
*>          FACT is CHARACTER*1
*>          Specifies whether or not the factored form of the matrix A is
*>          supplied on entry, and if not, whether the matrix A should be
*>          equilibrated before it is factored.
*>          = 'F':  On entry, AFB contains the factored form of A.
*>                  If EQUED = 'Y', the matrix A has been equilibrated
*>                  with scaling factors given by S.  AB and AFB will not
*>                  be modified.
*>          = 'N':  The matrix A will be copied to AFB and factored.
*>          = 'E':  The matrix A will be equilibrated if necessary, then
*>                  copied to AFB and factored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of linear equations, i.e., the order of the
*>          matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right-hand sides, i.e., the number of columns
*>          of the matrices B and X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          On entry, the upper or lower triangle of the symmetric band
*>          matrix A, stored in the first KD+1 rows of the array, except
*>          if FACT = 'F' and EQUED = 'Y', then A must contain the
*>          equilibrated matrix diag(S)*A*diag(S).  The j-th column of A
*>          is stored in the j-th column of the array AB as follows:
*>          if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(N,j+KD).
*>          See below for further details.
*>
*>          On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
*>          diag(S)*A*diag(S).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array A.  LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in,out] AFB
*> \verbatim
*>          AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
*>          If FACT = 'F', then AFB is an input argument and on entry
*>          contains the triangular factor U or L from the Cholesky
*>          factorization A = U**T*U or A = L*L**T of the band matrix
*>          A, in the same storage format as A (see AB).  If EQUED = 'Y',
*>          then AFB is the factored form of the equilibrated matrix A.
*>
*>          If FACT = 'N', then AFB is an output argument and on exit
*>          returns the triangular factor U or L from the Cholesky
*>          factorization A = U**T*U or A = L*L**T.
*>
*>          If FACT = 'E', then AFB is an output argument and on exit
*>          returns the triangular factor U or L from the Cholesky
*>          factorization A = U**T*U or A = L*L**T of the equilibrated
*>          matrix A (see the description of A for the form of the
*>          equilibrated matrix).
*> \endverbatim
*>
*> \param[in] LDAFB
*> \verbatim
*>          LDAFB is INTEGER
*>          The leading dimension of the array AFB.  LDAFB >= KD+1.
*> \endverbatim
*>
*> \param[in,out] EQUED
*> \verbatim
*>          EQUED is CHARACTER*1
*>          Specifies the form of equilibration that was done.
*>          = 'N':  No equilibration (always true if FACT = 'N').
*>          = 'Y':  Equilibration was done, i.e., A has been replaced by
*>                  diag(S) * A * diag(S).
*>          EQUED is an input argument if FACT = 'F'; otherwise, it is an
*>          output argument.
*> \endverbatim
*>
*> \param[in,out] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (N)
*>          The scale factors for A; not accessed if EQUED = 'N'.  S is
*>          an input argument if FACT = 'F'; otherwise, S is an output
*>          argument.  If FACT = 'F' and EQUED = 'Y', each element of S
*>          must be positive.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the N-by-NRHS right hand side matrix B.
*>          On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
*>          B is overwritten by diag(S) * B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
*>          the original system of equations.  Note that if EQUED = 'Y',
*>          A and B are modified on exit, and the solution to the
*>          equilibrated system is inv(diag(S))*X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The estimate of the reciprocal condition number of the matrix
*>          A after equilibration (if done).  If RCOND is less than the
*>          machine precision (in particular, if RCOND = 0), the matrix
*>          is singular to working precision.  This condition is
*>          indicated by a return code of INFO > 0.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The estimated forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).  The estimate is as reliable as
*>          the estimate for RCOND, and is almost always a slight
*>          overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in
*>          any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, and i is
*>                <= N:  the leading principal minor of order i of A
*>                       is not positive, so the factorization could not
*>                       be completed, and the solution has not been
*>                       computed. RCOND = 0 is returned.
*>                = N+1: U is nonsingular, but RCOND is less than machine
*>                       precision, meaning that the matrix is singular
*>                       to working precision.  Nevertheless, the
*>                       solution and error bounds are computed because
*>                       there are a number of situations where the
*>                       computed solution can be more accurate than the
*>                       value of RCOND would suggest.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pbsvx
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The band storage scheme is illustrated by the following example, when
*>  N = 6, KD = 2, and UPLO = 'U':
*>
*>  Two-dimensional storage of the symmetric matrix A:
*>
*>     a11  a12  a13
*>          a22  a23  a24
*>               a33  a34  a35
*>                    a44  a45  a46
*>                         a55  a56
*>     (aij=conjg(aji))         a66
*>
*>  Band storage of the upper triangle of A:
*>
*>      *    *   a13  a24  a35  a46
*>      *   a12  a23  a34  a45  a56
*>     a11  a22  a33  a44  a55  a66
*>
*>  Similarly, if UPLO = 'L' the format of A is as follows:
*>
*>     a11  a22  a33  a44  a55  a66
*>     a21  a32  a43  a54  a65   *
*>     a31  a42  a53  a64   *    *
*>
*>  Array elements marked * are not used by the routine.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB,
     $                   LDAFB,
     $                   EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
     $                   WORK, IWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          EQUED, FACT, UPLO
      INTEGER            INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
     $                   BERR( * ), FERR( * ), S( * ), WORK( * ),
     $                   X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            EQUIL, NOFACT, RCEQU, UPPER
      INTEGER            I, INFEQU, J, J1, J2
      DOUBLE PRECISION   AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANSB
      EXTERNAL           LSAME, DLAMCH, DLANSB
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLACPY, DLAQSB, DPBCON, DPBEQU,
     $                   DPBRFS,
     $                   DPBTRF, DPBTRS, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      NOFACT = LSAME( FACT, 'N' )
      EQUIL = LSAME( FACT, 'E' )
      UPPER = LSAME( UPLO, 'U' )
      IF( NOFACT .OR. EQUIL ) THEN
         EQUED = 'N'
         RCEQU = .FALSE.
      ELSE
         RCEQU = LSAME( EQUED, 'Y' )
         SMLNUM = DLAMCH( 'Safe minimum' )
         BIGNUM = ONE / SMLNUM
      END IF
*
*     Test the input parameters.
*
      IF( .NOT.NOFACT .AND.
     $    .NOT.EQUIL .AND.
     $    .NOT.LSAME( FACT, 'F' ) )
     $     THEN
         INFO = -1
      ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( KD.LT.0 ) THEN
         INFO = -4
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDAB.LT.KD+1 ) THEN
         INFO = -7
      ELSE IF( LDAFB.LT.KD+1 ) THEN
         INFO = -9
      ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
     $         ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
         INFO = -10
      ELSE
         IF( RCEQU ) THEN
            SMIN = BIGNUM
            SMAX = ZERO
            DO 10 J = 1, N
               SMIN = MIN( SMIN, S( J ) )
               SMAX = MAX( SMAX, S( J ) )
   10       CONTINUE
            IF( SMIN.LE.ZERO ) THEN
               INFO = -11
            ELSE IF( N.GT.0 ) THEN
               SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
            ELSE
               SCOND = ONE
            END IF
         END IF
         IF( INFO.EQ.0 ) THEN
            IF( LDB.LT.MAX( 1, N ) ) THEN
               INFO = -13
            ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
               INFO = -15
            END IF
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPBSVX', -INFO )
         RETURN
      END IF
*
      IF( EQUIL ) THEN
*
*        Compute row and column scalings to equilibrate the matrix A.
*
         CALL DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFEQU )
         IF( INFEQU.EQ.0 ) THEN
*
*           Equilibrate the matrix.
*
            CALL DLAQSB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX,
     $                   EQUED )
            RCEQU = LSAME( EQUED, 'Y' )
         END IF
      END IF
*
*     Scale the right-hand side.
*
      IF( RCEQU ) THEN
         DO 30 J = 1, NRHS
            DO 20 I = 1, N
               B( I, J ) = S( I )*B( I, J )
   20       CONTINUE
   30    CONTINUE
      END IF
*
      IF( NOFACT .OR. EQUIL ) THEN
*
*        Compute the Cholesky factorization A = U**T *U or A = L*L**T.
*
         IF( UPPER ) THEN
            DO 40 J = 1, N
               J1 = MAX( J-KD, 1 )
               CALL DCOPY( J-J1+1, AB( KD+1-J+J1, J ), 1,
     $                     AFB( KD+1-J+J1, J ), 1 )
   40       CONTINUE
         ELSE
            DO 50 J = 1, N
               J2 = MIN( J+KD, N )
               CALL DCOPY( J2-J+1, AB( 1, J ), 1, AFB( 1, J ), 1 )
   50       CONTINUE
         END IF
*
         CALL DPBTRF( UPLO, N, KD, AFB, LDAFB, INFO )
*
*        Return if INFO is non-zero.
*
         IF( INFO.GT.0 )THEN
            RCOND = ZERO
            RETURN
         END IF
      END IF
*
*     Compute the norm of the matrix A.
*
      ANORM = DLANSB( '1', UPLO, N, KD, AB, LDAB, WORK )
*
*     Compute the reciprocal of the condition number of A.
*
      CALL DPBCON( UPLO, N, KD, AFB, LDAFB, ANORM, RCOND, WORK,
     $             IWORK,
     $             INFO )
*
*     Compute the solution matrix X.
*
      CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
      CALL DPBTRS( UPLO, N, KD, NRHS, AFB, LDAFB, X, LDX, INFO )
*
*     Use iterative refinement to improve the computed solution and
*     compute error bounds and backward error estimates for it.
*
      CALL DPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB,
     $             X,
     $             LDX, FERR, BERR, WORK, IWORK, INFO )
*
*     Transform the solution matrix X to a solution of the original
*     system.
*
      IF( RCEQU ) THEN
         DO 70 J = 1, NRHS
            DO 60 I = 1, N
               X( I, J ) = S( I )*X( I, J )
   60       CONTINUE
   70    CONTINUE
         DO 80 J = 1, NRHS
            FERR( J ) = FERR( J ) / SCOND
   80    CONTINUE
      END IF
*
*     Set INFO = N+1 if the matrix is singular to working precision.
*
      IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
     $   INFO = N + 1
*
      RETURN
*
*     End of DPBSVX
*
      END
*> \brief \b DPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPBTF2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbtf2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbtf2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbtf2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPBTF2( UPLO, N, KD, AB, LDAB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, KD, LDAB, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPBTF2 computes the Cholesky factorization of a real symmetric
*> positive definite band matrix A.
*>
*> The factorization has the form
*>    A = U**T * U ,  if UPLO = 'U', or
*>    A = L  * L**T,  if UPLO = 'L',
*> where U is an upper triangular matrix, U**T is the transpose of U, and
*> L is lower triangular.
*>
*> This is the unblocked version of the algorithm, calling Level 2 BLAS.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the upper or lower triangular part of the
*>          symmetric matrix A is stored:
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of super-diagonals of the matrix A if UPLO = 'U',
*>          or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          On entry, the upper or lower triangle of the symmetric band
*>          matrix A, stored in the first KD+1 rows of the array.  The
*>          j-th column of A is stored in the j-th column of the array AB
*>          as follows:
*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*>
*>          On exit, if INFO = 0, the triangular factor U or L from the
*>          Cholesky factorization A = U**T*U or A = L*L**T of the band
*>          matrix A, in the same storage format as A.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD+1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -k, the k-th argument had an illegal value
*>          > 0: if INFO = k, the leading principal minor of order k
*>               is not positive, and the factorization could not be
*>               completed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pbtf2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The band storage scheme is illustrated by the following example, when
*>  N = 6, KD = 2, and UPLO = 'U':
*>
*>  On entry:                       On exit:
*>
*>      *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
*>      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
*>     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
*>
*>  Similarly, if UPLO = 'L' the format of A is as follows:
*>
*>  On entry:                       On exit:
*>
*>     a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
*>     a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
*>     a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *
*>
*>  Array elements marked * are not used by the routine.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DPBTF2( UPLO, N, KD, AB, LDAB, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, KD, LDAB, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            J, KLD, KN
      DOUBLE PRECISION   AJJ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL, DSYR, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( KD.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDAB.LT.KD+1 ) THEN
         INFO = -5
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPBTF2', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      KLD = MAX( 1, LDAB-1 )
*
      IF( UPPER ) THEN
*
*        Compute the Cholesky factorization A = U**T*U.
*
         DO 10 J = 1, N
*
*           Compute U(J,J) and test for non-positive-definiteness.
*
            AJJ = AB( KD+1, J )
            IF( AJJ.LE.ZERO )
     $         GO TO 30
            AJJ = SQRT( AJJ )
            AB( KD+1, J ) = AJJ
*
*           Compute elements J+1:J+KN of row J and update the
*           trailing submatrix within the band.
*
            KN = MIN( KD, N-J )
            IF( KN.GT.0 ) THEN
               CALL DSCAL( KN, ONE / AJJ, AB( KD, J+1 ), KLD )
               CALL DSYR( 'Upper', KN, -ONE, AB( KD, J+1 ), KLD,
     $                    AB( KD+1, J+1 ), KLD )
            END IF
   10    CONTINUE
      ELSE
*
*        Compute the Cholesky factorization A = L*L**T.
*
         DO 20 J = 1, N
*
*           Compute L(J,J) and test for non-positive-definiteness.
*
            AJJ = AB( 1, J )
            IF( AJJ.LE.ZERO )
     $         GO TO 30
            AJJ = SQRT( AJJ )
            AB( 1, J ) = AJJ
*
*           Compute elements J+1:J+KN of column J and update the
*           trailing submatrix within the band.
*
            KN = MIN( KD, N-J )
            IF( KN.GT.0 ) THEN
               CALL DSCAL( KN, ONE / AJJ, AB( 2, J ), 1 )
               CALL DSYR( 'Lower', KN, -ONE, AB( 2, J ), 1,
     $                    AB( 1, J+1 ), KLD )
            END IF
   20    CONTINUE
      END IF
      RETURN
*
   30 CONTINUE
      INFO = J
      RETURN
*
*     End of DPBTF2
*
      END
*> \brief \b DPBTRF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPBTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbtrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbtrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbtrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPBTRF( UPLO, N, KD, AB, LDAB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, KD, LDAB, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPBTRF computes the Cholesky factorization of a real symmetric
*> positive definite band matrix A.
*>
*> The factorization has the form
*>    A = U**T * U,  if UPLO = 'U', or
*>    A = L  * L**T,  if UPLO = 'L',
*> where U is an upper triangular matrix and L is lower triangular.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          On entry, the upper or lower triangle of the symmetric band
*>          matrix A, stored in the first KD+1 rows of the array.  The
*>          j-th column of A is stored in the j-th column of the array AB
*>          as follows:
*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*>
*>          On exit, if INFO = 0, the triangular factor U or L from the
*>          Cholesky factorization A = U**T*U or A = L*L**T of the band
*>          matrix A, in the same storage format as A.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD+1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the leading principal minor of order i
*>                is not positive, and the factorization could not be
*>                completed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pbtrf
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The band storage scheme is illustrated by the following example, when
*>  N = 6, KD = 2, and UPLO = 'U':
*>
*>  On entry:                       On exit:
*>
*>      *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
*>      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
*>     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
*>
*>  Similarly, if UPLO = 'L' the format of A is as follows:
*>
*>  On entry:                       On exit:
*>
*>     a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
*>     a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
*>     a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *
*>
*>  Array elements marked * are not used by the routine.
*> \endverbatim
*
*> \par Contributors:
*  ==================
*>
*>  Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989
*
*  =====================================================================
      SUBROUTINE DPBTRF( UPLO, N, KD, AB, LDAB, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, KD, LDAB, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
      INTEGER            NBMAX, LDWORK
      PARAMETER          ( NBMAX = 32, LDWORK = NBMAX+1 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, I2, I3, IB, II, J, JJ, NB
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   WORK( LDWORK, NBMAX )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMM, DPBTF2, DPOTF2, DSYRK, DTRSM,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF( ( .NOT.LSAME( UPLO, 'U' ) ) .AND.
     $    ( .NOT.LSAME( UPLO, 'L' ) ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( KD.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDAB.LT.KD+1 ) THEN
         INFO = -5
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPBTRF', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Determine the block size for this environment
*
      NB = ILAENV( 1, 'DPBTRF', UPLO, N, KD, -1, -1 )
*
*     The block size must not exceed the semi-bandwidth KD, and must not
*     exceed the limit set by the size of the local array WORK.
*
      NB = MIN( NB, NBMAX )
*
      IF( NB.LE.1 .OR. NB.GT.KD ) THEN
*
*        Use unblocked code
*
         CALL DPBTF2( UPLO, N, KD, AB, LDAB, INFO )
      ELSE
*
*        Use blocked code
*
         IF( LSAME( UPLO, 'U' ) ) THEN
*
*           Compute the Cholesky factorization of a symmetric band
*           matrix, given the upper triangle of the matrix in band
*           storage.
*
*           Zero the upper triangle of the work array.
*
            DO 20 J = 1, NB
               DO 10 I = 1, J - 1
                  WORK( I, J ) = ZERO
   10          CONTINUE
   20       CONTINUE
*
*           Process the band matrix one diagonal block at a time.
*
            DO 70 I = 1, N, NB
               IB = MIN( NB, N-I+1 )
*
*              Factorize the diagonal block
*
               CALL DPOTF2( UPLO, IB, AB( KD+1, I ), LDAB-1, II )
               IF( II.NE.0 ) THEN
                  INFO = I + II - 1
                  GO TO 150
               END IF
               IF( I+IB.LE.N ) THEN
*
*                 Update the relevant part of the trailing submatrix.
*                 If A11 denotes the diagonal block which has just been
*                 factorized, then we need to update the remaining
*                 blocks in the diagram:
*
*                    A11   A12   A13
*                          A22   A23
*                                A33
*
*                 The numbers of rows and columns in the partitioning
*                 are IB, I2, I3 respectively. The blocks A12, A22 and
*                 A23 are empty if IB = KD. The upper triangle of A13
*                 lies outside the band.
*
                  I2 = MIN( KD-IB, N-I-IB+1 )
                  I3 = MIN( IB, N-I-KD+1 )
*
                  IF( I2.GT.0 ) THEN
*
*                    Update A12
*
                     CALL DTRSM( 'Left', 'Upper', 'Transpose',
     $                           'Non-unit', IB, I2, ONE, AB( KD+1, I ),
     $                           LDAB-1, AB( KD+1-IB, I+IB ), LDAB-1 )
*
*                    Update A22
*
                     CALL DSYRK( 'Upper', 'Transpose', I2, IB, -ONE,
     $                           AB( KD+1-IB, I+IB ), LDAB-1, ONE,
     $                           AB( KD+1, I+IB ), LDAB-1 )
                  END IF
*
                  IF( I3.GT.0 ) THEN
*
*                    Copy the lower triangle of A13 into the work array.
*
                     DO 40 JJ = 1, I3
                        DO 30 II = JJ, IB
                           WORK( II, JJ ) = AB( II-JJ+1, JJ+I+KD-1 )
   30                   CONTINUE
   40                CONTINUE
*
*                    Update A13 (in the work array).
*
                     CALL DTRSM( 'Left', 'Upper', 'Transpose',
     $                           'Non-unit', IB, I3, ONE, AB( KD+1, I ),
     $                           LDAB-1, WORK, LDWORK )
*
*                    Update A23
*
                     IF( I2.GT.0 )
     $                  CALL DGEMM( 'Transpose', 'No Transpose', I2,
     $                              I3,
     $                              IB, -ONE, AB( KD+1-IB, I+IB ),
     $                              LDAB-1, WORK, LDWORK, ONE,
     $                              AB( 1+IB, I+KD ), LDAB-1 )
*
*                    Update A33
*
                     CALL DSYRK( 'Upper', 'Transpose', I3, IB, -ONE,
     $                           WORK, LDWORK, ONE, AB( KD+1, I+KD ),
     $                           LDAB-1 )
*
*                    Copy the lower triangle of A13 back into place.
*
                     DO 60 JJ = 1, I3
                        DO 50 II = JJ, IB
                           AB( II-JJ+1, JJ+I+KD-1 ) = WORK( II, JJ )
   50                   CONTINUE
   60                CONTINUE
                  END IF
               END IF
   70       CONTINUE
         ELSE
*
*           Compute the Cholesky factorization of a symmetric band
*           matrix, given the lower triangle of the matrix in band
*           storage.
*
*           Zero the lower triangle of the work array.
*
            DO 90 J = 1, NB
               DO 80 I = J + 1, NB
                  WORK( I, J ) = ZERO
   80          CONTINUE
   90       CONTINUE
*
*           Process the band matrix one diagonal block at a time.
*
            DO 140 I = 1, N, NB
               IB = MIN( NB, N-I+1 )
*
*              Factorize the diagonal block
*
               CALL DPOTF2( UPLO, IB, AB( 1, I ), LDAB-1, II )
               IF( II.NE.0 ) THEN
                  INFO = I + II - 1
                  GO TO 150
               END IF
               IF( I+IB.LE.N ) THEN
*
*                 Update the relevant part of the trailing submatrix.
*                 If A11 denotes the diagonal block which has just been
*                 factorized, then we need to update the remaining
*                 blocks in the diagram:
*
*                    A11
*                    A21   A22
*                    A31   A32   A33
*
*                 The numbers of rows and columns in the partitioning
*                 are IB, I2, I3 respectively. The blocks A21, A22 and
*                 A32 are empty if IB = KD. The lower triangle of A31
*                 lies outside the band.
*
                  I2 = MIN( KD-IB, N-I-IB+1 )
                  I3 = MIN( IB, N-I-KD+1 )
*
                  IF( I2.GT.0 ) THEN
*
*                    Update A21
*
                     CALL DTRSM( 'Right', 'Lower', 'Transpose',
     $                           'Non-unit', I2, IB, ONE, AB( 1, I ),
     $                           LDAB-1, AB( 1+IB, I ), LDAB-1 )
*
*                    Update A22
*
                     CALL DSYRK( 'Lower', 'No Transpose', I2, IB,
     $                           -ONE,
     $                           AB( 1+IB, I ), LDAB-1, ONE,
     $                           AB( 1, I+IB ), LDAB-1 )
                  END IF
*
                  IF( I3.GT.0 ) THEN
*
*                    Copy the upper triangle of A31 into the work array.
*
                     DO 110 JJ = 1, IB
                        DO 100 II = 1, MIN( JJ, I3 )
                           WORK( II, JJ ) = AB( KD+1-JJ+II, JJ+I-1 )
  100                   CONTINUE
  110                CONTINUE
*
*                    Update A31 (in the work array).
*
                     CALL DTRSM( 'Right', 'Lower', 'Transpose',
     $                           'Non-unit', I3, IB, ONE, AB( 1, I ),
     $                           LDAB-1, WORK, LDWORK )
*
*                    Update A32
*
                     IF( I2.GT.0 )
     $                  CALL DGEMM( 'No transpose', 'Transpose', I3,
     $                              I2,
     $                              IB, -ONE, WORK, LDWORK,
     $                              AB( 1+IB, I ), LDAB-1, ONE,
     $                              AB( 1+KD-IB, I+IB ), LDAB-1 )
*
*                    Update A33
*
                     CALL DSYRK( 'Lower', 'No Transpose', I3, IB,
     $                           -ONE,
     $                           WORK, LDWORK, ONE, AB( 1, I+KD ),
     $                           LDAB-1 )
*
*                    Copy the upper triangle of A31 back into place.
*
                     DO 130 JJ = 1, IB
                        DO 120 II = 1, MIN( JJ, I3 )
                           AB( KD+1-JJ+II, JJ+I-1 ) = WORK( II, JJ )
  120                   CONTINUE
  130                CONTINUE
                  END IF
               END IF
  140       CONTINUE
         END IF
      END IF
      RETURN
*
  150 CONTINUE
      RETURN
*
*     End of DPBTRF
*
      END
*> \brief \b DPBTRS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPBTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbtrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbtrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbtrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPBTRS( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, KD, LDAB, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPBTRS solves a system of linear equations A*X = B with a symmetric
*> positive definite band matrix A using the Cholesky factorization
*> A = U**T*U or A = L*L**T computed by DPBTRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangular factor stored in AB;
*>          = 'L':  Lower triangular factor stored in AB.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          The triangular factor U or L from the Cholesky factorization
*>          A = U**T*U or A = L*L**T of the band matrix A, stored in the
*>          first KD+1 rows of the array.  The j-th column of U or L is
*>          stored in the j-th column of the array AB as follows:
*>          if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO ='L', AB(1+i-j,j)    = L(i,j) for j<=i<=min(n,j+kd).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the right hand side matrix B.
*>          On exit, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pbtrs
*
*  =====================================================================
      SUBROUTINE DPBTRS( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, KD, LDAB, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            J
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DTBSV, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( KD.LT.0 ) THEN
         INFO = -3
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDAB.LT.KD+1 ) THEN
         INFO = -6
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -8
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPBTRS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 )
     $   RETURN
*
      IF( UPPER ) THEN
*
*        Solve A*X = B where A = U**T *U.
*
         DO 10 J = 1, NRHS
*
*           Solve U**T *X = B, overwriting B with X.
*
            CALL DTBSV( 'Upper', 'Transpose', 'Non-unit', N, KD, AB,
     $                  LDAB, B( 1, J ), 1 )
*
*           Solve U*X = B, overwriting B with X.
*
            CALL DTBSV( 'Upper', 'No transpose', 'Non-unit', N, KD,
     $                  AB,
     $                  LDAB, B( 1, J ), 1 )
   10    CONTINUE
      ELSE
*
*        Solve A*X = B where A = L*L**T.
*
         DO 20 J = 1, NRHS
*
*           Solve L*X = B, overwriting B with X.
*
            CALL DTBSV( 'Lower', 'No transpose', 'Non-unit', N, KD,
     $                  AB,
     $                  LDAB, B( 1, J ), 1 )
*
*           Solve L**T *X = B, overwriting B with X.
*
            CALL DTBSV( 'Lower', 'Transpose', 'Non-unit', N, KD, AB,
     $                  LDAB, B( 1, J ), 1 )
   20    CONTINUE
      END IF
*
      RETURN
*
*     End of DPBTRS
*
      END
*> \brief \b DPFTRF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPFTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpftrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpftrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpftrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPFTRF( TRANSR, UPLO, N, A, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANSR, UPLO
*       INTEGER            N, INFO
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( 0: * )
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPFTRF computes the Cholesky factorization of a real symmetric
*> positive definite matrix A.
*>
*> The factorization has the form
*>    A = U**T * U,  if UPLO = 'U', or
*>    A = L  * L**T,  if UPLO = 'L',
*> where U is an upper triangular matrix and L is lower triangular.
*>
*> This is the block version of the algorithm, calling Level 3 BLAS.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANSR
*> \verbatim
*>          TRANSR is CHARACTER*1
*>          = 'N':  The Normal TRANSR of RFP A is stored;
*>          = 'T':  The Transpose TRANSR of RFP A is stored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of RFP A is stored;
*>          = 'L':  Lower triangle of RFP A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 );
*>          On entry, the symmetric matrix A in RFP format. RFP format is
*>          described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
*>          then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
*>          (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
*>          the transpose of RFP A as defined when
*>          TRANSR = 'N'. The contents of RFP A are defined by UPLO as
*>          follows: If UPLO = 'U' the RFP A contains the NT elements of
*>          upper packed A. If UPLO = 'L' the RFP A contains the elements
*>          of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
*>          'T'. When TRANSR is 'N' the LDA is N+1 when N is even and N
*>          is odd. See the Note below for more details.
*>
*>          On exit, if INFO = 0, the factor U or L from the Cholesky
*>          factorization RFP A = U**T*U or RFP A = L*L**T.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the leading principal minor of order i
*>                is not positive, and the factorization could not be
*>                completed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pftrf
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  We first consider Rectangular Full Packed (RFP) Format when N is
*>  even. We give an example where N = 6.
*>
*>      AP is Upper             AP is Lower
*>
*>   00 01 02 03 04 05       00
*>      11 12 13 14 15       10 11
*>         22 23 24 25       20 21 22
*>            33 34 35       30 31 32 33
*>               44 45       40 41 42 43 44
*>                  55       50 51 52 53 54 55
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*>  the transpose of the first three columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*>  the transpose of the last three columns of AP lower.
*>  This covers the case N even and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>        03 04 05                33 43 53
*>        13 14 15                00 44 54
*>        23 24 25                10 11 55
*>        33 34 35                20 21 22
*>        00 44 45                30 31 32
*>        01 11 55                40 41 42
*>        02 12 22                50 51 52
*>
*>  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*>  transpose of RFP A above. One therefore gets:
*>
*>
*>           RFP A                   RFP A
*>
*>     03 13 23 33 00 01 02    33 00 10 20 30 40 50
*>     04 14 24 34 44 11 12    43 44 11 21 31 41 51
*>     05 15 25 35 45 55 22    53 54 55 22 32 42 52
*>
*>
*>  We then consider Rectangular Full Packed (RFP) Format when N is
*>  odd. We give an example where N = 5.
*>
*>     AP is Upper                 AP is Lower
*>
*>   00 01 02 03 04              00
*>      11 12 13 14              10 11
*>         22 23 24              20 21 22
*>            33 34              30 31 32 33
*>               44              40 41 42 43 44
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*>  the transpose of the first two columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*>  the transpose of the last two columns of AP lower.
*>  This covers the case N odd and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>        02 03 04                00 33 43
*>        12 13 14                10 11 44
*>        22 23 24                20 21 22
*>        00 33 34                30 31 32
*>        01 11 44                40 41 42
*>
*>  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*>  transpose of RFP A above. One therefore gets:
*>
*>           RFP A                   RFP A
*>
*>     02 12 22 00 01             00 10 20 30 40 50
*>     03 13 23 33 11             33 11 21 31 41 51
*>     04 14 24 34 44             43 44 22 32 42 52
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DPFTRF( TRANSR, UPLO, N, A, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANSR, UPLO
      INTEGER            N, INFO
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( 0: * )
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER, NISODD, NORMALTRANSR
      INTEGER            N1, N2, K
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA, DSYRK, DPOTRF, DTRSM
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MOD
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NORMALTRANSR = LSAME( TRANSR, 'N' )
      LOWER = LSAME( UPLO, 'L' )
      IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPFTRF', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     If N is odd, set NISODD = .TRUE.
*     If N is even, set K = N/2 and NISODD = .FALSE.
*
      IF( MOD( N, 2 ).EQ.0 ) THEN
         K = N / 2
         NISODD = .FALSE.
      ELSE
         NISODD = .TRUE.
      END IF
*
*     Set N1 and N2 depending on LOWER
*
      IF( LOWER ) THEN
         N2 = N / 2
         N1 = N - N2
      ELSE
         N1 = N / 2
         N2 = N - N1
      END IF
*
*     start execution: there are eight cases
*
      IF( NISODD ) THEN
*
*        N is odd
*
         IF( NORMALTRANSR ) THEN
*
*           N is odd and TRANSR = 'N'
*
            IF( LOWER ) THEN
*
*             SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:n1-1) )
*             T1 -> a(0,0), T2 -> a(0,1), S -> a(n1,0)
*             T1 -> a(0), T2 -> a(n), S -> a(n1)
*
               CALL DPOTRF( 'L', N1, A( 0 ), N, INFO )
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRSM( 'R', 'L', 'T', 'N', N2, N1, ONE, A( 0 ),
     $                     N,
     $                     A( N1 ), N )
               CALL DSYRK( 'U', 'N', N2, N1, -ONE, A( N1 ), N, ONE,
     $                     A( N ), N )
               CALL DPOTRF( 'U', N2, A( N ), N, INFO )
               IF( INFO.GT.0 )
     $            INFO = INFO + N1
*
            ELSE
*
*             SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:n2-1)
*             T1 -> a(n1+1,0), T2 -> a(n1,0), S -> a(0,0)
*             T1 -> a(n2), T2 -> a(n1), S -> a(0)
*
               CALL DPOTRF( 'L', N1, A( N2 ), N, INFO )
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRSM( 'L', 'L', 'N', 'N', N1, N2, ONE, A( N2 ),
     $                     N,
     $                     A( 0 ), N )
               CALL DSYRK( 'U', 'T', N2, N1, -ONE, A( 0 ), N, ONE,
     $                     A( N1 ), N )
               CALL DPOTRF( 'U', N2, A( N1 ), N, INFO )
               IF( INFO.GT.0 )
     $            INFO = INFO + N1
*
            END IF
*
         ELSE
*
*           N is odd and TRANSR = 'T'
*
            IF( LOWER ) THEN
*
*              SRPA for LOWER, TRANSPOSE and N is odd
*              T1 -> A(0,0) , T2 -> A(1,0) , S -> A(0,n1)
*              T1 -> a(0+0) , T2 -> a(1+0) , S -> a(0+n1*n1); lda=n1
*
               CALL DPOTRF( 'U', N1, A( 0 ), N1, INFO )
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRSM( 'L', 'U', 'T', 'N', N1, N2, ONE, A( 0 ),
     $                     N1,
     $                     A( N1*N1 ), N1 )
               CALL DSYRK( 'L', 'T', N2, N1, -ONE, A( N1*N1 ), N1,
     $                     ONE,
     $                     A( 1 ), N1 )
               CALL DPOTRF( 'L', N2, A( 1 ), N1, INFO )
               IF( INFO.GT.0 )
     $            INFO = INFO + N1
*
            ELSE
*
*              SRPA for UPPER, TRANSPOSE and N is odd
*              T1 -> A(0,n1+1), T2 -> A(0,n1), S -> A(0,0)
*              T1 -> a(n2*n2), T2 -> a(n1*n2), S -> a(0); lda = n2
*
               CALL DPOTRF( 'U', N1, A( N2*N2 ), N2, INFO )
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRSM( 'R', 'U', 'N', 'N', N2, N1, ONE,
     $                     A( N2*N2 ),
     $                     N2, A( 0 ), N2 )
               CALL DSYRK( 'L', 'N', N2, N1, -ONE, A( 0 ), N2, ONE,
     $                     A( N1*N2 ), N2 )
               CALL DPOTRF( 'L', N2, A( N1*N2 ), N2, INFO )
               IF( INFO.GT.0 )
     $            INFO = INFO + N1
*
            END IF
*
         END IF
*
      ELSE
*
*        N is even
*
         IF( NORMALTRANSR ) THEN
*
*           N is even and TRANSR = 'N'
*
            IF( LOWER ) THEN
*
*              SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
*              T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
*              T1 -> a(1), T2 -> a(0), S -> a(k+1)
*
               CALL DPOTRF( 'L', K, A( 1 ), N+1, INFO )
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRSM( 'R', 'L', 'T', 'N', K, K, ONE, A( 1 ),
     $                     N+1,
     $                     A( K+1 ), N+1 )
               CALL DSYRK( 'U', 'N', K, K, -ONE, A( K+1 ), N+1, ONE,
     $                     A( 0 ), N+1 )
               CALL DPOTRF( 'U', K, A( 0 ), N+1, INFO )
               IF( INFO.GT.0 )
     $            INFO = INFO + K
*
            ELSE
*
*              SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
*              T1 -> a(k+1,0) ,  T2 -> a(k,0),   S -> a(0,0)
*              T1 -> a(k+1), T2 -> a(k), S -> a(0)
*
               CALL DPOTRF( 'L', K, A( K+1 ), N+1, INFO )
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRSM( 'L', 'L', 'N', 'N', K, K, ONE, A( K+1 ),
     $                     N+1, A( 0 ), N+1 )
               CALL DSYRK( 'U', 'T', K, K, -ONE, A( 0 ), N+1, ONE,
     $                     A( K ), N+1 )
               CALL DPOTRF( 'U', K, A( K ), N+1, INFO )
               IF( INFO.GT.0 )
     $            INFO = INFO + K
*
            END IF
*
         ELSE
*
*           N is even and TRANSR = 'T'
*
            IF( LOWER ) THEN
*
*              SRPA for LOWER, TRANSPOSE and N is even (see paper)
*              T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1)
*              T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
*
               CALL DPOTRF( 'U', K, A( 0+K ), K, INFO )
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRSM( 'L', 'U', 'T', 'N', K, K, ONE, A( K ), N1,
     $                     A( K*( K+1 ) ), K )
               CALL DSYRK( 'L', 'T', K, K, -ONE, A( K*( K+1 ) ), K,
     $                     ONE,
     $                     A( 0 ), K )
               CALL DPOTRF( 'L', K, A( 0 ), K, INFO )
               IF( INFO.GT.0 )
     $            INFO = INFO + K
*
            ELSE
*
*              SRPA for UPPER, TRANSPOSE and N is even (see paper)
*              T1 -> B(0,k+1),     T2 -> B(0,k),   S -> B(0,0)
*              T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
*
               CALL DPOTRF( 'U', K, A( K*( K+1 ) ), K, INFO )
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRSM( 'R', 'U', 'N', 'N', K, K, ONE,
     $                     A( K*( K+1 ) ), K, A( 0 ), K )
               CALL DSYRK( 'L', 'N', K, K, -ONE, A( 0 ), K, ONE,
     $                     A( K*K ), K )
               CALL DPOTRF( 'L', K, A( K*K ), K, INFO )
               IF( INFO.GT.0 )
     $            INFO = INFO + K
*
            END IF
*
         END IF
*
      END IF
*
      RETURN
*
*     End of DPFTRF
*
      END
*> \brief \b DPFTRI
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPFTRI + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpftri.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpftri.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpftri.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPFTRI( TRANSR, UPLO, N, A, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANSR, UPLO
*       INTEGER            INFO, N
*       .. Array Arguments ..
*       DOUBLE PRECISION         A( 0: * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPFTRI computes the inverse of a (real) symmetric positive definite
*> matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
*> computed by DPFTRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANSR
*> \verbatim
*>          TRANSR is CHARACTER*1
*>          = 'N':  The Normal TRANSR of RFP A is stored;
*>          = 'T':  The Transpose TRANSR of RFP A is stored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 )
*>          On entry, the symmetric matrix A in RFP format. RFP format is
*>          described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
*>          then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
*>          (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
*>          the transpose of RFP A as defined when
*>          TRANSR = 'N'. The contents of RFP A are defined by UPLO as
*>          follows: If UPLO = 'U' the RFP A contains the nt elements of
*>          upper packed A. If UPLO = 'L' the RFP A contains the elements
*>          of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
*>          'T'. When TRANSR is 'N' the LDA is N+1 when N is even and N
*>          is odd. See the Note below for more details.
*>
*>          On exit, the symmetric inverse of the original matrix, in the
*>          same storage format.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the (i,i) element of the factor U or L is
*>                zero, and the inverse could not be computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pftri
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  We first consider Rectangular Full Packed (RFP) Format when N is
*>  even. We give an example where N = 6.
*>
*>      AP is Upper             AP is Lower
*>
*>   00 01 02 03 04 05       00
*>      11 12 13 14 15       10 11
*>         22 23 24 25       20 21 22
*>            33 34 35       30 31 32 33
*>               44 45       40 41 42 43 44
*>                  55       50 51 52 53 54 55
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*>  the transpose of the first three columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*>  the transpose of the last three columns of AP lower.
*>  This covers the case N even and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>        03 04 05                33 43 53
*>        13 14 15                00 44 54
*>        23 24 25                10 11 55
*>        33 34 35                20 21 22
*>        00 44 45                30 31 32
*>        01 11 55                40 41 42
*>        02 12 22                50 51 52
*>
*>  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*>  transpose of RFP A above. One therefore gets:
*>
*>
*>           RFP A                   RFP A
*>
*>     03 13 23 33 00 01 02    33 00 10 20 30 40 50
*>     04 14 24 34 44 11 12    43 44 11 21 31 41 51
*>     05 15 25 35 45 55 22    53 54 55 22 32 42 52
*>
*>
*>  We then consider Rectangular Full Packed (RFP) Format when N is
*>  odd. We give an example where N = 5.
*>
*>     AP is Upper                 AP is Lower
*>
*>   00 01 02 03 04              00
*>      11 12 13 14              10 11
*>         22 23 24              20 21 22
*>            33 34              30 31 32 33
*>               44              40 41 42 43 44
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*>  the transpose of the first two columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*>  the transpose of the last two columns of AP lower.
*>  This covers the case N odd and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>        02 03 04                00 33 43
*>        12 13 14                10 11 44
*>        22 23 24                20 21 22
*>        00 33 34                30 31 32
*>        01 11 44                40 41 42
*>
*>  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*>  transpose of RFP A above. One therefore gets:
*>
*>           RFP A                   RFP A
*>
*>     02 12 22 00 01             00 10 20 30 40 50
*>     03 13 23 33 11             33 11 21 31 41 51
*>     04 14 24 34 44             43 44 22 32 42 52
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DPFTRI( TRANSR, UPLO, N, A, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANSR, UPLO
      INTEGER            INFO, N
*     .. Array Arguments ..
      DOUBLE PRECISION         A( 0: * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER, NISODD, NORMALTRANSR
      INTEGER            N1, N2, K
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA, DTFTRI, DLAUUM, DTRMM,
     $                   DSYRK
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MOD
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NORMALTRANSR = LSAME( TRANSR, 'N' )
      LOWER = LSAME( UPLO, 'L' )
      IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPFTRI', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Invert the triangular Cholesky factor U or L.
*
      CALL DTFTRI( TRANSR, UPLO, 'N', N, A, INFO )
      IF( INFO.GT.0 )
     $   RETURN
*
*     If N is odd, set NISODD = .TRUE.
*     If N is even, set K = N/2 and NISODD = .FALSE.
*
      IF( MOD( N, 2 ).EQ.0 ) THEN
         K = N / 2
         NISODD = .FALSE.
      ELSE
         NISODD = .TRUE.
      END IF
*
*     Set N1 and N2 depending on LOWER
*
      IF( LOWER ) THEN
         N2 = N / 2
         N1 = N - N2
      ELSE
         N1 = N / 2
         N2 = N - N1
      END IF
*
*     Start execution of triangular matrix multiply: inv(U)*inv(U)^C or
*     inv(L)^C*inv(L). There are eight cases.
*
      IF( NISODD ) THEN
*
*        N is odd
*
         IF( NORMALTRANSR ) THEN
*
*           N is odd and TRANSR = 'N'
*
            IF( LOWER ) THEN
*
*              SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:N1-1) )
*              T1 -> a(0,0), T2 -> a(0,1), S -> a(N1,0)
*              T1 -> a(0), T2 -> a(n), S -> a(N1)
*
               CALL DLAUUM( 'L', N1, A( 0 ), N, INFO )
               CALL DSYRK( 'L', 'T', N1, N2, ONE, A( N1 ), N, ONE,
     $                     A( 0 ), N )
               CALL DTRMM( 'L', 'U', 'N', 'N', N2, N1, ONE, A( N ),
     $                     N,
     $                     A( N1 ), N )
               CALL DLAUUM( 'U', N2, A( N ), N, INFO )
*
            ELSE
*
*              SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:N2-1)
*              T1 -> a(N1+1,0), T2 -> a(N1,0), S -> a(0,0)
*              T1 -> a(N2), T2 -> a(N1), S -> a(0)
*
               CALL DLAUUM( 'L', N1, A( N2 ), N, INFO )
               CALL DSYRK( 'L', 'N', N1, N2, ONE, A( 0 ), N, ONE,
     $                     A( N2 ), N )
               CALL DTRMM( 'R', 'U', 'T', 'N', N1, N2, ONE, A( N1 ),
     $                     N,
     $                     A( 0 ), N )
               CALL DLAUUM( 'U', N2, A( N1 ), N, INFO )
*
            END IF
*
         ELSE
*
*           N is odd and TRANSR = 'T'
*
            IF( LOWER ) THEN
*
*              SRPA for LOWER, TRANSPOSE, and N is odd
*              T1 -> a(0), T2 -> a(1), S -> a(0+N1*N1)
*
               CALL DLAUUM( 'U', N1, A( 0 ), N1, INFO )
               CALL DSYRK( 'U', 'N', N1, N2, ONE, A( N1*N1 ), N1,
     $                     ONE,
     $                     A( 0 ), N1 )
               CALL DTRMM( 'R', 'L', 'N', 'N', N1, N2, ONE, A( 1 ),
     $                     N1,
     $                     A( N1*N1 ), N1 )
               CALL DLAUUM( 'L', N2, A( 1 ), N1, INFO )
*
            ELSE
*
*              SRPA for UPPER, TRANSPOSE, and N is odd
*              T1 -> a(0+N2*N2), T2 -> a(0+N1*N2), S -> a(0)
*
               CALL DLAUUM( 'U', N1, A( N2*N2 ), N2, INFO )
               CALL DSYRK( 'U', 'T', N1, N2, ONE, A( 0 ), N2, ONE,
     $                     A( N2*N2 ), N2 )
               CALL DTRMM( 'L', 'L', 'T', 'N', N2, N1, ONE,
     $                     A( N1*N2 ),
     $                     N2, A( 0 ), N2 )
               CALL DLAUUM( 'L', N2, A( N1*N2 ), N2, INFO )
*
            END IF
*
         END IF
*
      ELSE
*
*        N is even
*
         IF( NORMALTRANSR ) THEN
*
*           N is even and TRANSR = 'N'
*
            IF( LOWER ) THEN
*
*              SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
*              T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
*              T1 -> a(1), T2 -> a(0), S -> a(k+1)
*
               CALL DLAUUM( 'L', K, A( 1 ), N+1, INFO )
               CALL DSYRK( 'L', 'T', K, K, ONE, A( K+1 ), N+1, ONE,
     $                     A( 1 ), N+1 )
               CALL DTRMM( 'L', 'U', 'N', 'N', K, K, ONE, A( 0 ),
     $                     N+1,
     $                     A( K+1 ), N+1 )
               CALL DLAUUM( 'U', K, A( 0 ), N+1, INFO )
*
            ELSE
*
*              SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
*              T1 -> a(k+1,0) ,  T2 -> a(k,0),   S -> a(0,0)
*              T1 -> a(k+1), T2 -> a(k), S -> a(0)
*
               CALL DLAUUM( 'L', K, A( K+1 ), N+1, INFO )
               CALL DSYRK( 'L', 'N', K, K, ONE, A( 0 ), N+1, ONE,
     $                     A( K+1 ), N+1 )
               CALL DTRMM( 'R', 'U', 'T', 'N', K, K, ONE, A( K ),
     $                     N+1,
     $                     A( 0 ), N+1 )
               CALL DLAUUM( 'U', K, A( K ), N+1, INFO )
*
            END IF
*
         ELSE
*
*           N is even and TRANSR = 'T'
*
            IF( LOWER ) THEN
*
*              SRPA for LOWER, TRANSPOSE, and N is even (see paper)
*              T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1),
*              T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
*
               CALL DLAUUM( 'U', K, A( K ), K, INFO )
               CALL DSYRK( 'U', 'N', K, K, ONE, A( K*( K+1 ) ), K,
     $                     ONE,
     $                     A( K ), K )
               CALL DTRMM( 'R', 'L', 'N', 'N', K, K, ONE, A( 0 ), K,
     $                     A( K*( K+1 ) ), K )
               CALL DLAUUM( 'L', K, A( 0 ), K, INFO )
*
            ELSE
*
*              SRPA for UPPER, TRANSPOSE, and N is even (see paper)
*              T1 -> B(0,k+1),     T2 -> B(0,k),   S -> B(0,0),
*              T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
*
               CALL DLAUUM( 'U', K, A( K*( K+1 ) ), K, INFO )
               CALL DSYRK( 'U', 'T', K, K, ONE, A( 0 ), K, ONE,
     $                     A( K*( K+1 ) ), K )
               CALL DTRMM( 'L', 'L', 'T', 'N', K, K, ONE, A( K*K ),
     $                     K,
     $                     A( 0 ), K )
               CALL DLAUUM( 'L', K, A( K*K ), K, INFO )
*
            END IF
*
         END IF
*
      END IF
*
      RETURN
*
*     End of DPFTRI
*
      END
*> \brief \b DPFTRS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPFTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpftrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpftrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpftrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPFTRS( TRANSR, UPLO, N, NRHS, A, B, LDB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANSR, UPLO
*       INTEGER            INFO, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( 0: * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPFTRS solves a system of linear equations A*X = B with a symmetric
*> positive definite matrix A using the Cholesky factorization
*> A = U**T*U or A = L*L**T computed by DPFTRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANSR
*> \verbatim
*>          TRANSR is CHARACTER*1
*>          = 'N':  The Normal TRANSR of RFP A is stored;
*>          = 'T':  The Transpose TRANSR of RFP A is stored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of RFP A is stored;
*>          = 'L':  Lower triangle of RFP A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ).
*>          The triangular factor U or L from the Cholesky factorization
*>          of RFP A = U**T*U or RFP A = L*L**T, as computed by DPFTRF.
*>          See note below for more details about RFP A.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the right hand side matrix B.
*>          On exit, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pftrs
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  We first consider Rectangular Full Packed (RFP) Format when N is
*>  even. We give an example where N = 6.
*>
*>      AP is Upper             AP is Lower
*>
*>   00 01 02 03 04 05       00
*>      11 12 13 14 15       10 11
*>         22 23 24 25       20 21 22
*>            33 34 35       30 31 32 33
*>               44 45       40 41 42 43 44
*>                  55       50 51 52 53 54 55
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*>  the transpose of the first three columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*>  the transpose of the last three columns of AP lower.
*>  This covers the case N even and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>        03 04 05                33 43 53
*>        13 14 15                00 44 54
*>        23 24 25                10 11 55
*>        33 34 35                20 21 22
*>        00 44 45                30 31 32
*>        01 11 55                40 41 42
*>        02 12 22                50 51 52
*>
*>  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*>  transpose of RFP A above. One therefore gets:
*>
*>
*>           RFP A                   RFP A
*>
*>     03 13 23 33 00 01 02    33 00 10 20 30 40 50
*>     04 14 24 34 44 11 12    43 44 11 21 31 41 51
*>     05 15 25 35 45 55 22    53 54 55 22 32 42 52
*>
*>
*>  We then consider Rectangular Full Packed (RFP) Format when N is
*>  odd. We give an example where N = 5.
*>
*>     AP is Upper                 AP is Lower
*>
*>   00 01 02 03 04              00
*>      11 12 13 14              10 11
*>         22 23 24              20 21 22
*>            33 34              30 31 32 33
*>               44              40 41 42 43 44
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*>  the transpose of the first two columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*>  the transpose of the last two columns of AP lower.
*>  This covers the case N odd and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>        02 03 04                00 33 43
*>        12 13 14                10 11 44
*>        22 23 24                20 21 22
*>        00 33 34                30 31 32
*>        01 11 44                40 41 42
*>
*>  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*>  transpose of RFP A above. One therefore gets:
*>
*>           RFP A                   RFP A
*>
*>     02 12 22 00 01             00 10 20 30 40 50
*>     03 13 23 33 11             33 11 21 31 41 51
*>     04 14 24 34 44             43 44 22 32 42 52
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DPFTRS( TRANSR, UPLO, N, NRHS, A, B, LDB, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANSR, UPLO
      INTEGER            INFO, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( 0: * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER, NORMALTRANSR
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA, DTFSM
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NORMALTRANSR = LSAME( TRANSR, 'N' )
      LOWER = LSAME( UPLO, 'L' )
      IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPFTRS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 )
     $   RETURN
*
*     start execution: there are two triangular solves
*
      IF( LOWER ) THEN
         CALL DTFSM( TRANSR, 'L', UPLO, 'N', 'N', N, NRHS, ONE, A, B,
     $               LDB )
         CALL DTFSM( TRANSR, 'L', UPLO, 'T', 'N', N, NRHS, ONE, A, B,
     $               LDB )
      ELSE
         CALL DTFSM( TRANSR, 'L', UPLO, 'T', 'N', N, NRHS, ONE, A, B,
     $               LDB )
         CALL DTFSM( TRANSR, 'L', UPLO, 'N', 'N', N, NRHS, ONE, A, B,
     $               LDB )
      END IF
*
      RETURN
*
*     End of DPFTRS
*
      END
*> \brief \b DPOCON
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPOCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpocon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpocon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpocon.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPOCON( UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK,
*                          INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, N
*       DOUBLE PRECISION   ANORM, RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPOCON estimates the reciprocal of the condition number (in the
*> 1-norm) of a real symmetric positive definite matrix using the
*> Cholesky factorization A = U**T*U or A = L*L**T computed by DPOTRF.
*>
*> An estimate is obtained for norm(inv(A)), and the reciprocal of the
*> condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The triangular factor U or L from the Cholesky factorization
*>          A = U**T*U or A = L*L**T, as computed by DPOTRF.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*>          ANORM is DOUBLE PRECISION
*>          The 1-norm (or infinity-norm) of the symmetric matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The reciprocal of the condition number of the matrix A,
*>          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
*>          estimate of the 1-norm of inv(A) computed in this routine.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pocon
*
*  =====================================================================
      SUBROUTINE DPOCON( UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N
      DOUBLE PRECISION   ANORM, RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      CHARACTER          NORMIN
      INTEGER            IX, KASE
      DOUBLE PRECISION   AINVNM, SCALE, SCALEL, SCALEU, SMLNUM
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, IDAMAX, DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACN2, DLATRS, DRSCL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      ELSE IF( ANORM.LT.ZERO ) THEN
         INFO = -5
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPOCON', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      RCOND = ZERO
      IF( N.EQ.0 ) THEN
         RCOND = ONE
         RETURN
      ELSE IF( ANORM.EQ.ZERO ) THEN
         RETURN
      END IF
*
      SMLNUM = DLAMCH( 'Safe minimum' )
*
*     Estimate the 1-norm of inv(A).
*
      KASE = 0
      NORMIN = 'N'
   10 CONTINUE
      CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
      IF( KASE.NE.0 ) THEN
         IF( UPPER ) THEN
*
*           Multiply by inv(U**T).
*
            CALL DLATRS( 'Upper', 'Transpose', 'Non-unit', NORMIN, N,
     $                   A,
     $                   LDA, WORK, SCALEL, WORK( 2*N+1 ), INFO )
            NORMIN = 'Y'
*
*           Multiply by inv(U).
*
            CALL DLATRS( 'Upper', 'No transpose', 'Non-unit', NORMIN,
     $                   N,
     $                   A, LDA, WORK, SCALEU, WORK( 2*N+1 ), INFO )
         ELSE
*
*           Multiply by inv(L).
*
            CALL DLATRS( 'Lower', 'No transpose', 'Non-unit', NORMIN,
     $                   N,
     $                   A, LDA, WORK, SCALEL, WORK( 2*N+1 ), INFO )
            NORMIN = 'Y'
*
*           Multiply by inv(L**T).
*
            CALL DLATRS( 'Lower', 'Transpose', 'Non-unit', NORMIN, N,
     $                   A,
     $                   LDA, WORK, SCALEU, WORK( 2*N+1 ), INFO )
         END IF
*
*        Multiply by 1/SCALE if doing so will not cause overflow.
*
         SCALE = SCALEL*SCALEU
         IF( SCALE.NE.ONE ) THEN
            IX = IDAMAX( N, WORK, 1 )
            IF( SCALE.LT.ABS( WORK( IX ) )*SMLNUM .OR. SCALE.EQ.ZERO )
     $         GO TO 20
            CALL DRSCL( N, SCALE, WORK, 1 )
         END IF
         GO TO 10
      END IF
*
*     Compute the estimate of the reciprocal condition number.
*
      IF( AINVNM.NE.ZERO )
     $   RCOND = ( ONE / AINVNM ) / ANORM
*
   20 CONTINUE
      RETURN
*
*     End of DPOCON
*
      END
*> \brief \b DPOEQU
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPOEQU + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpoequ.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpoequ.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpoequ.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPOEQU( N, A, LDA, S, SCOND, AMAX, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, N
*       DOUBLE PRECISION   AMAX, SCOND
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), S( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPOEQU computes row and column scalings intended to equilibrate a
*> symmetric positive definite matrix A and reduce its condition number
*> (with respect to the two-norm).  S contains the scale factors,
*> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
*> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
*> choice of S puts the condition number of B within a factor N of the
*> smallest possible condition number over all possible diagonal
*> scalings.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The N-by-N symmetric positive definite matrix whose scaling
*>          factors are to be computed.  Only the diagonal elements of A
*>          are referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, S contains the scale factors for A.
*> \endverbatim
*>
*> \param[out] SCOND
*> \verbatim
*>          SCOND is DOUBLE PRECISION
*>          If INFO = 0, S contains the ratio of the smallest S(i) to
*>          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
*>          large nor too small, it is not worth scaling by S.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*>          AMAX is DOUBLE PRECISION
*>          Absolute value of largest matrix element.  If AMAX is very
*>          close to overflow or very close to underflow, the matrix
*>          should be scaled.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup poequ
*
*  =====================================================================
      SUBROUTINE DPOEQU( N, A, LDA, S, SCOND, AMAX, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, N
      DOUBLE PRECISION   AMAX, SCOND
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), S( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
      DOUBLE PRECISION   SMIN
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF( N.LT.0 ) THEN
         INFO = -1
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -3
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPOEQU', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 ) THEN
         SCOND = ONE
         AMAX = ZERO
         RETURN
      END IF
*
*     Find the minimum and maximum diagonal elements.
*
      S( 1 ) = A( 1, 1 )
      SMIN = S( 1 )
      AMAX = S( 1 )
      DO 10 I = 2, N
         S( I ) = A( I, I )
         SMIN = MIN( SMIN, S( I ) )
         AMAX = MAX( AMAX, S( I ) )
   10 CONTINUE
*
      IF( SMIN.LE.ZERO ) THEN
*
*        Find the first non-positive diagonal element and return.
*
         DO 20 I = 1, N
            IF( S( I ).LE.ZERO ) THEN
               INFO = I
               RETURN
            END IF
   20    CONTINUE
      ELSE
*
*        Set the scale factors to the reciprocals
*        of the diagonal elements.
*
         DO 30 I = 1, N
            S( I ) = ONE / SQRT( S( I ) )
   30    CONTINUE
*
*        Compute SCOND = min(S(I)) / max(S(I))
*
         SCOND = SQRT( SMIN ) / SQRT( AMAX )
      END IF
      RETURN
*
*     End of DPOEQU
*
      END
*> \brief \b DPOEQUB
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPOEQUB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpoequb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpoequb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpoequb.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, N
*       DOUBLE PRECISION   AMAX, SCOND
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), S( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPOEQUB computes row and column scalings intended to equilibrate a
*> symmetric positive definite matrix A and reduce its condition number
*> (with respect to the two-norm).  S contains the scale factors,
*> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
*> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
*> choice of S puts the condition number of B within a factor N of the
*> smallest possible condition number over all possible diagonal
*> scalings.
*>
*> This routine differs from DPOEQU by restricting the scaling factors
*> to a power of the radix.  Barring over- and underflow, scaling by
*> these factors introduces no additional rounding errors.  However, the
*> scaled diagonal entries are no longer approximately 1 but lie
*> between sqrt(radix) and 1/sqrt(radix).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The N-by-N symmetric positive definite matrix whose scaling
*>          factors are to be computed.  Only the diagonal elements of A
*>          are referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, S contains the scale factors for A.
*> \endverbatim
*>
*> \param[out] SCOND
*> \verbatim
*>          SCOND is DOUBLE PRECISION
*>          If INFO = 0, S contains the ratio of the smallest S(i) to
*>          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
*>          large nor too small, it is not worth scaling by S.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*>          AMAX is DOUBLE PRECISION
*>          Absolute value of largest matrix element.  If AMAX is very
*>          close to overflow or very close to underflow, the matrix
*>          should be scaled.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup poequb
*
*  =====================================================================
      SUBROUTINE DPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, N
      DOUBLE PRECISION   AMAX, SCOND
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), S( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
      DOUBLE PRECISION   SMIN, BASE, TMP
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT, LOG, INT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
*     Positive definite only performs 1 pass of equilibration.
*
      INFO = 0
      IF( N.LT.0 ) THEN
         INFO = -1
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -3
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPOEQUB', -INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( N.EQ.0 ) THEN
         SCOND = ONE
         AMAX = ZERO
         RETURN
      END IF

      BASE = DLAMCH( 'B' )
      TMP = -0.5D+0 / LOG ( BASE )
*
*     Find the minimum and maximum diagonal elements.
*
      S( 1 ) = A( 1, 1 )
      SMIN = S( 1 )
      AMAX = S( 1 )
      DO 10 I = 2, N
         S( I ) = A( I, I )
         SMIN = MIN( SMIN, S( I ) )
         AMAX = MAX( AMAX, S( I ) )
   10 CONTINUE
*
      IF( SMIN.LE.ZERO ) THEN
*
*        Find the first non-positive diagonal element and return.
*
         DO 20 I = 1, N
            IF( S( I ).LE.ZERO ) THEN
               INFO = I
               RETURN
            END IF
   20    CONTINUE
      ELSE
*
*        Set the scale factors to the reciprocals
*        of the diagonal elements.
*
         DO 30 I = 1, N
            S( I ) = BASE ** INT( TMP * LOG( S( I ) ) )
   30    CONTINUE
*
*        Compute SCOND = min(S(I)) / max(S(I)).
*
         SCOND = SQRT( SMIN ) / SQRT( AMAX )
      END IF
*
      RETURN
*
*     End of DPOEQUB
*
      END
*> \brief \b DPORFS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPORFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dporfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dporfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dporfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
*                          LDX, FERR, BERR, WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
*      $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPORFS improves the computed solution to a system of linear
*> equations when the coefficient matrix is symmetric positive definite,
*> and provides error bounds and backward error estimates for the
*> solution.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices B and X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The symmetric matrix A.  If UPLO = 'U', the leading N-by-N
*>          upper triangular part of A contains the upper triangular part
*>          of the matrix A, and the strictly lower triangular part of A
*>          is not referenced.  If UPLO = 'L', the leading N-by-N lower
*>          triangular part of A contains the lower triangular part of
*>          the matrix A, and the strictly upper triangular part of A is
*>          not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] AF
*> \verbatim
*>          AF is DOUBLE PRECISION array, dimension (LDAF,N)
*>          The triangular factor U or L from the Cholesky factorization
*>          A = U**T*U or A = L*L**T, as computed by DPOTRF.
*> \endverbatim
*>
*> \param[in] LDAF
*> \verbatim
*>          LDAF is INTEGER
*>          The leading dimension of the array AF.  LDAF >= max(1,N).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          On entry, the solution matrix X, as computed by DPOTRS.
*>          On exit, the improved solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The estimated forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).  The estimate is as reliable as
*>          the estimate for RCOND, and is almost always a slight
*>          overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in
*>          any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  ITMAX is the maximum number of steps of iterative refinement.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup porfs
*
*  =====================================================================
      SUBROUTINE DPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
     $                   LDX, FERR, BERR, WORK, IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
     $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            ITMAX
      PARAMETER          ( ITMAX = 5 )
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
      DOUBLE PRECISION   TWO
      PARAMETER          ( TWO = 2.0D+0 )
      DOUBLE PRECISION   THREE
      PARAMETER          ( THREE = 3.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            COUNT, I, J, K, KASE, NZ
      DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DLACN2, DPOTRS, DSYMV,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -9
      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
         INFO = -11
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPORFS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
         DO 10 J = 1, NRHS
            FERR( J ) = ZERO
            BERR( J ) = ZERO
   10    CONTINUE
         RETURN
      END IF
*
*     NZ = maximum number of nonzero elements in each row of A, plus 1
*
      NZ = N + 1
      EPS = DLAMCH( 'Epsilon' )
      SAFMIN = DLAMCH( 'Safe minimum' )
      SAFE1 = NZ*SAFMIN
      SAFE2 = SAFE1 / EPS
*
*     Do for each right hand side
*
      DO 140 J = 1, NRHS
*
         COUNT = 1
         LSTRES = THREE
   20    CONTINUE
*
*        Loop until stopping criterion is satisfied.
*
*        Compute residual R = B - A * X
*
         CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
         CALL DSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
     $               WORK( N+1 ), 1 )
*
*        Compute componentwise relative backward error from formula
*
*        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
*
*        where abs(Z) is the componentwise absolute value of the matrix
*        or vector Z.  If the i-th component of the denominator is less
*        than SAFE2, then SAFE1 is added to the i-th components of the
*        numerator and denominator before dividing.
*
         DO 30 I = 1, N
            WORK( I ) = ABS( B( I, J ) )
   30    CONTINUE
*
*        Compute abs(A)*abs(X) + abs(B).
*
         IF( UPPER ) THEN
            DO 50 K = 1, N
               S = ZERO
               XK = ABS( X( K, J ) )
               DO 40 I = 1, K - 1
                  WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
                  S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
   40          CONTINUE
               WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK + S
   50       CONTINUE
         ELSE
            DO 70 K = 1, N
               S = ZERO
               XK = ABS( X( K, J ) )
               WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK
               DO 60 I = K + 1, N
                  WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
                  S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
   60          CONTINUE
               WORK( K ) = WORK( K ) + S
   70       CONTINUE
         END IF
         S = ZERO
         DO 80 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
            ELSE
               S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
     $             ( WORK( I )+SAFE1 ) )
            END IF
   80    CONTINUE
         BERR( J ) = S
*
*        Test stopping criterion. Continue iterating if
*           1) The residual BERR(J) is larger than machine epsilon, and
*           2) BERR(J) decreased by at least a factor of 2 during the
*              last iteration, and
*           3) At most ITMAX iterations tried.
*
         IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
     $       COUNT.LE.ITMAX ) THEN
*
*           Update solution and try again.
*
            CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
            CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
            LSTRES = BERR( J )
            COUNT = COUNT + 1
            GO TO 20
         END IF
*
*        Bound error from formula
*
*        norm(X - XTRUE) / norm(X) .le. FERR =
*        norm( abs(inv(A))*
*           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
*
*        where
*          norm(Z) is the magnitude of the largest component of Z
*          inv(A) is the inverse of A
*          abs(Z) is the componentwise absolute value of the matrix or
*             vector Z
*          NZ is the maximum number of nonzeros in any row of A, plus 1
*          EPS is machine epsilon
*
*        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
*        is incremented by SAFE1 if the i-th component of
*        abs(A)*abs(X) + abs(B) is less than SAFE2.
*
*        Use DLACN2 to estimate the infinity-norm of the matrix
*           inv(A) * diag(W),
*        where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
*
         DO 90 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
            ELSE
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
            END IF
   90    CONTINUE
*
         KASE = 0
  100    CONTINUE
         CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
     $                FERR( J ),
     $                KASE, ISAVE )
         IF( KASE.NE.0 ) THEN
            IF( KASE.EQ.1 ) THEN
*
*              Multiply by diag(W)*inv(A**T).
*
               CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N,
     $                      INFO )
               DO 110 I = 1, N
                  WORK( N+I ) = WORK( I )*WORK( N+I )
  110          CONTINUE
            ELSE IF( KASE.EQ.2 ) THEN
*
*              Multiply by inv(A)*diag(W).
*
               DO 120 I = 1, N
                  WORK( N+I ) = WORK( I )*WORK( N+I )
  120          CONTINUE
               CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N,
     $                      INFO )
            END IF
            GO TO 100
         END IF
*
*        Normalize error.
*
         LSTRES = ZERO
         DO 130 I = 1, N
            LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
  130    CONTINUE
         IF( LSTRES.NE.ZERO )
     $      FERR( J ) = FERR( J ) / LSTRES
*
  140 CONTINUE
*
      RETURN
*
*     End of DPORFS
*
      END
*> \brief <b> DPOSV computes the solution to system of linear equations A * X = B for PO matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPOSV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dposv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dposv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dposv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPOSV( UPLO, N, NRHS, A, LDA, B, LDB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPOSV computes the solution to a real system of linear equations
*>    A * X = B,
*> where A is an N-by-N symmetric positive definite matrix and X and B
*> are N-by-NRHS matrices.
*>
*> The Cholesky decomposition is used to factor A as
*>    A = U**T* U,  if UPLO = 'U', or
*>    A = L * L**T,  if UPLO = 'L',
*> where U is an upper triangular matrix and L is a lower triangular
*> matrix.  The factored form of A is then used to solve the system of
*> equations A * X = B.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of linear equations, i.e., the order of the
*>          matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*>          N-by-N upper triangular part of A contains the upper
*>          triangular part of the matrix A, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading N-by-N lower triangular part of A contains the lower
*>          triangular part of the matrix A, and the strictly upper
*>          triangular part of A is not referenced.
*>
*>          On exit, if INFO = 0, the factor U or L from the Cholesky
*>          factorization A = U**T*U or A = L*L**T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the N-by-NRHS right hand side matrix B.
*>          On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the leading principal minor of order i
*>                of A is not positive, so the factorization could not
*>                be completed, and the solution has not been computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup posv
*
*  =====================================================================
      SUBROUTINE DPOSV( UPLO, N, NRHS, A, LDA, B, LDB, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DPOTRF, DPOTRS, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF( .NOT.LSAME( UPLO, 'U' ) .AND.
     $    .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPOSV ', -INFO )
         RETURN
      END IF
*
*     Compute the Cholesky factorization A = U**T*U or A = L*L**T.
*
      CALL DPOTRF( UPLO, N, A, LDA, INFO )
      IF( INFO.EQ.0 ) THEN
*
*        Solve the system A*X = B, overwriting B with X.
*
         CALL DPOTRS( UPLO, N, NRHS, A, LDA, B, LDB, INFO )
*
      END IF
      RETURN
*
*     End of DPOSV
*
      END
*> \brief <b> DPOSVX computes the solution to system of linear equations A * X = B for PO matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPOSVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dposvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dposvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dposvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
*                          S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
*                          IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          EQUED, FACT, UPLO
*       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
*       DOUBLE PRECISION   RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
*      $                   BERR( * ), FERR( * ), S( * ), WORK( * ),
*      $                   X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
*> compute the solution to a real system of linear equations
*>    A * X = B,
*> where A is an N-by-N symmetric positive definite matrix and X and B
*> are N-by-NRHS matrices.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
*  =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
*>    the system:
*>       diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
*>    Whether or not the system will be equilibrated depends on the
*>    scaling of the matrix A, but if equilibration is used, A is
*>    overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
*>
*> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
*>    factor the matrix A (after equilibration if FACT = 'E') as
*>       A = U**T* U,  if UPLO = 'U', or
*>       A = L * L**T,  if UPLO = 'L',
*>    where U is an upper triangular matrix and L is a lower triangular
*>    matrix.
*>
*> 3. If the leading principal minor of order i is not positive,
*>    then the routine returns with INFO = i. Otherwise, the factored
*>    form of A is used to estimate the condition number of the matrix
*>    A.  If the reciprocal of the condition number is less than machine
*>    precision, INFO = N+1 is returned as a warning, but the routine
*>    still goes on to solve for X and compute error bounds as
*>    described below.
*>
*> 4. The system of equations is solved for X using the factored form
*>    of A.
*>
*> 5. Iterative refinement is applied to improve the computed solution
*>    matrix and calculate error bounds and backward error estimates
*>    for it.
*>
*> 6. If equilibration was used, the matrix X is premultiplied by
*>    diag(S) so that it solves the original system before
*>    equilibration.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] FACT
*> \verbatim
*>          FACT is CHARACTER*1
*>          Specifies whether or not the factored form of the matrix A is
*>          supplied on entry, and if not, whether the matrix A should be
*>          equilibrated before it is factored.
*>          = 'F':  On entry, AF contains the factored form of A.
*>                  If EQUED = 'Y', the matrix A has been equilibrated
*>                  with scaling factors given by S.  A and AF will not
*>                  be modified.
*>          = 'N':  The matrix A will be copied to AF and factored.
*>          = 'E':  The matrix A will be equilibrated if necessary, then
*>                  copied to AF and factored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of linear equations, i.e., the order of the
*>          matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices B and X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the symmetric matrix A, except if FACT = 'F' and
*>          EQUED = 'Y', then A must contain the equilibrated matrix
*>          diag(S)*A*diag(S).  If UPLO = 'U', the leading
*>          N-by-N upper triangular part of A contains the upper
*>          triangular part of the matrix A, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading N-by-N lower triangular part of A contains the lower
*>          triangular part of the matrix A, and the strictly upper
*>          triangular part of A is not referenced.  A is not modified if
*>          FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
*>
*>          On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
*>          diag(S)*A*diag(S).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] AF
*> \verbatim
*>          AF is DOUBLE PRECISION array, dimension (LDAF,N)
*>          If FACT = 'F', then AF is an input argument and on entry
*>          contains the triangular factor U or L from the Cholesky
*>          factorization A = U**T*U or A = L*L**T, in the same storage
*>          format as A.  If EQUED .ne. 'N', then AF is the factored form
*>          of the equilibrated matrix diag(S)*A*diag(S).
*>
*>          If FACT = 'N', then AF is an output argument and on exit
*>          returns the triangular factor U or L from the Cholesky
*>          factorization A = U**T*U or A = L*L**T of the original
*>          matrix A.
*>
*>          If FACT = 'E', then AF is an output argument and on exit
*>          returns the triangular factor U or L from the Cholesky
*>          factorization A = U**T*U or A = L*L**T of the equilibrated
*>          matrix A (see the description of A for the form of the
*>          equilibrated matrix).
*> \endverbatim
*>
*> \param[in] LDAF
*> \verbatim
*>          LDAF is INTEGER
*>          The leading dimension of the array AF.  LDAF >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] EQUED
*> \verbatim
*>          EQUED is CHARACTER*1
*>          Specifies the form of equilibration that was done.
*>          = 'N':  No equilibration (always true if FACT = 'N').
*>          = 'Y':  Equilibration was done, i.e., A has been replaced by
*>                  diag(S) * A * diag(S).
*>          EQUED is an input argument if FACT = 'F'; otherwise, it is an
*>          output argument.
*> \endverbatim
*>
*> \param[in,out] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (N)
*>          The scale factors for A; not accessed if EQUED = 'N'.  S is
*>          an input argument if FACT = 'F'; otherwise, S is an output
*>          argument.  If FACT = 'F' and EQUED = 'Y', each element of S
*>          must be positive.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the N-by-NRHS right hand side matrix B.
*>          On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
*>          B is overwritten by diag(S) * B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
*>          the original system of equations.  Note that if EQUED = 'Y',
*>          A and B are modified on exit, and the solution to the
*>          equilibrated system is inv(diag(S))*X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The estimate of the reciprocal condition number of the matrix
*>          A after equilibration (if done).  If RCOND is less than the
*>          machine precision (in particular, if RCOND = 0), the matrix
*>          is singular to working precision.  This condition is
*>          indicated by a return code of INFO > 0.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The estimated forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).  The estimate is as reliable as
*>          the estimate for RCOND, and is almost always a slight
*>          overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in
*>          any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*>          > 0: if INFO = i, and i is
*>                <= N:  the leading principal minor of order i of A
*>                       is not positive, so the factorization could not
*>                       be completed, and the solution has not been
*>                       computed. RCOND = 0 is returned.
*>                = N+1: U is nonsingular, but RCOND is less than machine
*>                       precision, meaning that the matrix is singular
*>                       to working precision.  Nevertheless, the
*>                       solution and error bounds are computed because
*>                       there are a number of situations where the
*>                       computed solution can be more accurate than the
*>                       value of RCOND would suggest.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup posvx
*
*  =====================================================================
      SUBROUTINE DPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF,
     $                   EQUED,
     $                   S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
     $                   IWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          EQUED, FACT, UPLO
      INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
     $                   BERR( * ), FERR( * ), S( * ), WORK( * ),
     $                   X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            EQUIL, NOFACT, RCEQU
      INTEGER            I, INFEQU, J
      DOUBLE PRECISION   AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANSY
      EXTERNAL           LSAME, DLAMCH, DLANSY
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACPY, DLAQSY, DPOCON, DPOEQU, DPORFS,
     $                   DPOTRF,
     $                   DPOTRS, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      NOFACT = LSAME( FACT, 'N' )
      EQUIL = LSAME( FACT, 'E' )
      IF( NOFACT .OR. EQUIL ) THEN
         EQUED = 'N'
         RCEQU = .FALSE.
      ELSE
         RCEQU = LSAME( EQUED, 'Y' )
         SMLNUM = DLAMCH( 'Safe minimum' )
         BIGNUM = ONE / SMLNUM
      END IF
*
*     Test the input parameters.
*
      IF( .NOT.NOFACT .AND.
     $    .NOT.EQUIL .AND.
     $    .NOT.LSAME( FACT, 'F' ) )
     $     THEN
         INFO = -1
      ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND.
     $         .NOT.LSAME( UPLO, 'L' ) )
     $          THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -6
      ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
         INFO = -8
      ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
     $         ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
         INFO = -9
      ELSE
         IF( RCEQU ) THEN
            SMIN = BIGNUM
            SMAX = ZERO
            DO 10 J = 1, N
               SMIN = MIN( SMIN, S( J ) )
               SMAX = MAX( SMAX, S( J ) )
   10       CONTINUE
            IF( SMIN.LE.ZERO ) THEN
               INFO = -10
            ELSE IF( N.GT.0 ) THEN
               SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
            ELSE
               SCOND = ONE
            END IF
         END IF
         IF( INFO.EQ.0 ) THEN
            IF( LDB.LT.MAX( 1, N ) ) THEN
               INFO = -12
            ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
               INFO = -14
            END IF
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPOSVX', -INFO )
         RETURN
      END IF
*
      IF( EQUIL ) THEN
*
*        Compute row and column scalings to equilibrate the matrix A.
*
         CALL DPOEQU( N, A, LDA, S, SCOND, AMAX, INFEQU )
         IF( INFEQU.EQ.0 ) THEN
*
*           Equilibrate the matrix.
*
            CALL DLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
            RCEQU = LSAME( EQUED, 'Y' )
         END IF
      END IF
*
*     Scale the right hand side.
*
      IF( RCEQU ) THEN
         DO 30 J = 1, NRHS
            DO 20 I = 1, N
               B( I, J ) = S( I )*B( I, J )
   20       CONTINUE
   30    CONTINUE
      END IF
*
      IF( NOFACT .OR. EQUIL ) THEN
*
*        Compute the Cholesky factorization A = U**T *U or A = L*L**T.
*
         CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF )
         CALL DPOTRF( UPLO, N, AF, LDAF, INFO )
*
*        Return if INFO is non-zero.
*
         IF( INFO.GT.0 )THEN
            RCOND = ZERO
            RETURN
         END IF
      END IF
*
*     Compute the norm of the matrix A.
*
      ANORM = DLANSY( '1', UPLO, N, A, LDA, WORK )
*
*     Compute the reciprocal of the condition number of A.
*
      CALL DPOCON( UPLO, N, AF, LDAF, ANORM, RCOND, WORK, IWORK,
     $             INFO )
*
*     Compute the solution matrix X.
*
      CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
      CALL DPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO )
*
*     Use iterative refinement to improve the computed solution and
*     compute error bounds and backward error estimates for it.
*
      CALL DPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX,
     $             FERR, BERR, WORK, IWORK, INFO )
*
*     Transform the solution matrix X to a solution of the original
*     system.
*
      IF( RCEQU ) THEN
         DO 50 J = 1, NRHS
            DO 40 I = 1, N
               X( I, J ) = S( I )*X( I, J )
   40       CONTINUE
   50    CONTINUE
         DO 60 J = 1, NRHS
            FERR( J ) = FERR( J ) / SCOND
   60    CONTINUE
      END IF
*
*     Set INFO = N+1 if the matrix is singular to working precision.
*
      IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
     $   INFO = N + 1
*
      RETURN
*
*     End of DPOSVX
*
      END
*> \brief \b DPOTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPOTF2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpotf2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpotf2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpotf2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPOTF2( UPLO, N, A, LDA, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPOTF2 computes the Cholesky factorization of a real symmetric
*> positive definite matrix A.
*>
*> The factorization has the form
*>    A = U**T * U ,  if UPLO = 'U', or
*>    A = L  * L**T,  if UPLO = 'L',
*> where U is an upper triangular matrix and L is lower triangular.
*>
*> This is the unblocked version of the algorithm, calling Level 2 BLAS.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the upper or lower triangular part of the
*>          symmetric matrix A is stored.
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*>          n by n upper triangular part of A contains the upper
*>          triangular part of the matrix A, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading n by n lower triangular part of A contains the lower
*>          triangular part of the matrix A, and the strictly upper
*>          triangular part of A is not referenced.
*>
*>          On exit, if INFO = 0, the factor U or L from the Cholesky
*>          factorization A = U**T *U  or A = L*L**T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -k, the k-th argument had an illegal value
*>          > 0: if INFO = k, the leading principal minor of order k
*>               is not positive, and the factorization could not be
*>               completed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup potf2
*
*  =====================================================================
      SUBROUTINE DPOTF2( UPLO, N, A, LDA, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            J
      DOUBLE PRECISION   AJJ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      DOUBLE PRECISION   DDOT
      EXTERNAL           LSAME, DDOT, DISNAN
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMV, DSCAL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPOTF2', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( UPPER ) THEN
*
*        Compute the Cholesky factorization A = U**T *U.
*
         DO 10 J = 1, N
*
*           Compute U(J,J) and test for non-positive-definiteness.
*
            AJJ = A( J, J ) - DDOT( J-1, A( 1, J ), 1, A( 1, J ), 1 )
            IF( AJJ.LE.ZERO.OR.DISNAN( AJJ ) ) THEN
               A( J, J ) = AJJ
               GO TO 30
            END IF
            AJJ = SQRT( AJJ )
            A( J, J ) = AJJ
*
*           Compute elements J+1:N of row J.
*
            IF( J.LT.N ) THEN
               CALL DGEMV( 'Transpose', J-1, N-J, -ONE, A( 1, J+1 ),
     $                     LDA, A( 1, J ), 1, ONE, A( J, J+1 ), LDA )
               CALL DSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA )
            END IF
   10    CONTINUE
      ELSE
*
*        Compute the Cholesky factorization A = L*L**T.
*
         DO 20 J = 1, N
*
*           Compute L(J,J) and test for non-positive-definiteness.
*
            AJJ = A( J, J ) - DDOT( J-1, A( J, 1 ), LDA, A( J, 1 ),
     $            LDA )
            IF( AJJ.LE.ZERO.OR.DISNAN( AJJ ) ) THEN
               A( J, J ) = AJJ
               GO TO 30
            END IF
            AJJ = SQRT( AJJ )
            A( J, J ) = AJJ
*
*           Compute elements J+1:N of column J.
*
            IF( J.LT.N ) THEN
               CALL DGEMV( 'No transpose', N-J, J-1, -ONE, A( J+1,
     $                     1 ),
     $                     LDA, A( J, 1 ), LDA, ONE, A( J+1, J ), 1 )
               CALL DSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 )
            END IF
   20    CONTINUE
      END IF
      GO TO 40
*
   30 CONTINUE
      INFO = J
*
   40 CONTINUE
      RETURN
*
*     End of DPOTF2
*
      END
*> \brief \b DPOTRF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPOTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpotrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpotrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpotrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPOTRF( UPLO, N, A, LDA, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPOTRF computes the Cholesky factorization of a real symmetric
*> positive definite matrix A.
*>
*> The factorization has the form
*>    A = U**T * U,  if UPLO = 'U', or
*>    A = L  * L**T,  if UPLO = 'L',
*> where U is an upper triangular matrix and L is lower triangular.
*>
*> This is the block version of the algorithm, calling Level 3 BLAS.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*>          N-by-N upper triangular part of A contains the upper
*>          triangular part of the matrix A, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading N-by-N lower triangular part of A contains the lower
*>          triangular part of the matrix A, and the strictly upper
*>          triangular part of A is not referenced.
*>
*>          On exit, if INFO = 0, the factor U or L from the Cholesky
*>          factorization A = U**T*U or A = L*L**T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the leading principal minor of order i
*>                is not positive, and the factorization could not be
*>                completed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup potrf
*
*  =====================================================================
      SUBROUTINE DPOTRF( UPLO, N, A, LDA, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            J, JB, NB
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMM, DPOTRF2, DSYRK, DTRSM,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPOTRF', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Determine the block size for this environment.
*
      NB = ILAENV( 1, 'DPOTRF', UPLO, N, -1, -1, -1 )
      IF( NB.LE.1 .OR. NB.GE.N ) THEN
*
*        Use unblocked code.
*
         CALL DPOTRF2( UPLO, N, A, LDA, INFO )
      ELSE
*
*        Use blocked code.
*
         IF( UPPER ) THEN
*
*           Compute the Cholesky factorization A = U**T*U.
*
            DO 10 J = 1, N, NB
*
*              Update and factorize the current diagonal block and test
*              for non-positive-definiteness.
*
               JB = MIN( NB, N-J+1 )
               CALL DSYRK( 'Upper', 'Transpose', JB, J-1, -ONE,
     $                     A( 1, J ), LDA, ONE, A( J, J ), LDA )
               CALL DPOTRF2( 'Upper', JB, A( J, J ), LDA, INFO )
               IF( INFO.NE.0 )
     $            GO TO 30
               IF( J+JB.LE.N ) THEN
*
*                 Compute the current block row.
*
                  CALL DGEMM( 'Transpose', 'No transpose', JB,
     $                        N-J-JB+1,
     $                        J-1, -ONE, A( 1, J ), LDA, A( 1, J+JB ),
     $                        LDA, ONE, A( J, J+JB ), LDA )
                  CALL DTRSM( 'Left', 'Upper', 'Transpose',
     $                        'Non-unit',
     $                        JB, N-J-JB+1, ONE, A( J, J ), LDA,
     $                        A( J, J+JB ), LDA )
               END IF
   10       CONTINUE
*
         ELSE
*
*           Compute the Cholesky factorization A = L*L**T.
*
            DO 20 J = 1, N, NB
*
*              Update and factorize the current diagonal block and test
*              for non-positive-definiteness.
*
               JB = MIN( NB, N-J+1 )
               CALL DSYRK( 'Lower', 'No transpose', JB, J-1, -ONE,
     $                     A( J, 1 ), LDA, ONE, A( J, J ), LDA )
               CALL DPOTRF2( 'Lower', JB, A( J, J ), LDA, INFO )
               IF( INFO.NE.0 )
     $            GO TO 30
               IF( J+JB.LE.N ) THEN
*
*                 Compute the current block column.
*
                  CALL DGEMM( 'No transpose', 'Transpose', N-J-JB+1,
     $                        JB,
     $                        J-1, -ONE, A( J+JB, 1 ), LDA, A( J, 1 ),
     $                        LDA, ONE, A( J+JB, J ), LDA )
                  CALL DTRSM( 'Right', 'Lower', 'Transpose',
     $                        'Non-unit',
     $                        N-J-JB+1, JB, ONE, A( J, J ), LDA,
     $                        A( J+JB, J ), LDA )
               END IF
   20       CONTINUE
         END IF
      END IF
      GO TO 40
*
   30 CONTINUE
      INFO = INFO + J - 1
*
   40 CONTINUE
      RETURN
*
*     End of DPOTRF
*
      END
*> \brief \b DPOTRF2
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*  Definition:
*  ===========
*
*       RECURSIVE SUBROUTINE DPOTRF2( UPLO, N, A, LDA, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, N
*       ..
*       .. Array Arguments ..
*       REAL               A( LDA, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPOTRF2 computes the Cholesky factorization of a real symmetric
*> positive definite matrix A using the recursive algorithm.
*>
*> The factorization has the form
*>    A = U**T * U,  if UPLO = 'U', or
*>    A = L  * L**T,  if UPLO = 'L',
*> where U is an upper triangular matrix and L is lower triangular.
*>
*> This is the recursive version of the algorithm. It divides
*> the matrix into four submatrices:
*>
*>        [  A11 | A12  ]  where A11 is n1 by n1 and A22 is n2 by n2
*>    A = [ -----|----- ]  with n1 = n/2
*>        [  A21 | A22  ]       n2 = n-n1
*>
*> The subroutine calls itself to factor A11. Update and scale A21
*> or A12, update A22 then calls itself to factor A22.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*>          N-by-N upper triangular part of A contains the upper
*>          triangular part of the matrix A, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading N-by-N lower triangular part of A contains the lower
*>          triangular part of the matrix A, and the strictly upper
*>          triangular part of A is not referenced.
*>
*>          On exit, if INFO = 0, the factor U or L from the Cholesky
*>          factorization A = U**T*U or A = L*L**T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the leading principal minor of order i
*>                is not positive, and the factorization could not be
*>                completed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup potrf2
*
*  =====================================================================
      RECURSIVE SUBROUTINE DPOTRF2( UPLO, N, A, LDA, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            N1, N2, IINFO
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      EXTERNAL           LSAME, DISNAN
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSYRK, DTRSM, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPOTRF2', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     N=1 case
*
      IF( N.EQ.1 ) THEN
*
*        Test for non-positive-definiteness
*
         IF( A( 1, 1 ).LE.ZERO.OR.DISNAN( A( 1, 1 ) ) ) THEN
            INFO = 1
            RETURN
         END IF
*
*        Factor
*
         A( 1, 1 ) = SQRT( A( 1, 1 ) )
*
*     Use recursive code
*
      ELSE
         N1 = N/2
         N2 = N-N1
*
*        Factor A11
*
         CALL DPOTRF2( UPLO, N1, A( 1, 1 ), LDA, IINFO )
         IF ( IINFO.NE.0 ) THEN
            INFO = IINFO
            RETURN
         END IF
*
*        Compute the Cholesky factorization A = U**T*U
*
         IF( UPPER ) THEN
*
*           Update and scale A12
*
            CALL DTRSM( 'L', 'U', 'T', 'N', N1, N2, ONE,
     $                  A( 1, 1 ), LDA, A( 1, N1+1 ), LDA )
*
*           Update and factor A22
*
            CALL DSYRK( UPLO, 'T', N2, N1, -ONE, A( 1, N1+1 ), LDA,
     $                  ONE, A( N1+1, N1+1 ), LDA )
            CALL DPOTRF2( UPLO, N2, A( N1+1, N1+1 ), LDA, IINFO )
            IF ( IINFO.NE.0 ) THEN
               INFO = IINFO + N1
               RETURN
            END IF
*
*        Compute the Cholesky factorization A = L*L**T
*
         ELSE
*
*           Update and scale A21
*
            CALL DTRSM( 'R', 'L', 'T', 'N', N2, N1, ONE,
     $                  A( 1, 1 ), LDA, A( N1+1, 1 ), LDA )
*
*           Update and factor A22
*
            CALL DSYRK( UPLO, 'N', N2, N1, -ONE, A( N1+1, 1 ), LDA,
     $                  ONE, A( N1+1, N1+1 ), LDA )
            CALL DPOTRF2( UPLO, N2, A( N1+1, N1+1 ), LDA, IINFO )
            IF ( IINFO.NE.0 ) THEN
               INFO = IINFO + N1
               RETURN
            END IF
         END IF
      END IF
      RETURN
*
*     End of DPOTRF2
*
      END
*> \brief \b DPOTRI
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPOTRI + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpotri.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpotri.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpotri.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPOTRI( UPLO, N, A, LDA, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPOTRI computes the inverse of a real symmetric positive definite
*> matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
*> computed by DPOTRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the triangular factor U or L from the Cholesky
*>          factorization A = U**T*U or A = L*L**T, as computed by
*>          DPOTRF.
*>          On exit, the upper or lower triangle of the (symmetric)
*>          inverse of A, overwriting the input factor U or L.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the (i,i) element of the factor U or L is
*>                zero, and the inverse could not be computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup potri
*
*  =====================================================================
      SUBROUTINE DPOTRI( UPLO, N, A, LDA, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  =====================================================================
*
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLAUUM, DTRTRI, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF( .NOT.LSAME( UPLO, 'U' ) .AND.
     $    .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPOTRI', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Invert the triangular Cholesky factor U or L.
*
      CALL DTRTRI( UPLO, 'Non-unit', N, A, LDA, INFO )
      IF( INFO.GT.0 )
     $   RETURN
*
*     Form inv(U) * inv(U)**T or inv(L)**T * inv(L).
*
      CALL DLAUUM( UPLO, N, A, LDA, INFO )
*
      RETURN
*
*     End of DPOTRI
*
      END
*> \brief \b DPOTRS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPOTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpotrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpotrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpotrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPOTRS( UPLO, N, NRHS, A, LDA, B, LDB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPOTRS solves a system of linear equations A*X = B with a symmetric
*> positive definite matrix A using the Cholesky factorization
*> A = U**T*U or A = L*L**T computed by DPOTRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The triangular factor U or L from the Cholesky factorization
*>          A = U**T*U or A = L*L**T, as computed by DPOTRF.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the right hand side matrix B.
*>          On exit, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup potrs
*
*  =====================================================================
      SUBROUTINE DPOTRS( UPLO, N, NRHS, A, LDA, B, LDB, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DTRSM, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPOTRS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 )
     $   RETURN
*
      IF( UPPER ) THEN
*
*        Solve A*X = B where A = U**T *U.
*
*        Solve U**T *X = B, overwriting B with X.
*
         CALL DTRSM( 'Left', 'Upper', 'Transpose', 'Non-unit', N,
     $               NRHS,
     $               ONE, A, LDA, B, LDB )
*
*        Solve U*X = B, overwriting B with X.
*
         CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', N,
     $               NRHS, ONE, A, LDA, B, LDB )
      ELSE
*
*        Solve A*X = B where A = L*L**T.
*
*        Solve L*X = B, overwriting B with X.
*
         CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Non-unit', N,
     $               NRHS, ONE, A, LDA, B, LDB )
*
*        Solve L**T *X = B, overwriting B with X.
*
         CALL DTRSM( 'Left', 'Lower', 'Transpose', 'Non-unit', N,
     $               NRHS,
     $               ONE, A, LDA, B, LDB )
      END IF
*
      RETURN
*
*     End of DPOTRS
*
      END
*> \brief \b DPPCON
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPPCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dppcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dppcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dppcon.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPPCON( UPLO, N, AP, ANORM, RCOND, WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, N
*       DOUBLE PRECISION   ANORM, RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   AP( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPPCON estimates the reciprocal of the condition number (in the
*> 1-norm) of a real symmetric positive definite packed matrix using
*> the Cholesky factorization A = U**T*U or A = L*L**T computed by
*> DPPTRF.
*>
*> An estimate is obtained for norm(inv(A)), and the reciprocal of the
*> condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          The triangular factor U or L from the Cholesky factorization
*>          A = U**T*U or A = L*L**T, packed columnwise in a linear
*>          array.  The j-th column of U or L is stored in the array AP
*>          as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*>          ANORM is DOUBLE PRECISION
*>          The 1-norm (or infinity-norm) of the symmetric matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The reciprocal of the condition number of the matrix A,
*>          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
*>          estimate of the 1-norm of inv(A) computed in this routine.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ppcon
*
*  =====================================================================
      SUBROUTINE DPPCON( UPLO, N, AP, ANORM, RCOND, WORK, IWORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, N
      DOUBLE PRECISION   ANORM, RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   AP( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      CHARACTER          NORMIN
      INTEGER            IX, KASE
      DOUBLE PRECISION   AINVNM, SCALE, SCALEL, SCALEU, SMLNUM
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, IDAMAX, DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACN2, DLATPS, DRSCL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( ANORM.LT.ZERO ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPPCON', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      RCOND = ZERO
      IF( N.EQ.0 ) THEN
         RCOND = ONE
         RETURN
      ELSE IF( ANORM.EQ.ZERO ) THEN
         RETURN
      END IF
*
      SMLNUM = DLAMCH( 'Safe minimum' )
*
*     Estimate the 1-norm of the inverse.
*
      KASE = 0
      NORMIN = 'N'
   10 CONTINUE
      CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
      IF( KASE.NE.0 ) THEN
         IF( UPPER ) THEN
*
*           Multiply by inv(U**T).
*
            CALL DLATPS( 'Upper', 'Transpose', 'Non-unit', NORMIN, N,
     $                   AP, WORK, SCALEL, WORK( 2*N+1 ), INFO )
            NORMIN = 'Y'
*
*           Multiply by inv(U).
*
            CALL DLATPS( 'Upper', 'No transpose', 'Non-unit', NORMIN,
     $                   N,
     $                   AP, WORK, SCALEU, WORK( 2*N+1 ), INFO )
         ELSE
*
*           Multiply by inv(L).
*
            CALL DLATPS( 'Lower', 'No transpose', 'Non-unit', NORMIN,
     $                   N,
     $                   AP, WORK, SCALEL, WORK( 2*N+1 ), INFO )
            NORMIN = 'Y'
*
*           Multiply by inv(L**T).
*
            CALL DLATPS( 'Lower', 'Transpose', 'Non-unit', NORMIN, N,
     $                   AP, WORK, SCALEU, WORK( 2*N+1 ), INFO )
         END IF
*
*        Multiply by 1/SCALE if doing so will not cause overflow.
*
         SCALE = SCALEL*SCALEU
         IF( SCALE.NE.ONE ) THEN
            IX = IDAMAX( N, WORK, 1 )
            IF( SCALE.LT.ABS( WORK( IX ) )*SMLNUM .OR. SCALE.EQ.ZERO )
     $         GO TO 20
            CALL DRSCL( N, SCALE, WORK, 1 )
         END IF
         GO TO 10
      END IF
*
*     Compute the estimate of the reciprocal condition number.
*
      IF( AINVNM.NE.ZERO )
     $   RCOND = ( ONE / AINVNM ) / ANORM
*
   20 CONTINUE
      RETURN
*
*     End of DPPCON
*
      END
*> \brief \b DPPEQU
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPPEQU + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dppequ.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dppequ.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dppequ.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, N
*       DOUBLE PRECISION   AMAX, SCOND
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AP( * ), S( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPPEQU computes row and column scalings intended to equilibrate a
*> symmetric positive definite matrix A in packed storage and reduce
*> its condition number (with respect to the two-norm).  S contains the
*> scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix
*> B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
*> This choice of S puts the condition number of B within a factor N of
*> the smallest possible condition number over all possible diagonal
*> scalings.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          The upper or lower triangle of the symmetric matrix A, packed
*>          columnwise in a linear array.  The j-th column of A is stored
*>          in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, S contains the scale factors for A.
*> \endverbatim
*>
*> \param[out] SCOND
*> \verbatim
*>          SCOND is DOUBLE PRECISION
*>          If INFO = 0, S contains the ratio of the smallest S(i) to
*>          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
*>          large nor too small, it is not worth scaling by S.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*>          AMAX is DOUBLE PRECISION
*>          Absolute value of largest matrix element.  If AMAX is very
*>          close to overflow or very close to underflow, the matrix
*>          should be scaled.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ppequ
*
*  =====================================================================
      SUBROUTINE DPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, N
      DOUBLE PRECISION   AMAX, SCOND
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( * ), S( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I, JJ
      DOUBLE PRECISION   SMIN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPPEQU', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 ) THEN
         SCOND = ONE
         AMAX = ZERO
         RETURN
      END IF
*
*     Initialize SMIN and AMAX.
*
      S( 1 ) = AP( 1 )
      SMIN = S( 1 )
      AMAX = S( 1 )
*
      IF( UPPER ) THEN
*
*        UPLO = 'U':  Upper triangle of A is stored.
*        Find the minimum and maximum diagonal elements.
*
         JJ = 1
         DO 10 I = 2, N
            JJ = JJ + I
            S( I ) = AP( JJ )
            SMIN = MIN( SMIN, S( I ) )
            AMAX = MAX( AMAX, S( I ) )
   10    CONTINUE
*
      ELSE
*
*        UPLO = 'L':  Lower triangle of A is stored.
*        Find the minimum and maximum diagonal elements.
*
         JJ = 1
         DO 20 I = 2, N
            JJ = JJ + N - I + 2
            S( I ) = AP( JJ )
            SMIN = MIN( SMIN, S( I ) )
            AMAX = MAX( AMAX, S( I ) )
   20    CONTINUE
      END IF
*
      IF( SMIN.LE.ZERO ) THEN
*
*        Find the first non-positive diagonal element and return.
*
         DO 30 I = 1, N
            IF( S( I ).LE.ZERO ) THEN
               INFO = I
               RETURN
            END IF
   30    CONTINUE
      ELSE
*
*        Set the scale factors to the reciprocals
*        of the diagonal elements.
*
         DO 40 I = 1, N
            S( I ) = ONE / SQRT( S( I ) )
   40    CONTINUE
*
*        Compute SCOND = min(S(I)) / max(S(I))
*
         SCOND = SQRT( SMIN ) / SQRT( AMAX )
      END IF
      RETURN
*
*     End of DPPEQU
*
      END
*> \brief \b DPPRFS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPPRFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpprfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpprfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpprfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
*                          BERR, WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDB, LDX, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
*      $                   FERR( * ), WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPPRFS improves the computed solution to a system of linear
*> equations when the coefficient matrix is symmetric positive definite
*> and packed, and provides error bounds and backward error estimates
*> for the solution.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices B and X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          The upper or lower triangle of the symmetric matrix A, packed
*>          columnwise in a linear array.  The j-th column of A is stored
*>          in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[in] AFP
*> \verbatim
*>          AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          The triangular factor U or L from the Cholesky factorization
*>          A = U**T*U or A = L*L**T, as computed by DPPTRF/ZPPTRF,
*>          packed columnwise in a linear array in the same format as A
*>          (see AP).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          On entry, the solution matrix X, as computed by DPPTRS.
*>          On exit, the improved solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The estimated forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).  The estimate is as reliable as
*>          the estimate for RCOND, and is almost always a slight
*>          overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in
*>          any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  ITMAX is the maximum number of steps of iterative refinement.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pprfs
*
*  =====================================================================
      SUBROUTINE DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX,
     $                   FERR,
     $                   BERR, WORK, IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDB, LDX, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
     $                   FERR( * ), WORK( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            ITMAX
      PARAMETER          ( ITMAX = 5 )
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
      DOUBLE PRECISION   TWO
      PARAMETER          ( TWO = 2.0D+0 )
      DOUBLE PRECISION   THREE
      PARAMETER          ( THREE = 3.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            COUNT, I, IK, J, K, KASE, KK, NZ
      DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DLACN2, DPPTRS, DSPMV,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
         INFO = -9
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPPRFS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
         DO 10 J = 1, NRHS
            FERR( J ) = ZERO
            BERR( J ) = ZERO
   10    CONTINUE
         RETURN
      END IF
*
*     NZ = maximum number of nonzero elements in each row of A, plus 1
*
      NZ = N + 1
      EPS = DLAMCH( 'Epsilon' )
      SAFMIN = DLAMCH( 'Safe minimum' )
      SAFE1 = NZ*SAFMIN
      SAFE2 = SAFE1 / EPS
*
*     Do for each right hand side
*
      DO 140 J = 1, NRHS
*
         COUNT = 1
         LSTRES = THREE
   20    CONTINUE
*
*        Loop until stopping criterion is satisfied.
*
*        Compute residual R = B - A * X
*
         CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
         CALL DSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE,
     $               WORK( N+1 ),
     $               1 )
*
*        Compute componentwise relative backward error from formula
*
*        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
*
*        where abs(Z) is the componentwise absolute value of the matrix
*        or vector Z.  If the i-th component of the denominator is less
*        than SAFE2, then SAFE1 is added to the i-th components of the
*        numerator and denominator before dividing.
*
         DO 30 I = 1, N
            WORK( I ) = ABS( B( I, J ) )
   30    CONTINUE
*
*        Compute abs(A)*abs(X) + abs(B).
*
         KK = 1
         IF( UPPER ) THEN
            DO 50 K = 1, N
               S = ZERO
               XK = ABS( X( K, J ) )
               IK = KK
               DO 40 I = 1, K - 1
                  WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
                  S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
                  IK = IK + 1
   40          CONTINUE
               WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S
               KK = KK + K
   50       CONTINUE
         ELSE
            DO 70 K = 1, N
               S = ZERO
               XK = ABS( X( K, J ) )
               WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK
               IK = KK + 1
               DO 60 I = K + 1, N
                  WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
                  S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
                  IK = IK + 1
   60          CONTINUE
               WORK( K ) = WORK( K ) + S
               KK = KK + ( N-K+1 )
   70       CONTINUE
         END IF
         S = ZERO
         DO 80 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
            ELSE
               S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
     $             ( WORK( I )+SAFE1 ) )
            END IF
   80    CONTINUE
         BERR( J ) = S
*
*        Test stopping criterion. Continue iterating if
*           1) The residual BERR(J) is larger than machine epsilon, and
*           2) BERR(J) decreased by at least a factor of 2 during the
*              last iteration, and
*           3) At most ITMAX iterations tried.
*
         IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
     $       COUNT.LE.ITMAX ) THEN
*
*           Update solution and try again.
*
            CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO )
            CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
            LSTRES = BERR( J )
            COUNT = COUNT + 1
            GO TO 20
         END IF
*
*        Bound error from formula
*
*        norm(X - XTRUE) / norm(X) .le. FERR =
*        norm( abs(inv(A))*
*           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
*
*        where
*          norm(Z) is the magnitude of the largest component of Z
*          inv(A) is the inverse of A
*          abs(Z) is the componentwise absolute value of the matrix or
*             vector Z
*          NZ is the maximum number of nonzeros in any row of A, plus 1
*          EPS is machine epsilon
*
*        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
*        is incremented by SAFE1 if the i-th component of
*        abs(A)*abs(X) + abs(B) is less than SAFE2.
*
*        Use DLACN2 to estimate the infinity-norm of the matrix
*           inv(A) * diag(W),
*        where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
*
         DO 90 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
            ELSE
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
            END IF
   90    CONTINUE
*
         KASE = 0
  100    CONTINUE
         CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
     $                FERR( J ),
     $                KASE, ISAVE )
         IF( KASE.NE.0 ) THEN
            IF( KASE.EQ.1 ) THEN
*
*              Multiply by diag(W)*inv(A**T).
*
               CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO )
               DO 110 I = 1, N
                  WORK( N+I ) = WORK( I )*WORK( N+I )
  110          CONTINUE
            ELSE IF( KASE.EQ.2 ) THEN
*
*              Multiply by inv(A)*diag(W).
*
               DO 120 I = 1, N
                  WORK( N+I ) = WORK( I )*WORK( N+I )
  120          CONTINUE
               CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO )
            END IF
            GO TO 100
         END IF
*
*        Normalize error.
*
         LSTRES = ZERO
         DO 130 I = 1, N
            LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
  130    CONTINUE
         IF( LSTRES.NE.ZERO )
     $      FERR( J ) = FERR( J ) / LSTRES
*
  140 CONTINUE
*
      RETURN
*
*     End of DPPRFS
*
      END
*> \brief <b> DPPSV computes the solution to system of linear equations A * X = B for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPPSV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dppsv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dppsv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dppsv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPPSV( UPLO, N, NRHS, AP, B, LDB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AP( * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPPSV computes the solution to a real system of linear equations
*>    A * X = B,
*> where A is an N-by-N symmetric positive definite matrix stored in
*> packed format and X and B are N-by-NRHS matrices.
*>
*> The Cholesky decomposition is used to factor A as
*>    A = U**T* U,  if UPLO = 'U', or
*>    A = L * L**T,  if UPLO = 'L',
*> where U is an upper triangular matrix and L is a lower triangular
*> matrix.  The factored form of A is then used to solve the system of
*> equations A * X = B.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of linear equations, i.e., the order of the
*>          matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the symmetric matrix
*>          A, packed columnwise in a linear array.  The j-th column of A
*>          is stored in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*>          See below for further details.
*>
*>          On exit, if INFO = 0, the factor U or L from the Cholesky
*>          factorization A = U**T*U or A = L*L**T, in the same storage
*>          format as A.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the N-by-NRHS right hand side matrix B.
*>          On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the leading principal minor of order i
*>                of A is not positive, so the factorization could not
*>                be completed, and the solution has not been computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ppsv
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The packed storage scheme is illustrated by the following example
*>  when N = 4, UPLO = 'U':
*>
*>  Two-dimensional storage of the symmetric matrix A:
*>
*>     a11 a12 a13 a14
*>         a22 a23 a24
*>             a33 a34     (aij = conjg(aji))
*>                 a44
*>
*>  Packed storage of the upper triangle of A:
*>
*>  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DPPSV( UPLO, N, NRHS, AP, B, LDB, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DPPTRF, DPPTRS, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF( .NOT.LSAME( UPLO, 'U' ) .AND.
     $    .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPPSV ', -INFO )
         RETURN
      END IF
*
*     Compute the Cholesky factorization A = U**T*U or A = L*L**T.
*
      CALL DPPTRF( UPLO, N, AP, INFO )
      IF( INFO.EQ.0 ) THEN
*
*        Solve the system A*X = B, overwriting B with X.
*
         CALL DPPTRS( UPLO, N, NRHS, AP, B, LDB, INFO )
*
      END IF
      RETURN
*
*     End of DPPSV
*
      END
*> \brief <b> DPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPPSVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dppsvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dppsvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dppsvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB,
*                          X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          EQUED, FACT, UPLO
*       INTEGER            INFO, LDB, LDX, N, NRHS
*       DOUBLE PRECISION   RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
*      $                   FERR( * ), S( * ), WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
*> compute the solution to a real system of linear equations
*>    A * X = B,
*> where A is an N-by-N symmetric positive definite matrix stored in
*> packed format and X and B are N-by-NRHS matrices.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
*  =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
*>    the system:
*>       diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
*>    Whether or not the system will be equilibrated depends on the
*>    scaling of the matrix A, but if equilibration is used, A is
*>    overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
*>
*> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
*>    factor the matrix A (after equilibration if FACT = 'E') as
*>       A = U**T* U,  if UPLO = 'U', or
*>       A = L * L**T,  if UPLO = 'L',
*>    where U is an upper triangular matrix and L is a lower triangular
*>    matrix.
*>
*> 3. If the leading principal minor of order i is not positive,
*>    then the routine returns with INFO = i. Otherwise, the factored
*>    form of A is used to estimate the condition number of the matrix
*>    A.  If the reciprocal of the condition number is less than machine
*>    precision, INFO = N+1 is returned as a warning, but the routine
*>    still goes on to solve for X and compute error bounds as
*>    described below.
*>
*> 4. The system of equations is solved for X using the factored form
*>    of A.
*>
*> 5. Iterative refinement is applied to improve the computed solution
*>    matrix and calculate error bounds and backward error estimates
*>    for it.
*>
*> 6. If equilibration was used, the matrix X is premultiplied by
*>    diag(S) so that it solves the original system before
*>    equilibration.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] FACT
*> \verbatim
*>          FACT is CHARACTER*1
*>          Specifies whether or not the factored form of the matrix A is
*>          supplied on entry, and if not, whether the matrix A should be
*>          equilibrated before it is factored.
*>          = 'F':  On entry, AFP contains the factored form of A.
*>                  If EQUED = 'Y', the matrix A has been equilibrated
*>                  with scaling factors given by S.  AP and AFP will not
*>                  be modified.
*>          = 'N':  The matrix A will be copied to AFP and factored.
*>          = 'E':  The matrix A will be equilibrated if necessary, then
*>                  copied to AFP and factored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of linear equations, i.e., the order of the
*>          matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices B and X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the symmetric matrix
*>          A, packed columnwise in a linear array, except if FACT = 'F'
*>          and EQUED = 'Y', then A must contain the equilibrated matrix
*>          diag(S)*A*diag(S).  The j-th column of A is stored in the
*>          array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*>          See below for further details.  A is not modified if
*>          FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
*>
*>          On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
*>          diag(S)*A*diag(S).
*> \endverbatim
*>
*> \param[in,out] AFP
*> \verbatim
*>          AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          If FACT = 'F', then AFP is an input argument and on entry
*>          contains the triangular factor U or L from the Cholesky
*>          factorization A = U**T*U or A = L*L**T, in the same storage
*>          format as A.  If EQUED .ne. 'N', then AFP is the factored
*>          form of the equilibrated matrix A.
*>
*>          If FACT = 'N', then AFP is an output argument and on exit
*>          returns the triangular factor U or L from the Cholesky
*>          factorization A = U**T * U or A = L * L**T of the original
*>          matrix A.
*>
*>          If FACT = 'E', then AFP is an output argument and on exit
*>          returns the triangular factor U or L from the Cholesky
*>          factorization A = U**T * U or A = L * L**T of the equilibrated
*>          matrix A (see the description of AP for the form of the
*>          equilibrated matrix).
*> \endverbatim
*>
*> \param[in,out] EQUED
*> \verbatim
*>          EQUED is CHARACTER*1
*>          Specifies the form of equilibration that was done.
*>          = 'N':  No equilibration (always true if FACT = 'N').
*>          = 'Y':  Equilibration was done, i.e., A has been replaced by
*>                  diag(S) * A * diag(S).
*>          EQUED is an input argument if FACT = 'F'; otherwise, it is an
*>          output argument.
*> \endverbatim
*>
*> \param[in,out] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (N)
*>          The scale factors for A; not accessed if EQUED = 'N'.  S is
*>          an input argument if FACT = 'F'; otherwise, S is an output
*>          argument.  If FACT = 'F' and EQUED = 'Y', each element of S
*>          must be positive.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the N-by-NRHS right hand side matrix B.
*>          On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
*>          B is overwritten by diag(S) * B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
*>          the original system of equations.  Note that if EQUED = 'Y',
*>          A and B are modified on exit, and the solution to the
*>          equilibrated system is inv(diag(S))*X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The estimate of the reciprocal condition number of the matrix
*>          A after equilibration (if done).  If RCOND is less than the
*>          machine precision (in particular, if RCOND = 0), the matrix
*>          is singular to working precision.  This condition is
*>          indicated by a return code of INFO > 0.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The estimated forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).  The estimate is as reliable as
*>          the estimate for RCOND, and is almost always a slight
*>          overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in
*>          any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, and i is
*>                <= N:  the leading principal minor of order i of A
*>                       is not positive, so the factorization could not
*>                       be completed, and the solution has not been
*>                       computed. RCOND = 0 is returned.
*>                = N+1: U is nonsingular, but RCOND is less than machine
*>                       precision, meaning that the matrix is singular
*>                       to working precision.  Nevertheless, the
*>                       solution and error bounds are computed because
*>                       there are a number of situations where the
*>                       computed solution can be more accurate than the
*>                       value of RCOND would suggest.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ppsvx
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The packed storage scheme is illustrated by the following example
*>  when N = 4, UPLO = 'U':
*>
*>  Two-dimensional storage of the symmetric matrix A:
*>
*>     a11 a12 a13 a14
*>         a22 a23 a24
*>             a33 a34     (aij = conjg(aji))
*>                 a44
*>
*>  Packed storage of the upper triangle of A:
*>
*>  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B,
     $                   LDB,
     $                   X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          EQUED, FACT, UPLO
      INTEGER            INFO, LDB, LDX, N, NRHS
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
     $                   FERR( * ), S( * ), WORK( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            EQUIL, NOFACT, RCEQU
      INTEGER            I, INFEQU, J
      DOUBLE PRECISION   AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANSP
      EXTERNAL           LSAME, DLAMCH, DLANSP
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLACPY, DLAQSP, DPPCON, DPPEQU,
     $                   DPPRFS,
     $                   DPPTRF, DPPTRS, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      NOFACT = LSAME( FACT, 'N' )
      EQUIL = LSAME( FACT, 'E' )
      IF( NOFACT .OR. EQUIL ) THEN
         EQUED = 'N'
         RCEQU = .FALSE.
      ELSE
         RCEQU = LSAME( EQUED, 'Y' )
         SMLNUM = DLAMCH( 'Safe minimum' )
         BIGNUM = ONE / SMLNUM
      END IF
*
*     Test the input parameters.
*
      IF( .NOT.NOFACT .AND.
     $    .NOT.EQUIL .AND.
     $    .NOT.LSAME( FACT, 'F' ) )
     $     THEN
         INFO = -1
      ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND.
     $         .NOT.LSAME( UPLO, 'L' ) )
     $          THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -4
      ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
     $         ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
         INFO = -7
      ELSE
         IF( RCEQU ) THEN
            SMIN = BIGNUM
            SMAX = ZERO
            DO 10 J = 1, N
               SMIN = MIN( SMIN, S( J ) )
               SMAX = MAX( SMAX, S( J ) )
   10       CONTINUE
            IF( SMIN.LE.ZERO ) THEN
               INFO = -8
            ELSE IF( N.GT.0 ) THEN
               SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
            ELSE
               SCOND = ONE
            END IF
         END IF
         IF( INFO.EQ.0 ) THEN
            IF( LDB.LT.MAX( 1, N ) ) THEN
               INFO = -10
            ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
               INFO = -12
            END IF
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPPSVX', -INFO )
         RETURN
      END IF
*
      IF( EQUIL ) THEN
*
*        Compute row and column scalings to equilibrate the matrix A.
*
         CALL DPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFEQU )
         IF( INFEQU.EQ.0 ) THEN
*
*           Equilibrate the matrix.
*
            CALL DLAQSP( UPLO, N, AP, S, SCOND, AMAX, EQUED )
            RCEQU = LSAME( EQUED, 'Y' )
         END IF
      END IF
*
*     Scale the right-hand side.
*
      IF( RCEQU ) THEN
         DO 30 J = 1, NRHS
            DO 20 I = 1, N
               B( I, J ) = S( I )*B( I, J )
   20       CONTINUE
   30    CONTINUE
      END IF
*
      IF( NOFACT .OR. EQUIL ) THEN
*
*        Compute the Cholesky factorization A = U**T * U or A = L * L**T.
*
         CALL DCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 )
         CALL DPPTRF( UPLO, N, AFP, INFO )
*
*        Return if INFO is non-zero.
*
         IF( INFO.GT.0 )THEN
            RCOND = ZERO
            RETURN
         END IF
      END IF
*
*     Compute the norm of the matrix A.
*
      ANORM = DLANSP( 'I', UPLO, N, AP, WORK )
*
*     Compute the reciprocal of the condition number of A.
*
      CALL DPPCON( UPLO, N, AFP, ANORM, RCOND, WORK, IWORK, INFO )
*
*     Compute the solution matrix X.
*
      CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
      CALL DPPTRS( UPLO, N, NRHS, AFP, X, LDX, INFO )
*
*     Use iterative refinement to improve the computed solution and
*     compute error bounds and backward error estimates for it.
*
      CALL DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
     $             BERR,
     $             WORK, IWORK, INFO )
*
*     Transform the solution matrix X to a solution of the original
*     system.
*
      IF( RCEQU ) THEN
         DO 50 J = 1, NRHS
            DO 40 I = 1, N
               X( I, J ) = S( I )*X( I, J )
   40       CONTINUE
   50    CONTINUE
         DO 60 J = 1, NRHS
            FERR( J ) = FERR( J ) / SCOND
   60    CONTINUE
      END IF
*
*     Set INFO = N+1 if the matrix is singular to working precision.
*
      IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
     $   INFO = N + 1
*
      RETURN
*
*     End of DPPSVX
*
      END
*> \brief \b DPPTRF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPPTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpptrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpptrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpptrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPPTRF( UPLO, N, AP, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AP( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPPTRF computes the Cholesky factorization of a real symmetric
*> positive definite matrix A stored in packed format.
*>
*> The factorization has the form
*>    A = U**T * U,  if UPLO = 'U', or
*>    A = L  * L**T,  if UPLO = 'L',
*> where U is an upper triangular matrix and L is lower triangular.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the symmetric matrix
*>          A, packed columnwise in a linear array.  The j-th column of A
*>          is stored in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*>          See below for further details.
*>
*>          On exit, if INFO = 0, the triangular factor U or L from the
*>          Cholesky factorization A = U**T*U or A = L*L**T, in the same
*>          storage format as A.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the leading principal minor of order i
*>                is not positive, and the factorization could not be
*>                completed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pptrf
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The packed storage scheme is illustrated by the following example
*>  when N = 4, UPLO = 'U':
*>
*>  Two-dimensional storage of the symmetric matrix A:
*>
*>     a11 a12 a13 a14
*>         a22 a23 a24
*>             a33 a34     (aij = aji)
*>                 a44
*>
*>  Packed storage of the upper triangle of A:
*>
*>  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DPPTRF( UPLO, N, AP, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            J, JC, JJ
      DOUBLE PRECISION   AJJ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DDOT
      EXTERNAL           LSAME, DDOT
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL, DSPR, DTPSV, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPPTRF', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( UPPER ) THEN
*
*        Compute the Cholesky factorization A = U**T*U.
*
         JJ = 0
         DO 10 J = 1, N
            JC = JJ + 1
            JJ = JJ + J
*
*           Compute elements 1:J-1 of column J.
*
            IF( J.GT.1 )
     $         CALL DTPSV( 'Upper', 'Transpose', 'Non-unit', J-1, AP,
     $                     AP( JC ), 1 )
*
*           Compute U(J,J) and test for non-positive-definiteness.
*
            AJJ = AP( JJ ) - DDOT( J-1, AP( JC ), 1, AP( JC ), 1 )
            IF( AJJ.LE.ZERO ) THEN
               AP( JJ ) = AJJ
               GO TO 30
            END IF
            AP( JJ ) = SQRT( AJJ )
   10    CONTINUE
      ELSE
*
*        Compute the Cholesky factorization A = L*L**T.
*
         JJ = 1
         DO 20 J = 1, N
*
*           Compute L(J,J) and test for non-positive-definiteness.
*
            AJJ = AP( JJ )
            IF( AJJ.LE.ZERO ) THEN
               AP( JJ ) = AJJ
               GO TO 30
            END IF
            AJJ = SQRT( AJJ )
            AP( JJ ) = AJJ
*
*           Compute elements J+1:N of column J and update the trailing
*           submatrix.
*
            IF( J.LT.N ) THEN
               CALL DSCAL( N-J, ONE / AJJ, AP( JJ+1 ), 1 )
               CALL DSPR( 'Lower', N-J, -ONE, AP( JJ+1 ), 1,
     $                    AP( JJ+N-J+1 ) )
               JJ = JJ + N - J + 1
            END IF
   20    CONTINUE
      END IF
      GO TO 40
*
   30 CONTINUE
      INFO = J
*
   40 CONTINUE
      RETURN
*
*     End of DPPTRF
*
      END
*> \brief \b DPPTRI
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPPTRI + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpptri.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpptri.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpptri.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPPTRI( UPLO, N, AP, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AP( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPPTRI computes the inverse of a real symmetric positive definite
*> matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
*> computed by DPPTRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangular factor is stored in AP;
*>          = 'L':  Lower triangular factor is stored in AP.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the triangular factor U or L from the Cholesky
*>          factorization A = U**T*U or A = L*L**T, packed columnwise as
*>          a linear array.  The j-th column of U or L is stored in the
*>          array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.
*>
*>          On exit, the upper or lower triangle of the (symmetric)
*>          inverse of A, overwriting the input factor U or L.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the (i,i) element of the factor U or L is
*>                zero, and the inverse could not be computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pptri
*
*  =====================================================================
      SUBROUTINE DPPTRI( UPLO, N, AP, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            J, JC, JJ, JJN
      DOUBLE PRECISION   AJJ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DDOT
      EXTERNAL           LSAME, DDOT
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL, DSPR, DTPMV, DTPTRI, XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPPTRI', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Invert the triangular Cholesky factor U or L.
*
      CALL DTPTRI( UPLO, 'Non-unit', N, AP, INFO )
      IF( INFO.GT.0 )
     $   RETURN
*
      IF( UPPER ) THEN
*
*        Compute the product inv(U) * inv(U)**T.
*
         JJ = 0
         DO 10 J = 1, N
            JC = JJ + 1
            JJ = JJ + J
            IF( J.GT.1 )
     $         CALL DSPR( 'Upper', J-1, ONE, AP( JC ), 1, AP )
            AJJ = AP( JJ )
            CALL DSCAL( J, AJJ, AP( JC ), 1 )
   10    CONTINUE
*
      ELSE
*
*        Compute the product inv(L)**T * inv(L).
*
         JJ = 1
         DO 20 J = 1, N
            JJN = JJ + N - J + 1
            AP( JJ ) = DDOT( N-J+1, AP( JJ ), 1, AP( JJ ), 1 )
            IF( J.LT.N )
     $         CALL DTPMV( 'Lower', 'Transpose', 'Non-unit', N-J,
     $                     AP( JJN ), AP( JJ+1 ), 1 )
            JJ = JJN
   20    CONTINUE
      END IF
*
      RETURN
*
*     End of DPPTRI
*
      END
*> \brief \b DPPTRS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPPTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpptrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpptrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpptrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPPTRS( UPLO, N, NRHS, AP, B, LDB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AP( * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPPTRS solves a system of linear equations A*X = B with a symmetric
*> positive definite matrix A in packed storage using the Cholesky
*> factorization A = U**T*U or A = L*L**T computed by DPPTRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          The triangular factor U or L from the Cholesky factorization
*>          A = U**T*U or A = L*L**T, packed columnwise in a linear
*>          array.  The j-th column of U or L is stored in the array AP
*>          as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the right hand side matrix B.
*>          On exit, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pptrs
*
*  =====================================================================
      SUBROUTINE DPPTRS( UPLO, N, NRHS, AP, B, LDB, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DTPSV, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPPTRS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 )
     $   RETURN
*
      IF( UPPER ) THEN
*
*        Solve A*X = B where A = U**T * U.
*
         DO 10 I = 1, NRHS
*
*           Solve U**T *X = B, overwriting B with X.
*
            CALL DTPSV( 'Upper', 'Transpose', 'Non-unit', N, AP,
     $                  B( 1, I ), 1 )
*
*           Solve U*X = B, overwriting B with X.
*
            CALL DTPSV( 'Upper', 'No transpose', 'Non-unit', N, AP,
     $                  B( 1, I ), 1 )
   10    CONTINUE
      ELSE
*
*        Solve A*X = B where A = L * L**T.
*
         DO 20 I = 1, NRHS
*
*           Solve L*Y = B, overwriting B with X.
*
            CALL DTPSV( 'Lower', 'No transpose', 'Non-unit', N, AP,
     $                  B( 1, I ), 1 )
*
*           Solve L**T *X = Y, overwriting B with X.
*
            CALL DTPSV( 'Lower', 'Transpose', 'Non-unit', N, AP,
     $                  B( 1, I ), 1 )
   20    CONTINUE
      END IF
*
      RETURN
*
*     End of DPPTRS
*
      END
*> \brief \b DPSTF2 computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPSTF2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpstf2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpstf2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpstf2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPSTF2( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
*
*       .. Scalar Arguments ..
*       DOUBLE PRECISION   TOL
*       INTEGER            INFO, LDA, N, RANK
*       CHARACTER          UPLO
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), WORK( 2*N )
*       INTEGER            PIV( N )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPSTF2 computes the Cholesky factorization with complete
*> pivoting of a real symmetric positive semidefinite matrix A.
*>
*> The factorization has the form
*>    P**T * A * P = U**T * U ,  if UPLO = 'U',
*>    P**T * A * P = L  * L**T,  if UPLO = 'L',
*> where U is an upper triangular matrix and L is lower triangular, and
*> P is stored as vector PIV.
*>
*> This algorithm does not attempt to check that A is positive
*> semidefinite. This version of the algorithm calls level 2 BLAS.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the upper or lower triangular part of the
*>          symmetric matrix A is stored.
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*>          n by n upper triangular part of A contains the upper
*>          triangular part of the matrix A, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading n by n lower triangular part of A contains the lower
*>          triangular part of the matrix A, and the strictly upper
*>          triangular part of A is not referenced.
*>
*>          On exit, if INFO = 0, the factor U or L from the Cholesky
*>          factorization as above.
*> \endverbatim
*>
*> \param[out] PIV
*> \verbatim
*>          PIV is INTEGER array, dimension (N)
*>          PIV is such that the nonzero entries are P( PIV(K), K ) = 1.
*> \endverbatim
*>
*> \param[out] RANK
*> \verbatim
*>          RANK is INTEGER
*>          The rank of A given by the number of steps the algorithm
*>          completed.
*> \endverbatim
*>
*> \param[in] TOL
*> \verbatim
*>          TOL is DOUBLE PRECISION
*>          User defined tolerance. If TOL < 0, then N*U*MAX( A( K,K ) )
*>          will be used. The algorithm terminates at the (K-1)st step
*>          if the pivot <= TOL.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (2*N)
*>          Work space.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          < 0: If INFO = -K, the K-th argument had an illegal value,
*>          = 0: algorithm completed successfully, and
*>          > 0: the matrix A is either rank deficient with computed rank
*>               as returned in RANK, or is not positive semidefinite. See
*>               Section 7 of LAPACK Working Note #161 for further
*>               information.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pstf2
*
*  =====================================================================
      SUBROUTINE DPSTF2( UPLO, N, A, LDA, PIV, RANK, TOL, WORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   TOL
      INTEGER            INFO, LDA, N, RANK
      CHARACTER          UPLO
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), WORK( 2*N )
      INTEGER            PIV( N )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   AJJ, DSTOP, DTEMP
      INTEGER            I, ITEMP, J, PVT
      LOGICAL            UPPER
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      LOGICAL            LSAME, DISNAN
      EXTERNAL           DLAMCH, LSAME, DISNAN
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMV, DSCAL, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, SQRT, MAXLOC
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPSTF2', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Initialize PIV
*
      DO 100 I = 1, N
         PIV( I ) = I
  100 CONTINUE
*
*     Compute stopping value
*
      PVT = 1
      AJJ = A( PVT, PVT )
      DO I = 2, N
         IF( A( I, I ).GT.AJJ ) THEN
            PVT = I
            AJJ = A( PVT, PVT )
         END IF
      END DO
      IF( AJJ.LE.ZERO.OR.DISNAN( AJJ ) ) THEN
         RANK = 0
         INFO = 1
         GO TO 170
      END IF
*
*     Compute stopping value if not supplied
*
      IF( TOL.LT.ZERO ) THEN
         DSTOP = N * DLAMCH( 'Epsilon' ) * AJJ
      ELSE
         DSTOP = TOL
      END IF
*
*     Set first half of WORK to zero, holds dot products
*
      DO 110 I = 1, N
         WORK( I ) = 0
  110 CONTINUE
*
      IF( UPPER ) THEN
*
*        Compute the Cholesky factorization P**T * A * P = U**T * U
*
         DO 130 J = 1, N
*
*        Find pivot, test for exit, else swap rows and columns
*        Update dot products, compute possible pivots which are
*        stored in the second half of WORK
*
            DO 120 I = J, N
*
               IF( J.GT.1 ) THEN
                  WORK( I ) = WORK( I ) + A( J-1, I )**2
               END IF
               WORK( N+I ) = A( I, I ) - WORK( I )
*
  120       CONTINUE
*
            IF( J.GT.1 ) THEN
               ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
               PVT = ITEMP + J - 1
               AJJ = WORK( N+PVT )
               IF( AJJ.LE.DSTOP.OR.DISNAN( AJJ ) ) THEN
                  A( J, J ) = AJJ
                  GO TO 160
               END IF
            END IF
*
            IF( J.NE.PVT ) THEN
*
*              Pivot OK, so can now swap pivot rows and columns
*
               A( PVT, PVT ) = A( J, J )
               CALL DSWAP( J-1, A( 1, J ), 1, A( 1, PVT ), 1 )
               IF( PVT.LT.N )
     $            CALL DSWAP( N-PVT, A( J, PVT+1 ), LDA,
     $                        A( PVT, PVT+1 ), LDA )
               CALL DSWAP( PVT-J-1, A( J, J+1 ), LDA, A( J+1, PVT ),
     $                     1 )
*
*              Swap dot products and PIV
*
               DTEMP = WORK( J )
               WORK( J ) = WORK( PVT )
               WORK( PVT ) = DTEMP
               ITEMP = PIV( PVT )
               PIV( PVT ) = PIV( J )
               PIV( J ) = ITEMP
            END IF
*
            AJJ = SQRT( AJJ )
            A( J, J ) = AJJ
*
*           Compute elements J+1:N of row J
*
            IF( J.LT.N ) THEN
               CALL DGEMV( 'Trans', J-1, N-J, -ONE, A( 1, J+1 ), LDA,
     $                     A( 1, J ), 1, ONE, A( J, J+1 ), LDA )
               CALL DSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA )
            END IF
*
  130    CONTINUE
*
      ELSE
*
*        Compute the Cholesky factorization P**T * A * P = L * L**T
*
         DO 150 J = 1, N
*
*        Find pivot, test for exit, else swap rows and columns
*        Update dot products, compute possible pivots which are
*        stored in the second half of WORK
*
            DO 140 I = J, N
*
               IF( J.GT.1 ) THEN
                  WORK( I ) = WORK( I ) + A( I, J-1 )**2
               END IF
               WORK( N+I ) = A( I, I ) - WORK( I )
*
  140       CONTINUE
*
            IF( J.GT.1 ) THEN
               ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
               PVT = ITEMP + J - 1
               AJJ = WORK( N+PVT )
               IF( AJJ.LE.DSTOP.OR.DISNAN( AJJ ) ) THEN
                  A( J, J ) = AJJ
                  GO TO 160
               END IF
            END IF
*
            IF( J.NE.PVT ) THEN
*
*              Pivot OK, so can now swap pivot rows and columns
*
               A( PVT, PVT ) = A( J, J )
               CALL DSWAP( J-1, A( J, 1 ), LDA, A( PVT, 1 ), LDA )
               IF( PVT.LT.N )
     $            CALL DSWAP( N-PVT, A( PVT+1, J ), 1, A( PVT+1,
     $                        PVT ),
     $                        1 )
               CALL DSWAP( PVT-J-1, A( J+1, J ), 1, A( PVT, J+1 ),
     $                     LDA )
*
*              Swap dot products and PIV
*
               DTEMP = WORK( J )
               WORK( J ) = WORK( PVT )
               WORK( PVT ) = DTEMP
               ITEMP = PIV( PVT )
               PIV( PVT ) = PIV( J )
               PIV( J ) = ITEMP
            END IF
*
            AJJ = SQRT( AJJ )
            A( J, J ) = AJJ
*
*           Compute elements J+1:N of column J
*
            IF( J.LT.N ) THEN
               CALL DGEMV( 'No Trans', N-J, J-1, -ONE, A( J+1, 1 ),
     $                     LDA,
     $                     A( J, 1 ), LDA, ONE, A( J+1, J ), 1 )
               CALL DSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 )
            END IF
*
  150    CONTINUE
*
      END IF
*
*     Ran to completion, A has full rank
*
      RANK = N
*
      GO TO 170
  160 CONTINUE
*
*     Rank is number of steps completed.  Set INFO = 1 to signal
*     that the factorization cannot be used to solve a system.
*
      RANK = J - 1
      INFO = 1
*
  170 CONTINUE
      RETURN
*
*     End of DPSTF2
*
      END
*> \brief \b DPSTRF computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
*
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPSTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpstrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpstrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpstrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPSTRF( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
*
*       .. Scalar Arguments ..
*       DOUBLE PRECISION   TOL
*       INTEGER            INFO, LDA, N, RANK
*       CHARACTER          UPLO
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), WORK( 2*N )
*       INTEGER            PIV( N )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPSTRF computes the Cholesky factorization with complete
*> pivoting of a real symmetric positive semidefinite matrix A.
*>
*> The factorization has the form
*>    P**T * A * P = U**T * U ,  if UPLO = 'U',
*>    P**T * A * P = L  * L**T,  if UPLO = 'L',
*> where U is an upper triangular matrix and L is lower triangular, and
*> P is stored as vector PIV.
*>
*> This algorithm does not attempt to check that A is positive
*> semidefinite. This version of the algorithm calls level 3 BLAS.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the upper or lower triangular part of the
*>          symmetric matrix A is stored.
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*>          n by n upper triangular part of A contains the upper
*>          triangular part of the matrix A, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading n by n lower triangular part of A contains the lower
*>          triangular part of the matrix A, and the strictly upper
*>          triangular part of A is not referenced.
*>
*>          On exit, if INFO = 0, the factor U or L from the Cholesky
*>          factorization as above.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] PIV
*> \verbatim
*>          PIV is INTEGER array, dimension (N)
*>          PIV is such that the nonzero entries are P( PIV(K), K ) = 1.
*> \endverbatim
*>
*> \param[out] RANK
*> \verbatim
*>          RANK is INTEGER
*>          The rank of A given by the number of steps the algorithm
*>          completed.
*> \endverbatim
*>
*> \param[in] TOL
*> \verbatim
*>          TOL is DOUBLE PRECISION
*>          User defined tolerance. If TOL < 0, then N*U*MAX( A(K,K) )
*>          will be used. The algorithm terminates at the (K-1)st step
*>          if the pivot <= TOL.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (2*N)
*>          Work space.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          < 0: If INFO = -K, the K-th argument had an illegal value,
*>          = 0: algorithm completed successfully, and
*>          > 0: the matrix A is either rank deficient with computed rank
*>               as returned in RANK, or is not positive semidefinite. See
*>               Section 7 of LAPACK Working Note #161 for further
*>               information.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pstrf
*
*  =====================================================================
      SUBROUTINE DPSTRF( UPLO, N, A, LDA, PIV, RANK, TOL, WORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   TOL
      INTEGER            INFO, LDA, N, RANK
      CHARACTER          UPLO
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), WORK( 2*N )
      INTEGER            PIV( N )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   AJJ, DSTOP, DTEMP
      INTEGER            I, ITEMP, J, JB, K, NB, PVT
      LOGICAL            UPPER
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      INTEGER            ILAENV
      LOGICAL            LSAME, DISNAN
      EXTERNAL           DLAMCH, ILAENV, LSAME, DISNAN
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMV, DPSTF2, DSCAL, DSWAP, DSYRK,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT, MAXLOC
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPSTRF', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Get block size
*
      NB = ILAENV( 1, 'DPOTRF', UPLO, N, -1, -1, -1 )
      IF( NB.LE.1 .OR. NB.GE.N ) THEN
*
*        Use unblocked code
*
         CALL DPSTF2( UPLO, N, A( 1, 1 ), LDA, PIV, RANK, TOL, WORK,
     $                INFO )
         GO TO 200
*
      ELSE
*
*     Initialize PIV
*
         DO 100 I = 1, N
            PIV( I ) = I
  100    CONTINUE
*
*     Compute stopping value
*
         PVT = 1
         AJJ = A( PVT, PVT )
         DO I = 2, N
            IF( A( I, I ).GT.AJJ ) THEN
               PVT = I
               AJJ = A( PVT, PVT )
            END IF
         END DO
         IF( AJJ.LE.ZERO.OR.DISNAN( AJJ ) ) THEN
            RANK = 0
            INFO = 1
            GO TO 200
         END IF
*
*     Compute stopping value if not supplied
*
         IF( TOL.LT.ZERO ) THEN
            DSTOP = N * DLAMCH( 'Epsilon' ) * AJJ
         ELSE
            DSTOP = TOL
         END IF
*
*
         IF( UPPER ) THEN
*
*           Compute the Cholesky factorization P**T * A * P = U**T * U
*
            DO 140 K = 1, N, NB
*
*              Account for last block not being NB wide
*
               JB = MIN( NB, N-K+1 )
*
*              Set relevant part of first half of WORK to zero,
*              holds dot products
*
               DO 110 I = K, N
                  WORK( I ) = 0
  110          CONTINUE
*
               DO 130 J = K, K + JB - 1
*
*              Find pivot, test for exit, else swap rows and columns
*              Update dot products, compute possible pivots which are
*              stored in the second half of WORK
*
                  DO 120 I = J, N
*
                     IF( J.GT.K ) THEN
                        WORK( I ) = WORK( I ) + A( J-1, I )**2
                     END IF
                     WORK( N+I ) = A( I, I ) - WORK( I )
*
  120             CONTINUE
*
                  IF( J.GT.1 ) THEN
                     ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
                     PVT = ITEMP + J - 1
                     AJJ = WORK( N+PVT )
                     IF( AJJ.LE.DSTOP.OR.DISNAN( AJJ ) ) THEN
                        A( J, J ) = AJJ
                        GO TO 190
                     END IF
                  END IF
*
                  IF( J.NE.PVT ) THEN
*
*                    Pivot OK, so can now swap pivot rows and columns
*
                     A( PVT, PVT ) = A( J, J )
                     CALL DSWAP( J-1, A( 1, J ), 1, A( 1, PVT ), 1 )
                     IF( PVT.LT.N )
     $                  CALL DSWAP( N-PVT, A( J, PVT+1 ), LDA,
     $                              A( PVT, PVT+1 ), LDA )
                     CALL DSWAP( PVT-J-1, A( J, J+1 ), LDA,
     $                           A( J+1, PVT ), 1 )
*
*                    Swap dot products and PIV
*
                     DTEMP = WORK( J )
                     WORK( J ) = WORK( PVT )
                     WORK( PVT ) = DTEMP
                     ITEMP = PIV( PVT )
                     PIV( PVT ) = PIV( J )
                     PIV( J ) = ITEMP
                  END IF
*
                  AJJ = SQRT( AJJ )
                  A( J, J ) = AJJ
*
*                 Compute elements J+1:N of row J.
*
                  IF( J.LT.N ) THEN
                     CALL DGEMV( 'Trans', J-K, N-J, -ONE, A( K,
     $                           J+1 ),
     $                           LDA, A( K, J ), 1, ONE, A( J, J+1 ),
     $                           LDA )
                     CALL DSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA )
                  END IF
*
  130          CONTINUE
*
*              Update trailing matrix, J already incremented
*
               IF( K+JB.LE.N ) THEN
                  CALL DSYRK( 'Upper', 'Trans', N-J+1, JB, -ONE,
     $                        A( K, J ), LDA, ONE, A( J, J ), LDA )
               END IF
*
  140       CONTINUE
*
         ELSE
*
*        Compute the Cholesky factorization P**T * A * P = L * L**T
*
            DO 180 K = 1, N, NB
*
*              Account for last block not being NB wide
*
               JB = MIN( NB, N-K+1 )
*
*              Set relevant part of first half of WORK to zero,
*              holds dot products
*
               DO 150 I = K, N
                  WORK( I ) = 0
  150          CONTINUE
*
               DO 170 J = K, K + JB - 1
*
*              Find pivot, test for exit, else swap rows and columns
*              Update dot products, compute possible pivots which are
*              stored in the second half of WORK
*
                  DO 160 I = J, N
*
                     IF( J.GT.K ) THEN
                        WORK( I ) = WORK( I ) + A( I, J-1 )**2
                     END IF
                     WORK( N+I ) = A( I, I ) - WORK( I )
*
  160             CONTINUE
*
                  IF( J.GT.1 ) THEN
                     ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
                     PVT = ITEMP + J - 1
                     AJJ = WORK( N+PVT )
                     IF( AJJ.LE.DSTOP.OR.DISNAN( AJJ ) ) THEN
                        A( J, J ) = AJJ
                        GO TO 190
                     END IF
                  END IF
*
                  IF( J.NE.PVT ) THEN
*
*                    Pivot OK, so can now swap pivot rows and columns
*
                     A( PVT, PVT ) = A( J, J )
                     CALL DSWAP( J-1, A( J, 1 ), LDA, A( PVT, 1 ),
     $                           LDA )
                     IF( PVT.LT.N )
     $                  CALL DSWAP( N-PVT, A( PVT+1, J ), 1,
     $                              A( PVT+1, PVT ), 1 )
                     CALL DSWAP( PVT-J-1, A( J+1, J ), 1, A( PVT,
     $                           J+1 ),
     $                           LDA )
*
*                    Swap dot products and PIV
*
                     DTEMP = WORK( J )
                     WORK( J ) = WORK( PVT )
                     WORK( PVT ) = DTEMP
                     ITEMP = PIV( PVT )
                     PIV( PVT ) = PIV( J )
                     PIV( J ) = ITEMP
                  END IF
*
                  AJJ = SQRT( AJJ )
                  A( J, J ) = AJJ
*
*                 Compute elements J+1:N of column J.
*
                  IF( J.LT.N ) THEN
                     CALL DGEMV( 'No Trans', N-J, J-K, -ONE,
     $                           A( J+1, K ), LDA, A( J, K ), LDA, ONE,
     $                           A( J+1, J ), 1 )
                     CALL DSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 )
                  END IF
*
  170          CONTINUE
*
*              Update trailing matrix, J already incremented
*
               IF( K+JB.LE.N ) THEN
                  CALL DSYRK( 'Lower', 'No Trans', N-J+1, JB, -ONE,
     $                        A( J, K ), LDA, ONE, A( J, J ), LDA )
               END IF
*
  180       CONTINUE
*
         END IF
      END IF
*
*     Ran to completion, A has full rank
*
      RANK = N
*
      GO TO 200
  190 CONTINUE
*
*     Rank is the number of steps completed.  Set INFO = 1 to signal
*     that the factorization cannot be used to solve a system.
*
      RANK = J - 1
      INFO = 1
*
  200 CONTINUE
      RETURN
*
*     End of DPSTRF
*
      END
*> \brief \b DPTCON
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPTCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dptcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dptcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dptcon.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPTCON( N, D, E, ANORM, RCOND, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, N
*       DOUBLE PRECISION   ANORM, RCOND
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), E( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPTCON computes the reciprocal of the condition number (in the
*> 1-norm) of a real symmetric positive definite tridiagonal matrix
*> using the factorization A = L*D*L**T or A = U**T*D*U computed by
*> DPTTRF.
*>
*> Norm(inv(A)) is computed by a direct method, and the reciprocal of
*> the condition number is computed as
*>              RCOND = 1 / (ANORM * norm(inv(A))).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The n diagonal elements of the diagonal matrix D from the
*>          factorization of A, as computed by DPTTRF.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) off-diagonal elements of the unit bidiagonal factor
*>          U or L from the factorization of A,  as computed by DPTTRF.
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*>          ANORM is DOUBLE PRECISION
*>          The 1-norm of the original matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The reciprocal of the condition number of the matrix A,
*>          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is the
*>          1-norm of inv(A) computed in this routine.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ptcon
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The method used is described in Nicholas J. Higham, "Efficient
*>  Algorithms for Computing the Condition Number of a Tridiagonal
*>  Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DPTCON( N, D, E, ANORM, RCOND, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, N
      DOUBLE PRECISION   ANORM, RCOND
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), E( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, IX
      DOUBLE PRECISION   AINVNM
*     ..
*     .. External Functions ..
      INTEGER            IDAMAX
      EXTERNAL           IDAMAX
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments.
*
      INFO = 0
      IF( N.LT.0 ) THEN
         INFO = -1
      ELSE IF( ANORM.LT.ZERO ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPTCON', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      RCOND = ZERO
      IF( N.EQ.0 ) THEN
         RCOND = ONE
         RETURN
      ELSE IF( ANORM.EQ.ZERO ) THEN
         RETURN
      END IF
*
*     Check that D(1:N) is positive.
*
      DO 10 I = 1, N
         IF( D( I ).LE.ZERO )
     $      RETURN
   10 CONTINUE
*
*     Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
*
*        m(i,j) =  abs(A(i,j)), i = j,
*        m(i,j) = -abs(A(i,j)), i .ne. j,
*
*     and e = [ 1, 1, ..., 1 ]**T.  Note M(A) = M(L)*D*M(L)**T.
*
*     Solve M(L) * x = e.
*
      WORK( 1 ) = ONE
      DO 20 I = 2, N
         WORK( I ) = ONE + WORK( I-1 )*ABS( E( I-1 ) )
   20 CONTINUE
*
*     Solve D * M(L)**T * x = b.
*
      WORK( N ) = WORK( N ) / D( N )
      DO 30 I = N - 1, 1, -1
         WORK( I ) = WORK( I ) / D( I ) + WORK( I+1 )*ABS( E( I ) )
   30 CONTINUE
*
*     Compute AINVNM = max(x(i)), 1<=i<=n.
*
      IX = IDAMAX( N, WORK, 1 )
      AINVNM = ABS( WORK( IX ) )
*
*     Compute the reciprocal condition number.
*
      IF( AINVNM.NE.ZERO )
     $   RCOND = ( ONE / AINVNM ) / ANORM
*
      RETURN
*
*     End of DPTCON
*
      END
*> \brief \b DPTEQR
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPTEQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpteqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpteqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpteqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          COMPZ
*       INTEGER            INFO, LDZ, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), E( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPTEQR computes all eigenvalues and, optionally, eigenvectors of a
*> symmetric positive definite tridiagonal matrix by first factoring the
*> matrix using DPTTRF, and then calling DBDSQR to compute the singular
*> values of the bidiagonal factor.
*>
*> This routine computes the eigenvalues of the positive definite
*> tridiagonal matrix to high relative accuracy.  This means that if the
*> eigenvalues range over many orders of magnitude in size, then the
*> small eigenvalues and corresponding eigenvectors will be computed
*> more accurately than, for example, with the standard QR method.
*>
*> The eigenvectors of a full or band symmetric positive definite matrix
*> can also be found if DSYTRD, DSPTRD, or DSBTRD has been used to
*> reduce this matrix to tridiagonal form. (The reduction to tridiagonal
*> form, however, may preclude the possibility of obtaining high
*> relative accuracy in the small eigenvalues of the original matrix, if
*> these eigenvalues range over many orders of magnitude.)
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] COMPZ
*> \verbatim
*>          COMPZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only.
*>          = 'V':  Compute eigenvectors of original symmetric
*>                  matrix also.  Array Z contains the orthogonal
*>                  matrix used to reduce the original matrix to
*>                  tridiagonal form.
*>          = 'I':  Compute eigenvectors of tridiagonal matrix also.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, the n diagonal elements of the tridiagonal
*>          matrix.
*>          On normal exit, D contains the eigenvalues, in descending
*>          order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          On entry, the (n-1) subdiagonal elements of the tridiagonal
*>          matrix.
*>          On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
*>          On entry, if COMPZ = 'V', the orthogonal matrix used in the
*>          reduction to tridiagonal form.
*>          On exit, if COMPZ = 'V', the orthonormal eigenvectors of the
*>          original symmetric matrix;
*>          if COMPZ = 'I', the orthonormal eigenvectors of the
*>          tridiagonal matrix.
*>          If INFO > 0 on exit, Z contains the eigenvectors associated
*>          with only the stored eigenvalues.
*>          If  COMPZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          COMPZ = 'V' or 'I', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = i, and i is:
*>                <= N  the Cholesky factorization of the matrix could
*>                      not be performed because the leading principal
*>                      minor of order i was not positive.
*>                > N   the SVD algorithm failed to converge;
*>                      if INFO = N+i, i off-diagonal elements of the
*>                      bidiagonal factor did not converge to zero.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pteqr
*
*  =====================================================================
      SUBROUTINE DPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          COMPZ
      INTEGER            INFO, LDZ, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), E( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DBDSQR, DLASET, DPTTRF, XERBLA
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   C( 1, 1 ), VT( 1, 1 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, ICOMPZ, NRU
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IF( LSAME( COMPZ, 'N' ) ) THEN
         ICOMPZ = 0
      ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
         ICOMPZ = 1
      ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
         ICOMPZ = 2
      ELSE
         ICOMPZ = -1
      END IF
      IF( ICOMPZ.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( ( LDZ.LT.1 ) .OR. ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1,
     $         N ) ) ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPTEQR', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( N.EQ.1 ) THEN
         IF( ICOMPZ.GT.0 )
     $      Z( 1, 1 ) = ONE
         RETURN
      END IF
      IF( ICOMPZ.EQ.2 )
     $   CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
*
*     Call DPTTRF to factor the matrix.
*
      CALL DPTTRF( N, D, E, INFO )
      IF( INFO.NE.0 )
     $   RETURN
      DO 10 I = 1, N
         D( I ) = SQRT( D( I ) )
   10 CONTINUE
      DO 20 I = 1, N - 1
         E( I ) = E( I )*D( I )
   20 CONTINUE
*
*     Call DBDSQR to compute the singular values/vectors of the
*     bidiagonal factor.
*
      IF( ICOMPZ.GT.0 ) THEN
         NRU = N
      ELSE
         NRU = 0
      END IF
      CALL DBDSQR( 'Lower', N, 0, NRU, 0, D, E, VT, 1, Z, LDZ, C, 1,
     $             WORK, INFO )
*
*     Square the singular values.
*
      IF( INFO.EQ.0 ) THEN
         DO 30 I = 1, N
            D( I ) = D( I )*D( I )
   30    CONTINUE
      ELSE
         INFO = N + INFO
      END IF
*
      RETURN
*
*     End of DPTEQR
*
      END
*> \brief \b DPTRFS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPTRFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dptrfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dptrfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dptrfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,
*                          BERR, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDB, LDX, N, NRHS
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   B( LDB, * ), BERR( * ), D( * ), DF( * ),
*      $                   E( * ), EF( * ), FERR( * ), WORK( * ),
*      $                   X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPTRFS improves the computed solution to a system of linear
*> equations when the coefficient matrix is symmetric positive definite
*> and tridiagonal, and provides error bounds and backward error
*> estimates for the solution.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The n diagonal elements of the tridiagonal matrix A.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) subdiagonal elements of the tridiagonal matrix A.
*> \endverbatim
*>
*> \param[in] DF
*> \verbatim
*>          DF is DOUBLE PRECISION array, dimension (N)
*>          The n diagonal elements of the diagonal matrix D from the
*>          factorization computed by DPTTRF.
*> \endverbatim
*>
*> \param[in] EF
*> \verbatim
*>          EF is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) subdiagonal elements of the unit bidiagonal factor
*>          L from the factorization computed by DPTTRF.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          On entry, the solution matrix X, as computed by DPTTRS.
*>          On exit, the improved solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in
*>          any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  ITMAX is the maximum number of steps of iterative refinement.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ptrfs
*
*  =====================================================================
      SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,
     $                   BERR, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDB, LDX, N, NRHS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   B( LDB, * ), BERR( * ), D( * ), DF( * ),
     $                   E( * ), EF( * ), FERR( * ), WORK( * ),
     $                   X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            ITMAX
      PARAMETER          ( ITMAX = 5 )
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
      DOUBLE PRECISION   TWO
      PARAMETER          ( TWO = 2.0D+0 )
      DOUBLE PRECISION   THREE
      PARAMETER          ( THREE = 3.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            COUNT, I, IX, J, NZ
      DOUBLE PRECISION   BI, CX, DX, EPS, EX, LSTRES, S, SAFE1, SAFE2,
     $                   SAFMIN
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DPTTRS, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. External Functions ..
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           IDAMAX, DLAMCH
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF( N.LT.0 ) THEN
         INFO = -1
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -8
      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
         INFO = -10
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPTRFS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
         DO 10 J = 1, NRHS
            FERR( J ) = ZERO
            BERR( J ) = ZERO
   10    CONTINUE
         RETURN
      END IF
*
*     NZ = maximum number of nonzero elements in each row of A, plus 1
*
      NZ = 4
      EPS = DLAMCH( 'Epsilon' )
      SAFMIN = DLAMCH( 'Safe minimum' )
      SAFE1 = NZ*SAFMIN
      SAFE2 = SAFE1 / EPS
*
*     Do for each right hand side
*
      DO 90 J = 1, NRHS
*
         COUNT = 1
         LSTRES = THREE
   20    CONTINUE
*
*        Loop until stopping criterion is satisfied.
*
*        Compute residual R = B - A * X.  Also compute
*        abs(A)*abs(x) + abs(b) for use in the backward error bound.
*
         IF( N.EQ.1 ) THEN
            BI = B( 1, J )
            DX = D( 1 )*X( 1, J )
            WORK( N+1 ) = BI - DX
            WORK( 1 ) = ABS( BI ) + ABS( DX )
         ELSE
            BI = B( 1, J )
            DX = D( 1 )*X( 1, J )
            EX = E( 1 )*X( 2, J )
            WORK( N+1 ) = BI - DX - EX
            WORK( 1 ) = ABS( BI ) + ABS( DX ) + ABS( EX )
            DO 30 I = 2, N - 1
               BI = B( I, J )
               CX = E( I-1 )*X( I-1, J )
               DX = D( I )*X( I, J )
               EX = E( I )*X( I+1, J )
               WORK( N+I ) = BI - CX - DX - EX
               WORK( I ) = ABS( BI ) + ABS( CX ) + ABS( DX ) + ABS( EX )
   30       CONTINUE
            BI = B( N, J )
            CX = E( N-1 )*X( N-1, J )
            DX = D( N )*X( N, J )
            WORK( N+N ) = BI - CX - DX
            WORK( N ) = ABS( BI ) + ABS( CX ) + ABS( DX )
         END IF
*
*        Compute componentwise relative backward error from formula
*
*        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
*
*        where abs(Z) is the componentwise absolute value of the matrix
*        or vector Z.  If the i-th component of the denominator is less
*        than SAFE2, then SAFE1 is added to the i-th components of the
*        numerator and denominator before dividing.
*
         S = ZERO
         DO 40 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
            ELSE
               S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
     $             ( WORK( I )+SAFE1 ) )
            END IF
   40    CONTINUE
         BERR( J ) = S
*
*        Test stopping criterion. Continue iterating if
*           1) The residual BERR(J) is larger than machine epsilon, and
*           2) BERR(J) decreased by at least a factor of 2 during the
*              last iteration, and
*           3) At most ITMAX iterations tried.
*
         IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
     $       COUNT.LE.ITMAX ) THEN
*
*           Update solution and try again.
*
            CALL DPTTRS( N, 1, DF, EF, WORK( N+1 ), N, INFO )
            CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
            LSTRES = BERR( J )
            COUNT = COUNT + 1
            GO TO 20
         END IF
*
*        Bound error from formula
*
*        norm(X - XTRUE) / norm(X) .le. FERR =
*        norm( abs(inv(A))*
*           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
*
*        where
*          norm(Z) is the magnitude of the largest component of Z
*          inv(A) is the inverse of A
*          abs(Z) is the componentwise absolute value of the matrix or
*             vector Z
*          NZ is the maximum number of nonzeros in any row of A, plus 1
*          EPS is machine epsilon
*
*        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
*        is incremented by SAFE1 if the i-th component of
*        abs(A)*abs(X) + abs(B) is less than SAFE2.
*
         DO 50 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
            ELSE
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
            END IF
   50    CONTINUE
         IX = IDAMAX( N, WORK, 1 )
         FERR( J ) = WORK( IX )
*
*        Estimate the norm of inv(A).
*
*        Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
*
*           m(i,j) =  abs(A(i,j)), i = j,
*           m(i,j) = -abs(A(i,j)), i .ne. j,
*
*        and e = [ 1, 1, ..., 1 ]**T.  Note M(A) = M(L)*D*M(L)**T.
*
*        Solve M(L) * x = e.
*
         WORK( 1 ) = ONE
         DO 60 I = 2, N
            WORK( I ) = ONE + WORK( I-1 )*ABS( EF( I-1 ) )
   60    CONTINUE
*
*        Solve D * M(L)**T * x = b.
*
         WORK( N ) = WORK( N ) / DF( N )
         DO 70 I = N - 1, 1, -1
            WORK( I ) = WORK( I ) / DF( I ) + WORK( I+1 )*ABS( EF( I ) )
   70    CONTINUE
*
*        Compute norm(inv(A)) = max(x(i)), 1<=i<=n.
*
         IX = IDAMAX( N, WORK, 1 )
         FERR( J ) = FERR( J )*ABS( WORK( IX ) )
*
*        Normalize error.
*
         LSTRES = ZERO
         DO 80 I = 1, N
            LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
   80    CONTINUE
         IF( LSTRES.NE.ZERO )
     $      FERR( J ) = FERR( J ) / LSTRES
*
   90 CONTINUE
*
      RETURN
*
*     End of DPTRFS
*
      END
*> \brief <b> DPTSV computes the solution to system of linear equations A * X = B for PT matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPTSV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dptsv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dptsv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dptsv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPTSV( N, NRHS, D, E, B, LDB, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   B( LDB, * ), D( * ), E( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPTSV computes the solution to a real system of linear equations
*> A*X = B, where A is an N-by-N symmetric positive definite tridiagonal
*> matrix, and X and B are N-by-NRHS matrices.
*>
*> A is factored as A = L*D*L**T, and the factored form of A is then
*> used to solve the system of equations.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, the n diagonal elements of the tridiagonal matrix
*>          A.  On exit, the n diagonal elements of the diagonal matrix
*>          D from the factorization A = L*D*L**T.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          On entry, the (n-1) subdiagonal elements of the tridiagonal
*>          matrix A.  On exit, the (n-1) subdiagonal elements of the
*>          unit bidiagonal factor L from the L*D*L**T factorization of
*>          A.  (E can also be regarded as the superdiagonal of the unit
*>          bidiagonal factor U from the U**T*D*U factorization of A.)
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the N-by-NRHS right hand side matrix B.
*>          On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the leading principal minor of order i
*>                is not positive, and the solution has not been
*>                computed.  The factorization has not been completed
*>                unless i = N.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ptsv
*
*  =====================================================================
      SUBROUTINE DPTSV( N, NRHS, D, E, B, LDB, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   B( LDB, * ), D( * ), E( * )
*     ..
*
*  =====================================================================
*
*     .. External Subroutines ..
      EXTERNAL           DPTTRF, DPTTRS, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF( N.LT.0 ) THEN
         INFO = -1
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPTSV ', -INFO )
         RETURN
      END IF
*
*     Compute the L*D*L**T (or U**T*D*U) factorization of A.
*
      CALL DPTTRF( N, D, E, INFO )
      IF( INFO.EQ.0 ) THEN
*
*        Solve the system A*X = B, overwriting B with X.
*
         CALL DPTTRS( N, NRHS, D, E, B, LDB, INFO )
      END IF
      RETURN
*
*     End of DPTSV
*
      END
*> \brief <b> DPTSVX computes the solution to system of linear equations A * X = B for PT matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPTSVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dptsvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dptsvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dptsvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
*                          RCOND, FERR, BERR, WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          FACT
*       INTEGER            INFO, LDB, LDX, N, NRHS
*       DOUBLE PRECISION   RCOND
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   B( LDB, * ), BERR( * ), D( * ), DF( * ),
*      $                   E( * ), EF( * ), FERR( * ), WORK( * ),
*      $                   X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPTSVX uses the factorization A = L*D*L**T to compute the solution
*> to a real system of linear equations A*X = B, where A is an N-by-N
*> symmetric positive definite tridiagonal matrix and X and B are
*> N-by-NRHS matrices.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
*  =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'N', the matrix A is factored as A = L*D*L**T, where L
*>    is a unit lower bidiagonal matrix and D is diagonal.  The
*>    factorization can also be regarded as having the form
*>    A = U**T*D*U.
*>
*> 2. If the leading principal minor of order i is not positive,
*>    then the routine returns with INFO = i. Otherwise, the factored
*>    form of A is used to estimate the condition number of the matrix
*>    A.  If the reciprocal of the condition number is less than machine
*>    precision, INFO = N+1 is returned as a warning, but the routine
*>    still goes on to solve for X and compute error bounds as
*>    described below.
*>
*> 3. The system of equations is solved for X using the factored form
*>    of A.
*>
*> 4. Iterative refinement is applied to improve the computed solution
*>    matrix and calculate error bounds and backward error estimates
*>    for it.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] FACT
*> \verbatim
*>          FACT is CHARACTER*1
*>          Specifies whether or not the factored form of A has been
*>          supplied on entry.
*>          = 'F':  On entry, DF and EF contain the factored form of A.
*>                  D, E, DF, and EF will not be modified.
*>          = 'N':  The matrix A will be copied to DF and EF and
*>                  factored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices B and X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The n diagonal elements of the tridiagonal matrix A.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) subdiagonal elements of the tridiagonal matrix A.
*> \endverbatim
*>
*> \param[in,out] DF
*> \verbatim
*>          DF is DOUBLE PRECISION array, dimension (N)
*>          If FACT = 'F', then DF is an input argument and on entry
*>          contains the n diagonal elements of the diagonal matrix D
*>          from the L*D*L**T factorization of A.
*>          If FACT = 'N', then DF is an output argument and on exit
*>          contains the n diagonal elements of the diagonal matrix D
*>          from the L*D*L**T factorization of A.
*> \endverbatim
*>
*> \param[in,out] EF
*> \verbatim
*>          EF is DOUBLE PRECISION array, dimension (N-1)
*>          If FACT = 'F', then EF is an input argument and on entry
*>          contains the (n-1) subdiagonal elements of the unit
*>          bidiagonal factor L from the L*D*L**T factorization of A.
*>          If FACT = 'N', then EF is an output argument and on exit
*>          contains the (n-1) subdiagonal elements of the unit
*>          bidiagonal factor L from the L*D*L**T factorization of A.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          The N-by-NRHS right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          If INFO = 0 of INFO = N+1, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The reciprocal condition number of the matrix A.  If RCOND
*>          is less than the machine precision (in particular, if
*>          RCOND = 0), the matrix is singular to working precision.
*>          This condition is indicated by a return code of INFO > 0.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in any
*>          element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, and i is
*>                <= N:  the leading principal minor of order i of A
*>                       is not positive, so the factorization could not
*>                       be completed, and the solution has not been
*>                       computed. RCOND = 0 is returned.
*>                = N+1: U is nonsingular, but RCOND is less than machine
*>                       precision, meaning that the matrix is singular
*>                       to working precision.  Nevertheless, the
*>                       solution and error bounds are computed because
*>                       there are a number of situations where the
*>                       computed solution can be more accurate than the
*>                       value of RCOND would suggest.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ptsvx
*
*  =====================================================================
      SUBROUTINE DPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
     $                   RCOND, FERR, BERR, WORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          FACT
      INTEGER            INFO, LDB, LDX, N, NRHS
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   B( LDB, * ), BERR( * ), D( * ), DF( * ),
     $                   E( * ), EF( * ), FERR( * ), WORK( * ),
     $                   X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOFACT
      DOUBLE PRECISION   ANORM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANST
      EXTERNAL           LSAME, DLAMCH, DLANST
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLACPY, DPTCON, DPTRFS, DPTTRF,
     $                   DPTTRS,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NOFACT = LSAME( FACT, 'N' )
      IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -9
      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
         INFO = -11
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPTSVX', -INFO )
         RETURN
      END IF
*
      IF( NOFACT ) THEN
*
*        Compute the L*D*L**T (or U**T*D*U) factorization of A.
*
         CALL DCOPY( N, D, 1, DF, 1 )
         IF( N.GT.1 )
     $      CALL DCOPY( N-1, E, 1, EF, 1 )
         CALL DPTTRF( N, DF, EF, INFO )
*
*        Return if INFO is non-zero.
*
         IF( INFO.GT.0 )THEN
            RCOND = ZERO
            RETURN
         END IF
      END IF
*
*     Compute the norm of the matrix A.
*
      ANORM = DLANST( '1', N, D, E )
*
*     Compute the reciprocal of the condition number of A.
*
      CALL DPTCON( N, DF, EF, ANORM, RCOND, WORK, INFO )
*
*     Compute the solution vectors X.
*
      CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
      CALL DPTTRS( N, NRHS, DF, EF, X, LDX, INFO )
*
*     Use iterative refinement to improve the computed solutions and
*     compute error bounds and backward error estimates for them.
*
      CALL DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR,
     $             WORK, INFO )
*
*     Set INFO = N+1 if the matrix is singular to working precision.
*
      IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
     $   INFO = N + 1
*
      RETURN
*
*     End of DPTSVX
*
      END
*> \brief \b DPTTRF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPTTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpttrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpttrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpttrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPTTRF( N, D, E, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), E( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPTTRF computes the L*D*L**T factorization of a real symmetric
*> positive definite tridiagonal matrix A.  The factorization may also
*> be regarded as having the form A = U**T*D*U.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, the n diagonal elements of the tridiagonal matrix
*>          A.  On exit, the n diagonal elements of the diagonal matrix
*>          D from the L*D*L**T factorization of A.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          On entry, the (n-1) subdiagonal elements of the tridiagonal
*>          matrix A.  On exit, the (n-1) subdiagonal elements of the
*>          unit bidiagonal factor L from the L*D*L**T factorization of A.
*>          E can also be regarded as the superdiagonal of the unit
*>          bidiagonal factor U from the U**T*D*U factorization of A.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -k, the k-th argument had an illegal value
*>          > 0: if INFO = k, the leading principal minor of order k
*>               is not positive; if k < N, the factorization could not
*>               be completed, while if k = N, the factorization was
*>               completed, but D(N) <= 0.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pttrf
*
*  =====================================================================
      SUBROUTINE DPTTRF( N, D, E, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), E( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, I4
      DOUBLE PRECISION   EI
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MOD
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF( N.LT.0 ) THEN
         INFO = -1
         CALL XERBLA( 'DPTTRF', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Compute the L*D*L**T (or U**T*D*U) factorization of A.
*
      I4 = MOD( N-1, 4 )
      DO 10 I = 1, I4
         IF( D( I ).LE.ZERO ) THEN
            INFO = I
            GO TO 30
         END IF
         EI = E( I )
         E( I ) = EI / D( I )
         D( I+1 ) = D( I+1 ) - E( I )*EI
   10 CONTINUE
*
      DO 20 I = I4 + 1, N - 4, 4
*
*        Drop out of the loop if d(i) <= 0: the matrix is not positive
*        definite.
*
         IF( D( I ).LE.ZERO ) THEN
            INFO = I
            GO TO 30
         END IF
*
*        Solve for e(i) and d(i+1).
*
         EI = E( I )
         E( I ) = EI / D( I )
         D( I+1 ) = D( I+1 ) - E( I )*EI
*
         IF( D( I+1 ).LE.ZERO ) THEN
            INFO = I + 1
            GO TO 30
         END IF
*
*        Solve for e(i+1) and d(i+2).
*
         EI = E( I+1 )
         E( I+1 ) = EI / D( I+1 )
         D( I+2 ) = D( I+2 ) - E( I+1 )*EI
*
         IF( D( I+2 ).LE.ZERO ) THEN
            INFO = I + 2
            GO TO 30
         END IF
*
*        Solve for e(i+2) and d(i+3).
*
         EI = E( I+2 )
         E( I+2 ) = EI / D( I+2 )
         D( I+3 ) = D( I+3 ) - E( I+2 )*EI
*
         IF( D( I+3 ).LE.ZERO ) THEN
            INFO = I + 3
            GO TO 30
         END IF
*
*        Solve for e(i+3) and d(i+4).
*
         EI = E( I+3 )
         E( I+3 ) = EI / D( I+3 )
         D( I+4 ) = D( I+4 ) - E( I+3 )*EI
   20 CONTINUE
*
*     Check d(n) for positive definiteness.
*
      IF( D( N ).LE.ZERO )
     $   INFO = N
*
   30 CONTINUE
      RETURN
*
*     End of DPTTRF
*
      END
*> \brief \b DPTTRS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPTTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpttrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpttrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpttrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPTTRS( N, NRHS, D, E, B, LDB, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   B( LDB, * ), D( * ), E( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPTTRS solves a tridiagonal system of the form
*>    A * X = B
*> using the L*D*L**T factorization of A computed by DPTTRF.  D is a
*> diagonal matrix specified in the vector D, L is a unit bidiagonal
*> matrix whose subdiagonal is specified in the vector E, and X and B
*> are N by NRHS matrices.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the tridiagonal matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The n diagonal elements of the diagonal matrix D from the
*>          L*D*L**T factorization of A.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) subdiagonal elements of the unit bidiagonal factor
*>          L from the L*D*L**T factorization of A.  E can also be regarded
*>          as the superdiagonal of the unit bidiagonal factor U from the
*>          factorization A = U**T*D*U.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the right hand side vectors B for the system of
*>          linear equations.
*>          On exit, the solution vectors, X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -k, the k-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pttrs
*
*  =====================================================================
      SUBROUTINE DPTTRS( N, NRHS, D, E, B, LDB, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   B( LDB, * ), D( * ), E( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            J, JB, NB
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DPTTS2, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments.
*
      INFO = 0
      IF( N.LT.0 ) THEN
         INFO = -1
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPTTRS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 )
     $   RETURN
*
*     Determine the number of right-hand sides to solve at a time.
*
      IF( NRHS.EQ.1 ) THEN
         NB = 1
      ELSE
         NB = MAX( 1, ILAENV( 1, 'DPTTRS', ' ', N, NRHS, -1, -1 ) )
      END IF
*
      IF( NB.GE.NRHS ) THEN
         CALL DPTTS2( N, NRHS, D, E, B, LDB )
      ELSE
         DO 10 J = 1, NRHS, NB
            JB = MIN( NRHS-J+1, NB )
            CALL DPTTS2( N, JB, D, E, B( 1, J ), LDB )
   10    CONTINUE
      END IF
*
      RETURN
*
*     End of DPTTRS
*
      END
*> \brief \b DPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPTTS2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dptts2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dptts2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dptts2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DPTTS2( N, NRHS, D, E, B, LDB )
*
*       .. Scalar Arguments ..
*       INTEGER            LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   B( LDB, * ), D( * ), E( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DPTTS2 solves a tridiagonal system of the form
*>    A * X = B
*> using the L*D*L**T factorization of A computed by DPTTRF.  D is a
*> diagonal matrix specified in the vector D, L is a unit bidiagonal
*> matrix whose subdiagonal is specified in the vector E, and X and B
*> are N by NRHS matrices.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the tridiagonal matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The n diagonal elements of the diagonal matrix D from the
*>          L*D*L**T factorization of A.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) subdiagonal elements of the unit bidiagonal factor
*>          L from the L*D*L**T factorization of A.  E can also be regarded
*>          as the superdiagonal of the unit bidiagonal factor U from the
*>          factorization A = U**T*D*U.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the right hand side vectors B for the system of
*>          linear equations.
*>          On exit, the solution vectors, X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ptts2
*
*  =====================================================================
      SUBROUTINE DPTTS2( N, NRHS, D, E, B, LDB )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   B( LDB, * ), D( * ), E( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            I, J
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( N.LE.1 ) THEN
         IF( N.EQ.1 )
     $      CALL DSCAL( NRHS, 1.D0 / D( 1 ), B, LDB )
         RETURN
      END IF
*
*     Solve A * X = B using the factorization A = L*D*L**T,
*     overwriting each right hand side vector with its solution.
*
      DO 30 J = 1, NRHS
*
*           Solve L * x = b.
*
         DO 10 I = 2, N
            B( I, J ) = B( I, J ) - B( I-1, J )*E( I-1 )
   10    CONTINUE
*
*           Solve D * L**T * x = b.
*
         B( N, J ) = B( N, J ) / D( N )
         DO 20 I = N - 1, 1, -1
            B( I, J ) = B( I, J ) / D( I ) - B( I+1, J )*E( I )
   20    CONTINUE
   30 CONTINUE
*
      RETURN
*
*     End of DPTTS2
*
      END
*> \brief \b DRSCL multiplies a vector by the reciprocal of a real scalar.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DRSCL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/drscl.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/drscl.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/drscl.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DRSCL( N, SA, SX, INCX )
*
*       .. Scalar Arguments ..
*       INTEGER            INCX, N
*       DOUBLE PRECISION   SA
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   SX( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DRSCL multiplies an n-element real vector x by the real scalar 1/a.
*> This is done without overflow or underflow as long as
*> the final result x/a does not overflow or underflow.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of components of the vector x.
*> \endverbatim
*>
*> \param[in] SA
*> \verbatim
*>          SA is DOUBLE PRECISION
*>          The scalar a which is used to divide each component of x.
*>          SA must be >= 0, or the subroutine will divide by zero.
*> \endverbatim
*>
*> \param[in,out] SX
*> \verbatim
*>          SX is DOUBLE PRECISION array, dimension
*>                         (1+(N-1)*abs(INCX))
*>          The n-element vector x.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*>          INCX is INTEGER
*>          The increment between successive values of the vector SX.
*>          > 0:  SX(1) = X(1) and SX(1+(i-1)*INCX) = x(i),     1< i<= n
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup rscl
*
*  =====================================================================
      SUBROUTINE DRSCL( N, SA, SX, INCX )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INCX, N
      DOUBLE PRECISION   SA
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   SX( * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            DONE
      DOUBLE PRECISION   BIGNUM, CDEN, CDEN1, CNUM, CNUM1, MUL, SMLNUM
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( N.LE.0 )
     $   RETURN
*
*     Get machine parameters
*
      SMLNUM = DLAMCH( 'S' )
      BIGNUM = ONE / SMLNUM
*
*     Initialize the denominator to SA and the numerator to 1.
*
      CDEN = SA
      CNUM = ONE
*
   10 CONTINUE
      CDEN1 = CDEN*SMLNUM
      CNUM1 = CNUM / BIGNUM
      IF( ABS( CDEN1 ).GT.ABS( CNUM ) .AND. CNUM.NE.ZERO ) THEN
*
*        Pre-multiply X by SMLNUM if CDEN is large compared to CNUM.
*
         MUL = SMLNUM
         DONE = .FALSE.
         CDEN = CDEN1
      ELSE IF( ABS( CNUM1 ).GT.ABS( CDEN ) ) THEN
*
*        Pre-multiply X by BIGNUM if CDEN is small compared to CNUM.
*
         MUL = BIGNUM
         DONE = .FALSE.
         CNUM = CNUM1
      ELSE
*
*        Multiply X by CNUM / CDEN and return.
*
         MUL = CNUM / CDEN
         DONE = .TRUE.
      END IF
*
*     Scale the vector X by MUL
*
      CALL DSCAL( N, MUL, SX, INCX )
*
      IF( .NOT.DONE )
     $   GO TO 10
*
      RETURN
*
*     End of DRSCL
*
      END
*> \brief <b> DSBEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSBEV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbev.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbev.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbev.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSBEV( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK,
*                         INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, UPLO
*       INTEGER            INFO, KD, LDAB, LDZ, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSBEV computes all the eigenvalues and, optionally, eigenvectors of
*> a real symmetric band matrix A.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB, N)
*>          On entry, the upper or lower triangle of the symmetric band
*>          matrix A, stored in the first KD+1 rows of the array.  The
*>          j-th column of A is stored in the j-th column of the array AB
*>          as follows:
*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*>
*>          On exit, AB is overwritten by values generated during the
*>          reduction to tridiagonal form.  If UPLO = 'U', the first
*>          superdiagonal and the diagonal of the tridiagonal matrix T
*>          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
*>          the diagonal and first subdiagonal of T are returned in the
*>          first two rows of AB.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD + 1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
*>          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
*>          eigenvectors of the matrix A, with the i-th column of Z
*>          holding the eigenvector associated with W(i).
*>          If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (max(1,3*N-2))
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the algorithm failed to converge; i
*>                off-diagonal elements of an intermediate tridiagonal
*>                form did not converge to zero.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hbev
*
*  =====================================================================
      SUBROUTINE DSBEV( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK,
     $                  INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, UPLO
      INTEGER            INFO, KD, LDAB, LDZ, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER, WANTZ
      INTEGER            IINFO, IMAX, INDE, INDWRK, ISCALE
      DOUBLE PRECISION   ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
     $                   SMLNUM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANSB
      EXTERNAL           LSAME, DLAMCH, DLANSB
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASCL, DSBTRD, DSCAL, DSTEQR, DSTERF,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
      LOWER = LSAME( UPLO, 'L' )
*
      INFO = 0
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( KD.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDAB.LT.KD+1 ) THEN
         INFO = -6
      ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
         INFO = -9
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSBEV ', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( N.EQ.1 ) THEN
         IF( LOWER ) THEN
            W( 1 ) = AB( 1, 1 )
         ELSE
            W( 1 ) = AB( KD+1, 1 )
         END IF
         IF( WANTZ )
     $      Z( 1, 1 ) = ONE
         RETURN
      END IF
*
*     Get machine constants.
*
      SAFMIN = DLAMCH( 'Safe minimum' )
      EPS = DLAMCH( 'Precision' )
      SMLNUM = SAFMIN / EPS
      BIGNUM = ONE / SMLNUM
      RMIN = SQRT( SMLNUM )
      RMAX = SQRT( BIGNUM )
*
*     Scale matrix to allowable range, if necessary.
*
      ANRM = DLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK )
      ISCALE = 0
      IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
         ISCALE = 1
         SIGMA = RMIN / ANRM
      ELSE IF( ANRM.GT.RMAX ) THEN
         ISCALE = 1
         SIGMA = RMAX / ANRM
      END IF
      IF( ISCALE.EQ.1 ) THEN
         IF( LOWER ) THEN
            CALL DLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB,
     $                   INFO )
         ELSE
            CALL DLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB,
     $                   INFO )
         END IF
      END IF
*
*     Call DSBTRD to reduce symmetric band matrix to tridiagonal form.
*
      INDE = 1
      INDWRK = INDE + N
      CALL DSBTRD( JOBZ, UPLO, N, KD, AB, LDAB, W, WORK( INDE ), Z,
     $             LDZ,
     $             WORK( INDWRK ), IINFO )
*
*     For eigenvalues only, call DSTERF.  For eigenvectors, call SSTEQR.
*
      IF( .NOT.WANTZ ) THEN
         CALL DSTERF( N, W, WORK( INDE ), INFO )
      ELSE
         CALL DSTEQR( JOBZ, N, W, WORK( INDE ), Z, LDZ,
     $                WORK( INDWRK ),
     $                INFO )
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
      IF( ISCALE.EQ.1 ) THEN
         IF( INFO.EQ.0 ) THEN
            IMAX = N
         ELSE
            IMAX = INFO - 1
         END IF
         CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
      END IF
*
      RETURN
*
*     End of DSBEV
*
      END
*> \brief <b> DSBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSBEVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbevd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbevd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbevd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK,
*                          LWORK, IWORK, LIWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, UPLO
*       INTEGER            INFO, KD, LDAB, LDZ, LIWORK, LWORK, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSBEVD computes all the eigenvalues and, optionally, eigenvectors of
*> a real symmetric band matrix A. If eigenvectors are desired, it uses
*> a divide and conquer algorithm.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB, N)
*>          On entry, the upper or lower triangle of the symmetric band
*>          matrix A, stored in the first KD+1 rows of the array.  The
*>          j-th column of A is stored in the j-th column of the array AB
*>          as follows:
*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*>
*>          On exit, AB is overwritten by values generated during the
*>          reduction to tridiagonal form.  If UPLO = 'U', the first
*>          superdiagonal and the diagonal of the tridiagonal matrix T
*>          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
*>          the diagonal and first subdiagonal of T are returned in the
*>          first two rows of AB.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD + 1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
*>          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
*>          eigenvectors of the matrix A, with the i-th column of Z
*>          holding the eigenvector associated with W(i).
*>          If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array,
*>                                         dimension (LWORK)
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          IF N <= 1,                LWORK must be at least 1.
*>          If JOBZ  = 'N' and N > 2, LWORK must be at least 2*N.
*>          If JOBZ  = 'V' and N > 2, LWORK must be at least
*>                         ( 1 + 5*N + 2*N**2 ).
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal sizes of the WORK and IWORK
*>          arrays, returns these values as the first entries of the WORK
*>          and IWORK arrays, and no error message related to LWORK or
*>          LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK.
*>          If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
*>          If JOBZ  = 'V' and N > 2, LIWORK must be at least 3 + 5*N.
*>
*>          If LIWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the optimal sizes of the WORK and
*>          IWORK arrays, returns these values as the first entries of
*>          the WORK and IWORK arrays, and no error message related to
*>          LWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the algorithm failed to converge; i
*>                off-diagonal elements of an intermediate tridiagonal
*>                form did not converge to zero.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hbevd
*
*  =====================================================================
      SUBROUTINE DSBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ,
     $                   WORK,
     $                   LWORK, IWORK, LIWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, UPLO
      INTEGER            INFO, KD, LDAB, LDZ, LIWORK, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER, LQUERY, WANTZ
      INTEGER            IINFO, INDE, INDWK2, INDWRK, ISCALE, LIWMIN,
     $                   LLWRK2, LWMIN
      DOUBLE PRECISION   ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
     $                   SMLNUM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANSB
      EXTERNAL           LSAME, DLAMCH, DLANSB
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMM, DLACPY, DLASCL, DSBTRD, DSCAL,
     $                   DSTEDC,
     $                   DSTERF, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
      LOWER = LSAME( UPLO, 'L' )
      LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
      INFO = 0
      IF( N.LE.1 ) THEN
         LIWMIN = 1
         LWMIN = 1
      ELSE
         IF( WANTZ ) THEN
            LIWMIN = 3 + 5*N
            LWMIN = 1 + 5*N + 2*N**2
         ELSE
            LIWMIN = 1
            LWMIN = 2*N
         END IF
      END IF
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( KD.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDAB.LT.KD+1 ) THEN
         INFO = -6
      ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
         INFO = -9
      END IF
*
      IF( INFO.EQ.0 ) THEN
         WORK( 1 ) = LWMIN
         IWORK( 1 ) = LIWMIN
*
         IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -11
         ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -13
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSBEVD', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( N.EQ.1 ) THEN
         W( 1 ) = AB( 1, 1 )
         IF( WANTZ )
     $      Z( 1, 1 ) = ONE
         RETURN
      END IF
*
*     Get machine constants.
*
      SAFMIN = DLAMCH( 'Safe minimum' )
      EPS = DLAMCH( 'Precision' )
      SMLNUM = SAFMIN / EPS
      BIGNUM = ONE / SMLNUM
      RMIN = SQRT( SMLNUM )
      RMAX = SQRT( BIGNUM )
*
*     Scale matrix to allowable range, if necessary.
*
      ANRM = DLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK )
      ISCALE = 0
      IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
         ISCALE = 1
         SIGMA = RMIN / ANRM
      ELSE IF( ANRM.GT.RMAX ) THEN
         ISCALE = 1
         SIGMA = RMAX / ANRM
      END IF
      IF( ISCALE.EQ.1 ) THEN
         IF( LOWER ) THEN
            CALL DLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB,
     $                   INFO )
         ELSE
            CALL DLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB,
     $                   INFO )
         END IF
      END IF
*
*     Call DSBTRD to reduce symmetric band matrix to tridiagonal form.
*
      INDE = 1
      INDWRK = INDE + N
      INDWK2 = INDWRK + N*N
      LLWRK2 = LWORK - INDWK2 + 1
      CALL DSBTRD( JOBZ, UPLO, N, KD, AB, LDAB, W, WORK( INDE ), Z,
     $             LDZ,
     $             WORK( INDWRK ), IINFO )
*
*     For eigenvalues only, call DSTERF.  For eigenvectors, call SSTEDC.
*
      IF( .NOT.WANTZ ) THEN
         CALL DSTERF( N, W, WORK( INDE ), INFO )
      ELSE
         CALL DSTEDC( 'I', N, W, WORK( INDE ), WORK( INDWRK ), N,
     $                WORK( INDWK2 ), LLWRK2, IWORK, LIWORK, INFO )
         CALL DGEMM( 'N', 'N', N, N, N, ONE, Z, LDZ, WORK( INDWRK ),
     $               N,
     $               ZERO, WORK( INDWK2 ), N )
         CALL DLACPY( 'A', N, N, WORK( INDWK2 ), N, Z, LDZ )
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
      IF( ISCALE.EQ.1 )
     $   CALL DSCAL( N, ONE / SIGMA, W, 1 )
*
      WORK( 1 ) = LWMIN
      IWORK( 1 ) = LIWMIN
      RETURN
*
*     End of DSBEVD
*
      END
*> \brief <b> DSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSBEVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbevx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbevx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbevx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL,
*                          VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
*                          IFAIL, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, RANGE, UPLO
*       INTEGER            IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N
*       DOUBLE PRECISION   ABSTOL, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            IFAIL( * ), IWORK( * )
*       DOUBLE PRECISION   AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ),
*      $                   Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSBEVX computes selected eigenvalues and, optionally, eigenvectors
*> of a real symmetric band matrix A.  Eigenvalues and eigenvectors can
*> be selected by specifying either a range of values or a range of
*> indices for the desired eigenvalues.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*>          RANGE is CHARACTER*1
*>          = 'A': all eigenvalues will be found;
*>          = 'V': all eigenvalues in the half-open interval (VL,VU]
*>                 will be found;
*>          = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB, N)
*>          On entry, the upper or lower triangle of the symmetric band
*>          matrix A, stored in the first KD+1 rows of the array.  The
*>          j-th column of A is stored in the j-th column of the array AB
*>          as follows:
*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*>
*>          On exit, AB is overwritten by values generated during the
*>          reduction to tridiagonal form.  If UPLO = 'U', the first
*>          superdiagonal and the diagonal of the tridiagonal matrix T
*>          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
*>          the diagonal and first subdiagonal of T are returned in the
*>          first two rows of AB.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD + 1.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ, N)
*>          If JOBZ = 'V', the N-by-N orthogonal matrix used in the
*>                         reduction to tridiagonal form.
*>          If JOBZ = 'N', the array Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q.  If JOBZ = 'V', then
*>          LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is DOUBLE PRECISION
*>          If RANGE='V', the lower bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*>          VU is DOUBLE PRECISION
*>          If RANGE='V', the upper bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*>          IL is INTEGER
*>          If RANGE='I', the index of the
*>          smallest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*>          IU is INTEGER
*>          If RANGE='I', the index of the
*>          largest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*>          ABSTOL is DOUBLE PRECISION
*>          The absolute error tolerance for the eigenvalues.
*>          An approximate eigenvalue is accepted as converged
*>          when it is determined to lie in an interval [a,b]
*>          of width less than or equal to
*>
*>                  ABSTOL + EPS *   max( |a|,|b| ) ,
*>
*>          where EPS is the machine precision.  If ABSTOL is less than
*>          or equal to zero, then  EPS*|T|  will be used in its place,
*>          where |T| is the 1-norm of the tridiagonal matrix obtained
*>          by reducing AB to tridiagonal form.
*>
*>          Eigenvalues will be computed most accurately when ABSTOL is
*>          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*>          If this routine returns with INFO>0, indicating that some
*>          eigenvectors did not converge, try setting ABSTOL to
*>          2*DLAMCH('S').
*>
*>          See "Computing Small Singular Values of Bidiagonal Matrices
*>          with Guaranteed High Relative Accuracy," by Demmel and
*>          Kahan, LAPACK Working Note #3.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The total number of eigenvalues found.  0 <= M <= N.
*>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          The first M elements contain the selected eigenvalues in
*>          ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
*>          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*>          contain the orthonormal eigenvectors of the matrix A
*>          corresponding to the selected eigenvalues, with the i-th
*>          column of Z holding the eigenvector associated with W(i).
*>          If an eigenvector fails to converge, then that column of Z
*>          contains the latest approximation to the eigenvector, and the
*>          index of the eigenvector is returned in IFAIL.
*>          If JOBZ = 'N', then Z is not referenced.
*>          Note: the user must ensure that at least max(1,M) columns are
*>          supplied in the array Z; if RANGE = 'V', the exact value of M
*>          is not known in advance and an upper bound must be used.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (7*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*>          IFAIL is INTEGER array, dimension (N)
*>          If JOBZ = 'V', then if INFO = 0, the first M elements of
*>          IFAIL are zero.  If INFO > 0, then IFAIL contains the
*>          indices of the eigenvectors that failed to converge.
*>          If JOBZ = 'N', then IFAIL is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = i, then i eigenvectors failed to converge.
*>                Their indices are stored in array IFAIL.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hbevx
*
*  =====================================================================
      SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ,
     $                   VL,
     $                   VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
     $                   IFAIL, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, RANGE, UPLO
      INTEGER            IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N
      DOUBLE PRECISION   ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            IFAIL( * ), IWORK( * )
      DOUBLE PRECISION   AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ),
     $                   Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLEIG, INDEIG, LOWER, TEST, VALEIG, WANTZ
      CHARACTER          ORDER
      INTEGER            I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
     $                   INDISP, INDIWO, INDWRK, ISCALE, ITMP1, J, JJ,
     $                   NSPLIT
      DOUBLE PRECISION   ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
     $                   SIGMA, SMLNUM, TMP1, VLL, VUU
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANSB
      EXTERNAL           LSAME, DLAMCH, DLANSB
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGEMV, DLACPY, DLASCL, DSBTRD,
     $                   DSCAL,
     $                   DSTEBZ, DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
      ALLEIG = LSAME( RANGE, 'A' )
      VALEIG = LSAME( RANGE, 'V' )
      INDEIG = LSAME( RANGE, 'I' )
      LOWER = LSAME( UPLO, 'L' )
*
      INFO = 0
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
         INFO = -2
      ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( KD.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDAB.LT.KD+1 ) THEN
         INFO = -7
      ELSE IF( WANTZ .AND. LDQ.LT.MAX( 1, N ) ) THEN
         INFO = -9
      ELSE
         IF( VALEIG ) THEN
            IF( N.GT.0 .AND. VU.LE.VL )
     $         INFO = -11
         ELSE IF( INDEIG ) THEN
            IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
               INFO = -12
            ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
               INFO = -13
            END IF
         END IF
      END IF
      IF( INFO.EQ.0 ) THEN
         IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
     $      INFO = -18
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSBEVX', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      M = 0
      IF( N.EQ.0 )
     $   RETURN
*
      IF( N.EQ.1 ) THEN
         M = 1
         IF( LOWER ) THEN
            TMP1 = AB( 1, 1 )
         ELSE
            TMP1 = AB( KD+1, 1 )
         END IF
         IF( VALEIG ) THEN
            IF( .NOT.( VL.LT.TMP1 .AND. VU.GE.TMP1 ) )
     $         M = 0
         END IF
         IF( M.EQ.1 ) THEN
            W( 1 ) = TMP1
            IF( WANTZ )
     $         Z( 1, 1 ) = ONE
         END IF
         RETURN
      END IF
*
*     Get machine constants.
*
      SAFMIN = DLAMCH( 'Safe minimum' )
      EPS = DLAMCH( 'Precision' )
      SMLNUM = SAFMIN / EPS
      BIGNUM = ONE / SMLNUM
      RMIN = SQRT( SMLNUM )
      RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
*     Scale matrix to allowable range, if necessary.
*
      ISCALE = 0
      ABSTLL = ABSTOL
      IF( VALEIG ) THEN
         VLL = VL
         VUU = VU
      ELSE
         VLL = ZERO
         VUU = ZERO
      END IF
      ANRM = DLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK )
      IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
         ISCALE = 1
         SIGMA = RMIN / ANRM
      ELSE IF( ANRM.GT.RMAX ) THEN
         ISCALE = 1
         SIGMA = RMAX / ANRM
      END IF
      IF( ISCALE.EQ.1 ) THEN
         IF( LOWER ) THEN
            CALL DLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB,
     $                   INFO )
         ELSE
            CALL DLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB,
     $                   INFO )
         END IF
         IF( ABSTOL.GT.0 )
     $      ABSTLL = ABSTOL*SIGMA
         IF( VALEIG ) THEN
            VLL = VL*SIGMA
            VUU = VU*SIGMA
         END IF
      END IF
*
*     Call DSBTRD to reduce symmetric band matrix to tridiagonal form.
*
      INDD = 1
      INDE = INDD + N
      INDWRK = INDE + N
      CALL DSBTRD( JOBZ, UPLO, N, KD, AB, LDAB, WORK( INDD ),
     $             WORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO )
*
*     If all eigenvalues are desired and ABSTOL is less than or equal
*     to zero, then call DSTERF or SSTEQR.  If this fails for some
*     eigenvalue, then try DSTEBZ.
*
      TEST = .FALSE.
      IF (INDEIG) THEN
         IF (IL.EQ.1 .AND. IU.EQ.N) THEN
            TEST = .TRUE.
         END IF
      END IF
      IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
         CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
         INDEE = INDWRK + 2*N
         IF( .NOT.WANTZ ) THEN
            CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
            CALL DSTERF( N, W, WORK( INDEE ), INFO )
         ELSE
            CALL DLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
            CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
            CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
     $                   WORK( INDWRK ), INFO )
            IF( INFO.EQ.0 ) THEN
               DO 10 I = 1, N
                  IFAIL( I ) = 0
   10          CONTINUE
            END IF
         END IF
         IF( INFO.EQ.0 ) THEN
            M = N
            GO TO 30
         END IF
         INFO = 0
      END IF
*
*     Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
*
      IF( WANTZ ) THEN
         ORDER = 'B'
      ELSE
         ORDER = 'E'
      END IF
      INDIBL = 1
      INDISP = INDIBL + N
      INDIWO = INDISP + N
      CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
     $             WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
     $             IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
     $             IWORK( INDIWO ), INFO )
*
      IF( WANTZ ) THEN
         CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
     $                IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
     $                WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
*
*        Apply orthogonal matrix used in reduction to tridiagonal
*        form to eigenvectors returned by DSTEIN.
*
         DO 20 J = 1, M
            CALL DCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
            CALL DGEMV( 'N', N, N, ONE, Q, LDQ, WORK, 1, ZERO,
     $                  Z( 1, J ), 1 )
   20    CONTINUE
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
   30 CONTINUE
      IF( ISCALE.EQ.1 ) THEN
         IF( INFO.EQ.0 ) THEN
            IMAX = M
         ELSE
            IMAX = INFO - 1
         END IF
         CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
      END IF
*
*     If eigenvalues are not in order, then sort them, along with
*     eigenvectors.
*
      IF( WANTZ ) THEN
         DO 50 J = 1, M - 1
            I = 0
            TMP1 = W( J )
            DO 40 JJ = J + 1, M
               IF( W( JJ ).LT.TMP1 ) THEN
                  I = JJ
                  TMP1 = W( JJ )
               END IF
   40       CONTINUE
*
            IF( I.NE.0 ) THEN
               ITMP1 = IWORK( INDIBL+I-1 )
               W( I ) = W( J )
               IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
               W( J ) = TMP1
               IWORK( INDIBL+J-1 ) = ITMP1
               CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
               IF( INFO.NE.0 ) THEN
                  ITMP1 = IFAIL( I )
                  IFAIL( I ) = IFAIL( J )
                  IFAIL( J ) = ITMP1
               END IF
            END IF
   50    CONTINUE
      END IF
*
      RETURN
*
*     End of DSBEVX
*
      END
*> \brief \b DSBGST
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSBGST + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbgst.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbgst.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbgst.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSBGST( VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, X,
*                          LDX, WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO, VECT
*       INTEGER            INFO, KA, KB, LDAB, LDBB, LDX, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * ), BB( LDBB, * ), WORK( * ),
*      $                   X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSBGST reduces a real symmetric-definite banded generalized
*> eigenproblem  A*x = lambda*B*x  to standard form  C*y = lambda*y,
*> such that C has the same bandwidth as A.
*>
*> B must have been previously factorized as S**T*S by DPBSTF, using a
*> split Cholesky factorization. A is overwritten by C = X**T*A*X, where
*> X = S**(-1)*Q and Q is an orthogonal matrix chosen to preserve the
*> bandwidth of A.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] VECT
*> \verbatim
*>          VECT is CHARACTER*1
*>          = 'N':  do not form the transformation matrix X;
*>          = 'V':  form X.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A and B.  N >= 0.
*> \endverbatim
*>
*> \param[in] KA
*> \verbatim
*>          KA is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KA >= 0.
*> \endverbatim
*>
*> \param[in] KB
*> \verbatim
*>          KB is INTEGER
*>          The number of superdiagonals of the matrix B if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KA >= KB >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          On entry, the upper or lower triangle of the symmetric band
*>          matrix A, stored in the first ka+1 rows of the array.  The
*>          j-th column of A is stored in the j-th column of the array AB
*>          as follows:
*>          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
*>
*>          On exit, the transformed matrix X**T*A*X, stored in the same
*>          format as A.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KA+1.
*> \endverbatim
*>
*> \param[in] BB
*> \verbatim
*>          BB is DOUBLE PRECISION array, dimension (LDBB,N)
*>          The banded factor S from the split Cholesky factorization of
*>          B, as returned by DPBSTF, stored in the first KB+1 rows of
*>          the array.
*> \endverbatim
*>
*> \param[in] LDBB
*> \verbatim
*>          LDBB is INTEGER
*>          The leading dimension of the array BB.  LDBB >= KB+1.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,N)
*>          If VECT = 'V', the n-by-n matrix X.
*>          If VECT = 'N', the array X is not referenced.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.
*>          LDX >= max(1,N) if VECT = 'V'; LDX >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hbgst
*
*  =====================================================================
      SUBROUTINE DSBGST( VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB,
     $                   X,
     $                   LDX, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO, VECT
      INTEGER            INFO, KA, KB, LDAB, LDBB, LDX, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * ), BB( LDBB, * ), WORK( * ),
     $                   X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPDATE, UPPER, WANTX
      INTEGER            I, I0, I1, I2, INCA, J, J1, J1T, J2, J2T, K,
     $                   KA1, KB1, KBT, L, M, NR, NRT, NX
      DOUBLE PRECISION   BII, RA, RA1, T
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGER, DLAR2V, DLARGV, DLARTG, DLARTV,
     $                   DLASET,
     $                   DROT, DSCAL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      WANTX = LSAME( VECT, 'V' )
      UPPER = LSAME( UPLO, 'U' )
      KA1 = KA + 1
      KB1 = KB + 1
      INFO = 0
      IF( .NOT.WANTX .AND. .NOT.LSAME( VECT, 'N' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( KA.LT.0 ) THEN
         INFO = -4
      ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN
         INFO = -5
      ELSE IF( LDAB.LT.KA+1 ) THEN
         INFO = -7
      ELSE IF( LDBB.LT.KB+1 ) THEN
         INFO = -9
      ELSE IF( LDX.LT.1 .OR. WANTX .AND. LDX.LT.MAX( 1, N ) ) THEN
         INFO = -11
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSBGST', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      INCA = LDAB*KA1
*
*     Initialize X to the unit matrix, if needed
*
      IF( WANTX )
     $   CALL DLASET( 'Full', N, N, ZERO, ONE, X, LDX )
*
*     Set M to the splitting point m. It must be the same value as is
*     used in DPBSTF. The chosen value allows the arrays WORK and RWORK
*     to be of dimension (N).
*
      M = ( N+KB ) / 2
*
*     The routine works in two phases, corresponding to the two halves
*     of the split Cholesky factorization of B as S**T*S where
*
*     S = ( U    )
*         ( M  L )
*
*     with U upper triangular of order m, and L lower triangular of
*     order n-m. S has the same bandwidth as B.
*
*     S is treated as a product of elementary matrices:
*
*     S = S(m)*S(m-1)*...*S(2)*S(1)*S(m+1)*S(m+2)*...*S(n-1)*S(n)
*
*     where S(i) is determined by the i-th row of S.
*
*     In phase 1, the index i takes the values n, n-1, ... , m+1;
*     in phase 2, it takes the values 1, 2, ... , m.
*
*     For each value of i, the current matrix A is updated by forming
*     inv(S(i))**T*A*inv(S(i)). This creates a triangular bulge outside
*     the band of A. The bulge is then pushed down toward the bottom of
*     A in phase 1, and up toward the top of A in phase 2, by applying
*     plane rotations.
*
*     There are kb*(kb+1)/2 elements in the bulge, but at most 2*kb-1
*     of them are linearly independent, so annihilating a bulge requires
*     only 2*kb-1 plane rotations. The rotations are divided into a 1st
*     set of kb-1 rotations, and a 2nd set of kb rotations.
*
*     Wherever possible, rotations are generated and applied in vector
*     operations of length NR between the indices J1 and J2 (sometimes
*     replaced by modified values NRT, J1T or J2T).
*
*     The cosines and sines of the rotations are stored in the array
*     WORK. The cosines of the 1st set of rotations are stored in
*     elements n+2:n+m-kb-1 and the sines of the 1st set in elements
*     2:m-kb-1; the cosines of the 2nd set are stored in elements
*     n+m-kb+1:2*n and the sines of the second set in elements m-kb+1:n.
*
*     The bulges are not formed explicitly; nonzero elements outside the
*     band are created only when they are required for generating new
*     rotations; they are stored in the array WORK, in positions where
*     they are later overwritten by the sines of the rotations which
*     annihilate them.
*
*     **************************** Phase 1 *****************************
*
*     The logical structure of this phase is:
*
*     UPDATE = .TRUE.
*     DO I = N, M + 1, -1
*        use S(i) to update A and create a new bulge
*        apply rotations to push all bulges KA positions downward
*     END DO
*     UPDATE = .FALSE.
*     DO I = M + KA + 1, N - 1
*        apply rotations to push all bulges KA positions downward
*     END DO
*
*     To avoid duplicating code, the two loops are merged.
*
      UPDATE = .TRUE.
      I = N + 1
   10 CONTINUE
      IF( UPDATE ) THEN
         I = I - 1
         KBT = MIN( KB, I-1 )
         I0 = I - 1
         I1 = MIN( N, I+KA )
         I2 = I - KBT + KA1
         IF( I.LT.M+1 ) THEN
            UPDATE = .FALSE.
            I = I + 1
            I0 = M
            IF( KA.EQ.0 )
     $         GO TO 480
            GO TO 10
         END IF
      ELSE
         I = I + KA
         IF( I.GT.N-1 )
     $      GO TO 480
      END IF
*
      IF( UPPER ) THEN
*
*        Transform A, working with the upper triangle
*
         IF( UPDATE ) THEN
*
*           Form  inv(S(i))**T * A * inv(S(i))
*
            BII = BB( KB1, I )
            DO 20 J = I, I1
               AB( I-J+KA1, J ) = AB( I-J+KA1, J ) / BII
   20       CONTINUE
            DO 30 J = MAX( 1, I-KA ), I
               AB( J-I+KA1, I ) = AB( J-I+KA1, I ) / BII
   30       CONTINUE
            DO 60 K = I - KBT, I - 1
               DO 40 J = I - KBT, K
                  AB( J-K+KA1, K ) = AB( J-K+KA1, K ) -
     $                               BB( J-I+KB1, I )*AB( K-I+KA1, I ) -
     $                               BB( K-I+KB1, I )*AB( J-I+KA1, I ) +
     $                               AB( KA1, I )*BB( J-I+KB1, I )*
     $                               BB( K-I+KB1, I )
   40          CONTINUE
               DO 50 J = MAX( 1, I-KA ), I - KBT - 1
                  AB( J-K+KA1, K ) = AB( J-K+KA1, K ) -
     $                               BB( K-I+KB1, I )*AB( J-I+KA1, I )
   50          CONTINUE
   60       CONTINUE
            DO 80 J = I, I1
               DO 70 K = MAX( J-KA, I-KBT ), I - 1
                  AB( K-J+KA1, J ) = AB( K-J+KA1, J ) -
     $                               BB( K-I+KB1, I )*AB( I-J+KA1, J )
   70          CONTINUE
   80       CONTINUE
*
            IF( WANTX ) THEN
*
*              post-multiply X by inv(S(i))
*
               CALL DSCAL( N-M, ONE / BII, X( M+1, I ), 1 )
               IF( KBT.GT.0 )
     $            CALL DGER( N-M, KBT, -ONE, X( M+1, I ), 1,
     $                       BB( KB1-KBT, I ), 1, X( M+1, I-KBT ), LDX )
            END IF
*
*           store a(i,i1) in RA1 for use in next loop over K
*
            RA1 = AB( I-I1+KA1, I1 )
         END IF
*
*        Generate and apply vectors of rotations to chase all the
*        existing bulges KA positions down toward the bottom of the
*        band
*
         DO 130 K = 1, KB - 1
            IF( UPDATE ) THEN
*
*              Determine the rotations which would annihilate the bulge
*              which has in theory just been created
*
               IF( I-K+KA.LT.N .AND. I-K.GT.1 ) THEN
*
*                 generate rotation to annihilate a(i,i-k+ka+1)
*
                  CALL DLARTG( AB( K+1, I-K+KA ), RA1,
     $                         WORK( N+I-K+KA-M ), WORK( I-K+KA-M ),
     $                         RA )
*
*                 create nonzero element a(i-k,i-k+ka+1) outside the
*                 band and store it in WORK(i-k)
*
                  T = -BB( KB1-K, I )*RA1
                  WORK( I-K ) = WORK( N+I-K+KA-M )*T -
     $                          WORK( I-K+KA-M )*AB( 1, I-K+KA )
                  AB( 1, I-K+KA ) = WORK( I-K+KA-M )*T +
     $                              WORK( N+I-K+KA-M )*AB( 1, I-K+KA )
                  RA1 = RA
               END IF
            END IF
            J2 = I - K - 1 + MAX( 1, K-I0+2 )*KA1
            NR = ( N-J2+KA ) / KA1
            J1 = J2 + ( NR-1 )*KA1
            IF( UPDATE ) THEN
               J2T = MAX( J2, I+2*KA-K+1 )
            ELSE
               J2T = J2
            END IF
            NRT = ( N-J2T+KA ) / KA1
            DO 90 J = J2T, J1, KA1
*
*              create nonzero element a(j-ka,j+1) outside the band
*              and store it in WORK(j-m)
*
               WORK( J-M ) = WORK( J-M )*AB( 1, J+1 )
               AB( 1, J+1 ) = WORK( N+J-M )*AB( 1, J+1 )
   90       CONTINUE
*
*           generate rotations in 1st set to annihilate elements which
*           have been created outside the band
*
            IF( NRT.GT.0 )
     $         CALL DLARGV( NRT, AB( 1, J2T ), INCA, WORK( J2T-M ),
     $                      KA1,
     $                      WORK( N+J2T-M ), KA1 )
            IF( NR.GT.0 ) THEN
*
*              apply rotations in 1st set from the right
*
               DO 100 L = 1, KA - 1
                  CALL DLARTV( NR, AB( KA1-L, J2 ), INCA,
     $                         AB( KA-L, J2+1 ), INCA, WORK( N+J2-M ),
     $                         WORK( J2-M ), KA1 )
  100          CONTINUE
*
*              apply rotations in 1st set from both sides to diagonal
*              blocks
*
               CALL DLAR2V( NR, AB( KA1, J2 ), AB( KA1, J2+1 ),
     $                      AB( KA, J2+1 ), INCA, WORK( N+J2-M ),
     $                      WORK( J2-M ), KA1 )
*
            END IF
*
*           start applying rotations in 1st set from the left
*
            DO 110 L = KA - 1, KB - K + 1, -1
               NRT = ( N-J2+L ) / KA1
               IF( NRT.GT.0 )
     $            CALL DLARTV( NRT, AB( L, J2+KA1-L ), INCA,
     $                         AB( L+1, J2+KA1-L ), INCA,
     $                         WORK( N+J2-M ), WORK( J2-M ), KA1 )
  110       CONTINUE
*
            IF( WANTX ) THEN
*
*              post-multiply X by product of rotations in 1st set
*
               DO 120 J = J2, J1, KA1
                  CALL DROT( N-M, X( M+1, J ), 1, X( M+1, J+1 ), 1,
     $                       WORK( N+J-M ), WORK( J-M ) )
  120          CONTINUE
            END IF
  130    CONTINUE
*
         IF( UPDATE ) THEN
            IF( I2.LE.N .AND. KBT.GT.0 ) THEN
*
*              create nonzero element a(i-kbt,i-kbt+ka+1) outside the
*              band and store it in WORK(i-kbt)
*
               WORK( I-KBT ) = -BB( KB1-KBT, I )*RA1
            END IF
         END IF
*
         DO 170 K = KB, 1, -1
            IF( UPDATE ) THEN
               J2 = I - K - 1 + MAX( 2, K-I0+1 )*KA1
            ELSE
               J2 = I - K - 1 + MAX( 1, K-I0+1 )*KA1
            END IF
*
*           finish applying rotations in 2nd set from the left
*
            DO 140 L = KB - K, 1, -1
               NRT = ( N-J2+KA+L ) / KA1
               IF( NRT.GT.0 )
     $            CALL DLARTV( NRT, AB( L, J2-L+1 ), INCA,
     $                         AB( L+1, J2-L+1 ), INCA, WORK( N+J2-KA ),
     $                         WORK( J2-KA ), KA1 )
  140       CONTINUE
            NR = ( N-J2+KA ) / KA1
            J1 = J2 + ( NR-1 )*KA1
            DO 150 J = J1, J2, -KA1
               WORK( J ) = WORK( J-KA )
               WORK( N+J ) = WORK( N+J-KA )
  150       CONTINUE
            DO 160 J = J2, J1, KA1
*
*              create nonzero element a(j-ka,j+1) outside the band
*              and store it in WORK(j)
*
               WORK( J ) = WORK( J )*AB( 1, J+1 )
               AB( 1, J+1 ) = WORK( N+J )*AB( 1, J+1 )
  160       CONTINUE
            IF( UPDATE ) THEN
               IF( I-K.LT.N-KA .AND. K.LE.KBT )
     $            WORK( I-K+KA ) = WORK( I-K )
            END IF
  170    CONTINUE
*
         DO 210 K = KB, 1, -1
            J2 = I - K - 1 + MAX( 1, K-I0+1 )*KA1
            NR = ( N-J2+KA ) / KA1
            J1 = J2 + ( NR-1 )*KA1
            IF( NR.GT.0 ) THEN
*
*              generate rotations in 2nd set to annihilate elements
*              which have been created outside the band
*
               CALL DLARGV( NR, AB( 1, J2 ), INCA, WORK( J2 ), KA1,
     $                      WORK( N+J2 ), KA1 )
*
*              apply rotations in 2nd set from the right
*
               DO 180 L = 1, KA - 1
                  CALL DLARTV( NR, AB( KA1-L, J2 ), INCA,
     $                         AB( KA-L, J2+1 ), INCA, WORK( N+J2 ),
     $                         WORK( J2 ), KA1 )
  180          CONTINUE
*
*              apply rotations in 2nd set from both sides to diagonal
*              blocks
*
               CALL DLAR2V( NR, AB( KA1, J2 ), AB( KA1, J2+1 ),
     $                      AB( KA, J2+1 ), INCA, WORK( N+J2 ),
     $                      WORK( J2 ), KA1 )
*
            END IF
*
*           start applying rotations in 2nd set from the left
*
            DO 190 L = KA - 1, KB - K + 1, -1
               NRT = ( N-J2+L ) / KA1
               IF( NRT.GT.0 )
     $            CALL DLARTV( NRT, AB( L, J2+KA1-L ), INCA,
     $                         AB( L+1, J2+KA1-L ), INCA, WORK( N+J2 ),
     $                         WORK( J2 ), KA1 )
  190       CONTINUE
*
            IF( WANTX ) THEN
*
*              post-multiply X by product of rotations in 2nd set
*
               DO 200 J = J2, J1, KA1
                  CALL DROT( N-M, X( M+1, J ), 1, X( M+1, J+1 ), 1,
     $                       WORK( N+J ), WORK( J ) )
  200          CONTINUE
            END IF
  210    CONTINUE
*
         DO 230 K = 1, KB - 1
            J2 = I - K - 1 + MAX( 1, K-I0+2 )*KA1
*
*           finish applying rotations in 1st set from the left
*
            DO 220 L = KB - K, 1, -1
               NRT = ( N-J2+L ) / KA1
               IF( NRT.GT.0 )
     $            CALL DLARTV( NRT, AB( L, J2+KA1-L ), INCA,
     $                         AB( L+1, J2+KA1-L ), INCA,
     $                         WORK( N+J2-M ), WORK( J2-M ), KA1 )
  220       CONTINUE
  230    CONTINUE
*
         IF( KB.GT.1 ) THEN
            DO 240 J = N - 1, I - KB + 2*KA + 1, -1
               WORK( N+J-M ) = WORK( N+J-KA-M )
               WORK( J-M ) = WORK( J-KA-M )
  240       CONTINUE
         END IF
*
      ELSE
*
*        Transform A, working with the lower triangle
*
         IF( UPDATE ) THEN
*
*           Form  inv(S(i))**T * A * inv(S(i))
*
            BII = BB( 1, I )
            DO 250 J = I, I1
               AB( J-I+1, I ) = AB( J-I+1, I ) / BII
  250       CONTINUE
            DO 260 J = MAX( 1, I-KA ), I
               AB( I-J+1, J ) = AB( I-J+1, J ) / BII
  260       CONTINUE
            DO 290 K = I - KBT, I - 1
               DO 270 J = I - KBT, K
                  AB( K-J+1, J ) = AB( K-J+1, J ) -
     $                             BB( I-J+1, J )*AB( I-K+1, K ) -
     $                             BB( I-K+1, K )*AB( I-J+1, J ) +
     $                             AB( 1, I )*BB( I-J+1, J )*
     $                             BB( I-K+1, K )
  270          CONTINUE
               DO 280 J = MAX( 1, I-KA ), I - KBT - 1
                  AB( K-J+1, J ) = AB( K-J+1, J ) -
     $                             BB( I-K+1, K )*AB( I-J+1, J )
  280          CONTINUE
  290       CONTINUE
            DO 310 J = I, I1
               DO 300 K = MAX( J-KA, I-KBT ), I - 1
                  AB( J-K+1, K ) = AB( J-K+1, K ) -
     $                             BB( I-K+1, K )*AB( J-I+1, I )
  300          CONTINUE
  310       CONTINUE
*
            IF( WANTX ) THEN
*
*              post-multiply X by inv(S(i))
*
               CALL DSCAL( N-M, ONE / BII, X( M+1, I ), 1 )
               IF( KBT.GT.0 )
     $            CALL DGER( N-M, KBT, -ONE, X( M+1, I ), 1,
     $                       BB( KBT+1, I-KBT ), LDBB-1,
     $                       X( M+1, I-KBT ), LDX )
            END IF
*
*           store a(i1,i) in RA1 for use in next loop over K
*
            RA1 = AB( I1-I+1, I )
         END IF
*
*        Generate and apply vectors of rotations to chase all the
*        existing bulges KA positions down toward the bottom of the
*        band
*
         DO 360 K = 1, KB - 1
            IF( UPDATE ) THEN
*
*              Determine the rotations which would annihilate the bulge
*              which has in theory just been created
*
               IF( I-K+KA.LT.N .AND. I-K.GT.1 ) THEN
*
*                 generate rotation to annihilate a(i-k+ka+1,i)
*
                  CALL DLARTG( AB( KA1-K, I ), RA1,
     $                         WORK( N+I-K+KA-M ),
     $                         WORK( I-K+KA-M ), RA )
*
*                 create nonzero element a(i-k+ka+1,i-k) outside the
*                 band and store it in WORK(i-k)
*
                  T = -BB( K+1, I-K )*RA1
                  WORK( I-K ) = WORK( N+I-K+KA-M )*T -
     $                          WORK( I-K+KA-M )*AB( KA1, I-K )
                  AB( KA1, I-K ) = WORK( I-K+KA-M )*T +
     $                             WORK( N+I-K+KA-M )*AB( KA1, I-K )
                  RA1 = RA
               END IF
            END IF
            J2 = I - K - 1 + MAX( 1, K-I0+2 )*KA1
            NR = ( N-J2+KA ) / KA1
            J1 = J2 + ( NR-1 )*KA1
            IF( UPDATE ) THEN
               J2T = MAX( J2, I+2*KA-K+1 )
            ELSE
               J2T = J2
            END IF
            NRT = ( N-J2T+KA ) / KA1
            DO 320 J = J2T, J1, KA1
*
*              create nonzero element a(j+1,j-ka) outside the band
*              and store it in WORK(j-m)
*
               WORK( J-M ) = WORK( J-M )*AB( KA1, J-KA+1 )
               AB( KA1, J-KA+1 ) = WORK( N+J-M )*AB( KA1, J-KA+1 )
  320       CONTINUE
*
*           generate rotations in 1st set to annihilate elements which
*           have been created outside the band
*
            IF( NRT.GT.0 )
     $         CALL DLARGV( NRT, AB( KA1, J2T-KA ), INCA,
     $                      WORK( J2T-M ),
     $                      KA1, WORK( N+J2T-M ), KA1 )
            IF( NR.GT.0 ) THEN
*
*              apply rotations in 1st set from the left
*
               DO 330 L = 1, KA - 1
                  CALL DLARTV( NR, AB( L+1, J2-L ), INCA,
     $                         AB( L+2, J2-L ), INCA, WORK( N+J2-M ),
     $                         WORK( J2-M ), KA1 )
  330          CONTINUE
*
*              apply rotations in 1st set from both sides to diagonal
*              blocks
*
               CALL DLAR2V( NR, AB( 1, J2 ), AB( 1, J2+1 ), AB( 2,
     $                      J2 ),
     $                      INCA, WORK( N+J2-M ), WORK( J2-M ), KA1 )
*
            END IF
*
*           start applying rotations in 1st set from the right
*
            DO 340 L = KA - 1, KB - K + 1, -1
               NRT = ( N-J2+L ) / KA1
               IF( NRT.GT.0 )
     $            CALL DLARTV( NRT, AB( KA1-L+1, J2 ), INCA,
     $                         AB( KA1-L, J2+1 ), INCA, WORK( N+J2-M ),
     $                         WORK( J2-M ), KA1 )
  340       CONTINUE
*
            IF( WANTX ) THEN
*
*              post-multiply X by product of rotations in 1st set
*
               DO 350 J = J2, J1, KA1
                  CALL DROT( N-M, X( M+1, J ), 1, X( M+1, J+1 ), 1,
     $                       WORK( N+J-M ), WORK( J-M ) )
  350          CONTINUE
            END IF
  360    CONTINUE
*
         IF( UPDATE ) THEN
            IF( I2.LE.N .AND. KBT.GT.0 ) THEN
*
*              create nonzero element a(i-kbt+ka+1,i-kbt) outside the
*              band and store it in WORK(i-kbt)
*
               WORK( I-KBT ) = -BB( KBT+1, I-KBT )*RA1
            END IF
         END IF
*
         DO 400 K = KB, 1, -1
            IF( UPDATE ) THEN
               J2 = I - K - 1 + MAX( 2, K-I0+1 )*KA1
            ELSE
               J2 = I - K - 1 + MAX( 1, K-I0+1 )*KA1
            END IF
*
*           finish applying rotations in 2nd set from the right
*
            DO 370 L = KB - K, 1, -1
               NRT = ( N-J2+KA+L ) / KA1
               IF( NRT.GT.0 )
     $            CALL DLARTV( NRT, AB( KA1-L+1, J2-KA ), INCA,
     $                         AB( KA1-L, J2-KA+1 ), INCA,
     $                         WORK( N+J2-KA ), WORK( J2-KA ), KA1 )
  370       CONTINUE
            NR = ( N-J2+KA ) / KA1
            J1 = J2 + ( NR-1 )*KA1
            DO 380 J = J1, J2, -KA1
               WORK( J ) = WORK( J-KA )
               WORK( N+J ) = WORK( N+J-KA )
  380       CONTINUE
            DO 390 J = J2, J1, KA1
*
*              create nonzero element a(j+1,j-ka) outside the band
*              and store it in WORK(j)
*
               WORK( J ) = WORK( J )*AB( KA1, J-KA+1 )
               AB( KA1, J-KA+1 ) = WORK( N+J )*AB( KA1, J-KA+1 )
  390       CONTINUE
            IF( UPDATE ) THEN
               IF( I-K.LT.N-KA .AND. K.LE.KBT )
     $            WORK( I-K+KA ) = WORK( I-K )
            END IF
  400    CONTINUE
*
         DO 440 K = KB, 1, -1
            J2 = I - K - 1 + MAX( 1, K-I0+1 )*KA1
            NR = ( N-J2+KA ) / KA1
            J1 = J2 + ( NR-1 )*KA1
            IF( NR.GT.0 ) THEN
*
*              generate rotations in 2nd set to annihilate elements
*              which have been created outside the band
*
               CALL DLARGV( NR, AB( KA1, J2-KA ), INCA, WORK( J2 ),
     $                      KA1,
     $                      WORK( N+J2 ), KA1 )
*
*              apply rotations in 2nd set from the left
*
               DO 410 L = 1, KA - 1
                  CALL DLARTV( NR, AB( L+1, J2-L ), INCA,
     $                         AB( L+2, J2-L ), INCA, WORK( N+J2 ),
     $                         WORK( J2 ), KA1 )
  410          CONTINUE
*
*              apply rotations in 2nd set from both sides to diagonal
*              blocks
*
               CALL DLAR2V( NR, AB( 1, J2 ), AB( 1, J2+1 ), AB( 2,
     $                      J2 ),
     $                      INCA, WORK( N+J2 ), WORK( J2 ), KA1 )
*
            END IF
*
*           start applying rotations in 2nd set from the right
*
            DO 420 L = KA - 1, KB - K + 1, -1
               NRT = ( N-J2+L ) / KA1
               IF( NRT.GT.0 )
     $            CALL DLARTV( NRT, AB( KA1-L+1, J2 ), INCA,
     $                         AB( KA1-L, J2+1 ), INCA, WORK( N+J2 ),
     $                         WORK( J2 ), KA1 )
  420       CONTINUE
*
            IF( WANTX ) THEN
*
*              post-multiply X by product of rotations in 2nd set
*
               DO 430 J = J2, J1, KA1
                  CALL DROT( N-M, X( M+1, J ), 1, X( M+1, J+1 ), 1,
     $                       WORK( N+J ), WORK( J ) )
  430          CONTINUE
            END IF
  440    CONTINUE
*
         DO 460 K = 1, KB - 1
            J2 = I - K - 1 + MAX( 1, K-I0+2 )*KA1
*
*           finish applying rotations in 1st set from the right
*
            DO 450 L = KB - K, 1, -1
               NRT = ( N-J2+L ) / KA1
               IF( NRT.GT.0 )
     $            CALL DLARTV( NRT, AB( KA1-L+1, J2 ), INCA,
     $                         AB( KA1-L, J2+1 ), INCA, WORK( N+J2-M ),
     $                         WORK( J2-M ), KA1 )
  450       CONTINUE
  460    CONTINUE
*
         IF( KB.GT.1 ) THEN
            DO 470 J = N - 1, I - KB + 2*KA + 1, -1
               WORK( N+J-M ) = WORK( N+J-KA-M )
               WORK( J-M ) = WORK( J-KA-M )
  470       CONTINUE
         END IF
*
      END IF
*
      GO TO 10
*
  480 CONTINUE
*
*     **************************** Phase 2 *****************************
*
*     The logical structure of this phase is:
*
*     UPDATE = .TRUE.
*     DO I = 1, M
*        use S(i) to update A and create a new bulge
*        apply rotations to push all bulges KA positions upward
*     END DO
*     UPDATE = .FALSE.
*     DO I = M - KA - 1, 2, -1
*        apply rotations to push all bulges KA positions upward
*     END DO
*
*     To avoid duplicating code, the two loops are merged.
*
      UPDATE = .TRUE.
      I = 0
  490 CONTINUE
      IF( UPDATE ) THEN
         I = I + 1
         KBT = MIN( KB, M-I )
         I0 = I + 1
         I1 = MAX( 1, I-KA )
         I2 = I + KBT - KA1
         IF( I.GT.M ) THEN
            UPDATE = .FALSE.
            I = I - 1
            I0 = M + 1
            IF( KA.EQ.0 )
     $         RETURN
            GO TO 490
         END IF
      ELSE
         I = I - KA
         IF( I.LT.2 )
     $      RETURN
      END IF
*
      IF( I.LT.M-KBT ) THEN
         NX = M
      ELSE
         NX = N
      END IF
*
      IF( UPPER ) THEN
*
*        Transform A, working with the upper triangle
*
         IF( UPDATE ) THEN
*
*           Form  inv(S(i))**T * A * inv(S(i))
*
            BII = BB( KB1, I )
            DO 500 J = I1, I
               AB( J-I+KA1, I ) = AB( J-I+KA1, I ) / BII
  500       CONTINUE
            DO 510 J = I, MIN( N, I+KA )
               AB( I-J+KA1, J ) = AB( I-J+KA1, J ) / BII
  510       CONTINUE
            DO 540 K = I + 1, I + KBT
               DO 520 J = K, I + KBT
                  AB( K-J+KA1, J ) = AB( K-J+KA1, J ) -
     $                               BB( I-J+KB1, J )*AB( I-K+KA1, K ) -
     $                               BB( I-K+KB1, K )*AB( I-J+KA1, J ) +
     $                               AB( KA1, I )*BB( I-J+KB1, J )*
     $                               BB( I-K+KB1, K )
  520          CONTINUE
               DO 530 J = I + KBT + 1, MIN( N, I+KA )
                  AB( K-J+KA1, J ) = AB( K-J+KA1, J ) -
     $                               BB( I-K+KB1, K )*AB( I-J+KA1, J )
  530          CONTINUE
  540       CONTINUE
            DO 560 J = I1, I
               DO 550 K = I + 1, MIN( J+KA, I+KBT )
                  AB( J-K+KA1, K ) = AB( J-K+KA1, K ) -
     $                               BB( I-K+KB1, K )*AB( J-I+KA1, I )
  550          CONTINUE
  560       CONTINUE
*
            IF( WANTX ) THEN
*
*              post-multiply X by inv(S(i))
*
               CALL DSCAL( NX, ONE / BII, X( 1, I ), 1 )
               IF( KBT.GT.0 )
     $            CALL DGER( NX, KBT, -ONE, X( 1, I ), 1, BB( KB,
     $                       I+1 ),
     $                       LDBB-1, X( 1, I+1 ), LDX )
            END IF
*
*           store a(i1,i) in RA1 for use in next loop over K
*
            RA1 = AB( I1-I+KA1, I )
         END IF
*
*        Generate and apply vectors of rotations to chase all the
*        existing bulges KA positions up toward the top of the band
*
         DO 610 K = 1, KB - 1
            IF( UPDATE ) THEN
*
*              Determine the rotations which would annihilate the bulge
*              which has in theory just been created
*
               IF( I+K-KA1.GT.0 .AND. I+K.LT.M ) THEN
*
*                 generate rotation to annihilate a(i+k-ka-1,i)
*
                  CALL DLARTG( AB( K+1, I ), RA1, WORK( N+I+K-KA ),
     $                         WORK( I+K-KA ), RA )
*
*                 create nonzero element a(i+k-ka-1,i+k) outside the
*                 band and store it in WORK(m-kb+i+k)
*
                  T = -BB( KB1-K, I+K )*RA1
                  WORK( M-KB+I+K ) = WORK( N+I+K-KA )*T -
     $                               WORK( I+K-KA )*AB( 1, I+K )
                  AB( 1, I+K ) = WORK( I+K-KA )*T +
     $                           WORK( N+I+K-KA )*AB( 1, I+K )
                  RA1 = RA
               END IF
            END IF
            J2 = I + K + 1 - MAX( 1, K+I0-M+1 )*KA1
            NR = ( J2+KA-1 ) / KA1
            J1 = J2 - ( NR-1 )*KA1
            IF( UPDATE ) THEN
               J2T = MIN( J2, I-2*KA+K-1 )
            ELSE
               J2T = J2
            END IF
            NRT = ( J2T+KA-1 ) / KA1
            DO 570 J = J1, J2T, KA1
*
*              create nonzero element a(j-1,j+ka) outside the band
*              and store it in WORK(j)
*
               WORK( J ) = WORK( J )*AB( 1, J+KA-1 )
               AB( 1, J+KA-1 ) = WORK( N+J )*AB( 1, J+KA-1 )
  570       CONTINUE
*
*           generate rotations in 1st set to annihilate elements which
*           have been created outside the band
*
            IF( NRT.GT.0 )
     $         CALL DLARGV( NRT, AB( 1, J1+KA ), INCA, WORK( J1 ),
     $                      KA1,
     $                      WORK( N+J1 ), KA1 )
            IF( NR.GT.0 ) THEN
*
*              apply rotations in 1st set from the left
*
               DO 580 L = 1, KA - 1
                  CALL DLARTV( NR, AB( KA1-L, J1+L ), INCA,
     $                         AB( KA-L, J1+L ), INCA, WORK( N+J1 ),
     $                         WORK( J1 ), KA1 )
  580          CONTINUE
*
*              apply rotations in 1st set from both sides to diagonal
*              blocks
*
               CALL DLAR2V( NR, AB( KA1, J1 ), AB( KA1, J1-1 ),
     $                      AB( KA, J1 ), INCA, WORK( N+J1 ),
     $                      WORK( J1 ), KA1 )
*
            END IF
*
*           start applying rotations in 1st set from the right
*
            DO 590 L = KA - 1, KB - K + 1, -1
               NRT = ( J2+L-1 ) / KA1
               J1T = J2 - ( NRT-1 )*KA1
               IF( NRT.GT.0 )
     $            CALL DLARTV( NRT, AB( L, J1T ), INCA,
     $                         AB( L+1, J1T-1 ), INCA, WORK( N+J1T ),
     $                         WORK( J1T ), KA1 )
  590       CONTINUE
*
            IF( WANTX ) THEN
*
*              post-multiply X by product of rotations in 1st set
*
               DO 600 J = J1, J2, KA1
                  CALL DROT( NX, X( 1, J ), 1, X( 1, J-1 ), 1,
     $                       WORK( N+J ), WORK( J ) )
  600          CONTINUE
            END IF
  610    CONTINUE
*
         IF( UPDATE ) THEN
            IF( I2.GT.0 .AND. KBT.GT.0 ) THEN
*
*              create nonzero element a(i+kbt-ka-1,i+kbt) outside the
*              band and store it in WORK(m-kb+i+kbt)
*
               WORK( M-KB+I+KBT ) = -BB( KB1-KBT, I+KBT )*RA1
            END IF
         END IF
*
         DO 650 K = KB, 1, -1
            IF( UPDATE ) THEN
               J2 = I + K + 1 - MAX( 2, K+I0-M )*KA1
            ELSE
               J2 = I + K + 1 - MAX( 1, K+I0-M )*KA1
            END IF
*
*           finish applying rotations in 2nd set from the right
*
            DO 620 L = KB - K, 1, -1
               NRT = ( J2+KA+L-1 ) / KA1
               J1T = J2 - ( NRT-1 )*KA1
               IF( NRT.GT.0 )
     $            CALL DLARTV( NRT, AB( L, J1T+KA ), INCA,
     $                         AB( L+1, J1T+KA-1 ), INCA,
     $                         WORK( N+M-KB+J1T+KA ),
     $                         WORK( M-KB+J1T+KA ), KA1 )
  620       CONTINUE
            NR = ( J2+KA-1 ) / KA1
            J1 = J2 - ( NR-1 )*KA1
            DO 630 J = J1, J2, KA1
               WORK( M-KB+J ) = WORK( M-KB+J+KA )
               WORK( N+M-KB+J ) = WORK( N+M-KB+J+KA )
  630       CONTINUE
            DO 640 J = J1, J2, KA1
*
*              create nonzero element a(j-1,j+ka) outside the band
*              and store it in WORK(m-kb+j)
*
               WORK( M-KB+J ) = WORK( M-KB+J )*AB( 1, J+KA-1 )
               AB( 1, J+KA-1 ) = WORK( N+M-KB+J )*AB( 1, J+KA-1 )
  640       CONTINUE
            IF( UPDATE ) THEN
               IF( I+K.GT.KA1 .AND. K.LE.KBT )
     $            WORK( M-KB+I+K-KA ) = WORK( M-KB+I+K )
            END IF
  650    CONTINUE
*
         DO 690 K = KB, 1, -1
            J2 = I + K + 1 - MAX( 1, K+I0-M )*KA1
            NR = ( J2+KA-1 ) / KA1
            J1 = J2 - ( NR-1 )*KA1
            IF( NR.GT.0 ) THEN
*
*              generate rotations in 2nd set to annihilate elements
*              which have been created outside the band
*
               CALL DLARGV( NR, AB( 1, J1+KA ), INCA,
     $                      WORK( M-KB+J1 ),
     $                      KA1, WORK( N+M-KB+J1 ), KA1 )
*
*              apply rotations in 2nd set from the left
*
               DO 660 L = 1, KA - 1
                  CALL DLARTV( NR, AB( KA1-L, J1+L ), INCA,
     $                         AB( KA-L, J1+L ), INCA,
     $                         WORK( N+M-KB+J1 ), WORK( M-KB+J1 ), KA1 )
  660          CONTINUE
*
*              apply rotations in 2nd set from both sides to diagonal
*              blocks
*
               CALL DLAR2V( NR, AB( KA1, J1 ), AB( KA1, J1-1 ),
     $                      AB( KA, J1 ), INCA, WORK( N+M-KB+J1 ),
     $                      WORK( M-KB+J1 ), KA1 )
*
            END IF
*
*           start applying rotations in 2nd set from the right
*
            DO 670 L = KA - 1, KB - K + 1, -1
               NRT = ( J2+L-1 ) / KA1
               J1T = J2 - ( NRT-1 )*KA1
               IF( NRT.GT.0 )
     $            CALL DLARTV( NRT, AB( L, J1T ), INCA,
     $                         AB( L+1, J1T-1 ), INCA,
     $                         WORK( N+M-KB+J1T ), WORK( M-KB+J1T ),
     $                         KA1 )
  670       CONTINUE
*
            IF( WANTX ) THEN
*
*              post-multiply X by product of rotations in 2nd set
*
               DO 680 J = J1, J2, KA1
                  CALL DROT( NX, X( 1, J ), 1, X( 1, J-1 ), 1,
     $                       WORK( N+M-KB+J ), WORK( M-KB+J ) )
  680          CONTINUE
            END IF
  690    CONTINUE
*
         DO 710 K = 1, KB - 1
            J2 = I + K + 1 - MAX( 1, K+I0-M+1 )*KA1
*
*           finish applying rotations in 1st set from the right
*
            DO 700 L = KB - K, 1, -1
               NRT = ( J2+L-1 ) / KA1
               J1T = J2 - ( NRT-1 )*KA1
               IF( NRT.GT.0 )
     $            CALL DLARTV( NRT, AB( L, J1T ), INCA,
     $                         AB( L+1, J1T-1 ), INCA, WORK( N+J1T ),
     $                         WORK( J1T ), KA1 )
  700       CONTINUE
  710    CONTINUE
*
         IF( KB.GT.1 ) THEN
            DO 720 J = 2, MIN( I+KB, M ) - 2*KA - 1
               WORK( N+J ) = WORK( N+J+KA )
               WORK( J ) = WORK( J+KA )
  720       CONTINUE
         END IF
*
      ELSE
*
*        Transform A, working with the lower triangle
*
         IF( UPDATE ) THEN
*
*           Form  inv(S(i))**T * A * inv(S(i))
*
            BII = BB( 1, I )
            DO 730 J = I1, I
               AB( I-J+1, J ) = AB( I-J+1, J ) / BII
  730       CONTINUE
            DO 740 J = I, MIN( N, I+KA )
               AB( J-I+1, I ) = AB( J-I+1, I ) / BII
  740       CONTINUE
            DO 770 K = I + 1, I + KBT
               DO 750 J = K, I + KBT
                  AB( J-K+1, K ) = AB( J-K+1, K ) -
     $                             BB( J-I+1, I )*AB( K-I+1, I ) -
     $                             BB( K-I+1, I )*AB( J-I+1, I ) +
     $                             AB( 1, I )*BB( J-I+1, I )*
     $                             BB( K-I+1, I )
  750          CONTINUE
               DO 760 J = I + KBT + 1, MIN( N, I+KA )
                  AB( J-K+1, K ) = AB( J-K+1, K ) -
     $                             BB( K-I+1, I )*AB( J-I+1, I )
  760          CONTINUE
  770       CONTINUE
            DO 790 J = I1, I
               DO 780 K = I + 1, MIN( J+KA, I+KBT )
                  AB( K-J+1, J ) = AB( K-J+1, J ) -
     $                             BB( K-I+1, I )*AB( I-J+1, J )
  780          CONTINUE
  790       CONTINUE
*
            IF( WANTX ) THEN
*
*              post-multiply X by inv(S(i))
*
               CALL DSCAL( NX, ONE / BII, X( 1, I ), 1 )
               IF( KBT.GT.0 )
     $            CALL DGER( NX, KBT, -ONE, X( 1, I ), 1, BB( 2, I ),
     $                       1,
     $                       X( 1, I+1 ), LDX )
            END IF
*
*           store a(i,i1) in RA1 for use in next loop over K
*
            RA1 = AB( I-I1+1, I1 )
         END IF
*
*        Generate and apply vectors of rotations to chase all the
*        existing bulges KA positions up toward the top of the band
*
         DO 840 K = 1, KB - 1
            IF( UPDATE ) THEN
*
*              Determine the rotations which would annihilate the bulge
*              which has in theory just been created
*
               IF( I+K-KA1.GT.0 .AND. I+K.LT.M ) THEN
*
*                 generate rotation to annihilate a(i,i+k-ka-1)
*
                  CALL DLARTG( AB( KA1-K, I+K-KA ), RA1,
     $                         WORK( N+I+K-KA ), WORK( I+K-KA ), RA )
*
*                 create nonzero element a(i+k,i+k-ka-1) outside the
*                 band and store it in WORK(m-kb+i+k)
*
                  T = -BB( K+1, I )*RA1
                  WORK( M-KB+I+K ) = WORK( N+I+K-KA )*T -
     $                               WORK( I+K-KA )*AB( KA1, I+K-KA )
                  AB( KA1, I+K-KA ) = WORK( I+K-KA )*T +
     $                                WORK( N+I+K-KA )*AB( KA1, I+K-KA )
                  RA1 = RA
               END IF
            END IF
            J2 = I + K + 1 - MAX( 1, K+I0-M+1 )*KA1
            NR = ( J2+KA-1 ) / KA1
            J1 = J2 - ( NR-1 )*KA1
            IF( UPDATE ) THEN
               J2T = MIN( J2, I-2*KA+K-1 )
            ELSE
               J2T = J2
            END IF
            NRT = ( J2T+KA-1 ) / KA1
            DO 800 J = J1, J2T, KA1
*
*              create nonzero element a(j+ka,j-1) outside the band
*              and store it in WORK(j)
*
               WORK( J ) = WORK( J )*AB( KA1, J-1 )
               AB( KA1, J-1 ) = WORK( N+J )*AB( KA1, J-1 )
  800       CONTINUE
*
*           generate rotations in 1st set to annihilate elements which
*           have been created outside the band
*
            IF( NRT.GT.0 )
     $         CALL DLARGV( NRT, AB( KA1, J1 ), INCA, WORK( J1 ),
     $                      KA1,
     $                      WORK( N+J1 ), KA1 )
            IF( NR.GT.0 ) THEN
*
*              apply rotations in 1st set from the right
*
               DO 810 L = 1, KA - 1
                  CALL DLARTV( NR, AB( L+1, J1 ), INCA, AB( L+2,
     $                         J1-1 ),
     $                         INCA, WORK( N+J1 ), WORK( J1 ), KA1 )
  810          CONTINUE
*
*              apply rotations in 1st set from both sides to diagonal
*              blocks
*
               CALL DLAR2V( NR, AB( 1, J1 ), AB( 1, J1-1 ),
     $                      AB( 2, J1-1 ), INCA, WORK( N+J1 ),
     $                      WORK( J1 ), KA1 )
*
            END IF
*
*           start applying rotations in 1st set from the left
*
            DO 820 L = KA - 1, KB - K + 1, -1
               NRT = ( J2+L-1 ) / KA1
               J1T = J2 - ( NRT-1 )*KA1
               IF( NRT.GT.0 )
     $            CALL DLARTV( NRT, AB( KA1-L+1, J1T-KA1+L ), INCA,
     $                         AB( KA1-L, J1T-KA1+L ), INCA,
     $                         WORK( N+J1T ), WORK( J1T ), KA1 )
  820       CONTINUE
*
            IF( WANTX ) THEN
*
*              post-multiply X by product of rotations in 1st set
*
               DO 830 J = J1, J2, KA1
                  CALL DROT( NX, X( 1, J ), 1, X( 1, J-1 ), 1,
     $                       WORK( N+J ), WORK( J ) )
  830          CONTINUE
            END IF
  840    CONTINUE
*
         IF( UPDATE ) THEN
            IF( I2.GT.0 .AND. KBT.GT.0 ) THEN
*
*              create nonzero element a(i+kbt,i+kbt-ka-1) outside the
*              band and store it in WORK(m-kb+i+kbt)
*
               WORK( M-KB+I+KBT ) = -BB( KBT+1, I )*RA1
            END IF
         END IF
*
         DO 880 K = KB, 1, -1
            IF( UPDATE ) THEN
               J2 = I + K + 1 - MAX( 2, K+I0-M )*KA1
            ELSE
               J2 = I + K + 1 - MAX( 1, K+I0-M )*KA1
            END IF
*
*           finish applying rotations in 2nd set from the left
*
            DO 850 L = KB - K, 1, -1
               NRT = ( J2+KA+L-1 ) / KA1
               J1T = J2 - ( NRT-1 )*KA1
               IF( NRT.GT.0 )
     $            CALL DLARTV( NRT, AB( KA1-L+1, J1T+L-1 ), INCA,
     $                         AB( KA1-L, J1T+L-1 ), INCA,
     $                         WORK( N+M-KB+J1T+KA ),
     $                         WORK( M-KB+J1T+KA ), KA1 )
  850       CONTINUE
            NR = ( J2+KA-1 ) / KA1
            J1 = J2 - ( NR-1 )*KA1
            DO 860 J = J1, J2, KA1
               WORK( M-KB+J ) = WORK( M-KB+J+KA )
               WORK( N+M-KB+J ) = WORK( N+M-KB+J+KA )
  860       CONTINUE
            DO 870 J = J1, J2, KA1
*
*              create nonzero element a(j+ka,j-1) outside the band
*              and store it in WORK(m-kb+j)
*
               WORK( M-KB+J ) = WORK( M-KB+J )*AB( KA1, J-1 )
               AB( KA1, J-1 ) = WORK( N+M-KB+J )*AB( KA1, J-1 )
  870       CONTINUE
            IF( UPDATE ) THEN
               IF( I+K.GT.KA1 .AND. K.LE.KBT )
     $            WORK( M-KB+I+K-KA ) = WORK( M-KB+I+K )
            END IF
  880    CONTINUE
*
         DO 920 K = KB, 1, -1
            J2 = I + K + 1 - MAX( 1, K+I0-M )*KA1
            NR = ( J2+KA-1 ) / KA1
            J1 = J2 - ( NR-1 )*KA1
            IF( NR.GT.0 ) THEN
*
*              generate rotations in 2nd set to annihilate elements
*              which have been created outside the band
*
               CALL DLARGV( NR, AB( KA1, J1 ), INCA, WORK( M-KB+J1 ),
     $                      KA1, WORK( N+M-KB+J1 ), KA1 )
*
*              apply rotations in 2nd set from the right
*
               DO 890 L = 1, KA - 1
                  CALL DLARTV( NR, AB( L+1, J1 ), INCA, AB( L+2,
     $                         J1-1 ),
     $                         INCA, WORK( N+M-KB+J1 ), WORK( M-KB+J1 ),
     $                         KA1 )
  890          CONTINUE
*
*              apply rotations in 2nd set from both sides to diagonal
*              blocks
*
               CALL DLAR2V( NR, AB( 1, J1 ), AB( 1, J1-1 ),
     $                      AB( 2, J1-1 ), INCA, WORK( N+M-KB+J1 ),
     $                      WORK( M-KB+J1 ), KA1 )
*
            END IF
*
*           start applying rotations in 2nd set from the left
*
            DO 900 L = KA - 1, KB - K + 1, -1
               NRT = ( J2+L-1 ) / KA1
               J1T = J2 - ( NRT-1 )*KA1
               IF( NRT.GT.0 )
     $            CALL DLARTV( NRT, AB( KA1-L+1, J1T-KA1+L ), INCA,
     $                         AB( KA1-L, J1T-KA1+L ), INCA,
     $                         WORK( N+M-KB+J1T ), WORK( M-KB+J1T ),
     $                         KA1 )
  900       CONTINUE
*
            IF( WANTX ) THEN
*
*              post-multiply X by product of rotations in 2nd set
*
               DO 910 J = J1, J2, KA1
                  CALL DROT( NX, X( 1, J ), 1, X( 1, J-1 ), 1,
     $                       WORK( N+M-KB+J ), WORK( M-KB+J ) )
  910          CONTINUE
            END IF
  920    CONTINUE
*
         DO 940 K = 1, KB - 1
            J2 = I + K + 1 - MAX( 1, K+I0-M+1 )*KA1
*
*           finish applying rotations in 1st set from the left
*
            DO 930 L = KB - K, 1, -1
               NRT = ( J2+L-1 ) / KA1
               J1T = J2 - ( NRT-1 )*KA1
               IF( NRT.GT.0 )
     $            CALL DLARTV( NRT, AB( KA1-L+1, J1T-KA1+L ), INCA,
     $                         AB( KA1-L, J1T-KA1+L ), INCA,
     $                         WORK( N+J1T ), WORK( J1T ), KA1 )
  930       CONTINUE
  940    CONTINUE
*
         IF( KB.GT.1 ) THEN
            DO 950 J = 2, MIN( I+KB, M ) - 2*KA - 1
               WORK( N+J ) = WORK( N+J+KA )
               WORK( J ) = WORK( J+KA )
  950       CONTINUE
         END IF
*
      END IF
*
      GO TO 490
*
*     End of DSBGST
*
      END
*> \brief \b DSBGV
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSBGV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbgv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbgv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbgv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSBGV( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z,
*                         LDZ, WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, UPLO
*       INTEGER            INFO, KA, KB, LDAB, LDBB, LDZ, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * ), BB( LDBB, * ), W( * ),
*      $                   WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSBGV computes all the eigenvalues, and optionally, the eigenvectors
*> of a real generalized symmetric-definite banded eigenproblem, of
*> the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric
*> and banded, and B is also positive definite.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangles of A and B are stored;
*>          = 'L':  Lower triangles of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A and B.  N >= 0.
*> \endverbatim
*>
*> \param[in] KA
*> \verbatim
*>          KA is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'. KA >= 0.
*> \endverbatim
*>
*> \param[in] KB
*> \verbatim
*>          KB is INTEGER
*>          The number of superdiagonals of the matrix B if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'. KB >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB, N)
*>          On entry, the upper or lower triangle of the symmetric band
*>          matrix A, stored in the first ka+1 rows of the array.  The
*>          j-th column of A is stored in the j-th column of the array AB
*>          as follows:
*>          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
*>
*>          On exit, the contents of AB are destroyed.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KA+1.
*> \endverbatim
*>
*> \param[in,out] BB
*> \verbatim
*>          BB is DOUBLE PRECISION array, dimension (LDBB, N)
*>          On entry, the upper or lower triangle of the symmetric band
*>          matrix B, stored in the first kb+1 rows of the array.  The
*>          j-th column of B is stored in the j-th column of the array BB
*>          as follows:
*>          if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
*>          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).
*>
*>          On exit, the factor S from the split Cholesky factorization
*>          B = S**T*S, as returned by DPBSTF.
*> \endverbatim
*>
*> \param[in] LDBB
*> \verbatim
*>          LDBB is INTEGER
*>          The leading dimension of the array BB.  LDBB >= KB+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
*>          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
*>          eigenvectors, with the i-th column of Z holding the
*>          eigenvector associated with W(i). The eigenvectors are
*>          normalized so that Z**T*B*Z = I.
*>          If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, and i is:
*>             <= N:  the algorithm failed to converge:
*>                    i off-diagonal elements of an intermediate
*>                    tridiagonal form did not converge to zero;
*>             > N:   if INFO = N + i, for 1 <= i <= N, then DPBSTF
*>                    returned INFO = i: B is not positive definite.
*>                    The factorization of B could not be completed and
*>                    no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hbgv
*
*  =====================================================================
      SUBROUTINE DSBGV( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W,
     $                  Z,
     $                  LDZ, WORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, UPLO
      INTEGER            INFO, KA, KB, LDAB, LDBB, LDZ, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * ), BB( LDBB, * ), W( * ),
     $                   WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            UPPER, WANTZ
      CHARACTER          VECT
      INTEGER            IINFO, INDE, INDWRK
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DPBSTF, DSBGST, DSBTRD, DSTEQR, DSTERF,
     $                   XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
      UPPER = LSAME( UPLO, 'U' )
*
      INFO = 0
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( KA.LT.0 ) THEN
         INFO = -4
      ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN
         INFO = -5
      ELSE IF( LDAB.LT.KA+1 ) THEN
         INFO = -7
      ELSE IF( LDBB.LT.KB+1 ) THEN
         INFO = -9
      ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
         INFO = -12
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSBGV', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Form a split Cholesky factorization of B.
*
      CALL DPBSTF( UPLO, N, KB, BB, LDBB, INFO )
      IF( INFO.NE.0 ) THEN
         INFO = N + INFO
         RETURN
      END IF
*
*     Transform problem to standard eigenvalue problem.
*
      INDE = 1
      INDWRK = INDE + N
      CALL DSBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Z, LDZ,
     $             WORK( INDWRK ), IINFO )
*
*     Reduce to tridiagonal form.
*
      IF( WANTZ ) THEN
         VECT = 'U'
      ELSE
         VECT = 'N'
      END IF
      CALL DSBTRD( VECT, UPLO, N, KA, AB, LDAB, W, WORK( INDE ), Z,
     $             LDZ,
     $             WORK( INDWRK ), IINFO )
*
*     For eigenvalues only, call DSTERF.  For eigenvectors, call SSTEQR.
*
      IF( .NOT.WANTZ ) THEN
         CALL DSTERF( N, W, WORK( INDE ), INFO )
      ELSE
         CALL DSTEQR( JOBZ, N, W, WORK( INDE ), Z, LDZ,
     $                WORK( INDWRK ),
     $                INFO )
      END IF
      RETURN
*
*     End of DSBGV
*
      END
*> \brief \b DSBGVD
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSBGVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbgvd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbgvd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbgvd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W,
*                          Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, UPLO
*       INTEGER            INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LWORK, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   AB( LDAB, * ), BB( LDBB, * ), W( * ),
*      $                   WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSBGVD computes all the eigenvalues, and optionally, the eigenvectors
*> of a real generalized symmetric-definite banded eigenproblem, of the
*> form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric and
*> banded, and B is also positive definite.  If eigenvectors are
*> desired, it uses a divide and conquer algorithm.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangles of A and B are stored;
*>          = 'L':  Lower triangles of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A and B.  N >= 0.
*> \endverbatim
*>
*> \param[in] KA
*> \verbatim
*>          KA is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KA >= 0.
*> \endverbatim
*>
*> \param[in] KB
*> \verbatim
*>          KB is INTEGER
*>          The number of superdiagonals of the matrix B if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KB >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB, N)
*>          On entry, the upper or lower triangle of the symmetric band
*>          matrix A, stored in the first ka+1 rows of the array.  The
*>          j-th column of A is stored in the j-th column of the array AB
*>          as follows:
*>          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
*>
*>          On exit, the contents of AB are destroyed.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KA+1.
*> \endverbatim
*>
*> \param[in,out] BB
*> \verbatim
*>          BB is DOUBLE PRECISION array, dimension (LDBB, N)
*>          On entry, the upper or lower triangle of the symmetric band
*>          matrix B, stored in the first kb+1 rows of the array.  The
*>          j-th column of B is stored in the j-th column of the array BB
*>          as follows:
*>          if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
*>          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).
*>
*>          On exit, the factor S from the split Cholesky factorization
*>          B = S**T*S, as returned by DPBSTF.
*> \endverbatim
*>
*> \param[in] LDBB
*> \verbatim
*>          LDBB is INTEGER
*>          The leading dimension of the array BB.  LDBB >= KB+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
*>          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
*>          eigenvectors, with the i-th column of Z holding the
*>          eigenvector associated with W(i).  The eigenvectors are
*>          normalized so Z**T*B*Z = I.
*>          If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If N <= 1,               LWORK >= 1.
*>          If JOBZ = 'N' and N > 1, LWORK >= 2*N.
*>          If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal sizes of the WORK and IWORK
*>          arrays, returns these values as the first entries of the WORK
*>          and IWORK arrays, and no error message related to LWORK or
*>          LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*>          On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK.
*>          If JOBZ  = 'N' or N <= 1, LIWORK >= 1.
*>          If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.
*>
*>          If LIWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the optimal sizes of the WORK and
*>          IWORK arrays, returns these values as the first entries of
*>          the WORK and IWORK arrays, and no error message related to
*>          LWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, and i is:
*>             <= N:  the algorithm failed to converge:
*>                    i off-diagonal elements of an intermediate
*>                    tridiagonal form did not converge to zero;
*>             > N:   if INFO = N + i, for 1 <= i <= N, then DPBSTF
*>                    returned INFO = i: B is not positive definite.
*>                    The factorization of B could not be completed and
*>                    no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hbgvd
*
*> \par Contributors:
*  ==================
*>
*>     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
*  =====================================================================
      SUBROUTINE DSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB,
     $                   W,
     $                   Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, UPLO
      INTEGER            INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   AB( LDAB, * ), BB( LDBB, * ), W( * ),
     $                   WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, UPPER, WANTZ
      CHARACTER          VECT
      INTEGER            IINFO, INDE, INDWK2, INDWRK, LIWMIN, LLWRK2,
     $                   LWMIN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMM, DLACPY, DPBSTF, DSBGST, DSBTRD,
     $                   DSTEDC,
     $                   DSTERF, XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
      UPPER = LSAME( UPLO, 'U' )
      LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
      INFO = 0
      IF( N.LE.1 ) THEN
         LIWMIN = 1
         LWMIN = 1
      ELSE IF( WANTZ ) THEN
         LIWMIN = 3 + 5*N
         LWMIN = 1 + 5*N + 2*N**2
      ELSE
         LIWMIN = 1
         LWMIN = 2*N
      END IF
*
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( KA.LT.0 ) THEN
         INFO = -4
      ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN
         INFO = -5
      ELSE IF( LDAB.LT.KA+1 ) THEN
         INFO = -7
      ELSE IF( LDBB.LT.KB+1 ) THEN
         INFO = -9
      ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
         INFO = -12
      END IF
*
      IF( INFO.EQ.0 ) THEN
         WORK( 1 ) = LWMIN
         IWORK( 1 ) = LIWMIN
*
         IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -14
         ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -16
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSBGVD', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Form a split Cholesky factorization of B.
*
      CALL DPBSTF( UPLO, N, KB, BB, LDBB, INFO )
      IF( INFO.NE.0 ) THEN
         INFO = N + INFO
         RETURN
      END IF
*
*     Transform problem to standard eigenvalue problem.
*
      INDE = 1
      INDWRK = INDE + N
      INDWK2 = INDWRK + N*N
      LLWRK2 = LWORK - INDWK2 + 1
      CALL DSBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Z, LDZ,
     $             WORK, IINFO )
*
*     Reduce to tridiagonal form.
*
      IF( WANTZ ) THEN
         VECT = 'U'
      ELSE
         VECT = 'N'
      END IF
      CALL DSBTRD( VECT, UPLO, N, KA, AB, LDAB, W, WORK( INDE ), Z,
     $             LDZ,
     $             WORK( INDWRK ), IINFO )
*
*     For eigenvalues only, call DSTERF. For eigenvectors, call SSTEDC.
*
      IF( .NOT.WANTZ ) THEN
         CALL DSTERF( N, W, WORK( INDE ), INFO )
      ELSE
         CALL DSTEDC( 'I', N, W, WORK( INDE ), WORK( INDWRK ), N,
     $                WORK( INDWK2 ), LLWRK2, IWORK, LIWORK, INFO )
         CALL DGEMM( 'N', 'N', N, N, N, ONE, Z, LDZ, WORK( INDWRK ),
     $               N,
     $               ZERO, WORK( INDWK2 ), N )
         CALL DLACPY( 'A', N, N, WORK( INDWK2 ), N, Z, LDZ )
      END IF
*
      WORK( 1 ) = LWMIN
      IWORK( 1 ) = LIWMIN
*
      RETURN
*
*     End of DSBGVD
*
      END
*> \brief \b DSBGVX
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSBGVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbgvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbgvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbgvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
*                          LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
*                          LDZ, WORK, IWORK, IFAIL, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, RANGE, UPLO
*       INTEGER            IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
*      $                   N
*       DOUBLE PRECISION   ABSTOL, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            IFAIL( * ), IWORK( * )
*       DOUBLE PRECISION   AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
*      $                   W( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSBGVX computes selected eigenvalues, and optionally, eigenvectors
*> of a real generalized symmetric-definite banded eigenproblem, of
*> the form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric
*> and banded, and B is also positive definite.  Eigenvalues and
*> eigenvectors can be selected by specifying either all eigenvalues,
*> a range of values or a range of indices for the desired eigenvalues.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*>          RANGE is CHARACTER*1
*>          = 'A': all eigenvalues will be found.
*>          = 'V': all eigenvalues in the half-open interval (VL,VU]
*>                 will be found.
*>          = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangles of A and B are stored;
*>          = 'L':  Lower triangles of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A and B.  N >= 0.
*> \endverbatim
*>
*> \param[in] KA
*> \verbatim
*>          KA is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KA >= 0.
*> \endverbatim
*>
*> \param[in] KB
*> \verbatim
*>          KB is INTEGER
*>          The number of superdiagonals of the matrix B if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KB >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB, N)
*>          On entry, the upper or lower triangle of the symmetric band
*>          matrix A, stored in the first ka+1 rows of the array.  The
*>          j-th column of A is stored in the j-th column of the array AB
*>          as follows:
*>          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
*>
*>          On exit, the contents of AB are destroyed.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KA+1.
*> \endverbatim
*>
*> \param[in,out] BB
*> \verbatim
*>          BB is DOUBLE PRECISION array, dimension (LDBB, N)
*>          On entry, the upper or lower triangle of the symmetric band
*>          matrix B, stored in the first kb+1 rows of the array.  The
*>          j-th column of B is stored in the j-th column of the array BB
*>          as follows:
*>          if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
*>          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).
*>
*>          On exit, the factor S from the split Cholesky factorization
*>          B = S**T*S, as returned by DPBSTF.
*> \endverbatim
*>
*> \param[in] LDBB
*> \verbatim
*>          LDBB is INTEGER
*>          The leading dimension of the array BB.  LDBB >= KB+1.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ, N)
*>          If JOBZ = 'V', the n-by-n matrix used in the reduction of
*>          A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,
*>          and consequently C to tridiagonal form.
*>          If JOBZ = 'N', the array Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q.  If JOBZ = 'N',
*>          LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is DOUBLE PRECISION
*>
*>          If RANGE='V', the lower bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*>          VU is DOUBLE PRECISION
*>
*>          If RANGE='V', the upper bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*>          IL is INTEGER
*>
*>          If RANGE='I', the index of the
*>          smallest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*>          IU is INTEGER
*>
*>          If RANGE='I', the index of the
*>          largest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*>          ABSTOL is DOUBLE PRECISION
*>          The absolute error tolerance for the eigenvalues.
*>          An approximate eigenvalue is accepted as converged
*>          when it is determined to lie in an interval [a,b]
*>          of width less than or equal to
*>
*>                  ABSTOL + EPS *   max( |a|,|b| ) ,
*>
*>          where EPS is the machine precision.  If ABSTOL is less than
*>          or equal to zero, then  EPS*|T|  will be used in its place,
*>          where |T| is the 1-norm of the tridiagonal matrix obtained
*>          by reducing A to tridiagonal form.
*>
*>          Eigenvalues will be computed most accurately when ABSTOL is
*>          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*>          If this routine returns with INFO>0, indicating that some
*>          eigenvectors did not converge, try setting ABSTOL to
*>          2*DLAMCH('S').
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The total number of eigenvalues found.  0 <= M <= N.
*>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
*>          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
*>          eigenvectors, with the i-th column of Z holding the
*>          eigenvector associated with W(i).  The eigenvectors are
*>          normalized so Z**T*B*Z = I.
*>          If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (7*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*>          IFAIL is INTEGER array, dimension (M)
*>          If JOBZ = 'V', then if INFO = 0, the first M elements of
*>          IFAIL are zero.  If INFO > 0, then IFAIL contains the
*>          indices of the eigenvalues that failed to converge.
*>          If JOBZ = 'N', then IFAIL is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          <= N: if INFO = i, then i eigenvectors failed to converge.
*>                  Their indices are stored in IFAIL.
*>          > N:  DPBSTF returned an error code; i.e.,
*>                if INFO = N + i, for 1 <= i <= N, then the leading
*>                principal minor of order i of B is not positive.
*>                The factorization of B could not be completed and
*>                no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hbgvx
*
*> \par Contributors:
*  ==================
*>
*>     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
*  =====================================================================
      SUBROUTINE DSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
     $                   LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
     $                   LDZ, WORK, IWORK, IFAIL, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, RANGE, UPLO
      INTEGER            IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
     $                   N
      DOUBLE PRECISION   ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            IFAIL( * ), IWORK( * )
      DOUBLE PRECISION   AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
     $                   W( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLEIG, INDEIG, TEST, UPPER, VALEIG, WANTZ
      CHARACTER          ORDER, VECT
      INTEGER            I, IINFO, INDD, INDE, INDEE, INDISP,
     $                   INDIWO, INDWRK, ITMP1, J, JJ, NSPLIT
      DOUBLE PRECISION   TMP1
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGEMV, DLACPY, DPBSTF, DSBGST,
     $                   DSBTRD,
     $                   DSTEBZ, DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
      UPPER = LSAME( UPLO, 'U' )
      ALLEIG = LSAME( RANGE, 'A' )
      VALEIG = LSAME( RANGE, 'V' )
      INDEIG = LSAME( RANGE, 'I' )
*
      INFO = 0
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
         INFO = -2
      ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( KA.LT.0 ) THEN
         INFO = -5
      ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN
         INFO = -6
      ELSE IF( LDAB.LT.KA+1 ) THEN
         INFO = -8
      ELSE IF( LDBB.LT.KB+1 ) THEN
         INFO = -10
      ELSE IF( LDQ.LT.1 .OR. ( WANTZ .AND. LDQ.LT.N ) ) THEN
         INFO = -12
      ELSE
         IF( VALEIG ) THEN
            IF( N.GT.0 .AND. VU.LE.VL )
     $         INFO = -14
         ELSE IF( INDEIG ) THEN
            IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
               INFO = -15
            ELSE IF ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
               INFO = -16
            END IF
         END IF
      END IF
      IF( INFO.EQ.0) THEN
         IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
            INFO = -21
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSBGVX', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      M = 0
      IF( N.EQ.0 )
     $   RETURN
*
*     Form a split Cholesky factorization of B.
*
      CALL DPBSTF( UPLO, N, KB, BB, LDBB, INFO )
      IF( INFO.NE.0 ) THEN
         INFO = N + INFO
         RETURN
      END IF
*
*     Transform problem to standard eigenvalue problem.
*
      CALL DSBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ,
     $             WORK, IINFO )
*
*     Reduce symmetric band matrix to tridiagonal form.
*
      INDD = 1
      INDE = INDD + N
      INDWRK = INDE + N
      IF( WANTZ ) THEN
         VECT = 'U'
      ELSE
         VECT = 'N'
      END IF
      CALL DSBTRD( VECT, UPLO, N, KA, AB, LDAB, WORK( INDD ),
     $             WORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO )
*
*     If all eigenvalues are desired and ABSTOL is less than or equal
*     to zero, then call DSTERF or SSTEQR.  If this fails for some
*     eigenvalue, then try DSTEBZ.
*
      TEST = .FALSE.
      IF( INDEIG ) THEN
         IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
            TEST = .TRUE.
         END IF
      END IF
      IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
         CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
         INDEE = INDWRK + 2*N
         CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
         IF( .NOT.WANTZ ) THEN
            CALL DSTERF( N, W, WORK( INDEE ), INFO )
         ELSE
            CALL DLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
            CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
     $                   WORK( INDWRK ), INFO )
            IF( INFO.EQ.0 ) THEN
               DO 10 I = 1, N
                  IFAIL( I ) = 0
   10          CONTINUE
            END IF
         END IF
         IF( INFO.EQ.0 ) THEN
            M = N
            GO TO 30
         END IF
         INFO = 0
      END IF
*
*     Otherwise, call DSTEBZ and, if eigenvectors are desired,
*     call DSTEIN.
*
      IF( WANTZ ) THEN
         ORDER = 'B'
      ELSE
         ORDER = 'E'
      END IF
      INDISP = 1 + N
      INDIWO = INDISP + N
      CALL DSTEBZ( RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL,
     $             WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
     $             IWORK( 1 ), IWORK( INDISP ), WORK( INDWRK ),
     $             IWORK( INDIWO ), INFO )
*
      IF( WANTZ ) THEN
         CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
     $                IWORK( 1 ), IWORK( INDISP ), Z, LDZ,
     $                WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
*
*        Apply transformation matrix used in reduction to tridiagonal
*        form to eigenvectors returned by DSTEIN.
*
         DO 20 J = 1, M
            CALL DCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
            CALL DGEMV( 'N', N, N, ONE, Q, LDQ, WORK, 1, ZERO,
     $                  Z( 1, J ), 1 )
   20    CONTINUE
      END IF
*
   30 CONTINUE
*
*     If eigenvalues are not in order, then sort them, along with
*     eigenvectors.
*
      IF( WANTZ ) THEN
         DO 50 J = 1, M - 1
            I = 0
            TMP1 = W( J )
            DO 40 JJ = J + 1, M
               IF( W( JJ ).LT.TMP1 ) THEN
                  I = JJ
                  TMP1 = W( JJ )
               END IF
   40       CONTINUE
*
            IF( I.NE.0 ) THEN
               ITMP1 = IWORK( 1 + I-1 )
               W( I ) = W( J )
               IWORK( 1 + I-1 ) = IWORK( 1 + J-1 )
               W( J ) = TMP1
               IWORK( 1 + J-1 ) = ITMP1
               CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
               IF( INFO.NE.0 ) THEN
                  ITMP1 = IFAIL( I )
                  IFAIL( I ) = IFAIL( J )
                  IFAIL( J ) = ITMP1
               END IF
            END IF
   50    CONTINUE
      END IF
*
      RETURN
*
*     End of DSBGVX
*
      END
*> \brief \b DSBTRD
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSBTRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbtrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbtrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbtrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSBTRD( VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ,
*                          WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO, VECT
*       INTEGER            INFO, KD, LDAB, LDQ, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * ), D( * ), E( * ), Q( LDQ, * ),
*      $                   WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSBTRD reduces a real symmetric band matrix A to symmetric
*> tridiagonal form T by an orthogonal similarity transformation:
*> Q**T * A * Q = T.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] VECT
*> \verbatim
*>          VECT is CHARACTER*1
*>          = 'N':  do not form Q;
*>          = 'V':  form Q;
*>          = 'U':  update a matrix X, by forming X*Q.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          On entry, the upper or lower triangle of the symmetric band
*>          matrix A, stored in the first KD+1 rows of the array.  The
*>          j-th column of A is stored in the j-th column of the array AB
*>          as follows:
*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*>          On exit, the diagonal elements of AB are overwritten by the
*>          diagonal elements of the tridiagonal matrix T; if KD > 0, the
*>          elements on the first superdiagonal (if UPLO = 'U') or the
*>          first subdiagonal (if UPLO = 'L') are overwritten by the
*>          off-diagonal elements of T; the rest of AB is overwritten by
*>          values generated during the reduction.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD+1.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          The off-diagonal elements of the tridiagonal matrix T:
*>          E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ,N)
*>          On entry, if VECT = 'U', then Q must contain an N-by-N
*>          matrix X; if VECT = 'N' or 'V', then Q need not be set.
*>
*>          On exit:
*>          if VECT = 'V', Q contains the N-by-N orthogonal matrix Q;
*>          if VECT = 'U', Q contains the product X*Q;
*>          if VECT = 'N', the array Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q.
*>          LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hbtrd
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  Modified by Linda Kaufman, Bell Labs.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DSBTRD( VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ,
     $                   WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO, VECT
      INTEGER            INFO, KD, LDAB, LDQ, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * ), D( * ), E( * ), Q( LDQ, * ),
     $                   WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            INITQ, UPPER, WANTQ
      INTEGER            I, I2, IBL, INCA, INCX, IQAEND, IQB, IQEND, J,
     $                   J1, J1END, J1INC, J2, JEND, JIN, JINC, K, KD1,
     $                   KDM1, KDN, L, LAST, LEND, NQ, NR, NRT
      DOUBLE PRECISION   TEMP
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLAR2V, DLARGV, DLARTG, DLARTV, DLASET,
     $                   DROT,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INITQ = LSAME( VECT, 'V' )
      WANTQ = INITQ .OR. LSAME( VECT, 'U' )
      UPPER = LSAME( UPLO, 'U' )
      KD1 = KD + 1
      KDM1 = KD - 1
      INCX = LDAB - 1
      IQEND = 1
*
      INFO = 0
      IF( .NOT.WANTQ .AND. .NOT.LSAME( VECT, 'N' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( KD.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDAB.LT.KD1 ) THEN
         INFO = -6
      ELSE IF( LDQ.LT.MAX( 1, N ) .AND. WANTQ ) THEN
         INFO = -10
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSBTRD', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Initialize Q to the unit matrix, if needed
*
      IF( INITQ )
     $   CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
*
*     Wherever possible, plane rotations are generated and applied in
*     vector operations of length NR over the index set J1:J2:KD1.
*
*     The cosines and sines of the plane rotations are stored in the
*     arrays D and WORK.
*
      INCA = KD1*LDAB
      KDN = MIN( N-1, KD )
      IF( UPPER ) THEN
*
         IF( KD.GT.1 ) THEN
*
*           Reduce to tridiagonal form, working with upper triangle
*
            NR = 0
            J1 = KDN + 2
            J2 = 1
*
            DO 90 I = 1, N - 2
*
*              Reduce i-th row of matrix to tridiagonal form
*
               DO 80 K = KDN + 1, 2, -1
                  J1 = J1 + KDN
                  J2 = J2 + KDN
*
                  IF( NR.GT.0 ) THEN
*
*                    generate plane rotations to annihilate nonzero
*                    elements which have been created outside the band
*
                     CALL DLARGV( NR, AB( 1, J1-1 ), INCA,
     $                            WORK( J1 ),
     $                            KD1, D( J1 ), KD1 )
*
*                    apply rotations from the right
*
*
*                    Dependent on the the number of diagonals either
*                    DLARTV or DROT is used
*
                     IF( NR.GE.2*KD-1 ) THEN
                        DO 10 L = 1, KD - 1
                           CALL DLARTV( NR, AB( L+1, J1-1 ), INCA,
     $                                  AB( L, J1 ), INCA, D( J1 ),
     $                                  WORK( J1 ), KD1 )
   10                   CONTINUE
*
                     ELSE
                        JEND = J1 + ( NR-1 )*KD1
                        DO 20 JINC = J1, JEND, KD1
                           CALL DROT( KDM1, AB( 2, JINC-1 ), 1,
     $                                AB( 1, JINC ), 1, D( JINC ),
     $                                WORK( JINC ) )
   20                   CONTINUE
                     END IF
                  END IF
*
*
                  IF( K.GT.2 ) THEN
                     IF( K.LE.N-I+1 ) THEN
*
*                       generate plane rotation to annihilate a(i,i+k-1)
*                       within the band
*
                        CALL DLARTG( AB( KD-K+3, I+K-2 ),
     $                               AB( KD-K+2, I+K-1 ), D( I+K-1 ),
     $                               WORK( I+K-1 ), TEMP )
                        AB( KD-K+3, I+K-2 ) = TEMP
*
*                       apply rotation from the right
*
                        CALL DROT( K-3, AB( KD-K+4, I+K-2 ), 1,
     $                             AB( KD-K+3, I+K-1 ), 1, D( I+K-1 ),
     $                             WORK( I+K-1 ) )
                     END IF
                     NR = NR + 1
                     J1 = J1 - KDN - 1
                  END IF
*
*                 apply plane rotations from both sides to diagonal
*                 blocks
*
                  IF( NR.GT.0 )
     $               CALL DLAR2V( NR, AB( KD1, J1-1 ), AB( KD1, J1 ),
     $                            AB( KD, J1 ), INCA, D( J1 ),
     $                            WORK( J1 ), KD1 )
*
*                 apply plane rotations from the left
*
                  IF( NR.GT.0 ) THEN
                     IF( 2*KD-1.LT.NR ) THEN
*
*                    Dependent on the the number of diagonals either
*                    DLARTV or DROT is used
*
                        DO 30 L = 1, KD - 1
                           IF( J2+L.GT.N ) THEN
                              NRT = NR - 1
                           ELSE
                              NRT = NR
                           END IF
                           IF( NRT.GT.0 )
     $                        CALL DLARTV( NRT, AB( KD-L, J1+L ),
     $                                     INCA,
     $                                     AB( KD-L+1, J1+L ), INCA,
     $                                     D( J1 ), WORK( J1 ), KD1 )
   30                   CONTINUE
                     ELSE
                        J1END = J1 + KD1*( NR-2 )
                        IF( J1END.GE.J1 ) THEN
                           DO 40 JIN = J1, J1END, KD1
                              CALL DROT( KD-1, AB( KD-1, JIN+1 ),
     $                                   INCX,
     $                                   AB( KD, JIN+1 ), INCX,
     $                                   D( JIN ), WORK( JIN ) )
   40                      CONTINUE
                        END IF
                        LEND = MIN( KDM1, N-J2 )
                        LAST = J1END + KD1
                        IF( LEND.GT.0 )
     $                     CALL DROT( LEND, AB( KD-1, LAST+1 ), INCX,
     $                                AB( KD, LAST+1 ), INCX, D( LAST ),
     $                                WORK( LAST ) )
                     END IF
                  END IF
*
                  IF( WANTQ ) THEN
*
*                    accumulate product of plane rotations in Q
*
                     IF( INITQ ) THEN
*
*                 take advantage of the fact that Q was
*                 initially the Identity matrix
*
                        IQEND = MAX( IQEND, J2 )
                        I2 = MAX( 0, K-3 )
                        IQAEND = 1 + I*KD
                        IF( K.EQ.2 )
     $                     IQAEND = IQAEND + KD
                        IQAEND = MIN( IQAEND, IQEND )
                        DO 50 J = J1, J2, KD1
                           IBL = I - I2 / KDM1
                           I2 = I2 + 1
                           IQB = MAX( 1, J-IBL )
                           NQ = 1 + IQAEND - IQB
                           IQAEND = MIN( IQAEND+KD, IQEND )
                           CALL DROT( NQ, Q( IQB, J-1 ), 1, Q( IQB,
     $                                J ),
     $                                1, D( J ), WORK( J ) )
   50                   CONTINUE
                     ELSE
*
                        DO 60 J = J1, J2, KD1
                           CALL DROT( N, Q( 1, J-1 ), 1, Q( 1, J ),
     $                                1,
     $                                D( J ), WORK( J ) )
   60                   CONTINUE
                     END IF
*
                  END IF
*
                  IF( J2+KDN.GT.N ) THEN
*
*                    adjust J2 to keep within the bounds of the matrix
*
                     NR = NR - 1
                     J2 = J2 - KDN - 1
                  END IF
*
                  DO 70 J = J1, J2, KD1
*
*                    create nonzero element a(j-1,j+kd) outside the band
*                    and store it in WORK
*
                     WORK( J+KD ) = WORK( J )*AB( 1, J+KD )
                     AB( 1, J+KD ) = D( J )*AB( 1, J+KD )
   70             CONTINUE
   80          CONTINUE
   90       CONTINUE
         END IF
*
         IF( KD.GT.0 ) THEN
*
*           copy off-diagonal elements to E
*
            DO 100 I = 1, N - 1
               E( I ) = AB( KD, I+1 )
  100       CONTINUE
         ELSE
*
*           set E to zero if original matrix was diagonal
*
            DO 110 I = 1, N - 1
               E( I ) = ZERO
  110       CONTINUE
         END IF
*
*        copy diagonal elements to D
*
         DO 120 I = 1, N
            D( I ) = AB( KD1, I )
  120    CONTINUE
*
      ELSE
*
         IF( KD.GT.1 ) THEN
*
*           Reduce to tridiagonal form, working with lower triangle
*
            NR = 0
            J1 = KDN + 2
            J2 = 1
*
            DO 210 I = 1, N - 2
*
*              Reduce i-th column of matrix to tridiagonal form
*
               DO 200 K = KDN + 1, 2, -1
                  J1 = J1 + KDN
                  J2 = J2 + KDN
*
                  IF( NR.GT.0 ) THEN
*
*                    generate plane rotations to annihilate nonzero
*                    elements which have been created outside the band
*
                     CALL DLARGV( NR, AB( KD1, J1-KD1 ), INCA,
     $                            WORK( J1 ), KD1, D( J1 ), KD1 )
*
*                    apply plane rotations from one side
*
*
*                    Dependent on the the number of diagonals either
*                    DLARTV or DROT is used
*
                     IF( NR.GT.2*KD-1 ) THEN
                        DO 130 L = 1, KD - 1
                           CALL DLARTV( NR, AB( KD1-L, J1-KD1+L ),
     $                                  INCA,
     $                                  AB( KD1-L+1, J1-KD1+L ), INCA,
     $                                  D( J1 ), WORK( J1 ), KD1 )
  130                   CONTINUE
                     ELSE
                        JEND = J1 + KD1*( NR-1 )
                        DO 140 JINC = J1, JEND, KD1
                           CALL DROT( KDM1, AB( KD, JINC-KD ), INCX,
     $                                AB( KD1, JINC-KD ), INCX,
     $                                D( JINC ), WORK( JINC ) )
  140                   CONTINUE
                     END IF
*
                  END IF
*
                  IF( K.GT.2 ) THEN
                     IF( K.LE.N-I+1 ) THEN
*
*                       generate plane rotation to annihilate a(i+k-1,i)
*                       within the band
*
                        CALL DLARTG( AB( K-1, I ), AB( K, I ),
     $                               D( I+K-1 ), WORK( I+K-1 ), TEMP )
                        AB( K-1, I ) = TEMP
*
*                       apply rotation from the left
*
                        CALL DROT( K-3, AB( K-2, I+1 ), LDAB-1,
     $                             AB( K-1, I+1 ), LDAB-1, D( I+K-1 ),
     $                             WORK( I+K-1 ) )
                     END IF
                     NR = NR + 1
                     J1 = J1 - KDN - 1
                  END IF
*
*                 apply plane rotations from both sides to diagonal
*                 blocks
*
                  IF( NR.GT.0 )
     $               CALL DLAR2V( NR, AB( 1, J1-1 ), AB( 1, J1 ),
     $                            AB( 2, J1-1 ), INCA, D( J1 ),
     $                            WORK( J1 ), KD1 )
*
*                 apply plane rotations from the right
*
*
*                    Dependent on the the number of diagonals either
*                    DLARTV or DROT is used
*
                  IF( NR.GT.0 ) THEN
                     IF( NR.GT.2*KD-1 ) THEN
                        DO 150 L = 1, KD - 1
                           IF( J2+L.GT.N ) THEN
                              NRT = NR - 1
                           ELSE
                              NRT = NR
                           END IF
                           IF( NRT.GT.0 )
     $                        CALL DLARTV( NRT, AB( L+2, J1-1 ),
     $                                     INCA,
     $                                     AB( L+1, J1 ), INCA, D( J1 ),
     $                                     WORK( J1 ), KD1 )
  150                   CONTINUE
                     ELSE
                        J1END = J1 + KD1*( NR-2 )
                        IF( J1END.GE.J1 ) THEN
                           DO 160 J1INC = J1, J1END, KD1
                              CALL DROT( KDM1, AB( 3, J1INC-1 ), 1,
     $                                   AB( 2, J1INC ), 1, D( J1INC ),
     $                                   WORK( J1INC ) )
  160                      CONTINUE
                        END IF
                        LEND = MIN( KDM1, N-J2 )
                        LAST = J1END + KD1
                        IF( LEND.GT.0 )
     $                     CALL DROT( LEND, AB( 3, LAST-1 ), 1,
     $                                AB( 2, LAST ), 1, D( LAST ),
     $                                WORK( LAST ) )
                     END IF
                  END IF
*
*
*
                  IF( WANTQ ) THEN
*
*                    accumulate product of plane rotations in Q
*
                     IF( INITQ ) THEN
*
*                 take advantage of the fact that Q was
*                 initially the Identity matrix
*
                        IQEND = MAX( IQEND, J2 )
                        I2 = MAX( 0, K-3 )
                        IQAEND = 1 + I*KD
                        IF( K.EQ.2 )
     $                     IQAEND = IQAEND + KD
                        IQAEND = MIN( IQAEND, IQEND )
                        DO 170 J = J1, J2, KD1
                           IBL = I - I2 / KDM1
                           I2 = I2 + 1
                           IQB = MAX( 1, J-IBL )
                           NQ = 1 + IQAEND - IQB
                           IQAEND = MIN( IQAEND+KD, IQEND )
                           CALL DROT( NQ, Q( IQB, J-1 ), 1, Q( IQB,
     $                                J ),
     $                                1, D( J ), WORK( J ) )
  170                   CONTINUE
                     ELSE
*
                        DO 180 J = J1, J2, KD1
                           CALL DROT( N, Q( 1, J-1 ), 1, Q( 1, J ),
     $                                1,
     $                                D( J ), WORK( J ) )
  180                   CONTINUE
                     END IF
                  END IF
*
                  IF( J2+KDN.GT.N ) THEN
*
*                    adjust J2 to keep within the bounds of the matrix
*
                     NR = NR - 1
                     J2 = J2 - KDN - 1
                  END IF
*
                  DO 190 J = J1, J2, KD1
*
*                    create nonzero element a(j+kd,j-1) outside the
*                    band and store it in WORK
*
                     WORK( J+KD ) = WORK( J )*AB( KD1, J )
                     AB( KD1, J ) = D( J )*AB( KD1, J )
  190             CONTINUE
  200          CONTINUE
  210       CONTINUE
         END IF
*
         IF( KD.GT.0 ) THEN
*
*           copy off-diagonal elements to E
*
            DO 220 I = 1, N - 1
               E( I ) = AB( 2, I )
  220       CONTINUE
         ELSE
*
*           set E to zero if original matrix was diagonal
*
            DO 230 I = 1, N - 1
               E( I ) = ZERO
  230       CONTINUE
         END IF
*
*        copy diagonal elements to D
*
         DO 240 I = 1, N
            D( I ) = AB( 1, I )
  240    CONTINUE
      END IF
*
      RETURN
*
*     End of DSBTRD
*
      END
*> \brief \b DSFRK performs a symmetric rank-k operation for matrix in RFP format.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSFRK + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsfrk.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsfrk.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsfrk.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSFRK( TRANSR, UPLO, TRANS, N, K, ALPHA, A, LDA, BETA,
*                         C )
*
*       .. Scalar Arguments ..
*       DOUBLE PRECISION   ALPHA, BETA
*       INTEGER            K, LDA, N
*       CHARACTER          TRANS, TRANSR, UPLO
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), C( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> Level 3 BLAS like routine for C in RFP Format.
*>
*> DSFRK performs one of the symmetric rank--k operations
*>
*>    C := alpha*A*A**T + beta*C,
*>
*> or
*>
*>    C := alpha*A**T*A + beta*C,
*>
*> where alpha and beta are real scalars, C is an n--by--n symmetric
*> matrix and A is an n--by--k matrix in the first case and a k--by--n
*> matrix in the second case.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANSR
*> \verbatim
*>          TRANSR is CHARACTER*1
*>          = 'N':  The Normal Form of RFP A is stored;
*>          = 'T':  The Transpose Form of RFP A is stored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>           On  entry, UPLO specifies whether the upper or lower
*>           triangular part of the array C is to be referenced as
*>           follows:
*>
*>              UPLO = 'U' or 'u'   Only the upper triangular part of C
*>                                  is to be referenced.
*>
*>              UPLO = 'L' or 'l'   Only the lower triangular part of C
*>                                  is to be referenced.
*>
*>           Unchanged on exit.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>           On entry, TRANS specifies the operation to be performed as
*>           follows:
*>
*>              TRANS = 'N' or 'n'   C := alpha*A*A**T + beta*C.
*>
*>              TRANS = 'T' or 't'   C := alpha*A**T*A + beta*C.
*>
*>           Unchanged on exit.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>           On entry, N specifies the order of the matrix C. N must be
*>           at least zero.
*>           Unchanged on exit.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>           On entry with TRANS = 'N' or 'n', K specifies the number
*>           of  columns of the matrix A, and on entry with TRANS = 'T'
*>           or 't', K specifies the number of rows of the matrix A. K
*>           must be at least zero.
*>           Unchanged on exit.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*>          ALPHA is DOUBLE PRECISION
*>           On entry, ALPHA specifies the scalar alpha.
*>           Unchanged on exit.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,ka)
*>           where KA
*>           is K  when TRANS = 'N' or 'n', and is N otherwise. Before
*>           entry with TRANS = 'N' or 'n', the leading N--by--K part of
*>           the array A must contain the matrix A, otherwise the leading
*>           K--by--N part of the array A must contain the matrix A.
*>           Unchanged on exit.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>           On entry, LDA specifies the first dimension of A as declared
*>           in  the  calling  (sub)  program.   When  TRANS = 'N' or 'n'
*>           then  LDA must be at least  max( 1, n ), otherwise  LDA must
*>           be at least  max( 1, k ).
*>           Unchanged on exit.
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION
*>           On entry, BETA specifies the scalar beta.
*>           Unchanged on exit.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (NT)
*>           NT = N*(N+1)/2. On entry, the symmetric matrix C in RFP
*>           Format. RFP Format is described by TRANSR, UPLO and N.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hfrk
*
*  =====================================================================
      SUBROUTINE DSFRK( TRANSR, UPLO, TRANS, N, K, ALPHA, A, LDA,
     $                  BETA,
     $                  C )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   ALPHA, BETA
      INTEGER            K, LDA, N
      CHARACTER          TRANS, TRANSR, UPLO
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( * )
*     ..
*
*  =====================================================================
*
*     ..
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER, NORMALTRANSR, NISODD, NOTRANS
      INTEGER            INFO, NROWA, J, NK, N1, N2
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA, DGEMM, DSYRK
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NORMALTRANSR = LSAME( TRANSR, 'N' )
      LOWER = LSAME( UPLO, 'L' )
      NOTRANS = LSAME( TRANS, 'N' )
*
      IF( NOTRANS ) THEN
         NROWA = N
      ELSE
         NROWA = K
      END IF
*
      IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
         INFO = -2
      ELSE IF( .NOT.NOTRANS .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( K.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, NROWA ) ) THEN
         INFO = -8
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSFRK ', -INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
*     The quick return case: ((ALPHA.EQ.0).AND.(BETA.NE.ZERO)) is not
*     done (it is in DSYRK for example) and left in the general case.
*
      IF( ( N.EQ.0 ) .OR. ( ( ( ALPHA.EQ.ZERO ) .OR. ( K.EQ.0 ) ) .AND.
     $    ( BETA.EQ.ONE ) ) )RETURN
*
      IF( ( ALPHA.EQ.ZERO ) .AND. ( BETA.EQ.ZERO ) ) THEN
         DO J = 1, ( ( N*( N+1 ) ) / 2 )
            C( J ) = ZERO
         END DO
         RETURN
      END IF
*
*     C is N-by-N.
*     If N is odd, set NISODD = .TRUE., and N1 and N2.
*     If N is even, NISODD = .FALSE., and NK.
*
      IF( MOD( N, 2 ).EQ.0 ) THEN
         NISODD = .FALSE.
         NK = N / 2
      ELSE
         NISODD = .TRUE.
         IF( LOWER ) THEN
            N2 = N / 2
            N1 = N - N2
         ELSE
            N1 = N / 2
            N2 = N - N1
         END IF
      END IF
*
      IF( NISODD ) THEN
*
*        N is odd
*
         IF( NORMALTRANSR ) THEN
*
*           N is odd and TRANSR = 'N'
*
            IF( LOWER ) THEN
*
*              N is odd, TRANSR = 'N', and UPLO = 'L'
*
               IF( NOTRANS ) THEN
*
*                 N is odd, TRANSR = 'N', UPLO = 'L', and TRANS = 'N'
*
                  CALL DSYRK( 'L', 'N', N1, K, ALPHA, A( 1, 1 ), LDA,
     $                        BETA, C( 1 ), N )
                  CALL DSYRK( 'U', 'N', N2, K, ALPHA, A( N1+1, 1 ),
     $                        LDA,
     $                        BETA, C( N+1 ), N )
                  CALL DGEMM( 'N', 'T', N2, N1, K, ALPHA, A( N1+1,
     $                        1 ),
     $                        LDA, A( 1, 1 ), LDA, BETA, C( N1+1 ), N )
*
               ELSE
*
*                 N is odd, TRANSR = 'N', UPLO = 'L', and TRANS = 'T'
*
                  CALL DSYRK( 'L', 'T', N1, K, ALPHA, A( 1, 1 ), LDA,
     $                        BETA, C( 1 ), N )
                  CALL DSYRK( 'U', 'T', N2, K, ALPHA, A( 1, N1+1 ),
     $                        LDA,
     $                        BETA, C( N+1 ), N )
                  CALL DGEMM( 'T', 'N', N2, N1, K, ALPHA, A( 1,
     $                        N1+1 ),
     $                        LDA, A( 1, 1 ), LDA, BETA, C( N1+1 ), N )
*
               END IF
*
            ELSE
*
*              N is odd, TRANSR = 'N', and UPLO = 'U'
*
               IF( NOTRANS ) THEN
*
*                 N is odd, TRANSR = 'N', UPLO = 'U', and TRANS = 'N'
*
                  CALL DSYRK( 'L', 'N', N1, K, ALPHA, A( 1, 1 ), LDA,
     $                        BETA, C( N2+1 ), N )
                  CALL DSYRK( 'U', 'N', N2, K, ALPHA, A( N2, 1 ),
     $                        LDA,
     $                        BETA, C( N1+1 ), N )
                  CALL DGEMM( 'N', 'T', N1, N2, K, ALPHA, A( 1, 1 ),
     $                        LDA, A( N2, 1 ), LDA, BETA, C( 1 ), N )
*
               ELSE
*
*                 N is odd, TRANSR = 'N', UPLO = 'U', and TRANS = 'T'
*
                  CALL DSYRK( 'L', 'T', N1, K, ALPHA, A( 1, 1 ), LDA,
     $                        BETA, C( N2+1 ), N )
                  CALL DSYRK( 'U', 'T', N2, K, ALPHA, A( 1, N2 ),
     $                        LDA,
     $                        BETA, C( N1+1 ), N )
                  CALL DGEMM( 'T', 'N', N1, N2, K, ALPHA, A( 1, 1 ),
     $                        LDA, A( 1, N2 ), LDA, BETA, C( 1 ), N )
*
               END IF
*
            END IF
*
         ELSE
*
*           N is odd, and TRANSR = 'T'
*
            IF( LOWER ) THEN
*
*              N is odd, TRANSR = 'T', and UPLO = 'L'
*
               IF( NOTRANS ) THEN
*
*                 N is odd, TRANSR = 'T', UPLO = 'L', and TRANS = 'N'
*
                  CALL DSYRK( 'U', 'N', N1, K, ALPHA, A( 1, 1 ), LDA,
     $                        BETA, C( 1 ), N1 )
                  CALL DSYRK( 'L', 'N', N2, K, ALPHA, A( N1+1, 1 ),
     $                        LDA,
     $                        BETA, C( 2 ), N1 )
                  CALL DGEMM( 'N', 'T', N1, N2, K, ALPHA, A( 1, 1 ),
     $                        LDA, A( N1+1, 1 ), LDA, BETA,
     $                        C( N1*N1+1 ), N1 )
*
               ELSE
*
*                 N is odd, TRANSR = 'T', UPLO = 'L', and TRANS = 'T'
*
                  CALL DSYRK( 'U', 'T', N1, K, ALPHA, A( 1, 1 ), LDA,
     $                        BETA, C( 1 ), N1 )
                  CALL DSYRK( 'L', 'T', N2, K, ALPHA, A( 1, N1+1 ),
     $                        LDA,
     $                        BETA, C( 2 ), N1 )
                  CALL DGEMM( 'T', 'N', N1, N2, K, ALPHA, A( 1, 1 ),
     $                        LDA, A( 1, N1+1 ), LDA, BETA,
     $                        C( N1*N1+1 ), N1 )
*
               END IF
*
            ELSE
*
*              N is odd, TRANSR = 'T', and UPLO = 'U'
*
               IF( NOTRANS ) THEN
*
*                 N is odd, TRANSR = 'T', UPLO = 'U', and TRANS = 'N'
*
                  CALL DSYRK( 'U', 'N', N1, K, ALPHA, A( 1, 1 ), LDA,
     $                        BETA, C( N2*N2+1 ), N2 )
                  CALL DSYRK( 'L', 'N', N2, K, ALPHA, A( N1+1, 1 ),
     $                        LDA,
     $                        BETA, C( N1*N2+1 ), N2 )
                  CALL DGEMM( 'N', 'T', N2, N1, K, ALPHA, A( N1+1,
     $                        1 ),
     $                        LDA, A( 1, 1 ), LDA, BETA, C( 1 ), N2 )
*
               ELSE
*
*                 N is odd, TRANSR = 'T', UPLO = 'U', and TRANS = 'T'
*
                  CALL DSYRK( 'U', 'T', N1, K, ALPHA, A( 1, 1 ), LDA,
     $                        BETA, C( N2*N2+1 ), N2 )
                  CALL DSYRK( 'L', 'T', N2, K, ALPHA, A( 1, N1+1 ),
     $                        LDA,
     $                        BETA, C( N1*N2+1 ), N2 )
                  CALL DGEMM( 'T', 'N', N2, N1, K, ALPHA, A( 1,
     $                        N1+1 ),
     $                        LDA, A( 1, 1 ), LDA, BETA, C( 1 ), N2 )
*
               END IF
*
            END IF
*
         END IF
*
      ELSE
*
*        N is even
*
         IF( NORMALTRANSR ) THEN
*
*           N is even and TRANSR = 'N'
*
            IF( LOWER ) THEN
*
*              N is even, TRANSR = 'N', and UPLO = 'L'
*
               IF( NOTRANS ) THEN
*
*                 N is even, TRANSR = 'N', UPLO = 'L', and TRANS = 'N'
*
                  CALL DSYRK( 'L', 'N', NK, K, ALPHA, A( 1, 1 ), LDA,
     $                        BETA, C( 2 ), N+1 )
                  CALL DSYRK( 'U', 'N', NK, K, ALPHA, A( NK+1, 1 ),
     $                        LDA,
     $                        BETA, C( 1 ), N+1 )
                  CALL DGEMM( 'N', 'T', NK, NK, K, ALPHA, A( NK+1,
     $                        1 ),
     $                        LDA, A( 1, 1 ), LDA, BETA, C( NK+2 ),
     $                        N+1 )
*
               ELSE
*
*                 N is even, TRANSR = 'N', UPLO = 'L', and TRANS = 'T'
*
                  CALL DSYRK( 'L', 'T', NK, K, ALPHA, A( 1, 1 ), LDA,
     $                        BETA, C( 2 ), N+1 )
                  CALL DSYRK( 'U', 'T', NK, K, ALPHA, A( 1, NK+1 ),
     $                        LDA,
     $                        BETA, C( 1 ), N+1 )
                  CALL DGEMM( 'T', 'N', NK, NK, K, ALPHA, A( 1,
     $                        NK+1 ),
     $                        LDA, A( 1, 1 ), LDA, BETA, C( NK+2 ),
     $                        N+1 )
*
               END IF
*
            ELSE
*
*              N is even, TRANSR = 'N', and UPLO = 'U'
*
               IF( NOTRANS ) THEN
*
*                 N is even, TRANSR = 'N', UPLO = 'U', and TRANS = 'N'
*
                  CALL DSYRK( 'L', 'N', NK, K, ALPHA, A( 1, 1 ), LDA,
     $                        BETA, C( NK+2 ), N+1 )
                  CALL DSYRK( 'U', 'N', NK, K, ALPHA, A( NK+1, 1 ),
     $                        LDA,
     $                        BETA, C( NK+1 ), N+1 )
                  CALL DGEMM( 'N', 'T', NK, NK, K, ALPHA, A( 1, 1 ),
     $                        LDA, A( NK+1, 1 ), LDA, BETA, C( 1 ),
     $                        N+1 )
*
               ELSE
*
*                 N is even, TRANSR = 'N', UPLO = 'U', and TRANS = 'T'
*
                  CALL DSYRK( 'L', 'T', NK, K, ALPHA, A( 1, 1 ), LDA,
     $                        BETA, C( NK+2 ), N+1 )
                  CALL DSYRK( 'U', 'T', NK, K, ALPHA, A( 1, NK+1 ),
     $                        LDA,
     $                        BETA, C( NK+1 ), N+1 )
                  CALL DGEMM( 'T', 'N', NK, NK, K, ALPHA, A( 1, 1 ),
     $                        LDA, A( 1, NK+1 ), LDA, BETA, C( 1 ),
     $                        N+1 )
*
               END IF
*
            END IF
*
         ELSE
*
*           N is even, and TRANSR = 'T'
*
            IF( LOWER ) THEN
*
*              N is even, TRANSR = 'T', and UPLO = 'L'
*
               IF( NOTRANS ) THEN
*
*                 N is even, TRANSR = 'T', UPLO = 'L', and TRANS = 'N'
*
                  CALL DSYRK( 'U', 'N', NK, K, ALPHA, A( 1, 1 ), LDA,
     $                        BETA, C( NK+1 ), NK )
                  CALL DSYRK( 'L', 'N', NK, K, ALPHA, A( NK+1, 1 ),
     $                        LDA,
     $                        BETA, C( 1 ), NK )
                  CALL DGEMM( 'N', 'T', NK, NK, K, ALPHA, A( 1, 1 ),
     $                        LDA, A( NK+1, 1 ), LDA, BETA,
     $                        C( ( ( NK+1 )*NK )+1 ), NK )
*
               ELSE
*
*                 N is even, TRANSR = 'T', UPLO = 'L', and TRANS = 'T'
*
                  CALL DSYRK( 'U', 'T', NK, K, ALPHA, A( 1, 1 ), LDA,
     $                        BETA, C( NK+1 ), NK )
                  CALL DSYRK( 'L', 'T', NK, K, ALPHA, A( 1, NK+1 ),
     $                        LDA,
     $                        BETA, C( 1 ), NK )
                  CALL DGEMM( 'T', 'N', NK, NK, K, ALPHA, A( 1, 1 ),
     $                        LDA, A( 1, NK+1 ), LDA, BETA,
     $                        C( ( ( NK+1 )*NK )+1 ), NK )
*
               END IF
*
            ELSE
*
*              N is even, TRANSR = 'T', and UPLO = 'U'
*
               IF( NOTRANS ) THEN
*
*                 N is even, TRANSR = 'T', UPLO = 'U', and TRANS = 'N'
*
                  CALL DSYRK( 'U', 'N', NK, K, ALPHA, A( 1, 1 ), LDA,
     $                        BETA, C( NK*( NK+1 )+1 ), NK )
                  CALL DSYRK( 'L', 'N', NK, K, ALPHA, A( NK+1, 1 ),
     $                        LDA,
     $                        BETA, C( NK*NK+1 ), NK )
                  CALL DGEMM( 'N', 'T', NK, NK, K, ALPHA, A( NK+1,
     $                        1 ),
     $                        LDA, A( 1, 1 ), LDA, BETA, C( 1 ), NK )
*
               ELSE
*
*                 N is even, TRANSR = 'T', UPLO = 'U', and TRANS = 'T'
*
                  CALL DSYRK( 'U', 'T', NK, K, ALPHA, A( 1, 1 ), LDA,
     $                        BETA, C( NK*( NK+1 )+1 ), NK )
                  CALL DSYRK( 'L', 'T', NK, K, ALPHA, A( 1, NK+1 ),
     $                        LDA,
     $                        BETA, C( NK*NK+1 ), NK )
                  CALL DGEMM( 'T', 'N', NK, NK, K, ALPHA, A( 1,
     $                        NK+1 ),
     $                        LDA, A( 1, 1 ), LDA, BETA, C( 1 ), NK )
*
               END IF
*
            END IF
*
         END IF
*
      END IF
*
      RETURN
*
*     End of DSFRK
*
      END
*> \brief \b DSPCON
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspcon.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSPCON( UPLO, N, AP, IPIV, ANORM, RCOND, WORK, IWORK,
*                          INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, N
*       DOUBLE PRECISION   ANORM, RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), IWORK( * )
*       DOUBLE PRECISION   AP( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSPCON estimates the reciprocal of the condition number (in the
*> 1-norm) of a real symmetric packed matrix A using the factorization
*> A = U*D*U**T or A = L*D*L**T computed by DSPTRF.
*>
*> An estimate is obtained for norm(inv(A)), and the reciprocal of the
*> condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the details of the factorization are stored
*>          as an upper or lower triangular matrix.
*>          = 'U':  Upper triangular, form is A = U*D*U**T;
*>          = 'L':  Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          The block diagonal matrix D and the multipliers used to
*>          obtain the factor U or L as computed by DSPTRF, stored as a
*>          packed triangular matrix.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the block structure of D
*>          as determined by DSPTRF.
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*>          ANORM is DOUBLE PRECISION
*>          The 1-norm of the original matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The reciprocal of the condition number of the matrix A,
*>          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
*>          estimate of the 1-norm of inv(A) computed in this routine.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpcon
*
*  =====================================================================
      SUBROUTINE DSPCON( UPLO, N, AP, IPIV, ANORM, RCOND, WORK,
     $                   IWORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, N
      DOUBLE PRECISION   ANORM, RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), IWORK( * )
      DOUBLE PRECISION   AP( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I, IP, KASE
      DOUBLE PRECISION   AINVNM
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACN2, DSPTRS, XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( ANORM.LT.ZERO ) THEN
         INFO = -5
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSPCON', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      RCOND = ZERO
      IF( N.EQ.0 ) THEN
         RCOND = ONE
         RETURN
      ELSE IF( ANORM.LE.ZERO ) THEN
         RETURN
      END IF
*
*     Check that the diagonal matrix D is nonsingular.
*
      IF( UPPER ) THEN
*
*        Upper triangular storage: examine D from bottom to top
*
         IP = N*( N+1 ) / 2
         DO 10 I = N, 1, -1
            IF( IPIV( I ).GT.0 .AND. AP( IP ).EQ.ZERO )
     $         RETURN
            IP = IP - I
   10    CONTINUE
      ELSE
*
*        Lower triangular storage: examine D from top to bottom.
*
         IP = 1
         DO 20 I = 1, N
            IF( IPIV( I ).GT.0 .AND. AP( IP ).EQ.ZERO )
     $         RETURN
            IP = IP + N - I + 1
   20    CONTINUE
      END IF
*
*     Estimate the 1-norm of the inverse.
*
      KASE = 0
   30 CONTINUE
      CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
      IF( KASE.NE.0 ) THEN
*
*        Multiply by inv(L*D*L**T) or inv(U*D*U**T).
*
         CALL DSPTRS( UPLO, N, 1, AP, IPIV, WORK, N, INFO )
         GO TO 30
      END IF
*
*     Compute the estimate of the reciprocal condition number.
*
      IF( AINVNM.NE.ZERO )
     $   RCOND = ( ONE / AINVNM ) / ANORM
*
      RETURN
*
*     End of DSPCON
*
      END
*> \brief <b> DSPEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPEV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspev.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspev.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspev.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSPEV( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, UPLO
*       INTEGER            INFO, LDZ, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AP( * ), W( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSPEV computes all the eigenvalues and, optionally, eigenvectors of a
*> real symmetric matrix A in packed storage.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the symmetric matrix
*>          A, packed columnwise in a linear array.  The j-th column of A
*>          is stored in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*>
*>          On exit, AP is overwritten by values generated during the
*>          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
*>          and first superdiagonal of the tridiagonal matrix T overwrite
*>          the corresponding elements of A, and if UPLO = 'L', the
*>          diagonal and first subdiagonal of T overwrite the
*>          corresponding elements of A.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
*>          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
*>          eigenvectors of the matrix A, with the i-th column of Z
*>          holding the eigenvector associated with W(i).
*>          If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = i, the algorithm failed to converge; i
*>                off-diagonal elements of an intermediate tridiagonal
*>                form did not converge to zero.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpev
*
*  =====================================================================
      SUBROUTINE DSPEV( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, UPLO
      INTEGER            INFO, LDZ, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( * ), W( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            WANTZ
      INTEGER            IINFO, IMAX, INDE, INDTAU, INDWRK, ISCALE
      DOUBLE PRECISION   ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
     $                   SMLNUM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANSP
      EXTERNAL           LSAME, DLAMCH, DLANSP
*     ..
*     .. External Subroutines ..
      EXTERNAL           DOPGTR, DSCAL, DSPTRD, DSTEQR, DSTERF,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
*
      INFO = 0
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( LSAME( UPLO, 'U' ) .OR.
     $         LSAME( UPLO, 'L' ) ) )
     $          THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
         INFO = -7
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSPEV ', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( N.EQ.1 ) THEN
         W( 1 ) = AP( 1 )
         IF( WANTZ )
     $      Z( 1, 1 ) = ONE
         RETURN
      END IF
*
*     Get machine constants.
*
      SAFMIN = DLAMCH( 'Safe minimum' )
      EPS = DLAMCH( 'Precision' )
      SMLNUM = SAFMIN / EPS
      BIGNUM = ONE / SMLNUM
      RMIN = SQRT( SMLNUM )
      RMAX = SQRT( BIGNUM )
*
*     Scale matrix to allowable range, if necessary.
*
      ANRM = DLANSP( 'M', UPLO, N, AP, WORK )
      ISCALE = 0
      IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
         ISCALE = 1
         SIGMA = RMIN / ANRM
      ELSE IF( ANRM.GT.RMAX ) THEN
         ISCALE = 1
         SIGMA = RMAX / ANRM
      END IF
      IF( ISCALE.EQ.1 ) THEN
         CALL DSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
      END IF
*
*     Call DSPTRD to reduce symmetric packed matrix to tridiagonal form.
*
      INDE = 1
      INDTAU = INDE + N
      CALL DSPTRD( UPLO, N, AP, W, WORK( INDE ), WORK( INDTAU ),
     $             IINFO )
*
*     For eigenvalues only, call DSTERF.  For eigenvectors, first call
*     DOPGTR to generate the orthogonal matrix, then call DSTEQR.
*
      IF( .NOT.WANTZ ) THEN
         CALL DSTERF( N, W, WORK( INDE ), INFO )
      ELSE
         INDWRK = INDTAU + N
         CALL DOPGTR( UPLO, N, AP, WORK( INDTAU ), Z, LDZ,
     $                WORK( INDWRK ), IINFO )
         CALL DSTEQR( JOBZ, N, W, WORK( INDE ), Z, LDZ,
     $                WORK( INDTAU ),
     $                INFO )
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
      IF( ISCALE.EQ.1 ) THEN
         IF( INFO.EQ.0 ) THEN
            IMAX = N
         ELSE
            IMAX = INFO - 1
         END IF
         CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
      END IF
*
      RETURN
*
*     End of DSPEV
*
      END
*> \brief <b> DSPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPEVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspevd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspevd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspevd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK,
*                          IWORK, LIWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, UPLO
*       INTEGER            INFO, LDZ, LIWORK, LWORK, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   AP( * ), W( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSPEVD computes all the eigenvalues and, optionally, eigenvectors
*> of a real symmetric matrix A in packed storage. If eigenvectors are
*> desired, it uses a divide and conquer algorithm.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the symmetric matrix
*>          A, packed columnwise in a linear array.  The j-th column of A
*>          is stored in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*>
*>          On exit, AP is overwritten by values generated during the
*>          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
*>          and first superdiagonal of the tridiagonal matrix T overwrite
*>          the corresponding elements of A, and if UPLO = 'L', the
*>          diagonal and first subdiagonal of T overwrite the
*>          corresponding elements of A.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
*>          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
*>          eigenvectors of the matrix A, with the i-th column of Z
*>          holding the eigenvector associated with W(i).
*>          If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the required LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If N <= 1,               LWORK must be at least 1.
*>          If JOBZ = 'N' and N > 1, LWORK must be at least 2*N.
*>          If JOBZ = 'V' and N > 1, LWORK must be at least
*>                                                 1 + 6*N + N**2.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the required sizes of the WORK and IWORK
*>          arrays, returns these values as the first entries of the WORK
*>          and IWORK arrays, and no error message related to LWORK or
*>          LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*>          On exit, if INFO = 0, IWORK(1) returns the required LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK.
*>          If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
*>          If JOBZ  = 'V' and N > 1, LIWORK must be at least 3 + 5*N.
*>
*>          If LIWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the required sizes of the WORK and
*>          IWORK arrays, returns these values as the first entries of
*>          the WORK and IWORK arrays, and no error message related to
*>          LWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = i, the algorithm failed to converge; i
*>                off-diagonal elements of an intermediate tridiagonal
*>                form did not converge to zero.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpevd
*
*  =====================================================================
      SUBROUTINE DSPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK,
     $                   IWORK, LIWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, UPLO
      INTEGER            INFO, LDZ, LIWORK, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   AP( * ), W( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, WANTZ
      INTEGER            IINFO, INDE, INDTAU, INDWRK, ISCALE, LIWMIN,
     $                   LLWORK, LWMIN
      DOUBLE PRECISION   ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
     $                   SMLNUM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANSP
      EXTERNAL           LSAME, DLAMCH, DLANSP
*     ..
*     .. External Subroutines ..
      EXTERNAL           DOPMTR, DSCAL, DSPTRD, DSTEDC, DSTERF,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
      LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
      INFO = 0
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( LSAME( UPLO, 'U' ) .OR.
     $         LSAME( UPLO, 'L' ) ) )
     $          THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
         INFO = -7
      END IF
*
      IF( INFO.EQ.0 ) THEN
         IF( N.LE.1 ) THEN
            LIWMIN = 1
            LWMIN = 1
         ELSE
            IF( WANTZ ) THEN
               LIWMIN = 3 + 5*N
               LWMIN = 1 + 6*N + N**2
            ELSE
               LIWMIN = 1
               LWMIN = 2*N
            END IF
         END IF
         IWORK( 1 ) = LIWMIN
         WORK( 1 ) = LWMIN
*
         IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -9
         ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -11
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSPEVD', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( N.EQ.1 ) THEN
         W( 1 ) = AP( 1 )
         IF( WANTZ )
     $      Z( 1, 1 ) = ONE
         RETURN
      END IF
*
*     Get machine constants.
*
      SAFMIN = DLAMCH( 'Safe minimum' )
      EPS = DLAMCH( 'Precision' )
      SMLNUM = SAFMIN / EPS
      BIGNUM = ONE / SMLNUM
      RMIN = SQRT( SMLNUM )
      RMAX = SQRT( BIGNUM )
*
*     Scale matrix to allowable range, if necessary.
*
      ANRM = DLANSP( 'M', UPLO, N, AP, WORK )
      ISCALE = 0
      IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
         ISCALE = 1
         SIGMA = RMIN / ANRM
      ELSE IF( ANRM.GT.RMAX ) THEN
         ISCALE = 1
         SIGMA = RMAX / ANRM
      END IF
      IF( ISCALE.EQ.1 ) THEN
         CALL DSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
      END IF
*
*     Call DSPTRD to reduce symmetric packed matrix to tridiagonal form.
*
      INDE = 1
      INDTAU = INDE + N
      CALL DSPTRD( UPLO, N, AP, W, WORK( INDE ), WORK( INDTAU ),
     $             IINFO )
*
*     For eigenvalues only, call DSTERF.  For eigenvectors, first call
*     DSTEDC to generate the eigenvector matrix, WORK(INDWRK), of the
*     tridiagonal matrix, then call DOPMTR to multiply it by the
*     Householder transformations represented in AP.
*
      IF( .NOT.WANTZ ) THEN
         CALL DSTERF( N, W, WORK( INDE ), INFO )
      ELSE
         INDWRK = INDTAU + N
         LLWORK = LWORK - INDWRK + 1
         CALL DSTEDC( 'I', N, W, WORK( INDE ), Z, LDZ,
     $                WORK( INDWRK ),
     $                LLWORK, IWORK, LIWORK, INFO )
         CALL DOPMTR( 'L', UPLO, 'N', N, N, AP, WORK( INDTAU ), Z,
     $                LDZ,
     $                WORK( INDWRK ), IINFO )
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
      IF( ISCALE.EQ.1 )
     $   CALL DSCAL( N, ONE / SIGMA, W, 1 )
*
      WORK( 1 ) = LWMIN
      IWORK( 1 ) = LIWMIN
      RETURN
*
*     End of DSPEVD
*
      END
*> \brief <b> DSPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPEVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspevx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspevx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspevx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
*                          ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL,
*                          INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, RANGE, UPLO
*       INTEGER            IL, INFO, IU, LDZ, M, N
*       DOUBLE PRECISION   ABSTOL, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            IFAIL( * ), IWORK( * )
*       DOUBLE PRECISION   AP( * ), W( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSPEVX computes selected eigenvalues and, optionally, eigenvectors
*> of a real symmetric matrix A in packed storage.  Eigenvalues/vectors
*> can be selected by specifying either a range of values or a range of
*> indices for the desired eigenvalues.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*>          RANGE is CHARACTER*1
*>          = 'A': all eigenvalues will be found;
*>          = 'V': all eigenvalues in the half-open interval (VL,VU]
*>                 will be found;
*>          = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the symmetric matrix
*>          A, packed columnwise in a linear array.  The j-th column of A
*>          is stored in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*>
*>          On exit, AP is overwritten by values generated during the
*>          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
*>          and first superdiagonal of the tridiagonal matrix T overwrite
*>          the corresponding elements of A, and if UPLO = 'L', the
*>          diagonal and first subdiagonal of T overwrite the
*>          corresponding elements of A.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is DOUBLE PRECISION
*>          If RANGE='V', the lower bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*>          VU is DOUBLE PRECISION
*>          If RANGE='V', the upper bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*>          IL is INTEGER
*>          If RANGE='I', the index of the
*>          smallest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*>          IU is INTEGER
*>          If RANGE='I', the index of the
*>          largest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*>          ABSTOL is DOUBLE PRECISION
*>          The absolute error tolerance for the eigenvalues.
*>          An approximate eigenvalue is accepted as converged
*>          when it is determined to lie in an interval [a,b]
*>          of width less than or equal to
*>
*>                  ABSTOL + EPS *   max( |a|,|b| ) ,
*>
*>          where EPS is the machine precision.  If ABSTOL is less than
*>          or equal to zero, then  EPS*|T|  will be used in its place,
*>          where |T| is the 1-norm of the tridiagonal matrix obtained
*>          by reducing AP to tridiagonal form.
*>
*>          Eigenvalues will be computed most accurately when ABSTOL is
*>          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*>          If this routine returns with INFO>0, indicating that some
*>          eigenvectors did not converge, try setting ABSTOL to
*>          2*DLAMCH('S').
*>
*>          See "Computing Small Singular Values of Bidiagonal Matrices
*>          with Guaranteed High Relative Accuracy," by Demmel and
*>          Kahan, LAPACK Working Note #3.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The total number of eigenvalues found.  0 <= M <= N.
*>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, the selected eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
*>          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*>          contain the orthonormal eigenvectors of the matrix A
*>          corresponding to the selected eigenvalues, with the i-th
*>          column of Z holding the eigenvector associated with W(i).
*>          If an eigenvector fails to converge, then that column of Z
*>          contains the latest approximation to the eigenvector, and the
*>          index of the eigenvector is returned in IFAIL.
*>          If JOBZ = 'N', then Z is not referenced.
*>          Note: the user must ensure that at least max(1,M) columns are
*>          supplied in the array Z; if RANGE = 'V', the exact value of M
*>          is not known in advance and an upper bound must be used.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (8*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*>          IFAIL is INTEGER array, dimension (N)
*>          If JOBZ = 'V', then if INFO = 0, the first M elements of
*>          IFAIL are zero.  If INFO > 0, then IFAIL contains the
*>          indices of the eigenvectors that failed to converge.
*>          If JOBZ = 'N', then IFAIL is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, then i eigenvectors failed to converge.
*>                Their indices are stored in array IFAIL.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpevx
*
*  =====================================================================
      SUBROUTINE DSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
     $                   ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL,
     $                   INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, RANGE, UPLO
      INTEGER            IL, INFO, IU, LDZ, M, N
      DOUBLE PRECISION   ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            IFAIL( * ), IWORK( * )
      DOUBLE PRECISION   AP( * ), W( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLEIG, INDEIG, TEST, VALEIG, WANTZ
      CHARACTER          ORDER
      INTEGER            I, IINFO, IMAX, INDD, INDE, INDEE,
     $                   INDISP, INDIWO, INDTAU, INDWRK, ISCALE, ITMP1,
     $                   J, JJ, NSPLIT
      DOUBLE PRECISION   ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
     $                   SIGMA, SMLNUM, TMP1, VLL, VUU
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANSP
      EXTERNAL           LSAME, DLAMCH, DLANSP
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DOPGTR, DOPMTR, DSCAL, DSPTRD,
     $                   DSTEBZ,
     $                   DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
      ALLEIG = LSAME( RANGE, 'A' )
      VALEIG = LSAME( RANGE, 'V' )
      INDEIG = LSAME( RANGE, 'I' )
*
      INFO = 0
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
         INFO = -2
      ELSE IF( .NOT.( LSAME( UPLO, 'L' ) .OR.
     $         LSAME( UPLO, 'U' ) ) )
     $          THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE
         IF( VALEIG ) THEN
            IF( N.GT.0 .AND. VU.LE.VL )
     $         INFO = -7
         ELSE IF( INDEIG ) THEN
            IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
               INFO = -8
            ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
               INFO = -9
            END IF
         END IF
      END IF
      IF( INFO.EQ.0 ) THEN
         IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
     $      INFO = -14
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSPEVX', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      M = 0
      IF( N.EQ.0 )
     $   RETURN
*
      IF( N.EQ.1 ) THEN
         IF( ALLEIG .OR. INDEIG ) THEN
            M = 1
            W( 1 ) = AP( 1 )
         ELSE
            IF( VL.LT.AP( 1 ) .AND. VU.GE.AP( 1 ) ) THEN
               M = 1
               W( 1 ) = AP( 1 )
            END IF
         END IF
         IF( WANTZ )
     $      Z( 1, 1 ) = ONE
         RETURN
      END IF
*
*     Get machine constants.
*
      SAFMIN = DLAMCH( 'Safe minimum' )
      EPS = DLAMCH( 'Precision' )
      SMLNUM = SAFMIN / EPS
      BIGNUM = ONE / SMLNUM
      RMIN = SQRT( SMLNUM )
      RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
*     Scale matrix to allowable range, if necessary.
*
      ISCALE = 0
      ABSTLL = ABSTOL
      IF( VALEIG ) THEN
         VLL = VL
         VUU = VU
      ELSE
         VLL = ZERO
         VUU = ZERO
      END IF
      ANRM = DLANSP( 'M', UPLO, N, AP, WORK )
      IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
         ISCALE = 1
         SIGMA = RMIN / ANRM
      ELSE IF( ANRM.GT.RMAX ) THEN
         ISCALE = 1
         SIGMA = RMAX / ANRM
      END IF
      IF( ISCALE.EQ.1 ) THEN
         CALL DSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
         IF( ABSTOL.GT.0 )
     $      ABSTLL = ABSTOL*SIGMA
         IF( VALEIG ) THEN
            VLL = VL*SIGMA
            VUU = VU*SIGMA
         END IF
      END IF
*
*     Call DSPTRD to reduce symmetric packed matrix to tridiagonal form.
*
      INDTAU = 1
      INDE = INDTAU + N
      INDD = INDE + N
      INDWRK = INDD + N
      CALL DSPTRD( UPLO, N, AP, WORK( INDD ), WORK( INDE ),
     $             WORK( INDTAU ), IINFO )
*
*     If all eigenvalues are desired and ABSTOL is less than or equal
*     to zero, then call DSTERF or DOPGTR and SSTEQR.  If this fails
*     for some eigenvalue, then try DSTEBZ.
*
      TEST = .FALSE.
      IF (INDEIG) THEN
         IF (IL.EQ.1 .AND. IU.EQ.N) THEN
            TEST = .TRUE.
         END IF
      END IF
      IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
         CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
         INDEE = INDWRK + 2*N
         IF( .NOT.WANTZ ) THEN
            CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
            CALL DSTERF( N, W, WORK( INDEE ), INFO )
         ELSE
            CALL DOPGTR( UPLO, N, AP, WORK( INDTAU ), Z, LDZ,
     $                   WORK( INDWRK ), IINFO )
            CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
            CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
     $                   WORK( INDWRK ), INFO )
            IF( INFO.EQ.0 ) THEN
               DO 10 I = 1, N
                  IFAIL( I ) = 0
   10          CONTINUE
            END IF
         END IF
         IF( INFO.EQ.0 ) THEN
            M = N
            GO TO 20
         END IF
         INFO = 0
      END IF
*
*     Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
*
      IF( WANTZ ) THEN
         ORDER = 'B'
      ELSE
         ORDER = 'E'
      END IF
      INDISP = 1 + N
      INDIWO = INDISP + N
      CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
     $             WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
     $             IWORK( 1 ), IWORK( INDISP ), WORK( INDWRK ),
     $             IWORK( INDIWO ), INFO )
*
      IF( WANTZ ) THEN
         CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
     $                IWORK( 1 ), IWORK( INDISP ), Z, LDZ,
     $                WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
*
*        Apply orthogonal matrix used in reduction to tridiagonal
*        form to eigenvectors returned by DSTEIN.
*
         CALL DOPMTR( 'L', UPLO, 'N', N, M, AP, WORK( INDTAU ), Z,
     $                LDZ,
     $                WORK( INDWRK ), IINFO )
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
   20 CONTINUE
      IF( ISCALE.EQ.1 ) THEN
         IF( INFO.EQ.0 ) THEN
            IMAX = M
         ELSE
            IMAX = INFO - 1
         END IF
         CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
      END IF
*
*     If eigenvalues are not in order, then sort them, along with
*     eigenvectors.
*
      IF( WANTZ ) THEN
         DO 40 J = 1, M - 1
            I = 0
            TMP1 = W( J )
            DO 30 JJ = J + 1, M
               IF( W( JJ ).LT.TMP1 ) THEN
                  I = JJ
                  TMP1 = W( JJ )
               END IF
   30       CONTINUE
*
            IF( I.NE.0 ) THEN
               ITMP1 = IWORK( 1 + I-1 )
               W( I ) = W( J )
               IWORK( 1 + I-1 ) = IWORK( 1 + J-1 )
               W( J ) = TMP1
               IWORK( 1 + J-1 ) = ITMP1
               CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
               IF( INFO.NE.0 ) THEN
                  ITMP1 = IFAIL( I )
                  IFAIL( I ) = IFAIL( J )
                  IFAIL( J ) = ITMP1
               END IF
            END IF
   40    CONTINUE
      END IF
*
      RETURN
*
*     End of DSPEVX
*
      END
*> \brief \b DSPGST
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPGST + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspgst.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspgst.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspgst.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, ITYPE, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AP( * ), BP( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSPGST reduces a real symmetric-definite generalized eigenproblem
*> to standard form, using packed storage.
*>
*> If ITYPE = 1, the problem is A*x = lambda*B*x,
*> and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
*>
*> If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
*> B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.
*>
*> B must have been previously factorized as U**T*U or L*L**T by DPPTRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ITYPE
*> \verbatim
*>          ITYPE is INTEGER
*>          = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
*>          = 2 or 3: compute U*A*U**T or L**T*A*L.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored and B is factored as
*>                  U**T*U;
*>          = 'L':  Lower triangle of A is stored and B is factored as
*>                  L*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A and B.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the symmetric matrix
*>          A, packed columnwise in a linear array.  The j-th column of A
*>          is stored in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*>
*>          On exit, if INFO = 0, the transformed matrix, stored in the
*>          same format as A.
*> \endverbatim
*>
*> \param[in] BP
*> \verbatim
*>          BP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          The triangular factor from the Cholesky factorization of B,
*>          stored in the same format as A, as returned by DPPTRF.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpgst
*
*  =====================================================================
      SUBROUTINE DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, ITYPE, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( * ), BP( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, HALF
      PARAMETER          ( ONE = 1.0D0, HALF = 0.5D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            J, J1, J1J1, JJ, K, K1, K1K1, KK
      DOUBLE PRECISION   AJJ, AKK, BJJ, BKK, CT
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DSCAL, DSPMV, DSPR2, DTPMV,
     $                   DTPSV,
     $                   XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DDOT
      EXTERNAL           LSAME, DDOT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
         INFO = -1
      ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSPGST', -INFO )
         RETURN
      END IF
*
      IF( ITYPE.EQ.1 ) THEN
         IF( UPPER ) THEN
*
*           Compute inv(U**T)*A*inv(U)
*
*           J1 and JJ are the indices of A(1,j) and A(j,j)
*
            JJ = 0
            DO 10 J = 1, N
               J1 = JJ + 1
               JJ = JJ + J
*
*              Compute the j-th column of the upper triangle of A
*
               BJJ = BP( JJ )
               CALL DTPSV( UPLO, 'Transpose', 'Nonunit', J, BP,
     $                     AP( J1 ), 1 )
               CALL DSPMV( UPLO, J-1, -ONE, AP, BP( J1 ), 1, ONE,
     $                     AP( J1 ), 1 )
               CALL DSCAL( J-1, ONE / BJJ, AP( J1 ), 1 )
               AP( JJ ) = ( AP( JJ )-DDOT( J-1, AP( J1 ), 1,
     $             BP( J1 ),
     $                    1 ) ) / BJJ
   10       CONTINUE
         ELSE
*
*           Compute inv(L)*A*inv(L**T)
*
*           KK and K1K1 are the indices of A(k,k) and A(k+1,k+1)
*
            KK = 1
            DO 20 K = 1, N
               K1K1 = KK + N - K + 1
*
*              Update the lower triangle of A(k:n,k:n)
*
               AKK = AP( KK )
               BKK = BP( KK )
               AKK = AKK / BKK**2
               AP( KK ) = AKK
               IF( K.LT.N ) THEN
                  CALL DSCAL( N-K, ONE / BKK, AP( KK+1 ), 1 )
                  CT = -HALF*AKK
                  CALL DAXPY( N-K, CT, BP( KK+1 ), 1, AP( KK+1 ), 1 )
                  CALL DSPR2( UPLO, N-K, -ONE, AP( KK+1 ), 1,
     $                        BP( KK+1 ), 1, AP( K1K1 ) )
                  CALL DAXPY( N-K, CT, BP( KK+1 ), 1, AP( KK+1 ), 1 )
                  CALL DTPSV( UPLO, 'No transpose', 'Non-unit', N-K,
     $                        BP( K1K1 ), AP( KK+1 ), 1 )
               END IF
               KK = K1K1
   20       CONTINUE
         END IF
      ELSE
         IF( UPPER ) THEN
*
*           Compute U*A*U**T
*
*           K1 and KK are the indices of A(1,k) and A(k,k)
*
            KK = 0
            DO 30 K = 1, N
               K1 = KK + 1
               KK = KK + K
*
*              Update the upper triangle of A(1:k,1:k)
*
               AKK = AP( KK )
               BKK = BP( KK )
               CALL DTPMV( UPLO, 'No transpose', 'Non-unit', K-1, BP,
     $                     AP( K1 ), 1 )
               CT = HALF*AKK
               CALL DAXPY( K-1, CT, BP( K1 ), 1, AP( K1 ), 1 )
               CALL DSPR2( UPLO, K-1, ONE, AP( K1 ), 1, BP( K1 ), 1,
     $                     AP )
               CALL DAXPY( K-1, CT, BP( K1 ), 1, AP( K1 ), 1 )
               CALL DSCAL( K-1, BKK, AP( K1 ), 1 )
               AP( KK ) = AKK*BKK**2
   30       CONTINUE
         ELSE
*
*           Compute L**T *A*L
*
*           JJ and J1J1 are the indices of A(j,j) and A(j+1,j+1)
*
            JJ = 1
            DO 40 J = 1, N
               J1J1 = JJ + N - J + 1
*
*              Compute the j-th column of the lower triangle of A
*
               AJJ = AP( JJ )
               BJJ = BP( JJ )
               AP( JJ ) = AJJ*BJJ + DDOT( N-J, AP( JJ+1 ), 1,
     $                    BP( JJ+1 ), 1 )
               CALL DSCAL( N-J, BJJ, AP( JJ+1 ), 1 )
               CALL DSPMV( UPLO, N-J, ONE, AP( J1J1 ), BP( JJ+1 ), 1,
     $                     ONE, AP( JJ+1 ), 1 )
               CALL DTPMV( UPLO, 'Transpose', 'Non-unit', N-J+1,
     $                     BP( JJ ), AP( JJ ), 1 )
               JJ = J1J1
   40       CONTINUE
         END IF
      END IF
      RETURN
*
*     End of DSPGST
*
      END
*> \brief \b DSPGV
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPGV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspgv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspgv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspgv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSPGV( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK,
*                         INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, UPLO
*       INTEGER            INFO, ITYPE, LDZ, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AP( * ), BP( * ), W( * ), WORK( * ),
*      $                   Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSPGV computes all the eigenvalues and, optionally, the eigenvectors
*> of a real generalized symmetric-definite eigenproblem, of the form
*> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
*> Here A and B are assumed to be symmetric, stored in packed format,
*> and B is also positive definite.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ITYPE
*> \verbatim
*>          ITYPE is INTEGER
*>          Specifies the problem type to be solved:
*>          = 1:  A*x = (lambda)*B*x
*>          = 2:  A*B*x = (lambda)*x
*>          = 3:  B*A*x = (lambda)*x
*> \endverbatim
*>
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangles of A and B are stored;
*>          = 'L':  Lower triangles of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A and B.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the symmetric matrix
*>          A, packed columnwise in a linear array.  The j-th column of A
*>          is stored in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*>
*>          On exit, the contents of AP are destroyed.
*> \endverbatim
*>
*> \param[in,out] BP
*> \verbatim
*>          BP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the symmetric matrix
*>          B, packed columnwise in a linear array.  The j-th column of B
*>          is stored in the array BP as follows:
*>          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
*>          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
*>
*>          On exit, the triangular factor U or L from the Cholesky
*>          factorization B = U**T*U or B = L*L**T, in the same storage
*>          format as B.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
*>          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
*>          eigenvectors.  The eigenvectors are normalized as follows:
*>          if ITYPE = 1 or 2, Z**T*B*Z = I;
*>          if ITYPE = 3, Z**T*inv(B)*Z = I.
*>          If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  DPPTRF or DSPEV returned an error code:
*>             <= N:  if INFO = i, DSPEV failed to converge;
*>                    i off-diagonal elements of an intermediate
*>                    tridiagonal form did not converge to zero.
*>             > N:   if INFO = n + i, for 1 <= i <= n, then the leading
*>                    principal minor of order i of B is not positive.
*>                    The factorization of B could not be completed and
*>                    no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpgv
*
*  =====================================================================
      SUBROUTINE DSPGV( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ,
     $                  WORK,
     $                  INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, UPLO
      INTEGER            INFO, ITYPE, LDZ, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( * ), BP( * ), W( * ), WORK( * ),
     $                   Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            UPPER, WANTZ
      CHARACTER          TRANS
      INTEGER            J, NEIG
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DPPTRF, DSPEV, DSPGST, DTPMV, DTPSV,
     $                   XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
      UPPER = LSAME( UPLO, 'U' )
*
      INFO = 0
      IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
         INFO = -1
      ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -2
      ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
         INFO = -9
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSPGV ', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Form a Cholesky factorization of B.
*
      CALL DPPTRF( UPLO, N, BP, INFO )
      IF( INFO.NE.0 ) THEN
         INFO = N + INFO
         RETURN
      END IF
*
*     Transform problem to standard eigenvalue problem and solve.
*
      CALL DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
      CALL DSPEV( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, INFO )
*
      IF( WANTZ ) THEN
*
*        Backtransform eigenvectors to the original problem.
*
         NEIG = N
         IF( INFO.GT.0 )
     $      NEIG = INFO - 1
         IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
*
*           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
*           backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
*
            IF( UPPER ) THEN
               TRANS = 'N'
            ELSE
               TRANS = 'T'
            END IF
*
            DO 10 J = 1, NEIG
               CALL DTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
     $                     1 )
   10       CONTINUE
*
         ELSE IF( ITYPE.EQ.3 ) THEN
*
*           For B*A*x=(lambda)*x;
*           backtransform eigenvectors: x = L*y or U**T*y
*
            IF( UPPER ) THEN
               TRANS = 'T'
            ELSE
               TRANS = 'N'
            END IF
*
            DO 20 J = 1, NEIG
               CALL DTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
     $                     1 )
   20       CONTINUE
         END IF
      END IF
      RETURN
*
*     End of DSPGV
*
      END
*> \brief \b DSPGVD
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPGVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspgvd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspgvd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspgvd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSPGVD( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK,
*                          LWORK, IWORK, LIWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, UPLO
*       INTEGER            INFO, ITYPE, LDZ, LIWORK, LWORK, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   AP( * ), BP( * ), W( * ), WORK( * ),
*      $                   Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSPGVD computes all the eigenvalues, and optionally, the eigenvectors
*> of a real generalized symmetric-definite eigenproblem, of the form
*> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
*> B are assumed to be symmetric, stored in packed format, and B is also
*> positive definite.
*> If eigenvectors are desired, it uses a divide and conquer algorithm.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ITYPE
*> \verbatim
*>          ITYPE is INTEGER
*>          Specifies the problem type to be solved:
*>          = 1:  A*x = (lambda)*B*x
*>          = 2:  A*B*x = (lambda)*x
*>          = 3:  B*A*x = (lambda)*x
*> \endverbatim
*>
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangles of A and B are stored;
*>          = 'L':  Lower triangles of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A and B.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the symmetric matrix
*>          A, packed columnwise in a linear array.  The j-th column of A
*>          is stored in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*>
*>          On exit, the contents of AP are destroyed.
*> \endverbatim
*>
*> \param[in,out] BP
*> \verbatim
*>          BP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the symmetric matrix
*>          B, packed columnwise in a linear array.  The j-th column of B
*>          is stored in the array BP as follows:
*>          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
*>          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
*>
*>          On exit, the triangular factor U or L from the Cholesky
*>          factorization B = U**T*U or B = L*L**T, in the same storage
*>          format as B.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
*>          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
*>          eigenvectors.  The eigenvectors are normalized as follows:
*>          if ITYPE = 1 or 2, Z**T*B*Z = I;
*>          if ITYPE = 3, Z**T*inv(B)*Z = I.
*>          If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the required LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If N <= 1,               LWORK >= 1.
*>          If JOBZ = 'N' and N > 1, LWORK >= 2*N.
*>          If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the required sizes of the WORK and IWORK
*>          arrays, returns these values as the first entries of the WORK
*>          and IWORK arrays, and no error message related to LWORK or
*>          LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*>          On exit, if INFO = 0, IWORK(1) returns the required LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK.
*>          If JOBZ  = 'N' or N <= 1, LIWORK >= 1.
*>          If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.
*>
*>          If LIWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the required sizes of the WORK and
*>          IWORK arrays, returns these values as the first entries of
*>          the WORK and IWORK arrays, and no error message related to
*>          LWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  DPPTRF or DSPEVD returned an error code:
*>             <= N:  if INFO = i, DSPEVD failed to converge;
*>                    i off-diagonal elements of an intermediate
*>                    tridiagonal form did not converge to zero;
*>             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
*>                    principal minor of order i of B is not positive.
*>                    The factorization of B could not be completed and
*>                    no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpgvd
*
*> \par Contributors:
*  ==================
*>
*>     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
*  =====================================================================
      SUBROUTINE DSPGVD( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ,
     $                   WORK,
     $                   LWORK, IWORK, LIWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, UPLO
      INTEGER            INFO, ITYPE, LDZ, LIWORK, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   AP( * ), BP( * ), W( * ), WORK( * ),
     $                   Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            LQUERY, UPPER, WANTZ
      CHARACTER          TRANS
      INTEGER            J, LIWMIN, LWMIN, NEIG
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DPPTRF, DSPEVD, DSPGST, DTPMV, DTPSV,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE, MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
      UPPER = LSAME( UPLO, 'U' )
      LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
      INFO = 0
      IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
         INFO = -1
      ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -2
      ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
         INFO = -9
      END IF
*
      IF( INFO.EQ.0 ) THEN
         IF( N.LE.1 ) THEN
            LIWMIN = 1
            LWMIN = 1
         ELSE
            IF( WANTZ ) THEN
               LIWMIN = 3 + 5*N
               LWMIN = 1 + 6*N + 2*N**2
            ELSE
               LIWMIN = 1
               LWMIN = 2*N
            END IF
         END IF
         WORK( 1 ) = LWMIN
         IWORK( 1 ) = LIWMIN
         IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -11
         ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -13
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSPGVD', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Form a Cholesky factorization of BP.
*
      CALL DPPTRF( UPLO, N, BP, INFO )
      IF( INFO.NE.0 ) THEN
         INFO = N + INFO
         RETURN
      END IF
*
*     Transform problem to standard eigenvalue problem and solve.
*
      CALL DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
      CALL DSPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK, IWORK,
     $             LIWORK, INFO )
      LWMIN = INT( MAX( DBLE( LWMIN ), DBLE( WORK( 1 ) ) ) )
      LIWMIN = INT( MAX( DBLE( LIWMIN ), DBLE( IWORK( 1 ) ) ) )
*
      IF( WANTZ ) THEN
*
*        Backtransform eigenvectors to the original problem.
*
         NEIG = N
         IF( INFO.GT.0 )
     $      NEIG = INFO - 1
         IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
*
*           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
*           backtransform eigenvectors: x = inv(L)**T *y or inv(U)*y
*
            IF( UPPER ) THEN
               TRANS = 'N'
            ELSE
               TRANS = 'T'
            END IF
*
            DO 10 J = 1, NEIG
               CALL DTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
     $                     1 )
   10       CONTINUE
*
         ELSE IF( ITYPE.EQ.3 ) THEN
*
*           For B*A*x=(lambda)*x;
*           backtransform eigenvectors: x = L*y or U**T *y
*
            IF( UPPER ) THEN
               TRANS = 'T'
            ELSE
               TRANS = 'N'
            END IF
*
            DO 20 J = 1, NEIG
               CALL DTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
     $                     1 )
   20       CONTINUE
         END IF
      END IF
*
      WORK( 1 ) = LWMIN
      IWORK( 1 ) = LIWMIN
*
      RETURN
*
*     End of DSPGVD
*
      END
*> \brief \b DSPGVX
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPGVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspgvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspgvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspgvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
*                          IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
*                          IFAIL, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, RANGE, UPLO
*       INTEGER            IL, INFO, ITYPE, IU, LDZ, M, N
*       DOUBLE PRECISION   ABSTOL, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            IFAIL( * ), IWORK( * )
*       DOUBLE PRECISION   AP( * ), BP( * ), W( * ), WORK( * ),
*      $                   Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSPGVX computes selected eigenvalues, and optionally, eigenvectors
*> of a real generalized symmetric-definite eigenproblem, of the form
*> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A
*> and B are assumed to be symmetric, stored in packed storage, and B
*> is also positive definite.  Eigenvalues and eigenvectors can be
*> selected by specifying either a range of values or a range of indices
*> for the desired eigenvalues.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ITYPE
*> \verbatim
*>          ITYPE is INTEGER
*>          Specifies the problem type to be solved:
*>          = 1:  A*x = (lambda)*B*x
*>          = 2:  A*B*x = (lambda)*x
*>          = 3:  B*A*x = (lambda)*x
*> \endverbatim
*>
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*>          RANGE is CHARACTER*1
*>          = 'A': all eigenvalues will be found.
*>          = 'V': all eigenvalues in the half-open interval (VL,VU]
*>                 will be found.
*>          = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A and B are stored;
*>          = 'L':  Lower triangle of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix pencil (A,B).  N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the symmetric matrix
*>          A, packed columnwise in a linear array.  The j-th column of A
*>          is stored in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*>
*>          On exit, the contents of AP are destroyed.
*> \endverbatim
*>
*> \param[in,out] BP
*> \verbatim
*>          BP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the symmetric matrix
*>          B, packed columnwise in a linear array.  The j-th column of B
*>          is stored in the array BP as follows:
*>          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
*>          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
*>
*>          On exit, the triangular factor U or L from the Cholesky
*>          factorization B = U**T*U or B = L*L**T, in the same storage
*>          format as B.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is DOUBLE PRECISION
*>
*>          If RANGE='V', the lower bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*>          VU is DOUBLE PRECISION
*>
*>          If RANGE='V', the upper bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*>          IL is INTEGER
*>
*>          If RANGE='I', the index of the
*>          smallest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*>          IU is INTEGER
*>
*>          If RANGE='I', the index of the
*>          largest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*>          ABSTOL is DOUBLE PRECISION
*>          The absolute error tolerance for the eigenvalues.
*>          An approximate eigenvalue is accepted as converged
*>          when it is determined to lie in an interval [a,b]
*>          of width less than or equal to
*>
*>                  ABSTOL + EPS *   max( |a|,|b| ) ,
*>
*>          where EPS is the machine precision.  If ABSTOL is less than
*>          or equal to zero, then  EPS*|T|  will be used in its place,
*>          where |T| is the 1-norm of the tridiagonal matrix obtained
*>          by reducing A to tridiagonal form.
*>
*>          Eigenvalues will be computed most accurately when ABSTOL is
*>          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*>          If this routine returns with INFO>0, indicating that some
*>          eigenvectors did not converge, try setting ABSTOL to
*>          2*DLAMCH('S').
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The total number of eigenvalues found.  0 <= M <= N.
*>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          On normal exit, the first M elements contain the selected
*>          eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
*>          If JOBZ = 'N', then Z is not referenced.
*>          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*>          contain the orthonormal eigenvectors of the matrix A
*>          corresponding to the selected eigenvalues, with the i-th
*>          column of Z holding the eigenvector associated with W(i).
*>          The eigenvectors are normalized as follows:
*>          if ITYPE = 1 or 2, Z**T*B*Z = I;
*>          if ITYPE = 3, Z**T*inv(B)*Z = I.
*>
*>          If an eigenvector fails to converge, then that column of Z
*>          contains the latest approximation to the eigenvector, and the
*>          index of the eigenvector is returned in IFAIL.
*>          Note: the user must ensure that at least max(1,M) columns are
*>          supplied in the array Z; if RANGE = 'V', the exact value of M
*>          is not known in advance and an upper bound must be used.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (8*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*>          IFAIL is INTEGER array, dimension (N)
*>          If JOBZ = 'V', then if INFO = 0, the first M elements of
*>          IFAIL are zero.  If INFO > 0, then IFAIL contains the
*>          indices of the eigenvectors that failed to converge.
*>          If JOBZ = 'N', then IFAIL is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  DPPTRF or DSPEVX returned an error code:
*>             <= N:  if INFO = i, DSPEVX failed to converge;
*>                    i eigenvectors failed to converge.  Their indices
*>                    are stored in array IFAIL.
*>             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
*>                    principal minor of order i of B is not positive.
*>                    The factorization of B could not be completed and
*>                    no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpgvx
*
*> \par Contributors:
*  ==================
*>
*>     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
*  =====================================================================
      SUBROUTINE DSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
     $                   IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
     $                   IFAIL, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, RANGE, UPLO
      INTEGER            IL, INFO, ITYPE, IU, LDZ, M, N
      DOUBLE PRECISION   ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            IFAIL( * ), IWORK( * )
      DOUBLE PRECISION   AP( * ), BP( * ), W( * ), WORK( * ),
     $                   Z( LDZ, * )
*     ..
*
* =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            ALLEIG, INDEIG, UPPER, VALEIG, WANTZ
      CHARACTER          TRANS
      INTEGER            J
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DPPTRF, DSPEVX, DSPGST, DTPMV, DTPSV,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      UPPER = LSAME( UPLO, 'U' )
      WANTZ = LSAME( JOBZ, 'V' )
      ALLEIG = LSAME( RANGE, 'A' )
      VALEIG = LSAME( RANGE, 'V' )
      INDEIG = LSAME( RANGE, 'I' )
*
      INFO = 0
      IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
         INFO = -1
      ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -2
      ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
         INFO = -3
      ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
         INFO = -4
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      ELSE
         IF( VALEIG ) THEN
            IF( N.GT.0 .AND. VU.LE.VL ) THEN
               INFO = -9
            END IF
         ELSE IF( INDEIG ) THEN
            IF( IL.LT.1 ) THEN
               INFO = -10
            ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
               INFO = -11
            END IF
         END IF
      END IF
      IF( INFO.EQ.0 ) THEN
         IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
            INFO = -16
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSPGVX', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      M = 0
      IF( N.EQ.0 )
     $   RETURN
*
*     Form a Cholesky factorization of B.
*
      CALL DPPTRF( UPLO, N, BP, INFO )
      IF( INFO.NE.0 ) THEN
         INFO = N + INFO
         RETURN
      END IF
*
*     Transform problem to standard eigenvalue problem and solve.
*
      CALL DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
      CALL DSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL,
     $             M,
     $             W, Z, LDZ, WORK, IWORK, IFAIL, INFO )
*
      IF( WANTZ ) THEN
*
*        Backtransform eigenvectors to the original problem.
*
         IF( INFO.GT.0 )
     $      M = INFO - 1
         IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
*
*           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
*           backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
*
            IF( UPPER ) THEN
               TRANS = 'N'
            ELSE
               TRANS = 'T'
            END IF
*
            DO 10 J = 1, M
               CALL DTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
     $                     1 )
   10       CONTINUE
*
         ELSE IF( ITYPE.EQ.3 ) THEN
*
*           For B*A*x=(lambda)*x;
*           backtransform eigenvectors: x = L*y or U**T*y
*
            IF( UPPER ) THEN
               TRANS = 'T'
            ELSE
               TRANS = 'N'
            END IF
*
            DO 20 J = 1, M
               CALL DTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
     $                     1 )
   20       CONTINUE
         END IF
      END IF
*
      RETURN
*
*     End of DSPGVX
*
      END
*> \brief \b DSPRFS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPRFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsprfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsprfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsprfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
*                          FERR, BERR, WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDB, LDX, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), IWORK( * )
*       DOUBLE PRECISION   AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
*      $                   FERR( * ), WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSPRFS improves the computed solution to a system of linear
*> equations when the coefficient matrix is symmetric indefinite
*> and packed, and provides error bounds and backward error estimates
*> for the solution.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices B and X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          The upper or lower triangle of the symmetric matrix A, packed
*>          columnwise in a linear array.  The j-th column of A is stored
*>          in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[in] AFP
*> \verbatim
*>          AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          The factored form of the matrix A.  AFP contains the block
*>          diagonal matrix D and the multipliers used to obtain the
*>          factor U or L from the factorization A = U*D*U**T or
*>          A = L*D*L**T as computed by DSPTRF, stored as a packed
*>          triangular matrix.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the block structure of D
*>          as determined by DSPTRF.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          On entry, the solution matrix X, as computed by DSPTRS.
*>          On exit, the improved solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The estimated forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).  The estimate is as reliable as
*>          the estimate for RCOND, and is almost always a slight
*>          overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in
*>          any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  ITMAX is the maximum number of steps of iterative refinement.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hprfs
*
*  =====================================================================
      SUBROUTINE DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,
     $                   LDX,
     $                   FERR, BERR, WORK, IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDB, LDX, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), IWORK( * )
      DOUBLE PRECISION   AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
     $                   FERR( * ), WORK( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            ITMAX
      PARAMETER          ( ITMAX = 5 )
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
      DOUBLE PRECISION   TWO
      PARAMETER          ( TWO = 2.0D+0 )
      DOUBLE PRECISION   THREE
      PARAMETER          ( THREE = 3.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            COUNT, I, IK, J, K, KASE, KK, NZ
      DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DLACN2, DSPMV, DSPTRS,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -8
      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
         INFO = -10
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSPRFS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
         DO 10 J = 1, NRHS
            FERR( J ) = ZERO
            BERR( J ) = ZERO
   10    CONTINUE
         RETURN
      END IF
*
*     NZ = maximum number of nonzero elements in each row of A, plus 1
*
      NZ = N + 1
      EPS = DLAMCH( 'Epsilon' )
      SAFMIN = DLAMCH( 'Safe minimum' )
      SAFE1 = NZ*SAFMIN
      SAFE2 = SAFE1 / EPS
*
*     Do for each right hand side
*
      DO 140 J = 1, NRHS
*
         COUNT = 1
         LSTRES = THREE
   20    CONTINUE
*
*        Loop until stopping criterion is satisfied.
*
*        Compute residual R = B - A * X
*
         CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
         CALL DSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE,
     $               WORK( N+1 ),
     $               1 )
*
*        Compute componentwise relative backward error from formula
*
*        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
*
*        where abs(Z) is the componentwise absolute value of the matrix
*        or vector Z.  If the i-th component of the denominator is less
*        than SAFE2, then SAFE1 is added to the i-th components of the
*        numerator and denominator before dividing.
*
         DO 30 I = 1, N
            WORK( I ) = ABS( B( I, J ) )
   30    CONTINUE
*
*        Compute abs(A)*abs(X) + abs(B).
*
         KK = 1
         IF( UPPER ) THEN
            DO 50 K = 1, N
               S = ZERO
               XK = ABS( X( K, J ) )
               IK = KK
               DO 40 I = 1, K - 1
                  WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
                  S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
                  IK = IK + 1
   40          CONTINUE
               WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S
               KK = KK + K
   50       CONTINUE
         ELSE
            DO 70 K = 1, N
               S = ZERO
               XK = ABS( X( K, J ) )
               WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK
               IK = KK + 1
               DO 60 I = K + 1, N
                  WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
                  S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
                  IK = IK + 1
   60          CONTINUE
               WORK( K ) = WORK( K ) + S
               KK = KK + ( N-K+1 )
   70       CONTINUE
         END IF
         S = ZERO
         DO 80 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
            ELSE
               S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
     $             ( WORK( I )+SAFE1 ) )
            END IF
   80    CONTINUE
         BERR( J ) = S
*
*        Test stopping criterion. Continue iterating if
*           1) The residual BERR(J) is larger than machine epsilon, and
*           2) BERR(J) decreased by at least a factor of 2 during the
*              last iteration, and
*           3) At most ITMAX iterations tried.
*
         IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
     $       COUNT.LE.ITMAX ) THEN
*
*           Update solution and try again.
*
            CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N,
     $                   INFO )
            CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
            LSTRES = BERR( J )
            COUNT = COUNT + 1
            GO TO 20
         END IF
*
*        Bound error from formula
*
*        norm(X - XTRUE) / norm(X) .le. FERR =
*        norm( abs(inv(A))*
*           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
*
*        where
*          norm(Z) is the magnitude of the largest component of Z
*          inv(A) is the inverse of A
*          abs(Z) is the componentwise absolute value of the matrix or
*             vector Z
*          NZ is the maximum number of nonzeros in any row of A, plus 1
*          EPS is machine epsilon
*
*        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
*        is incremented by SAFE1 if the i-th component of
*        abs(A)*abs(X) + abs(B) is less than SAFE2.
*
*        Use DLACN2 to estimate the infinity-norm of the matrix
*           inv(A) * diag(W),
*        where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
*
         DO 90 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
            ELSE
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
            END IF
   90    CONTINUE
*
         KASE = 0
  100    CONTINUE
         CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
     $                FERR( J ),
     $                KASE, ISAVE )
         IF( KASE.NE.0 ) THEN
            IF( KASE.EQ.1 ) THEN
*
*              Multiply by diag(W)*inv(A**T).
*
               CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N,
     $                      INFO )
               DO 110 I = 1, N
                  WORK( N+I ) = WORK( I )*WORK( N+I )
  110          CONTINUE
            ELSE IF( KASE.EQ.2 ) THEN
*
*              Multiply by inv(A)*diag(W).
*
               DO 120 I = 1, N
                  WORK( N+I ) = WORK( I )*WORK( N+I )
  120          CONTINUE
               CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N,
     $                      INFO )
            END IF
            GO TO 100
         END IF
*
*        Normalize error.
*
         LSTRES = ZERO
         DO 130 I = 1, N
            LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
  130    CONTINUE
         IF( LSTRES.NE.ZERO )
     $      FERR( J ) = FERR( J ) / LSTRES
*
  140 CONTINUE
*
      RETURN
*
*     End of DSPRFS
*
      END
*> \brief <b> DSPSV computes the solution to system of linear equations A * X = B for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPSV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspsv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspsv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspsv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSPSV( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   AP( * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSPSV computes the solution to a real system of linear equations
*>    A * X = B,
*> where A is an N-by-N symmetric matrix stored in packed format and X
*> and B are N-by-NRHS matrices.
*>
*> The diagonal pivoting method is used to factor A as
*>    A = U * D * U**T,  if UPLO = 'U', or
*>    A = L * D * L**T,  if UPLO = 'L',
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, D is symmetric and block diagonal with 1-by-1
*> and 2-by-2 diagonal blocks.  The factored form of A is then used to
*> solve the system of equations A * X = B.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of linear equations, i.e., the order of the
*>          matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the symmetric matrix
*>          A, packed columnwise in a linear array.  The j-th column of A
*>          is stored in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*>          See below for further details.
*>
*>          On exit, the block diagonal matrix D and the multipliers used
*>          to obtain the factor U or L from the factorization
*>          A = U*D*U**T or A = L*D*L**T as computed by DSPTRF, stored as
*>          a packed triangular matrix in the same storage format as A.
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the block structure of D, as
*>          determined by DSPTRF.  If IPIV(k) > 0, then rows and columns
*>          k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1
*>          diagonal block.  If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0,
*>          then rows and columns k-1 and -IPIV(k) were interchanged and
*>          D(k-1:k,k-1:k) is a 2-by-2 diagonal block.  If UPLO = 'L' and
*>          IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and
*>          -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
*>          diagonal block.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the N-by-NRHS right hand side matrix B.
*>          On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, D(i,i) is exactly zero.  The factorization
*>                has been completed, but the block diagonal matrix D is
*>                exactly singular, so the solution could not be
*>                computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpsv
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The packed storage scheme is illustrated by the following example
*>  when N = 4, UPLO = 'U':
*>
*>  Two-dimensional storage of the symmetric matrix A:
*>
*>     a11 a12 a13 a14
*>         a22 a23 a24
*>             a33 a34     (aij = aji)
*>                 a44
*>
*>  Packed storage of the upper triangle of A:
*>
*>  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DSPSV( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   AP( * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSPTRF, DSPTRS, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF( .NOT.LSAME( UPLO, 'U' ) .AND.
     $    .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSPSV ', -INFO )
         RETURN
      END IF
*
*     Compute the factorization A = U*D*U**T or A = L*D*L**T.
*
      CALL DSPTRF( UPLO, N, AP, IPIV, INFO )
      IF( INFO.EQ.0 ) THEN
*
*        Solve the system A*X = B, overwriting B with X.
*
         CALL DSPTRS( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO )
*
      END IF
      RETURN
*
*     End of DSPSV
*
      END
*> \brief <b> DSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPSVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspsvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspsvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspsvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,
*                          LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          FACT, UPLO
*       INTEGER            INFO, LDB, LDX, N, NRHS
*       DOUBLE PRECISION   RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), IWORK( * )
*       DOUBLE PRECISION   AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
*      $                   FERR( * ), WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
*> A = L*D*L**T to compute the solution to a real system of linear
*> equations A * X = B, where A is an N-by-N symmetric matrix stored
*> in packed format and X and B are N-by-NRHS matrices.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
*  =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'N', the diagonal pivoting method is used to factor A as
*>       A = U * D * U**T,  if UPLO = 'U', or
*>       A = L * D * L**T,  if UPLO = 'L',
*>    where U (or L) is a product of permutation and unit upper (lower)
*>    triangular matrices and D is symmetric and block diagonal with
*>    1-by-1 and 2-by-2 diagonal blocks.
*>
*> 2. If some D(i,i)=0, so that D is exactly singular, then the routine
*>    returns with INFO = i. Otherwise, the factored form of A is used
*>    to estimate the condition number of the matrix A.  If the
*>    reciprocal of the condition number is less than machine precision,
*>    INFO = N+1 is returned as a warning, but the routine still goes on
*>    to solve for X and compute error bounds as described below.
*>
*> 3. The system of equations is solved for X using the factored form
*>    of A.
*>
*> 4. Iterative refinement is applied to improve the computed solution
*>    matrix and calculate error bounds and backward error estimates
*>    for it.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] FACT
*> \verbatim
*>          FACT is CHARACTER*1
*>          Specifies whether or not the factored form of A has been
*>          supplied on entry.
*>          = 'F':  On entry, AFP and IPIV contain the factored form of
*>                  A.  AP, AFP and IPIV will not be modified.
*>          = 'N':  The matrix A will be copied to AFP and factored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of linear equations, i.e., the order of the
*>          matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices B and X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          The upper or lower triangle of the symmetric matrix A, packed
*>          columnwise in a linear array.  The j-th column of A is stored
*>          in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*>          See below for further details.
*> \endverbatim
*>
*> \param[in,out] AFP
*> \verbatim
*>          AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          If FACT = 'F', then AFP is an input argument and on entry
*>          contains the block diagonal matrix D and the multipliers used
*>          to obtain the factor U or L from the factorization
*>          A = U*D*U**T or A = L*D*L**T as computed by DSPTRF, stored as
*>          a packed triangular matrix in the same storage format as A.
*>
*>          If FACT = 'N', then AFP is an output argument and on exit
*>          contains the block diagonal matrix D and the multipliers used
*>          to obtain the factor U or L from the factorization
*>          A = U*D*U**T or A = L*D*L**T as computed by DSPTRF, stored as
*>          a packed triangular matrix in the same storage format as A.
*> \endverbatim
*>
*> \param[in,out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          If FACT = 'F', then IPIV is an input argument and on entry
*>          contains details of the interchanges and the block structure
*>          of D, as determined by DSPTRF.
*>          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*>          interchanged and D(k,k) is a 1-by-1 diagonal block.
*>          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
*>          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
*>          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
*>          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
*>          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*>
*>          If FACT = 'N', then IPIV is an output argument and on exit
*>          contains details of the interchanges and the block structure
*>          of D, as determined by DSPTRF.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          The N-by-NRHS right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The estimate of the reciprocal condition number of the matrix
*>          A.  If RCOND is less than the machine precision (in
*>          particular, if RCOND = 0), the matrix is singular to working
*>          precision.  This condition is indicated by a return code of
*>          INFO > 0.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The estimated forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).  The estimate is as reliable as
*>          the estimate for RCOND, and is almost always a slight
*>          overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in
*>          any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, and i is
*>                <= N:  D(i,i) is exactly zero.  The factorization
*>                       has been completed but the factor D is exactly
*>                       singular, so the solution and error bounds could
*>                       not be computed. RCOND = 0 is returned.
*>                = N+1: D is nonsingular, but RCOND is less than machine
*>                       precision, meaning that the matrix is singular
*>                       to working precision.  Nevertheless, the
*>                       solution and error bounds are computed because
*>                       there are a number of situations where the
*>                       computed solution can be more accurate than the
*>                       value of RCOND would suggest.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpsvx
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The packed storage scheme is illustrated by the following example
*>  when N = 4, UPLO = 'U':
*>
*>  Two-dimensional storage of the symmetric matrix A:
*>
*>     a11 a12 a13 a14
*>         a22 a23 a24
*>             a33 a34     (aij = aji)
*>                 a44
*>
*>  Packed storage of the upper triangle of A:
*>
*>  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB,
     $                   X,
     $                   LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          FACT, UPLO
      INTEGER            INFO, LDB, LDX, N, NRHS
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), IWORK( * )
      DOUBLE PRECISION   AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
     $                   FERR( * ), WORK( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOFACT
      DOUBLE PRECISION   ANORM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANSP
      EXTERNAL           LSAME, DLAMCH, DLANSP
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLACPY, DSPCON, DSPRFS, DSPTRF,
     $                   DSPTRS,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NOFACT = LSAME( FACT, 'N' )
      IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND.
     $         .NOT.LSAME( UPLO, 'L' ) )
     $          THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -9
      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
         INFO = -11
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSPSVX', -INFO )
         RETURN
      END IF
*
      IF( NOFACT ) THEN
*
*        Compute the factorization A = U*D*U**T or A = L*D*L**T.
*
         CALL DCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 )
         CALL DSPTRF( UPLO, N, AFP, IPIV, INFO )
*
*        Return if INFO is non-zero.
*
         IF( INFO.GT.0 )THEN
            RCOND = ZERO
            RETURN
         END IF
      END IF
*
*     Compute the norm of the matrix A.
*
      ANORM = DLANSP( 'I', UPLO, N, AP, WORK )
*
*     Compute the reciprocal of the condition number of A.
*
      CALL DSPCON( UPLO, N, AFP, IPIV, ANORM, RCOND, WORK, IWORK,
     $             INFO )
*
*     Compute the solution vectors X.
*
      CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
      CALL DSPTRS( UPLO, N, NRHS, AFP, IPIV, X, LDX, INFO )
*
*     Use iterative refinement to improve the computed solutions and
*     compute error bounds and backward error estimates for them.
*
      CALL DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
     $             FERR,
     $             BERR, WORK, IWORK, INFO )
*
*     Set INFO = N+1 if the matrix is singular to working precision.
*
      IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
     $   INFO = N + 1
*
      RETURN
*
*     End of DSPSVX
*
      END
*> \brief \b DSPTRD
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPTRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsptrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsptrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsptrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AP( * ), D( * ), E( * ), TAU( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSPTRD reduces a real symmetric matrix A stored in packed form to
*> symmetric tridiagonal form T by an orthogonal similarity
*> transformation: Q**T * A * Q = T.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the symmetric matrix
*>          A, packed columnwise in a linear array.  The j-th column of A
*>          is stored in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*>          On exit, if UPLO = 'U', the diagonal and first superdiagonal
*>          of A are overwritten by the corresponding elements of the
*>          tridiagonal matrix T, and the elements above the first
*>          superdiagonal, with the array TAU, represent the orthogonal
*>          matrix Q as a product of elementary reflectors; if UPLO
*>          = 'L', the diagonal and first subdiagonal of A are over-
*>          written by the corresponding elements of the tridiagonal
*>          matrix T, and the elements below the first subdiagonal, with
*>          the array TAU, represent the orthogonal matrix Q as a product
*>          of elementary reflectors. See Further Details.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The diagonal elements of the tridiagonal matrix T:
*>          D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          The off-diagonal elements of the tridiagonal matrix T:
*>          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (N-1)
*>          The scalar factors of the elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hptrd
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  If UPLO = 'U', the matrix Q is represented as a product of elementary
*>  reflectors
*>
*>     Q = H(n-1) . . . H(2) H(1).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
*>  overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
*>
*>  If UPLO = 'L', the matrix Q is represented as a product of elementary
*>  reflectors
*>
*>     Q = H(1) H(2) . . . H(n-1).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
*>  overwriting A(i+2:n,i), and tau is stored in TAU(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( * ), D( * ), E( * ), TAU( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO, HALF
      PARAMETER          ( ONE = 1.0D0, ZERO = 0.0D0,
     $                   HALF = 1.0D0 / 2.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I, I1, I1I1, II
      DOUBLE PRECISION   ALPHA, TAUI
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DLARFG, DSPMV, DSPR2, XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DDOT
      EXTERNAL           LSAME, DDOT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSPTRD', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.LE.0 )
     $   RETURN
*
      IF( UPPER ) THEN
*
*        Reduce the upper triangle of A.
*        I1 is the index in AP of A(1,I+1).
*
         I1 = N*( N-1 ) / 2 + 1
         DO 10 I = N - 1, 1, -1
*
*           Generate elementary reflector H(i) = I - tau * v * v**T
*           to annihilate A(1:i-1,i+1)
*
            CALL DLARFG( I, AP( I1+I-1 ), AP( I1 ), 1, TAUI )
            E( I ) = AP( I1+I-1 )
*
            IF( TAUI.NE.ZERO ) THEN
*
*              Apply H(i) from both sides to A(1:i,1:i)
*
               AP( I1+I-1 ) = ONE
*
*              Compute  y := tau * A * v  storing y in TAU(1:i)
*
               CALL DSPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU,
     $                     1 )
*
*              Compute  w := y - 1/2 * tau * (y**T *v) * v
*
               ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, AP( I1 ), 1 )
               CALL DAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 )
*
*              Apply the transformation as a rank-2 update:
*                 A := A - v * w**T - w * v**T
*
               CALL DSPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP )
*
               AP( I1+I-1 ) = E( I )
            END IF
            D( I+1 ) = AP( I1+I )
            TAU( I ) = TAUI
            I1 = I1 - I
   10    CONTINUE
         D( 1 ) = AP( 1 )
      ELSE
*
*        Reduce the lower triangle of A. II is the index in AP of
*        A(i,i) and I1I1 is the index of A(i+1,i+1).
*
         II = 1
         DO 20 I = 1, N - 1
            I1I1 = II + N - I + 1
*
*           Generate elementary reflector H(i) = I - tau * v * v**T
*           to annihilate A(i+2:n,i)
*
            CALL DLARFG( N-I, AP( II+1 ), AP( II+2 ), 1, TAUI )
            E( I ) = AP( II+1 )
*
            IF( TAUI.NE.ZERO ) THEN
*
*              Apply H(i) from both sides to A(i+1:n,i+1:n)
*
               AP( II+1 ) = ONE
*
*              Compute  y := tau * A * v  storing y in TAU(i:n-1)
*
               CALL DSPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ),
     $                     1,
     $                     ZERO, TAU( I ), 1 )
*
*              Compute  w := y - 1/2 * tau * (y**T *v) * v
*
               ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, AP( II+1 ),
     $                 1 )
               CALL DAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 )
*
*              Apply the transformation as a rank-2 update:
*                 A := A - v * w**T - w * v**T
*
               CALL DSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ),
     $                     1,
     $                     AP( I1I1 ) )
*
               AP( II+1 ) = E( I )
            END IF
            D( I ) = AP( II )
            TAU( I ) = TAUI
            II = I1I1
   20    CONTINUE
         D( N ) = AP( II )
      END IF
*
      RETURN
*
*     End of DSPTRD
*
      END
*> \brief \b DSPTRF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsptrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsptrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsptrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSPTRF( UPLO, N, AP, IPIV, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   AP( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSPTRF computes the factorization of a real symmetric matrix A stored
*> in packed format using the Bunch-Kaufman diagonal pivoting method:
*>
*>    A = U*D*U**T  or  A = L*D*L**T
*>
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, and D is symmetric and block diagonal with
*> 1-by-1 and 2-by-2 diagonal blocks.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the symmetric matrix
*>          A, packed columnwise in a linear array.  The j-th column of A
*>          is stored in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*>
*>          On exit, the block diagonal matrix D and the multipliers used
*>          to obtain the factor U or L, stored as a packed triangular
*>          matrix overwriting A (see below for further details).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the block structure of D.
*>          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*>          interchanged and D(k,k) is a 1-by-1 diagonal block.
*>          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
*>          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
*>          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
*>          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
*>          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*>          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization
*>               has been completed, but the block diagonal matrix D is
*>               exactly singular, and division by zero will occur if it
*>               is used to solve a system of equations.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hptrf
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  If UPLO = 'U', then A = U*D*U**T, where
*>     U = P(n)*U(n)* ... *P(k)U(k)* ...,
*>  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
*>  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
*>  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
*>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
*>
*>             (   I    v    0   )   k-s
*>     U(k) =  (   0    I    0   )   s
*>             (   0    0    I   )   n-k
*>                k-s   s   n-k
*>
*>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
*>  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
*>  and A(k,k), and v overwrites A(1:k-2,k-1:k).
*>
*>  If UPLO = 'L', then A = L*D*L**T, where
*>     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
*>  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
*>  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
*>  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
*>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
*>
*>             (   I    0     0   )  k-1
*>     L(k) =  (   0    I     0   )  s
*>             (   0    v     I   )  n-k-s+1
*>                k-1   s  n-k-s+1
*>
*>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
*>  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
*>  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
*> \endverbatim
*
*> \par Contributors:
*  ==================
*>
*>  J. Lewis, Boeing Computer Services Company
*>
*  =====================================================================
      SUBROUTINE DSPTRF( UPLO, N, AP, IPIV, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   AP( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
      DOUBLE PRECISION   EIGHT, SEVTEN
      PARAMETER          ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I, IMAX, J, JMAX, K, KC, KK, KNC, KP, KPC,
     $                   KSTEP, KX, NPP
      DOUBLE PRECISION   ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22, R1,
     $                   ROWMAX, T, WK, WKM1, WKP1
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX
      EXTERNAL           LSAME, IDAMAX
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL, DSPR, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSPTRF', -INFO )
         RETURN
      END IF
*
*     Initialize ALPHA for use in choosing pivot block size.
*
      ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
*
      IF( UPPER ) THEN
*
*        Factorize A as U*D*U**T using the upper triangle of A
*
*        K is the main loop index, decreasing from N to 1 in steps of
*        1 or 2
*
         K = N
         KC = ( N-1 )*N / 2 + 1
   10    CONTINUE
         KNC = KC
*
*        If K < 1, exit from loop
*
         IF( K.LT.1 )
     $      GO TO 110
         KSTEP = 1
*
*        Determine rows and columns to be interchanged and whether
*        a 1-by-1 or 2-by-2 pivot block will be used
*
         ABSAKK = ABS( AP( KC+K-1 ) )
*
*        IMAX is the row-index of the largest off-diagonal element in
*        column K, and COLMAX is its absolute value
*
         IF( K.GT.1 ) THEN
            IMAX = IDAMAX( K-1, AP( KC ), 1 )
            COLMAX = ABS( AP( KC+IMAX-1 ) )
         ELSE
            COLMAX = ZERO
         END IF
*
         IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
*
*           Column K is zero: set INFO and continue
*
            IF( INFO.EQ.0 )
     $         INFO = K
            KP = K
         ELSE
            IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
*
*              no interchange, use 1-by-1 pivot block
*
               KP = K
            ELSE
*
               ROWMAX = ZERO
               JMAX = IMAX
               KX = IMAX*( IMAX+1 ) / 2 + IMAX
               DO 20 J = IMAX + 1, K
                  IF( ABS( AP( KX ) ).GT.ROWMAX ) THEN
                     ROWMAX = ABS( AP( KX ) )
                     JMAX = J
                  END IF
                  KX = KX + J
   20          CONTINUE
               KPC = ( IMAX-1 )*IMAX / 2 + 1
               IF( IMAX.GT.1 ) THEN
                  JMAX = IDAMAX( IMAX-1, AP( KPC ), 1 )
                  ROWMAX = MAX( ROWMAX, ABS( AP( KPC+JMAX-1 ) ) )
               END IF
*
               IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
*
*                 no interchange, use 1-by-1 pivot block
*
                  KP = K
               ELSE IF( ABS( AP( KPC+IMAX-1 ) ).GE.ALPHA*ROWMAX ) THEN
*
*                 interchange rows and columns K and IMAX, use 1-by-1
*                 pivot block
*
                  KP = IMAX
               ELSE
*
*                 interchange rows and columns K-1 and IMAX, use 2-by-2
*                 pivot block
*
                  KP = IMAX
                  KSTEP = 2
               END IF
            END IF
*
            KK = K - KSTEP + 1
            IF( KSTEP.EQ.2 )
     $         KNC = KNC - K + 1
            IF( KP.NE.KK ) THEN
*
*              Interchange rows and columns KK and KP in the leading
*              submatrix A(1:k,1:k)
*
               CALL DSWAP( KP-1, AP( KNC ), 1, AP( KPC ), 1 )
               KX = KPC + KP - 1
               DO 30 J = KP + 1, KK - 1
                  KX = KX + J - 1
                  T = AP( KNC+J-1 )
                  AP( KNC+J-1 ) = AP( KX )
                  AP( KX ) = T
   30          CONTINUE
               T = AP( KNC+KK-1 )
               AP( KNC+KK-1 ) = AP( KPC+KP-1 )
               AP( KPC+KP-1 ) = T
               IF( KSTEP.EQ.2 ) THEN
                  T = AP( KC+K-2 )
                  AP( KC+K-2 ) = AP( KC+KP-1 )
                  AP( KC+KP-1 ) = T
               END IF
            END IF
*
*           Update the leading submatrix
*
            IF( KSTEP.EQ.1 ) THEN
*
*              1-by-1 pivot block D(k): column k now holds
*
*              W(k) = U(k)*D(k)
*
*              where U(k) is the k-th column of U
*
*              Perform a rank-1 update of A(1:k-1,1:k-1) as
*
*              A := A - U(k)*D(k)*U(k)**T = A - W(k)*1/D(k)*W(k)**T
*
               R1 = ONE / AP( KC+K-1 )
               CALL DSPR( UPLO, K-1, -R1, AP( KC ), 1, AP )
*
*              Store U(k) in column k
*
               CALL DSCAL( K-1, R1, AP( KC ), 1 )
            ELSE
*
*              2-by-2 pivot block D(k): columns k and k-1 now hold
*
*              ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
*
*              where U(k) and U(k-1) are the k-th and (k-1)-th columns
*              of U
*
*              Perform a rank-2 update of A(1:k-2,1:k-2) as
*
*              A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
*                 = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T
*
               IF( K.GT.2 ) THEN
*
                  D12 = AP( K-1+( K-1 )*K / 2 )
                  D22 = AP( K-1+( K-2 )*( K-1 ) / 2 ) / D12
                  D11 = AP( K+( K-1 )*K / 2 ) / D12
                  T = ONE / ( D11*D22-ONE )
                  D12 = T / D12
*
                  DO 50 J = K - 2, 1, -1
                     WKM1 = D12*( D11*AP( J+( K-2 )*( K-1 ) / 2 )-
     $                      AP( J+( K-1 )*K / 2 ) )
                     WK = D12*( D22*AP( J+( K-1 )*K / 2 )-
     $                    AP( J+( K-2 )*( K-1 ) / 2 ) )
                     DO 40 I = J, 1, -1
                        AP( I+( J-1 )*J / 2 ) = AP( I+( J-1 )*J / 2 ) -
     $                     AP( I+( K-1 )*K / 2 )*WK -
     $                     AP( I+( K-2 )*( K-1 ) / 2 )*WKM1
   40                CONTINUE
                     AP( J+( K-1 )*K / 2 ) = WK
                     AP( J+( K-2 )*( K-1 ) / 2 ) = WKM1
   50             CONTINUE
*
               END IF
*
            END IF
         END IF
*
*        Store details of the interchanges in IPIV
*
         IF( KSTEP.EQ.1 ) THEN
            IPIV( K ) = KP
         ELSE
            IPIV( K ) = -KP
            IPIV( K-1 ) = -KP
         END IF
*
*        Decrease K and return to the start of the main loop
*
         K = K - KSTEP
         KC = KNC - K
         GO TO 10
*
      ELSE
*
*        Factorize A as L*D*L**T using the lower triangle of A
*
*        K is the main loop index, increasing from 1 to N in steps of
*        1 or 2
*
         K = 1
         KC = 1
         NPP = N*( N+1 ) / 2
   60    CONTINUE
         KNC = KC
*
*        If K > N, exit from loop
*
         IF( K.GT.N )
     $      GO TO 110
         KSTEP = 1
*
*        Determine rows and columns to be interchanged and whether
*        a 1-by-1 or 2-by-2 pivot block will be used
*
         ABSAKK = ABS( AP( KC ) )
*
*        IMAX is the row-index of the largest off-diagonal element in
*        column K, and COLMAX is its absolute value
*
         IF( K.LT.N ) THEN
            IMAX = K + IDAMAX( N-K, AP( KC+1 ), 1 )
            COLMAX = ABS( AP( KC+IMAX-K ) )
         ELSE
            COLMAX = ZERO
         END IF
*
         IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
*
*           Column K is zero: set INFO and continue
*
            IF( INFO.EQ.0 )
     $         INFO = K
            KP = K
         ELSE
            IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
*
*              no interchange, use 1-by-1 pivot block
*
               KP = K
            ELSE
*
*              JMAX is the column-index of the largest off-diagonal
*              element in row IMAX, and ROWMAX is its absolute value
*
               ROWMAX = ZERO
               KX = KC + IMAX - K
               DO 70 J = K, IMAX - 1
                  IF( ABS( AP( KX ) ).GT.ROWMAX ) THEN
                     ROWMAX = ABS( AP( KX ) )
                     JMAX = J
                  END IF
                  KX = KX + N - J
   70          CONTINUE
               KPC = NPP - ( N-IMAX+1 )*( N-IMAX+2 ) / 2 + 1
               IF( IMAX.LT.N ) THEN
                  JMAX = IMAX + IDAMAX( N-IMAX, AP( KPC+1 ), 1 )
                  ROWMAX = MAX( ROWMAX, ABS( AP( KPC+JMAX-IMAX ) ) )
               END IF
*
               IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
*
*                 no interchange, use 1-by-1 pivot block
*
                  KP = K
               ELSE IF( ABS( AP( KPC ) ).GE.ALPHA*ROWMAX ) THEN
*
*                 interchange rows and columns K and IMAX, use 1-by-1
*                 pivot block
*
                  KP = IMAX
               ELSE
*
*                 interchange rows and columns K+1 and IMAX, use 2-by-2
*                 pivot block
*
                  KP = IMAX
                  KSTEP = 2
               END IF
            END IF
*
            KK = K + KSTEP - 1
            IF( KSTEP.EQ.2 )
     $         KNC = KNC + N - K + 1
            IF( KP.NE.KK ) THEN
*
*              Interchange rows and columns KK and KP in the trailing
*              submatrix A(k:n,k:n)
*
               IF( KP.LT.N )
     $            CALL DSWAP( N-KP, AP( KNC+KP-KK+1 ), 1,
     $                        AP( KPC+1 ),
     $                        1 )
               KX = KNC + KP - KK
               DO 80 J = KK + 1, KP - 1
                  KX = KX + N - J + 1
                  T = AP( KNC+J-KK )
                  AP( KNC+J-KK ) = AP( KX )
                  AP( KX ) = T
   80          CONTINUE
               T = AP( KNC )
               AP( KNC ) = AP( KPC )
               AP( KPC ) = T
               IF( KSTEP.EQ.2 ) THEN
                  T = AP( KC+1 )
                  AP( KC+1 ) = AP( KC+KP-K )
                  AP( KC+KP-K ) = T
               END IF
            END IF
*
*           Update the trailing submatrix
*
            IF( KSTEP.EQ.1 ) THEN
*
*              1-by-1 pivot block D(k): column k now holds
*
*              W(k) = L(k)*D(k)
*
*              where L(k) is the k-th column of L
*
               IF( K.LT.N ) THEN
*
*                 Perform a rank-1 update of A(k+1:n,k+1:n) as
*
*                 A := A - L(k)*D(k)*L(k)**T = A - W(k)*(1/D(k))*W(k)**T
*
                  R1 = ONE / AP( KC )
                  CALL DSPR( UPLO, N-K, -R1, AP( KC+1 ), 1,
     $                       AP( KC+N-K+1 ) )
*
*                 Store L(k) in column K
*
                  CALL DSCAL( N-K, R1, AP( KC+1 ), 1 )
               END IF
            ELSE
*
*              2-by-2 pivot block D(k): columns K and K+1 now hold
*
*              ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
*
*              where L(k) and L(k+1) are the k-th and (k+1)-th columns
*              of L
*
               IF( K.LT.N-1 ) THEN
*
*                 Perform a rank-2 update of A(k+2:n,k+2:n) as
*
*                 A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**T
*                    = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**T
*
*                 where L(k) and L(k+1) are the k-th and (k+1)-th
*                 columns of L
*
                  D21 = AP( K+1+( K-1 )*( 2*N-K ) / 2 )
                  D11 = AP( K+1+K*( 2*N-K-1 ) / 2 ) / D21
                  D22 = AP( K+( K-1 )*( 2*N-K ) / 2 ) / D21
                  T = ONE / ( D11*D22-ONE )
                  D21 = T / D21
*
                  DO 100 J = K + 2, N
                     WK = D21*( D11*AP( J+( K-1 )*( 2*N-K ) / 2 )-
     $                    AP( J+K*( 2*N-K-1 ) / 2 ) )
                     WKP1 = D21*( D22*AP( J+K*( 2*N-K-1 ) / 2 )-
     $                      AP( J+( K-1 )*( 2*N-K ) / 2 ) )
*
                     DO 90 I = J, N
                        AP( I+( J-1 )*( 2*N-J ) / 2 ) = AP( I+( J-1 )*
     $                     ( 2*N-J ) / 2 ) - AP( I+( K-1 )*( 2*N-K ) /
     $                     2 )*WK - AP( I+K*( 2*N-K-1 ) / 2 )*WKP1
   90                CONTINUE
*
                     AP( J+( K-1 )*( 2*N-K ) / 2 ) = WK
                     AP( J+K*( 2*N-K-1 ) / 2 ) = WKP1
*
  100             CONTINUE
               END IF
            END IF
         END IF
*
*        Store details of the interchanges in IPIV
*
         IF( KSTEP.EQ.1 ) THEN
            IPIV( K ) = KP
         ELSE
            IPIV( K ) = -KP
            IPIV( K+1 ) = -KP
         END IF
*
*        Increase K and return to the start of the main loop
*
         K = K + KSTEP
         KC = KNC + N - K + 2
         GO TO 60
*
      END IF
*
  110 CONTINUE
      RETURN
*
*     End of DSPTRF
*
      END
*> \brief \b DSPTRI
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPTRI + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsptri.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsptri.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsptri.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSPTRI( UPLO, N, AP, IPIV, WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   AP( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSPTRI computes the inverse of a real symmetric indefinite matrix
*> A in packed storage using the factorization A = U*D*U**T or
*> A = L*D*L**T computed by DSPTRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the details of the factorization are stored
*>          as an upper or lower triangular matrix.
*>          = 'U':  Upper triangular, form is A = U*D*U**T;
*>          = 'L':  Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the block diagonal matrix D and the multipliers
*>          used to obtain the factor U or L as computed by DSPTRF,
*>          stored as a packed triangular matrix.
*>
*>          On exit, if INFO = 0, the (symmetric) inverse of the original
*>          matrix, stored as a packed triangular matrix. The j-th column
*>          of inv(A) is stored in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = inv(A)(i,j) for 1<=i<=j;
*>          if UPLO = 'L',
*>             AP(i + (j-1)*(2n-j)/2) = inv(A)(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the block structure of D
*>          as determined by DSPTRF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*>          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
*>               inverse could not be computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hptri
*
*  =====================================================================
      SUBROUTINE DSPTRI( UPLO, N, AP, IPIV, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   AP( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            J, K, KC, KCNEXT, KP, KPC, KSTEP, KX, NPP
      DOUBLE PRECISION   AK, AKKP1, AKP1, D, T, TEMP
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DDOT
      EXTERNAL           LSAME, DDOT
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DSPMV, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSPTRI', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Check that the diagonal matrix D is nonsingular.
*
      IF( UPPER ) THEN
*
*        Upper triangular storage: examine D from bottom to top
*
         KP = N*( N+1 ) / 2
         DO 10 INFO = N, 1, -1
            IF( IPIV( INFO ).GT.0 .AND. AP( KP ).EQ.ZERO )
     $         RETURN
            KP = KP - INFO
   10    CONTINUE
      ELSE
*
*        Lower triangular storage: examine D from top to bottom.
*
         KP = 1
         DO 20 INFO = 1, N
            IF( IPIV( INFO ).GT.0 .AND. AP( KP ).EQ.ZERO )
     $         RETURN
            KP = KP + N - INFO + 1
   20    CONTINUE
      END IF
      INFO = 0
*
      IF( UPPER ) THEN
*
*        Compute inv(A) from the factorization A = U*D*U**T.
*
*        K is the main loop index, increasing from 1 to N in steps of
*        1 or 2, depending on the size of the diagonal blocks.
*
         K = 1
         KC = 1
   30    CONTINUE
*
*        If K > N, exit from loop.
*
         IF( K.GT.N )
     $      GO TO 50
*
         KCNEXT = KC + K
         IF( IPIV( K ).GT.0 ) THEN
*
*           1 x 1 diagonal block
*
*           Invert the diagonal block.
*
            AP( KC+K-1 ) = ONE / AP( KC+K-1 )
*
*           Compute column K of the inverse.
*
            IF( K.GT.1 ) THEN
               CALL DCOPY( K-1, AP( KC ), 1, WORK, 1 )
               CALL DSPMV( UPLO, K-1, -ONE, AP, WORK, 1, ZERO,
     $                     AP( KC ),
     $                     1 )
               AP( KC+K-1 ) = AP( KC+K-1 ) -
     $                        DDOT( K-1, WORK, 1, AP( KC ), 1 )
            END IF
            KSTEP = 1
         ELSE
*
*           2 x 2 diagonal block
*
*           Invert the diagonal block.
*
            T = ABS( AP( KCNEXT+K-1 ) )
            AK = AP( KC+K-1 ) / T
            AKP1 = AP( KCNEXT+K ) / T
            AKKP1 = AP( KCNEXT+K-1 ) / T
            D = T*( AK*AKP1-ONE )
            AP( KC+K-1 ) = AKP1 / D
            AP( KCNEXT+K ) = AK / D
            AP( KCNEXT+K-1 ) = -AKKP1 / D
*
*           Compute columns K and K+1 of the inverse.
*
            IF( K.GT.1 ) THEN
               CALL DCOPY( K-1, AP( KC ), 1, WORK, 1 )
               CALL DSPMV( UPLO, K-1, -ONE, AP, WORK, 1, ZERO,
     $                     AP( KC ),
     $                     1 )
               AP( KC+K-1 ) = AP( KC+K-1 ) -
     $                        DDOT( K-1, WORK, 1, AP( KC ), 1 )
               AP( KCNEXT+K-1 ) = AP( KCNEXT+K-1 ) -
     $                            DDOT( K-1, AP( KC ), 1,
     $                                  AP( KCNEXT ),
     $                            1 )
               CALL DCOPY( K-1, AP( KCNEXT ), 1, WORK, 1 )
               CALL DSPMV( UPLO, K-1, -ONE, AP, WORK, 1, ZERO,
     $                     AP( KCNEXT ), 1 )
               AP( KCNEXT+K ) = AP( KCNEXT+K ) -
     $                          DDOT( K-1, WORK, 1, AP( KCNEXT ), 1 )
            END IF
            KSTEP = 2
            KCNEXT = KCNEXT + K + 1
         END IF
*
         KP = ABS( IPIV( K ) )
         IF( KP.NE.K ) THEN
*
*           Interchange rows and columns K and KP in the leading
*           submatrix A(1:k+1,1:k+1)
*
            KPC = ( KP-1 )*KP / 2 + 1
            CALL DSWAP( KP-1, AP( KC ), 1, AP( KPC ), 1 )
            KX = KPC + KP - 1
            DO 40 J = KP + 1, K - 1
               KX = KX + J - 1
               TEMP = AP( KC+J-1 )
               AP( KC+J-1 ) = AP( KX )
               AP( KX ) = TEMP
   40       CONTINUE
            TEMP = AP( KC+K-1 )
            AP( KC+K-1 ) = AP( KPC+KP-1 )
            AP( KPC+KP-1 ) = TEMP
            IF( KSTEP.EQ.2 ) THEN
               TEMP = AP( KC+K+K-1 )
               AP( KC+K+K-1 ) = AP( KC+K+KP-1 )
               AP( KC+K+KP-1 ) = TEMP
            END IF
         END IF
*
         K = K + KSTEP
         KC = KCNEXT
         GO TO 30
   50    CONTINUE
*
      ELSE
*
*        Compute inv(A) from the factorization A = L*D*L**T.
*
*        K is the main loop index, increasing from 1 to N in steps of
*        1 or 2, depending on the size of the diagonal blocks.
*
         NPP = N*( N+1 ) / 2
         K = N
         KC = NPP
   60    CONTINUE
*
*        If K < 1, exit from loop.
*
         IF( K.LT.1 )
     $      GO TO 80
*
         KCNEXT = KC - ( N-K+2 )
         IF( IPIV( K ).GT.0 ) THEN
*
*           1 x 1 diagonal block
*
*           Invert the diagonal block.
*
            AP( KC ) = ONE / AP( KC )
*
*           Compute column K of the inverse.
*
            IF( K.LT.N ) THEN
               CALL DCOPY( N-K, AP( KC+1 ), 1, WORK, 1 )
               CALL DSPMV( UPLO, N-K, -ONE, AP( KC+N-K+1 ), WORK, 1,
     $                     ZERO, AP( KC+1 ), 1 )
               AP( KC ) = AP( KC ) - DDOT( N-K, WORK, 1, AP( KC+1 ),
     $             1 )
            END IF
            KSTEP = 1
         ELSE
*
*           2 x 2 diagonal block
*
*           Invert the diagonal block.
*
            T = ABS( AP( KCNEXT+1 ) )
            AK = AP( KCNEXT ) / T
            AKP1 = AP( KC ) / T
            AKKP1 = AP( KCNEXT+1 ) / T
            D = T*( AK*AKP1-ONE )
            AP( KCNEXT ) = AKP1 / D
            AP( KC ) = AK / D
            AP( KCNEXT+1 ) = -AKKP1 / D
*
*           Compute columns K-1 and K of the inverse.
*
            IF( K.LT.N ) THEN
               CALL DCOPY( N-K, AP( KC+1 ), 1, WORK, 1 )
               CALL DSPMV( UPLO, N-K, -ONE, AP( KC+( N-K+1 ) ), WORK,
     $                     1,
     $                     ZERO, AP( KC+1 ), 1 )
               AP( KC ) = AP( KC ) - DDOT( N-K, WORK, 1, AP( KC+1 ),
     $             1 )
               AP( KCNEXT+1 ) = AP( KCNEXT+1 ) -
     $                          DDOT( N-K, AP( KC+1 ), 1,
     $                          AP( KCNEXT+2 ), 1 )
               CALL DCOPY( N-K, AP( KCNEXT+2 ), 1, WORK, 1 )
               CALL DSPMV( UPLO, N-K, -ONE, AP( KC+( N-K+1 ) ), WORK,
     $                     1,
     $                     ZERO, AP( KCNEXT+2 ), 1 )
               AP( KCNEXT ) = AP( KCNEXT ) -
     $                        DDOT( N-K, WORK, 1, AP( KCNEXT+2 ), 1 )
            END IF
            KSTEP = 2
            KCNEXT = KCNEXT - ( N-K+3 )
         END IF
*
         KP = ABS( IPIV( K ) )
         IF( KP.NE.K ) THEN
*
*           Interchange rows and columns K and KP in the trailing
*           submatrix A(k-1:n,k-1:n)
*
            KPC = NPP - ( N-KP+1 )*( N-KP+2 ) / 2 + 1
            IF( KP.LT.N )
     $         CALL DSWAP( N-KP, AP( KC+KP-K+1 ), 1, AP( KPC+1 ), 1 )
            KX = KC + KP - K
            DO 70 J = K + 1, KP - 1
               KX = KX + N - J + 1
               TEMP = AP( KC+J-K )
               AP( KC+J-K ) = AP( KX )
               AP( KX ) = TEMP
   70       CONTINUE
            TEMP = AP( KC )
            AP( KC ) = AP( KPC )
            AP( KPC ) = TEMP
            IF( KSTEP.EQ.2 ) THEN
               TEMP = AP( KC-N+K-1 )
               AP( KC-N+K-1 ) = AP( KC-N+KP-1 )
               AP( KC-N+KP-1 ) = TEMP
            END IF
         END IF
*
         K = K - KSTEP
         KC = KCNEXT
         GO TO 60
   80    CONTINUE
      END IF
*
      RETURN
*
*     End of DSPTRI
*
      END
*> \brief \b DSPTRS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsptrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsptrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsptrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSPTRS( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   AP( * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSPTRS solves a system of linear equations A*X = B with a real
*> symmetric matrix A stored in packed format using the factorization
*> A = U*D*U**T or A = L*D*L**T computed by DSPTRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the details of the factorization are stored
*>          as an upper or lower triangular matrix.
*>          = 'U':  Upper triangular, form is A = U*D*U**T;
*>          = 'L':  Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          The block diagonal matrix D and the multipliers used to
*>          obtain the factor U or L as computed by DSPTRF, stored as a
*>          packed triangular matrix.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the block structure of D
*>          as determined by DSPTRF.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the right hand side matrix B.
*>          On exit, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hptrs
*
*  =====================================================================
      SUBROUTINE DSPTRS( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   AP( * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            J, K, KC, KP
      DOUBLE PRECISION   AK, AKM1, AKM1K, BK, BKM1, DENOM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMV, DGER, DSCAL, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSPTRS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 )
     $   RETURN
*
      IF( UPPER ) THEN
*
*        Solve A*X = B, where A = U*D*U**T.
*
*        First solve U*D*X = B, overwriting B with X.
*
*        K is the main loop index, decreasing from N to 1 in steps of
*        1 or 2, depending on the size of the diagonal blocks.
*
         K = N
         KC = N*( N+1 ) / 2 + 1
   10    CONTINUE
*
*        If K < 1, exit from loop.
*
         IF( K.LT.1 )
     $      GO TO 30
*
         KC = KC - K
         IF( IPIV( K ).GT.0 ) THEN
*
*           1 x 1 diagonal block
*
*           Interchange rows K and IPIV(K).
*
            KP = IPIV( K )
            IF( KP.NE.K )
     $         CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
*
*           Multiply by inv(U(K)), where U(K) is the transformation
*           stored in column K of A.
*
            CALL DGER( K-1, NRHS, -ONE, AP( KC ), 1, B( K, 1 ), LDB,
     $                 B( 1, 1 ), LDB )
*
*           Multiply by the inverse of the diagonal block.
*
            CALL DSCAL( NRHS, ONE / AP( KC+K-1 ), B( K, 1 ), LDB )
            K = K - 1
         ELSE
*
*           2 x 2 diagonal block
*
*           Interchange rows K-1 and -IPIV(K).
*
            KP = -IPIV( K )
            IF( KP.NE.K-1 )
     $         CALL DSWAP( NRHS, B( K-1, 1 ), LDB, B( KP, 1 ), LDB )
*
*           Multiply by inv(U(K)), where U(K) is the transformation
*           stored in columns K-1 and K of A.
*
            CALL DGER( K-2, NRHS, -ONE, AP( KC ), 1, B( K, 1 ), LDB,
     $                 B( 1, 1 ), LDB )
            CALL DGER( K-2, NRHS, -ONE, AP( KC-( K-1 ) ), 1,
     $                 B( K-1, 1 ), LDB, B( 1, 1 ), LDB )
*
*           Multiply by the inverse of the diagonal block.
*
            AKM1K = AP( KC+K-2 )
            AKM1 = AP( KC-1 ) / AKM1K
            AK = AP( KC+K-1 ) / AKM1K
            DENOM = AKM1*AK - ONE
            DO 20 J = 1, NRHS
               BKM1 = B( K-1, J ) / AKM1K
               BK = B( K, J ) / AKM1K
               B( K-1, J ) = ( AK*BKM1-BK ) / DENOM
               B( K, J ) = ( AKM1*BK-BKM1 ) / DENOM
   20       CONTINUE
            KC = KC - K + 1
            K = K - 2
         END IF
*
         GO TO 10
   30    CONTINUE
*
*        Next solve U**T*X = B, overwriting B with X.
*
*        K is the main loop index, increasing from 1 to N in steps of
*        1 or 2, depending on the size of the diagonal blocks.
*
         K = 1
         KC = 1
   40    CONTINUE
*
*        If K > N, exit from loop.
*
         IF( K.GT.N )
     $      GO TO 50
*
         IF( IPIV( K ).GT.0 ) THEN
*
*           1 x 1 diagonal block
*
*           Multiply by inv(U**T(K)), where U(K) is the transformation
*           stored in column K of A.
*
            CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB,
     $                  AP( KC ),
     $                  1, ONE, B( K, 1 ), LDB )
*
*           Interchange rows K and IPIV(K).
*
            KP = IPIV( K )
            IF( KP.NE.K )
     $         CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
            KC = KC + K
            K = K + 1
         ELSE
*
*           2 x 2 diagonal block
*
*           Multiply by inv(U**T(K+1)), where U(K+1) is the transformation
*           stored in columns K and K+1 of A.
*
            CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB,
     $                  AP( KC ),
     $                  1, ONE, B( K, 1 ), LDB )
            CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB,
     $                  AP( KC+K ), 1, ONE, B( K+1, 1 ), LDB )
*
*           Interchange rows K and -IPIV(K).
*
            KP = -IPIV( K )
            IF( KP.NE.K )
     $         CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
            KC = KC + 2*K + 1
            K = K + 2
         END IF
*
         GO TO 40
   50    CONTINUE
*
      ELSE
*
*        Solve A*X = B, where A = L*D*L**T.
*
*        First solve L*D*X = B, overwriting B with X.
*
*        K is the main loop index, increasing from 1 to N in steps of
*        1 or 2, depending on the size of the diagonal blocks.
*
         K = 1
         KC = 1
   60    CONTINUE
*
*        If K > N, exit from loop.
*
         IF( K.GT.N )
     $      GO TO 80
*
         IF( IPIV( K ).GT.0 ) THEN
*
*           1 x 1 diagonal block
*
*           Interchange rows K and IPIV(K).
*
            KP = IPIV( K )
            IF( KP.NE.K )
     $         CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
*
*           Multiply by inv(L(K)), where L(K) is the transformation
*           stored in column K of A.
*
            IF( K.LT.N )
     $         CALL DGER( N-K, NRHS, -ONE, AP( KC+1 ), 1, B( K, 1 ),
     $                    LDB, B( K+1, 1 ), LDB )
*
*           Multiply by the inverse of the diagonal block.
*
            CALL DSCAL( NRHS, ONE / AP( KC ), B( K, 1 ), LDB )
            KC = KC + N - K + 1
            K = K + 1
         ELSE
*
*           2 x 2 diagonal block
*
*           Interchange rows K+1 and -IPIV(K).
*
            KP = -IPIV( K )
            IF( KP.NE.K+1 )
     $         CALL DSWAP( NRHS, B( K+1, 1 ), LDB, B( KP, 1 ), LDB )
*
*           Multiply by inv(L(K)), where L(K) is the transformation
*           stored in columns K and K+1 of A.
*
            IF( K.LT.N-1 ) THEN
               CALL DGER( N-K-1, NRHS, -ONE, AP( KC+2 ), 1, B( K,
     $                    1 ),
     $                    LDB, B( K+2, 1 ), LDB )
               CALL DGER( N-K-1, NRHS, -ONE, AP( KC+N-K+2 ), 1,
     $                    B( K+1, 1 ), LDB, B( K+2, 1 ), LDB )
            END IF
*
*           Multiply by the inverse of the diagonal block.
*
            AKM1K = AP( KC+1 )
            AKM1 = AP( KC ) / AKM1K
            AK = AP( KC+N-K+1 ) / AKM1K
            DENOM = AKM1*AK - ONE
            DO 70 J = 1, NRHS
               BKM1 = B( K, J ) / AKM1K
               BK = B( K+1, J ) / AKM1K
               B( K, J ) = ( AK*BKM1-BK ) / DENOM
               B( K+1, J ) = ( AKM1*BK-BKM1 ) / DENOM
   70       CONTINUE
            KC = KC + 2*( N-K ) + 1
            K = K + 2
         END IF
*
         GO TO 60
   80    CONTINUE
*
*        Next solve L**T*X = B, overwriting B with X.
*
*        K is the main loop index, decreasing from N to 1 in steps of
*        1 or 2, depending on the size of the diagonal blocks.
*
         K = N
         KC = N*( N+1 ) / 2 + 1
   90    CONTINUE
*
*        If K < 1, exit from loop.
*
         IF( K.LT.1 )
     $      GO TO 100
*
         KC = KC - ( N-K+1 )
         IF( IPIV( K ).GT.0 ) THEN
*
*           1 x 1 diagonal block
*
*           Multiply by inv(L**T(K)), where L(K) is the transformation
*           stored in column K of A.
*
            IF( K.LT.N )
     $         CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ),
     $                     LDB, AP( KC+1 ), 1, ONE, B( K, 1 ), LDB )
*
*           Interchange rows K and IPIV(K).
*
            KP = IPIV( K )
            IF( KP.NE.K )
     $         CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
            K = K - 1
         ELSE
*
*           2 x 2 diagonal block
*
*           Multiply by inv(L**T(K-1)), where L(K-1) is the transformation
*           stored in columns K-1 and K of A.
*
            IF( K.LT.N ) THEN
               CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ),
     $                     LDB, AP( KC+1 ), 1, ONE, B( K, 1 ), LDB )
               CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ),
     $                     LDB, AP( KC-( N-K ) ), 1, ONE, B( K-1, 1 ),
     $                     LDB )
            END IF
*
*           Interchange rows K and -IPIV(K).
*
            KP = -IPIV( K )
            IF( KP.NE.K )
     $         CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
            KC = KC - ( N-K+2 )
            K = K - 2
         END IF
*
         GO TO 90
  100    CONTINUE
      END IF
*
      RETURN
*
*     End of DSPTRS
*
      END
*> \brief \b DSTEBZ
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEBZ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstebz.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstebz.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstebz.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSTEBZ( RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E,
*                          M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK,
*                          INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          ORDER, RANGE
*       INTEGER            IL, INFO, IU, M, N, NSPLIT
*       DOUBLE PRECISION   ABSTOL, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            IBLOCK( * ), ISPLIT( * ), IWORK( * )
*       DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSTEBZ computes the eigenvalues of a symmetric tridiagonal
*> matrix T.  The user may ask for all eigenvalues, all eigenvalues
*> in the half-open interval (VL, VU], or the IL-th through IU-th
*> eigenvalues.
*>
*> To avoid overflow, the matrix must be scaled so that its
*> largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
*> accuracy, it should not be much smaller than that.
*>
*> See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
*> Matrix", Report CS41, Computer Science Dept., Stanford
*> University, July 21, 1966.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] RANGE
*> \verbatim
*>          RANGE is CHARACTER*1
*>          = 'A': ("All")   all eigenvalues will be found.
*>          = 'V': ("Value") all eigenvalues in the half-open interval
*>                           (VL, VU] will be found.
*>          = 'I': ("Index") the IL-th through IU-th eigenvalues (of the
*>                           entire matrix) will be found.
*> \endverbatim
*>
*> \param[in] ORDER
*> \verbatim
*>          ORDER is CHARACTER*1
*>          = 'B': ("By Block") the eigenvalues will be grouped by
*>                              split-off block (see IBLOCK, ISPLIT) and
*>                              ordered from smallest to largest within
*>                              the block.
*>          = 'E': ("Entire matrix")
*>                              the eigenvalues for the entire matrix
*>                              will be ordered from smallest to
*>                              largest.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the tridiagonal matrix T.  N >= 0.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is DOUBLE PRECISION
*>
*>          If RANGE='V', the lower bound of the interval to
*>          be searched for eigenvalues.  Eigenvalues less than or equal
*>          to VL, or greater than VU, will not be returned.  VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*>          VU is DOUBLE PRECISION
*>
*>          If RANGE='V', the upper bound of the interval to
*>          be searched for eigenvalues.  Eigenvalues less than or equal
*>          to VL, or greater than VU, will not be returned.  VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*>          IL is INTEGER
*>
*>          If RANGE='I', the index of the
*>          smallest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*>          IU is INTEGER
*>
*>          If RANGE='I', the index of the
*>          largest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*>          ABSTOL is DOUBLE PRECISION
*>          The absolute tolerance for the eigenvalues.  An eigenvalue
*>          (or cluster) is considered to be located if it has been
*>          determined to lie in an interval whose width is ABSTOL or
*>          less.  If ABSTOL is less than or equal to zero, then ULP*|T|
*>          will be used, where |T| means the 1-norm of T.
*>
*>          Eigenvalues will be computed most accurately when ABSTOL is
*>          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The n diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) off-diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The actual number of eigenvalues found. 0 <= M <= N.
*>          (See also the description of INFO=2,3.)
*> \endverbatim
*>
*> \param[out] NSPLIT
*> \verbatim
*>          NSPLIT is INTEGER
*>          The number of diagonal blocks in the matrix T.
*>          1 <= NSPLIT <= N.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          On exit, the first M elements of W will contain the
*>          eigenvalues.  (DSTEBZ may use the remaining N-M elements as
*>          workspace.)
*> \endverbatim
*>
*> \param[out] IBLOCK
*> \verbatim
*>          IBLOCK is INTEGER array, dimension (N)
*>          At each row/column j where E(j) is zero or small, the
*>          matrix T is considered to split into a block diagonal
*>          matrix.  On exit, if INFO = 0, IBLOCK(i) specifies to which
*>          block (from 1 to the number of blocks) the eigenvalue W(i)
*>          belongs.  (DSTEBZ may use the remaining N-M elements as
*>          workspace.)
*> \endverbatim
*>
*> \param[out] ISPLIT
*> \verbatim
*>          ISPLIT is INTEGER array, dimension (N)
*>          The splitting points, at which T breaks up into submatrices.
*>          The first submatrix consists of rows/columns 1 to ISPLIT(1),
*>          the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
*>          etc., and the NSPLIT-th consists of rows/columns
*>          ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
*>          (Only the first NSPLIT elements will actually be used, but
*>          since the user cannot know a priori what value NSPLIT will
*>          have, N words must be reserved for ISPLIT.)
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  some or all of the eigenvalues failed to converge or
*>                were not computed:
*>                =1 or 3: Bisection failed to converge for some
*>                        eigenvalues; these eigenvalues are flagged by a
*>                        negative block number.  The effect is that the
*>                        eigenvalues may not be as accurate as the
*>                        absolute and relative tolerances.  This is
*>                        generally caused by unexpectedly inaccurate
*>                        arithmetic.
*>                =2 or 3: RANGE='I' only: Not all of the eigenvalues
*>                        IL:IU were found.
*>                        Effect: M < IU+1-IL
*>                        Cause:  non-monotonic arithmetic, causing the
*>                                Sturm sequence to be non-monotonic.
*>                        Cure:   recalculate, using RANGE='A', and pick
*>                                out eigenvalues IL:IU.  In some cases,
*>                                increasing the PARAMETER "FUDGE" may
*>                                make things work.
*>                = 4:    RANGE='I', and the Gershgorin interval
*>                        initially used was too small.  No eigenvalues
*>                        were computed.
*>                        Probable cause: your machine has sloppy
*>                                        floating-point arithmetic.
*>                        Cure: Increase the PARAMETER "FUDGE",
*>                              recompile, and try again.
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  RELFAC  DOUBLE PRECISION, default = 2.0e0
*>          The relative tolerance.  An interval (a,b] lies within
*>          "relative tolerance" if  b-a < RELFAC*ulp*max(|a|,|b|),
*>          where "ulp" is the machine precision (distance from 1 to
*>          the next larger floating point number.)
*>
*>  FUDGE   DOUBLE PRECISION, default = 2
*>          A "fudge factor" to widen the Gershgorin intervals.  Ideally,
*>          a value of 1 should work, but on machines with sloppy
*>          arithmetic, this needs to be larger.  The default for
*>          publicly released versions should be large enough to handle
*>          the worst machine around.  Note that this has no effect
*>          on accuracy of the solution.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup stebz
*
*  =====================================================================
      SUBROUTINE DSTEBZ( RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D,
     $                   E,
     $                   M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          ORDER, RANGE
      INTEGER            IL, INFO, IU, M, N, NSPLIT
      DOUBLE PRECISION   ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            IBLOCK( * ), ISPLIT( * ), IWORK( * )
      DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO, HALF
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
     $                   HALF = 1.0D0 / TWO )
      DOUBLE PRECISION   FUDGE, RELFAC
      PARAMETER          ( FUDGE = 2.1D0, RELFAC = 2.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NCNVRG, TOOFEW
      INTEGER            IB, IBEGIN, IDISCL, IDISCU, IE, IEND, IINFO,
     $                   IM, IN, IOFF, IORDER, IOUT, IRANGE, ITMAX,
     $                   ITMP1, IW, IWOFF, J, JB, JDISC, JE, NB, NWL,
     $                   NWU
      DOUBLE PRECISION   ATOLI, BNORM, GL, GU, PIVMIN, RTOLI, SAFEMN,
     $                   TMP1, TMP2, TNORM, ULP, WKILL, WL, WLU, WU, WUL
*     ..
*     .. Local Arrays ..
      INTEGER            IDUMMA( 1 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, ILAENV, DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLAEBZ, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, INT, LOG, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
      INFO = 0
*
*     Decode RANGE
*
      IF( LSAME( RANGE, 'A' ) ) THEN
         IRANGE = 1
      ELSE IF( LSAME( RANGE, 'V' ) ) THEN
         IRANGE = 2
      ELSE IF( LSAME( RANGE, 'I' ) ) THEN
         IRANGE = 3
      ELSE
         IRANGE = 0
      END IF
*
*     Decode ORDER
*
      IF( LSAME( ORDER, 'B' ) ) THEN
         IORDER = 2
      ELSE IF( LSAME( ORDER, 'E' ) ) THEN
         IORDER = 1
      ELSE
         IORDER = 0
      END IF
*
*     Check for Errors
*
      IF( IRANGE.LE.0 ) THEN
         INFO = -1
      ELSE IF( IORDER.LE.0 ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( IRANGE.EQ.2 ) THEN
         IF( VL.GE.VU )
     $      INFO = -5
      ELSE IF( IRANGE.EQ.3 .AND. ( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) )
     $          THEN
         INFO = -6
      ELSE IF( IRANGE.EQ.3 .AND. ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) )
     $          THEN
         INFO = -7
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSTEBZ', -INFO )
         RETURN
      END IF
*
*     Initialize error flags
*
      INFO = 0
      NCNVRG = .FALSE.
      TOOFEW = .FALSE.
*
*     Quick return if possible
*
      M = 0
      IF( N.EQ.0 )
     $   RETURN
*
*     Simplifications:
*
      IF( IRANGE.EQ.3 .AND. IL.EQ.1 .AND. IU.EQ.N )
     $   IRANGE = 1
*
*     Get machine constants
*     NB is the minimum vector length for vector bisection, or 0
*     if only scalar is to be done.
*
      SAFEMN = DLAMCH( 'S' )
      ULP = DLAMCH( 'P' )
      RTOLI = ULP*RELFAC
      NB = ILAENV( 1, 'DSTEBZ', ' ', N, -1, -1, -1 )
      IF( NB.LE.1 )
     $   NB = 0
*
*     Special Case when N=1
*
      IF( N.EQ.1 ) THEN
         NSPLIT = 1
         ISPLIT( 1 ) = 1
         IF( IRANGE.EQ.2 .AND. ( VL.GE.D( 1 ) .OR. VU.LT.D( 1 ) ) ) THEN
            M = 0
         ELSE
            W( 1 ) = D( 1 )
            IBLOCK( 1 ) = 1
            M = 1
         END IF
         RETURN
      END IF
*
*     Compute Splitting Points
*
      NSPLIT = 1
      WORK( N ) = ZERO
      PIVMIN = ONE
*
      DO 10 J = 2, N
         TMP1 = E( J-1 )**2
         IF( ABS( D( J )*D( J-1 ) )*ULP**2+SAFEMN.GT.TMP1 ) THEN
            ISPLIT( NSPLIT ) = J - 1
            NSPLIT = NSPLIT + 1
            WORK( J-1 ) = ZERO
         ELSE
            WORK( J-1 ) = TMP1
            PIVMIN = MAX( PIVMIN, TMP1 )
         END IF
   10 CONTINUE
      ISPLIT( NSPLIT ) = N
      PIVMIN = PIVMIN*SAFEMN
*
*     Compute Interval and ATOLI
*
      IF( IRANGE.EQ.3 ) THEN
*
*        RANGE='I': Compute the interval containing eigenvalues
*                   IL through IU.
*
*        Compute Gershgorin interval for entire (split) matrix
*        and use it as the initial interval
*
         GU = D( 1 )
         GL = D( 1 )
         TMP1 = ZERO
*
         DO 20 J = 1, N - 1
            TMP2 = SQRT( WORK( J ) )
            GU = MAX( GU, D( J )+TMP1+TMP2 )
            GL = MIN( GL, D( J )-TMP1-TMP2 )
            TMP1 = TMP2
   20    CONTINUE
*
         GU = MAX( GU, D( N )+TMP1 )
         GL = MIN( GL, D( N )-TMP1 )
         TNORM = MAX( ABS( GL ), ABS( GU ) )
         GL = GL - FUDGE*TNORM*ULP*N - FUDGE*TWO*PIVMIN
         GU = GU + FUDGE*TNORM*ULP*N + FUDGE*PIVMIN
*
*        Compute Iteration parameters
*
         ITMAX = INT( ( LOG( TNORM+PIVMIN )-LOG( PIVMIN ) ) /
     $           LOG( TWO ) ) + 2
         IF( ABSTOL.LE.ZERO ) THEN
            ATOLI = ULP*TNORM
         ELSE
            ATOLI = ABSTOL
         END IF
*
         WORK( N+1 ) = GL
         WORK( N+2 ) = GL
         WORK( N+3 ) = GU
         WORK( N+4 ) = GU
         WORK( N+5 ) = GL
         WORK( N+6 ) = GU
         IWORK( 1 ) = -1
         IWORK( 2 ) = -1
         IWORK( 3 ) = N + 1
         IWORK( 4 ) = N + 1
         IWORK( 5 ) = IL - 1
         IWORK( 6 ) = IU
*
         CALL DLAEBZ( 3, ITMAX, N, 2, 2, NB, ATOLI, RTOLI, PIVMIN, D,
     $                E,
     $                WORK, IWORK( 5 ), WORK( N+1 ), WORK( N+5 ), IOUT,
     $                IWORK, W, IBLOCK, IINFO )
*
         IF( IWORK( 6 ).EQ.IU ) THEN
            WL = WORK( N+1 )
            WLU = WORK( N+3 )
            NWL = IWORK( 1 )
            WU = WORK( N+4 )
            WUL = WORK( N+2 )
            NWU = IWORK( 4 )
         ELSE
            WL = WORK( N+2 )
            WLU = WORK( N+4 )
            NWL = IWORK( 2 )
            WU = WORK( N+3 )
            WUL = WORK( N+1 )
            NWU = IWORK( 3 )
         END IF
*
         IF( NWL.LT.0 .OR. NWL.GE.N .OR. NWU.LT.1 .OR. NWU.GT.N ) THEN
            INFO = 4
            RETURN
         END IF
      ELSE
*
*        RANGE='A' or 'V' -- Set ATOLI
*
         TNORM = MAX( ABS( D( 1 ) )+ABS( E( 1 ) ),
     $           ABS( D( N ) )+ABS( E( N-1 ) ) )
*
         DO 30 J = 2, N - 1
            TNORM = MAX( TNORM, ABS( D( J ) )+ABS( E( J-1 ) )+
     $              ABS( E( J ) ) )
   30    CONTINUE
*
         IF( ABSTOL.LE.ZERO ) THEN
            ATOLI = ULP*TNORM
         ELSE
            ATOLI = ABSTOL
         END IF
*
         IF( IRANGE.EQ.2 ) THEN
            WL = VL
            WU = VU
         ELSE
            WL = ZERO
            WU = ZERO
         END IF
      END IF
*
*     Find Eigenvalues -- Loop Over Blocks and recompute NWL and NWU.
*     NWL accumulates the number of eigenvalues .le. WL,
*     NWU accumulates the number of eigenvalues .le. WU
*
      M = 0
      IEND = 0
      INFO = 0
      NWL = 0
      NWU = 0
*
      DO 70 JB = 1, NSPLIT
         IOFF = IEND
         IBEGIN = IOFF + 1
         IEND = ISPLIT( JB )
         IN = IEND - IOFF
*
         IF( IN.EQ.1 ) THEN
*
*           Special Case -- IN=1
*
            IF( IRANGE.EQ.1 .OR. WL.GE.D( IBEGIN )-PIVMIN )
     $         NWL = NWL + 1
            IF( IRANGE.EQ.1 .OR. WU.GE.D( IBEGIN )-PIVMIN )
     $         NWU = NWU + 1
            IF( IRANGE.EQ.1 .OR. ( WL.LT.D( IBEGIN )-PIVMIN .AND. WU.GE.
     $          D( IBEGIN )-PIVMIN ) ) THEN
               M = M + 1
               W( M ) = D( IBEGIN )
               IBLOCK( M ) = JB
            END IF
         ELSE
*
*           General Case -- IN > 1
*
*           Compute Gershgorin Interval
*           and use it as the initial interval
*
            GU = D( IBEGIN )
            GL = D( IBEGIN )
            TMP1 = ZERO
*
            DO 40 J = IBEGIN, IEND - 1
               TMP2 = ABS( E( J ) )
               GU = MAX( GU, D( J )+TMP1+TMP2 )
               GL = MIN( GL, D( J )-TMP1-TMP2 )
               TMP1 = TMP2
   40       CONTINUE
*
            GU = MAX( GU, D( IEND )+TMP1 )
            GL = MIN( GL, D( IEND )-TMP1 )
            BNORM = MAX( ABS( GL ), ABS( GU ) )
            GL = GL - FUDGE*BNORM*ULP*IN - FUDGE*PIVMIN
            GU = GU + FUDGE*BNORM*ULP*IN + FUDGE*PIVMIN
*
*           Compute ATOLI for the current submatrix
*
            IF( ABSTOL.LE.ZERO ) THEN
               ATOLI = ULP*MAX( ABS( GL ), ABS( GU ) )
            ELSE
               ATOLI = ABSTOL
            END IF
*
            IF( IRANGE.GT.1 ) THEN
               IF( GU.LT.WL ) THEN
                  NWL = NWL + IN
                  NWU = NWU + IN
                  GO TO 70
               END IF
               GL = MAX( GL, WL )
               GU = MIN( GU, WU )
               IF( GL.GE.GU )
     $            GO TO 70
            END IF
*
*           Set Up Initial Interval
*
            WORK( N+1 ) = GL
            WORK( N+IN+1 ) = GU
            CALL DLAEBZ( 1, 0, IN, IN, 1, NB, ATOLI, RTOLI, PIVMIN,
     $                   D( IBEGIN ), E( IBEGIN ), WORK( IBEGIN ),
     $                   IDUMMA, WORK( N+1 ), WORK( N+2*IN+1 ), IM,
     $                   IWORK, W( M+1 ), IBLOCK( M+1 ), IINFO )
*
            NWL = NWL + IWORK( 1 )
            NWU = NWU + IWORK( IN+1 )
            IWOFF = M - IWORK( 1 )
*
*           Compute Eigenvalues
*
            ITMAX = INT( ( LOG( GU-GL+PIVMIN )-LOG( PIVMIN ) ) /
     $              LOG( TWO ) ) + 2
            CALL DLAEBZ( 2, ITMAX, IN, IN, 1, NB, ATOLI, RTOLI,
     $                   PIVMIN,
     $                   D( IBEGIN ), E( IBEGIN ), WORK( IBEGIN ),
     $                   IDUMMA, WORK( N+1 ), WORK( N+2*IN+1 ), IOUT,
     $                   IWORK, W( M+1 ), IBLOCK( M+1 ), IINFO )
*
*           Copy Eigenvalues Into W and IBLOCK
*           Use -JB for block number for unconverged eigenvalues.
*
            DO 60 J = 1, IOUT
               TMP1 = HALF*( WORK( J+N )+WORK( J+IN+N ) )
*
*              Flag non-convergence.
*
               IF( J.GT.IOUT-IINFO ) THEN
                  NCNVRG = .TRUE.
                  IB = -JB
               ELSE
                  IB = JB
               END IF
               DO 50 JE = IWORK( J ) + 1 + IWOFF,
     $                 IWORK( J+IN ) + IWOFF
                  W( JE ) = TMP1
                  IBLOCK( JE ) = IB
   50          CONTINUE
   60       CONTINUE
*
            M = M + IM
         END IF
   70 CONTINUE
*
*     If RANGE='I', then (WL,WU) contains eigenvalues NWL+1,...,NWU
*     If NWL+1 < IL or NWU > IU, discard extra eigenvalues.
*
      IF( IRANGE.EQ.3 ) THEN
         IM = 0
         IDISCL = IL - 1 - NWL
         IDISCU = NWU - IU
*
         IF( IDISCL.GT.0 .OR. IDISCU.GT.0 ) THEN
            DO 80 JE = 1, M
               IF( W( JE ).LE.WLU .AND. IDISCL.GT.0 ) THEN
                  IDISCL = IDISCL - 1
               ELSE IF( W( JE ).GE.WUL .AND. IDISCU.GT.0 ) THEN
                  IDISCU = IDISCU - 1
               ELSE
                  IM = IM + 1
                  W( IM ) = W( JE )
                  IBLOCK( IM ) = IBLOCK( JE )
               END IF
   80       CONTINUE
            M = IM
         END IF
         IF( IDISCL.GT.0 .OR. IDISCU.GT.0 ) THEN
*
*           Code to deal with effects of bad arithmetic:
*           Some low eigenvalues to be discarded are not in (WL,WLU],
*           or high eigenvalues to be discarded are not in (WUL,WU]
*           so just kill off the smallest IDISCL/largest IDISCU
*           eigenvalues, by simply finding the smallest/largest
*           eigenvalue(s).
*
*           (If N(w) is monotone non-decreasing, this should never
*               happen.)
*
            IF( IDISCL.GT.0 ) THEN
               WKILL = WU
               DO 100 JDISC = 1, IDISCL
                  IW = 0
                  DO 90 JE = 1, M
                     IF( IBLOCK( JE ).NE.0 .AND.
     $                   ( W( JE ).LT.WKILL .OR. IW.EQ.0 ) ) THEN
                        IW = JE
                        WKILL = W( JE )
                     END IF
   90             CONTINUE
                  IBLOCK( IW ) = 0
  100          CONTINUE
            END IF
            IF( IDISCU.GT.0 ) THEN
*
               WKILL = WL
               DO 120 JDISC = 1, IDISCU
                  IW = 0
                  DO 110 JE = 1, M
                     IF( IBLOCK( JE ).NE.0 .AND.
     $                   ( W( JE ).GT.WKILL .OR. IW.EQ.0 ) ) THEN
                        IW = JE
                        WKILL = W( JE )
                     END IF
  110             CONTINUE
                  IBLOCK( IW ) = 0
  120          CONTINUE
            END IF
            IM = 0
            DO 130 JE = 1, M
               IF( IBLOCK( JE ).NE.0 ) THEN
                  IM = IM + 1
                  W( IM ) = W( JE )
                  IBLOCK( IM ) = IBLOCK( JE )
               END IF
  130       CONTINUE
            M = IM
         END IF
         IF( IDISCL.LT.0 .OR. IDISCU.LT.0 ) THEN
            TOOFEW = .TRUE.
         END IF
      END IF
*
*     If ORDER='B', do nothing -- the eigenvalues are already sorted
*        by block.
*     If ORDER='E', sort the eigenvalues from smallest to largest
*
      IF( IORDER.EQ.1 .AND. NSPLIT.GT.1 ) THEN
         DO 150 JE = 1, M - 1
            IE = 0
            TMP1 = W( JE )
            DO 140 J = JE + 1, M
               IF( W( J ).LT.TMP1 ) THEN
                  IE = J
                  TMP1 = W( J )
               END IF
  140       CONTINUE
*
            IF( IE.NE.0 ) THEN
               ITMP1 = IBLOCK( IE )
               W( IE ) = W( JE )
               IBLOCK( IE ) = IBLOCK( JE )
               W( JE ) = TMP1
               IBLOCK( JE ) = ITMP1
            END IF
  150    CONTINUE
      END IF
*
      INFO = 0
      IF( NCNVRG )
     $   INFO = INFO + 1
      IF( TOOFEW )
     $   INFO = INFO + 2
      RETURN
*
*     End of DSTEBZ
*
      END
*> \brief \b DSTEDC
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEDC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstedc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstedc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstedc.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSTEDC( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK,
*                          LIWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          COMPZ
*       INTEGER            INFO, LDZ, LIWORK, LWORK, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   D( * ), E( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSTEDC computes all eigenvalues and, optionally, eigenvectors of a
*> symmetric tridiagonal matrix using the divide and conquer method.
*> The eigenvectors of a full or band real symmetric matrix can also be
*> found if DSYTRD or DSPTRD or DSBTRD has been used to reduce this
*> matrix to tridiagonal form.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] COMPZ
*> \verbatim
*>          COMPZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only.
*>          = 'I':  Compute eigenvectors of tridiagonal matrix also.
*>          = 'V':  Compute eigenvectors of original dense symmetric
*>                  matrix also.  On entry, Z contains the orthogonal
*>                  matrix used to reduce the original matrix to
*>                  tridiagonal form.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The dimension of the symmetric tridiagonal matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, the diagonal elements of the tridiagonal matrix.
*>          On exit, if INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          On entry, the subdiagonal elements of the tridiagonal matrix.
*>          On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ,N)
*>          On entry, if COMPZ = 'V', then Z contains the orthogonal
*>          matrix used in the reduction to tridiagonal form.
*>          On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
*>          orthonormal eigenvectors of the original symmetric matrix,
*>          and if COMPZ = 'I', Z contains the orthonormal eigenvectors
*>          of the symmetric tridiagonal matrix.
*>          If  COMPZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1.
*>          If eigenvectors are desired, then LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If COMPZ = 'N' or N <= 1 then LWORK must be at least 1.
*>          If COMPZ = 'V' and N > 1 then LWORK must be at least
*>                         ( 1 + 3*N + 2*N*lg N + 4*N**2 ),
*>                         where lg( N ) = smallest integer k such
*>                         that 2**k >= N.
*>          If COMPZ = 'I' and N > 1 then LWORK must be at least
*>                         ( 1 + 4*N + N**2 ).
*>          Note that for COMPZ = 'I' or 'V', then if N is less than or
*>          equal to the minimum divide size, usually 25, then LWORK need
*>          only be max(1,2*(N-1)).
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK.
*>          If COMPZ = 'N' or N <= 1 then LIWORK must be at least 1.
*>          If COMPZ = 'V' and N > 1 then LIWORK must be at least
*>                         ( 6 + 6*N + 5*N*lg N ).
*>          If COMPZ = 'I' and N > 1 then LIWORK must be at least
*>                         ( 3 + 5*N ).
*>          Note that for COMPZ = 'I' or 'V', then if N is less than or
*>          equal to the minimum divide size, usually 25, then LIWORK
*>          need only be 1.
*>
*>          If LIWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the optimal size of the IWORK array,
*>          returns this value as the first entry of the IWORK array, and
*>          no error message related to LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  The algorithm failed to compute an eigenvalue while
*>                working on the submatrix lying in rows and columns
*>                INFO/(N+1) through mod(INFO,N+1).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup stedc
*
*> \par Contributors:
*  ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA \n
*>  Modified by Francoise Tisseur, University of Tennessee
*>
*  =====================================================================
      SUBROUTINE DSTEDC( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK,
     $                   LIWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          COMPZ
      INTEGER            INFO, LDZ, LIWORK, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   D( * ), E( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            FINISH, I, ICOMPZ, II, J, K, LGN, LIWMIN,
     $                   LWMIN, M, SMLSIZ, START, STOREZ, STRTRW
      DOUBLE PRECISION   EPS, ORGNRM, P, TINY
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANST
      EXTERNAL           LSAME, ILAENV, DLAMCH, DLANST
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMM, DLACPY, DLAED0, DLASCL, DLASET,
     $                   DLASRT,
     $                   DSTEQR, DSTERF, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, INT, LOG, MAX, MOD, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
      IF( LSAME( COMPZ, 'N' ) ) THEN
         ICOMPZ = 0
      ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
         ICOMPZ = 1
      ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
         ICOMPZ = 2
      ELSE
         ICOMPZ = -1
      END IF
      IF( ICOMPZ.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( ( LDZ.LT.1 ) .OR.
     $         ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1, N ) ) ) THEN
         INFO = -6
      END IF
*
      IF( INFO.EQ.0 ) THEN
*
*        Compute the workspace requirements
*
         SMLSIZ = ILAENV( 9, 'DSTEDC', ' ', 0, 0, 0, 0 )
         IF( N.LE.1 .OR. ICOMPZ.EQ.0 ) THEN
            LIWMIN = 1
            LWMIN = 1
         ELSE IF( N.LE.SMLSIZ ) THEN
            LIWMIN = 1
            LWMIN = 2*( N - 1 )
         ELSE
            LGN = INT( LOG( DBLE( N ) )/LOG( TWO ) )
            IF( 2**LGN.LT.N )
     $         LGN = LGN + 1
            IF( 2**LGN.LT.N )
     $         LGN = LGN + 1
            IF( ICOMPZ.EQ.1 ) THEN
               LWMIN = 1 + 3*N + 2*N*LGN + 4*N**2
               LIWMIN = 6 + 6*N + 5*N*LGN
            ELSE IF( ICOMPZ.EQ.2 ) THEN
               LWMIN = 1 + 4*N + N**2
               LIWMIN = 3 + 5*N
            END IF
         END IF
         WORK( 1 ) = LWMIN
         IWORK( 1 ) = LIWMIN
*
         IF( LWORK.LT.LWMIN .AND. .NOT. LQUERY ) THEN
            INFO = -8
         ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT. LQUERY ) THEN
            INFO = -10
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSTEDC', -INFO )
         RETURN
      ELSE IF (LQUERY) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
      IF( N.EQ.1 ) THEN
         IF( ICOMPZ.NE.0 )
     $      Z( 1, 1 ) = ONE
         RETURN
      END IF
*
*     If the following conditional clause is removed, then the routine
*     will use the Divide and Conquer routine to compute only the
*     eigenvalues, which requires (3N + 3N**2) real workspace and
*     (2 + 5N + 2N lg(N)) integer workspace.
*     Since on many architectures DSTERF is much faster than any other
*     algorithm for finding eigenvalues only, it is used here
*     as the default. If the conditional clause is removed, then
*     information on the size of workspace needs to be changed.
*
*     If COMPZ = 'N', use DSTERF to compute the eigenvalues.
*
      IF( ICOMPZ.EQ.0 ) THEN
         CALL DSTERF( N, D, E, INFO )
         GO TO 50
      END IF
*
*     If N is smaller than the minimum divide size (SMLSIZ+1), then
*     solve the problem with another solver.
*
      IF( N.LE.SMLSIZ ) THEN
*
         CALL DSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
      ELSE
*
*        If COMPZ = 'V', the Z matrix must be stored elsewhere for later
*        use.
*
         IF( ICOMPZ.EQ.1 ) THEN
            STOREZ = 1 + N*N
         ELSE
            STOREZ = 1
         END IF
*
         IF( ICOMPZ.EQ.2 ) THEN
            CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
         END IF
*
*        Scale.
*
         ORGNRM = DLANST( 'M', N, D, E )
         IF( ORGNRM.EQ.ZERO )
     $      GO TO 50
*
         EPS = DLAMCH( 'Epsilon' )
*
         START = 1
*
*        while ( START <= N )
*
   10    CONTINUE
         IF( START.LE.N ) THEN
*
*           Let FINISH be the position of the next subdiagonal entry
*           such that E( FINISH ) <= TINY or FINISH = N if no such
*           subdiagonal exists.  The matrix identified by the elements
*           between START and FINISH constitutes an independent
*           sub-problem.
*
            FINISH = START
   20       CONTINUE
            IF( FINISH.LT.N ) THEN
               TINY = EPS*SQRT( ABS( D( FINISH ) ) )*
     $                    SQRT( ABS( D( FINISH+1 ) ) )
               IF( ABS( E( FINISH ) ).GT.TINY ) THEN
                  FINISH = FINISH + 1
                  GO TO 20
               END IF
            END IF
*
*           (Sub) Problem determined.  Compute its size and solve it.
*
            M = FINISH - START + 1
            IF( M.EQ.1 ) THEN
               START = FINISH + 1
               GO TO 10
            END IF
            IF( M.GT.SMLSIZ ) THEN
*
*              Scale.
*
               ORGNRM = DLANST( 'M', M, D( START ), E( START ) )
               CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, M, 1, D( START ),
     $                      M,
     $                      INFO )
               CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, M-1, 1,
     $                      E( START ),
     $                      M-1, INFO )
*
               IF( ICOMPZ.EQ.1 ) THEN
                  STRTRW = 1
               ELSE
                  STRTRW = START
               END IF
               CALL DLAED0( ICOMPZ, N, M, D( START ), E( START ),
     $                      Z( STRTRW, START ), LDZ, WORK( 1 ), N,
     $                      WORK( STOREZ ), IWORK, INFO )
               IF( INFO.NE.0 ) THEN
                  INFO = ( INFO / ( M+1 )+START-1 )*( N+1 ) +
     $                   MOD( INFO, ( M+1 ) ) + START - 1
                  GO TO 50
               END IF
*
*              Scale back.
*
               CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, M, 1, D( START ),
     $                      M,
     $                      INFO )
*
            ELSE
               IF( ICOMPZ.EQ.1 ) THEN
*
*                 Since QR won't update a Z matrix which is larger than
*                 the length of D, we must solve the sub-problem in a
*                 workspace and then multiply back into Z.
*
                  CALL DSTEQR( 'I', M, D( START ), E( START ), WORK,
     $                         M,
     $                         WORK( M*M+1 ), INFO )
                  CALL DLACPY( 'A', N, M, Z( 1, START ), LDZ,
     $                         WORK( STOREZ ), N )
                  CALL DGEMM( 'N', 'N', N, M, M, ONE,
     $                        WORK( STOREZ ), N, WORK, M, ZERO,
     $                        Z( 1, START ), LDZ )
               ELSE IF( ICOMPZ.EQ.2 ) THEN
                  CALL DSTEQR( 'I', M, D( START ), E( START ),
     $                         Z( START, START ), LDZ, WORK, INFO )
               ELSE
                  CALL DSTERF( M, D( START ), E( START ), INFO )
               END IF
               IF( INFO.NE.0 ) THEN
                  INFO = START*( N+1 ) + FINISH
                  GO TO 50
               END IF
            END IF
*
            START = FINISH + 1
            GO TO 10
         END IF
*
*        endwhile
*
         IF( ICOMPZ.EQ.0 ) THEN
*
*          Use Quick Sort
*
           CALL DLASRT( 'I', N, D, INFO )
*
         ELSE
*
*          Use Selection Sort to minimize swaps of eigenvectors
*
           DO 40 II = 2, N
              I = II - 1
              K = I
              P = D( I )
              DO 30 J = II, N
                 IF( D( J ).LT.P ) THEN
                    K = J
                    P = D( J )
                 END IF
   30         CONTINUE
              IF( K.NE.I ) THEN
                 D( K ) = D( I )
                 D( I ) = P
                 CALL DSWAP( N, Z( 1, I ), 1, Z( 1, K ), 1 )
              END IF
   40      CONTINUE
         END IF
      END IF
*
   50 CONTINUE
      WORK( 1 ) = LWMIN
      IWORK( 1 ) = LIWMIN
*
      RETURN
*
*     End of DSTEDC
*
      END
*> \brief \b DSTEGR
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEGR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstegr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstegr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstegr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSTEGR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
*                  ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK,
*                  LIWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, RANGE
*       INTEGER            IL, INFO, IU, LDZ, LIWORK, LWORK, M, N
*       DOUBLE PRECISION ABSTOL, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            ISUPPZ( * ), IWORK( * )
*       DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * )
*       DOUBLE PRECISION   Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSTEGR computes selected eigenvalues and, optionally, eigenvectors
*> of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
*> a well defined set of pairwise different real eigenvalues, the corresponding
*> real eigenvectors are pairwise orthogonal.
*>
*> The spectrum may be computed either completely or partially by specifying
*> either an interval (VL,VU] or a range of indices IL:IU for the desired
*> eigenvalues.
*>
*> DSTEGR is a compatibility wrapper around the improved DSTEMR routine.
*> See DSTEMR for further details.
*>
*> One important change is that the ABSTOL parameter no longer provides any
*> benefit and hence is no longer used.
*>
*> Note : DSTEGR and DSTEMR work only on machines which follow
*> IEEE-754 floating-point standard in their handling of infinities and
*> NaNs.  Normal execution may create these exceptional values and hence
*> may abort due to a floating point exception in environments which
*> do not conform to the IEEE-754 standard.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*>          RANGE is CHARACTER*1
*>          = 'A': all eigenvalues will be found.
*>          = 'V': all eigenvalues in the half-open interval (VL,VU]
*>                 will be found.
*>          = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, the N diagonal elements of the tridiagonal matrix
*>          T. On exit, D is overwritten.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N)
*>          On entry, the (N-1) subdiagonal elements of the tridiagonal
*>          matrix T in elements 1 to N-1 of E. E(N) need not be set on
*>          input, but is used internally as workspace.
*>          On exit, E is overwritten.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is DOUBLE PRECISION
*>
*>          If RANGE='V', the lower bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*>          VU is DOUBLE PRECISION
*>
*>          If RANGE='V', the upper bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*>          IL is INTEGER
*>
*>          If RANGE='I', the index of the
*>          smallest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*>          IU is INTEGER
*>
*>          If RANGE='I', the index of the
*>          largest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*>          ABSTOL is DOUBLE PRECISION
*>          Unused.  Was the absolute error tolerance for the
*>          eigenvalues/eigenvectors in previous versions.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The total number of eigenvalues found.  0 <= M <= N.
*>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          The first M elements contain the selected eigenvalues in
*>          ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
*>          If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
*>          contain the orthonormal eigenvectors of the matrix T
*>          corresponding to the selected eigenvalues, with the i-th
*>          column of Z holding the eigenvector associated with W(i).
*>          If JOBZ = 'N', then Z is not referenced.
*>          Note: the user must ensure that at least max(1,M) columns are
*>          supplied in the array Z; if RANGE = 'V', the exact value of M
*>          is not known in advance and an upper bound must be used.
*>          Supplying N columns is always safe.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', then LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] ISUPPZ
*> \verbatim
*>          ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
*>          The support of the eigenvectors in Z, i.e., the indices
*>          indicating the nonzero elements in Z. The i-th computed eigenvector
*>          is nonzero only in elements ISUPPZ( 2*i-1 ) through
*>          ISUPPZ( 2*i ). This is relevant in the case when the matrix
*>          is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (LWORK)
*>          On exit, if INFO = 0, WORK(1) returns the optimal
*>          (and minimal) LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= max(1,18*N)
*>          if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (LIWORK)
*>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK.  LIWORK >= max(1,10*N)
*>          if the eigenvectors are desired, and LIWORK >= max(1,8*N)
*>          if only the eigenvalues are to be computed.
*>          If LIWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the optimal size of the IWORK array,
*>          returns this value as the first entry of the IWORK array, and
*>          no error message related to LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          On exit, INFO
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = 1X, internal error in DLARRE,
*>                if INFO = 2X, internal error in DLARRV.
*>                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
*>                the nonzero error code returned by DLARRE or
*>                DLARRV, respectively.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup stegr
*
*> \par Contributors:
*  ==================
*>
*> Inderjit Dhillon, IBM Almaden, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, LBNL/NERSC, USA \n
*
*  =====================================================================
      SUBROUTINE DSTEGR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
     $           ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK,
     $           LIWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, RANGE
      INTEGER            IL, INFO, IU, LDZ, LIWORK, LWORK, M, N
      DOUBLE PRECISION ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            ISUPPZ( * ), IWORK( * )
      DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * )
      DOUBLE PRECISION   Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL TRYRAC
*     ..
*     .. External Subroutines ..
      EXTERNAL DSTEMR
*     ..
*     .. Executable Statements ..
      INFO = 0
      TRYRAC = .FALSE.

      CALL DSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
     $                   M, W, Z, LDZ, N, ISUPPZ, TRYRAC, WORK, LWORK,
     $                   IWORK, LIWORK, INFO )
*
*     End of DSTEGR
*
      END
*> \brief \b DSTEIN
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEIN + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstein.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstein.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstein.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK,
*                          IWORK, IFAIL, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDZ, M, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IBLOCK( * ), IFAIL( * ), ISPLIT( * ),
*      $                   IWORK( * )
*       DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSTEIN computes the eigenvectors of a real symmetric tridiagonal
*> matrix T corresponding to specified eigenvalues, using inverse
*> iteration.
*>
*> The maximum number of iterations allowed for each eigenvector is
*> specified by an internal parameter MAXITS (currently set to 5).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The n diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          The (n-1) subdiagonal elements of the tridiagonal matrix
*>          T, in elements 1 to N-1.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of eigenvectors to be found.  0 <= M <= N.
*> \endverbatim
*>
*> \param[in] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          The first M elements of W contain the eigenvalues for
*>          which eigenvectors are to be computed.  The eigenvalues
*>          should be grouped by split-off block and ordered from
*>          smallest to largest within the block.  ( The output array
*>          W from DSTEBZ with ORDER = 'B' is expected here. )
*> \endverbatim
*>
*> \param[in] IBLOCK
*> \verbatim
*>          IBLOCK is INTEGER array, dimension (N)
*>          The submatrix indices associated with the corresponding
*>          eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to
*>          the first submatrix from the top, =2 if W(i) belongs to
*>          the second submatrix, etc.  ( The output array IBLOCK
*>          from DSTEBZ is expected here. )
*> \endverbatim
*>
*> \param[in] ISPLIT
*> \verbatim
*>          ISPLIT is INTEGER array, dimension (N)
*>          The splitting points, at which T breaks up into submatrices.
*>          The first submatrix consists of rows/columns 1 to
*>          ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
*>          through ISPLIT( 2 ), etc.
*>          ( The output array ISPLIT from DSTEBZ is expected here. )
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, M)
*>          The computed eigenvectors.  The eigenvector associated
*>          with the eigenvalue W(i) is stored in the i-th column of
*>          Z.  Any vector which fails to converge is set to its current
*>          iterate after MAXITS iterations.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*>          IFAIL is INTEGER array, dimension (M)
*>          On normal exit, all elements of IFAIL are zero.
*>          If one or more eigenvectors fail to converge after
*>          MAXITS iterations, then their indices are stored in
*>          array IFAIL.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit.
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*>          > 0: if INFO = i, then i eigenvectors failed to converge
*>               in MAXITS iterations.  Their indices are stored in
*>               array IFAIL.
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  MAXITS  INTEGER, default = 5
*>          The maximum number of iterations performed.
*>
*>  EXTRA   INTEGER, default = 2
*>          The number of iterations performed after norm growth
*>          criterion is satisfied, should be at least 1.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup stein
*
*  =====================================================================
      SUBROUTINE DSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK,
     $                   IWORK, IFAIL, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDZ, M, N
*     ..
*     .. Array Arguments ..
      INTEGER            IBLOCK( * ), IFAIL( * ), ISPLIT( * ),
     $                   IWORK( * )
      DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TEN, ODM3, ODM1
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TEN = 1.0D+1,
     $                   ODM3 = 1.0D-3, ODM1 = 1.0D-1 )
      INTEGER            MAXITS, EXTRA
      PARAMETER          ( MAXITS = 5, EXTRA = 2 )
*     ..
*     .. Local Scalars ..
      INTEGER            B1, BLKSIZ, BN, GPIND, I, IINFO, INDRV1,
     $                   INDRV2, INDRV3, INDRV4, INDRV5, ITS, J, J1,
     $                   JBLK, JMAX, NBLK, NRMCHK
      DOUBLE PRECISION   DTPCRT, EPS, EPS1, NRM, ONENRM, ORTOL, PERTOL,
     $                   SCL, SEP, TOL, XJ, XJM, ZTR
*     ..
*     .. Local Arrays ..
      INTEGER            ISEED( 4 )
*     ..
*     .. External Functions ..
      INTEGER            IDAMAX
      DOUBLE PRECISION   DDOT, DLAMCH, DNRM2
      EXTERNAL           IDAMAX, DDOT, DLAMCH, DNRM2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DLAGTF, DLAGTS, DLARNV,
     $                   DSCAL,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      DO 10 I = 1, M
         IFAIL( I ) = 0
   10 CONTINUE
*
      IF( N.LT.0 ) THEN
         INFO = -1
      ELSE IF( M.LT.0 .OR. M.GT.N ) THEN
         INFO = -4
      ELSE IF( LDZ.LT.MAX( 1, N ) ) THEN
         INFO = -9
      ELSE
         DO 20 J = 2, M
            IF( IBLOCK( J ).LT.IBLOCK( J-1 ) ) THEN
               INFO = -6
               GO TO 30
            END IF
            IF( IBLOCK( J ).EQ.IBLOCK( J-1 ) .AND. W( J ).LT.W( J-1 ) )
     $           THEN
               INFO = -5
               GO TO 30
            END IF
   20    CONTINUE
   30    CONTINUE
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSTEIN', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. M.EQ.0 ) THEN
         RETURN
      ELSE IF( N.EQ.1 ) THEN
         Z( 1, 1 ) = ONE
         RETURN
      END IF
*
*     Get machine constants.
*
      EPS = DLAMCH( 'Precision' )
*
*     Initialize seed for random number generator DLARNV.
*
      DO 40 I = 1, 4
         ISEED( I ) = 1
   40 CONTINUE
*
*     Initialize pointers.
*
      INDRV1 = 0
      INDRV2 = INDRV1 + N
      INDRV3 = INDRV2 + N
      INDRV4 = INDRV3 + N
      INDRV5 = INDRV4 + N
*
*     Compute eigenvectors of matrix blocks.
*
      J1 = 1
      DO 160 NBLK = 1, IBLOCK( M )
*
*        Find starting and ending indices of block nblk.
*
         IF( NBLK.EQ.1 ) THEN
            B1 = 1
         ELSE
            B1 = ISPLIT( NBLK-1 ) + 1
         END IF
         BN = ISPLIT( NBLK )
         BLKSIZ = BN - B1 + 1
         IF( BLKSIZ.EQ.1 )
     $      GO TO 60
         GPIND = J1
*
*        Compute reorthogonalization criterion and stopping criterion.
*
         ONENRM = ABS( D( B1 ) ) + ABS( E( B1 ) )
         ONENRM = MAX( ONENRM, ABS( D( BN ) )+ABS( E( BN-1 ) ) )
         DO 50 I = B1 + 1, BN - 1
            ONENRM = MAX( ONENRM, ABS( D( I ) )+ABS( E( I-1 ) )+
     $               ABS( E( I ) ) )
   50    CONTINUE
         ORTOL = ODM3*ONENRM
*
         DTPCRT = SQRT( ODM1 / BLKSIZ )
*
*        Loop through eigenvalues of block nblk.
*
   60    CONTINUE
         JBLK = 0
         DO 150 J = J1, M
            IF( IBLOCK( J ).NE.NBLK ) THEN
               J1 = J
               GO TO 160
            END IF
            JBLK = JBLK + 1
            XJ = W( J )
*
*           Skip all the work if the block size is one.
*
            IF( BLKSIZ.EQ.1 ) THEN
               WORK( INDRV1+1 ) = ONE
               GO TO 120
            END IF
*
*           If eigenvalues j and j-1 are too close, add a relatively
*           small perturbation.
*
            IF( JBLK.GT.1 ) THEN
               EPS1 = ABS( EPS*XJ )
               PERTOL = TEN*EPS1
               SEP = XJ - XJM
               IF( SEP.LT.PERTOL )
     $            XJ = XJM + PERTOL
            END IF
*
            ITS = 0
            NRMCHK = 0
*
*           Get random starting vector.
*
            CALL DLARNV( 2, ISEED, BLKSIZ, WORK( INDRV1+1 ) )
*
*           Copy the matrix T so it won't be destroyed in factorization.
*
            CALL DCOPY( BLKSIZ, D( B1 ), 1, WORK( INDRV4+1 ), 1 )
            CALL DCOPY( BLKSIZ-1, E( B1 ), 1, WORK( INDRV2+2 ), 1 )
            CALL DCOPY( BLKSIZ-1, E( B1 ), 1, WORK( INDRV3+1 ), 1 )
*
*           Compute LU factors with partial pivoting  ( PT = LU )
*
            TOL = ZERO
            CALL DLAGTF( BLKSIZ, WORK( INDRV4+1 ), XJ,
     $                   WORK( INDRV2+2 ),
     $                   WORK( INDRV3+1 ), TOL, WORK( INDRV5+1 ), IWORK,
     $                   IINFO )
*
*           Update iteration count.
*
   70       CONTINUE
            ITS = ITS + 1
            IF( ITS.GT.MAXITS )
     $         GO TO 100
*
*           Normalize and scale the righthand side vector Pb.
*
            JMAX = IDAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
            SCL = BLKSIZ*ONENRM*MAX( EPS,
     $            ABS( WORK( INDRV4+BLKSIZ ) ) ) /
     $            ABS( WORK( INDRV1+JMAX ) )
            CALL DSCAL( BLKSIZ, SCL, WORK( INDRV1+1 ), 1 )
*
*           Solve the system LU = Pb.
*
            CALL DLAGTS( -1, BLKSIZ, WORK( INDRV4+1 ),
     $                   WORK( INDRV2+2 ),
     $                   WORK( INDRV3+1 ), WORK( INDRV5+1 ), IWORK,
     $                   WORK( INDRV1+1 ), TOL, IINFO )
*
*           Reorthogonalize by modified Gram-Schmidt if eigenvalues are
*           close enough.
*
            IF( JBLK.EQ.1 )
     $         GO TO 90
            IF( ABS( XJ-XJM ).GT.ORTOL )
     $         GPIND = J
            IF( GPIND.NE.J ) THEN
               DO 80 I = GPIND, J - 1
                  ZTR = -DDOT( BLKSIZ, WORK( INDRV1+1 ), 1, Z( B1,
     $                         I ),
     $                  1 )
                  CALL DAXPY( BLKSIZ, ZTR, Z( B1, I ), 1,
     $                        WORK( INDRV1+1 ), 1 )
   80          CONTINUE
            END IF
*
*           Check the infinity norm of the iterate.
*
   90       CONTINUE
            JMAX = IDAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
            NRM = ABS( WORK( INDRV1+JMAX ) )
*
*           Continue for additional iterations after norm reaches
*           stopping criterion.
*
            IF( NRM.LT.DTPCRT )
     $         GO TO 70
            NRMCHK = NRMCHK + 1
            IF( NRMCHK.LT.EXTRA+1 )
     $         GO TO 70
*
            GO TO 110
*
*           If stopping criterion was not satisfied, update info and
*           store eigenvector number in array ifail.
*
  100       CONTINUE
            INFO = INFO + 1
            IFAIL( INFO ) = J
*
*           Accept iterate as jth eigenvector.
*
  110       CONTINUE
            SCL = ONE / DNRM2( BLKSIZ, WORK( INDRV1+1 ), 1 )
            JMAX = IDAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
            IF( WORK( INDRV1+JMAX ).LT.ZERO )
     $         SCL = -SCL
            CALL DSCAL( BLKSIZ, SCL, WORK( INDRV1+1 ), 1 )
  120       CONTINUE
            DO 130 I = 1, N
               Z( I, J ) = ZERO
  130       CONTINUE
            DO 140 I = 1, BLKSIZ
               Z( B1+I-1, J ) = WORK( INDRV1+I )
  140       CONTINUE
*
*           Save the shift to check eigenvalue spacing at next
*           iteration.
*
            XJM = XJ
*
  150    CONTINUE
  160 CONTINUE
*
      RETURN
*
*     End of DSTEIN
*
      END
*> \brief \b DSTEMR
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEMR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstemr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstemr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstemr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
*                          M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
*                          IWORK, LIWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, RANGE
*       LOGICAL            TRYRAC
*       INTEGER            IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
*       DOUBLE PRECISION VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            ISUPPZ( * ), IWORK( * )
*       DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * )
*       DOUBLE PRECISION   Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSTEMR computes selected eigenvalues and, optionally, eigenvectors
*> of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
*> a well defined set of pairwise different real eigenvalues, the corresponding
*> real eigenvectors are pairwise orthogonal.
*>
*> The spectrum may be computed either completely or partially by specifying
*> either an interval (VL,VU] or a range of indices IL:IU for the desired
*> eigenvalues.
*>
*> Depending on the number of desired eigenvalues, these are computed either
*> by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
*> computed by the use of various suitable L D L^T factorizations near clusters
*> of close eigenvalues (referred to as RRRs, Relatively Robust
*> Representations). An informal sketch of the algorithm follows.
*>
*> For each unreduced block (submatrix) of T,
*>    (a) Compute T - sigma I  = L D L^T, so that L and D
*>        define all the wanted eigenvalues to high relative accuracy.
*>        This means that small relative changes in the entries of D and L
*>        cause only small relative changes in the eigenvalues and
*>        eigenvectors. The standard (unfactored) representation of the
*>        tridiagonal matrix T does not have this property in general.
*>    (b) Compute the eigenvalues to suitable accuracy.
*>        If the eigenvectors are desired, the algorithm attains full
*>        accuracy of the computed eigenvalues only right before
*>        the corresponding vectors have to be computed, see steps c) and d).
*>    (c) For each cluster of close eigenvalues, select a new
*>        shift close to the cluster, find a new factorization, and refine
*>        the shifted eigenvalues to suitable accuracy.
*>    (d) For each eigenvalue with a large enough relative separation compute
*>        the corresponding eigenvector by forming a rank revealing twisted
*>        factorization. Go back to (c) for any clusters that remain.
*>
*> For more details, see:
*> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
*>   to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
*>   Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
*> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
*>   Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
*>   2004.  Also LAPACK Working Note 154.
*> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
*>   tridiagonal eigenvalue/eigenvector problem",
*>   Computer Science Division Technical Report No. UCB/CSD-97-971,
*>   UC Berkeley, May 1997.
*>
*> Further Details
*> 1.DSTEMR works only on machines which follow IEEE-754
*> floating-point standard in their handling of infinities and NaNs.
*> This permits the use of efficient inner loops avoiding a check for
*> zero divisors.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*>          RANGE is CHARACTER*1
*>          = 'A': all eigenvalues will be found.
*>          = 'V': all eigenvalues in the half-open interval (VL,VU]
*>                 will be found.
*>          = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, the N diagonal elements of the tridiagonal matrix
*>          T. On exit, D is overwritten.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N)
*>          On entry, the (N-1) subdiagonal elements of the tridiagonal
*>          matrix T in elements 1 to N-1 of E. E(N) need not be set on
*>          input, but is used internally as workspace.
*>          On exit, E is overwritten.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is DOUBLE PRECISION
*>
*>          If RANGE='V', the lower bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*>          VU is DOUBLE PRECISION
*>
*>          If RANGE='V', the upper bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*>          IL is INTEGER
*>
*>          If RANGE='I', the index of the
*>          smallest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*>          IU is INTEGER
*>
*>          If RANGE='I', the index of the
*>          largest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The total number of eigenvalues found.  0 <= M <= N.
*>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          The first M elements contain the selected eigenvalues in
*>          ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
*>          If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
*>          contain the orthonormal eigenvectors of the matrix T
*>          corresponding to the selected eigenvalues, with the i-th
*>          column of Z holding the eigenvector associated with W(i).
*>          If JOBZ = 'N', then Z is not referenced.
*>          Note: the user must ensure that at least max(1,M) columns are
*>          supplied in the array Z; if RANGE = 'V', the exact value of M
*>          is not known in advance and can be computed with a workspace
*>          query by setting NZC = -1, see below.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', then LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[in] NZC
*> \verbatim
*>          NZC is INTEGER
*>          The number of eigenvectors to be held in the array Z.
*>          If RANGE = 'A', then NZC >= max(1,N).
*>          If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
*>          If RANGE = 'I', then NZC >= IU-IL+1.
*>          If NZC = -1, then a workspace query is assumed; the
*>          routine calculates the number of columns of the array Z that
*>          are needed to hold the eigenvectors.
*>          This value is returned as the first entry of the Z array, and
*>          no error message related to NZC is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] ISUPPZ
*> \verbatim
*>          ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
*>          The support of the eigenvectors in Z, i.e., the indices
*>          indicating the nonzero elements in Z. The i-th computed eigenvector
*>          is nonzero only in elements ISUPPZ( 2*i-1 ) through
*>          ISUPPZ( 2*i ). This is relevant in the case when the matrix
*>          is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
*> \endverbatim
*>
*> \param[in,out] TRYRAC
*> \verbatim
*>          TRYRAC is LOGICAL
*>          If TRYRAC = .TRUE., indicates that the code should check whether
*>          the tridiagonal matrix defines its eigenvalues to high relative
*>          accuracy.  If so, the code uses relative-accuracy preserving
*>          algorithms that might be (a bit) slower depending on the matrix.
*>          If the matrix does not define its eigenvalues to high relative
*>          accuracy, the code can uses possibly faster algorithms.
*>          If TRYRAC = .FALSE., the code is not required to guarantee
*>          relatively accurate eigenvalues and can use the fastest possible
*>          techniques.
*>          On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
*>          does not define its eigenvalues to high relative accuracy.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (LWORK)
*>          On exit, if INFO = 0, WORK(1) returns the optimal
*>          (and minimal) LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= max(1,18*N)
*>          if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (LIWORK)
*>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK.  LIWORK >= max(1,10*N)
*>          if the eigenvectors are desired, and LIWORK >= max(1,8*N)
*>          if only the eigenvalues are to be computed.
*>          If LIWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the optimal size of the IWORK array,
*>          returns this value as the first entry of the IWORK array, and
*>          no error message related to LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          On exit, INFO
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = 1X, internal error in DLARRE,
*>                if INFO = 2X, internal error in DLARRV.
*>                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
*>                the nonzero error code returned by DLARRE or
*>                DLARRV, respectively.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup stemr
*
*> \par Contributors:
*  ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA \n
*> Aravindh Krishnamoorthy, FAU, Erlangen, Germany \n
*
*  =====================================================================
      SUBROUTINE DSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
     $                   M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
     $                   IWORK, LIWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, RANGE
      LOGICAL            TRYRAC
      INTEGER            IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
      DOUBLE PRECISION VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            ISUPPZ( * ), IWORK( * )
      DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * )
      DOUBLE PRECISION   Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, FOUR, MINRGP
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0,
     $                     FOUR = 4.0D0,
     $                     MINRGP = 1.0D-3 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY,
     $                   LAESWAP
      INTEGER            I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW,
     $                   IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD,
     $                   INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP,
     $                   ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT,
     $                   NZCMIN, OFFSET, WBEGIN, WEND
      DOUBLE PRECISION   BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN,
     $                   RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN,
     $                   THRESH, TMP, TNRM, WL, WU
*     ..
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANST
      EXTERNAL           LSAME, DLAMCH, DLANST
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLAE2, DLAEV2, DLARRC, DLARRE,
     $                   DLARRJ,
     $                   DLARRR, DLARRV, DLASRT, DSCAL, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT


*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
      ALLEIG = LSAME( RANGE, 'A' )
      VALEIG = LSAME( RANGE, 'V' )
      INDEIG = LSAME( RANGE, 'I' )
*
      LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) )
      ZQUERY = ( NZC.EQ.-1 )
      LAESWAP = .FALSE.

*     DSTEMR needs WORK of size 6*N, IWORK of size 3*N.
*     In addition, DLARRE needs WORK of size 6*N, IWORK of size 5*N.
*     Furthermore, DLARRV needs WORK of size 12*N, IWORK of size 7*N.
      IF( WANTZ ) THEN
         LWMIN = 18*N
         LIWMIN = 10*N
      ELSE
*        need less workspace if only the eigenvalues are wanted
         LWMIN = 12*N
         LIWMIN = 8*N
      ENDIF

      WL = ZERO
      WU = ZERO
      IIL = 0
      IIU = 0
      NSPLIT = 0

      IF( VALEIG ) THEN
*        We do not reference VL, VU in the cases RANGE = 'I','A'
*        The interval (WL, WU] contains all the wanted eigenvalues.
*        It is either given by the user or computed in DLARRE.
         WL = VL
         WU = VU
      ELSEIF( INDEIG ) THEN
*        We do not reference IL, IU in the cases RANGE = 'V','A'
         IIL = IL
         IIU = IU
      ENDIF
*
      INFO = 0
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN
         INFO = -7
      ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN
         INFO = -8
      ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN
         INFO = -9
      ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
         INFO = -13
      ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
         INFO = -17
      ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
         INFO = -19
      END IF
*
*     Get machine constants.
*
      SAFMIN = DLAMCH( 'Safe minimum' )
      EPS = DLAMCH( 'Precision' )
      SMLNUM = SAFMIN / EPS
      BIGNUM = ONE / SMLNUM
      RMIN = SQRT( SMLNUM )
      RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
      IF( INFO.EQ.0 ) THEN
         WORK( 1 ) = LWMIN
         IWORK( 1 ) = LIWMIN
*
         IF( WANTZ .AND. ALLEIG ) THEN
            NZCMIN = N
         ELSE IF( WANTZ .AND. VALEIG ) THEN
            CALL DLARRC( 'T', N, VL, VU, D, E, SAFMIN,
     $                            NZCMIN, ITMP, ITMP2, INFO )
         ELSE IF( WANTZ .AND. INDEIG ) THEN
            NZCMIN = IIU-IIL+1
         ELSE
*           WANTZ .EQ. FALSE.
            NZCMIN = 0
         ENDIF
         IF( ZQUERY .AND. INFO.EQ.0 ) THEN
            Z( 1,1 ) = NZCMIN
         ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN
            INFO = -14
         END IF
      END IF

      IF( INFO.NE.0 ) THEN
*
         CALL XERBLA( 'DSTEMR', -INFO )
*
         RETURN
      ELSE IF( LQUERY .OR. ZQUERY ) THEN
         RETURN
      END IF
*
*     Handle N = 0, 1, and 2 cases immediately
*
      M = 0
      IF( N.EQ.0 )
     $   RETURN
*
      IF( N.EQ.1 ) THEN
         IF( ALLEIG .OR. INDEIG ) THEN
            M = 1
            W( 1 ) = D( 1 )
         ELSE
            IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN
               M = 1
               W( 1 ) = D( 1 )
            END IF
         END IF
         IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
            Z( 1, 1 ) = ONE
            ISUPPZ(1) = 1
            ISUPPZ(2) = 1
         END IF
         RETURN
      END IF
*
      IF( N.EQ.2 ) THEN
         IF( .NOT.WANTZ ) THEN
            CALL DLAE2( D(1), E(1), D(2), R1, R2 )
         ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
            CALL DLAEV2( D(1), E(1), D(2), R1, R2, CS, SN )
         END IF
*        D/S/LAE2 and D/S/LAEV2 outputs satisfy |R1| >= |R2|. However,
*        the following code requires R1 >= R2. Hence, we correct
*        the order of R1, R2, CS, SN if R1 < R2 before further processing.
         IF( R1.LT.R2 ) THEN
            E(2) = R1
            R1 = R2
            R2 = E(2)
            LAESWAP = .TRUE.
         ENDIF
         IF( ALLEIG.OR.
     $      (VALEIG.AND.(R2.GT.WL).AND.
     $                  (R2.LE.WU)).OR.
     $      (INDEIG.AND.(IIL.EQ.1)) ) THEN
            M = M+1
            W( M ) = R2
            IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
               IF( LAESWAP ) THEN
                  Z( 1, M ) = CS
                  Z( 2, M ) = SN
               ELSE
                  Z( 1, M ) = -SN
                  Z( 2, M ) = CS
               ENDIF
*              Note: At most one of SN and CS can be zero.
               IF (SN.NE.ZERO) THEN
                  IF (CS.NE.ZERO) THEN
                     ISUPPZ(2*M-1) = 1
                     ISUPPZ(2*M) = 2
                  ELSE
                     ISUPPZ(2*M-1) = 1
                     ISUPPZ(2*M) = 1
                  END IF
               ELSE
                  ISUPPZ(2*M-1) = 2
                  ISUPPZ(2*M) = 2
               END IF
            ENDIF
         ENDIF
         IF( ALLEIG.OR.
     $      (VALEIG.AND.(R1.GT.WL).AND.
     $                  (R1.LE.WU)).OR.
     $      (INDEIG.AND.(IIU.EQ.2)) ) THEN
            M = M+1
            W( M ) = R1
            IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
               IF( LAESWAP ) THEN
                  Z( 1, M ) = -SN
                  Z( 2, M ) = CS
               ELSE
                  Z( 1, M ) = CS
                  Z( 2, M ) = SN
               ENDIF
*              Note: At most one of SN and CS can be zero.
               IF (SN.NE.ZERO) THEN
                  IF (CS.NE.ZERO) THEN
                     ISUPPZ(2*M-1) = 1
                     ISUPPZ(2*M) = 2
                  ELSE
                     ISUPPZ(2*M-1) = 1
                     ISUPPZ(2*M) = 1
                  END IF
               ELSE
                  ISUPPZ(2*M-1) = 2
                  ISUPPZ(2*M) = 2
               END IF
            ENDIF
         ENDIF

      ELSE

*     Continue with general N

         INDGRS = 1
         INDERR = 2*N + 1
         INDGP = 3*N + 1
         INDD = 4*N + 1
         INDE2 = 5*N + 1
         INDWRK = 6*N + 1
*
         IINSPL = 1
         IINDBL = N + 1
         IINDW = 2*N + 1
         IINDWK = 3*N + 1
*
*        Scale matrix to allowable range, if necessary.
*        The allowable range is related to the PIVMIN parameter; see the
*        comments in DLARRD.  The preference for scaling small values
*        up is heuristic; we expect users' matrices not to be close to the
*        RMAX threshold.
*
         SCALE = ONE
         TNRM = DLANST( 'M', N, D, E )
         IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
            SCALE = RMIN / TNRM
         ELSE IF( TNRM.GT.RMAX ) THEN
            SCALE = RMAX / TNRM
         END IF
         IF( SCALE.NE.ONE ) THEN
            CALL DSCAL( N, SCALE, D, 1 )
            CALL DSCAL( N-1, SCALE, E, 1 )
            TNRM = TNRM*SCALE
            IF( VALEIG ) THEN
*              If eigenvalues in interval have to be found,
*              scale (WL, WU] accordingly
               WL = WL*SCALE
               WU = WU*SCALE
            ENDIF
         END IF
*
*        Compute the desired eigenvalues of the tridiagonal after splitting
*        into smaller subblocks if the corresponding off-diagonal elements
*        are small
*        THRESH is the splitting parameter for DLARRE
*        A negative THRESH forces the old splitting criterion based on the
*        size of the off-diagonal. A positive THRESH switches to splitting
*        which preserves relative accuracy.
*
         IF( TRYRAC ) THEN
*           Test whether the matrix warrants the more expensive relative approach.
            CALL DLARRR( N, D, E, IINFO )
         ELSE
*           The user does not care about relative accurately eigenvalues
            IINFO = -1
         ENDIF
*        Set the splitting criterion
         IF (IINFO.EQ.0) THEN
            THRESH = EPS
         ELSE
            THRESH = -EPS
*           relative accuracy is desired but T does not guarantee it
            TRYRAC = .FALSE.
         ENDIF
*
         IF( TRYRAC ) THEN
*           Copy original diagonal, needed to guarantee relative accuracy
            CALL DCOPY(N,D,1,WORK(INDD),1)
         ENDIF
*        Store the squares of the offdiagonal values of T
         DO 5 J = 1, N-1
            WORK( INDE2+J-1 ) = E(J)**2
 5       CONTINUE

*        Set the tolerance parameters for bisection
         IF( .NOT.WANTZ ) THEN
*           DLARRE computes the eigenvalues to full precision.
            RTOL1 = FOUR * EPS
            RTOL2 = FOUR * EPS
         ELSE
*           DLARRE computes the eigenvalues to less than full precision.
*           DLARRV will refine the eigenvalue approximations, and we can
*           need less accurate initial bisection in DLARRE.
*           Note: these settings do only affect the subset case and DLARRE
            RTOL1 = SQRT(EPS)
            RTOL2 = MAX( SQRT(EPS)*5.0D-3, FOUR * EPS )
         ENDIF
         CALL DLARRE( RANGE, N, WL, WU, IIL, IIU, D, E,
     $             WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT,
     $             IWORK( IINSPL ), M, W, WORK( INDERR ),
     $             WORK( INDGP ), IWORK( IINDBL ),
     $             IWORK( IINDW ), WORK( INDGRS ), PIVMIN,
     $             WORK( INDWRK ), IWORK( IINDWK ), IINFO )
         IF( IINFO.NE.0 ) THEN
            INFO = 10 + ABS( IINFO )
            RETURN
         END IF
*        Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired
*        part of the spectrum. All desired eigenvalues are contained in
*        (WL,WU]


         IF( WANTZ ) THEN
*
*           Compute the desired eigenvectors corresponding to the computed
*           eigenvalues
*
            CALL DLARRV( N, WL, WU, D, E,
     $                PIVMIN, IWORK( IINSPL ), M,
     $                1, M, MINRGP, RTOL1, RTOL2,
     $                W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ),
     $                IWORK( IINDW ), WORK( INDGRS ), Z, LDZ,
     $                ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO )
            IF( IINFO.NE.0 ) THEN
               INFO = 20 + ABS( IINFO )
               RETURN
            END IF
         ELSE
*           DLARRE computes eigenvalues of the (shifted) root representation
*           DLARRV returns the eigenvalues of the unshifted matrix.
*           However, if the eigenvectors are not desired by the user, we need
*           to apply the corresponding shifts from DLARRE to obtain the
*           eigenvalues of the original matrix.
            DO 20 J = 1, M
               ITMP = IWORK( IINDBL+J-1 )
               W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) )
 20         CONTINUE
         END IF
*

         IF ( TRYRAC ) THEN
*           Refine computed eigenvalues so that they are relatively accurate
*           with respect to the original matrix T.
            IBEGIN = 1
            WBEGIN = 1
            DO 39  JBLK = 1, IWORK( IINDBL+M-1 )
               IEND = IWORK( IINSPL+JBLK-1 )
               IN = IEND - IBEGIN + 1
               WEND = WBEGIN - 1
*              check if any eigenvalues have to be refined in this block
 36            CONTINUE
               IF( WEND.LT.M ) THEN
                  IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN
                     WEND = WEND + 1
                     GO TO 36
                  END IF
               END IF
               IF( WEND.LT.WBEGIN ) THEN
                  IBEGIN = IEND + 1
                  GO TO 39
               END IF

               OFFSET = IWORK(IINDW+WBEGIN-1)-1
               IFIRST = IWORK(IINDW+WBEGIN-1)
               ILAST = IWORK(IINDW+WEND-1)
               RTOL2 = FOUR * EPS
               CALL DLARRJ( IN,
     $                   WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1),
     $                   IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN),
     $                   WORK( INDERR+WBEGIN-1 ),
     $                   WORK( INDWRK ), IWORK( IINDWK ), PIVMIN,
     $                   TNRM, IINFO )
               IBEGIN = IEND + 1
               WBEGIN = WEND + 1
 39         CONTINUE
         ENDIF
*
*        If matrix was scaled, then rescale eigenvalues appropriately.
*
         IF( SCALE.NE.ONE ) THEN
            CALL DSCAL( M, ONE / SCALE, W, 1 )
         END IF

      END IF

*
*     If eigenvalues are not in increasing order, then sort them,
*     possibly along with eigenvectors.
*
      IF( NSPLIT.GT.1 .OR. N.EQ.2 ) THEN
         IF( .NOT. WANTZ ) THEN
            CALL DLASRT( 'I', M, W, IINFO )
            IF( IINFO.NE.0 ) THEN
               INFO = 3
               RETURN
            END IF
         ELSE
            DO 60 J = 1, M - 1
               I = 0
               TMP = W( J )
               DO 50 JJ = J + 1, M
                  IF( W( JJ ).LT.TMP ) THEN
                     I = JJ
                     TMP = W( JJ )
                  END IF
 50            CONTINUE
               IF( I.NE.0 ) THEN
                  W( I ) = W( J )
                  W( J ) = TMP
                  IF( WANTZ ) THEN
                     CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
                     ITMP = ISUPPZ( 2*I-1 )
                     ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 )
                     ISUPPZ( 2*J-1 ) = ITMP
                     ITMP = ISUPPZ( 2*I )
                     ISUPPZ( 2*I ) = ISUPPZ( 2*J )
                     ISUPPZ( 2*J ) = ITMP
                  END IF
               END IF
 60         CONTINUE
         END IF
      ENDIF
*
*
      WORK( 1 ) = LWMIN
      IWORK( 1 ) = LIWMIN
      RETURN
*
*     End of DSTEMR
*
      END
*> \brief \b DSTEQR
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsteqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsteqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsteqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          COMPZ
*       INTEGER            INFO, LDZ, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), E( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSTEQR computes all eigenvalues and, optionally, eigenvectors of a
*> symmetric tridiagonal matrix using the implicit QL or QR method.
*> The eigenvectors of a full or band symmetric matrix can also be found
*> if DSYTRD or DSPTRD or DSBTRD has been used to reduce this matrix to
*> tridiagonal form.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] COMPZ
*> \verbatim
*>          COMPZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only.
*>          = 'V':  Compute eigenvalues and eigenvectors of the original
*>                  symmetric matrix.  On entry, Z must contain the
*>                  orthogonal matrix used to reduce the original matrix
*>                  to tridiagonal form.
*>          = 'I':  Compute eigenvalues and eigenvectors of the
*>                  tridiagonal matrix.  Z is initialized to the identity
*>                  matrix.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, the diagonal elements of the tridiagonal matrix.
*>          On exit, if INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          On entry, the (n-1) subdiagonal elements of the tridiagonal
*>          matrix.
*>          On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
*>          On entry, if  COMPZ = 'V', then Z contains the orthogonal
*>          matrix used in the reduction to tridiagonal form.
*>          On exit, if INFO = 0, then if  COMPZ = 'V', Z contains the
*>          orthonormal eigenvectors of the original symmetric matrix,
*>          and if COMPZ = 'I', Z contains the orthonormal eigenvectors
*>          of the symmetric tridiagonal matrix.
*>          If COMPZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          eigenvectors are desired, then  LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (max(1,2*N-2))
*>          If COMPZ = 'N', then WORK is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  the algorithm has failed to find all the eigenvalues in
*>                a total of 30*N iterations; if INFO = i, then i
*>                elements of E have not converged to zero; on exit, D
*>                and E contain the elements of a symmetric tridiagonal
*>                matrix which is orthogonally similar to the original
*>                matrix.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup steqr
*
*  =====================================================================
      SUBROUTINE DSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          COMPZ
      INTEGER            INFO, LDZ, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), E( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO, THREE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
     $                   THREE = 3.0D0 )
      INTEGER            MAXIT
      PARAMETER          ( MAXIT = 30 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, ICOMPZ, II, ISCALE, J, JTOT, K, L, L1, LEND,
     $                   LENDM1, LENDP1, LENDSV, LM1, LSV, M, MM, MM1,
     $                   NM1, NMAXIT
      DOUBLE PRECISION   ANORM, B, C, EPS, EPS2, F, G, P, R, RT1, RT2,
     $                   S, SAFMAX, SAFMIN, SSFMAX, SSFMIN, TST
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANST, DLAPY2
      EXTERNAL           LSAME, DLAMCH, DLANST, DLAPY2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLAE2, DLAEV2, DLARTG, DLASCL, DLASET,
     $                   DLASR,
     $                   DLASRT, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SIGN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IF( LSAME( COMPZ, 'N' ) ) THEN
         ICOMPZ = 0
      ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
         ICOMPZ = 1
      ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
         ICOMPZ = 2
      ELSE
         ICOMPZ = -1
      END IF
      IF( ICOMPZ.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( ( LDZ.LT.1 ) .OR. ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1,
     $         N ) ) ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSTEQR', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( N.EQ.1 ) THEN
         IF( ICOMPZ.EQ.2 )
     $      Z( 1, 1 ) = ONE
         RETURN
      END IF
*
*     Determine the unit roundoff and over/underflow thresholds.
*
      EPS = DLAMCH( 'E' )
      EPS2 = EPS**2
      SAFMIN = DLAMCH( 'S' )
      SAFMAX = ONE / SAFMIN
      SSFMAX = SQRT( SAFMAX ) / THREE
      SSFMIN = SQRT( SAFMIN ) / EPS2
*
*     Compute the eigenvalues and eigenvectors of the tridiagonal
*     matrix.
*
      IF( ICOMPZ.EQ.2 )
     $   CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
*
      NMAXIT = N*MAXIT
      JTOT = 0
*
*     Determine where the matrix splits and choose QL or QR iteration
*     for each block, according to whether top or bottom diagonal
*     element is smaller.
*
      L1 = 1
      NM1 = N - 1
*
   10 CONTINUE
      IF( L1.GT.N )
     $   GO TO 160
      IF( L1.GT.1 )
     $   E( L1-1 ) = ZERO
      IF( L1.LE.NM1 ) THEN
         DO 20 M = L1, NM1
            TST = ABS( E( M ) )
            IF( TST.EQ.ZERO )
     $         GO TO 30
            IF( TST.LE.( SQRT( ABS( D( M ) ) )*SQRT( ABS( D( M+
     $          1 ) ) ) )*EPS ) THEN
               E( M ) = ZERO
               GO TO 30
            END IF
   20    CONTINUE
      END IF
      M = N
*
   30 CONTINUE
      L = L1
      LSV = L
      LEND = M
      LENDSV = LEND
      L1 = M + 1
      IF( LEND.EQ.L )
     $   GO TO 10
*
*     Scale submatrix in rows and columns L to LEND
*
      ANORM = DLANST( 'M', LEND-L+1, D( L ), E( L ) )
      ISCALE = 0
      IF( ANORM.EQ.ZERO )
     $   GO TO 10
      IF( ANORM.GT.SSFMAX ) THEN
         ISCALE = 1
         CALL DLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L+1, 1, D( L ),
     $                N,
     $                INFO )
         CALL DLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L, 1, E( L ), N,
     $                INFO )
      ELSE IF( ANORM.LT.SSFMIN ) THEN
         ISCALE = 2
         CALL DLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L+1, 1, D( L ),
     $                N,
     $                INFO )
         CALL DLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L, 1, E( L ), N,
     $                INFO )
      END IF
*
*     Choose between QL and QR iteration
*
      IF( ABS( D( LEND ) ).LT.ABS( D( L ) ) ) THEN
         LEND = LSV
         L = LENDSV
      END IF
*
      IF( LEND.GT.L ) THEN
*
*        QL Iteration
*
*        Look for small subdiagonal element.
*
   40    CONTINUE
         IF( L.NE.LEND ) THEN
            LENDM1 = LEND - 1
            DO 50 M = L, LENDM1
               TST = ABS( E( M ) )**2
               IF( TST.LE.( EPS2*ABS( D( M ) ) )*ABS( D( M+1 ) )+
     $             SAFMIN )GO TO 60
   50       CONTINUE
         END IF
*
         M = LEND
*
   60    CONTINUE
         IF( M.LT.LEND )
     $      E( M ) = ZERO
         P = D( L )
         IF( M.EQ.L )
     $      GO TO 80
*
*        If remaining matrix is 2-by-2, use DLAE2 or SLAEV2
*        to compute its eigensystem.
*
         IF( M.EQ.L+1 ) THEN
            IF( ICOMPZ.GT.0 ) THEN
               CALL DLAEV2( D( L ), E( L ), D( L+1 ), RT1, RT2, C,
     $                      S )
               WORK( L ) = C
               WORK( N-1+L ) = S
               CALL DLASR( 'R', 'V', 'B', N, 2, WORK( L ),
     $                     WORK( N-1+L ), Z( 1, L ), LDZ )
            ELSE
               CALL DLAE2( D( L ), E( L ), D( L+1 ), RT1, RT2 )
            END IF
            D( L ) = RT1
            D( L+1 ) = RT2
            E( L ) = ZERO
            L = L + 2
            IF( L.LE.LEND )
     $         GO TO 40
            GO TO 140
         END IF
*
         IF( JTOT.EQ.NMAXIT )
     $      GO TO 140
         JTOT = JTOT + 1
*
*        Form shift.
*
         G = ( D( L+1 )-P ) / ( TWO*E( L ) )
         R = DLAPY2( G, ONE )
         G = D( M ) - P + ( E( L ) / ( G+SIGN( R, G ) ) )
*
         S = ONE
         C = ONE
         P = ZERO
*
*        Inner loop
*
         MM1 = M - 1
         DO 70 I = MM1, L, -1
            F = S*E( I )
            B = C*E( I )
            CALL DLARTG( G, F, C, S, R )
            IF( I.NE.M-1 )
     $         E( I+1 ) = R
            G = D( I+1 ) - P
            R = ( D( I )-G )*S + TWO*C*B
            P = S*R
            D( I+1 ) = G + P
            G = C*R - B
*
*           If eigenvectors are desired, then save rotations.
*
            IF( ICOMPZ.GT.0 ) THEN
               WORK( I ) = C
               WORK( N-1+I ) = -S
            END IF
*
   70    CONTINUE
*
*        If eigenvectors are desired, then apply saved rotations.
*
         IF( ICOMPZ.GT.0 ) THEN
            MM = M - L + 1
            CALL DLASR( 'R', 'V', 'B', N, MM, WORK( L ),
     $                  WORK( N-1+L ),
     $                  Z( 1, L ), LDZ )
         END IF
*
         D( L ) = D( L ) - P
         E( L ) = G
         GO TO 40
*
*        Eigenvalue found.
*
   80    CONTINUE
         D( L ) = P
*
         L = L + 1
         IF( L.LE.LEND )
     $      GO TO 40
         GO TO 140
*
      ELSE
*
*        QR Iteration
*
*        Look for small superdiagonal element.
*
   90    CONTINUE
         IF( L.NE.LEND ) THEN
            LENDP1 = LEND + 1
            DO 100 M = L, LENDP1, -1
               TST = ABS( E( M-1 ) )**2
               IF( TST.LE.( EPS2*ABS( D( M ) ) )*ABS( D( M-1 ) )+
     $             SAFMIN )GO TO 110
  100       CONTINUE
         END IF
*
         M = LEND
*
  110    CONTINUE
         IF( M.GT.LEND )
     $      E( M-1 ) = ZERO
         P = D( L )
         IF( M.EQ.L )
     $      GO TO 130
*
*        If remaining matrix is 2-by-2, use DLAE2 or SLAEV2
*        to compute its eigensystem.
*
         IF( M.EQ.L-1 ) THEN
            IF( ICOMPZ.GT.0 ) THEN
               CALL DLAEV2( D( L-1 ), E( L-1 ), D( L ), RT1, RT2, C,
     $                      S )
               WORK( M ) = C
               WORK( N-1+M ) = S
               CALL DLASR( 'R', 'V', 'F', N, 2, WORK( M ),
     $                     WORK( N-1+M ), Z( 1, L-1 ), LDZ )
            ELSE
               CALL DLAE2( D( L-1 ), E( L-1 ), D( L ), RT1, RT2 )
            END IF
            D( L-1 ) = RT1
            D( L ) = RT2
            E( L-1 ) = ZERO
            L = L - 2
            IF( L.GE.LEND )
     $         GO TO 90
            GO TO 140
         END IF
*
         IF( JTOT.EQ.NMAXIT )
     $      GO TO 140
         JTOT = JTOT + 1
*
*        Form shift.
*
         G = ( D( L-1 )-P ) / ( TWO*E( L-1 ) )
         R = DLAPY2( G, ONE )
         G = D( M ) - P + ( E( L-1 ) / ( G+SIGN( R, G ) ) )
*
         S = ONE
         C = ONE
         P = ZERO
*
*        Inner loop
*
         LM1 = L - 1
         DO 120 I = M, LM1
            F = S*E( I )
            B = C*E( I )
            CALL DLARTG( G, F, C, S, R )
            IF( I.NE.M )
     $         E( I-1 ) = R
            G = D( I ) - P
            R = ( D( I+1 )-G )*S + TWO*C*B
            P = S*R
            D( I ) = G + P
            G = C*R - B
*
*           If eigenvectors are desired, then save rotations.
*
            IF( ICOMPZ.GT.0 ) THEN
               WORK( I ) = C
               WORK( N-1+I ) = S
            END IF
*
  120    CONTINUE
*
*        If eigenvectors are desired, then apply saved rotations.
*
         IF( ICOMPZ.GT.0 ) THEN
            MM = L - M + 1
            CALL DLASR( 'R', 'V', 'F', N, MM, WORK( M ),
     $                  WORK( N-1+M ),
     $                  Z( 1, M ), LDZ )
         END IF
*
         D( L ) = D( L ) - P
         E( LM1 ) = G
         GO TO 90
*
*        Eigenvalue found.
*
  130    CONTINUE
         D( L ) = P
*
         L = L - 1
         IF( L.GE.LEND )
     $      GO TO 90
         GO TO 140
*
      END IF
*
*     Undo scaling if necessary
*
  140 CONTINUE
      IF( ISCALE.EQ.1 ) THEN
         CALL DLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV+1, 1,
     $                D( LSV ), N, INFO )
         CALL DLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV, 1,
     $                E( LSV ),
     $                N, INFO )
      ELSE IF( ISCALE.EQ.2 ) THEN
         CALL DLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV+1, 1,
     $                D( LSV ), N, INFO )
         CALL DLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV, 1,
     $                E( LSV ),
     $                N, INFO )
      END IF
*
*     Check for no convergence to an eigenvalue after a total
*     of N*MAXIT iterations.
*
      IF( JTOT.LT.NMAXIT )
     $   GO TO 10
      DO 150 I = 1, N - 1
         IF( E( I ).NE.ZERO )
     $      INFO = INFO + 1
  150 CONTINUE
      GO TO 190
*
*     Order eigenvalues and eigenvectors.
*
  160 CONTINUE
      IF( ICOMPZ.EQ.0 ) THEN
*
*        Use Quick Sort
*
         CALL DLASRT( 'I', N, D, INFO )
*
      ELSE
*
*        Use Selection Sort to minimize swaps of eigenvectors
*
         DO 180 II = 2, N
            I = II - 1
            K = I
            P = D( I )
            DO 170 J = II, N
               IF( D( J ).LT.P ) THEN
                  K = J
                  P = D( J )
               END IF
  170       CONTINUE
            IF( K.NE.I ) THEN
               D( K ) = D( I )
               D( I ) = P
               CALL DSWAP( N, Z( 1, I ), 1, Z( 1, K ), 1 )
            END IF
  180    CONTINUE
      END IF
*
  190 CONTINUE
      RETURN
*
*     End of DSTEQR
*
      END
*> \brief \b DSTERF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTERF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsterf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsterf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsterf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSTERF( N, D, E, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), E( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSTERF computes all eigenvalues of a symmetric tridiagonal matrix
*> using the Pal-Walker-Kahan variant of the QL or QR algorithm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, the n diagonal elements of the tridiagonal matrix.
*>          On exit, if INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          On entry, the (n-1) subdiagonal elements of the tridiagonal
*>          matrix.
*>          On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  the algorithm failed to find all of the eigenvalues in
*>                a total of 30*N iterations; if INFO = i, then i
*>                elements of E have not converged to zero.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup sterf
*
*  =====================================================================
      SUBROUTINE DSTERF( N, D, E, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), E( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO, THREE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
     $                   THREE = 3.0D0 )
      INTEGER            MAXIT
      PARAMETER          ( MAXIT = 30 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, ISCALE, JTOT, L, L1, LEND, LENDSV, LSV, M,
     $                   NMAXIT
      DOUBLE PRECISION   ALPHA, ANORM, BB, C, EPS, EPS2, GAMMA, OLDC,
     $                   OLDGAM, P, R, RT1, RT2, RTE, S, SAFMAX, SAFMIN,
     $                   SIGMA, SSFMAX, SSFMIN, RMAX
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH, DLANST, DLAPY2
      EXTERNAL           DLAMCH, DLANST, DLAPY2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLAE2, DLASCL, DLASRT, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, SIGN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
*     Quick return if possible
*
      IF( N.LT.0 ) THEN
         INFO = -1
         CALL XERBLA( 'DSTERF', -INFO )
         RETURN
      END IF
      IF( N.LE.1 )
     $   RETURN
*
*     Determine the unit roundoff for this environment.
*
      EPS = DLAMCH( 'E' )
      EPS2 = EPS**2
      SAFMIN = DLAMCH( 'S' )
      SAFMAX = ONE / SAFMIN
      SSFMAX = SQRT( SAFMAX ) / THREE
      SSFMIN = SQRT( SAFMIN ) / EPS2
      RMAX = DLAMCH( 'O' )
*
*     Compute the eigenvalues of the tridiagonal matrix.
*
      NMAXIT = N*MAXIT
      SIGMA = ZERO
      JTOT = 0
*
*     Determine where the matrix splits and choose QL or QR iteration
*     for each block, according to whether top or bottom diagonal
*     element is smaller.
*
      L1 = 1
*
   10 CONTINUE
      IF( L1.GT.N )
     $   GO TO 170
      IF( L1.GT.1 )
     $   E( L1-1 ) = ZERO
      DO 20 M = L1, N - 1
         IF( ABS( E( M ) ).LE.( SQRT( ABS( D( M ) ) )*SQRT( ABS( D( M+
     $       1 ) ) ) )*EPS ) THEN
            E( M ) = ZERO
            GO TO 30
         END IF
   20 CONTINUE
      M = N
*
   30 CONTINUE
      L = L1
      LSV = L
      LEND = M
      LENDSV = LEND
      L1 = M + 1
      IF( LEND.EQ.L )
     $   GO TO 10
*
*     Scale submatrix in rows and columns L to LEND
*
      ANORM = DLANST( 'M', LEND-L+1, D( L ), E( L ) )
      ISCALE = 0
      IF( ANORM.EQ.ZERO )
     $   GO TO 10
      IF( (ANORM.GT.SSFMAX) ) THEN
         ISCALE = 1
         CALL DLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L+1, 1, D( L ),
     $                N,
     $                INFO )
         CALL DLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L, 1, E( L ), N,
     $                INFO )
      ELSE IF( ANORM.LT.SSFMIN ) THEN
         ISCALE = 2
         CALL DLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L+1, 1, D( L ),
     $                N,
     $                INFO )
         CALL DLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L, 1, E( L ), N,
     $                INFO )
      END IF
*
      DO 40 I = L, LEND - 1
         E( I ) = E( I )**2
   40 CONTINUE
*
*     Choose between QL and QR iteration
*
      IF( ABS( D( LEND ) ).LT.ABS( D( L ) ) ) THEN
         LEND = LSV
         L = LENDSV
      END IF
*
      IF( LEND.GE.L ) THEN
*
*        QL Iteration
*
*        Look for small subdiagonal element.
*
   50    CONTINUE
         IF( L.NE.LEND ) THEN
            DO 60 M = L, LEND - 1
               IF( ABS( E( M ) ).LE.EPS2*ABS( D( M )*D( M+1 ) ) )
     $            GO TO 70
   60       CONTINUE
         END IF
         M = LEND
*
   70    CONTINUE
         IF( M.LT.LEND )
     $      E( M ) = ZERO
         P = D( L )
         IF( M.EQ.L )
     $      GO TO 90
*
*        If remaining matrix is 2 by 2, use DLAE2 to compute its
*        eigenvalues.
*
         IF( M.EQ.L+1 ) THEN
            RTE = SQRT( E( L ) )
            CALL DLAE2( D( L ), RTE, D( L+1 ), RT1, RT2 )
            D( L ) = RT1
            D( L+1 ) = RT2
            E( L ) = ZERO
            L = L + 2
            IF( L.LE.LEND )
     $         GO TO 50
            GO TO 150
         END IF
*
         IF( JTOT.EQ.NMAXIT )
     $      GO TO 150
         JTOT = JTOT + 1
*
*        Form shift.
*
         RTE = SQRT( E( L ) )
         SIGMA = ( D( L+1 )-P ) / ( TWO*RTE )
         R = DLAPY2( SIGMA, ONE )
         SIGMA = P - ( RTE / ( SIGMA+SIGN( R, SIGMA ) ) )
*
         C = ONE
         S = ZERO
         GAMMA = D( M ) - SIGMA
         P = GAMMA*GAMMA
*
*        Inner loop
*
         DO 80 I = M - 1, L, -1
            BB = E( I )
            R = P + BB
            IF( I.NE.M-1 )
     $         E( I+1 ) = S*R
            OLDC = C
            C = P / R
            S = BB / R
            OLDGAM = GAMMA
            ALPHA = D( I )
            GAMMA = C*( ALPHA-SIGMA ) - S*OLDGAM
            D( I+1 ) = OLDGAM + ( ALPHA-GAMMA )
            IF( C.NE.ZERO ) THEN
               P = ( GAMMA*GAMMA ) / C
            ELSE
               P = OLDC*BB
            END IF
   80    CONTINUE
*
         E( L ) = S*P
         D( L ) = SIGMA + GAMMA
         GO TO 50
*
*        Eigenvalue found.
*
   90    CONTINUE
         D( L ) = P
*
         L = L + 1
         IF( L.LE.LEND )
     $      GO TO 50
         GO TO 150
*
      ELSE
*
*        QR Iteration
*
*        Look for small superdiagonal element.
*
  100    CONTINUE
         DO 110 M = L, LEND + 1, -1
            IF( ABS( E( M-1 ) ).LE.EPS2*ABS( D( M )*D( M-1 ) ) )
     $         GO TO 120
  110    CONTINUE
         M = LEND
*
  120    CONTINUE
         IF( M.GT.LEND )
     $      E( M-1 ) = ZERO
         P = D( L )
         IF( M.EQ.L )
     $      GO TO 140
*
*        If remaining matrix is 2 by 2, use DLAE2 to compute its
*        eigenvalues.
*
         IF( M.EQ.L-1 ) THEN
            RTE = SQRT( E( L-1 ) )
            CALL DLAE2( D( L ), RTE, D( L-1 ), RT1, RT2 )
            D( L ) = RT1
            D( L-1 ) = RT2
            E( L-1 ) = ZERO
            L = L - 2
            IF( L.GE.LEND )
     $         GO TO 100
            GO TO 150
         END IF
*
         IF( JTOT.EQ.NMAXIT )
     $      GO TO 150
         JTOT = JTOT + 1
*
*        Form shift.
*
         RTE = SQRT( E( L-1 ) )
         SIGMA = ( D( L-1 )-P ) / ( TWO*RTE )
         R = DLAPY2( SIGMA, ONE )
         SIGMA = P - ( RTE / ( SIGMA+SIGN( R, SIGMA ) ) )
*
         C = ONE
         S = ZERO
         GAMMA = D( M ) - SIGMA
         P = GAMMA*GAMMA
*
*        Inner loop
*
         DO 130 I = M, L - 1
            BB = E( I )
            R = P + BB
            IF( I.NE.M )
     $         E( I-1 ) = S*R
            OLDC = C
            C = P / R
            S = BB / R
            OLDGAM = GAMMA
            ALPHA = D( I+1 )
            GAMMA = C*( ALPHA-SIGMA ) - S*OLDGAM
            D( I ) = OLDGAM + ( ALPHA-GAMMA )
            IF( C.NE.ZERO ) THEN
               P = ( GAMMA*GAMMA ) / C
            ELSE
               P = OLDC*BB
            END IF
  130    CONTINUE
*
         E( L-1 ) = S*P
         D( L ) = SIGMA + GAMMA
         GO TO 100
*
*        Eigenvalue found.
*
  140    CONTINUE
         D( L ) = P
*
         L = L - 1
         IF( L.GE.LEND )
     $      GO TO 100
         GO TO 150
*
      END IF
*
*     Undo scaling if necessary
*
  150 CONTINUE
      IF( ISCALE.EQ.1 )
     $   CALL DLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV+1, 1,
     $                D( LSV ), N, INFO )
      IF( ISCALE.EQ.2 )
     $   CALL DLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV+1, 1,
     $                D( LSV ), N, INFO )
*
*     Check for no convergence to an eigenvalue after a total
*     of N*MAXIT iterations.
*
      IF( JTOT.LT.NMAXIT )
     $   GO TO 10
      DO 160 I = 1, N - 1
         IF( E( I ).NE.ZERO )
     $      INFO = INFO + 1
  160 CONTINUE
      GO TO 180
*
*     Sort eigenvalues in increasing order.
*
  170 CONTINUE
      CALL DLASRT( 'I', N, D, INFO )
*
  180 CONTINUE
      RETURN
*
*     End of DSTERF
*
      END
*> \brief <b> DSTEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstev.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstev.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstev.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSTEV( JOBZ, N, D, E, Z, LDZ, WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ
*       INTEGER            INFO, LDZ, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), E( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSTEV computes all eigenvalues and, optionally, eigenvectors of a
*> real symmetric tridiagonal matrix A.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, the n diagonal elements of the tridiagonal matrix
*>          A.
*>          On exit, if INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          On entry, the (n-1) subdiagonal elements of the tridiagonal
*>          matrix A, stored in elements 1 to N-1 of E.
*>          On exit, the contents of E are destroyed.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
*>          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
*>          eigenvectors of the matrix A, with the i-th column of Z
*>          holding the eigenvector associated with D(i).
*>          If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (max(1,2*N-2))
*>          If JOBZ = 'N', WORK is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the algorithm failed to converge; i
*>                off-diagonal elements of E did not converge to zero.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup stev
*
*  =====================================================================
      SUBROUTINE DSTEV( JOBZ, N, D, E, Z, LDZ, WORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ
      INTEGER            INFO, LDZ, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), E( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            WANTZ
      INTEGER            IMAX, ISCALE
      DOUBLE PRECISION   BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM,
     $                   TNRM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANST
      EXTERNAL           LSAME, DLAMCH, DLANST
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL, DSTEQR, DSTERF, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
*
      INFO = 0
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
         INFO = -6
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSTEV ', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( N.EQ.1 ) THEN
         IF( WANTZ )
     $      Z( 1, 1 ) = ONE
         RETURN
      END IF
*
*     Get machine constants.
*
      SAFMIN = DLAMCH( 'Safe minimum' )
      EPS = DLAMCH( 'Precision' )
      SMLNUM = SAFMIN / EPS
      BIGNUM = ONE / SMLNUM
      RMIN = SQRT( SMLNUM )
      RMAX = SQRT( BIGNUM )
*
*     Scale matrix to allowable range, if necessary.
*
      ISCALE = 0
      TNRM = DLANST( 'M', N, D, E )
      IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
         ISCALE = 1
         SIGMA = RMIN / TNRM
      ELSE IF( TNRM.GT.RMAX ) THEN
         ISCALE = 1
         SIGMA = RMAX / TNRM
      END IF
      IF( ISCALE.EQ.1 ) THEN
         CALL DSCAL( N, SIGMA, D, 1 )
         CALL DSCAL( N-1, SIGMA, E( 1 ), 1 )
      END IF
*
*     For eigenvalues only, call DSTERF.  For eigenvalues and
*     eigenvectors, call DSTEQR.
*
      IF( .NOT.WANTZ ) THEN
         CALL DSTERF( N, D, E, INFO )
      ELSE
         CALL DSTEQR( 'I', N, D, E, Z, LDZ, WORK, INFO )
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
      IF( ISCALE.EQ.1 ) THEN
         IF( INFO.EQ.0 ) THEN
            IMAX = N
         ELSE
            IMAX = INFO - 1
         END IF
         CALL DSCAL( IMAX, ONE / SIGMA, D, 1 )
      END IF
*
      RETURN
*
*     End of DSTEV
*
      END
*> \brief <b> DSTEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstevd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstevd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstevd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSTEVD( JOBZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK,
*                          LIWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ
*       INTEGER            INFO, LDZ, LIWORK, LWORK, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   D( * ), E( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSTEVD computes all eigenvalues and, optionally, eigenvectors of a
*> real symmetric tridiagonal matrix. If eigenvectors are desired, it
*> uses a divide and conquer algorithm.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, the n diagonal elements of the tridiagonal matrix
*>          A.
*>          On exit, if INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          On entry, the (n-1) subdiagonal elements of the tridiagonal
*>          matrix A, stored in elements 1 to N-1 of E.
*>          On exit, the contents of E are destroyed.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
*>          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
*>          eigenvectors of the matrix A, with the i-th column of Z
*>          holding the eigenvector associated with D(i).
*>          If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array,
*>                                         dimension (LWORK)
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If JOBZ  = 'N' or N <= 1 then LWORK must be at least 1.
*>          If JOBZ  = 'V' and N > 1 then LWORK must be at least
*>                         ( 1 + 4*N + N**2 ).
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal sizes of the WORK and IWORK
*>          arrays, returns these values as the first entries of the WORK
*>          and IWORK arrays, and no error message related to LWORK or
*>          LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK.
*>          If JOBZ  = 'N' or N <= 1 then LIWORK must be at least 1.
*>          If JOBZ  = 'V' and N > 1 then LIWORK must be at least 3+5*N.
*>
*>          If LIWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the optimal sizes of the WORK and
*>          IWORK arrays, returns these values as the first entries of
*>          the WORK and IWORK arrays, and no error message related to
*>          LWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the algorithm failed to converge; i
*>                off-diagonal elements of E did not converge to zero.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup stevd
*
*  =====================================================================
      SUBROUTINE DSTEVD( JOBZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK,
     $                   LIWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ
      INTEGER            INFO, LDZ, LIWORK, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   D( * ), E( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, WANTZ
      INTEGER            ISCALE, LIWMIN, LWMIN
      DOUBLE PRECISION   BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM,
     $                   TNRM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANST
      EXTERNAL           LSAME, DLAMCH, DLANST
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL, DSTEDC, DSTERF, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
      LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
      INFO = 0
      LIWMIN = 1
      LWMIN = 1
      IF( N.GT.1 .AND. WANTZ ) THEN
         LWMIN = 1 + 4*N + N**2
         LIWMIN = 3 + 5*N
      END IF
*
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
         INFO = -6
      END IF
*
      IF( INFO.EQ.0 ) THEN
         WORK( 1 ) = LWMIN
         IWORK( 1 ) = LIWMIN
*
         IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -8
         ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -10
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSTEVD', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( N.EQ.1 ) THEN
         IF( WANTZ )
     $      Z( 1, 1 ) = ONE
         RETURN
      END IF
*
*     Get machine constants.
*
      SAFMIN = DLAMCH( 'Safe minimum' )
      EPS = DLAMCH( 'Precision' )
      SMLNUM = SAFMIN / EPS
      BIGNUM = ONE / SMLNUM
      RMIN = SQRT( SMLNUM )
      RMAX = SQRT( BIGNUM )
*
*     Scale matrix to allowable range, if necessary.
*
      ISCALE = 0
      TNRM = DLANST( 'M', N, D, E )
      IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
         ISCALE = 1
         SIGMA = RMIN / TNRM
      ELSE IF( TNRM.GT.RMAX ) THEN
         ISCALE = 1
         SIGMA = RMAX / TNRM
      END IF
      IF( ISCALE.EQ.1 ) THEN
         CALL DSCAL( N, SIGMA, D, 1 )
         CALL DSCAL( N-1, SIGMA, E( 1 ), 1 )
      END IF
*
*     For eigenvalues only, call DSTERF.  For eigenvalues and
*     eigenvectors, call DSTEDC.
*
      IF( .NOT.WANTZ ) THEN
         CALL DSTERF( N, D, E, INFO )
      ELSE
         CALL DSTEDC( 'I', N, D, E, Z, LDZ, WORK, LWORK, IWORK,
     $                LIWORK,
     $                INFO )
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
      IF( ISCALE.EQ.1 )
     $   CALL DSCAL( N, ONE / SIGMA, D, 1 )
*
      WORK( 1 ) = LWMIN
      IWORK( 1 ) = LIWMIN
*
      RETURN
*
*     End of DSTEVD
*
      END
*> \brief <b> DSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEVR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstevr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstevr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstevr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
*                          M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK,
*                          LIWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, RANGE
*       INTEGER            IL, INFO, IU, LDZ, LIWORK, LWORK, M, N
*       DOUBLE PRECISION   ABSTOL, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            ISUPPZ( * ), IWORK( * )
*       DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSTEVR computes selected eigenvalues and, optionally, eigenvectors
*> of a real symmetric tridiagonal matrix T.  Eigenvalues and
*> eigenvectors can be selected by specifying either a range of values
*> or a range of indices for the desired eigenvalues.
*>
*> Whenever possible, DSTEVR calls DSTEMR to compute the
*> eigenspectrum using Relatively Robust Representations.  DSTEMR
*> computes eigenvalues by the dqds algorithm, while orthogonal
*> eigenvectors are computed from various "good" L D L^T representations
*> (also known as Relatively Robust Representations). Gram-Schmidt
*> orthogonalization is avoided as far as possible. More specifically,
*> the various steps of the algorithm are as follows. For the i-th
*> unreduced block of T,
*>    (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
*>         is a relatively robust representation,
*>    (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
*>        relative accuracy by the dqds algorithm,
*>    (c) If there is a cluster of close eigenvalues, "choose" sigma_i
*>        close to the cluster, and go to step (a),
*>    (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
*>        compute the corresponding eigenvector by forming a
*>        rank-revealing twisted factorization.
*> The desired accuracy of the output can be specified by the input
*> parameter ABSTOL.
*>
*> For more details, see "A new O(n^2) algorithm for the symmetric
*> tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon,
*> Computer Science Division Technical Report No. UCB//CSD-97-971,
*> UC Berkeley, May 1997.
*>
*>
*> Note 1 : DSTEVR calls DSTEMR when the full spectrum is requested
*> on machines which conform to the ieee-754 floating point standard.
*> DSTEVR calls DSTEBZ and DSTEIN on non-ieee machines and
*> when partial spectrum requests are made.
*>
*> Normal execution of DSTEMR may create NaNs and infinities and
*> hence may abort due to a floating point exception in environments
*> which do not handle NaNs and infinities in the ieee standard default
*> manner.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*>          RANGE is CHARACTER*1
*>          = 'A': all eigenvalues will be found.
*>          = 'V': all eigenvalues in the half-open interval (VL,VU]
*>                 will be found.
*>          = 'I': the IL-th through IU-th eigenvalues will be found.
*>          For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
*>          DSTEIN are called
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, the n diagonal elements of the tridiagonal matrix
*>          A.
*>          On exit, D may be multiplied by a constant factor chosen
*>          to avoid over/underflow in computing the eigenvalues.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (max(1,N-1))
*>          On entry, the (n-1) subdiagonal elements of the tridiagonal
*>          matrix A in elements 1 to N-1 of E.
*>          On exit, E may be multiplied by a constant factor chosen
*>          to avoid over/underflow in computing the eigenvalues.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is DOUBLE PRECISION
*>          If RANGE='V', the lower bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*>          VU is DOUBLE PRECISION
*>          If RANGE='V', the upper bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*>          IL is INTEGER
*>          If RANGE='I', the index of the
*>          smallest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*>          IU is INTEGER
*>          If RANGE='I', the index of the
*>          largest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*>          ABSTOL is DOUBLE PRECISION
*>          The absolute error tolerance for the eigenvalues.
*>          An approximate eigenvalue is accepted as converged
*>          when it is determined to lie in an interval [a,b]
*>          of width less than or equal to
*>
*>                  ABSTOL + EPS *   max( |a|,|b| ) ,
*>
*>          where EPS is the machine precision.  If ABSTOL is less than
*>          or equal to zero, then  EPS*|T|  will be used in its place,
*>          where |T| is the 1-norm of the tridiagonal matrix obtained
*>          by reducing A to tridiagonal form.
*>
*>          See "Computing Small Singular Values of Bidiagonal Matrices
*>          with Guaranteed High Relative Accuracy," by Demmel and
*>          Kahan, LAPACK Working Note #3.
*>
*>          If high relative accuracy is important, set ABSTOL to
*>          DLAMCH( 'Safe minimum' ).  Doing so will guarantee that
*>          eigenvalues are computed to high relative accuracy when
*>          possible in future releases.  The current code does not
*>          make any guarantees about high relative accuracy, but
*>          future releases will. See J. Barlow and J. Demmel,
*>          "Computing Accurate Eigensystems of Scaled Diagonally
*>          Dominant Matrices", LAPACK Working Note #7, for a discussion
*>          of which matrices define their eigenvalues to high relative
*>          accuracy.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The total number of eigenvalues found.  0 <= M <= N.
*>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          The first M elements contain the selected eigenvalues in
*>          ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
*>          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*>          contain the orthonormal eigenvectors of the matrix A
*>          corresponding to the selected eigenvalues, with the i-th
*>          column of Z holding the eigenvector associated with W(i).
*>          Note: the user must ensure that at least max(1,M) columns are
*>          supplied in the array Z; if RANGE = 'V', the exact value of M
*>          is not known in advance and an upper bound must be used.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] ISUPPZ
*> \verbatim
*>          ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
*>          The support of the eigenvectors in Z, i.e., the indices
*>          indicating the nonzero elements in Z. The i-th eigenvector
*>          is nonzero only in elements ISUPPZ( 2*i-1 ) through
*>          ISUPPZ( 2*i ).
*>          Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal (and
*>          minimal) LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.  LWORK >= max(1,20*N).
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal sizes of the WORK and IWORK
*>          arrays, returns these values as the first entries of the WORK
*>          and IWORK arrays, and no error message related to LWORK or
*>          LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*>          On exit, if INFO = 0, IWORK(1) returns the optimal (and
*>          minimal) LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK.  LIWORK >= max(1,10*N).
*>
*>          If LIWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the optimal sizes of the WORK and
*>          IWORK arrays, returns these values as the first entries of
*>          the WORK and IWORK arrays, and no error message related to
*>          LWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  Internal error
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup stevr
*
*> \par Contributors:
*  ==================
*>
*>     Inderjit Dhillon, IBM Almaden, USA \n
*>     Osni Marques, LBNL/NERSC, USA \n
*>     Ken Stanley, Computer Science Division, University of
*>       California at Berkeley, USA \n
*>
*  =====================================================================
      SUBROUTINE DSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
     $                   ABSTOL,
     $                   M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK,
     $                   LIWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, RANGE
      INTEGER            IL, INFO, IU, LDZ, LIWORK, LWORK, M, N
      DOUBLE PRECISION   ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            ISUPPZ( * ), IWORK( * )
      DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLEIG, INDEIG, TEST, LQUERY, VALEIG, WANTZ,
     $                   TRYRAC
      CHARACTER          ORDER
      INTEGER            I, IEEEOK, IMAX, INDIBL, INDIFL, INDISP,
     $                   INDIWO, ISCALE, ITMP1, J, JJ, LIWMIN, LWMIN,
     $                   NSPLIT
      DOUBLE PRECISION   BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM,
     $                   TMP1, TNRM, VLL, VUU
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANST
      EXTERNAL           LSAME, ILAENV, DLAMCH, DLANST
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DSCAL, DSTEBZ, DSTEMR, DSTEIN,
     $                   DSTERF,
     $                   DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*
*     Test the input parameters.
*
      IEEEOK = ILAENV( 10, 'DSTEVR', 'N', 1, 2, 3, 4 )
*
      WANTZ = LSAME( JOBZ, 'V' )
      ALLEIG = LSAME( RANGE, 'A' )
      VALEIG = LSAME( RANGE, 'V' )
      INDEIG = LSAME( RANGE, 'I' )
*
      LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) )
      LWMIN = MAX( 1, 20*N )
      LIWMIN = MAX( 1, 10*N )
*
*
      INFO = 0
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE
         IF( VALEIG ) THEN
            IF( N.GT.0 .AND. VU.LE.VL )
     $         INFO = -7
         ELSE IF( INDEIG ) THEN
            IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
               INFO = -8
            ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
               INFO = -9
            END IF
         END IF
      END IF
      IF( INFO.EQ.0 ) THEN
         IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
            INFO = -14
         END IF
      END IF
*
      IF( INFO.EQ.0 ) THEN
         WORK( 1 ) = LWMIN
         IWORK( 1 ) = LIWMIN
*
         IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -17
         ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -19
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSTEVR', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      M = 0
      IF( N.EQ.0 )
     $   RETURN
*
      IF( N.EQ.1 ) THEN
         IF( ALLEIG .OR. INDEIG ) THEN
            M = 1
            W( 1 ) = D( 1 )
         ELSE
            IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN
               M = 1
               W( 1 ) = D( 1 )
            END IF
         END IF
         IF( WANTZ )
     $      Z( 1, 1 ) = ONE
         RETURN
      END IF
*
*     Get machine constants.
*
      SAFMIN = DLAMCH( 'Safe minimum' )
      EPS = DLAMCH( 'Precision' )
      SMLNUM = SAFMIN / EPS
      BIGNUM = ONE / SMLNUM
      RMIN = SQRT( SMLNUM )
      RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
*
*     Scale matrix to allowable range, if necessary.
*
      ISCALE = 0
      IF( VALEIG ) THEN
         VLL = VL
         VUU = VU
      END IF
*
      TNRM = DLANST( 'M', N, D, E )
      IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
         ISCALE = 1
         SIGMA = RMIN / TNRM
      ELSE IF( TNRM.GT.RMAX ) THEN
         ISCALE = 1
         SIGMA = RMAX / TNRM
      END IF
      IF( ISCALE.EQ.1 ) THEN
         CALL DSCAL( N, SIGMA, D, 1 )
         CALL DSCAL( N-1, SIGMA, E( 1 ), 1 )
         IF( VALEIG ) THEN
            VLL = VL*SIGMA
            VUU = VU*SIGMA
         END IF
      END IF

*     Initialize indices into workspaces.  Note: These indices are used only
*     if DSTERF or DSTEMR fail.

*     IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and
*     stores the block indices of each of the M<=N eigenvalues.
      INDIBL = 1
*     IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and
*     stores the starting and finishing indices of each block.
      INDISP = INDIBL + N
*     IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
*     that corresponding to eigenvectors that fail to converge in
*     DSTEIN.  This information is discarded; if any fail, the driver
*     returns INFO > 0.
      INDIFL = INDISP + N
*     INDIWO is the offset of the remaining integer workspace.
      INDIWO = INDISP + N
*
*     If all eigenvalues are desired, then
*     call DSTERF or DSTEMR.  If this fails for some eigenvalue, then
*     try DSTEBZ.
*
*
      TEST = .FALSE.
      IF( INDEIG ) THEN
         IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
            TEST = .TRUE.
         END IF
      END IF
      IF( ( ALLEIG .OR. TEST ) .AND. IEEEOK.EQ.1 ) THEN
         CALL DCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 )
         IF( .NOT.WANTZ ) THEN
            CALL DCOPY( N, D, 1, W, 1 )
            CALL DSTERF( N, W, WORK, INFO )
         ELSE
            CALL DCOPY( N, D, 1, WORK( N+1 ), 1 )
            IF (ABSTOL .LE. TWO*N*EPS) THEN
               TRYRAC = .TRUE.
            ELSE
               TRYRAC = .FALSE.
            END IF
            CALL DSTEMR( JOBZ, 'A', N, WORK( N+1 ), WORK, VL, VU, IL,
     $                   IU, M, W, Z, LDZ, N, ISUPPZ, TRYRAC,
     $                   WORK( 2*N+1 ), LWORK-2*N, IWORK, LIWORK, INFO )
*
         END IF
         IF( INFO.EQ.0 ) THEN
            M = N
            GO TO 10
         END IF
         INFO = 0
      END IF
*
*     Otherwise, call DSTEBZ and, if eigenvectors are desired, DSTEIN.
*
      IF( WANTZ ) THEN
         ORDER = 'B'
      ELSE
         ORDER = 'E'
      END IF

      CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E,
     $             M,
     $             NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), WORK,
     $             IWORK( INDIWO ), INFO )
*
      IF( WANTZ ) THEN
         CALL DSTEIN( N, D, E, M, W, IWORK( INDIBL ),
     $                IWORK( INDISP ),
     $                Z, LDZ, WORK, IWORK( INDIWO ), IWORK( INDIFL ),
     $                INFO )
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
   10 CONTINUE
      IF( ISCALE.EQ.1 ) THEN
         IF( INFO.EQ.0 ) THEN
            IMAX = M
         ELSE
            IMAX = INFO - 1
         END IF
         CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
      END IF
*
*     If eigenvalues are not in order, then sort them, along with
*     eigenvectors.
*
      IF( WANTZ ) THEN
         DO 30 J = 1, M - 1
            I = 0
            TMP1 = W( J )
            DO 20 JJ = J + 1, M
               IF( W( JJ ).LT.TMP1 ) THEN
                  I = JJ
                  TMP1 = W( JJ )
               END IF
   20       CONTINUE
*
            IF( I.NE.0 ) THEN
               ITMP1 = IWORK( I )
               W( I ) = W( J )
               IWORK( I ) = IWORK( J )
               W( J ) = TMP1
               IWORK( J ) = ITMP1
               CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
            END IF
   30    CONTINUE
      END IF
*
*      Causes problems with tests 19 & 20:
*      IF (wantz .and. INDEIG ) Z( 1,1) = Z(1,1) / 1.002 + .002
*
*
      WORK( 1 ) = LWMIN
      IWORK( 1 ) = LIWMIN
      RETURN
*
*     End of DSTEVR
*
      END
*> \brief <b> DSTEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstevx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstevx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstevx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSTEVX( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
*                          M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, RANGE
*       INTEGER            IL, INFO, IU, LDZ, M, N
*       DOUBLE PRECISION   ABSTOL, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            IFAIL( * ), IWORK( * )
*       DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSTEVX computes selected eigenvalues and, optionally, eigenvectors
*> of a real symmetric tridiagonal matrix A.  Eigenvalues and
*> eigenvectors can be selected by specifying either a range of values
*> or a range of indices for the desired eigenvalues.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*>          RANGE is CHARACTER*1
*>          = 'A': all eigenvalues will be found.
*>          = 'V': all eigenvalues in the half-open interval (VL,VU]
*>                 will be found.
*>          = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, the n diagonal elements of the tridiagonal matrix
*>          A.
*>          On exit, D may be multiplied by a constant factor chosen
*>          to avoid over/underflow in computing the eigenvalues.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (max(1,N-1))
*>          On entry, the (n-1) subdiagonal elements of the tridiagonal
*>          matrix A in elements 1 to N-1 of E.
*>          On exit, E may be multiplied by a constant factor chosen
*>          to avoid over/underflow in computing the eigenvalues.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is DOUBLE PRECISION
*>          If RANGE='V', the lower bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*>          VU is DOUBLE PRECISION
*>          If RANGE='V', the upper bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*>          IL is INTEGER
*>          If RANGE='I', the index of the
*>          smallest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*>          IU is INTEGER
*>          If RANGE='I', the index of the
*>          largest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*>          ABSTOL is DOUBLE PRECISION
*>          The absolute error tolerance for the eigenvalues.
*>          An approximate eigenvalue is accepted as converged
*>          when it is determined to lie in an interval [a,b]
*>          of width less than or equal to
*>
*>                  ABSTOL + EPS *   max( |a|,|b| ) ,
*>
*>          where EPS is the machine precision.  If ABSTOL is less
*>          than or equal to zero, then  EPS*|T|  will be used in
*>          its place, where |T| is the 1-norm of the tridiagonal
*>          matrix.
*>
*>          Eigenvalues will be computed most accurately when ABSTOL is
*>          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*>          If this routine returns with INFO>0, indicating that some
*>          eigenvectors did not converge, try setting ABSTOL to
*>          2*DLAMCH('S').
*>
*>          See "Computing Small Singular Values of Bidiagonal Matrices
*>          with Guaranteed High Relative Accuracy," by Demmel and
*>          Kahan, LAPACK Working Note #3.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The total number of eigenvalues found.  0 <= M <= N.
*>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          The first M elements contain the selected eigenvalues in
*>          ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
*>          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*>          contain the orthonormal eigenvectors of the matrix A
*>          corresponding to the selected eigenvalues, with the i-th
*>          column of Z holding the eigenvector associated with W(i).
*>          If an eigenvector fails to converge (INFO > 0), then that
*>          column of Z contains the latest approximation to the
*>          eigenvector, and the index of the eigenvector is returned
*>          in IFAIL.  If JOBZ = 'N', then Z is not referenced.
*>          Note: the user must ensure that at least max(1,M) columns are
*>          supplied in the array Z; if RANGE = 'V', the exact value of M
*>          is not known in advance and an upper bound must be used.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*>          IFAIL is INTEGER array, dimension (N)
*>          If JOBZ = 'V', then if INFO = 0, the first M elements of
*>          IFAIL are zero.  If INFO > 0, then IFAIL contains the
*>          indices of the eigenvectors that failed to converge.
*>          If JOBZ = 'N', then IFAIL is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, then i eigenvectors failed to converge.
*>                Their indices are stored in array IFAIL.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup stevx
*
*  =====================================================================
      SUBROUTINE DSTEVX( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
     $                   ABSTOL,
     $                   M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, RANGE
      INTEGER            IL, INFO, IU, LDZ, M, N
      DOUBLE PRECISION   ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            IFAIL( * ), IWORK( * )
      DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLEIG, INDEIG, TEST, VALEIG, WANTZ
      CHARACTER          ORDER
      INTEGER            I, IMAX, INDISP, INDIWO, INDWRK,
     $                   ISCALE, ITMP1, J, JJ, NSPLIT
      DOUBLE PRECISION   BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM,
     $                   TMP1, TNRM, VLL, VUU
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANST
      EXTERNAL           LSAME, DLAMCH, DLANST
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DSCAL, DSTEBZ, DSTEIN, DSTEQR,
     $                   DSTERF,
     $                   DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
      ALLEIG = LSAME( RANGE, 'A' )
      VALEIG = LSAME( RANGE, 'V' )
      INDEIG = LSAME( RANGE, 'I' )
*
      INFO = 0
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE
         IF( VALEIG ) THEN
            IF( N.GT.0 .AND. VU.LE.VL )
     $         INFO = -7
         ELSE IF( INDEIG ) THEN
            IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
               INFO = -8
            ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
               INFO = -9
            END IF
         END IF
      END IF
      IF( INFO.EQ.0 ) THEN
         IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
     $      INFO = -14
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSTEVX', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      M = 0
      IF( N.EQ.0 )
     $   RETURN
*
      IF( N.EQ.1 ) THEN
         IF( ALLEIG .OR. INDEIG ) THEN
            M = 1
            W( 1 ) = D( 1 )
         ELSE
            IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN
               M = 1
               W( 1 ) = D( 1 )
            END IF
         END IF
         IF( WANTZ )
     $      Z( 1, 1 ) = ONE
         RETURN
      END IF
*
*     Get machine constants.
*
      SAFMIN = DLAMCH( 'Safe minimum' )
      EPS = DLAMCH( 'Precision' )
      SMLNUM = SAFMIN / EPS
      BIGNUM = ONE / SMLNUM
      RMIN = SQRT( SMLNUM )
      RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
*     Scale matrix to allowable range, if necessary.
*
      ISCALE = 0
      IF( VALEIG ) THEN
         VLL = VL
         VUU = VU
      ELSE
         VLL = ZERO
         VUU = ZERO
      END IF
      TNRM = DLANST( 'M', N, D, E )
      IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
         ISCALE = 1
         SIGMA = RMIN / TNRM
      ELSE IF( TNRM.GT.RMAX ) THEN
         ISCALE = 1
         SIGMA = RMAX / TNRM
      END IF
      IF( ISCALE.EQ.1 ) THEN
         CALL DSCAL( N, SIGMA, D, 1 )
         CALL DSCAL( N-1, SIGMA, E( 1 ), 1 )
         IF( VALEIG ) THEN
            VLL = VL*SIGMA
            VUU = VU*SIGMA
         END IF
      END IF
*
*     If all eigenvalues are desired and ABSTOL is less than zero, then
*     call DSTERF or SSTEQR.  If this fails for some eigenvalue, then
*     try DSTEBZ.
*
      TEST = .FALSE.
      IF( INDEIG ) THEN
         IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
            TEST = .TRUE.
         END IF
      END IF
      IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
         CALL DCOPY( N, D, 1, W, 1 )
         CALL DCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 )
         INDWRK = N + 1
         IF( .NOT.WANTZ ) THEN
            CALL DSTERF( N, W, WORK, INFO )
         ELSE
            CALL DSTEQR( 'I', N, W, WORK, Z, LDZ, WORK( INDWRK ),
     $                   INFO )
            IF( INFO.EQ.0 ) THEN
               DO 10 I = 1, N
                  IFAIL( I ) = 0
   10          CONTINUE
            END IF
         END IF
         IF( INFO.EQ.0 ) THEN
            M = N
            GO TO 20
         END IF
         INFO = 0
      END IF
*
*     Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
*
      IF( WANTZ ) THEN
         ORDER = 'B'
      ELSE
         ORDER = 'E'
      END IF
      INDWRK = 1
      INDISP = 1 + N
      INDIWO = INDISP + N
      CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E,
     $             M,
     $             NSPLIT, W, IWORK( 1 ), IWORK( INDISP ),
     $             WORK( INDWRK ), IWORK( INDIWO ), INFO )
*
      IF( WANTZ ) THEN
         CALL DSTEIN( N, D, E, M, W, IWORK( 1 ), IWORK( INDISP ),
     $                Z, LDZ, WORK( INDWRK ), IWORK( INDIWO ), IFAIL,
     $                INFO )
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
   20 CONTINUE
      IF( ISCALE.EQ.1 ) THEN
         IF( INFO.EQ.0 ) THEN
            IMAX = M
         ELSE
            IMAX = INFO - 1
         END IF
         CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
      END IF
*
*     If eigenvalues are not in order, then sort them, along with
*     eigenvectors.
*
      IF( WANTZ ) THEN
         DO 40 J = 1, M - 1
            I = 0
            TMP1 = W( J )
            DO 30 JJ = J + 1, M
               IF( W( JJ ).LT.TMP1 ) THEN
                  I = JJ
                  TMP1 = W( JJ )
               END IF
   30       CONTINUE
*
            IF( I.NE.0 ) THEN
               ITMP1 = IWORK( 1 + I-1 )
               W( I ) = W( J )
               IWORK( 1 + I-1 ) = IWORK( 1 + J-1 )
               W( J ) = TMP1
               IWORK( 1 + J-1 ) = ITMP1
               CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
               IF( INFO.NE.0 ) THEN
                  ITMP1 = IFAIL( I )
                  IFAIL( I ) = IFAIL( J )
                  IFAIL( J ) = ITMP1
               END IF
            END IF
   40    CONTINUE
      END IF
*
      RETURN
*
*     End of DSTEVX
*
      END
*> \brief \b DSYCON
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsycon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsycon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsycon.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYCON( UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK,
*                          IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, N
*       DOUBLE PRECISION   ANORM, RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYCON estimates the reciprocal of the condition number (in the
*> 1-norm) of a real symmetric matrix A using the factorization
*> A = U*D*U**T or A = L*D*L**T computed by DSYTRF.
*>
*> An estimate is obtained for norm(inv(A)), and the reciprocal of the
*> condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the details of the factorization are stored
*>          as an upper or lower triangular matrix.
*>          = 'U':  Upper triangular, form is A = U*D*U**T;
*>          = 'L':  Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The block diagonal matrix D and the multipliers used to
*>          obtain the factor U or L as computed by DSYTRF.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the block structure of D
*>          as determined by DSYTRF.
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*>          ANORM is DOUBLE PRECISION
*>          The 1-norm of the original matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The reciprocal of the condition number of the matrix A,
*>          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
*>          estimate of the 1-norm of inv(A) computed in this routine.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hecon
*
*  =====================================================================
      SUBROUTINE DSYCON( UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK,
     $                   IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N
      DOUBLE PRECISION   ANORM, RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I, KASE
      DOUBLE PRECISION   AINVNM
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACN2, DSYTRS, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      ELSE IF( ANORM.LT.ZERO ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYCON', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      RCOND = ZERO
      IF( N.EQ.0 ) THEN
         RCOND = ONE
         RETURN
      ELSE IF( ANORM.LE.ZERO ) THEN
         RETURN
      END IF
*
*     Check that the diagonal matrix D is nonsingular.
*
      IF( UPPER ) THEN
*
*        Upper triangular storage: examine D from bottom to top
*
         DO 10 I = N, 1, -1
            IF( IPIV( I ).GT.0 .AND. A( I, I ).EQ.ZERO )
     $         RETURN
   10    CONTINUE
      ELSE
*
*        Lower triangular storage: examine D from top to bottom.
*
         DO 20 I = 1, N
            IF( IPIV( I ).GT.0 .AND. A( I, I ).EQ.ZERO )
     $         RETURN
   20    CONTINUE
      END IF
*
*     Estimate the 1-norm of the inverse.
*
      KASE = 0
   30 CONTINUE
      CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
      IF( KASE.NE.0 ) THEN
*
*        Multiply by inv(L*D*L**T) or inv(U*D*U**T).
*
         CALL DSYTRS( UPLO, N, 1, A, LDA, IPIV, WORK, N, INFO )
         GO TO 30
      END IF
*
*     Compute the estimate of the reciprocal condition number.
*
      IF( AINVNM.NE.ZERO )
     $   RCOND = ( ONE / AINVNM ) / ANORM
*
      RETURN
*
*     End of DSYCON
*
      END
*> \brief \b DSYCONV
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYCONV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyconv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyconv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyconv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYCONV( UPLO, WAY, N, A, LDA, IPIV, E, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO, WAY
*       INTEGER            INFO, LDA, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   A( LDA, * ), E( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYCONV convert A given by TRF into L and D and vice-versa.
*> Get Non-diag elements of D (returned in workspace) and
*> apply or reverse permutation done in TRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the details of the factorization are stored
*>          as an upper or lower triangular matrix.
*>          = 'U':  Upper triangular, form is A = U*D*U**T;
*>          = 'L':  Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] WAY
*> \verbatim
*>          WAY is CHARACTER*1
*>          = 'C': Convert
*>          = 'R': Revert
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The block diagonal matrix D and the multipliers used to
*>          obtain the factor U or L as computed by DSYTRF.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the block structure of D
*>          as determined by DSYTRF.
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N)
*>          E stores the supdiagonal/subdiagonal of the symmetric 1-by-1
*>          or 2-by-2 block diagonal matrix D in LDLT.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup syconv
*
*  =====================================================================
      SUBROUTINE DSYCONV( UPLO, WAY, N, A, LDA, IPIV, E, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO, WAY
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * ), E( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     .. Local Scalars ..
      LOGICAL            UPPER, CONVERT
      INTEGER            I, IP, J
      DOUBLE PRECISION   TEMP
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      CONVERT = LSAME( WAY, 'C' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.CONVERT .AND. .NOT.LSAME( WAY, 'R' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5

      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYCONV', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( UPPER ) THEN
*
*      A is UPPER
*
*      Convert A (A is upper)
*
*        Convert VALUE
*
         IF ( CONVERT ) THEN
            I=N
            E(1)=ZERO
            DO WHILE ( I .GT. 1 )
               IF( IPIV(I) .LT. 0 ) THEN
                  E(I)=A(I-1,I)
                  E(I-1)=ZERO
                  A(I-1,I)=ZERO
                  I=I-1
               ELSE
                  E(I)=ZERO
               ENDIF
               I=I-1
            END DO
*
*        Convert PERMUTATIONS
*
         I=N
         DO WHILE ( I .GE. 1 )
            IF( IPIV(I) .GT. 0) THEN
               IP=IPIV(I)
               IF( I .LT. N) THEN
                  DO 12 J= I+1,N
                    TEMP=A(IP,J)
                    A(IP,J)=A(I,J)
                    A(I,J)=TEMP
 12            CONTINUE
               ENDIF
            ELSE
              IP=-IPIV(I)
               IF( I .LT. N) THEN
             DO 13 J= I+1,N
                 TEMP=A(IP,J)
                 A(IP,J)=A(I-1,J)
                 A(I-1,J)=TEMP
 13            CONTINUE
                ENDIF
                I=I-1
           ENDIF
           I=I-1
        END DO

         ELSE
*
*      Revert A (A is upper)
*
*
*        Revert PERMUTATIONS
*
            I=1
            DO WHILE ( I .LE. N )
               IF( IPIV(I) .GT. 0 ) THEN
                  IP=IPIV(I)
                  IF( I .LT. N) THEN
                  DO J= I+1,N
                    TEMP=A(IP,J)
                    A(IP,J)=A(I,J)
                    A(I,J)=TEMP
                  END DO
                  ENDIF
               ELSE
                 IP=-IPIV(I)
                 I=I+1
                 IF( I .LT. N) THEN
                    DO J= I+1,N
                       TEMP=A(IP,J)
                       A(IP,J)=A(I-1,J)
                       A(I-1,J)=TEMP
                    END DO
                 ENDIF
               ENDIF
               I=I+1
            END DO
*
*        Revert VALUE
*
            I=N
            DO WHILE ( I .GT. 1 )
               IF( IPIV(I) .LT. 0 ) THEN
                  A(I-1,I)=E(I)
                  I=I-1
               ENDIF
               I=I-1
            END DO
         END IF
      ELSE
*
*      A is LOWER
*
         IF ( CONVERT ) THEN
*
*      Convert A (A is lower)
*
*
*        Convert VALUE
*
            I=1
            E(N)=ZERO
            DO WHILE ( I .LE. N )
               IF( I.LT.N .AND. IPIV(I) .LT. 0 ) THEN
                  E(I)=A(I+1,I)
                  E(I+1)=ZERO
                  A(I+1,I)=ZERO
                  I=I+1
               ELSE
                  E(I)=ZERO
               ENDIF
               I=I+1
            END DO
*
*        Convert PERMUTATIONS
*
         I=1
         DO WHILE ( I .LE. N )
            IF( IPIV(I) .GT. 0 ) THEN
               IP=IPIV(I)
               IF (I .GT. 1) THEN
               DO 22 J= 1,I-1
                 TEMP=A(IP,J)
                 A(IP,J)=A(I,J)
                 A(I,J)=TEMP
 22            CONTINUE
               ENDIF
            ELSE
              IP=-IPIV(I)
              IF (I .GT. 1) THEN
              DO 23 J= 1,I-1
                 TEMP=A(IP,J)
                 A(IP,J)=A(I+1,J)
                 A(I+1,J)=TEMP
 23           CONTINUE
              ENDIF
              I=I+1
           ENDIF
           I=I+1
        END DO
         ELSE
*
*      Revert A (A is lower)
*
*
*        Revert PERMUTATIONS
*
            I=N
            DO WHILE ( I .GE. 1 )
               IF( IPIV(I) .GT. 0 ) THEN
                  IP=IPIV(I)
                  IF (I .GT. 1) THEN
                     DO J= 1,I-1
                        TEMP=A(I,J)
                        A(I,J)=A(IP,J)
                        A(IP,J)=TEMP
                     END DO
                  ENDIF
               ELSE
                  IP=-IPIV(I)
                  I=I-1
                  IF (I .GT. 1) THEN
                     DO J= 1,I-1
                        TEMP=A(I+1,J)
                        A(I+1,J)=A(IP,J)
                        A(IP,J)=TEMP
                     END DO
                  ENDIF
               ENDIF
               I=I-1
            END DO
*
*        Revert VALUE
*
            I=1
            DO WHILE ( I .LE. N-1 )
               IF( IPIV(I) .LT. 0 ) THEN
                  A(I+1,I)=E(I)
                  I=I+1
               ENDIF
               I=I+1
            END DO
         END IF
      END IF

      RETURN
*
*     End of DSYCONV
*
      END
*> \brief \b DSYEQUB
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYEQUB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyequb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyequb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyequb.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, N
*       DOUBLE PRECISION   AMAX, SCOND
*       CHARACTER          UPLO
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), S( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYEQUB computes row and column scalings intended to equilibrate a
*> symmetric matrix A (with respect to the Euclidean norm) and reduce
*> its condition number. The scale factors S are computed by the BIN
*> algorithm (see references) so that the scaled matrix B with elements
*> B(i,j) = S(i)*A(i,j)*S(j) has a condition number within a factor N of
*> the smallest possible condition number over all possible diagonal
*> scalings.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The N-by-N symmetric matrix whose scaling factors are to be
*>          computed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, S contains the scale factors for A.
*> \endverbatim
*>
*> \param[out] SCOND
*> \verbatim
*>          SCOND is DOUBLE PRECISION
*>          If INFO = 0, S contains the ratio of the smallest S(i) to
*>          the largest S(i). If SCOND >= 0.1 and AMAX is neither too
*>          large nor too small, it is not worth scaling by S.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*>          AMAX is DOUBLE PRECISION
*>          Largest absolute value of any matrix element. If AMAX is
*>          very close to overflow or very close to underflow, the
*>          matrix should be scaled.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup heequb
*
*> \par References:
*  ================
*>
*>  Livne, O.E. and Golub, G.H., "Scaling by Binormalization", \n
*>  Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004. \n
*>  DOI 10.1023/B:NUMA.0000016606.32820.69 \n
*>  Tech report version: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.1679
*>
*  =====================================================================
      SUBROUTINE DSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK,
     $                    INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, N
      DOUBLE PRECISION   AMAX, SCOND
      CHARACTER          UPLO
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), S( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D0, ZERO = 0.0D0 )
      INTEGER            MAX_ITER
      PARAMETER          ( MAX_ITER = 100 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J, ITER
      DOUBLE PRECISION   AVG, STD, TOL, C0, C1, C2, T, U, SI, D, BASE,
     $                   SMIN, SMAX, SMLNUM, BIGNUM, SCALE, SUMSQ
      LOGICAL            UP
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      LOGICAL            LSAME
      EXTERNAL           DLAMCH, LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASSQ, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, INT, LOG, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF ( .NOT. ( LSAME( UPLO, 'U' ) .OR.
     $     LSAME( UPLO, 'L' ) ) ) THEN
         INFO = -1
      ELSE IF ( N .LT. 0 ) THEN
         INFO = -2
      ELSE IF ( LDA .LT. MAX( 1, N ) ) THEN
         INFO = -4
      END IF
      IF ( INFO .NE. 0 ) THEN
         CALL XERBLA( 'DSYEQUB', -INFO )
         RETURN
      END IF

      UP = LSAME( UPLO, 'U' )
      AMAX = ZERO
*
*     Quick return if possible.
*
      IF ( N .EQ. 0 ) THEN
         SCOND = ONE
         RETURN
      END IF

      DO I = 1, N
         S( I ) = ZERO
      END DO

      AMAX = ZERO
      IF ( UP ) THEN
         DO J = 1, N
            DO I = 1, J-1
               S( I ) = MAX( S( I ), ABS( A( I, J ) ) )
               S( J ) = MAX( S( J ), ABS( A( I, J ) ) )
               AMAX = MAX( AMAX, ABS( A( I, J ) ) )
            END DO
            S( J ) = MAX( S( J ), ABS( A( J, J ) ) )
            AMAX = MAX( AMAX, ABS( A( J, J ) ) )
         END DO
      ELSE
         DO J = 1, N
            S( J ) = MAX( S( J ), ABS( A( J, J ) ) )
            AMAX = MAX( AMAX, ABS( A( J, J ) ) )
            DO I = J+1, N
               S( I ) = MAX( S( I ), ABS( A( I, J ) ) )
               S( J ) = MAX( S( J ), ABS( A( I, J ) ) )
               AMAX = MAX( AMAX, ABS( A( I, J ) ) )
            END DO
         END DO
      END IF
      DO J = 1, N
         S( J ) = 1.0D0 / S( J )
      END DO

      TOL = ONE / SQRT( 2.0D0 * N )

      DO ITER = 1, MAX_ITER
         SCALE = 0.0D0
         SUMSQ = 0.0D0
*        beta = |A|s
         DO I = 1, N
            WORK( I ) = ZERO
         END DO
         IF ( UP ) THEN
            DO J = 1, N
               DO I = 1, J-1
                  WORK( I ) = WORK( I ) + ABS( A( I, J ) ) * S( J )
                  WORK( J ) = WORK( J ) + ABS( A( I, J ) ) * S( I )
               END DO
               WORK( J ) = WORK( J ) + ABS( A( J, J ) ) * S( J )
            END DO
         ELSE
            DO J = 1, N
               WORK( J ) = WORK( J ) + ABS( A( J, J ) ) * S( J )
               DO I = J+1, N
                  WORK( I ) = WORK( I ) + ABS( A( I, J ) ) * S( J )
                  WORK( J ) = WORK( J ) + ABS( A( I, J ) ) * S( I )
               END DO
            END DO
         END IF

*        avg = s^T beta / n
         AVG = 0.0D0
         DO I = 1, N
            AVG = AVG + S( I )*WORK( I )
         END DO
         AVG = AVG / N

         STD = 0.0D0
         DO I = N+1, 2*N
            WORK( I ) = S( I-N ) * WORK( I-N ) - AVG
         END DO
         CALL DLASSQ( N, WORK( N+1 ), 1, SCALE, SUMSQ )
         STD = SCALE * SQRT( SUMSQ / N )

         IF ( STD .LT. TOL * AVG ) GOTO 999

         DO I = 1, N
            T = ABS( A( I, I ) )
            SI = S( I )
            C2 = ( N-1 ) * T
            C1 = ( N-2 ) * ( WORK( I ) - T*SI )
            C0 = -(T*SI)*SI + 2*WORK( I )*SI - N*AVG
            D = C1*C1 - 4*C0*C2

            IF ( D .LE. 0 ) THEN
               INFO = -1
               RETURN
            END IF
            SI = -2*C0 / ( C1 + SQRT( D ) )

            D = SI - S( I )
            U = ZERO
            IF ( UP ) THEN
               DO J = 1, I
                  T = ABS( A( J, I ) )
                  U = U + S( J )*T
                  WORK( J ) = WORK( J ) + D*T
               END DO
               DO J = I+1,N
                  T = ABS( A( I, J ) )
                  U = U + S( J )*T
                  WORK( J ) = WORK( J ) + D*T
               END DO
            ELSE
               DO J = 1, I
                  T = ABS( A( I, J ) )
                  U = U + S( J )*T
                  WORK( J ) = WORK( J ) + D*T
               END DO
               DO J = I+1,N
                  T = ABS( A( J, I ) )
                  U = U + S( J )*T
                  WORK( J ) = WORK( J ) + D*T
               END DO
            END IF

            AVG = AVG + ( U + WORK( I ) ) * D / N
            S( I ) = SI
         END DO
      END DO

 999  CONTINUE

      SMLNUM = DLAMCH( 'SAFEMIN' )
      BIGNUM = ONE / SMLNUM
      SMIN = BIGNUM
      SMAX = ZERO
      T = ONE / SQRT( AVG )
      BASE = DLAMCH( 'B' )
      U = ONE / LOG( BASE )
      DO I = 1, N
         S( I ) = BASE ** INT( U * LOG( S( I ) * T ) )
         SMIN = MIN( SMIN, S( I ) )
         SMAX = MAX( SMAX, S( I ) )
      END DO
      SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
*
      END
*> \brief <b> DSYEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYEV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyev.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyev.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyev.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYEV( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, UPLO
*       INTEGER            INFO, LDA, LWORK, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), W( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYEV computes all eigenvalues and, optionally, eigenvectors of a
*> real symmetric matrix A.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the
*>          leading N-by-N upper triangular part of A contains the
*>          upper triangular part of the matrix A.  If UPLO = 'L',
*>          the leading N-by-N lower triangular part of A contains
*>          the lower triangular part of the matrix A.
*>          On exit, if JOBZ = 'V', then if INFO = 0, A contains the
*>          orthonormal eigenvectors of the matrix A.
*>          If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')
*>          or the upper triangle (if UPLO='U') of A, including the
*>          diagonal, is destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The length of the array WORK.  LWORK >= max(1,3*N-1).
*>          For optimal efficiency, LWORK >= (NB+2)*N,
*>          where NB is the blocksize for DSYTRD returned by ILAENV.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the algorithm failed to converge; i
*>                off-diagonal elements of an intermediate tridiagonal
*>                form did not converge to zero.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup heev
*
*  =====================================================================
      SUBROUTINE DSYEV( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, UPLO
      INTEGER            INFO, LDA, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), W( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER, LQUERY, WANTZ
      INTEGER            IINFO, IMAX, INDE, INDTAU, INDWRK, ISCALE,
     $                   LLWORK, LWKOPT, NB
      DOUBLE PRECISION   ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
     $                   SMLNUM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANSY
      EXTERNAL           LSAME, ILAENV, DLAMCH, DLANSY
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASCL, DORGTR, DSCAL, DSTEQR, DSTERF,
     $                   DSYTRD,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
      LOWER = LSAME( UPLO, 'L' )
      LQUERY = ( LWORK.EQ.-1 )
*
      INFO = 0
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      END IF
*
      IF( INFO.EQ.0 ) THEN
         NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
         LWKOPT = MAX( 1, ( NB+2 )*N )
         WORK( 1 ) = LWKOPT
*
         IF( LWORK.LT.MAX( 1, 3*N-1 ) .AND. .NOT.LQUERY )
     $      INFO = -8
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYEV ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 ) THEN
         RETURN
      END IF
*
      IF( N.EQ.1 ) THEN
         W( 1 ) = A( 1, 1 )
         WORK( 1 ) = 2
         IF( WANTZ )
     $      A( 1, 1 ) = ONE
         RETURN
      END IF
*
*     Get machine constants.
*
      SAFMIN = DLAMCH( 'Safe minimum' )
      EPS = DLAMCH( 'Precision' )
      SMLNUM = SAFMIN / EPS
      BIGNUM = ONE / SMLNUM
      RMIN = SQRT( SMLNUM )
      RMAX = SQRT( BIGNUM )
*
*     Scale matrix to allowable range, if necessary.
*
      ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK )
      ISCALE = 0
      IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
         ISCALE = 1
         SIGMA = RMIN / ANRM
      ELSE IF( ANRM.GT.RMAX ) THEN
         ISCALE = 1
         SIGMA = RMAX / ANRM
      END IF
      IF( ISCALE.EQ.1 )
     $   CALL DLASCL( UPLO, 0, 0, ONE, SIGMA, N, N, A, LDA, INFO )
*
*     Call DSYTRD to reduce symmetric matrix to tridiagonal form.
*
      INDE = 1
      INDTAU = INDE + N
      INDWRK = INDTAU + N
      LLWORK = LWORK - INDWRK + 1
      CALL DSYTRD( UPLO, N, A, LDA, W, WORK( INDE ), WORK( INDTAU ),
     $             WORK( INDWRK ), LLWORK, IINFO )
*
*     For eigenvalues only, call DSTERF.  For eigenvectors, first call
*     DORGTR to generate the orthogonal matrix, then call DSTEQR.
*
      IF( .NOT.WANTZ ) THEN
         CALL DSTERF( N, W, WORK( INDE ), INFO )
      ELSE
         CALL DORGTR( UPLO, N, A, LDA, WORK( INDTAU ),
     $                WORK( INDWRK ),
     $                LLWORK, IINFO )
         CALL DSTEQR( JOBZ, N, W, WORK( INDE ), A, LDA,
     $                WORK( INDTAU ),
     $                INFO )
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
      IF( ISCALE.EQ.1 ) THEN
         IF( INFO.EQ.0 ) THEN
            IMAX = N
         ELSE
            IMAX = INFO - 1
         END IF
         CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
      END IF
*
*     Set WORK(1) to optimal workspace size.
*
      WORK( 1 ) = LWKOPT
*
      RETURN
*
*     End of DSYEV
*
      END
*> \brief <b> DSYEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYEVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyevd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyevd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyevd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYEVD( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK,
*                          LIWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, UPLO
*       INTEGER            INFO, LDA, LIWORK, LWORK, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), W( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYEVD computes all eigenvalues and, optionally, eigenvectors of a
*> real symmetric matrix A. If eigenvectors are desired, it uses a
*> divide and conquer algorithm.
*>
*> Because of large use of BLAS of level 3, DSYEVD needs N**2 more
*> workspace than DSYEVX.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the
*>          leading N-by-N upper triangular part of A contains the
*>          upper triangular part of the matrix A.  If UPLO = 'L',
*>          the leading N-by-N lower triangular part of A contains
*>          the lower triangular part of the matrix A.
*>          On exit, if JOBZ = 'V', then if INFO = 0, A contains the
*>          orthonormal eigenvectors of the matrix A.
*>          If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')
*>          or the upper triangle (if UPLO='U') of A, including the
*>          diagonal, is destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If N <= 1,               LWORK must be at least 1.
*>          If JOBZ = 'N' and N > 1, LWORK must be at least 2*N+1.
*>          If JOBZ = 'V' and N > 1, LWORK must be at least
*>                                                1 + 6*N + 2*N**2.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal sizes of the WORK and IWORK
*>          arrays, returns these values as the first entries of the WORK
*>          and IWORK arrays, and no error message related to LWORK or
*>          LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK.
*>          If N <= 1,                LIWORK must be at least 1.
*>          If JOBZ  = 'N' and N > 1, LIWORK must be at least 1.
*>          If JOBZ  = 'V' and N > 1, LIWORK must be at least 3 + 5*N.
*>
*>          If LIWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the optimal sizes of the WORK and
*>          IWORK arrays, returns these values as the first entries of
*>          the WORK and IWORK arrays, and no error message related to
*>          LWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i and JOBZ = 'N', then the algorithm failed
*>                to converge; i off-diagonal elements of an intermediate
*>                tridiagonal form did not converge to zero;
*>                if INFO = i and JOBZ = 'V', then the algorithm failed
*>                to compute an eigenvalue while working on the submatrix
*>                lying in rows and columns INFO/(N+1) through
*>                mod(INFO,N+1).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup heevd
*
*> \par Contributors:
*  ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA \n
*>  Modified by Francoise Tisseur, University of Tennessee \n
*>  Modified description of INFO. Sven, 16 Feb 05. \n


*>
*  =====================================================================
      SUBROUTINE DSYEVD( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK,
     $                   IWORK,
     $                   LIWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, UPLO
      INTEGER            INFO, LDA, LIWORK, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), W( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
*
      LOGICAL            LOWER, LQUERY, WANTZ
      INTEGER            IINFO, INDE, INDTAU, INDWK2, INDWRK, ISCALE,
     $                   LIOPT, LIWMIN, LLWORK, LLWRK2, LOPT, LWMIN
      DOUBLE PRECISION   ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
     $                   SMLNUM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANSY
      EXTERNAL           LSAME, DLAMCH, DLANSY, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACPY, DLASCL, DORMTR, DSCAL, DSTEDC,
     $                   DSTERF,
     $                   DSYTRD, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
      LOWER = LSAME( UPLO, 'L' )
      LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
      INFO = 0
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      END IF
*
      IF( INFO.EQ.0 ) THEN
         IF( N.LE.1 ) THEN
            LIWMIN = 1
            LWMIN = 1
            LOPT = LWMIN
            LIOPT = LIWMIN
         ELSE
            IF( WANTZ ) THEN
               LIWMIN = 3 + 5*N
               LWMIN = 1 + 6*N + 2*N**2
            ELSE
               LIWMIN = 1
               LWMIN = 2*N + 1
            END IF
            LOPT = MAX( LWMIN, 2*N +
     $                  N*ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1,
     $                            -1 ) )
            LIOPT = LIWMIN
         END IF
         WORK( 1 ) = LOPT
         IWORK( 1 ) = LIOPT
*
         IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -8
         ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -10
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYEVD', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( N.EQ.1 ) THEN
         W( 1 ) = A( 1, 1 )
         IF( WANTZ )
     $      A( 1, 1 ) = ONE
         RETURN
      END IF
*
*     Get machine constants.
*
      SAFMIN = DLAMCH( 'Safe minimum' )
      EPS = DLAMCH( 'Precision' )
      SMLNUM = SAFMIN / EPS
      BIGNUM = ONE / SMLNUM
      RMIN = SQRT( SMLNUM )
      RMAX = SQRT( BIGNUM )
*
*     Scale matrix to allowable range, if necessary.
*
      ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK )
      ISCALE = 0
      IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
         ISCALE = 1
         SIGMA = RMIN / ANRM
      ELSE IF( ANRM.GT.RMAX ) THEN
         ISCALE = 1
         SIGMA = RMAX / ANRM
      END IF
      IF( ISCALE.EQ.1 )
     $   CALL DLASCL( UPLO, 0, 0, ONE, SIGMA, N, N, A, LDA, INFO )
*
*     Call DSYTRD to reduce symmetric matrix to tridiagonal form.
*
      INDE = 1
      INDTAU = INDE + N
      INDWRK = INDTAU + N
      LLWORK = LWORK - INDWRK + 1
      INDWK2 = INDWRK + N*N
      LLWRK2 = LWORK - INDWK2 + 1
*
      CALL DSYTRD( UPLO, N, A, LDA, W, WORK( INDE ), WORK( INDTAU ),
     $             WORK( INDWRK ), LLWORK, IINFO )
*
*     For eigenvalues only, call DSTERF.  For eigenvectors, first call
*     DSTEDC to generate the eigenvector matrix, WORK(INDWRK), of the
*     tridiagonal matrix, then call DORMTR to multiply it by the
*     Householder transformations stored in A.
*
      IF( .NOT.WANTZ ) THEN
         CALL DSTERF( N, W, WORK( INDE ), INFO )
      ELSE
         CALL DSTEDC( 'I', N, W, WORK( INDE ), WORK( INDWRK ), N,
     $                WORK( INDWK2 ), LLWRK2, IWORK, LIWORK, INFO )
         CALL DORMTR( 'L', UPLO, 'N', N, N, A, LDA, WORK( INDTAU ),
     $                WORK( INDWRK ), N, WORK( INDWK2 ), LLWRK2, IINFO )
         CALL DLACPY( 'A', N, N, WORK( INDWRK ), N, A, LDA )
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
      IF( ISCALE.EQ.1 )
     $   CALL DSCAL( N, ONE / SIGMA, W, 1 )
*
      WORK( 1 ) = LOPT
      IWORK( 1 ) = LIOPT
*
      RETURN
*
*     End of DSYEVD
*
      END
*> \brief <b> DSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYEVR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyevr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyevr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyevr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
*                          ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK,
*                          IWORK, LIWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, RANGE, UPLO
*       INTEGER            IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N
*       DOUBLE PRECISION   ABSTOL, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            ISUPPZ( * ), IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYEVR computes selected eigenvalues and, optionally, eigenvectors
*> of a real symmetric matrix A. Eigenvalues and eigenvectors can be
*> selected by specifying either a range of values or a range of indices
*> for the desired eigenvalues. Invocations with different choices for
*> these parameters may result in the computation of slightly different
*> eigenvalues and/or eigenvectors for the same matrix. The reason for
*> this behavior is that there exists a variety of algorithms (each
*> performing best for a particular set of options) with DSYEVR
*> attempting to select the best based on the various parameters. In all
*> cases, the computed values are accurate within the limits of finite
*> precision arithmetic.
*>
*> DSYEVR first reduces the matrix A to tridiagonal form T with a call
*> to DSYTRD.  Then, whenever possible, DSYEVR calls DSTEMR to compute
*> the eigenspectrum using Relatively Robust Representations.  DSTEMR
*> computes eigenvalues by the dqds algorithm, while orthogonal
*> eigenvectors are computed from various "good" L D L^T representations
*> (also known as Relatively Robust Representations). Gram-Schmidt
*> orthogonalization is avoided as far as possible. More specifically,
*> the various steps of the algorithm are as follows.
*>
*> For each unreduced block (submatrix) of T,
*>    (a) Compute T - sigma I  = L D L^T, so that L and D
*>        define all the wanted eigenvalues to high relative accuracy.
*>        This means that small relative changes in the entries of D and L
*>        cause only small relative changes in the eigenvalues and
*>        eigenvectors. The standard (unfactored) representation of the
*>        tridiagonal matrix T does not have this property in general.
*>    (b) Compute the eigenvalues to suitable accuracy.
*>        If the eigenvectors are desired, the algorithm attains full
*>        accuracy of the computed eigenvalues only right before
*>        the corresponding vectors have to be computed, see steps c) and d).
*>    (c) For each cluster of close eigenvalues, select a new
*>        shift close to the cluster, find a new factorization, and refine
*>        the shifted eigenvalues to suitable accuracy.
*>    (d) For each eigenvalue with a large enough relative separation compute
*>        the corresponding eigenvector by forming a rank revealing twisted
*>        factorization. Go back to (c) for any clusters that remain.
*>
*> The desired accuracy of the output can be specified by the input
*> parameter ABSTOL.
*>
*> For more details, see DSTEMR's documentation and:
*> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
*>   to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
*>   Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
*> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
*>   Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
*>   2004.  Also LAPACK Working Note 154.
*> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
*>   tridiagonal eigenvalue/eigenvector problem",
*>   Computer Science Division Technical Report No. UCB/CSD-97-971,
*>   UC Berkeley, May 1997.
*>
*>
*> Note 1 : DSYEVR calls DSTEMR when the full spectrum is requested
*> on machines which conform to the ieee-754 floating point standard.
*> DSYEVR calls DSTEBZ and DSTEIN on non-ieee machines and
*> when partial spectrum requests are made.
*>
*> Normal execution of DSTEMR may create NaNs and infinities and
*> hence may abort due to a floating point exception in environments
*> which do not handle NaNs and infinities in the ieee standard default
*> manner.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*>
*>          This parameter influences the choice of the algorithm and
*>          may alter the computed values.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*>          RANGE is CHARACTER*1
*>          = 'A': all eigenvalues will be found.
*>          = 'V': all eigenvalues in the half-open interval (VL,VU]
*>                 will be found.
*>          = 'I': the IL-th through IU-th eigenvalues will be found.
*>          For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
*>          DSTEIN are called
*>
*>          This parameter influences the choice of the algorithm and
*>          may alter the computed values.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the
*>          leading N-by-N upper triangular part of A contains the
*>          upper triangular part of the matrix A.  If UPLO = 'L',
*>          the leading N-by-N lower triangular part of A contains
*>          the lower triangular part of the matrix A.
*>          On exit, the lower triangle (if UPLO='L') or the upper
*>          triangle (if UPLO='U') of A, including the diagonal, is
*>          destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is DOUBLE PRECISION
*>          If RANGE='V', the lower bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*>          VU is DOUBLE PRECISION
*>          If RANGE='V', the upper bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*>          IL is INTEGER
*>          If RANGE='I', the index of the
*>          smallest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*>          IU is INTEGER
*>          If RANGE='I', the index of the
*>          largest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*>          ABSTOL is DOUBLE PRECISION
*>          The absolute error tolerance for the eigenvalues.
*>          An approximate eigenvalue is accepted as converged
*>          when it is determined to lie in an interval [a,b]
*>          of width less than or equal to
*>
*>                  ABSTOL + EPS *   max( |a|,|b| ) ,
*>
*>          where EPS is the machine precision.  If ABSTOL is less than
*>          or equal to zero, then  EPS*|T|  will be used in its place,
*>          where |T| is the 1-norm of the tridiagonal matrix obtained
*>          by reducing A to tridiagonal form.
*>
*>          See "Computing Small Singular Values of Bidiagonal Matrices
*>          with Guaranteed High Relative Accuracy," by Demmel and
*>          Kahan, LAPACK Working Note #3.
*>
*>          If high relative accuracy is important, set ABSTOL to
*>          DLAMCH( 'Safe minimum' ).  Doing so will guarantee that
*>          eigenvalues are computed to high relative accuracy when
*>          possible in future releases.  The current code does not
*>          make any guarantees about high relative accuracy, but
*>          future releases will. See J. Barlow and J. Demmel,
*>          "Computing Accurate Eigensystems of Scaled Diagonally
*>          Dominant Matrices", LAPACK Working Note #7, for a discussion
*>          of which matrices define their eigenvalues to high relative
*>          accuracy.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The total number of eigenvalues found.  0 <= M <= N.
*>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          The first M elements contain the selected eigenvalues in
*>          ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
*>          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*>          contain the orthonormal eigenvectors of the matrix A
*>          corresponding to the selected eigenvalues, with the i-th
*>          column of Z holding the eigenvector associated with W(i).
*>          If JOBZ = 'N', then Z is not referenced.
*>          Note: the user must ensure that at least max(1,M) columns are
*>          supplied in the array Z; if RANGE = 'V', the exact value of M
*>          is not known in advance and an upper bound must be used.
*>          Supplying N columns is always safe.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] ISUPPZ
*> \verbatim
*>          ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
*>          The support of the eigenvectors in Z, i.e., the indices
*>          indicating the nonzero elements in Z. The i-th eigenvector
*>          is nonzero only in elements ISUPPZ( 2*i-1 ) through
*>          ISUPPZ( 2*i ). This is an output of DSTEMR (tridiagonal
*>          matrix). The support of the eigenvectors of A is typically
*>          1:N because of the orthogonal transformations applied by DORMTR.
*>          Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If N <= 1, LWORK >= 1, else LWORK >= 26*N.
*>          For optimal efficiency, LWORK >= (NB+6)*N,
*>          where NB is the max of the blocksize for DSYTRD and DORMTR
*>          returned by ILAENV.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK.
*>          If N <= 1, LIWORK >= 1, else LIWORK >= 10*N.
*>
*>          If LIWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the optimal size of the IWORK array,
*>          returns this value as the first entry of the IWORK array, and
*>          no error message related to LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  Internal error
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup heevr
*
*> \par Contributors:
*  ==================
*>
*>     Inderjit Dhillon, IBM Almaden, USA \n
*>     Osni Marques, LBNL/NERSC, USA \n
*>     Ken Stanley, Computer Science Division, University of
*>       California at Berkeley, USA \n
*>     Jason Riedy, Computer Science Division, University of
*>       California at Berkeley, USA \n
*>
*  =====================================================================
      SUBROUTINE DSYEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL,
     $                   IU,
     $                   ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK,
     $                   IWORK, LIWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, RANGE, UPLO
      INTEGER            IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N
      DOUBLE PRECISION   ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            ISUPPZ( * ), IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLEIG, INDEIG, LOWER, LQUERY, VALEIG, WANTZ,
     $                   TRYRAC
      CHARACTER          ORDER
      INTEGER            I, IEEEOK, IINFO, IMAX, INDD, INDDD, INDE,
     $                   INDEE, INDIBL, INDIFL, INDISP, INDIWO, INDTAU,
     $                   INDWK, INDWKN, ISCALE, J, JJ, LIWMIN,
     $                   LLWORK, LLWRKN, LWKOPT, LWMIN, NB, NSPLIT
      DOUBLE PRECISION   ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
     $                   SIGMA, SMLNUM, TMP1, VLL, VUU
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANSY
      EXTERNAL           LSAME, ILAENV, DLAMCH, DLANSY
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DORMTR, DSCAL, DSTEBZ, DSTEMR,
     $                   DSTEIN,
     $                   DSTERF, DSWAP, DSYTRD, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      IEEEOK = ILAENV( 10, 'DSYEVR', 'N', 1, 2, 3, 4 )
*
      LOWER = LSAME( UPLO, 'L' )
      WANTZ = LSAME( JOBZ, 'V' )
      ALLEIG = LSAME( RANGE, 'A' )
      VALEIG = LSAME( RANGE, 'V' )
      INDEIG = LSAME( RANGE, 'I' )
*
      LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) )
*
      IF( N.LE.1 ) THEN
         LWMIN  = 1
         LIWMIN = 1
      ELSE
         LWMIN  = 26*N
         LIWMIN = 10*N
      END IF
*
      INFO = 0
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
         INFO = -2
      ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -6
      ELSE
         IF( VALEIG ) THEN
            IF( N.GT.0 .AND. VU.LE.VL )
     $         INFO = -8
         ELSE IF( INDEIG ) THEN
            IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
               INFO = -9
            ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
               INFO = -10
            END IF
         END IF
      END IF
      IF( INFO.EQ.0 ) THEN
         IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
            INFO = -15
         ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -18
         ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -20
         END IF
      END IF
*
      IF( INFO.EQ.0 ) THEN
         NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
         NB = MAX( NB, ILAENV( 1, 'DORMTR', UPLO, N, -1, -1, -1 ) )
         LWKOPT = MAX( ( NB+1 )*N, LWMIN )
         WORK( 1 ) = LWKOPT
         IWORK( 1 ) = LIWMIN
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYEVR', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      M = 0
      IF( N.EQ.0 ) THEN
         WORK( 1 ) = 1
         RETURN
      END IF
*
      IF( N.EQ.1 ) THEN
         WORK( 1 ) = 1
         IF( ALLEIG .OR. INDEIG ) THEN
            M = 1
            W( 1 ) = A( 1, 1 )
         ELSE
            IF( VL.LT.A( 1, 1 ) .AND. VU.GE.A( 1, 1 ) ) THEN
               M = 1
               W( 1 ) = A( 1, 1 )
            END IF
         END IF
         IF( WANTZ ) THEN
            Z( 1, 1 ) = ONE
            ISUPPZ( 1 ) = 1
            ISUPPZ( 2 ) = 1
         END IF
         RETURN
      END IF
*
*     Get machine constants.
*
      SAFMIN = DLAMCH( 'Safe minimum' )
      EPS = DLAMCH( 'Precision' )
      SMLNUM = SAFMIN / EPS
      BIGNUM = ONE / SMLNUM
      RMIN = SQRT( SMLNUM )
      RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
*     Scale matrix to allowable range, if necessary.
*
      ISCALE = 0
      ABSTLL = ABSTOL
      IF (VALEIG) THEN
         VLL = VL
         VUU = VU
      END IF
      ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK )
      IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
         ISCALE = 1
         SIGMA = RMIN / ANRM
      ELSE IF( ANRM.GT.RMAX ) THEN
         ISCALE = 1
         SIGMA = RMAX / ANRM
      END IF
      IF( ISCALE.EQ.1 ) THEN
         IF( LOWER ) THEN
            DO 10 J = 1, N
               CALL DSCAL( N-J+1, SIGMA, A( J, J ), 1 )
   10       CONTINUE
         ELSE
            DO 20 J = 1, N
               CALL DSCAL( J, SIGMA, A( 1, J ), 1 )
   20       CONTINUE
         END IF
         IF( ABSTOL.GT.0 )
     $      ABSTLL = ABSTOL*SIGMA
         IF( VALEIG ) THEN
            VLL = VL*SIGMA
            VUU = VU*SIGMA
         END IF
      END IF

*     Initialize indices into workspaces.  Note: The IWORK indices are
*     used only if DSTERF or DSTEMR fail.

*     WORK(INDTAU:INDTAU+N-1) stores the scalar factors of the
*     elementary reflectors used in DSYTRD.
      INDTAU = 1
*     WORK(INDD:INDD+N-1) stores the tridiagonal's diagonal entries.
      INDD = INDTAU + N
*     WORK(INDE:INDE+N-1) stores the off-diagonal entries of the
*     tridiagonal matrix from DSYTRD.
      INDE = INDD + N
*     WORK(INDDD:INDDD+N-1) is a copy of the diagonal entries over
*     -written by DSTEMR (the DSTERF path copies the diagonal to W).
      INDDD = INDE + N
*     WORK(INDEE:INDEE+N-1) is a copy of the off-diagonal entries over
*     -written while computing the eigenvalues in DSTERF and DSTEMR.
      INDEE = INDDD + N
*     INDWK is the starting offset of the left-over workspace, and
*     LLWORK is the remaining workspace size.
      INDWK = INDEE + N
      LLWORK = LWORK - INDWK + 1

*     IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and
*     stores the block indices of each of the M<=N eigenvalues.
      INDIBL = 1
*     IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and
*     stores the starting and finishing indices of each block.
      INDISP = INDIBL + N
*     IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
*     that corresponding to eigenvectors that fail to converge in
*     DSTEIN.  This information is discarded; if any fail, the driver
*     returns INFO > 0.
      INDIFL = INDISP + N
*     INDIWO is the offset of the remaining integer workspace.
      INDIWO = INDIFL + N

*
*     Call DSYTRD to reduce symmetric matrix to tridiagonal form.
*
      CALL DSYTRD( UPLO, N, A, LDA, WORK( INDD ), WORK( INDE ),
     $             WORK( INDTAU ), WORK( INDWK ), LLWORK, IINFO )
*
*     If all eigenvalues are desired
*     then call DSTERF or DSTEMR and DORMTR.
*
      IF( ( ALLEIG .OR. ( INDEIG .AND. IL.EQ.1 .AND. IU.EQ.N ) ) .AND.
     $    IEEEOK.EQ.1 ) THEN
         IF( .NOT.WANTZ ) THEN
            CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
            CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
            CALL DSTERF( N, W, WORK( INDEE ), INFO )
         ELSE
            CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
            CALL DCOPY( N, WORK( INDD ), 1, WORK( INDDD ), 1 )
*
            IF (ABSTOL .LE. TWO*N*EPS) THEN
               TRYRAC = .TRUE.
            ELSE
               TRYRAC = .FALSE.
            END IF
            CALL DSTEMR( JOBZ, 'A', N, WORK( INDDD ), WORK( INDEE ),
     $                   VL, VU, IL, IU, M, W, Z, LDZ, N, ISUPPZ,
     $                   TRYRAC, WORK( INDWK ), LWORK, IWORK, LIWORK,
     $                   INFO )
*
*
*
*        Apply orthogonal matrix used in reduction to tridiagonal
*        form to eigenvectors returned by DSTEMR.
*
            IF( WANTZ .AND. INFO.EQ.0 ) THEN
               INDWKN = INDE
               LLWRKN = LWORK - INDWKN + 1
               CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA,
     $                      WORK( INDTAU ), Z, LDZ, WORK( INDWKN ),
     $                      LLWRKN, IINFO )
            END IF
         END IF
*
*
         IF( INFO.EQ.0 ) THEN
*           Everything worked.  Skip DSTEBZ/DSTEIN.  IWORK(:) are
*           undefined.
            M = N
            GO TO 30
         END IF
         INFO = 0
      END IF
*
*     Otherwise, call DSTEBZ and, if eigenvectors are desired, DSTEIN.
*     Also call DSTEBZ and DSTEIN if DSTEMR fails.
*
      IF( WANTZ ) THEN
         ORDER = 'B'
      ELSE
         ORDER = 'E'
      END IF

      CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
     $             WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
     $             IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWK ),
     $             IWORK( INDIWO ), INFO )
*
      IF( WANTZ ) THEN
         CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
     $                IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
     $                WORK( INDWK ), IWORK( INDIWO ), IWORK( INDIFL ),
     $                INFO )
*
*        Apply orthogonal matrix used in reduction to tridiagonal
*        form to eigenvectors returned by DSTEIN.
*
         INDWKN = INDE
         LLWRKN = LWORK - INDWKN + 1
         CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ),
     $                Z,
     $                LDZ, WORK( INDWKN ), LLWRKN, IINFO )
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
*  Jump here if DSTEMR/DSTEIN succeeded.
   30 CONTINUE
      IF( ISCALE.EQ.1 ) THEN
         IF( INFO.EQ.0 ) THEN
            IMAX = M
         ELSE
            IMAX = INFO - 1
         END IF
         CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
      END IF
*
*     If eigenvalues are not in order, then sort them, along with
*     eigenvectors.  Note: We do not sort the IFAIL portion of IWORK.
*     It may not be initialized (if DSTEMR/DSTEIN succeeded), and we do
*     not return this detailed information to the user.
*
      IF( WANTZ ) THEN
         DO 50 J = 1, M - 1
            I = 0
            TMP1 = W( J )
            DO 40 JJ = J + 1, M
               IF( W( JJ ).LT.TMP1 ) THEN
                  I = JJ
                  TMP1 = W( JJ )
               END IF
   40       CONTINUE
*
            IF( I.NE.0 ) THEN
               W( I ) = W( J )
               W( J ) = TMP1
               CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
            END IF
   50    CONTINUE
      END IF
*
*     Set WORK(1) to optimal workspace size.
*
      WORK( 1 ) = LWKOPT
      IWORK( 1 ) = LIWMIN
*
      RETURN
*
*     End of DSYEVR
*
      END
*> \brief <b> DSYEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYEVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyevx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyevx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyevx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
*                          ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK,
*                          IFAIL, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, RANGE, UPLO
*       INTEGER            IL, INFO, IU, LDA, LDZ, LWORK, M, N
*       DOUBLE PRECISION   ABSTOL, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            IFAIL( * ), IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYEVX computes selected eigenvalues and, optionally, eigenvectors
*> of a real symmetric matrix A.  Eigenvalues and eigenvectors can be
*> selected by specifying either a range of values or a range of indices
*> for the desired eigenvalues.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*>          RANGE is CHARACTER*1
*>          = 'A': all eigenvalues will be found.
*>          = 'V': all eigenvalues in the half-open interval (VL,VU]
*>                 will be found.
*>          = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the
*>          leading N-by-N upper triangular part of A contains the
*>          upper triangular part of the matrix A.  If UPLO = 'L',
*>          the leading N-by-N lower triangular part of A contains
*>          the lower triangular part of the matrix A.
*>          On exit, the lower triangle (if UPLO='L') or the upper
*>          triangle (if UPLO='U') of A, including the diagonal, is
*>          destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is DOUBLE PRECISION
*>          If RANGE='V', the lower bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*>          VU is DOUBLE PRECISION
*>          If RANGE='V', the upper bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*>          IL is INTEGER
*>          If RANGE='I', the index of the
*>          smallest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*>          IU is INTEGER
*>          If RANGE='I', the index of the
*>          largest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*>          ABSTOL is DOUBLE PRECISION
*>          The absolute error tolerance for the eigenvalues.
*>          An approximate eigenvalue is accepted as converged
*>          when it is determined to lie in an interval [a,b]
*>          of width less than or equal to
*>
*>                  ABSTOL + EPS *   max( |a|,|b| ) ,
*>
*>          where EPS is the machine precision.  If ABSTOL is less than
*>          or equal to zero, then  EPS*|T|  will be used in its place,
*>          where |T| is the 1-norm of the tridiagonal matrix obtained
*>          by reducing A to tridiagonal form.
*>
*>          Eigenvalues will be computed most accurately when ABSTOL is
*>          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*>          If this routine returns with INFO>0, indicating that some
*>          eigenvectors did not converge, try setting ABSTOL to
*>          2*DLAMCH('S').
*>
*>          See "Computing Small Singular Values of Bidiagonal Matrices
*>          with Guaranteed High Relative Accuracy," by Demmel and
*>          Kahan, LAPACK Working Note #3.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The total number of eigenvalues found.  0 <= M <= N.
*>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          On normal exit, the first M elements contain the selected
*>          eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
*>          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*>          contain the orthonormal eigenvectors of the matrix A
*>          corresponding to the selected eigenvalues, with the i-th
*>          column of Z holding the eigenvector associated with W(i).
*>          If an eigenvector fails to converge, then that column of Z
*>          contains the latest approximation to the eigenvector, and the
*>          index of the eigenvector is returned in IFAIL.
*>          If JOBZ = 'N', then Z is not referenced.
*>          Note: the user must ensure that at least max(1,M) columns are
*>          supplied in the array Z; if RANGE = 'V', the exact value of M
*>          is not known in advance and an upper bound must be used.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The length of the array WORK.  LWORK >= 1, when N <= 1;
*>          otherwise 8*N.
*>          For optimal efficiency, LWORK >= (NB+3)*N,
*>          where NB is the max of the blocksize for DSYTRD and DORMTR
*>          returned by ILAENV.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*>          IFAIL is INTEGER array, dimension (N)
*>          If JOBZ = 'V', then if INFO = 0, the first M elements of
*>          IFAIL are zero.  If INFO > 0, then IFAIL contains the
*>          indices of the eigenvectors that failed to converge.
*>          If JOBZ = 'N', then IFAIL is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, then i eigenvectors failed to converge.
*>                Their indices are stored in array IFAIL.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup heevx
*
*  =====================================================================
      SUBROUTINE DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL,
     $                   IU,
     $                   ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK,
     $                   IFAIL, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, RANGE, UPLO
      INTEGER            IL, INFO, IU, LDA, LDZ, LWORK, M, N
      DOUBLE PRECISION   ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            IFAIL( * ), IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
     $                   WANTZ
      CHARACTER          ORDER
      INTEGER            I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
     $                   INDISP, INDIWO, INDTAU, INDWKN, INDWRK, ISCALE,
     $                   ITMP1, J, JJ, LLWORK, LLWRKN, LWKMIN,
     $                   LWKOPT, NB, NSPLIT
      DOUBLE PRECISION   ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
     $                   SIGMA, SMLNUM, TMP1, VLL, VUU
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANSY
      EXTERNAL           LSAME, ILAENV, DLAMCH, DLANSY
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLACPY, DORGTR, DORMTR, DSCAL,
     $                   DSTEBZ,
     $                   DSTEIN, DSTEQR, DSTERF, DSWAP, DSYTRD, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      LOWER = LSAME( UPLO, 'L' )
      WANTZ = LSAME( JOBZ, 'V' )
      ALLEIG = LSAME( RANGE, 'A' )
      VALEIG = LSAME( RANGE, 'V' )
      INDEIG = LSAME( RANGE, 'I' )
      LQUERY = ( LWORK.EQ.-1 )
*
      INFO = 0
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
         INFO = -2
      ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -6
      ELSE
         IF( VALEIG ) THEN
            IF( N.GT.0 .AND. VU.LE.VL )
     $         INFO = -8
         ELSE IF( INDEIG ) THEN
            IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
               INFO = -9
            ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
               INFO = -10
            END IF
         END IF
      END IF
      IF( INFO.EQ.0 ) THEN
         IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
            INFO = -15
         END IF
      END IF
*
      IF( INFO.EQ.0 ) THEN
         IF( N.LE.1 ) THEN
            LWKMIN = 1
            LWKOPT = 1
         ELSE
            LWKMIN = 8*N
            NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
            NB = MAX( NB, ILAENV( 1, 'DORMTR', UPLO, N, -1, -1,
     $                -1 ) )
            LWKOPT = MAX( LWKMIN, ( NB + 3 )*N )
         END IF
         WORK( 1 ) = LWKOPT
*
         IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY )
     $      INFO = -17
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYEVX', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      M = 0
      IF( N.EQ.0 ) THEN
         RETURN
      END IF
*
      IF( N.EQ.1 ) THEN
         IF( ALLEIG .OR. INDEIG ) THEN
            M = 1
            W( 1 ) = A( 1, 1 )
         ELSE
            IF( VL.LT.A( 1, 1 ) .AND. VU.GE.A( 1, 1 ) ) THEN
               M = 1
               W( 1 ) = A( 1, 1 )
            END IF
         END IF
         IF( WANTZ )
     $      Z( 1, 1 ) = ONE
         RETURN
      END IF
*
*     Get machine constants.
*
      SAFMIN = DLAMCH( 'Safe minimum' )
      EPS = DLAMCH( 'Precision' )
      SMLNUM = SAFMIN / EPS
      BIGNUM = ONE / SMLNUM
      RMIN = SQRT( SMLNUM )
      RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
*     Scale matrix to allowable range, if necessary.
*
      ISCALE = 0
      ABSTLL = ABSTOL
      IF( VALEIG ) THEN
         VLL = VL
         VUU = VU
      END IF
      ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK )
      IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
         ISCALE = 1
         SIGMA = RMIN / ANRM
      ELSE IF( ANRM.GT.RMAX ) THEN
         ISCALE = 1
         SIGMA = RMAX / ANRM
      END IF
      IF( ISCALE.EQ.1 ) THEN
         IF( LOWER ) THEN
            DO 10 J = 1, N
               CALL DSCAL( N-J+1, SIGMA, A( J, J ), 1 )
   10       CONTINUE
         ELSE
            DO 20 J = 1, N
               CALL DSCAL( J, SIGMA, A( 1, J ), 1 )
   20       CONTINUE
         END IF
         IF( ABSTOL.GT.0 )
     $      ABSTLL = ABSTOL*SIGMA
         IF( VALEIG ) THEN
            VLL = VL*SIGMA
            VUU = VU*SIGMA
         END IF
      END IF
*
*     Call DSYTRD to reduce symmetric matrix to tridiagonal form.
*
      INDTAU = 1
      INDE = INDTAU + N
      INDD = INDE + N
      INDWRK = INDD + N
      LLWORK = LWORK - INDWRK + 1
      CALL DSYTRD( UPLO, N, A, LDA, WORK( INDD ), WORK( INDE ),
     $             WORK( INDTAU ), WORK( INDWRK ), LLWORK, IINFO )
*
*     If all eigenvalues are desired and ABSTOL is less than or equal to
*     zero, then call DSTERF or DORGTR and SSTEQR.  If this fails for
*     some eigenvalue, then try DSTEBZ.
*
      TEST = .FALSE.
      IF( INDEIG ) THEN
         IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
            TEST = .TRUE.
         END IF
      END IF
      IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
         CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
         INDEE = INDWRK + 2*N
         IF( .NOT.WANTZ ) THEN
            CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
            CALL DSTERF( N, W, WORK( INDEE ), INFO )
         ELSE
            CALL DLACPY( 'A', N, N, A, LDA, Z, LDZ )
            CALL DORGTR( UPLO, N, Z, LDZ, WORK( INDTAU ),
     $                   WORK( INDWRK ), LLWORK, IINFO )
            CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
            CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
     $                   WORK( INDWRK ), INFO )
            IF( INFO.EQ.0 ) THEN
               DO 30 I = 1, N
                  IFAIL( I ) = 0
   30          CONTINUE
            END IF
         END IF
         IF( INFO.EQ.0 ) THEN
            M = N
            GO TO 40
         END IF
         INFO = 0
      END IF
*
*     Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
*
      IF( WANTZ ) THEN
         ORDER = 'B'
      ELSE
         ORDER = 'E'
      END IF
      INDIBL = 1
      INDISP = INDIBL + N
      INDIWO = INDISP + N
      CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
     $             WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
     $             IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
     $             IWORK( INDIWO ), INFO )
*
      IF( WANTZ ) THEN
         CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
     $                IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
     $                WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
*
*        Apply orthogonal matrix used in reduction to tridiagonal
*        form to eigenvectors returned by DSTEIN.
*
         INDWKN = INDE
         LLWRKN = LWORK - INDWKN + 1
         CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ),
     $                Z,
     $                LDZ, WORK( INDWKN ), LLWRKN, IINFO )
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
   40 CONTINUE
      IF( ISCALE.EQ.1 ) THEN
         IF( INFO.EQ.0 ) THEN
            IMAX = M
         ELSE
            IMAX = INFO - 1
         END IF
         CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
      END IF
*
*     If eigenvalues are not in order, then sort them, along with
*     eigenvectors.
*
      IF( WANTZ ) THEN
         DO 60 J = 1, M - 1
            I = 0
            TMP1 = W( J )
            DO 50 JJ = J + 1, M
               IF( W( JJ ).LT.TMP1 ) THEN
                  I = JJ
                  TMP1 = W( JJ )
               END IF
   50       CONTINUE
*
            IF( I.NE.0 ) THEN
               ITMP1 = IWORK( INDIBL+I-1 )
               W( I ) = W( J )
               IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
               W( J ) = TMP1
               IWORK( INDIBL+J-1 ) = ITMP1
               CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
               IF( INFO.NE.0 ) THEN
                  ITMP1 = IFAIL( I )
                  IFAIL( I ) = IFAIL( J )
                  IFAIL( J ) = ITMP1
               END IF
            END IF
   60    CONTINUE
      END IF
*
*     Set WORK(1) to optimal workspace size.
*
      WORK( 1 ) = LWKOPT
*
      RETURN
*
*     End of DSYEVX
*
      END
*> \brief \b DSYGS2 reduces a symmetric definite generalized eigenproblem to standard form, using the factorization results obtained from spotrf (unblocked algorithm).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYGS2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsygs2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsygs2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsygs2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYGS2( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, ITYPE, LDA, LDB, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYGS2 reduces a real symmetric-definite generalized eigenproblem
*> to standard form.
*>
*> If ITYPE = 1, the problem is A*x = lambda*B*x,
*> and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
*>
*> If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
*> B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T *A*L.
*>
*> B must have been previously factorized as U**T *U or L*L**T by DPOTRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ITYPE
*> \verbatim
*>          ITYPE is INTEGER
*>          = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
*>          = 2 or 3: compute U*A*U**T or L**T *A*L.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the upper or lower triangular part of the
*>          symmetric matrix A is stored, and how B has been factorized.
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A and B.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*>          n by n upper triangular part of A contains the upper
*>          triangular part of the matrix A, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading n by n lower triangular part of A contains the lower
*>          triangular part of the matrix A, and the strictly upper
*>          triangular part of A is not referenced.
*>
*>          On exit, if INFO = 0, the transformed matrix, stored in the
*>          same format as A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,N)
*>          The triangular factor from the Cholesky factorization of B,
*>          as returned by DPOTRF.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hegs2
*
*  =====================================================================
      SUBROUTINE DSYGS2( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, ITYPE, LDA, LDB, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, HALF
      PARAMETER          ( ONE = 1.0D0, HALF = 0.5D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            K
      DOUBLE PRECISION   AKK, BKK, CT
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DSCAL, DSYR2, DTRMV, DTRSV,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
         INFO = -1
      ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYGS2', -INFO )
         RETURN
      END IF
*
      IF( ITYPE.EQ.1 ) THEN
         IF( UPPER ) THEN
*
*           Compute inv(U**T)*A*inv(U)
*
            DO 10 K = 1, N
*
*              Update the upper triangle of A(k:n,k:n)
*
               AKK = A( K, K )
               BKK = B( K, K )
               AKK = AKK / BKK**2
               A( K, K ) = AKK
               IF( K.LT.N ) THEN
                  CALL DSCAL( N-K, ONE / BKK, A( K, K+1 ), LDA )
                  CT = -HALF*AKK
                  CALL DAXPY( N-K, CT, B( K, K+1 ), LDB, A( K, K+1 ),
     $                        LDA )
                  CALL DSYR2( UPLO, N-K, -ONE, A( K, K+1 ), LDA,
     $                        B( K, K+1 ), LDB, A( K+1, K+1 ), LDA )
                  CALL DAXPY( N-K, CT, B( K, K+1 ), LDB, A( K, K+1 ),
     $                        LDA )
                  CALL DTRSV( UPLO, 'Transpose', 'Non-unit', N-K,
     $                        B( K+1, K+1 ), LDB, A( K, K+1 ), LDA )
               END IF
   10       CONTINUE
         ELSE
*
*           Compute inv(L)*A*inv(L**T)
*
            DO 20 K = 1, N
*
*              Update the lower triangle of A(k:n,k:n)
*
               AKK = A( K, K )
               BKK = B( K, K )
               AKK = AKK / BKK**2
               A( K, K ) = AKK
               IF( K.LT.N ) THEN
                  CALL DSCAL( N-K, ONE / BKK, A( K+1, K ), 1 )
                  CT = -HALF*AKK
                  CALL DAXPY( N-K, CT, B( K+1, K ), 1, A( K+1, K ),
     $                        1 )
                  CALL DSYR2( UPLO, N-K, -ONE, A( K+1, K ), 1,
     $                        B( K+1, K ), 1, A( K+1, K+1 ), LDA )
                  CALL DAXPY( N-K, CT, B( K+1, K ), 1, A( K+1, K ),
     $                        1 )
                  CALL DTRSV( UPLO, 'No transpose', 'Non-unit', N-K,
     $                        B( K+1, K+1 ), LDB, A( K+1, K ), 1 )
               END IF
   20       CONTINUE
         END IF
      ELSE
         IF( UPPER ) THEN
*
*           Compute U*A*U**T
*
            DO 30 K = 1, N
*
*              Update the upper triangle of A(1:k,1:k)
*
               AKK = A( K, K )
               BKK = B( K, K )
               CALL DTRMV( UPLO, 'No transpose', 'Non-unit', K-1, B,
     $                     LDB, A( 1, K ), 1 )
               CT = HALF*AKK
               CALL DAXPY( K-1, CT, B( 1, K ), 1, A( 1, K ), 1 )
               CALL DSYR2( UPLO, K-1, ONE, A( 1, K ), 1, B( 1, K ),
     $                     1,
     $                     A, LDA )
               CALL DAXPY( K-1, CT, B( 1, K ), 1, A( 1, K ), 1 )
               CALL DSCAL( K-1, BKK, A( 1, K ), 1 )
               A( K, K ) = AKK*BKK**2
   30       CONTINUE
         ELSE
*
*           Compute L**T *A*L
*
            DO 40 K = 1, N
*
*              Update the lower triangle of A(1:k,1:k)
*
               AKK = A( K, K )
               BKK = B( K, K )
               CALL DTRMV( UPLO, 'Transpose', 'Non-unit', K-1, B,
     $                     LDB,
     $                     A( K, 1 ), LDA )
               CT = HALF*AKK
               CALL DAXPY( K-1, CT, B( K, 1 ), LDB, A( K, 1 ), LDA )
               CALL DSYR2( UPLO, K-1, ONE, A( K, 1 ), LDA, B( K, 1 ),
     $                     LDB, A, LDA )
               CALL DAXPY( K-1, CT, B( K, 1 ), LDB, A( K, 1 ), LDA )
               CALL DSCAL( K-1, BKK, A( K, 1 ), LDA )
               A( K, K ) = AKK*BKK**2
   40       CONTINUE
         END IF
      END IF
      RETURN
*
*     End of DSYGS2
*
      END
*> \brief \b DSYGST
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYGST + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsygst.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsygst.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsygst.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, ITYPE, LDA, LDB, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYGST reduces a real symmetric-definite generalized eigenproblem
*> to standard form.
*>
*> If ITYPE = 1, the problem is A*x = lambda*B*x,
*> and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
*>
*> If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
*> B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.
*>
*> B must have been previously factorized as U**T*U or L*L**T by DPOTRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ITYPE
*> \verbatim
*>          ITYPE is INTEGER
*>          = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
*>          = 2 or 3: compute U*A*U**T or L**T*A*L.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored and B is factored as
*>                  U**T*U;
*>          = 'L':  Lower triangle of A is stored and B is factored as
*>                  L*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A and B.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*>          N-by-N upper triangular part of A contains the upper
*>          triangular part of the matrix A, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading N-by-N lower triangular part of A contains the lower
*>          triangular part of the matrix A, and the strictly upper
*>          triangular part of A is not referenced.
*>
*>          On exit, if INFO = 0, the transformed matrix, stored in the
*>          same format as A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,N)
*>          The triangular factor from the Cholesky factorization of B,
*>          as returned by DPOTRF.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hegst
*
*  =====================================================================
      SUBROUTINE DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, ITYPE, LDA, LDB, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, HALF
      PARAMETER          ( ONE = 1.0D0, HALF = 0.5D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            K, KB, NB
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSYGS2, DSYMM, DSYR2K, DTRMM, DTRSM,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
         INFO = -1
      ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYGST', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Determine the block size for this environment.
*
      NB = ILAENV( 1, 'DSYGST', UPLO, N, -1, -1, -1 )
*
      IF( NB.LE.1 .OR. NB.GE.N ) THEN
*
*        Use unblocked code
*
         CALL DSYGS2( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
      ELSE
*
*        Use blocked code
*
         IF( ITYPE.EQ.1 ) THEN
            IF( UPPER ) THEN
*
*              Compute inv(U**T)*A*inv(U)
*
               DO 10 K = 1, N, NB
                  KB = MIN( N-K+1, NB )
*
*                 Update the upper triangle of A(k:n,k:n)
*
                  CALL DSYGS2( ITYPE, UPLO, KB, A( K, K ), LDA,
     $                         B( K, K ), LDB, INFO )
                  IF( K+KB.LE.N ) THEN
                     CALL DTRSM( 'Left', UPLO, 'Transpose',
     $                           'Non-unit',
     $                           KB, N-K-KB+1, ONE, B( K, K ), LDB,
     $                           A( K, K+KB ), LDA )
                     CALL DSYMM( 'Left', UPLO, KB, N-K-KB+1, -HALF,
     $                           A( K, K ), LDA, B( K, K+KB ), LDB, ONE,
     $                           A( K, K+KB ), LDA )
                     CALL DSYR2K( UPLO, 'Transpose', N-K-KB+1, KB,
     $                            -ONE,
     $                            A( K, K+KB ), LDA, B( K, K+KB ), LDB,
     $                            ONE, A( K+KB, K+KB ), LDA )
                     CALL DSYMM( 'Left', UPLO, KB, N-K-KB+1, -HALF,
     $                           A( K, K ), LDA, B( K, K+KB ), LDB, ONE,
     $                           A( K, K+KB ), LDA )
                     CALL DTRSM( 'Right', UPLO, 'No transpose',
     $                           'Non-unit', KB, N-K-KB+1, ONE,
     $                           B( K+KB, K+KB ), LDB, A( K, K+KB ),
     $                           LDA )
                  END IF
   10          CONTINUE
            ELSE
*
*              Compute inv(L)*A*inv(L**T)
*
               DO 20 K = 1, N, NB
                  KB = MIN( N-K+1, NB )
*
*                 Update the lower triangle of A(k:n,k:n)
*
                  CALL DSYGS2( ITYPE, UPLO, KB, A( K, K ), LDA,
     $                         B( K, K ), LDB, INFO )
                  IF( K+KB.LE.N ) THEN
                     CALL DTRSM( 'Right', UPLO, 'Transpose',
     $                           'Non-unit',
     $                           N-K-KB+1, KB, ONE, B( K, K ), LDB,
     $                           A( K+KB, K ), LDA )
                     CALL DSYMM( 'Right', UPLO, N-K-KB+1, KB, -HALF,
     $                           A( K, K ), LDA, B( K+KB, K ), LDB, ONE,
     $                           A( K+KB, K ), LDA )
                     CALL DSYR2K( UPLO, 'No transpose', N-K-KB+1, KB,
     $                            -ONE, A( K+KB, K ), LDA, B( K+KB, K ),
     $                            LDB, ONE, A( K+KB, K+KB ), LDA )
                     CALL DSYMM( 'Right', UPLO, N-K-KB+1, KB, -HALF,
     $                           A( K, K ), LDA, B( K+KB, K ), LDB, ONE,
     $                           A( K+KB, K ), LDA )
                     CALL DTRSM( 'Left', UPLO, 'No transpose',
     $                           'Non-unit', N-K-KB+1, KB, ONE,
     $                           B( K+KB, K+KB ), LDB, A( K+KB, K ),
     $                           LDA )
                  END IF
   20          CONTINUE
            END IF
         ELSE
            IF( UPPER ) THEN
*
*              Compute U*A*U**T
*
               DO 30 K = 1, N, NB
                  KB = MIN( N-K+1, NB )
*
*                 Update the upper triangle of A(1:k+kb-1,1:k+kb-1)
*
                  CALL DTRMM( 'Left', UPLO, 'No transpose',
     $                        'Non-unit',
     $                        K-1, KB, ONE, B, LDB, A( 1, K ), LDA )
                  CALL DSYMM( 'Right', UPLO, K-1, KB, HALF, A( K,
     $                        K ),
     $                        LDA, B( 1, K ), LDB, ONE, A( 1, K ), LDA )
                  CALL DSYR2K( UPLO, 'No transpose', K-1, KB, ONE,
     $                         A( 1, K ), LDA, B( 1, K ), LDB, ONE, A,
     $                         LDA )
                  CALL DSYMM( 'Right', UPLO, K-1, KB, HALF, A( K,
     $                        K ),
     $                        LDA, B( 1, K ), LDB, ONE, A( 1, K ), LDA )
                  CALL DTRMM( 'Right', UPLO, 'Transpose', 'Non-unit',
     $                        K-1, KB, ONE, B( K, K ), LDB, A( 1, K ),
     $                        LDA )
                  CALL DSYGS2( ITYPE, UPLO, KB, A( K, K ), LDA,
     $                         B( K, K ), LDB, INFO )
   30          CONTINUE
            ELSE
*
*              Compute L**T*A*L
*
               DO 40 K = 1, N, NB
                  KB = MIN( N-K+1, NB )
*
*                 Update the lower triangle of A(1:k+kb-1,1:k+kb-1)
*
                  CALL DTRMM( 'Right', UPLO, 'No transpose',
     $                        'Non-unit',
     $                        KB, K-1, ONE, B, LDB, A( K, 1 ), LDA )
                  CALL DSYMM( 'Left', UPLO, KB, K-1, HALF, A( K, K ),
     $                        LDA, B( K, 1 ), LDB, ONE, A( K, 1 ), LDA )
                  CALL DSYR2K( UPLO, 'Transpose', K-1, KB, ONE,
     $                         A( K, 1 ), LDA, B( K, 1 ), LDB, ONE, A,
     $                         LDA )
                  CALL DSYMM( 'Left', UPLO, KB, K-1, HALF, A( K, K ),
     $                        LDA, B( K, 1 ), LDB, ONE, A( K, 1 ), LDA )
                  CALL DTRMM( 'Left', UPLO, 'Transpose', 'Non-unit',
     $                        KB,
     $                        K-1, ONE, B( K, K ), LDB, A( K, 1 ), LDA )
                  CALL DSYGS2( ITYPE, UPLO, KB, A( K, K ), LDA,
     $                         B( K, K ), LDB, INFO )
   40          CONTINUE
            END IF
         END IF
      END IF
      RETURN
*
*     End of DSYGST
*
      END
*> \brief \b DSYGV
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYGV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsygv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsygv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsygv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYGV( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,
*                         LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, UPLO
*       INTEGER            INFO, ITYPE, LDA, LDB, LWORK, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), W( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYGV computes all the eigenvalues, and optionally, the eigenvectors
*> of a real generalized symmetric-definite eigenproblem, of the form
*> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
*> Here A and B are assumed to be symmetric and B is also
*> positive definite.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ITYPE
*> \verbatim
*>          ITYPE is INTEGER
*>          Specifies the problem type to be solved:
*>          = 1:  A*x = (lambda)*B*x
*>          = 2:  A*B*x = (lambda)*x
*>          = 3:  B*A*x = (lambda)*x
*> \endverbatim
*>
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangles of A and B are stored;
*>          = 'L':  Lower triangles of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A and B.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the
*>          leading N-by-N upper triangular part of A contains the
*>          upper triangular part of the matrix A.  If UPLO = 'L',
*>          the leading N-by-N lower triangular part of A contains
*>          the lower triangular part of the matrix A.
*>
*>          On exit, if JOBZ = 'V', then if INFO = 0, A contains the
*>          matrix Z of eigenvectors.  The eigenvectors are normalized
*>          as follows:
*>          if ITYPE = 1 or 2, Z**T*B*Z = I;
*>          if ITYPE = 3, Z**T*inv(B)*Z = I.
*>          If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
*>          or the lower triangle (if UPLO='L') of A, including the
*>          diagonal, is destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB, N)
*>          On entry, the symmetric positive definite matrix B.
*>          If UPLO = 'U', the leading N-by-N upper triangular part of B
*>          contains the upper triangular part of the matrix B.
*>          If UPLO = 'L', the leading N-by-N lower triangular part of B
*>          contains the lower triangular part of the matrix B.
*>
*>          On exit, if INFO <= N, the part of B containing the matrix is
*>          overwritten by the triangular factor U or L from the Cholesky
*>          factorization B = U**T*U or B = L*L**T.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The length of the array WORK.  LWORK >= max(1,3*N-1).
*>          For optimal efficiency, LWORK >= (NB+2)*N,
*>          where NB is the blocksize for DSYTRD returned by ILAENV.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  DPOTRF or DSYEV returned an error code:
*>             <= N:  if INFO = i, DSYEV failed to converge;
*>                    i off-diagonal elements of an intermediate
*>                    tridiagonal form did not converge to zero;
*>             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
*>                    principal minor of order i of B is not positive.
*>                    The factorization of B could not be completed and
*>                    no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hegv
*
*  =====================================================================
      SUBROUTINE DSYGV( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W,
     $                  WORK,
     $                  LWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, UPLO
      INTEGER            INFO, ITYPE, LDA, LDB, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), W( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, UPPER, WANTZ
      CHARACTER          TRANS
      INTEGER            LWKMIN, LWKOPT, NB, NEIG
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DPOTRF, DSYEV, DSYGST, DTRMM, DTRSM,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
      UPPER = LSAME( UPLO, 'U' )
      LQUERY = ( LWORK.EQ.-1 )
*
      INFO = 0
      IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
         INFO = -1
      ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -2
      ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -6
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -8
      END IF
*
      IF( INFO.EQ.0 ) THEN
         LWKMIN = MAX( 1, 3*N - 1 )
         NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
         LWKOPT = MAX( LWKMIN, ( NB + 2 )*N )
         WORK( 1 ) = LWKOPT
*
         IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
            INFO = -11
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYGV ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Form a Cholesky factorization of B.
*
      CALL DPOTRF( UPLO, N, B, LDB, INFO )
      IF( INFO.NE.0 ) THEN
         INFO = N + INFO
         RETURN
      END IF
*
*     Transform problem to standard eigenvalue problem and solve.
*
      CALL DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
      CALL DSYEV( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, INFO )
*
      IF( WANTZ ) THEN
*
*        Backtransform eigenvectors to the original problem.
*
         NEIG = N
         IF( INFO.GT.0 )
     $      NEIG = INFO - 1
         IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
*
*           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
*           backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
*
            IF( UPPER ) THEN
               TRANS = 'N'
            ELSE
               TRANS = 'T'
            END IF
*
            CALL DTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, NEIG,
     $                  ONE,
     $                  B, LDB, A, LDA )
*
         ELSE IF( ITYPE.EQ.3 ) THEN
*
*           For B*A*x=(lambda)*x;
*           backtransform eigenvectors: x = L*y or U**T*y
*
            IF( UPPER ) THEN
               TRANS = 'T'
            ELSE
               TRANS = 'N'
            END IF
*
            CALL DTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, NEIG,
     $                  ONE,
     $                  B, LDB, A, LDA )
         END IF
      END IF
*
      WORK( 1 ) = LWKOPT
      RETURN
*
*     End of DSYGV
*
      END
*> \brief \b DSYGVD
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYGVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsygvd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsygvd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsygvd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYGVD( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,
*                          LWORK, IWORK, LIWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, UPLO
*       INTEGER            INFO, ITYPE, LDA, LDB, LIWORK, LWORK, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), W( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYGVD computes all the eigenvalues, and optionally, the eigenvectors
*> of a real generalized symmetric-definite eigenproblem, of the form
*> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
*> B are assumed to be symmetric and B is also positive definite.
*> If eigenvectors are desired, it uses a divide and conquer algorithm.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ITYPE
*> \verbatim
*>          ITYPE is INTEGER
*>          Specifies the problem type to be solved:
*>          = 1:  A*x = (lambda)*B*x
*>          = 2:  A*B*x = (lambda)*x
*>          = 3:  B*A*x = (lambda)*x
*> \endverbatim
*>
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangles of A and B are stored;
*>          = 'L':  Lower triangles of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A and B.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the
*>          leading N-by-N upper triangular part of A contains the
*>          upper triangular part of the matrix A.  If UPLO = 'L',
*>          the leading N-by-N lower triangular part of A contains
*>          the lower triangular part of the matrix A.
*>
*>          On exit, if JOBZ = 'V', then if INFO = 0, A contains the
*>          matrix Z of eigenvectors.  The eigenvectors are normalized
*>          as follows:
*>          if ITYPE = 1 or 2, Z**T*B*Z = I;
*>          if ITYPE = 3, Z**T*inv(B)*Z = I.
*>          If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
*>          or the lower triangle (if UPLO='L') of A, including the
*>          diagonal, is destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB, N)
*>          On entry, the symmetric matrix B.  If UPLO = 'U', the
*>          leading N-by-N upper triangular part of B contains the
*>          upper triangular part of the matrix B.  If UPLO = 'L',
*>          the leading N-by-N lower triangular part of B contains
*>          the lower triangular part of the matrix B.
*>
*>          On exit, if INFO <= N, the part of B containing the matrix is
*>          overwritten by the triangular factor U or L from the Cholesky
*>          factorization B = U**T*U or B = L*L**T.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If N <= 1,               LWORK >= 1.
*>          If JOBZ = 'N' and N > 1, LWORK >= 2*N+1.
*>          If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal sizes of the WORK and IWORK
*>          arrays, returns these values as the first entries of the WORK
*>          and IWORK arrays, and no error message related to LWORK or
*>          LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK.
*>          If N <= 1,                LIWORK >= 1.
*>          If JOBZ  = 'N' and N > 1, LIWORK >= 1.
*>          If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.
*>
*>          If LIWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the optimal sizes of the WORK and
*>          IWORK arrays, returns these values as the first entries of
*>          the WORK and IWORK arrays, and no error message related to
*>          LWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  DPOTRF or DSYEVD returned an error code:
*>             <= N:  if INFO = i and JOBZ = 'N', then the algorithm
*>                    failed to converge; i off-diagonal elements of an
*>                    intermediate tridiagonal form did not converge to
*>                    zero;
*>                    if INFO = i and JOBZ = 'V', then the algorithm
*>                    failed to compute an eigenvalue while working on
*>                    the submatrix lying in rows and columns INFO/(N+1)
*>                    through mod(INFO,N+1);
*>             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
*>                    principal minor of order i of B is not positive.
*>                    The factorization of B could not be completed and
*>                    no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hegvd
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  Modified so that no backsubstitution is performed if DSYEVD fails to
*>  converge (NEIG in old code could be greater than N causing out of
*>  bounds reference to A - reported by Ralf Meyer).  Also corrected the
*>  description of INFO and the test on ITYPE. Sven, 16 Feb 05.
*> \endverbatim
*
*> \par Contributors:
*  ==================
*>
*>     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*>
*  =====================================================================
      SUBROUTINE DSYGVD( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W,
     $                   WORK,
     $                   LWORK, IWORK, LIWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, UPLO
      INTEGER            INFO, ITYPE, LDA, LDB, LIWORK, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), W( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, UPPER, WANTZ
      CHARACTER          TRANS
      INTEGER            LIOPT, LIWMIN, LOPT, LWMIN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DPOTRF, DSYEVD, DSYGST, DTRMM, DTRSM,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE, MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
      UPPER = LSAME( UPLO, 'U' )
      LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
      INFO = 0
      IF( N.LE.1 ) THEN
         LIWMIN = 1
         LWMIN = 1
      ELSE IF( WANTZ ) THEN
         LIWMIN = 3 + 5*N
         LWMIN = 1 + 6*N + 2*N**2
      ELSE
         LIWMIN = 1
         LWMIN = 2*N + 1
      END IF
      LOPT = LWMIN
      LIOPT = LIWMIN
      IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
         INFO = -1
      ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -2
      ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -6
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -8
      END IF
*
      IF( INFO.EQ.0 ) THEN
         WORK( 1 ) = LOPT
         IWORK( 1 ) = LIOPT
*
         IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -11
         ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -13
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYGVD', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Form a Cholesky factorization of B.
*
      CALL DPOTRF( UPLO, N, B, LDB, INFO )
      IF( INFO.NE.0 ) THEN
         INFO = N + INFO
         RETURN
      END IF
*
*     Transform problem to standard eigenvalue problem and solve.
*
      CALL DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
      CALL DSYEVD( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK,
     $             LIWORK,
     $             INFO )
      LOPT = INT( MAX( DBLE( LOPT ), DBLE( WORK( 1 ) ) ) )
      LIOPT = INT( MAX( DBLE( LIOPT ), DBLE( IWORK( 1 ) ) ) )
*
      IF( WANTZ .AND. INFO.EQ.0 ) THEN
*
*        Backtransform eigenvectors to the original problem.
*
         IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
*
*           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
*           backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
*
            IF( UPPER ) THEN
               TRANS = 'N'
            ELSE
               TRANS = 'T'
            END IF
*
            CALL DTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, N, ONE,
     $                  B, LDB, A, LDA )
*
         ELSE IF( ITYPE.EQ.3 ) THEN
*
*           For B*A*x=(lambda)*x;
*           backtransform eigenvectors: x = L*y or U**T*y
*
            IF( UPPER ) THEN
               TRANS = 'T'
            ELSE
               TRANS = 'N'
            END IF
*
            CALL DTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, N, ONE,
     $                  B, LDB, A, LDA )
         END IF
      END IF
*
      WORK( 1 ) = LOPT
      IWORK( 1 ) = LIOPT
*
      RETURN
*
*     End of DSYGVD
*
      END
*> \brief \b DSYGVX
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYGVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsygvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsygvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsygvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
*                          VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
*                          LWORK, IWORK, IFAIL, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, RANGE, UPLO
*       INTEGER            IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
*       DOUBLE PRECISION   ABSTOL, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            IFAIL( * ), IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), W( * ), WORK( * ),
*      $                   Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYGVX computes selected eigenvalues, and optionally, eigenvectors
*> of a real generalized symmetric-definite eigenproblem, of the form
*> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A
*> and B are assumed to be symmetric and B is also positive definite.
*> Eigenvalues and eigenvectors can be selected by specifying either a
*> range of values or a range of indices for the desired eigenvalues.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ITYPE
*> \verbatim
*>          ITYPE is INTEGER
*>          Specifies the problem type to be solved:
*>          = 1:  A*x = (lambda)*B*x
*>          = 2:  A*B*x = (lambda)*x
*>          = 3:  B*A*x = (lambda)*x
*> \endverbatim
*>
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*>          RANGE is CHARACTER*1
*>          = 'A': all eigenvalues will be found.
*>          = 'V': all eigenvalues in the half-open interval (VL,VU]
*>                 will be found.
*>          = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A and B are stored;
*>          = 'L':  Lower triangle of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix pencil (A,B).  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the
*>          leading N-by-N upper triangular part of A contains the
*>          upper triangular part of the matrix A.  If UPLO = 'L',
*>          the leading N-by-N lower triangular part of A contains
*>          the lower triangular part of the matrix A.
*>
*>          On exit, the lower triangle (if UPLO='L') or the upper
*>          triangle (if UPLO='U') of A, including the diagonal, is
*>          destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB, N)
*>          On entry, the symmetric matrix B.  If UPLO = 'U', the
*>          leading N-by-N upper triangular part of B contains the
*>          upper triangular part of the matrix B.  If UPLO = 'L',
*>          the leading N-by-N lower triangular part of B contains
*>          the lower triangular part of the matrix B.
*>
*>          On exit, if INFO <= N, the part of B containing the matrix is
*>          overwritten by the triangular factor U or L from the Cholesky
*>          factorization B = U**T*U or B = L*L**T.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is DOUBLE PRECISION
*>          If RANGE='V', the lower bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*>          VU is DOUBLE PRECISION
*>          If RANGE='V', the upper bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*>          IL is INTEGER
*>          If RANGE='I', the index of the
*>          smallest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*>          IU is INTEGER
*>          If RANGE='I', the index of the
*>          largest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*>          ABSTOL is DOUBLE PRECISION
*>          The absolute error tolerance for the eigenvalues.
*>          An approximate eigenvalue is accepted as converged
*>          when it is determined to lie in an interval [a,b]
*>          of width less than or equal to
*>
*>                  ABSTOL + EPS *   max( |a|,|b| ) ,
*>
*>          where EPS is the machine precision.  If ABSTOL is less than
*>          or equal to zero, then  EPS*|T|  will be used in its place,
*>          where |T| is the 1-norm of the tridiagonal matrix obtained
*>          by reducing C to tridiagonal form, where C is the symmetric
*>          matrix of the standard symmetric problem to which the
*>          generalized problem is transformed.
*>
*>          Eigenvalues will be computed most accurately when ABSTOL is
*>          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*>          If this routine returns with INFO>0, indicating that some
*>          eigenvectors did not converge, try setting ABSTOL to
*>          2*DLAMCH('S').
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The total number of eigenvalues found.  0 <= M <= N.
*>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          On normal exit, the first M elements contain the selected
*>          eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
*>          If JOBZ = 'N', then Z is not referenced.
*>          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*>          contain the orthonormal eigenvectors of the matrix A
*>          corresponding to the selected eigenvalues, with the i-th
*>          column of Z holding the eigenvector associated with W(i).
*>          The eigenvectors are normalized as follows:
*>          if ITYPE = 1 or 2, Z**T*B*Z = I;
*>          if ITYPE = 3, Z**T*inv(B)*Z = I.
*>
*>          If an eigenvector fails to converge, then that column of Z
*>          contains the latest approximation to the eigenvector, and the
*>          index of the eigenvector is returned in IFAIL.
*>          Note: the user must ensure that at least max(1,M) columns are
*>          supplied in the array Z; if RANGE = 'V', the exact value of M
*>          is not known in advance and an upper bound must be used.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The length of the array WORK.  LWORK >= max(1,8*N).
*>          For optimal efficiency, LWORK >= (NB+3)*N,
*>          where NB is the blocksize for DSYTRD returned by ILAENV.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*>          IFAIL is INTEGER array, dimension (N)
*>          If JOBZ = 'V', then if INFO = 0, the first M elements of
*>          IFAIL are zero.  If INFO > 0, then IFAIL contains the
*>          indices of the eigenvectors that failed to converge.
*>          If JOBZ = 'N', then IFAIL is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  DPOTRF or DSYEVX returned an error code:
*>             <= N:  if INFO = i, DSYEVX failed to converge;
*>                    i eigenvectors failed to converge.  Their indices
*>                    are stored in array IFAIL.
*>             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
*>                    principal minor of order i of B is not positive.
*>                    The factorization of B could not be completed and
*>                    no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hegvx
*
*> \par Contributors:
*  ==================
*>
*>     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
*  =====================================================================
      SUBROUTINE DSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
     $                   VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
     $                   LWORK, IWORK, IFAIL, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, RANGE, UPLO
      INTEGER            IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
      DOUBLE PRECISION   ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            IFAIL( * ), IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), W( * ), WORK( * ),
     $                   Z( LDZ, * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ
      CHARACTER          TRANS
      INTEGER            LWKMIN, LWKOPT, NB
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DPOTRF, DSYEVX, DSYGST, DTRMM, DTRSM,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      UPPER = LSAME( UPLO, 'U' )
      WANTZ = LSAME( JOBZ, 'V' )
      ALLEIG = LSAME( RANGE, 'A' )
      VALEIG = LSAME( RANGE, 'V' )
      INDEIG = LSAME( RANGE, 'I' )
      LQUERY = ( LWORK.EQ.-1 )
*
      INFO = 0
      IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
         INFO = -1
      ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -2
      ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
         INFO = -3
      ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
         INFO = -4
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -9
      ELSE
         IF( VALEIG ) THEN
            IF( N.GT.0 .AND. VU.LE.VL )
     $         INFO = -11
         ELSE IF( INDEIG ) THEN
            IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
               INFO = -12
            ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
               INFO = -13
            END IF
         END IF
      END IF
      IF (INFO.EQ.0) THEN
         IF (LDZ.LT.1 .OR. (WANTZ .AND. LDZ.LT.N)) THEN
            INFO = -18
         END IF
      END IF
*
      IF( INFO.EQ.0 ) THEN
         LWKMIN = MAX( 1, 8*N )
         NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
         LWKOPT = MAX( LWKMIN, ( NB + 3 )*N )
         WORK( 1 ) = LWKOPT
*
         IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
            INFO = -20
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYGVX', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      M = 0
      IF( N.EQ.0 ) THEN
         RETURN
      END IF
*
*     Form a Cholesky factorization of B.
*
      CALL DPOTRF( UPLO, N, B, LDB, INFO )
      IF( INFO.NE.0 ) THEN
         INFO = N + INFO
         RETURN
      END IF
*
*     Transform problem to standard eigenvalue problem and solve.
*
      CALL DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
      CALL DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
     $             ABSTOL,
     $             M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO )
*
      IF( WANTZ ) THEN
*
*        Backtransform eigenvectors to the original problem.
*
         IF( INFO.GT.0 )
     $      M = INFO - 1
         IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
*
*           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
*           backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
*
            IF( UPPER ) THEN
               TRANS = 'N'
            ELSE
               TRANS = 'T'
            END IF
*
            CALL DTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, M, ONE,
     $                  B,
     $                  LDB, Z, LDZ )
*
         ELSE IF( ITYPE.EQ.3 ) THEN
*
*           For B*A*x=(lambda)*x;
*           backtransform eigenvectors: x = L*y or U**T*y
*
            IF( UPPER ) THEN
               TRANS = 'T'
            ELSE
               TRANS = 'N'
            END IF
*
            CALL DTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, M, ONE,
     $                  B,
     $                  LDB, Z, LDZ )
         END IF
      END IF
*
*     Set WORK(1) to optimal workspace size.
*
      WORK( 1 ) = LWKOPT
*
      RETURN
*
*     End of DSYGVX
*
      END
*> \brief \b DSYRFS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYRFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyrfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyrfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyrfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
*                          X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
*      $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYRFS improves the computed solution to a system of linear
*> equations when the coefficient matrix is symmetric indefinite, and
*> provides error bounds and backward error estimates for the solution.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices B and X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The symmetric matrix A.  If UPLO = 'U', the leading N-by-N
*>          upper triangular part of A contains the upper triangular part
*>          of the matrix A, and the strictly lower triangular part of A
*>          is not referenced.  If UPLO = 'L', the leading N-by-N lower
*>          triangular part of A contains the lower triangular part of
*>          the matrix A, and the strictly upper triangular part of A is
*>          not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] AF
*> \verbatim
*>          AF is DOUBLE PRECISION array, dimension (LDAF,N)
*>          The factored form of the matrix A.  AF contains the block
*>          diagonal matrix D and the multipliers used to obtain the
*>          factor U or L from the factorization A = U*D*U**T or
*>          A = L*D*L**T as computed by DSYTRF.
*> \endverbatim
*>
*> \param[in] LDAF
*> \verbatim
*>          LDAF is INTEGER
*>          The leading dimension of the array AF.  LDAF >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the block structure of D
*>          as determined by DSYTRF.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          On entry, the solution matrix X, as computed by DSYTRS.
*>          On exit, the improved solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The estimated forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).  The estimate is as reliable as
*>          the estimate for RCOND, and is almost always a slight
*>          overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in
*>          any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  ITMAX is the maximum number of steps of iterative refinement.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup herfs
*
*  =====================================================================
      SUBROUTINE DSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B,
     $                   LDB,
     $                   X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
     $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            ITMAX
      PARAMETER          ( ITMAX = 5 )
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
      DOUBLE PRECISION   TWO
      PARAMETER          ( TWO = 2.0D+0 )
      DOUBLE PRECISION   THREE
      PARAMETER          ( THREE = 3.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            COUNT, I, J, K, KASE, NZ
      DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DLACN2, DSYMV, DSYTRS,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -10
      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
         INFO = -12
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYRFS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
         DO 10 J = 1, NRHS
            FERR( J ) = ZERO
            BERR( J ) = ZERO
   10    CONTINUE
         RETURN
      END IF
*
*     NZ = maximum number of nonzero elements in each row of A, plus 1
*
      NZ = N + 1
      EPS = DLAMCH( 'Epsilon' )
      SAFMIN = DLAMCH( 'Safe minimum' )
      SAFE1 = NZ*SAFMIN
      SAFE2 = SAFE1 / EPS
*
*     Do for each right hand side
*
      DO 140 J = 1, NRHS
*
         COUNT = 1
         LSTRES = THREE
   20    CONTINUE
*
*        Loop until stopping criterion is satisfied.
*
*        Compute residual R = B - A * X
*
         CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
         CALL DSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
     $               WORK( N+1 ), 1 )
*
*        Compute componentwise relative backward error from formula
*
*        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
*
*        where abs(Z) is the componentwise absolute value of the matrix
*        or vector Z.  If the i-th component of the denominator is less
*        than SAFE2, then SAFE1 is added to the i-th components of the
*        numerator and denominator before dividing.
*
         DO 30 I = 1, N
            WORK( I ) = ABS( B( I, J ) )
   30    CONTINUE
*
*        Compute abs(A)*abs(X) + abs(B).
*
         IF( UPPER ) THEN
            DO 50 K = 1, N
               S = ZERO
               XK = ABS( X( K, J ) )
               DO 40 I = 1, K - 1
                  WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
                  S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
   40          CONTINUE
               WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK + S
   50       CONTINUE
         ELSE
            DO 70 K = 1, N
               S = ZERO
               XK = ABS( X( K, J ) )
               WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK
               DO 60 I = K + 1, N
                  WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
                  S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
   60          CONTINUE
               WORK( K ) = WORK( K ) + S
   70       CONTINUE
         END IF
         S = ZERO
         DO 80 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
            ELSE
               S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
     $             ( WORK( I )+SAFE1 ) )
            END IF
   80    CONTINUE
         BERR( J ) = S
*
*        Test stopping criterion. Continue iterating if
*           1) The residual BERR(J) is larger than machine epsilon, and
*           2) BERR(J) decreased by at least a factor of 2 during the
*              last iteration, and
*           3) At most ITMAX iterations tried.
*
         IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
     $       COUNT.LE.ITMAX ) THEN
*
*           Update solution and try again.
*
            CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
     $                   INFO )
            CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
            LSTRES = BERR( J )
            COUNT = COUNT + 1
            GO TO 20
         END IF
*
*        Bound error from formula
*
*        norm(X - XTRUE) / norm(X) .le. FERR =
*        norm( abs(inv(A))*
*           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
*
*        where
*          norm(Z) is the magnitude of the largest component of Z
*          inv(A) is the inverse of A
*          abs(Z) is the componentwise absolute value of the matrix or
*             vector Z
*          NZ is the maximum number of nonzeros in any row of A, plus 1
*          EPS is machine epsilon
*
*        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
*        is incremented by SAFE1 if the i-th component of
*        abs(A)*abs(X) + abs(B) is less than SAFE2.
*
*        Use DLACN2 to estimate the infinity-norm of the matrix
*           inv(A) * diag(W),
*        where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
*
         DO 90 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
            ELSE
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
            END IF
   90    CONTINUE
*
         KASE = 0
  100    CONTINUE
         CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
     $                FERR( J ),
     $                KASE, ISAVE )
         IF( KASE.NE.0 ) THEN
            IF( KASE.EQ.1 ) THEN
*
*              Multiply by diag(W)*inv(A**T).
*
               CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ),
     $                      N,
     $                      INFO )
               DO 110 I = 1, N
                  WORK( N+I ) = WORK( I )*WORK( N+I )
  110          CONTINUE
            ELSE IF( KASE.EQ.2 ) THEN
*
*              Multiply by inv(A)*diag(W).
*
               DO 120 I = 1, N
                  WORK( N+I ) = WORK( I )*WORK( N+I )
  120          CONTINUE
               CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ),
     $                      N,
     $                      INFO )
            END IF
            GO TO 100
         END IF
*
*        Normalize error.
*
         LSTRES = ZERO
         DO 130 I = 1, N
            LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
  130    CONTINUE
         IF( LSTRES.NE.ZERO )
     $      FERR( J ) = FERR( J ) / LSTRES
*
  140 CONTINUE
*
      RETURN
*
*     End of DSYRFS
*
      END
*> \brief <b> DSYSV computes the solution to system of linear equations A * X = B for SY matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYSV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsysv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsysv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsysv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYSV( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK,
*                         LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, LDB, LWORK, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYSV computes the solution to a real system of linear equations
*>    A * X = B,
*> where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
*> matrices.
*>
*> The diagonal pivoting method is used to factor A as
*>    A = U * D * U**T,  if UPLO = 'U', or
*>    A = L * D * L**T,  if UPLO = 'L',
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, and D is symmetric and block diagonal with
*> 1-by-1 and 2-by-2 diagonal blocks.  The factored form of A is then
*> used to solve the system of equations A * X = B.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of linear equations, i.e., the order of the
*>          matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*>          N-by-N upper triangular part of A contains the upper
*>          triangular part of the matrix A, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading N-by-N lower triangular part of A contains the lower
*>          triangular part of the matrix A, and the strictly upper
*>          triangular part of A is not referenced.
*>
*>          On exit, if INFO = 0, the block diagonal matrix D and the
*>          multipliers used to obtain the factor U or L from the
*>          factorization A = U*D*U**T or A = L*D*L**T as computed by
*>          DSYTRF.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the block structure of D, as
*>          determined by DSYTRF.  If IPIV(k) > 0, then rows and columns
*>          k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1
*>          diagonal block.  If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0,
*>          then rows and columns k-1 and -IPIV(k) were interchanged and
*>          D(k-1:k,k-1:k) is a 2-by-2 diagonal block.  If UPLO = 'L' and
*>          IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and
*>          -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
*>          diagonal block.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the N-by-NRHS right hand side matrix B.
*>          On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The length of WORK.  LWORK >= 1, and for best performance
*>          LWORK >= max(1,N*NB), where NB is the optimal blocksize for
*>          DSYTRF.
*>          for LWORK < N, TRS will be done with Level BLAS 2
*>          for LWORK >= N, TRS will be done with Level BLAS 3
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*>          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization
*>               has been completed, but the block diagonal matrix D is
*>               exactly singular, so the solution could not be computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hesv
*
*  =====================================================================
      SUBROUTINE DSYSV( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK,
     $                  LWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, LDB, LWORK, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            LWKOPT
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA, DSYTRF, DSYTRS, DSYTRS2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      IF( .NOT.LSAME( UPLO, 'U' ) .AND.
     $    .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -8
      ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
         INFO = -10
      END IF
*
      IF( INFO.EQ.0 ) THEN
         IF( N.EQ.0 ) THEN
            LWKOPT = 1
         ELSE
            CALL DSYTRF( UPLO, N, A, LDA, IPIV, WORK, -1, INFO )
            LWKOPT = INT( WORK( 1 ) )
         END IF
         WORK( 1 ) = LWKOPT
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYSV ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Compute the factorization A = U*D*U**T or A = L*D*L**T.
*
      CALL DSYTRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
      IF( INFO.EQ.0 ) THEN
*
*        Solve the system A*X = B, overwriting B with X.
*
         IF ( LWORK.LT.N ) THEN
*
*        Solve with TRS ( Use Level BLAS 2)
*
            CALL DSYTRS( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
*
         ELSE
*
*        Solve with TRS2 ( Use Level BLAS 3)
*
            CALL DSYTRS2( UPLO,N,NRHS,A,LDA,IPIV,B,LDB,WORK,INFO )
*
         END IF
*
      END IF
*
      WORK( 1 ) = LWKOPT
*
      RETURN
*
*     End of DSYSV
*
      END
*> \brief <b> DSYSVX computes the solution to system of linear equations A * X = B for SY matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYSVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsysvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsysvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsysvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B,
*                          LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK,
*                          IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          FACT, UPLO
*       INTEGER            INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
*       DOUBLE PRECISION   RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
*      $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYSVX uses the diagonal pivoting factorization to compute the
*> solution to a real system of linear equations A * X = B,
*> where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
*> matrices.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
*  =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'N', the diagonal pivoting method is used to factor A.
*>    The form of the factorization is
*>       A = U * D * U**T,  if UPLO = 'U', or
*>       A = L * D * L**T,  if UPLO = 'L',
*>    where U (or L) is a product of permutation and unit upper (lower)
*>    triangular matrices, and D is symmetric and block diagonal with
*>    1-by-1 and 2-by-2 diagonal blocks.
*>
*> 2. If some D(i,i)=0, so that D is exactly singular, then the routine
*>    returns with INFO = i. Otherwise, the factored form of A is used
*>    to estimate the condition number of the matrix A.  If the
*>    reciprocal of the condition number is less than machine precision,
*>    INFO = N+1 is returned as a warning, but the routine still goes on
*>    to solve for X and compute error bounds as described below.
*>
*> 3. The system of equations is solved for X using the factored form
*>    of A.
*>
*> 4. Iterative refinement is applied to improve the computed solution
*>    matrix and calculate error bounds and backward error estimates
*>    for it.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] FACT
*> \verbatim
*>          FACT is CHARACTER*1
*>          Specifies whether or not the factored form of A has been
*>          supplied on entry.
*>          = 'F':  On entry, AF and IPIV contain the factored form of
*>                  A.  AF and IPIV will not be modified.
*>          = 'N':  The matrix A will be copied to AF and factored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of linear equations, i.e., the order of the
*>          matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices B and X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The symmetric matrix A.  If UPLO = 'U', the leading N-by-N
*>          upper triangular part of A contains the upper triangular part
*>          of the matrix A, and the strictly lower triangular part of A
*>          is not referenced.  If UPLO = 'L', the leading N-by-N lower
*>          triangular part of A contains the lower triangular part of
*>          the matrix A, and the strictly upper triangular part of A is
*>          not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] AF
*> \verbatim
*>          AF is DOUBLE PRECISION array, dimension (LDAF,N)
*>          If FACT = 'F', then AF is an input argument and on entry
*>          contains the block diagonal matrix D and the multipliers used
*>          to obtain the factor U or L from the factorization
*>          A = U*D*U**T or A = L*D*L**T as computed by DSYTRF.
*>
*>          If FACT = 'N', then AF is an output argument and on exit
*>          returns the block diagonal matrix D and the multipliers used
*>          to obtain the factor U or L from the factorization
*>          A = U*D*U**T or A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] LDAF
*> \verbatim
*>          LDAF is INTEGER
*>          The leading dimension of the array AF.  LDAF >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          If FACT = 'F', then IPIV is an input argument and on entry
*>          contains details of the interchanges and the block structure
*>          of D, as determined by DSYTRF.
*>          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*>          interchanged and D(k,k) is a 1-by-1 diagonal block.
*>          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
*>          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
*>          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
*>          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
*>          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*>
*>          If FACT = 'N', then IPIV is an output argument and on exit
*>          contains details of the interchanges and the block structure
*>          of D, as determined by DSYTRF.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          The N-by-NRHS right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The estimate of the reciprocal condition number of the matrix
*>          A.  If RCOND is less than the machine precision (in
*>          particular, if RCOND = 0), the matrix is singular to working
*>          precision.  This condition is indicated by a return code of
*>          INFO > 0.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The estimated forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).  The estimate is as reliable as
*>          the estimate for RCOND, and is almost always a slight
*>          overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in
*>          any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The length of WORK.  LWORK >= max(1,3*N), and for best
*>          performance, when FACT = 'N', LWORK >= max(1,3*N,N*NB), where
*>          NB is the optimal blocksize for DSYTRF.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*>          > 0: if INFO = i, and i is
*>                <= N:  D(i,i) is exactly zero.  The factorization
*>                       has been completed but the factor D is exactly
*>                       singular, so the solution and error bounds could
*>                       not be computed. RCOND = 0 is returned.
*>                = N+1: D is nonsingular, but RCOND is less than machine
*>                       precision, meaning that the matrix is singular
*>                       to working precision.  Nevertheless, the
*>                       solution and error bounds are computed because
*>                       there are a number of situations where the
*>                       computed solution can be more accurate than the
*>                       value of RCOND would suggest.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hesvx
*
*  =====================================================================
      SUBROUTINE DSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV,
     $                   B,
     $                   LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK,
     $                   IWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          FACT, UPLO
      INTEGER            INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
     $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, NOFACT
      INTEGER            LWKMIN, LWKOPT, NB
      DOUBLE PRECISION   ANORM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANSY
      EXTERNAL           LSAME, ILAENV, DLAMCH, DLANSY
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACPY, DSYCON, DSYRFS, DSYTRF, DSYTRS,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NOFACT = LSAME( FACT, 'N' )
      LQUERY = ( LWORK.EQ.-1 )
      LWKMIN = MAX( 1, 3*N )
      IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND.
     $         .NOT.LSAME( UPLO, 'L' ) )
     $          THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -6
      ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
         INFO = -8
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -11
      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
         INFO = -13
      ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
         INFO = -18
      END IF
*
      IF( INFO.EQ.0 ) THEN
         LWKOPT = LWKMIN
         IF( NOFACT ) THEN
            NB = ILAENV( 1, 'DSYTRF', UPLO, N, -1, -1, -1 )
            LWKOPT = MAX( LWKOPT, N*NB )
         END IF
         WORK( 1 ) = LWKOPT
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYSVX', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
      IF( NOFACT ) THEN
*
*        Compute the factorization A = U*D*U**T or A = L*D*L**T.
*
         CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF )
         CALL DSYTRF( UPLO, N, AF, LDAF, IPIV, WORK, LWORK, INFO )
*
*        Return if INFO is non-zero.
*
         IF( INFO.GT.0 )THEN
            RCOND = ZERO
            RETURN
         END IF
      END IF
*
*     Compute the norm of the matrix A.
*
      ANORM = DLANSY( 'I', UPLO, N, A, LDA, WORK )
*
*     Compute the reciprocal of the condition number of A.
*
      CALL DSYCON( UPLO, N, AF, LDAF, IPIV, ANORM, RCOND, WORK,
     $             IWORK,
     $             INFO )
*
*     Compute the solution vectors X.
*
      CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
      CALL DSYTRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
*
*     Use iterative refinement to improve the computed solutions and
*     compute error bounds and backward error estimates for them.
*
      CALL DSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
     $             LDX, FERR, BERR, WORK, IWORK, INFO )
*
*     Set INFO = N+1 if the matrix is singular to working precision.
*
      IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
     $   INFO = N + 1
*
      WORK( 1 ) = LWKOPT
*
      RETURN
*
*     End of DSYSVX
*
      END
*> \brief \b DSYSWAPR applies an elementary permutation on the rows and columns of a symmetric matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYSWAPR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyswapr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyswapr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyswapr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYSWAPR( UPLO, N, A, LDA, I1, I2)
*
*       .. Scalar Arguments ..
*       CHARACTER        UPLO
*       INTEGER          I1, I2, LDA, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION A( LDA, N )
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYSWAPR applies an elementary permutation on the rows and the columns of
*> a symmetric matrix.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the details of the factorization are stored
*>          as an upper or lower triangular matrix.
*>          = 'U':  Upper triangular, form is A = U*D*U**T;
*>          = 'L':  Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,*)
*>          On entry, the N-by-N matrix A. On exit, the permuted matrix
*>          where the rows I1 and I2 and columns I1 and I2 are interchanged.
*>          If UPLO = 'U', the interchanges are applied to the upper
*>          triangular part and the strictly lower triangular part of A is
*>          not referenced; if UPLO = 'L', the interchanges are applied to
*>          the lower triangular part and the part of A above the diagonal
*>          is not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] I1
*> \verbatim
*>          I1 is INTEGER
*>          Index of the first row to swap
*> \endverbatim
*>
*> \param[in] I2
*> \verbatim
*>          I2 is INTEGER
*>          Index of the second row to swap
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup heswapr
*
*  =====================================================================
      SUBROUTINE DSYSWAPR( UPLO, N, A, LDA, I1, I2)
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER        UPLO
      INTEGER          I1, I2, LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION A( LDA, * )
*
*  =====================================================================
*
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      DOUBLE PRECISION   TMP
*
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL         DSWAP
*     ..
*     .. Executable Statements ..
*
      UPPER = LSAME( UPLO, 'U' )
      IF (UPPER) THEN
*
*         UPPER
*         first swap
*          - swap column I1 and I2 from I1 to I1-1
         CALL DSWAP( I1-1, A(1,I1), 1, A(1,I2), 1 )
*
*          second swap :
*          - swap A(I1,I1) and A(I2,I2)
*          - swap row I1 from I1+1 to I2-1 with col I2 from I1+1 to I2-1
         TMP=A(I1,I1)
         A(I1,I1)=A(I2,I2)
         A(I2,I2)=TMP
*
         CALL DSWAP( I2-I1-1, A(I1,I1+1), LDA, A(I1+1,I2), 1 )
*
*          third swap
*          - swap row I1 and I2 from I2+1 to N
         IF ( I2.LT.N )
     $      CALL DSWAP( N-I2, A(I1,I2+1), LDA, A(I2,I2+1), LDA )
*
        ELSE
*
*         LOWER
*         first swap
*          - swap row I1 and I2 from I1 to I1-1
         CALL DSWAP( I1-1, A(I1,1), LDA, A(I2,1), LDA )
*
*         second swap :
*          - swap A(I1,I1) and A(I2,I2)
*          - swap col I1 from I1+1 to I2-1 with row I2 from I1+1 to I2-1
          TMP=A(I1,I1)
          A(I1,I1)=A(I2,I2)
          A(I2,I2)=TMP
*
          CALL DSWAP( I2-I1-1, A(I1+1,I1), 1, A(I2,I1+1), LDA )
*
*         third swap
*          - swap col I1 and I2 from I2+1 to N
         IF ( I2.LT.N )
     $      CALL DSWAP( N-I2, A(I2+1,I1), 1, A(I2+1,I2), 1 )
*
      ENDIF
      END SUBROUTINE DSYSWAPR

*> \brief \b DSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTD2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytd2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytd2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytd2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAU( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal
*> form T by an orthogonal similarity transformation: Q**T * A * Q = T.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the upper or lower triangular part of the
*>          symmetric matrix A is stored:
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*>          n-by-n upper triangular part of A contains the upper
*>          triangular part of the matrix A, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading n-by-n lower triangular part of A contains the lower
*>          triangular part of the matrix A, and the strictly upper
*>          triangular part of A is not referenced.
*>          On exit, if UPLO = 'U', the diagonal and first superdiagonal
*>          of A are overwritten by the corresponding elements of the
*>          tridiagonal matrix T, and the elements above the first
*>          superdiagonal, with the array TAU, represent the orthogonal
*>          matrix Q as a product of elementary reflectors; if UPLO
*>          = 'L', the diagonal and first subdiagonal of A are over-
*>          written by the corresponding elements of the tridiagonal
*>          matrix T, and the elements below the first subdiagonal, with
*>          the array TAU, represent the orthogonal matrix Q as a product
*>          of elementary reflectors. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The diagonal elements of the tridiagonal matrix T:
*>          D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          The off-diagonal elements of the tridiagonal matrix T:
*>          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (N-1)
*>          The scalar factors of the elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetd2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  If UPLO = 'U', the matrix Q is represented as a product of elementary
*>  reflectors
*>
*>     Q = H(n-1) . . . H(2) H(1).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
*>  A(1:i-1,i+1), and tau in TAU(i).
*>
*>  If UPLO = 'L', the matrix Q is represented as a product of elementary
*>  reflectors
*>
*>     Q = H(1) H(2) . . . H(n-1).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
*>  and tau in TAU(i).
*>
*>  The contents of A on exit are illustrated by the following examples
*>  with n = 5:
*>
*>  if UPLO = 'U':                       if UPLO = 'L':
*>
*>    (  d   e   v2  v3  v4 )              (  d                  )
*>    (      d   e   v3  v4 )              (  e   d              )
*>    (          d   e   v4 )              (  v1  e   d          )
*>    (              d   e  )              (  v1  v2  e   d      )
*>    (                  d  )              (  v1  v2  v3  e   d  )
*>
*>  where d and e denote diagonal and off-diagonal elements of T, and vi
*>  denotes an element of the vector defining H(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAU( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO, HALF
      PARAMETER          ( ONE = 1.0D0, ZERO = 0.0D0,
     $                   HALF = 1.0D0 / 2.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I
      DOUBLE PRECISION   ALPHA, TAUI
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DLARFG, DSYMV, DSYR2, XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DDOT
      EXTERNAL           LSAME, DDOT
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYTD2', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.LE.0 )
     $   RETURN
*
      IF( UPPER ) THEN
*
*        Reduce the upper triangle of A
*
         DO 10 I = N - 1, 1, -1
*
*           Generate elementary reflector H(i) = I - tau * v * v**T
*           to annihilate A(1:i-1,i+1)
*
            CALL DLARFG( I, A( I, I+1 ), A( 1, I+1 ), 1, TAUI )
            E( I ) = A( I, I+1 )
*
            IF( TAUI.NE.ZERO ) THEN
*
*              Apply H(i) from both sides to A(1:i,1:i)
*
               A( I, I+1 ) = ONE
*
*              Compute  x := tau * A * v  storing x in TAU(1:i)
*
               CALL DSYMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1,
     $                     ZERO,
     $                     TAU, 1 )
*
*              Compute  w := x - 1/2 * tau * (x**T * v) * v
*
               ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, A( 1, I+1 ), 1 )
               CALL DAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 )
*
*              Apply the transformation as a rank-2 update:
*                 A := A - v * w**T - w * v**T
*
               CALL DSYR2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A,
     $                     LDA )
*
               A( I, I+1 ) = E( I )
            END IF
            D( I+1 ) = A( I+1, I+1 )
            TAU( I ) = TAUI
   10    CONTINUE
         D( 1 ) = A( 1, 1 )
      ELSE
*
*        Reduce the lower triangle of A
*
         DO 20 I = 1, N - 1
*
*           Generate elementary reflector H(i) = I - tau * v * v**T
*           to annihilate A(i+2:n,i)
*
            CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
     $                   TAUI )
            E( I ) = A( I+1, I )
*
            IF( TAUI.NE.ZERO ) THEN
*
*              Apply H(i) from both sides to A(i+1:n,i+1:n)
*
               A( I+1, I ) = ONE
*
*              Compute  x := tau * A * v  storing y in TAU(i:n-1)
*
               CALL DSYMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA,
     $                     A( I+1, I ), 1, ZERO, TAU( I ), 1 )
*
*              Compute  w := x - 1/2 * tau * (x**T * v) * v
*
               ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, A( I+1,
     $                                  I ),
     $                 1 )
               CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 )
*
*              Apply the transformation as a rank-2 update:
*                 A := A - v * w**T - w * v**T
*
               CALL DSYR2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ),
     $                     1,
     $                     A( I+1, I+1 ), LDA )
*
               A( I+1, I ) = E( I )
            END IF
            D( I ) = A( I, I )
            TAU( I ) = TAUI
   20    CONTINUE
         D( N ) = A( N, N )
      END IF
*
      RETURN
*
*     End of DSYTD2
*
      END
*> \brief \b DSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTF2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytf2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytf2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytf2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYTF2( UPLO, N, A, LDA, IPIV, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   A( LDA, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYTF2 computes the factorization of a real symmetric matrix A using
*> the Bunch-Kaufman diagonal pivoting method:
*>
*>    A = U*D*U**T  or  A = L*D*L**T
*>
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, U**T is the transpose of U, and D is symmetric and
*> block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
*>
*> This is the unblocked version of the algorithm, calling Level 2 BLAS.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the upper or lower triangular part of the
*>          symmetric matrix A is stored:
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*>          n-by-n upper triangular part of A contains the upper
*>          triangular part of the matrix A, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading n-by-n lower triangular part of A contains the lower
*>          triangular part of the matrix A, and the strictly upper
*>          triangular part of A is not referenced.
*>
*>          On exit, the block diagonal matrix D and the multipliers used
*>          to obtain the factor U or L (see below for further details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the block structure of D.
*>
*>          If UPLO = 'U':
*>             If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*>             interchanged and D(k,k) is a 1-by-1 diagonal block.
*>
*>             If IPIV(k) = IPIV(k-1) < 0, then rows and columns
*>             k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
*>             is a 2-by-2 diagonal block.
*>
*>          If UPLO = 'L':
*>             If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*>             interchanged and D(k,k) is a 1-by-1 diagonal block.
*>
*>             If IPIV(k) = IPIV(k+1) < 0, then rows and columns
*>             k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
*>             is a 2-by-2 diagonal block.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -k, the k-th argument had an illegal value
*>          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
*>               has been completed, but the block diagonal matrix D is
*>               exactly singular, and division by zero will occur if it
*>               is used to solve a system of equations.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetf2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  If UPLO = 'U', then A = U*D*U**T, where
*>     U = P(n)*U(n)* ... *P(k)U(k)* ...,
*>  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
*>  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
*>  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
*>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
*>
*>             (   I    v    0   )   k-s
*>     U(k) =  (   0    I    0   )   s
*>             (   0    0    I   )   n-k
*>                k-s   s   n-k
*>
*>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
*>  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
*>  and A(k,k), and v overwrites A(1:k-2,k-1:k).
*>
*>  If UPLO = 'L', then A = L*D*L**T, where
*>     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
*>  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
*>  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
*>  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
*>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
*>
*>             (   I    0     0   )  k-1
*>     L(k) =  (   0    I     0   )  s
*>             (   0    v     I   )  n-k-s+1
*>                k-1   s  n-k-s+1
*>
*>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
*>  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
*>  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
*> \endverbatim
*
*> \par Contributors:
*  ==================
*>
*> \verbatim
*>
*>  09-29-06 - patch from
*>    Bobby Cheng, MathWorks
*>
*>    Replace l.204 and l.372
*>         IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
*>    by
*>         IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
*>
*>  01-01-96 - Based on modifications by
*>    J. Lewis, Boeing Computer Services Company
*>    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*>  1-96 - Based on modifications by J. Lewis, Boeing Computer Services
*>         Company
*> \endverbatim
*
*  =====================================================================
      SUBROUTINE DSYTF2( UPLO, N, A, LDA, IPIV, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
      DOUBLE PRECISION   EIGHT, SEVTEN
      PARAMETER          ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I, IMAX, J, JMAX, K, KK, KP, KSTEP
      DOUBLE PRECISION   ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22, R1,
     $                   ROWMAX, T, WK, WKM1, WKP1
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      INTEGER            IDAMAX
      EXTERNAL           LSAME, IDAMAX, DISNAN
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL, DSWAP, DSYR, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYTF2', -INFO )
         RETURN
      END IF
*
*     Initialize ALPHA for use in choosing pivot block size.
*
      ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
*
      IF( UPPER ) THEN
*
*        Factorize A as U*D*U**T using the upper triangle of A
*
*        K is the main loop index, decreasing from N to 1 in steps of
*        1 or 2
*
         K = N
   10    CONTINUE
*
*        If K < 1, exit from loop
*
         IF( K.LT.1 )
     $      GO TO 70
         KSTEP = 1
*
*        Determine rows and columns to be interchanged and whether
*        a 1-by-1 or 2-by-2 pivot block will be used
*
         ABSAKK = ABS( A( K, K ) )
*
*        IMAX is the row-index of the largest off-diagonal element in
*        column K, and COLMAX is its absolute value.
*        Determine both COLMAX and IMAX.
*
         IF( K.GT.1 ) THEN
            IMAX = IDAMAX( K-1, A( 1, K ), 1 )
            COLMAX = ABS( A( IMAX, K ) )
         ELSE
            COLMAX = ZERO
         END IF
*
         IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR.
     $       DISNAN(ABSAKK) ) THEN
*
*           Column K is zero or underflow, or contains a NaN:
*           set INFO and continue
*
            IF( INFO.EQ.0 )
     $         INFO = K
            KP = K
         ELSE
            IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
*
*              no interchange, use 1-by-1 pivot block
*
               KP = K
            ELSE
*
*              JMAX is the column-index of the largest off-diagonal
*              element in row IMAX, and ROWMAX is its absolute value
*
               JMAX = IMAX + IDAMAX( K-IMAX, A( IMAX, IMAX+1 ), LDA )
               ROWMAX = ABS( A( IMAX, JMAX ) )
               IF( IMAX.GT.1 ) THEN
                  JMAX = IDAMAX( IMAX-1, A( 1, IMAX ), 1 )
                  ROWMAX = MAX( ROWMAX, ABS( A( JMAX, IMAX ) ) )
               END IF
*
               IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
*
*                 no interchange, use 1-by-1 pivot block
*
                  KP = K
               ELSE IF( ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
*
*                 interchange rows and columns K and IMAX, use 1-by-1
*                 pivot block
*
                  KP = IMAX
               ELSE
*
*                 interchange rows and columns K-1 and IMAX, use 2-by-2
*                 pivot block
*
                  KP = IMAX
                  KSTEP = 2
               END IF
            END IF
*
            KK = K - KSTEP + 1
            IF( KP.NE.KK ) THEN
*
*              Interchange rows and columns KK and KP in the leading
*              submatrix A(1:k,1:k)
*
               CALL DSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
               CALL DSWAP( KK-KP-1, A( KP+1, KK ), 1, A( KP, KP+1 ),
     $                     LDA )
               T = A( KK, KK )
               A( KK, KK ) = A( KP, KP )
               A( KP, KP ) = T
               IF( KSTEP.EQ.2 ) THEN
                  T = A( K-1, K )
                  A( K-1, K ) = A( KP, K )
                  A( KP, K ) = T
               END IF
            END IF
*
*           Update the leading submatrix
*
            IF( KSTEP.EQ.1 ) THEN
*
*              1-by-1 pivot block D(k): column k now holds
*
*              W(k) = U(k)*D(k)
*
*              where U(k) is the k-th column of U
*
*              Perform a rank-1 update of A(1:k-1,1:k-1) as
*
*              A := A - U(k)*D(k)*U(k)**T = A - W(k)*1/D(k)*W(k)**T
*
               R1 = ONE / A( K, K )
               CALL DSYR( UPLO, K-1, -R1, A( 1, K ), 1, A, LDA )
*
*              Store U(k) in column k
*
               CALL DSCAL( K-1, R1, A( 1, K ), 1 )
            ELSE
*
*              2-by-2 pivot block D(k): columns k and k-1 now hold
*
*              ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
*
*              where U(k) and U(k-1) are the k-th and (k-1)-th columns
*              of U
*
*              Perform a rank-2 update of A(1:k-2,1:k-2) as
*
*              A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
*                 = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T
*
               IF( K.GT.2 ) THEN
*
                  D12 = A( K-1, K )
                  D22 = A( K-1, K-1 ) / D12
                  D11 = A( K, K ) / D12
                  T = ONE / ( D11*D22-ONE )
                  D12 = T / D12
*
                  DO 30 J = K - 2, 1, -1
                     WKM1 = D12*( D11*A( J, K-1 )-A( J, K ) )
                     WK = D12*( D22*A( J, K )-A( J, K-1 ) )
                     DO 20 I = J, 1, -1
                        A( I, J ) = A( I, J ) - A( I, K )*WK -
     $                              A( I, K-1 )*WKM1
   20                CONTINUE
                     A( J, K ) = WK
                     A( J, K-1 ) = WKM1
   30             CONTINUE
*
               END IF
*
            END IF
         END IF
*
*        Store details of the interchanges in IPIV
*
         IF( KSTEP.EQ.1 ) THEN
            IPIV( K ) = KP
         ELSE
            IPIV( K ) = -KP
            IPIV( K-1 ) = -KP
         END IF
*
*        Decrease K and return to the start of the main loop
*
         K = K - KSTEP
         GO TO 10
*
      ELSE
*
*        Factorize A as L*D*L**T using the lower triangle of A
*
*        K is the main loop index, increasing from 1 to N in steps of
*        1 or 2
*
         K = 1
   40    CONTINUE
*
*        If K > N, exit from loop
*
         IF( K.GT.N )
     $      GO TO 70
         KSTEP = 1
*
*        Determine rows and columns to be interchanged and whether
*        a 1-by-1 or 2-by-2 pivot block will be used
*
         ABSAKK = ABS( A( K, K ) )
*
*        IMAX is the row-index of the largest off-diagonal element in
*        column K, and COLMAX is its absolute value.
*        Determine both COLMAX and IMAX.
*
         IF( K.LT.N ) THEN
            IMAX = K + IDAMAX( N-K, A( K+1, K ), 1 )
            COLMAX = ABS( A( IMAX, K ) )
         ELSE
            COLMAX = ZERO
         END IF
*
         IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR.
     $       DISNAN(ABSAKK) ) THEN
*
*           Column K is zero or underflow, or contains a NaN:
*           set INFO and continue
*
            IF( INFO.EQ.0 )
     $         INFO = K
            KP = K
         ELSE
            IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
*
*              no interchange, use 1-by-1 pivot block
*
               KP = K
            ELSE
*
*              JMAX is the column-index of the largest off-diagonal
*              element in row IMAX, and ROWMAX is its absolute value
*
               JMAX = K - 1 + IDAMAX( IMAX-K, A( IMAX, K ), LDA )
               ROWMAX = ABS( A( IMAX, JMAX ) )
               IF( IMAX.LT.N ) THEN
                  JMAX = IMAX + IDAMAX( N-IMAX, A( IMAX+1, IMAX ),
     $                                  1 )
                  ROWMAX = MAX( ROWMAX, ABS( A( JMAX, IMAX ) ) )
               END IF
*
               IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
*
*                 no interchange, use 1-by-1 pivot block
*
                  KP = K
               ELSE IF( ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
*
*                 interchange rows and columns K and IMAX, use 1-by-1
*                 pivot block
*
                  KP = IMAX
               ELSE
*
*                 interchange rows and columns K+1 and IMAX, use 2-by-2
*                 pivot block
*
                  KP = IMAX
                  KSTEP = 2
               END IF
            END IF
*
            KK = K + KSTEP - 1
            IF( KP.NE.KK ) THEN
*
*              Interchange rows and columns KK and KP in the trailing
*              submatrix A(k:n,k:n)
*
               IF( KP.LT.N )
     $            CALL DSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ),
     $                        1 )
               CALL DSWAP( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
     $                     LDA )
               T = A( KK, KK )
               A( KK, KK ) = A( KP, KP )
               A( KP, KP ) = T
               IF( KSTEP.EQ.2 ) THEN
                  T = A( K+1, K )
                  A( K+1, K ) = A( KP, K )
                  A( KP, K ) = T
               END IF
            END IF
*
*           Update the trailing submatrix
*
            IF( KSTEP.EQ.1 ) THEN
*
*              1-by-1 pivot block D(k): column k now holds
*
*              W(k) = L(k)*D(k)
*
*              where L(k) is the k-th column of L
*
               IF( K.LT.N ) THEN
*
*                 Perform a rank-1 update of A(k+1:n,k+1:n) as
*
*                 A := A - L(k)*D(k)*L(k)**T = A - W(k)*(1/D(k))*W(k)**T
*
                  D11 = ONE / A( K, K )
                  CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
     $                       A( K+1, K+1 ), LDA )
*
*                 Store L(k) in column K
*
                  CALL DSCAL( N-K, D11, A( K+1, K ), 1 )
               END IF
            ELSE
*
*              2-by-2 pivot block D(k)
*
               IF( K.LT.N-1 ) THEN
*
*                 Perform a rank-2 update of A(k+2:n,k+2:n) as
*
*                 A := A - ( (A(k) A(k+1))*D(k)**(-1) ) * (A(k) A(k+1))**T
*
*                 where L(k) and L(k+1) are the k-th and (k+1)-th
*                 columns of L
*
                  D21 = A( K+1, K )
                  D11 = A( K+1, K+1 ) / D21
                  D22 = A( K, K ) / D21
                  T = ONE / ( D11*D22-ONE )
                  D21 = T / D21
*
                  DO 60 J = K + 2, N
*
                     WK = D21*( D11*A( J, K )-A( J, K+1 ) )
                     WKP1 = D21*( D22*A( J, K+1 )-A( J, K ) )
*
                     DO 50 I = J, N
                        A( I, J ) = A( I, J ) - A( I, K )*WK -
     $                              A( I, K+1 )*WKP1
   50                CONTINUE
*
                     A( J, K ) = WK
                     A( J, K+1 ) = WKP1
*
   60             CONTINUE
               END IF
            END IF
         END IF
*
*        Store details of the interchanges in IPIV
*
         IF( KSTEP.EQ.1 ) THEN
            IPIV( K ) = KP
         ELSE
            IPIV( K ) = -KP
            IPIV( K+1 ) = -KP
         END IF
*
*        Increase K and return to the start of the main loop
*
         K = K + KSTEP
         GO TO 40
*
      END IF
*
   70 CONTINUE
*
      RETURN
*
*     End of DSYTF2
*
      END
*> \brief \b DSYTRD
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, LWORK, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAU( * ),
*      $                   WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYTRD reduces a real symmetric matrix A to real symmetric
*> tridiagonal form T by an orthogonal similarity transformation:
*> Q**T * A * Q = T.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*>          N-by-N upper triangular part of A contains the upper
*>          triangular part of the matrix A, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading N-by-N lower triangular part of A contains the lower
*>          triangular part of the matrix A, and the strictly upper
*>          triangular part of A is not referenced.
*>          On exit, if UPLO = 'U', the diagonal and first superdiagonal
*>          of A are overwritten by the corresponding elements of the
*>          tridiagonal matrix T, and the elements above the first
*>          superdiagonal, with the array TAU, represent the orthogonal
*>          matrix Q as a product of elementary reflectors; if UPLO
*>          = 'L', the diagonal and first subdiagonal of A are over-
*>          written by the corresponding elements of the tridiagonal
*>          matrix T, and the elements below the first subdiagonal, with
*>          the array TAU, represent the orthogonal matrix Q as a product
*>          of elementary reflectors. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          The diagonal elements of the tridiagonal matrix T:
*>          D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          The off-diagonal elements of the tridiagonal matrix T:
*>          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (N-1)
*>          The scalar factors of the elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.  LWORK >= 1.
*>          For optimum performance LWORK >= N*NB, where NB is the
*>          optimal blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetrd
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  If UPLO = 'U', the matrix Q is represented as a product of elementary
*>  reflectors
*>
*>     Q = H(n-1) . . . H(2) H(1).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
*>  A(1:i-1,i+1), and tau in TAU(i).
*>
*>  If UPLO = 'L', the matrix Q is represented as a product of elementary
*>  reflectors
*>
*>     Q = H(1) H(2) . . . H(n-1).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real vector with
*>  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
*>  and tau in TAU(i).
*>
*>  The contents of A on exit are illustrated by the following examples
*>  with n = 5:
*>
*>  if UPLO = 'U':                       if UPLO = 'L':
*>
*>    (  d   e   v2  v3  v4 )              (  d                  )
*>    (      d   e   v3  v4 )              (  e   d              )
*>    (          d   e   v4 )              (  v1  e   d          )
*>    (              d   e  )              (  v1  v2  e   d      )
*>    (                  d  )              (  v1  v2  v3  e   d  )
*>
*>  where d and e denote diagonal and off-diagonal elements of T, and vi
*>  denotes an element of the vector defining H(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAU( * ),
     $                   WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, UPPER
      INTEGER            I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
     $                   NBMIN, NX
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLATRD, DSYR2K, DSYTD2, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      LQUERY = ( LWORK.EQ.-1 )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
         INFO = -9
      END IF
*
      IF( INFO.EQ.0 ) THEN
*
*        Determine the block size.
*
         NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
         LWKOPT = MAX( 1, N*NB )
         WORK( 1 ) = LWKOPT
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYTRD', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 ) THEN
         WORK( 1 ) = 1
         RETURN
      END IF
*
      NX = N
      IWS = 1
      IF( NB.GT.1 .AND. NB.LT.N ) THEN
*
*        Determine when to cross over from blocked to unblocked code
*        (last block is always handled by unblocked code).
*
         NX = MAX( NB, ILAENV( 3, 'DSYTRD', UPLO, N, -1, -1, -1 ) )
         IF( NX.LT.N ) THEN
*
*           Determine if workspace is large enough for blocked code.
*
            LDWORK = N
            IWS = LDWORK*NB
            IF( LWORK.LT.IWS ) THEN
*
*              Not enough workspace to use optimal NB:  determine the
*              minimum value of NB, and reduce NB or force use of
*              unblocked code by setting NX = N.
*
               NB = MAX( LWORK / LDWORK, 1 )
               NBMIN = ILAENV( 2, 'DSYTRD', UPLO, N, -1, -1, -1 )
               IF( NB.LT.NBMIN )
     $            NX = N
            END IF
         ELSE
            NX = N
         END IF
      ELSE
         NB = 1
      END IF
*
      IF( UPPER ) THEN
*
*        Reduce the upper triangle of A.
*        Columns 1:kk are handled by the unblocked method.
*
         KK = N - ( ( N-NX+NB-1 ) / NB )*NB
         DO 20 I = N - NB + 1, KK + 1, -NB
*
*           Reduce columns i:i+nb-1 to tridiagonal form and form the
*           matrix W which is needed to update the unreduced part of
*           the matrix
*
            CALL DLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK,
     $                   LDWORK )
*
*           Update the unreduced submatrix A(1:i-1,1:i-1), using an
*           update of the form:  A := A - V*W**T - W*V**T
*
            CALL DSYR2K( UPLO, 'No transpose', I-1, NB, -ONE, A( 1,
     $                   I ),
     $                   LDA, WORK, LDWORK, ONE, A, LDA )
*
*           Copy superdiagonal elements back into A, and diagonal
*           elements into D
*
            DO 10 J = I, I + NB - 1
               A( J-1, J ) = E( J-1 )
               D( J ) = A( J, J )
   10       CONTINUE
   20    CONTINUE
*
*        Use unblocked code to reduce the last or only block
*
         CALL DSYTD2( UPLO, KK, A, LDA, D, E, TAU, IINFO )
      ELSE
*
*        Reduce the lower triangle of A
*
         DO 40 I = 1, N - NX, NB
*
*           Reduce columns i:i+nb-1 to tridiagonal form and form the
*           matrix W which is needed to update the unreduced part of
*           the matrix
*
            CALL DLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ),
     $                   TAU( I ), WORK, LDWORK )
*
*           Update the unreduced submatrix A(i+ib:n,i+ib:n), using
*           an update of the form:  A := A - V*W**T - W*V**T
*
            CALL DSYR2K( UPLO, 'No transpose', N-I-NB+1, NB, -ONE,
     $                   A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE,
     $                   A( I+NB, I+NB ), LDA )
*
*           Copy subdiagonal elements back into A, and diagonal
*           elements into D
*
            DO 30 J = I, I + NB - 1
               A( J+1, J ) = E( J )
               D( J ) = A( J, J )
   30       CONTINUE
   40    CONTINUE
*
*        Use unblocked code to reduce the last or only block
*
         CALL DSYTD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ),
     $                TAU( I ), IINFO )
      END IF
*
      WORK( 1 ) = LWKOPT
      RETURN
*
*     End of DSYTRD
*
      END
*> \brief \b DSYTRF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYTRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, LWORK, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   A( LDA, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYTRF computes the factorization of a real symmetric matrix A using
*> the Bunch-Kaufman diagonal pivoting method.  The form of the
*> factorization is
*>
*>    A = U**T*D*U  or  A = L*D*L**T
*>
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, and D is symmetric and block diagonal with
*> 1-by-1 and 2-by-2 diagonal blocks.
*>
*> This is the blocked version of the algorithm, calling Level 3 BLAS.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*>          N-by-N upper triangular part of A contains the upper
*>          triangular part of the matrix A, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading N-by-N lower triangular part of A contains the lower
*>          triangular part of the matrix A, and the strictly upper
*>          triangular part of A is not referenced.
*>
*>          On exit, the block diagonal matrix D and the multipliers used
*>          to obtain the factor U or L (see below for further details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the block structure of D.
*>          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*>          interchanged and D(k,k) is a 1-by-1 diagonal block.
*>          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
*>          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
*>          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
*>          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
*>          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The length of WORK.  LWORK >= 1.  For best performance
*>          LWORK >= N*NB, where NB is the block size returned by ILAENV.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, D(i,i) is exactly zero.  The factorization
*>                has been completed, but the block diagonal matrix D is
*>                exactly singular, and division by zero will occur if it
*>                is used to solve a system of equations.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetrf
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  If UPLO = 'U', then A = U**T*D*U, where
*>     U = P(n)*U(n)* ... *P(k)U(k)* ...,
*>  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
*>  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
*>  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
*>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
*>
*>             (   I    v    0   )   k-s
*>     U(k) =  (   0    I    0   )   s
*>             (   0    0    I   )   n-k
*>                k-s   s   n-k
*>
*>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
*>  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
*>  and A(k,k), and v overwrites A(1:k-2,k-1:k).
*>
*>  If UPLO = 'L', then A = L*D*L**T, where
*>     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
*>  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
*>  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
*>  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
*>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
*>
*>             (   I    0     0   )  k-1
*>     L(k) =  (   0    I     0   )  s
*>             (   0    v     I   )  n-k-s+1
*>                k-1   s  n-k-s+1
*>
*>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
*>  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
*>  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DSYTRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            LQUERY, UPPER
      INTEGER            IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLASYF, DSYTF2, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      LQUERY = ( LWORK.EQ.-1 )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
         INFO = -7
      END IF
*
      IF( INFO.EQ.0 ) THEN
*
*        Determine the block size
*
         NB = ILAENV( 1, 'DSYTRF', UPLO, N, -1, -1, -1 )
         LWKOPT = MAX( 1, N*NB )
         WORK( 1 ) = LWKOPT
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYTRF', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
      NBMIN = 2
      LDWORK = N
      IF( NB.GT.1 .AND. NB.LT.N ) THEN
         IWS = LDWORK*NB
         IF( LWORK.LT.IWS ) THEN
            NB = MAX( LWORK / LDWORK, 1 )
            NBMIN = MAX( 2, ILAENV( 2, 'DSYTRF', UPLO, N, -1, -1,
     $                   -1 ) )
         END IF
      ELSE
         IWS = 1
      END IF
      IF( NB.LT.NBMIN )
     $   NB = N
*
      IF( UPPER ) THEN
*
*        Factorize A as U**T*D*U using the upper triangle of A
*
*        K is the main loop index, decreasing from N to 1 in steps of
*        KB, where KB is the number of columns factorized by DLASYF;
*        KB is either NB or NB-1, or K for the last block
*
         K = N
   10    CONTINUE
*
*        If K < 1, exit from loop
*
         IF( K.LT.1 )
     $      GO TO 40
*
         IF( K.GT.NB ) THEN
*
*           Factorize columns k-kb+1:k of A and use blocked code to
*           update columns 1:k-kb
*
            CALL DLASYF( UPLO, K, NB, KB, A, LDA, IPIV, WORK, LDWORK,
     $                   IINFO )
         ELSE
*
*           Use unblocked code to factorize columns 1:k of A
*
            CALL DSYTF2( UPLO, K, A, LDA, IPIV, IINFO )
            KB = K
         END IF
*
*        Set INFO on the first occurrence of a zero pivot
*
         IF( INFO.EQ.0 .AND. IINFO.GT.0 )
     $      INFO = IINFO
*
*        Decrease K and return to the start of the main loop
*
         K = K - KB
         GO TO 10
*
      ELSE
*
*        Factorize A as L*D*L**T using the lower triangle of A
*
*        K is the main loop index, increasing from 1 to N in steps of
*        KB, where KB is the number of columns factorized by DLASYF;
*        KB is either NB or NB-1, or N-K+1 for the last block
*
         K = 1
   20    CONTINUE
*
*        If K > N, exit from loop
*
         IF( K.GT.N )
     $      GO TO 40
*
         IF( K.LE.N-NB ) THEN
*
*           Factorize columns k:k+kb-1 of A and use blocked code to
*           update columns k+kb:n
*
            CALL DLASYF( UPLO, N-K+1, NB, KB, A( K, K ), LDA,
     $                   IPIV( K ),
     $                   WORK, LDWORK, IINFO )
         ELSE
*
*           Use unblocked code to factorize columns k:n of A
*
            CALL DSYTF2( UPLO, N-K+1, A( K, K ), LDA, IPIV( K ),
     $                   IINFO )
            KB = N - K + 1
         END IF
*
*        Set INFO on the first occurrence of a zero pivot
*
         IF( INFO.EQ.0 .AND. IINFO.GT.0 )
     $      INFO = IINFO + K - 1
*
*        Adjust IPIV
*
         DO 30 J = K, K + KB - 1
            IF( IPIV( J ).GT.0 ) THEN
               IPIV( J ) = IPIV( J ) + K - 1
            ELSE
               IPIV( J ) = IPIV( J ) - K + 1
            END IF
   30    CONTINUE
*
*        Increase K and return to the start of the main loop
*
         K = K + KB
         GO TO 20
*
      END IF
*
   40 CONTINUE
*
      WORK( 1 ) = LWKOPT
      RETURN
*
*     End of DSYTRF
*
      END
*> \brief \b DSYTRI
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTRI + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytri.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytri.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytri.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYTRI( UPLO, N, A, LDA, IPIV, WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   A( LDA, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYTRI computes the inverse of a real symmetric indefinite matrix
*> A using the factorization A = U*D*U**T or A = L*D*L**T computed by
*> DSYTRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the details of the factorization are stored
*>          as an upper or lower triangular matrix.
*>          = 'U':  Upper triangular, form is A = U*D*U**T;
*>          = 'L':  Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the block diagonal matrix D and the multipliers
*>          used to obtain the factor U or L as computed by DSYTRF.
*>
*>          On exit, if INFO = 0, the (symmetric) inverse of the original
*>          matrix.  If UPLO = 'U', the upper triangular part of the
*>          inverse is formed and the part of A below the diagonal is not
*>          referenced; if UPLO = 'L' the lower triangular part of the
*>          inverse is formed and the part of A above the diagonal is
*>          not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the block structure of D
*>          as determined by DSYTRF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*>          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
*>               inverse could not be computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetri
*
*  =====================================================================
      SUBROUTINE DSYTRI( UPLO, N, A, LDA, IPIV, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            K, KP, KSTEP
      DOUBLE PRECISION   AK, AKKP1, AKP1, D, T, TEMP
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DDOT
      EXTERNAL           LSAME, DDOT
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DSWAP, DSYMV, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYTRI', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Check that the diagonal matrix D is nonsingular.
*
      IF( UPPER ) THEN
*
*        Upper triangular storage: examine D from bottom to top
*
         DO 10 INFO = N, 1, -1
            IF( IPIV( INFO ).GT.0 .AND. A( INFO, INFO ).EQ.ZERO )
     $         RETURN
   10    CONTINUE
      ELSE
*
*        Lower triangular storage: examine D from top to bottom.
*
         DO 20 INFO = 1, N
            IF( IPIV( INFO ).GT.0 .AND. A( INFO, INFO ).EQ.ZERO )
     $         RETURN
   20    CONTINUE
      END IF
      INFO = 0
*
      IF( UPPER ) THEN
*
*        Compute inv(A) from the factorization A = U*D*U**T.
*
*        K is the main loop index, increasing from 1 to N in steps of
*        1 or 2, depending on the size of the diagonal blocks.
*
         K = 1
   30    CONTINUE
*
*        If K > N, exit from loop.
*
         IF( K.GT.N )
     $      GO TO 40
*
         IF( IPIV( K ).GT.0 ) THEN
*
*           1 x 1 diagonal block
*
*           Invert the diagonal block.
*
            A( K, K ) = ONE / A( K, K )
*
*           Compute column K of the inverse.
*
            IF( K.GT.1 ) THEN
               CALL DCOPY( K-1, A( 1, K ), 1, WORK, 1 )
               CALL DSYMV( UPLO, K-1, -ONE, A, LDA, WORK, 1, ZERO,
     $                     A( 1, K ), 1 )
               A( K, K ) = A( K, K ) - DDOT( K-1, WORK, 1, A( 1, K ),
     $                     1 )
            END IF
            KSTEP = 1
         ELSE
*
*           2 x 2 diagonal block
*
*           Invert the diagonal block.
*
            T = ABS( A( K, K+1 ) )
            AK = A( K, K ) / T
            AKP1 = A( K+1, K+1 ) / T
            AKKP1 = A( K, K+1 ) / T
            D = T*( AK*AKP1-ONE )
            A( K, K ) = AKP1 / D
            A( K+1, K+1 ) = AK / D
            A( K, K+1 ) = -AKKP1 / D
*
*           Compute columns K and K+1 of the inverse.
*
            IF( K.GT.1 ) THEN
               CALL DCOPY( K-1, A( 1, K ), 1, WORK, 1 )
               CALL DSYMV( UPLO, K-1, -ONE, A, LDA, WORK, 1, ZERO,
     $                     A( 1, K ), 1 )
               A( K, K ) = A( K, K ) - DDOT( K-1, WORK, 1, A( 1, K ),
     $                     1 )
               A( K, K+1 ) = A( K, K+1 ) -
     $                       DDOT( K-1, A( 1, K ), 1, A( 1, K+1 ),
     $                             1 )
               CALL DCOPY( K-1, A( 1, K+1 ), 1, WORK, 1 )
               CALL DSYMV( UPLO, K-1, -ONE, A, LDA, WORK, 1, ZERO,
     $                     A( 1, K+1 ), 1 )
               A( K+1, K+1 ) = A( K+1, K+1 ) -
     $                         DDOT( K-1, WORK, 1, A( 1, K+1 ), 1 )
            END IF
            KSTEP = 2
         END IF
*
         KP = ABS( IPIV( K ) )
         IF( KP.NE.K ) THEN
*
*           Interchange rows and columns K and KP in the leading
*           submatrix A(1:k+1,1:k+1)
*
            CALL DSWAP( KP-1, A( 1, K ), 1, A( 1, KP ), 1 )
            CALL DSWAP( K-KP-1, A( KP+1, K ), 1, A( KP, KP+1 ), LDA )
            TEMP = A( K, K )
            A( K, K ) = A( KP, KP )
            A( KP, KP ) = TEMP
            IF( KSTEP.EQ.2 ) THEN
               TEMP = A( K, K+1 )
               A( K, K+1 ) = A( KP, K+1 )
               A( KP, K+1 ) = TEMP
            END IF
         END IF
*
         K = K + KSTEP
         GO TO 30
   40    CONTINUE
*
      ELSE
*
*        Compute inv(A) from the factorization A = L*D*L**T.
*
*        K is the main loop index, increasing from 1 to N in steps of
*        1 or 2, depending on the size of the diagonal blocks.
*
         K = N
   50    CONTINUE
*
*        If K < 1, exit from loop.
*
         IF( K.LT.1 )
     $      GO TO 60
*
         IF( IPIV( K ).GT.0 ) THEN
*
*           1 x 1 diagonal block
*
*           Invert the diagonal block.
*
            A( K, K ) = ONE / A( K, K )
*
*           Compute column K of the inverse.
*
            IF( K.LT.N ) THEN
               CALL DCOPY( N-K, A( K+1, K ), 1, WORK, 1 )
               CALL DSYMV( UPLO, N-K, -ONE, A( K+1, K+1 ), LDA, WORK,
     $                     1,
     $                     ZERO, A( K+1, K ), 1 )
               A( K, K ) = A( K, K ) - DDOT( N-K, WORK, 1, A( K+1,
     $            K ),
     $                     1 )
            END IF
            KSTEP = 1
         ELSE
*
*           2 x 2 diagonal block
*
*           Invert the diagonal block.
*
            T = ABS( A( K, K-1 ) )
            AK = A( K-1, K-1 ) / T
            AKP1 = A( K, K ) / T
            AKKP1 = A( K, K-1 ) / T
            D = T*( AK*AKP1-ONE )
            A( K-1, K-1 ) = AKP1 / D
            A( K, K ) = AK / D
            A( K, K-1 ) = -AKKP1 / D
*
*           Compute columns K-1 and K of the inverse.
*
            IF( K.LT.N ) THEN
               CALL DCOPY( N-K, A( K+1, K ), 1, WORK, 1 )
               CALL DSYMV( UPLO, N-K, -ONE, A( K+1, K+1 ), LDA, WORK,
     $                     1,
     $                     ZERO, A( K+1, K ), 1 )
               A( K, K ) = A( K, K ) - DDOT( N-K, WORK, 1, A( K+1,
     $            K ),
     $                     1 )
               A( K, K-1 ) = A( K, K-1 ) -
     $                       DDOT( N-K, A( K+1, K ), 1, A( K+1,
     $                             K-1 ),
     $                       1 )
               CALL DCOPY( N-K, A( K+1, K-1 ), 1, WORK, 1 )
               CALL DSYMV( UPLO, N-K, -ONE, A( K+1, K+1 ), LDA, WORK,
     $                     1,
     $                     ZERO, A( K+1, K-1 ), 1 )
               A( K-1, K-1 ) = A( K-1, K-1 ) -
     $                         DDOT( N-K, WORK, 1, A( K+1, K-1 ), 1 )
            END IF
            KSTEP = 2
         END IF
*
         KP = ABS( IPIV( K ) )
         IF( KP.NE.K ) THEN
*
*           Interchange rows and columns K and KP in the trailing
*           submatrix A(k-1:n,k-1:n)
*
            IF( KP.LT.N )
     $         CALL DSWAP( N-KP, A( KP+1, K ), 1, A( KP+1, KP ), 1 )
            CALL DSWAP( KP-K-1, A( K+1, K ), 1, A( KP, K+1 ), LDA )
            TEMP = A( K, K )
            A( K, K ) = A( KP, KP )
            A( KP, KP ) = TEMP
            IF( KSTEP.EQ.2 ) THEN
               TEMP = A( K, K-1 )
               A( K, K-1 ) = A( KP, K-1 )
               A( KP, K-1 ) = TEMP
            END IF
         END IF
*
         K = K - KSTEP
         GO TO 50
   60    CONTINUE
      END IF
*
      RETURN
*
*     End of DSYTRI
*
      END
*> \brief \b DSYTRI2
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTRI2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytri2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytri2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytri2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYTRI2( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, LWORK, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   A( LDA, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYTRI2 computes the inverse of a DOUBLE PRECISION symmetric indefinite matrix
*> A using the factorization A = U*D*U**T or A = L*D*L**T computed by
*> DSYTRF. DSYTRI2 sets the LEADING DIMENSION of the workspace
*> before calling DSYTRI2X that actually computes the inverse.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the details of the factorization are stored
*>          as an upper or lower triangular matrix.
*>          = 'U':  Upper triangular, form is A = U*D*U**T;
*>          = 'L':  Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the block diagonal matrix D and the multipliers
*>          used to obtain the factor U or L as computed by DSYTRF.
*>
*>          On exit, if INFO = 0, the (symmetric) inverse of the original
*>          matrix.  If UPLO = 'U', the upper triangular part of the
*>          inverse is formed and the part of A below the diagonal is not
*>          referenced; if UPLO = 'L' the lower triangular part of the
*>          inverse is formed and the part of A above the diagonal is
*>          not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the block structure of D
*>          as determined by DSYTRF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If N = 0, LWORK >= 1, else LWORK >= (N+NB+1)*(NB+3).
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          calculates:
*>              - the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array,
*>              - and no error message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*>          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
*>               inverse could not be computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetri2
*
*  =====================================================================
      SUBROUTINE DSYTRI2( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            UPPER, LQUERY
      INTEGER            MINSIZE, NBMAX
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSYTRI, DSYTRI2X, XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      LQUERY = ( LWORK.EQ.-1 )
*
*     Get blocksize
*
      NBMAX = ILAENV( 1, 'DSYTRI2', UPLO, N, -1, -1, -1 )
      IF( N.EQ.0 ) THEN
         MINSIZE = 1
      ELSE IF( NBMAX.GE.N ) THEN
         MINSIZE = N
      ELSE
         MINSIZE = (N+NBMAX+1)*(NBMAX+3)
      END IF
*
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      ELSE IF( LWORK.LT.MINSIZE .AND. .NOT.LQUERY ) THEN
         INFO = -7
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYTRI2', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         WORK( 1 ) = MINSIZE
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN

      IF( NBMAX.GE.N ) THEN
         CALL DSYTRI( UPLO, N, A, LDA, IPIV, WORK, INFO )
      ELSE
         CALL DSYTRI2X( UPLO, N, A, LDA, IPIV, WORK, NBMAX, INFO )
      END IF
*
      RETURN
*
*     End of DSYTRI2
*
      END
*> \brief \b DSYTRI2X
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTRI2X + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytri2x.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytri2x.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytri2x.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYTRI2X( UPLO, N, A, LDA, IPIV, WORK, NB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, N, NB
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   A( LDA, * ), WORK( N+NB+1,* )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYTRI2X computes the inverse of a real symmetric indefinite matrix
*> A using the factorization A = U*D*U**T or A = L*D*L**T computed by
*> DSYTRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the details of the factorization are stored
*>          as an upper or lower triangular matrix.
*>          = 'U':  Upper triangular, form is A = U*D*U**T;
*>          = 'L':  Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the NNB diagonal matrix D and the multipliers
*>          used to obtain the factor U or L as computed by DSYTRF.
*>
*>          On exit, if INFO = 0, the (symmetric) inverse of the original
*>          matrix.  If UPLO = 'U', the upper triangular part of the
*>          inverse is formed and the part of A below the diagonal is not
*>          referenced; if UPLO = 'L' the lower triangular part of the
*>          inverse is formed and the part of A above the diagonal is
*>          not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the NNB structure of D
*>          as determined by DSYTRF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N+NB+1,NB+3)
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*>          NB is INTEGER
*>          Block size
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*>          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
*>               inverse could not be computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetri2x
*
*  =====================================================================
      SUBROUTINE DSYTRI2X( UPLO, N, A, LDA, IPIV, WORK, NB, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N, NB
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * ), WORK( N+NB+1,* )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I, IINFO, IP, K, CUT, NNB
      INTEGER            COUNT
      INTEGER            J, U11, INVD

      DOUBLE PRECISION   AK, AKKP1, AKP1, D, T
      DOUBLE PRECISION   U01_I_J, U01_IP1_J
      DOUBLE PRECISION   U11_I_J, U11_IP1_J
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSYCONV, XERBLA, DTRTRI
      EXTERNAL           DGEMM, DTRMM, DSYSWAPR
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      END IF
*
*     Quick return if possible
*
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYTRI2X', -INFO )
         RETURN
      END IF
      IF( N.EQ.0 )
     $   RETURN
*
*     Convert A
*     Workspace got Non-diag elements of D
*
      CALL DSYCONV( UPLO, 'C', N, A, LDA, IPIV, WORK, IINFO )
*
*     Check that the diagonal matrix D is nonsingular.
*
      IF( UPPER ) THEN
*
*        Upper triangular storage: examine D from bottom to top
*
         DO INFO = N, 1, -1
            IF( IPIV( INFO ).GT.0 .AND. A( INFO, INFO ).EQ.ZERO )
     $         RETURN
         END DO
      ELSE
*
*        Lower triangular storage: examine D from top to bottom.
*
         DO INFO = 1, N
            IF( IPIV( INFO ).GT.0 .AND. A( INFO, INFO ).EQ.ZERO )
     $         RETURN
         END DO
      END IF
      INFO = 0
*
*  Splitting Workspace
*     U01 is a block (N,NB+1)
*     The first element of U01 is in WORK(1,1)
*     U11 is a block (NB+1,NB+1)
*     The first element of U11 is in WORK(N+1,1)
      U11 = N
*     INVD is a block (N,2)
*     The first element of INVD is in WORK(1,INVD)
      INVD = NB+2

      IF( UPPER ) THEN
*
*        invA = P * inv(U**T)*inv(D)*inv(U)*P**T.
*
        CALL DTRTRI( UPLO, 'U', N, A, LDA, INFO )
*
*       inv(D) and inv(D)*inv(U)
*
        K=1
        DO WHILE ( K .LE. N )
         IF( IPIV( K ).GT.0 ) THEN
*           1 x 1 diagonal NNB
             WORK(K,INVD) = ONE /  A( K, K )
             WORK(K,INVD+1) = 0
            K=K+1
         ELSE
*           2 x 2 diagonal NNB
             T = WORK(K+1,1)
             AK = A( K, K ) / T
             AKP1 = A( K+1, K+1 ) / T
             AKKP1 = WORK(K+1,1)  / T
             D = T*( AK*AKP1-ONE )
             WORK(K,INVD) = AKP1 / D
             WORK(K+1,INVD+1) = AK / D
             WORK(K,INVD+1) = -AKKP1 / D
             WORK(K+1,INVD) = -AKKP1 / D
            K=K+2
         END IF
        END DO
*
*       inv(U**T) = (inv(U))**T
*
*       inv(U**T)*inv(D)*inv(U)
*
        CUT=N
        DO WHILE (CUT .GT. 0)
           NNB=NB
           IF (CUT .LE. NNB) THEN
              NNB=CUT
           ELSE
              COUNT = 0
*             count negative elements,
              DO I=CUT+1-NNB,CUT
                  IF (IPIV(I) .LT. 0) COUNT=COUNT+1
              END DO
*             need a even number for a clear cut
              IF (MOD(COUNT,2) .EQ. 1) NNB=NNB+1
           END IF

           CUT=CUT-NNB
*
*          U01 Block
*
           DO I=1,CUT
             DO J=1,NNB
              WORK(I,J)=A(I,CUT+J)
             END DO
           END DO
*
*          U11 Block
*
           DO I=1,NNB
             WORK(U11+I,I)=ONE
             DO J=1,I-1
                WORK(U11+I,J)=ZERO
             END DO
             DO J=I+1,NNB
                WORK(U11+I,J)=A(CUT+I,CUT+J)
             END DO
           END DO
*
*          invD*U01
*
           I=1
           DO WHILE (I .LE. CUT)
             IF (IPIV(I) > 0) THEN
                DO J=1,NNB
                    WORK(I,J)=WORK(I,INVD)*WORK(I,J)
                END DO
                I=I+1
             ELSE
                DO J=1,NNB
                   U01_I_J = WORK(I,J)
                   U01_IP1_J = WORK(I+1,J)
                   WORK(I,J)=WORK(I,INVD)*U01_I_J+
     $                      WORK(I,INVD+1)*U01_IP1_J
                   WORK(I+1,J)=WORK(I+1,INVD)*U01_I_J+
     $                      WORK(I+1,INVD+1)*U01_IP1_J
                END DO
                I=I+2
             END IF
           END DO
*
*        invD1*U11
*
           I=1
           DO WHILE (I .LE. NNB)
             IF (IPIV(CUT+I) > 0) THEN
                DO J=I,NNB
                    WORK(U11+I,J)=WORK(CUT+I,INVD)*WORK(U11+I,J)
                END DO
                I=I+1
             ELSE
                DO J=I,NNB
                   U11_I_J = WORK(U11+I,J)
                   U11_IP1_J = WORK(U11+I+1,J)
                WORK(U11+I,J)=WORK(CUT+I,INVD)*WORK(U11+I,J) +
     $                      WORK(CUT+I,INVD+1)*WORK(U11+I+1,J)
                WORK(U11+I+1,J)=WORK(CUT+I+1,INVD)*U11_I_J+
     $                      WORK(CUT+I+1,INVD+1)*U11_IP1_J
                END DO
                I=I+2
             END IF
           END DO
*
*       U11**T*invD1*U11->U11
*
        CALL DTRMM('L','U','T','U',NNB, NNB,
     $             ONE,A(CUT+1,CUT+1),LDA,WORK(U11+1,1),N+NB+1)
*
         DO I=1,NNB
            DO J=I,NNB
              A(CUT+I,CUT+J)=WORK(U11+I,J)
            END DO
         END DO
*
*          U01**T*invD*U01->A(CUT+I,CUT+J)
*
         CALL DGEMM('T','N',NNB,NNB,CUT,ONE,A(1,CUT+1),LDA,
     $              WORK,N+NB+1, ZERO, WORK(U11+1,1), N+NB+1)

*
*        U11 =  U11**T*invD1*U11 + U01**T*invD*U01
*
         DO I=1,NNB
            DO J=I,NNB
              A(CUT+I,CUT+J)=A(CUT+I,CUT+J)+WORK(U11+I,J)
            END DO
         END DO
*
*        U01 =  U00**T*invD0*U01
*
         CALL DTRMM('L',UPLO,'T','U',CUT, NNB,
     $             ONE,A,LDA,WORK,N+NB+1)

*
*        Update U01
*
         DO I=1,CUT
           DO J=1,NNB
            A(I,CUT+J)=WORK(I,J)
           END DO
         END DO
*
*      Next Block
*
       END DO
*
*        Apply PERMUTATIONS P and P**T: P * inv(U**T)*inv(D)*inv(U) *P**T
*
            I=1
            DO WHILE ( I .LE. N )
               IF( IPIV(I) .GT. 0 ) THEN
                  IP=IPIV(I)
                 IF (I .LT. IP) CALL DSYSWAPR( UPLO, N, A, LDA, I ,
     $                IP )
                 IF (I .GT. IP) CALL DSYSWAPR( UPLO, N, A, LDA, IP ,
     $                I )
               ELSE
                 IP=-IPIV(I)
                 I=I+1
                 IF ( (I-1) .LT. IP)
     $                  CALL DSYSWAPR( UPLO, N, A, LDA, I-1 ,IP )
                 IF ( (I-1) .GT. IP)
     $                  CALL DSYSWAPR( UPLO, N, A, LDA, IP ,I-1 )
              ENDIF
               I=I+1
            END DO
      ELSE
*
*        LOWER...
*
*        invA = P * inv(U**T)*inv(D)*inv(U)*P**T.
*
         CALL DTRTRI( UPLO, 'U', N, A, LDA, INFO )
*
*       inv(D) and inv(D)*inv(U)
*
        K=N
        DO WHILE ( K .GE. 1 )
         IF( IPIV( K ).GT.0 ) THEN
*           1 x 1 diagonal NNB
             WORK(K,INVD) = ONE /  A( K, K )
             WORK(K,INVD+1) = 0
            K=K-1
         ELSE
*           2 x 2 diagonal NNB
             T = WORK(K-1,1)
             AK = A( K-1, K-1 ) / T
             AKP1 = A( K, K ) / T
             AKKP1 = WORK(K-1,1) / T
             D = T*( AK*AKP1-ONE )
             WORK(K-1,INVD) = AKP1 / D
             WORK(K,INVD) = AK / D
             WORK(K,INVD+1) = -AKKP1 / D
             WORK(K-1,INVD+1) = -AKKP1 / D
            K=K-2
         END IF
        END DO
*
*       inv(U**T) = (inv(U))**T
*
*       inv(U**T)*inv(D)*inv(U)
*
        CUT=0
        DO WHILE (CUT .LT. N)
           NNB=NB
           IF (CUT + NNB .GT. N) THEN
              NNB=N-CUT
           ELSE
              COUNT = 0
*             count negative elements,
              DO I=CUT+1,CUT+NNB
                  IF (IPIV(I) .LT. 0) COUNT=COUNT+1
              END DO
*             need a even number for a clear cut
              IF (MOD(COUNT,2) .EQ. 1) NNB=NNB+1
           END IF
*     L21 Block
           DO I=1,N-CUT-NNB
             DO J=1,NNB
              WORK(I,J)=A(CUT+NNB+I,CUT+J)
             END DO
           END DO
*     L11 Block
           DO I=1,NNB
             WORK(U11+I,I)=ONE
             DO J=I+1,NNB
                WORK(U11+I,J)=ZERO
             END DO
             DO J=1,I-1
                WORK(U11+I,J)=A(CUT+I,CUT+J)
             END DO
           END DO
*
*          invD*L21
*
           I=N-CUT-NNB
           DO WHILE (I .GE. 1)
             IF (IPIV(CUT+NNB+I) > 0) THEN
                DO J=1,NNB
                    WORK(I,J)=WORK(CUT+NNB+I,INVD)*WORK(I,J)
                END DO
                I=I-1
             ELSE
                DO J=1,NNB
                   U01_I_J = WORK(I,J)
                   U01_IP1_J = WORK(I-1,J)
                   WORK(I,J)=WORK(CUT+NNB+I,INVD)*U01_I_J+
     $                        WORK(CUT+NNB+I,INVD+1)*U01_IP1_J
                   WORK(I-1,J)=WORK(CUT+NNB+I-1,INVD+1)*U01_I_J+
     $                        WORK(CUT+NNB+I-1,INVD)*U01_IP1_J
                END DO
                I=I-2
             END IF
           END DO
*
*        invD1*L11
*
           I=NNB
           DO WHILE (I .GE. 1)
             IF (IPIV(CUT+I) > 0) THEN
                DO J=1,NNB
                    WORK(U11+I,J)=WORK(CUT+I,INVD)*WORK(U11+I,J)
                END DO
                I=I-1
             ELSE
                DO J=1,NNB
                   U11_I_J = WORK(U11+I,J)
                   U11_IP1_J = WORK(U11+I-1,J)
                WORK(U11+I,J)=WORK(CUT+I,INVD)*WORK(U11+I,J) +
     $                      WORK(CUT+I,INVD+1)*U11_IP1_J
                WORK(U11+I-1,J)=WORK(CUT+I-1,INVD+1)*U11_I_J+
     $                      WORK(CUT+I-1,INVD)*U11_IP1_J
                END DO
                I=I-2
             END IF
           END DO
*
*       L11**T*invD1*L11->L11
*
        CALL DTRMM('L',UPLO,'T','U',NNB, NNB,
     $             ONE,A(CUT+1,CUT+1),LDA,WORK(U11+1,1),N+NB+1)

*
         DO I=1,NNB
            DO J=1,I
              A(CUT+I,CUT+J)=WORK(U11+I,J)
            END DO
         END DO
*
        IF ( (CUT+NNB) .LT. N ) THEN
*
*          L21**T*invD2*L21->A(CUT+I,CUT+J)
*
         CALL DGEMM('T','N',NNB,NNB,N-NNB-CUT,ONE,A(CUT+NNB+1,CUT+1)
     $             ,LDA,WORK,N+NB+1, ZERO, WORK(U11+1,1), N+NB+1)

*
*        L11 =  L11**T*invD1*L11 + U01**T*invD*U01
*
         DO I=1,NNB
            DO J=1,I
              A(CUT+I,CUT+J)=A(CUT+I,CUT+J)+WORK(U11+I,J)
            END DO
         END DO
*
*        L01 =  L22**T*invD2*L21
*
         CALL DTRMM('L',UPLO,'T','U', N-NNB-CUT, NNB,
     $             ONE,A(CUT+NNB+1,CUT+NNB+1),LDA,WORK,N+NB+1)
*
*      Update L21
*
         DO I=1,N-CUT-NNB
           DO J=1,NNB
              A(CUT+NNB+I,CUT+J)=WORK(I,J)
           END DO
         END DO

       ELSE
*
*        L11 =  L11**T*invD1*L11
*
         DO I=1,NNB
            DO J=1,I
              A(CUT+I,CUT+J)=WORK(U11+I,J)
            END DO
         END DO
       END IF
*
*      Next Block
*
           CUT=CUT+NNB
       END DO
*
*        Apply PERMUTATIONS P and P**T: P * inv(U**T)*inv(D)*inv(U) *P**T
*
            I=N
            DO WHILE ( I .GE. 1 )
               IF( IPIV(I) .GT. 0 ) THEN
                  IP=IPIV(I)
                 IF (I .LT. IP) CALL DSYSWAPR( UPLO, N, A, LDA, I ,
     $                IP  )
                 IF (I .GT. IP) CALL DSYSWAPR( UPLO, N, A, LDA, IP ,
     $                I )
               ELSE
                 IP=-IPIV(I)
                 IF ( I .LT. IP) CALL DSYSWAPR( UPLO, N, A, LDA, I ,
     $                IP )
                 IF ( I .GT. IP) CALL DSYSWAPR( UPLO, N, A, LDA, IP,
     $                I )
                 I=I-1
               ENDIF
               I=I-1
            END DO
      END IF
*
      RETURN
*
*     End of DSYTRI2X
*
      END

*> \brief \b DSYTRS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYTRS( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYTRS solves a system of linear equations A*X = B with a real
*> symmetric matrix A using the factorization A = U*D*U**T or
*> A = L*D*L**T computed by DSYTRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the details of the factorization are stored
*>          as an upper or lower triangular matrix.
*>          = 'U':  Upper triangular, form is A = U*D*U**T;
*>          = 'L':  Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The block diagonal matrix D and the multipliers used to
*>          obtain the factor U or L as computed by DSYTRF.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the block structure of D
*>          as determined by DSYTRF.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the right hand side matrix B.
*>          On exit, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetrs
*
*  =====================================================================
      SUBROUTINE DSYTRS( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            J, K, KP
      DOUBLE PRECISION   AK, AKM1, AKM1K, BK, BKM1, DENOM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMV, DGER, DSCAL, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -8
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYTRS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 )
     $   RETURN
*
      IF( UPPER ) THEN
*
*        Solve A*X = B, where A = U*D*U**T.
*
*        First solve U*D*X = B, overwriting B with X.
*
*        K is the main loop index, decreasing from N to 1 in steps of
*        1 or 2, depending on the size of the diagonal blocks.
*
         K = N
   10    CONTINUE
*
*        If K < 1, exit from loop.
*
         IF( K.LT.1 )
     $      GO TO 30
*
         IF( IPIV( K ).GT.0 ) THEN
*
*           1 x 1 diagonal block
*
*           Interchange rows K and IPIV(K).
*
            KP = IPIV( K )
            IF( KP.NE.K )
     $         CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
*
*           Multiply by inv(U(K)), where U(K) is the transformation
*           stored in column K of A.
*
            CALL DGER( K-1, NRHS, -ONE, A( 1, K ), 1, B( K, 1 ), LDB,
     $                 B( 1, 1 ), LDB )
*
*           Multiply by the inverse of the diagonal block.
*
            CALL DSCAL( NRHS, ONE / A( K, K ), B( K, 1 ), LDB )
            K = K - 1
         ELSE
*
*           2 x 2 diagonal block
*
*           Interchange rows K-1 and -IPIV(K).
*
            KP = -IPIV( K )
            IF( KP.NE.K-1 )
     $         CALL DSWAP( NRHS, B( K-1, 1 ), LDB, B( KP, 1 ), LDB )
*
*           Multiply by inv(U(K)), where U(K) is the transformation
*           stored in columns K-1 and K of A.
*
            CALL DGER( K-2, NRHS, -ONE, A( 1, K ), 1, B( K, 1 ), LDB,
     $                 B( 1, 1 ), LDB )
            CALL DGER( K-2, NRHS, -ONE, A( 1, K-1 ), 1, B( K-1, 1 ),
     $                 LDB, B( 1, 1 ), LDB )
*
*           Multiply by the inverse of the diagonal block.
*
            AKM1K = A( K-1, K )
            AKM1 = A( K-1, K-1 ) / AKM1K
            AK = A( K, K ) / AKM1K
            DENOM = AKM1*AK - ONE
            DO 20 J = 1, NRHS
               BKM1 = B( K-1, J ) / AKM1K
               BK = B( K, J ) / AKM1K
               B( K-1, J ) = ( AK*BKM1-BK ) / DENOM
               B( K, J ) = ( AKM1*BK-BKM1 ) / DENOM
   20       CONTINUE
            K = K - 2
         END IF
*
         GO TO 10
   30    CONTINUE
*
*        Next solve U**T *X = B, overwriting B with X.
*
*        K is the main loop index, increasing from 1 to N in steps of
*        1 or 2, depending on the size of the diagonal blocks.
*
         K = 1
   40    CONTINUE
*
*        If K > N, exit from loop.
*
         IF( K.GT.N )
     $      GO TO 50
*
         IF( IPIV( K ).GT.0 ) THEN
*
*           1 x 1 diagonal block
*
*           Multiply by inv(U**T(K)), where U(K) is the transformation
*           stored in column K of A.
*
            CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB, A( 1,
     $                  K ),
     $                  1, ONE, B( K, 1 ), LDB )
*
*           Interchange rows K and IPIV(K).
*
            KP = IPIV( K )
            IF( KP.NE.K )
     $         CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
            K = K + 1
         ELSE
*
*           2 x 2 diagonal block
*
*           Multiply by inv(U**T(K+1)), where U(K+1) is the transformation
*           stored in columns K and K+1 of A.
*
            CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB, A( 1,
     $                  K ),
     $                  1, ONE, B( K, 1 ), LDB )
            CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB,
     $                  A( 1, K+1 ), 1, ONE, B( K+1, 1 ), LDB )
*
*           Interchange rows K and -IPIV(K).
*
            KP = -IPIV( K )
            IF( KP.NE.K )
     $         CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
            K = K + 2
         END IF
*
         GO TO 40
   50    CONTINUE
*
      ELSE
*
*        Solve A*X = B, where A = L*D*L**T.
*
*        First solve L*D*X = B, overwriting B with X.
*
*        K is the main loop index, increasing from 1 to N in steps of
*        1 or 2, depending on the size of the diagonal blocks.
*
         K = 1
   60    CONTINUE
*
*        If K > N, exit from loop.
*
         IF( K.GT.N )
     $      GO TO 80
*
         IF( IPIV( K ).GT.0 ) THEN
*
*           1 x 1 diagonal block
*
*           Interchange rows K and IPIV(K).
*
            KP = IPIV( K )
            IF( KP.NE.K )
     $         CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
*
*           Multiply by inv(L(K)), where L(K) is the transformation
*           stored in column K of A.
*
            IF( K.LT.N )
     $         CALL DGER( N-K, NRHS, -ONE, A( K+1, K ), 1, B( K, 1 ),
     $                    LDB, B( K+1, 1 ), LDB )
*
*           Multiply by the inverse of the diagonal block.
*
            CALL DSCAL( NRHS, ONE / A( K, K ), B( K, 1 ), LDB )
            K = K + 1
         ELSE
*
*           2 x 2 diagonal block
*
*           Interchange rows K+1 and -IPIV(K).
*
            KP = -IPIV( K )
            IF( KP.NE.K+1 )
     $         CALL DSWAP( NRHS, B( K+1, 1 ), LDB, B( KP, 1 ), LDB )
*
*           Multiply by inv(L(K)), where L(K) is the transformation
*           stored in columns K and K+1 of A.
*
            IF( K.LT.N-1 ) THEN
               CALL DGER( N-K-1, NRHS, -ONE, A( K+2, K ), 1, B( K,
     $                    1 ),
     $                    LDB, B( K+2, 1 ), LDB )
               CALL DGER( N-K-1, NRHS, -ONE, A( K+2, K+1 ), 1,
     $                    B( K+1, 1 ), LDB, B( K+2, 1 ), LDB )
            END IF
*
*           Multiply by the inverse of the diagonal block.
*
            AKM1K = A( K+1, K )
            AKM1 = A( K, K ) / AKM1K
            AK = A( K+1, K+1 ) / AKM1K
            DENOM = AKM1*AK - ONE
            DO 70 J = 1, NRHS
               BKM1 = B( K, J ) / AKM1K
               BK = B( K+1, J ) / AKM1K
               B( K, J ) = ( AK*BKM1-BK ) / DENOM
               B( K+1, J ) = ( AKM1*BK-BKM1 ) / DENOM
   70       CONTINUE
            K = K + 2
         END IF
*
         GO TO 60
   80    CONTINUE
*
*        Next solve L**T *X = B, overwriting B with X.
*
*        K is the main loop index, decreasing from N to 1 in steps of
*        1 or 2, depending on the size of the diagonal blocks.
*
         K = N
   90    CONTINUE
*
*        If K < 1, exit from loop.
*
         IF( K.LT.1 )
     $      GO TO 100
*
         IF( IPIV( K ).GT.0 ) THEN
*
*           1 x 1 diagonal block
*
*           Multiply by inv(L**T(K)), where L(K) is the transformation
*           stored in column K of A.
*
            IF( K.LT.N )
     $         CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ),
     $                     LDB, A( K+1, K ), 1, ONE, B( K, 1 ), LDB )
*
*           Interchange rows K and IPIV(K).
*
            KP = IPIV( K )
            IF( KP.NE.K )
     $         CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
            K = K - 1
         ELSE
*
*           2 x 2 diagonal block
*
*           Multiply by inv(L**T(K-1)), where L(K-1) is the transformation
*           stored in columns K-1 and K of A.
*
            IF( K.LT.N ) THEN
               CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ),
     $                     LDB, A( K+1, K ), 1, ONE, B( K, 1 ), LDB )
               CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ),
     $                     LDB, A( K+1, K-1 ), 1, ONE, B( K-1, 1 ),
     $                     LDB )
            END IF
*
*           Interchange rows K and -IPIV(K).
*
            KP = -IPIV( K )
            IF( KP.NE.K )
     $         CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
            K = K - 2
         END IF
*
         GO TO 90
  100    CONTINUE
      END IF
*
      RETURN
*
*     End of DSYTRS
*
      END
*> \brief \b DSYTRS2
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTRS2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytrs2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytrs2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytrs2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYTRS2( UPLO, N, NRHS, A, LDA, IPIV, B, LDB,
*                           WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSYTRS2 solves a system of linear equations A*X = B with a real
*> symmetric matrix A using the factorization A = U*D*U**T or
*> A = L*D*L**T computed by DSYTRF and converted by DSYCONV.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the details of the factorization are stored
*>          as an upper or lower triangular matrix.
*>          = 'U':  Upper triangular, form is A = U*D*U**T;
*>          = 'L':  Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The block diagonal matrix D and the multipliers used to
*>          obtain the factor U or L as computed by DSYTRF.
*>          Note that A is input / output. This might be counter-intuitive,
*>          and one may think that A is input only. A is input / output. This
*>          is because, at the start of the subroutine, we permute A in a
*>          "better" form and then we permute A back to its original form at
*>          the end.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the block structure of D
*>          as determined by DSYTRF.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the right hand side matrix B.
*>          On exit, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetrs2
*
*  =====================================================================
      SUBROUTINE DSYTRS2( UPLO, N, NRHS, A, LDA, IPIV, B, LDB,
     $                    WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I, IINFO, J, K, KP
      DOUBLE PRECISION   AK, AKM1, AKM1K, BK, BKM1, DENOM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL, DSYCONV, DSWAP, DTRSM,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -8
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYTRS2', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 )
     $   RETURN
*
*     Convert A
*
      CALL DSYCONV( UPLO, 'C', N, A, LDA, IPIV, WORK, IINFO )
*
      IF( UPPER ) THEN
*
*        Solve A*X = B, where A = U*D*U**T.
*
*       P**T * B
        K=N
        DO WHILE ( K .GE. 1 )
         IF( IPIV( K ).GT.0 ) THEN
*           1 x 1 diagonal block
*           Interchange rows K and IPIV(K).
            KP = IPIV( K )
            IF( KP.NE.K )
     $         CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
            K=K-1
         ELSE
*           2 x 2 diagonal block
*           Interchange rows K-1 and -IPIV(K).
            KP = -IPIV( K )
            IF( KP.EQ.-IPIV( K-1 ) )
     $         CALL DSWAP( NRHS, B( K-1, 1 ), LDB, B( KP, 1 ), LDB )
            K=K-2
         END IF
        END DO
*
*  Compute (U \P**T * B) -> B    [ (U \P**T * B) ]
*
        CALL DTRSM('L','U','N','U',N,NRHS,ONE,A,LDA,B,LDB)
*
*  Compute D \ B -> B   [ D \ (U \P**T * B) ]
*
         I=N
         DO WHILE ( I .GE. 1 )
            IF( IPIV(I) .GT. 0 ) THEN
              CALL DSCAL( NRHS, ONE / A( I, I ), B( I, 1 ), LDB )
            ELSEIF ( I .GT. 1) THEN
               IF ( IPIV(I-1) .EQ. IPIV(I) ) THEN
                  AKM1K = WORK(I)
                  AKM1 = A( I-1, I-1 ) / AKM1K
                  AK = A( I, I ) / AKM1K
                  DENOM = AKM1*AK - ONE
                  DO 15 J = 1, NRHS
                     BKM1 = B( I-1, J ) / AKM1K
                     BK = B( I, J ) / AKM1K
                     B( I-1, J ) = ( AK*BKM1-BK ) / DENOM
                     B( I, J ) = ( AKM1*BK-BKM1 ) / DENOM
 15              CONTINUE
               I = I - 1
               ENDIF
            ENDIF
            I = I - 1
         END DO
*
*      Compute (U**T \ B) -> B   [ U**T \ (D \ (U \P**T * B) ) ]
*
         CALL DTRSM('L','U','T','U',N,NRHS,ONE,A,LDA,B,LDB)
*
*       P * B  [ P * (U**T \ (D \ (U \P**T * B) )) ]
*
        K=1
        DO WHILE ( K .LE. N )
         IF( IPIV( K ).GT.0 ) THEN
*           1 x 1 diagonal block
*           Interchange rows K and IPIV(K).
            KP = IPIV( K )
            IF( KP.NE.K )
     $         CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
            K=K+1
         ELSE
*           2 x 2 diagonal block
*           Interchange rows K-1 and -IPIV(K).
            KP = -IPIV( K )
            IF( K .LT. N .AND. KP.EQ.-IPIV( K+1 ) )
     $         CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
            K=K+2
         ENDIF
        END DO
*
      ELSE
*
*        Solve A*X = B, where A = L*D*L**T.
*
*       P**T * B
        K=1
        DO WHILE ( K .LE. N )
         IF( IPIV( K ).GT.0 ) THEN
*           1 x 1 diagonal block
*           Interchange rows K and IPIV(K).
            KP = IPIV( K )
            IF( KP.NE.K )
     $         CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
            K=K+1
         ELSE
*           2 x 2 diagonal block
*           Interchange rows K and -IPIV(K+1).
            KP = -IPIV( K+1 )
            IF( KP.EQ.-IPIV( K ) )
     $         CALL DSWAP( NRHS, B( K+1, 1 ), LDB, B( KP, 1 ), LDB )
            K=K+2
         ENDIF
        END DO
*
*  Compute (L \P**T * B) -> B    [ (L \P**T * B) ]
*
        CALL DTRSM('L','L','N','U',N,NRHS,ONE,A,LDA,B,LDB)
*
*  Compute D \ B -> B   [ D \ (L \P**T * B) ]
*
         I=1
         DO WHILE ( I .LE. N )
            IF( IPIV(I) .GT. 0 ) THEN
              CALL DSCAL( NRHS, ONE / A( I, I ), B( I, 1 ), LDB )
            ELSE
                  AKM1K = WORK(I)
                  AKM1 = A( I, I ) / AKM1K
                  AK = A( I+1, I+1 ) / AKM1K
                  DENOM = AKM1*AK - ONE
                  DO 25 J = 1, NRHS
                     BKM1 = B( I, J ) / AKM1K
                     BK = B( I+1, J ) / AKM1K
                     B( I, J ) = ( AK*BKM1-BK ) / DENOM
                     B( I+1, J ) = ( AKM1*BK-BKM1 ) / DENOM
 25              CONTINUE
                  I = I + 1
            ENDIF
            I = I + 1
         END DO
*
*  Compute (L**T \ B) -> B   [ L**T \ (D \ (L \P**T * B) ) ]
*
        CALL DTRSM('L','L','T','U',N,NRHS,ONE,A,LDA,B,LDB)
*
*       P * B  [ P * (L**T \ (D \ (L \P**T * B) )) ]
*
        K=N
        DO WHILE ( K .GE. 1 )
         IF( IPIV( K ).GT.0 ) THEN
*           1 x 1 diagonal block
*           Interchange rows K and IPIV(K).
            KP = IPIV( K )
            IF( KP.NE.K )
     $         CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
            K=K-1
         ELSE
*           2 x 2 diagonal block
*           Interchange rows K-1 and -IPIV(K).
            KP = -IPIV( K )
            IF( K.GT.1 .AND. KP.EQ.-IPIV( K-1 ) )
     $         CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
            K=K-2
         ENDIF
        END DO
*
      END IF
*
*     Revert A
*
      CALL DSYCONV( UPLO, 'R', N, A, LDA, IPIV, WORK, IINFO )
*
      RETURN
*
*     End of DSYTRS2
*
      END
*> \brief \b DTBCON
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTBCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtbcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtbcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtbcon.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTBCON( NORM, UPLO, DIAG, N, KD, AB, LDAB, RCOND, WORK,
*                          IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, NORM, UPLO
*       INTEGER            INFO, KD, LDAB, N
*       DOUBLE PRECISION   RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   AB( LDAB, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTBCON estimates the reciprocal of the condition number of a
*> triangular band matrix A, in either the 1-norm or the infinity-norm.
*>
*> The norm of A is computed and an estimate is obtained for
*> norm(inv(A)), then the reciprocal of the condition number is
*> computed as
*>    RCOND = 1 / ( norm(A) * norm(inv(A)) ).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NORM
*> \verbatim
*>          NORM is CHARACTER*1
*>          Specifies whether the 1-norm condition number or the
*>          infinity-norm condition number is required:
*>          = '1' or 'O':  1-norm;
*>          = 'I':         Infinity-norm.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  A is upper triangular;
*>          = 'L':  A is lower triangular.
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          = 'N':  A is non-unit triangular;
*>          = 'U':  A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals or subdiagonals of the
*>          triangular band matrix A.  KD >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          The upper or lower triangular band matrix A, stored in the
*>          first kd+1 rows of the array. The j-th column of A is stored
*>          in the j-th column of the array AB as follows:
*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*>          If DIAG = 'U', the diagonal elements of A are not referenced
*>          and are assumed to be 1.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD+1.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The reciprocal of the condition number of the matrix A,
*>          computed as RCOND = 1/(norm(A) * norm(inv(A))).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tbcon
*
*  =====================================================================
      SUBROUTINE DTBCON( NORM, UPLO, DIAG, N, KD, AB, LDAB, RCOND,
     $                   WORK,
     $                   IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, NORM, UPLO
      INTEGER            INFO, KD, LDAB, N
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   AB( LDAB, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOUNIT, ONENRM, UPPER
      CHARACTER          NORMIN
      INTEGER            IX, KASE, KASE1
      DOUBLE PRECISION   AINVNM, ANORM, SCALE, SMLNUM, XNORM
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH, DLANTB
      EXTERNAL           LSAME, IDAMAX, DLAMCH, DLANTB
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACN2, DLATBS, DRSCL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' )
      NOUNIT = LSAME( DIAG, 'N' )
*
      IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -2
      ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( KD.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDAB.LT.KD+1 ) THEN
         INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTBCON', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 ) THEN
         RCOND = ONE
         RETURN
      END IF
*
      RCOND = ZERO
      SMLNUM = DLAMCH( 'Safe minimum' )*DBLE( MAX( 1, N ) )
*
*     Compute the norm of the triangular matrix A.
*
      ANORM = DLANTB( NORM, UPLO, DIAG, N, KD, AB, LDAB, WORK )
*
*     Continue only if ANORM > 0.
*
      IF( ANORM.GT.ZERO ) THEN
*
*        Estimate the norm of the inverse of A.
*
         AINVNM = ZERO
         NORMIN = 'N'
         IF( ONENRM ) THEN
            KASE1 = 1
         ELSE
            KASE1 = 2
         END IF
         KASE = 0
   10    CONTINUE
         CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE,
     $                ISAVE )
         IF( KASE.NE.0 ) THEN
            IF( KASE.EQ.KASE1 ) THEN
*
*              Multiply by inv(A).
*
               CALL DLATBS( UPLO, 'No transpose', DIAG, NORMIN, N,
     $                      KD,
     $                      AB, LDAB, WORK, SCALE, WORK( 2*N+1 ), INFO )
            ELSE
*
*              Multiply by inv(A**T).
*
               CALL DLATBS( UPLO, 'Transpose', DIAG, NORMIN, N, KD,
     $                      AB,
     $                      LDAB, WORK, SCALE, WORK( 2*N+1 ), INFO )
            END IF
            NORMIN = 'Y'
*
*           Multiply by 1/SCALE if doing so will not cause overflow.
*
            IF( SCALE.NE.ONE ) THEN
               IX = IDAMAX( N, WORK, 1 )
               XNORM = ABS( WORK( IX ) )
               IF( SCALE.LT.XNORM*SMLNUM .OR. SCALE.EQ.ZERO )
     $            GO TO 20
               CALL DRSCL( N, SCALE, WORK, 1 )
            END IF
            GO TO 10
         END IF
*
*        Compute the estimate of the reciprocal condition number.
*
         IF( AINVNM.NE.ZERO )
     $      RCOND = ( ONE / ANORM ) / AINVNM
      END IF
*
   20 CONTINUE
      RETURN
*
*     End of DTBCON
*
      END
*> \brief \b DTBRFS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTBRFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtbrfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtbrfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtbrfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
*                          LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, TRANS, UPLO
*       INTEGER            INFO, KD, LDAB, LDB, LDX, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * ), BERR( * ),
*      $                   FERR( * ), WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTBRFS provides error bounds and backward error estimates for the
*> solution to a system of linear equations with a triangular band
*> coefficient matrix.
*>
*> The solution matrix X must be computed by DTBTRS or some other
*> means before entering this routine.  DTBRFS does not do iterative
*> refinement because doing so cannot improve the backward error.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  A is upper triangular;
*>          = 'L':  A is lower triangular.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the form of the system of equations:
*>          = 'N':  A * X = B  (No transpose)
*>          = 'T':  A**T * X = B  (Transpose)
*>          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          = 'N':  A is non-unit triangular;
*>          = 'U':  A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals or subdiagonals of the
*>          triangular band matrix A.  KD >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices B and X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          The upper or lower triangular band matrix A, stored in the
*>          first kd+1 rows of the array. The j-th column of A is stored
*>          in the j-th column of the array AB as follows:
*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*>          If DIAG = 'U', the diagonal elements of A are not referenced
*>          and are assumed to be 1.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          The solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The estimated forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).  The estimate is as reliable as
*>          the estimate for RCOND, and is almost always a slight
*>          overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in
*>          any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tbrfs
*
*  =====================================================================
      SUBROUTINE DTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
     $                   LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, TRANS, UPLO
      INTEGER            INFO, KD, LDAB, LDB, LDX, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * ), BERR( * ),
     $                   FERR( * ), WORK( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOTRAN, NOUNIT, UPPER
      CHARACTER          TRANST
      INTEGER            I, J, K, KASE, NZ
      DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DLACN2, DTBMV, DTBSV,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      NOTRAN = LSAME( TRANS, 'N' )
      NOUNIT = LSAME( DIAG, 'N' )
*
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
     $         LSAME( TRANS, 'C' ) ) THEN
         INFO = -2
      ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( KD.LT.0 ) THEN
         INFO = -5
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -6
      ELSE IF( LDAB.LT.KD+1 ) THEN
         INFO = -8
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -10
      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
         INFO = -12
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTBRFS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
         DO 10 J = 1, NRHS
            FERR( J ) = ZERO
            BERR( J ) = ZERO
   10    CONTINUE
         RETURN
      END IF
*
      IF( NOTRAN ) THEN
         TRANST = 'T'
      ELSE
         TRANST = 'N'
      END IF
*
*     NZ = maximum number of nonzero elements in each row of A, plus 1
*
      NZ = KD + 2
      EPS = DLAMCH( 'Epsilon' )
      SAFMIN = DLAMCH( 'Safe minimum' )
      SAFE1 = NZ*SAFMIN
      SAFE2 = SAFE1 / EPS
*
*     Do for each right hand side
*
      DO 250 J = 1, NRHS
*
*        Compute residual R = B - op(A) * X,
*        where op(A) = A or A**T, depending on TRANS.
*
         CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 )
         CALL DTBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK( N+1 ),
     $               1 )
         CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 )
*
*        Compute componentwise relative backward error from formula
*
*        max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
*
*        where abs(Z) is the componentwise absolute value of the matrix
*        or vector Z.  If the i-th component of the denominator is less
*        than SAFE2, then SAFE1 is added to the i-th components of the
*        numerator and denominator before dividing.
*
         DO 20 I = 1, N
            WORK( I ) = ABS( B( I, J ) )
   20    CONTINUE
*
         IF( NOTRAN ) THEN
*
*           Compute abs(A)*abs(X) + abs(B).
*
            IF( UPPER ) THEN
               IF( NOUNIT ) THEN
                  DO 40 K = 1, N
                     XK = ABS( X( K, J ) )
                     DO 30 I = MAX( 1, K-KD ), K
                        WORK( I ) = WORK( I ) +
     $                              ABS( AB( KD+1+I-K, K ) )*XK
   30                CONTINUE
   40             CONTINUE
               ELSE
                  DO 60 K = 1, N
                     XK = ABS( X( K, J ) )
                     DO 50 I = MAX( 1, K-KD ), K - 1
                        WORK( I ) = WORK( I ) +
     $                              ABS( AB( KD+1+I-K, K ) )*XK
   50                CONTINUE
                     WORK( K ) = WORK( K ) + XK
   60             CONTINUE
               END IF
            ELSE
               IF( NOUNIT ) THEN
                  DO 80 K = 1, N
                     XK = ABS( X( K, J ) )
                     DO 70 I = K, MIN( N, K+KD )
                        WORK( I ) = WORK( I ) + ABS( AB( 1+I-K, K ) )*XK
   70                CONTINUE
   80             CONTINUE
               ELSE
                  DO 100 K = 1, N
                     XK = ABS( X( K, J ) )
                     DO 90 I = K + 1, MIN( N, K+KD )
                        WORK( I ) = WORK( I ) + ABS( AB( 1+I-K, K ) )*XK
   90                CONTINUE
                     WORK( K ) = WORK( K ) + XK
  100             CONTINUE
               END IF
            END IF
         ELSE
*
*           Compute abs(A**T)*abs(X) + abs(B).
*
            IF( UPPER ) THEN
               IF( NOUNIT ) THEN
                  DO 120 K = 1, N
                     S = ZERO
                     DO 110 I = MAX( 1, K-KD ), K
                        S = S + ABS( AB( KD+1+I-K, K ) )*
     $                      ABS( X( I, J ) )
  110                CONTINUE
                     WORK( K ) = WORK( K ) + S
  120             CONTINUE
               ELSE
                  DO 140 K = 1, N
                     S = ABS( X( K, J ) )
                     DO 130 I = MAX( 1, K-KD ), K - 1
                        S = S + ABS( AB( KD+1+I-K, K ) )*
     $                      ABS( X( I, J ) )
  130                CONTINUE
                     WORK( K ) = WORK( K ) + S
  140             CONTINUE
               END IF
            ELSE
               IF( NOUNIT ) THEN
                  DO 160 K = 1, N
                     S = ZERO
                     DO 150 I = K, MIN( N, K+KD )
                        S = S + ABS( AB( 1+I-K, K ) )*ABS( X( I, J ) )
  150                CONTINUE
                     WORK( K ) = WORK( K ) + S
  160             CONTINUE
               ELSE
                  DO 180 K = 1, N
                     S = ABS( X( K, J ) )
                     DO 170 I = K + 1, MIN( N, K+KD )
                        S = S + ABS( AB( 1+I-K, K ) )*ABS( X( I, J ) )
  170                CONTINUE
                     WORK( K ) = WORK( K ) + S
  180             CONTINUE
               END IF
            END IF
         END IF
         S = ZERO
         DO 190 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
            ELSE
               S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
     $             ( WORK( I )+SAFE1 ) )
            END IF
  190    CONTINUE
         BERR( J ) = S
*
*        Bound error from formula
*
*        norm(X - XTRUE) / norm(X) .le. FERR =
*        norm( abs(inv(op(A)))*
*           ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
*
*        where
*          norm(Z) is the magnitude of the largest component of Z
*          inv(op(A)) is the inverse of op(A)
*          abs(Z) is the componentwise absolute value of the matrix or
*             vector Z
*          NZ is the maximum number of nonzeros in any row of A, plus 1
*          EPS is machine epsilon
*
*        The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
*        is incremented by SAFE1 if the i-th component of
*        abs(op(A))*abs(X) + abs(B) is less than SAFE2.
*
*        Use DLACN2 to estimate the infinity-norm of the matrix
*           inv(op(A)) * diag(W),
*        where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
*
         DO 200 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
            ELSE
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
            END IF
  200    CONTINUE
*
         KASE = 0
  210    CONTINUE
         CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
     $                FERR( J ),
     $                KASE, ISAVE )
         IF( KASE.NE.0 ) THEN
            IF( KASE.EQ.1 ) THEN
*
*              Multiply by diag(W)*inv(op(A)**T).
*
               CALL DTBSV( UPLO, TRANST, DIAG, N, KD, AB, LDAB,
     $                     WORK( N+1 ), 1 )
               DO 220 I = 1, N
                  WORK( N+I ) = WORK( I )*WORK( N+I )
  220          CONTINUE
            ELSE
*
*              Multiply by inv(op(A))*diag(W).
*
               DO 230 I = 1, N
                  WORK( N+I ) = WORK( I )*WORK( N+I )
  230          CONTINUE
               CALL DTBSV( UPLO, TRANS, DIAG, N, KD, AB, LDAB,
     $                     WORK( N+1 ), 1 )
            END IF
            GO TO 210
         END IF
*
*        Normalize error.
*
         LSTRES = ZERO
         DO 240 I = 1, N
            LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
  240    CONTINUE
         IF( LSTRES.NE.ZERO )
     $      FERR( J ) = FERR( J ) / LSTRES
*
  250 CONTINUE
*
      RETURN
*
*     End of DTBRFS
*
      END
*> \brief \b DTBTRS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTBTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtbtrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtbtrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtbtrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTBTRS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
*                          LDB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, TRANS, UPLO
*       INTEGER            INFO, KD, LDAB, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTBTRS solves a triangular system of the form
*>
*>    A * X = B  or  A**T * X = B,
*>
*> where A is a triangular band matrix of order N, and B is an N-by-NRHS matrix.
*>
*> This subroutine verifies that A is nonsingular, but callers should note that only exact
*> singularity is detected. It is conceivable for one or more diagonal elements of A to be
*> subnormally tiny numbers without this subroutine signalling an error.
*>
*> If a possible loss of numerical precision due to near-singular matrices is a concern, the
*> caller should verify that A is nonsingular within some tolerance before calling this subroutine.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  A is upper triangular;
*>          = 'L':  A is lower triangular.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the form the system of equations:
*>          = 'N':  A * X = B  (No transpose)
*>          = 'T':  A**T * X = B  (Transpose)
*>          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          = 'N':  A is non-unit triangular;
*>          = 'U':  A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals or subdiagonals of the
*>          triangular band matrix A.  KD >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          The upper or lower triangular band matrix A, stored in the
*>          first kd+1 rows of AB.  The j-th column of A is stored
*>          in the j-th column of the array AB as follows:
*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*>          If DIAG = 'U', the diagonal elements of A are not referenced
*>          and are assumed to be 1.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the right hand side matrix B.
*>          On exit, if INFO = 0, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the i-th diagonal element of A is exactly zero,
*>                indicating that the matrix is singular and the
*>                solutions X have not been computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tbtrs
*
*  =====================================================================
      SUBROUTINE DTBTRS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
     $                   LDB, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, TRANS, UPLO
      INTEGER            INFO, KD, LDAB, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOUNIT, UPPER
      INTEGER            J
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DTBSV, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NOUNIT = LSAME( DIAG, 'N' )
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.LSAME( TRANS, 'N' ) .AND. .NOT.
     $         LSAME( TRANS, 'T' ) .AND.
     $                .NOT.LSAME( TRANS, 'C' ) ) THEN
         INFO = -2
      ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( KD.LT.0 ) THEN
         INFO = -5
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -6
      ELSE IF( LDAB.LT.KD+1 ) THEN
         INFO = -8
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -10
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTBTRS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Check for singularity.
*
      IF( NOUNIT ) THEN
         IF( UPPER ) THEN
            DO 10 INFO = 1, N
               IF( AB( KD+1, INFO ).EQ.ZERO )
     $            RETURN
   10       CONTINUE
         ELSE
            DO 20 INFO = 1, N
               IF( AB( 1, INFO ).EQ.ZERO )
     $            RETURN
   20       CONTINUE
         END IF
      END IF
      INFO = 0
*
*     Solve A * X = B  or  A**T * X = B.
*
      DO 30 J = 1, NRHS
         CALL DTBSV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, B( 1, J ),
     $               1 )
   30 CONTINUE
*
      RETURN
*
*     End of DTBTRS
*
      END
*> \brief \b DTFSM solves a matrix equation (one operand is a triangular matrix in RFP format).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTFSM + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtfsm.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtfsm.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtfsm.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTFSM( TRANSR, SIDE, UPLO, TRANS, DIAG, M, N, ALPHA, A,
*                         B, LDB )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANSR, DIAG, SIDE, TRANS, UPLO
*       INTEGER            LDB, M, N
*       DOUBLE PRECISION   ALPHA
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( 0: * ), B( 0: LDB-1, 0: * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> Level 3 BLAS like routine for A in RFP Format.
*>
*> DTFSM  solves the matrix equation
*>
*>    op( A )*X = alpha*B  or  X*op( A ) = alpha*B
*>
*> where alpha is a scalar, X and B are m by n matrices, A is a unit, or
*> non-unit,  upper or lower triangular matrix  and  op( A )  is one  of
*>
*>    op( A ) = A   or   op( A ) = A**T.
*>
*> A is in Rectangular Full Packed (RFP) Format.
*>
*> The matrix X is overwritten on B.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANSR
*> \verbatim
*>          TRANSR is CHARACTER*1
*>          = 'N':  The Normal Form of RFP A is stored;
*>          = 'T':  The Transpose Form of RFP A is stored.
*> \endverbatim
*>
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>           On entry, SIDE specifies whether op( A ) appears on the left
*>           or right of X as follows:
*>
*>              SIDE = 'L' or 'l'   op( A )*X = alpha*B.
*>
*>              SIDE = 'R' or 'r'   X*op( A ) = alpha*B.
*>
*>           Unchanged on exit.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>           On entry, UPLO specifies whether the RFP matrix A came from
*>           an upper or lower triangular matrix as follows:
*>           UPLO = 'U' or 'u' RFP A came from an upper triangular matrix
*>           UPLO = 'L' or 'l' RFP A came from a  lower triangular matrix
*>
*>           Unchanged on exit.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>           On entry, TRANS  specifies the form of op( A ) to be used
*>           in the matrix multiplication as follows:
*>
*>              TRANS  = 'N' or 'n'   op( A ) = A.
*>
*>              TRANS  = 'T' or 't'   op( A ) = A'.
*>
*>           Unchanged on exit.
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>           On entry, DIAG specifies whether or not RFP A is unit
*>           triangular as follows:
*>
*>              DIAG = 'U' or 'u'   A is assumed to be unit triangular.
*>
*>              DIAG = 'N' or 'n'   A is not assumed to be unit
*>                                  triangular.
*>
*>           Unchanged on exit.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>           On entry, M specifies the number of rows of B. M must be at
*>           least zero.
*>           Unchanged on exit.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>           On entry, N specifies the number of columns of B.  N must be
*>           at least zero.
*>           Unchanged on exit.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*>          ALPHA is DOUBLE PRECISION
*>           On entry,  ALPHA specifies the scalar  alpha. When  alpha is
*>           zero then  A is not referenced and  B need not be set before
*>           entry.
*>           Unchanged on exit.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (NT)
*>           NT = N*(N+1)/2 if SIDE='R' and NT = M*(M+1)/2 otherwise.
*>           On entry, the matrix A in RFP Format.
*>           RFP Format is described by TRANSR, UPLO and N as follows:
*>           If TRANSR='N' then RFP A is (0:N,0:K-1) when N is even;
*>           K=N/2. RFP A is (0:N-1,0:K) when N is odd; K=N/2. If
*>           TRANSR = 'T' then RFP is the transpose of RFP A as
*>           defined when TRANSR = 'N'. The contents of RFP A are defined
*>           by UPLO as follows: If UPLO = 'U' the RFP A contains the NT
*>           elements of upper packed A either in normal or
*>           transpose Format. If UPLO = 'L' the RFP A contains
*>           the NT elements of lower packed A either in normal or
*>           transpose Format. The LDA of RFP A is (N+1)/2 when
*>           TRANSR = 'T'. When TRANSR is 'N' the LDA is N+1 when N is
*>           even and is N when is odd.
*>           See the Note below for more details. Unchanged on exit.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,N)
*>           Before entry,  the leading  m by n part of the array  B must
*>           contain  the  right-hand  side  matrix  B,  and  on exit  is
*>           overwritten by the solution matrix  X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>           On entry, LDB specifies the first dimension of B as declared
*>           in  the  calling  (sub)  program.   LDB  must  be  at  least
*>           max( 1, m ).
*>           Unchanged on exit.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tfsm
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  We first consider Rectangular Full Packed (RFP) Format when N is
*>  even. We give an example where N = 6.
*>
*>      AP is Upper             AP is Lower
*>
*>   00 01 02 03 04 05       00
*>      11 12 13 14 15       10 11
*>         22 23 24 25       20 21 22
*>            33 34 35       30 31 32 33
*>               44 45       40 41 42 43 44
*>                  55       50 51 52 53 54 55
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*>  the transpose of the first three columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*>  the transpose of the last three columns of AP lower.
*>  This covers the case N even and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>        03 04 05                33 43 53
*>        13 14 15                00 44 54
*>        23 24 25                10 11 55
*>        33 34 35                20 21 22
*>        00 44 45                30 31 32
*>        01 11 55                40 41 42
*>        02 12 22                50 51 52
*>
*>  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*>  transpose of RFP A above. One therefore gets:
*>
*>
*>           RFP A                   RFP A
*>
*>     03 13 23 33 00 01 02    33 00 10 20 30 40 50
*>     04 14 24 34 44 11 12    43 44 11 21 31 41 51
*>     05 15 25 35 45 55 22    53 54 55 22 32 42 52
*>
*>
*>  We then consider Rectangular Full Packed (RFP) Format when N is
*>  odd. We give an example where N = 5.
*>
*>     AP is Upper                 AP is Lower
*>
*>   00 01 02 03 04              00
*>      11 12 13 14              10 11
*>         22 23 24              20 21 22
*>            33 34              30 31 32 33
*>               44              40 41 42 43 44
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*>  the transpose of the first two columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*>  the transpose of the last two columns of AP lower.
*>  This covers the case N odd and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>        02 03 04                00 33 43
*>        12 13 14                10 11 44
*>        22 23 24                20 21 22
*>        00 33 34                30 31 32
*>        01 11 44                40 41 42
*>
*>  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*>  transpose of RFP A above. One therefore gets:
*>
*>           RFP A                   RFP A
*>
*>     02 12 22 00 01             00 10 20 30 40 50
*>     03 13 23 33 11             33 11 21 31 41 51
*>     04 14 24 34 44             43 44 22 32 42 52
*> \endverbatim
*
*  =====================================================================
      SUBROUTINE DTFSM( TRANSR, SIDE, UPLO, TRANS, DIAG, M, N, ALPHA,
     $                  A,
     $                  B, LDB )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANSR, DIAG, SIDE, TRANS, UPLO
      INTEGER            LDB, M, N
      DOUBLE PRECISION   ALPHA
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( 0: * ), B( 0: LDB-1, 0: * )
*     ..
*
*  =====================================================================
*
*     ..
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER, LSIDE, MISODD, NISODD, NORMALTRANSR,
     $                   NOTRANS
      INTEGER            M1, M2, N1, N2, K, INFO, I, J
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA, DGEMM, DTRSM
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MOD
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NORMALTRANSR = LSAME( TRANSR, 'N' )
      LSIDE = LSAME( SIDE, 'L' )
      LOWER = LSAME( UPLO, 'L' )
      NOTRANS = LSAME( TRANS, 'N' )
      IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.LSIDE .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
         INFO = -2
      ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
         INFO = -3
      ELSE IF( .NOT.NOTRANS .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
         INFO = -4
      ELSE IF( .NOT.LSAME( DIAG, 'N' ) .AND.
     $         .NOT.LSAME( DIAG, 'U' ) )
     $         THEN
         INFO = -5
      ELSE IF( M.LT.0 ) THEN
         INFO = -6
      ELSE IF( N.LT.0 ) THEN
         INFO = -7
      ELSE IF( LDB.LT.MAX( 1, M ) ) THEN
         INFO = -11
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTFSM ', -INFO )
         RETURN
      END IF
*
*     Quick return when ( (N.EQ.0).OR.(M.EQ.0) )
*
      IF( ( M.EQ.0 ) .OR. ( N.EQ.0 ) )
     $   RETURN
*
*     Quick return when ALPHA.EQ.(0D+0)
*
      IF( ALPHA.EQ.ZERO ) THEN
         DO 20 J = 0, N - 1
            DO 10 I = 0, M - 1
               B( I, J ) = ZERO
   10       CONTINUE
   20    CONTINUE
         RETURN
      END IF
*
      IF( LSIDE ) THEN
*
*        SIDE = 'L'
*
*        A is M-by-M.
*        If M is odd, set NISODD = .TRUE., and M1 and M2.
*        If M is even, NISODD = .FALSE., and M.
*
         IF( MOD( M, 2 ).EQ.0 ) THEN
            MISODD = .FALSE.
            K = M / 2
         ELSE
            MISODD = .TRUE.
            IF( LOWER ) THEN
               M2 = M / 2
               M1 = M - M2
            ELSE
               M1 = M / 2
               M2 = M - M1
            END IF
         END IF
*
*
         IF( MISODD ) THEN
*
*           SIDE = 'L' and N is odd
*
            IF( NORMALTRANSR ) THEN
*
*              SIDE = 'L', N is odd, and TRANSR = 'N'
*
               IF( LOWER ) THEN
*
*                 SIDE  ='L', N is odd, TRANSR = 'N', and UPLO = 'L'
*
                  IF( NOTRANS ) THEN
*
*                    SIDE  ='L', N is odd, TRANSR = 'N', UPLO = 'L', and
*                    TRANS = 'N'
*
                     IF( M.EQ.1 ) THEN
                        CALL DTRSM( 'L', 'L', 'N', DIAG, M1, N,
     $                              ALPHA,
     $                              A, M, B, LDB )
                     ELSE
                        CALL DTRSM( 'L', 'L', 'N', DIAG, M1, N,
     $                              ALPHA,
     $                              A( 0 ), M, B, LDB )
                        CALL DGEMM( 'N', 'N', M2, N, M1, -ONE,
     $                              A( M1 ),
     $                              M, B, LDB, ALPHA, B( M1, 0 ), LDB )
                        CALL DTRSM( 'L', 'U', 'T', DIAG, M2, N, ONE,
     $                              A( M ), M, B( M1, 0 ), LDB )
                     END IF
*
                  ELSE
*
*                    SIDE  ='L', N is odd, TRANSR = 'N', UPLO = 'L', and
*                    TRANS = 'T'
*
                     IF( M.EQ.1 ) THEN
                        CALL DTRSM( 'L', 'L', 'T', DIAG, M1, N,
     $                              ALPHA,
     $                              A( 0 ), M, B, LDB )
                     ELSE
                        CALL DTRSM( 'L', 'U', 'N', DIAG, M2, N,
     $                              ALPHA,
     $                              A( M ), M, B( M1, 0 ), LDB )
                        CALL DGEMM( 'T', 'N', M1, N, M2, -ONE,
     $                              A( M1 ),
     $                              M, B( M1, 0 ), LDB, ALPHA, B, LDB )
                        CALL DTRSM( 'L', 'L', 'T', DIAG, M1, N, ONE,
     $                              A( 0 ), M, B, LDB )
                     END IF
*
                  END IF
*
               ELSE
*
*                 SIDE  ='L', N is odd, TRANSR = 'N', and UPLO = 'U'
*
                  IF( .NOT.NOTRANS ) THEN
*
*                    SIDE  ='L', N is odd, TRANSR = 'N', UPLO = 'U', and
*                    TRANS = 'N'
*
                     CALL DTRSM( 'L', 'L', 'N', DIAG, M1, N, ALPHA,
     $                           A( M2 ), M, B, LDB )
                     CALL DGEMM( 'T', 'N', M2, N, M1, -ONE, A( 0 ),
     $                           M,
     $                           B, LDB, ALPHA, B( M1, 0 ), LDB )
                     CALL DTRSM( 'L', 'U', 'T', DIAG, M2, N, ONE,
     $                           A( M1 ), M, B( M1, 0 ), LDB )
*
                  ELSE
*
*                    SIDE  ='L', N is odd, TRANSR = 'N', UPLO = 'U', and
*                    TRANS = 'T'
*
                     CALL DTRSM( 'L', 'U', 'N', DIAG, M2, N, ALPHA,
     $                           A( M1 ), M, B( M1, 0 ), LDB )
                     CALL DGEMM( 'N', 'N', M1, N, M2, -ONE, A( 0 ),
     $                           M,
     $                           B( M1, 0 ), LDB, ALPHA, B, LDB )
                     CALL DTRSM( 'L', 'L', 'T', DIAG, M1, N, ONE,
     $                           A( M2 ), M, B, LDB )
*
                  END IF
*
               END IF
*
            ELSE
*
*              SIDE = 'L', N is odd, and TRANSR = 'T'
*
               IF( LOWER ) THEN
*
*                 SIDE  ='L', N is odd, TRANSR = 'T', and UPLO = 'L'
*
                  IF( NOTRANS ) THEN
*
*                    SIDE  ='L', N is odd, TRANSR = 'T', UPLO = 'L', and
*                    TRANS = 'N'
*
                     IF( M.EQ.1 ) THEN
                        CALL DTRSM( 'L', 'U', 'T', DIAG, M1, N,
     $                              ALPHA,
     $                              A( 0 ), M1, B, LDB )
                     ELSE
                        CALL DTRSM( 'L', 'U', 'T', DIAG, M1, N,
     $                              ALPHA,
     $                              A( 0 ), M1, B, LDB )
                        CALL DGEMM( 'T', 'N', M2, N, M1, -ONE,
     $                              A( M1*M1 ), M1, B, LDB, ALPHA,
     $                              B( M1, 0 ), LDB )
                        CALL DTRSM( 'L', 'L', 'N', DIAG, M2, N, ONE,
     $                              A( 1 ), M1, B( M1, 0 ), LDB )
                     END IF
*
                  ELSE
*
*                    SIDE  ='L', N is odd, TRANSR = 'T', UPLO = 'L', and
*                    TRANS = 'T'
*
                     IF( M.EQ.1 ) THEN
                        CALL DTRSM( 'L', 'U', 'N', DIAG, M1, N,
     $                              ALPHA,
     $                              A( 0 ), M1, B, LDB )
                     ELSE
                        CALL DTRSM( 'L', 'L', 'T', DIAG, M2, N,
     $                              ALPHA,
     $                              A( 1 ), M1, B( M1, 0 ), LDB )
                        CALL DGEMM( 'N', 'N', M1, N, M2, -ONE,
     $                              A( M1*M1 ), M1, B( M1, 0 ), LDB,
     $                              ALPHA, B, LDB )
                        CALL DTRSM( 'L', 'U', 'N', DIAG, M1, N, ONE,
     $                              A( 0 ), M1, B, LDB )
                     END IF
*
                  END IF
*
               ELSE
*
*                 SIDE  ='L', N is odd, TRANSR = 'T', and UPLO = 'U'
*
                  IF( .NOT.NOTRANS ) THEN
*
*                    SIDE  ='L', N is odd, TRANSR = 'T', UPLO = 'U', and
*                    TRANS = 'N'
*
                     CALL DTRSM( 'L', 'U', 'T', DIAG, M1, N, ALPHA,
     $                           A( M2*M2 ), M2, B, LDB )
                     CALL DGEMM( 'N', 'N', M2, N, M1, -ONE, A( 0 ),
     $                           M2,
     $                           B, LDB, ALPHA, B( M1, 0 ), LDB )
                     CALL DTRSM( 'L', 'L', 'N', DIAG, M2, N, ONE,
     $                           A( M1*M2 ), M2, B( M1, 0 ), LDB )
*
                  ELSE
*
*                    SIDE  ='L', N is odd, TRANSR = 'T', UPLO = 'U', and
*                    TRANS = 'T'
*
                     CALL DTRSM( 'L', 'L', 'T', DIAG, M2, N, ALPHA,
     $                           A( M1*M2 ), M2, B( M1, 0 ), LDB )
                     CALL DGEMM( 'T', 'N', M1, N, M2, -ONE, A( 0 ),
     $                           M2,
     $                           B( M1, 0 ), LDB, ALPHA, B, LDB )
                     CALL DTRSM( 'L', 'U', 'N', DIAG, M1, N, ONE,
     $                           A( M2*M2 ), M2, B, LDB )
*
                  END IF
*
               END IF
*
            END IF
*
         ELSE
*
*           SIDE = 'L' and N is even
*
            IF( NORMALTRANSR ) THEN
*
*              SIDE = 'L', N is even, and TRANSR = 'N'
*
               IF( LOWER ) THEN
*
*                 SIDE  ='L', N is even, TRANSR = 'N', and UPLO = 'L'
*
                  IF( NOTRANS ) THEN
*
*                    SIDE  ='L', N is even, TRANSR = 'N', UPLO = 'L',
*                    and TRANS = 'N'
*
                     CALL DTRSM( 'L', 'L', 'N', DIAG, K, N, ALPHA,
     $                           A( 1 ), M+1, B, LDB )
                     CALL DGEMM( 'N', 'N', K, N, K, -ONE, A( K+1 ),
     $                           M+1, B, LDB, ALPHA, B( K, 0 ), LDB )
                     CALL DTRSM( 'L', 'U', 'T', DIAG, K, N, ONE,
     $                           A( 0 ), M+1, B( K, 0 ), LDB )
*
                  ELSE
*
*                    SIDE  ='L', N is even, TRANSR = 'N', UPLO = 'L',
*                    and TRANS = 'T'
*
                     CALL DTRSM( 'L', 'U', 'N', DIAG, K, N, ALPHA,
     $                           A( 0 ), M+1, B( K, 0 ), LDB )
                     CALL DGEMM( 'T', 'N', K, N, K, -ONE, A( K+1 ),
     $                           M+1, B( K, 0 ), LDB, ALPHA, B, LDB )
                     CALL DTRSM( 'L', 'L', 'T', DIAG, K, N, ONE,
     $                           A( 1 ), M+1, B, LDB )
*
                  END IF
*
               ELSE
*
*                 SIDE  ='L', N is even, TRANSR = 'N', and UPLO = 'U'
*
                  IF( .NOT.NOTRANS ) THEN
*
*                    SIDE  ='L', N is even, TRANSR = 'N', UPLO = 'U',
*                    and TRANS = 'N'
*
                     CALL DTRSM( 'L', 'L', 'N', DIAG, K, N, ALPHA,
     $                           A( K+1 ), M+1, B, LDB )
                     CALL DGEMM( 'T', 'N', K, N, K, -ONE, A( 0 ),
     $                           M+1,
     $                           B, LDB, ALPHA, B( K, 0 ), LDB )
                     CALL DTRSM( 'L', 'U', 'T', DIAG, K, N, ONE,
     $                           A( K ), M+1, B( K, 0 ), LDB )
*
                  ELSE
*
*                    SIDE  ='L', N is even, TRANSR = 'N', UPLO = 'U',
*                    and TRANS = 'T'
                     CALL DTRSM( 'L', 'U', 'N', DIAG, K, N, ALPHA,
     $                           A( K ), M+1, B( K, 0 ), LDB )
                     CALL DGEMM( 'N', 'N', K, N, K, -ONE, A( 0 ),
     $                           M+1,
     $                           B( K, 0 ), LDB, ALPHA, B, LDB )
                     CALL DTRSM( 'L', 'L', 'T', DIAG, K, N, ONE,
     $                           A( K+1 ), M+1, B, LDB )
*
                  END IF
*
               END IF
*
            ELSE
*
*              SIDE = 'L', N is even, and TRANSR = 'T'
*
               IF( LOWER ) THEN
*
*                 SIDE  ='L', N is even, TRANSR = 'T', and UPLO = 'L'
*
                  IF( NOTRANS ) THEN
*
*                    SIDE  ='L', N is even, TRANSR = 'T', UPLO = 'L',
*                    and TRANS = 'N'
*
                     CALL DTRSM( 'L', 'U', 'T', DIAG, K, N, ALPHA,
     $                           A( K ), K, B, LDB )
                     CALL DGEMM( 'T', 'N', K, N, K, -ONE,
     $                           A( K*( K+1 ) ), K, B, LDB, ALPHA,
     $                           B( K, 0 ), LDB )
                     CALL DTRSM( 'L', 'L', 'N', DIAG, K, N, ONE,
     $                           A( 0 ), K, B( K, 0 ), LDB )
*
                  ELSE
*
*                    SIDE  ='L', N is even, TRANSR = 'T', UPLO = 'L',
*                    and TRANS = 'T'
*
                     CALL DTRSM( 'L', 'L', 'T', DIAG, K, N, ALPHA,
     $                           A( 0 ), K, B( K, 0 ), LDB )
                     CALL DGEMM( 'N', 'N', K, N, K, -ONE,
     $                           A( K*( K+1 ) ), K, B( K, 0 ), LDB,
     $                           ALPHA, B, LDB )
                     CALL DTRSM( 'L', 'U', 'N', DIAG, K, N, ONE,
     $                           A( K ), K, B, LDB )
*
                  END IF
*
               ELSE
*
*                 SIDE  ='L', N is even, TRANSR = 'T', and UPLO = 'U'
*
                  IF( .NOT.NOTRANS ) THEN
*
*                    SIDE  ='L', N is even, TRANSR = 'T', UPLO = 'U',
*                    and TRANS = 'N'
*
                     CALL DTRSM( 'L', 'U', 'T', DIAG, K, N, ALPHA,
     $                           A( K*( K+1 ) ), K, B, LDB )
                     CALL DGEMM( 'N', 'N', K, N, K, -ONE, A( 0 ), K,
     $                           B,
     $                           LDB, ALPHA, B( K, 0 ), LDB )
                     CALL DTRSM( 'L', 'L', 'N', DIAG, K, N, ONE,
     $                           A( K*K ), K, B( K, 0 ), LDB )
*
                  ELSE
*
*                    SIDE  ='L', N is even, TRANSR = 'T', UPLO = 'U',
*                    and TRANS = 'T'
*
                     CALL DTRSM( 'L', 'L', 'T', DIAG, K, N, ALPHA,
     $                           A( K*K ), K, B( K, 0 ), LDB )
                     CALL DGEMM( 'T', 'N', K, N, K, -ONE, A( 0 ), K,
     $                           B( K, 0 ), LDB, ALPHA, B, LDB )
                     CALL DTRSM( 'L', 'U', 'N', DIAG, K, N, ONE,
     $                           A( K*( K+1 ) ), K, B, LDB )
*
                  END IF
*
               END IF
*
            END IF
*
         END IF
*
      ELSE
*
*        SIDE = 'R'
*
*        A is N-by-N.
*        If N is odd, set NISODD = .TRUE., and N1 and N2.
*        If N is even, NISODD = .FALSE., and K.
*
         IF( MOD( N, 2 ).EQ.0 ) THEN
            NISODD = .FALSE.
            K = N / 2
         ELSE
            NISODD = .TRUE.
            IF( LOWER ) THEN
               N2 = N / 2
               N1 = N - N2
            ELSE
               N1 = N / 2
               N2 = N - N1
            END IF
         END IF
*
         IF( NISODD ) THEN
*
*           SIDE = 'R' and N is odd
*
            IF( NORMALTRANSR ) THEN
*
*              SIDE = 'R', N is odd, and TRANSR = 'N'
*
               IF( LOWER ) THEN
*
*                 SIDE  ='R', N is odd, TRANSR = 'N', and UPLO = 'L'
*
                  IF( NOTRANS ) THEN
*
*                    SIDE  ='R', N is odd, TRANSR = 'N', UPLO = 'L', and
*                    TRANS = 'N'
*
                     CALL DTRSM( 'R', 'U', 'T', DIAG, M, N2, ALPHA,
     $                           A( N ), N, B( 0, N1 ), LDB )
                     CALL DGEMM( 'N', 'N', M, N1, N2, -ONE, B( 0,
     $                           N1 ),
     $                           LDB, A( N1 ), N, ALPHA, B( 0, 0 ),
     $                           LDB )
                     CALL DTRSM( 'R', 'L', 'N', DIAG, M, N1, ONE,
     $                           A( 0 ), N, B( 0, 0 ), LDB )
*
                  ELSE
*
*                    SIDE  ='R', N is odd, TRANSR = 'N', UPLO = 'L', and
*                    TRANS = 'T'
*
                     CALL DTRSM( 'R', 'L', 'T', DIAG, M, N1, ALPHA,
     $                           A( 0 ), N, B( 0, 0 ), LDB )
                     CALL DGEMM( 'N', 'T', M, N2, N1, -ONE, B( 0,
     $                           0 ),
     $                           LDB, A( N1 ), N, ALPHA, B( 0, N1 ),
     $                           LDB )
                     CALL DTRSM( 'R', 'U', 'N', DIAG, M, N2, ONE,
     $                           A( N ), N, B( 0, N1 ), LDB )
*
                  END IF
*
               ELSE
*
*                 SIDE  ='R', N is odd, TRANSR = 'N', and UPLO = 'U'
*
                  IF( NOTRANS ) THEN
*
*                    SIDE  ='R', N is odd, TRANSR = 'N', UPLO = 'U', and
*                    TRANS = 'N'
*
                     CALL DTRSM( 'R', 'L', 'T', DIAG, M, N1, ALPHA,
     $                           A( N2 ), N, B( 0, 0 ), LDB )
                     CALL DGEMM( 'N', 'N', M, N2, N1, -ONE, B( 0,
     $                           0 ),
     $                           LDB, A( 0 ), N, ALPHA, B( 0, N1 ),
     $                           LDB )
                     CALL DTRSM( 'R', 'U', 'N', DIAG, M, N2, ONE,
     $                           A( N1 ), N, B( 0, N1 ), LDB )
*
                  ELSE
*
*                    SIDE  ='R', N is odd, TRANSR = 'N', UPLO = 'U', and
*                    TRANS = 'T'
*
                     CALL DTRSM( 'R', 'U', 'T', DIAG, M, N2, ALPHA,
     $                           A( N1 ), N, B( 0, N1 ), LDB )
                     CALL DGEMM( 'N', 'T', M, N1, N2, -ONE, B( 0,
     $                           N1 ),
     $                           LDB, A( 0 ), N, ALPHA, B( 0, 0 ), LDB )
                     CALL DTRSM( 'R', 'L', 'N', DIAG, M, N1, ONE,
     $                           A( N2 ), N, B( 0, 0 ), LDB )
*
                  END IF
*
               END IF
*
            ELSE
*
*              SIDE = 'R', N is odd, and TRANSR = 'T'
*
               IF( LOWER ) THEN
*
*                 SIDE  ='R', N is odd, TRANSR = 'T', and UPLO = 'L'
*
                  IF( NOTRANS ) THEN
*
*                    SIDE  ='R', N is odd, TRANSR = 'T', UPLO = 'L', and
*                    TRANS = 'N'
*
                     CALL DTRSM( 'R', 'L', 'N', DIAG, M, N2, ALPHA,
     $                           A( 1 ), N1, B( 0, N1 ), LDB )
                     CALL DGEMM( 'N', 'T', M, N1, N2, -ONE, B( 0,
     $                           N1 ),
     $                           LDB, A( N1*N1 ), N1, ALPHA, B( 0, 0 ),
     $                           LDB )
                     CALL DTRSM( 'R', 'U', 'T', DIAG, M, N1, ONE,
     $                           A( 0 ), N1, B( 0, 0 ), LDB )
*
                  ELSE
*
*                    SIDE  ='R', N is odd, TRANSR = 'T', UPLO = 'L', and
*                    TRANS = 'T'
*
                     CALL DTRSM( 'R', 'U', 'N', DIAG, M, N1, ALPHA,
     $                           A( 0 ), N1, B( 0, 0 ), LDB )
                     CALL DGEMM( 'N', 'N', M, N2, N1, -ONE, B( 0,
     $                           0 ),
     $                           LDB, A( N1*N1 ), N1, ALPHA, B( 0, N1 ),
     $                           LDB )
                     CALL DTRSM( 'R', 'L', 'T', DIAG, M, N2, ONE,
     $                           A( 1 ), N1, B( 0, N1 ), LDB )
*
                  END IF
*
               ELSE
*
*                 SIDE  ='R', N is odd, TRANSR = 'T', and UPLO = 'U'
*
                  IF( NOTRANS ) THEN
*
*                    SIDE  ='R', N is odd, TRANSR = 'T', UPLO = 'U', and
*                    TRANS = 'N'
*
                     CALL DTRSM( 'R', 'U', 'N', DIAG, M, N1, ALPHA,
     $                           A( N2*N2 ), N2, B( 0, 0 ), LDB )
                     CALL DGEMM( 'N', 'T', M, N2, N1, -ONE, B( 0,
     $                           0 ),
     $                           LDB, A( 0 ), N2, ALPHA, B( 0, N1 ),
     $                           LDB )
                     CALL DTRSM( 'R', 'L', 'T', DIAG, M, N2, ONE,
     $                           A( N1*N2 ), N2, B( 0, N1 ), LDB )
*
                  ELSE
*
*                    SIDE  ='R', N is odd, TRANSR = 'T', UPLO = 'U', and
*                    TRANS = 'T'
*
                     CALL DTRSM( 'R', 'L', 'N', DIAG, M, N2, ALPHA,
     $                           A( N1*N2 ), N2, B( 0, N1 ), LDB )
                     CALL DGEMM( 'N', 'N', M, N1, N2, -ONE, B( 0,
     $                           N1 ),
     $                           LDB, A( 0 ), N2, ALPHA, B( 0, 0 ),
     $                           LDB )
                     CALL DTRSM( 'R', 'U', 'T', DIAG, M, N1, ONE,
     $                           A( N2*N2 ), N2, B( 0, 0 ), LDB )
*
                  END IF
*
               END IF
*
            END IF
*
         ELSE
*
*           SIDE = 'R' and N is even
*
            IF( NORMALTRANSR ) THEN
*
*              SIDE = 'R', N is even, and TRANSR = 'N'
*
               IF( LOWER ) THEN
*
*                 SIDE  ='R', N is even, TRANSR = 'N', and UPLO = 'L'
*
                  IF( NOTRANS ) THEN
*
*                    SIDE  ='R', N is even, TRANSR = 'N', UPLO = 'L',
*                    and TRANS = 'N'
*
                     CALL DTRSM( 'R', 'U', 'T', DIAG, M, K, ALPHA,
     $                           A( 0 ), N+1, B( 0, K ), LDB )
                     CALL DGEMM( 'N', 'N', M, K, K, -ONE, B( 0, K ),
     $                           LDB, A( K+1 ), N+1, ALPHA, B( 0, 0 ),
     $                           LDB )
                     CALL DTRSM( 'R', 'L', 'N', DIAG, M, K, ONE,
     $                           A( 1 ), N+1, B( 0, 0 ), LDB )
*
                  ELSE
*
*                    SIDE  ='R', N is even, TRANSR = 'N', UPLO = 'L',
*                    and TRANS = 'T'
*
                     CALL DTRSM( 'R', 'L', 'T', DIAG, M, K, ALPHA,
     $                           A( 1 ), N+1, B( 0, 0 ), LDB )
                     CALL DGEMM( 'N', 'T', M, K, K, -ONE, B( 0, 0 ),
     $                           LDB, A( K+1 ), N+1, ALPHA, B( 0, K ),
     $                           LDB )
                     CALL DTRSM( 'R', 'U', 'N', DIAG, M, K, ONE,
     $                           A( 0 ), N+1, B( 0, K ), LDB )
*
                  END IF
*
               ELSE
*
*                 SIDE  ='R', N is even, TRANSR = 'N', and UPLO = 'U'
*
                  IF( NOTRANS ) THEN
*
*                    SIDE  ='R', N is even, TRANSR = 'N', UPLO = 'U',
*                    and TRANS = 'N'
*
                     CALL DTRSM( 'R', 'L', 'T', DIAG, M, K, ALPHA,
     $                           A( K+1 ), N+1, B( 0, 0 ), LDB )
                     CALL DGEMM( 'N', 'N', M, K, K, -ONE, B( 0, 0 ),
     $                           LDB, A( 0 ), N+1, ALPHA, B( 0, K ),
     $                           LDB )
                     CALL DTRSM( 'R', 'U', 'N', DIAG, M, K, ONE,
     $                           A( K ), N+1, B( 0, K ), LDB )
*
                  ELSE
*
*                    SIDE  ='R', N is even, TRANSR = 'N', UPLO = 'U',
*                    and TRANS = 'T'
*
                     CALL DTRSM( 'R', 'U', 'T', DIAG, M, K, ALPHA,
     $                           A( K ), N+1, B( 0, K ), LDB )
                     CALL DGEMM( 'N', 'T', M, K, K, -ONE, B( 0, K ),
     $                           LDB, A( 0 ), N+1, ALPHA, B( 0, 0 ),
     $                           LDB )
                     CALL DTRSM( 'R', 'L', 'N', DIAG, M, K, ONE,
     $                           A( K+1 ), N+1, B( 0, 0 ), LDB )
*
                  END IF
*
               END IF
*
            ELSE
*
*              SIDE = 'R', N is even, and TRANSR = 'T'
*
               IF( LOWER ) THEN
*
*                 SIDE  ='R', N is even, TRANSR = 'T', and UPLO = 'L'
*
                  IF( NOTRANS ) THEN
*
*                    SIDE  ='R', N is even, TRANSR = 'T', UPLO = 'L',
*                    and TRANS = 'N'
*
                     CALL DTRSM( 'R', 'L', 'N', DIAG, M, K, ALPHA,
     $                           A( 0 ), K, B( 0, K ), LDB )
                     CALL DGEMM( 'N', 'T', M, K, K, -ONE, B( 0, K ),
     $                           LDB, A( ( K+1 )*K ), K, ALPHA,
     $                           B( 0, 0 ), LDB )
                     CALL DTRSM( 'R', 'U', 'T', DIAG, M, K, ONE,
     $                           A( K ), K, B( 0, 0 ), LDB )
*
                  ELSE
*
*                    SIDE  ='R', N is even, TRANSR = 'T', UPLO = 'L',
*                    and TRANS = 'T'
*
                     CALL DTRSM( 'R', 'U', 'N', DIAG, M, K, ALPHA,
     $                           A( K ), K, B( 0, 0 ), LDB )
                     CALL DGEMM( 'N', 'N', M, K, K, -ONE, B( 0, 0 ),
     $                           LDB, A( ( K+1 )*K ), K, ALPHA,
     $                           B( 0, K ), LDB )
                     CALL DTRSM( 'R', 'L', 'T', DIAG, M, K, ONE,
     $                           A( 0 ), K, B( 0, K ), LDB )
*
                  END IF
*
               ELSE
*
*                 SIDE  ='R', N is even, TRANSR = 'T', and UPLO = 'U'
*
                  IF( NOTRANS ) THEN
*
*                    SIDE  ='R', N is even, TRANSR = 'T', UPLO = 'U',
*                    and TRANS = 'N'
*
                     CALL DTRSM( 'R', 'U', 'N', DIAG, M, K, ALPHA,
     $                           A( ( K+1 )*K ), K, B( 0, 0 ), LDB )
                     CALL DGEMM( 'N', 'T', M, K, K, -ONE, B( 0, 0 ),
     $                           LDB, A( 0 ), K, ALPHA, B( 0, K ), LDB )
                     CALL DTRSM( 'R', 'L', 'T', DIAG, M, K, ONE,
     $                           A( K*K ), K, B( 0, K ), LDB )
*
                  ELSE
*
*                    SIDE  ='R', N is even, TRANSR = 'T', UPLO = 'U',
*                    and TRANS = 'T'
*
                     CALL DTRSM( 'R', 'L', 'N', DIAG, M, K, ALPHA,
     $                           A( K*K ), K, B( 0, K ), LDB )
                     CALL DGEMM( 'N', 'N', M, K, K, -ONE, B( 0, K ),
     $                           LDB, A( 0 ), K, ALPHA, B( 0, 0 ), LDB )
                     CALL DTRSM( 'R', 'U', 'T', DIAG, M, K, ONE,
     $                           A( ( K+1 )*K ), K, B( 0, 0 ), LDB )
*
                  END IF
*
               END IF
*
            END IF
*
         END IF
      END IF
*
      RETURN
*
*     End of DTFSM
*
      END
*> \brief \b DTFTRI
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTFTRI + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtftri.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtftri.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtftri.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTFTRI( TRANSR, UPLO, DIAG, N, A, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANSR, UPLO, DIAG
*       INTEGER            INFO, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( 0: * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTFTRI computes the inverse of a triangular matrix A stored in RFP
*> format.
*>
*> This is a Level 3 BLAS version of the algorithm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANSR
*> \verbatim
*>          TRANSR is CHARACTER*1
*>          = 'N':  The Normal TRANSR of RFP A is stored;
*>          = 'T':  The Transpose TRANSR of RFP A is stored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  A is upper triangular;
*>          = 'L':  A is lower triangular.
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          = 'N':  A is non-unit triangular;
*>          = 'U':  A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (0:nt-1);
*>          nt=N*(N+1)/2. On entry, the triangular factor of a Hermitian
*>          Positive Definite matrix A in RFP format. RFP format is
*>          described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
*>          then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
*>          (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
*>          the transpose of RFP A as defined when
*>          TRANSR = 'N'. The contents of RFP A are defined by UPLO as
*>          follows: If UPLO = 'U' the RFP A contains the nt elements of
*>          upper packed A; If UPLO = 'L' the RFP A contains the nt
*>          elements of lower packed A. The LDA of RFP A is (N+1)/2 when
*>          TRANSR = 'T'. When TRANSR is 'N' the LDA is N+1 when N is
*>          even and N is odd. See the Note below for more details.
*>
*>          On exit, the (triangular) inverse of the original matrix, in
*>          the same storage format.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*>          > 0: if INFO = i, A(i,i) is exactly zero.  The triangular
*>               matrix is singular and its inverse can not be computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tftri
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  We first consider Rectangular Full Packed (RFP) Format when N is
*>  even. We give an example where N = 6.
*>
*>      AP is Upper             AP is Lower
*>
*>   00 01 02 03 04 05       00
*>      11 12 13 14 15       10 11
*>         22 23 24 25       20 21 22
*>            33 34 35       30 31 32 33
*>               44 45       40 41 42 43 44
*>                  55       50 51 52 53 54 55
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*>  the transpose of the first three columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*>  the transpose of the last three columns of AP lower.
*>  This covers the case N even and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>        03 04 05                33 43 53
*>        13 14 15                00 44 54
*>        23 24 25                10 11 55
*>        33 34 35                20 21 22
*>        00 44 45                30 31 32
*>        01 11 55                40 41 42
*>        02 12 22                50 51 52
*>
*>  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*>  transpose of RFP A above. One therefore gets:
*>
*>
*>           RFP A                   RFP A
*>
*>     03 13 23 33 00 01 02    33 00 10 20 30 40 50
*>     04 14 24 34 44 11 12    43 44 11 21 31 41 51
*>     05 15 25 35 45 55 22    53 54 55 22 32 42 52
*>
*>
*>  We then consider Rectangular Full Packed (RFP) Format when N is
*>  odd. We give an example where N = 5.
*>
*>     AP is Upper                 AP is Lower
*>
*>   00 01 02 03 04              00
*>      11 12 13 14              10 11
*>         22 23 24              20 21 22
*>            33 34              30 31 32 33
*>               44              40 41 42 43 44
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*>  the transpose of the first two columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*>  the transpose of the last two columns of AP lower.
*>  This covers the case N odd and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>        02 03 04                00 33 43
*>        12 13 14                10 11 44
*>        22 23 24                20 21 22
*>        00 33 34                30 31 32
*>        01 11 44                40 41 42
*>
*>  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*>  transpose of RFP A above. One therefore gets:
*>
*>           RFP A                   RFP A
*>
*>     02 12 22 00 01             00 10 20 30 40 50
*>     03 13 23 33 11             33 11 21 31 41 51
*>     04 14 24 34 44             43 44 22 32 42 52
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTFTRI( TRANSR, UPLO, DIAG, N, A, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANSR, UPLO, DIAG
      INTEGER            INFO, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( 0: * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER, NISODD, NORMALTRANSR
      INTEGER            N1, N2, K
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA, DTRMM, DTRTRI
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MOD
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NORMALTRANSR = LSAME( TRANSR, 'N' )
      LOWER = LSAME( UPLO, 'L' )
      IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
         INFO = -2
      ELSE IF( .NOT.LSAME( DIAG, 'N' ) .AND.
     $         .NOT.LSAME( DIAG, 'U' ) )
     $         THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTFTRI', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     If N is odd, set NISODD = .TRUE.
*     If N is even, set K = N/2 and NISODD = .FALSE.
*
      IF( MOD( N, 2 ).EQ.0 ) THEN
         K = N / 2
         NISODD = .FALSE.
      ELSE
         NISODD = .TRUE.
      END IF
*
*     Set N1 and N2 depending on LOWER
*
      IF( LOWER ) THEN
         N2 = N / 2
         N1 = N - N2
      ELSE
         N1 = N / 2
         N2 = N - N1
      END IF
*
*
*     start execution: there are eight cases
*
      IF( NISODD ) THEN
*
*        N is odd
*
         IF( NORMALTRANSR ) THEN
*
*           N is odd and TRANSR = 'N'
*
            IF( LOWER ) THEN
*
*             SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:n1-1) )
*             T1 -> a(0,0), T2 -> a(0,1), S -> a(n1,0)
*             T1 -> a(0), T2 -> a(n), S -> a(n1)
*
               CALL DTRTRI( 'L', DIAG, N1, A( 0 ), N, INFO )
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRMM( 'R', 'L', 'N', DIAG, N2, N1, -ONE, A( 0 ),
     $                     N, A( N1 ), N )
               CALL DTRTRI( 'U', DIAG, N2, A( N ), N, INFO )
               IF( INFO.GT.0 )
     $            INFO = INFO + N1
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRMM( 'L', 'U', 'T', DIAG, N2, N1, ONE, A( N ),
     $                     N,
     $                     A( N1 ), N )
*
            ELSE
*
*             SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:n2-1)
*             T1 -> a(n1+1,0), T2 -> a(n1,0), S -> a(0,0)
*             T1 -> a(n2), T2 -> a(n1), S -> a(0)
*
               CALL DTRTRI( 'L', DIAG, N1, A( N2 ), N, INFO )
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRMM( 'L', 'L', 'T', DIAG, N1, N2, -ONE,
     $                     A( N2 ),
     $                     N, A( 0 ), N )
               CALL DTRTRI( 'U', DIAG, N2, A( N1 ), N, INFO )
               IF( INFO.GT.0 )
     $            INFO = INFO + N1
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRMM( 'R', 'U', 'N', DIAG, N1, N2, ONE, A( N1 ),
     $                     N, A( 0 ), N )
*
            END IF
*
         ELSE
*
*           N is odd and TRANSR = 'T'
*
            IF( LOWER ) THEN
*
*              SRPA for LOWER, TRANSPOSE and N is odd
*              T1 -> a(0), T2 -> a(1), S -> a(0+n1*n1)
*
               CALL DTRTRI( 'U', DIAG, N1, A( 0 ), N1, INFO )
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRMM( 'L', 'U', 'N', DIAG, N1, N2, -ONE, A( 0 ),
     $                     N1, A( N1*N1 ), N1 )
               CALL DTRTRI( 'L', DIAG, N2, A( 1 ), N1, INFO )
               IF( INFO.GT.0 )
     $            INFO = INFO + N1
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRMM( 'R', 'L', 'T', DIAG, N1, N2, ONE, A( 1 ),
     $                     N1, A( N1*N1 ), N1 )
*
            ELSE
*
*              SRPA for UPPER, TRANSPOSE and N is odd
*              T1 -> a(0+n2*n2), T2 -> a(0+n1*n2), S -> a(0)
*
               CALL DTRTRI( 'U', DIAG, N1, A( N2*N2 ), N2, INFO )
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRMM( 'R', 'U', 'T', DIAG, N2, N1, -ONE,
     $                     A( N2*N2 ), N2, A( 0 ), N2 )
               CALL DTRTRI( 'L', DIAG, N2, A( N1*N2 ), N2, INFO )
               IF( INFO.GT.0 )
     $            INFO = INFO + N1
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRMM( 'L', 'L', 'N', DIAG, N2, N1, ONE,
     $                     A( N1*N2 ), N2, A( 0 ), N2 )
            END IF
*
         END IF
*
      ELSE
*
*        N is even
*
         IF( NORMALTRANSR ) THEN
*
*           N is even and TRANSR = 'N'
*
            IF( LOWER ) THEN
*
*              SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
*              T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
*              T1 -> a(1), T2 -> a(0), S -> a(k+1)
*
               CALL DTRTRI( 'L', DIAG, K, A( 1 ), N+1, INFO )
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRMM( 'R', 'L', 'N', DIAG, K, K, -ONE, A( 1 ),
     $                     N+1, A( K+1 ), N+1 )
               CALL DTRTRI( 'U', DIAG, K, A( 0 ), N+1, INFO )
               IF( INFO.GT.0 )
     $            INFO = INFO + K
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRMM( 'L', 'U', 'T', DIAG, K, K, ONE, A( 0 ),
     $                     N+1,
     $                     A( K+1 ), N+1 )
*
            ELSE
*
*              SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
*              T1 -> a(k+1,0) ,  T2 -> a(k,0),   S -> a(0,0)
*              T1 -> a(k+1), T2 -> a(k), S -> a(0)
*
               CALL DTRTRI( 'L', DIAG, K, A( K+1 ), N+1, INFO )
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRMM( 'L', 'L', 'T', DIAG, K, K, -ONE, A( K+1 ),
     $                     N+1, A( 0 ), N+1 )
               CALL DTRTRI( 'U', DIAG, K, A( K ), N+1, INFO )
               IF( INFO.GT.0 )
     $            INFO = INFO + K
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRMM( 'R', 'U', 'N', DIAG, K, K, ONE, A( K ),
     $                     N+1,
     $                     A( 0 ), N+1 )
            END IF
         ELSE
*
*           N is even and TRANSR = 'T'
*
            IF( LOWER ) THEN
*
*              SRPA for LOWER, TRANSPOSE and N is even (see paper)
*              T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1)
*              T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
*
               CALL DTRTRI( 'U', DIAG, K, A( K ), K, INFO )
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRMM( 'L', 'U', 'N', DIAG, K, K, -ONE, A( K ),
     $                     K,
     $                     A( K*( K+1 ) ), K )
               CALL DTRTRI( 'L', DIAG, K, A( 0 ), K, INFO )
               IF( INFO.GT.0 )
     $            INFO = INFO + K
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRMM( 'R', 'L', 'T', DIAG, K, K, ONE, A( 0 ), K,
     $                     A( K*( K+1 ) ), K )
            ELSE
*
*              SRPA for UPPER, TRANSPOSE and N is even (see paper)
*              T1 -> B(0,k+1),     T2 -> B(0,k),   S -> B(0,0)
*              T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
*
               CALL DTRTRI( 'U', DIAG, K, A( K*( K+1 ) ), K, INFO )
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRMM( 'R', 'U', 'T', DIAG, K, K, -ONE,
     $                     A( K*( K+1 ) ), K, A( 0 ), K )
               CALL DTRTRI( 'L', DIAG, K, A( K*K ), K, INFO )
               IF( INFO.GT.0 )
     $            INFO = INFO + K
               IF( INFO.GT.0 )
     $            RETURN
               CALL DTRMM( 'L', 'L', 'N', DIAG, K, K, ONE, A( K*K ),
     $                     K,
     $                     A( 0 ), K )
            END IF
         END IF
      END IF
*
      RETURN
*
*     End of DTFTRI
*
      END
*> \brief \b DTFTTP copies a triangular matrix from the rectangular full packed format (TF) to the standard packed format (TP).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTFTTP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtfttp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtfttp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtfttp.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTFTTP( TRANSR, UPLO, N, ARF, AP, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANSR, UPLO
*       INTEGER            INFO, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AP( 0: * ), ARF( 0: * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTFTTP copies a triangular matrix A from rectangular full packed
*> format (TF) to standard packed format (TP).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANSR
*> \verbatim
*>          TRANSR is CHARACTER*1
*>          = 'N':  ARF is in Normal format;
*>          = 'T':  ARF is in Transpose format;
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  A is upper triangular;
*>          = 'L':  A is lower triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] ARF
*> \verbatim
*>          ARF is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ),
*>          On entry, the upper or lower triangular matrix A stored in
*>          RFP format. For a further discussion see Notes below.
*> \endverbatim
*>
*> \param[out] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ),
*>          On exit, the upper or lower triangular matrix A, packed
*>          columnwise in a linear array. The j-th column of A is stored
*>          in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tfttp
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  We first consider Rectangular Full Packed (RFP) Format when N is
*>  even. We give an example where N = 6.
*>
*>      AP is Upper             AP is Lower
*>
*>   00 01 02 03 04 05       00
*>      11 12 13 14 15       10 11
*>         22 23 24 25       20 21 22
*>            33 34 35       30 31 32 33
*>               44 45       40 41 42 43 44
*>                  55       50 51 52 53 54 55
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*>  the transpose of the first three columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*>  the transpose of the last three columns of AP lower.
*>  This covers the case N even and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>        03 04 05                33 43 53
*>        13 14 15                00 44 54
*>        23 24 25                10 11 55
*>        33 34 35                20 21 22
*>        00 44 45                30 31 32
*>        01 11 55                40 41 42
*>        02 12 22                50 51 52
*>
*>  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*>  transpose of RFP A above. One therefore gets:
*>
*>
*>           RFP A                   RFP A
*>
*>     03 13 23 33 00 01 02    33 00 10 20 30 40 50
*>     04 14 24 34 44 11 12    43 44 11 21 31 41 51
*>     05 15 25 35 45 55 22    53 54 55 22 32 42 52
*>
*>
*>  We then consider Rectangular Full Packed (RFP) Format when N is
*>  odd. We give an example where N = 5.
*>
*>     AP is Upper                 AP is Lower
*>
*>   00 01 02 03 04              00
*>      11 12 13 14              10 11
*>         22 23 24              20 21 22
*>            33 34              30 31 32 33
*>               44              40 41 42 43 44
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*>  the transpose of the first two columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*>  the transpose of the last two columns of AP lower.
*>  This covers the case N odd and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>        02 03 04                00 33 43
*>        12 13 14                10 11 44
*>        22 23 24                20 21 22
*>        00 33 34                30 31 32
*>        01 11 44                40 41 42
*>
*>  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*>  transpose of RFP A above. One therefore gets:
*>
*>           RFP A                   RFP A
*>
*>     02 12 22 00 01             00 10 20 30 40 50
*>     03 13 23 33 11             33 11 21 31 41 51
*>     04 14 24 34 44             43 44 22 32 42 52
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTFTTP( TRANSR, UPLO, N, ARF, AP, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANSR, UPLO
      INTEGER            INFO, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( 0: * ), ARF( 0: * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER, NISODD, NORMALTRANSR
      INTEGER            N1, N2, K, NT
      INTEGER            I, J, IJ
      INTEGER            IJP, JP, LDA, JS
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NORMALTRANSR = LSAME( TRANSR, 'N' )
      LOWER = LSAME( UPLO, 'L' )
      IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTFTTP', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( N.EQ.1 ) THEN
         IF( NORMALTRANSR ) THEN
            AP( 0 ) = ARF( 0 )
         ELSE
            AP( 0 ) = ARF( 0 )
         END IF
         RETURN
      END IF
*
*     Size of array ARF(0:NT-1)
*
      NT = N*( N+1 ) / 2
*
*     Set N1 and N2 depending on LOWER
*
      IF( LOWER ) THEN
         N2 = N / 2
         N1 = N - N2
      ELSE
         N1 = N / 2
         N2 = N - N1
      END IF
*
*     If N is odd, set NISODD = .TRUE.
*     If N is even, set K = N/2 and NISODD = .FALSE.
*
*     set lda of ARF^C; ARF^C is (0:(N+1)/2-1,0:N-noe)
*     where noe = 0 if n is even, noe = 1 if n is odd
*
      IF( MOD( N, 2 ).EQ.0 ) THEN
         K = N / 2
         NISODD = .FALSE.
         LDA = N + 1
      ELSE
         NISODD = .TRUE.
         LDA = N
      END IF
*
*     ARF^C has lda rows and n+1-noe cols
*
      IF( .NOT.NORMALTRANSR )
     $   LDA = ( N+1 ) / 2
*
*     start execution: there are eight cases
*
      IF( NISODD ) THEN
*
*        N is odd
*
         IF( NORMALTRANSR ) THEN
*
*           N is odd and TRANSR = 'N'
*
            IF( LOWER ) THEN
*
*             SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:n1-1) )
*             T1 -> a(0,0), T2 -> a(0,1), S -> a(n1,0)
*             T1 -> a(0), T2 -> a(n), S -> a(n1); lda = n
*
               IJP = 0
               JP = 0
               DO J = 0, N2
                  DO I = J, N - 1
                     IJ = I + JP
                     AP( IJP ) = ARF( IJ )
                     IJP = IJP + 1
                  END DO
                  JP = JP + LDA
               END DO
               DO I = 0, N2 - 1
                  DO J = 1 + I, N2
                     IJ = I + J*LDA
                     AP( IJP ) = ARF( IJ )
                     IJP = IJP + 1
                  END DO
               END DO
*
            ELSE
*
*             SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:n2-1)
*             T1 -> a(n1+1,0), T2 -> a(n1,0), S -> a(0,0)
*             T1 -> a(n2), T2 -> a(n1), S -> a(0)
*
               IJP = 0
               DO J = 0, N1 - 1
                  IJ = N2 + J
                  DO I = 0, J
                     AP( IJP ) = ARF( IJ )
                     IJP = IJP + 1
                     IJ = IJ + LDA
                  END DO
               END DO
               JS = 0
               DO J = N1, N - 1
                  IJ = JS
                  DO IJ = JS, JS + J
                     AP( IJP ) = ARF( IJ )
                     IJP = IJP + 1
                  END DO
                  JS = JS + LDA
               END DO
*
            END IF
*
         ELSE
*
*           N is odd and TRANSR = 'T'
*
            IF( LOWER ) THEN
*
*              SRPA for LOWER, TRANSPOSE and N is odd
*              T1 -> A(0,0) , T2 -> A(1,0) , S -> A(0,n1)
*              T1 -> a(0+0) , T2 -> a(1+0) , S -> a(0+n1*n1); lda=n1
*
               IJP = 0
               DO I = 0, N2
                  DO IJ = I*( LDA+1 ), N*LDA - 1, LDA
                     AP( IJP ) = ARF( IJ )
                     IJP = IJP + 1
                  END DO
               END DO
               JS = 1
               DO J = 0, N2 - 1
                  DO IJ = JS, JS + N2 - J - 1
                     AP( IJP ) = ARF( IJ )
                     IJP = IJP + 1
                  END DO
                  JS = JS + LDA + 1
               END DO
*
            ELSE
*
*              SRPA for UPPER, TRANSPOSE and N is odd
*              T1 -> A(0,n1+1), T2 -> A(0,n1), S -> A(0,0)
*              T1 -> a(n2*n2), T2 -> a(n1*n2), S -> a(0); lda = n2
*
               IJP = 0
               JS = N2*LDA
               DO J = 0, N1 - 1
                  DO IJ = JS, JS + J
                     AP( IJP ) = ARF( IJ )
                     IJP = IJP + 1
                  END DO
                  JS = JS + LDA
               END DO
               DO I = 0, N1
                  DO IJ = I, I + ( N1+I )*LDA, LDA
                     AP( IJP ) = ARF( IJ )
                     IJP = IJP + 1
                  END DO
               END DO
*
            END IF
*
         END IF
*
      ELSE
*
*        N is even
*
         IF( NORMALTRANSR ) THEN
*
*           N is even and TRANSR = 'N'
*
            IF( LOWER ) THEN
*
*              SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
*              T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
*              T1 -> a(1), T2 -> a(0), S -> a(k+1)
*
               IJP = 0
               JP = 0
               DO J = 0, K - 1
                  DO I = J, N - 1
                     IJ = 1 + I + JP
                     AP( IJP ) = ARF( IJ )
                     IJP = IJP + 1
                  END DO
                  JP = JP + LDA
               END DO
               DO I = 0, K - 1
                  DO J = I, K - 1
                     IJ = I + J*LDA
                     AP( IJP ) = ARF( IJ )
                     IJP = IJP + 1
                  END DO
               END DO
*
            ELSE
*
*              SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
*              T1 -> a(k+1,0) ,  T2 -> a(k,0),   S -> a(0,0)
*              T1 -> a(k+1), T2 -> a(k), S -> a(0)
*
               IJP = 0
               DO J = 0, K - 1
                  IJ = K + 1 + J
                  DO I = 0, J
                     AP( IJP ) = ARF( IJ )
                     IJP = IJP + 1
                     IJ = IJ + LDA
                  END DO
               END DO
               JS = 0
               DO J = K, N - 1
                  IJ = JS
                  DO IJ = JS, JS + J
                     AP( IJP ) = ARF( IJ )
                     IJP = IJP + 1
                  END DO
                  JS = JS + LDA
               END DO
*
            END IF
*
         ELSE
*
*           N is even and TRANSR = 'T'
*
            IF( LOWER ) THEN
*
*              SRPA for LOWER, TRANSPOSE and N is even (see paper)
*              T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1)
*              T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
*
               IJP = 0
               DO I = 0, K - 1
                  DO IJ = I + ( I+1 )*LDA, ( N+1 )*LDA - 1, LDA
                     AP( IJP ) = ARF( IJ )
                     IJP = IJP + 1
                  END DO
               END DO
               JS = 0
               DO J = 0, K - 1
                  DO IJ = JS, JS + K - J - 1
                     AP( IJP ) = ARF( IJ )
                     IJP = IJP + 1
                  END DO
                  JS = JS + LDA + 1
               END DO
*
            ELSE
*
*              SRPA for UPPER, TRANSPOSE and N is even (see paper)
*              T1 -> B(0,k+1),     T2 -> B(0,k),   S -> B(0,0)
*              T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
*
               IJP = 0
               JS = ( K+1 )*LDA
               DO J = 0, K - 1
                  DO IJ = JS, JS + J
                     AP( IJP ) = ARF( IJ )
                     IJP = IJP + 1
                  END DO
                  JS = JS + LDA
               END DO
               DO I = 0, K - 1
                  DO IJ = I, I + ( K+I )*LDA, LDA
                     AP( IJP ) = ARF( IJ )
                     IJP = IJP + 1
                  END DO
               END DO
*
            END IF
*
         END IF
*
      END IF
*
      RETURN
*
*     End of DTFTTP
*
      END
*> \brief \b DTFTTR copies a triangular matrix from the rectangular full packed format (TF) to the standard full format (TR).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTFTTR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtfttr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtfttr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtfttr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTFTTR( TRANSR, UPLO, N, ARF, A, LDA, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANSR, UPLO
*       INTEGER            INFO, N, LDA
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( 0: LDA-1, 0: * ), ARF( 0: * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTFTTR copies a triangular matrix A from rectangular full packed
*> format (TF) to standard full format (TR).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANSR
*> \verbatim
*>          TRANSR is CHARACTER*1
*>          = 'N':  ARF is in Normal format;
*>          = 'T':  ARF is in Transpose format.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  A is upper triangular;
*>          = 'L':  A is lower triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices ARF and A. N >= 0.
*> \endverbatim
*>
*> \param[in] ARF
*> \verbatim
*>          ARF is DOUBLE PRECISION array, dimension (N*(N+1)/2).
*>          On entry, the upper (if UPLO = 'U') or lower (if UPLO = 'L')
*>          matrix A in RFP format. See the "Notes" below for more
*>          details.
*> \endverbatim
*>
*> \param[out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On exit, the triangular matrix A.  If UPLO = 'U', the
*>          leading N-by-N upper triangular part of the array A contains
*>          the upper triangular matrix, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading N-by-N lower triangular part of the array A contains
*>          the lower triangular matrix, and the strictly upper
*>          triangular part of A is not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tfttr
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  We first consider Rectangular Full Packed (RFP) Format when N is
*>  even. We give an example where N = 6.
*>
*>      AP is Upper             AP is Lower
*>
*>   00 01 02 03 04 05       00
*>      11 12 13 14 15       10 11
*>         22 23 24 25       20 21 22
*>            33 34 35       30 31 32 33
*>               44 45       40 41 42 43 44
*>                  55       50 51 52 53 54 55
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*>  the transpose of the first three columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*>  the transpose of the last three columns of AP lower.
*>  This covers the case N even and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>        03 04 05                33 43 53
*>        13 14 15                00 44 54
*>        23 24 25                10 11 55
*>        33 34 35                20 21 22
*>        00 44 45                30 31 32
*>        01 11 55                40 41 42
*>        02 12 22                50 51 52
*>
*>  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*>  transpose of RFP A above. One therefore gets:
*>
*>
*>           RFP A                   RFP A
*>
*>     03 13 23 33 00 01 02    33 00 10 20 30 40 50
*>     04 14 24 34 44 11 12    43 44 11 21 31 41 51
*>     05 15 25 35 45 55 22    53 54 55 22 32 42 52
*>
*>
*>  We then consider Rectangular Full Packed (RFP) Format when N is
*>  odd. We give an example where N = 5.
*>
*>     AP is Upper                 AP is Lower
*>
*>   00 01 02 03 04              00
*>      11 12 13 14              10 11
*>         22 23 24              20 21 22
*>            33 34              30 31 32 33
*>               44              40 41 42 43 44
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*>  the transpose of the first two columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*>  the transpose of the last two columns of AP lower.
*>  This covers the case N odd and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>        02 03 04                00 33 43
*>        12 13 14                10 11 44
*>        22 23 24                20 21 22
*>        00 33 34                30 31 32
*>        01 11 44                40 41 42
*>
*>  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*>  transpose of RFP A above. One therefore gets:
*>
*>           RFP A                   RFP A
*>
*>     02 12 22 00 01             00 10 20 30 40 50
*>     03 13 23 33 11             33 11 21 31 41 51
*>     04 14 24 34 44             43 44 22 32 42 52
*> \endverbatim
*
*  =====================================================================
      SUBROUTINE DTFTTR( TRANSR, UPLO, N, ARF, A, LDA, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANSR, UPLO
      INTEGER            INFO, N, LDA
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( 0: LDA-1, 0: * ), ARF( 0: * )
*     ..
*
*  =====================================================================
*
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER, NISODD, NORMALTRANSR
      INTEGER            N1, N2, K, NT, NX2, NP1X2
      INTEGER            I, J, L, IJ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MOD
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NORMALTRANSR = LSAME( TRANSR, 'N' )
      LOWER = LSAME( UPLO, 'L' )
      IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTFTTR', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.LE.1 ) THEN
         IF( N.EQ.1 ) THEN
            A( 0, 0 ) = ARF( 0 )
         END IF
         RETURN
      END IF
*
*     Size of array ARF(0:nt-1)
*
      NT = N*( N+1 ) / 2
*
*     set N1 and N2 depending on LOWER: for N even N1=N2=K
*
      IF( LOWER ) THEN
         N2 = N / 2
         N1 = N - N2
      ELSE
         N1 = N / 2
         N2 = N - N1
      END IF
*
*     If N is odd, set NISODD = .TRUE., LDA=N+1 and A is (N+1)--by--K2.
*     If N is even, set K = N/2 and NISODD = .FALSE., LDA=N and A is
*     N--by--(N+1)/2.
*
      IF( MOD( N, 2 ).EQ.0 ) THEN
         K = N / 2
         NISODD = .FALSE.
         IF( .NOT.LOWER )
     $      NP1X2 = N + N + 2
      ELSE
         NISODD = .TRUE.
         IF( .NOT.LOWER )
     $      NX2 = N + N
      END IF
*
      IF( NISODD ) THEN
*
*        N is odd
*
         IF( NORMALTRANSR ) THEN
*
*           N is odd and TRANSR = 'N'
*
            IF( LOWER ) THEN
*
*              N is odd, TRANSR = 'N', and UPLO = 'L'
*
               IJ = 0
               DO J = 0, N2
                  DO I = N1, N2 + J
                     A( N2+J, I ) = ARF( IJ )
                     IJ = IJ + 1
                  END DO
                  DO I = J, N - 1
                     A( I, J ) = ARF( IJ )
                     IJ = IJ + 1
                  END DO
               END DO
*
            ELSE
*
*              N is odd, TRANSR = 'N', and UPLO = 'U'
*
               IJ = NT - N
               DO J = N - 1, N1, -1
                  DO I = 0, J
                     A( I, J ) = ARF( IJ )
                     IJ = IJ + 1
                  END DO
                  DO L = J - N1, N1 - 1
                     A( J-N1, L ) = ARF( IJ )
                     IJ = IJ + 1
                  END DO
                  IJ = IJ - NX2
               END DO
*
            END IF
*
         ELSE
*
*           N is odd and TRANSR = 'T'
*
            IF( LOWER ) THEN
*
*              N is odd, TRANSR = 'T', and UPLO = 'L'
*
               IJ = 0
               DO J = 0, N2 - 1
                  DO I = 0, J
                     A( J, I ) = ARF( IJ )
                     IJ = IJ + 1
                  END DO
                  DO I = N1 + J, N - 1
                     A( I, N1+J ) = ARF( IJ )
                     IJ = IJ + 1
                  END DO
               END DO
               DO J = N2, N - 1
                  DO I = 0, N1 - 1
                     A( J, I ) = ARF( IJ )
                     IJ = IJ + 1
                  END DO
               END DO
*
            ELSE
*
*              N is odd, TRANSR = 'T', and UPLO = 'U'
*
               IJ = 0
               DO J = 0, N1
                  DO I = N1, N - 1
                     A( J, I ) = ARF( IJ )
                     IJ = IJ + 1
                  END DO
               END DO
               DO J = 0, N1 - 1
                  DO I = 0, J
                     A( I, J ) = ARF( IJ )
                     IJ = IJ + 1
                  END DO
                  DO L = N2 + J, N - 1
                     A( N2+J, L ) = ARF( IJ )
                     IJ = IJ + 1
                  END DO
               END DO
*
            END IF
*
         END IF
*
      ELSE
*
*        N is even
*
         IF( NORMALTRANSR ) THEN
*
*           N is even and TRANSR = 'N'
*
            IF( LOWER ) THEN
*
*              N is even, TRANSR = 'N', and UPLO = 'L'
*
               IJ = 0
               DO J = 0, K - 1
                  DO I = K, K + J
                     A( K+J, I ) = ARF( IJ )
                     IJ = IJ + 1
                  END DO
                  DO I = J, N - 1
                     A( I, J ) = ARF( IJ )
                     IJ = IJ + 1
                  END DO
               END DO
*
            ELSE
*
*              N is even, TRANSR = 'N', and UPLO = 'U'
*
               IJ = NT - N - 1
               DO J = N - 1, K, -1
                  DO I = 0, J
                     A( I, J ) = ARF( IJ )
                     IJ = IJ + 1
                  END DO
                  DO L = J - K, K - 1
                     A( J-K, L ) = ARF( IJ )
                     IJ = IJ + 1
                  END DO
                  IJ = IJ - NP1X2
               END DO
*
            END IF
*
         ELSE
*
*           N is even and TRANSR = 'T'
*
            IF( LOWER ) THEN
*
*              N is even, TRANSR = 'T', and UPLO = 'L'
*
               IJ = 0
               J = K
               DO I = K, N - 1
                  A( I, J ) = ARF( IJ )
                  IJ = IJ + 1
               END DO
               DO J = 0, K - 2
                  DO I = 0, J
                     A( J, I ) = ARF( IJ )
                     IJ = IJ + 1
                  END DO
                  DO I = K + 1 + J, N - 1
                     A( I, K+1+J ) = ARF( IJ )
                     IJ = IJ + 1
                  END DO
               END DO
               DO J = K - 1, N - 1
                  DO I = 0, K - 1
                     A( J, I ) = ARF( IJ )
                     IJ = IJ + 1
                  END DO
               END DO
*
            ELSE
*
*              N is even, TRANSR = 'T', and UPLO = 'U'
*
               IJ = 0
               DO J = 0, K
                  DO I = K, N - 1
                     A( J, I ) = ARF( IJ )
                     IJ = IJ + 1
                  END DO
               END DO
               DO J = 0, K - 2
                  DO I = 0, J
                     A( I, J ) = ARF( IJ )
                     IJ = IJ + 1
                  END DO
                  DO L = K + 1 + J, N - 1
                     A( K+1+J, L ) = ARF( IJ )
                     IJ = IJ + 1
                  END DO
               END DO
*              Note that here, on exit of the loop, J = K-1
               DO I = 0, J
                  A( I, J ) = ARF( IJ )
                  IJ = IJ + 1
               END DO
*
            END IF
*
         END IF
*
      END IF
*
      RETURN
*
*     End of DTFTTR
*
      END
*> \brief \b DTGEVC
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTGEVC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgevc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgevc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgevc.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTGEVC( SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL,
*                          LDVL, VR, LDVR, MM, M, WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          HOWMNY, SIDE
*       INTEGER            INFO, LDP, LDS, LDVL, LDVR, M, MM, N
*       ..
*       .. Array Arguments ..
*       LOGICAL            SELECT( * )
*       DOUBLE PRECISION   P( LDP, * ), S( LDS, * ), VL( LDVL, * ),
*      $                   VR( LDVR, * ), WORK( * )
*       ..
*
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTGEVC computes some or all of the right and/or left eigenvectors of
*> a pair of real matrices (S,P), where S is a quasi-triangular matrix
*> and P is upper triangular.  Matrix pairs of this type are produced by
*> the generalized Schur factorization of a matrix pair (A,B):
*>
*>    A = Q*S*Z**T,  B = Q*P*Z**T
*>
*> as computed by DGGHRD + DHGEQZ.
*>
*> The right eigenvector x and the left eigenvector y of (S,P)
*> corresponding to an eigenvalue w are defined by:
*>
*>    S*x = w*P*x,  (y**H)*S = w*(y**H)*P,
*>
*> where y**H denotes the conjugate transpose of y.
*> The eigenvalues are not input to this routine, but are computed
*> directly from the diagonal blocks of S and P.
*>
*> This routine returns the matrices X and/or Y of right and left
*> eigenvectors of (S,P), or the products Z*X and/or Q*Y,
*> where Z and Q are input matrices.
*> If Q and Z are the orthogonal factors from the generalized Schur
*> factorization of a matrix pair (A,B), then Z*X and Q*Y
*> are the matrices of right and left eigenvectors of (A,B).
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'R': compute right eigenvectors only;
*>          = 'L': compute left eigenvectors only;
*>          = 'B': compute both right and left eigenvectors.
*> \endverbatim
*>
*> \param[in] HOWMNY
*> \verbatim
*>          HOWMNY is CHARACTER*1
*>          = 'A': compute all right and/or left eigenvectors;
*>          = 'B': compute all right and/or left eigenvectors,
*>                 backtransformed by the matrices in VR and/or VL;
*>          = 'S': compute selected right and/or left eigenvectors,
*>                 specified by the logical array SELECT.
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*>          SELECT is LOGICAL array, dimension (N)
*>          If HOWMNY='S', SELECT specifies the eigenvectors to be
*>          computed.  If w(j) is a real eigenvalue, the corresponding
*>          real eigenvector is computed if SELECT(j) is .TRUE..
*>          If w(j) and w(j+1) are the real and imaginary parts of a
*>          complex eigenvalue, the corresponding complex eigenvector
*>          is computed if either SELECT(j) or SELECT(j+1) is .TRUE.,
*>          and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is
*>          set to .FALSE..
*>          Not referenced if HOWMNY = 'A' or 'B'.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices S and P.  N >= 0.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (LDS,N)
*>          The upper quasi-triangular matrix S from a generalized Schur
*>          factorization, as computed by DHGEQZ.
*> \endverbatim
*>
*> \param[in] LDS
*> \verbatim
*>          LDS is INTEGER
*>          The leading dimension of array S.  LDS >= max(1,N).
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*>          P is DOUBLE PRECISION array, dimension (LDP,N)
*>          The upper triangular matrix P from a generalized Schur
*>          factorization, as computed by DHGEQZ.
*>          2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks
*>          of S must be in positive diagonal form.
*> \endverbatim
*>
*> \param[in] LDP
*> \verbatim
*>          LDP is INTEGER
*>          The leading dimension of array P.  LDP >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*>          VL is DOUBLE PRECISION array, dimension (LDVL,MM)
*>          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
*>          contain an N-by-N matrix Q (usually the orthogonal matrix Q
*>          of left Schur vectors returned by DHGEQZ).
*>          On exit, if SIDE = 'L' or 'B', VL contains:
*>          if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
*>          if HOWMNY = 'B', the matrix Q*Y;
*>          if HOWMNY = 'S', the left eigenvectors of (S,P) specified by
*>                      SELECT, stored consecutively in the columns of
*>                      VL, in the same order as their eigenvalues.
*>
*>          A complex eigenvector corresponding to a complex eigenvalue
*>          is stored in two consecutive columns, the first holding the
*>          real part, and the second the imaginary part.
*>
*>          Not referenced if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*>          LDVL is INTEGER
*>          The leading dimension of array VL.  LDVL >= 1, and if
*>          SIDE = 'L' or 'B', LDVL >= N.
*> \endverbatim
*>
*> \param[in,out] VR
*> \verbatim
*>          VR is DOUBLE PRECISION array, dimension (LDVR,MM)
*>          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
*>          contain an N-by-N matrix Z (usually the orthogonal matrix Z
*>          of right Schur vectors returned by DHGEQZ).
*>
*>          On exit, if SIDE = 'R' or 'B', VR contains:
*>          if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
*>          if HOWMNY = 'B' or 'b', the matrix Z*X;
*>          if HOWMNY = 'S' or 's', the right eigenvectors of (S,P)
*>                      specified by SELECT, stored consecutively in the
*>                      columns of VR, in the same order as their
*>                      eigenvalues.
*>
*>          A complex eigenvector corresponding to a complex eigenvalue
*>          is stored in two consecutive columns, the first holding the
*>          real part and the second the imaginary part.
*>
*>          Not referenced if SIDE = 'L'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*>          LDVR is INTEGER
*>          The leading dimension of the array VR.  LDVR >= 1, and if
*>          SIDE = 'R' or 'B', LDVR >= N.
*> \endverbatim
*>
*> \param[in] MM
*> \verbatim
*>          MM is INTEGER
*>          The number of columns in the arrays VL and/or VR. MM >= M.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The number of columns in the arrays VL and/or VR actually
*>          used to store the eigenvectors.  If HOWMNY = 'A' or 'B', M
*>          is set to N.  Each selected real eigenvector occupies one
*>          column and each selected complex eigenvector occupies two
*>          columns.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (6*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  the 2-by-2 block (INFO:INFO+1) does not have a complex
*>                eigenvalue.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tgevc
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  Allocation of workspace:
*>  ---------- -- ---------
*>
*>     WORK( j ) = 1-norm of j-th column of A, above the diagonal
*>     WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
*>     WORK( 2*N+1:3*N ) = real part of eigenvector
*>     WORK( 3*N+1:4*N ) = imaginary part of eigenvector
*>     WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
*>     WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector
*>
*>  Rowwise vs. columnwise solution methods:
*>  ------- --  ---------- -------- -------
*>
*>  Finding a generalized eigenvector consists basically of solving the
*>  singular triangular system
*>
*>   (A - w B) x = 0     (for right) or:   (A - w B)**H y = 0  (for left)
*>
*>  Consider finding the i-th right eigenvector (assume all eigenvalues
*>  are real). The equation to be solved is:
*>       n                   i
*>  0 = sum  C(j,k) v(k)  = sum  C(j,k) v(k)     for j = i,. . .,1
*>      k=j                 k=j
*>
*>  where  C = (A - w B)  (The components v(i+1:n) are 0.)
*>
*>  The "rowwise" method is:
*>
*>  (1)  v(i) := 1
*>  for j = i-1,. . .,1:
*>                          i
*>      (2) compute  s = - sum C(j,k) v(k)   and
*>                        k=j+1
*>
*>      (3) v(j) := s / C(j,j)
*>
*>  Step 2 is sometimes called the "dot product" step, since it is an
*>  inner product between the j-th row and the portion of the eigenvector
*>  that has been computed so far.
*>
*>  The "columnwise" method consists basically in doing the sums
*>  for all the rows in parallel.  As each v(j) is computed, the
*>  contribution of v(j) times the j-th column of C is added to the
*>  partial sums.  Since FORTRAN arrays are stored columnwise, this has
*>  the advantage that at each step, the elements of C that are accessed
*>  are adjacent to one another, whereas with the rowwise method, the
*>  elements accessed at a step are spaced LDS (and LDP) words apart.
*>
*>  When finding left eigenvectors, the matrix in question is the
*>  transpose of the one in storage, so the rowwise method then
*>  actually accesses columns of A and B at each step, and so is the
*>  preferred method.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTGEVC( SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL,
     $                   LDVL, VR, LDVR, MM, M, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          HOWMNY, SIDE
      INTEGER            INFO, LDP, LDS, LDVL, LDVR, M, MM, N
*     ..
*     .. Array Arguments ..
      LOGICAL            SELECT( * )
      DOUBLE PRECISION   P( LDP, * ), S( LDS, * ), VL( LDVL, * ),
     $                   VR( LDVR, * ), WORK( * )
*     ..
*
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, SAFETY
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0,
     $                   SAFETY = 1.0D+2 )
*     ..
*     .. Local Scalars ..
      LOGICAL            COMPL, COMPR, IL2BY2, ILABAD, ILALL, ILBACK,
     $                   ILBBAD, ILCOMP, ILCPLX, LSA, LSB
      INTEGER            I, IBEG, IEIG, IEND, IHWMNY, IINFO, IM, ISIDE,
     $                   J, JA, JC, JE, JR, JW, NA, NW
      DOUBLE PRECISION   ACOEF, ACOEFA, ANORM, ASCALE, BCOEFA, BCOEFI,
     $                   BCOEFR, BIG, BIGNUM, BNORM, BSCALE, CIM2A,
     $                   CIM2B, CIMAGA, CIMAGB, CRE2A, CRE2B, CREALA,
     $                   CREALB, DMIN, SAFMIN, SALFAR, SBETA, SCALE,
     $                   SMALL, TEMP, TEMP2, TEMP2I, TEMP2R, ULP, XMAX,
     $                   XSCALE
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   BDIAG( 2 ), SUM( 2, 2 ), SUMS( 2, 2 ),
     $                   SUMP( 2, 2 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMV, DLACPY, DLAG2, DLALN2,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Decode and Test the input parameters
*
      IF( LSAME( HOWMNY, 'A' ) ) THEN
         IHWMNY = 1
         ILALL = .TRUE.
         ILBACK = .FALSE.
      ELSE IF( LSAME( HOWMNY, 'S' ) ) THEN
         IHWMNY = 2
         ILALL = .FALSE.
         ILBACK = .FALSE.
      ELSE IF( LSAME( HOWMNY, 'B' ) ) THEN
         IHWMNY = 3
         ILALL = .TRUE.
         ILBACK = .TRUE.
      ELSE
         IHWMNY = -1
         ILALL = .TRUE.
      END IF
*
      IF( LSAME( SIDE, 'R' ) ) THEN
         ISIDE = 1
         COMPL = .FALSE.
         COMPR = .TRUE.
      ELSE IF( LSAME( SIDE, 'L' ) ) THEN
         ISIDE = 2
         COMPL = .TRUE.
         COMPR = .FALSE.
      ELSE IF( LSAME( SIDE, 'B' ) ) THEN
         ISIDE = 3
         COMPL = .TRUE.
         COMPR = .TRUE.
      ELSE
         ISIDE = -1
      END IF
*
      INFO = 0
      IF( ISIDE.LT.0 ) THEN
         INFO = -1
      ELSE IF( IHWMNY.LT.0 ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDS.LT.MAX( 1, N ) ) THEN
         INFO = -6
      ELSE IF( LDP.LT.MAX( 1, N ) ) THEN
         INFO = -8
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTGEVC', -INFO )
         RETURN
      END IF
*
*     Count the number of eigenvectors to be computed
*
      IF( .NOT.ILALL ) THEN
         IM = 0
         ILCPLX = .FALSE.
         DO 10 J = 1, N
            IF( ILCPLX ) THEN
               ILCPLX = .FALSE.
               GO TO 10
            END IF
            IF( J.LT.N ) THEN
               IF( S( J+1, J ).NE.ZERO )
     $            ILCPLX = .TRUE.
            END IF
            IF( ILCPLX ) THEN
               IF( SELECT( J ) .OR. SELECT( J+1 ) )
     $            IM = IM + 2
            ELSE
               IF( SELECT( J ) )
     $            IM = IM + 1
            END IF
   10    CONTINUE
      ELSE
         IM = N
      END IF
*
*     Check 2-by-2 diagonal blocks of A, B
*
      ILABAD = .FALSE.
      ILBBAD = .FALSE.
      DO 20 J = 1, N - 1
         IF( S( J+1, J ).NE.ZERO ) THEN
            IF( P( J, J ).EQ.ZERO .OR. P( J+1, J+1 ).EQ.ZERO .OR.
     $          P( J, J+1 ).NE.ZERO )ILBBAD = .TRUE.
            IF( J.LT.N-1 ) THEN
               IF( S( J+2, J+1 ).NE.ZERO )
     $            ILABAD = .TRUE.
            END IF
         END IF
   20 CONTINUE
*
      IF( ILABAD ) THEN
         INFO = -5
      ELSE IF( ILBBAD ) THEN
         INFO = -7
      ELSE IF( COMPL .AND. LDVL.LT.N .OR. LDVL.LT.1 ) THEN
         INFO = -10
      ELSE IF( COMPR .AND. LDVR.LT.N .OR. LDVR.LT.1 ) THEN
         INFO = -12
      ELSE IF( MM.LT.IM ) THEN
         INFO = -13
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTGEVC', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      M = IM
      IF( N.EQ.0 )
     $   RETURN
*
*     Machine Constants
*
      SAFMIN = DLAMCH( 'Safe minimum' )
      BIG = ONE / SAFMIN
      ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' )
      SMALL = SAFMIN*N / ULP
      BIG = ONE / SMALL
      BIGNUM = ONE / ( SAFMIN*N )
*
*     Compute the 1-norm of each column of the strictly upper triangular
*     part (i.e., excluding all elements belonging to the diagonal
*     blocks) of A and B to check for possible overflow in the
*     triangular solver.
*
      ANORM = ABS( S( 1, 1 ) )
      IF( N.GT.1 )
     $   ANORM = ANORM + ABS( S( 2, 1 ) )
      BNORM = ABS( P( 1, 1 ) )
      WORK( 1 ) = ZERO
      WORK( N+1 ) = ZERO
*
      DO 50 J = 2, N
         TEMP = ZERO
         TEMP2 = ZERO
         IF( S( J, J-1 ).EQ.ZERO ) THEN
            IEND = J - 1
         ELSE
            IEND = J - 2
         END IF
         DO 30 I = 1, IEND
            TEMP = TEMP + ABS( S( I, J ) )
            TEMP2 = TEMP2 + ABS( P( I, J ) )
   30    CONTINUE
         WORK( J ) = TEMP
         WORK( N+J ) = TEMP2
         DO 40 I = IEND + 1, MIN( J+1, N )
            TEMP = TEMP + ABS( S( I, J ) )
            TEMP2 = TEMP2 + ABS( P( I, J ) )
   40    CONTINUE
         ANORM = MAX( ANORM, TEMP )
         BNORM = MAX( BNORM, TEMP2 )
   50 CONTINUE
*
      ASCALE = ONE / MAX( ANORM, SAFMIN )
      BSCALE = ONE / MAX( BNORM, SAFMIN )
*
*     Left eigenvectors
*
      IF( COMPL ) THEN
         IEIG = 0
*
*        Main loop over eigenvalues
*
         ILCPLX = .FALSE.
         DO 220 JE = 1, N
*
*           Skip this iteration if (a) HOWMNY='S' and SELECT=.FALSE., or
*           (b) this would be the second of a complex pair.
*           Check for complex eigenvalue, so as to be sure of which
*           entry(-ies) of SELECT to look at.
*
            IF( ILCPLX ) THEN
               ILCPLX = .FALSE.
               GO TO 220
            END IF
            NW = 1
            IF( JE.LT.N ) THEN
               IF( S( JE+1, JE ).NE.ZERO ) THEN
                  ILCPLX = .TRUE.
                  NW = 2
               END IF
            END IF
            IF( ILALL ) THEN
               ILCOMP = .TRUE.
            ELSE IF( ILCPLX ) THEN
               ILCOMP = SELECT( JE ) .OR. SELECT( JE+1 )
            ELSE
               ILCOMP = SELECT( JE )
            END IF
            IF( .NOT.ILCOMP )
     $         GO TO 220
*
*           Decide if (a) singular pencil, (b) real eigenvalue, or
*           (c) complex eigenvalue.
*
            IF( .NOT.ILCPLX ) THEN
               IF( ABS( S( JE, JE ) ).LE.SAFMIN .AND.
     $             ABS( P( JE, JE ) ).LE.SAFMIN ) THEN
*
*                 Singular matrix pencil -- return unit eigenvector
*
                  IEIG = IEIG + 1
                  DO 60 JR = 1, N
                     VL( JR, IEIG ) = ZERO
   60             CONTINUE
                  VL( IEIG, IEIG ) = ONE
                  GO TO 220
               END IF
            END IF
*
*           Clear vector
*
            DO 70 JR = 1, NW*N
               WORK( 2*N+JR ) = ZERO
   70       CONTINUE
*                                                 T
*           Compute coefficients in  ( a A - b B )  y = 0
*              a  is  ACOEF
*              b  is  BCOEFR + i*BCOEFI
*
            IF( .NOT.ILCPLX ) THEN
*
*              Real eigenvalue
*
               TEMP = ONE / MAX( ABS( S( JE, JE ) )*ASCALE,
     $                ABS( P( JE, JE ) )*BSCALE, SAFMIN )
               SALFAR = ( TEMP*S( JE, JE ) )*ASCALE
               SBETA = ( TEMP*P( JE, JE ) )*BSCALE
               ACOEF = SBETA*ASCALE
               BCOEFR = SALFAR*BSCALE
               BCOEFI = ZERO
*
*              Scale to avoid underflow
*
               SCALE = ONE
               LSA = ABS( SBETA ).GE.SAFMIN .AND. ABS( ACOEF ).LT.SMALL
               LSB = ABS( SALFAR ).GE.SAFMIN .AND. ABS( BCOEFR ).LT.
     $               SMALL
               IF( LSA )
     $            SCALE = ( SMALL / ABS( SBETA ) )*MIN( ANORM, BIG )
               IF( LSB )
     $            SCALE = MAX( SCALE, ( SMALL / ABS( SALFAR ) )*
     $                    MIN( BNORM, BIG ) )
               IF( LSA .OR. LSB ) THEN
                  SCALE = MIN( SCALE, ONE /
     $                    ( SAFMIN*MAX( ONE, ABS( ACOEF ),
     $                    ABS( BCOEFR ) ) ) )
                  IF( LSA ) THEN
                     ACOEF = ASCALE*( SCALE*SBETA )
                  ELSE
                     ACOEF = SCALE*ACOEF
                  END IF
                  IF( LSB ) THEN
                     BCOEFR = BSCALE*( SCALE*SALFAR )
                  ELSE
                     BCOEFR = SCALE*BCOEFR
                  END IF
               END IF
               ACOEFA = ABS( ACOEF )
               BCOEFA = ABS( BCOEFR )
*
*              First component is 1
*
               WORK( 2*N+JE ) = ONE
               XMAX = ONE
            ELSE
*
*              Complex eigenvalue
*
               CALL DLAG2( S( JE, JE ), LDS, P( JE, JE ), LDP,
     $                     SAFMIN*SAFETY, ACOEF, TEMP, BCOEFR, TEMP2,
     $                     BCOEFI )
               BCOEFI = -BCOEFI
               IF( BCOEFI.EQ.ZERO ) THEN
                  INFO = JE
                  RETURN
               END IF
*
*              Scale to avoid over/underflow
*
               ACOEFA = ABS( ACOEF )
               BCOEFA = ABS( BCOEFR ) + ABS( BCOEFI )
               SCALE = ONE
               IF( ACOEFA*ULP.LT.SAFMIN .AND. ACOEFA.GE.SAFMIN )
     $            SCALE = ( SAFMIN / ULP ) / ACOEFA
               IF( BCOEFA*ULP.LT.SAFMIN .AND. BCOEFA.GE.SAFMIN )
     $            SCALE = MAX( SCALE, ( SAFMIN / ULP ) / BCOEFA )
               IF( SAFMIN*ACOEFA.GT.ASCALE )
     $            SCALE = ASCALE / ( SAFMIN*ACOEFA )
               IF( SAFMIN*BCOEFA.GT.BSCALE )
     $            SCALE = MIN( SCALE, BSCALE / ( SAFMIN*BCOEFA ) )
               IF( SCALE.NE.ONE ) THEN
                  ACOEF = SCALE*ACOEF
                  ACOEFA = ABS( ACOEF )
                  BCOEFR = SCALE*BCOEFR
                  BCOEFI = SCALE*BCOEFI
                  BCOEFA = ABS( BCOEFR ) + ABS( BCOEFI )
               END IF
*
*              Compute first two components of eigenvector
*
               TEMP = ACOEF*S( JE+1, JE )
               TEMP2R = ACOEF*S( JE, JE ) - BCOEFR*P( JE, JE )
               TEMP2I = -BCOEFI*P( JE, JE )
               IF( ABS( TEMP ).GT.ABS( TEMP2R )+ABS( TEMP2I ) ) THEN
                  WORK( 2*N+JE ) = ONE
                  WORK( 3*N+JE ) = ZERO
                  WORK( 2*N+JE+1 ) = -TEMP2R / TEMP
                  WORK( 3*N+JE+1 ) = -TEMP2I / TEMP
               ELSE
                  WORK( 2*N+JE+1 ) = ONE
                  WORK( 3*N+JE+1 ) = ZERO
                  TEMP = ACOEF*S( JE, JE+1 )
                  WORK( 2*N+JE ) = ( BCOEFR*P( JE+1, JE+1 )-ACOEF*
     $                             S( JE+1, JE+1 ) ) / TEMP
                  WORK( 3*N+JE ) = BCOEFI*P( JE+1, JE+1 ) / TEMP
               END IF
               XMAX = MAX( ABS( WORK( 2*N+JE ) )+ABS( WORK( 3*N+JE ) ),
     $                ABS( WORK( 2*N+JE+1 ) )+ABS( WORK( 3*N+JE+1 ) ) )
            END IF
*
            DMIN = MAX( ULP*ACOEFA*ANORM, ULP*BCOEFA*BNORM, SAFMIN )
*
*                                           T
*           Triangular solve of  (a A - b B)  y = 0
*
*                                   T
*           (rowwise in  (a A - b B) , or columnwise in (a A - b B) )
*
            IL2BY2 = .FALSE.
*
            DO 160 J = JE + NW, N
               IF( IL2BY2 ) THEN
                  IL2BY2 = .FALSE.
                  GO TO 160
               END IF
*
               NA = 1
               BDIAG( 1 ) = P( J, J )
               IF( J.LT.N ) THEN
                  IF( S( J+1, J ).NE.ZERO ) THEN
                     IL2BY2 = .TRUE.
                     BDIAG( 2 ) = P( J+1, J+1 )
                     NA = 2
                  END IF
               END IF
*
*              Check whether scaling is necessary for dot products
*
               XSCALE = ONE / MAX( ONE, XMAX )
               TEMP = MAX( WORK( J ), WORK( N+J ),
     $                ACOEFA*WORK( J )+BCOEFA*WORK( N+J ) )
               IF( IL2BY2 )
     $            TEMP = MAX( TEMP, WORK( J+1 ), WORK( N+J+1 ),
     $                   ACOEFA*WORK( J+1 )+BCOEFA*WORK( N+J+1 ) )
               IF( TEMP.GT.BIGNUM*XSCALE ) THEN
                  DO 90 JW = 0, NW - 1
                     DO 80 JR = JE, J - 1
                        WORK( ( JW+2 )*N+JR ) = XSCALE*
     $                     WORK( ( JW+2 )*N+JR )
   80                CONTINUE
   90             CONTINUE
                  XMAX = XMAX*XSCALE
               END IF
*
*              Compute dot products
*
*                    j-1
*              SUM = sum  conjg( a*S(k,j) - b*P(k,j) )*x(k)
*                    k=je
*
*              To reduce the op count, this is done as
*
*              _        j-1                  _        j-1
*              a*conjg( sum  S(k,j)*x(k) ) - b*conjg( sum  P(k,j)*x(k) )
*                       k=je                          k=je
*
*              which may cause underflow problems if A or B are close
*              to underflow.  (E.g., less than SMALL.)
*
*
               DO 120 JW = 1, NW
                  DO 110 JA = 1, NA
                     SUMS( JA, JW ) = ZERO
                     SUMP( JA, JW ) = ZERO
*
                     DO 100 JR = JE, J - 1
                        SUMS( JA, JW ) = SUMS( JA, JW ) +
     $                                   S( JR, J+JA-1 )*
     $                                   WORK( ( JW+1 )*N+JR )
                        SUMP( JA, JW ) = SUMP( JA, JW ) +
     $                                   P( JR, J+JA-1 )*
     $                                   WORK( ( JW+1 )*N+JR )
  100                CONTINUE
  110             CONTINUE
  120          CONTINUE
*
               DO 130 JA = 1, NA
                  IF( ILCPLX ) THEN
                     SUM( JA, 1 ) = -ACOEF*SUMS( JA, 1 ) +
     $                              BCOEFR*SUMP( JA, 1 ) -
     $                              BCOEFI*SUMP( JA, 2 )
                     SUM( JA, 2 ) = -ACOEF*SUMS( JA, 2 ) +
     $                              BCOEFR*SUMP( JA, 2 ) +
     $                              BCOEFI*SUMP( JA, 1 )
                  ELSE
                     SUM( JA, 1 ) = -ACOEF*SUMS( JA, 1 ) +
     $                              BCOEFR*SUMP( JA, 1 )
                  END IF
  130          CONTINUE
*
*                                  T
*              Solve  ( a A - b B )  y = SUM(,)
*              with scaling and perturbation of the denominator
*
               CALL DLALN2( .TRUE., NA, NW, DMIN, ACOEF, S( J, J ),
     $                      LDS,
     $                      BDIAG( 1 ), BDIAG( 2 ), SUM, 2, BCOEFR,
     $                      BCOEFI, WORK( 2*N+J ), N, SCALE, TEMP,
     $                      IINFO )
               IF( SCALE.LT.ONE ) THEN
                  DO 150 JW = 0, NW - 1
                     DO 140 JR = JE, J - 1
                        WORK( ( JW+2 )*N+JR ) = SCALE*
     $                     WORK( ( JW+2 )*N+JR )
  140                CONTINUE
  150             CONTINUE
                  XMAX = SCALE*XMAX
               END IF
               XMAX = MAX( XMAX, TEMP )
  160       CONTINUE
*
*           Copy eigenvector to VL, back transforming if
*           HOWMNY='B'.
*
            IEIG = IEIG + 1
            IF( ILBACK ) THEN
               DO 170 JW = 0, NW - 1
                  CALL DGEMV( 'N', N, N+1-JE, ONE, VL( 1, JE ), LDVL,
     $                        WORK( ( JW+2 )*N+JE ), 1, ZERO,
     $                        WORK( ( JW+4 )*N+1 ), 1 )
  170          CONTINUE
               CALL DLACPY( ' ', N, NW, WORK( 4*N+1 ), N, VL( 1,
     $                      JE ),
     $                      LDVL )
               IBEG = 1
            ELSE
               CALL DLACPY( ' ', N, NW, WORK( 2*N+1 ), N, VL( 1,
     $                      IEIG ),
     $                      LDVL )
               IBEG = JE
            END IF
*
*           Scale eigenvector
*
            XMAX = ZERO
            IF( ILCPLX ) THEN
               DO 180 J = IBEG, N
                  XMAX = MAX( XMAX, ABS( VL( J, IEIG ) )+
     $                   ABS( VL( J, IEIG+1 ) ) )
  180          CONTINUE
            ELSE
               DO 190 J = IBEG, N
                  XMAX = MAX( XMAX, ABS( VL( J, IEIG ) ) )
  190          CONTINUE
            END IF
*
            IF( XMAX.GT.SAFMIN ) THEN
               XSCALE = ONE / XMAX
*
               DO 210 JW = 0, NW - 1
                  DO 200 JR = IBEG, N
                     VL( JR, IEIG+JW ) = XSCALE*VL( JR, IEIG+JW )
  200             CONTINUE
  210          CONTINUE
            END IF
            IEIG = IEIG + NW - 1
*
  220    CONTINUE
      END IF
*
*     Right eigenvectors
*
      IF( COMPR ) THEN
         IEIG = IM + 1
*
*        Main loop over eigenvalues
*
         ILCPLX = .FALSE.
         DO 500 JE = N, 1, -1
*
*           Skip this iteration if (a) HOWMNY='S' and SELECT=.FALSE., or
*           (b) this would be the second of a complex pair.
*           Check for complex eigenvalue, so as to be sure of which
*           entry(-ies) of SELECT to look at -- if complex, SELECT(JE)
*           or SELECT(JE-1).
*           If this is a complex pair, the 2-by-2 diagonal block
*           corresponding to the eigenvalue is in rows/columns JE-1:JE
*
            IF( ILCPLX ) THEN
               ILCPLX = .FALSE.
               GO TO 500
            END IF
            NW = 1
            IF( JE.GT.1 ) THEN
               IF( S( JE, JE-1 ).NE.ZERO ) THEN
                  ILCPLX = .TRUE.
                  NW = 2
               END IF
            END IF
            IF( ILALL ) THEN
               ILCOMP = .TRUE.
            ELSE IF( ILCPLX ) THEN
               ILCOMP = SELECT( JE ) .OR. SELECT( JE-1 )
            ELSE
               ILCOMP = SELECT( JE )
            END IF
            IF( .NOT.ILCOMP )
     $         GO TO 500
*
*           Decide if (a) singular pencil, (b) real eigenvalue, or
*           (c) complex eigenvalue.
*
            IF( .NOT.ILCPLX ) THEN
               IF( ABS( S( JE, JE ) ).LE.SAFMIN .AND.
     $             ABS( P( JE, JE ) ).LE.SAFMIN ) THEN
*
*                 Singular matrix pencil -- unit eigenvector
*
                  IEIG = IEIG - 1
                  DO 230 JR = 1, N
                     VR( JR, IEIG ) = ZERO
  230             CONTINUE
                  VR( IEIG, IEIG ) = ONE
                  GO TO 500
               END IF
            END IF
*
*           Clear vector
*
            DO 250 JW = 0, NW - 1
               DO 240 JR = 1, N
                  WORK( ( JW+2 )*N+JR ) = ZERO
  240          CONTINUE
  250       CONTINUE
*
*           Compute coefficients in  ( a A - b B ) x = 0
*              a  is  ACOEF
*              b  is  BCOEFR + i*BCOEFI
*
            IF( .NOT.ILCPLX ) THEN
*
*              Real eigenvalue
*
               TEMP = ONE / MAX( ABS( S( JE, JE ) )*ASCALE,
     $                ABS( P( JE, JE ) )*BSCALE, SAFMIN )
               SALFAR = ( TEMP*S( JE, JE ) )*ASCALE
               SBETA = ( TEMP*P( JE, JE ) )*BSCALE
               ACOEF = SBETA*ASCALE
               BCOEFR = SALFAR*BSCALE
               BCOEFI = ZERO
*
*              Scale to avoid underflow
*
               SCALE = ONE
               LSA = ABS( SBETA ).GE.SAFMIN .AND. ABS( ACOEF ).LT.SMALL
               LSB = ABS( SALFAR ).GE.SAFMIN .AND. ABS( BCOEFR ).LT.
     $               SMALL
               IF( LSA )
     $            SCALE = ( SMALL / ABS( SBETA ) )*MIN( ANORM, BIG )
               IF( LSB )
     $            SCALE = MAX( SCALE, ( SMALL / ABS( SALFAR ) )*
     $                    MIN( BNORM, BIG ) )
               IF( LSA .OR. LSB ) THEN
                  SCALE = MIN( SCALE, ONE /
     $                    ( SAFMIN*MAX( ONE, ABS( ACOEF ),
     $                    ABS( BCOEFR ) ) ) )
                  IF( LSA ) THEN
                     ACOEF = ASCALE*( SCALE*SBETA )
                  ELSE
                     ACOEF = SCALE*ACOEF
                  END IF
                  IF( LSB ) THEN
                     BCOEFR = BSCALE*( SCALE*SALFAR )
                  ELSE
                     BCOEFR = SCALE*BCOEFR
                  END IF
               END IF
               ACOEFA = ABS( ACOEF )
               BCOEFA = ABS( BCOEFR )
*
*              First component is 1
*
               WORK( 2*N+JE ) = ONE
               XMAX = ONE
*
*              Compute contribution from column JE of A and B to sum
*              (See "Further Details", above.)
*
               DO 260 JR = 1, JE - 1
                  WORK( 2*N+JR ) = BCOEFR*P( JR, JE ) -
     $                             ACOEF*S( JR, JE )
  260          CONTINUE
            ELSE
*
*              Complex eigenvalue
*
               CALL DLAG2( S( JE-1, JE-1 ), LDS, P( JE-1, JE-1 ),
     $                     LDP,
     $                     SAFMIN*SAFETY, ACOEF, TEMP, BCOEFR, TEMP2,
     $                     BCOEFI )
               IF( BCOEFI.EQ.ZERO ) THEN
                  INFO = JE - 1
                  RETURN
               END IF
*
*              Scale to avoid over/underflow
*
               ACOEFA = ABS( ACOEF )
               BCOEFA = ABS( BCOEFR ) + ABS( BCOEFI )
               SCALE = ONE
               IF( ACOEFA*ULP.LT.SAFMIN .AND. ACOEFA.GE.SAFMIN )
     $            SCALE = ( SAFMIN / ULP ) / ACOEFA
               IF( BCOEFA*ULP.LT.SAFMIN .AND. BCOEFA.GE.SAFMIN )
     $            SCALE = MAX( SCALE, ( SAFMIN / ULP ) / BCOEFA )
               IF( SAFMIN*ACOEFA.GT.ASCALE )
     $            SCALE = ASCALE / ( SAFMIN*ACOEFA )
               IF( SAFMIN*BCOEFA.GT.BSCALE )
     $            SCALE = MIN( SCALE, BSCALE / ( SAFMIN*BCOEFA ) )
               IF( SCALE.NE.ONE ) THEN
                  ACOEF = SCALE*ACOEF
                  ACOEFA = ABS( ACOEF )
                  BCOEFR = SCALE*BCOEFR
                  BCOEFI = SCALE*BCOEFI
                  BCOEFA = ABS( BCOEFR ) + ABS( BCOEFI )
               END IF
*
*              Compute first two components of eigenvector
*              and contribution to sums
*
               TEMP = ACOEF*S( JE, JE-1 )
               TEMP2R = ACOEF*S( JE, JE ) - BCOEFR*P( JE, JE )
               TEMP2I = -BCOEFI*P( JE, JE )
               IF( ABS( TEMP ).GE.ABS( TEMP2R )+ABS( TEMP2I ) ) THEN
                  WORK( 2*N+JE ) = ONE
                  WORK( 3*N+JE ) = ZERO
                  WORK( 2*N+JE-1 ) = -TEMP2R / TEMP
                  WORK( 3*N+JE-1 ) = -TEMP2I / TEMP
               ELSE
                  WORK( 2*N+JE-1 ) = ONE
                  WORK( 3*N+JE-1 ) = ZERO
                  TEMP = ACOEF*S( JE-1, JE )
                  WORK( 2*N+JE ) = ( BCOEFR*P( JE-1, JE-1 )-ACOEF*
     $                             S( JE-1, JE-1 ) ) / TEMP
                  WORK( 3*N+JE ) = BCOEFI*P( JE-1, JE-1 ) / TEMP
               END IF
*
               XMAX = MAX( ABS( WORK( 2*N+JE ) )+ABS( WORK( 3*N+JE ) ),
     $                ABS( WORK( 2*N+JE-1 ) )+ABS( WORK( 3*N+JE-1 ) ) )
*
*              Compute contribution from columns JE and JE-1
*              of A and B to the sums.
*
               CREALA = ACOEF*WORK( 2*N+JE-1 )
               CIMAGA = ACOEF*WORK( 3*N+JE-1 )
               CREALB = BCOEFR*WORK( 2*N+JE-1 ) -
     $                  BCOEFI*WORK( 3*N+JE-1 )
               CIMAGB = BCOEFI*WORK( 2*N+JE-1 ) +
     $                  BCOEFR*WORK( 3*N+JE-1 )
               CRE2A = ACOEF*WORK( 2*N+JE )
               CIM2A = ACOEF*WORK( 3*N+JE )
               CRE2B = BCOEFR*WORK( 2*N+JE ) - BCOEFI*WORK( 3*N+JE )
               CIM2B = BCOEFI*WORK( 2*N+JE ) + BCOEFR*WORK( 3*N+JE )
               DO 270 JR = 1, JE - 2
                  WORK( 2*N+JR ) = -CREALA*S( JR, JE-1 ) +
     $                             CREALB*P( JR, JE-1 ) -
     $                             CRE2A*S( JR, JE ) + CRE2B*P( JR, JE )
                  WORK( 3*N+JR ) = -CIMAGA*S( JR, JE-1 ) +
     $                             CIMAGB*P( JR, JE-1 ) -
     $                             CIM2A*S( JR, JE ) + CIM2B*P( JR, JE )
  270          CONTINUE
            END IF
*
            DMIN = MAX( ULP*ACOEFA*ANORM, ULP*BCOEFA*BNORM, SAFMIN )
*
*           Columnwise triangular solve of  (a A - b B)  x = 0
*
            IL2BY2 = .FALSE.
            DO 370 J = JE - NW, 1, -1
*
*              If a 2-by-2 block, is in position j-1:j, wait until
*              next iteration to process it (when it will be j:j+1)
*
               IF( .NOT.IL2BY2 .AND. J.GT.1 ) THEN
                  IF( S( J, J-1 ).NE.ZERO ) THEN
                     IL2BY2 = .TRUE.
                     GO TO 370
                  END IF
               END IF
               BDIAG( 1 ) = P( J, J )
               IF( IL2BY2 ) THEN
                  NA = 2
                  BDIAG( 2 ) = P( J+1, J+1 )
               ELSE
                  NA = 1
               END IF
*
*              Compute x(j) (and x(j+1), if 2-by-2 block)
*
               CALL DLALN2( .FALSE., NA, NW, DMIN, ACOEF, S( J, J ),
     $                      LDS, BDIAG( 1 ), BDIAG( 2 ), WORK( 2*N+J ),
     $                      N, BCOEFR, BCOEFI, SUM, 2, SCALE, TEMP,
     $                      IINFO )
               IF( SCALE.LT.ONE ) THEN
*
                  DO 290 JW = 0, NW - 1
                     DO 280 JR = 1, JE
                        WORK( ( JW+2 )*N+JR ) = SCALE*
     $                     WORK( ( JW+2 )*N+JR )
  280                CONTINUE
  290             CONTINUE
               END IF
               XMAX = MAX( SCALE*XMAX, TEMP )
*
               DO 310 JW = 1, NW
                  DO 300 JA = 1, NA
                     WORK( ( JW+1 )*N+J+JA-1 ) = SUM( JA, JW )
  300             CONTINUE
  310          CONTINUE
*
*              w = w + x(j)*(a S(*,j) - b P(*,j) ) with scaling
*
               IF( J.GT.1 ) THEN
*
*                 Check whether scaling is necessary for sum.
*
                  XSCALE = ONE / MAX( ONE, XMAX )
                  TEMP = ACOEFA*WORK( J ) + BCOEFA*WORK( N+J )
                  IF( IL2BY2 )
     $               TEMP = MAX( TEMP, ACOEFA*WORK( J+1 )+BCOEFA*
     $                      WORK( N+J+1 ) )
                  TEMP = MAX( TEMP, ACOEFA, BCOEFA )
                  IF( TEMP.GT.BIGNUM*XSCALE ) THEN
*
                     DO 330 JW = 0, NW - 1
                        DO 320 JR = 1, JE
                           WORK( ( JW+2 )*N+JR ) = XSCALE*
     $                        WORK( ( JW+2 )*N+JR )
  320                   CONTINUE
  330                CONTINUE
                     XMAX = XMAX*XSCALE
                  END IF
*
*                 Compute the contributions of the off-diagonals of
*                 column j (and j+1, if 2-by-2 block) of A and B to the
*                 sums.
*
*
                  DO 360 JA = 1, NA
                     IF( ILCPLX ) THEN
                        CREALA = ACOEF*WORK( 2*N+J+JA-1 )
                        CIMAGA = ACOEF*WORK( 3*N+J+JA-1 )
                        CREALB = BCOEFR*WORK( 2*N+J+JA-1 ) -
     $                           BCOEFI*WORK( 3*N+J+JA-1 )
                        CIMAGB = BCOEFI*WORK( 2*N+J+JA-1 ) +
     $                           BCOEFR*WORK( 3*N+J+JA-1 )
                        DO 340 JR = 1, J - 1
                           WORK( 2*N+JR ) = WORK( 2*N+JR ) -
     $                                      CREALA*S( JR, J+JA-1 ) +
     $                                      CREALB*P( JR, J+JA-1 )
                           WORK( 3*N+JR ) = WORK( 3*N+JR ) -
     $                                      CIMAGA*S( JR, J+JA-1 ) +
     $                                      CIMAGB*P( JR, J+JA-1 )
  340                   CONTINUE
                     ELSE
                        CREALA = ACOEF*WORK( 2*N+J+JA-1 )
                        CREALB = BCOEFR*WORK( 2*N+J+JA-1 )
                        DO 350 JR = 1, J - 1
                           WORK( 2*N+JR ) = WORK( 2*N+JR ) -
     $                                      CREALA*S( JR, J+JA-1 ) +
     $                                      CREALB*P( JR, J+JA-1 )
  350                   CONTINUE
                     END IF
  360             CONTINUE
               END IF
*
               IL2BY2 = .FALSE.
  370       CONTINUE
*
*           Copy eigenvector to VR, back transforming if
*           HOWMNY='B'.
*
            IEIG = IEIG - NW
            IF( ILBACK ) THEN
*
               DO 410 JW = 0, NW - 1
                  DO 380 JR = 1, N
                     WORK( ( JW+4 )*N+JR ) = WORK( ( JW+2 )*N+1 )*
     $                                       VR( JR, 1 )
  380             CONTINUE
*
*                 A series of compiler directives to defeat
*                 vectorization for the next loop
*
*
                  DO 400 JC = 2, JE
                     DO 390 JR = 1, N
                        WORK( ( JW+4 )*N+JR ) = WORK( ( JW+4 )*N+JR ) +
     $                     WORK( ( JW+2 )*N+JC )*VR( JR, JC )
  390                CONTINUE
  400             CONTINUE
  410          CONTINUE
*
               DO 430 JW = 0, NW - 1
                  DO 420 JR = 1, N
                     VR( JR, IEIG+JW ) = WORK( ( JW+4 )*N+JR )
  420             CONTINUE
  430          CONTINUE
*
               IEND = N
            ELSE
               DO 450 JW = 0, NW - 1
                  DO 440 JR = 1, N
                     VR( JR, IEIG+JW ) = WORK( ( JW+2 )*N+JR )
  440             CONTINUE
  450          CONTINUE
*
               IEND = JE
            END IF
*
*           Scale eigenvector
*
            XMAX = ZERO
            IF( ILCPLX ) THEN
               DO 460 J = 1, IEND
                  XMAX = MAX( XMAX, ABS( VR( J, IEIG ) )+
     $                   ABS( VR( J, IEIG+1 ) ) )
  460          CONTINUE
            ELSE
               DO 470 J = 1, IEND
                  XMAX = MAX( XMAX, ABS( VR( J, IEIG ) ) )
  470          CONTINUE
            END IF
*
            IF( XMAX.GT.SAFMIN ) THEN
               XSCALE = ONE / XMAX
               DO 490 JW = 0, NW - 1
                  DO 480 JR = 1, IEND
                     VR( JR, IEIG+JW ) = XSCALE*VR( JR, IEIG+JW )
  480             CONTINUE
  490          CONTINUE
            END IF
  500    CONTINUE
      END IF
*
      RETURN
*
*     End of DTGEVC
*
      END
*> \brief \b DTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTGEX2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgex2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgex2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgex2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
*                          LDZ, J1, N1, N2, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       LOGICAL            WANTQ, WANTZ
*       INTEGER            INFO, J1, LDA, LDB, LDQ, LDZ, LWORK, N, N1, N2
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
*      $                   WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22)
*> of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair
*> (A, B) by an orthogonal equivalence transformation.
*>
*> (A, B) must be in generalized real Schur canonical form (as returned
*> by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2
*> diagonal blocks. B is upper triangular.
*>
*> Optionally, the matrices Q and Z of generalized Schur vectors are
*> updated.
*>
*>        Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T
*>        Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] WANTQ
*> \verbatim
*>          WANTQ is LOGICAL
*>          .TRUE. : update the left transformation matrix Q;
*>          .FALSE.: do not update Q.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*>          WANTZ is LOGICAL
*>          .TRUE. : update the right transformation matrix Z;
*>          .FALSE.: do not update Z.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimensions (LDA,N)
*>          On entry, the matrix A in the pair (A, B).
*>          On exit, the updated matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimensions (LDB,N)
*>          On entry, the matrix B in the pair (A, B).
*>          On exit, the updated matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ,N)
*>          On entry, if WANTQ = .TRUE., the orthogonal matrix Q.
*>          On exit, the updated matrix Q.
*>          Not referenced if WANTQ = .FALSE..
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q. LDQ >= 1.
*>          If WANTQ = .TRUE., LDQ >= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ,N)
*>          On entry, if WANTZ =.TRUE., the orthogonal matrix Z.
*>          On exit, the updated matrix Z.
*>          Not referenced if WANTZ = .FALSE..
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z. LDZ >= 1.
*>          If WANTZ = .TRUE., LDZ >= N.
*> \endverbatim
*>
*> \param[in] J1
*> \verbatim
*>          J1 is INTEGER
*>          The index to the first block (A11, B11). 1 <= J1 <= N.
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*>          N1 is INTEGER
*>          The order of the first block (A11, B11). N1 = 0, 1 or 2.
*> \endverbatim
*>
*> \param[in] N2
*> \verbatim
*>          N2 is INTEGER
*>          The order of the second block (A22, B22). N2 = 0, 1 or 2.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)).
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          LWORK >=  MAX( 1, N*(N2+N1), (N2+N1)*(N2+N1)*2 )
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>            =0: Successful exit
*>            >0: If INFO = 1, the transformed matrix (A, B) would be
*>                too far from generalized Schur form; the blocks are
*>                not swapped and (A, B) and (Q, Z) are unchanged.
*>                The problem of swapping is too ill-conditioned.
*>            <0: If INFO = -16: LWORK is too small. Appropriate value
*>                for LWORK is returned in WORK(1).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tgex2
*
*> \par Further Details:
*  =====================
*>
*>  In the current code both weak and strong stability tests are
*>  performed. The user can omit the strong stability test by changing
*>  the internal logical parameter WANDS to .FALSE.. See ref. [2] for
*>  details.
*
*> \par Contributors:
*  ==================
*>
*>     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*>     Umea University, S-901 87 Umea, Sweden.
*
*> \par References:
*  ================
*>
*> \verbatim
*>
*>  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
*>      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
*>      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
*>      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
*>
*>  [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
*>      Eigenvalues of a Regular Matrix Pair (A, B) and Condition
*>      Estimation: Theory, Algorithms and Software,
*>      Report UMINF - 94.04, Department of Computing Science, Umea
*>      University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
*>      Note 87. To appear in Numerical Algorithms, 1996.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
     $                   LDZ, J1, N1, N2, WORK, LWORK, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      LOGICAL            WANTQ, WANTZ
      INTEGER            INFO, J1, LDA, LDB, LDQ, LDZ, LWORK, N, N1, N2
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
     $                   WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*  Replaced various illegal calls to DCOPY by calls to DLASET, or by DO
*  loops. Sven Hammarling, 1/5/02.
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
      DOUBLE PRECISION   TWENTY
      PARAMETER          ( TWENTY = 2.0D+01 )
      INTEGER            LDST
      PARAMETER          ( LDST = 4 )
      LOGICAL            WANDS
      PARAMETER          ( WANDS = .TRUE. )
*     ..
*     .. Local Scalars ..
      LOGICAL            STRONG, WEAK
      INTEGER            I, IDUM, LINFO, M
      DOUBLE PRECISION   BQRA21, BRQA21, DDUM, DNORMA, DNORMB, DSCALE,
     $                   DSUM, EPS, F, G, SA, SB, SCALE, SMLNUM,
     $                   THRESHA, THRESHB
*     ..
*     .. Local Arrays ..
      INTEGER            IWORK( LDST + 2 )
      DOUBLE PRECISION   AI( 2 ), AR( 2 ), BE( 2 ), IR( LDST, LDST ),
     $                   IRCOP( LDST, LDST ), LI( LDST, LDST ),
     $                   LICOP( LDST, LDST ), S( LDST, LDST ),
     $                   SCPY( LDST, LDST ), T( LDST, LDST ),
     $                   TAUL( LDST ), TAUR( LDST ), TCPY( LDST, LDST )
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMM, DGEQR2, DGERQ2, DLACPY, DLAGV2,
     $                   DLARTG,
     $                   DLASET, DLASSQ, DORG2R, DORGR2, DORM2R, DORMR2,
     $                   DROT, DSCAL, DTGSY2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
      INFO = 0
*
*     Quick return if possible
*
      IF( N.LE.1 .OR. N1.LE.0 .OR. N2.LE.0 )
     $   RETURN
      IF( N1.GT.N .OR. ( J1+N1 ).GT.N )
     $   RETURN
      M = N1 + N2
      IF( LWORK.LT.MAX( 1, N*M, M*M*2 ) ) THEN
         INFO = -16
         WORK( 1 ) = MAX( 1, N*M, M*M*2 )
         RETURN
      END IF
*
      WEAK = .FALSE.
      STRONG = .FALSE.
*
*     Make a local copy of selected block
*
      CALL DLASET( 'Full', LDST, LDST, ZERO, ZERO, LI, LDST )
      CALL DLASET( 'Full', LDST, LDST, ZERO, ZERO, IR, LDST )
      CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, S, LDST )
      CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, T, LDST )
*
*     Compute threshold for testing acceptance of swapping.
*
      EPS = DLAMCH( 'P' )
      SMLNUM = DLAMCH( 'S' ) / EPS
      DSCALE = ZERO
      DSUM = ONE
      CALL DLACPY( 'Full', M, M, S, LDST, WORK, M )
      CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM )
      DNORMA = DSCALE*SQRT( DSUM )
      DSCALE = ZERO
      DSUM = ONE
      CALL DLACPY( 'Full', M, M, T, LDST, WORK, M )
      CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM )
      DNORMB = DSCALE*SQRT( DSUM )
*
*     THRES has been changed from
*        THRESH = MAX( TEN*EPS*SA, SMLNUM )
*     to
*        THRESH = MAX( TWENTY*EPS*SA, SMLNUM )
*     on 04/01/10.
*     "Bug" reported by Ondra Kamenik, confirmed by Julie Langou, fixed by
*     Jim Demmel and Guillaume Revy. See forum post 1783.
*
      THRESHA = MAX( TWENTY*EPS*DNORMA, SMLNUM )
      THRESHB = MAX( TWENTY*EPS*DNORMB, SMLNUM )
*
      IF( M.EQ.2 ) THEN
*
*        CASE 1: Swap 1-by-1 and 1-by-1 blocks.
*
*        Compute orthogonal QL and RQ that swap 1-by-1 and 1-by-1 blocks
*        using Givens rotations and perform the swap tentatively.
*
         F = S( 2, 2 )*T( 1, 1 ) - T( 2, 2 )*S( 1, 1 )
         G = S( 2, 2 )*T( 1, 2 ) - T( 2, 2 )*S( 1, 2 )
         SA = ABS( S( 2, 2 ) ) * ABS( T( 1, 1 ) )
         SB = ABS( S( 1, 1 ) ) * ABS( T( 2, 2 ) )
         CALL DLARTG( F, G, IR( 1, 2 ), IR( 1, 1 ), DDUM )
         IR( 2, 1 ) = -IR( 1, 2 )
         IR( 2, 2 ) = IR( 1, 1 )
         CALL DROT( 2, S( 1, 1 ), 1, S( 1, 2 ), 1, IR( 1, 1 ),
     $              IR( 2, 1 ) )
         CALL DROT( 2, T( 1, 1 ), 1, T( 1, 2 ), 1, IR( 1, 1 ),
     $              IR( 2, 1 ) )
         IF( SA.GE.SB ) THEN
            CALL DLARTG( S( 1, 1 ), S( 2, 1 ), LI( 1, 1 ), LI( 2,
     $                   1 ),
     $                   DDUM )
         ELSE
            CALL DLARTG( T( 1, 1 ), T( 2, 1 ), LI( 1, 1 ), LI( 2,
     $                   1 ),
     $                   DDUM )
         END IF
         CALL DROT( 2, S( 1, 1 ), LDST, S( 2, 1 ), LDST, LI( 1, 1 ),
     $              LI( 2, 1 ) )
         CALL DROT( 2, T( 1, 1 ), LDST, T( 2, 1 ), LDST, LI( 1, 1 ),
     $              LI( 2, 1 ) )
         LI( 2, 2 ) = LI( 1, 1 )
         LI( 1, 2 ) = -LI( 2, 1 )
*
*        Weak stability test: |S21| <= O(EPS F-norm((A)))
*                           and  |T21| <= O(EPS F-norm((B)))
*
         WEAK = ABS( S( 2, 1 ) ) .LE. THRESHA .AND.
     $      ABS( T( 2, 1 ) ) .LE. THRESHB
         IF( .NOT.WEAK )
     $      GO TO 70
*
         IF( WANDS ) THEN
*
*           Strong stability test:
*               F-norm((A-QL**H*S*QR)) <= O(EPS*F-norm((A)))
*               and
*               F-norm((B-QL**H*T*QR)) <= O(EPS*F-norm((B)))
*
            CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA,
     $                   WORK( M*M+1 ),
     $                   M )
            CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, S, LDST,
     $                  ZERO,
     $                  WORK, M )
            CALL DGEMM( 'N', 'T', M, M, M, -ONE, WORK, M, IR, LDST,
     $                  ONE,
     $                  WORK( M*M+1 ), M )
            DSCALE = ZERO
            DSUM = ONE
            CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
            SA = DSCALE*SQRT( DSUM )
*
            CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB,
     $                   WORK( M*M+1 ),
     $                   M )
            CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, T, LDST,
     $                  ZERO,
     $                  WORK, M )
            CALL DGEMM( 'N', 'T', M, M, M, -ONE, WORK, M, IR, LDST,
     $                  ONE,
     $                  WORK( M*M+1 ), M )
            DSCALE = ZERO
            DSUM = ONE
            CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
            SB = DSCALE*SQRT( DSUM )
            STRONG = SA.LE.THRESHA .AND. SB.LE.THRESHB
            IF( .NOT.STRONG )
     $         GO TO 70
         END IF
*
*        Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and
*               (A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)).
*
         CALL DROT( J1+1, A( 1, J1 ), 1, A( 1, J1+1 ), 1, IR( 1, 1 ),
     $              IR( 2, 1 ) )
         CALL DROT( J1+1, B( 1, J1 ), 1, B( 1, J1+1 ), 1, IR( 1, 1 ),
     $              IR( 2, 1 ) )
         CALL DROT( N-J1+1, A( J1, J1 ), LDA, A( J1+1, J1 ), LDA,
     $              LI( 1, 1 ), LI( 2, 1 ) )
         CALL DROT( N-J1+1, B( J1, J1 ), LDB, B( J1+1, J1 ), LDB,
     $              LI( 1, 1 ), LI( 2, 1 ) )
*
*        Set  N1-by-N2 (2,1) - blocks to ZERO.
*
         A( J1+1, J1 ) = ZERO
         B( J1+1, J1 ) = ZERO
*
*        Accumulate transformations into Q and Z if requested.
*
         IF( WANTZ )
     $      CALL DROT( N, Z( 1, J1 ), 1, Z( 1, J1+1 ), 1, IR( 1, 1 ),
     $                 IR( 2, 1 ) )
         IF( WANTQ )
     $      CALL DROT( N, Q( 1, J1 ), 1, Q( 1, J1+1 ), 1, LI( 1, 1 ),
     $                 LI( 2, 1 ) )
*
*        Exit with INFO = 0 if swap was successfully performed.
*
         RETURN
*
      ELSE
*
*        CASE 2: Swap 1-by-1 and 2-by-2 blocks, or 2-by-2
*                and 2-by-2 blocks.
*
*        Solve the generalized Sylvester equation
*                 S11 * R - L * S22 = SCALE * S12
*                 T11 * R - L * T22 = SCALE * T12
*        for R and L. Solutions in LI and IR.
*
         CALL DLACPY( 'Full', N1, N2, T( 1, N1+1 ), LDST, LI, LDST )
         CALL DLACPY( 'Full', N1, N2, S( 1, N1+1 ), LDST,
     $                IR( N2+1, N1+1 ), LDST )
         CALL DTGSY2( 'N', 0, N1, N2, S, LDST, S( N1+1, N1+1 ), LDST,
     $                IR( N2+1, N1+1 ), LDST, T, LDST, T( N1+1, N1+1 ),
     $                LDST, LI, LDST, SCALE, DSUM, DSCALE, IWORK, IDUM,
     $                LINFO )
         IF( LINFO.NE.0 )
     $      GO TO 70
*
*        Compute orthogonal matrix QL:
*
*                    QL**T * LI = [ TL ]
*                                 [ 0  ]
*        where
*                    LI =  [      -L              ]
*                          [ SCALE * identity(N2) ]
*
         DO 10 I = 1, N2
            CALL DSCAL( N1, -ONE, LI( 1, I ), 1 )
            LI( N1+I, I ) = SCALE
   10    CONTINUE
         CALL DGEQR2( M, N2, LI, LDST, TAUL, WORK, LINFO )
         IF( LINFO.NE.0 )
     $      GO TO 70
         CALL DORG2R( M, M, N2, LI, LDST, TAUL, WORK, LINFO )
         IF( LINFO.NE.0 )
     $      GO TO 70
*
*        Compute orthogonal matrix RQ:
*
*                    IR * RQ**T =   [ 0  TR],
*
*         where IR = [ SCALE * identity(N1), R ]
*
         DO 20 I = 1, N1
            IR( N2+I, I ) = SCALE
   20    CONTINUE
         CALL DGERQ2( N1, M, IR( N2+1, 1 ), LDST, TAUR, WORK, LINFO )
         IF( LINFO.NE.0 )
     $      GO TO 70
         CALL DORGR2( M, M, N1, IR, LDST, TAUR, WORK, LINFO )
         IF( LINFO.NE.0 )
     $      GO TO 70
*
*        Perform the swapping tentatively:
*
         CALL DGEMM( 'T', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO,
     $               WORK, M )
         CALL DGEMM( 'N', 'T', M, M, M, ONE, WORK, M, IR, LDST, ZERO,
     $               S,
     $               LDST )
         CALL DGEMM( 'T', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO,
     $               WORK, M )
         CALL DGEMM( 'N', 'T', M, M, M, ONE, WORK, M, IR, LDST, ZERO,
     $               T,
     $               LDST )
         CALL DLACPY( 'F', M, M, S, LDST, SCPY, LDST )
         CALL DLACPY( 'F', M, M, T, LDST, TCPY, LDST )
         CALL DLACPY( 'F', M, M, IR, LDST, IRCOP, LDST )
         CALL DLACPY( 'F', M, M, LI, LDST, LICOP, LDST )
*
*        Triangularize the B-part by an RQ factorization.
*        Apply transformation (from left) to A-part, giving S.
*
         CALL DGERQ2( M, M, T, LDST, TAUR, WORK, LINFO )
         IF( LINFO.NE.0 )
     $      GO TO 70
         CALL DORMR2( 'R', 'T', M, M, M, T, LDST, TAUR, S, LDST,
     $                WORK,
     $                LINFO )
         IF( LINFO.NE.0 )
     $      GO TO 70
         CALL DORMR2( 'L', 'N', M, M, M, T, LDST, TAUR, IR, LDST,
     $                WORK,
     $                LINFO )
         IF( LINFO.NE.0 )
     $      GO TO 70
*
*        Compute F-norm(S21) in BRQA21. (T21 is 0.)
*
         DSCALE = ZERO
         DSUM = ONE
         DO 30 I = 1, N2
            CALL DLASSQ( N1, S( N2+1, I ), 1, DSCALE, DSUM )
   30    CONTINUE
         BRQA21 = DSCALE*SQRT( DSUM )
*
*        Triangularize the B-part by a QR factorization.
*        Apply transformation (from right) to A-part, giving S.
*
         CALL DGEQR2( M, M, TCPY, LDST, TAUL, WORK, LINFO )
         IF( LINFO.NE.0 )
     $      GO TO 70
         CALL DORM2R( 'L', 'T', M, M, M, TCPY, LDST, TAUL, SCPY,
     $                LDST,
     $                WORK, INFO )
         CALL DORM2R( 'R', 'N', M, M, M, TCPY, LDST, TAUL, LICOP,
     $                LDST,
     $                WORK, INFO )
         IF( LINFO.NE.0 )
     $      GO TO 70
*
*        Compute F-norm(S21) in BQRA21. (T21 is 0.)
*
         DSCALE = ZERO
         DSUM = ONE
         DO 40 I = 1, N2
            CALL DLASSQ( N1, SCPY( N2+1, I ), 1, DSCALE, DSUM )
   40    CONTINUE
         BQRA21 = DSCALE*SQRT( DSUM )
*
*        Decide which method to use.
*          Weak stability test:
*             F-norm(S21) <= O(EPS * F-norm((S)))
*
         IF( BQRA21.LE.BRQA21 .AND. BQRA21.LE.THRESHA ) THEN
            CALL DLACPY( 'F', M, M, SCPY, LDST, S, LDST )
            CALL DLACPY( 'F', M, M, TCPY, LDST, T, LDST )
            CALL DLACPY( 'F', M, M, IRCOP, LDST, IR, LDST )
            CALL DLACPY( 'F', M, M, LICOP, LDST, LI, LDST )
         ELSE IF( BRQA21.GE.THRESHA ) THEN
            GO TO 70
         END IF
*
*        Set lower triangle of B-part to zero
*
         CALL DLASET( 'Lower', M-1, M-1, ZERO, ZERO, T(2,1), LDST )
*
         IF( WANDS ) THEN
*
*           Strong stability test:
*               F-norm((A-QL**H*S*QR)) <= O(EPS*F-norm((A)))
*               and
*               F-norm((B-QL**H*T*QR)) <= O(EPS*F-norm((B)))
*
            CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA,
     $                   WORK( M*M+1 ),
     $                   M )
            CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, S, LDST,
     $                  ZERO,
     $                  WORK, M )
            CALL DGEMM( 'N', 'N', M, M, M, -ONE, WORK, M, IR, LDST,
     $                  ONE,
     $                  WORK( M*M+1 ), M )
            DSCALE = ZERO
            DSUM = ONE
            CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
            SA = DSCALE*SQRT( DSUM )
*
            CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB,
     $                   WORK( M*M+1 ),
     $                   M )
            CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, T, LDST,
     $                  ZERO,
     $                  WORK, M )
            CALL DGEMM( 'N', 'N', M, M, M, -ONE, WORK, M, IR, LDST,
     $                  ONE,
     $                  WORK( M*M+1 ), M )
            DSCALE = ZERO
            DSUM = ONE
            CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
            SB = DSCALE*SQRT( DSUM )
            STRONG = SA.LE.THRESHA .AND. SB.LE.THRESHB
            IF( .NOT.STRONG )
     $         GO TO 70
*
         END IF
*
*        If the swap is accepted ("weakly" and "strongly"), apply the
*        transformations and set N1-by-N2 (2,1)-block to zero.
*
         CALL DLASET( 'Full', N1, N2, ZERO, ZERO, S(N2+1,1), LDST )
*
*        copy back M-by-M diagonal block starting at index J1 of (A, B)
*
         CALL DLACPY( 'F', M, M, S, LDST, A( J1, J1 ), LDA )
         CALL DLACPY( 'F', M, M, T, LDST, B( J1, J1 ), LDB )
         CALL DLASET( 'Full', LDST, LDST, ZERO, ZERO, T, LDST )
*
*        Standardize existing 2-by-2 blocks.
*
         CALL DLASET( 'Full', M, M, ZERO, ZERO, WORK, M )
         WORK( 1 ) = ONE
         T( 1, 1 ) = ONE
         IDUM = LWORK - M*M - 2
         IF( N2.GT.1 ) THEN
            CALL DLAGV2( A( J1, J1 ), LDA, B( J1, J1 ), LDB, AR, AI,
     $                   BE,
     $                   WORK( 1 ), WORK( 2 ), T( 1, 1 ), T( 2, 1 ) )
            WORK( M+1 ) = -WORK( 2 )
            WORK( M+2 ) = WORK( 1 )
            T( N2, N2 ) = T( 1, 1 )
            T( 1, 2 ) = -T( 2, 1 )
         END IF
         WORK( M*M ) = ONE
         T( M, M ) = ONE
*
         IF( N1.GT.1 ) THEN
            CALL DLAGV2( A( J1+N2, J1+N2 ), LDA, B( J1+N2, J1+N2 ),
     $                   LDB,
     $                   TAUR, TAUL, WORK( M*M+1 ), WORK( N2*M+N2+1 ),
     $                   WORK( N2*M+N2+2 ), T( N2+1, N2+1 ),
     $                   T( M, M-1 ) )
            WORK( M*M ) = WORK( N2*M+N2+1 )
            WORK( M*M-1 ) = -WORK( N2*M+N2+2 )
            T( M, M ) = T( N2+1, N2+1 )
            T( M-1, M ) = -T( M, M-1 )
         END IF
         CALL DGEMM( 'T', 'N', N2, N1, N2, ONE, WORK, M, A( J1,
     $               J1+N2 ),
     $               LDA, ZERO, WORK( M*M+1 ), N2 )
         CALL DLACPY( 'Full', N2, N1, WORK( M*M+1 ), N2, A( J1,
     $                J1+N2 ),
     $                LDA )
         CALL DGEMM( 'T', 'N', N2, N1, N2, ONE, WORK, M, B( J1,
     $               J1+N2 ),
     $               LDB, ZERO, WORK( M*M+1 ), N2 )
         CALL DLACPY( 'Full', N2, N1, WORK( M*M+1 ), N2, B( J1,
     $                J1+N2 ),
     $                LDB )
         CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, WORK, M, ZERO,
     $               WORK( M*M+1 ), M )
         CALL DLACPY( 'Full', M, M, WORK( M*M+1 ), M, LI, LDST )
         CALL DGEMM( 'N', 'N', N2, N1, N1, ONE, A( J1, J1+N2 ), LDA,
     $               T( N2+1, N2+1 ), LDST, ZERO, WORK, N2 )
         CALL DLACPY( 'Full', N2, N1, WORK, N2, A( J1, J1+N2 ), LDA )
         CALL DGEMM( 'N', 'N', N2, N1, N1, ONE, B( J1, J1+N2 ), LDB,
     $               T( N2+1, N2+1 ), LDST, ZERO, WORK, N2 )
         CALL DLACPY( 'Full', N2, N1, WORK, N2, B( J1, J1+N2 ), LDB )
         CALL DGEMM( 'T', 'N', M, M, M, ONE, IR, LDST, T, LDST, ZERO,
     $               WORK, M )
         CALL DLACPY( 'Full', M, M, WORK, M, IR, LDST )
*
*        Accumulate transformations into Q and Z if requested.
*
         IF( WANTQ ) THEN
            CALL DGEMM( 'N', 'N', N, M, M, ONE, Q( 1, J1 ), LDQ, LI,
     $                  LDST, ZERO, WORK, N )
            CALL DLACPY( 'Full', N, M, WORK, N, Q( 1, J1 ), LDQ )
*
         END IF
*
         IF( WANTZ ) THEN
            CALL DGEMM( 'N', 'N', N, M, M, ONE, Z( 1, J1 ), LDZ, IR,
     $                  LDST, ZERO, WORK, N )
            CALL DLACPY( 'Full', N, M, WORK, N, Z( 1, J1 ), LDZ )
*
         END IF
*
*        Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and
*                (A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)).
*
         I = J1 + M
         IF( I.LE.N ) THEN
            CALL DGEMM( 'T', 'N', M, N-I+1, M, ONE, LI, LDST,
     $                  A( J1, I ), LDA, ZERO, WORK, M )
            CALL DLACPY( 'Full', M, N-I+1, WORK, M, A( J1, I ), LDA )
            CALL DGEMM( 'T', 'N', M, N-I+1, M, ONE, LI, LDST,
     $                  B( J1, I ), LDB, ZERO, WORK, M )
            CALL DLACPY( 'Full', M, N-I+1, WORK, M, B( J1, I ), LDB )
         END IF
         I = J1 - 1
         IF( I.GT.0 ) THEN
            CALL DGEMM( 'N', 'N', I, M, M, ONE, A( 1, J1 ), LDA, IR,
     $                  LDST, ZERO, WORK, I )
            CALL DLACPY( 'Full', I, M, WORK, I, A( 1, J1 ), LDA )
            CALL DGEMM( 'N', 'N', I, M, M, ONE, B( 1, J1 ), LDB, IR,
     $                  LDST, ZERO, WORK, I )
            CALL DLACPY( 'Full', I, M, WORK, I, B( 1, J1 ), LDB )
         END IF
*
*        Exit with INFO = 0 if swap was successfully performed.
*
         RETURN
*
      END IF
*
*     Exit with INFO = 1 if swap was rejected.
*
   70 CONTINUE
*
      INFO = 1
      RETURN
*
*     End of DTGEX2
*
      END
*> \brief \b DTGEXC
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTGEXC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgexc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgexc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgexc.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTGEXC( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
*                          LDZ, IFST, ILST, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       LOGICAL            WANTQ, WANTZ
*       INTEGER            IFST, ILST, INFO, LDA, LDB, LDQ, LDZ, LWORK, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
*      $                   WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTGEXC reorders the generalized real Schur decomposition of a real
*> matrix pair (A,B) using an orthogonal equivalence transformation
*>
*>                (A, B) = Q * (A, B) * Z**T,
*>
*> so that the diagonal block of (A, B) with row index IFST is moved
*> to row ILST.
*>
*> (A, B) must be in generalized real Schur canonical form (as returned
*> by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2
*> diagonal blocks. B is upper triangular.
*>
*> Optionally, the matrices Q and Z of generalized Schur vectors are
*> updated.
*>
*>        Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T
*>        Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] WANTQ
*> \verbatim
*>          WANTQ is LOGICAL
*>          .TRUE. : update the left transformation matrix Q;
*>          .FALSE.: do not update Q.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*>          WANTZ is LOGICAL
*>          .TRUE. : update the right transformation matrix Z;
*>          .FALSE.: do not update Z.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the matrix A in generalized real Schur canonical
*>          form.
*>          On exit, the updated matrix A, again in generalized
*>          real Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,N)
*>          On entry, the matrix B in generalized real Schur canonical
*>          form (A,B).
*>          On exit, the updated matrix B, again in generalized
*>          real Schur canonical form (A,B).
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ,N)
*>          On entry, if WANTQ = .TRUE., the orthogonal matrix Q.
*>          On exit, the updated matrix Q.
*>          If WANTQ = .FALSE., Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q. LDQ >= 1.
*>          If WANTQ = .TRUE., LDQ >= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ,N)
*>          On entry, if WANTZ = .TRUE., the orthogonal matrix Z.
*>          On exit, the updated matrix Z.
*>          If WANTZ = .FALSE., Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z. LDZ >= 1.
*>          If WANTZ = .TRUE., LDZ >= N.
*> \endverbatim
*>
*> \param[in,out] IFST
*> \verbatim
*>          IFST is INTEGER
*> \endverbatim
*>
*> \param[in,out] ILST
*> \verbatim
*>          ILST is INTEGER
*>          Specify the reordering of the diagonal blocks of (A, B).
*>          The block with row index IFST is moved to row ILST, by a
*>          sequence of swapping between adjacent blocks.
*>          On exit, if IFST pointed on entry to the second row of
*>          a 2-by-2 block, it is changed to point to the first row;
*>          ILST always points to the first row of the block in its
*>          final position (which may differ from its input value by
*>          +1 or -1). 1 <= IFST, ILST <= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          LWORK >= 1 when N <= 1, otherwise LWORK >= 4*N + 16.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>           =0:  successful exit.
*>           <0:  if INFO = -i, the i-th argument had an illegal value.
*>           =1:  The transformed matrix pair (A, B) would be too far
*>                from generalized Schur form; the problem is ill-
*>                conditioned. (A, B) may have been partially reordered,
*>                and ILST points to the first row of the current
*>                position of the block being moved.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tgexc
*
*> \par Contributors:
*  ==================
*>
*>     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*>     Umea University, S-901 87 Umea, Sweden.
*
*> \par References:
*  ================
*>
*> \verbatim
*>
*>  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
*>      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
*>      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
*>      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTGEXC( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
     $                   LDZ, IFST, ILST, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      LOGICAL            WANTQ, WANTZ
      INTEGER            IFST, ILST, INFO, LDA, LDB, LDQ, LDZ, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
     $                   WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            HERE, LWMIN, NBF, NBL, NBNEXT
*     ..
*     .. External Subroutines ..
      EXTERNAL           DTGEX2, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Decode and test input arguments.
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDQ.LT.1 .OR. WANTQ .AND. ( LDQ.LT.MAX( 1, N ) ) ) THEN
         INFO = -9
      ELSE IF( LDZ.LT.1 .OR. WANTZ .AND. ( LDZ.LT.MAX( 1, N ) ) ) THEN
         INFO = -11
      ELSE IF( IFST.LT.1 .OR. IFST.GT.N ) THEN
         INFO = -12
      ELSE IF( ILST.LT.1 .OR. ILST.GT.N ) THEN
         INFO = -13
      END IF
*
      IF( INFO.EQ.0 ) THEN
         IF( N.LE.1 ) THEN
            LWMIN = 1
         ELSE
            LWMIN = 4*N + 16
         END IF
         WORK(1) = LWMIN
*
         IF (LWORK.LT.LWMIN .AND. .NOT.LQUERY) THEN
            INFO = -15
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTGEXC', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.LE.1 )
     $   RETURN
*
*     Determine the first row of the specified block and find out
*     if it is 1-by-1 or 2-by-2.
*
      IF( IFST.GT.1 ) THEN
         IF( A( IFST, IFST-1 ).NE.ZERO )
     $      IFST = IFST - 1
      END IF
      NBF = 1
      IF( IFST.LT.N ) THEN
         IF( A( IFST+1, IFST ).NE.ZERO )
     $      NBF = 2
      END IF
*
*     Determine the first row of the final block
*     and find out if it is 1-by-1 or 2-by-2.
*
      IF( ILST.GT.1 ) THEN
         IF( A( ILST, ILST-1 ).NE.ZERO )
     $      ILST = ILST - 1
      END IF
      NBL = 1
      IF( ILST.LT.N ) THEN
         IF( A( ILST+1, ILST ).NE.ZERO )
     $      NBL = 2
      END IF
      IF( IFST.EQ.ILST )
     $   RETURN
*
      IF( IFST.LT.ILST ) THEN
*
*        Update ILST.
*
         IF( NBF.EQ.2 .AND. NBL.EQ.1 )
     $      ILST = ILST - 1
         IF( NBF.EQ.1 .AND. NBL.EQ.2 )
     $      ILST = ILST + 1
*
         HERE = IFST
*
   10    CONTINUE
*
*        Swap with next one below.
*
         IF( NBF.EQ.1 .OR. NBF.EQ.2 ) THEN
*
*           Current block either 1-by-1 or 2-by-2.
*
            NBNEXT = 1
            IF( HERE+NBF+1.LE.N ) THEN
               IF( A( HERE+NBF+1, HERE+NBF ).NE.ZERO )
     $            NBNEXT = 2
            END IF
            CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
     $                   LDZ, HERE, NBF, NBNEXT, WORK, LWORK, INFO )
            IF( INFO.NE.0 ) THEN
               ILST = HERE
               RETURN
            END IF
            HERE = HERE + NBNEXT
*
*           Test if 2-by-2 block breaks into two 1-by-1 blocks.
*
            IF( NBF.EQ.2 ) THEN
               IF( A( HERE+1, HERE ).EQ.ZERO )
     $            NBF = 3
            END IF
*
         ELSE
*
*           Current block consists of two 1-by-1 blocks, each of which
*           must be swapped individually.
*
            NBNEXT = 1
            IF( HERE+3.LE.N ) THEN
               IF( A( HERE+3, HERE+2 ).NE.ZERO )
     $            NBNEXT = 2
            END IF
            CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
     $                   LDZ, HERE+1, 1, NBNEXT, WORK, LWORK, INFO )
            IF( INFO.NE.0 ) THEN
               ILST = HERE
               RETURN
            END IF
            IF( NBNEXT.EQ.1 ) THEN
*
*              Swap two 1-by-1 blocks.
*
               CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ,
     $                      Z,
     $                      LDZ, HERE, 1, 1, WORK, LWORK, INFO )
               IF( INFO.NE.0 ) THEN
                  ILST = HERE
                  RETURN
               END IF
               HERE = HERE + 1
*
            ELSE
*
*              Recompute NBNEXT in case of 2-by-2 split.
*
               IF( A( HERE+2, HERE+1 ).EQ.ZERO )
     $            NBNEXT = 1
               IF( NBNEXT.EQ.2 ) THEN
*
*                 2-by-2 block did not split.
*
                  CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q,
     $                         LDQ,
     $                         Z, LDZ, HERE, 1, NBNEXT, WORK, LWORK,
     $                         INFO )
                  IF( INFO.NE.0 ) THEN
                     ILST = HERE
                     RETURN
                  END IF
                  HERE = HERE + 2
               ELSE
*
*                 2-by-2 block did split.
*
                  CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q,
     $                         LDQ,
     $                         Z, LDZ, HERE, 1, 1, WORK, LWORK, INFO )
                  IF( INFO.NE.0 ) THEN
                     ILST = HERE
                     RETURN
                  END IF
                  HERE = HERE + 1
                  CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q,
     $                         LDQ,
     $                         Z, LDZ, HERE, 1, 1, WORK, LWORK, INFO )
                  IF( INFO.NE.0 ) THEN
                     ILST = HERE
                     RETURN
                  END IF
                  HERE = HERE + 1
               END IF
*
            END IF
         END IF
         IF( HERE.LT.ILST )
     $      GO TO 10
      ELSE
         HERE = IFST
*
   20    CONTINUE
*
*        Swap with next one below.
*
         IF( NBF.EQ.1 .OR. NBF.EQ.2 ) THEN
*
*           Current block either 1-by-1 or 2-by-2.
*
            NBNEXT = 1
            IF( HERE.GE.3 ) THEN
               IF( A( HERE-1, HERE-2 ).NE.ZERO )
     $            NBNEXT = 2
            END IF
            CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
     $                   LDZ, HERE-NBNEXT, NBNEXT, NBF, WORK, LWORK,
     $                   INFO )
            IF( INFO.NE.0 ) THEN
               ILST = HERE
               RETURN
            END IF
            HERE = HERE - NBNEXT
*
*           Test if 2-by-2 block breaks into two 1-by-1 blocks.
*
            IF( NBF.EQ.2 ) THEN
               IF( A( HERE+1, HERE ).EQ.ZERO )
     $            NBF = 3
            END IF
*
         ELSE
*
*           Current block consists of two 1-by-1 blocks, each of which
*           must be swapped individually.
*
            NBNEXT = 1
            IF( HERE.GE.3 ) THEN
               IF( A( HERE-1, HERE-2 ).NE.ZERO )
     $            NBNEXT = 2
            END IF
            CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
     $                   LDZ, HERE-NBNEXT, NBNEXT, 1, WORK, LWORK,
     $                   INFO )
            IF( INFO.NE.0 ) THEN
               ILST = HERE
               RETURN
            END IF
            IF( NBNEXT.EQ.1 ) THEN
*
*              Swap two 1-by-1 blocks.
*
               CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ,
     $                      Z,
     $                      LDZ, HERE, NBNEXT, 1, WORK, LWORK, INFO )
               IF( INFO.NE.0 ) THEN
                  ILST = HERE
                  RETURN
               END IF
               HERE = HERE - 1
            ELSE
*
*             Recompute NBNEXT in case of 2-by-2 split.
*
               IF( A( HERE, HERE-1 ).EQ.ZERO )
     $            NBNEXT = 1
               IF( NBNEXT.EQ.2 ) THEN
*
*                 2-by-2 block did not split.
*
                  CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q,
     $                         LDQ,
     $                         Z, LDZ, HERE-1, 2, 1, WORK, LWORK, INFO )
                  IF( INFO.NE.0 ) THEN
                     ILST = HERE
                     RETURN
                  END IF
                  HERE = HERE - 2
               ELSE
*
*                 2-by-2 block did split.
*
                  CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q,
     $                         LDQ,
     $                         Z, LDZ, HERE, 1, 1, WORK, LWORK, INFO )
                  IF( INFO.NE.0 ) THEN
                     ILST = HERE
                     RETURN
                  END IF
                  HERE = HERE - 1
                  CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q,
     $                         LDQ,
     $                         Z, LDZ, HERE, 1, 1, WORK, LWORK, INFO )
                  IF( INFO.NE.0 ) THEN
                     ILST = HERE
                     RETURN
                  END IF
                  HERE = HERE - 1
               END IF
            END IF
         END IF
         IF( HERE.GT.ILST )
     $      GO TO 20
      END IF
      ILST = HERE
      WORK( 1 ) = LWMIN
      RETURN
*
*     End of DTGEXC
*
      END
*> \brief \b DTGSEN
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTGSEN + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgsen.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgsen.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgsen.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB,
*                          ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, M, PL,
*                          PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO )
*
*       .. Scalar Arguments ..
*       LOGICAL            WANTQ, WANTZ
*       INTEGER            IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK,
*      $                   M, N
*       DOUBLE PRECISION   PL, PR
*       ..
*       .. Array Arguments ..
*       LOGICAL            SELECT( * )
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
*      $                   B( LDB, * ), BETA( * ), DIF( * ), Q( LDQ, * ),
*      $                   WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTGSEN reorders the generalized real Schur decomposition of a real
*> matrix pair (A, B) (in terms of an orthonormal equivalence trans-
*> formation Q**T * (A, B) * Z), so that a selected cluster of eigenvalues
*> appears in the leading diagonal blocks of the upper quasi-triangular
*> matrix A and the upper triangular B. The leading columns of Q and
*> Z form orthonormal bases of the corresponding left and right eigen-
*> spaces (deflating subspaces). (A, B) must be in generalized real
*> Schur canonical form (as returned by DGGES), i.e. A is block upper
*> triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper
*> triangular.
*>
*> DTGSEN also computes the generalized eigenvalues
*>
*>             w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)
*>
*> of the reordered matrix pair (A, B).
*>
*> Optionally, DTGSEN computes the estimates of reciprocal condition
*> numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
*> (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
*> between the matrix pairs (A11, B11) and (A22,B22) that correspond to
*> the selected cluster and the eigenvalues outside the cluster, resp.,
*> and norms of "projections" onto left and right eigenspaces w.r.t.
*> the selected cluster in the (1,1)-block.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] IJOB
*> \verbatim
*>          IJOB is INTEGER
*>          Specifies whether condition numbers are required for the
*>          cluster of eigenvalues (PL and PR) or the deflating subspaces
*>          (Difu and Difl):
*>           =0: Only reorder w.r.t. SELECT. No extras.
*>           =1: Reciprocal of norms of "projections" onto left and right
*>               eigenspaces w.r.t. the selected cluster (PL and PR).
*>           =2: Upper bounds on Difu and Difl. F-norm-based estimate
*>               (DIF(1:2)).
*>           =3: Estimate of Difu and Difl. 1-norm-based estimate
*>               (DIF(1:2)).
*>               About 5 times as expensive as IJOB = 2.
*>           =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
*>               version to get it all.
*>           =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)
*> \endverbatim
*>
*> \param[in] WANTQ
*> \verbatim
*>          WANTQ is LOGICAL
*>          .TRUE. : update the left transformation matrix Q;
*>          .FALSE.: do not update Q.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*>          WANTZ is LOGICAL
*>          .TRUE. : update the right transformation matrix Z;
*>          .FALSE.: do not update Z.
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*>          SELECT is LOGICAL array, dimension (N)
*>          SELECT specifies the eigenvalues in the selected cluster.
*>          To select a real eigenvalue w(j), SELECT(j) must be set to
*>          .TRUE.. To select a complex conjugate pair of eigenvalues
*>          w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
*>          either SELECT(j) or SELECT(j+1) or both must be set to
*>          .TRUE.; a complex conjugate pair of eigenvalues must be
*>          either both included in the cluster or both excluded.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension(LDA,N)
*>          On entry, the upper quasi-triangular matrix A, with (A, B) in
*>          generalized real Schur canonical form.
*>          On exit, A is overwritten by the reordered matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension(LDB,N)
*>          On entry, the upper triangular matrix B, with (A, B) in
*>          generalized real Schur canonical form.
*>          On exit, B is overwritten by the reordered matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*>          ALPHAR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*>          ALPHAI is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION array, dimension (N)
*>
*>          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
*>          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i
*>          and BETA(j),j=1,...,N  are the diagonals of the complex Schur
*>          form (S,T) that would result if the 2-by-2 diagonal blocks of
*>          the real generalized Schur form of (A,B) were further reduced
*>          to triangular form using complex unitary transformations.
*>          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
*>          positive, then the j-th and (j+1)-st eigenvalues are a
*>          complex conjugate pair, with ALPHAI(j+1) negative.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ,N)
*>          On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.
*>          On exit, Q has been postmultiplied by the left orthogonal
*>          transformation matrix which reorder (A, B); The leading M
*>          columns of Q form orthonormal bases for the specified pair of
*>          left eigenspaces (deflating subspaces).
*>          If WANTQ = .FALSE., Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q.  LDQ >= 1;
*>          and if WANTQ = .TRUE., LDQ >= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ,N)
*>          On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.
*>          On exit, Z has been postmultiplied by the left orthogonal
*>          transformation matrix which reorder (A, B); The leading M
*>          columns of Z form orthonormal bases for the specified pair of
*>          left eigenspaces (deflating subspaces).
*>          If WANTZ = .FALSE., Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z. LDZ >= 1;
*>          If WANTZ = .TRUE., LDZ >= N.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The dimension of the specified pair of left and right eigen-
*>          spaces (deflating subspaces). 0 <= M <= N.
*> \endverbatim
*>
*> \param[out] PL
*> \verbatim
*>          PL is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] PR
*> \verbatim
*>          PR is DOUBLE PRECISION
*>
*>          If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
*>          reciprocal of the norm of "projections" onto left and right
*>          eigenspaces with respect to the selected cluster.
*>          0 < PL, PR <= 1.
*>          If M = 0 or M = N, PL = PR  = 1.
*>          If IJOB = 0, 2 or 3, PL and PR are not referenced.
*> \endverbatim
*>
*> \param[out] DIF
*> \verbatim
*>          DIF is DOUBLE PRECISION array, dimension (2).
*>          If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
*>          If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
*>          Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
*>          estimates of Difu and Difl.
*>          If M = 0 or N, DIF(1:2) = F-norm([A, B]).
*>          If IJOB = 0 or 1, DIF is not referenced.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >=  4*N+16.
*>          If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)).
*>          If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(N-M)).
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK. LIWORK >= 1.
*>          If IJOB = 1, 2 or 4, LIWORK >=  N+6.
*>          If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6).
*>
*>          If LIWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the optimal size of the IWORK array,
*>          returns this value as the first entry of the IWORK array, and
*>          no error message related to LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>            =0: Successful exit.
*>            <0: If INFO = -i, the i-th argument had an illegal value.
*>            =1: Reordering of (A, B) failed because the transformed
*>                matrix pair (A, B) would be too far from generalized
*>                Schur form; the problem is very ill-conditioned.
*>                (A, B) may have been partially reordered.
*>                If requested, 0 is returned in DIF(*), PL and PR.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tgsen
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  DTGSEN first collects the selected eigenvalues by computing
*>  orthogonal U and W that move them to the top left corner of (A, B).
*>  In other words, the selected eigenvalues are the eigenvalues of
*>  (A11, B11) in:
*>
*>              U**T*(A, B)*W = (A11 A12) (B11 B12) n1
*>                              ( 0  A22),( 0  B22) n2
*>                                n1  n2    n1  n2
*>
*>  where N = n1+n2 and U**T means the transpose of U. The first n1 columns
*>  of U and W span the specified pair of left and right eigenspaces
*>  (deflating subspaces) of (A, B).
*>
*>  If (A, B) has been obtained from the generalized real Schur
*>  decomposition of a matrix pair (C, D) = Q*(A, B)*Z**T, then the
*>  reordered generalized real Schur form of (C, D) is given by
*>
*>           (C, D) = (Q*U)*(U**T*(A, B)*W)*(Z*W)**T,
*>
*>  and the first n1 columns of Q*U and Z*W span the corresponding
*>  deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).
*>
*>  Note that if the selected eigenvalue is sufficiently ill-conditioned,
*>  then its value may differ significantly from its value before
*>  reordering.
*>
*>  The reciprocal condition numbers of the left and right eigenspaces
*>  spanned by the first n1 columns of U and W (or Q*U and Z*W) may
*>  be returned in DIF(1:2), corresponding to Difu and Difl, resp.
*>
*>  The Difu and Difl are defined as:
*>
*>       Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
*>  and
*>       Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],
*>
*>  where sigma-min(Zu) is the smallest singular value of the
*>  (2*n1*n2)-by-(2*n1*n2) matrix
*>
*>       Zu = [ kron(In2, A11)  -kron(A22**T, In1) ]
*>            [ kron(In2, B11)  -kron(B22**T, In1) ].
*>
*>  Here, Inx is the identity matrix of size nx and A22**T is the
*>  transpose of A22. kron(X, Y) is the Kronecker product between
*>  the matrices X and Y.
*>
*>  When DIF(2) is small, small changes in (A, B) can cause large changes
*>  in the deflating subspace. An approximate (asymptotic) bound on the
*>  maximum angular error in the computed deflating subspaces is
*>
*>       EPS * norm((A, B)) / DIF(2),
*>
*>  where EPS is the machine precision.
*>
*>  The reciprocal norm of the projectors on the left and right
*>  eigenspaces associated with (A11, B11) may be returned in PL and PR.
*>  They are computed as follows. First we compute L and R so that
*>  P*(A, B)*Q is block diagonal, where
*>
*>       P = ( I -L ) n1           Q = ( I R ) n1
*>           ( 0  I ) n2    and        ( 0 I ) n2
*>             n1 n2                    n1 n2
*>
*>  and (L, R) is the solution to the generalized Sylvester equation
*>
*>       A11*R - L*A22 = -A12
*>       B11*R - L*B22 = -B12
*>
*>  Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
*>  An approximate (asymptotic) bound on the average absolute error of
*>  the selected eigenvalues is
*>
*>       EPS * norm((A, B)) / PL.
*>
*>  There are also global error bounds which valid for perturbations up
*>  to a certain restriction:  A lower bound (x) on the smallest
*>  F-norm(E,F) for which an eigenvalue of (A11, B11) may move and
*>  coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),
*>  (i.e. (A + E, B + F), is
*>
*>   x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).
*>
*>  An approximate bound on x can be computed from DIF(1:2), PL and PR.
*>
*>  If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed
*>  (L', R') and unperturbed (L, R) left and right deflating subspaces
*>  associated with the selected cluster in the (1,1)-blocks can be
*>  bounded as
*>
*>   max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
*>   max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))
*>
*>  See LAPACK User's Guide section 4.11 or the following references
*>  for more information.
*>
*>  Note that if the default method for computing the Frobenius-norm-
*>  based estimate DIF is not wanted (see DLATDF), then the parameter
*>  IDIFJB (see below) should be changed from 3 to 4 (routine DLATDF
*>  (IJOB = 2 will be used)). See DTGSYL for more details.
*> \endverbatim
*
*> \par Contributors:
*  ==================
*>
*>     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*>     Umea University, S-901 87 Umea, Sweden.
*
*> \par References:
*  ================
*>
*> \verbatim
*>
*>  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
*>      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
*>      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
*>      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
*>
*>  [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
*>      Eigenvalues of a Regular Matrix Pair (A, B) and Condition
*>      Estimation: Theory, Algorithms and Software,
*>      Report UMINF - 94.04, Department of Computing Science, Umea
*>      University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
*>      Note 87. To appear in Numerical Algorithms, 1996.
*>
*>  [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
*>      for Solving the Generalized Sylvester Equation and Estimating the
*>      Separation between Regular Matrix Pairs, Report UMINF - 93.23,
*>      Department of Computing Science, Umea University, S-901 87 Umea,
*>      Sweden, December 1993, Revised April 1994, Also as LAPACK Working
*>      Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1,
*>      1996.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B,
     $                   LDB,
     $                   ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, M, PL,
     $                   PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      LOGICAL            WANTQ, WANTZ
      INTEGER            IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK,
     $                   M, N
      DOUBLE PRECISION   PL, PR
*     ..
*     .. Array Arguments ..
      LOGICAL            SELECT( * )
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
     $                   B( LDB, * ), BETA( * ), DIF( * ), Q( LDQ, * ),
     $                   WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            IDIFJB
      PARAMETER          ( IDIFJB = 3 )
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, PAIR, SWAP, WANTD, WANTD1, WANTD2,
     $                   WANTP
      INTEGER            I, IERR, IJB, K, KASE, KK, KS, LIWMIN, LWMIN,
     $                   MN2, N1, N2
      DOUBLE PRECISION   DSCALE, DSUM, EPS, RDSCAL, SMLNUM
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACN2, DLACPY, DLAG2, DLASSQ, DTGEXC,
     $                   DTGSYL,
     $                   XERBLA
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, SIGN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Decode and test the input parameters
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
      IF( IJOB.LT.0 .OR. IJOB.GT.5 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -9
      ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
         INFO = -14
      ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
         INFO = -16
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTGSEN', -INFO )
         RETURN
      END IF
*
*     Get machine constants
*
      EPS = DLAMCH( 'P' )
      SMLNUM = DLAMCH( 'S' ) / EPS
      IERR = 0
*
      WANTP = IJOB.EQ.1 .OR. IJOB.GE.4
      WANTD1 = IJOB.EQ.2 .OR. IJOB.EQ.4
      WANTD2 = IJOB.EQ.3 .OR. IJOB.EQ.5
      WANTD = WANTD1 .OR. WANTD2
*
*     Set M to the dimension of the specified pair of deflating
*     subspaces.
*
      M = 0
      PAIR = .FALSE.
      IF( .NOT.LQUERY .OR. IJOB.NE.0 ) THEN
      DO 10 K = 1, N
         IF( PAIR ) THEN
            PAIR = .FALSE.
         ELSE
            IF( K.LT.N ) THEN
               IF( A( K+1, K ).EQ.ZERO ) THEN
                  IF( SELECT( K ) )
     $               M = M + 1
               ELSE
                  PAIR = .TRUE.
                  IF( SELECT( K ) .OR. SELECT( K+1 ) )
     $               M = M + 2
               END IF
            ELSE
               IF( SELECT( N ) )
     $            M = M + 1
            END IF
         END IF
   10 CONTINUE
      END IF
*
      IF( IJOB.EQ.1 .OR. IJOB.EQ.2 .OR. IJOB.EQ.4 ) THEN
         LWMIN = MAX( 1, 4*N+16, 2*M*( N-M ) )
         LIWMIN = MAX( 1, N+6 )
      ELSE IF( IJOB.EQ.3 .OR. IJOB.EQ.5 ) THEN
         LWMIN = MAX( 1, 4*N+16, 4*M*( N-M ) )
         LIWMIN = MAX( 1, 2*M*( N-M ), N+6 )
      ELSE
         LWMIN = MAX( 1, 4*N+16 )
         LIWMIN = 1
      END IF
*
      WORK( 1 ) = LWMIN
      IWORK( 1 ) = LIWMIN
*
      IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
         INFO = -22
      ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
         INFO = -24
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTGSEN', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( M.EQ.N .OR. M.EQ.0 ) THEN
         IF( WANTP ) THEN
            PL = ONE
            PR = ONE
         END IF
         IF( WANTD ) THEN
            DSCALE = ZERO
            DSUM = ONE
            DO 20 I = 1, N
               CALL DLASSQ( N, A( 1, I ), 1, DSCALE, DSUM )
               CALL DLASSQ( N, B( 1, I ), 1, DSCALE, DSUM )
   20       CONTINUE
            DIF( 1 ) = DSCALE*SQRT( DSUM )
            DIF( 2 ) = DIF( 1 )
         END IF
         GO TO 60
      END IF
*
*     Collect the selected blocks at the top-left corner of (A, B).
*
      KS = 0
      PAIR = .FALSE.
      DO 30 K = 1, N
         IF( PAIR ) THEN
            PAIR = .FALSE.
         ELSE
*
            SWAP = SELECT( K )
            IF( K.LT.N ) THEN
               IF( A( K+1, K ).NE.ZERO ) THEN
                  PAIR = .TRUE.
                  SWAP = SWAP .OR. SELECT( K+1 )
               END IF
            END IF
*
            IF( SWAP ) THEN
               KS = KS + 1
*
*              Swap the K-th block to position KS.
*              Perform the reordering of diagonal blocks in (A, B)
*              by orthogonal transformation matrices and update
*              Q and Z accordingly (if requested):
*
               KK = K
               IF( K.NE.KS )
     $            CALL DTGEXC( WANTQ, WANTZ, N, A, LDA, B, LDB, Q,
     $                         LDQ,
     $                         Z, LDZ, KK, KS, WORK, LWORK, IERR )
*
               IF( IERR.GT.0 ) THEN
*
*                 Swap is rejected: exit.
*
                  INFO = 1
                  IF( WANTP ) THEN
                     PL = ZERO
                     PR = ZERO
                  END IF
                  IF( WANTD ) THEN
                     DIF( 1 ) = ZERO
                     DIF( 2 ) = ZERO
                  END IF
                  GO TO 60
               END IF
*
               IF( PAIR )
     $            KS = KS + 1
            END IF
         END IF
   30 CONTINUE
      IF( WANTP ) THEN
*
*        Solve generalized Sylvester equation for R and L
*        and compute PL and PR.
*
         N1 = M
         N2 = N - M
         I = N1 + 1
         IJB = 0
         CALL DLACPY( 'Full', N1, N2, A( 1, I ), LDA, WORK, N1 )
         CALL DLACPY( 'Full', N1, N2, B( 1, I ), LDB,
     $                WORK( N1*N2+1 ),
     $                N1 )
         CALL DTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA, WORK,
     $                N1, B, LDB, B( I, I ), LDB, WORK( N1*N2+1 ), N1,
     $                DSCALE, DIF( 1 ), WORK( N1*N2*2+1 ),
     $                LWORK-2*N1*N2, IWORK, IERR )
*
*        Estimate the reciprocal of norms of "projections" onto left
*        and right eigenspaces.
*
         RDSCAL = ZERO
         DSUM = ONE
         CALL DLASSQ( N1*N2, WORK, 1, RDSCAL, DSUM )
         PL = RDSCAL*SQRT( DSUM )
         IF( PL.EQ.ZERO ) THEN
            PL = ONE
         ELSE
            PL = DSCALE / ( SQRT( DSCALE*DSCALE / PL+PL )*SQRT( PL ) )
         END IF
         RDSCAL = ZERO
         DSUM = ONE
         CALL DLASSQ( N1*N2, WORK( N1*N2+1 ), 1, RDSCAL, DSUM )
         PR = RDSCAL*SQRT( DSUM )
         IF( PR.EQ.ZERO ) THEN
            PR = ONE
         ELSE
            PR = DSCALE / ( SQRT( DSCALE*DSCALE / PR+PR )*SQRT( PR ) )
         END IF
      END IF
*
      IF( WANTD ) THEN
*
*        Compute estimates of Difu and Difl.
*
         IF( WANTD1 ) THEN
            N1 = M
            N2 = N - M
            I = N1 + 1
            IJB = IDIFJB
*
*           Frobenius norm-based Difu-estimate.
*
            CALL DTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA,
     $                   WORK,
     $                   N1, B, LDB, B( I, I ), LDB, WORK( N1*N2+1 ),
     $                   N1, DSCALE, DIF( 1 ), WORK( 2*N1*N2+1 ),
     $                   LWORK-2*N1*N2, IWORK, IERR )
*
*           Frobenius norm-based Difl-estimate.
*
            CALL DTGSYL( 'N', IJB, N2, N1, A( I, I ), LDA, A, LDA,
     $                   WORK,
     $                   N2, B( I, I ), LDB, B, LDB, WORK( N1*N2+1 ),
     $                   N2, DSCALE, DIF( 2 ), WORK( 2*N1*N2+1 ),
     $                   LWORK-2*N1*N2, IWORK, IERR )
         ELSE
*
*
*           Compute 1-norm-based estimates of Difu and Difl using
*           reversed communication with DLACN2. In each step a
*           generalized Sylvester equation or a transposed variant
*           is solved.
*
            KASE = 0
            N1 = M
            N2 = N - M
            I = N1 + 1
            IJB = 0
            MN2 = 2*N1*N2
*
*           1-norm-based estimate of Difu.
*
   40       CONTINUE
            CALL DLACN2( MN2, WORK( MN2+1 ), WORK, IWORK, DIF( 1 ),
     $                   KASE, ISAVE )
            IF( KASE.NE.0 ) THEN
               IF( KASE.EQ.1 ) THEN
*
*                 Solve generalized Sylvester equation.
*
                  CALL DTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ),
     $                         LDA,
     $                         WORK, N1, B, LDB, B( I, I ), LDB,
     $                         WORK( N1*N2+1 ), N1, DSCALE, DIF( 1 ),
     $                         WORK( 2*N1*N2+1 ), LWORK-2*N1*N2, IWORK,
     $                         IERR )
               ELSE
*
*                 Solve the transposed variant.
*
                  CALL DTGSYL( 'T', IJB, N1, N2, A, LDA, A( I, I ),
     $                         LDA,
     $                         WORK, N1, B, LDB, B( I, I ), LDB,
     $                         WORK( N1*N2+1 ), N1, DSCALE, DIF( 1 ),
     $                         WORK( 2*N1*N2+1 ), LWORK-2*N1*N2, IWORK,
     $                         IERR )
               END IF
               GO TO 40
            END IF
            DIF( 1 ) = DSCALE / DIF( 1 )
*
*           1-norm-based estimate of Difl.
*
   50       CONTINUE
            CALL DLACN2( MN2, WORK( MN2+1 ), WORK, IWORK, DIF( 2 ),
     $                   KASE, ISAVE )
            IF( KASE.NE.0 ) THEN
               IF( KASE.EQ.1 ) THEN
*
*                 Solve generalized Sylvester equation.
*
                  CALL DTGSYL( 'N', IJB, N2, N1, A( I, I ), LDA, A,
     $                         LDA,
     $                         WORK, N2, B( I, I ), LDB, B, LDB,
     $                         WORK( N1*N2+1 ), N2, DSCALE, DIF( 2 ),
     $                         WORK( 2*N1*N2+1 ), LWORK-2*N1*N2, IWORK,
     $                         IERR )
               ELSE
*
*                 Solve the transposed variant.
*
                  CALL DTGSYL( 'T', IJB, N2, N1, A( I, I ), LDA, A,
     $                         LDA,
     $                         WORK, N2, B( I, I ), LDB, B, LDB,
     $                         WORK( N1*N2+1 ), N2, DSCALE, DIF( 2 ),
     $                         WORK( 2*N1*N2+1 ), LWORK-2*N1*N2, IWORK,
     $                         IERR )
               END IF
               GO TO 50
            END IF
            DIF( 2 ) = DSCALE / DIF( 2 )
*
         END IF
      END IF
*
   60 CONTINUE
*
*     Compute generalized eigenvalues of reordered pair (A, B) and
*     normalize the generalized Schur form.
*
      PAIR = .FALSE.
      DO 80 K = 1, N
         IF( PAIR ) THEN
            PAIR = .FALSE.
         ELSE
*
            IF( K.LT.N ) THEN
               IF( A( K+1, K ).NE.ZERO ) THEN
                  PAIR = .TRUE.
               END IF
            END IF
*
            IF( PAIR ) THEN
*
*             Compute the eigenvalue(s) at position K.
*
               WORK( 1 ) = A( K, K )
               WORK( 2 ) = A( K+1, K )
               WORK( 3 ) = A( K, K+1 )
               WORK( 4 ) = A( K+1, K+1 )
               WORK( 5 ) = B( K, K )
               WORK( 6 ) = B( K+1, K )
               WORK( 7 ) = B( K, K+1 )
               WORK( 8 ) = B( K+1, K+1 )
               CALL DLAG2( WORK, 2, WORK( 5 ), 2, SMLNUM*EPS,
     $                     BETA( K ),
     $                     BETA( K+1 ), ALPHAR( K ), ALPHAR( K+1 ),
     $                     ALPHAI( K ) )
               ALPHAI( K+1 ) = -ALPHAI( K )
*
            ELSE
*
               IF( SIGN( ONE, B( K, K ) ).LT.ZERO ) THEN
*
*                 If B(K,K) is negative, make it positive
*
                  DO 70 I = 1, N
                     A( K, I ) = -A( K, I )
                     B( K, I ) = -B( K, I )
                     IF( WANTQ ) Q( I, K ) = -Q( I, K )
   70             CONTINUE
               END IF
*
               ALPHAR( K ) = A( K, K )
               ALPHAI( K ) = ZERO
               BETA( K ) = B( K, K )
*
            END IF
         END IF
   80 CONTINUE
*
      WORK( 1 ) = LWMIN
      IWORK( 1 ) = LIWMIN
*
      RETURN
*
*     End of DTGSEN
*
      END
*> \brief \b DTGSJA
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTGSJA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgsja.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgsja.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgsja.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B,
*                          LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV,
*                          Q, LDQ, WORK, NCYCLE, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBQ, JOBU, JOBV
*       INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
*      $                   NCYCLE, P
*       DOUBLE PRECISION   TOLA, TOLB
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), ALPHA( * ), B( LDB, * ),
*      $                   BETA( * ), Q( LDQ, * ), U( LDU, * ),
*      $                   V( LDV, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTGSJA computes the generalized singular value decomposition (GSVD)
*> of two real upper triangular (or trapezoidal) matrices A and B.
*>
*> On entry, it is assumed that matrices A and B have the following
*> forms, which may be obtained by the preprocessing subroutine DGGSVP
*> from a general M-by-N matrix A and P-by-N matrix B:
*>
*>              N-K-L  K    L
*>    A =    K ( 0    A12  A13 ) if M-K-L >= 0;
*>           L ( 0     0   A23 )
*>       M-K-L ( 0     0    0  )
*>
*>            N-K-L  K    L
*>    A =  K ( 0    A12  A13 ) if M-K-L < 0;
*>       M-K ( 0     0   A23 )
*>
*>            N-K-L  K    L
*>    B =  L ( 0     0   B13 )
*>       P-L ( 0     0    0  )
*>
*> where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
*> upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
*> otherwise A23 is (M-K)-by-L upper trapezoidal.
*>
*> On exit,
*>
*>        U**T *A*Q = D1*( 0 R ),    V**T *B*Q = D2*( 0 R ),
*>
*> where U, V and Q are orthogonal matrices.
*> R is a nonsingular upper triangular matrix, and D1 and D2 are
*> ``diagonal'' matrices, which are of the following structures:
*>
*> If M-K-L >= 0,
*>
*>                     K  L
*>        D1 =     K ( I  0 )
*>                 L ( 0  C )
*>             M-K-L ( 0  0 )
*>
*>                   K  L
*>        D2 = L   ( 0  S )
*>             P-L ( 0  0 )
*>
*>                N-K-L  K    L
*>   ( 0 R ) = K (  0   R11  R12 ) K
*>             L (  0    0   R22 ) L
*>
*> where
*>
*>   C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
*>   S = diag( BETA(K+1),  ... , BETA(K+L) ),
*>   C**2 + S**2 = I.
*>
*>   R is stored in A(1:K+L,N-K-L+1:N) on exit.
*>
*> If M-K-L < 0,
*>
*>                K M-K K+L-M
*>     D1 =   K ( I  0    0   )
*>          M-K ( 0  C    0   )
*>
*>                  K M-K K+L-M
*>     D2 =   M-K ( 0  S    0   )
*>          K+L-M ( 0  0    I   )
*>            P-L ( 0  0    0   )
*>
*>                N-K-L  K   M-K  K+L-M
*> ( 0 R ) =    K ( 0    R11  R12  R13  )
*>           M-K ( 0     0   R22  R23  )
*>         K+L-M ( 0     0    0   R33  )
*>
*> where
*> C = diag( ALPHA(K+1), ... , ALPHA(M) ),
*> S = diag( BETA(K+1),  ... , BETA(M) ),
*> C**2 + S**2 = I.
*>
*> R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
*>     (  0  R22 R23 )
*> in B(M-K+1:L,N+M-K-L+1:N) on exit.
*>
*> The computation of the orthogonal transformation matrices U, V or Q
*> is optional.  These matrices may either be formed explicitly, or they
*> may be postmultiplied into input matrices U1, V1, or Q1.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBU
*> \verbatim
*>          JOBU is CHARACTER*1
*>          = 'U':  U must contain an orthogonal matrix U1 on entry, and
*>                  the product U1*U is returned;
*>          = 'I':  U is initialized to the unit matrix, and the
*>                  orthogonal matrix U is returned;
*>          = 'N':  U is not computed.
*> \endverbatim
*>
*> \param[in] JOBV
*> \verbatim
*>          JOBV is CHARACTER*1
*>          = 'V':  V must contain an orthogonal matrix V1 on entry, and
*>                  the product V1*V is returned;
*>          = 'I':  V is initialized to the unit matrix, and the
*>                  orthogonal matrix V is returned;
*>          = 'N':  V is not computed.
*> \endverbatim
*>
*> \param[in] JOBQ
*> \verbatim
*>          JOBQ is CHARACTER*1
*>          = 'Q':  Q must contain an orthogonal matrix Q1 on entry, and
*>                  the product Q1*Q is returned;
*>          = 'I':  Q is initialized to the unit matrix, and the
*>                  orthogonal matrix Q is returned;
*>          = 'N':  Q is not computed.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*>          P is INTEGER
*>          The number of rows of the matrix B.  P >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrices A and B.  N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*>          L is INTEGER
*>
*>          K and L specify the subblocks in the input matrices A and B:
*>          A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
*>          of A and B, whose GSVD is going to be computed by DTGSJA.
*>          See Further Details.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
*>          matrix R or part of R.  See Purpose for details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,N)
*>          On entry, the P-by-N matrix B.
*>          On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
*>          a part of R.  See Purpose for details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,P).
*> \endverbatim
*>
*> \param[in] TOLA
*> \verbatim
*>          TOLA is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] TOLB
*> \verbatim
*>          TOLB is DOUBLE PRECISION
*>
*>          TOLA and TOLB are the convergence criteria for the Jacobi-
*>          Kogbetliantz iteration procedure. Generally, they are the
*>          same as used in the preprocessing step, say
*>              TOLA = max(M,N)*norm(A)*MAZHEPS,
*>              TOLB = max(P,N)*norm(B)*MAZHEPS.
*> \endverbatim
*>
*> \param[out] ALPHA
*> \verbatim
*>          ALPHA is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION array, dimension (N)
*>
*>          On exit, ALPHA and BETA contain the generalized singular
*>          value pairs of A and B;
*>            ALPHA(1:K) = 1,
*>            BETA(1:K)  = 0,
*>          and if M-K-L >= 0,
*>            ALPHA(K+1:K+L) = diag(C),
*>            BETA(K+1:K+L)  = diag(S),
*>          or if M-K-L < 0,
*>            ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
*>            BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
*>          Furthermore, if K+L < N,
*>            ALPHA(K+L+1:N) = 0 and
*>            BETA(K+L+1:N)  = 0.
*> \endverbatim
*>
*> \param[in,out] U
*> \verbatim
*>          U is DOUBLE PRECISION array, dimension (LDU,M)
*>          On entry, if JOBU = 'U', U must contain a matrix U1 (usually
*>          the orthogonal matrix returned by DGGSVP).
*>          On exit,
*>          if JOBU = 'I', U contains the orthogonal matrix U;
*>          if JOBU = 'U', U contains the product U1*U.
*>          If JOBU = 'N', U is not referenced.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>          The leading dimension of the array U. LDU >= max(1,M) if
*>          JOBU = 'U'; LDU >= 1 otherwise.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension (LDV,P)
*>          On entry, if JOBV = 'V', V must contain a matrix V1 (usually
*>          the orthogonal matrix returned by DGGSVP).
*>          On exit,
*>          if JOBV = 'I', V contains the orthogonal matrix V;
*>          if JOBV = 'V', V contains the product V1*V.
*>          If JOBV = 'N', V is not referenced.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of the array V. LDV >= max(1,P) if
*>          JOBV = 'V'; LDV >= 1 otherwise.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ,N)
*>          On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
*>          the orthogonal matrix returned by DGGSVP).
*>          On exit,
*>          if JOBQ = 'I', Q contains the orthogonal matrix Q;
*>          if JOBQ = 'Q', Q contains the product Q1*Q.
*>          If JOBQ = 'N', Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q. LDQ >= max(1,N) if
*>          JOBQ = 'Q'; LDQ >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] NCYCLE
*> \verbatim
*>          NCYCLE is INTEGER
*>          The number of cycles required for convergence.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          = 1:  the procedure does not converge after MAXIT cycles.
*> \endverbatim
*>
*> \verbatim
*>  Internal Parameters
*>  ===================
*>
*>  MAXIT   INTEGER
*>          MAXIT specifies the total loops that the iterative procedure
*>          may take. If after MAXIT cycles, the routine fails to
*>          converge, we return INFO = 1.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tgsja
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
*>  min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
*>  matrix B13 to the form:
*>
*>           U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,
*>
*>  where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose
*>  of Z.  C1 and S1 are diagonal matrices satisfying
*>
*>                C1**2 + S1**2 = I,
*>
*>  and R1 is an L-by-L nonsingular upper triangular matrix.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B,
     $                   LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV,
     $                   Q, LDQ, WORK, NCYCLE, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBQ, JOBU, JOBV
      INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
     $                   NCYCLE, P
      DOUBLE PRECISION   TOLA, TOLB
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), ALPHA( * ), B( LDB, * ),
     $                   BETA( * ), Q( LDQ, * ), U( LDU, * ),
     $                   V( LDV, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            MAXIT
      PARAMETER          ( MAXIT = 40 )
      DOUBLE PRECISION   ZERO, ONE, HUGENUM
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
*
      LOGICAL            INITQ, INITU, INITV, UPPER, WANTQ, WANTU, WANTV
      INTEGER            I, J, KCYCLE
      DOUBLE PRECISION   A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, ERROR,
     $                   GAMMA, RWK, SNQ, SNU, SNV, SSMIN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLAGS2, DLAPLL, DLARTG, DLASET,
     $                   DROT,
     $                   DSCAL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, HUGE
      PARAMETER          ( HUGENUM = HUGE(ZERO) )
*     ..
*     .. Executable Statements ..
*
*     Decode and test the input parameters
*
      INITU = LSAME( JOBU, 'I' )
      WANTU = INITU .OR. LSAME( JOBU, 'U' )
*
      INITV = LSAME( JOBV, 'I' )
      WANTV = INITV .OR. LSAME( JOBV, 'V' )
*
      INITQ = LSAME( JOBQ, 'I' )
      WANTQ = INITQ .OR. LSAME( JOBQ, 'Q' )
*
      INFO = 0
      IF( .NOT.( INITU .OR. WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( INITV .OR.
     $         WANTV .OR.
     $         LSAME( JOBV, 'N' ) ) ) THEN
         INFO = -2
      ELSE IF( .NOT.( INITQ .OR.
     $         WANTQ .OR.
     $         LSAME( JOBQ, 'N' ) ) ) THEN
         INFO = -3
      ELSE IF( M.LT.0 ) THEN
         INFO = -4
      ELSE IF( P.LT.0 ) THEN
         INFO = -5
      ELSE IF( N.LT.0 ) THEN
         INFO = -6
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -10
      ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
         INFO = -12
      ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
         INFO = -18
      ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
         INFO = -20
      ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
         INFO = -22
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTGSJA', -INFO )
         RETURN
      END IF
*
*     Initialize U, V and Q, if necessary
*
      IF( INITU )
     $   CALL DLASET( 'Full', M, M, ZERO, ONE, U, LDU )
      IF( INITV )
     $   CALL DLASET( 'Full', P, P, ZERO, ONE, V, LDV )
      IF( INITQ )
     $   CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
*
*     Loop until convergence
*
      UPPER = .FALSE.
      DO 40 KCYCLE = 1, MAXIT
*
         UPPER = .NOT.UPPER
*
         DO 20 I = 1, L - 1
            DO 10 J = I + 1, L
*
               A1 = ZERO
               A2 = ZERO
               A3 = ZERO
               IF( K+I.LE.M )
     $            A1 = A( K+I, N-L+I )
               IF( K+J.LE.M )
     $            A3 = A( K+J, N-L+J )
*
               B1 = B( I, N-L+I )
               B3 = B( J, N-L+J )
*
               IF( UPPER ) THEN
                  IF( K+I.LE.M )
     $               A2 = A( K+I, N-L+J )
                  B2 = B( I, N-L+J )
               ELSE
                  IF( K+J.LE.M )
     $               A2 = A( K+J, N-L+I )
                  B2 = B( J, N-L+I )
               END IF
*
               CALL DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU,
     $                      CSV, SNV, CSQ, SNQ )
*
*              Update (K+I)-th and (K+J)-th rows of matrix A: U**T *A
*
               IF( K+J.LE.M )
     $            CALL DROT( L, A( K+J, N-L+1 ), LDA, A( K+I,
     $                       N-L+1 ),
     $                       LDA, CSU, SNU )
*
*              Update I-th and J-th rows of matrix B: V**T *B
*
               CALL DROT( L, B( J, N-L+1 ), LDB, B( I, N-L+1 ), LDB,
     $                    CSV, SNV )
*
*              Update (N-L+I)-th and (N-L+J)-th columns of matrices
*              A and B: A*Q and B*Q
*
               CALL DROT( MIN( K+L, M ), A( 1, N-L+J ), 1,
     $                    A( 1, N-L+I ), 1, CSQ, SNQ )
*
               CALL DROT( L, B( 1, N-L+J ), 1, B( 1, N-L+I ), 1, CSQ,
     $                    SNQ )
*
               IF( UPPER ) THEN
                  IF( K+I.LE.M )
     $               A( K+I, N-L+J ) = ZERO
                  B( I, N-L+J ) = ZERO
               ELSE
                  IF( K+J.LE.M )
     $               A( K+J, N-L+I ) = ZERO
                  B( J, N-L+I ) = ZERO
               END IF
*
*              Update orthogonal matrices U, V, Q, if desired.
*
               IF( WANTU .AND. K+J.LE.M )
     $            CALL DROT( M, U( 1, K+J ), 1, U( 1, K+I ), 1, CSU,
     $                       SNU )
*
               IF( WANTV )
     $            CALL DROT( P, V( 1, J ), 1, V( 1, I ), 1, CSV,
     $                       SNV )
*
               IF( WANTQ )
     $            CALL DROT( N, Q( 1, N-L+J ), 1, Q( 1, N-L+I ), 1,
     $                       CSQ,
     $                       SNQ )
*
   10       CONTINUE
   20    CONTINUE
*
         IF( .NOT.UPPER ) THEN
*
*           The matrices A13 and B13 were lower triangular at the start
*           of the cycle, and are now upper triangular.
*
*           Convergence test: test the parallelism of the corresponding
*           rows of A and B.
*
            ERROR = ZERO
            DO 30 I = 1, MIN( L, M-K )
               CALL DCOPY( L-I+1, A( K+I, N-L+I ), LDA, WORK, 1 )
               CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, WORK( L+1 ),
     $                     1 )
               CALL DLAPLL( L-I+1, WORK, 1, WORK( L+1 ), 1, SSMIN )
               ERROR = MAX( ERROR, SSMIN )
   30       CONTINUE
*
            IF( ABS( ERROR ).LE.MIN( TOLA, TOLB ) )
     $         GO TO 50
         END IF
*
*        End of cycle loop
*
   40 CONTINUE
*
*     The algorithm has not converged after MAXIT cycles.
*
      INFO = 1
      GO TO 100
*
   50 CONTINUE
*
*     If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged.
*     Compute the generalized singular value pairs (ALPHA, BETA), and
*     set the triangular matrix R to array A.
*
      DO 60 I = 1, K
         ALPHA( I ) = ONE
         BETA( I ) = ZERO
   60 CONTINUE
*
      DO 70 I = 1, MIN( L, M-K )
*
         A1 = A( K+I, N-L+I )
         B1 = B( I, N-L+I )
         GAMMA = B1 / A1
*
         IF( (GAMMA.LE.HUGENUM).AND.(GAMMA.GE.-HUGENUM) ) THEN
*
*           change sign if necessary
*
            IF( GAMMA.LT.ZERO ) THEN
               CALL DSCAL( L-I+1, -ONE, B( I, N-L+I ), LDB )
               IF( WANTV )
     $            CALL DSCAL( P, -ONE, V( 1, I ), 1 )
            END IF
*
            CALL DLARTG( ABS( GAMMA ), ONE, BETA( K+I ),
     $                   ALPHA( K+I ),
     $                   RWK )
*
            IF( ALPHA( K+I ).GE.BETA( K+I ) ) THEN
               CALL DSCAL( L-I+1, ONE / ALPHA( K+I ), A( K+I,
     $                     N-L+I ),
     $                     LDA )
            ELSE
               CALL DSCAL( L-I+1, ONE / BETA( K+I ), B( I, N-L+I ),
     $                     LDB )
               CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I,
     $                     N-L+I ),
     $                     LDA )
            END IF
*
         ELSE
*
            ALPHA( K+I ) = ZERO
            BETA( K+I ) = ONE
            CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
     $                  LDA )
*
         END IF
*
   70 CONTINUE
*
*     Post-assignment
*
      DO 80 I = M + 1, K + L
         ALPHA( I ) = ZERO
         BETA( I ) = ONE
   80 CONTINUE
*
      IF( K+L.LT.N ) THEN
         DO 90 I = K + L + 1, N
            ALPHA( I ) = ZERO
            BETA( I ) = ZERO
   90    CONTINUE
      END IF
*
  100 CONTINUE
      NCYCLE = KCYCLE
      RETURN
*
*     End of DTGSJA
*
      END
*> \brief \b DTGSNA
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTGSNA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgsna.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgsna.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgsna.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,
*                          LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK,
*                          IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          HOWMNY, JOB
*       INTEGER            INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
*       ..
*       .. Array Arguments ..
*       LOGICAL            SELECT( * )
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), DIF( * ), S( * ),
*      $                   VL( LDVL, * ), VR( LDVR, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTGSNA estimates reciprocal condition numbers for specified
*> eigenvalues and/or eigenvectors of a matrix pair (A, B) in
*> generalized real Schur canonical form (or of any matrix pair
*> (Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where
*> Z**T denotes the transpose of Z.
*>
*> (A, B) must be in generalized real Schur form (as returned by DGGES),
*> i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal
*> blocks. B is upper triangular.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOB
*> \verbatim
*>          JOB is CHARACTER*1
*>          Specifies whether condition numbers are required for
*>          eigenvalues (S) or eigenvectors (DIF):
*>          = 'E': for eigenvalues only (S);
*>          = 'V': for eigenvectors only (DIF);
*>          = 'B': for both eigenvalues and eigenvectors (S and DIF).
*> \endverbatim
*>
*> \param[in] HOWMNY
*> \verbatim
*>          HOWMNY is CHARACTER*1
*>          = 'A': compute condition numbers for all eigenpairs;
*>          = 'S': compute condition numbers for selected eigenpairs
*>                 specified by the array SELECT.
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*>          SELECT is LOGICAL array, dimension (N)
*>          If HOWMNY = 'S', SELECT specifies the eigenpairs for which
*>          condition numbers are required. To select condition numbers
*>          for the eigenpair corresponding to a real eigenvalue w(j),
*>          SELECT(j) must be set to .TRUE.. To select condition numbers
*>          corresponding to a complex conjugate pair of eigenvalues w(j)
*>          and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
*>          set to .TRUE..
*>          If HOWMNY = 'A', SELECT is not referenced.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the square matrix pair (A, B). N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The upper quasi-triangular matrix A in the pair (A,B).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,N)
*>          The upper triangular matrix B in the pair (A,B).
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is DOUBLE PRECISION array, dimension (LDVL,M)
*>          If JOB = 'E' or 'B', VL must contain left eigenvectors of
*>          (A, B), corresponding to the eigenpairs specified by HOWMNY
*>          and SELECT. The eigenvectors must be stored in consecutive
*>          columns of VL, as returned by DTGEVC.
*>          If JOB = 'V', VL is not referenced.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*>          LDVL is INTEGER
*>          The leading dimension of the array VL. LDVL >= 1.
*>          If JOB = 'E' or 'B', LDVL >= N.
*> \endverbatim
*>
*> \param[in] VR
*> \verbatim
*>          VR is DOUBLE PRECISION array, dimension (LDVR,M)
*>          If JOB = 'E' or 'B', VR must contain right eigenvectors of
*>          (A, B), corresponding to the eigenpairs specified by HOWMNY
*>          and SELECT. The eigenvectors must be stored in consecutive
*>          columns ov VR, as returned by DTGEVC.
*>          If JOB = 'V', VR is not referenced.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*>          LDVR is INTEGER
*>          The leading dimension of the array VR. LDVR >= 1.
*>          If JOB = 'E' or 'B', LDVR >= N.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (MM)
*>          If JOB = 'E' or 'B', the reciprocal condition numbers of the
*>          selected eigenvalues, stored in consecutive elements of the
*>          array. For a complex conjugate pair of eigenvalues two
*>          consecutive elements of S are set to the same value. Thus
*>          S(j), DIF(j), and the j-th columns of VL and VR all
*>          correspond to the same eigenpair (but not in general the
*>          j-th eigenpair, unless all eigenpairs are selected).
*>          If JOB = 'V', S is not referenced.
*> \endverbatim
*>
*> \param[out] DIF
*> \verbatim
*>          DIF is DOUBLE PRECISION array, dimension (MM)
*>          If JOB = 'V' or 'B', the estimated reciprocal condition
*>          numbers of the selected eigenvectors, stored in consecutive
*>          elements of the array. For a complex eigenvector two
*>          consecutive elements of DIF are set to the same value. If
*>          the eigenvalues cannot be reordered to compute DIF(j), DIF(j)
*>          is set to 0; this can only occur when the true value would be
*>          very small anyway.
*>          If JOB = 'E', DIF is not referenced.
*> \endverbatim
*>
*> \param[in] MM
*> \verbatim
*>          MM is INTEGER
*>          The number of elements in the arrays S and DIF. MM >= M.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The number of elements of the arrays S and DIF used to store
*>          the specified condition numbers; for each selected real
*>          eigenvalue one element is used, and for each selected complex
*>          conjugate pair of eigenvalues, two elements are used.
*>          If HOWMNY = 'A', M is set to N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= max(1,N).
*>          If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N + 6)
*>          If JOB = 'E', IWORK is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          =0: Successful exit
*>          <0: If INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tgsna
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The reciprocal of the condition number of a generalized eigenvalue
*>  w = (a, b) is defined as
*>
*>       S(w) = (|u**TAv|**2 + |u**TBv|**2)**(1/2) / (norm(u)*norm(v))
*>
*>  where u and v are the left and right eigenvectors of (A, B)
*>  corresponding to w; |z| denotes the absolute value of the complex
*>  number, and norm(u) denotes the 2-norm of the vector u.
*>  The pair (a, b) corresponds to an eigenvalue w = a/b (= u**TAv/u**TBv)
*>  of the matrix pair (A, B). If both a and b equal zero, then (A B) is
*>  singular and S(I) = -1 is returned.
*>
*>  An approximate error bound on the chordal distance between the i-th
*>  computed generalized eigenvalue w and the corresponding exact
*>  eigenvalue lambda is
*>
*>       chord(w, lambda) <= EPS * norm(A, B) / S(I)
*>
*>  where EPS is the machine precision.
*>
*>  The reciprocal of the condition number DIF(i) of right eigenvector u
*>  and left eigenvector v corresponding to the generalized eigenvalue w
*>  is defined as follows:
*>
*>  a) If the i-th eigenvalue w = (a,b) is real
*>
*>     Suppose U and V are orthogonal transformations such that
*>
*>              U**T*(A, B)*V  = (S, T) = ( a   *  ) ( b  *  )  1
*>                                        ( 0  S22 ),( 0 T22 )  n-1
*>                                          1  n-1     1 n-1
*>
*>     Then the reciprocal condition number DIF(i) is
*>
*>                Difl((a, b), (S22, T22)) = sigma-min( Zl ),
*>
*>     where sigma-min(Zl) denotes the smallest singular value of the
*>     2(n-1)-by-2(n-1) matrix
*>
*>         Zl = [ kron(a, In-1)  -kron(1, S22) ]
*>              [ kron(b, In-1)  -kron(1, T22) ] .
*>
*>     Here In-1 is the identity matrix of size n-1. kron(X, Y) is the
*>     Kronecker product between the matrices X and Y.
*>
*>     Note that if the default method for computing DIF(i) is wanted
*>     (see DLATDF), then the parameter DIFDRI (see below) should be
*>     changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)).
*>     See DTGSYL for more details.
*>
*>  b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,
*>
*>     Suppose U and V are orthogonal transformations such that
*>
*>              U**T*(A, B)*V = (S, T) = ( S11  *   ) ( T11  *  )  2
*>                                       ( 0    S22 ),( 0    T22) n-2
*>                                         2    n-2     2    n-2
*>
*>     and (S11, T11) corresponds to the complex conjugate eigenvalue
*>     pair (w, conjg(w)). There exist unitary matrices U1 and V1 such
*>     that
*>
*>       U1**T*S11*V1 = ( s11 s12 ) and U1**T*T11*V1 = ( t11 t12 )
*>                      (  0  s22 )                    (  0  t22 )
*>
*>     where the generalized eigenvalues w = s11/t11 and
*>     conjg(w) = s22/t22.
*>
*>     Then the reciprocal condition number DIF(i) is bounded by
*>
*>         min( d1, max( 1, |real(s11)/real(s22)| )*d2 )
*>
*>     where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where
*>     Z1 is the complex 2-by-2 matrix
*>
*>              Z1 =  [ s11  -s22 ]
*>                    [ t11  -t22 ],
*>
*>     This is done by computing (using real arithmetic) the
*>     roots of the characteristical polynomial det(Z1**T * Z1 - lambda I),
*>     where Z1**T denotes the transpose of Z1 and det(X) denotes
*>     the determinant of X.
*>
*>     and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an
*>     upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)
*>
*>              Z2 = [ kron(S11**T, In-2)  -kron(I2, S22) ]
*>                   [ kron(T11**T, In-2)  -kron(I2, T22) ]
*>
*>     Note that if the default method for computing DIF is wanted (see
*>     DLATDF), then the parameter DIFDRI (see below) should be changed
*>     from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL
*>     for more details.
*>
*>  For each eigenvalue/vector specified by SELECT, DIF stores a
*>  Frobenius norm-based estimate of Difl.
*>
*>  An approximate error bound for the i-th computed eigenvector VL(i) or
*>  VR(i) is given by
*>
*>             EPS * norm(A, B) / DIF(i).
*>
*>  See ref. [2-3] for more details and further references.
*> \endverbatim
*
*> \par Contributors:
*  ==================
*>
*>     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*>     Umea University, S-901 87 Umea, Sweden.
*
*> \par References:
*  ================
*>
*> \verbatim
*>
*>  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
*>      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
*>      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
*>      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
*>
*>  [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
*>      Eigenvalues of a Regular Matrix Pair (A, B) and Condition
*>      Estimation: Theory, Algorithms and Software,
*>      Report UMINF - 94.04, Department of Computing Science, Umea
*>      University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
*>      Note 87. To appear in Numerical Algorithms, 1996.
*>
*>  [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
*>      for Solving the Generalized Sylvester Equation and Estimating the
*>      Separation between Regular Matrix Pairs, Report UMINF - 93.23,
*>      Department of Computing Science, Umea University, S-901 87 Umea,
*>      Sweden, December 1993, Revised April 1994, Also as LAPACK Working
*>      Note 75.  To appear in ACM Trans. on Math. Software, Vol 22,
*>      No 1, 1996.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,
     $                   LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK,
     $                   IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          HOWMNY, JOB
      INTEGER            INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
*     ..
*     .. Array Arguments ..
      LOGICAL            SELECT( * )
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), DIF( * ), S( * ),
     $                   VL( LDVL, * ), VR( LDVR, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            DIFDRI
      PARAMETER          ( DIFDRI = 3 )
      DOUBLE PRECISION   ZERO, ONE, TWO, FOUR
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0,
     $                   FOUR = 4.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, PAIR, SOMCON, WANTBH, WANTDF, WANTS
      INTEGER            I, IERR, IFST, ILST, IZ, K, KS, LWMIN, N1, N2
      DOUBLE PRECISION   ALPHAI, ALPHAR, ALPRQT, BETA, C1, C2, COND,
     $                   EPS, LNRM, RNRM, ROOT1, ROOT2, SCALE, SMLNUM,
     $                   TMPII, TMPIR, TMPRI, TMPRR, UHAV, UHAVI, UHBV,
     $                   UHBVI
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   DUMMY( 1 ), DUMMY1( 1 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DDOT, DLAMCH, DLAPY2, DNRM2
      EXTERNAL           LSAME, DDOT, DLAMCH, DLAPY2, DNRM2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMV, DLACPY, DLAG2, DTGEXC, DTGSYL,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Decode and test the input parameters
*
      WANTBH = LSAME( JOB, 'B' )
      WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
      WANTDF = LSAME( JOB, 'V' ) .OR. WANTBH
*
      SOMCON = LSAME( HOWMNY, 'S' )
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
*
      IF( .NOT.WANTS .AND. .NOT.WANTDF ) THEN
         INFO = -1
      ELSE IF( .NOT.LSAME( HOWMNY, 'A' ) .AND. .NOT.SOMCON ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -6
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -8
      ELSE IF( WANTS .AND. LDVL.LT.N ) THEN
         INFO = -10
      ELSE IF( WANTS .AND. LDVR.LT.N ) THEN
         INFO = -12
      ELSE
*
*        Set M to the number of eigenpairs for which condition numbers
*        are required, and test MM.
*
         IF( SOMCON ) THEN
            M = 0
            PAIR = .FALSE.
            DO 10 K = 1, N
               IF( PAIR ) THEN
                  PAIR = .FALSE.
               ELSE
                  IF( K.LT.N ) THEN
                     IF( A( K+1, K ).EQ.ZERO ) THEN
                        IF( SELECT( K ) )
     $                     M = M + 1
                     ELSE
                        PAIR = .TRUE.
                        IF( SELECT( K ) .OR. SELECT( K+1 ) )
     $                     M = M + 2
                     END IF
                  ELSE
                     IF( SELECT( N ) )
     $                  M = M + 1
                  END IF
               END IF
   10       CONTINUE
         ELSE
            M = N
         END IF
*
         IF( N.EQ.0 ) THEN
            LWMIN = 1
         ELSE IF( LSAME( JOB, 'V' ) .OR. LSAME( JOB, 'B' ) ) THEN
            LWMIN = 2*N*( N + 2 ) + 16
         ELSE
            LWMIN = N
         END IF
         WORK( 1 ) = LWMIN
*
         IF( MM.LT.M ) THEN
            INFO = -15
         ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -18
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTGSNA', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Get machine constants
*
      EPS = DLAMCH( 'P' )
      SMLNUM = DLAMCH( 'S' ) / EPS
      KS = 0
      PAIR = .FALSE.
*
      DO 20 K = 1, N
*
*        Determine whether A(k,k) begins a 1-by-1 or 2-by-2 block.
*
         IF( PAIR ) THEN
            PAIR = .FALSE.
            GO TO 20
         ELSE
            IF( K.LT.N )
     $         PAIR = A( K+1, K ).NE.ZERO
         END IF
*
*        Determine whether condition numbers are required for the k-th
*        eigenpair.
*
         IF( SOMCON ) THEN
            IF( PAIR ) THEN
               IF( .NOT.SELECT( K ) .AND. .NOT.SELECT( K+1 ) )
     $            GO TO 20
            ELSE
               IF( .NOT.SELECT( K ) )
     $            GO TO 20
            END IF
         END IF
*
         KS = KS + 1
*
         IF( WANTS ) THEN
*
*           Compute the reciprocal condition number of the k-th
*           eigenvalue.
*
            IF( PAIR ) THEN
*
*              Complex eigenvalue pair.
*
               RNRM = DLAPY2( DNRM2( N, VR( 1, KS ), 1 ),
     $                DNRM2( N, VR( 1, KS+1 ), 1 ) )
               LNRM = DLAPY2( DNRM2( N, VL( 1, KS ), 1 ),
     $                DNRM2( N, VL( 1, KS+1 ), 1 ) )
               CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS ), 1,
     $                     ZERO,
     $                     WORK, 1 )
               TMPRR = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
               TMPRI = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
               CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS+1 ), 1,
     $                     ZERO, WORK, 1 )
               TMPII = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
               TMPIR = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
               UHAV = TMPRR + TMPII
               UHAVI = TMPIR - TMPRI
               CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS ), 1,
     $                     ZERO,
     $                     WORK, 1 )
               TMPRR = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
               TMPRI = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
               CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS+1 ), 1,
     $                     ZERO, WORK, 1 )
               TMPII = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
               TMPIR = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
               UHBV = TMPRR + TMPII
               UHBVI = TMPIR - TMPRI
               UHAV = DLAPY2( UHAV, UHAVI )
               UHBV = DLAPY2( UHBV, UHBVI )
               COND = DLAPY2( UHAV, UHBV )
               S( KS ) = COND / ( RNRM*LNRM )
               S( KS+1 ) = S( KS )
*
            ELSE
*
*              Real eigenvalue.
*
               RNRM = DNRM2( N, VR( 1, KS ), 1 )
               LNRM = DNRM2( N, VL( 1, KS ), 1 )
               CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS ), 1,
     $                     ZERO,
     $                     WORK, 1 )
               UHAV = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
               CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS ), 1,
     $                     ZERO,
     $                     WORK, 1 )
               UHBV = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
               COND = DLAPY2( UHAV, UHBV )
               IF( COND.EQ.ZERO ) THEN
                  S( KS ) = -ONE
               ELSE
                  S( KS ) = COND / ( RNRM*LNRM )
               END IF
            END IF
         END IF
*
         IF( WANTDF ) THEN
            IF( N.EQ.1 ) THEN
               DIF( KS ) = DLAPY2( A( 1, 1 ), B( 1, 1 ) )
               GO TO 20
            END IF
*
*           Estimate the reciprocal condition number of the k-th
*           eigenvectors.
            IF( PAIR ) THEN
*
*              Copy the  2-by 2 pencil beginning at (A(k,k), B(k, k)).
*              Compute the eigenvalue(s) at position K.
*
               WORK( 1 ) = A( K, K )
               WORK( 2 ) = A( K+1, K )
               WORK( 3 ) = A( K, K+1 )
               WORK( 4 ) = A( K+1, K+1 )
               WORK( 5 ) = B( K, K )
               WORK( 6 ) = B( K+1, K )
               WORK( 7 ) = B( K, K+1 )
               WORK( 8 ) = B( K+1, K+1 )
               CALL DLAG2( WORK, 2, WORK( 5 ), 2, SMLNUM*EPS, BETA,
     $                     DUMMY1( 1 ), ALPHAR, DUMMY( 1 ), ALPHAI )
               ALPRQT = ONE
               C1 = TWO*( ALPHAR*ALPHAR+ALPHAI*ALPHAI+BETA*BETA )
               C2 = FOUR*BETA*BETA*ALPHAI*ALPHAI
               ROOT1 = C1 + SQRT( C1*C1-4.0D0*C2 )
               ROOT1 = ROOT1 / TWO
               ROOT2 = C2 / ROOT1
               COND = MIN( SQRT( ROOT1 ), SQRT( ROOT2 ) )
            END IF
*
*           Copy the matrix (A, B) to the array WORK and swap the
*           diagonal block beginning at A(k,k) to the (1,1) position.
*
            CALL DLACPY( 'Full', N, N, A, LDA, WORK, N )
            CALL DLACPY( 'Full', N, N, B, LDB, WORK( N*N+1 ), N )
            IFST = K
            ILST = 1
*
            CALL DTGEXC( .FALSE., .FALSE., N, WORK, N, WORK( N*N+1 ),
     $                   N,
     $                   DUMMY, 1, DUMMY1, 1, IFST, ILST,
     $                   WORK( N*N*2+1 ), LWORK-2*N*N, IERR )
*
            IF( IERR.GT.0 ) THEN
*
*              Ill-conditioned problem - swap rejected.
*
               DIF( KS ) = ZERO
            ELSE
*
*              Reordering successful, solve generalized Sylvester
*              equation for R and L,
*                         A22 * R - L * A11 = A12
*                         B22 * R - L * B11 = B12,
*              and compute estimate of Difl((A11,B11), (A22, B22)).
*
               N1 = 1
               IF( WORK( 2 ).NE.ZERO )
     $            N1 = 2
               N2 = N - N1
               IF( N2.EQ.0 ) THEN
                  DIF( KS ) = COND
               ELSE
                  I = N*N + 1
                  IZ = 2*N*N + 1
                  CALL DTGSYL( 'N', DIFDRI, N2, N1,
     $                         WORK( N*N1+N1+1 ),
     $                         N, WORK, N, WORK( N1+1 ), N,
     $                         WORK( N*N1+N1+I ), N, WORK( I ), N,
     $                         WORK( N1+I ), N, SCALE, DIF( KS ),
     $                         WORK( IZ+1 ), LWORK-2*N*N, IWORK, IERR )
*
                  IF( PAIR )
     $               DIF( KS ) = MIN( MAX( ONE, ALPRQT )*DIF( KS ),
     $                           COND )
               END IF
            END IF
            IF( PAIR )
     $         DIF( KS+1 ) = DIF( KS )
         END IF
         IF( PAIR )
     $      KS = KS + 1
*
   20 CONTINUE
      WORK( 1 ) = LWMIN
      RETURN
*
*     End of DTGSNA
*
      END
*> \brief \b DTGSY2 solves the generalized Sylvester equation (unblocked algorithm).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTGSY2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgsy2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgsy2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgsy2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
*                          LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL,
*                          IWORK, PQ, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANS
*       INTEGER            IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N,
*      $                   PQ
*       DOUBLE PRECISION   RDSCAL, RDSUM, SCALE
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * ),
*      $                   D( LDD, * ), E( LDE, * ), F( LDF, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTGSY2 solves the generalized Sylvester equation:
*>
*>             A * R - L * B = scale * C                (1)
*>             D * R - L * E = scale * F,
*>
*> using Level 1 and 2 BLAS. where R and L are unknown M-by-N matrices,
*> (A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M,
*> N-by-N and M-by-N, respectively, with real entries. (A, D) and (B, E)
*> must be in generalized Schur canonical form, i.e. A, B are upper
*> quasi triangular and D, E are upper triangular. The solution (R, L)
*> overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor
*> chosen to avoid overflow.
*>
*> In matrix notation solving equation (1) corresponds to solve
*> Z*x = scale*b, where Z is defined as
*>
*>        Z = [ kron(In, A)  -kron(B**T, Im) ]             (2)
*>            [ kron(In, D)  -kron(E**T, Im) ],
*>
*> Ik is the identity matrix of size k and X**T is the transpose of X.
*> kron(X, Y) is the Kronecker product between the matrices X and Y.
*> In the process of solving (1), we solve a number of such systems
*> where Dim(In), Dim(In) = 1 or 2.
*>
*> If TRANS = 'T', solve the transposed system Z**T*y = scale*b for y,
*> which is equivalent to solve for R and L in
*>
*>             A**T * R  + D**T * L   = scale * C           (3)
*>             R  * B**T + L  * E**T  = scale * -F
*>
*> This case is used to compute an estimate of Dif[(A, D), (B, E)] =
*> sigma_min(Z) using reverse communication with DLACON.
*>
*> DTGSY2 also (IJOB >= 1) contributes to the computation in DTGSYL
*> of an upper bound on the separation between to matrix pairs. Then
*> the input (A, D), (B, E) are sub-pencils of the matrix pair in
*> DTGSYL. See DTGSYL for details.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N': solve the generalized Sylvester equation (1).
*>          = 'T': solve the 'transposed' system (3).
*> \endverbatim
*>
*> \param[in] IJOB
*> \verbatim
*>          IJOB is INTEGER
*>          Specifies what kind of functionality to be performed.
*>          = 0: solve (1) only.
*>          = 1: A contribution from this subsystem to a Frobenius
*>               norm-based estimate of the separation between two matrix
*>               pairs is computed. (look ahead strategy is used).
*>          = 2: A contribution from this subsystem to a Frobenius
*>               norm-based estimate of the separation between two matrix
*>               pairs is computed. (DGECON on sub-systems is used.)
*>          Not referenced if TRANS = 'T'.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          On entry, M specifies the order of A and D, and the row
*>          dimension of C, F, R and L.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          On entry, N specifies the order of B and E, and the column
*>          dimension of C, F, R and L.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, M)
*>          On entry, A contains an upper quasi triangular matrix.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the matrix A. LDA >= max(1, M).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB, N)
*>          On entry, B contains an upper quasi triangular matrix.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the matrix B. LDB >= max(1, N).
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC, N)
*>          On entry, C contains the right-hand-side of the first matrix
*>          equation in (1).
*>          On exit, if IJOB = 0, C has been overwritten by the
*>          solution R.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the matrix C. LDC >= max(1, M).
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (LDD, M)
*>          On entry, D contains an upper triangular matrix.
*> \endverbatim
*>
*> \param[in] LDD
*> \verbatim
*>          LDD is INTEGER
*>          The leading dimension of the matrix D. LDD >= max(1, M).
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (LDE, N)
*>          On entry, E contains an upper triangular matrix.
*> \endverbatim
*>
*> \param[in] LDE
*> \verbatim
*>          LDE is INTEGER
*>          The leading dimension of the matrix E. LDE >= max(1, N).
*> \endverbatim
*>
*> \param[in,out] F
*> \verbatim
*>          F is DOUBLE PRECISION array, dimension (LDF, N)
*>          On entry, F contains the right-hand-side of the second matrix
*>          equation in (1).
*>          On exit, if IJOB = 0, F has been overwritten by the
*>          solution L.
*> \endverbatim
*>
*> \param[in] LDF
*> \verbatim
*>          LDF is INTEGER
*>          The leading dimension of the matrix F. LDF >= max(1, M).
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*>          SCALE is DOUBLE PRECISION
*>          On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions
*>          R and L (C and F on entry) will hold the solutions to a
*>          slightly perturbed system but the input matrices A, B, D and
*>          E have not been changed. If SCALE = 0, R and L will hold the
*>          solutions to the homogeneous system with C = F = 0. Normally,
*>          SCALE = 1.
*> \endverbatim
*>
*> \param[in,out] RDSUM
*> \verbatim
*>          RDSUM is DOUBLE PRECISION
*>          On entry, the sum of squares of computed contributions to
*>          the Dif-estimate under computation by DTGSYL, where the
*>          scaling factor RDSCAL (see below) has been factored out.
*>          On exit, the corresponding sum of squares updated with the
*>          contributions from the current sub-system.
*>          If TRANS = 'T' RDSUM is not touched.
*>          NOTE: RDSUM only makes sense when DTGSY2 is called by DTGSYL.
*> \endverbatim
*>
*> \param[in,out] RDSCAL
*> \verbatim
*>          RDSCAL is DOUBLE PRECISION
*>          On entry, scaling factor used to prevent overflow in RDSUM.
*>          On exit, RDSCAL is updated w.r.t. the current contributions
*>          in RDSUM.
*>          If TRANS = 'T', RDSCAL is not touched.
*>          NOTE: RDSCAL only makes sense when DTGSY2 is called by
*>                DTGSYL.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (M+N+2)
*> \endverbatim
*>
*> \param[out] PQ
*> \verbatim
*>          PQ is INTEGER
*>          On exit, the number of subsystems (of size 2-by-2, 4-by-4 and
*>          8-by-8) solved by this routine.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          On exit, if INFO is set to
*>            =0: Successful exit
*>            <0: If INFO = -i, the i-th argument had an illegal value.
*>            >0: The matrix pairs (A, D) and (B, E) have common or very
*>                close eigenvalues.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tgsy2
*
*> \par Contributors:
*  ==================
*>
*>     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*>     Umea University, S-901 87 Umea, Sweden.
*
*  =====================================================================
      SUBROUTINE DTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC,
     $                   D,
     $                   LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL,
     $                   IWORK, PQ, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS
      INTEGER            IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N,
     $                   PQ
      DOUBLE PRECISION   RDSCAL, RDSUM, SCALE
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * ),
     $                   D( LDD, * ), E( LDE, * ), F( LDF, * )
*     ..
*
*  =====================================================================
*  Replaced various illegal calls to DCOPY by calls to DLASET.
*  Sven Hammarling, 27/5/02.
*
*     .. Parameters ..
      INTEGER            LDZ
      PARAMETER          ( LDZ = 8 )
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOTRAN
      INTEGER            I, IE, IERR, II, IS, ISP1, J, JE, JJ, JS, JSP1,
     $                   K, MB, NB, P, Q, ZDIM
      DOUBLE PRECISION   ALPHA, SCALOC
*     ..
*     .. Local Arrays ..
      INTEGER            IPIV( LDZ ), JPIV( LDZ )
      DOUBLE PRECISION   RHS( LDZ ), Z( LDZ, LDZ )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DGEMM, DGEMV, DGER,
     $                   DGESC2,
     $                   DGETC2, DLASET, DLATDF, DSCAL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Decode and test input parameters
*
      INFO = 0
      IERR = 0
      NOTRAN = LSAME( TRANS, 'N' )
      IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
         INFO = -1
      ELSE IF( NOTRAN ) THEN
         IF( ( IJOB.LT.0 ) .OR. ( IJOB.GT.2 ) ) THEN
            INFO = -2
         END IF
      END IF
      IF( INFO.EQ.0 ) THEN
         IF( M.LE.0 ) THEN
            INFO = -3
         ELSE IF( N.LE.0 ) THEN
            INFO = -4
         ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
            INFO = -6
         ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
            INFO = -8
         ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
            INFO = -10
         ELSE IF( LDD.LT.MAX( 1, M ) ) THEN
            INFO = -12
         ELSE IF( LDE.LT.MAX( 1, N ) ) THEN
            INFO = -14
         ELSE IF( LDF.LT.MAX( 1, M ) ) THEN
            INFO = -16
         END IF
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTGSY2', -INFO )
         RETURN
      END IF
*
*     Determine block structure of A
*
      PQ = 0
      P = 0
      I = 1
   10 CONTINUE
      IF( I.GT.M )
     $   GO TO 20
      P = P + 1
      IWORK( P ) = I
      IF( I.EQ.M )
     $   GO TO 20
      IF( A( I+1, I ).NE.ZERO ) THEN
         I = I + 2
      ELSE
         I = I + 1
      END IF
      GO TO 10
   20 CONTINUE
      IWORK( P+1 ) = M + 1
*
*     Determine block structure of B
*
      Q = P + 1
      J = 1
   30 CONTINUE
      IF( J.GT.N )
     $   GO TO 40
      Q = Q + 1
      IWORK( Q ) = J
      IF( J.EQ.N )
     $   GO TO 40
      IF( B( J+1, J ).NE.ZERO ) THEN
         J = J + 2
      ELSE
         J = J + 1
      END IF
      GO TO 30
   40 CONTINUE
      IWORK( Q+1 ) = N + 1
      PQ = P*( Q-P-1 )
*
      IF( NOTRAN ) THEN
*
*        Solve (I, J) - subsystem
*           A(I, I) * R(I, J) - L(I, J) * B(J, J) = C(I, J)
*           D(I, I) * R(I, J) - L(I, J) * E(J, J) = F(I, J)
*        for I = P, P - 1, ..., 1; J = 1, 2, ..., Q
*
         SCALE = ONE
         SCALOC = ONE
         DO 120 J = P + 2, Q
            JS = IWORK( J )
            JSP1 = JS + 1
            JE = IWORK( J+1 ) - 1
            NB = JE - JS + 1
            DO 110 I = P, 1, -1
*
               IS = IWORK( I )
               ISP1 = IS + 1
               IE = IWORK( I+1 ) - 1
               MB = IE - IS + 1
               ZDIM = MB*NB*2
*
               IF( ( MB.EQ.1 ) .AND. ( NB.EQ.1 ) ) THEN
*
*                 Build a 2-by-2 system Z * x = RHS
*
                  Z( 1, 1 ) = A( IS, IS )
                  Z( 2, 1 ) = D( IS, IS )
                  Z( 1, 2 ) = -B( JS, JS )
                  Z( 2, 2 ) = -E( JS, JS )
*
*                 Set up right hand side(s)
*
                  RHS( 1 ) = C( IS, JS )
                  RHS( 2 ) = F( IS, JS )
*
*                 Solve Z * x = RHS
*
                  CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
                  IF( IERR.GT.0 )
     $               INFO = IERR
*
                  IF( IJOB.EQ.0 ) THEN
                     CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
     $                            SCALOC )
                     IF( SCALOC.NE.ONE ) THEN
                        DO 50 K = 1, N
                           CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
                           CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
   50                   CONTINUE
                        SCALE = SCALE*SCALOC
                     END IF
                  ELSE
                     CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM,
     $                            RDSCAL, IPIV, JPIV )
                  END IF
*
*                 Unpack solution vector(s)
*
                  C( IS, JS ) = RHS( 1 )
                  F( IS, JS ) = RHS( 2 )
*
*                 Substitute R(I, J) and L(I, J) into remaining
*                 equation.
*
                  IF( I.GT.1 ) THEN
                     ALPHA = -RHS( 1 )
                     CALL DAXPY( IS-1, ALPHA, A( 1, IS ), 1, C( 1,
     $                           JS ),
     $                           1 )
                     CALL DAXPY( IS-1, ALPHA, D( 1, IS ), 1, F( 1,
     $                           JS ),
     $                           1 )
                  END IF
                  IF( J.LT.Q ) THEN
                     CALL DAXPY( N-JE, RHS( 2 ), B( JS, JE+1 ), LDB,
     $                           C( IS, JE+1 ), LDC )
                     CALL DAXPY( N-JE, RHS( 2 ), E( JS, JE+1 ), LDE,
     $                           F( IS, JE+1 ), LDF )
                  END IF
*
               ELSE IF( ( MB.EQ.1 ) .AND. ( NB.EQ.2 ) ) THEN
*
*                 Build a 4-by-4 system Z * x = RHS
*
                  Z( 1, 1 ) = A( IS, IS )
                  Z( 2, 1 ) = ZERO
                  Z( 3, 1 ) = D( IS, IS )
                  Z( 4, 1 ) = ZERO
*
                  Z( 1, 2 ) = ZERO
                  Z( 2, 2 ) = A( IS, IS )
                  Z( 3, 2 ) = ZERO
                  Z( 4, 2 ) = D( IS, IS )
*
                  Z( 1, 3 ) = -B( JS, JS )
                  Z( 2, 3 ) = -B( JS, JSP1 )
                  Z( 3, 3 ) = -E( JS, JS )
                  Z( 4, 3 ) = -E( JS, JSP1 )
*
                  Z( 1, 4 ) = -B( JSP1, JS )
                  Z( 2, 4 ) = -B( JSP1, JSP1 )
                  Z( 3, 4 ) = ZERO
                  Z( 4, 4 ) = -E( JSP1, JSP1 )
*
*                 Set up right hand side(s)
*
                  RHS( 1 ) = C( IS, JS )
                  RHS( 2 ) = C( IS, JSP1 )
                  RHS( 3 ) = F( IS, JS )
                  RHS( 4 ) = F( IS, JSP1 )
*
*                 Solve Z * x = RHS
*
                  CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
                  IF( IERR.GT.0 )
     $               INFO = IERR
*
                  IF( IJOB.EQ.0 ) THEN
                     CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
     $                            SCALOC )
                     IF( SCALOC.NE.ONE ) THEN
                        DO 60 K = 1, N
                           CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
                           CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
   60                   CONTINUE
                        SCALE = SCALE*SCALOC
                     END IF
                  ELSE
                     CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM,
     $                            RDSCAL, IPIV, JPIV )
                  END IF
*
*                 Unpack solution vector(s)
*
                  C( IS, JS ) = RHS( 1 )
                  C( IS, JSP1 ) = RHS( 2 )
                  F( IS, JS ) = RHS( 3 )
                  F( IS, JSP1 ) = RHS( 4 )
*
*                 Substitute R(I, J) and L(I, J) into remaining
*                 equation.
*
                  IF( I.GT.1 ) THEN
                     CALL DGER( IS-1, NB, -ONE, A( 1, IS ), 1,
     $                          RHS( 1 ),
     $                          1, C( 1, JS ), LDC )
                     CALL DGER( IS-1, NB, -ONE, D( 1, IS ), 1,
     $                          RHS( 1 ),
     $                          1, F( 1, JS ), LDF )
                  END IF
                  IF( J.LT.Q ) THEN
                     CALL DAXPY( N-JE, RHS( 3 ), B( JS, JE+1 ), LDB,
     $                           C( IS, JE+1 ), LDC )
                     CALL DAXPY( N-JE, RHS( 3 ), E( JS, JE+1 ), LDE,
     $                           F( IS, JE+1 ), LDF )
                     CALL DAXPY( N-JE, RHS( 4 ), B( JSP1, JE+1 ),
     $                           LDB,
     $                           C( IS, JE+1 ), LDC )
                     CALL DAXPY( N-JE, RHS( 4 ), E( JSP1, JE+1 ),
     $                           LDE,
     $                           F( IS, JE+1 ), LDF )
                  END IF
*
               ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.1 ) ) THEN
*
*                 Build a 4-by-4 system Z * x = RHS
*
                  Z( 1, 1 ) = A( IS, IS )
                  Z( 2, 1 ) = A( ISP1, IS )
                  Z( 3, 1 ) = D( IS, IS )
                  Z( 4, 1 ) = ZERO
*
                  Z( 1, 2 ) = A( IS, ISP1 )
                  Z( 2, 2 ) = A( ISP1, ISP1 )
                  Z( 3, 2 ) = D( IS, ISP1 )
                  Z( 4, 2 ) = D( ISP1, ISP1 )
*
                  Z( 1, 3 ) = -B( JS, JS )
                  Z( 2, 3 ) = ZERO
                  Z( 3, 3 ) = -E( JS, JS )
                  Z( 4, 3 ) = ZERO
*
                  Z( 1, 4 ) = ZERO
                  Z( 2, 4 ) = -B( JS, JS )
                  Z( 3, 4 ) = ZERO
                  Z( 4, 4 ) = -E( JS, JS )
*
*                 Set up right hand side(s)
*
                  RHS( 1 ) = C( IS, JS )
                  RHS( 2 ) = C( ISP1, JS )
                  RHS( 3 ) = F( IS, JS )
                  RHS( 4 ) = F( ISP1, JS )
*
*                 Solve Z * x = RHS
*
                  CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
                  IF( IERR.GT.0 )
     $               INFO = IERR
                  IF( IJOB.EQ.0 ) THEN
                     CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
     $                            SCALOC )
                     IF( SCALOC.NE.ONE ) THEN
                        DO 70 K = 1, N
                           CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
                           CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
   70                   CONTINUE
                        SCALE = SCALE*SCALOC
                     END IF
                  ELSE
                     CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM,
     $                            RDSCAL, IPIV, JPIV )
                  END IF
*
*                 Unpack solution vector(s)
*
                  C( IS, JS ) = RHS( 1 )
                  C( ISP1, JS ) = RHS( 2 )
                  F( IS, JS ) = RHS( 3 )
                  F( ISP1, JS ) = RHS( 4 )
*
*                 Substitute R(I, J) and L(I, J) into remaining
*                 equation.
*
                  IF( I.GT.1 ) THEN
                     CALL DGEMV( 'N', IS-1, MB, -ONE, A( 1, IS ),
     $                           LDA,
     $                           RHS( 1 ), 1, ONE, C( 1, JS ), 1 )
                     CALL DGEMV( 'N', IS-1, MB, -ONE, D( 1, IS ),
     $                           LDD,
     $                           RHS( 1 ), 1, ONE, F( 1, JS ), 1 )
                  END IF
                  IF( J.LT.Q ) THEN
                     CALL DGER( MB, N-JE, ONE, RHS( 3 ), 1,
     $                          B( JS, JE+1 ), LDB, C( IS, JE+1 ), LDC )
                     CALL DGER( MB, N-JE, ONE, RHS( 3 ), 1,
     $                          E( JS, JE+1 ), LDE, F( IS, JE+1 ), LDF )
                  END IF
*
               ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.2 ) ) THEN
*
*                 Build an 8-by-8 system Z * x = RHS
*
                  CALL DLASET( 'F', LDZ, LDZ, ZERO, ZERO, Z, LDZ )
*
                  Z( 1, 1 ) = A( IS, IS )
                  Z( 2, 1 ) = A( ISP1, IS )
                  Z( 5, 1 ) = D( IS, IS )
*
                  Z( 1, 2 ) = A( IS, ISP1 )
                  Z( 2, 2 ) = A( ISP1, ISP1 )
                  Z( 5, 2 ) = D( IS, ISP1 )
                  Z( 6, 2 ) = D( ISP1, ISP1 )
*
                  Z( 3, 3 ) = A( IS, IS )
                  Z( 4, 3 ) = A( ISP1, IS )
                  Z( 7, 3 ) = D( IS, IS )
*
                  Z( 3, 4 ) = A( IS, ISP1 )
                  Z( 4, 4 ) = A( ISP1, ISP1 )
                  Z( 7, 4 ) = D( IS, ISP1 )
                  Z( 8, 4 ) = D( ISP1, ISP1 )
*
                  Z( 1, 5 ) = -B( JS, JS )
                  Z( 3, 5 ) = -B( JS, JSP1 )
                  Z( 5, 5 ) = -E( JS, JS )
                  Z( 7, 5 ) = -E( JS, JSP1 )
*
                  Z( 2, 6 ) = -B( JS, JS )
                  Z( 4, 6 ) = -B( JS, JSP1 )
                  Z( 6, 6 ) = -E( JS, JS )
                  Z( 8, 6 ) = -E( JS, JSP1 )
*
                  Z( 1, 7 ) = -B( JSP1, JS )
                  Z( 3, 7 ) = -B( JSP1, JSP1 )
                  Z( 7, 7 ) = -E( JSP1, JSP1 )
*
                  Z( 2, 8 ) = -B( JSP1, JS )
                  Z( 4, 8 ) = -B( JSP1, JSP1 )
                  Z( 8, 8 ) = -E( JSP1, JSP1 )
*
*                 Set up right hand side(s)
*
                  K = 1
                  II = MB*NB + 1
                  DO 80 JJ = 0, NB - 1
                     CALL DCOPY( MB, C( IS, JS+JJ ), 1, RHS( K ), 1 )
                     CALL DCOPY( MB, F( IS, JS+JJ ), 1, RHS( II ),
     $                           1 )
                     K = K + MB
                     II = II + MB
   80             CONTINUE
*
*                 Solve Z * x = RHS
*
                  CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
                  IF( IERR.GT.0 )
     $               INFO = IERR
                  IF( IJOB.EQ.0 ) THEN
                     CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
     $                            SCALOC )
                     IF( SCALOC.NE.ONE ) THEN
                        DO 90 K = 1, N
                           CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
                           CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
   90                   CONTINUE
                        SCALE = SCALE*SCALOC
                     END IF
                  ELSE
                     CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM,
     $                            RDSCAL, IPIV, JPIV )
                  END IF
*
*                 Unpack solution vector(s)
*
                  K = 1
                  II = MB*NB + 1
                  DO 100 JJ = 0, NB - 1
                     CALL DCOPY( MB, RHS( K ), 1, C( IS, JS+JJ ), 1 )
                     CALL DCOPY( MB, RHS( II ), 1, F( IS, JS+JJ ),
     $                           1 )
                     K = K + MB
                     II = II + MB
  100             CONTINUE
*
*                 Substitute R(I, J) and L(I, J) into remaining
*                 equation.
*
                  IF( I.GT.1 ) THEN
                     CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
     $                           A( 1, IS ), LDA, RHS( 1 ), MB, ONE,
     $                           C( 1, JS ), LDC )
                     CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
     $                           D( 1, IS ), LDD, RHS( 1 ), MB, ONE,
     $                           F( 1, JS ), LDF )
                  END IF
                  IF( J.LT.Q ) THEN
                     K = MB*NB + 1
                     CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE,
     $                           RHS( K ),
     $                           MB, B( JS, JE+1 ), LDB, ONE,
     $                           C( IS, JE+1 ), LDC )
                     CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE,
     $                           RHS( K ),
     $                           MB, E( JS, JE+1 ), LDE, ONE,
     $                           F( IS, JE+1 ), LDF )
                  END IF
*
               END IF
*
  110       CONTINUE
  120    CONTINUE
      ELSE
*
*        Solve (I, J) - subsystem
*             A(I, I)**T * R(I, J) + D(I, I)**T * L(J, J)  =  C(I, J)
*             R(I, I)  * B(J, J) + L(I, J)  * E(J, J)  = -F(I, J)
*        for I = 1, 2, ..., P, J = Q, Q - 1, ..., 1
*
         SCALE = ONE
         SCALOC = ONE
         DO 200 I = 1, P
*
            IS = IWORK( I )
            ISP1 = IS + 1
            IE = IWORK ( I+1 ) - 1
            MB = IE - IS + 1
            DO 190 J = Q, P + 2, -1
*
               JS = IWORK( J )
               JSP1 = JS + 1
               JE = IWORK( J+1 ) - 1
               NB = JE - JS + 1
               ZDIM = MB*NB*2
               IF( ( MB.EQ.1 ) .AND. ( NB.EQ.1 ) ) THEN
*
*                 Build a 2-by-2 system Z**T * x = RHS
*
                  Z( 1, 1 ) = A( IS, IS )
                  Z( 2, 1 ) = -B( JS, JS )
                  Z( 1, 2 ) = D( IS, IS )
                  Z( 2, 2 ) = -E( JS, JS )
*
*                 Set up right hand side(s)
*
                  RHS( 1 ) = C( IS, JS )
                  RHS( 2 ) = F( IS, JS )
*
*                 Solve Z**T * x = RHS
*
                  CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
                  IF( IERR.GT.0 )
     $               INFO = IERR
*
                  CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
     $                         SCALOC )
                  IF( SCALOC.NE.ONE ) THEN
                     DO 130 K = 1, N
                        CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
                        CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
  130                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
*
*                 Unpack solution vector(s)
*
                  C( IS, JS ) = RHS( 1 )
                  F( IS, JS ) = RHS( 2 )
*
*                 Substitute R(I, J) and L(I, J) into remaining
*                 equation.
*
                  IF( J.GT.P+2 ) THEN
                     ALPHA = RHS( 1 )
                     CALL DAXPY( JS-1, ALPHA, B( 1, JS ), 1, F( IS,
     $                           1 ),
     $                           LDF )
                     ALPHA = RHS( 2 )
                     CALL DAXPY( JS-1, ALPHA, E( 1, JS ), 1, F( IS,
     $                           1 ),
     $                           LDF )
                  END IF
                  IF( I.LT.P ) THEN
                     ALPHA = -RHS( 1 )
                     CALL DAXPY( M-IE, ALPHA, A( IS, IE+1 ), LDA,
     $                           C( IE+1, JS ), 1 )
                     ALPHA = -RHS( 2 )
                     CALL DAXPY( M-IE, ALPHA, D( IS, IE+1 ), LDD,
     $                           C( IE+1, JS ), 1 )
                  END IF
*
               ELSE IF( ( MB.EQ.1 ) .AND. ( NB.EQ.2 ) ) THEN
*
*                 Build a 4-by-4 system Z**T * x = RHS
*
                  Z( 1, 1 ) = A( IS, IS )
                  Z( 2, 1 ) = ZERO
                  Z( 3, 1 ) = -B( JS, JS )
                  Z( 4, 1 ) = -B( JSP1, JS )
*
                  Z( 1, 2 ) = ZERO
                  Z( 2, 2 ) = A( IS, IS )
                  Z( 3, 2 ) = -B( JS, JSP1 )
                  Z( 4, 2 ) = -B( JSP1, JSP1 )
*
                  Z( 1, 3 ) = D( IS, IS )
                  Z( 2, 3 ) = ZERO
                  Z( 3, 3 ) = -E( JS, JS )
                  Z( 4, 3 ) = ZERO
*
                  Z( 1, 4 ) = ZERO
                  Z( 2, 4 ) = D( IS, IS )
                  Z( 3, 4 ) = -E( JS, JSP1 )
                  Z( 4, 4 ) = -E( JSP1, JSP1 )
*
*                 Set up right hand side(s)
*
                  RHS( 1 ) = C( IS, JS )
                  RHS( 2 ) = C( IS, JSP1 )
                  RHS( 3 ) = F( IS, JS )
                  RHS( 4 ) = F( IS, JSP1 )
*
*                 Solve Z**T * x = RHS
*
                  CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
                  IF( IERR.GT.0 )
     $               INFO = IERR
                  CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
     $                         SCALOC )
                  IF( SCALOC.NE.ONE ) THEN
                     DO 140 K = 1, N
                        CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
                        CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
  140                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
*
*                 Unpack solution vector(s)
*
                  C( IS, JS ) = RHS( 1 )
                  C( IS, JSP1 ) = RHS( 2 )
                  F( IS, JS ) = RHS( 3 )
                  F( IS, JSP1 ) = RHS( 4 )
*
*                 Substitute R(I, J) and L(I, J) into remaining
*                 equation.
*
                  IF( J.GT.P+2 ) THEN
                     CALL DAXPY( JS-1, RHS( 1 ), B( 1, JS ), 1,
     $                           F( IS, 1 ), LDF )
                     CALL DAXPY( JS-1, RHS( 2 ), B( 1, JSP1 ), 1,
     $                           F( IS, 1 ), LDF )
                     CALL DAXPY( JS-1, RHS( 3 ), E( 1, JS ), 1,
     $                           F( IS, 1 ), LDF )
                     CALL DAXPY( JS-1, RHS( 4 ), E( 1, JSP1 ), 1,
     $                           F( IS, 1 ), LDF )
                  END IF
                  IF( I.LT.P ) THEN
                     CALL DGER( M-IE, NB, -ONE, A( IS, IE+1 ), LDA,
     $                          RHS( 1 ), 1, C( IE+1, JS ), LDC )
                     CALL DGER( M-IE, NB, -ONE, D( IS, IE+1 ), LDD,
     $                          RHS( 3 ), 1, C( IE+1, JS ), LDC )
                  END IF
*
               ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.1 ) ) THEN
*
*                 Build a 4-by-4 system Z**T * x = RHS
*
                  Z( 1, 1 ) = A( IS, IS )
                  Z( 2, 1 ) = A( IS, ISP1 )
                  Z( 3, 1 ) = -B( JS, JS )
                  Z( 4, 1 ) = ZERO
*
                  Z( 1, 2 ) = A( ISP1, IS )
                  Z( 2, 2 ) = A( ISP1, ISP1 )
                  Z( 3, 2 ) = ZERO
                  Z( 4, 2 ) = -B( JS, JS )
*
                  Z( 1, 3 ) = D( IS, IS )
                  Z( 2, 3 ) = D( IS, ISP1 )
                  Z( 3, 3 ) = -E( JS, JS )
                  Z( 4, 3 ) = ZERO
*
                  Z( 1, 4 ) = ZERO
                  Z( 2, 4 ) = D( ISP1, ISP1 )
                  Z( 3, 4 ) = ZERO
                  Z( 4, 4 ) = -E( JS, JS )
*
*                 Set up right hand side(s)
*
                  RHS( 1 ) = C( IS, JS )
                  RHS( 2 ) = C( ISP1, JS )
                  RHS( 3 ) = F( IS, JS )
                  RHS( 4 ) = F( ISP1, JS )
*
*                 Solve Z**T * x = RHS
*
                  CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
                  IF( IERR.GT.0 )
     $               INFO = IERR
*
                  CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
     $                         SCALOC )
                  IF( SCALOC.NE.ONE ) THEN
                     DO 150 K = 1, N
                        CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
                        CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
  150                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
*
*                 Unpack solution vector(s)
*
                  C( IS, JS ) = RHS( 1 )
                  C( ISP1, JS ) = RHS( 2 )
                  F( IS, JS ) = RHS( 3 )
                  F( ISP1, JS ) = RHS( 4 )
*
*                 Substitute R(I, J) and L(I, J) into remaining
*                 equation.
*
                  IF( J.GT.P+2 ) THEN
                     CALL DGER( MB, JS-1, ONE, RHS( 1 ), 1, B( 1,
     $                          JS ),
     $                          1, F( IS, 1 ), LDF )
                     CALL DGER( MB, JS-1, ONE, RHS( 3 ), 1, E( 1,
     $                          JS ),
     $                          1, F( IS, 1 ), LDF )
                  END IF
                  IF( I.LT.P ) THEN
                     CALL DGEMV( 'T', MB, M-IE, -ONE, A( IS, IE+1 ),
     $                           LDA, RHS( 1 ), 1, ONE, C( IE+1, JS ),
     $                           1 )
                     CALL DGEMV( 'T', MB, M-IE, -ONE, D( IS, IE+1 ),
     $                           LDD, RHS( 3 ), 1, ONE, C( IE+1, JS ),
     $                           1 )
                  END IF
*
               ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.2 ) ) THEN
*
*                 Build an 8-by-8 system Z**T * x = RHS
*
                  CALL DLASET( 'F', LDZ, LDZ, ZERO, ZERO, Z, LDZ )
*
                  Z( 1, 1 ) = A( IS, IS )
                  Z( 2, 1 ) = A( IS, ISP1 )
                  Z( 5, 1 ) = -B( JS, JS )
                  Z( 7, 1 ) = -B( JSP1, JS )
*
                  Z( 1, 2 ) = A( ISP1, IS )
                  Z( 2, 2 ) = A( ISP1, ISP1 )
                  Z( 6, 2 ) = -B( JS, JS )
                  Z( 8, 2 ) = -B( JSP1, JS )
*
                  Z( 3, 3 ) = A( IS, IS )
                  Z( 4, 3 ) = A( IS, ISP1 )
                  Z( 5, 3 ) = -B( JS, JSP1 )
                  Z( 7, 3 ) = -B( JSP1, JSP1 )
*
                  Z( 3, 4 ) = A( ISP1, IS )
                  Z( 4, 4 ) = A( ISP1, ISP1 )
                  Z( 6, 4 ) = -B( JS, JSP1 )
                  Z( 8, 4 ) = -B( JSP1, JSP1 )
*
                  Z( 1, 5 ) = D( IS, IS )
                  Z( 2, 5 ) = D( IS, ISP1 )
                  Z( 5, 5 ) = -E( JS, JS )
*
                  Z( 2, 6 ) = D( ISP1, ISP1 )
                  Z( 6, 6 ) = -E( JS, JS )
*
                  Z( 3, 7 ) = D( IS, IS )
                  Z( 4, 7 ) = D( IS, ISP1 )
                  Z( 5, 7 ) = -E( JS, JSP1 )
                  Z( 7, 7 ) = -E( JSP1, JSP1 )
*
                  Z( 4, 8 ) = D( ISP1, ISP1 )
                  Z( 6, 8 ) = -E( JS, JSP1 )
                  Z( 8, 8 ) = -E( JSP1, JSP1 )
*
*                 Set up right hand side(s)
*
                  K = 1
                  II = MB*NB + 1
                  DO 160 JJ = 0, NB - 1
                     CALL DCOPY( MB, C( IS, JS+JJ ), 1, RHS( K ), 1 )
                     CALL DCOPY( MB, F( IS, JS+JJ ), 1, RHS( II ),
     $                           1 )
                     K = K + MB
                     II = II + MB
  160             CONTINUE
*
*
*                 Solve Z**T * x = RHS
*
                  CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
                  IF( IERR.GT.0 )
     $               INFO = IERR
*
                  CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
     $                         SCALOC )
                  IF( SCALOC.NE.ONE ) THEN
                     DO 170 K = 1, N
                        CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
                        CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
  170                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
*
*                 Unpack solution vector(s)
*
                  K = 1
                  II = MB*NB + 1
                  DO 180 JJ = 0, NB - 1
                     CALL DCOPY( MB, RHS( K ), 1, C( IS, JS+JJ ), 1 )
                     CALL DCOPY( MB, RHS( II ), 1, F( IS, JS+JJ ),
     $                           1 )
                     K = K + MB
                     II = II + MB
  180             CONTINUE
*
*                 Substitute R(I, J) and L(I, J) into remaining
*                 equation.
*
                  IF( J.GT.P+2 ) THEN
                     CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE,
     $                           C( IS, JS ), LDC, B( 1, JS ), LDB, ONE,
     $                           F( IS, 1 ), LDF )
                     CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE,
     $                           F( IS, JS ), LDF, E( 1, JS ), LDE, ONE,
     $                           F( IS, 1 ), LDF )
                  END IF
                  IF( I.LT.P ) THEN
                     CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE,
     $                           A( IS, IE+1 ), LDA, C( IS, JS ), LDC,
     $                           ONE, C( IE+1, JS ), LDC )
                     CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE,
     $                           D( IS, IE+1 ), LDD, F( IS, JS ), LDF,
     $                           ONE, C( IE+1, JS ), LDC )
                  END IF
*
               END IF
*
  190       CONTINUE
  200    CONTINUE
*
      END IF
      RETURN
*
*     End of DTGSY2
*
      END
*> \brief \b DTGSYL
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTGSYL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgsyl.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgsyl.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgsyl.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTGSYL( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
*                          LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK,
*                          IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANS
*       INTEGER            IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF,
*      $                   LWORK, M, N
*       DOUBLE PRECISION   DIF, SCALE
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * ),
*      $                   D( LDD, * ), E( LDE, * ), F( LDF, * ),
*      $                   WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTGSYL solves the generalized Sylvester equation:
*>
*>             A * R - L * B = scale * C                 (1)
*>             D * R - L * E = scale * F
*>
*> where R and L are unknown m-by-n matrices, (A, D), (B, E) and
*> (C, F) are given matrix pairs of size m-by-m, n-by-n and m-by-n,
*> respectively, with real entries. (A, D) and (B, E) must be in
*> generalized (real) Schur canonical form, i.e. A, B are upper quasi
*> triangular and D, E are upper triangular.
*>
*> The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output
*> scaling factor chosen to avoid overflow.
*>
*> In matrix notation (1) is equivalent to solve  Zx = scale b, where
*> Z is defined as
*>
*>            Z = [ kron(In, A)  -kron(B**T, Im) ]         (2)
*>                [ kron(In, D)  -kron(E**T, Im) ].
*>
*> Here Ik is the identity matrix of size k and X**T is the transpose of
*> X. kron(X, Y) is the Kronecker product between the matrices X and Y.
*>
*> If TRANS = 'T', DTGSYL solves the transposed system Z**T*y = scale*b,
*> which is equivalent to solve for R and L in
*>
*>             A**T * R + D**T * L = scale * C           (3)
*>             R * B**T + L * E**T = scale * -F
*>
*> This case (TRANS = 'T') is used to compute an one-norm-based estimate
*> of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)
*> and (B,E), using DLACON.
*>
*> If IJOB >= 1, DTGSYL computes a Frobenius norm-based estimate
*> of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the
*> reciprocal of the smallest singular value of Z. See [1-2] for more
*> information.
*>
*> This is a level 3 BLAS algorithm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N': solve the generalized Sylvester equation (1).
*>          = 'T': solve the 'transposed' system (3).
*> \endverbatim
*>
*> \param[in] IJOB
*> \verbatim
*>          IJOB is INTEGER
*>          Specifies what kind of functionality to be performed.
*>          = 0: solve (1) only.
*>          = 1: The functionality of 0 and 3.
*>          = 2: The functionality of 0 and 4.
*>          = 3: Only an estimate of Dif[(A,D), (B,E)] is computed.
*>               (look ahead strategy IJOB  = 1 is used).
*>          = 4: Only an estimate of Dif[(A,D), (B,E)] is computed.
*>               ( DGECON on sub-systems is used ).
*>          Not referenced if TRANS = 'T'.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The order of the matrices A and D, and the row dimension of
*>          the matrices C, F, R and L.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices B and E, and the column dimension
*>          of the matrices C, F, R and L.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, M)
*>          The upper quasi triangular matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1, M).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB, N)
*>          The upper quasi triangular matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1, N).
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC, N)
*>          On entry, C contains the right-hand-side of the first matrix
*>          equation in (1) or (3).
*>          On exit, if IJOB = 0, 1 or 2, C has been overwritten by
*>          the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,
*>          the solution achieved during the computation of the
*>          Dif-estimate.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1, M).
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (LDD, M)
*>          The upper triangular matrix D.
*> \endverbatim
*>
*> \param[in] LDD
*> \verbatim
*>          LDD is INTEGER
*>          The leading dimension of the array D. LDD >= max(1, M).
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (LDE, N)
*>          The upper triangular matrix E.
*> \endverbatim
*>
*> \param[in] LDE
*> \verbatim
*>          LDE is INTEGER
*>          The leading dimension of the array E. LDE >= max(1, N).
*> \endverbatim
*>
*> \param[in,out] F
*> \verbatim
*>          F is DOUBLE PRECISION array, dimension (LDF, N)
*>          On entry, F contains the right-hand-side of the second matrix
*>          equation in (1) or (3).
*>          On exit, if IJOB = 0, 1 or 2, F has been overwritten by
*>          the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,
*>          the solution achieved during the computation of the
*>          Dif-estimate.
*> \endverbatim
*>
*> \param[in] LDF
*> \verbatim
*>          LDF is INTEGER
*>          The leading dimension of the array F. LDF >= max(1, M).
*> \endverbatim
*>
*> \param[out] DIF
*> \verbatim
*>          DIF is DOUBLE PRECISION
*>          On exit DIF is the reciprocal of a lower bound of the
*>          reciprocal of the Dif-function, i.e. DIF is an upper bound of
*>          Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2).
*>          IF IJOB = 0 or TRANS = 'T', DIF is not touched.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*>          SCALE is DOUBLE PRECISION
*>          On exit SCALE is the scaling factor in (1) or (3).
*>          If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,
*>          to a slightly perturbed system but the input matrices A, B, D
*>          and E have not been changed. If SCALE = 0, C and F hold the
*>          solutions R and L, respectively, to the homogeneous system
*>          with C = F = 0. Normally, SCALE = 1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK > = 1.
*>          If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (M+N+6)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>            =0: successful exit
*>            <0: If INFO = -i, the i-th argument had an illegal value.
*>            >0: (A, D) and (B, E) have common or close eigenvalues.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tgsyl
*
*> \par Contributors:
*  ==================
*>
*>     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*>     Umea University, S-901 87 Umea, Sweden.
*
*> \par References:
*  ================
*>
*> \verbatim
*>
*>  [1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
*>      for Solving the Generalized Sylvester Equation and Estimating the
*>      Separation between Regular Matrix Pairs, Report UMINF - 93.23,
*>      Department of Computing Science, Umea University, S-901 87 Umea,
*>      Sweden, December 1993, Revised April 1994, Also as LAPACK Working
*>      Note 75.  To appear in ACM Trans. on Math. Software, Vol 22,
*>      No 1, 1996.
*>
*>  [2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
*>      Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.
*>      Appl., 15(4):1045-1060, 1994
*>
*>  [3] B. Kagstrom and L. Westin, Generalized Schur Methods with
*>      Condition Estimators for Solving the Generalized Sylvester
*>      Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,
*>      July 1989, pp 745-751.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTGSYL( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC,
     $                   D,
     $                   LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK,
     $                   IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS
      INTEGER            IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF,
     $                   LWORK, M, N
      DOUBLE PRECISION   DIF, SCALE
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * ),
     $                   D( LDD, * ), E( LDE, * ), F( LDF, * ),
     $                   WORK( * )
*     ..
*
*  =====================================================================
*  Replaced various illegal calls to DCOPY by calls to DLASET.
*  Sven Hammarling, 1/5/02.
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, NOTRAN
      INTEGER            I, IE, IFUNC, IROUND, IS, ISOLVE, J, JE, JS, K,
     $                   LINFO, LWMIN, MB, NB, P, PPQQ, PQ, Q
      DOUBLE PRECISION   DSCALE, DSUM, SCALE2, SCALOC
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMM, DLACPY, DLASET, DSCAL, DTGSY2,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Decode and test input parameters
*
      INFO = 0
      NOTRAN = LSAME( TRANS, 'N' )
      LQUERY = ( LWORK.EQ.-1 )
*
      IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
         INFO = -1
      ELSE IF( NOTRAN ) THEN
         IF( ( IJOB.LT.0 ) .OR. ( IJOB.GT.4 ) ) THEN
            INFO = -2
         END IF
      END IF
      IF( INFO.EQ.0 ) THEN
         IF( M.LE.0 ) THEN
            INFO = -3
         ELSE IF( N.LE.0 ) THEN
            INFO = -4
         ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
            INFO = -6
         ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
            INFO = -8
         ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
            INFO = -10
         ELSE IF( LDD.LT.MAX( 1, M ) ) THEN
            INFO = -12
         ELSE IF( LDE.LT.MAX( 1, N ) ) THEN
            INFO = -14
         ELSE IF( LDF.LT.MAX( 1, M ) ) THEN
            INFO = -16
         END IF
      END IF
*
      IF( INFO.EQ.0 ) THEN
         IF( NOTRAN ) THEN
            IF( IJOB.EQ.1 .OR. IJOB.EQ.2 ) THEN
               LWMIN = MAX( 1, 2*M*N )
            ELSE
               LWMIN = 1
            END IF
         ELSE
            LWMIN = 1
         END IF
         WORK( 1 ) = LWMIN
*
         IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -20
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTGSYL', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 ) THEN
         SCALE = 1
         IF( NOTRAN ) THEN
            IF( IJOB.NE.0 ) THEN
               DIF = 0
            END IF
         END IF
         RETURN
      END IF
*
*     Determine optimal block sizes MB and NB
*
      MB = ILAENV( 2, 'DTGSYL', TRANS, M, N, -1, -1 )
      NB = ILAENV( 5, 'DTGSYL', TRANS, M, N, -1, -1 )
*
      ISOLVE = 1
      IFUNC = 0
      IF( NOTRAN ) THEN
         IF( IJOB.GE.3 ) THEN
            IFUNC = IJOB - 2
            CALL DLASET( 'F', M, N, ZERO, ZERO, C, LDC )
            CALL DLASET( 'F', M, N, ZERO, ZERO, F, LDF )
         ELSE IF( IJOB.GE.1 ) THEN
            ISOLVE = 2
         END IF
      END IF
*
      IF( ( MB.LE.1 .AND. NB.LE.1 ) .OR. ( MB.GE.M .AND. NB.GE.N ) )
     $     THEN
*
         DO 30 IROUND = 1, ISOLVE
*
*           Use unblocked Level 2 solver
*
            DSCALE = ZERO
            DSUM = ONE
            PQ = 0
            CALL DTGSY2( TRANS, IFUNC, M, N, A, LDA, B, LDB, C, LDC,
     $                   D,
     $                   LDD, E, LDE, F, LDF, SCALE, DSUM, DSCALE,
     $                   IWORK, PQ, INFO )
            IF( DSCALE.NE.ZERO ) THEN
               IF( IJOB.EQ.1 .OR. IJOB.EQ.3 ) THEN
                  DIF = SQRT( DBLE( 2*M*N ) ) / ( DSCALE*SQRT( DSUM ) )
               ELSE
                  DIF = SQRT( DBLE( PQ ) ) / ( DSCALE*SQRT( DSUM ) )
               END IF
            END IF
*
            IF( ISOLVE.EQ.2 .AND. IROUND.EQ.1 ) THEN
               IF( NOTRAN ) THEN
                  IFUNC = IJOB
               END IF
               SCALE2 = SCALE
               CALL DLACPY( 'F', M, N, C, LDC, WORK, M )
               CALL DLACPY( 'F', M, N, F, LDF, WORK( M*N+1 ), M )
               CALL DLASET( 'F', M, N, ZERO, ZERO, C, LDC )
               CALL DLASET( 'F', M, N, ZERO, ZERO, F, LDF )
            ELSE IF( ISOLVE.EQ.2 .AND. IROUND.EQ.2 ) THEN
               CALL DLACPY( 'F', M, N, WORK, M, C, LDC )
               CALL DLACPY( 'F', M, N, WORK( M*N+1 ), M, F, LDF )
               SCALE = SCALE2
            END IF
   30    CONTINUE
*
         RETURN
      END IF
*
*     Determine block structure of A
*
      P = 0
      I = 1
   40 CONTINUE
      IF( I.GT.M )
     $   GO TO 50
      P = P + 1
      IWORK( P ) = I
      I = I + MB
      IF( I.GE.M )
     $   GO TO 50
      IF( A( I, I-1 ).NE.ZERO )
     $   I = I + 1
      GO TO 40
   50 CONTINUE
*
      IWORK( P+1 ) = M + 1
      IF( IWORK( P ).EQ.IWORK( P+1 ) )
     $   P = P - 1
*
*     Determine block structure of B
*
      Q = P + 1
      J = 1
   60 CONTINUE
      IF( J.GT.N )
     $   GO TO 70
      Q = Q + 1
      IWORK( Q ) = J
      J = J + NB
      IF( J.GE.N )
     $   GO TO 70
      IF( B( J, J-1 ).NE.ZERO )
     $   J = J + 1
      GO TO 60
   70 CONTINUE
*
      IWORK( Q+1 ) = N + 1
      IF( IWORK( Q ).EQ.IWORK( Q+1 ) )
     $   Q = Q - 1
*
      IF( NOTRAN ) THEN
*
         DO 150 IROUND = 1, ISOLVE
*
*           Solve (I, J)-subsystem
*               A(I, I) * R(I, J) - L(I, J) * B(J, J) = C(I, J)
*               D(I, I) * R(I, J) - L(I, J) * E(J, J) = F(I, J)
*           for I = P, P - 1,..., 1; J = 1, 2,..., Q
*
            DSCALE = ZERO
            DSUM = ONE
            PQ = 0
            SCALE = ONE
            DO 130 J = P + 2, Q
               JS = IWORK( J )
               JE = IWORK( J+1 ) - 1
               NB = JE - JS + 1
               DO 120 I = P, 1, -1
                  IS = IWORK( I )
                  IE = IWORK( I+1 ) - 1
                  MB = IE - IS + 1
                  PPQQ = 0
                  CALL DTGSY2( TRANS, IFUNC, MB, NB, A( IS, IS ),
     $                         LDA,
     $                         B( JS, JS ), LDB, C( IS, JS ), LDC,
     $                         D( IS, IS ), LDD, E( JS, JS ), LDE,
     $                         F( IS, JS ), LDF, SCALOC, DSUM, DSCALE,
     $                         IWORK( Q+2 ), PPQQ, LINFO )
                  IF( LINFO.GT.0 )
     $               INFO = LINFO
*
                  PQ = PQ + PPQQ
                  IF( SCALOC.NE.ONE ) THEN
                     DO 80 K = 1, JS - 1
                        CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
                        CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
   80                CONTINUE
                     DO 90 K = JS, JE
                        CALL DSCAL( IS-1, SCALOC, C( 1, K ), 1 )
                        CALL DSCAL( IS-1, SCALOC, F( 1, K ), 1 )
   90                CONTINUE
                     DO 100 K = JS, JE
                        CALL DSCAL( M-IE, SCALOC, C( IE+1, K ), 1 )
                        CALL DSCAL( M-IE, SCALOC, F( IE+1, K ), 1 )
  100                CONTINUE
                     DO 110 K = JE + 1, N
                        CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
                        CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
  110                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
*
*                 Substitute R(I, J) and L(I, J) into remaining
*                 equation.
*
                  IF( I.GT.1 ) THEN
                     CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
     $                           A( 1, IS ), LDA, C( IS, JS ), LDC, ONE,
     $                           C( 1, JS ), LDC )
                     CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
     $                           D( 1, IS ), LDD, C( IS, JS ), LDC, ONE,
     $                           F( 1, JS ), LDF )
                  END IF
                  IF( J.LT.Q ) THEN
                     CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE,
     $                           F( IS, JS ), LDF, B( JS, JE+1 ), LDB,
     $                           ONE, C( IS, JE+1 ), LDC )
                     CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE,
     $                           F( IS, JS ), LDF, E( JS, JE+1 ), LDE,
     $                           ONE, F( IS, JE+1 ), LDF )
                  END IF
  120          CONTINUE
  130       CONTINUE
            IF( DSCALE.NE.ZERO ) THEN
               IF( IJOB.EQ.1 .OR. IJOB.EQ.3 ) THEN
                  DIF = SQRT( DBLE( 2*M*N ) ) / ( DSCALE*SQRT( DSUM ) )
               ELSE
                  DIF = SQRT( DBLE( PQ ) ) / ( DSCALE*SQRT( DSUM ) )
               END IF
            END IF
            IF( ISOLVE.EQ.2 .AND. IROUND.EQ.1 ) THEN
               IF( NOTRAN ) THEN
                  IFUNC = IJOB
               END IF
               SCALE2 = SCALE
               CALL DLACPY( 'F', M, N, C, LDC, WORK, M )
               CALL DLACPY( 'F', M, N, F, LDF, WORK( M*N+1 ), M )
               CALL DLASET( 'F', M, N, ZERO, ZERO, C, LDC )
               CALL DLASET( 'F', M, N, ZERO, ZERO, F, LDF )
            ELSE IF( ISOLVE.EQ.2 .AND. IROUND.EQ.2 ) THEN
               CALL DLACPY( 'F', M, N, WORK, M, C, LDC )
               CALL DLACPY( 'F', M, N, WORK( M*N+1 ), M, F, LDF )
               SCALE = SCALE2
            END IF
  150    CONTINUE
*
      ELSE
*
*        Solve transposed (I, J)-subsystem
*             A(I, I)**T * R(I, J)  + D(I, I)**T * L(I, J)  =  C(I, J)
*             R(I, J)  * B(J, J)**T + L(I, J)  * E(J, J)**T = -F(I, J)
*        for I = 1,2,..., P; J = Q, Q-1,..., 1
*
         SCALE = ONE
         DO 210 I = 1, P
            IS = IWORK( I )
            IE = IWORK( I+1 ) - 1
            MB = IE - IS + 1
            DO 200 J = Q, P + 2, -1
               JS = IWORK( J )
               JE = IWORK( J+1 ) - 1
               NB = JE - JS + 1
               CALL DTGSY2( TRANS, IFUNC, MB, NB, A( IS, IS ), LDA,
     $                      B( JS, JS ), LDB, C( IS, JS ), LDC,
     $                      D( IS, IS ), LDD, E( JS, JS ), LDE,
     $                      F( IS, JS ), LDF, SCALOC, DSUM, DSCALE,
     $                      IWORK( Q+2 ), PPQQ, LINFO )
               IF( LINFO.GT.0 )
     $            INFO = LINFO
               IF( SCALOC.NE.ONE ) THEN
                  DO 160 K = 1, JS - 1
                     CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
                     CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
  160             CONTINUE
                  DO 170 K = JS, JE
                     CALL DSCAL( IS-1, SCALOC, C( 1, K ), 1 )
                     CALL DSCAL( IS-1, SCALOC, F( 1, K ), 1 )
  170             CONTINUE
                  DO 180 K = JS, JE
                     CALL DSCAL( M-IE, SCALOC, C( IE+1, K ), 1 )
                     CALL DSCAL( M-IE, SCALOC, F( IE+1, K ), 1 )
  180             CONTINUE
                  DO 190 K = JE + 1, N
                     CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
                     CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
  190             CONTINUE
                  SCALE = SCALE*SCALOC
               END IF
*
*              Substitute R(I, J) and L(I, J) into remaining equation.
*
               IF( J.GT.P+2 ) THEN
                  CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE, C( IS,
     $                        JS ),
     $                        LDC, B( 1, JS ), LDB, ONE, F( IS, 1 ),
     $                        LDF )
                  CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE, F( IS,
     $                        JS ),
     $                        LDF, E( 1, JS ), LDE, ONE, F( IS, 1 ),
     $                        LDF )
               END IF
               IF( I.LT.P ) THEN
                  CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE,
     $                        A( IS, IE+1 ), LDA, C( IS, JS ), LDC, ONE,
     $                        C( IE+1, JS ), LDC )
                  CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE,
     $                        D( IS, IE+1 ), LDD, F( IS, JS ), LDF, ONE,
     $                        C( IE+1, JS ), LDC )
               END IF
  200       CONTINUE
  210    CONTINUE
*
      END IF
*
      WORK( 1 ) = LWMIN
*
      RETURN
*
*     End of DTGSYL
*
      END
*> \brief \b DTPCON
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTPCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtpcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtpcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtpcon.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTPCON( NORM, UPLO, DIAG, N, AP, RCOND, WORK, IWORK,
*                          INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, NORM, UPLO
*       INTEGER            INFO, N
*       DOUBLE PRECISION   RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   AP( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTPCON estimates the reciprocal of the condition number of a packed
*> triangular matrix A, in either the 1-norm or the infinity-norm.
*>
*> The norm of A is computed and an estimate is obtained for
*> norm(inv(A)), then the reciprocal of the condition number is
*> computed as
*>    RCOND = 1 / ( norm(A) * norm(inv(A)) ).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NORM
*> \verbatim
*>          NORM is CHARACTER*1
*>          Specifies whether the 1-norm condition number or the
*>          infinity-norm condition number is required:
*>          = '1' or 'O':  1-norm;
*>          = 'I':         Infinity-norm.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  A is upper triangular;
*>          = 'L':  A is lower triangular.
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          = 'N':  A is non-unit triangular;
*>          = 'U':  A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          The upper or lower triangular matrix A, packed columnwise in
*>          a linear array.  The j-th column of A is stored in the array
*>          AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*>          If DIAG = 'U', the diagonal elements of A are not referenced
*>          and are assumed to be 1.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The reciprocal of the condition number of the matrix A,
*>          computed as RCOND = 1/(norm(A) * norm(inv(A))).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tpcon
*
*  =====================================================================
      SUBROUTINE DTPCON( NORM, UPLO, DIAG, N, AP, RCOND, WORK, IWORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, NORM, UPLO
      INTEGER            INFO, N
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   AP( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOUNIT, ONENRM, UPPER
      CHARACTER          NORMIN
      INTEGER            IX, KASE, KASE1
      DOUBLE PRECISION   AINVNM, ANORM, SCALE, SMLNUM, XNORM
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH, DLANTP
      EXTERNAL           LSAME, IDAMAX, DLAMCH, DLANTP
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACN2, DLATPS, DRSCL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' )
      NOUNIT = LSAME( DIAG, 'N' )
*
      IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -2
      ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTPCON', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 ) THEN
         RCOND = ONE
         RETURN
      END IF
*
      RCOND = ZERO
      SMLNUM = DLAMCH( 'Safe minimum' )*DBLE( MAX( 1, N ) )
*
*     Compute the norm of the triangular matrix A.
*
      ANORM = DLANTP( NORM, UPLO, DIAG, N, AP, WORK )
*
*     Continue only if ANORM > 0.
*
      IF( ANORM.GT.ZERO ) THEN
*
*        Estimate the norm of the inverse of A.
*
         AINVNM = ZERO
         NORMIN = 'N'
         IF( ONENRM ) THEN
            KASE1 = 1
         ELSE
            KASE1 = 2
         END IF
         KASE = 0
   10    CONTINUE
         CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE,
     $                ISAVE )
         IF( KASE.NE.0 ) THEN
            IF( KASE.EQ.KASE1 ) THEN
*
*              Multiply by inv(A).
*
               CALL DLATPS( UPLO, 'No transpose', DIAG, NORMIN, N,
     $                      AP,
     $                      WORK, SCALE, WORK( 2*N+1 ), INFO )
            ELSE
*
*              Multiply by inv(A**T).
*
               CALL DLATPS( UPLO, 'Transpose', DIAG, NORMIN, N, AP,
     $                      WORK, SCALE, WORK( 2*N+1 ), INFO )
            END IF
            NORMIN = 'Y'
*
*           Multiply by 1/SCALE if doing so will not cause overflow.
*
            IF( SCALE.NE.ONE ) THEN
               IX = IDAMAX( N, WORK, 1 )
               XNORM = ABS( WORK( IX ) )
               IF( SCALE.LT.XNORM*SMLNUM .OR. SCALE.EQ.ZERO )
     $            GO TO 20
               CALL DRSCL( N, SCALE, WORK, 1 )
            END IF
            GO TO 10
         END IF
*
*        Compute the estimate of the reciprocal condition number.
*
         IF( AINVNM.NE.ZERO )
     $      RCOND = ( ONE / ANORM ) / AINVNM
      END IF
*
   20 CONTINUE
      RETURN
*
*     End of DTPCON
*
      END
*> \brief \b DTPMQRT
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTPMQRT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtpmqrt.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtpmqrt.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtpmqrt.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTPMQRT( SIDE, TRANS, M, N, K, L, NB, V, LDV, T, LDT,
*                           A, LDA, B, LDB, WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER SIDE, TRANS
*       INTEGER   INFO, K, LDV, LDA, LDB, M, N, L, NB, LDT
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   V( LDV, * ), A( LDA, * ), B( LDB, * ),
*      $                   T( LDT, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTPMQRT applies a real orthogonal matrix Q obtained from a
*> "triangular-pentagonal" real block reflector H to a general
*> real matrix C, which consists of two blocks A and B.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': apply Q or Q**T from the Left;
*>          = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N':  No transpose, apply Q;
*>          = 'T':  Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix B. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix B. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines
*>          the matrix Q.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*>          L is INTEGER
*>          The order of the trapezoidal part of V.
*>          K >= L >= 0.  See Further Details.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*>          NB is INTEGER
*>          The block size used for the storage of T.  K >= NB >= 1.
*>          This must be the same value of NB used to generate T
*>          in CTPQRT.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension (LDV,K)
*>          The i-th column must contain the vector which defines the
*>          elementary reflector H(i), for i = 1,2,...,k, as returned by
*>          CTPQRT in B.  See Further Details.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of the array V.
*>          If SIDE = 'L', LDV >= max(1,M);
*>          if SIDE = 'R', LDV >= max(1,N).
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,K)
*>          The upper triangular factors of the block reflectors
*>          as returned by CTPQRT, stored as a NB-by-K matrix.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T.  LDT >= NB.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension
*>          (LDA,N) if SIDE = 'L' or
*>          (LDA,K) if SIDE = 'R'
*>          On entry, the K-by-N or M-by-K matrix A.
*>          On exit, A is overwritten by the corresponding block of
*>          Q*C or Q**T*C or C*Q or C*Q**T.  See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.
*>          If SIDE = 'L', LDC >= max(1,K);
*>          If SIDE = 'R', LDC >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,N)
*>          On entry, the M-by-N matrix B.
*>          On exit, B is overwritten by the corresponding block of
*>          Q*C or Q**T*C or C*Q or C*Q**T.  See Further Details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.
*>          LDB >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array. The dimension of WORK is
*>           N*NB if SIDE = 'L', or  M*NB if SIDE = 'R'.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tpmqrt
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The columns of the pentagonal matrix V contain the elementary reflectors
*>  H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a
*>  trapezoidal block V2:
*>
*>        V = [V1]
*>            [V2].
*>
*>  The size of the trapezoidal block V2 is determined by the parameter L,
*>  where 0 <= L <= K; V2 is upper trapezoidal, consisting of the first L
*>  rows of a K-by-K upper triangular matrix.  If L=K, V2 is upper triangular;
*>  if L=0, there is no trapezoidal block, hence V = V1 is rectangular.
*>
*>  If SIDE = 'L':  C = [A]  where A is K-by-N,  B is M-by-N and V is M-by-K.
*>                      [B]
*>
*>  If SIDE = 'R':  C = [A B]  where A is M-by-K, B is M-by-N and V is N-by-K.
*>
*>  The real orthogonal matrix Q is formed from V and T.
*>
*>  If TRANS='N' and SIDE='L', C is on exit replaced with Q * C.
*>
*>  If TRANS='T' and SIDE='L', C is on exit replaced with Q**T * C.
*>
*>  If TRANS='N' and SIDE='R', C is on exit replaced with C * Q.
*>
*>  If TRANS='T' and SIDE='R', C is on exit replaced with C * Q**T.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTPMQRT( SIDE, TRANS, M, N, K, L, NB, V, LDV, T,
     $                    LDT,
     $                    A, LDA, B, LDB, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER SIDE, TRANS
      INTEGER   INFO, K, LDV, LDA, LDB, M, N, L, NB, LDT
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   V( LDV, * ), A( LDA, * ), B( LDB, * ),
     $                   T( LDT, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     ..
*     .. Local Scalars ..
      LOGICAL            LEFT, RIGHT, TRAN, NOTRAN
      INTEGER            I, IB, MB, LB, KF, LDAQ, LDVQ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DTPRFB, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     .. Test the input arguments ..
*
      INFO   = 0
      LEFT   = LSAME( SIDE,  'L' )
      RIGHT  = LSAME( SIDE,  'R' )
      TRAN   = LSAME( TRANS, 'T' )
      NOTRAN = LSAME( TRANS, 'N' )
*
      IF ( LEFT ) THEN
         LDVQ = MAX( 1, M )
         LDAQ = MAX( 1, K )
      ELSE IF ( RIGHT ) THEN
         LDVQ = MAX( 1, N )
         LDAQ = MAX( 1, M )
      END IF
      IF( .NOT.LEFT .AND. .NOT.RIGHT ) THEN
         INFO = -1
      ELSE IF( .NOT.TRAN .AND. .NOT.NOTRAN ) THEN
         INFO = -2
      ELSE IF( M.LT.0 ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( K.LT.0 ) THEN
         INFO = -5
      ELSE IF( L.LT.0 .OR. L.GT.K ) THEN
         INFO = -6
      ELSE IF( NB.LT.1 .OR. (NB.GT.K .AND. K.GT.0) ) THEN
         INFO = -7
      ELSE IF( LDV.LT.LDVQ ) THEN
         INFO = -9
      ELSE IF( LDT.LT.NB ) THEN
         INFO = -11
      ELSE IF( LDA.LT.LDAQ ) THEN
         INFO = -13
      ELSE IF( LDB.LT.MAX( 1, M ) ) THEN
         INFO = -15
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTPMQRT', -INFO )
         RETURN
      END IF
*
*     .. Quick return if possible ..
*
      IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) RETURN
*
      IF( LEFT .AND. TRAN ) THEN
*
         DO I = 1, K, NB
            IB = MIN( NB, K-I+1 )
            MB = MIN( M-L+I+IB-1, M )
            IF( I.GE.L ) THEN
               LB = 0
            ELSE
               LB = MB-M+L-I+1
            END IF
            CALL DTPRFB( 'L', 'T', 'F', 'C', MB, N, IB, LB,
     $                   V( 1, I ), LDV, T( 1, I ), LDT,
     $                   A( I, 1 ), LDA, B, LDB, WORK, IB )
         END DO
*
      ELSE IF( RIGHT .AND. NOTRAN ) THEN
*
         DO I = 1, K, NB
            IB = MIN( NB, K-I+1 )
            MB = MIN( N-L+I+IB-1, N )
            IF( I.GE.L ) THEN
               LB = 0
            ELSE
               LB = MB-N+L-I+1
            END IF
            CALL DTPRFB( 'R', 'N', 'F', 'C', M, MB, IB, LB,
     $                   V( 1, I ), LDV, T( 1, I ), LDT,
     $                   A( 1, I ), LDA, B, LDB, WORK, M )
         END DO
*
      ELSE IF( LEFT .AND. NOTRAN ) THEN
*
         KF = ((K-1)/NB)*NB+1
         DO I = KF, 1, -NB
            IB = MIN( NB, K-I+1 )
            MB = MIN( M-L+I+IB-1, M )
            IF( I.GE.L ) THEN
               LB = 0
            ELSE
               LB = MB-M+L-I+1
            END IF
            CALL DTPRFB( 'L', 'N', 'F', 'C', MB, N, IB, LB,
     $                   V( 1, I ), LDV, T( 1, I ), LDT,
     $                   A( I, 1 ), LDA, B, LDB, WORK, IB )
         END DO
*
      ELSE IF( RIGHT .AND. TRAN ) THEN
*
         KF = ((K-1)/NB)*NB+1
         DO I = KF, 1, -NB
            IB = MIN( NB, K-I+1 )
            MB = MIN( N-L+I+IB-1, N )
            IF( I.GE.L ) THEN
               LB = 0
            ELSE
               LB = MB-N+L-I+1
            END IF
            CALL DTPRFB( 'R', 'T', 'F', 'C', M, MB, IB, LB,
     $                   V( 1, I ), LDV, T( 1, I ), LDT,
     $                   A( 1, I ), LDA, B, LDB, WORK, M )
         END DO
*
      END IF
*
      RETURN
*
*     End of DTPMQRT
*
      END
*> \brief \b DTPQRT
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTPQRT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtpqrt.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtpqrt.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtpqrt.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTPQRT( M, N, L, NB, A, LDA, B, LDB, T, LDT, WORK,
*                          INFO )
*
*       .. Scalar Arguments ..
*       INTEGER INFO, LDA, LDB, LDT, N, M, L, NB
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION A( LDA, * ), B( LDB, * ), T( LDT, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTPQRT computes a blocked QR factorization of a real
*> "triangular-pentagonal" matrix C, which is composed of a
*> triangular block A and pentagonal block B, using the compact
*> WY representation for Q.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix B.
*>          M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix B, and the order of the
*>          triangular matrix A.
*>          N >= 0.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*>          L is INTEGER
*>          The number of rows of the upper trapezoidal part of B.
*>          MIN(M,N) >= L >= 0.  See Further Details.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*>          NB is INTEGER
*>          The block size to be used in the blocked QR.  N >= NB >= 1.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the upper triangular N-by-N matrix A.
*>          On exit, the elements on and above the diagonal of the array
*>          contain the upper triangular matrix R.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,N)
*>          On entry, the pentagonal M-by-N matrix B.  The first M-L rows
*>          are rectangular, and the last L rows are upper trapezoidal.
*>          On exit, B contains the pentagonal matrix V.  See Further Details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,M).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,N)
*>          The upper triangular block reflectors stored in compact form
*>          as a sequence of upper triangular blocks.  See Further Details.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T.  LDT >= NB.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (NB*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tpqrt
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The input matrix C is a (N+M)-by-N matrix
*>
*>               C = [ A ]
*>                   [ B ]
*>
*>  where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal
*>  matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N
*>  upper trapezoidal matrix B2:
*>
*>               B = [ B1 ]  <- (M-L)-by-N rectangular
*>                   [ B2 ]  <-     L-by-N upper trapezoidal.
*>
*>  The upper trapezoidal matrix B2 consists of the first L rows of a
*>  N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
*>  B is rectangular M-by-N; if M=L=N, B is upper triangular.
*>
*>  The matrix W stores the elementary reflectors H(i) in the i-th column
*>  below the diagonal (of A) in the (N+M)-by-N input matrix C
*>
*>               C = [ A ]  <- upper triangular N-by-N
*>                   [ B ]  <- M-by-N pentagonal
*>
*>  so that W can be represented as
*>
*>               W = [ I ]  <- identity, N-by-N
*>                   [ V ]  <- M-by-N, same form as B.
*>
*>  Thus, all of information needed for W is contained on exit in B, which
*>  we call V above.  Note that V has the same form as B; that is,
*>
*>               V = [ V1 ] <- (M-L)-by-N rectangular
*>                   [ V2 ] <-     L-by-N upper trapezoidal.
*>
*>  The columns of V represent the vectors which define the H(i)'s.
*>
*>  The number of blocks is B = ceiling(N/NB), where each
*>  block is of order NB except for the last block, which is of order
*>  IB = N - (B-1)*NB.  For each of the B blocks, a upper triangular block
*>  reflector factor is computed: T1, T2, ..., TB.  The NB-by-NB (and IB-by-IB
*>  for the last block) T's are stored in the NB-by-N matrix T as
*>
*>               T = [T1 T2 ... TB].
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTPQRT( M, N, L, NB, A, LDA, B, LDB, T, LDT, WORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER INFO, LDA, LDB, LDT, N, M, L, NB
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION A( LDA, * ), B( LDB, * ), T( LDT, * ), WORK( * )
*     ..
*
* =====================================================================
*
*     ..
*     .. Local Scalars ..
      INTEGER    I, IB, LB, MB, IINFO
*     ..
*     .. External Subroutines ..
      EXTERNAL   DTPQRT2, DTPRFB, XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( L.LT.0 .OR. (L.GT.MIN(M,N) .AND. MIN(M,N).GE.0)) THEN
         INFO = -3
      ELSE IF( NB.LT.1 .OR. (NB.GT.N .AND. N.GT.0)) THEN
         INFO = -4
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -6
      ELSE IF( LDB.LT.MAX( 1, M ) ) THEN
         INFO = -8
      ELSE IF( LDT.LT.NB ) THEN
         INFO = -10
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTPQRT', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 ) RETURN
*
      DO I = 1, N, NB
*
*     Compute the QR factorization of the current block
*
         IB = MIN( N-I+1, NB )
         MB = MIN( M-L+I+IB-1, M )
         IF( I.GE.L ) THEN
            LB = 0
         ELSE
            LB = MB-M+L-I+1
         END IF
*
         CALL DTPQRT2( MB, IB, LB, A(I,I), LDA, B( 1, I ), LDB,
     $                 T(1, I ), LDT, IINFO )
*
*     Update by applying H**T to B(:,I+IB:N) from the left
*
         IF( I+IB.LE.N ) THEN
            CALL DTPRFB( 'L', 'T', 'F', 'C', MB, N-I-IB+1, IB, LB,
     $                    B( 1, I ), LDB, T( 1, I ), LDT,
     $                    A( I, I+IB ), LDA, B( 1, I+IB ), LDB,
     $                    WORK, IB )
         END IF
      END DO
      RETURN
*
*     End of DTPQRT
*
      END
*> \brief \b DTPQRT2 computes a QR factorization of a real or complex "triangular-pentagonal" matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTPQRT2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtpqrt2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtpqrt2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtpqrt2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTPQRT2( M, N, L, A, LDA, B, LDB, T, LDT, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER   INFO, LDA, LDB, LDT, N, M, L
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), T( LDT, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTPQRT2 computes a QR factorization of a real "triangular-pentagonal"
*> matrix C, which is composed of a triangular block A and pentagonal block B,
*> using the compact WY representation for Q.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The total number of rows of the matrix B.
*>          M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix B, and the order of
*>          the triangular matrix A.
*>          N >= 0.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*>          L is INTEGER
*>          The number of rows of the upper trapezoidal part of B.
*>          MIN(M,N) >= L >= 0.  See Further Details.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the upper triangular N-by-N matrix A.
*>          On exit, the elements on and above the diagonal of the array
*>          contain the upper triangular matrix R.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,N)
*>          On entry, the pentagonal M-by-N matrix B.  The first M-L rows
*>          are rectangular, and the last L rows are upper trapezoidal.
*>          On exit, B contains the pentagonal matrix V.  See Further Details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,M).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,N)
*>          The N-by-N upper triangular factor T of the block reflector.
*>          See Further Details.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T.  LDT >= max(1,N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tpqrt2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The input matrix C is a (N+M)-by-N matrix
*>
*>               C = [ A ]
*>                   [ B ]
*>
*>  where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal
*>  matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N
*>  upper trapezoidal matrix B2:
*>
*>               B = [ B1 ]  <- (M-L)-by-N rectangular
*>                   [ B2 ]  <-     L-by-N upper trapezoidal.
*>
*>  The upper trapezoidal matrix B2 consists of the first L rows of a
*>  N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
*>  B is rectangular M-by-N; if M=L=N, B is upper triangular.
*>
*>  The matrix W stores the elementary reflectors H(i) in the i-th column
*>  below the diagonal (of A) in the (N+M)-by-N input matrix C
*>
*>               C = [ A ]  <- upper triangular N-by-N
*>                   [ B ]  <- M-by-N pentagonal
*>
*>  so that W can be represented as
*>
*>               W = [ I ]  <- identity, N-by-N
*>                   [ V ]  <- M-by-N, same form as B.
*>
*>  Thus, all of information needed for W is contained on exit in B, which
*>  we call V above.  Note that V has the same form as B; that is,
*>
*>               V = [ V1 ] <- (M-L)-by-N rectangular
*>                   [ V2 ] <-     L-by-N upper trapezoidal.
*>
*>  The columns of V represent the vectors which define the H(i)'s.
*>  The (M+N)-by-(M+N) block reflector H is then given by
*>
*>               H = I - W * T * W**T
*>
*>  where W^H is the conjugate transpose of W and T is the upper triangular
*>  factor of the block reflector.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTPQRT2( M, N, L, A, LDA, B, LDB, T, LDT, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER   INFO, LDA, LDB, LDT, N, M, L
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), T( LDT, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION  ONE, ZERO
      PARAMETER( ONE = 1.0, ZERO = 0.0 )
*     ..
*     .. Local Scalars ..
      INTEGER   I, J, P, MP, NP
      DOUBLE PRECISION   ALPHA
*     ..
*     .. External Subroutines ..
      EXTERNAL  DLARFG, DGEMV, DGER, DTRMV, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( L.LT.0 .OR. L.GT.MIN(M,N) ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, M ) ) THEN
         INFO = -7
      ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
         INFO = -9
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTPQRT2', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. M.EQ.0 ) RETURN
*
      DO I = 1, N
*
*        Generate elementary reflector H(I) to annihilate B(:,I)
*
         P = M-L+MIN( L, I )
         CALL DLARFG( P+1, A( I, I ), B( 1, I ), 1, T( I, 1 ) )
         IF( I.LT.N ) THEN
*
*           W(1:N-I) := C(I:M,I+1:N)^H * C(I:M,I) [use W = T(:,N)]
*
            DO J = 1, N-I
               T( J, N ) = (A( I, I+J ))
            END DO
            CALL DGEMV( 'T', P, N-I, ONE, B( 1, I+1 ), LDB,
     $                  B( 1, I ), 1, ONE, T( 1, N ), 1 )
*
*           C(I:M,I+1:N) = C(I:m,I+1:N) + alpha*C(I:M,I)*W(1:N-1)^H
*
            ALPHA = -(T( I, 1 ))
            DO J = 1, N-I
               A( I, I+J ) = A( I, I+J ) + ALPHA*(T( J, N ))
            END DO
            CALL DGER( P, N-I, ALPHA, B( 1, I ), 1,
     $           T( 1, N ), 1, B( 1, I+1 ), LDB )
         END IF
      END DO
*
      DO I = 2, N
*
*        T(1:I-1,I) := C(I:M,1:I-1)^H * (alpha * C(I:M,I))
*
         ALPHA = -T( I, 1 )

         DO J = 1, I-1
            T( J, I ) = ZERO
         END DO
         P = MIN( I-1, L )
         MP = MIN( M-L+1, M )
         NP = MIN( P+1, N )
*
*        Triangular part of B2
*
         DO J = 1, P
            T( J, I ) = ALPHA*B( M-L+J, I )
         END DO
         CALL DTRMV( 'U', 'T', 'N', P, B( MP, 1 ), LDB,
     $               T( 1, I ), 1 )
*
*        Rectangular part of B2
*
         CALL DGEMV( 'T', L, I-1-P, ALPHA, B( MP, NP ), LDB,
     $               B( MP, I ), 1, ZERO, T( NP, I ), 1 )
*
*        B1
*
         CALL DGEMV( 'T', M-L, I-1, ALPHA, B, LDB, B( 1, I ), 1,
     $               ONE, T( 1, I ), 1 )
*
*        T(1:I-1,I) := T(1:I-1,1:I-1) * T(1:I-1,I)
*
         CALL DTRMV( 'U', 'N', 'N', I-1, T, LDT, T( 1, I ), 1 )
*
*        T(I,I) = tau(I)
*
         T( I, I ) = T( I, 1 )
         T( I, 1 ) = ZERO
      END DO

*
*     End of DTPQRT2
*
      END
*> \brief \b DTPRFB applies a real "triangular-pentagonal" block reflector to a real matrix, which is composed of two blocks.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTPRFB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtprfb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtprfb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtprfb.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTPRFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, L,
*                          V, LDV, T, LDT, A, LDA, B, LDB, WORK, LDWORK )
*
*       .. Scalar Arguments ..
*       CHARACTER DIRECT, SIDE, STOREV, TRANS
*       INTEGER   K, L, LDA, LDB, LDT, LDV, LDWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), T( LDT, * ),
*      $          V( LDV, * ), WORK( LDWORK, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTPRFB applies a real "triangular-pentagonal" block reflector H or its
*> transpose H**T to a real matrix C, which is composed of two
*> blocks A and B, either from the left or right.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'L': apply H or H**T from the Left
*>          = 'R': apply H or H**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N': apply H (No transpose)
*>          = 'T': apply H**T (Transpose)
*> \endverbatim
*>
*> \param[in] DIRECT
*> \verbatim
*>          DIRECT is CHARACTER*1
*>          Indicates how H is formed from a product of elementary
*>          reflectors
*>          = 'F': H = H(1) H(2) . . . H(k) (Forward)
*>          = 'B': H = H(k) . . . H(2) H(1) (Backward)
*> \endverbatim
*>
*> \param[in] STOREV
*> \verbatim
*>          STOREV is CHARACTER*1
*>          Indicates how the vectors which define the elementary
*>          reflectors are stored:
*>          = 'C': Columns
*>          = 'R': Rows
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix B.
*>          M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix B.
*>          N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The order of the matrix T, i.e. the number of elementary
*>          reflectors whose product defines the block reflector.
*>          K >= 0.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*>          L is INTEGER
*>          The order of the trapezoidal part of V.
*>          K >= L >= 0.  See Further Details.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*>          V is DOUBLE PRECISION array, dimension
*>                                (LDV,K) if STOREV = 'C'
*>                                (LDV,M) if STOREV = 'R' and SIDE = 'L'
*>                                (LDV,N) if STOREV = 'R' and SIDE = 'R'
*>          The pentagonal matrix V, which contains the elementary reflectors
*>          H(1), H(2), ..., H(K).  See Further Details.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of the array V.
*>          If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);
*>          if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);
*>          if STOREV = 'R', LDV >= K.
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,K)
*>          The triangular K-by-K matrix T in the representation of the
*>          block reflector.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T.
*>          LDT >= K.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension
*>          (LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R'
*>          On entry, the K-by-N or M-by-K matrix A.
*>          On exit, A is overwritten by the corresponding block of
*>          H*C or H**T*C or C*H or C*H**T.  See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.
*>          If SIDE = 'L', LDA >= max(1,K);
*>          If SIDE = 'R', LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,N)
*>          On entry, the M-by-N matrix B.
*>          On exit, B is overwritten by the corresponding block of
*>          H*C or H**T*C or C*H or C*H**T.  See Further Details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.
*>          LDB >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension
*>          (LDWORK,N) if SIDE = 'L',
*>          (LDWORK,K) if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDWORK
*> \verbatim
*>          LDWORK is INTEGER
*>          The leading dimension of the array WORK.
*>          If SIDE = 'L', LDWORK >= K;
*>          if SIDE = 'R', LDWORK >= M.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tprfb
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix C is a composite matrix formed from blocks A and B.
*>  The block B is of size M-by-N; if SIDE = 'R', A is of size M-by-K,
*>  and if SIDE = 'L', A is of size K-by-N.
*>
*>  If SIDE = 'R' and DIRECT = 'F', C = [A B].
*>
*>  If SIDE = 'L' and DIRECT = 'F', C = [A]
*>                                      [B].
*>
*>  If SIDE = 'R' and DIRECT = 'B', C = [B A].
*>
*>  If SIDE = 'L' and DIRECT = 'B', C = [B]
*>                                      [A].
*>
*>  The pentagonal matrix V is composed of a rectangular block V1 and a
*>  trapezoidal block V2.  The size of the trapezoidal block is determined by
*>  the parameter L, where 0<=L<=K.  If L=K, the V2 block of V is triangular;
*>  if L=0, there is no trapezoidal block, thus V = V1 is rectangular.
*>
*>  If DIRECT = 'F' and STOREV = 'C':  V = [V1]
*>                                         [V2]
*>     - V2 is upper trapezoidal (first L rows of K-by-K upper triangular)
*>
*>  If DIRECT = 'F' and STOREV = 'R':  V = [V1 V2]
*>
*>     - V2 is lower trapezoidal (first L columns of K-by-K lower triangular)
*>
*>  If DIRECT = 'B' and STOREV = 'C':  V = [V2]
*>                                         [V1]
*>     - V2 is lower trapezoidal (last L rows of K-by-K lower triangular)
*>
*>  If DIRECT = 'B' and STOREV = 'R':  V = [V2 V1]
*>
*>     - V2 is upper trapezoidal (last L columns of K-by-K upper triangular)
*>
*>  If STOREV = 'C' and SIDE = 'L', V is M-by-K with V2 L-by-K.
*>
*>  If STOREV = 'C' and SIDE = 'R', V is N-by-K with V2 L-by-K.
*>
*>  If STOREV = 'R' and SIDE = 'L', V is K-by-M with V2 K-by-L.
*>
*>  If STOREV = 'R' and SIDE = 'R', V is K-by-N with V2 K-by-L.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTPRFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, L,
     $                   V, LDV, T, LDT, A, LDA, B, LDB, WORK, LDWORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER DIRECT, SIDE, STOREV, TRANS
      INTEGER   K, L, LDA, LDB, LDT, LDV, LDWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), T( LDT, * ),
     $          V( LDV, * ), WORK( LDWORK, * )
*     ..
*
*  ==========================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER ( ONE = 1.0, ZERO = 0.0 )
*     ..
*     .. Local Scalars ..
      INTEGER   I, J, MP, NP, KP
      LOGICAL   LEFT, FORWARD, COLUMN, RIGHT, BACKWARD, ROW
*     ..
*     .. External Functions ..
      LOGICAL   LSAME
      EXTERNAL  LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL  DGEMM, DTRMM
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( M.LE.0 .OR. N.LE.0 .OR. K.LE.0 .OR. L.LT.0 ) RETURN
*
      IF( LSAME( STOREV, 'C' ) ) THEN
         COLUMN = .TRUE.
         ROW = .FALSE.
      ELSE IF ( LSAME( STOREV, 'R' ) ) THEN
         COLUMN = .FALSE.
         ROW = .TRUE.
      ELSE
         COLUMN = .FALSE.
         ROW = .FALSE.
      END IF
*
      IF( LSAME( SIDE, 'L' ) ) THEN
         LEFT = .TRUE.
         RIGHT = .FALSE.
      ELSE IF( LSAME( SIDE, 'R' ) ) THEN
         LEFT = .FALSE.
         RIGHT = .TRUE.
      ELSE
         LEFT = .FALSE.
         RIGHT = .FALSE.
      END IF
*
      IF( LSAME( DIRECT, 'F' ) ) THEN
         FORWARD = .TRUE.
         BACKWARD = .FALSE.
      ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
         FORWARD = .FALSE.
         BACKWARD = .TRUE.
      ELSE
         FORWARD = .FALSE.
         BACKWARD = .FALSE.
      END IF
*
* ---------------------------------------------------------------------------
*
      IF( COLUMN .AND. FORWARD .AND. LEFT  ) THEN
*
* ---------------------------------------------------------------------------
*
*        Let  W =  [ I ]    (K-by-K)
*                  [ V ]    (M-by-K)
*
*        Form  H C  or  H**T C  where  C = [ A ]  (K-by-N)
*                                          [ B ]  (M-by-N)
*
*        H = I - W T W**T          or  H**T = I - W T**T W**T
*
*        A = A -   T (A + V**T B)  or  A = A -   T**T (A + V**T B)
*        B = B - V T (A + V**T B)  or  B = B - V T**T (A + V**T B)
*
* ---------------------------------------------------------------------------
*
         MP = MIN( M-L+1, M )
         KP = MIN( L+1, K )
*
         DO J = 1, N
            DO I = 1, L
               WORK( I, J ) = B( M-L+I, J )
            END DO
         END DO
         CALL DTRMM( 'L', 'U', 'T', 'N', L, N, ONE, V( MP, 1 ), LDV,
     $               WORK, LDWORK )
         CALL DGEMM( 'T', 'N', L, N, M-L, ONE, V, LDV, B, LDB,
     $               ONE, WORK, LDWORK )
         CALL DGEMM( 'T', 'N', K-L, N, M, ONE, V( 1, KP ), LDV,
     $               B, LDB, ZERO, WORK( KP, 1 ), LDWORK )
*
         DO J = 1, N
            DO I = 1, K
               WORK( I, J ) = WORK( I, J ) + A( I, J )
            END DO
         END DO
*
         CALL DTRMM( 'L', 'U', TRANS, 'N', K, N, ONE, T, LDT,
     $               WORK, LDWORK )
*
         DO J = 1, N
            DO I = 1, K
               A( I, J ) = A( I, J ) - WORK( I, J )
            END DO
         END DO
*
         CALL DGEMM( 'N', 'N', M-L, N, K, -ONE, V, LDV, WORK, LDWORK,
     $               ONE, B, LDB )
         CALL DGEMM( 'N', 'N', L, N, K-L, -ONE, V( MP, KP ), LDV,
     $               WORK( KP, 1 ), LDWORK, ONE, B( MP, 1 ),  LDB )
         CALL DTRMM( 'L', 'U', 'N', 'N', L, N, ONE, V( MP, 1 ), LDV,
     $               WORK, LDWORK )
         DO J = 1, N
            DO I = 1, L
               B( M-L+I, J ) = B( M-L+I, J ) - WORK( I, J )
            END DO
         END DO
*
* ---------------------------------------------------------------------------
*
      ELSE IF( COLUMN .AND. FORWARD .AND. RIGHT ) THEN
*
* ---------------------------------------------------------------------------
*
*        Let  W =  [ I ]    (K-by-K)
*                  [ V ]    (N-by-K)
*
*        Form  C H or  C H**T  where  C = [ A B ] (A is M-by-K, B is M-by-N)
*
*        H = I - W T W**T          or  H**T = I - W T**T W**T
*
*        A = A - (A + B V) T      or  A = A - (A + B V) T**T
*        B = B - (A + B V) T V**T  or  B = B - (A + B V) T**T V**T
*
* ---------------------------------------------------------------------------
*
         NP = MIN( N-L+1, N )
         KP = MIN( L+1, K )
*
         DO J = 1, L
            DO I = 1, M
               WORK( I, J ) = B( I, N-L+J )
            END DO
         END DO
         CALL DTRMM( 'R', 'U', 'N', 'N', M, L, ONE, V( NP, 1 ), LDV,
     $               WORK, LDWORK )
         CALL DGEMM( 'N', 'N', M, L, N-L, ONE, B, LDB,
     $               V, LDV, ONE, WORK, LDWORK )
         CALL DGEMM( 'N', 'N', M, K-L, N, ONE, B, LDB,
     $               V( 1, KP ), LDV, ZERO, WORK( 1, KP ), LDWORK )
*
         DO J = 1, K
            DO I = 1, M
               WORK( I, J ) = WORK( I, J ) + A( I, J )
            END DO
         END DO
*
         CALL DTRMM( 'R', 'U', TRANS, 'N', M, K, ONE, T, LDT,
     $               WORK, LDWORK )
*
         DO J = 1, K
            DO I = 1, M
               A( I, J ) = A( I, J ) - WORK( I, J )
            END DO
         END DO
*
         CALL DGEMM( 'N', 'T', M, N-L, K, -ONE, WORK, LDWORK,
     $               V, LDV, ONE, B, LDB )
         CALL DGEMM( 'N', 'T', M, L, K-L, -ONE, WORK( 1, KP ),
     $               LDWORK,
     $               V( NP, KP ), LDV, ONE, B( 1, NP ), LDB )
         CALL DTRMM( 'R', 'U', 'T', 'N', M, L, ONE, V( NP, 1 ), LDV,
     $               WORK, LDWORK )
         DO J = 1, L
            DO I = 1, M
               B( I, N-L+J ) = B( I, N-L+J ) - WORK( I, J )
            END DO
         END DO
*
* ---------------------------------------------------------------------------
*
      ELSE IF( COLUMN .AND. BACKWARD .AND. LEFT ) THEN
*
* ---------------------------------------------------------------------------
*
*        Let  W =  [ V ]    (M-by-K)
*                  [ I ]    (K-by-K)
*
*        Form  H C  or  H**T C  where  C = [ B ]  (M-by-N)
*                                          [ A ]  (K-by-N)
*
*        H = I - W T W**T          or  H**T = I - W T**T W**T
*
*        A = A -   T (A + V**T B)  or  A = A -   T**T (A + V**T B)
*        B = B - V T (A + V**T B)  or  B = B - V T**T (A + V**T B)
*
* ---------------------------------------------------------------------------
*
         MP = MIN( L+1, M )
         KP = MIN( K-L+1, K )
*
         DO J = 1, N
            DO I = 1, L
               WORK( K-L+I, J ) = B( I, J )
            END DO
         END DO
*
         CALL DTRMM( 'L', 'L', 'T', 'N', L, N, ONE, V( 1, KP ), LDV,
     $               WORK( KP, 1 ), LDWORK )
         CALL DGEMM( 'T', 'N', L, N, M-L, ONE, V( MP, KP ), LDV,
     $               B( MP, 1 ), LDB, ONE, WORK( KP, 1 ), LDWORK )
         CALL DGEMM( 'T', 'N', K-L, N, M, ONE, V, LDV,
     $               B, LDB, ZERO, WORK, LDWORK )
*
         DO J = 1, N
            DO I = 1, K
               WORK( I, J ) = WORK( I, J ) + A( I, J )
            END DO
         END DO
*
         CALL DTRMM( 'L', 'L', TRANS, 'N', K, N, ONE, T, LDT,
     $               WORK, LDWORK )
*
         DO J = 1, N
            DO I = 1, K
               A( I, J ) = A( I, J ) - WORK( I, J )
            END DO
         END DO
*
         CALL DGEMM( 'N', 'N', M-L, N, K, -ONE, V( MP, 1 ), LDV,
     $               WORK, LDWORK, ONE, B( MP, 1 ), LDB )
         CALL DGEMM( 'N', 'N', L, N, K-L, -ONE, V, LDV,
     $               WORK, LDWORK, ONE, B,  LDB )
         CALL DTRMM( 'L', 'L', 'N', 'N', L, N, ONE, V( 1, KP ), LDV,
     $               WORK( KP, 1 ), LDWORK )
         DO J = 1, N
            DO I = 1, L
               B( I, J ) = B( I, J ) - WORK( K-L+I, J )
            END DO
         END DO
*
* ---------------------------------------------------------------------------
*
      ELSE IF( COLUMN .AND. BACKWARD .AND. RIGHT ) THEN
*
* ---------------------------------------------------------------------------
*
*        Let  W =  [ V ]    (N-by-K)
*                  [ I ]    (K-by-K)
*
*        Form  C H  or  C H**T  where  C = [ B A ] (B is M-by-N, A is M-by-K)
*
*        H = I - W T W**T          or  H**T = I - W T**T W**T
*
*        A = A - (A + B V) T      or  A = A - (A + B V) T**T
*        B = B - (A + B V) T V**T  or  B = B - (A + B V) T**T V**T
*
* ---------------------------------------------------------------------------
*
         NP = MIN( L+1, N )
         KP = MIN( K-L+1, K )
*
         DO J = 1, L
            DO I = 1, M
               WORK( I, K-L+J ) = B( I, J )
            END DO
         END DO
         CALL DTRMM( 'R', 'L', 'N', 'N', M, L, ONE, V( 1, KP ), LDV,
     $               WORK( 1, KP ), LDWORK )
         CALL DGEMM( 'N', 'N', M, L, N-L, ONE, B( 1, NP ), LDB,
     $               V( NP, KP ), LDV, ONE, WORK( 1, KP ), LDWORK )
         CALL DGEMM( 'N', 'N', M, K-L, N, ONE, B, LDB,
     $               V, LDV, ZERO, WORK, LDWORK )
*
         DO J = 1, K
            DO I = 1, M
               WORK( I, J ) = WORK( I, J ) + A( I, J )
            END DO
         END DO
*
         CALL DTRMM( 'R', 'L', TRANS, 'N', M, K, ONE, T, LDT,
     $               WORK, LDWORK )
*
         DO J = 1, K
            DO I = 1, M
               A( I, J ) = A( I, J ) - WORK( I, J )
            END DO
         END DO
*
         CALL DGEMM( 'N', 'T', M, N-L, K, -ONE, WORK, LDWORK,
     $               V( NP, 1 ), LDV, ONE, B( 1, NP ), LDB )
         CALL DGEMM( 'N', 'T', M, L, K-L, -ONE, WORK, LDWORK,
     $               V, LDV, ONE, B, LDB )
         CALL DTRMM( 'R', 'L', 'T', 'N', M, L, ONE, V( 1, KP ), LDV,
     $               WORK( 1, KP ), LDWORK )
         DO J = 1, L
            DO I = 1, M
               B( I, J ) = B( I, J ) - WORK( I, K-L+J )
            END DO
         END DO
*
* ---------------------------------------------------------------------------
*
      ELSE IF( ROW .AND. FORWARD .AND. LEFT ) THEN
*
* ---------------------------------------------------------------------------
*
*        Let  W =  [ I V ] ( I is K-by-K, V is K-by-M )
*
*        Form  H C  or  H**T C  where  C = [ A ]  (K-by-N)
*                                          [ B ]  (M-by-N)
*
*        H = I - W**T T W          or  H**T = I - W**T T**T W
*
*        A = A -     T (A + V B)  or  A = A -     T**T (A + V B)
*        B = B - V**T T (A + V B)  or  B = B - V**T T**T (A + V B)
*
* ---------------------------------------------------------------------------
*
         MP = MIN( M-L+1, M )
         KP = MIN( L+1, K )
*
         DO J = 1, N
            DO I = 1, L
               WORK( I, J ) = B( M-L+I, J )
            END DO
         END DO
         CALL DTRMM( 'L', 'L', 'N', 'N', L, N, ONE, V( 1, MP ), LDV,
     $               WORK, LDB )
         CALL DGEMM( 'N', 'N', L, N, M-L, ONE, V, LDV,B, LDB,
     $               ONE, WORK, LDWORK )
         CALL DGEMM( 'N', 'N', K-L, N, M, ONE, V( KP, 1 ), LDV,
     $               B, LDB, ZERO, WORK( KP, 1 ), LDWORK )
*
         DO J = 1, N
            DO I = 1, K
               WORK( I, J ) = WORK( I, J ) + A( I, J )
            END DO
         END DO
*
         CALL DTRMM( 'L', 'U', TRANS, 'N', K, N, ONE, T, LDT,
     $               WORK, LDWORK )
*
         DO J = 1, N
            DO I = 1, K
               A( I, J ) = A( I, J ) - WORK( I, J )
            END DO
         END DO
*
         CALL DGEMM( 'T', 'N', M-L, N, K, -ONE, V, LDV, WORK, LDWORK,
     $               ONE, B, LDB )
         CALL DGEMM( 'T', 'N', L, N, K-L, -ONE, V( KP, MP ), LDV,
     $               WORK( KP, 1 ), LDWORK, ONE, B( MP, 1 ), LDB )
         CALL DTRMM( 'L', 'L', 'T', 'N', L, N, ONE, V( 1, MP ), LDV,
     $               WORK, LDWORK )
         DO J = 1, N
            DO I = 1, L
               B( M-L+I, J ) = B( M-L+I, J ) - WORK( I, J )
            END DO
         END DO
*
* ---------------------------------------------------------------------------
*
      ELSE IF( ROW .AND. FORWARD .AND. RIGHT ) THEN
*
* ---------------------------------------------------------------------------
*
*        Let  W =  [ I V ] ( I is K-by-K, V is K-by-N )
*
*        Form  C H  or  C H**T  where  C = [ A B ] (A is M-by-K, B is M-by-N)
*
*        H = I - W**T T W            or  H**T = I - W**T T**T W
*
*        A = A - (A + B V**T) T      or  A = A - (A + B V**T) T**T
*        B = B - (A + B V**T) T V    or  B = B - (A + B V**T) T**T V
*
* ---------------------------------------------------------------------------
*
         NP = MIN( N-L+1, N )
         KP = MIN( L+1, K )
*
         DO J = 1, L
            DO I = 1, M
               WORK( I, J ) = B( I, N-L+J )
            END DO
         END DO
         CALL DTRMM( 'R', 'L', 'T', 'N', M, L, ONE, V( 1, NP ), LDV,
     $               WORK, LDWORK )
         CALL DGEMM( 'N', 'T', M, L, N-L, ONE, B, LDB, V, LDV,
     $               ONE, WORK, LDWORK )
         CALL DGEMM( 'N', 'T', M, K-L, N, ONE, B, LDB,
     $               V( KP, 1 ), LDV, ZERO, WORK( 1, KP ), LDWORK )
*
         DO J = 1, K
            DO I = 1, M
               WORK( I, J ) = WORK( I, J ) + A( I, J )
            END DO
         END DO
*
         CALL DTRMM( 'R', 'U', TRANS, 'N', M, K, ONE, T, LDT,
     $               WORK, LDWORK )
*
         DO J = 1, K
            DO I = 1, M
               A( I, J ) = A( I, J ) - WORK( I, J )
            END DO
         END DO
*
         CALL DGEMM( 'N', 'N', M, N-L, K, -ONE, WORK, LDWORK,
     $               V, LDV, ONE, B, LDB )
         CALL DGEMM( 'N', 'N', M, L, K-L, -ONE, WORK( 1, KP ),
     $               LDWORK,
     $               V( KP, NP ), LDV, ONE, B( 1, NP ), LDB )
         CALL DTRMM( 'R', 'L', 'N', 'N', M, L, ONE, V( 1, NP ), LDV,
     $               WORK, LDWORK )
         DO J = 1, L
            DO I = 1, M
               B( I, N-L+J ) = B( I, N-L+J ) - WORK( I, J )
            END DO
         END DO
*
* ---------------------------------------------------------------------------
*
      ELSE IF( ROW .AND. BACKWARD .AND. LEFT ) THEN
*
* ---------------------------------------------------------------------------
*
*        Let  W =  [ V I ] ( I is K-by-K, V is K-by-M )
*
*        Form  H C  or  H**T C  where  C = [ B ]  (M-by-N)
*                                          [ A ]  (K-by-N)
*
*        H = I - W**T T W          or  H**T = I - W**T T**T W
*
*        A = A -     T (A + V B)  or  A = A -     T**T (A + V B)
*        B = B - V**T T (A + V B)  or  B = B - V**T T**T (A + V B)
*
* ---------------------------------------------------------------------------
*
         MP = MIN( L+1, M )
         KP = MIN( K-L+1, K )
*
         DO J = 1, N
            DO I = 1, L
               WORK( K-L+I, J ) = B( I, J )
            END DO
         END DO
         CALL DTRMM( 'L', 'U', 'N', 'N', L, N, ONE, V( KP, 1 ), LDV,
     $               WORK( KP, 1 ), LDWORK )
         CALL DGEMM( 'N', 'N', L, N, M-L, ONE, V( KP, MP ), LDV,
     $               B( MP, 1 ), LDB, ONE, WORK( KP, 1 ), LDWORK )
         CALL DGEMM( 'N', 'N', K-L, N, M, ONE, V, LDV, B, LDB,
     $               ZERO, WORK, LDWORK )
*
         DO J = 1, N
            DO I = 1, K
               WORK( I, J ) = WORK( I, J ) + A( I, J )
            END DO
         END DO
*
         CALL DTRMM( 'L', 'L ', TRANS, 'N', K, N, ONE, T, LDT,
     $               WORK, LDWORK )
*
         DO J = 1, N
            DO I = 1, K
               A( I, J ) = A( I, J ) - WORK( I, J )
            END DO
         END DO
*
         CALL DGEMM( 'T', 'N', M-L, N, K, -ONE, V( 1, MP ), LDV,
     $               WORK, LDWORK, ONE, B( MP, 1 ), LDB )
         CALL DGEMM( 'T', 'N', L, N, K-L, -ONE, V, LDV,
     $               WORK, LDWORK, ONE, B, LDB )
         CALL DTRMM( 'L', 'U', 'T', 'N', L, N, ONE, V( KP, 1 ), LDV,
     $               WORK( KP, 1 ), LDWORK )
         DO J = 1, N
            DO I = 1, L
               B( I, J ) = B( I, J ) - WORK( K-L+I, J )
            END DO
         END DO
*
* ---------------------------------------------------------------------------
*
      ELSE IF( ROW .AND. BACKWARD .AND. RIGHT ) THEN
*
* ---------------------------------------------------------------------------
*
*        Let  W =  [ V I ] ( I is K-by-K, V is K-by-N )
*
*        Form  C H  or  C H**T  where  C = [ B A ] (A is M-by-K, B is M-by-N)
*
*        H = I - W**T T W            or  H**T = I - W**T T**T W
*
*        A = A - (A + B V**T) T      or  A = A - (A + B V**T) T**T
*        B = B - (A + B V**T) T V    or  B = B - (A + B V**T) T**T V
*
* ---------------------------------------------------------------------------
*
         NP = MIN( L+1, N )
         KP = MIN( K-L+1, K )
*
         DO J = 1, L
            DO I = 1, M
               WORK( I, K-L+J ) = B( I, J )
            END DO
         END DO
         CALL DTRMM( 'R', 'U', 'T', 'N', M, L, ONE, V( KP, 1 ), LDV,
     $               WORK( 1, KP ), LDWORK )
         CALL DGEMM( 'N', 'T', M, L, N-L, ONE, B( 1, NP ), LDB,
     $               V( KP, NP ), LDV, ONE, WORK( 1, KP ), LDWORK )
         CALL DGEMM( 'N', 'T', M, K-L, N, ONE, B, LDB, V, LDV,
     $               ZERO, WORK, LDWORK )
*
         DO J = 1, K
            DO I = 1, M
               WORK( I, J ) = WORK( I, J ) + A( I, J )
            END DO
         END DO
*
         CALL DTRMM( 'R', 'L', TRANS, 'N', M, K, ONE, T, LDT,
     $               WORK, LDWORK )
*
         DO J = 1, K
            DO I = 1, M
               A( I, J ) = A( I, J ) - WORK( I, J )
            END DO
         END DO
*
         CALL DGEMM( 'N', 'N', M, N-L, K, -ONE, WORK, LDWORK,
     $               V( 1, NP ), LDV, ONE, B( 1, NP ), LDB )
         CALL DGEMM( 'N', 'N', M, L, K-L , -ONE, WORK, LDWORK,
     $               V, LDV, ONE, B, LDB )
         CALL DTRMM( 'R', 'U', 'N', 'N', M, L, ONE, V( KP, 1 ), LDV,
     $               WORK( 1, KP ), LDWORK )
         DO J = 1, L
            DO I = 1, M
               B( I, J ) = B( I, J ) - WORK( I, K-L+J )
            END DO
         END DO
*
      END IF
*
      RETURN
*
*     End of DTPRFB
*
      END
*> \brief \b DTPRFS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTPRFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtprfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtprfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtprfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX,
*                          FERR, BERR, WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, TRANS, UPLO
*       INTEGER            INFO, LDB, LDX, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   AP( * ), B( LDB, * ), BERR( * ), FERR( * ),
*      $                   WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTPRFS provides error bounds and backward error estimates for the
*> solution to a system of linear equations with a triangular packed
*> coefficient matrix.
*>
*> The solution matrix X must be computed by DTPTRS or some other
*> means before entering this routine.  DTPRFS does not do iterative
*> refinement because doing so cannot improve the backward error.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  A is upper triangular;
*>          = 'L':  A is lower triangular.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the form of the system of equations:
*>          = 'N':  A * X = B  (No transpose)
*>          = 'T':  A**T * X = B  (Transpose)
*>          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          = 'N':  A is non-unit triangular;
*>          = 'U':  A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices B and X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          The upper or lower triangular matrix A, packed columnwise in
*>          a linear array.  The j-th column of A is stored in the array
*>          AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*>          If DIAG = 'U', the diagonal elements of A are not referenced
*>          and are assumed to be 1.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          The solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The estimated forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).  The estimate is as reliable as
*>          the estimate for RCOND, and is almost always a slight
*>          overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in
*>          any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tprfs
*
*  =====================================================================
      SUBROUTINE DTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X,
     $                   LDX,
     $                   FERR, BERR, WORK, IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, TRANS, UPLO
      INTEGER            INFO, LDB, LDX, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   AP( * ), B( LDB, * ), BERR( * ), FERR( * ),
     $                   WORK( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOTRAN, NOUNIT, UPPER
      CHARACTER          TRANST
      INTEGER            I, J, K, KASE, KC, NZ
      DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DLACN2, DTPMV, DTPSV,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      NOTRAN = LSAME( TRANS, 'N' )
      NOUNIT = LSAME( DIAG, 'N' )
*
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
     $         LSAME( TRANS, 'C' ) ) THEN
         INFO = -2
      ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -8
      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
         INFO = -10
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTPRFS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
         DO 10 J = 1, NRHS
            FERR( J ) = ZERO
            BERR( J ) = ZERO
   10    CONTINUE
         RETURN
      END IF
*
      IF( NOTRAN ) THEN
         TRANST = 'T'
      ELSE
         TRANST = 'N'
      END IF
*
*     NZ = maximum number of nonzero elements in each row of A, plus 1
*
      NZ = N + 1
      EPS = DLAMCH( 'Epsilon' )
      SAFMIN = DLAMCH( 'Safe minimum' )
      SAFE1 = NZ*SAFMIN
      SAFE2 = SAFE1 / EPS
*
*     Do for each right hand side
*
      DO 250 J = 1, NRHS
*
*        Compute residual R = B - op(A) * X,
*        where op(A) = A or A**T, depending on TRANS.
*
         CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 )
         CALL DTPMV( UPLO, TRANS, DIAG, N, AP, WORK( N+1 ), 1 )
         CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 )
*
*        Compute componentwise relative backward error from formula
*
*        max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
*
*        where abs(Z) is the componentwise absolute value of the matrix
*        or vector Z.  If the i-th component of the denominator is less
*        than SAFE2, then SAFE1 is added to the i-th components of the
*        numerator and denominator before dividing.
*
         DO 20 I = 1, N
            WORK( I ) = ABS( B( I, J ) )
   20    CONTINUE
*
         IF( NOTRAN ) THEN
*
*           Compute abs(A)*abs(X) + abs(B).
*
            IF( UPPER ) THEN
               KC = 1
               IF( NOUNIT ) THEN
                  DO 40 K = 1, N
                     XK = ABS( X( K, J ) )
                     DO 30 I = 1, K
                        WORK( I ) = WORK( I ) + ABS( AP( KC+I-1 ) )*XK
   30                CONTINUE
                     KC = KC + K
   40             CONTINUE
               ELSE
                  DO 60 K = 1, N
                     XK = ABS( X( K, J ) )
                     DO 50 I = 1, K - 1
                        WORK( I ) = WORK( I ) + ABS( AP( KC+I-1 ) )*XK
   50                CONTINUE
                     WORK( K ) = WORK( K ) + XK
                     KC = KC + K
   60             CONTINUE
               END IF
            ELSE
               KC = 1
               IF( NOUNIT ) THEN
                  DO 80 K = 1, N
                     XK = ABS( X( K, J ) )
                     DO 70 I = K, N
                        WORK( I ) = WORK( I ) + ABS( AP( KC+I-K ) )*XK
   70                CONTINUE
                     KC = KC + N - K + 1
   80             CONTINUE
               ELSE
                  DO 100 K = 1, N
                     XK = ABS( X( K, J ) )
                     DO 90 I = K + 1, N
                        WORK( I ) = WORK( I ) + ABS( AP( KC+I-K ) )*XK
   90                CONTINUE
                     WORK( K ) = WORK( K ) + XK
                     KC = KC + N - K + 1
  100             CONTINUE
               END IF
            END IF
         ELSE
*
*           Compute abs(A**T)*abs(X) + abs(B).
*
            IF( UPPER ) THEN
               KC = 1
               IF( NOUNIT ) THEN
                  DO 120 K = 1, N
                     S = ZERO
                     DO 110 I = 1, K
                        S = S + ABS( AP( KC+I-1 ) )*ABS( X( I, J ) )
  110                CONTINUE
                     WORK( K ) = WORK( K ) + S
                     KC = KC + K
  120             CONTINUE
               ELSE
                  DO 140 K = 1, N
                     S = ABS( X( K, J ) )
                     DO 130 I = 1, K - 1
                        S = S + ABS( AP( KC+I-1 ) )*ABS( X( I, J ) )
  130                CONTINUE
                     WORK( K ) = WORK( K ) + S
                     KC = KC + K
  140             CONTINUE
               END IF
            ELSE
               KC = 1
               IF( NOUNIT ) THEN
                  DO 160 K = 1, N
                     S = ZERO
                     DO 150 I = K, N
                        S = S + ABS( AP( KC+I-K ) )*ABS( X( I, J ) )
  150                CONTINUE
                     WORK( K ) = WORK( K ) + S
                     KC = KC + N - K + 1
  160             CONTINUE
               ELSE
                  DO 180 K = 1, N
                     S = ABS( X( K, J ) )
                     DO 170 I = K + 1, N
                        S = S + ABS( AP( KC+I-K ) )*ABS( X( I, J ) )
  170                CONTINUE
                     WORK( K ) = WORK( K ) + S
                     KC = KC + N - K + 1
  180             CONTINUE
               END IF
            END IF
         END IF
         S = ZERO
         DO 190 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
            ELSE
               S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
     $             ( WORK( I )+SAFE1 ) )
            END IF
  190    CONTINUE
         BERR( J ) = S
*
*        Bound error from formula
*
*        norm(X - XTRUE) / norm(X) .le. FERR =
*        norm( abs(inv(op(A)))*
*           ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
*
*        where
*          norm(Z) is the magnitude of the largest component of Z
*          inv(op(A)) is the inverse of op(A)
*          abs(Z) is the componentwise absolute value of the matrix or
*             vector Z
*          NZ is the maximum number of nonzeros in any row of A, plus 1
*          EPS is machine epsilon
*
*        The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
*        is incremented by SAFE1 if the i-th component of
*        abs(op(A))*abs(X) + abs(B) is less than SAFE2.
*
*        Use DLACN2 to estimate the infinity-norm of the matrix
*           inv(op(A)) * diag(W),
*        where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
*
         DO 200 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
            ELSE
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
            END IF
  200    CONTINUE
*
         KASE = 0
  210    CONTINUE
         CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
     $                FERR( J ),
     $                KASE, ISAVE )
         IF( KASE.NE.0 ) THEN
            IF( KASE.EQ.1 ) THEN
*
*              Multiply by diag(W)*inv(op(A)**T).
*
               CALL DTPSV( UPLO, TRANST, DIAG, N, AP, WORK( N+1 ),
     $                     1 )
               DO 220 I = 1, N
                  WORK( N+I ) = WORK( I )*WORK( N+I )
  220          CONTINUE
            ELSE
*
*              Multiply by inv(op(A))*diag(W).
*
               DO 230 I = 1, N
                  WORK( N+I ) = WORK( I )*WORK( N+I )
  230          CONTINUE
               CALL DTPSV( UPLO, TRANS, DIAG, N, AP, WORK( N+1 ), 1 )
            END IF
            GO TO 210
         END IF
*
*        Normalize error.
*
         LSTRES = ZERO
         DO 240 I = 1, N
            LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
  240    CONTINUE
         IF( LSTRES.NE.ZERO )
     $      FERR( J ) = FERR( J ) / LSTRES
*
  250 CONTINUE
*
      RETURN
*
*     End of DTPRFS
*
      END
*> \brief \b DTPTRI
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTPTRI + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtptri.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtptri.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtptri.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTPTRI( UPLO, DIAG, N, AP, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, UPLO
*       INTEGER            INFO, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AP( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTPTRI computes the inverse of a real upper or lower triangular
*> matrix A stored in packed format.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  A is upper triangular;
*>          = 'L':  A is lower triangular.
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          = 'N':  A is non-unit triangular;
*>          = 'U':  A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangular matrix A, stored
*>          columnwise in a linear array.  The j-th column of A is stored
*>          in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*((2*n-j)/2) = A(i,j) for j<=i<=n.
*>          See below for further details.
*>          On exit, the (triangular) inverse of the original matrix, in
*>          the same packed storage format.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, A(i,i) is exactly zero.  The triangular
*>                matrix is singular and its inverse can not be computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tptri
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  A triangular matrix A can be transferred to packed storage using one
*>  of the following program segments:
*>
*>  UPLO = 'U':                      UPLO = 'L':
*>
*>        JC = 1                           JC = 1
*>        DO 2 J = 1, N                    DO 2 J = 1, N
*>           DO 1 I = 1, J                    DO 1 I = J, N
*>              AP(JC+I-1) = A(I,J)              AP(JC+I-J) = A(I,J)
*>      1    CONTINUE                    1    CONTINUE
*>           JC = JC + J                      JC = JC + N - J + 1
*>      2 CONTINUE                       2 CONTINUE
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTPTRI( UPLO, DIAG, N, AP, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, UPLO
      INTEGER            INFO, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOUNIT, UPPER
      INTEGER            J, JC, JCLAST, JJ
      DOUBLE PRECISION   AJJ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL, DTPMV, XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      NOUNIT = LSAME( DIAG, 'N' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTPTRI', -INFO )
         RETURN
      END IF
*
*     Check for singularity if non-unit.
*
      IF( NOUNIT ) THEN
         IF( UPPER ) THEN
            JJ = 0
            DO 10 INFO = 1, N
               JJ = JJ + INFO
               IF( AP( JJ ).EQ.ZERO )
     $            RETURN
   10       CONTINUE
         ELSE
            JJ = 1
            DO 20 INFO = 1, N
               IF( AP( JJ ).EQ.ZERO )
     $            RETURN
               JJ = JJ + N - INFO + 1
   20       CONTINUE
         END IF
         INFO = 0
      END IF
*
      IF( UPPER ) THEN
*
*        Compute inverse of upper triangular matrix.
*
         JC = 1
         DO 30 J = 1, N
            IF( NOUNIT ) THEN
               AP( JC+J-1 ) = ONE / AP( JC+J-1 )
               AJJ = -AP( JC+J-1 )
            ELSE
               AJJ = -ONE
            END IF
*
*           Compute elements 1:j-1 of j-th column.
*
            CALL DTPMV( 'Upper', 'No transpose', DIAG, J-1, AP,
     $                  AP( JC ), 1 )
            CALL DSCAL( J-1, AJJ, AP( JC ), 1 )
            JC = JC + J
   30    CONTINUE
*
      ELSE
*
*        Compute inverse of lower triangular matrix.
*
         JC = N*( N+1 ) / 2
         DO 40 J = N, 1, -1
            IF( NOUNIT ) THEN
               AP( JC ) = ONE / AP( JC )
               AJJ = -AP( JC )
            ELSE
               AJJ = -ONE
            END IF
            IF( J.LT.N ) THEN
*
*              Compute elements j+1:n of j-th column.
*
               CALL DTPMV( 'Lower', 'No transpose', DIAG, N-J,
     $                     AP( JCLAST ), AP( JC+1 ), 1 )
               CALL DSCAL( N-J, AJJ, AP( JC+1 ), 1 )
            END IF
            JCLAST = JC
            JC = JC - N + J - 2
   40    CONTINUE
      END IF
*
      RETURN
*
*     End of DTPTRI
*
      END
*> \brief \b DTPTRS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTPTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtptrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtptrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtptrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTPTRS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, TRANS, UPLO
*       INTEGER            INFO, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AP( * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTPTRS solves a triangular system of the form
*>
*>    A * X = B  or  A**T * X = B,
*>
*> where A is a triangular matrix of order N stored in packed format, and B is an N-by-NRHS matrix.
*>
*> This subroutine verifies that A is nonsingular, but callers should note that only exact
*> singularity is detected. It is conceivable for one or more diagonal elements of A to be
*> subnormally tiny numbers without this subroutine signalling an error.
*>
*> If a possible loss of numerical precision due to near-singular matrices is a concern, the
*> caller should verify that A is nonsingular within some tolerance before calling this subroutine.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  A is upper triangular;
*>          = 'L':  A is lower triangular.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the form of the system of equations:
*>          = 'N':  A * X = B  (No transpose)
*>          = 'T':  A**T * X = B  (Transpose)
*>          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          = 'N':  A is non-unit triangular;
*>          = 'U':  A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          The upper or lower triangular matrix A, packed columnwise in
*>          a linear array.  The j-th column of A is stored in the array
*>          AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the right hand side matrix B.
*>          On exit, if INFO = 0, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the i-th diagonal element of A is exactly zero,
*>                indicating that the matrix is singular and the
*>                solutions X have not been computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tptrs
*
*  =====================================================================
      SUBROUTINE DTPTRS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, TRANS, UPLO
      INTEGER            INFO, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOUNIT, UPPER
      INTEGER            J, JC
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DTPSV, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      NOUNIT = LSAME( DIAG, 'N' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.LSAME( TRANS, 'N' ) .AND. .NOT.
     $         LSAME( TRANS, 'T' ) .AND.
     $                .NOT.LSAME( TRANS, 'C' ) ) THEN
         INFO = -2
      ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -8
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTPTRS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Check for singularity.
*
      IF( NOUNIT ) THEN
         IF( UPPER ) THEN
            JC = 1
            DO 10 INFO = 1, N
               IF( AP( JC+INFO-1 ).EQ.ZERO )
     $            RETURN
               JC = JC + INFO
   10       CONTINUE
         ELSE
            JC = 1
            DO 20 INFO = 1, N
               IF( AP( JC ).EQ.ZERO )
     $            RETURN
               JC = JC + N - INFO + 1
   20       CONTINUE
         END IF
      END IF
      INFO = 0
*
*     Solve A * x = b  or  A**T * x = b.
*
      DO 30 J = 1, NRHS
         CALL DTPSV( UPLO, TRANS, DIAG, N, AP, B( 1, J ), 1 )
   30 CONTINUE
*
      RETURN
*
*     End of DTPTRS
*
      END
*> \brief \b DTPTTF copies a triangular matrix from the standard packed format (TP) to the rectangular full packed format (TF).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTPTTF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtpttf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtpttf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtpttf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTPTTF( TRANSR, UPLO, N, AP, ARF, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANSR, UPLO
*       INTEGER            INFO, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AP( 0: * ), ARF( 0: * )
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTPTTF copies a triangular matrix A from standard packed format (TP)
*> to rectangular full packed format (TF).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANSR
*> \verbatim
*>          TRANSR is CHARACTER*1
*>          = 'N':  ARF in Normal format is wanted;
*>          = 'T':  ARF in Conjugate-transpose format is wanted.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  A is upper triangular;
*>          = 'L':  A is lower triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ),
*>          On entry, the upper or lower triangular matrix A, packed
*>          columnwise in a linear array. The j-th column of A is stored
*>          in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[out] ARF
*> \verbatim
*>          ARF is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ),
*>          On exit, the upper or lower triangular matrix A stored in
*>          RFP format. For a further discussion see Notes below.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tpttf
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  We first consider Rectangular Full Packed (RFP) Format when N is
*>  even. We give an example where N = 6.
*>
*>      AP is Upper             AP is Lower
*>
*>   00 01 02 03 04 05       00
*>      11 12 13 14 15       10 11
*>         22 23 24 25       20 21 22
*>            33 34 35       30 31 32 33
*>               44 45       40 41 42 43 44
*>                  55       50 51 52 53 54 55
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*>  the transpose of the first three columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*>  the transpose of the last three columns of AP lower.
*>  This covers the case N even and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>        03 04 05                33 43 53
*>        13 14 15                00 44 54
*>        23 24 25                10 11 55
*>        33 34 35                20 21 22
*>        00 44 45                30 31 32
*>        01 11 55                40 41 42
*>        02 12 22                50 51 52
*>
*>  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*>  transpose of RFP A above. One therefore gets:
*>
*>
*>           RFP A                   RFP A
*>
*>     03 13 23 33 00 01 02    33 00 10 20 30 40 50
*>     04 14 24 34 44 11 12    43 44 11 21 31 41 51
*>     05 15 25 35 45 55 22    53 54 55 22 32 42 52
*>
*>
*>  We then consider Rectangular Full Packed (RFP) Format when N is
*>  odd. We give an example where N = 5.
*>
*>     AP is Upper                 AP is Lower
*>
*>   00 01 02 03 04              00
*>      11 12 13 14              10 11
*>         22 23 24              20 21 22
*>            33 34              30 31 32 33
*>               44              40 41 42 43 44
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*>  the transpose of the first two columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*>  the transpose of the last two columns of AP lower.
*>  This covers the case N odd and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>        02 03 04                00 33 43
*>        12 13 14                10 11 44
*>        22 23 24                20 21 22
*>        00 33 34                30 31 32
*>        01 11 44                40 41 42
*>
*>  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*>  transpose of RFP A above. One therefore gets:
*>
*>           RFP A                   RFP A
*>
*>     02 12 22 00 01             00 10 20 30 40 50
*>     03 13 23 33 11             33 11 21 31 41 51
*>     04 14 24 34 44             43 44 22 32 42 52
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTPTTF( TRANSR, UPLO, N, AP, ARF, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANSR, UPLO
      INTEGER            INFO, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( 0: * ), ARF( 0: * )
*
*  =====================================================================
*
*     .. Parameters ..
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER, NISODD, NORMALTRANSR
      INTEGER            N1, N2, K, NT
      INTEGER            I, J, IJ
      INTEGER            IJP, JP, LDA, JS
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MOD
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NORMALTRANSR = LSAME( TRANSR, 'N' )
      LOWER = LSAME( UPLO, 'L' )
      IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTPTTF', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( N.EQ.1 ) THEN
         IF( NORMALTRANSR ) THEN
            ARF( 0 ) = AP( 0 )
         ELSE
            ARF( 0 ) = AP( 0 )
         END IF
         RETURN
      END IF
*
*     Size of array ARF(0:NT-1)
*
      NT = N*( N+1 ) / 2
*
*     Set N1 and N2 depending on LOWER
*
      IF( LOWER ) THEN
         N2 = N / 2
         N1 = N - N2
      ELSE
         N1 = N / 2
         N2 = N - N1
      END IF
*
*     If N is odd, set NISODD = .TRUE.
*     If N is even, set K = N/2 and NISODD = .FALSE.
*
*     set lda of ARF^C; ARF^C is (0:(N+1)/2-1,0:N-noe)
*     where noe = 0 if n is even, noe = 1 if n is odd
*
      IF( MOD( N, 2 ).EQ.0 ) THEN
         K = N / 2
         NISODD = .FALSE.
         LDA = N + 1
      ELSE
         NISODD = .TRUE.
         LDA = N
      END IF
*
*     ARF^C has lda rows and n+1-noe cols
*
      IF( .NOT.NORMALTRANSR )
     $   LDA = ( N+1 ) / 2
*
*     start execution: there are eight cases
*
      IF( NISODD ) THEN
*
*        N is odd
*
         IF( NORMALTRANSR ) THEN
*
*           N is odd and TRANSR = 'N'
*
            IF( LOWER ) THEN
*
*              N is odd, TRANSR = 'N', and UPLO = 'L'
*
               IJP = 0
               JP = 0
               DO J = 0, N2
                  DO I = J, N - 1
                     IJ = I + JP
                     ARF( IJ ) = AP( IJP )
                     IJP = IJP + 1
                  END DO
                  JP = JP + LDA
               END DO
               DO I = 0, N2 - 1
                  DO J = 1 + I, N2
                     IJ = I + J*LDA
                     ARF( IJ ) = AP( IJP )
                     IJP = IJP + 1
                  END DO
               END DO
*
            ELSE
*
*              N is odd, TRANSR = 'N', and UPLO = 'U'
*
               IJP = 0
               DO J = 0, N1 - 1
                  IJ = N2 + J
                  DO I = 0, J
                     ARF( IJ ) = AP( IJP )
                     IJP = IJP + 1
                     IJ = IJ + LDA
                  END DO
               END DO
               JS = 0
               DO J = N1, N - 1
                  IJ = JS
                  DO IJ = JS, JS + J
                     ARF( IJ ) = AP( IJP )
                     IJP = IJP + 1
                  END DO
                  JS = JS + LDA
               END DO
*
            END IF
*
         ELSE
*
*           N is odd and TRANSR = 'T'
*
            IF( LOWER ) THEN
*
*              N is odd, TRANSR = 'T', and UPLO = 'L'
*
               IJP = 0
               DO I = 0, N2
                  DO IJ = I*( LDA+1 ), N*LDA - 1, LDA
                     ARF( IJ ) = AP( IJP )
                     IJP = IJP + 1
                  END DO
               END DO
               JS = 1
               DO J = 0, N2 - 1
                  DO IJ = JS, JS + N2 - J - 1
                     ARF( IJ ) = AP( IJP )
                     IJP = IJP + 1
                  END DO
                  JS = JS + LDA + 1
               END DO
*
            ELSE
*
*              N is odd, TRANSR = 'T', and UPLO = 'U'
*
               IJP = 0
               JS = N2*LDA
               DO J = 0, N1 - 1
                  DO IJ = JS, JS + J
                     ARF( IJ ) = AP( IJP )
                     IJP = IJP + 1
                  END DO
                  JS = JS + LDA
               END DO
               DO I = 0, N1
                  DO IJ = I, I + ( N1+I )*LDA, LDA
                     ARF( IJ ) = AP( IJP )
                     IJP = IJP + 1
                  END DO
               END DO
*
            END IF
*
         END IF
*
      ELSE
*
*        N is even
*
         IF( NORMALTRANSR ) THEN
*
*           N is even and TRANSR = 'N'
*
            IF( LOWER ) THEN
*
*              N is even, TRANSR = 'N', and UPLO = 'L'
*
               IJP = 0
               JP = 0
               DO J = 0, K - 1
                  DO I = J, N - 1
                     IJ = 1 + I + JP
                     ARF( IJ ) = AP( IJP )
                     IJP = IJP + 1
                  END DO
                  JP = JP + LDA
               END DO
               DO I = 0, K - 1
                  DO J = I, K - 1
                     IJ = I + J*LDA
                     ARF( IJ ) = AP( IJP )
                     IJP = IJP + 1
                  END DO
               END DO
*
            ELSE
*
*              N is even, TRANSR = 'N', and UPLO = 'U'
*
               IJP = 0
               DO J = 0, K - 1
                  IJ = K + 1 + J
                  DO I = 0, J
                     ARF( IJ ) = AP( IJP )
                     IJP = IJP + 1
                     IJ = IJ + LDA
                  END DO
               END DO
               JS = 0
               DO J = K, N - 1
                  IJ = JS
                  DO IJ = JS, JS + J
                     ARF( IJ ) = AP( IJP )
                     IJP = IJP + 1
                  END DO
                  JS = JS + LDA
               END DO
*
            END IF
*
         ELSE
*
*           N is even and TRANSR = 'T'
*
            IF( LOWER ) THEN
*
*              N is even, TRANSR = 'T', and UPLO = 'L'
*
               IJP = 0
               DO I = 0, K - 1
                  DO IJ = I + ( I+1 )*LDA, ( N+1 )*LDA - 1, LDA
                     ARF( IJ ) = AP( IJP )
                     IJP = IJP + 1
                  END DO
               END DO
               JS = 0
               DO J = 0, K - 1
                  DO IJ = JS, JS + K - J - 1
                     ARF( IJ ) = AP( IJP )
                     IJP = IJP + 1
                  END DO
                  JS = JS + LDA + 1
               END DO
*
            ELSE
*
*              N is even, TRANSR = 'T', and UPLO = 'U'
*
               IJP = 0
               JS = ( K+1 )*LDA
               DO J = 0, K - 1
                  DO IJ = JS, JS + J
                     ARF( IJ ) = AP( IJP )
                     IJP = IJP + 1
                  END DO
                  JS = JS + LDA
               END DO
               DO I = 0, K - 1
                  DO IJ = I, I + ( K+I )*LDA, LDA
                     ARF( IJ ) = AP( IJP )
                     IJP = IJP + 1
                  END DO
               END DO
*
            END IF
*
         END IF
*
      END IF
*
      RETURN
*
*     End of DTPTTF
*
      END
*> \brief \b DTPTTR copies a triangular matrix from the standard packed format (TP) to the standard full format (TR).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTPTTR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtpttr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtpttr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtpttr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTPTTR( UPLO, N, AP, A, LDA, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, N, LDA
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), AP( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTPTTR copies a triangular matrix A from standard packed format (TP)
*> to standard full format (TR).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  A is upper triangular.
*>          = 'L':  A is lower triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ),
*>          On entry, the upper or lower triangular matrix A, packed
*>          columnwise in a linear array. The j-th column of A is stored
*>          in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension ( LDA, N )
*>          On exit, the triangular matrix A.  If UPLO = 'U', the leading
*>          N-by-N upper triangular part of A contains the upper
*>          triangular part of the matrix A, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading N-by-N lower triangular part of A contains the lower
*>          triangular part of the matrix A, and the strictly upper
*>          triangular part of A is not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tpttr
*
*  =====================================================================
      SUBROUTINE DTPTTR( UPLO, N, AP, A, LDA, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, N, LDA
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), AP( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER
      INTEGER            I, J, K
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      LOWER = LSAME( UPLO, 'L' )
      IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTPTTR', -INFO )
         RETURN
      END IF
*
      IF( LOWER ) THEN
         K = 0
         DO J = 1, N
            DO I = J, N
               K = K + 1
               A( I, J ) = AP( K )
            END DO
         END DO
      ELSE
         K = 0
         DO J = 1, N
            DO I = 1, J
               K = K + 1
               A( I, J ) = AP( K )
            END DO
         END DO
      END IF
*
*
      RETURN
*
*     End of DTPTTR
*
      END
*> \brief \b DTRCON
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrcon.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTRCON( NORM, UPLO, DIAG, N, A, LDA, RCOND, WORK,
*                          IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, NORM, UPLO
*       INTEGER            INFO, LDA, N
*       DOUBLE PRECISION   RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTRCON estimates the reciprocal of the condition number of a
*> triangular matrix A, in either the 1-norm or the infinity-norm.
*>
*> The norm of A is computed and an estimate is obtained for
*> norm(inv(A)), then the reciprocal of the condition number is
*> computed as
*>    RCOND = 1 / ( norm(A) * norm(inv(A)) ).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NORM
*> \verbatim
*>          NORM is CHARACTER*1
*>          Specifies whether the 1-norm condition number or the
*>          infinity-norm condition number is required:
*>          = '1' or 'O':  1-norm;
*>          = 'I':         Infinity-norm.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  A is upper triangular;
*>          = 'L':  A is lower triangular.
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          = 'N':  A is non-unit triangular;
*>          = 'U':  A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The triangular matrix A.  If UPLO = 'U', the leading N-by-N
*>          upper triangular part of the array A contains the upper
*>          triangular matrix, and the strictly lower triangular part of
*>          A is not referenced.  If UPLO = 'L', the leading N-by-N lower
*>          triangular part of the array A contains the lower triangular
*>          matrix, and the strictly upper triangular part of A is not
*>          referenced.  If DIAG = 'U', the diagonal elements of A are
*>          also not referenced and are assumed to be 1.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          The reciprocal of the condition number of the matrix A,
*>          computed as RCOND = 1/(norm(A) * norm(inv(A))).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trcon
*
*  =====================================================================
      SUBROUTINE DTRCON( NORM, UPLO, DIAG, N, A, LDA, RCOND, WORK,
     $                   IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, NORM, UPLO
      INTEGER            INFO, LDA, N
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOUNIT, ONENRM, UPPER
      CHARACTER          NORMIN
      INTEGER            IX, KASE, KASE1
      DOUBLE PRECISION   AINVNM, ANORM, SCALE, SMLNUM, XNORM
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH, DLANTR
      EXTERNAL           LSAME, IDAMAX, DLAMCH, DLANTR
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACN2, DLATRS, DRSCL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' )
      NOUNIT = LSAME( DIAG, 'N' )
*
      IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -2
      ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTRCON', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 ) THEN
         RCOND = ONE
         RETURN
      END IF
*
      RCOND = ZERO
      SMLNUM = DLAMCH( 'Safe minimum' )*DBLE( MAX( 1, N ) )
*
*     Compute the norm of the triangular matrix A.
*
      ANORM = DLANTR( NORM, UPLO, DIAG, N, N, A, LDA, WORK )
*
*     Continue only if ANORM > 0.
*
      IF( ANORM.GT.ZERO ) THEN
*
*        Estimate the norm of the inverse of A.
*
         AINVNM = ZERO
         NORMIN = 'N'
         IF( ONENRM ) THEN
            KASE1 = 1
         ELSE
            KASE1 = 2
         END IF
         KASE = 0
   10    CONTINUE
         CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE,
     $                ISAVE )
         IF( KASE.NE.0 ) THEN
            IF( KASE.EQ.KASE1 ) THEN
*
*              Multiply by inv(A).
*
               CALL DLATRS( UPLO, 'No transpose', DIAG, NORMIN, N, A,
     $                      LDA, WORK, SCALE, WORK( 2*N+1 ), INFO )
            ELSE
*
*              Multiply by inv(A**T).
*
               CALL DLATRS( UPLO, 'Transpose', DIAG, NORMIN, N, A,
     $                      LDA,
     $                      WORK, SCALE, WORK( 2*N+1 ), INFO )
            END IF
            NORMIN = 'Y'
*
*           Multiply by 1/SCALE if doing so will not cause overflow.
*
            IF( SCALE.NE.ONE ) THEN
               IX = IDAMAX( N, WORK, 1 )
               XNORM = ABS( WORK( IX ) )
               IF( SCALE.LT.XNORM*SMLNUM .OR. SCALE.EQ.ZERO )
     $            GO TO 20
               CALL DRSCL( N, SCALE, WORK, 1 )
            END IF
            GO TO 10
         END IF
*
*        Compute the estimate of the reciprocal condition number.
*
         IF( AINVNM.NE.ZERO )
     $      RCOND = ( ONE / ANORM ) / AINVNM
      END IF
*
   20 CONTINUE
      RETURN
*
*     End of DTRCON
*
      END
*> \brief \b DTREVC
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTREVC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrevc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrevc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrevc.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
*                          LDVR, MM, M, WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          HOWMNY, SIDE
*       INTEGER            INFO, LDT, LDVL, LDVR, M, MM, N
*       ..
*       .. Array Arguments ..
*       LOGICAL            SELECT( * )
*       DOUBLE PRECISION   T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
*      $                   WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTREVC computes some or all of the right and/or left eigenvectors of
*> a real upper quasi-triangular matrix T.
*> Matrices of this type are produced by the Schur factorization of
*> a real general matrix:  A = Q*T*Q**T, as computed by DHSEQR.
*>
*> The right eigenvector x and the left eigenvector y of T corresponding
*> to an eigenvalue w are defined by:
*>
*>    T*x = w*x,     (y**H)*T = w*(y**H)
*>
*> where y**H denotes the conjugate transpose of y.
*> The eigenvalues are not input to this routine, but are read directly
*> from the diagonal blocks of T.
*>
*> This routine returns the matrices X and/or Y of right and left
*> eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
*> input matrix.  If Q is the orthogonal factor that reduces a matrix
*> A to Schur form T, then Q*X and Q*Y are the matrices of right and
*> left eigenvectors of A.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'R':  compute right eigenvectors only;
*>          = 'L':  compute left eigenvectors only;
*>          = 'B':  compute both right and left eigenvectors.
*> \endverbatim
*>
*> \param[in] HOWMNY
*> \verbatim
*>          HOWMNY is CHARACTER*1
*>          = 'A':  compute all right and/or left eigenvectors;
*>          = 'B':  compute all right and/or left eigenvectors,
*>                  backtransformed by the matrices in VR and/or VL;
*>          = 'S':  compute selected right and/or left eigenvectors,
*>                  as indicated by the logical array SELECT.
*> \endverbatim
*>
*> \param[in,out] SELECT
*> \verbatim
*>          SELECT is LOGICAL array, dimension (N)
*>          If HOWMNY = 'S', SELECT specifies the eigenvectors to be
*>          computed.
*>          If w(j) is a real eigenvalue, the corresponding real
*>          eigenvector is computed if SELECT(j) is .TRUE..
*>          If w(j) and w(j+1) are the real and imaginary parts of a
*>          complex eigenvalue, the corresponding complex eigenvector is
*>          computed if either SELECT(j) or SELECT(j+1) is .TRUE., and
*>          on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to
*>          .FALSE..
*>          Not referenced if HOWMNY = 'A' or 'B'.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix T. N >= 0.
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,N)
*>          The upper quasi-triangular matrix T in Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*>          VL is DOUBLE PRECISION array, dimension (LDVL,MM)
*>          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
*>          contain an N-by-N matrix Q (usually the orthogonal matrix Q
*>          of Schur vectors returned by DHSEQR).
*>          On exit, if SIDE = 'L' or 'B', VL contains:
*>          if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
*>          if HOWMNY = 'B', the matrix Q*Y;
*>          if HOWMNY = 'S', the left eigenvectors of T specified by
*>                           SELECT, stored consecutively in the columns
*>                           of VL, in the same order as their
*>                           eigenvalues.
*>          A complex eigenvector corresponding to a complex eigenvalue
*>          is stored in two consecutive columns, the first holding the
*>          real part, and the second the imaginary part.
*>          Not referenced if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*>          LDVL is INTEGER
*>          The leading dimension of the array VL.  LDVL >= 1, and if
*>          SIDE = 'L' or 'B', LDVL >= N.
*> \endverbatim
*>
*> \param[in,out] VR
*> \verbatim
*>          VR is DOUBLE PRECISION array, dimension (LDVR,MM)
*>          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
*>          contain an N-by-N matrix Q (usually the orthogonal matrix Q
*>          of Schur vectors returned by DHSEQR).
*>          On exit, if SIDE = 'R' or 'B', VR contains:
*>          if HOWMNY = 'A', the matrix X of right eigenvectors of T;
*>          if HOWMNY = 'B', the matrix Q*X;
*>          if HOWMNY = 'S', the right eigenvectors of T specified by
*>                           SELECT, stored consecutively in the columns
*>                           of VR, in the same order as their
*>                           eigenvalues.
*>          A complex eigenvector corresponding to a complex eigenvalue
*>          is stored in two consecutive columns, the first holding the
*>          real part and the second the imaginary part.
*>          Not referenced if SIDE = 'L'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*>          LDVR is INTEGER
*>          The leading dimension of the array VR.  LDVR >= 1, and if
*>          SIDE = 'R' or 'B', LDVR >= N.
*> \endverbatim
*>
*> \param[in] MM
*> \verbatim
*>          MM is INTEGER
*>          The number of columns in the arrays VL and/or VR. MM >= M.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The number of columns in the arrays VL and/or VR actually
*>          used to store the eigenvectors.
*>          If HOWMNY = 'A' or 'B', M is set to N.
*>          Each selected real eigenvector occupies one column and each
*>          selected complex eigenvector occupies two columns.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trevc
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The algorithm used in this program is basically backward (forward)
*>  substitution, with scaling to make the the code robust against
*>  possible overflow.
*>
*>  Each eigenvector is normalized so that the element of largest
*>  magnitude has magnitude 1; here the magnitude of a complex number
*>  (x,y) is taken to be |x| + |y|.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL,
     $                   VR,
     $                   LDVR, MM, M, WORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          HOWMNY, SIDE
      INTEGER            INFO, LDT, LDVL, LDVR, M, MM, N
*     ..
*     .. Array Arguments ..
      LOGICAL            SELECT( * )
      DOUBLE PRECISION   T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
     $                   WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLV, BOTHV, LEFTV, OVER, PAIR, RIGHTV, SOMEV
      INTEGER            I, IERR, II, IP, IS, J, J1, J2, JNXT, K, KI, N2
      DOUBLE PRECISION   BETA, BIGNUM, EMAX, OVFL, REC, REMAX, SCALE,
     $                   SMIN, SMLNUM, ULP, UNFL, VCRIT, VMAX, WI, WR,
     $                   XNORM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX
      DOUBLE PRECISION   DDOT, DLAMCH
      EXTERNAL           LSAME, IDAMAX, DDOT, DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DGEMV, DLALN2, DSCAL,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   X( 2, 2 )
*     ..
*     .. Executable Statements ..
*
*     Decode and test the input parameters
*
      BOTHV = LSAME( SIDE, 'B' )
      RIGHTV = LSAME( SIDE, 'R' ) .OR. BOTHV
      LEFTV = LSAME( SIDE, 'L' ) .OR. BOTHV
*
      ALLV = LSAME( HOWMNY, 'A' )
      OVER = LSAME( HOWMNY, 'B' )
      SOMEV = LSAME( HOWMNY, 'S' )
*
      INFO = 0
      IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
         INFO = -1
      ELSE IF( .NOT.ALLV .AND. .NOT.OVER .AND. .NOT.SOMEV ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
         INFO = -6
      ELSE IF( LDVL.LT.1 .OR. ( LEFTV .AND. LDVL.LT.N ) ) THEN
         INFO = -8
      ELSE IF( LDVR.LT.1 .OR. ( RIGHTV .AND. LDVR.LT.N ) ) THEN
         INFO = -10
      ELSE
*
*        Set M to the number of columns required to store the selected
*        eigenvectors, standardize the array SELECT if necessary, and
*        test MM.
*
         IF( SOMEV ) THEN
            M = 0
            PAIR = .FALSE.
            DO 10 J = 1, N
               IF( PAIR ) THEN
                  PAIR = .FALSE.
                  SELECT( J ) = .FALSE.
               ELSE
                  IF( J.LT.N ) THEN
                     IF( T( J+1, J ).EQ.ZERO ) THEN
                        IF( SELECT( J ) )
     $                     M = M + 1
                     ELSE
                        PAIR = .TRUE.
                        IF( SELECT( J ) .OR. SELECT( J+1 ) ) THEN
                           SELECT( J ) = .TRUE.
                           M = M + 2
                        END IF
                     END IF
                  ELSE
                     IF( SELECT( N ) )
     $                  M = M + 1
                  END IF
               END IF
   10       CONTINUE
         ELSE
            M = N
         END IF
*
         IF( MM.LT.M ) THEN
            INFO = -11
         END IF
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTREVC', -INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Set the constants to control overflow.
*
      UNFL = DLAMCH( 'Safe minimum' )
      OVFL = ONE / UNFL
      ULP = DLAMCH( 'Precision' )
      SMLNUM = UNFL*( N / ULP )
      BIGNUM = ( ONE-ULP ) / SMLNUM
*
*     Compute 1-norm of each column of strictly upper triangular
*     part of T to control overflow in triangular solver.
*
      WORK( 1 ) = ZERO
      DO 30 J = 2, N
         WORK( J ) = ZERO
         DO 20 I = 1, J - 1
            WORK( J ) = WORK( J ) + ABS( T( I, J ) )
   20    CONTINUE
   30 CONTINUE
*
*     Index IP is used to specify the real or complex eigenvalue:
*       IP = 0, real eigenvalue,
*            1, first of conjugate complex pair: (wr,wi)
*           -1, second of conjugate complex pair: (wr,wi)
*
      N2 = 2*N
*
      IF( RIGHTV ) THEN
*
*        Compute right eigenvectors.
*
         IP = 0
         IS = M
         DO 140 KI = N, 1, -1
*
            IF( IP.EQ.1 )
     $         GO TO 130
            IF( KI.EQ.1 )
     $         GO TO 40
            IF( T( KI, KI-1 ).EQ.ZERO )
     $         GO TO 40
            IP = -1
*
   40       CONTINUE
            IF( SOMEV ) THEN
               IF( IP.EQ.0 ) THEN
                  IF( .NOT.SELECT( KI ) )
     $               GO TO 130
               ELSE
                  IF( .NOT.SELECT( KI-1 ) )
     $               GO TO 130
               END IF
            END IF
*
*           Compute the KI-th eigenvalue (WR,WI).
*
            WR = T( KI, KI )
            WI = ZERO
            IF( IP.NE.0 )
     $         WI = SQRT( ABS( T( KI, KI-1 ) ) )*
     $              SQRT( ABS( T( KI-1, KI ) ) )
            SMIN = MAX( ULP*( ABS( WR )+ABS( WI ) ), SMLNUM )
*
            IF( IP.EQ.0 ) THEN
*
*              Real right eigenvector
*
               WORK( KI+N ) = ONE
*
*              Form right-hand side
*
               DO 50 K = 1, KI - 1
                  WORK( K+N ) = -T( K, KI )
   50          CONTINUE
*
*              Solve the upper quasi-triangular system:
*                 (T(1:KI-1,1:KI-1) - WR)*X = SCALE*WORK.
*
               JNXT = KI - 1
               DO 60 J = KI - 1, 1, -1
                  IF( J.GT.JNXT )
     $               GO TO 60
                  J1 = J
                  J2 = J
                  JNXT = J - 1
                  IF( J.GT.1 ) THEN
                     IF( T( J, J-1 ).NE.ZERO ) THEN
                        J1 = J - 1
                        JNXT = J - 2
                     END IF
                  END IF
*
                  IF( J1.EQ.J2 ) THEN
*
*                    1-by-1 diagonal block
*
                     CALL DLALN2( .FALSE., 1, 1, SMIN, ONE, T( J,
     $                            J ),
     $                            LDT, ONE, ONE, WORK( J+N ), N, WR,
     $                            ZERO, X, 2, SCALE, XNORM, IERR )
*
*                    Scale X(1,1) to avoid overflow when updating
*                    the right-hand side.
*
                     IF( XNORM.GT.ONE ) THEN
                        IF( WORK( J ).GT.BIGNUM / XNORM ) THEN
                           X( 1, 1 ) = X( 1, 1 ) / XNORM
                           SCALE = SCALE / XNORM
                        END IF
                     END IF
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE )
     $                  CALL DSCAL( KI, SCALE, WORK( 1+N ), 1 )
                     WORK( J+N ) = X( 1, 1 )
*
*                    Update right-hand side
*
                     CALL DAXPY( J-1, -X( 1, 1 ), T( 1, J ), 1,
     $                           WORK( 1+N ), 1 )
*
                  ELSE
*
*                    2-by-2 diagonal block
*
                     CALL DLALN2( .FALSE., 2, 1, SMIN, ONE,
     $                            T( J-1, J-1 ), LDT, ONE, ONE,
     $                            WORK( J-1+N ), N, WR, ZERO, X, 2,
     $                            SCALE, XNORM, IERR )
*
*                    Scale X(1,1) and X(2,1) to avoid overflow when
*                    updating the right-hand side.
*
                     IF( XNORM.GT.ONE ) THEN
                        BETA = MAX( WORK( J-1 ), WORK( J ) )
                        IF( BETA.GT.BIGNUM / XNORM ) THEN
                           X( 1, 1 ) = X( 1, 1 ) / XNORM
                           X( 2, 1 ) = X( 2, 1 ) / XNORM
                           SCALE = SCALE / XNORM
                        END IF
                     END IF
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE )
     $                  CALL DSCAL( KI, SCALE, WORK( 1+N ), 1 )
                     WORK( J-1+N ) = X( 1, 1 )
                     WORK( J+N ) = X( 2, 1 )
*
*                    Update right-hand side
*
                     CALL DAXPY( J-2, -X( 1, 1 ), T( 1, J-1 ), 1,
     $                           WORK( 1+N ), 1 )
                     CALL DAXPY( J-2, -X( 2, 1 ), T( 1, J ), 1,
     $                           WORK( 1+N ), 1 )
                  END IF
   60          CONTINUE
*
*              Copy the vector x or Q*x to VR and normalize.
*
               IF( .NOT.OVER ) THEN
                  CALL DCOPY( KI, WORK( 1+N ), 1, VR( 1, IS ), 1 )
*
                  II = IDAMAX( KI, VR( 1, IS ), 1 )
                  REMAX = ONE / ABS( VR( II, IS ) )
                  CALL DSCAL( KI, REMAX, VR( 1, IS ), 1 )
*
                  DO 70 K = KI + 1, N
                     VR( K, IS ) = ZERO
   70             CONTINUE
               ELSE
                  IF( KI.GT.1 )
     $               CALL DGEMV( 'N', N, KI-1, ONE, VR, LDVR,
     $                           WORK( 1+N ), 1, WORK( KI+N ),
     $                           VR( 1, KI ), 1 )
*
                  II = IDAMAX( N, VR( 1, KI ), 1 )
                  REMAX = ONE / ABS( VR( II, KI ) )
                  CALL DSCAL( N, REMAX, VR( 1, KI ), 1 )
               END IF
*
            ELSE
*
*              Complex right eigenvector.
*
*              Initial solve
*                [ (T(KI-1,KI-1) T(KI-1,KI) ) - (WR + I* WI)]*X = 0.
*                [ (T(KI,KI-1)   T(KI,KI)   )               ]
*
               IF( ABS( T( KI-1, KI ) ).GE.ABS( T( KI, KI-1 ) ) ) THEN
                  WORK( KI-1+N ) = ONE
                  WORK( KI+N2 ) = WI / T( KI-1, KI )
               ELSE
                  WORK( KI-1+N ) = -WI / T( KI, KI-1 )
                  WORK( KI+N2 ) = ONE
               END IF
               WORK( KI+N ) = ZERO
               WORK( KI-1+N2 ) = ZERO
*
*              Form right-hand side
*
               DO 80 K = 1, KI - 2
                  WORK( K+N ) = -WORK( KI-1+N )*T( K, KI-1 )
                  WORK( K+N2 ) = -WORK( KI+N2 )*T( K, KI )
   80          CONTINUE
*
*              Solve upper quasi-triangular system:
*              (T(1:KI-2,1:KI-2) - (WR+i*WI))*X = SCALE*(WORK+i*WORK2)
*
               JNXT = KI - 2
               DO 90 J = KI - 2, 1, -1
                  IF( J.GT.JNXT )
     $               GO TO 90
                  J1 = J
                  J2 = J
                  JNXT = J - 1
                  IF( J.GT.1 ) THEN
                     IF( T( J, J-1 ).NE.ZERO ) THEN
                        J1 = J - 1
                        JNXT = J - 2
                     END IF
                  END IF
*
                  IF( J1.EQ.J2 ) THEN
*
*                    1-by-1 diagonal block
*
                     CALL DLALN2( .FALSE., 1, 2, SMIN, ONE, T( J,
     $                            J ),
     $                            LDT, ONE, ONE, WORK( J+N ), N, WR, WI,
     $                            X, 2, SCALE, XNORM, IERR )
*
*                    Scale X(1,1) and X(1,2) to avoid overflow when
*                    updating the right-hand side.
*
                     IF( XNORM.GT.ONE ) THEN
                        IF( WORK( J ).GT.BIGNUM / XNORM ) THEN
                           X( 1, 1 ) = X( 1, 1 ) / XNORM
                           X( 1, 2 ) = X( 1, 2 ) / XNORM
                           SCALE = SCALE / XNORM
                        END IF
                     END IF
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE ) THEN
                        CALL DSCAL( KI, SCALE, WORK( 1+N ), 1 )
                        CALL DSCAL( KI, SCALE, WORK( 1+N2 ), 1 )
                     END IF
                     WORK( J+N ) = X( 1, 1 )
                     WORK( J+N2 ) = X( 1, 2 )
*
*                    Update the right-hand side
*
                     CALL DAXPY( J-1, -X( 1, 1 ), T( 1, J ), 1,
     $                           WORK( 1+N ), 1 )
                     CALL DAXPY( J-1, -X( 1, 2 ), T( 1, J ), 1,
     $                           WORK( 1+N2 ), 1 )
*
                  ELSE
*
*                    2-by-2 diagonal block
*
                     CALL DLALN2( .FALSE., 2, 2, SMIN, ONE,
     $                            T( J-1, J-1 ), LDT, ONE, ONE,
     $                            WORK( J-1+N ), N, WR, WI, X, 2, SCALE,
     $                            XNORM, IERR )
*
*                    Scale X to avoid overflow when updating
*                    the right-hand side.
*
                     IF( XNORM.GT.ONE ) THEN
                        BETA = MAX( WORK( J-1 ), WORK( J ) )
                        IF( BETA.GT.BIGNUM / XNORM ) THEN
                           REC = ONE / XNORM
                           X( 1, 1 ) = X( 1, 1 )*REC
                           X( 1, 2 ) = X( 1, 2 )*REC
                           X( 2, 1 ) = X( 2, 1 )*REC
                           X( 2, 2 ) = X( 2, 2 )*REC
                           SCALE = SCALE*REC
                        END IF
                     END IF
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE ) THEN
                        CALL DSCAL( KI, SCALE, WORK( 1+N ), 1 )
                        CALL DSCAL( KI, SCALE, WORK( 1+N2 ), 1 )
                     END IF
                     WORK( J-1+N ) = X( 1, 1 )
                     WORK( J+N ) = X( 2, 1 )
                     WORK( J-1+N2 ) = X( 1, 2 )
                     WORK( J+N2 ) = X( 2, 2 )
*
*                    Update the right-hand side
*
                     CALL DAXPY( J-2, -X( 1, 1 ), T( 1, J-1 ), 1,
     $                           WORK( 1+N ), 1 )
                     CALL DAXPY( J-2, -X( 2, 1 ), T( 1, J ), 1,
     $                           WORK( 1+N ), 1 )
                     CALL DAXPY( J-2, -X( 1, 2 ), T( 1, J-1 ), 1,
     $                           WORK( 1+N2 ), 1 )
                     CALL DAXPY( J-2, -X( 2, 2 ), T( 1, J ), 1,
     $                           WORK( 1+N2 ), 1 )
                  END IF
   90          CONTINUE
*
*              Copy the vector x or Q*x to VR and normalize.
*
               IF( .NOT.OVER ) THEN
                  CALL DCOPY( KI, WORK( 1+N ), 1, VR( 1, IS-1 ), 1 )
                  CALL DCOPY( KI, WORK( 1+N2 ), 1, VR( 1, IS ), 1 )
*
                  EMAX = ZERO
                  DO 100 K = 1, KI
                     EMAX = MAX( EMAX, ABS( VR( K, IS-1 ) )+
     $                      ABS( VR( K, IS ) ) )
  100             CONTINUE
*
                  REMAX = ONE / EMAX
                  CALL DSCAL( KI, REMAX, VR( 1, IS-1 ), 1 )
                  CALL DSCAL( KI, REMAX, VR( 1, IS ), 1 )
*
                  DO 110 K = KI + 1, N
                     VR( K, IS-1 ) = ZERO
                     VR( K, IS ) = ZERO
  110             CONTINUE
*
               ELSE
*
                  IF( KI.GT.2 ) THEN
                     CALL DGEMV( 'N', N, KI-2, ONE, VR, LDVR,
     $                           WORK( 1+N ), 1, WORK( KI-1+N ),
     $                           VR( 1, KI-1 ), 1 )
                     CALL DGEMV( 'N', N, KI-2, ONE, VR, LDVR,
     $                           WORK( 1+N2 ), 1, WORK( KI+N2 ),
     $                           VR( 1, KI ), 1 )
                  ELSE
                     CALL DSCAL( N, WORK( KI-1+N ), VR( 1, KI-1 ),
     $                           1 )
                     CALL DSCAL( N, WORK( KI+N2 ), VR( 1, KI ), 1 )
                  END IF
*
                  EMAX = ZERO
                  DO 120 K = 1, N
                     EMAX = MAX( EMAX, ABS( VR( K, KI-1 ) )+
     $                      ABS( VR( K, KI ) ) )
  120             CONTINUE
                  REMAX = ONE / EMAX
                  CALL DSCAL( N, REMAX, VR( 1, KI-1 ), 1 )
                  CALL DSCAL( N, REMAX, VR( 1, KI ), 1 )
               END IF
            END IF
*
            IS = IS - 1
            IF( IP.NE.0 )
     $         IS = IS - 1
  130       CONTINUE
            IF( IP.EQ.1 )
     $         IP = 0
            IF( IP.EQ.-1 )
     $         IP = 1
  140    CONTINUE
      END IF
*
      IF( LEFTV ) THEN
*
*        Compute left eigenvectors.
*
         IP = 0
         IS = 1
         DO 260 KI = 1, N
*
            IF( IP.EQ.-1 )
     $         GO TO 250
            IF( KI.EQ.N )
     $         GO TO 150
            IF( T( KI+1, KI ).EQ.ZERO )
     $         GO TO 150
            IP = 1
*
  150       CONTINUE
            IF( SOMEV ) THEN
               IF( .NOT.SELECT( KI ) )
     $            GO TO 250
            END IF
*
*           Compute the KI-th eigenvalue (WR,WI).
*
            WR = T( KI, KI )
            WI = ZERO
            IF( IP.NE.0 )
     $         WI = SQRT( ABS( T( KI, KI+1 ) ) )*
     $              SQRT( ABS( T( KI+1, KI ) ) )
            SMIN = MAX( ULP*( ABS( WR )+ABS( WI ) ), SMLNUM )
*
            IF( IP.EQ.0 ) THEN
*
*              Real left eigenvector.
*
               WORK( KI+N ) = ONE
*
*              Form right-hand side
*
               DO 160 K = KI + 1, N
                  WORK( K+N ) = -T( KI, K )
  160          CONTINUE
*
*              Solve the quasi-triangular system:
*                 (T(KI+1:N,KI+1:N) - WR)**T*X = SCALE*WORK
*
               VMAX = ONE
               VCRIT = BIGNUM
*
               JNXT = KI + 1
               DO 170 J = KI + 1, N
                  IF( J.LT.JNXT )
     $               GO TO 170
                  J1 = J
                  J2 = J
                  JNXT = J + 1
                  IF( J.LT.N ) THEN
                     IF( T( J+1, J ).NE.ZERO ) THEN
                        J2 = J + 1
                        JNXT = J + 2
                     END IF
                  END IF
*
                  IF( J1.EQ.J2 ) THEN
*
*                    1-by-1 diagonal block
*
*                    Scale if necessary to avoid overflow when forming
*                    the right-hand side.
*
                     IF( WORK( J ).GT.VCRIT ) THEN
                        REC = ONE / VMAX
                        CALL DSCAL( N-KI+1, REC, WORK( KI+N ), 1 )
                        VMAX = ONE
                        VCRIT = BIGNUM
                     END IF
*
                     WORK( J+N ) = WORK( J+N ) -
     $                             DDOT( J-KI-1, T( KI+1, J ), 1,
     $                             WORK( KI+1+N ), 1 )
*
*                    Solve (T(J,J)-WR)**T*X = WORK
*
                     CALL DLALN2( .FALSE., 1, 1, SMIN, ONE, T( J,
     $                            J ),
     $                            LDT, ONE, ONE, WORK( J+N ), N, WR,
     $                            ZERO, X, 2, SCALE, XNORM, IERR )
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE )
     $                  CALL DSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 )
                     WORK( J+N ) = X( 1, 1 )
                     VMAX = MAX( ABS( WORK( J+N ) ), VMAX )
                     VCRIT = BIGNUM / VMAX
*
                  ELSE
*
*                    2-by-2 diagonal block
*
*                    Scale if necessary to avoid overflow when forming
*                    the right-hand side.
*
                     BETA = MAX( WORK( J ), WORK( J+1 ) )
                     IF( BETA.GT.VCRIT ) THEN
                        REC = ONE / VMAX
                        CALL DSCAL( N-KI+1, REC, WORK( KI+N ), 1 )
                        VMAX = ONE
                        VCRIT = BIGNUM
                     END IF
*
                     WORK( J+N ) = WORK( J+N ) -
     $                             DDOT( J-KI-1, T( KI+1, J ), 1,
     $                             WORK( KI+1+N ), 1 )
*
                     WORK( J+1+N ) = WORK( J+1+N ) -
     $                               DDOT( J-KI-1, T( KI+1, J+1 ), 1,
     $                               WORK( KI+1+N ), 1 )
*
*                    Solve
*                      [T(J,J)-WR   T(J,J+1)     ]**T * X = SCALE*( WORK1 )
*                      [T(J+1,J)    T(J+1,J+1)-WR]                ( WORK2 )
*
                     CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, T( J, J ),
     $                            LDT, ONE, ONE, WORK( J+N ), N, WR,
     $                            ZERO, X, 2, SCALE, XNORM, IERR )
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE )
     $                  CALL DSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 )
                     WORK( J+N ) = X( 1, 1 )
                     WORK( J+1+N ) = X( 2, 1 )
*
                     VMAX = MAX( ABS( WORK( J+N ) ),
     $                      ABS( WORK( J+1+N ) ), VMAX )
                     VCRIT = BIGNUM / VMAX
*
                  END IF
  170          CONTINUE
*
*              Copy the vector x or Q*x to VL and normalize.
*
               IF( .NOT.OVER ) THEN
                  CALL DCOPY( N-KI+1, WORK( KI+N ), 1, VL( KI, IS ),
     $                        1 )
*
                  II = IDAMAX( N-KI+1, VL( KI, IS ), 1 ) + KI - 1
                  REMAX = ONE / ABS( VL( II, IS ) )
                  CALL DSCAL( N-KI+1, REMAX, VL( KI, IS ), 1 )
*
                  DO 180 K = 1, KI - 1
                     VL( K, IS ) = ZERO
  180             CONTINUE
*
               ELSE
*
                  IF( KI.LT.N )
     $               CALL DGEMV( 'N', N, N-KI, ONE, VL( 1, KI+1 ),
     $                           LDVL,
     $                           WORK( KI+1+N ), 1, WORK( KI+N ),
     $                           VL( 1, KI ), 1 )
*
                  II = IDAMAX( N, VL( 1, KI ), 1 )
                  REMAX = ONE / ABS( VL( II, KI ) )
                  CALL DSCAL( N, REMAX, VL( 1, KI ), 1 )
*
               END IF
*
            ELSE
*
*              Complex left eigenvector.
*
*               Initial solve:
*                 ((T(KI,KI)    T(KI,KI+1) )**T - (WR - I* WI))*X = 0.
*                 ((T(KI+1,KI) T(KI+1,KI+1))                )
*
               IF( ABS( T( KI, KI+1 ) ).GE.ABS( T( KI+1, KI ) ) ) THEN
                  WORK( KI+N ) = WI / T( KI, KI+1 )
                  WORK( KI+1+N2 ) = ONE
               ELSE
                  WORK( KI+N ) = ONE
                  WORK( KI+1+N2 ) = -WI / T( KI+1, KI )
               END IF
               WORK( KI+1+N ) = ZERO
               WORK( KI+N2 ) = ZERO
*
*              Form right-hand side
*
               DO 190 K = KI + 2, N
                  WORK( K+N ) = -WORK( KI+N )*T( KI, K )
                  WORK( K+N2 ) = -WORK( KI+1+N2 )*T( KI+1, K )
  190          CONTINUE
*
*              Solve complex quasi-triangular system:
*              ( T(KI+2,N:KI+2,N) - (WR-i*WI) )*X = WORK1+i*WORK2
*
               VMAX = ONE
               VCRIT = BIGNUM
*
               JNXT = KI + 2
               DO 200 J = KI + 2, N
                  IF( J.LT.JNXT )
     $               GO TO 200
                  J1 = J
                  J2 = J
                  JNXT = J + 1
                  IF( J.LT.N ) THEN
                     IF( T( J+1, J ).NE.ZERO ) THEN
                        J2 = J + 1
                        JNXT = J + 2
                     END IF
                  END IF
*
                  IF( J1.EQ.J2 ) THEN
*
*                    1-by-1 diagonal block
*
*                    Scale if necessary to avoid overflow when
*                    forming the right-hand side elements.
*
                     IF( WORK( J ).GT.VCRIT ) THEN
                        REC = ONE / VMAX
                        CALL DSCAL( N-KI+1, REC, WORK( KI+N ), 1 )
                        CALL DSCAL( N-KI+1, REC, WORK( KI+N2 ), 1 )
                        VMAX = ONE
                        VCRIT = BIGNUM
                     END IF
*
                     WORK( J+N ) = WORK( J+N ) -
     $                             DDOT( J-KI-2, T( KI+2, J ), 1,
     $                             WORK( KI+2+N ), 1 )
                     WORK( J+N2 ) = WORK( J+N2 ) -
     $                              DDOT( J-KI-2, T( KI+2, J ), 1,
     $                              WORK( KI+2+N2 ), 1 )
*
*                    Solve (T(J,J)-(WR-i*WI))*(X11+i*X12)= WK+I*WK2
*
                     CALL DLALN2( .FALSE., 1, 2, SMIN, ONE, T( J,
     $                            J ),
     $                            LDT, ONE, ONE, WORK( J+N ), N, WR,
     $                            -WI, X, 2, SCALE, XNORM, IERR )
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE ) THEN
                        CALL DSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 )
                        CALL DSCAL( N-KI+1, SCALE, WORK( KI+N2 ), 1 )
                     END IF
                     WORK( J+N ) = X( 1, 1 )
                     WORK( J+N2 ) = X( 1, 2 )
                     VMAX = MAX( ABS( WORK( J+N ) ),
     $                      ABS( WORK( J+N2 ) ), VMAX )
                     VCRIT = BIGNUM / VMAX
*
                  ELSE
*
*                    2-by-2 diagonal block
*
*                    Scale if necessary to avoid overflow when forming
*                    the right-hand side elements.
*
                     BETA = MAX( WORK( J ), WORK( J+1 ) )
                     IF( BETA.GT.VCRIT ) THEN
                        REC = ONE / VMAX
                        CALL DSCAL( N-KI+1, REC, WORK( KI+N ), 1 )
                        CALL DSCAL( N-KI+1, REC, WORK( KI+N2 ), 1 )
                        VMAX = ONE
                        VCRIT = BIGNUM
                     END IF
*
                     WORK( J+N ) = WORK( J+N ) -
     $                             DDOT( J-KI-2, T( KI+2, J ), 1,
     $                             WORK( KI+2+N ), 1 )
*
                     WORK( J+N2 ) = WORK( J+N2 ) -
     $                              DDOT( J-KI-2, T( KI+2, J ), 1,
     $                              WORK( KI+2+N2 ), 1 )
*
                     WORK( J+1+N ) = WORK( J+1+N ) -
     $                               DDOT( J-KI-2, T( KI+2, J+1 ), 1,
     $                               WORK( KI+2+N ), 1 )
*
                     WORK( J+1+N2 ) = WORK( J+1+N2 ) -
     $                                DDOT( J-KI-2, T( KI+2, J+1 ),
     $                                      1,
     $                                WORK( KI+2+N2 ), 1 )
*
*                    Solve 2-by-2 complex linear equation
*                      ([T(j,j)   T(j,j+1)  ]**T-(wr-i*wi)*I)*X = SCALE*B
*                      ([T(j+1,j) T(j+1,j+1)]               )
*
                     CALL DLALN2( .TRUE., 2, 2, SMIN, ONE, T( J, J ),
     $                            LDT, ONE, ONE, WORK( J+N ), N, WR,
     $                            -WI, X, 2, SCALE, XNORM, IERR )
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE ) THEN
                        CALL DSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 )
                        CALL DSCAL( N-KI+1, SCALE, WORK( KI+N2 ), 1 )
                     END IF
                     WORK( J+N ) = X( 1, 1 )
                     WORK( J+N2 ) = X( 1, 2 )
                     WORK( J+1+N ) = X( 2, 1 )
                     WORK( J+1+N2 ) = X( 2, 2 )
                     VMAX = MAX( ABS( X( 1, 1 ) ), ABS( X( 1, 2 ) ),
     $                      ABS( X( 2, 1 ) ), ABS( X( 2, 2 ) ), VMAX )
                     VCRIT = BIGNUM / VMAX
*
                  END IF
  200          CONTINUE
*
*              Copy the vector x or Q*x to VL and normalize.
*
               IF( .NOT.OVER ) THEN
                  CALL DCOPY( N-KI+1, WORK( KI+N ), 1, VL( KI, IS ),
     $                        1 )
                  CALL DCOPY( N-KI+1, WORK( KI+N2 ), 1, VL( KI,
     $                        IS+1 ),
     $                        1 )
*
                  EMAX = ZERO
                  DO 220 K = KI, N
                     EMAX = MAX( EMAX, ABS( VL( K, IS ) )+
     $                      ABS( VL( K, IS+1 ) ) )
  220             CONTINUE
                  REMAX = ONE / EMAX
                  CALL DSCAL( N-KI+1, REMAX, VL( KI, IS ), 1 )
                  CALL DSCAL( N-KI+1, REMAX, VL( KI, IS+1 ), 1 )
*
                  DO 230 K = 1, KI - 1
                     VL( K, IS ) = ZERO
                     VL( K, IS+1 ) = ZERO
  230             CONTINUE
               ELSE
                  IF( KI.LT.N-1 ) THEN
                     CALL DGEMV( 'N', N, N-KI-1, ONE, VL( 1, KI+2 ),
     $                           LDVL, WORK( KI+2+N ), 1, WORK( KI+N ),
     $                           VL( 1, KI ), 1 )
                     CALL DGEMV( 'N', N, N-KI-1, ONE, VL( 1, KI+2 ),
     $                           LDVL, WORK( KI+2+N2 ), 1,
     $                           WORK( KI+1+N2 ), VL( 1, KI+1 ), 1 )
                  ELSE
                     CALL DSCAL( N, WORK( KI+N ), VL( 1, KI ), 1 )
                     CALL DSCAL( N, WORK( KI+1+N2 ), VL( 1, KI+1 ),
     $                           1 )
                  END IF
*
                  EMAX = ZERO
                  DO 240 K = 1, N
                     EMAX = MAX( EMAX, ABS( VL( K, KI ) )+
     $                      ABS( VL( K, KI+1 ) ) )
  240             CONTINUE
                  REMAX = ONE / EMAX
                  CALL DSCAL( N, REMAX, VL( 1, KI ), 1 )
                  CALL DSCAL( N, REMAX, VL( 1, KI+1 ), 1 )
*
               END IF
*
            END IF
*
            IS = IS + 1
            IF( IP.NE.0 )
     $         IS = IS + 1
  250       CONTINUE
            IF( IP.EQ.-1 )
     $         IP = 0
            IF( IP.EQ.1 )
     $         IP = -1
*
  260    CONTINUE
*
      END IF
*
      RETURN
*
*     End of DTREVC
*
      END
*> \brief \b DTREVC3
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTREVC3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrevc3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrevc3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrevc3.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTREVC3( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL,
*                           VR, LDVR, MM, M, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          HOWMNY, SIDE
*       INTEGER            INFO, LDT, LDVL, LDVR, LWORK, M, MM, N
*       ..
*       .. Array Arguments ..
*       LOGICAL            SELECT( * )
*       DOUBLE PRECISION   T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
*      $                   WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTREVC3 computes some or all of the right and/or left eigenvectors of
*> a real upper quasi-triangular matrix T.
*> Matrices of this type are produced by the Schur factorization of
*> a real general matrix:  A = Q*T*Q**T, as computed by DHSEQR.
*>
*> The right eigenvector x and the left eigenvector y of T corresponding
*> to an eigenvalue w are defined by:
*>
*>    T*x = w*x,     (y**T)*T = w*(y**T)
*>
*> where y**T denotes the transpose of the vector y.
*> The eigenvalues are not input to this routine, but are read directly
*> from the diagonal blocks of T.
*>
*> This routine returns the matrices X and/or Y of right and left
*> eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
*> input matrix. If Q is the orthogonal factor that reduces a matrix
*> A to Schur form T, then Q*X and Q*Y are the matrices of right and
*> left eigenvectors of A.
*>
*> This uses a Level 3 BLAS version of the back transformation.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'R':  compute right eigenvectors only;
*>          = 'L':  compute left eigenvectors only;
*>          = 'B':  compute both right and left eigenvectors.
*> \endverbatim
*>
*> \param[in] HOWMNY
*> \verbatim
*>          HOWMNY is CHARACTER*1
*>          = 'A':  compute all right and/or left eigenvectors;
*>          = 'B':  compute all right and/or left eigenvectors,
*>                  backtransformed by the matrices in VR and/or VL;
*>          = 'S':  compute selected right and/or left eigenvectors,
*>                  as indicated by the logical array SELECT.
*> \endverbatim
*>
*> \param[in,out] SELECT
*> \verbatim
*>          SELECT is LOGICAL array, dimension (N)
*>          If HOWMNY = 'S', SELECT specifies the eigenvectors to be
*>          computed.
*>          If w(j) is a real eigenvalue, the corresponding real
*>          eigenvector is computed if SELECT(j) is .TRUE..
*>          If w(j) and w(j+1) are the real and imaginary parts of a
*>          complex eigenvalue, the corresponding complex eigenvector is
*>          computed if either SELECT(j) or SELECT(j+1) is .TRUE., and
*>          on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to
*>          .FALSE..
*>          Not referenced if HOWMNY = 'A' or 'B'.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix T. N >= 0.
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,N)
*>          The upper quasi-triangular matrix T in Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*>          VL is DOUBLE PRECISION array, dimension (LDVL,MM)
*>          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
*>          contain an N-by-N matrix Q (usually the orthogonal matrix Q
*>          of Schur vectors returned by DHSEQR).
*>          On exit, if SIDE = 'L' or 'B', VL contains:
*>          if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
*>          if HOWMNY = 'B', the matrix Q*Y;
*>          if HOWMNY = 'S', the left eigenvectors of T specified by
*>                           SELECT, stored consecutively in the columns
*>                           of VL, in the same order as their
*>                           eigenvalues.
*>          A complex eigenvector corresponding to a complex eigenvalue
*>          is stored in two consecutive columns, the first holding the
*>          real part, and the second the imaginary part.
*>          Not referenced if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*>          LDVL is INTEGER
*>          The leading dimension of the array VL.
*>          LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N.
*> \endverbatim
*>
*> \param[in,out] VR
*> \verbatim
*>          VR is DOUBLE PRECISION array, dimension (LDVR,MM)
*>          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
*>          contain an N-by-N matrix Q (usually the orthogonal matrix Q
*>          of Schur vectors returned by DHSEQR).
*>          On exit, if SIDE = 'R' or 'B', VR contains:
*>          if HOWMNY = 'A', the matrix X of right eigenvectors of T;
*>          if HOWMNY = 'B', the matrix Q*X;
*>          if HOWMNY = 'S', the right eigenvectors of T specified by
*>                           SELECT, stored consecutively in the columns
*>                           of VR, in the same order as their
*>                           eigenvalues.
*>          A complex eigenvector corresponding to a complex eigenvalue
*>          is stored in two consecutive columns, the first holding the
*>          real part and the second the imaginary part.
*>          Not referenced if SIDE = 'L'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*>          LDVR is INTEGER
*>          The leading dimension of the array VR.
*>          LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N.
*> \endverbatim
*>
*> \param[in] MM
*> \verbatim
*>          MM is INTEGER
*>          The number of columns in the arrays VL and/or VR. MM >= M.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The number of columns in the arrays VL and/or VR actually
*>          used to store the eigenvectors.
*>          If HOWMNY = 'A' or 'B', M is set to N.
*>          Each selected real eigenvector occupies one column and each
*>          selected complex eigenvector occupies two columns.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of array WORK. LWORK >= max(1,3*N).
*>          For optimum performance, LWORK >= N + 2*N*NB, where NB is
*>          the optimal blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trevc3
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The algorithm used in this program is basically backward (forward)
*>  substitution, with scaling to make the the code robust against
*>  possible overflow.
*>
*>  Each eigenvector is normalized so that the element of largest
*>  magnitude has magnitude 1; here the magnitude of a complex number
*>  (x,y) is taken to be |x| + |y|.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTREVC3( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL,
     $                    VR, LDVR, MM, M, WORK, LWORK, INFO )
      IMPLICIT NONE
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          HOWMNY, SIDE
      INTEGER            INFO, LDT, LDVL, LDVR, LWORK, M, MM, N
*     ..
*     .. Array Arguments ..
      LOGICAL            SELECT( * )
      DOUBLE PRECISION   T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
     $                   WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
      INTEGER            NBMIN, NBMAX
      PARAMETER          ( NBMIN = 8, NBMAX = 128 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLV, BOTHV, LEFTV, LQUERY, OVER, PAIR,
     $                   RIGHTV, SOMEV
      INTEGER            I, IERR, II, IP, IS, J, J1, J2, JNXT, K, KI,
     $                   IV, MAXWRK, NB, KI2
      DOUBLE PRECISION   BETA, BIGNUM, EMAX, OVFL, REC, REMAX, SCALE,
     $                   SMIN, SMLNUM, ULP, UNFL, VCRIT, VMAX, WI, WR,
     $                   XNORM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX, ILAENV
      DOUBLE PRECISION   DDOT, DLAMCH
      EXTERNAL           LSAME, IDAMAX, ILAENV, DDOT, DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DGEMV, DLALN2, DSCAL,
     $                   XERBLA,
     $                   DGEMM, DLASET, DLACPY
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   X( 2, 2 )
      INTEGER            ISCOMPLEX( NBMAX )
*     ..
*     .. Executable Statements ..
*
*     Decode and test the input parameters
*
      BOTHV  = LSAME( SIDE, 'B' )
      RIGHTV = LSAME( SIDE, 'R' ) .OR. BOTHV
      LEFTV  = LSAME( SIDE, 'L' ) .OR. BOTHV
*
      ALLV  = LSAME( HOWMNY, 'A' )
      OVER  = LSAME( HOWMNY, 'B' )
      SOMEV = LSAME( HOWMNY, 'S' )
*
      INFO = 0
      NB = ILAENV( 1, 'DTREVC', SIDE // HOWMNY, N, -1, -1, -1 )
      MAXWRK = MAX( 1, N + 2*N*NB )
      WORK(1) = MAXWRK
      LQUERY = ( LWORK.EQ.-1 )
      IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
         INFO = -1
      ELSE IF( .NOT.ALLV .AND. .NOT.OVER .AND. .NOT.SOMEV ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
         INFO = -6
      ELSE IF( LDVL.LT.1 .OR. ( LEFTV .AND. LDVL.LT.N ) ) THEN
         INFO = -8
      ELSE IF( LDVR.LT.1 .OR. ( RIGHTV .AND. LDVR.LT.N ) ) THEN
         INFO = -10
      ELSE IF( LWORK.LT.MAX( 1, 3*N ) .AND. .NOT.LQUERY ) THEN
         INFO = -14
      ELSE
*
*        Set M to the number of columns required to store the selected
*        eigenvectors, standardize the array SELECT if necessary, and
*        test MM.
*
         IF( SOMEV ) THEN
            M = 0
            PAIR = .FALSE.
            DO 10 J = 1, N
               IF( PAIR ) THEN
                  PAIR = .FALSE.
                  SELECT( J ) = .FALSE.
               ELSE
                  IF( J.LT.N ) THEN
                     IF( T( J+1, J ).EQ.ZERO ) THEN
                        IF( SELECT( J ) )
     $                     M = M + 1
                     ELSE
                        PAIR = .TRUE.
                        IF( SELECT( J ) .OR. SELECT( J+1 ) ) THEN
                           SELECT( J ) = .TRUE.
                           M = M + 2
                        END IF
                     END IF
                  ELSE
                     IF( SELECT( N ) )
     $                  M = M + 1
                  END IF
               END IF
   10       CONTINUE
         ELSE
            M = N
         END IF
*
         IF( MM.LT.M ) THEN
            INFO = -11
         END IF
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTREVC3', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Use blocked version of back-transformation if sufficient workspace.
*     Zero-out the workspace to avoid potential NaN propagation.
*
      IF( OVER .AND. LWORK .GE. N + 2*N*NBMIN ) THEN
         NB = (LWORK - N) / (2*N)
         NB = MIN( NB, NBMAX )
         CALL DLASET( 'F', N, 1+2*NB, ZERO, ZERO, WORK, N )
      ELSE
         NB = 1
      END IF
*
*     Set the constants to control overflow.
*
      UNFL = DLAMCH( 'Safe minimum' )
      OVFL = ONE / UNFL
      ULP = DLAMCH( 'Precision' )
      SMLNUM = UNFL*( N / ULP )
      BIGNUM = ( ONE-ULP ) / SMLNUM
*
*     Compute 1-norm of each column of strictly upper triangular
*     part of T to control overflow in triangular solver.
*
      WORK( 1 ) = ZERO
      DO 30 J = 2, N
         WORK( J ) = ZERO
         DO 20 I = 1, J - 1
            WORK( J ) = WORK( J ) + ABS( T( I, J ) )
   20    CONTINUE
   30 CONTINUE
*
*     Index IP is used to specify the real or complex eigenvalue:
*       IP = 0, real eigenvalue,
*            1, first  of conjugate complex pair: (wr,wi)
*           -1, second of conjugate complex pair: (wr,wi)
*       ISCOMPLEX array stores IP for each column in current block.
*
      IF( RIGHTV ) THEN
*
*        ============================================================
*        Compute right eigenvectors.
*
*        IV is index of column in current block.
*        For complex right vector, uses IV-1 for real part and IV for complex part.
*        Non-blocked version always uses IV=2;
*        blocked     version starts with IV=NB, goes down to 1 or 2.
*        (Note the "0-th" column is used for 1-norms computed above.)
         IV = 2
         IF( NB.GT.2 ) THEN
            IV = NB
         END IF

         IP = 0
         IS = M
         DO 140 KI = N, 1, -1
            IF( IP.EQ.-1 ) THEN
*              previous iteration (ki+1) was second of conjugate pair,
*              so this ki is first of conjugate pair; skip to end of loop
               IP = 1
               GO TO 140
            ELSE IF( KI.EQ.1 ) THEN
*              last column, so this ki must be real eigenvalue
               IP = 0
            ELSE IF( T( KI, KI-1 ).EQ.ZERO ) THEN
*              zero on sub-diagonal, so this ki is real eigenvalue
               IP = 0
            ELSE
*              non-zero on sub-diagonal, so this ki is second of conjugate pair
               IP = -1
            END IF

            IF( SOMEV ) THEN
               IF( IP.EQ.0 ) THEN
                  IF( .NOT.SELECT( KI ) )
     $               GO TO 140
               ELSE
                  IF( .NOT.SELECT( KI-1 ) )
     $               GO TO 140
               END IF
            END IF
*
*           Compute the KI-th eigenvalue (WR,WI).
*
            WR = T( KI, KI )
            WI = ZERO
            IF( IP.NE.0 )
     $         WI = SQRT( ABS( T( KI, KI-1 ) ) )*
     $              SQRT( ABS( T( KI-1, KI ) ) )
            SMIN = MAX( ULP*( ABS( WR )+ABS( WI ) ), SMLNUM )
*
            IF( IP.EQ.0 ) THEN
*
*              --------------------------------------------------------
*              Real right eigenvector
*
               WORK( KI + IV*N ) = ONE
*
*              Form right-hand side.
*
               DO 50 K = 1, KI - 1
                  WORK( K + IV*N ) = -T( K, KI )
   50          CONTINUE
*
*              Solve upper quasi-triangular system:
*              [ T(1:KI-1,1:KI-1) - WR ]*X = SCALE*WORK.
*
               JNXT = KI - 1
               DO 60 J = KI - 1, 1, -1
                  IF( J.GT.JNXT )
     $               GO TO 60
                  J1 = J
                  J2 = J
                  JNXT = J - 1
                  IF( J.GT.1 ) THEN
                     IF( T( J, J-1 ).NE.ZERO ) THEN
                        J1   = J - 1
                        JNXT = J - 2
                     END IF
                  END IF
*
                  IF( J1.EQ.J2 ) THEN
*
*                    1-by-1 diagonal block
*
                     CALL DLALN2( .FALSE., 1, 1, SMIN, ONE, T( J,
     $                            J ),
     $                            LDT, ONE, ONE, WORK( J+IV*N ), N, WR,
     $                            ZERO, X, 2, SCALE, XNORM, IERR )
*
*                    Scale X(1,1) to avoid overflow when updating
*                    the right-hand side.
*
                     IF( XNORM.GT.ONE ) THEN
                        IF( WORK( J ).GT.BIGNUM / XNORM ) THEN
                           X( 1, 1 ) = X( 1, 1 ) / XNORM
                           SCALE = SCALE / XNORM
                        END IF
                     END IF
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE )
     $                  CALL DSCAL( KI, SCALE, WORK( 1+IV*N ), 1 )
                     WORK( J+IV*N ) = X( 1, 1 )
*
*                    Update right-hand side
*
                     CALL DAXPY( J-1, -X( 1, 1 ), T( 1, J ), 1,
     $                           WORK( 1+IV*N ), 1 )
*
                  ELSE
*
*                    2-by-2 diagonal block
*
                     CALL DLALN2( .FALSE., 2, 1, SMIN, ONE,
     $                            T( J-1, J-1 ), LDT, ONE, ONE,
     $                            WORK( J-1+IV*N ), N, WR, ZERO, X, 2,
     $                            SCALE, XNORM, IERR )
*
*                    Scale X(1,1) and X(2,1) to avoid overflow when
*                    updating the right-hand side.
*
                     IF( XNORM.GT.ONE ) THEN
                        BETA = MAX( WORK( J-1 ), WORK( J ) )
                        IF( BETA.GT.BIGNUM / XNORM ) THEN
                           X( 1, 1 ) = X( 1, 1 ) / XNORM
                           X( 2, 1 ) = X( 2, 1 ) / XNORM
                           SCALE = SCALE / XNORM
                        END IF
                     END IF
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE )
     $                  CALL DSCAL( KI, SCALE, WORK( 1+IV*N ), 1 )
                     WORK( J-1+IV*N ) = X( 1, 1 )
                     WORK( J  +IV*N ) = X( 2, 1 )
*
*                    Update right-hand side
*
                     CALL DAXPY( J-2, -X( 1, 1 ), T( 1, J-1 ), 1,
     $                           WORK( 1+IV*N ), 1 )
                     CALL DAXPY( J-2, -X( 2, 1 ), T( 1, J ), 1,
     $                           WORK( 1+IV*N ), 1 )
                  END IF
   60          CONTINUE
*
*              Copy the vector x or Q*x to VR and normalize.
*
               IF( .NOT.OVER ) THEN
*                 ------------------------------
*                 no back-transform: copy x to VR and normalize.
                  CALL DCOPY( KI, WORK( 1 + IV*N ), 1, VR( 1, IS ),
     $                        1 )
*
                  II = IDAMAX( KI, VR( 1, IS ), 1 )
                  REMAX = ONE / ABS( VR( II, IS ) )
                  CALL DSCAL( KI, REMAX, VR( 1, IS ), 1 )
*
                  DO 70 K = KI + 1, N
                     VR( K, IS ) = ZERO
   70             CONTINUE
*
               ELSE IF( NB.EQ.1 ) THEN
*                 ------------------------------
*                 version 1: back-transform each vector with GEMV, Q*x.
                  IF( KI.GT.1 )
     $               CALL DGEMV( 'N', N, KI-1, ONE, VR, LDVR,
     $                           WORK( 1 + IV*N ), 1, WORK( KI + IV*N ),
     $                           VR( 1, KI ), 1 )
*
                  II = IDAMAX( N, VR( 1, KI ), 1 )
                  REMAX = ONE / ABS( VR( II, KI ) )
                  CALL DSCAL( N, REMAX, VR( 1, KI ), 1 )
*
               ELSE
*                 ------------------------------
*                 version 2: back-transform block of vectors with GEMM
*                 zero out below vector
                  DO K = KI + 1, N
                     WORK( K + IV*N ) = ZERO
                  END DO
                  ISCOMPLEX( IV ) = IP
*                 back-transform and normalization is done below
               END IF
            ELSE
*
*              --------------------------------------------------------
*              Complex right eigenvector.
*
*              Initial solve
*              [ ( T(KI-1,KI-1) T(KI-1,KI) ) - (WR + I*WI) ]*X = 0.
*              [ ( T(KI,  KI-1) T(KI,  KI) )               ]
*
               IF( ABS( T( KI-1, KI ) ).GE.ABS( T( KI, KI-1 ) ) ) THEN
                  WORK( KI-1 + (IV-1)*N ) = ONE
                  WORK( KI   + (IV  )*N ) = WI / T( KI-1, KI )
               ELSE
                  WORK( KI-1 + (IV-1)*N ) = -WI / T( KI, KI-1 )
                  WORK( KI   + (IV  )*N ) = ONE
               END IF
               WORK( KI   + (IV-1)*N ) = ZERO
               WORK( KI-1 + (IV  )*N ) = ZERO
*
*              Form right-hand side.
*
               DO 80 K = 1, KI - 2
                  WORK( K+(IV-1)*N ) = -WORK( KI-1+(IV-1)*N )*T(K,KI-1)
                  WORK( K+(IV  )*N ) = -WORK( KI  +(IV  )*N )*T(K,KI  )
   80          CONTINUE
*
*              Solve upper quasi-triangular system:
*              [ T(1:KI-2,1:KI-2) - (WR+i*WI) ]*X = SCALE*(WORK+i*WORK2)
*
               JNXT = KI - 2
               DO 90 J = KI - 2, 1, -1
                  IF( J.GT.JNXT )
     $               GO TO 90
                  J1 = J
                  J2 = J
                  JNXT = J - 1
                  IF( J.GT.1 ) THEN
                     IF( T( J, J-1 ).NE.ZERO ) THEN
                        J1   = J - 1
                        JNXT = J - 2
                     END IF
                  END IF
*
                  IF( J1.EQ.J2 ) THEN
*
*                    1-by-1 diagonal block
*
                     CALL DLALN2( .FALSE., 1, 2, SMIN, ONE, T( J,
     $                            J ),
     $                            LDT, ONE, ONE, WORK( J+(IV-1)*N ), N,
     $                            WR, WI, X, 2, SCALE, XNORM, IERR )
*
*                    Scale X(1,1) and X(1,2) to avoid overflow when
*                    updating the right-hand side.
*
                     IF( XNORM.GT.ONE ) THEN
                        IF( WORK( J ).GT.BIGNUM / XNORM ) THEN
                           X( 1, 1 ) = X( 1, 1 ) / XNORM
                           X( 1, 2 ) = X( 1, 2 ) / XNORM
                           SCALE = SCALE / XNORM
                        END IF
                     END IF
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE ) THEN
                        CALL DSCAL( KI, SCALE, WORK( 1+(IV-1)*N ),
     $                              1 )
                        CALL DSCAL( KI, SCALE, WORK( 1+(IV  )*N ),
     $                              1 )
                     END IF
                     WORK( J+(IV-1)*N ) = X( 1, 1 )
                     WORK( J+(IV  )*N ) = X( 1, 2 )
*
*                    Update the right-hand side
*
                     CALL DAXPY( J-1, -X( 1, 1 ), T( 1, J ), 1,
     $                           WORK( 1+(IV-1)*N ), 1 )
                     CALL DAXPY( J-1, -X( 1, 2 ), T( 1, J ), 1,
     $                           WORK( 1+(IV  )*N ), 1 )
*
                  ELSE
*
*                    2-by-2 diagonal block
*
                     CALL DLALN2( .FALSE., 2, 2, SMIN, ONE,
     $                            T( J-1, J-1 ), LDT, ONE, ONE,
     $                            WORK( J-1+(IV-1)*N ), N, WR, WI, X, 2,
     $                            SCALE, XNORM, IERR )
*
*                    Scale X to avoid overflow when updating
*                    the right-hand side.
*
                     IF( XNORM.GT.ONE ) THEN
                        BETA = MAX( WORK( J-1 ), WORK( J ) )
                        IF( BETA.GT.BIGNUM / XNORM ) THEN
                           REC = ONE / XNORM
                           X( 1, 1 ) = X( 1, 1 )*REC
                           X( 1, 2 ) = X( 1, 2 )*REC
                           X( 2, 1 ) = X( 2, 1 )*REC
                           X( 2, 2 ) = X( 2, 2 )*REC
                           SCALE = SCALE*REC
                        END IF
                     END IF
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE ) THEN
                        CALL DSCAL( KI, SCALE, WORK( 1+(IV-1)*N ),
     $                              1 )
                        CALL DSCAL( KI, SCALE, WORK( 1+(IV  )*N ),
     $                              1 )
                     END IF
                     WORK( J-1+(IV-1)*N ) = X( 1, 1 )
                     WORK( J  +(IV-1)*N ) = X( 2, 1 )
                     WORK( J-1+(IV  )*N ) = X( 1, 2 )
                     WORK( J  +(IV  )*N ) = X( 2, 2 )
*
*                    Update the right-hand side
*
                     CALL DAXPY( J-2, -X( 1, 1 ), T( 1, J-1 ), 1,
     $                           WORK( 1+(IV-1)*N   ), 1 )
                     CALL DAXPY( J-2, -X( 2, 1 ), T( 1, J ), 1,
     $                           WORK( 1+(IV-1)*N   ), 1 )
                     CALL DAXPY( J-2, -X( 1, 2 ), T( 1, J-1 ), 1,
     $                           WORK( 1+(IV  )*N ), 1 )
                     CALL DAXPY( J-2, -X( 2, 2 ), T( 1, J ), 1,
     $                           WORK( 1+(IV  )*N ), 1 )
                  END IF
   90          CONTINUE
*
*              Copy the vector x or Q*x to VR and normalize.
*
               IF( .NOT.OVER ) THEN
*                 ------------------------------
*                 no back-transform: copy x to VR and normalize.
                  CALL DCOPY( KI, WORK( 1+(IV-1)*N ), 1, VR(1,IS-1),
     $                        1 )
                  CALL DCOPY( KI, WORK( 1+(IV  )*N ), 1, VR(1,IS  ),
     $                        1 )
*
                  EMAX = ZERO
                  DO 100 K = 1, KI
                     EMAX = MAX( EMAX, ABS( VR( K, IS-1 ) )+
     $                                 ABS( VR( K, IS   ) ) )
  100             CONTINUE
                  REMAX = ONE / EMAX
                  CALL DSCAL( KI, REMAX, VR( 1, IS-1 ), 1 )
                  CALL DSCAL( KI, REMAX, VR( 1, IS   ), 1 )
*
                  DO 110 K = KI + 1, N
                     VR( K, IS-1 ) = ZERO
                     VR( K, IS   ) = ZERO
  110             CONTINUE
*
               ELSE IF( NB.EQ.1 ) THEN
*                 ------------------------------
*                 version 1: back-transform each vector with GEMV, Q*x.
                  IF( KI.GT.2 ) THEN
                     CALL DGEMV( 'N', N, KI-2, ONE, VR, LDVR,
     $                           WORK( 1    + (IV-1)*N ), 1,
     $                           WORK( KI-1 + (IV-1)*N ), VR(1,KI-1), 1)
                     CALL DGEMV( 'N', N, KI-2, ONE, VR, LDVR,
     $                           WORK( 1  + (IV)*N ), 1,
     $                           WORK( KI + (IV)*N ), VR( 1, KI ), 1 )
                  ELSE
                     CALL DSCAL( N, WORK(KI-1+(IV-1)*N), VR(1,KI-1),
     $                           1)
                     CALL DSCAL( N, WORK(KI  +(IV  )*N), VR(1,KI  ),
     $                           1)
                  END IF
*
                  EMAX = ZERO
                  DO 120 K = 1, N
                     EMAX = MAX( EMAX, ABS( VR( K, KI-1 ) )+
     $                                 ABS( VR( K, KI   ) ) )
  120             CONTINUE
                  REMAX = ONE / EMAX
                  CALL DSCAL( N, REMAX, VR( 1, KI-1 ), 1 )
                  CALL DSCAL( N, REMAX, VR( 1, KI   ), 1 )
*
               ELSE
*                 ------------------------------
*                 version 2: back-transform block of vectors with GEMM
*                 zero out below vector
                  DO K = KI + 1, N
                     WORK( K + (IV-1)*N ) = ZERO
                     WORK( K + (IV  )*N ) = ZERO
                  END DO
                  ISCOMPLEX( IV-1 ) = -IP
                  ISCOMPLEX( IV   ) =  IP
                  IV = IV - 1
*                 back-transform and normalization is done below
               END IF
            END IF

            IF( NB.GT.1 ) THEN
*              --------------------------------------------------------
*              Blocked version of back-transform
*              For complex case, KI2 includes both vectors (KI-1 and KI)
               IF( IP.EQ.0 ) THEN
                  KI2 = KI
               ELSE
                  KI2 = KI - 1
               END IF

*              Columns IV:NB of work are valid vectors.
*              When the number of vectors stored reaches NB-1 or NB,
*              or if this was last vector, do the GEMM
               IF( (IV.LE.2) .OR. (KI2.EQ.1) ) THEN
                  CALL DGEMM( 'N', 'N', N, NB-IV+1, KI2+NB-IV, ONE,
     $                        VR, LDVR,
     $                        WORK( 1 + (IV)*N    ), N,
     $                        ZERO,
     $                        WORK( 1 + (NB+IV)*N ), N )
*                 normalize vectors
                  DO K = IV, NB
                     IF( ISCOMPLEX(K).EQ.0 ) THEN
*                       real eigenvector
                        II = IDAMAX( N, WORK( 1 + (NB+K)*N ), 1 )
                        REMAX = ONE / ABS( WORK( II + (NB+K)*N ) )
                     ELSE IF( ISCOMPLEX(K).EQ.1 ) THEN
*                       first eigenvector of conjugate pair
                        EMAX = ZERO
                        DO II = 1, N
                           EMAX = MAX( EMAX,
     $                                 ABS( WORK( II + (NB+K  )*N ) )+
     $                                 ABS( WORK( II + (NB+K+1)*N ) ) )
                        END DO
                        REMAX = ONE / EMAX
*                    else if ISCOMPLEX(K).EQ.-1
*                       second eigenvector of conjugate pair
*                       reuse same REMAX as previous K
                     END IF
                     CALL DSCAL( N, REMAX, WORK( 1 + (NB+K)*N ), 1 )
                  END DO
                  CALL DLACPY( 'F', N, NB-IV+1,
     $                         WORK( 1 + (NB+IV)*N ), N,
     $                         VR( 1, KI2 ), LDVR )
                  IV = NB
               ELSE
                  IV = IV - 1
               END IF
            END IF ! blocked back-transform
*
            IS = IS - 1
            IF( IP.NE.0 )
     $         IS = IS - 1
  140    CONTINUE
      END IF

      IF( LEFTV ) THEN
*
*        ============================================================
*        Compute left eigenvectors.
*
*        IV is index of column in current block.
*        For complex left vector, uses IV for real part and IV+1 for complex part.
*        Non-blocked version always uses IV=1;
*        blocked     version starts with IV=1, goes up to NB-1 or NB.
*        (Note the "0-th" column is used for 1-norms computed above.)
         IV = 1
         IP = 0
         IS = 1
         DO 260 KI = 1, N
            IF( IP.EQ.1 ) THEN
*              previous iteration (ki-1) was first of conjugate pair,
*              so this ki is second of conjugate pair; skip to end of loop
               IP = -1
               GO TO 260
            ELSE IF( KI.EQ.N ) THEN
*              last column, so this ki must be real eigenvalue
               IP = 0
            ELSE IF( T( KI+1, KI ).EQ.ZERO ) THEN
*              zero on sub-diagonal, so this ki is real eigenvalue
               IP = 0
            ELSE
*              non-zero on sub-diagonal, so this ki is first of conjugate pair
               IP = 1
            END IF
*
            IF( SOMEV ) THEN
               IF( .NOT.SELECT( KI ) )
     $            GO TO 260
            END IF
*
*           Compute the KI-th eigenvalue (WR,WI).
*
            WR = T( KI, KI )
            WI = ZERO
            IF( IP.NE.0 )
     $         WI = SQRT( ABS( T( KI, KI+1 ) ) )*
     $              SQRT( ABS( T( KI+1, KI ) ) )
            SMIN = MAX( ULP*( ABS( WR )+ABS( WI ) ), SMLNUM )
*
            IF( IP.EQ.0 ) THEN
*
*              --------------------------------------------------------
*              Real left eigenvector
*
               WORK( KI + IV*N ) = ONE
*
*              Form right-hand side.
*
               DO 160 K = KI + 1, N
                  WORK( K + IV*N ) = -T( KI, K )
  160          CONTINUE
*
*              Solve transposed quasi-triangular system:
*              [ T(KI+1:N,KI+1:N) - WR ]**T * X = SCALE*WORK
*
               VMAX = ONE
               VCRIT = BIGNUM
*
               JNXT = KI + 1
               DO 170 J = KI + 1, N
                  IF( J.LT.JNXT )
     $               GO TO 170
                  J1 = J
                  J2 = J
                  JNXT = J + 1
                  IF( J.LT.N ) THEN
                     IF( T( J+1, J ).NE.ZERO ) THEN
                        J2 = J + 1
                        JNXT = J + 2
                     END IF
                  END IF
*
                  IF( J1.EQ.J2 ) THEN
*
*                    1-by-1 diagonal block
*
*                    Scale if necessary to avoid overflow when forming
*                    the right-hand side.
*
                     IF( WORK( J ).GT.VCRIT ) THEN
                        REC = ONE / VMAX
                        CALL DSCAL( N-KI+1, REC, WORK( KI+IV*N ), 1 )
                        VMAX = ONE
                        VCRIT = BIGNUM
                     END IF
*
                     WORK( J+IV*N ) = WORK( J+IV*N ) -
     $                                DDOT( J-KI-1, T( KI+1, J ), 1,
     $                                      WORK( KI+1+IV*N ), 1 )
*
*                    Solve [ T(J,J) - WR ]**T * X = WORK
*
                     CALL DLALN2( .FALSE., 1, 1, SMIN, ONE, T( J,
     $                            J ),
     $                            LDT, ONE, ONE, WORK( J+IV*N ), N, WR,
     $                            ZERO, X, 2, SCALE, XNORM, IERR )
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE )
     $                  CALL DSCAL( N-KI+1, SCALE, WORK( KI+IV*N ),
     $                              1 )
                     WORK( J+IV*N ) = X( 1, 1 )
                     VMAX = MAX( ABS( WORK( J+IV*N ) ), VMAX )
                     VCRIT = BIGNUM / VMAX
*
                  ELSE
*
*                    2-by-2 diagonal block
*
*                    Scale if necessary to avoid overflow when forming
*                    the right-hand side.
*
                     BETA = MAX( WORK( J ), WORK( J+1 ) )
                     IF( BETA.GT.VCRIT ) THEN
                        REC = ONE / VMAX
                        CALL DSCAL( N-KI+1, REC, WORK( KI+IV*N ), 1 )
                        VMAX = ONE
                        VCRIT = BIGNUM
                     END IF
*
                     WORK( J+IV*N ) = WORK( J+IV*N ) -
     $                                DDOT( J-KI-1, T( KI+1, J ), 1,
     $                                      WORK( KI+1+IV*N ), 1 )
*
                     WORK( J+1+IV*N ) = WORK( J+1+IV*N ) -
     $                                  DDOT( J-KI-1, T( KI+1, J+1 ),
     $                                        1,
     $                                        WORK( KI+1+IV*N ), 1 )
*
*                    Solve
*                    [ T(J,J)-WR   T(J,J+1)      ]**T * X = SCALE*( WORK1 )
*                    [ T(J+1,J)    T(J+1,J+1)-WR ]                ( WORK2 )
*
                     CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, T( J, J ),
     $                            LDT, ONE, ONE, WORK( J+IV*N ), N, WR,
     $                            ZERO, X, 2, SCALE, XNORM, IERR )
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE )
     $                  CALL DSCAL( N-KI+1, SCALE, WORK( KI+IV*N ),
     $                              1 )
                     WORK( J  +IV*N ) = X( 1, 1 )
                     WORK( J+1+IV*N ) = X( 2, 1 )
*
                     VMAX = MAX( ABS( WORK( J  +IV*N ) ),
     $                           ABS( WORK( J+1+IV*N ) ), VMAX )
                     VCRIT = BIGNUM / VMAX
*
                  END IF
  170          CONTINUE
*
*              Copy the vector x or Q*x to VL and normalize.
*
               IF( .NOT.OVER ) THEN
*                 ------------------------------
*                 no back-transform: copy x to VL and normalize.
                  CALL DCOPY( N-KI+1, WORK( KI + IV*N ), 1,
     $                                VL( KI, IS ), 1 )
*
                  II = IDAMAX( N-KI+1, VL( KI, IS ), 1 ) + KI - 1
                  REMAX = ONE / ABS( VL( II, IS ) )
                  CALL DSCAL( N-KI+1, REMAX, VL( KI, IS ), 1 )
*
                  DO 180 K = 1, KI - 1
                     VL( K, IS ) = ZERO
  180             CONTINUE
*
               ELSE IF( NB.EQ.1 ) THEN
*                 ------------------------------
*                 version 1: back-transform each vector with GEMV, Q*x.
                  IF( KI.LT.N )
     $               CALL DGEMV( 'N', N, N-KI, ONE,
     $                           VL( 1, KI+1 ), LDVL,
     $                           WORK( KI+1 + IV*N ), 1,
     $                           WORK( KI   + IV*N ), VL( 1, KI ), 1 )
*
                  II = IDAMAX( N, VL( 1, KI ), 1 )
                  REMAX = ONE / ABS( VL( II, KI ) )
                  CALL DSCAL( N, REMAX, VL( 1, KI ), 1 )
*
               ELSE
*                 ------------------------------
*                 version 2: back-transform block of vectors with GEMM
*                 zero out above vector
*                 could go from KI-NV+1 to KI-1
                  DO K = 1, KI - 1
                     WORK( K + IV*N ) = ZERO
                  END DO
                  ISCOMPLEX( IV ) = IP
*                 back-transform and normalization is done below
               END IF
            ELSE
*
*              --------------------------------------------------------
*              Complex left eigenvector.
*
*              Initial solve:
*              [ ( T(KI,KI)    T(KI,KI+1)  )**T - (WR - I* WI) ]*X = 0.
*              [ ( T(KI+1,KI) T(KI+1,KI+1) )                   ]
*
               IF( ABS( T( KI, KI+1 ) ).GE.ABS( T( KI+1, KI ) ) ) THEN
                  WORK( KI   + (IV  )*N ) = WI / T( KI, KI+1 )
                  WORK( KI+1 + (IV+1)*N ) = ONE
               ELSE
                  WORK( KI   + (IV  )*N ) = ONE
                  WORK( KI+1 + (IV+1)*N ) = -WI / T( KI+1, KI )
               END IF
               WORK( KI+1 + (IV  )*N ) = ZERO
               WORK( KI   + (IV+1)*N ) = ZERO
*
*              Form right-hand side.
*
               DO 190 K = KI + 2, N
                  WORK( K+(IV  )*N ) = -WORK( KI  +(IV  )*N )*T(KI,  K)
                  WORK( K+(IV+1)*N ) = -WORK( KI+1+(IV+1)*N )*T(KI+1,K)
  190          CONTINUE
*
*              Solve transposed quasi-triangular system:
*              [ T(KI+2:N,KI+2:N)**T - (WR-i*WI) ]*X = WORK1+i*WORK2
*
               VMAX = ONE
               VCRIT = BIGNUM
*
               JNXT = KI + 2
               DO 200 J = KI + 2, N
                  IF( J.LT.JNXT )
     $               GO TO 200
                  J1 = J
                  J2 = J
                  JNXT = J + 1
                  IF( J.LT.N ) THEN
                     IF( T( J+1, J ).NE.ZERO ) THEN
                        J2 = J + 1
                        JNXT = J + 2
                     END IF
                  END IF
*
                  IF( J1.EQ.J2 ) THEN
*
*                    1-by-1 diagonal block
*
*                    Scale if necessary to avoid overflow when
*                    forming the right-hand side elements.
*
                     IF( WORK( J ).GT.VCRIT ) THEN
                        REC = ONE / VMAX
                        CALL DSCAL( N-KI+1, REC, WORK(KI+(IV  )*N),
     $                              1 )
                        CALL DSCAL( N-KI+1, REC, WORK(KI+(IV+1)*N),
     $                              1 )
                        VMAX = ONE
                        VCRIT = BIGNUM
                     END IF
*
                     WORK( J+(IV  )*N ) = WORK( J+(IV)*N ) -
     $                                  DDOT( J-KI-2, T( KI+2, J ),
     $                                        1,
     $                                        WORK( KI+2+(IV)*N ), 1 )
                     WORK( J+(IV+1)*N ) = WORK( J+(IV+1)*N ) -
     $                                  DDOT( J-KI-2, T( KI+2, J ),
     $                                        1,
     $                                        WORK( KI+2+(IV+1)*N ), 1 )
*
*                    Solve [ T(J,J)-(WR-i*WI) ]*(X11+i*X12)= WK+I*WK2
*
                     CALL DLALN2( .FALSE., 1, 2, SMIN, ONE, T( J,
     $                            J ),
     $                            LDT, ONE, ONE, WORK( J+IV*N ), N, WR,
     $                            -WI, X, 2, SCALE, XNORM, IERR )
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE ) THEN
                        CALL DSCAL( N-KI+1, SCALE, WORK(KI+(IV  )*N),
     $                              1)
                        CALL DSCAL( N-KI+1, SCALE, WORK(KI+(IV+1)*N),
     $                              1)
                     END IF
                     WORK( J+(IV  )*N ) = X( 1, 1 )
                     WORK( J+(IV+1)*N ) = X( 1, 2 )
                     VMAX = MAX( ABS( WORK( J+(IV  )*N ) ),
     $                           ABS( WORK( J+(IV+1)*N ) ), VMAX )
                     VCRIT = BIGNUM / VMAX
*
                  ELSE
*
*                    2-by-2 diagonal block
*
*                    Scale if necessary to avoid overflow when forming
*                    the right-hand side elements.
*
                     BETA = MAX( WORK( J ), WORK( J+1 ) )
                     IF( BETA.GT.VCRIT ) THEN
                        REC = ONE / VMAX
                        CALL DSCAL( N-KI+1, REC, WORK(KI+(IV  )*N),
     $                              1 )
                        CALL DSCAL( N-KI+1, REC, WORK(KI+(IV+1)*N),
     $                              1 )
                        VMAX = ONE
                        VCRIT = BIGNUM
                     END IF
*
                     WORK( J  +(IV  )*N ) = WORK( J+(IV)*N ) -
     $                                DDOT( J-KI-2, T( KI+2, J ), 1,
     $                                      WORK( KI+2+(IV)*N ), 1 )
*
                     WORK( J  +(IV+1)*N ) = WORK( J+(IV+1)*N ) -
     $                                DDOT( J-KI-2, T( KI+2, J ), 1,
     $                                      WORK( KI+2+(IV+1)*N ), 1 )
*
                     WORK( J+1+(IV  )*N ) = WORK( J+1+(IV)*N ) -
     $                                DDOT( J-KI-2, T( KI+2, J+1 ),
     $                                      1,
     $                                      WORK( KI+2+(IV)*N ), 1 )
*
                     WORK( J+1+(IV+1)*N ) = WORK( J+1+(IV+1)*N ) -
     $                                DDOT( J-KI-2, T( KI+2, J+1 ),
     $                                      1,
     $                                      WORK( KI+2+(IV+1)*N ), 1 )
*
*                    Solve 2-by-2 complex linear equation
*                    [ (T(j,j)   T(j,j+1)  )**T - (wr-i*wi)*I ]*X = SCALE*B
*                    [ (T(j+1,j) T(j+1,j+1))                  ]
*
                     CALL DLALN2( .TRUE., 2, 2, SMIN, ONE, T( J, J ),
     $                            LDT, ONE, ONE, WORK( J+IV*N ), N, WR,
     $                            -WI, X, 2, SCALE, XNORM, IERR )
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE ) THEN
                        CALL DSCAL( N-KI+1, SCALE, WORK(KI+(IV  )*N),
     $                              1)
                        CALL DSCAL( N-KI+1, SCALE, WORK(KI+(IV+1)*N),
     $                              1)
                     END IF
                     WORK( J  +(IV  )*N ) = X( 1, 1 )
                     WORK( J  +(IV+1)*N ) = X( 1, 2 )
                     WORK( J+1+(IV  )*N ) = X( 2, 1 )
                     WORK( J+1+(IV+1)*N ) = X( 2, 2 )
                     VMAX = MAX( ABS( X( 1, 1 ) ), ABS( X( 1, 2 ) ),
     $                           ABS( X( 2, 1 ) ), ABS( X( 2, 2 ) ),
     $                           VMAX )
                     VCRIT = BIGNUM / VMAX
*
                  END IF
  200          CONTINUE
*
*              Copy the vector x or Q*x to VL and normalize.
*
               IF( .NOT.OVER ) THEN
*                 ------------------------------
*                 no back-transform: copy x to VL and normalize.
                  CALL DCOPY( N-KI+1, WORK( KI + (IV  )*N ), 1,
     $                        VL( KI, IS   ), 1 )
                  CALL DCOPY( N-KI+1, WORK( KI + (IV+1)*N ), 1,
     $                        VL( KI, IS+1 ), 1 )
*
                  EMAX = ZERO
                  DO 220 K = KI, N
                     EMAX = MAX( EMAX, ABS( VL( K, IS   ) )+
     $                                 ABS( VL( K, IS+1 ) ) )
  220             CONTINUE
                  REMAX = ONE / EMAX
                  CALL DSCAL( N-KI+1, REMAX, VL( KI, IS   ), 1 )
                  CALL DSCAL( N-KI+1, REMAX, VL( KI, IS+1 ), 1 )
*
                  DO 230 K = 1, KI - 1
                     VL( K, IS   ) = ZERO
                     VL( K, IS+1 ) = ZERO
  230             CONTINUE
*
               ELSE IF( NB.EQ.1 ) THEN
*                 ------------------------------
*                 version 1: back-transform each vector with GEMV, Q*x.
                  IF( KI.LT.N-1 ) THEN
                     CALL DGEMV( 'N', N, N-KI-1, ONE,
     $                           VL( 1, KI+2 ), LDVL,
     $                           WORK( KI+2 + (IV)*N ), 1,
     $                           WORK( KI   + (IV)*N ),
     $                           VL( 1, KI ), 1 )
                     CALL DGEMV( 'N', N, N-KI-1, ONE,
     $                           VL( 1, KI+2 ), LDVL,
     $                           WORK( KI+2 + (IV+1)*N ), 1,
     $                           WORK( KI+1 + (IV+1)*N ),
     $                           VL( 1, KI+1 ), 1 )
                  ELSE
                     CALL DSCAL( N, WORK(KI+  (IV  )*N), VL(1, KI  ),
     $                           1)
                     CALL DSCAL( N, WORK(KI+1+(IV+1)*N), VL(1, KI+1),
     $                           1)
                  END IF
*
                  EMAX = ZERO
                  DO 240 K = 1, N
                     EMAX = MAX( EMAX, ABS( VL( K, KI   ) )+
     $                                 ABS( VL( K, KI+1 ) ) )
  240             CONTINUE
                  REMAX = ONE / EMAX
                  CALL DSCAL( N, REMAX, VL( 1, KI   ), 1 )
                  CALL DSCAL( N, REMAX, VL( 1, KI+1 ), 1 )
*
               ELSE
*                 ------------------------------
*                 version 2: back-transform block of vectors with GEMM
*                 zero out above vector
*                 could go from KI-NV+1 to KI-1
                  DO K = 1, KI - 1
                     WORK( K + (IV  )*N ) = ZERO
                     WORK( K + (IV+1)*N ) = ZERO
                  END DO
                  ISCOMPLEX( IV   ) =  IP
                  ISCOMPLEX( IV+1 ) = -IP
                  IV = IV + 1
*                 back-transform and normalization is done below
               END IF
            END IF

            IF( NB.GT.1 ) THEN
*              --------------------------------------------------------
*              Blocked version of back-transform
*              For complex case, KI2 includes both vectors (KI and KI+1)
               IF( IP.EQ.0 ) THEN
                  KI2 = KI
               ELSE
                  KI2 = KI + 1
               END IF

*              Columns 1:IV of work are valid vectors.
*              When the number of vectors stored reaches NB-1 or NB,
*              or if this was last vector, do the GEMM
               IF( (IV.GE.NB-1) .OR. (KI2.EQ.N) ) THEN
                  CALL DGEMM( 'N', 'N', N, IV, N-KI2+IV, ONE,
     $                        VL( 1, KI2-IV+1 ), LDVL,
     $                        WORK( KI2-IV+1 + (1)*N ), N,
     $                        ZERO,
     $                        WORK( 1 + (NB+1)*N ), N )
*                 normalize vectors
                  DO K = 1, IV
                     IF( ISCOMPLEX(K).EQ.0) THEN
*                       real eigenvector
                        II = IDAMAX( N, WORK( 1 + (NB+K)*N ), 1 )
                        REMAX = ONE / ABS( WORK( II + (NB+K)*N ) )
                     ELSE IF( ISCOMPLEX(K).EQ.1) THEN
*                       first eigenvector of conjugate pair
                        EMAX = ZERO
                        DO II = 1, N
                           EMAX = MAX( EMAX,
     $                                 ABS( WORK( II + (NB+K  )*N ) )+
     $                                 ABS( WORK( II + (NB+K+1)*N ) ) )
                        END DO
                        REMAX = ONE / EMAX
*                    else if ISCOMPLEX(K).EQ.-1
*                       second eigenvector of conjugate pair
*                       reuse same REMAX as previous K
                     END IF
                     CALL DSCAL( N, REMAX, WORK( 1 + (NB+K)*N ), 1 )
                  END DO
                  CALL DLACPY( 'F', N, IV,
     $                         WORK( 1 + (NB+1)*N ), N,
     $                         VL( 1, KI2-IV+1 ), LDVL )
                  IV = 1
               ELSE
                  IV = IV + 1
               END IF
            END IF ! blocked back-transform
*
            IS = IS + 1
            IF( IP.NE.0 )
     $         IS = IS + 1
  260    CONTINUE
      END IF
*
      RETURN
*
*     End of DTREVC3
*
      END
*> \brief \b DTREXC
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTREXC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrexc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrexc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrexc.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTREXC( COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, WORK,
*                          INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          COMPQ
*       INTEGER            IFST, ILST, INFO, LDQ, LDT, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   Q( LDQ, * ), T( LDT, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTREXC reorders the real Schur factorization of a real matrix
*> A = Q*T*Q**T, so that the diagonal block of T with row index IFST is
*> moved to row ILST.
*>
*> The real Schur form T is reordered by an orthogonal similarity
*> transformation Z**T*T*Z, and optionally the matrix Q of Schur vectors
*> is updated by postmultiplying it with Z.
*>
*> T must be in Schur canonical form (as returned by DHSEQR), that is,
*> block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
*> 2-by-2 diagonal block has its diagonal elements equal and its
*> off-diagonal elements of opposite sign.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] COMPQ
*> \verbatim
*>          COMPQ is CHARACTER*1
*>          = 'V':  update the matrix Q of Schur vectors;
*>          = 'N':  do not update Q.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix T. N >= 0.
*>          If N == 0 arguments ILST and IFST may be any value.
*> \endverbatim
*>
*> \param[in,out] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,N)
*>          On entry, the upper quasi-triangular matrix T, in Schur
*>          Schur canonical form.
*>          On exit, the reordered upper quasi-triangular matrix, again
*>          in Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ,N)
*>          On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
*>          On exit, if COMPQ = 'V', Q has been postmultiplied by the
*>          orthogonal transformation matrix Z which reorders T.
*>          If COMPQ = 'N', Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q.  LDQ >= 1, and if
*>          COMPQ = 'V', LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] IFST
*> \verbatim
*>          IFST is INTEGER
*> \endverbatim
*>
*> \param[in,out] ILST
*> \verbatim
*>          ILST is INTEGER
*>
*>          Specify the reordering of the diagonal blocks of T.
*>          The block with row index IFST is moved to row ILST, by a
*>          sequence of transpositions between adjacent blocks.
*>          On exit, if IFST pointed on entry to the second row of a
*>          2-by-2 block, it is changed to point to the first row; ILST
*>          always points to the first row of the block in its final
*>          position (which may differ from its input value by +1 or -1).
*>          1 <= IFST <= N; 1 <= ILST <= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          = 1:  two adjacent blocks were too close to swap (the problem
*>                is very ill-conditioned); T may have been partially
*>                reordered, and ILST points to the first row of the
*>                current position of the block being moved.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trexc
*
*  =====================================================================
      SUBROUTINE DTREXC( COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, WORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          COMPQ
      INTEGER            IFST, ILST, INFO, LDQ, LDT, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   Q( LDQ, * ), T( LDT, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            WANTQ
      INTEGER            HERE, NBF, NBL, NBNEXT
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLAEXC, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Decode and test the input arguments.
*
      INFO = 0
      WANTQ = LSAME( COMPQ, 'V' )
      IF( .NOT.WANTQ .AND. .NOT.LSAME( COMPQ, 'N' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
         INFO = -4
      ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.MAX( 1, N ) ) ) THEN
         INFO = -6
      ELSE IF(( IFST.LT.1 .OR. IFST.GT.N ).AND.( N.GT.0 )) THEN
         INFO = -7
      ELSE IF(( ILST.LT.1 .OR. ILST.GT.N ).AND.( N.GT.0 )) THEN
         INFO = -8
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTREXC', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.LE.1 )
     $   RETURN
*
*     Determine the first row of specified block
*     and find out it is 1 by 1 or 2 by 2.
*
      IF( IFST.GT.1 ) THEN
         IF( T( IFST, IFST-1 ).NE.ZERO )
     $      IFST = IFST - 1
      END IF
      NBF = 1
      IF( IFST.LT.N ) THEN
         IF( T( IFST+1, IFST ).NE.ZERO )
     $      NBF = 2
      END IF
*
*     Determine the first row of the final block
*     and find out it is 1 by 1 or 2 by 2.
*
      IF( ILST.GT.1 ) THEN
         IF( T( ILST, ILST-1 ).NE.ZERO )
     $      ILST = ILST - 1
      END IF
      NBL = 1
      IF( ILST.LT.N ) THEN
         IF( T( ILST+1, ILST ).NE.ZERO )
     $      NBL = 2
      END IF
*
      IF( IFST.EQ.ILST )
     $   RETURN
*
      IF( IFST.LT.ILST ) THEN
*
*        Update ILST
*
         IF( NBF.EQ.2 .AND. NBL.EQ.1 )
     $      ILST = ILST - 1
         IF( NBF.EQ.1 .AND. NBL.EQ.2 )
     $      ILST = ILST + 1
*
         HERE = IFST
*
   10    CONTINUE
*
*        Swap block with next one below
*
         IF( NBF.EQ.1 .OR. NBF.EQ.2 ) THEN
*
*           Current block either 1 by 1 or 2 by 2
*
            NBNEXT = 1
            IF( HERE+NBF+1.LE.N ) THEN
               IF( T( HERE+NBF+1, HERE+NBF ).NE.ZERO )
     $            NBNEXT = 2
            END IF
            CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, NBF, NBNEXT,
     $                   WORK, INFO )
            IF( INFO.NE.0 ) THEN
               ILST = HERE
               RETURN
            END IF
            HERE = HERE + NBNEXT
*
*           Test if 2 by 2 block breaks into two 1 by 1 blocks
*
            IF( NBF.EQ.2 ) THEN
               IF( T( HERE+1, HERE ).EQ.ZERO )
     $            NBF = 3
            END IF
*
         ELSE
*
*           Current block consists of two 1 by 1 blocks each of which
*           must be swapped individually
*
            NBNEXT = 1
            IF( HERE+3.LE.N ) THEN
               IF( T( HERE+3, HERE+2 ).NE.ZERO )
     $            NBNEXT = 2
            END IF
            CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE+1, 1, NBNEXT,
     $                   WORK, INFO )
            IF( INFO.NE.0 ) THEN
               ILST = HERE
               RETURN
            END IF
            IF( NBNEXT.EQ.1 ) THEN
*
*              Swap two 1 by 1 blocks, no problems possible
*
               CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, 1,
     $                      NBNEXT,
     $                      WORK, INFO )
               HERE = HERE + 1
            ELSE
*
*              Recompute NBNEXT in case 2 by 2 split
*
               IF( T( HERE+2, HERE+1 ).EQ.ZERO )
     $            NBNEXT = 1
               IF( NBNEXT.EQ.2 ) THEN
*
*                 2 by 2 Block did not split
*
                  CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, 1,
     $                         NBNEXT, WORK, INFO )
                  IF( INFO.NE.0 ) THEN
                     ILST = HERE
                     RETURN
                  END IF
                  HERE = HERE + 2
               ELSE
*
*                 2 by 2 Block did split
*
                  CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, 1, 1,
     $                         WORK, INFO )
                  CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE+1, 1,
     $                         1,
     $                         WORK, INFO )
                  HERE = HERE + 2
               END IF
            END IF
         END IF
         IF( HERE.LT.ILST )
     $      GO TO 10
*
      ELSE
*
         HERE = IFST
   20    CONTINUE
*
*        Swap block with next one above
*
         IF( NBF.EQ.1 .OR. NBF.EQ.2 ) THEN
*
*           Current block either 1 by 1 or 2 by 2
*
            NBNEXT = 1
            IF( HERE.GE.3 ) THEN
               IF( T( HERE-1, HERE-2 ).NE.ZERO )
     $            NBNEXT = 2
            END IF
            CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE-NBNEXT,
     $                   NBNEXT,
     $                   NBF, WORK, INFO )
            IF( INFO.NE.0 ) THEN
               ILST = HERE
               RETURN
            END IF
            HERE = HERE - NBNEXT
*
*           Test if 2 by 2 block breaks into two 1 by 1 blocks
*
            IF( NBF.EQ.2 ) THEN
               IF( T( HERE+1, HERE ).EQ.ZERO )
     $            NBF = 3
            END IF
*
         ELSE
*
*           Current block consists of two 1 by 1 blocks each of which
*           must be swapped individually
*
            NBNEXT = 1
            IF( HERE.GE.3 ) THEN
               IF( T( HERE-1, HERE-2 ).NE.ZERO )
     $            NBNEXT = 2
            END IF
            CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE-NBNEXT,
     $                   NBNEXT,
     $                   1, WORK, INFO )
            IF( INFO.NE.0 ) THEN
               ILST = HERE
               RETURN
            END IF
            IF( NBNEXT.EQ.1 ) THEN
*
*              Swap two 1 by 1 blocks, no problems possible
*
               CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, NBNEXT,
     $                      1,
     $                      WORK, INFO )
               HERE = HERE - 1
            ELSE
*
*              Recompute NBNEXT in case 2 by 2 split
*
               IF( T( HERE, HERE-1 ).EQ.ZERO )
     $            NBNEXT = 1
               IF( NBNEXT.EQ.2 ) THEN
*
*                 2 by 2 Block did not split
*
                  CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE-1, 2,
     $                         1,
     $                         WORK, INFO )
                  IF( INFO.NE.0 ) THEN
                     ILST = HERE
                     RETURN
                  END IF
                  HERE = HERE - 2
               ELSE
*
*                 2 by 2 Block did split
*
                  CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, 1, 1,
     $                         WORK, INFO )
                  CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE-1, 1,
     $                         1,
     $                         WORK, INFO )
                  HERE = HERE - 2
               END IF
            END IF
         END IF
         IF( HERE.GT.ILST )
     $      GO TO 20
      END IF
      ILST = HERE
*
      RETURN
*
*     End of DTREXC
*
      END
*> \brief \b DTRRFS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRRFS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrrfs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrrfs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrrfs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
*                          LDX, FERR, BERR, WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, TRANS, UPLO
*       INTEGER            INFO, LDA, LDB, LDX, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
*      $                   WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTRRFS provides error bounds and backward error estimates for the
*> solution to a system of linear equations with a triangular
*> coefficient matrix.
*>
*> The solution matrix X must be computed by DTRTRS or some other
*> means before entering this routine.  DTRRFS does not do iterative
*> refinement because doing so cannot improve the backward error.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  A is upper triangular;
*>          = 'L':  A is lower triangular.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the form of the system of equations:
*>          = 'N':  A * X = B  (No transpose)
*>          = 'T':  A**T * X = B  (Transpose)
*>          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          = 'N':  A is non-unit triangular;
*>          = 'U':  A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices B and X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The triangular matrix A.  If UPLO = 'U', the leading N-by-N
*>          upper triangular part of the array A contains the upper
*>          triangular matrix, and the strictly lower triangular part of
*>          A is not referenced.  If UPLO = 'L', the leading N-by-N lower
*>          triangular part of the array A contains the lower triangular
*>          matrix, and the strictly upper triangular part of A is not
*>          referenced.  If DIAG = 'U', the diagonal elements of A are
*>          also not referenced and are assumed to be 1.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          The solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The estimated forward error bound for each solution vector
*>          X(j) (the j-th column of the solution matrix X).
*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
*>          is an estimated upper bound for the magnitude of the largest
*>          element in (X(j) - XTRUE) divided by the magnitude of the
*>          largest element in X(j).  The estimate is as reliable as
*>          the estimate for RCOND, and is almost always a slight
*>          overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector X(j) (i.e., the smallest relative change in
*>          any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trrfs
*
*  =====================================================================
      SUBROUTINE DTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB,
     $                   X,
     $                   LDX, FERR, BERR, WORK, IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, TRANS, UPLO
      INTEGER            INFO, LDA, LDB, LDX, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
     $                   WORK( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOTRAN, NOUNIT, UPPER
      CHARACTER          TRANST
      INTEGER            I, J, K, KASE, NZ
      DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DLACN2, DTRMV, DTRSV,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      NOTRAN = LSAME( TRANS, 'N' )
      NOUNIT = LSAME( DIAG, 'N' )
*
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
     $         LSAME( TRANS, 'C' ) ) THEN
         INFO = -2
      ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -9
      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
         INFO = -11
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTRRFS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
         DO 10 J = 1, NRHS
            FERR( J ) = ZERO
            BERR( J ) = ZERO
   10    CONTINUE
         RETURN
      END IF
*
      IF( NOTRAN ) THEN
         TRANST = 'T'
      ELSE
         TRANST = 'N'
      END IF
*
*     NZ = maximum number of nonzero elements in each row of A, plus 1
*
      NZ = N + 1
      EPS = DLAMCH( 'Epsilon' )
      SAFMIN = DLAMCH( 'Safe minimum' )
      SAFE1 = NZ*SAFMIN
      SAFE2 = SAFE1 / EPS
*
*     Do for each right hand side
*
      DO 250 J = 1, NRHS
*
*        Compute residual R = B - op(A) * X,
*        where op(A) = A or A**T, depending on TRANS.
*
         CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 )
         CALL DTRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK( N+1 ), 1 )
         CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 )
*
*        Compute componentwise relative backward error from formula
*
*        max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
*
*        where abs(Z) is the componentwise absolute value of the matrix
*        or vector Z.  If the i-th component of the denominator is less
*        than SAFE2, then SAFE1 is added to the i-th components of the
*        numerator and denominator before dividing.
*
         DO 20 I = 1, N
            WORK( I ) = ABS( B( I, J ) )
   20    CONTINUE
*
         IF( NOTRAN ) THEN
*
*           Compute abs(A)*abs(X) + abs(B).
*
            IF( UPPER ) THEN
               IF( NOUNIT ) THEN
                  DO 40 K = 1, N
                     XK = ABS( X( K, J ) )
                     DO 30 I = 1, K
                        WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
   30                CONTINUE
   40             CONTINUE
               ELSE
                  DO 60 K = 1, N
                     XK = ABS( X( K, J ) )
                     DO 50 I = 1, K - 1
                        WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
   50                CONTINUE
                     WORK( K ) = WORK( K ) + XK
   60             CONTINUE
               END IF
            ELSE
               IF( NOUNIT ) THEN
                  DO 80 K = 1, N
                     XK = ABS( X( K, J ) )
                     DO 70 I = K, N
                        WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
   70                CONTINUE
   80             CONTINUE
               ELSE
                  DO 100 K = 1, N
                     XK = ABS( X( K, J ) )
                     DO 90 I = K + 1, N
                        WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
   90                CONTINUE
                     WORK( K ) = WORK( K ) + XK
  100             CONTINUE
               END IF
            END IF
         ELSE
*
*           Compute abs(A**T)*abs(X) + abs(B).
*
            IF( UPPER ) THEN
               IF( NOUNIT ) THEN
                  DO 120 K = 1, N
                     S = ZERO
                     DO 110 I = 1, K
                        S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
  110                CONTINUE
                     WORK( K ) = WORK( K ) + S
  120             CONTINUE
               ELSE
                  DO 140 K = 1, N
                     S = ABS( X( K, J ) )
                     DO 130 I = 1, K - 1
                        S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
  130                CONTINUE
                     WORK( K ) = WORK( K ) + S
  140             CONTINUE
               END IF
            ELSE
               IF( NOUNIT ) THEN
                  DO 160 K = 1, N
                     S = ZERO
                     DO 150 I = K, N
                        S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
  150                CONTINUE
                     WORK( K ) = WORK( K ) + S
  160             CONTINUE
               ELSE
                  DO 180 K = 1, N
                     S = ABS( X( K, J ) )
                     DO 170 I = K + 1, N
                        S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
  170                CONTINUE
                     WORK( K ) = WORK( K ) + S
  180             CONTINUE
               END IF
            END IF
         END IF
         S = ZERO
         DO 190 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
            ELSE
               S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
     $             ( WORK( I )+SAFE1 ) )
            END IF
  190    CONTINUE
         BERR( J ) = S
*
*        Bound error from formula
*
*        norm(X - XTRUE) / norm(X) .le. FERR =
*        norm( abs(inv(op(A)))*
*           ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
*
*        where
*          norm(Z) is the magnitude of the largest component of Z
*          inv(op(A)) is the inverse of op(A)
*          abs(Z) is the componentwise absolute value of the matrix or
*             vector Z
*          NZ is the maximum number of nonzeros in any row of A, plus 1
*          EPS is machine epsilon
*
*        The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
*        is incremented by SAFE1 if the i-th component of
*        abs(op(A))*abs(X) + abs(B) is less than SAFE2.
*
*        Use DLACN2 to estimate the infinity-norm of the matrix
*           inv(op(A)) * diag(W),
*        where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
*
         DO 200 I = 1, N
            IF( WORK( I ).GT.SAFE2 ) THEN
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
            ELSE
               WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
            END IF
  200    CONTINUE
*
         KASE = 0
  210    CONTINUE
         CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK,
     $                FERR( J ),
     $                KASE, ISAVE )
         IF( KASE.NE.0 ) THEN
            IF( KASE.EQ.1 ) THEN
*
*              Multiply by diag(W)*inv(op(A)**T).
*
               CALL DTRSV( UPLO, TRANST, DIAG, N, A, LDA,
     $                     WORK( N+1 ),
     $                     1 )
               DO 220 I = 1, N
                  WORK( N+I ) = WORK( I )*WORK( N+I )
  220          CONTINUE
            ELSE
*
*              Multiply by inv(op(A))*diag(W).
*
               DO 230 I = 1, N
                  WORK( N+I ) = WORK( I )*WORK( N+I )
  230          CONTINUE
               CALL DTRSV( UPLO, TRANS, DIAG, N, A, LDA, WORK( N+1 ),
     $                     1 )
            END IF
            GO TO 210
         END IF
*
*        Normalize error.
*
         LSTRES = ZERO
         DO 240 I = 1, N
            LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
  240    CONTINUE
         IF( LSTRES.NE.ZERO )
     $      FERR( J ) = FERR( J ) / LSTRES
*
  250 CONTINUE
*
      RETURN
*
*     End of DTRRFS
*
      END
*> \brief \b DTRSEN
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRSEN + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrsen.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrsen.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrsen.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI,
*                          M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          COMPQ, JOB
*       INTEGER            INFO, LDQ, LDT, LIWORK, LWORK, M, N
*       DOUBLE PRECISION   S, SEP
*       ..
*       .. Array Arguments ..
*       LOGICAL            SELECT( * )
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ),
*      $                   WR( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTRSEN reorders the real Schur factorization of a real matrix
*> A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
*> the leading diagonal blocks of the upper quasi-triangular matrix T,
*> and the leading columns of Q form an orthonormal basis of the
*> corresponding right invariant subspace.
*>
*> Optionally the routine computes the reciprocal condition numbers of
*> the cluster of eigenvalues and/or the invariant subspace.
*>
*> T must be in Schur canonical form (as returned by DHSEQR), that is,
*> block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
*> 2-by-2 diagonal block has its diagonal elements equal and its
*> off-diagonal elements of opposite sign.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOB
*> \verbatim
*>          JOB is CHARACTER*1
*>          Specifies whether condition numbers are required for the
*>          cluster of eigenvalues (S) or the invariant subspace (SEP):
*>          = 'N': none;
*>          = 'E': for eigenvalues only (S);
*>          = 'V': for invariant subspace only (SEP);
*>          = 'B': for both eigenvalues and invariant subspace (S and
*>                 SEP).
*> \endverbatim
*>
*> \param[in] COMPQ
*> \verbatim
*>          COMPQ is CHARACTER*1
*>          = 'V': update the matrix Q of Schur vectors;
*>          = 'N': do not update Q.
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*>          SELECT is LOGICAL array, dimension (N)
*>          SELECT specifies the eigenvalues in the selected cluster. To
*>          select a real eigenvalue w(j), SELECT(j) must be set to
*>          .TRUE.. To select a complex conjugate pair of eigenvalues
*>          w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
*>          either SELECT(j) or SELECT(j+1) or both must be set to
*>          .TRUE.; a complex conjugate pair of eigenvalues must be
*>          either both included in the cluster or both excluded.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix T. N >= 0.
*> \endverbatim
*>
*> \param[in,out] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,N)
*>          On entry, the upper quasi-triangular matrix T, in Schur
*>          canonical form.
*>          On exit, T is overwritten by the reordered matrix T, again in
*>          Schur canonical form, with the selected eigenvalues in the
*>          leading diagonal blocks.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ,N)
*>          On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
*>          On exit, if COMPQ = 'V', Q has been postmultiplied by the
*>          orthogonal transformation matrix which reorders T; the
*>          leading M columns of Q form an orthonormal basis for the
*>          specified invariant subspace.
*>          If COMPQ = 'N', Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q.
*>          LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*>          WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*> \param[out] WI
*> \verbatim
*>          WI is DOUBLE PRECISION array, dimension (N)
*>
*>          The real and imaginary parts, respectively, of the reordered
*>          eigenvalues of T. The eigenvalues are stored in the same
*>          order as on the diagonal of T, with WR(i) = T(i,i) and, if
*>          T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
*>          WI(i+1) = -WI(i). Note that if a complex eigenvalue is
*>          sufficiently ill-conditioned, then its value may differ
*>          significantly from its value before reordering.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The dimension of the specified invariant subspace.
*>          0 < = M <= N.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION
*>          If JOB = 'E' or 'B', S is a lower bound on the reciprocal
*>          condition number for the selected cluster of eigenvalues.
*>          S cannot underestimate the true reciprocal condition number
*>          by more than a factor of sqrt(N). If M = 0 or N, S = 1.
*>          If JOB = 'N' or 'V', S is not referenced.
*> \endverbatim
*>
*> \param[out] SEP
*> \verbatim
*>          SEP is DOUBLE PRECISION
*>          If JOB = 'V' or 'B', SEP is the estimated reciprocal
*>          condition number of the specified invariant subspace. If
*>          M = 0 or N, SEP = norm(T).
*>          If JOB = 'N' or 'E', SEP is not referenced.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If JOB = 'N', LWORK >= max(1,N);
*>          if JOB = 'E', LWORK >= max(1,M*(N-M));
*>          if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK.
*>          If JOB = 'N' or 'E', LIWORK >= 1;
*>          if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)).
*>
*>          If LIWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the optimal size of the IWORK array,
*>          returns this value as the first entry of the IWORK array, and
*>          no error message related to LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*>          = 1: reordering of T failed because some eigenvalues are too
*>               close to separate (the problem is very ill-conditioned);
*>               T may have been partially reordered, and WR and WI
*>               contain the eigenvalues in the same order as in T; S and
*>               SEP (if requested) are set to zero.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trsen
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  DTRSEN first collects the selected eigenvalues by computing an
*>  orthogonal transformation Z to move them to the top left corner of T.
*>  In other words, the selected eigenvalues are the eigenvalues of T11
*>  in:
*>
*>          Z**T * T * Z = ( T11 T12 ) n1
*>                         (  0  T22 ) n2
*>                            n1  n2
*>
*>  where N = n1+n2 and Z**T means the transpose of Z. The first n1 columns
*>  of Z span the specified invariant subspace of T.
*>
*>  If T has been obtained from the real Schur factorization of a matrix
*>  A = Q*T*Q**T, then the reordered real Schur factorization of A is given
*>  by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, and the first n1 columns of Q*Z span
*>  the corresponding invariant subspace of A.
*>
*>  The reciprocal condition number of the average of the eigenvalues of
*>  T11 may be returned in S. S lies between 0 (very badly conditioned)
*>  and 1 (very well conditioned). It is computed as follows. First we
*>  compute R so that
*>
*>                         P = ( I  R ) n1
*>                             ( 0  0 ) n2
*>                               n1 n2
*>
*>  is the projector on the invariant subspace associated with T11.
*>  R is the solution of the Sylvester equation:
*>
*>                        T11*R - R*T22 = T12.
*>
*>  Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
*>  the two-norm of M. Then S is computed as the lower bound
*>
*>                      (1 + F-norm(R)**2)**(-1/2)
*>
*>  on the reciprocal of 2-norm(P), the true reciprocal condition number.
*>  S cannot underestimate 1 / 2-norm(P) by more than a factor of
*>  sqrt(N).
*>
*>  An approximate error bound for the computed average of the
*>  eigenvalues of T11 is
*>
*>                         EPS * norm(T) / S
*>
*>  where EPS is the machine precision.
*>
*>  The reciprocal condition number of the right invariant subspace
*>  spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
*>  SEP is defined as the separation of T11 and T22:
*>
*>                     sep( T11, T22 ) = sigma-min( C )
*>
*>  where sigma-min(C) is the smallest singular value of the
*>  n1*n2-by-n1*n2 matrix
*>
*>     C  = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )
*>
*>  I(m) is an m by m identity matrix, and kprod denotes the Kronecker
*>  product. We estimate sigma-min(C) by the reciprocal of an estimate of
*>  the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
*>  cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).
*>
*>  When SEP is small, small changes in T can cause large changes in
*>  the invariant subspace. An approximate bound on the maximum angular
*>  error in the computed right invariant subspace is
*>
*>                      EPS * norm(T) / SEP
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR,
     $                   WI,
     $                   M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          COMPQ, JOB
      INTEGER            INFO, LDQ, LDT, LIWORK, LWORK, M, N
      DOUBLE PRECISION   S, SEP
*     ..
*     .. Array Arguments ..
      LOGICAL            SELECT( * )
      INTEGER            IWORK( * )
      DOUBLE PRECISION   Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ),
     $                   WR( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, PAIR, SWAP, WANTBH, WANTQ, WANTS,
     $                   WANTSP
      INTEGER            IERR, K, KASE, KK, KS, LIWMIN, LWMIN, N1, N2,
     $                   NN
      DOUBLE PRECISION   EST, RNORM, SCALE
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLANGE
      EXTERNAL           LSAME, DLANGE
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACN2, DLACPY, DTREXC, DTRSYL,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Decode and test the input parameters
*
      WANTBH = LSAME( JOB, 'B' )
      WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
      WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH
      WANTQ = LSAME( COMPQ, 'V' )
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.WANTS .AND. .NOT.WANTSP )
     $     THEN
         INFO = -1
      ELSE IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
         INFO = -6
      ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
         INFO = -8
      ELSE
*
*        Set M to the dimension of the specified invariant subspace,
*        and test LWORK and LIWORK.
*
         M = 0
         PAIR = .FALSE.
         DO 10 K = 1, N
            IF( PAIR ) THEN
               PAIR = .FALSE.
            ELSE
               IF( K.LT.N ) THEN
                  IF( T( K+1, K ).EQ.ZERO ) THEN
                     IF( SELECT( K ) )
     $                  M = M + 1
                  ELSE
                     PAIR = .TRUE.
                     IF( SELECT( K ) .OR. SELECT( K+1 ) )
     $                  M = M + 2
                  END IF
               ELSE
                  IF( SELECT( N ) )
     $               M = M + 1
               END IF
            END IF
   10    CONTINUE
*
         N1 = M
         N2 = N - M
         NN = N1*N2
*
         IF( WANTSP ) THEN
            LWMIN = MAX( 1, 2*NN )
            LIWMIN = MAX( 1, NN )
         ELSE IF( LSAME( JOB, 'N' ) ) THEN
            LWMIN = MAX( 1, N )
            LIWMIN = 1
         ELSE IF( LSAME( JOB, 'E' ) ) THEN
            LWMIN = MAX( 1, NN )
            LIWMIN = 1
         END IF
*
         IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -15
         ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -17
         END IF
      END IF
*
      IF( INFO.EQ.0 ) THEN
         WORK( 1 ) = LWMIN
         IWORK( 1 ) = LIWMIN
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTRSEN', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( M.EQ.N .OR. M.EQ.0 ) THEN
         IF( WANTS )
     $      S = ONE
         IF( WANTSP )
     $      SEP = DLANGE( '1', N, N, T, LDT, WORK )
         GO TO 40
      END IF
*
*     Collect the selected blocks at the top-left corner of T.
*
      KS = 0
      PAIR = .FALSE.
      DO 20 K = 1, N
         IF( PAIR ) THEN
            PAIR = .FALSE.
         ELSE
            SWAP = SELECT( K )
            IF( K.LT.N ) THEN
               IF( T( K+1, K ).NE.ZERO ) THEN
                  PAIR = .TRUE.
                  SWAP = SWAP .OR. SELECT( K+1 )
               END IF
            END IF
            IF( SWAP ) THEN
               KS = KS + 1
*
*              Swap the K-th block to position KS.
*
               IERR = 0
               KK = K
               IF( K.NE.KS )
     $            CALL DTREXC( COMPQ, N, T, LDT, Q, LDQ, KK, KS,
     $                         WORK,
     $                         IERR )
               IF( IERR.EQ.1 .OR. IERR.EQ.2 ) THEN
*
*                 Blocks too close to swap: exit.
*
                  INFO = 1
                  IF( WANTS )
     $               S = ZERO
                  IF( WANTSP )
     $               SEP = ZERO
                  GO TO 40
               END IF
               IF( PAIR )
     $            KS = KS + 1
            END IF
         END IF
   20 CONTINUE
*
      IF( WANTS ) THEN
*
*        Solve Sylvester equation for R:
*
*           T11*R - R*T22 = scale*T12
*
         CALL DLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 )
         CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT, T( N1+1, N1+1 ),
     $                LDT, WORK, N1, SCALE, IERR )
*
*        Estimate the reciprocal of the condition number of the cluster
*        of eigenvalues.
*
         RNORM = DLANGE( 'F', N1, N2, WORK, N1, WORK )
         IF( RNORM.EQ.ZERO ) THEN
            S = ONE
         ELSE
            S = SCALE / ( SQRT( SCALE*SCALE / RNORM+RNORM )*
     $          SQRT( RNORM ) )
         END IF
      END IF
*
      IF( WANTSP ) THEN
*
*        Estimate sep(T11,T22).
*
         EST = ZERO
         KASE = 0
   30    CONTINUE
         CALL DLACN2( NN, WORK( NN+1 ), WORK, IWORK, EST, KASE,
     $                ISAVE )
         IF( KASE.NE.0 ) THEN
            IF( KASE.EQ.1 ) THEN
*
*              Solve  T11*R - R*T22 = scale*X.
*
               CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT,
     $                      T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
     $                      IERR )
            ELSE
*
*              Solve T11**T*R - R*T22**T = scale*X.
*
               CALL DTRSYL( 'T', 'T', -1, N1, N2, T, LDT,
     $                      T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
     $                      IERR )
            END IF
            GO TO 30
         END IF
*
         SEP = SCALE / EST
      END IF
*
   40 CONTINUE
*
*     Store the output eigenvalues in WR and WI.
*
      DO 50 K = 1, N
         WR( K ) = T( K, K )
         WI( K ) = ZERO
   50 CONTINUE
      DO 60 K = 1, N - 1
         IF( T( K+1, K ).NE.ZERO ) THEN
            WI( K ) = SQRT( ABS( T( K, K+1 ) ) )*
     $                SQRT( ABS( T( K+1, K ) ) )
            WI( K+1 ) = -WI( K )
         END IF
   60 CONTINUE
*
      WORK( 1 ) = LWMIN
      IWORK( 1 ) = LIWMIN
*
      RETURN
*
*     End of DTRSEN
*
      END
*> \brief \b DTRSNA
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRSNA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrsna.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrsna.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrsna.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTRSNA( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
*                          LDVR, S, SEP, MM, M, WORK, LDWORK, IWORK,
*                          INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          HOWMNY, JOB
*       INTEGER            INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N
*       ..
*       .. Array Arguments ..
*       LOGICAL            SELECT( * )
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   S( * ), SEP( * ), T( LDT, * ), VL( LDVL, * ),
*      $                   VR( LDVR, * ), WORK( LDWORK, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTRSNA estimates reciprocal condition numbers for specified
*> eigenvalues and/or right eigenvectors of a real upper
*> quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q
*> orthogonal).
*>
*> T must be in Schur canonical form (as returned by DHSEQR), that is,
*> block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
*> 2-by-2 diagonal block has its diagonal elements equal and its
*> off-diagonal elements of opposite sign.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOB
*> \verbatim
*>          JOB is CHARACTER*1
*>          Specifies whether condition numbers are required for
*>          eigenvalues (S) or eigenvectors (SEP):
*>          = 'E': for eigenvalues only (S);
*>          = 'V': for eigenvectors only (SEP);
*>          = 'B': for both eigenvalues and eigenvectors (S and SEP).
*> \endverbatim
*>
*> \param[in] HOWMNY
*> \verbatim
*>          HOWMNY is CHARACTER*1
*>          = 'A': compute condition numbers for all eigenpairs;
*>          = 'S': compute condition numbers for selected eigenpairs
*>                 specified by the array SELECT.
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*>          SELECT is LOGICAL array, dimension (N)
*>          If HOWMNY = 'S', SELECT specifies the eigenpairs for which
*>          condition numbers are required. To select condition numbers
*>          for the eigenpair corresponding to a real eigenvalue w(j),
*>          SELECT(j) must be set to .TRUE.. To select condition numbers
*>          corresponding to a complex conjugate pair of eigenvalues w(j)
*>          and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
*>          set to .TRUE..
*>          If HOWMNY = 'A', SELECT is not referenced.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix T. N >= 0.
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,N)
*>          The upper quasi-triangular matrix T, in Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is DOUBLE PRECISION array, dimension (LDVL,M)
*>          If JOB = 'E' or 'B', VL must contain left eigenvectors of T
*>          (or of any Q*T*Q**T with Q orthogonal), corresponding to the
*>          eigenpairs specified by HOWMNY and SELECT. The eigenvectors
*>          must be stored in consecutive columns of VL, as returned by
*>          DHSEIN or DTREVC.
*>          If JOB = 'V', VL is not referenced.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*>          LDVL is INTEGER
*>          The leading dimension of the array VL.
*>          LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.
*> \endverbatim
*>
*> \param[in] VR
*> \verbatim
*>          VR is DOUBLE PRECISION array, dimension (LDVR,M)
*>          If JOB = 'E' or 'B', VR must contain right eigenvectors of T
*>          (or of any Q*T*Q**T with Q orthogonal), corresponding to the
*>          eigenpairs specified by HOWMNY and SELECT. The eigenvectors
*>          must be stored in consecutive columns of VR, as returned by
*>          DHSEIN or DTREVC.
*>          If JOB = 'V', VR is not referenced.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*>          LDVR is INTEGER
*>          The leading dimension of the array VR.
*>          LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (MM)
*>          If JOB = 'E' or 'B', the reciprocal condition numbers of the
*>          selected eigenvalues, stored in consecutive elements of the
*>          array. For a complex conjugate pair of eigenvalues two
*>          consecutive elements of S are set to the same value. Thus
*>          S(j), SEP(j), and the j-th columns of VL and VR all
*>          correspond to the same eigenpair (but not in general the
*>          j-th eigenpair, unless all eigenpairs are selected).
*>          If JOB = 'V', S is not referenced.
*> \endverbatim
*>
*> \param[out] SEP
*> \verbatim
*>          SEP is DOUBLE PRECISION array, dimension (MM)
*>          If JOB = 'V' or 'B', the estimated reciprocal condition
*>          numbers of the selected eigenvectors, stored in consecutive
*>          elements of the array. For a complex eigenvector two
*>          consecutive elements of SEP are set to the same value. If
*>          the eigenvalues cannot be reordered to compute SEP(j), SEP(j)
*>          is set to 0; this can only occur when the true value would be
*>          very small anyway.
*>          If JOB = 'E', SEP is not referenced.
*> \endverbatim
*>
*> \param[in] MM
*> \verbatim
*>          MM is INTEGER
*>          The number of elements in the arrays S (if JOB = 'E' or 'B')
*>           and/or SEP (if JOB = 'V' or 'B'). MM >= M.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The number of elements of the arrays S and/or SEP actually
*>          used to store the estimated condition numbers.
*>          If HOWMNY = 'A', M is set to N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (LDWORK,N+6)
*>          If JOB = 'E', WORK is not referenced.
*> \endverbatim
*>
*> \param[in] LDWORK
*> \verbatim
*>          LDWORK is INTEGER
*>          The leading dimension of the array WORK.
*>          LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (2*(N-1))
*>          If JOB = 'E', IWORK is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trsna
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The reciprocal of the condition number of an eigenvalue lambda is
*>  defined as
*>
*>          S(lambda) = |v**T*u| / (norm(u)*norm(v))
*>
*>  where u and v are the right and left eigenvectors of T corresponding
*>  to lambda; v**T denotes the transpose of v, and norm(u)
*>  denotes the Euclidean norm. These reciprocal condition numbers always
*>  lie between zero (very badly conditioned) and one (very well
*>  conditioned). If n = 1, S(lambda) is defined to be 1.
*>
*>  An approximate error bound for a computed eigenvalue W(i) is given by
*>
*>                      EPS * norm(T) / S(i)
*>
*>  where EPS is the machine precision.
*>
*>  The reciprocal of the condition number of the right eigenvector u
*>  corresponding to lambda is defined as follows. Suppose
*>
*>              T = ( lambda  c  )
*>                  (   0    T22 )
*>
*>  Then the reciprocal condition number is
*>
*>          SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )
*>
*>  where sigma-min denotes the smallest singular value. We approximate
*>  the smallest singular value by the reciprocal of an estimate of the
*>  one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is
*>  defined to be abs(T(1,1)).
*>
*>  An approximate error bound for a computed right eigenvector VR(i)
*>  is given by
*>
*>                      EPS * norm(T) / SEP(i)
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTRSNA( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL,
     $                   VR,
     $                   LDVR, S, SEP, MM, M, WORK, LDWORK, IWORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          HOWMNY, JOB
      INTEGER            INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N
*     ..
*     .. Array Arguments ..
      LOGICAL            SELECT( * )
      INTEGER            IWORK( * )
      DOUBLE PRECISION   S( * ), SEP( * ), T( LDT, * ), VL( LDVL, * ),
     $                   VR( LDVR, * ), WORK( LDWORK, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            PAIR, SOMCON, WANTBH, WANTS, WANTSP
      INTEGER            I, IERR, IFST, ILST, J, K, KASE, KS, N2, NN
      DOUBLE PRECISION   BIGNUM, COND, CS, DELTA, DUMM, EPS, EST, LNRM,
     $                   MU, PROD, PROD1, PROD2, RNRM, SCALE, SMLNUM, SN
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
      DOUBLE PRECISION   DUMMY( 1 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DDOT, DLAMCH, DLAPY2, DNRM2
      EXTERNAL           LSAME, DDOT, DLAMCH, DLAPY2, DNRM2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLACN2, DLACPY, DLAQTR, DTREXC,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Decode and test the input parameters
*
      WANTBH = LSAME( JOB, 'B' )
      WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
      WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH
*
      SOMCON = LSAME( HOWMNY, 'S' )
*
      INFO = 0
      IF( .NOT.WANTS .AND. .NOT.WANTSP ) THEN
         INFO = -1
      ELSE IF( .NOT.LSAME( HOWMNY, 'A' ) .AND. .NOT.SOMCON ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
         INFO = -6
      ELSE IF( LDVL.LT.1 .OR. ( WANTS .AND. LDVL.LT.N ) ) THEN
         INFO = -8
      ELSE IF( LDVR.LT.1 .OR. ( WANTS .AND. LDVR.LT.N ) ) THEN
         INFO = -10
      ELSE
*
*        Set M to the number of eigenpairs for which condition numbers
*        are required, and test MM.
*
         IF( SOMCON ) THEN
            M = 0
            PAIR = .FALSE.
            DO 10 K = 1, N
               IF( PAIR ) THEN
                  PAIR = .FALSE.
               ELSE
                  IF( K.LT.N ) THEN
                     IF( T( K+1, K ).EQ.ZERO ) THEN
                        IF( SELECT( K ) )
     $                     M = M + 1
                     ELSE
                        PAIR = .TRUE.
                        IF( SELECT( K ) .OR. SELECT( K+1 ) )
     $                     M = M + 2
                     END IF
                  ELSE
                     IF( SELECT( N ) )
     $                  M = M + 1
                  END IF
               END IF
   10       CONTINUE
         ELSE
            M = N
         END IF
*
         IF( MM.LT.M ) THEN
            INFO = -13
         ELSE IF( LDWORK.LT.1 .OR. ( WANTSP .AND. LDWORK.LT.N ) ) THEN
            INFO = -16
         END IF
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTRSNA', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( N.EQ.1 ) THEN
         IF( SOMCON ) THEN
            IF( .NOT.SELECT( 1 ) )
     $         RETURN
         END IF
         IF( WANTS )
     $      S( 1 ) = ONE
         IF( WANTSP )
     $      SEP( 1 ) = ABS( T( 1, 1 ) )
         RETURN
      END IF
*
*     Get machine constants
*
      EPS = DLAMCH( 'P' )
      SMLNUM = DLAMCH( 'S' ) / EPS
      BIGNUM = ONE / SMLNUM
*
      KS = 0
      PAIR = .FALSE.
      DO 60 K = 1, N
*
*        Determine whether T(k,k) begins a 1-by-1 or 2-by-2 block.
*
         IF( PAIR ) THEN
            PAIR = .FALSE.
            GO TO 60
         ELSE
            IF( K.LT.N )
     $         PAIR = T( K+1, K ).NE.ZERO
         END IF
*
*        Determine whether condition numbers are required for the k-th
*        eigenpair.
*
         IF( SOMCON ) THEN
            IF( PAIR ) THEN
               IF( .NOT.SELECT( K ) .AND. .NOT.SELECT( K+1 ) )
     $            GO TO 60
            ELSE
               IF( .NOT.SELECT( K ) )
     $            GO TO 60
            END IF
         END IF
*
         KS = KS + 1
*
         IF( WANTS ) THEN
*
*           Compute the reciprocal condition number of the k-th
*           eigenvalue.
*
            IF( .NOT.PAIR ) THEN
*
*              Real eigenvalue.
*
               PROD = DDOT( N, VR( 1, KS ), 1, VL( 1, KS ), 1 )
               RNRM = DNRM2( N, VR( 1, KS ), 1 )
               LNRM = DNRM2( N, VL( 1, KS ), 1 )
               S( KS ) = ABS( PROD ) / ( RNRM*LNRM )
            ELSE
*
*              Complex eigenvalue.
*
               PROD1 = DDOT( N, VR( 1, KS ), 1, VL( 1, KS ), 1 )
               PROD1 = PROD1 + DDOT( N, VR( 1, KS+1 ), 1, VL( 1,
     $                               KS+1 ),
     $                 1 )
               PROD2 = DDOT( N, VL( 1, KS ), 1, VR( 1, KS+1 ), 1 )
               PROD2 = PROD2 - DDOT( N, VL( 1, KS+1 ), 1, VR( 1,
     $                               KS ),
     $                 1 )
               RNRM = DLAPY2( DNRM2( N, VR( 1, KS ), 1 ),
     $                DNRM2( N, VR( 1, KS+1 ), 1 ) )
               LNRM = DLAPY2( DNRM2( N, VL( 1, KS ), 1 ),
     $                DNRM2( N, VL( 1, KS+1 ), 1 ) )
               COND = DLAPY2( PROD1, PROD2 ) / ( RNRM*LNRM )
               S( KS ) = COND
               S( KS+1 ) = COND
            END IF
         END IF
*
         IF( WANTSP ) THEN
*
*           Estimate the reciprocal condition number of the k-th
*           eigenvector.
*
*           Copy the matrix T to the array WORK and swap the diagonal
*           block beginning at T(k,k) to the (1,1) position.
*
            CALL DLACPY( 'Full', N, N, T, LDT, WORK, LDWORK )
            IFST = K
            ILST = 1
            CALL DTREXC( 'No Q', N, WORK, LDWORK, DUMMY, 1, IFST,
     $                   ILST,
     $                   WORK( 1, N+1 ), IERR )
*
            IF( IERR.EQ.1 .OR. IERR.EQ.2 ) THEN
*
*              Could not swap because blocks not well separated
*
               SCALE = ONE
               EST = BIGNUM
            ELSE
*
*              Reordering successful
*
               IF( WORK( 2, 1 ).EQ.ZERO ) THEN
*
*                 Form C = T22 - lambda*I in WORK(2:N,2:N).
*
                  DO 20 I = 2, N
                     WORK( I, I ) = WORK( I, I ) - WORK( 1, 1 )
   20             CONTINUE
                  N2 = 1
                  NN = N - 1
               ELSE
*
*                 Triangularize the 2 by 2 block by unitary
*                 transformation U = [  cs   i*ss ]
*                                    [ i*ss   cs  ].
*                 such that the (1,1) position of WORK is complex
*                 eigenvalue lambda with positive imaginary part. (2,2)
*                 position of WORK is the complex eigenvalue lambda
*                 with negative imaginary  part.
*
                  MU = SQRT( ABS( WORK( 1, 2 ) ) )*
     $                 SQRT( ABS( WORK( 2, 1 ) ) )
                  DELTA = DLAPY2( MU, WORK( 2, 1 ) )
                  CS = MU / DELTA
                  SN = -WORK( 2, 1 ) / DELTA
*
*                 Form
*
*                 C**T = WORK(2:N,2:N) + i*[rwork(1) ..... rwork(n-1) ]
*                                          [   mu                     ]
*                                          [         ..               ]
*                                          [             ..           ]
*                                          [                  mu      ]
*                 where C**T is transpose of matrix C,
*                 and RWORK is stored starting in the N+1-st column of
*                 WORK.
*
                  DO 30 J = 3, N
                     WORK( 2, J ) = CS*WORK( 2, J )
                     WORK( J, J ) = WORK( J, J ) - WORK( 1, 1 )
   30             CONTINUE
                  WORK( 2, 2 ) = ZERO
*
                  WORK( 1, N+1 ) = TWO*MU
                  DO 40 I = 2, N - 1
                     WORK( I, N+1 ) = SN*WORK( 1, I+1 )
   40             CONTINUE
                  N2 = 2
                  NN = 2*( N-1 )
               END IF
*
*              Estimate norm(inv(C**T))
*
               EST = ZERO
               KASE = 0
   50          CONTINUE
               CALL DLACN2( NN, WORK( 1, N+2 ), WORK( 1, N+4 ),
     $                      IWORK,
     $                      EST, KASE, ISAVE )
               IF( KASE.NE.0 ) THEN
                  IF( KASE.EQ.1 ) THEN
                     IF( N2.EQ.1 ) THEN
*
*                       Real eigenvalue: solve C**T*x = scale*c.
*
                        CALL DLAQTR( .TRUE., .TRUE., N-1, WORK( 2,
     $                               2 ),
     $                               LDWORK, DUMMY, DUMM, SCALE,
     $                               WORK( 1, N+4 ), WORK( 1, N+6 ),
     $                               IERR )
                     ELSE
*
*                       Complex eigenvalue: solve
*                       C**T*(p+iq) = scale*(c+id) in real arithmetic.
*
                        CALL DLAQTR( .TRUE., .FALSE., N-1, WORK( 2,
     $                               2 ),
     $                               LDWORK, WORK( 1, N+1 ), MU, SCALE,
     $                               WORK( 1, N+4 ), WORK( 1, N+6 ),
     $                               IERR )
                     END IF
                  ELSE
                     IF( N2.EQ.1 ) THEN
*
*                       Real eigenvalue: solve C*x = scale*c.
*
                        CALL DLAQTR( .FALSE., .TRUE., N-1, WORK( 2,
     $                               2 ),
     $                               LDWORK, DUMMY, DUMM, SCALE,
     $                               WORK( 1, N+4 ), WORK( 1, N+6 ),
     $                               IERR )
                     ELSE
*
*                       Complex eigenvalue: solve
*                       C*(p+iq) = scale*(c+id) in real arithmetic.
*
                        CALL DLAQTR( .FALSE., .FALSE., N-1,
     $                               WORK( 2, 2 ), LDWORK,
     $                               WORK( 1, N+1 ), MU, SCALE,
     $                               WORK( 1, N+4 ), WORK( 1, N+6 ),
     $                               IERR )
*
                     END IF
                  END IF
*
                  GO TO 50
               END IF
            END IF
*
            SEP( KS ) = SCALE / MAX( EST, SMLNUM )
            IF( PAIR )
     $         SEP( KS+1 ) = SEP( KS )
         END IF
*
         IF( PAIR )
     $      KS = KS + 1
*
   60 CONTINUE
      RETURN
*
*     End of DTRSNA
*
      END
*> \brief \b DTRSYL
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRSYL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrsyl.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrsyl.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrsyl.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTRSYL( TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C,
*                          LDC, SCALE, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANA, TRANB
*       INTEGER            INFO, ISGN, LDA, LDB, LDC, M, N
*       DOUBLE PRECISION   SCALE
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTRSYL solves the real Sylvester matrix equation:
*>
*>    op(A)*X + X*op(B) = scale*C or
*>    op(A)*X - X*op(B) = scale*C,
*>
*> where op(A) = A or A**T, and  A and B are both upper quasi-
*> triangular. A is M-by-M and B is N-by-N; the right hand side C and
*> the solution X are M-by-N; and scale is an output scale factor, set
*> <= 1 to avoid overflow in X.
*>
*> A and B must be in Schur canonical form (as returned by DHSEQR), that
*> is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks;
*> each 2-by-2 diagonal block has its diagonal elements equal and its
*> off-diagonal elements of opposite sign.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANA
*> \verbatim
*>          TRANA is CHARACTER*1
*>          Specifies the option op(A):
*>          = 'N': op(A) = A    (No transpose)
*>          = 'T': op(A) = A**T (Transpose)
*>          = 'C': op(A) = A**H (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] TRANB
*> \verbatim
*>          TRANB is CHARACTER*1
*>          Specifies the option op(B):
*>          = 'N': op(B) = B    (No transpose)
*>          = 'T': op(B) = B**T (Transpose)
*>          = 'C': op(B) = B**H (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] ISGN
*> \verbatim
*>          ISGN is INTEGER
*>          Specifies the sign in the equation:
*>          = +1: solve op(A)*X + X*op(B) = scale*C
*>          = -1: solve op(A)*X - X*op(B) = scale*C
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The order of the matrix A, and the number of rows in the
*>          matrices X and C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix B, and the number of columns in the
*>          matrices X and C. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,M)
*>          The upper quasi-triangular matrix A, in Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,N)
*>          The upper quasi-triangular matrix B, in Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC,N)
*>          On entry, the M-by-N right hand side matrix C.
*>          On exit, C is overwritten by the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C. LDC >= max(1,M)
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*>          SCALE is DOUBLE PRECISION
*>          The scale factor, scale, set <= 1 to avoid overflow in X.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*>          = 1: A and B have common or very close eigenvalues; perturbed
*>               values were used to solve the equation (but the matrices
*>               A and B are unchanged).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trsyl
*
*  =====================================================================
      SUBROUTINE DTRSYL( TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C,
     $                   LDC, SCALE, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANA, TRANB
      INTEGER            INFO, ISGN, LDA, LDB, LDC, M, N
      DOUBLE PRECISION   SCALE
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOTRNA, NOTRNB
      INTEGER            IERR, J, K, K1, K2, KNEXT, L, L1, L2, LNEXT
      DOUBLE PRECISION   A11, BIGNUM, DA11, DB, EPS, SCALOC, SGN, SMIN,
     $                   SMLNUM, SUML, SUMR, XNORM
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   DUM( 1 ), VEC( 2, 2 ), X( 2, 2 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DDOT, DLAMCH, DLANGE
      EXTERNAL           LSAME, DDOT, DLAMCH, DLANGE
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLALN2, DLASY2, DSCAL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Decode and Test input parameters
*
      NOTRNA = LSAME( TRANA, 'N' )
      NOTRNB = LSAME( TRANB, 'N' )
*
      INFO = 0
      IF( .NOT.NOTRNA .AND. .NOT.LSAME( TRANA, 'T' ) .AND. .NOT.
     $    LSAME( TRANA, 'C' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRNB .AND. .NOT.LSAME( TRANB, 'T' ) .AND. .NOT.
     $         LSAME( TRANB, 'C' ) ) THEN
         INFO = -2
      ELSE IF( ISGN.NE.1 .AND. ISGN.NE.-1 ) THEN
         INFO = -3
      ELSE IF( M.LT.0 ) THEN
         INFO = -4
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -7
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -9
      ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
         INFO = -11
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTRSYL', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      SCALE = ONE
      IF( M.EQ.0 .OR. N.EQ.0 )
     $   RETURN
*
*     Set constants to control overflow
*
      EPS = DLAMCH( 'P' )
      SMLNUM = DLAMCH( 'S' )
      BIGNUM = ONE / SMLNUM
      SMLNUM = SMLNUM*DBLE( M*N ) / EPS
      BIGNUM = ONE / SMLNUM
*
      SMIN = MAX( SMLNUM, EPS*DLANGE( 'M', M, M, A, LDA, DUM ),
     $       EPS*DLANGE( 'M', N, N, B, LDB, DUM ) )
*
      SGN = ISGN
*
      IF( NOTRNA .AND. NOTRNB ) THEN
*
*        Solve    A*X + ISGN*X*B = scale*C.
*
*        The (K,L)th block of X is determined starting from
*        bottom-left corner column by column by
*
*         A(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L)
*
*        Where
*                  M                         L-1
*        R(K,L) = SUM [A(K,I)*X(I,L)] + ISGN*SUM [X(K,J)*B(J,L)].
*                I=K+1                       J=1
*
*        Start column loop (index = L)
*        L1 (L2) : column index of the first (first) row of X(K,L).
*
         LNEXT = 1
         DO 60 L = 1, N
            IF( L.LT.LNEXT )
     $         GO TO 60
            IF( L.EQ.N ) THEN
               L1 = L
               L2 = L
            ELSE
               IF( B( L+1, L ).NE.ZERO ) THEN
                  L1 = L
                  L2 = L + 1
                  LNEXT = L + 2
               ELSE
                  L1 = L
                  L2 = L
                  LNEXT = L + 1
               END IF
            END IF
*
*           Start row loop (index = K)
*           K1 (K2): row index of the first (last) row of X(K,L).
*
            KNEXT = M
            DO 50 K = M, 1, -1
               IF( K.GT.KNEXT )
     $            GO TO 50
               IF( K.EQ.1 ) THEN
                  K1 = K
                  K2 = K
               ELSE
                  IF( A( K, K-1 ).NE.ZERO ) THEN
                     K1 = K - 1
                     K2 = K
                     KNEXT = K - 2
                  ELSE
                     K1 = K
                     K2 = K
                     KNEXT = K - 1
                  END IF
               END IF
*
               IF( L1.EQ.L2 .AND. K1.EQ.K2 ) THEN
                  SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA,
     $                   C( MIN( K1+1, M ), L1 ), 1 )
                  SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
                  VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
                  SCALOC = ONE
*
                  A11 = A( K1, K1 ) + SGN*B( L1, L1 )
                  DA11 = ABS( A11 )
                  IF( DA11.LE.SMIN ) THEN
                     A11 = SMIN
                     DA11 = SMIN
                     INFO = 1
                  END IF
                  DB = ABS( VEC( 1, 1 ) )
                  IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
                     IF( DB.GT.BIGNUM*DA11 )
     $                  SCALOC = ONE / DB
                  END IF
                  X( 1, 1 ) = ( VEC( 1, 1 )*SCALOC ) / A11
*
                  IF( SCALOC.NE.ONE ) THEN
                     DO 10 J = 1, N
                        CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
   10                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
                  C( K1, L1 ) = X( 1, 1 )
*
               ELSE IF( L1.EQ.L2 .AND. K1.NE.K2 ) THEN
*
                  SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA,
     $                   C( MIN( K2+1, M ), L1 ), 1 )
                  SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
                  VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
*
                  SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA,
     $                   C( MIN( K2+1, M ), L1 ), 1 )
                  SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L1 ), 1 )
                  VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
*
                  CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, A( K1, K1 ),
     $                         LDA, ONE, ONE, VEC, 2, -SGN*B( L1, L1 ),
     $                         ZERO, X, 2, SCALOC, XNORM, IERR )
                  IF( IERR.NE.0 )
     $               INFO = 1
*
                  IF( SCALOC.NE.ONE ) THEN
                     DO 20 J = 1, N
                        CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
   20                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
                  C( K1, L1 ) = X( 1, 1 )
                  C( K2, L1 ) = X( 2, 1 )
*
               ELSE IF( L1.NE.L2 .AND. K1.EQ.K2 ) THEN
*
                  SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA,
     $                   C( MIN( K1+1, M ), L1 ), 1 )
                  SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
                  VEC( 1, 1 ) = SGN*( C( K1, L1 )-( SUML+SGN*SUMR ) )
*
                  SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA,
     $                   C( MIN( K1+1, M ), L2 ), 1 )
                  SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L2 ), 1 )
                  VEC( 2, 1 ) = SGN*( C( K1, L2 )-( SUML+SGN*SUMR ) )
*
                  CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, B( L1, L1 ),
     $                         LDB, ONE, ONE, VEC, 2, -SGN*A( K1, K1 ),
     $                         ZERO, X, 2, SCALOC, XNORM, IERR )
                  IF( IERR.NE.0 )
     $               INFO = 1
*
                  IF( SCALOC.NE.ONE ) THEN
                     DO 30 J = 1, N
                        CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
   30                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
                  C( K1, L1 ) = X( 1, 1 )
                  C( K1, L2 ) = X( 2, 1 )
*
               ELSE IF( L1.NE.L2 .AND. K1.NE.K2 ) THEN
*
                  SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA,
     $                   C( MIN( K2+1, M ), L1 ), 1 )
                  SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
                  VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
*
                  SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA,
     $                   C( MIN( K2+1, M ), L2 ), 1 )
                  SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L2 ), 1 )
                  VEC( 1, 2 ) = C( K1, L2 ) - ( SUML+SGN*SUMR )
*
                  SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA,
     $                   C( MIN( K2+1, M ), L1 ), 1 )
                  SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L1 ), 1 )
                  VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
*
                  SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA,
     $                   C( MIN( K2+1, M ), L2 ), 1 )
                  SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L2 ), 1 )
                  VEC( 2, 2 ) = C( K2, L2 ) - ( SUML+SGN*SUMR )
*
                  CALL DLASY2( .FALSE., .FALSE., ISGN, 2, 2,
     $                         A( K1, K1 ), LDA, B( L1, L1 ), LDB, VEC,
     $                         2, SCALOC, X, 2, XNORM, IERR )
                  IF( IERR.NE.0 )
     $               INFO = 1
*
                  IF( SCALOC.NE.ONE ) THEN
                     DO 40 J = 1, N
                        CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
   40                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
                  C( K1, L1 ) = X( 1, 1 )
                  C( K1, L2 ) = X( 1, 2 )
                  C( K2, L1 ) = X( 2, 1 )
                  C( K2, L2 ) = X( 2, 2 )
               END IF
*
   50       CONTINUE
*
   60    CONTINUE
*
      ELSE IF( .NOT.NOTRNA .AND. NOTRNB ) THEN
*
*        Solve    A**T *X + ISGN*X*B = scale*C.
*
*        The (K,L)th block of X is determined starting from
*        upper-left corner column by column by
*
*          A(K,K)**T*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L)
*
*        Where
*                   K-1        T                    L-1
*          R(K,L) = SUM [A(I,K)**T*X(I,L)] +ISGN*SUM [X(K,J)*B(J,L)]
*                   I=1                          J=1
*
*        Start column loop (index = L)
*        L1 (L2): column index of the first (last) row of X(K,L)
*
         LNEXT = 1
         DO 120 L = 1, N
            IF( L.LT.LNEXT )
     $         GO TO 120
            IF( L.EQ.N ) THEN
               L1 = L
               L2 = L
            ELSE
               IF( B( L+1, L ).NE.ZERO ) THEN
                  L1 = L
                  L2 = L + 1
                  LNEXT = L + 2
               ELSE
                  L1 = L
                  L2 = L
                  LNEXT = L + 1
               END IF
            END IF
*
*           Start row loop (index = K)
*           K1 (K2): row index of the first (last) row of X(K,L)
*
            KNEXT = 1
            DO 110 K = 1, M
               IF( K.LT.KNEXT )
     $            GO TO 110
               IF( K.EQ.M ) THEN
                  K1 = K
                  K2 = K
               ELSE
                  IF( A( K+1, K ).NE.ZERO ) THEN
                     K1 = K
                     K2 = K + 1
                     KNEXT = K + 2
                  ELSE
                     K1 = K
                     K2 = K
                     KNEXT = K + 1
                  END IF
               END IF
*
               IF( L1.EQ.L2 .AND. K1.EQ.K2 ) THEN
                  SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
                  SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
                  VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
                  SCALOC = ONE
*
                  A11 = A( K1, K1 ) + SGN*B( L1, L1 )
                  DA11 = ABS( A11 )
                  IF( DA11.LE.SMIN ) THEN
                     A11 = SMIN
                     DA11 = SMIN
                     INFO = 1
                  END IF
                  DB = ABS( VEC( 1, 1 ) )
                  IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
                     IF( DB.GT.BIGNUM*DA11 )
     $                  SCALOC = ONE / DB
                  END IF
                  X( 1, 1 ) = ( VEC( 1, 1 )*SCALOC ) / A11
*
                  IF( SCALOC.NE.ONE ) THEN
                     DO 70 J = 1, N
                        CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
   70                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
                  C( K1, L1 ) = X( 1, 1 )
*
               ELSE IF( L1.EQ.L2 .AND. K1.NE.K2 ) THEN
*
                  SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
                  SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
                  VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
*
                  SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L1 ), 1 )
                  SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L1 ), 1 )
                  VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
*
                  CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, A( K1, K1 ),
     $                         LDA, ONE, ONE, VEC, 2, -SGN*B( L1, L1 ),
     $                         ZERO, X, 2, SCALOC, XNORM, IERR )
                  IF( IERR.NE.0 )
     $               INFO = 1
*
                  IF( SCALOC.NE.ONE ) THEN
                     DO 80 J = 1, N
                        CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
   80                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
                  C( K1, L1 ) = X( 1, 1 )
                  C( K2, L1 ) = X( 2, 1 )
*
               ELSE IF( L1.NE.L2 .AND. K1.EQ.K2 ) THEN
*
                  SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
                  SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
                  VEC( 1, 1 ) = SGN*( C( K1, L1 )-( SUML+SGN*SUMR ) )
*
                  SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L2 ), 1 )
                  SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L2 ), 1 )
                  VEC( 2, 1 ) = SGN*( C( K1, L2 )-( SUML+SGN*SUMR ) )
*
                  CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, B( L1, L1 ),
     $                         LDB, ONE, ONE, VEC, 2, -SGN*A( K1, K1 ),
     $                         ZERO, X, 2, SCALOC, XNORM, IERR )
                  IF( IERR.NE.0 )
     $               INFO = 1
*
                  IF( SCALOC.NE.ONE ) THEN
                     DO 90 J = 1, N
                        CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
   90                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
                  C( K1, L1 ) = X( 1, 1 )
                  C( K1, L2 ) = X( 2, 1 )
*
               ELSE IF( L1.NE.L2 .AND. K1.NE.K2 ) THEN
*
                  SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
                  SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
                  VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
*
                  SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L2 ), 1 )
                  SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L2 ), 1 )
                  VEC( 1, 2 ) = C( K1, L2 ) - ( SUML+SGN*SUMR )
*
                  SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L1 ), 1 )
                  SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L1 ), 1 )
                  VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
*
                  SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L2 ), 1 )
                  SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L2 ), 1 )
                  VEC( 2, 2 ) = C( K2, L2 ) - ( SUML+SGN*SUMR )
*
                  CALL DLASY2( .TRUE., .FALSE., ISGN, 2, 2, A( K1,
     $                         K1 ),
     $                         LDA, B( L1, L1 ), LDB, VEC, 2, SCALOC, X,
     $                         2, XNORM, IERR )
                  IF( IERR.NE.0 )
     $               INFO = 1
*
                  IF( SCALOC.NE.ONE ) THEN
                     DO 100 J = 1, N
                        CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
  100                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
                  C( K1, L1 ) = X( 1, 1 )
                  C( K1, L2 ) = X( 1, 2 )
                  C( K2, L1 ) = X( 2, 1 )
                  C( K2, L2 ) = X( 2, 2 )
               END IF
*
  110       CONTINUE
  120    CONTINUE
*
      ELSE IF( .NOT.NOTRNA .AND. .NOT.NOTRNB ) THEN
*
*        Solve    A**T*X + ISGN*X*B**T = scale*C.
*
*        The (K,L)th block of X is determined starting from
*        top-right corner column by column by
*
*           A(K,K)**T*X(K,L) + ISGN*X(K,L)*B(L,L)**T = C(K,L) - R(K,L)
*
*        Where
*                     K-1                            N
*            R(K,L) = SUM [A(I,K)**T*X(I,L)] + ISGN*SUM [X(K,J)*B(L,J)**T].
*                     I=1                          J=L+1
*
*        Start column loop (index = L)
*        L1 (L2): column index of the first (last) row of X(K,L)
*
         LNEXT = N
         DO 180 L = N, 1, -1
            IF( L.GT.LNEXT )
     $         GO TO 180
            IF( L.EQ.1 ) THEN
               L1 = L
               L2 = L
            ELSE
               IF( B( L, L-1 ).NE.ZERO ) THEN
                  L1 = L - 1
                  L2 = L
                  LNEXT = L - 2
               ELSE
                  L1 = L
                  L2 = L
                  LNEXT = L - 1
               END IF
            END IF
*
*           Start row loop (index = K)
*           K1 (K2): row index of the first (last) row of X(K,L)
*
            KNEXT = 1
            DO 170 K = 1, M
               IF( K.LT.KNEXT )
     $            GO TO 170
               IF( K.EQ.M ) THEN
                  K1 = K
                  K2 = K
               ELSE
                  IF( A( K+1, K ).NE.ZERO ) THEN
                     K1 = K
                     K2 = K + 1
                     KNEXT = K + 2
                  ELSE
                     K1 = K
                     K2 = K
                     KNEXT = K + 1
                  END IF
               END IF
*
               IF( L1.EQ.L2 .AND. K1.EQ.K2 ) THEN
                  SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
                  SUMR = DDOT( N-L1, C( K1, MIN( L1+1, N ) ), LDC,
     $                   B( L1, MIN( L1+1, N ) ), LDB )
                  VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
                  SCALOC = ONE
*
                  A11 = A( K1, K1 ) + SGN*B( L1, L1 )
                  DA11 = ABS( A11 )
                  IF( DA11.LE.SMIN ) THEN
                     A11 = SMIN
                     DA11 = SMIN
                     INFO = 1
                  END IF
                  DB = ABS( VEC( 1, 1 ) )
                  IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
                     IF( DB.GT.BIGNUM*DA11 )
     $                  SCALOC = ONE / DB
                  END IF
                  X( 1, 1 ) = ( VEC( 1, 1 )*SCALOC ) / A11
*
                  IF( SCALOC.NE.ONE ) THEN
                     DO 130 J = 1, N
                        CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
  130                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
                  C( K1, L1 ) = X( 1, 1 )
*
               ELSE IF( L1.EQ.L2 .AND. K1.NE.K2 ) THEN
*
                  SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
                  SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
     $                   B( L1, MIN( L2+1, N ) ), LDB )
                  VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
*
                  SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L1 ), 1 )
                  SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC,
     $                   B( L1, MIN( L2+1, N ) ), LDB )
                  VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
*
                  CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, A( K1, K1 ),
     $                         LDA, ONE, ONE, VEC, 2, -SGN*B( L1, L1 ),
     $                         ZERO, X, 2, SCALOC, XNORM, IERR )
                  IF( IERR.NE.0 )
     $               INFO = 1
*
                  IF( SCALOC.NE.ONE ) THEN
                     DO 140 J = 1, N
                        CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
  140                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
                  C( K1, L1 ) = X( 1, 1 )
                  C( K2, L1 ) = X( 2, 1 )
*
               ELSE IF( L1.NE.L2 .AND. K1.EQ.K2 ) THEN
*
                  SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
                  SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
     $                   B( L1, MIN( L2+1, N ) ), LDB )
                  VEC( 1, 1 ) = SGN*( C( K1, L1 )-( SUML+SGN*SUMR ) )
*
                  SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L2 ), 1 )
                  SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
     $                   B( L2, MIN( L2+1, N ) ), LDB )
                  VEC( 2, 1 ) = SGN*( C( K1, L2 )-( SUML+SGN*SUMR ) )
*
                  CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, B( L1, L1 ),
     $                         LDB, ONE, ONE, VEC, 2, -SGN*A( K1, K1 ),
     $                         ZERO, X, 2, SCALOC, XNORM, IERR )
                  IF( IERR.NE.0 )
     $               INFO = 1
*
                  IF( SCALOC.NE.ONE ) THEN
                     DO 150 J = 1, N
                        CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
  150                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
                  C( K1, L1 ) = X( 1, 1 )
                  C( K1, L2 ) = X( 2, 1 )
*
               ELSE IF( L1.NE.L2 .AND. K1.NE.K2 ) THEN
*
                  SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
                  SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
     $                   B( L1, MIN( L2+1, N ) ), LDB )
                  VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
*
                  SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L2 ), 1 )
                  SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
     $                   B( L2, MIN( L2+1, N ) ), LDB )
                  VEC( 1, 2 ) = C( K1, L2 ) - ( SUML+SGN*SUMR )
*
                  SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L1 ), 1 )
                  SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC,
     $                   B( L1, MIN( L2+1, N ) ), LDB )
                  VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
*
                  SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L2 ), 1 )
                  SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC,
     $                   B( L2, MIN( L2+1, N ) ), LDB )
                  VEC( 2, 2 ) = C( K2, L2 ) - ( SUML+SGN*SUMR )
*
                  CALL DLASY2( .TRUE., .TRUE., ISGN, 2, 2, A( K1,
     $                         K1 ),
     $                         LDA, B( L1, L1 ), LDB, VEC, 2, SCALOC, X,
     $                         2, XNORM, IERR )
                  IF( IERR.NE.0 )
     $               INFO = 1
*
                  IF( SCALOC.NE.ONE ) THEN
                     DO 160 J = 1, N
                        CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
  160                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
                  C( K1, L1 ) = X( 1, 1 )
                  C( K1, L2 ) = X( 1, 2 )
                  C( K2, L1 ) = X( 2, 1 )
                  C( K2, L2 ) = X( 2, 2 )
               END IF
*
  170       CONTINUE
  180    CONTINUE
*
      ELSE IF( NOTRNA .AND. .NOT.NOTRNB ) THEN
*
*        Solve    A*X + ISGN*X*B**T = scale*C.
*
*        The (K,L)th block of X is determined starting from
*        bottom-right corner column by column by
*
*            A(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L)**T = C(K,L) - R(K,L)
*
*        Where
*                      M                          N
*            R(K,L) = SUM [A(K,I)*X(I,L)] + ISGN*SUM [X(K,J)*B(L,J)**T].
*                    I=K+1                      J=L+1
*
*        Start column loop (index = L)
*        L1 (L2): column index of the first (last) row of X(K,L)
*
         LNEXT = N
         DO 240 L = N, 1, -1
            IF( L.GT.LNEXT )
     $         GO TO 240
            IF( L.EQ.1 ) THEN
               L1 = L
               L2 = L
            ELSE
               IF( B( L, L-1 ).NE.ZERO ) THEN
                  L1 = L - 1
                  L2 = L
                  LNEXT = L - 2
               ELSE
                  L1 = L
                  L2 = L
                  LNEXT = L - 1
               END IF
            END IF
*
*           Start row loop (index = K)
*           K1 (K2): row index of the first (last) row of X(K,L)
*
            KNEXT = M
            DO 230 K = M, 1, -1
               IF( K.GT.KNEXT )
     $            GO TO 230
               IF( K.EQ.1 ) THEN
                  K1 = K
                  K2 = K
               ELSE
                  IF( A( K, K-1 ).NE.ZERO ) THEN
                     K1 = K - 1
                     K2 = K
                     KNEXT = K - 2
                  ELSE
                     K1 = K
                     K2 = K
                     KNEXT = K - 1
                  END IF
               END IF
*
               IF( L1.EQ.L2 .AND. K1.EQ.K2 ) THEN
                  SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA,
     $                   C( MIN( K1+1, M ), L1 ), 1 )
                  SUMR = DDOT( N-L1, C( K1, MIN( L1+1, N ) ), LDC,
     $                   B( L1, MIN( L1+1, N ) ), LDB )
                  VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
                  SCALOC = ONE
*
                  A11 = A( K1, K1 ) + SGN*B( L1, L1 )
                  DA11 = ABS( A11 )
                  IF( DA11.LE.SMIN ) THEN
                     A11 = SMIN
                     DA11 = SMIN
                     INFO = 1
                  END IF
                  DB = ABS( VEC( 1, 1 ) )
                  IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
                     IF( DB.GT.BIGNUM*DA11 )
     $                  SCALOC = ONE / DB
                  END IF
                  X( 1, 1 ) = ( VEC( 1, 1 )*SCALOC ) / A11
*
                  IF( SCALOC.NE.ONE ) THEN
                     DO 190 J = 1, N
                        CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
  190                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
                  C( K1, L1 ) = X( 1, 1 )
*
               ELSE IF( L1.EQ.L2 .AND. K1.NE.K2 ) THEN
*
                  SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA,
     $                   C( MIN( K2+1, M ), L1 ), 1 )
                  SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
     $                   B( L1, MIN( L2+1, N ) ), LDB )
                  VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
*
                  SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA,
     $                   C( MIN( K2+1, M ), L1 ), 1 )
                  SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC,
     $                   B( L1, MIN( L2+1, N ) ), LDB )
                  VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
*
                  CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, A( K1, K1 ),
     $                         LDA, ONE, ONE, VEC, 2, -SGN*B( L1, L1 ),
     $                         ZERO, X, 2, SCALOC, XNORM, IERR )
                  IF( IERR.NE.0 )
     $               INFO = 1
*
                  IF( SCALOC.NE.ONE ) THEN
                     DO 200 J = 1, N
                        CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
  200                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
                  C( K1, L1 ) = X( 1, 1 )
                  C( K2, L1 ) = X( 2, 1 )
*
               ELSE IF( L1.NE.L2 .AND. K1.EQ.K2 ) THEN
*
                  SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA,
     $                   C( MIN( K1+1, M ), L1 ), 1 )
                  SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
     $                   B( L1, MIN( L2+1, N ) ), LDB )
                  VEC( 1, 1 ) = SGN*( C( K1, L1 )-( SUML+SGN*SUMR ) )
*
                  SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA,
     $                   C( MIN( K1+1, M ), L2 ), 1 )
                  SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
     $                   B( L2, MIN( L2+1, N ) ), LDB )
                  VEC( 2, 1 ) = SGN*( C( K1, L2 )-( SUML+SGN*SUMR ) )
*
                  CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, B( L1, L1 ),
     $                         LDB, ONE, ONE, VEC, 2, -SGN*A( K1, K1 ),
     $                         ZERO, X, 2, SCALOC, XNORM, IERR )
                  IF( IERR.NE.0 )
     $               INFO = 1
*
                  IF( SCALOC.NE.ONE ) THEN
                     DO 210 J = 1, N
                        CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
  210                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
                  C( K1, L1 ) = X( 1, 1 )
                  C( K1, L2 ) = X( 2, 1 )
*
               ELSE IF( L1.NE.L2 .AND. K1.NE.K2 ) THEN
*
                  SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA,
     $                   C( MIN( K2+1, M ), L1 ), 1 )
                  SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
     $                   B( L1, MIN( L2+1, N ) ), LDB )
                  VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
*
                  SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA,
     $                   C( MIN( K2+1, M ), L2 ), 1 )
                  SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
     $                   B( L2, MIN( L2+1, N ) ), LDB )
                  VEC( 1, 2 ) = C( K1, L2 ) - ( SUML+SGN*SUMR )
*
                  SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA,
     $                   C( MIN( K2+1, M ), L1 ), 1 )
                  SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC,
     $                   B( L1, MIN( L2+1, N ) ), LDB )
                  VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
*
                  SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA,
     $                   C( MIN( K2+1, M ), L2 ), 1 )
                  SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC,
     $                   B( L2, MIN( L2+1, N ) ), LDB )
                  VEC( 2, 2 ) = C( K2, L2 ) - ( SUML+SGN*SUMR )
*
                  CALL DLASY2( .FALSE., .TRUE., ISGN, 2, 2, A( K1,
     $                         K1 ),
     $                         LDA, B( L1, L1 ), LDB, VEC, 2, SCALOC, X,
     $                         2, XNORM, IERR )
                  IF( IERR.NE.0 )
     $               INFO = 1
*
                  IF( SCALOC.NE.ONE ) THEN
                     DO 220 J = 1, N
                        CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
  220                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
                  C( K1, L1 ) = X( 1, 1 )
                  C( K1, L2 ) = X( 1, 2 )
                  C( K2, L1 ) = X( 2, 1 )
                  C( K2, L2 ) = X( 2, 2 )
               END IF
*
  230       CONTINUE
  240    CONTINUE
*
      END IF
*
      RETURN
*
*     End of DTRSYL
*
      END
*> \brief \b DTRTI2 computes the inverse of a triangular matrix (unblocked algorithm).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRTI2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrti2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrti2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrti2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTRTI2( UPLO, DIAG, N, A, LDA, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, UPLO
*       INTEGER            INFO, LDA, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTRTI2 computes the inverse of a real upper or lower triangular
*> matrix.
*>
*> This is the Level 2 BLAS version of the algorithm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the matrix A is upper or lower triangular.
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          Specifies whether or not the matrix A is unit triangular.
*>          = 'N':  Non-unit triangular
*>          = 'U':  Unit triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the triangular matrix A.  If UPLO = 'U', the
*>          leading n by n upper triangular part of the array A contains
*>          the upper triangular matrix, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading n by n lower triangular part of the array A contains
*>          the lower triangular matrix, and the strictly upper
*>          triangular part of A is not referenced.  If DIAG = 'U', the
*>          diagonal elements of A are also not referenced and are
*>          assumed to be 1.
*>
*>          On exit, the (triangular) inverse of the original matrix, in
*>          the same storage format.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -k, the k-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trti2
*
*  =====================================================================
      SUBROUTINE DTRTI2( UPLO, DIAG, N, A, LDA, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, UPLO
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOUNIT, UPPER
      INTEGER            J
      DOUBLE PRECISION   AJJ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL, DTRMV, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      NOUNIT = LSAME( DIAG, 'N' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTRTI2', -INFO )
         RETURN
      END IF
*
      IF( UPPER ) THEN
*
*        Compute inverse of upper triangular matrix.
*
         DO 10 J = 1, N
            IF( NOUNIT ) THEN
               A( J, J ) = ONE / A( J, J )
               AJJ = -A( J, J )
            ELSE
               AJJ = -ONE
            END IF
*
*           Compute elements 1:j-1 of j-th column.
*
            CALL DTRMV( 'Upper', 'No transpose', DIAG, J-1, A, LDA,
     $                  A( 1, J ), 1 )
            CALL DSCAL( J-1, AJJ, A( 1, J ), 1 )
   10    CONTINUE
      ELSE
*
*        Compute inverse of lower triangular matrix.
*
         DO 20 J = N, 1, -1
            IF( NOUNIT ) THEN
               A( J, J ) = ONE / A( J, J )
               AJJ = -A( J, J )
            ELSE
               AJJ = -ONE
            END IF
            IF( J.LT.N ) THEN
*
*              Compute elements j+1:n of j-th column.
*
               CALL DTRMV( 'Lower', 'No transpose', DIAG, N-J,
     $                     A( J+1, J+1 ), LDA, A( J+1, J ), 1 )
               CALL DSCAL( N-J, AJJ, A( J+1, J ), 1 )
            END IF
   20    CONTINUE
      END IF
*
      RETURN
*
*     End of DTRTI2
*
      END
*> \brief \b DTRTRI
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRTRI + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrtri.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrtri.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrtri.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTRTRI( UPLO, DIAG, N, A, LDA, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, UPLO
*       INTEGER            INFO, LDA, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTRTRI computes the inverse of a real upper or lower triangular
*> matrix A.
*>
*> This is the Level 3 BLAS version of the algorithm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  A is upper triangular;
*>          = 'L':  A is lower triangular.
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          = 'N':  A is non-unit triangular;
*>          = 'U':  A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the triangular matrix A.  If UPLO = 'U', the
*>          leading N-by-N upper triangular part of the array A contains
*>          the upper triangular matrix, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading N-by-N lower triangular part of the array A contains
*>          the lower triangular matrix, and the strictly upper
*>          triangular part of A is not referenced.  If DIAG = 'U', the
*>          diagonal elements of A are also not referenced and are
*>          assumed to be 1.
*>          On exit, the (triangular) inverse of the original matrix, in
*>          the same storage format.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*>          > 0: if INFO = i, A(i,i) is exactly zero.  The triangular
*>               matrix is singular and its inverse can not be computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trtri
*
*  =====================================================================
      SUBROUTINE DTRTRI( UPLO, DIAG, N, A, LDA, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, UPLO
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOUNIT, UPPER
      INTEGER            J, JB, NB, NN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DTRMM, DTRSM, DTRTI2, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      NOUNIT = LSAME( DIAG, 'N' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTRTRI', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Check for singularity if non-unit.
*
      IF( NOUNIT ) THEN
         DO 10 INFO = 1, N
            IF( A( INFO, INFO ).EQ.ZERO )
     $         RETURN
   10    CONTINUE
         INFO = 0
      END IF
*
*     Determine the block size for this environment.
*
      NB = ILAENV( 1, 'DTRTRI', UPLO // DIAG, N, -1, -1, -1 )
      IF( NB.LE.1 .OR. NB.GE.N ) THEN
*
*        Use unblocked code
*
         CALL DTRTI2( UPLO, DIAG, N, A, LDA, INFO )
      ELSE
*
*        Use blocked code
*
         IF( UPPER ) THEN
*
*           Compute inverse of upper triangular matrix
*
            DO 20 J = 1, N, NB
               JB = MIN( NB, N-J+1 )
*
*              Compute rows 1:j-1 of current block column
*
               CALL DTRMM( 'Left', 'Upper', 'No transpose', DIAG,
     $                     J-1,
     $                     JB, ONE, A, LDA, A( 1, J ), LDA )
               CALL DTRSM( 'Right', 'Upper', 'No transpose', DIAG,
     $                     J-1,
     $                     JB, -ONE, A( J, J ), LDA, A( 1, J ), LDA )
*
*              Compute inverse of current diagonal block
*
               CALL DTRTI2( 'Upper', DIAG, JB, A( J, J ), LDA, INFO )
   20       CONTINUE
         ELSE
*
*           Compute inverse of lower triangular matrix
*
            NN = ( ( N-1 ) / NB )*NB + 1
            DO 30 J = NN, 1, -NB
               JB = MIN( NB, N-J+1 )
               IF( J+JB.LE.N ) THEN
*
*                 Compute rows j+jb:n of current block column
*
                  CALL DTRMM( 'Left', 'Lower', 'No transpose', DIAG,
     $                        N-J-JB+1, JB, ONE, A( J+JB, J+JB ), LDA,
     $                        A( J+JB, J ), LDA )
                  CALL DTRSM( 'Right', 'Lower', 'No transpose', DIAG,
     $                        N-J-JB+1, JB, -ONE, A( J, J ), LDA,
     $                        A( J+JB, J ), LDA )
               END IF
*
*              Compute inverse of current diagonal block
*
               CALL DTRTI2( 'Lower', DIAG, JB, A( J, J ), LDA, INFO )
   30       CONTINUE
         END IF
      END IF
*
      RETURN
*
*     End of DTRTRI
*
      END
*> \brief \b DTRTRS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrtrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrtrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrtrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTRTRS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB,
*                          INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, TRANS, UPLO
*       INTEGER            INFO, LDA, LDB, N, NRHS
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTRTRS solves a triangular system of the form
*>
*>    A * X = B  or  A**T * X = B,
*>
*> where A is a triangular matrix of order N, and B is an N-by-NRHS matrix.
*>
*> This subroutine verifies that A is nonsingular, but callers should note that only exact
*> singularity is detected. It is conceivable for one or more diagonal elements of A to be
*> subnormally tiny numbers without this subroutine signalling an error.
*>
*> If a possible loss of numerical precision due to near-singular matrices is a concern, the
*> caller should verify that A is nonsingular within some tolerance before calling this subroutine.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  A is upper triangular;
*>          = 'L':  A is lower triangular.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the form of the system of equations:
*>          = 'N':  A * X = B  (No transpose)
*>          = 'T':  A**T * X = B  (Transpose)
*>          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          = 'N':  A is non-unit triangular;
*>          = 'U':  A is unit triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The triangular matrix A.  If UPLO = 'U', the leading N-by-N
*>          upper triangular part of the array A contains the upper
*>          triangular matrix, and the strictly lower triangular part of
*>          A is not referenced.  If UPLO = 'L', the leading N-by-N lower
*>          triangular part of the array A contains the lower triangular
*>          matrix, and the strictly upper triangular part of A is not
*>          referenced.  If DIAG = 'U', the diagonal elements of A are
*>          also not referenced and are assumed to be 1.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the right hand side matrix B.
*>          On exit, if INFO = 0, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, the i-th diagonal element of A is exactly zero,
*>               indicating that the matrix is singular and the solutions
*>               X have not been computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trtrs
*
*  =====================================================================
      SUBROUTINE DTRTRS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, TRANS, UPLO
      INTEGER            INFO, LDA, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOUNIT
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DTRSM, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NOUNIT = LSAME( DIAG, 'N' )
      IF( .NOT.LSAME( UPLO, 'U' ) .AND.
     $    .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.LSAME( TRANS, 'N' ) .AND. .NOT.
     $         LSAME( TRANS, 'T' ) .AND.
     $                .NOT.LSAME( TRANS, 'C' ) ) THEN
         INFO = -2
      ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -9
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTRTRS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Check for singularity.
*
      IF( NOUNIT ) THEN
         DO 10 INFO = 1, N
            IF( A( INFO, INFO ).EQ.ZERO )
     $         RETURN
   10    CONTINUE
      END IF
      INFO = 0
*
*     Solve A * x = b  or  A**T * x = b.
*
      CALL DTRSM( 'Left', UPLO, TRANS, DIAG, N, NRHS, ONE, A, LDA, B,
     $            LDB )
*
      RETURN
*
*     End of DTRTRS
*
      END
*> \brief \b DTRTTF copies a triangular matrix from the standard full format (TR) to the rectangular full packed format (TF).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRTTF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrttf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrttf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrttf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTRTTF( TRANSR, UPLO, N, A, LDA, ARF, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANSR, UPLO
*       INTEGER            INFO, N, LDA
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( 0: LDA-1, 0: * ), ARF( 0: * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTRTTF copies a triangular matrix A from standard full format (TR)
*> to rectangular full packed format (TF) .
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANSR
*> \verbatim
*>          TRANSR is CHARACTER*1
*>          = 'N':  ARF in Normal form is wanted;
*>          = 'T':  ARF in Transpose form is wanted.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N).
*>          On entry, the triangular matrix A.  If UPLO = 'U', the
*>          leading N-by-N upper triangular part of the array A contains
*>          the upper triangular matrix, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading N-by-N lower triangular part of the array A contains
*>          the lower triangular matrix, and the strictly upper
*>          triangular part of A is not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the matrix A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] ARF
*> \verbatim
*>          ARF is DOUBLE PRECISION array, dimension (NT).
*>          NT=N*(N+1)/2. On exit, the triangular matrix A in RFP format.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trttf
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  We first consider Rectangular Full Packed (RFP) Format when N is
*>  even. We give an example where N = 6.
*>
*>      AP is Upper             AP is Lower
*>
*>   00 01 02 03 04 05       00
*>      11 12 13 14 15       10 11
*>         22 23 24 25       20 21 22
*>            33 34 35       30 31 32 33
*>               44 45       40 41 42 43 44
*>                  55       50 51 52 53 54 55
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*>  the transpose of the first three columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*>  the transpose of the last three columns of AP lower.
*>  This covers the case N even and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>        03 04 05                33 43 53
*>        13 14 15                00 44 54
*>        23 24 25                10 11 55
*>        33 34 35                20 21 22
*>        00 44 45                30 31 32
*>        01 11 55                40 41 42
*>        02 12 22                50 51 52
*>
*>  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*>  transpose of RFP A above. One therefore gets:
*>
*>
*>           RFP A                   RFP A
*>
*>     03 13 23 33 00 01 02    33 00 10 20 30 40 50
*>     04 14 24 34 44 11 12    43 44 11 21 31 41 51
*>     05 15 25 35 45 55 22    53 54 55 22 32 42 52
*>
*>
*>  We then consider Rectangular Full Packed (RFP) Format when N is
*>  odd. We give an example where N = 5.
*>
*>     AP is Upper                 AP is Lower
*>
*>   00 01 02 03 04              00
*>      11 12 13 14              10 11
*>         22 23 24              20 21 22
*>            33 34              30 31 32 33
*>               44              40 41 42 43 44
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*>  the transpose of the first two columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*>  the transpose of the last two columns of AP lower.
*>  This covers the case N odd and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>        02 03 04                00 33 43
*>        12 13 14                10 11 44
*>        22 23 24                20 21 22
*>        00 33 34                30 31 32
*>        01 11 44                40 41 42
*>
*>  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
*>  transpose of RFP A above. One therefore gets:
*>
*>           RFP A                   RFP A
*>
*>     02 12 22 00 01             00 10 20 30 40 50
*>     03 13 23 33 11             33 11 21 31 41 51
*>     04 14 24 34 44             43 44 22 32 42 52
*> \endverbatim
*
*  =====================================================================
      SUBROUTINE DTRTTF( TRANSR, UPLO, N, A, LDA, ARF, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANSR, UPLO
      INTEGER            INFO, N, LDA
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( 0: LDA-1, 0: * ), ARF( 0: * )
*     ..
*
*  =====================================================================
*
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER, NISODD, NORMALTRANSR
      INTEGER            I, IJ, J, K, L, N1, N2, NT, NX2, NP1X2
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MOD
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      NORMALTRANSR = LSAME( TRANSR, 'N' )
      LOWER = LSAME( UPLO, 'L' )
      IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTRTTF', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.LE.1 ) THEN
         IF( N.EQ.1 ) THEN
            ARF( 0 ) = A( 0, 0 )
         END IF
         RETURN
      END IF
*
*     Size of array ARF(0:nt-1)
*
      NT = N*( N+1 ) / 2
*
*     Set N1 and N2 depending on LOWER: for N even N1=N2=K
*
      IF( LOWER ) THEN
         N2 = N / 2
         N1 = N - N2
      ELSE
         N1 = N / 2
         N2 = N - N1
      END IF
*
*     If N is odd, set NISODD = .TRUE., LDA=N+1 and A is (N+1)--by--K2.
*     If N is even, set K = N/2 and NISODD = .FALSE., LDA=N and A is
*     N--by--(N+1)/2.
*
      IF( MOD( N, 2 ).EQ.0 ) THEN
         K = N / 2
         NISODD = .FALSE.
         IF( .NOT.LOWER )
     $      NP1X2 = N + N + 2
      ELSE
         NISODD = .TRUE.
         IF( .NOT.LOWER )
     $      NX2 = N + N
      END IF
*
      IF( NISODD ) THEN
*
*        N is odd
*
         IF( NORMALTRANSR ) THEN
*
*           N is odd and TRANSR = 'N'
*
            IF( LOWER ) THEN
*
*              N is odd, TRANSR = 'N', and UPLO = 'L'
*
               IJ = 0
               DO J = 0, N2
                  DO I = N1, N2 + J
                     ARF( IJ ) = A( N2+J, I )
                     IJ = IJ + 1
                  END DO
                  DO I = J, N - 1
                     ARF( IJ ) = A( I, J )
                     IJ = IJ + 1
                  END DO
               END DO
*
            ELSE
*
*              N is odd, TRANSR = 'N', and UPLO = 'U'
*
               IJ = NT - N
               DO J = N - 1, N1, -1
                  DO I = 0, J
                     ARF( IJ ) = A( I, J )
                     IJ = IJ + 1
                  END DO
                  DO L = J - N1, N1 - 1
                     ARF( IJ ) = A( J-N1, L )
                     IJ = IJ + 1
                  END DO
                  IJ = IJ - NX2
               END DO
*
            END IF
*
         ELSE
*
*           N is odd and TRANSR = 'T'
*
            IF( LOWER ) THEN
*
*              N is odd, TRANSR = 'T', and UPLO = 'L'
*
               IJ = 0
               DO J = 0, N2 - 1
                  DO I = 0, J
                     ARF( IJ ) = A( J, I )
                     IJ = IJ + 1
                  END DO
                  DO I = N1 + J, N - 1
                     ARF( IJ ) = A( I, N1+J )
                     IJ = IJ + 1
                  END DO
               END DO
               DO J = N2, N - 1
                  DO I = 0, N1 - 1
                     ARF( IJ ) = A( J, I )
                     IJ = IJ + 1
                  END DO
               END DO
*
            ELSE
*
*              N is odd, TRANSR = 'T', and UPLO = 'U'
*
               IJ = 0
               DO J = 0, N1
                  DO I = N1, N - 1
                     ARF( IJ ) = A( J, I )
                     IJ = IJ + 1
                  END DO
               END DO
               DO J = 0, N1 - 1
                  DO I = 0, J
                     ARF( IJ ) = A( I, J )
                     IJ = IJ + 1
                  END DO
                  DO L = N2 + J, N - 1
                     ARF( IJ ) = A( N2+J, L )
                     IJ = IJ + 1
                  END DO
               END DO
*
            END IF
*
         END IF
*
      ELSE
*
*        N is even
*
         IF( NORMALTRANSR ) THEN
*
*           N is even and TRANSR = 'N'
*
            IF( LOWER ) THEN
*
*              N is even, TRANSR = 'N', and UPLO = 'L'
*
               IJ = 0
               DO J = 0, K - 1
                  DO I = K, K + J
                     ARF( IJ ) = A( K+J, I )
                     IJ = IJ + 1
                  END DO
                  DO I = J, N - 1
                     ARF( IJ ) = A( I, J )
                     IJ = IJ + 1
                  END DO
               END DO
*
            ELSE
*
*              N is even, TRANSR = 'N', and UPLO = 'U'
*
               IJ = NT - N - 1
               DO J = N - 1, K, -1
                  DO I = 0, J
                     ARF( IJ ) = A( I, J )
                     IJ = IJ + 1
                  END DO
                  DO L = J - K, K - 1
                     ARF( IJ ) = A( J-K, L )
                     IJ = IJ + 1
                  END DO
                  IJ = IJ - NP1X2
               END DO
*
            END IF
*
         ELSE
*
*           N is even and TRANSR = 'T'
*
            IF( LOWER ) THEN
*
*              N is even, TRANSR = 'T', and UPLO = 'L'
*
               IJ = 0
               J = K
               DO I = K, N - 1
                  ARF( IJ ) = A( I, J )
                  IJ = IJ + 1
               END DO
               DO J = 0, K - 2
                  DO I = 0, J
                     ARF( IJ ) = A( J, I )
                     IJ = IJ + 1
                  END DO
                  DO I = K + 1 + J, N - 1
                     ARF( IJ ) = A( I, K+1+J )
                     IJ = IJ + 1
                  END DO
               END DO
               DO J = K - 1, N - 1
                  DO I = 0, K - 1
                     ARF( IJ ) = A( J, I )
                     IJ = IJ + 1
                  END DO
               END DO
*
            ELSE
*
*              N is even, TRANSR = 'T', and UPLO = 'U'
*
               IJ = 0
               DO J = 0, K
                  DO I = K, N - 1
                     ARF( IJ ) = A( J, I )
                     IJ = IJ + 1
                  END DO
               END DO
               DO J = 0, K - 2
                  DO I = 0, J
                     ARF( IJ ) = A( I, J )
                     IJ = IJ + 1
                  END DO
                  DO L = K + 1 + J, N - 1
                     ARF( IJ ) = A( K+1+J, L )
                     IJ = IJ + 1
                  END DO
               END DO
*              Note that here, on exit of the loop, J = K-1
               DO I = 0, J
                  ARF( IJ ) = A( I, J )
                  IJ = IJ + 1
               END DO
*
            END IF
*
         END IF
*
      END IF
*
      RETURN
*
*     End of DTRTTF
*
      END
*> \brief \b DTRTTP copies a triangular matrix from the standard full format (TR) to the standard packed format (TP).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTRTTP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrttp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrttp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrttp.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTRTTP( UPLO, N, A, LDA, AP, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, N, LDA
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), AP( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTRTTP copies a triangular matrix A from full format (TR) to standard
*> packed format (TP).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  A is upper triangular.
*>          = 'L':  A is lower triangular.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices AP and A.  N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On exit, the triangular matrix A.  If UPLO = 'U', the leading
*>          N-by-N upper triangular part of A contains the upper
*>          triangular part of the matrix A, and the strictly lower
*>          triangular part of A is not referenced.  If UPLO = 'L', the
*>          leading N-by-N lower triangular part of A contains the lower
*>          triangular part of the matrix A, and the strictly upper
*>          triangular part of A is not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On exit, the upper or lower triangular matrix A, packed
*>          columnwise in a linear array. The j-th column of A is stored
*>          in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup trttp
*
*  =====================================================================
      SUBROUTINE DTRTTP( UPLO, N, A, LDA, AP, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, N, LDA
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), AP( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER
      INTEGER            I, J, K
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      LOWER = LSAME( UPLO, 'L' )
      IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTRTTP', -INFO )
         RETURN
      END IF
*
      IF( LOWER ) THEN
         K = 0
         DO J = 1, N
            DO I = J, N
               K = K + 1
               AP( K ) = A( I, J )
            END DO
         END DO
      ELSE
         K = 0
         DO J = 1, N
            DO I = 1, J
               K = K + 1
               AP( K ) = A( I, J )
            END DO
         END DO
      END IF
*
*
      RETURN
*
*     End of DTRTTP
*
      END
*> \brief \b DTZRQF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTZRQF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtzrqf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtzrqf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtzrqf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTZRQF( M, N, A, LDA, TAU, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DTZRZF.
*>
*> DTZRQF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
*> to upper triangular form by means of orthogonal transformations.
*>
*> The upper trapezoidal matrix A is factored as
*>
*>    A = ( R  0 ) * Z,
*>
*> where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
*> triangular matrix.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= M.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the leading M-by-N upper trapezoidal part of the
*>          array A must contain the matrix to be factorized.
*>          On exit, the leading M-by-M upper triangular part of A
*>          contains the upper triangular matrix R, and elements M+1 to
*>          N of the first M rows of A, with the array TAU, represent the
*>          orthogonal matrix Z as a product of M elementary reflectors.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (M)
*>          The scalar factors of the elementary reflectors.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleOTHERcomputational
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The factorization is obtained by Householder's method.  The kth
*>  transformation matrix, Z( k ), which is used to introduce zeros into
*>  the ( m - k + 1 )th row of A, is given in the form
*>
*>     Z( k ) = ( I     0   ),
*>              ( 0  T( k ) )
*>
*>  where
*>
*>     T( k ) = I - tau*u( k )*u( k )**T,   u( k ) = (   1    ),
*>                                                   (   0    )
*>                                                   ( z( k ) )
*>
*>  tau is a scalar and z( k ) is an ( n - m ) element vector.
*>  tau and z( k ) are chosen to annihilate the elements of the kth row
*>  of X.
*>
*>  The scalar tau is returned in the kth element of TAU and the vector
*>  u( k ) in the kth row of A, such that the elements of z( k ) are
*>  in  a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in
*>  the upper triangular part of A.
*>
*>  Z is given by
*>
*>     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTZRQF( M, N, A, LDA, TAU, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, K, M1
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DGEMV, DGER, DLARFG, XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.M ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTZRQF', -INFO )
         RETURN
      END IF
*
*     Perform the factorization.
*
      IF( M.EQ.0 )
     $   RETURN
      IF( M.EQ.N ) THEN
         DO 10 I = 1, N
            TAU( I ) = ZERO
   10    CONTINUE
      ELSE
         M1 = MIN( M+1, N )
         DO 20 K = M, 1, -1
*
*           Use a Householder reflection to zero the kth row of A.
*           First set up the reflection.
*
            CALL DLARFG( N-M+1, A( K, K ), A( K, M1 ), LDA,
     $                   TAU( K ) )
*
            IF( ( TAU( K ).NE.ZERO ) .AND. ( K.GT.1 ) ) THEN
*
*              We now perform the operation  A := A*P( k ).
*
*              Use the first ( k - 1 ) elements of TAU to store  a( k ),
*              where  a( k ) consists of the first ( k - 1 ) elements of
*              the  kth column  of  A.  Also  let  B  denote  the  first
*              ( k - 1 ) rows of the last ( n - m ) columns of A.
*
               CALL DCOPY( K-1, A( 1, K ), 1, TAU, 1 )
*
*              Form   w = a( k ) + B*z( k )  in TAU.
*
               CALL DGEMV( 'No transpose', K-1, N-M, ONE, A( 1, M1 ),
     $                     LDA, A( K, M1 ), LDA, ONE, TAU, 1 )
*
*              Now form  a( k ) := a( k ) - tau*w
*              and       B      := B      - tau*w*z( k )**T.
*
               CALL DAXPY( K-1, -TAU( K ), TAU, 1, A( 1, K ), 1 )
               CALL DGER( K-1, N-M, -TAU( K ), TAU, 1, A( K, M1 ),
     $                    LDA, A( 1, M1 ), LDA )
            END IF
   20    CONTINUE
      END IF
*
      RETURN
*
*     End of DTZRQF
*
      END
*> \brief \b DTZRZF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTZRZF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtzrzf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtzrzf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtzrzf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTZRZF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTZRZF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
*> to upper triangular form by means of orthogonal transformations.
*>
*> The upper trapezoidal matrix A is factored as
*>
*>    A = ( R  0 ) * Z,
*>
*> where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
*> triangular matrix.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= M.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the leading M-by-N upper trapezoidal part of the
*>          array A must contain the matrix to be factorized.
*>          On exit, the leading M-by-M upper triangular part of A
*>          contains the upper triangular matrix R, and elements M+1 to
*>          N of the first M rows of A, with the array TAU, represent the
*>          orthogonal matrix Z as a product of M elementary reflectors.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (M)
*>          The scalar factors of the elementary reflectors.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.  LWORK >= max(1,M).
*>          For optimum performance LWORK >= M*NB, where NB is
*>          the optimal blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tzrzf
*
*> \par Contributors:
*  ==================
*>
*>    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The N-by-N matrix Z can be computed by
*>
*>     Z =  Z(1)*Z(2)* ... *Z(M)
*>
*>  where each N-by-N Z(k) is given by
*>
*>     Z(k) = I - tau(k)*v(k)*v(k)**T
*>
*>  with v(k) is the kth row vector of the M-by-N matrix
*>
*>     V = ( I   A(:,M+1:N) )
*>
*>  I is the M-by-M identity matrix, A(:,M+1:N)
*>  is the output stored in A on exit from DTZRZF,
*>  and tau(k) is the kth element of the array TAU.
*>
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTZRZF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IB, IWS, KI, KK, LDWORK, LWKMIN, LWKOPT,
     $                   M1, MU, NB, NBMIN, NX
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA, DLARZB, DLARZT, DLATRZ
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.M ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -4
      END IF
*
      IF( INFO.EQ.0 ) THEN
         IF( M.EQ.0 .OR. M.EQ.N ) THEN
            LWKOPT = 1
            LWKMIN = 1
         ELSE
*
*           Determine the block size.
*
            NB = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )
            LWKOPT = M*NB
            LWKMIN = MAX( 1, M )
         END IF
         WORK( 1 ) = LWKOPT
*
         IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
            INFO = -7
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTZRZF', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 ) THEN
         RETURN
      ELSE IF( M.EQ.N ) THEN
         DO 10 I = 1, N
            TAU( I ) = ZERO
   10    CONTINUE
         RETURN
      END IF
*
      NBMIN = 2
      NX = 1
      IWS = M
      IF( NB.GT.1 .AND. NB.LT.M ) THEN
*
*        Determine when to cross over from blocked to unblocked code.
*
         NX = MAX( 0, ILAENV( 3, 'DGERQF', ' ', M, N, -1, -1 ) )
         IF( NX.LT.M ) THEN
*
*           Determine if workspace is large enough for blocked code.
*
            LDWORK = M
            IWS = LDWORK*NB
            IF( LWORK.LT.IWS ) THEN
*
*              Not enough workspace to use optimal NB:  reduce NB and
*              determine the minimum value of NB.
*
               NB = LWORK / LDWORK
               NBMIN = MAX( 2, ILAENV( 2, 'DGERQF', ' ', M, N, -1,
     $                 -1 ) )
            END IF
         END IF
      END IF
*
      IF( NB.GE.NBMIN .AND. NB.LT.M .AND. NX.LT.M ) THEN
*
*        Use blocked code initially.
*        The last kk rows are handled by the block method.
*
         M1 = MIN( M+1, N )
         KI = ( ( M-NX-1 ) / NB )*NB
         KK = MIN( M, KI+NB )
*
         DO 20 I = M - KK + KI + 1, M - KK + 1, -NB
            IB = MIN( M-I+1, NB )
*
*           Compute the TZ factorization of the current block
*           A(i:i+ib-1,i:n)
*
            CALL DLATRZ( IB, N-I+1, N-M, A( I, I ), LDA, TAU( I ),
     $                   WORK )
            IF( I.GT.1 ) THEN
*
*              Form the triangular factor of the block reflector
*              H = H(i+ib-1) . . . H(i+1) H(i)
*
               CALL DLARZT( 'Backward', 'Rowwise', N-M, IB, A( I,
     $                      M1 ),
     $                      LDA, TAU( I ), WORK, LDWORK )
*
*              Apply H to A(1:i-1,i:n) from the right
*
               CALL DLARZB( 'Right', 'No transpose', 'Backward',
     $                      'Rowwise', I-1, N-I+1, IB, N-M, A( I, M1 ),
     $                      LDA, WORK, LDWORK, A( 1, I ), LDA,
     $                      WORK( IB+1 ), LDWORK )
            END IF
   20    CONTINUE
         MU = I + NB - 1
      ELSE
         MU = M
      END IF
*
*     Use unblocked code to factor the last or only block
*
      IF( MU.GT.0 )
     $   CALL DLATRZ( MU, N, N-M, A, LDA, TAU, WORK )
*
      WORK( 1 ) = LWKOPT
*
      RETURN
*
*     End of DTZRZF
*
      END
*> \brief \b IEEECK
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download IEEECK + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ieeeck.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ieeeck.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ieeeck.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       INTEGER          FUNCTION IEEECK( ISPEC, ZERO, ONE )
*
*       .. Scalar Arguments ..
*       INTEGER            ISPEC
*       REAL               ONE, ZERO
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> IEEECK is called from the ILAENV to verify that Infinity and
*> possibly NaN arithmetic is safe (i.e. will not trap).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ISPEC
*> \verbatim
*>          ISPEC is INTEGER
*>          Specifies whether to test just for infinity arithmetic
*>          or whether to test for infinity and NaN arithmetic.
*>          = 0: Verify infinity arithmetic only.
*>          = 1: Verify infinity and NaN arithmetic.
*> \endverbatim
*>
*> \param[in] ZERO
*> \verbatim
*>          ZERO is REAL
*>          Must contain the value 0.0
*>          This is passed to prevent the compiler from optimizing
*>          away this code.
*> \endverbatim
*>
*> \param[in] ONE
*> \verbatim
*>          ONE is REAL
*>          Must contain the value 1.0
*>          This is passed to prevent the compiler from optimizing
*>          away this code.
*>
*>  RETURN VALUE:  INTEGER
*>          = 0:  Arithmetic failed to produce the correct answers
*>          = 1:  Arithmetic produced the correct answers
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ieeeck
*
*  =====================================================================
      INTEGER          FUNCTION IEEECK( ISPEC, ZERO, ONE )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            ISPEC
      REAL               ONE, ZERO
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      REAL               NAN1, NAN2, NAN3, NAN4, NAN5, NAN6, NEGINF,
     $                   NEGZRO, NEWZRO, POSINF
*     ..
*     .. Executable Statements ..
      IEEECK = 1
*
      POSINF = ONE / ZERO
      IF( POSINF.LE.ONE ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      NEGINF = -ONE / ZERO
      IF( NEGINF.GE.ZERO ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      NEGZRO = ONE / ( NEGINF+ONE )
      IF( NEGZRO.NE.ZERO ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      NEGINF = ONE / NEGZRO
      IF( NEGINF.GE.ZERO ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      NEWZRO = NEGZRO + ZERO
      IF( NEWZRO.NE.ZERO ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      POSINF = ONE / NEWZRO
      IF( POSINF.LE.ONE ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      NEGINF = NEGINF*POSINF
      IF( NEGINF.GE.ZERO ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      POSINF = POSINF*POSINF
      IF( POSINF.LE.ONE ) THEN
         IEEECK = 0
         RETURN
      END IF
*
*
*
*
*     Return if we were only asked to check infinity arithmetic
*
      IF( ISPEC.EQ.0 )
     $   RETURN
*
      NAN1 = POSINF + NEGINF
*
      NAN2 = POSINF / NEGINF
*
      NAN3 = POSINF / POSINF
*
      NAN4 = POSINF*ZERO
*
      NAN5 = NEGINF*NEGZRO
*
      NAN6 = NAN5*ZERO
*
      IF( NAN1.EQ.NAN1 ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      IF( NAN2.EQ.NAN2 ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      IF( NAN3.EQ.NAN3 ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      IF( NAN4.EQ.NAN4 ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      IF( NAN5.EQ.NAN5 ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      IF( NAN6.EQ.NAN6 ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      RETURN
      END
*> \brief \b ILADLC scans a matrix for its last non-zero column.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ILADLC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/iladlc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/iladlc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/iladlc.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       INTEGER FUNCTION ILADLC( M, N, A, LDA )
*
*       .. Scalar Arguments ..
*       INTEGER            M, N, LDA
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ILADLC scans A for its last non-zero column.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The m by n matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ilalc
*
*  =====================================================================
      INTEGER FUNCTION ILADLC( M, N, A, LDA )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            M, N, LDA
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION ZERO
      PARAMETER ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER I
*     ..
*     .. Executable Statements ..
*
*     Quick test for the common case where one corner is non-zero.
      IF( N.EQ.0 ) THEN
         ILADLC = N
      ELSE IF( A(1, N).NE.ZERO .OR. A(M, N).NE.ZERO ) THEN
         ILADLC = N
      ELSE
*     Now scan each column from the end, returning with the first non-zero.
         DO ILADLC = N, 1, -1
            DO I = 1, M
               IF( A(I, ILADLC).NE.ZERO ) RETURN
            END DO
         END DO
      END IF
      RETURN
      END
*> \brief \b ILADLR scans a matrix for its last non-zero row.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ILADLR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/iladlr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/iladlr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/iladlr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       INTEGER FUNCTION ILADLR( M, N, A, LDA )
*
*       .. Scalar Arguments ..
*       INTEGER            M, N, LDA
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ILADLR scans A for its last non-zero row.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          The m by n matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ilalr
*
*  =====================================================================
      INTEGER FUNCTION ILADLR( M, N, A, LDA )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            M, N, LDA
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION ZERO
      PARAMETER ( ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER I, J
*     ..
*     .. Executable Statements ..
*
*     Quick test for the common case where one corner is non-zero.
      IF( M.EQ.0 ) THEN
         ILADLR = M
      ELSE IF( A(M, 1).NE.ZERO .OR. A(M, N).NE.ZERO ) THEN
         ILADLR = M
      ELSE
*     Scan up each column tracking the last zero row seen.
         ILADLR = 0
         DO J = 1, N
            I=M
            DO WHILE((A(MAX(I,1),J).EQ.ZERO).AND.(I.GE.1))
               I=I-1
            ENDDO
            ILADLR = MAX( ILADLR, I )
         END DO
      END IF
      RETURN
      END
*> \brief \b ILAENV
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ILAENV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ilaenv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ilaenv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ilaenv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       INTEGER FUNCTION ILAENV( ISPEC, NAME, OPTS, N1, N2, N3, N4 )
*
*       .. Scalar Arguments ..
*       CHARACTER*( * )    NAME, OPTS
*       INTEGER            ISPEC, N1, N2, N3, N4
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ILAENV is called from the LAPACK routines to choose problem-dependent
*> parameters for the local environment.  See ISPEC for a description of
*> the parameters.
*>
*> ILAENV returns an INTEGER
*> if ILAENV >= 0: ILAENV returns the value of the parameter specified by ISPEC
*> if ILAENV < 0:  if ILAENV = -k, the k-th argument had an illegal value.
*>
*> This version provides a set of parameters which should give good,
*> but not optimal, performance on many of the currently available
*> computers.  Users are encouraged to modify this subroutine to set
*> the tuning parameters for their particular machine using the option
*> and problem size information in the arguments.
*>
*> This routine will not function correctly if it is converted to all
*> lower case.  Converting it to all upper case is allowed.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ISPEC
*> \verbatim
*>          ISPEC is INTEGER
*>          Specifies the parameter to be returned as the value of
*>          ILAENV.
*>          = 1: the optimal blocksize; if this value is 1, an unblocked
*>               algorithm will give the best performance.
*>          = 2: the minimum block size for which the block routine
*>               should be used; if the usable block size is less than
*>               this value, an unblocked routine should be used.
*>          = 3: the crossover point (in a block routine, for N less
*>               than this value, an unblocked routine should be used)
*>          = 4: the number of shifts, used in the nonsymmetric
*>               eigenvalue routines (DEPRECATED)
*>          = 5: the minimum column dimension for blocking to be used;
*>               rectangular blocks must have dimension at least k by m,
*>               where k is given by ILAENV(2,...) and m by ILAENV(5,...)
*>          = 6: the crossover point for the SVD (when reducing an m by n
*>               matrix to bidiagonal form, if max(m,n)/min(m,n) exceeds
*>               this value, a QR factorization is used first to reduce
*>               the matrix to a triangular form.)
*>          = 7: the number of processors
*>          = 8: the crossover point for the multishift QR method
*>               for nonsymmetric eigenvalue problems (DEPRECATED)
*>          = 9: maximum size of the subproblems at the bottom of the
*>               computation tree in the divide-and-conquer algorithm
*>               (used by xGELSD and xGESDD)
*>          =10: ieee infinity and NaN arithmetic can be trusted not to trap
*>          =11: infinity arithmetic can be trusted not to trap
*>          12 <= ISPEC <= 17:
*>               xHSEQR or related subroutines,
*>               see IPARMQ for detailed explanation
*> \endverbatim
*>
*> \param[in] NAME
*> \verbatim
*>          NAME is CHARACTER*(*)
*>          The name of the calling subroutine, in either upper case or
*>          lower case.
*> \endverbatim
*>
*> \param[in] OPTS
*> \verbatim
*>          OPTS is CHARACTER*(*)
*>          The character options to the subroutine NAME, concatenated
*>          into a single character string.  For example, UPLO = 'U',
*>          TRANS = 'T', and DIAG = 'N' for a triangular routine would
*>          be specified as OPTS = 'UTN'.
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*>          N1 is INTEGER
*> \endverbatim
*>
*> \param[in] N2
*> \verbatim
*>          N2 is INTEGER
*> \endverbatim
*>
*> \param[in] N3
*> \verbatim
*>          N3 is INTEGER
*> \endverbatim
*>
*> \param[in] N4
*> \verbatim
*>          N4 is INTEGER
*>          Problem dimensions for the subroutine NAME; these may not all
*>          be required.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ilaenv
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The following conventions have been used when calling ILAENV from the
*>  LAPACK routines:
*>  1)  OPTS is a concatenation of all of the character options to
*>      subroutine NAME, in the same order that they appear in the
*>      argument list for NAME, even if they are not used in determining
*>      the value of the parameter specified by ISPEC.
*>  2)  The problem dimensions N1, N2, N3, N4 are specified in the order
*>      that they appear in the argument list for NAME.  N1 is used
*>      first, N2 second, and so on, and unused problem dimensions are
*>      passed a value of -1.
*>  3)  The parameter value returned by ILAENV is checked for validity in
*>      the calling subroutine.  For example, ILAENV is used to retrieve
*>      the optimal blocksize for STRTRI as follows:
*>
*>      NB = ILAENV( 1, 'STRTRI', UPLO // DIAG, N, -1, -1, -1 )
*>      IF( NB.LE.1 ) NB = MAX( 1, N )
*> \endverbatim
*>
*  =====================================================================
      INTEGER FUNCTION ILAENV( ISPEC, NAME, OPTS, N1, N2, N3, N4 )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER*( * )    NAME, OPTS
      INTEGER            ISPEC, N1, N2, N3, N4
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            I, IC, IZ, NB, NBMIN, NX
      LOGICAL            CNAME, SNAME, TWOSTAGE
      CHARACTER          C1*1, C2*2, C4*2, C3*3, SUBNAM*16
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          CHAR, ICHAR, INT, MIN, REAL
*     ..
*     .. External Functions ..
      INTEGER            IEEECK, IPARMQ, IPARAM2STAGE
      EXTERNAL           IEEECK, IPARMQ, IPARAM2STAGE
*     ..
*     .. Executable Statements ..
*
      GO TO ( 10, 10, 10, 80, 90, 100, 110, 120,
     $        130, 140, 150, 160, 160, 160, 160, 160, 160)ISPEC
*
*     Invalid value for ISPEC
*
      ILAENV = -1
      RETURN
*
   10 CONTINUE
*
*     Convert NAME to upper case if the first character is lower case.
*
      ILAENV = 1
      SUBNAM = NAME
      IC = ICHAR( SUBNAM( 1: 1 ) )
      IZ = ICHAR( 'Z' )
      IF( IZ.EQ.90 .OR. IZ.EQ.122 ) THEN
*
*        ASCII character set
*
         IF( IC.GE.97 .AND. IC.LE.122 ) THEN
            SUBNAM( 1: 1 ) = CHAR( IC-32 )
            DO 20 I = 2, 6
               IC = ICHAR( SUBNAM( I: I ) )
               IF( IC.GE.97 .AND. IC.LE.122 )
     $            SUBNAM( I: I ) = CHAR( IC-32 )
   20       CONTINUE
         END IF
*
      ELSE IF( IZ.EQ.233 .OR. IZ.EQ.169 ) THEN
*
*        EBCDIC character set
*
         IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
     $       ( IC.GE.145 .AND. IC.LE.153 ) .OR.
     $       ( IC.GE.162 .AND. IC.LE.169 ) ) THEN
            SUBNAM( 1: 1 ) = CHAR( IC+64 )
            DO 30 I = 2, 6
               IC = ICHAR( SUBNAM( I: I ) )
               IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
     $             ( IC.GE.145 .AND. IC.LE.153 ) .OR.
     $             ( IC.GE.162 .AND. IC.LE.169 ) )SUBNAM( I:
     $             I ) = CHAR( IC+64 )
   30       CONTINUE
         END IF
*
      ELSE IF( IZ.EQ.218 .OR. IZ.EQ.250 ) THEN
*
*        Prime machines:  ASCII+128
*
         IF( IC.GE.225 .AND. IC.LE.250 ) THEN
            SUBNAM( 1: 1 ) = CHAR( IC-32 )
            DO 40 I = 2, 6
               IC = ICHAR( SUBNAM( I: I ) )
               IF( IC.GE.225 .AND. IC.LE.250 )
     $            SUBNAM( I: I ) = CHAR( IC-32 )
   40       CONTINUE
         END IF
      END IF
*
      C1 = SUBNAM( 1: 1 )
      SNAME = C1.EQ.'S' .OR. C1.EQ.'D'
      CNAME = C1.EQ.'C' .OR. C1.EQ.'Z'
      IF( .NOT.( CNAME .OR. SNAME ) )
     $   RETURN
      C2 = SUBNAM( 2: 3 )
      C3 = SUBNAM( 4: 6 )
      C4 = C3( 2: 3 )
      TWOSTAGE = LEN( SUBNAM ).GE.11
     $           .AND. SUBNAM( 11: 11 ).EQ.'2'
*
      GO TO ( 50, 60, 70 )ISPEC
*
   50 CONTINUE
*
*     ISPEC = 1:  block size
*
*     In these examples, separate code is provided for setting NB for
*     real and complex.  We assume that NB will take the same value in
*     single or double precision.
*
      NB = 1
*
      IF( SUBNAM(2:6).EQ.'LAORH' ) THEN
*
*        This is for *LAORHR_GETRFNP routine
*
         IF( SNAME ) THEN
             NB = 32
         ELSE
             NB = 32
         END IF
      ELSE IF( C2.EQ.'GE' ) THEN
         IF( C3.EQ.'TRF' ) THEN
            IF( SNAME ) THEN
               NB = 64
            ELSE
               NB = 64
            END IF
         ELSE IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR.
     $            C3.EQ.'QLF' ) THEN
            IF( SNAME ) THEN
               NB = 32
            ELSE
               NB = 32
            END IF
         ELSE IF( C3.EQ.'QR ') THEN
            IF( N3 .EQ. 1) THEN
               IF( SNAME ) THEN
*     M*N
                  IF ((N1*N2.LE.131072).OR.(N1.LE.8192)) THEN
                     NB = N1
                  ELSE
                     NB = 32768/N2
                  END IF
               ELSE
                  IF ((N1*N2.LE.131072).OR.(N1.LE.8192)) THEN
                     NB = N1
                  ELSE
                     NB = 32768/N2
                  END IF
               END IF
            ELSE
               IF( SNAME ) THEN
                  NB = 1
               ELSE
                  NB = 1
               END IF
            END IF
         ELSE IF( C3.EQ.'LQ ') THEN
            IF( N3 .EQ. 2) THEN
               IF( SNAME ) THEN
*     M*N
                  IF ((N1*N2.LE.131072).OR.(N1.LE.8192)) THEN
                     NB = N1
                  ELSE
                     NB = 32768/N2
                  END IF
               ELSE
                  IF ((N1*N2.LE.131072).OR.(N1.LE.8192)) THEN
                     NB = N1
                  ELSE
                     NB = 32768/N2
                  END IF
               END IF
            ELSE
               IF( SNAME ) THEN
                  NB = 1
               ELSE
                  NB = 1
               END IF
            END IF
         ELSE IF( C3.EQ.'HRD' ) THEN
            IF( SNAME ) THEN
               NB = 32
            ELSE
               NB = 32
            END IF
         ELSE IF( C3.EQ.'BRD' ) THEN
            IF( SNAME ) THEN
               NB = 32
            ELSE
               NB = 32
            END IF
         ELSE IF( C3.EQ.'TRI' ) THEN
            IF( SNAME ) THEN
               NB = 64
            ELSE
               NB = 64
            END IF
         ELSE IF( SUBNAM( 4: 7 ).EQ.'QP3RK' ) THEN
            IF( SNAME ) THEN
               NB = 32
            ELSE
               NB = 32
            END IF
         END IF
      ELSE IF( C2.EQ.'PO' ) THEN
         IF( C3.EQ.'TRF' ) THEN
            IF( SNAME ) THEN
               NB = 64
            ELSE
               NB = 64
            END IF
         END IF
      ELSE IF( C2.EQ.'SY' ) THEN
         IF( C3.EQ.'TRF' ) THEN
            IF( SNAME ) THEN
               IF( TWOSTAGE ) THEN
                  NB = 192
               ELSE
                  NB = 64
               END IF
            ELSE
               IF( TWOSTAGE ) THEN
                  NB = 192
               ELSE
                  NB = 64
               END IF
            END IF
         ELSE IF( SNAME .AND. C3.EQ.'TRD' ) THEN
            NB = 32
         ELSE IF( SNAME .AND. C3.EQ.'GST' ) THEN
            NB = 64
         END IF
      ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
         IF( C3.EQ.'TRF' ) THEN
            IF( TWOSTAGE ) THEN
               NB = 192
            ELSE
               NB = 64
            END IF
         ELSE IF( C3.EQ.'TRD' ) THEN
            NB = 32
         ELSE IF( C3.EQ.'GST' ) THEN
            NB = 64
         END IF
      ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
         IF( C3( 1: 1 ).EQ.'G' ) THEN
            IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
     $          'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
     $           THEN
               NB = 32
            END IF
         ELSE IF( C3( 1: 1 ).EQ.'M' ) THEN
            IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
     $          'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
     $           THEN
               NB = 32
            END IF
         END IF
      ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
         IF( C3( 1: 1 ).EQ.'G' ) THEN
            IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
     $          'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
     $           THEN
               NB = 32
            END IF
         ELSE IF( C3( 1: 1 ).EQ.'M' ) THEN
            IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
     $          'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
     $           THEN
               NB = 32
            END IF
         END IF
      ELSE IF( C2.EQ.'GB' ) THEN
         IF( C3.EQ.'TRF' ) THEN
            IF( SNAME ) THEN
               IF( N4.LE.64 ) THEN
                  NB = 1
               ELSE
                  NB = 32
               END IF
            ELSE
               IF( N4.LE.64 ) THEN
                  NB = 1
               ELSE
                  NB = 32
               END IF
            END IF
         END IF
      ELSE IF( C2.EQ.'PB' ) THEN
         IF( C3.EQ.'TRF' ) THEN
            IF( SNAME ) THEN
               IF( N2.LE.64 ) THEN
                  NB = 1
               ELSE
                  NB = 32
               END IF
            ELSE
               IF( N2.LE.64 ) THEN
                  NB = 1
               ELSE
                  NB = 32
               END IF
            END IF
         END IF
      ELSE IF( C2.EQ.'TR' ) THEN
         IF( C3.EQ.'TRI' ) THEN
            IF( SNAME ) THEN
               NB = 64
            ELSE
               NB = 64
            END IF
         ELSE IF ( C3.EQ.'EVC' ) THEN
            IF( SNAME ) THEN
               NB = 64
            ELSE
               NB = 64
            END IF
         ELSE IF( C3.EQ.'SYL' ) THEN
*           The upper bound is to prevent overly aggressive scaling.
            IF( SNAME ) THEN
               NB = MIN( MAX( 48, INT( ( MIN( N1, N2 ) * 16 ) / 100) ),
     $                   240 )
            ELSE
               NB = MIN( MAX( 24, INT( ( MIN( N1, N2 ) * 8 ) / 100) ),
     $                   80 )
            END IF
         END IF
      ELSE IF( C2.EQ.'LA' ) THEN
         IF( C3.EQ.'UUM' ) THEN
            IF( SNAME ) THEN
               NB = 64
            ELSE
               NB = 64
            END IF
         ELSE IF( C3.EQ.'TRS' ) THEN
            IF( SNAME ) THEN
               NB = 32
            ELSE
               NB = 32
            END IF
         END IF
      ELSE IF( SNAME .AND. C2.EQ.'ST' ) THEN
         IF( C3.EQ.'EBZ' ) THEN
            NB = 1
         END IF
      ELSE IF( C2.EQ.'GG' ) THEN
         NB = 32
         IF( C3.EQ.'HD3' ) THEN
            IF( SNAME ) THEN
               NB = 32
            ELSE
               NB = 32
            END IF
         END IF
      END IF
      ILAENV = NB
      RETURN
*
   60 CONTINUE
*
*     ISPEC = 2:  minimum block size
*
      NBMIN = 2
      IF( C2.EQ.'GE' ) THEN
         IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR. C3.EQ.
     $       'QLF' ) THEN
            IF( SNAME ) THEN
               NBMIN = 2
            ELSE
               NBMIN = 2
            END IF
         ELSE IF( C3.EQ.'HRD' ) THEN
            IF( SNAME ) THEN
               NBMIN = 2
            ELSE
               NBMIN = 2
            END IF
         ELSE IF( C3.EQ.'BRD' ) THEN
            IF( SNAME ) THEN
               NBMIN = 2
            ELSE
               NBMIN = 2
            END IF
         ELSE IF( C3.EQ.'TRI' ) THEN
            IF( SNAME ) THEN
               NBMIN = 2
            ELSE
               NBMIN = 2
            END IF
         ELSE IF( SUBNAM( 4: 7 ).EQ.'QP3RK' ) THEN
            IF( SNAME ) THEN
               NBMIN = 2
            ELSE
               NBMIN = 2
            END IF
         END IF

      ELSE IF( C2.EQ.'SY' ) THEN
         IF( C3.EQ.'TRF' ) THEN
            IF( SNAME ) THEN
               NBMIN = 8
            ELSE
               NBMIN = 8
            END IF
         ELSE IF( SNAME .AND. C3.EQ.'TRD' ) THEN
            NBMIN = 2
         END IF
      ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
         IF( C3.EQ.'TRD' ) THEN
            NBMIN = 2
         END IF
      ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
         IF( C3( 1: 1 ).EQ.'G' ) THEN
            IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
     $          'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
     $           THEN
               NBMIN = 2
            END IF
         ELSE IF( C3( 1: 1 ).EQ.'M' ) THEN
            IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
     $          'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
     $           THEN
               NBMIN = 2
            END IF
         END IF
      ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
         IF( C3( 1: 1 ).EQ.'G' ) THEN
            IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
     $          'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
     $           THEN
               NBMIN = 2
            END IF
         ELSE IF( C3( 1: 1 ).EQ.'M' ) THEN
            IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
     $          'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
     $           THEN
               NBMIN = 2
            END IF
         END IF
      ELSE IF( C2.EQ.'GG' ) THEN
         NBMIN = 2
         IF( C3.EQ.'HD3' ) THEN
            NBMIN = 2
         END IF
      END IF
      ILAENV = NBMIN
      RETURN
*
   70 CONTINUE
*
*     ISPEC = 3:  crossover point
*
      NX = 0
      IF( C2.EQ.'GE' ) THEN
         IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR. C3.EQ.
     $       'QLF' ) THEN
            IF( SNAME ) THEN
               NX = 128
            ELSE
               NX = 128
            END IF
         ELSE IF( C3.EQ.'HRD' ) THEN
            IF( SNAME ) THEN
               NX = 128
            ELSE
               NX = 128
            END IF
         ELSE IF( C3.EQ.'BRD' ) THEN
            IF( SNAME ) THEN
               NX = 128
            ELSE
               NX = 128
            END IF
         ELSE IF( SUBNAM( 4: 7 ).EQ.'QP3RK' ) THEN
            IF( SNAME ) THEN
               NX = 128
            ELSE
               NX = 128
            END IF
         END IF
      ELSE IF( C2.EQ.'SY' ) THEN
         IF( SNAME .AND. C3.EQ.'TRD' ) THEN
            NX = 32
         END IF
      ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
         IF( C3.EQ.'TRD' ) THEN
            NX = 32
         END IF
      ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
         IF( C3( 1: 1 ).EQ.'G' ) THEN
            IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
     $          'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
     $           THEN
               NX = 128
            END IF
         END IF
      ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
         IF( C3( 1: 1 ).EQ.'G' ) THEN
            IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
     $          'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
     $           THEN
               NX = 128
            END IF
         END IF
      ELSE IF( C2.EQ.'GG' ) THEN
         NX = 128
         IF( C3.EQ.'HD3' ) THEN
            NX = 128
         END IF
      END IF
      ILAENV = NX
      RETURN
*
   80 CONTINUE
*
*     ISPEC = 4:  number of shifts (used by xHSEQR)
*
      ILAENV = 6
      RETURN
*
   90 CONTINUE
*
*     ISPEC = 5:  minimum column dimension (not used)
*
      ILAENV = 2
      RETURN
*
  100 CONTINUE
*
*     ISPEC = 6:  crossover point for SVD (used by xGELSS and xGESVD)
*
      ILAENV = INT( REAL( MIN( N1, N2 ) )*1.6E0 )
      RETURN
*
  110 CONTINUE
*
*     ISPEC = 7:  number of processors (not used)
*
      ILAENV = 1
      RETURN
*
  120 CONTINUE
*
*     ISPEC = 8:  crossover point for multishift (used by xHSEQR)
*
      ILAENV = 50
      RETURN
*
  130 CONTINUE
*
*     ISPEC = 9:  maximum size of the subproblems at the bottom of the
*                 computation tree in the divide-and-conquer algorithm
*                 (used by xGELSD and xGESDD)
*
      ILAENV = 25
      RETURN
*
  140 CONTINUE
*
*     ISPEC = 10: ieee and infinity NaN arithmetic can be trusted not to trap
*
*     ILAENV = 0
      ILAENV = 1
      IF( ILAENV.EQ.1 ) THEN
         ILAENV = IEEECK( 1, 0.0, 1.0 )
      END IF
      RETURN
*
  150 CONTINUE
*
*     ISPEC = 11: ieee infinity arithmetic can be trusted not to trap
*
*     ILAENV = 0
      ILAENV = 1
      IF( ILAENV.EQ.1 ) THEN
         ILAENV = IEEECK( 0, 0.0, 1.0 )
      END IF
      RETURN
*
  160 CONTINUE
*
*     12 <= ISPEC <= 17: xHSEQR or related subroutines.
*
      ILAENV = IPARMQ( ISPEC, NAME, OPTS, N1, N2, N3, N4 )
      RETURN
*
*     End of ILAENV
*
      END
*> \brief \b ILAPREC
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ILAPREC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ilaprec.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ilaprec.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ilaprec.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       INTEGER FUNCTION ILAPREC( PREC )
*
*       .. Scalar Arguments ..
*       CHARACTER          PREC
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> This subroutine translated from a character string specifying an
*> intermediate precision to the relevant BLAST-specified integer
*> constant.
*>
*> ILAPREC returns an INTEGER.  If ILAPREC < 0, then the input is not a
*> character indicating a supported intermediate precision.  Otherwise
*> ILAPREC returns the constant value corresponding to PREC.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ilaprec
*
*  =====================================================================
      INTEGER FUNCTION ILAPREC( PREC )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          PREC
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER BLAS_PREC_SINGLE, BLAS_PREC_DOUBLE, BLAS_PREC_INDIGENOUS,
     $           BLAS_PREC_EXTRA
      PARAMETER ( BLAS_PREC_SINGLE = 211, BLAS_PREC_DOUBLE = 212,
     $     BLAS_PREC_INDIGENOUS = 213, BLAS_PREC_EXTRA = 214 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. Executable Statements ..
      IF( LSAME( PREC, 'S' ) ) THEN
         ILAPREC = BLAS_PREC_SINGLE
      ELSE IF( LSAME( PREC, 'D' ) ) THEN
         ILAPREC = BLAS_PREC_DOUBLE
      ELSE IF( LSAME( PREC, 'I' ) ) THEN
         ILAPREC = BLAS_PREC_INDIGENOUS
      ELSE IF( LSAME( PREC, 'X' ) .OR. LSAME( PREC, 'E' ) ) THEN
         ILAPREC = BLAS_PREC_EXTRA
      ELSE
         ILAPREC = -1
      END IF
      RETURN
*
*     End of ILAPREC
*
      END
*> \brief \b ILATRANS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ILATRANS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ilatrans.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ilatrans.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ilatrans.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       INTEGER FUNCTION ILATRANS( TRANS )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANS
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> This subroutine translates from a character string specifying a
*> transposition operation to the relevant BLAST-specified integer
*> constant.
*>
*> ILATRANS returns an INTEGER.  If ILATRANS < 0, then the input is not
*> a character indicating a transposition operator.  Otherwise ILATRANS
*> returns the constant value corresponding to TRANS.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ilatrans
*
*  =====================================================================
      INTEGER FUNCTION ILATRANS( TRANS )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER BLAS_NO_TRANS, BLAS_TRANS, BLAS_CONJ_TRANS
      PARAMETER ( BLAS_NO_TRANS = 111, BLAS_TRANS = 112,
     $     BLAS_CONJ_TRANS = 113 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. Executable Statements ..
      IF( LSAME( TRANS, 'N' ) ) THEN
         ILATRANS = BLAS_NO_TRANS
      ELSE IF( LSAME( TRANS, 'T' ) ) THEN
         ILATRANS = BLAS_TRANS
      ELSE IF( LSAME( TRANS, 'C' ) ) THEN
         ILATRANS = BLAS_CONJ_TRANS
      ELSE
         ILATRANS = -1
      END IF
      RETURN
*
*     End of ILATRANS
*
      END
*> \brief \b ILAVER returns the LAPACK version.
**
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*  Definition:
*  ===========
*
*     SUBROUTINE ILAVER( VERS_MAJOR, VERS_MINOR, VERS_PATCH )
*
*     INTEGER VERS_MAJOR, VERS_MINOR, VERS_PATCH
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>  This subroutine returns the LAPACK version.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*>  \param[out] VERS_MAJOR
*>      VERS_MAJOR is INTEGER
*>      return the lapack major version
*>
*>  \param[out] VERS_MINOR
*>      VERS_MINOR is INTEGER
*>      return the lapack minor version from the major version
*>
*>  \param[out] VERS_PATCH
*>      VERS_PATCH is INTEGER
*>      return the lapack patch version from the minor version
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ilaver
*
*  =====================================================================
      SUBROUTINE ILAVER( VERS_MAJOR, VERS_MINOR, VERS_PATCH )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*  =====================================================================
*
      INTEGER VERS_MAJOR, VERS_MINOR, VERS_PATCH
*  =====================================================================
      VERS_MAJOR = 3
      VERS_MINOR = 12
      VERS_PATCH = 1
*  =====================================================================
*
      RETURN
      END
# 1 "SRC/iparam2stage.F"
# 1 "<built-in>" 1
# 1 "<built-in>" 3
# 423 "<built-in>" 3
# 1 "<command line>" 1
# 1 "<built-in>" 2
# 1 "SRC/iparam2stage.F" 2
*> \brief \b IPARAM2STAGE
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download IPARAM2STAGE + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/iparam2stage.F">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/iparam2stage.F">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/iparam2stage.F">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       INTEGER FUNCTION IPARAM2STAGE( ISPEC, NAME, OPTS,
*                                    NI, NBI, IBI, NXI )
*       #if defined(_OPENMP)
*           use omp_lib
*       #endif
*       IMPLICIT NONE
*
*       .. Scalar Arguments ..
*       CHARACTER*( * )    NAME, OPTS
*       INTEGER            ISPEC, NI, NBI, IBI, NXI
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>      This program sets problem and machine dependent parameters
*>      useful for xHETRD_2STAGE, xHETRD_HE2HB, xHETRD_HB2ST,
*>      xGEBRD_2STAGE, xGEBRD_GE2GB, xGEBRD_GB2BD
*>      and related subroutines for eigenvalue problems.
*>      It is called whenever ILAENV is called with 17 <= ISPEC <= 21.
*>      It is called whenever ILAENV2STAGE is called with 1 <= ISPEC <= 5
*>      with a direct conversion ISPEC + 16.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ISPEC
*> \verbatim
*>          ISPEC is integer scalar
*>              ISPEC specifies which tunable parameter IPARAM2STAGE should
*>              return.
*>
*>              ISPEC=17: the optimal blocksize nb for the reduction to
*>                        BAND
*>
*>              ISPEC=18: the optimal blocksize ib for the eigenvectors
*>                        singular vectors update routine
*>
*>              ISPEC=19: The length of the array that store the Housholder
*>                        representation for the second stage
*>                        Band to Tridiagonal or Bidiagonal
*>
*>              ISPEC=20: The workspace needed for the routine in input.
*>
*>              ISPEC=21: For future release.
*> \endverbatim
*>
*> \param[in] NAME
*> \verbatim
*>          NAME is character string
*>               Name of the calling subroutine
*> \endverbatim
*>
*> \param[in] OPTS
*> \verbatim
*>          OPTS is CHARACTER*(*)
*>          The character options to the subroutine NAME, concatenated
*>          into a single character string.  For example, UPLO = 'U',
*>          TRANS = 'T', and DIAG = 'N' for a triangular routine would
*>          be specified as OPTS = 'UTN'.
*> \endverbatim
*>
*> \param[in] NI
*> \verbatim
*>          NI is INTEGER which is the size of the matrix
*> \endverbatim
*>
*> \param[in] NBI
*> \verbatim
*>          NBI is INTEGER which is the used in the reduction,
*>          (e.g., the size of the band), needed to compute workspace
*>          and LHOUS2.
*> \endverbatim
*>
*> \param[in] IBI
*> \verbatim
*>          IBI is INTEGER which represent the IB of the reduction,
*>          needed to compute workspace and LHOUS2.
*> \endverbatim
*>
*> \param[in] NXI
*> \verbatim
*>          NXI is INTEGER needed in the future release.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup iparam2stage
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  Implemented by Azzam Haidar.
*>
*>  All detail are available on technical report, SC11, SC13 papers.
*>
*>  Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
*>  Parallel reduction to condensed forms for symmetric eigenvalue problems
*>  using aggregated fine-grained and memory-aware kernels. In Proceedings
*>  of 2011 International Conference for High Performance Computing,
*>  Networking, Storage and Analysis (SC '11), New York, NY, USA,
*>  Article 8 , 11 pages.
*>  http://doi.acm.org/10.1145/2063384.2063394
*>
*>  A. Haidar, J. Kurzak, P. Luszczek, 2013.
*>  An improved parallel singular value algorithm and its implementation
*>  for multicore hardware, In Proceedings of 2013 International Conference
*>  for High Performance Computing, Networking, Storage and Analysis (SC '13).
*>  Denver, Colorado, USA, 2013.
*>  Article 90, 12 pages.
*>  http://doi.acm.org/10.1145/2503210.2503292
*>
*>  A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
*>  A novel hybrid CPU-GPU generalized eigensolver for electronic structure
*>  calculations based on fine-grained memory aware tasks.
*>  International Journal of High Performance Computing Applications.
*>  Volume 28 Issue 2, Pages 196-209, May 2014.
*>  http://hpc.sagepub.com/content/28/2/196
*>
*> \endverbatim
*>
*  =====================================================================
      INTEGER FUNCTION IPARAM2STAGE( ISPEC, NAME, OPTS,
     $                              NI, NBI, IBI, NXI )



      IMPLICIT NONE
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER*( * )    NAME, OPTS
      INTEGER            ISPEC, NI, NBI, IBI, NXI
*
*  ================================================================
*     ..
*     .. Local Scalars ..
      INTEGER            I, IC, IZ, KD, IB, LHOUS, LWORK, NTHREADS,
     $                   FACTOPTNB, QROPTNB, LQOPTNB
      LOGICAL            RPREC, CPREC
      CHARACTER          PREC*1, ALGO*3, STAG*5, SUBNAM*12, VECT*1
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          CHAR, ICHAR, MAX
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      LOGICAL            LSAME
      EXTERNAL           ILAENV, LSAME
*     ..
*     .. Executable Statements ..
*
*     Invalid value for ISPEC
*
      IF( (ISPEC.LT.17).OR.(ISPEC.GT.21) ) THEN
          IPARAM2STAGE = -1
          RETURN
      ENDIF
*
*     Get the number of threads
*
      NTHREADS = 1





*      WRITE(*,*) 'IPARAM VOICI NTHREADS ISPEC ',NTHREADS, ISPEC
*
      IF( ISPEC .NE. 19 ) THEN
*
*        Convert NAME to upper case if the first character is lower case.
*
         IPARAM2STAGE = -1
         SUBNAM = NAME
         IC = ICHAR( SUBNAM( 1: 1 ) )
         IZ = ICHAR( 'Z' )
         IF( IZ.EQ.90 .OR. IZ.EQ.122 ) THEN
*
*           ASCII character set
*
            IF( IC.GE.97 .AND. IC.LE.122 ) THEN
               SUBNAM( 1: 1 ) = CHAR( IC-32 )
               DO 100 I = 2, 12
                  IC = ICHAR( SUBNAM( I: I ) )
                  IF( IC.GE.97 .AND. IC.LE.122 )
     $               SUBNAM( I: I ) = CHAR( IC-32 )
  100          CONTINUE
            END IF
*
         ELSE IF( IZ.EQ.233 .OR. IZ.EQ.169 ) THEN
*
*           EBCDIC character set
*
            IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
     $          ( IC.GE.145 .AND. IC.LE.153 ) .OR.
     $          ( IC.GE.162 .AND. IC.LE.169 ) ) THEN
               SUBNAM( 1: 1 ) = CHAR( IC+64 )
               DO 110 I = 2, 12
                  IC = ICHAR( SUBNAM( I: I ) )
                  IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
     $                ( IC.GE.145 .AND. IC.LE.153 ) .OR.
     $                ( IC.GE.162 .AND. IC.LE.169 ) )SUBNAM( I:
     $                I ) = CHAR( IC+64 )
  110          CONTINUE
            END IF
*
         ELSE IF( IZ.EQ.218 .OR. IZ.EQ.250 ) THEN
*
*           Prime machines:  ASCII+128
*
            IF( IC.GE.225 .AND. IC.LE.250 ) THEN
               SUBNAM( 1: 1 ) = CHAR( IC-32 )
               DO 120 I = 2, 12
                 IC = ICHAR( SUBNAM( I: I ) )
                 IF( IC.GE.225 .AND. IC.LE.250 )
     $             SUBNAM( I: I ) = CHAR( IC-32 )
  120          CONTINUE
            END IF
         END IF
*
         PREC  = SUBNAM( 1: 1 )
         ALGO  = SUBNAM( 4: 6 )
         STAG  = SUBNAM( 8:12 )
         RPREC = PREC.EQ.'S' .OR. PREC.EQ.'D'
         CPREC = PREC.EQ.'C' .OR. PREC.EQ.'Z'
*
*        Invalid value for PRECISION
*
         IF( .NOT.( RPREC .OR. CPREC ) ) THEN
             IPARAM2STAGE = -1
             RETURN
         ENDIF
      ENDIF
*      WRITE(*,*),'RPREC,CPREC ',RPREC,CPREC,
*     $           '   ALGO ',ALGO,'    STAGE ',STAG
*
*
      IF (( ISPEC .EQ. 17 ) .OR. ( ISPEC .EQ. 18 )) THEN
*
*     ISPEC = 17, 18:  block size KD, IB
*     Could be also dependent from N but for now it
*     depend only on sequential or parallel
*
         IF( NTHREADS.GT.4 ) THEN
            IF( CPREC ) THEN
               KD = 128
               IB = 32
            ELSE
               KD = 160
               IB = 40
            ENDIF
         ELSE IF( NTHREADS.GT.1 ) THEN
            IF( CPREC ) THEN
               KD = 64
               IB = 32
            ELSE
               KD = 64
               IB = 32
            ENDIF
         ELSE
            IF( CPREC ) THEN
               KD = 16
               IB = 16
            ELSE
               KD = 32
               IB = 16
            ENDIF
         ENDIF
         IF( ISPEC.EQ.17 ) IPARAM2STAGE = KD
         IF( ISPEC.EQ.18 ) IPARAM2STAGE = IB
*
      ELSE IF ( ISPEC .EQ. 19 ) THEN
*
*     ISPEC = 19:
*     LHOUS length of the Houselholder representation
*     matrix (V,T) of the second stage. should be >= 1.
*
*     Will add the VECT OPTION HERE next release
         VECT  = OPTS(1:1)
         IF( LSAME( VECT, 'N' ) ) THEN
            LHOUS = MAX( 1, 4*NI )
         ELSE
*           This is not correct, it need to call the ALGO and the stage2
            LHOUS = MAX( 1, 4*NI ) + IBI
         ENDIF
         IF( LHOUS.GE.0 ) THEN
            IPARAM2STAGE = LHOUS
         ELSE
            IPARAM2STAGE = -1
         ENDIF
*
      ELSE IF ( ISPEC .EQ. 20 ) THEN
*
*     ISPEC = 20: (21 for future use)
*     LWORK length of the workspace for
*     either or both stages for TRD and BRD. should be >= 1.
*     TRD:
*     TRD_stage 1: = LT + LW + LS1 + LS2
*                  = LDT*KD + N*KD + N*MAX(KD,FACTOPTNB) + LDS2*KD
*                    where LDT=LDS2=KD
*                  = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD
*     TRD_stage 2: = (2NB+1)*N + KD*NTHREADS
*     TRD_both   : = max(stage1,stage2) + AB ( AB=(KD+1)*N )
*                  = N*KD + N*max(KD+1,FACTOPTNB)
*                    + max(2*KD*KD, KD*NTHREADS)
*                    + (KD+1)*N
         LWORK        = -1
         SUBNAM(1:1)  = PREC
         SUBNAM(2:6)  = 'GEQRF'
         QROPTNB      = ILAENV( 1, SUBNAM, ' ', NI, NBI, -1, -1 )
         SUBNAM(2:6)  = 'GELQF'
         LQOPTNB      = ILAENV( 1, SUBNAM, ' ', NBI, NI, -1, -1 )
*        Could be QR or LQ for TRD and the max for BRD
         FACTOPTNB    = MAX(QROPTNB, LQOPTNB)
         IF( ALGO.EQ.'TRD' ) THEN
            IF( STAG.EQ.'2STAG' ) THEN
               LWORK = NI*NBI + NI*MAX(NBI+1,FACTOPTNB)
     $              + MAX(2*NBI*NBI, NBI*NTHREADS)
     $              + (NBI+1)*NI
            ELSE IF( (STAG.EQ.'HE2HB').OR.(STAG.EQ.'SY2SB') ) THEN
               LWORK = NI*NBI + NI*MAX(NBI,FACTOPTNB) + 2*NBI*NBI
            ELSE IF( (STAG.EQ.'HB2ST').OR.(STAG.EQ.'SB2ST') ) THEN
               LWORK = (2*NBI+1)*NI + NBI*NTHREADS
            ENDIF
         ELSE IF( ALGO.EQ.'BRD' ) THEN
            IF( STAG.EQ.'2STAG' ) THEN
               LWORK = 2*NI*NBI + NI*MAX(NBI+1,FACTOPTNB)
     $              + MAX(2*NBI*NBI, NBI*NTHREADS)
     $              + (NBI+1)*NI
            ELSE IF( STAG.EQ.'GE2GB' ) THEN
               LWORK = NI*NBI + NI*MAX(NBI,FACTOPTNB) + 2*NBI*NBI
            ELSE IF( STAG.EQ.'GB2BD' ) THEN
               LWORK = (3*NBI+1)*NI + NBI*NTHREADS
            ENDIF
         ENDIF
         LWORK = MAX ( 1, LWORK )

         IF( LWORK.GT.0 ) THEN
            IPARAM2STAGE = LWORK
         ELSE
            IPARAM2STAGE = -1
         ENDIF
*
      ELSE IF ( ISPEC .EQ. 21 ) THEN
*
*     ISPEC = 21 for future use
         IPARAM2STAGE = NXI
      ENDIF
*
*     ==== End of IPARAM2STAGE ====
*
      END

*> \brief \b IPARMQ
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download IPARMQ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/iparmq.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/iparmq.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/iparmq.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       INTEGER FUNCTION IPARMQ( ISPEC, NAME, OPTS, N, ILO, IHI, LWORK )
*
*       .. Scalar Arguments ..
*       INTEGER            IHI, ILO, ISPEC, LWORK, N
*       CHARACTER          NAME*( * ), OPTS*( * )
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>      This program sets problem and machine dependent parameters
*>      useful for xHSEQR and related subroutines for eigenvalue
*>      problems. It is called whenever
*>      IPARMQ is called with 12 <= ISPEC <= 16
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ISPEC
*> \verbatim
*>          ISPEC is INTEGER
*>              ISPEC specifies which tunable parameter IPARMQ should
*>              return.
*>
*>              ISPEC=12: (INMIN)  Matrices of order nmin or less
*>                        are sent directly to xLAHQR, the implicit
*>                        double shift QR algorithm.  NMIN must be
*>                        at least 11.
*>
*>              ISPEC=13: (INWIN)  Size of the deflation window.
*>                        This is best set greater than or equal to
*>                        the number of simultaneous shifts NS.
*>                        Larger matrices benefit from larger deflation
*>                        windows.
*>
*>              ISPEC=14: (INIBL) Determines when to stop nibbling and
*>                        invest in an (expensive) multi-shift QR sweep.
*>                        If the aggressive early deflation subroutine
*>                        finds LD converged eigenvalues from an order
*>                        NW deflation window and LD > (NW*NIBBLE)/100,
*>                        then the next QR sweep is skipped and early
*>                        deflation is applied immediately to the
*>                        remaining active diagonal block.  Setting
*>                        IPARMQ(ISPEC=14) = 0 causes TTQRE to skip a
*>                        multi-shift QR sweep whenever early deflation
*>                        finds a converged eigenvalue.  Setting
*>                        IPARMQ(ISPEC=14) greater than or equal to 100
*>                        prevents TTQRE from skipping a multi-shift
*>                        QR sweep.
*>
*>              ISPEC=15: (NSHFTS) The number of simultaneous shifts in
*>                        a multi-shift QR iteration.
*>
*>              ISPEC=16: (IACC22) IPARMQ is set to 0, 1 or 2 with the
*>                        following meanings.
*>                        0:  During the multi-shift QR/QZ sweep,
*>                            blocked eigenvalue reordering, blocked
*>                            Hessenberg-triangular reduction,
*>                            reflections and/or rotations are not
*>                            accumulated when updating the
*>                            far-from-diagonal matrix entries.
*>                        1:  During the multi-shift QR/QZ sweep,
*>                            blocked eigenvalue reordering, blocked
*>                            Hessenberg-triangular reduction,
*>                            reflections and/or rotations are
*>                            accumulated, and matrix-matrix
*>                            multiplication is used to update the
*>                            far-from-diagonal matrix entries.
*>                        2:  During the multi-shift QR/QZ sweep,
*>                            blocked eigenvalue reordering, blocked
*>                            Hessenberg-triangular reduction,
*>                            reflections and/or rotations are
*>                            accumulated, and 2-by-2 block structure
*>                            is exploited during matrix-matrix
*>                            multiplies.
*>                        (If xTRMM is slower than xGEMM, then
*>                        IPARMQ(ISPEC=16)=1 may be more efficient than
*>                        IPARMQ(ISPEC=16)=2 despite the greater level of
*>                        arithmetic work implied by the latter choice.)
*>
*>              ISPEC=17: (ICOST) An estimate of the relative cost of flops
*>                        within the near-the-diagonal shift chase compared
*>                        to flops within the BLAS calls of a QZ sweep.
*> \endverbatim
*>
*> \param[in] NAME
*> \verbatim
*>          NAME is CHARACTER string
*>               Name of the calling subroutine
*> \endverbatim
*>
*> \param[in] OPTS
*> \verbatim
*>          OPTS is CHARACTER string
*>               This is a concatenation of the string arguments to
*>               TTQRE.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>               N is the order of the Hessenberg matrix H.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*>          ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*>          IHI is INTEGER
*>               It is assumed that H is already upper triangular
*>               in rows and columns 1:ILO-1 and IHI+1:N.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>               The amount of workspace available.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup iparmq
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>       Little is known about how best to choose these parameters.
*>       It is possible to use different values of the parameters
*>       for each of CHSEQR, DHSEQR, SHSEQR and ZHSEQR.
*>
*>       It is probably best to choose different parameters for
*>       different matrices and different parameters at different
*>       times during the iteration, but this has not been
*>       implemented --- yet.
*>
*>
*>       The best choices of most of the parameters depend
*>       in an ill-understood way on the relative execution
*>       rate of xLAQR3 and xLAQR5 and on the nature of each
*>       particular eigenvalue problem.  Experiment may be the
*>       only practical way to determine which choices are most
*>       effective.
*>
*>       Following is a list of default values supplied by IPARMQ.
*>       These defaults may be adjusted in order to attain better
*>       performance in any particular computational environment.
*>
*>       IPARMQ(ISPEC=12) The xLAHQR vs xLAQR0 crossover point.
*>                        Default: 75. (Must be at least 11.)
*>
*>       IPARMQ(ISPEC=13) Recommended deflation window size.
*>                        This depends on ILO, IHI and NS, the
*>                        number of simultaneous shifts returned
*>                        by IPARMQ(ISPEC=15).  The default for
*>                        (IHI-ILO+1) <= 500 is NS.  The default
*>                        for (IHI-ILO+1) > 500 is 3*NS/2.
*>
*>       IPARMQ(ISPEC=14) Nibble crossover point.  Default: 14.
*>
*>       IPARMQ(ISPEC=15) Number of simultaneous shifts, NS.
*>                        a multi-shift QR iteration.
*>
*>                        If IHI-ILO+1 is ...
*>
*>                        greater than      ...but less    ... the
*>                        or equal to ...      than        default is
*>
*>                                0               30       NS =   2+
*>                               30               60       NS =   4+
*>                               60              150       NS =  10
*>                              150              590       NS =  **
*>                              590             3000       NS =  64
*>                             3000             6000       NS = 128
*>                             6000             infinity   NS = 256
*>
*>                    (+)  By default matrices of this order are
*>                         passed to the implicit double shift routine
*>                         xLAHQR.  See IPARMQ(ISPEC=12) above.   These
*>                         values of NS are used only in case of a rare
*>                         xLAHQR failure.
*>
*>                    (**) The asterisks (**) indicate an ad-hoc
*>                         function increasing from 10 to 64.
*>
*>       IPARMQ(ISPEC=16) Select structured matrix multiply.
*>                        (See ISPEC=16 above for details.)
*>                        Default: 3.
*>
*>       IPARMQ(ISPEC=17) Relative cost heuristic for blocksize selection.
*>                        Expressed as a percentage.
*>                        Default: 10.
*> \endverbatim
*>
*  =====================================================================
      INTEGER FUNCTION IPARMQ( ISPEC, NAME, OPTS, N, ILO, IHI,
     $                         LWORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            IHI, ILO, ISPEC, LWORK, N
      CHARACTER          NAME*( * ), OPTS*( * )
*
*  ================================================================
*     .. Parameters ..
      INTEGER            INMIN, INWIN, INIBL, ISHFTS, IACC22, ICOST
      PARAMETER          ( INMIN = 12, INWIN = 13, INIBL = 14,
     $                   ISHFTS = 15, IACC22 = 16, ICOST = 17 )
      INTEGER            NMIN, K22MIN, KACMIN, NIBBLE, KNWSWP, RCOST
      PARAMETER          ( NMIN = 75, K22MIN = 14, KACMIN = 14,
     $                   NIBBLE = 14, KNWSWP = 500, RCOST = 10 )
      REAL               TWO
      PARAMETER          ( TWO = 2.0 )
*     ..
*     .. Local Scalars ..
      INTEGER            NH, NS
      INTEGER            I, IC, IZ
      CHARACTER          SUBNAM*6
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          LOG, MAX, MOD, NINT, REAL
*     ..
*     .. Executable Statements ..
      IF( ( ISPEC.EQ.ISHFTS ) .OR. ( ISPEC.EQ.INWIN ) .OR.
     $    ( ISPEC.EQ.IACC22 ) ) THEN
*
*        ==== Set the number simultaneous shifts ====
*
         NH = IHI - ILO + 1
         NS = 2
         IF( NH.GE.30 )
     $      NS = 4
         IF( NH.GE.60 )
     $      NS = 10
         IF( NH.GE.150 )
     $      NS = MAX( 10, NH / NINT( LOG( REAL( NH ) ) / LOG( TWO ) ) )
         IF( NH.GE.590 )
     $      NS = 64
         IF( NH.GE.3000 )
     $      NS = 128
         IF( NH.GE.6000 )
     $      NS = 256
         NS = MAX( 2, NS-MOD( NS, 2 ) )
      END IF
*
      IF( ISPEC.EQ.INMIN ) THEN
*
*
*        ===== Matrices of order smaller than NMIN get sent
*        .     to xLAHQR, the classic double shift algorithm.
*        .     This must be at least 11. ====
*
         IPARMQ = NMIN
*
      ELSE IF( ISPEC.EQ.INIBL ) THEN
*
*        ==== INIBL: skip a multi-shift qr iteration and
*        .    whenever aggressive early deflation finds
*        .    at least (NIBBLE*(window size)/100) deflations. ====
*
         IPARMQ = NIBBLE
*
      ELSE IF( ISPEC.EQ.ISHFTS ) THEN
*
*        ==== NSHFTS: The number of simultaneous shifts =====
*
         IPARMQ = NS
*
      ELSE IF( ISPEC.EQ.INWIN ) THEN
*
*        ==== NW: deflation window size.  ====
*
         IF( NH.LE.KNWSWP ) THEN
            IPARMQ = NS
         ELSE
            IPARMQ = 3*NS / 2
         END IF
*
      ELSE IF( ISPEC.EQ.IACC22 ) THEN
*
*        ==== IACC22: Whether to accumulate reflections
*        .     before updating the far-from-diagonal elements
*        .     and whether to use 2-by-2 block structure while
*        .     doing it.  A small amount of work could be saved
*        .     by making this choice dependent also upon the
*        .     NH=IHI-ILO+1.
*
*
*        Convert NAME to upper case if the first character is lower case.
*
         IPARMQ = 0
         SUBNAM = NAME
         IC = ICHAR( SUBNAM( 1: 1 ) )
         IZ = ICHAR( 'Z' )
         IF( IZ.EQ.90 .OR. IZ.EQ.122 ) THEN
*
*           ASCII character set
*
            IF( IC.GE.97 .AND. IC.LE.122 ) THEN
               SUBNAM( 1: 1 ) = CHAR( IC-32 )
               DO I = 2, 6
                  IC = ICHAR( SUBNAM( I: I ) )
                  IF( IC.GE.97 .AND. IC.LE.122 )
     $               SUBNAM( I: I ) = CHAR( IC-32 )
               END DO
            END IF
*
         ELSE IF( IZ.EQ.233 .OR. IZ.EQ.169 ) THEN
*
*           EBCDIC character set
*
            IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
     $          ( IC.GE.145 .AND. IC.LE.153 ) .OR.
     $          ( IC.GE.162 .AND. IC.LE.169 ) ) THEN
               SUBNAM( 1: 1 ) = CHAR( IC+64 )
               DO I = 2, 6
                  IC = ICHAR( SUBNAM( I: I ) )
                  IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
     $                ( IC.GE.145 .AND. IC.LE.153 ) .OR.
     $                ( IC.GE.162 .AND. IC.LE.169 ) )SUBNAM( I:
     $                I ) = CHAR( IC+64 )
               END DO
            END IF
*
         ELSE IF( IZ.EQ.218 .OR. IZ.EQ.250 ) THEN
*
*           Prime machines:  ASCII+128
*
            IF( IC.GE.225 .AND. IC.LE.250 ) THEN
               SUBNAM( 1: 1 ) = CHAR( IC-32 )
               DO I = 2, 6
                  IC = ICHAR( SUBNAM( I: I ) )
                  IF( IC.GE.225 .AND. IC.LE.250 )
     $               SUBNAM( I: I ) = CHAR( IC-32 )
               END DO
            END IF
         END IF
*
         IF( SUBNAM( 2:6 ).EQ.'GGHRD' .OR.
     $       SUBNAM( 2:6 ).EQ.'GGHD3' ) THEN
            IPARMQ = 1
            IF( NH.GE.K22MIN )
     $         IPARMQ = 2
         ELSE IF ( SUBNAM( 4:6 ).EQ.'EXC' ) THEN
            IF( NH.GE.KACMIN )
     $         IPARMQ = 1
            IF( NH.GE.K22MIN )
     $         IPARMQ = 2
         ELSE IF ( SUBNAM( 2:6 ).EQ.'HSEQR' .OR.
     $             SUBNAM( 2:5 ).EQ.'LAQR' ) THEN
            IF( NS.GE.KACMIN )
     $         IPARMQ = 1
            IF( NS.GE.K22MIN )
     $         IPARMQ = 2
         END IF
*
      ELSE IF( ISPEC.EQ.ICOST ) THEN
*
*        === Relative cost of near-the-diagonal chase vs
*            BLAS updates ===
*
         IPARMQ = RCOST
      ELSE
*        ===== invalid value of ispec =====
         IPARMQ = -1
*
      END IF
*
*     ==== End of IPARMQ ====
*
      END
*> \brief \b DROUNDUP_LWORK
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*  Definition:
*  ===========
*
*      DOUBLE PRECISION FUNCTION DROUNDUP_LWORK( LWORK )
*
*     .. Scalar Arguments ..
*      INTEGER          LWORK
*     ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DROUNDUP_LWORK deals with a subtle bug with returning LWORK as a Float.
*> This routine guarantees it is rounded up instead of down by
*> multiplying LWORK by 1+eps when it is necessary, where eps is the relative machine precision.
*> E.g.,
*>
*>        float( 9007199254740993            ) == 9007199254740992
*>        float( 9007199254740993 ) * (1.+eps) == 9007199254740994
*>
*> \return DROUNDUP_LWORK
*> \verbatim
*>         DROUNDUP_LWORK >= LWORK.
*>         DROUNDUP_LWORK is guaranteed to have zero decimal part.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] LWORK Workspace size.
*
*  Authors:
*  ========
*
*> \author Weslley Pereira, University of Colorado Denver, USA
*
*> \ingroup roundup_lwork
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>  This routine was inspired in the method `magma_zmake_lwork` from MAGMA.
*>  \see https://bitbucket.org/icl/magma/src/master/control/magma_zauxiliary.cpp
*> \endverbatim
*
*  =====================================================================
      DOUBLE PRECISION FUNCTION DROUNDUP_LWORK( LWORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER          LWORK
*     ..
*
* =====================================================================
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC         EPSILON, DBLE, INT
*     ..
*     .. Executable Statements ..
*     ..
      DROUNDUP_LWORK = DBLE( LWORK )
*
      IF( INT( DROUNDUP_LWORK ) .LT. LWORK ) THEN
*         Force round up of LWORK
          DROUNDUP_LWORK = DROUNDUP_LWORK *
     $                     ( 1.0D+0 + EPSILON(0.0D+0) )
      ENDIF
*
      RETURN
*
*     End of DROUNDUP_LWORK
*
      END
