EfficientMaxEigenpair is a package for computing the maximal eigenpair of the matrices with non-negative off-diagonal elements (or as a dual, the minimal eigenvalue of the matrices with non-positive off-diagonal elements). This vignette is a simple guide to using the package. All the algorithms and examples provided in this vignette are available in the paper “Efficient initials for computing maximal eigenpair” by Mu-Fa Chen. The paper Chen (2016) is now included in the Vol 4, in the middle of the website:
\begin{center}
\url{http://math0.bnu.edu.cn/~chenmf/}
\end{center}
Let us install and require the package EfficientMaxEigenpair first.
require(EfficientMaxEigenpair)
Before moving to the details, let us illustrate the algorithm by two typical examples.
The matrices we considered for the tridiagonal case are in the form of \begin{center} \(Q=\begin{bmatrix} -1 & 1^2 & 0 & 0 & \dots & 0 \ 1^2 & -5 & 2^2 & 0 & \dots & 0 \ 0 & 2^2 & -13 & 3^2 & \dots & 0 \ \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \ 0 & \dots & \dots & (n-2)^2 & -(n-2)^2-(n-1)^2 & (n-1)^2 \ 0 & \dots & \dots & 0 & (n-1)^2 & -(n-1)^2-n^2 \end{bmatrix}\), \end{center} where \(n\) is the dimension of the matrix. For the infinite case, we have known that the maximal eigenvalue is \(-¼\). We now want to calulate the maximal eigenvalues of matrices in different dimensions by the Rayleigh quotient iteration.
# Define the dimension of matrix to be 100.
nn = 100
# Generate the corresponding tridiagonal matrix.
a = c(1:(nn - 1))^2
b = c(1:(nn - 1))^2
c = rep(0, length(a) + 1)
c[length(a) + 1] = nn^2
# Output the corresponding convergence maximal eigenpairs of the tridiagonal
# matrix.
eigenpair = eff.ini.maxeig.tri(a, b, c, xi = 7/8, improved = T)
The output here are the approximations \(z\) of the maximal eigenvalue of matrix \(Q\). However, in what follow, we often write -\(z\) instead of \(z\) for simplicity.
# The approximating sequence z_0, z_1, z_2 of the maximal eigenvalue.
eigenpair$z
## [1] -0.387333 -0.376393 -0.376383
The following figure shows the corresponding eigenvetors \(v\) up to a sign. To be consistent with the paper's notation, we accept the convention that the first element in vector start with \(v[0]\). From the first figure below, one sees that the eigenvectors \(-v[1]\) and \(v[2]\) are nearly the same. However, if we look at the last parts of the curves, it is clear that \(-v[1]\) and \(v[2]\) are actually different.
In the following, because of the space limitation, we do not report the eigenvector results and write -\(z\) instead of \(z\) for simplicity.
# Define the dimension of each matrix.
nn = c(100, 500, 1000, 5000)
zmat = matrix(0, length(nn), 4)
zmat[, 1] = nn
for (i in 1:length(nn)) {
# Generate the corresponding tridiagonal matrix for different dimensions.
a = c(1:(nn[i] - 1))^2
b = c(1:(nn[i] - 1))^2
c = rep(0, length(a) + 1)
c[length(a) + 1] = nn[i]^2
# Output the corresponding dual case results, i.e, the minimal eigenvalue of
# the tridiagonal matrix -Q with non-posivie off-diagonal elements.
zmat[i, -1] = -eff.ini.maxeig.tri(a, b, c, xi = 7/8, improved = T)$z[1:3]
}
colnames(zmat) = c("Dimension", "-z_0", "-z_1", "-z_2")
zmat
# The approximating sequence -z_0, -z_1, -z_2 of the maximal eigenvalue.
## Dimension -z_0 -z_1 -z_2
## [1,] 100 0.387333 0.376393 0.376383
## [2,] 500 0.349147 0.338342 0.338329
## [3,] 1000 0.338027 0.327254 0.327240
## [4,] 5000 0.319895 0.308550 0.308529
The matrix we considered for this general case is
# Generate the general matrix A
A = matrix(c(1, 3, 9, 5, 2, 14, 10, 6, 0, 11, 11, 7, 0, 0, 1, 8), 4, 4)
A
## [,1] [,2] [,3] [,4]
## [1,] 1 2 0 0
## [2,] 3 14 11 0
## [3,] 9 10 11 1
## [4,] 5 6 7 8
This matrix has complex eigenvalues:
# Calculate the corresponding eigenvalues of matrix A
eigen(A)$values
## [1] 24.029261+0.000000i 7.722536+0.000000i 1.124102+2.405217i
## [4] 1.124102-2.405217i
We have to be carefull to choice the initial eigenpairs \(z_0\) and \(v[0]\) for this matrix A. The following is one counterexample with dangerous initials. The approximating sequence converges to the next to maximal eigenvalue rather than the maximal eigenvalue.
# Calulate the approximating sequence of maximal eigenvalue by the Rayleigh
# quotient iteration.
eff.ini.maxeig.general(A, z0 = "Auto", digit.thresh = 4)$z
## [1] "m=31"
## [1] 13.7532 7.1098 7.6088 7.7219 7.7225
Here we fixed the problem by using improved algorithm and safer initials which will be illustrated in detail of the algorithm part.
# Calulate the approximating sequence of maximal eigenvalue by the Rayleigh
# quotient iteration.
eff.ini.maxeig.general(A, improved = T, xi = 0.65, digit.thresh = 4)$z
## [1] "m=31"
## [1] 24.1681 23.6847 24.0369 24.0293
Having illustrated the examples, we now introduce the package in details. It offers four main algorithms:
eff.ini.maxeig.tri(): Calculate the maximal eigenpair for the tridiagonal matrix. The coefficient xi is used in the improved initial \(z_0\) to form the convex combination of \(1/\delta_1\) and \((v_0,Av_0)_\mu\) (in the dual case \((v_0,-Av_0)_\mu\)), it should between 0 and 1. The inner product \((,)_\mu\) here is in the space \(L^{2}(\mu)\). The option improved = F indicates whether the improved algorithm to find the efficient initials are considered. Usually the option improved = T is used when the coefficient xi does not equal to 1. The choice xi=1 is very safe but less effective. When xi is a little away from 1, the algorithm becomes more effective. However, when xi is closed to 0, the algorithm becomes less effective again and is even dangerous. The “best” choice of xi depends on the model. For different models, one may choose different xi. In the present tridiagonal case, we suggest a common choice xi=7/8 (See Example 7 above/below).
digit.thresh = 6 which implies 1e-6 without any special requirement. Same for the following three algorithms and the examples shown in this vignette.eff.ini.maxeig.general(): Calculate the maximal eigenpair for the general matrix with non-negative off-diagonal elements. Let the general matrix \(A=(a_{ij}:i,j\in E)\).
v0_tilde, if v0_tilde=NULL, the one computed in Section 4.2 in the paper Chen (2016) is used. Typically, we use two different initial vectors \(v_0\), the uniformly distributed one (Section 4.1) and the one computed in Section 4.2, below (10). The first one is safer but less effective and we encourage to use the second one.z_0="fixed" corresponding to the choice I, i.e, set \(z_0=\sup_{i\in E}A_{i}\), where \(A_{i}=\sum_{j\in E}a_{ij}\). The option z_0="Auto" corresponding to the choice II, i.e, set \(z_0=v_0^{\ast}Av_{0}\). This simpler choice is used in the other (\(z_k:k\ge 1\)), but it may be dangerous as initial \(z_0\) as illustrated by Example 20 above/below. The option z_0="numeric" corresponding to the choice III, as mentioned in the first algorithm for diagonally dominant matrices.z0numeric is only used when the initial \(z_0\) is considered based on (11) in the paper Chen (2016).improved, xi and digit.thresh which are the same as defined in function eff.ini.maxeig.tri(). The improved algorithm is recommended. However, here we should choose a smaller xi, say 1/3 or 0.65 used in Examples 18-20 below.eff.ini.secondeig.tri(): Calculate the next to maximal eigenpair for the tridiagonal matrix. As mentioned in the cited paper, for simplicity, here we assume that the sums of each row of the input tridiagonal matrix should be 0, i.e, \(A_{i}=0\) for all \(i\in E\). Similarly, there are other options xi and digit.thresh which are the same as defined in function eff.ini.maxeig.tri().
eff.ini.secondeig.general(): Calculate the next to maximal eigenpair for the general conservative matrix where the conservativity of matrix means that the sums of each row are all 0, i.e, \(A_{i}=0\) for all \(i\in E\).
z_0="fixed" corresponding to the approximation to be \(z_0\approx\lambda_0(A_1)\), where \(A_1\) is an auxiliary matrix \(A\)-matrix given in below. The option z_0="Auto" corresponding to the approximation given in Section 6 in the paper Chen (2016). It may be necessary to use z_0="fixed", especially for large matrices.digit.thresh which is the same as defined in function eff.ini.maxeig.tri().There are two auxiliary functions tridia() and ray.quot() where:
tridiag(): Generate tridiagonal matrix \(A\) based on three input vectors.ray.quot(): Rayleigh quotient iteration algorithm for computing the maximal eigenpair of matrix \(A\).The research project is supported in part by the National Natural Science Foundation of China (No. 11131003), the “985'' project from the Ministry of Education in China, and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. The first author acknowledges the assisted one for his hard work in preparing this package in R.
We show most of the examples in the paper Chen (2016) in this section. For a convenient comparison, we keep the same subsection names and examples as the paper Chen (2016).
In this subsection, armed with eff.ini.maxeig.tri(), we can calculate the maximal eigenpair for the tridiagonal matrix. Sometimes we will report the minimal eigenvalue of \(-Q\) by the convention in the paper Chen (2016).
a = c(1:7)^2
b = c(1:7)^2
c = rep(0, length(a) + 1)
c[length(a) + 1] = 8^2
-eff.ini.maxeig.tri(a, b, c, xi = 1)$z
## [1] 0.485985 0.525313 0.525268
In the tridiagonal case, the improved algorithm is presented below Example 10 in Chen (2016).
nn = c(100, 500, 1000, 5000, 7500, 10^4)
for (i in 1:6) {
a = c(1:(nn[i] - 1))^2
b = c(1:(nn[i] - 1))^2
c = rep(0, length(a) + 1)
c[length(a) + 1] = nn[i]^2
print(-eff.ini.maxeig.tri(a, b, c, xi = 1)$z)
print(-eff.ini.maxeig.tri(a, b, c, xi = 7/8, improved = T)$z)
}
## [1] 0.348549 0.376437 0.376383
## [1] 0.387333 0.376393 0.376383
## [1] 0.310195 0.338402 0.338329
## [1] 0.349147 0.338342 0.338329
## [1] 0.299089 0.327320 0.327240
## [1] 0.338027 0.327254 0.327240
## [1] 0.281156 0.308623 0.308529
## [1] 0.319895 0.308550 0.308529
## [1] 0.277865 0.305016 0.304918
## [1] 0.316529 0.304942 0.304918
## [1] 0.275762 0.302660 0.302561
## [1] 0.314370 0.302586 0.302561
a = 14/100
b = 40/100
c = c(-25/100 - 40/100, -12/100 - 14/100)
eff.ini.maxeig.tri(a, b, c, xi = 1)$z
eff.ini.maxeig.tri(a, b, c, xi = 7/8, improved = T)$z
## [1] "Input vector c should be all nonnegative!"
## [1] "m=0.65"
## [1] 0.437923 0.430603 0.430408
## [1] "Input vector c should be all nonnegative!"
## [1] "m=0.65"
## [1] 0.436733 0.430407 0.430408
a = c(sqrt(10), sqrt(130)/11)
b = c(11/sqrt(10), 20 * sqrt(130)/143)
c = c(-1 - 11/sqrt(10), -25/11 - sqrt(10) - 20 * sqrt(130)/143, -8/11 - sqrt(130)/11)
eff.ini.maxeig.tri(a, b, c, xi = 1, digit.thresh = 5)$z
eff.ini.maxeig.tri(a, b, c, xi = 7/8, improved = T, digit.thresh = 5)$z
## [1] "Input vector c should be all nonnegative!"
## [1] "m=7.02965587709025"
## [1] 5.37523 5.23850 5.23607
## [1] "Input vector c should be all nonnegative!"
## [1] "m=7.02965587709025"
## [1] 5.31610 5.23598 5.23607
To get the maximal eigenpair for the general matrix \(Q\), there are several choices of the initial vector \(v_0\) and the initial \(z_0\). In this subsection, we study several combinations of the choices.
A = matrix(c(1, 1, 3, 2, 2, 2, 3, 1, 1), 3, 3)
eff.ini.maxeig.general(A, v0_tilde = rep(1, dim(A)[1]), z0 = "fixed", digit.thresh = 5)$z
## [1] "m=6"
## [1] 6.00000 5.27273 5.23639 5.23607
A = t(matrix(seq(1, 16), 4, 4))
eff.ini.maxeig.general(A, v0_tilde = rep(1, dim(A)[1]), z0 = "fixed", digit.thresh = 4)$z
## [1] "m=58"
## [1] 58.0000 37.3442 36.2674 36.2095 36.2094
A = matrix(c(1, 3, 9, 5, 2, 14, 10, 6, 0, 11, 11, 7, 0, 0, 1, 8), 4, 4)
eff.ini.maxeig.general(A, v0_tilde = rep(1, dim(A)[1]), z0 = "fixed", digit.thresh = 4)$z
## [1] "m=31"
## [1] 31.0000 24.4393 24.0385 24.0293
a = c(1:7)^2
b = c(1:7)^2
c = rep(0, length(a) + 1)
c[length(a) + 1] = 8^2
N = length(a)
Q = tridiag(b, a, -c(b[1] + c[1], a[1:N - 1] + b[2:N] + c[2:N], a[N] + c[N +
1]))
A = 113 * diag(1, (N + 1)) + Q
113 - eff.ini.maxeig.general(A, v0_tilde = rep(1, dim(A)[1]), z0 = "fixed")$z
## [1] "m=113"
## [1] 0.000000 0.602312 0.525463 0.525268
A = matrix(c(1, 1, 3, 2, 2, 2, 3, 1, 1), 3, 3)
eff.ini.maxeig.general(A, v0_tilde = rep(1, dim(A)[1]), z0 = "Auto", digit.thresh = 5)$z
## [1] "m=6"
## [1] 5.33333 5.24183 5.23608 5.23607
A = t(matrix(seq(1, 16), 4, 4))
eff.ini.maxeig.general(A, v0_tilde = rep(1, dim(A)[1]), z0 = "Auto", digit.thresh = 4)$z
## [1] "m=58"
## [1] 34.0000 35.8428 36.2127 36.2094
A = matrix(c(1, 3, 9, 5, 2, 14, 10, 6, 0, 11, 11, 7, 0, 0, 1, 8), 4, 4)
eff.ini.maxeig.general(A, v0_tilde = rep(1, dim(A)[1]), z0 = "Auto", digit.thresh = 4)$z
## [1] "m=31"
## [1] 22.0000 23.7316 24.0317 24.0293
Here we use the combination coefficient xi=7/8 which is a little different with Chen (2016).
A = matrix(c(1, 3, 9, 5, 2, 14, 10, 6, 0, 11, 11, 7, 0, 0, 1, 8), 4, 4)
S = (t(A) + A)/2
N = dim(S)[1]
a = diag(S[-1, -N])
b = diag(S[-N, -1])
c = rep(NA, N)
c[1] = -diag(S)[1] - b[1]
c[2:(N - 1)] = -diag(S)[2:(N - 1)] - b[2:(N - 1)] - a[1:(N - 2)]
c[N] = -diag(S)[N] - a[N - 1]
z0ini = eff.ini.maxeig.tri(a, b, c, xi = 7/8, improved = T)$z[1]
z0ini
eff.ini.maxeig.general(A, v0_tilde = rep(1, dim(A)[1]), z0 = "numeric", z0numeric = 28 -
z0ini, digit.thresh = 4)$z
## [1] "Input vector c should be all nonnegative!"
## [1] "m=27"
## [1] 23.61986
## [1] "m=31"
## [1] 26.6199 24.2643 24.0315 24.0293
A = matrix(c(1, 3, 9, 5, 2, 14, 10, 6, 0, 11, 11, 7, 0, 0, 1, 8), 4, 4)
S = A
N = dim(S)[1]
a = diag(S[-1, -N])
b = diag(S[-N, -1])
c[1] = -diag(S)[1] - b[1]
c[2:(N - 1)] = -diag(S)[2:(N - 1)] - b[2:(N - 1)] - a[1:(N - 2)]
c[N] = -diag(S)[N] - a[N - 1]
z0ini = eff.ini.maxeig.tri(a, b, c, xi = 7/8, improved = T)$z[1]
z0ini
eff.ini.maxeig.general(A, v0_tilde = rep(1, dim(A)[1]), z0 = "numeric", z0numeric = 31 -
z0ini, digit.thresh = 4)$z
## [1] "Input vector c should be all nonnegative!"
## [1] "m=28"
## [1] 23.31942
## [1] "m=31"
## [1] 23.3194 23.9388 24.0295 24.0293
####Efficient initial vector \(v_0\) The improved algorithm is presented at the end of Section 4 in Chen (2016). The outputs at the last line in Examples 18 and 19 are different from those presented in the cited paper, due to the reason that a slight different combination was used there.
A = matrix(c(1, 1, 3, 2, 2, 2, 3, 1, 1), 3, 3)
eff.ini.maxeig.general(A, z0 = "Auto", digit.thresh = 5)$z
eff.ini.maxeig.general(A, improved = T, xi = 1, digit.thresh = 5)$z
eff.ini.maxeig.general(A, improved = T, xi = 1/3, digit.thresh = 5)$z
## [1] "m=6"
## [1] 5.11616 5.23883 5.23607
## [1] "m=6"
## [1] 5.90218 5.21447 5.23617 5.23607
## [1] "m=6"
## [1] 5.34368 5.23217 5.23607
A = t(matrix(seq(1, 16), 4, 4))
eff.ini.maxeig.general(A, z0 = "Auto", digit.thresh = 4)$z
eff.ini.maxeig.general(A, improved = T, xi = 1, digit.thresh = 4)$z
eff.ini.maxeig.general(A, improved = T, xi = 1/3, digit.thresh = 4)$z
## [1] "m=58"
## [1] 34.4924 36.1469 36.2095 36.2094
## [1] "m=58"
## [1] 57.6110 35.1607 36.2415 36.2094
## [1] "m=58"
## [1] 41.4657 35.8529 36.2131 36.2094
A = matrix(c(1, 3, 9, 5, 2, 14, 10, 6, 0, 11, 11, 7, 0, 0, 1, 8), 4, 4)
eff.ini.maxeig.general(A, z0 = "Auto", digit.thresh = 4)$z
eff.ini.maxeig.general(A, improved = T, xi = 1, digit.thresh = 4)$z
eff.ini.maxeig.general(A, improved = T, xi = 0.65, digit.thresh = 4)$z
## [1] "m=31"
## [1] 13.7532 7.1098 7.6088 7.7219 7.7225
## [1] "m=31"
## [1] 30.6865 16.8617 37.4077 29.4527 25.1739 24.1007 24.0296 24.0293
## [1] "m=31"
## [1] 24.1681 23.6847 24.0369 24.0293
b4 = c(0.01, 1, 100)
digits = c(9, 7, 6)
for (i in 1:3) {
A = matrix(c(-3, 4, 0, 10, 0, 2, -7, 5, 0, 0, 0, 3, -5, 0, 0, 1, 0, 0, -16,
11, 0, 0, 0, 6, -11 - b4[i]), 5, 5)
print(-eff.ini.maxeig.general(A, z0 = "Auto", digit.thresh = digits[i])$z)
}
## [1] 0.002000000 0.000278151 0.000278686
## [1] 0.0990974 0.0236258 0.0245174 0.0245175
## [1] 1.666906 0.200058 0.182609 0.182819
b4 = c(0.01, 1, 100)
digits = c(9, 7, 6)
for (i in 1:3) {
A = matrix(c(-5, 3, 0, 0, 0, 5, -7, 2, 0, 0, 0, 4, -3, 10, 0, 0, 0, 1, -16,
11, 0, 0, 0, 6, -11 - b4[i]), 5, 5)
print(-eff.ini.maxeig.general(A, z0 = "Auto", digit.thresh = digits[i])$z)
}
## [1] 0.002000000 0.000278151 0.000278686
## [1] 0.1091040 0.0234222 0.0245174 0.0245175
## [1] 1.550170 0.133420 0.182541 0.182819
b4 = c(0.01, 1, 100)
a = c(3, 2, 10, 11)
b = c(5, 4, 1, 6)
b4 = c(0.01, 1, 100, 10^6)
digits = c(9, 7, 6, 6)
for (i in 1:4) {
c = c(rep(0, 4), b4[i])
print(-eff.ini.maxeig.tri(a, b, c, xi = 1, digit.thresh = digits[i])$z)
}
## [1] 0.000278670 0.000278686
## [1] 0.0244003 0.0245190 0.0245175
## [1] 0.179806 0.182912 0.182819
## [1] 0.191917 0.195239 0.195145
a = c(1:7)^2
b = c(1:7)^2
eff.ini.seceig.tri(a, b, xi = 0)$z
eff.ini.seceig.tri(a, b, xi = 1)$z
eff.ini.seceig.tri(a, b, xi = 2/5)$z
## [1] 0.902633 0.820614 0.820539
## [1] 0.456343 0.821600 0.820539
## [1] 0.724117 0.820629 0.820539
a = c(3, 2, 10, 11)
b = c(5, 4, 1, 6)
eff.ini.seceig.tri(a, b, xi = 0, digit.thresh = 5)$z
eff.ini.seceig.tri(a, b, xi = 1, digit.thresh = 5)$z
eff.ini.seceig.tri(a, b, xi = 2/5, digit.thresh = 5)$z
## [1] 3.84977 3.05196 3.03673
## [1] 1.72924 3.05715 3.03673
## [1] 3.00156 3.03675 3.03673
The results of two choices of the initial \(z_0\) are presented in the following.
Q = matrix(c(-30, 1/5, 11/28, 55/3291, 30, -17, 275/42, 330/1097, 0, 84/5, -20,
588/1097, 0, 0, 1097/84, -2809/3291), 4, 4)
eff.ini.seceig.general(Q, z0 = "Auto", digit.thresh = 5)$z
eff.ini.seceig.general(Q, z0 = "fixed", digit.thresh = 5)$z
## [1] 7.73666 8.15020 8.17129 8.17131
## [1] 7.34195 8.13214 8.17124 8.17131
Q = matrix(c(-57, 135/59, 243/91, 351/287, 118/27, -52, 590/91, 118/41, 91/9,
637/59, -47, 195/41, 1148/27, 2296/59, 492/13, -62/7), 4, 4)
eff.ini.seceig.general(Q, z0 = "Auto", digit.thresh = 4)$z
eff.ini.seceig.general(Q, z0 = "fixed", digit.thresh = 4)$z
## [1] 47.5318 47.5453 47.5454
## [1] 38.7143 47.5346 47.5453 47.5454