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Boost Math - Quadrature and Differentiation

Quadrature and Differentiation

The Quadrature and Differentiation section of the Boost Math library provides methods for numerical integration and differentiation of functions. These methods can be used directly in R without needing any additional compilation.

Trapezoidal Quadrature

# Trapezoidal rule integration of sin(x) from 0 to pi
trapezoidal(sin, 0, pi)
#> [1] 2

Gauss-Legendre Quadrature

# Gauss-Legendre integration of exp(x) from 0 to 1
gauss_legendre(exp, 0, 1, points = 7)
#> [1] 1.718282

Gauss-Kronrod Quadrature

# Adaptive Gauss-Kronrod integration of log(x) from 1 to 2
gauss_kronrod(log, 1, 2, points = 15, max_depth = 10)
#> [1] 0.3862944

# Non-adaptive Gauss-Kronrod integration of log(x) from 1 to 2
gauss_kronrod(log, 1, 2, points = 15, max_depth = 0)
#> [1] 0.3862944

Double-Exponential Quadrature

# Tanh-sinh quadrature of log(x) from 0 to 1
tanh_sinh(function(x) { log(x) * log1p(-x) }, a = 0, b = 1)
#> [1] 0.3550659
# Sinh-sinh quadrature of exp(-x^2)
sinh_sinh(function(x) { exp(-x * x) })
#> [1] 1.772454
# Exp-sinh quadrature of exp(-3*x) from 0 to Inf
exp_sinh(function(x) { exp(-3 * x) }, a = 0, b = Inf)
#> [1] 0.3333333

Fourier Integrals

# Fourier sine integral of sin(x) with omega = 1
ooura_fourier_sin(function(x) { 1 / x }, omega = 1)
#> [1] 1.570796
#> attr(,"relative_error")
#> [1] 1.26555e-11
# Fourier cosine integral of cos(x) with omega = 1
ooura_fourier_cos(function(x) { 1/ (x * x + 1) }, omega = 1)
#> [1] 0.5778637
#> attr(,"relative_error")
#> [1] 6.417739e-09

Numerical Differentiation

# Finite difference derivative of sin(x) at pi/4
finite_difference_derivative(sin, pi / 4)
#> [1] 0.7071068
# Complex step derivative of exp(x) at 1.7
complex_step_derivative(exp, 1.7)
#> [1] 5.473947

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