gauss.quad: Gaussian Quadrature

In statmod: Statistical Modeling

Gaussian Quadrature

Description

Calculate nodes and weights for Gaussian quadrature.

Usage

gauss.quad(n, kind = "legendre", alpha = 0, beta = 0)

Arguments

n	number of nodes and weights
kind	kind of Gaussian quadrature, one of "legendre", "chebyshev1", "chebyshev2", "hermite", "jacobi" or "laguerre"
alpha	parameter for Jacobi or Laguerre quadrature, must be greater than -1
beta	parameter for Jacobi quadrature, must be greater than -1

Details

The integral from a to b of w(x)*f(x) is approximated by sum(w*f(x)) where x is the vector of nodes and w is the vector of weights. The approximation is exact if f(x) is a polynomial of order no more than 2n-1. The possible choices for w(x), a and b are as follows:

Legendre quadrature: w(x)=1 on (-1,1).

Chebyshev quadrature of the 1st kind: w(x)=1/sqrt(1-x^2) on (-1,1).

Chebyshev quadrature of the 2nd kind: w(x)=sqrt(1-x^2) on (-1,1).

Hermite quadrature: w(x)=exp(-x^2) on (-Inf,Inf).

Jacobi quadrature: w(x)=(1-x)^alpha*(1+x)^beta on (-1,1). Note that Chebyshev quadrature is a special case of this.

Laguerre quadrature: w(x)=x^alpha*exp(-x) on (0,Inf).

The algorithm used to generated the nodes and weights is explained in Golub and Welsch (1969).

Value

A list containing the components

nodes	vector of values at which to evaluate the function
weights	vector of weights to give the function values

Author(s)

Gordon Smyth, using Netlib Fortran code https://netlib.org/go/gaussq.f, and including a suggestion from Stephane Laurent

References

Golub, G. H., and Welsch, J. H. (1969). Calculation of Gaussian quadrature rules. Mathematics of Computation 23, 221-230.

Golub, G. H. (1973). Some modified matrix eigenvalue problems. Siam Review 15, 318-334.

Smyth, G. K. (1998). Numerical integration. In: Encyclopedia of Biostatistics, P. Armitage and T. Colton (eds.), Wiley, London, pages 3088-3095. http://www.statsci.org/smyth/pubs/NumericalIntegration-Preprint.pdf

Smyth, G. K. (1998). Polynomial approximation. In: Encyclopedia of Biostatistics, P. Armitage and T. Colton (eds.), Wiley, London, pages 3425-3429. http://www.statsci.org/smyth/pubs/PolyApprox-Preprint.pdf

Stroud, AH, and Secrest, D (1966). Gaussian Quadrature Formulas. Prentice-Hall, Englewood Cliffs, N.J.

See Also

gauss.quad.prob, integrate

Examples

#  mean of gamma distribution with alpha=6
out <- gauss.quad(10,"laguerre",alpha=5)
sum(out$weights * out$nodes) / gamma(6)

statmod documentation built on Jan. 6, 2023, 5:14 p.m.