statmod (version 1.4.8)

gauss.quad: 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

Value

  • A list containing the components
  • nodesvector of values at which to evaluate the function
  • weightsvector of weights to give the function values

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 method is explained in Golub and Welsch (1969).

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

See Also

gauss.quad.prob, integrate

Examples

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

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