0th

Percentile

Calculate nodes and weights for Gaussian quadrature.

Keywords
math
##### 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 method is explained in Golub and Welsch (1969).

##### Value

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

##### Note

This function solves a dense nxn eigenvector problem and is therefore slow for large n. It could be made far more efficient by using an eigenvector function designed to compute the leading terms of the eigenvectors for tridiagonal matrices.

##### 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.

gauss.quad.prob, integrate

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