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 Computation23, 221-230.
Golub, G. H. (1973). Some modified matrix eigenvalue problems.
Siam Review15, 318-334.
Stroud and Secrest (1966). Gaussian Quadrature Formulas. Prentice-
Hall, Englewood Cliffs, N.J.