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 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

##### 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.

gauss.quad.prob, integrate

##### Aliases
# NOT RUN {
sum(out$weights * out$nodes) / gamma(6)