gauss.quad

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Gaussian Quadrature

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

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/EoB/ban042-.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/EoB/bap064-.pdf Stroud, AH, and Secrest, D (1966). Gaussian Quadrature Formulas. Prentice-Hall, Englewood Cliffs, N.J.

See Also

gauss.quad.prob, integrate

Aliases
  • gauss.quad
Examples
#  mean of gamma distribution with alpha=6
out <- gauss.quad(10,"laguerre",alpha=5)
sum(out$weights * out$nodes) / gamma(6)
Documentation reproduced from package statmod, version 1.4.20, License: GPL-2 | GPL-3

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