Gaussian Quadrature
Calculate nodes and weights for Gaussian quadrature.
gauss.quad(n, kind = "legendre", alpha = 0, beta = 0)
n |
number of nodes and weights |
kind |
kind of Gaussian quadrature, one of |
alpha |
parameter for Jacobi or Laguerre quadrature, must be greater than -1 |
beta |
parameter for Jacobi quadrature, must be greater than -1 |
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).
A list containing the components
nodes |
vector of values at which to evaluate the function |
weights |
vector of weights to give the function values |
Gordon Smyth, using Netlib Fortran code http://www.netlib.org/go/gaussq.f, and including a suggestion from Stephane Laurent
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.
# mean of gamma distribution with alpha=6 out <- gauss.quad(10,"laguerre",alpha=5) sum(out$weights * out$nodes) / gamma(6)
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