Gauss-Laguerre Quadrature
Description
Gauss-Laguerre quadrature formulas are used to integrate functions f(x) e - x from 0 to infinity. Let I ( f(x) e - x ) denote the integral of f(x) e - x over the positive x-axis. With respect to the inner product < f,g > = I (f(x) g(x) w(x)), the Laguerre polynomials L n(x) = e x ( d n / dx n )( x n e - x ) for n > 0, and L0(x) = 1 form an orthogonal family of polynomials with weight function w(x) = e - x on the positive x-axis.
For integrating f(x) e - x over the positive x-axis, the n -point Gauss-Laguerre quadrature formula, GLn ( f(x) ), is given by
GLn ( f(x) ) = A1 f(x1) + ··· + An f(xn),where xi , i = 1,...,n, are the zeros of Ln and Ai = (n!)² / (xi [ Ln-1 (xi) ] ² ), i = 1,...,n.
The truncation error is I[-1,1] ( f(x) ) - GLn ( f(x) ) = K f (2n)( c ) / ( 2n! ),where K is a constant, and c is some unknown number -1 < c < 1. The constant K is easily determined from K = I[-1,1] ( x 2n ) - GLn ( x 2n ).
Function List
Functions: For n = 2,3,4,5,6,7,8,9,10,12,14,16,20,24,28,32,40,48,56, and 64
- double Gauss_Laguerre_Integration_n pts(
double (*f)(double) )
Integrate (*f)(x) Exp(-x) from 0 to infinity where (*f)(x) is the user supplied function.
- void Gauss_Laguerre_Zeros_n pts( double nodes[])
Returns the zeros of the Laguerre polynomial Ln(x) in the user allocated array nodes of dimension n.
- void Gauss_Laguerre_Coefs_n pts( double wght[] )
Returns the weights of the Gauss-Laguerre formula in the user allocated array wght of dimension n. The i th element of the array wght is the weight which corresponds to the i th node in the array nodes as returned in Gauss_Laguerre_Zeros_n pts( double nodes[]).
C Source