Mathematics Source Library

Gauss-Laguerre Quadrature

Gauss-Laguerre Quadrature

Gauss-Laguerre quadrature formulas are used to integrate functions f(x) e - x over the positive x-axis.

With respect to the inner product < f,g > = 0 (f(x) g(x) w(x)) dx, 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.

The n-point Gauss-Laguerre quadrature formula, GLn ( f(x) ), for approximating the integral of f(x) e - x over [0,∞), 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.

Function List

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 ∞ 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 Code

  • The file, gauss_laguerre_2pts.c, contains the versions of Gauss_Laguerre_Integration_2pts( ), Gauss_Laguerre_Zeros_2pts and Gauss_Laguerre_Coefs_2pts( ) written in C.

  • The file, gauss_laguerre_3pts.c, contains the versions of Gauss_Laguerre_Integration_3pts( ), Gauss_Laguerre_Zeros_3pts and Gauss_Laguerre_Coefs_3pts( ) written in C.

  • The file, gauss_laguerre_4pts.c, contains the versions of Gauss_Laguerre_Integration_4pts( ), Gauss_Laguerre_Zeros_4pts and Gauss_Laguerre_Coefs_4pts( ) written in C.

  • The file, gauss_laguerre_5pts.c, contains the versions of Gauss_Laguerre_Integration_5pts( ), Gauss_Laguerre_Zeros_5pts and Gauss_Laguerre_Coefs_5pts( ) written in C.

  • The file, gauss_laguerre_6pts.c, contains the versions of Gauss_Laguerre_Integration_6pts( ), Gauss_Laguerre_Zeros_6pts and Gauss_Laguerre_Coefs_6pts( ) written in C.

  • The file, gauss_laguerre_7pts.c, contains the versions of Gauss_Laguerre_Integration_7pts( ), Gauss_Laguerre_Zeros_7pts and Gauss_Laguerre_Coefs_7pts( ) written in C.

  • The file, gauss_laguerre_8pts.c, contains the versions of Gauss_Laguerre_Integration_8pts( ), Gauss_Laguerre_Zeros_8pts and Gauss_Laguerre_Coefs_8pts( ) written in C.

  • The file, gauss_laguerre_9pts.c, contains the versions of Gauss_Laguerre_Integration_9pts( ), Gauss_Laguerre_Zeros_9pts and Gauss_Laguerre_Coefs_9pts( ) written in C.

  • The file, gauss_laguerre_10pts.c, contains the versions of Gauss_Laguerre_Integration_10pts( ), Gauss_Laguerre_Zeros_10pts and Gauss_Laguerre_Coefs_10pts( ) written in C.

  • The file, gauss_laguerre_12pts.c, contains the versions of Gauss_Laguerre_Integration_12pts( ), Gauss_Laguerre_Zeros_12pts and Gauss_Laguerre_Coefs_12pts( ) written in C.

  • The file, gauss_laguerre_14pts.c, contains the versions of Gauss_Laguerre_Integration_14pts( ), Gauss_Laguerre_Zeros_14pts and Gauss_Laguerre_Coefs_14pts( ) written in C.

  • The file, gauss_laguerre_16pts.c, contains the versions of Gauss_Laguerre_Integration_16pts( ), Gauss_Laguerre_Zeros_16pts and Gauss_Laguerre_Coefs_16pts( ) written in C.

  • The file, gauss_laguerre_20pts.c, contains the versions of Gauss_Laguerre_Integration_20pts( ), Gauss_Laguerre_Zeros_20pts and Gauss_Laguerre_Coefs_20pts( ) written in C.

  • The file, gauss_laguerre_24pts.c, contains the versions of Gauss_Laguerre_Integration_24pts( ), Gauss_Laguerre_Zeros_24pts and Gauss_Laguerre_Coefs_24pts( ) written in C.

  • The file, gauss_laguerre_28pts.c, contains the versions of Gauss_Laguerre_Integration_28pts( ), Gauss_Laguerre_Zeros_28pts and Gauss_Laguerre_Coefs_28pts( ) written in C.

  • The file, gauss_laguerre_32pts.c, contains the versions of Gauss_Laguerre_Integration_32pts( ), Gauss_Laguerre_Zeros_32pts and Gauss_Laguerre_Coefs_32pts( ) written in C.

  • The file, gauss_laguerre_40pts.c, contains the versions of Gauss_Laguerre_Integration_40pts( ), Gauss_Laguerre_Zeros_40pts and Gauss_Laguerre_Coefs_40pts( ) written in C.

  • The file, gauss_laguerre_48pts.c, contains the versions of Gauss_Laguerre_Integration_48pts( ), Gauss_Laguerre_Zeros_48pts and Gauss_Laguerre_Coefs_48pts( ) written in C.

  • The file, gauss_laguerre_56pts.c, contains the versions of Gauss_Laguerre_Integration_56pts( ), Gauss_Laguerre_Zeros_56pts and Gauss_Laguerre_Coefs_56pts( ) written in C.

  • The file, gauss_laguerre_64pts.c, contains the versions of Gauss_Laguerre_Integration_64pts( ), Gauss_Laguerre_Zeros_64pts and Gauss_Laguerre_Coefs_64pts( ) written in C.