## Gauss-Laguerre Quadrature

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

^{ - x}With respect to the inner product

*< f,g >*=

*∫*, the Laguerre polynomials

_{0}^{∞}(f(x) g(x) w(x)) dx*L*for

_{ n}(x) = e^{ x}( d^{ n}/ dx^{ n})( x^{ n}e^{ - x })*n > 0*, and

*L*form an orthogonal family of polynomials with weight function

_{0}(x) = 1*w(x) = e*on the positive x-axis.

^{ - x}The

*n*-point Gauss-Laguerre quadrature formula,

*GL*for approximating the integral of

_{n}( f(x) ),*f(x) e*over

^{ - x}*[0,∞)*, is given by

*GL*=

_{n}( f(x) )*A*,

_{1}f(x_{1}) + ··· + A_{n}f(x_{n})*x*are the zeros of

_{i}, i = 1,...,n,*L*and

_{n}*A*,

_{i}= (n!)² / (x_{i}[ L_{n-1}(x_{i}) ] 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*L*in the user allocated array nodes of dimension n._{n}(x)

- 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*element of the array wght is the weight which corresponds to the^{ th}*i*node in the array nodes as returned in Gauss_Laguerre_Zeros_^{ th}*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*.