Legendre Library

Legendre Polynomials Library

This library includes:

• Gauss-Legendre Quadrature
• Numerically stable, accurate and efficient calculation of an order n Legendre Polynomial at values -1<=x<=1
• Numerically stable calculation of the derivative of an order n Legendre Polynomial at x.
• Calculation of the zeros of Legendre polynomials.
• Calculation of the Legendre polynomial coefficients.

Gauss-Legendre Quadrature

Also called Gaussian Quadrature, is commonly used to compute definite integrals of a function F(x) with high accuracy using a small number of points where F(x) is evaluated. The method using n points is exact when F(x) is a polynomial of degree 2*n+1 or less, and is very accurate when F(x) can be accurately approximated by a polynomial of degree n (i.e., by its Taylor series with n terms).

Function Gauss_Quadrature_Pts(K, lb, ub)

The function used to compute a Gauss-Legendre quadrature. To use, supply and index, K:=1..n. The length of K is used for n, the number of points used. The function has two return values, both indexed by K:

• The points where you should evaluate your function.
• The weights

The bounds for the integration, «lb» and «ub» default to -1 to 1 if not specified. The following pattern illustrates how this function is used to integrate F(x) from a to b using n points:

Index K := 1..n;

Local (xi,wi) := Gauss_Quadrature(K, a, b) Do Sum( wi * F(xi), K )