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.
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
- 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: