MathTL::GaussRule Class Reference

#include <gauss_quadrature.h>

Inheritance diagram for MathTL::GaussRule:

List of all members.

Public Member Functions
	GaussRule (const OrthogonalPolynomial &P, const double a, const double b, const unsigned int N)
	GaussRule (const Array1D< double > &moments, const double a, const double b, const unsigned int N)
	GaussRule (const Array1D< double > &moments, const OrthogonalPolynomial &T, const double a, const double b, const unsigned int N)

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Detailed Description

N-point 1D Gauss rule for arbitrary weight functions, several constructors are available:

• from the coefficients a_k, b_k in the three-term recursion P_k(x) = (x-alpha_k) * P_{k-1}(x) - beta_k * P_{k-2}(x), k=1,2,... for the w-orthogonal polynomials P_k with leading coefficient 1, where p_{-1}(t)=0, p_0(t)=1
• from 2N (generalized) moments of the form = ^b T_k(x)w(x)dx where the polynomials T_k also fulfill a recurrence relation T_k(x) = (x-a_k) * T_{k-1}(x) - b_k * T_{k-2}(x), k=1,2,...

references: Sack/Donovan: An Algorithm for Gaussian Quadrature given Modified Moments, Numer. Math. 18(1972), 465-478 Golub/Gutknecht: Modified Moments for Indefinite Weight Functions, Numer. Math. 57(1990), 607-624

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Constructor & Destructor Documentation

MathTL::GaussRule::GaussRule	(	const OrthogonalPolynomial &	P,
		const double	a,
		const double	b,
		const unsigned int	N
	)

construct N-point Gauss rule from three-term recursion coefficients for orthogonal polynomials P on [a,b]

Parameters:

P	orth. polynomial, should provide a_1,...,a_N and b_1,...,b_N
N	number of quadrature points

MathTL::GaussRule::GaussRule	(	const Array1D< double > &	moments,
		const double	a,
		const double	b,
		const unsigned int	N
	)

construct N-point Gauss rule from (at least) 2N (monomial) moments of the weight function ^b x^k w(x)dx

MathTL::GaussRule::GaussRule	(	const Array1D< double > &	moments,
		const OrthogonalPolynomial &	T,
		const double	a,
		const double	b,
		const unsigned int	N
	)

construct N-point Gauss rule from (at least) 2N generalized moments of the weight function ^b T_k(x)w(x)dx

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The documentation for this class was generated from the following files:

• numerics/gauss_quadrature.h
• numerics/gauss_quadrature.cpp