MathTL::OrthogonalPolynomial Class Reference

#include <ortho_poly.h>

Inheritance diagram for MathTL::OrthogonalPolynomial:

List of all members.

Public Member Functions
virtual	~OrthogonalPolynomial ()
	virtual destructor
virtual double	a (const unsigned int k) const =0
virtual double	b (const unsigned int k) const =0
double	operator() (const unsigned int n, const double x) const
Polynomial< double >	assemble (const unsigned int n) const
double	forwardSummation (const Vector< double > &coeffs, const double x) const
double	adjointSummation (const Vector< double > &coeffs, const double x) const

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Detailed Description

Abstract base class for orthogonal polynomials that fulfill a (homogeneous) three-term recursion of the form

p_k(t) = (t-a_k) * p_{k-1}(t) - b_k * p_{k-2}(t), k=1,2,...

where p_{-1}(t)=0, p_0(t)=1.

Due to the specific shape of the recurrence relation, the p_k will have leading coefficient 1. Moreover, we have

a_k = <tp_{k-1}, p_{k-1}> / <p_{k-1}, p_{k-1}> b_k = <tp_{k-1}, p_{k-2}> / <p_{k-2}, p_{k-2}>

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Member Function Documentation

virtual double MathTL::OrthogonalPolynomial::a

(

const unsigned int

k

)

const [pure virtual]

the coefficients a_k

Implemented in MathTL::GenMomentsPolynomial, MathTL::LegendrePolynomial, MathTL::ChebyshevPolynomial, and MathTL::Monomial.

double MathTL::OrthogonalPolynomial::adjointSummation	(	const Vector< double > &	coeffs,
		const double	x
	)		const

adjoint summation of {k=0}^n * p_k(x) remarks:

• generalized Horner scheme (-> less multiplications than forward summation)
• numerically stable for dominant solutions of the three-term recursion
• potentially unstable for minimal solutions

Polynomial< double > MathTL::OrthogonalPolynomial::assemble

(

const unsigned int

n

)

const

assemble n-th orthogonal polynomial

virtual double MathTL::OrthogonalPolynomial::b

(

const unsigned int

k

)

const [pure virtual]

the coefficients b_k

Implemented in MathTL::GenMomentsPolynomial, MathTL::LegendrePolynomial, MathTL::ChebyshevPolynomial, and MathTL::Monomial.

double MathTL::OrthogonalPolynomial::forwardSummation	(	const Vector< double > &	coeffs,
		const double	x
	)		const

(trivial) forward summation of {k=0}^n * p_k(x)

double MathTL::OrthogonalPolynomial::operator()	(	const unsigned int	n,
		const double	x
	)		const

evaluate n-th orthogonal polynomial at x

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

• numerics/ortho_poly.h
• numerics/ortho_poly.cpp