Ortho_poly Class Reference
[Orthogonal polynomials]

Base class for orthogonal polynomials on [-1,1]. More...

#include <ortho_poly.h>

Inheritance diagram for Ortho_poly:


Public Member Functions
virtual	~Ortho_poly ()
	Destructor.
void	operator= (const Ortho_poly &)
	Assignment to another Ortho_poly.
int	n () const
	returns N, i.e. the maximum degree of the polynomials
virtual double	weight (double x) const =0
	Weight function .
virtual double	operator() (int i, double x) const =0
	Value of the polynomial .
virtual const Grid &	gauss_nodes () const =0
	Gauss nodes.
virtual double	gauss_weight (int i) const =0
	Gauss weights.
virtual double	gauss_gamma (int i) const =0
	Gamma factor $\gamma_i$ (square of the polynomial number i with respect to the discrete scalar product associated with the Gauss nodes).
virtual const Grid &	gauss_lobatto_nodes () const =0
	Gauss-Lobatto nodes.
virtual double	gauss_lobatto_weight (int i) const =0
	Gauss-Lobatto weights.
virtual double	gauss_lobatto_gamma (int i) const =0
	Gamma factor $\gamma_i$ (square of the polynomial number i with respect to the discrete scalar product associated with the Gauss-Lobatto nodes).
void	coef_interpolant_Gauss (double(f)(double), double cf) const
	Coefficients of the interpolant polynomial through the Gauss nodes.
void	coef_interpolant_GL (double(f)(double), double cf) const
	Coefficients of the interpolant polynomial through the Gauss-Lobatto nodes.
virtual void	coef_projection (double(f)(double), double cf) const =0
	Coefficients of the expansion over the basis polynomials of the orthogonal projection on the space of polynomials of maximum degree N.
double	series (const double *a, double x) const
	Value of some expansion over the polynomials: $\sum_{i=0}^N a_i\, p_i(x)$ .
Protected Member Functions
	Ortho_poly (const Ortho_poly &)
	Copy constructor.
	Ortho_poly (int ni)
	Constructor to be used only by derived classes (hence protected).
Protected Attributes
const int	nn
	N = maximum degree of the polynomials
const Grid *	p_gauss_nodes
	Pointer on the Gauss nodes.
double *	p_gauss_weights
	Pointer on the Gauss weights.
double *	p_gauss_gamma
	Pointer on the gamma factors (squares of the polynomials with respect to the discrete scalar product associated with the Gauss nodes).
const Grid *	p_gauss_lobatto_nodes
	Pointer on the Gauss-Lobatto nodes.
double *	p_gauss_lobatto_weights
	Pointer on the Gauss-Lobatto weights.
double *	p_gauss_lobatto_gamma
	Pointer on the gamma factors (square of the polynomials with respect to the discrete scalar product associated with the Gauss-Lobatto nodes).

Detailed Description

Base class for orthogonal polynomials on [-1,1].

() Storage of polynomials of an orthogonal family $(p_i)$ up to a given degree N. The class Ortho_poly is abstract: it cannot be instanciated.

Definition at line 52 of file ortho_poly.h.

Constructor & Destructor Documentation

Ortho_poly::Ortho_poly ( const Ortho_poly & ) [protected]

Copy constructor.

Definition at line 56 of file ortho_poly.C.
References p_gauss_gamma, p_gauss_lobatto_gamma, p_gauss_lobatto_nodes, p_gauss_lobatto_weights, p_gauss_nodes, and p_gauss_weights.

Ortho_poly::Ortho_poly ( int ni ) [protected]

Constructor to be used only by derived classes (hence protected).

Parameters:

ni maximum degree N of the polynomials

Definition at line 72 of file ortho_poly.C.
References p_gauss_gamma, p_gauss_lobatto_gamma, p_gauss_lobatto_nodes, p_gauss_lobatto_weights, p_gauss_nodes, and p_gauss_weights.

Ortho_poly::~Ortho_poly ( ) [virtual]

Destructor.

Definition at line 91 of file ortho_poly.C.
References p_gauss_gamma, p_gauss_lobatto_gamma, p_gauss_lobatto_nodes, p_gauss_lobatto_weights, p_gauss_nodes, and p_gauss_weights.

Member Function Documentation

void Ortho_poly::coef_interpolant_Gauss ( double(*)(double) f,

double * cf

) const

Coefficients of the interpolant polynomial through the Gauss nodes.

Parameters:

(*f)(double) function to be interpolated

cf [output] array containing the N+1 coefficients of the expansion of the interpolant on the basis polynomials. This array must be of size at least N+1 and must have been allocated by the user prior to the call of this method.

Definition at line 128 of file ortho_poly.C.
References gauss_nodes(), gauss_weight(), and nn.

void Ortho_poly::coef_interpolant_GL ( double(*)(double) f,

double * cf

) const

Coefficients of the interpolant polynomial through the Gauss-Lobatto nodes.

Parameters:

(*f)(double) function to be interpolated

cf [output] array containing the N+1 coefficients of the expansion of the interpolant on the basis polynomials. This array must be of size at least N+1 and must have been allocated by the user prior to the call of this method.

Definition at line 156 of file ortho_poly.C.
References gauss_lobatto_nodes(), gauss_lobatto_weight(), and nn.

virtual void Ortho_poly::coef_projection ( double(*)(double) f,

double * cf

) const [pure virtual]

Coefficients of the expansion over the basis polynomials of the orthogonal projection on the space of polynomials of maximum degree N.

Parameters:

(*f)(double) function to be projected

cf [output] array containing the N+1 coefficients of the expansion over the basis polynomials of the orthogonal projection on the space of polynomials of maximum degree N. This array must be of size at least N+1 and must have been allocated by the user prior to the call of this method.

Implemented in Chebyshev_poly, and Legendre_poly.

virtual double Ortho_poly::gauss_gamma ( int i ) const [pure virtual]

Gamma factor $\gamma_i$ (square of the polynomial number i with respect to the discrete scalar product associated with the Gauss nodes).

Implemented in Chebyshev_poly, and Legendre_poly.

virtual double Ortho_poly::gauss_lobatto_gamma ( int i ) const [pure virtual]

Gamma factor $\gamma_i$ (square of the polynomial number i with respect to the discrete scalar product associated with the Gauss-Lobatto nodes).

Implemented in Chebyshev_poly, and Legendre_poly.

virtual const Grid& Ortho_poly::gauss_lobatto_nodes ( ) const [pure virtual]

Gauss-Lobatto nodes.

Implemented in Chebyshev_poly, and Legendre_poly.

virtual double Ortho_poly::gauss_lobatto_weight ( int i ) const [pure virtual]

Gauss-Lobatto weights.

Parameters:

i index of the weight

Returns:
weight $w_i$

Implemented in Chebyshev_poly, and Legendre_poly.

virtual const Grid& Ortho_poly::gauss_nodes ( ) const [pure virtual]

Gauss nodes.

Implemented in Chebyshev_poly, and Legendre_poly.

virtual double Ortho_poly::gauss_weight ( int i ) const [pure virtual]

Gauss weights.

Parameters:

i index of the weight

Returns:
weight $w_i$

Implemented in Chebyshev_poly, and Legendre_poly.

int Ortho_poly::n ( ) const

returns N, i.e. the maximum degree of the polynomials

Definition at line 117 of file ortho_poly.C.
References nn.

virtual double Ortho_poly::operator() ( int i,

double x

) const [pure virtual]

Value of the polynomial $p_i$ .

Parameters:

i no. of the polynomial

x point in [-1,1] where the polynomial is to be evaluated

Returns:
value of $p_i(x)$

Implemented in Chebyshev_poly, and Legendre_poly.

void Ortho_poly::operator= ( const Ortho_poly & )

Assignment to another Ortho_poly.

Definition at line 106 of file ortho_poly.C.

double Ortho_poly::series ( const double * a,

double x

) const

Value of some expansion over the polynomials: $\sum_{i=0}^N a_i\, p_i(x)$ .

Parameters:

a array of size at least N+1 and storing the N+1 coefficients $a_i$ of the expansion on the basis polynomials $p_i(x)$ . For an interpolant polynomial, a is the array cf computed by the method coef_interpolant_Gauss (Gauss nodes) or coef_interpolant_GL (Gauss-Lobatto nodes)

x point in [-1,1] where the series is to be evaluated

Returns:
value of $\sum_{i=0}^N a_i\, p_i(x)$

Definition at line 184 of file ortho_poly.C.
References nn.

virtual double Ortho_poly::weight ( double x ) const [pure virtual]

Weight function $w$ .

Parameters:

x point in [-1,1] where the weight function is to be evaluated

Returns:
value of $w(x)$

Implemented in Chebyshev_poly, and Legendre_poly.

Member Data Documentation

const int Ortho_poly::nn [protected]

N = maximum degree of the polynomials

Definition at line 57 of file ortho_poly.h.

double* Ortho_poly::p_gauss_gamma [mutable, protected]

Pointer on the gamma factors (squares of the polynomials with respect to the discrete scalar product associated with the Gauss nodes).

Definition at line 71 of file ortho_poly.h.

double* Ortho_poly::p_gauss_lobatto_gamma [mutable, protected]

Pointer on the gamma factors (square of the polynomials with respect to the discrete scalar product associated with the Gauss-Lobatto nodes).

Definition at line 85 of file ortho_poly.h.

const Grid* Ortho_poly::p_gauss_lobatto_nodes [mutable, protected]

Pointer on the Gauss-Lobatto nodes.

Definition at line 75 of file ortho_poly.h.

double* Ortho_poly::p_gauss_lobatto_weights [mutable, protected]

Pointer on the Gauss-Lobatto weights.

Definition at line 79 of file ortho_poly.h.

const Grid* Ortho_poly::p_gauss_nodes [mutable, protected]

Pointer on the Gauss nodes.

Definition at line 61 of file ortho_poly.h.

double* Ortho_poly::p_gauss_weights [mutable, protected]

Pointer on the Gauss weights.

Definition at line 65 of file ortho_poly.h.

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

Generated on Tue Dec 6 14:48:44 2011 for POLYNOM by

1.4.6

Ortho_poly Class Reference [Orthogonal polynomials]

Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

Ortho_poly Class Reference
[Orthogonal polynomials]