Basic class for numerical integration. More...

#include <quadrature.hh>

Inheritance diagram for concepts::Quadrature< type >:

Public Member Functions
const Real *	abscissas () const
	Returns a pointer into the array of the abscissas. More...

uint	n () const
	Returns the number of quadrature points. More...

	Quadrature (uint n)
	Constructor. More...

const Real *	weights () const
	Returns a pointer into the array of the weights. More...

Static Public Member Functions
static std::string	storedPoints ()
	Lists the integration orders (points) which have been computed so far. More...

Protected Member Functions
virtual std::ostream &	info (std::ostream &os) const
	Returns information in an output stream. More...

Protected Attributes
Real *	abscissas_
	Abscissas. More...

Real *	weights_
	Weights. More...

Private Member Functions
bool	find_ ()

Private Attributes
uint	n_
	Number of quadrature points. More...

Static Private Attributes
static std::map< uint, Real * >	abscissas_H
	Hash of the already computed values of the abscissas of this rule. More...

static std::map< uint, Real * >	weights_H
	Hash of the already computed values of the weights of this rule. More...

Detailed Description

template<int type>
class concepts::Quadrature< type >

Basic class for numerical integration.

This class provides the datastructures for the weights and abscissas and the methods to get them.

The values are computed in the template instantiations.

Parameters

type

Template parameter: type of the quadrature rule. Can take the values

0: Gauss Lobatto quadrature. This rule includes both endpoints.
$\int_{-1}^1 f(x) \, dx \approx \sum_{i=0}^p w_i f(x_i)$
is exact for $f \in P_{2p-1}$ and n = p+1 points. n must be greater or equal to 2.
The abscissas are the zeros of $(1-x^2) P_{p-1}^{(1,1)}(x)$ and the weights are $w_i = 2/(p(p+1) (P_p^{(0,0)}(x_i))^2)$ .
1: Gauss Jacobi Lobatto quadrature. This rule includes both endpoints.
$\int_{-1}^1 f(x) (1-x) \, dx \approx \sum_{i=0}^p w_i f(x_i)$
is exact for $f \in P_{2p-1}$ and n = p+1 points. n must be greater or equal to 2.
The abscissas are the zeros of $(1-x^2) P_{p-1}^{(2,1)}(x)$ and the weights are $w_i = 4/(p(p+2) (P_p^{(1,0)}(x_i))^2)$ and $w_p = 8/(p(p+2) (P_p^{(1,0)}(x_i))^2)$ .
2: Gauss Jacobi Lobatto quadrature. This rule includes both endpoints.
$\int_{-1}^1 f(x) (1-x)^2 \, dx \approx \sum_{i=0}^p w_i f(x_i)$
is exact for $f \in P_{2p-1}$ and n = p+1 points. n must be greater or equal to 2.
The abscissas are the zeros of $(1-x^2) P_{p-1}^{(3,1)}(x)$ and the weights are $w_i = 8/(p(p+3) (P_p^{(2,0)}(x_i))^2)$ and $w_p = 24/(p(p+3) (P_p^{(2,0)}(x_i))^2)$ .
3: Gauss Radau Jacobi quadrature. This rule includes only one of the endpoints, ie. -1.
$\int_{-1}^1 f(x) (1-x) \, dx \approx \sum_{i=0}^p w_i f(x_i)$
is exact for $f \in P_{2p}$ and n = p+1 points. n must be greater or equal to 2.
The abscissas are the zeros of $(1+x) P_p^{(1,1)}(x)$ and the weights are $w_i = 2(1-x_i)/(p(p+1) (P_p^{(1,0)}(x_1))^2)$ .
4: Gauss Jacobi quadrature. This rule does not include the endpoints -1 and 1.
$\int_{-1}^1 f(x) \, dx \approx \sum_{i=0}^p w_i f(x_i)$
is exact for $f \in P_{2p+1}$ and n = p+1 points. n must be greater or equal to 1.
The abscissas are the zeros of $P_{p+1}^{(0,0)}(x)$ and the weights are
$w_i = \frac{2}{1-x_i^2} \left( \frac{d}{dx} \left. P^{(0,0)}_{p+1}(x) \right|_{x=x_i} \right)^{-2}.$
5: Trapeze quadrature. This rule include the endpoints -1 and 1.
$\int_{-1}^1 f(x) \, dx \approx \sum_{i=0}^{n-1} w_i f(x_i)$
is only exact for $f \in P_{1}$ . n must be greater or equal to 2.
The abscissas are equidistant and the weights are
$w_i = \frac{1}{n-1}\left\{\begin{array}{ll}1 & i \in \{0,n\}\\2 & \mbox{otherwise}\end{array}\right.$

Requesting the same quadrature rule with the same amount of integration points several times does not harm: the values are only stored once (internally).

Test:

test::QuadratureTest

test::KarniadakisTest

Author: Philipp Frauenfelder, 2000

Definition at line 97 of file quadrature.hh.

Constructor & Destructor Documentation

◆ Quadrature()

template<int type>

concepts::Quadrature< type >::Quadrature ( uint n )

Constructor.

Computes the quadrature points.

Parameters

n	Number of entries to be computed

Member Function Documentation

◆ abscissas()

template<int type>

const Real* concepts::Quadrature< type >::abscissas ( ) const

inline

Returns a pointer into the array of the abscissas.

Definition at line 105 of file quadrature.hh.

◆ find_()

template<int type>

bool concepts::Quadrature< type >::find_ ( )

private

◆ info()

template<int type>

virtual std::ostream& concepts::Quadrature< type >::info ( std::ostream & os ) const

protectedvirtual

Returns information in an output stream.

Reimplemented from concepts::OutputOperator.

◆ n()

template<int type>

uint concepts::Quadrature< type >::n ( ) const

inline

Returns the number of quadrature points.

Definition at line 109 of file quadrature.hh.

◆ storedPoints()

template<int type>

static std::string concepts::Quadrature< type >::storedPoints ( )

static

Lists the integration orders (points) which have been computed so far.

Returns: a string containing the computed orders.

◆ weights()

template<int type>

const Real* concepts::Quadrature< type >::weights ( ) const

inline

Returns a pointer into the array of the weights.

Definition at line 107 of file quadrature.hh.

Member Data Documentation

◆ abscissas_

template<int type>

Real* concepts::Quadrature< type >::abscissas_

protected

Abscissas.

Definition at line 119 of file quadrature.hh.

◆ abscissas_H

template<int type>

std::map<uint, Real*> concepts::Quadrature< type >::abscissas_H

staticprivate

Hash of the already computed values of the abscissas of this rule.

The hash is static, ie. the data is available to all instances of this class.

Definition at line 127 of file quadrature.hh.

◆ n_

template<int type>

uint concepts::Quadrature< type >::n_

private

Number of quadrature points.

Definition at line 136 of file quadrature.hh.

◆ weights_

template<int type>

Real* concepts::Quadrature< type >::weights_

protected

Weights.

Definition at line 121 of file quadrature.hh.

◆ weights_H

template<int type>

std::map<uint, Real*> concepts::Quadrature< type >::weights_H

staticprivate

Hash of the already computed values of the weights of this rule.

The hash is static, ie. the data is available to all instances of this class.

Definition at line 133 of file quadrature.hh.

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

integration/quadrature.hh

concepts::Quadrature< type > Class Template Reference

Public Member Functions

Static Public Member Functions

Protected Member Functions

Protected Attributes

Private Member Functions

Private Attributes

Static Private Attributes

Detailed Description

template<int type> class concepts::Quadrature< type >

Constructor & Destructor Documentation

◆ Quadrature()

Member Function Documentation

◆ abscissas()

◆ find_()

◆ info()

◆ n()

◆ storedPoints()

◆ weights()

Member Data Documentation

◆ abscissas_

◆ abscissas_H

◆ n_

◆ weights_

◆ weights_H

template<int type>
class concepts::Quadrature< type >