Implements the Gaussian quadrature. Numerical Integration Abstract class. More...

#include <tpzgaussrule.h>

Collaboration diagram for TPZGaussRule:

Public Types
enum	{ NRULESLOBATTO_ORDER, NRULESLEGENDRE_ORDER }

Public Member Functions
int	NInt () const
	Returns number of integration points. More...

int	Type ()
	Returns the gaussian quadrature type (Legendre, Lobatto, Jacobi) More...

long double	Loc (int i) const
	Returns location of the ith point. More...

long double	W (int i) const
	Returns weight for the ith point. More...

long double	Alpha ()
	Returns ALPHA: it is the exponent of (1-X) in the quadrature rule: $weight = (1-X)^\alpha * (1+X)^\beta$ . More...

long double	Beta ()
	Returns BETA: it is the exponent of (1-X) in the quadrature rule: . More...

int	Order ()
	return the order of the integration rule More...

void	SetParametersJacobi (long double alpha, long double beta)
	Sets alpha and beta, the parameters of the Jacobi polynomials. More...

void	SetType (int &type, int order)
	Sets a gaussian quadrature: 0 - Gauss Legendre, 1 - Gauss Lobatto, 2 - Gauss Jacobi with alpha = beta = 1., 3 - Gauss Chebyshev for alpha = beta = -0.5. More...

void	Print (std::ostream &out=std::cout)
	Prints the number of integration points, all points and weights (as one dimension) More...

Protected Member Functions
int	ComputingGaussLegendreQuadrature (int order)
	Computes the points and weights for Gauss Legendre Quadrature over the parametric 1D element [-1.0, 1.0]. More...

void	ComputingGaussLegendreQuadrature (int *npoints, TPZVec< long double > &Location, TPZVec< long double > &Weight)
	Computes the points and weights for Gauss Legendre Quadrature over the parametric 1D element [-1.0, 1.0] It is util to be called by another rule that need the Gaussian quadrature based on number of points not on order. More...

int	ComputingGaussLobattoQuadrature (int order)
	Computes the points and weights for Gauss Lobatto Quadrature over the parametric 1D element It is to integrate functions , but the first and last integration points are and respectively. More...

int	ComputingGaussJacobiQuadrature (int order, long double alpha, long double beta)
	Computes the points and weights for Gauss Lobatto Quadrature over the parametric 1D element It is to integrate functions , but one of integration point is or . More...

void	ComputingGaussJacobiQuadrature (int *npoints, long double alpha, long double beta, TPZVec< long double > &Location, TPZVec< long double > &Weight)
	Compute numerical integration points and weight for Gauss Jacobi quadrature over It is to integrate functions as $(1-x)^\alpha (1+x)^\beta f(x)$ . It is util to be called by another rule. Do not stores the integration points and weights in the current instance. More...

long double	JacobiPolinomial (long double x, int order, long double alpha, long double beta, TPZVec< long double > &b, TPZVec< long double > &c, long double dp2, long double p1)
	Evaluates the Jacobi polinomial for real x. More...

long double	JacobiPolinomial (long double x, int alpha, int beta, unsigned int n)
	Evaluates the polinomial Jacobi on x point. More...

int	ComputingGaussChebyshevQuadrature (int order)
	Computes the points and weights for Gauss Chebyshev Quadrature over the parametric 1D element [-1.0, 1.0] It is to integrate functions as $f(x)/\sqrt(1-x^2))$ . More...

Private Member Functions
	TPZGaussRule (int order, int type=0, long double alpha=0.0L, long double beta=0.0L)
	Constructor of quadrature rule. More...

	TPZGaussRule (const TPZGaussRule &copy)

TPZGaussRule &	operator= (const TPZGaussRule &copy)

	~TPZGaussRule ()
	Default destructor. More...

Private Attributes
int	fType
	fType = 1 (Gauss Lobatto quadrature), fType = 2 (Gauss Jacobi quadrature) for any alpha and beta the parameters of Jacobi polynomials, fType = 3 (Gauss Chebyshev) is a special Gauss Jacobi quadrature for alpha = -.5 and beta = -0.5 fType = 0(default) (Gauss Legendre quadrature) is a special Gauss Jacobi quadrature for alpha = beta = 0.0 More...

int	fNumInt
	Number of integration points for this object. More...

TPZVec< long double >	fLocation
	Location of the integration point. More...

TPZVec< long double >	fWeight
	Weight of the integration point. More...

long double	fAlpha
	ALPHA is the exponent of (1-X) in the quadrature rule: weight = (1-X)^ALPHA * (1+X)^BETA. More...

long double	fBeta
	BETA is the exponent of (1+X) in the quadrature rule: weight = (1-X)^ALPHA * (1+X)^BETA. More...

int	fOrder
	order of the integration rule More...

Friends
class	TPZIntRuleP3D
	A instance of the cubature rule for pyramid can access computing methods to construct its points and weights. More...

class	TPZIntRuleList
	The list can to access the constructor of the current class. More...

Detailed Description

Implements the Gaussian quadrature. Numerical Integration Abstract class.

Author: Philippe R. B. Devloo phil@.nosp@m.fec..nosp@m.unica.nosp@m.mp.b.nosp@m.r

Definition at line 19 of file tpzgaussrule.h.

Member Enumeration Documentation

◆ anonymous enum

anonymous enum

Enumerator
NRULESLOBATTO_ORDER
NRULESLEGENDRE_ORDER

Definition at line 67 of file tpzgaussrule.h.

Constructor & Destructor Documentation

◆ TPZGaussRule() [1/2]

TPZGaussRule::TPZGaussRule	(	int	order,
		int	type = `0`,
		long double	alpha = `0.0L`,
		long double	beta = `0.0L`
	)

private

Constructor of quadrature rule.

Parameters

order	Order of the polinomial will be integrated exactly with this cubature rule
type	Identifies the type of integration points constructed
alpha	Exponent of the factor (1 - ksi) to Jacobi type
beta	Exponent of the factor (1 + ksi) to Jacobi type

Note: If alpha = beta = 0.0L the Jacobi type is identical to Legendre type

Definition at line 22 of file tpzgaussrule.cpp.

References IsZero(), and PZError.

◆ TPZGaussRule() [2/2]

TPZGaussRule::TPZGaussRule ( const TPZGaussRule & copy )

private

Definition at line 66 of file tpzgaussrule.cpp.

◆ ~TPZGaussRule()

TPZGaussRule::~TPZGaussRule ( )

private

Default destructor.

Definition at line 88 of file tpzgaussrule.cpp.

References fLocation, fNumInt, fWeight, and TPZVec< T >::Resize().

Member Function Documentation

◆ Alpha()

long double TPZGaussRule::Alpha ( )

inline

Returns ALPHA: it is the exponent of (1-X) in the quadrature rule: $weight = (1-X)^\alpha * (1+X)^\beta$ .

Definition at line 81 of file tpzgaussrule.h.

References fAlpha.

◆ Beta()

long double TPZGaussRule::Beta ( )

inline

Returns BETA: it is the exponent of (1-X) in the quadrature rule: $ weight = (1-X)^ALPHA * (1+X)^BETA $ .

Definition at line 83 of file tpzgaussrule.h.

References fBeta.

◆ ComputingGaussChebyshevQuadrature()

int TPZGaussRule::ComputingGaussChebyshevQuadrature ( int order )

protected

Computes the points and weights for Gauss Chebyshev Quadrature over the parametric 1D element [-1.0, 1.0] It is to integrate functions as $f(x)/\sqrt(1-x^2))$ .

Parameters

order Order of the polinomial will be integrated exactly with this cubature rule

Definition at line 242 of file tpzgaussrule.cpp.

References fLocation, fNumInt, fWeight, and TPZVec< T >::Resize().

Referenced by SetParametersJacobi(), and SetType().

◆ ComputingGaussJacobiQuadrature() [1/2]

int TPZGaussRule::ComputingGaussJacobiQuadrature	(	int	order,
		long double	alpha,
		long double	beta
	)

protected

Computes the points and weights for Gauss Lobatto Quadrature over the parametric 1D element $ [-1.0, 1.0] $ It is to integrate functions $ f(x) $ , but one of integration point is $-1.0$ or $1.0$ .

Parameters

order	Order of the polinomial will be integrated exactly with this cubature rule Compute numerical integration points and weight for Gauss Jacobi quadrature over It is to integrate functions as $(1-x)^\alpha (1+x)^\beta f(x)$
order	Order of the polinomial will be integrated exactly with this cubature rule
alpha	Exponent of the factor (1 - ksi) to Jacobi type
beta	Exponent of the factor (1 + ksi) to Jacobi type

Gauss Jacobi quadrature

Definition at line 379 of file tpzgaussrule.cpp.

References fLocation, fNumInt, and fWeight.

Referenced by TPZIntRuleP3D::ComputingCubatureRuleForPyramid(), SetParametersJacobi(), and SetType().

◆ ComputingGaussJacobiQuadrature() [2/2]

void TPZGaussRule::ComputingGaussJacobiQuadrature	(	int *	npoints,
		long double	alpha,
		long double	beta,
		TPZVec< long double > &	Location,
		TPZVec< long double > &	Weight
	)

protected

Compute numerical integration points and weight for Gauss Jacobi quadrature over $ [-1,1] $ It is to integrate functions as $(1-x)^\alpha (1+x)^\beta f(x)$ . It is util to be called by another rule. Do not stores the integration points and weights in the current instance.

Parameters

npoints	Number of integration points
alpha	Exponent of factor (1+x)
beta	Exponent of factor (1-x)
Location	Vector of integration points ksi
Weight	Vector of corresponding weights

Gauss Jacobi quadrature

Definition at line 390 of file tpzgaussrule.cpp.

References gamma(), JacobiPolinomial(), machinePrecision(), pow(), and TPZVec< T >::Resize().

◆ ComputingGaussLegendreQuadrature() [1/2]

int TPZGaussRule::ComputingGaussLegendreQuadrature ( int order )

protected

Computes the points and weights for Gauss Legendre Quadrature over the parametric 1D element [-1.0, 1.0].

Parameters

order Order of the polinomial will be integrated exactly with this cubature rule

GAUSS LEGENDRE QUADRATURE

Definition at line 170 of file tpzgaussrule.cpp.

References fLocation, fNumInt, fWeight, and NRULESLEGENDRE_ORDER.

Referenced by TPZIntRuleP3D::ComputingCubatureRuleForPyramid(), SetParametersJacobi(), and SetType().

◆ ComputingGaussLegendreQuadrature() [2/2]

void TPZGaussRule::ComputingGaussLegendreQuadrature	(	int *	npoints,
		TPZVec< long double > &	Location,
		TPZVec< long double > &	Weight
	)

protected

Computes the points and weights for Gauss Legendre Quadrature over the parametric 1D element [-1.0, 1.0] It is util to be called by another rule that need the Gaussian quadrature based on number of points not on order.

Parameters

npoints	Number of integration points
Location	Vector of integration points ksi
Weight	Vector of corresponding weights

Definition at line 183 of file tpzgaussrule.cpp.

References fabs, m, machinePrecision(), PZINTEGRAL_MAXITERATIONS_ALLOWED, TPZVec< T >::Resize(), and pzgeom::tol.

◆ ComputingGaussLobattoQuadrature()

int TPZGaussRule::ComputingGaussLobattoQuadrature ( int order )

protected

Computes the points and weights for Gauss Lobatto Quadrature over the parametric 1D element $ [-1.0, 1.0] $ It is to integrate functions $ f(x) $ , but the first and last integration points are $ -1.0 $ and $1.0$ respectively.

Parameters

order Order of the polinomial will be integrated exactly with this cubature rule

Note: The implementation follow Karniadakis and Sherwin: "Spectral/hp element methods for computational fluid dynamics (Oxford University Press, 2005).

Computes the points and weights for Gauss Lobatto quadrature

Definition at line 288 of file tpzgaussrule.cpp.

References test::f, fabs, fLocation, fNumInt, fWeight, gamma(), JacobiPolinomial(), m, NRULESLOBATTO_ORDER, PZINTEGRAL_MAXITERATIONS_ALLOWED, TPZVec< T >::Resize(), and pzgeom::tol.

Referenced by SetParametersJacobi(), and SetType().

◆ JacobiPolinomial() [1/2]

long double TPZGaussRule::JacobiPolinomial	(	long double	x,
		int	order,
		long double	alpha,
		long double	beta,
		TPZVec< long double > &	b,
		TPZVec< long double > &	c,
		long double *	dp2,
		long double *	p1
	)

protected

Evaluates the Jacobi polinomial for real x.

Parameters

x	Real number to compute J(x)
order	Order of the polinomial
alpha	Exponent of factor (1+x)
beta	Exponent of factor (1-x)
b	Recursion coefficients (Jacobi matrix)
c	Recursion coefficients
dp2	Returns the value of J'(x)
p1	Returns the value of J[order-1](x), where J[order-1] represent the Jacobi polinomial to degree order-1.

Evaluate the Jacobi polinomial with alpha and beta real numbers

Definition at line 527 of file tpzgaussrule.cpp.

Referenced by ComputingGaussJacobiQuadrature(), ComputingGaussLobattoQuadrature(), and SetParametersJacobi().

◆ JacobiPolinomial() [2/2]

long double TPZGaussRule::JacobiPolinomial	(	long double	x,
		int	alpha,
		int	beta,
		unsigned int	n
	)

protected

Evaluates the polinomial Jacobi on x point.

Parameters

x	point on the polinomial Jacobi is evaluated, J(x)
n	is the order of the polinomial
alpha	is the exponent of the factor (1-x)
beta	is the exponent of the factor (1+x)

Note: The Jacobi polinomial is evaluated using a recursion formulae

Definition at line 262 of file tpzgaussrule.cpp.

◆ Loc()

long double TPZGaussRule::Loc ( int i ) const

Returns location of the ith point.

Definition at line 96 of file tpzgaussrule.cpp.

References fLocation, fNumInt, and PZError.

Referenced by TPZPrInteg< TFather >::Point(), and Type().

◆ NInt()

int TPZGaussRule::NInt ( ) const

inline

Returns number of integration points.

Definition at line 70 of file tpzgaussrule.h.

References fNumInt.

Referenced by TPZPrInteg< TFather >::NPoints(), and TPZPrInteg< TFather >::Point().

◆ operator=()

TPZGaussRule & TPZGaussRule::operator= ( const TPZGaussRule & copy )

private

Definition at line 72 of file tpzgaussrule.cpp.

References fAlpha, fBeta, fLocation, fNumInt, fOrder, fType, and fWeight.

◆ Order()

int TPZGaussRule::Order ( )

inline

return the order of the integration rule

Definition at line 86 of file tpzgaussrule.h.

References fOrder.

Referenced by TPZInt1d::SetOrder(), TPZIntCube3D::SetOrder(), TPZInt1d::TPZInt1d(), TPZIntCube3D::TPZIntCube3D(), and TPZIntQuad::TPZIntQuad().

◆ Print()

void TPZGaussRule::Print ( std::ostream & out = std::cout )

Prints the number of integration points, all points and weights (as one dimension)

Definition at line 116 of file tpzgaussrule.cpp.

References fLocation, fNumInt, fType, and fWeight.

Referenced by TPZInt1d::Print(), and SetParametersJacobi().

◆ SetParametersJacobi()

void TPZGaussRule::SetParametersJacobi	(	long double	alpha,
		long double	beta
	)

inline

Sets alpha and beta, the parameters of the Jacobi polynomials.

Parameters

alpha	Exponent of the factor (1 - ksi) to Jacobi type
beta	Exponent of the factor (1 + ksi) to Jacobi type

Definition at line 95 of file tpzgaussrule.h.

References ComputingGaussChebyshevQuadrature(), ComputingGaussJacobiQuadrature(), ComputingGaussLegendreQuadrature(), ComputingGaussLobattoQuadrature(), gamma(), JacobiPolinomial(), machinePrecision(), Print(), and SetType().

◆ SetType()

void TPZGaussRule::SetType	(	int &	type,
		int	order
	)

Sets a gaussian quadrature: 0 - Gauss Legendre, 1 - Gauss Lobatto, 2 - Gauss Jacobi with alpha = beta = 1., 3 - Gauss Chebyshev for alpha = beta = -0.5.

Parameters

order	Order of the polinomial will be integrated exactly with this cubature rule
type	Identifies the type of integration points constructed

Definition at line 131 of file tpzgaussrule.cpp.

References ComputingGaussChebyshevQuadrature(), ComputingGaussJacobiQuadrature(), ComputingGaussLegendreQuadrature(), ComputingGaussLobattoQuadrature(), DebugStop, fAlpha, fBeta, fLocation, fType, fWeight, and TPZVec< T >::Resize().

Referenced by SetParametersJacobi(), TPZInt1d::SetType(), TPZIntQuad::SetType(), and TPZIntCube3D::SetType().

◆ Type()

int TPZGaussRule::Type ( )

inline

Returns the gaussian quadrature type (Legendre, Lobatto, Jacobi)

Definition at line 73 of file tpzgaussrule.h.

References fType, Loc(), and W().

Referenced by TPZIntQuad::SetOrder().

◆ W()

long double TPZGaussRule::W ( int i ) const

Returns weight for the ith point.

Definition at line 107 of file tpzgaussrule.cpp.

References fNumInt, fWeight, and PZError.

Referenced by TPZPrInteg< TFather >::Point(), and Type().

Friends And Related Function Documentation

◆ TPZIntRuleList

friend class TPZIntRuleList

friend

The list can to access the constructor of the current class.

Definition at line 32 of file tpzgaussrule.h.

◆ TPZIntRuleP3D

friend class TPZIntRuleP3D

friend

A instance of the cubature rule for pyramid can access computing methods to construct its points and weights.

Definition at line 29 of file tpzgaussrule.h.

Member Data Documentation

◆ fAlpha

long double TPZGaussRule::fAlpha

private

ALPHA is the exponent of (1-X) in the quadrature rule: weight = (1-X)^ALPHA * (1+X)^BETA.

Definition at line 42 of file tpzgaussrule.h.

Referenced by Alpha(), operator=(), and SetType().

◆ fBeta

long double TPZGaussRule::fBeta

private

BETA is the exponent of (1+X) in the quadrature rule: weight = (1-X)^ALPHA * (1+X)^BETA.

Definition at line 44 of file tpzgaussrule.h.

Referenced by Beta(), operator=(), and SetType().

◆ fLocation

TPZVec<long double> TPZGaussRule::fLocation

private

Location of the integration point.

Definition at line 37 of file tpzgaussrule.h.

Referenced by ComputingGaussChebyshevQuadrature(), ComputingGaussJacobiQuadrature(), ComputingGaussLegendreQuadrature(), ComputingGaussLobattoQuadrature(), Loc(), operator=(), Print(), SetType(), and ~TPZGaussRule().

◆ fNumInt

int TPZGaussRule::fNumInt

private

Number of integration points for this object.

Definition at line 35 of file tpzgaussrule.h.

Referenced by ComputingGaussChebyshevQuadrature(), ComputingGaussJacobiQuadrature(), ComputingGaussLegendreQuadrature(), ComputingGaussLobattoQuadrature(), Loc(), NInt(), operator=(), Print(), W(), and ~TPZGaussRule().

◆ fOrder

int TPZGaussRule::fOrder

private

order of the integration rule

Definition at line 47 of file tpzgaussrule.h.

Referenced by operator=(), and Order().

◆ fType

int TPZGaussRule::fType

private

fType = 1 (Gauss Lobatto quadrature), fType = 2 (Gauss Jacobi quadrature) for any alpha and beta the parameters of Jacobi polynomials, fType = 3 (Gauss Chebyshev) is a special Gauss Jacobi quadrature for alpha = -.5 and beta = -0.5 fType = 0(default) (Gauss Legendre quadrature) is a special Gauss Jacobi quadrature for alpha = beta = 0.0

Definition at line 26 of file tpzgaussrule.h.

Referenced by operator=(), Print(), SetType(), and Type().

◆ fWeight

TPZVec<long double> TPZGaussRule::fWeight

private

Weight of the integration point.

Definition at line 39 of file tpzgaussrule.h.

Referenced by ComputingGaussChebyshevQuadrature(), ComputingGaussJacobiQuadrature(), ComputingGaussLegendreQuadrature(), ComputingGaussLobattoQuadrature(), operator=(), Print(), SetType(), W(), and ~TPZGaussRule().

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

Integral/tpzgaussrule.h
Integral/tpzgaussrule.cpp

Public Types

Public Member Functions

Protected Member Functions

Private Member Functions

Private Attributes

Friends

Detailed Description

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ TPZGaussRule() [1/2]

◆ TPZGaussRule() [2/2]

◆ ~TPZGaussRule()

Member Function Documentation

◆ Alpha()

◆ Beta()

◆ ComputingGaussChebyshevQuadrature()

◆ ComputingGaussJacobiQuadrature() [1/2]

◆ ComputingGaussJacobiQuadrature() [2/2]

◆ ComputingGaussLegendreQuadrature() [1/2]

◆ ComputingGaussLegendreQuadrature() [2/2]

◆ ComputingGaussLobattoQuadrature()

◆ JacobiPolinomial() [1/2]

◆ JacobiPolinomial() [2/2]

◆ Loc()

◆ NInt()

◆ operator=()

◆ Order()

◆ Print()

◆ SetParametersJacobi()

◆ SetType()

◆ Type()

◆ W()

Friends And Related Function Documentation

◆ TPZIntRuleList

◆ TPZIntRuleP3D

Member Data Documentation

◆ fAlpha

◆ fBeta

◆ fLocation

◆ fNumInt

◆ fOrder

◆ fType

◆ fWeight