Jacobi polynomial basis. More...

#include <Stokhos_JacobiBasis.hpp>

Inheritance diagram for Stokhos::JacobiBasis< ordinal_type, value_type >:

Collaboration diagram for Stokhos::JacobiBasis< ordinal_type, value_type >:

Public Member Functions
	JacobiBasis (ordinal_type p, value_type alphaIndex, value_type betaIndex, bool normalize=false, GrowthPolicy growth=SLOW_GROWTH)
	Constructor. More...

	~JacobiBasis ()
	Destructor.

Implementation of Stokhos::OneDOrthogPolyBasis methods
virtual Teuchos::RCP < OneDOrthogPolyBasis < ordinal_type, value_type > >	cloneWithOrder (ordinal_type p) const
	Clone this object with the option of building a higher order basis. More...

Public Member Functions inherited from Stokhos::RecurrenceBasis< ordinal_type, value_type >
virtual	~RecurrenceBasis ()
	Destructor.

virtual void	getRecurrenceCoefficients (Teuchos::Array< value_type > &alpha, Teuchos::Array< value_type > &beta, Teuchos::Array< value_type > &delta, Teuchos::Array< value_type > &gamma) const
	Return recurrence coefficients defined by above formula.

virtual void	evaluateBasesAndDerivatives (const value_type &point, Teuchos::Array< value_type > &vals, Teuchos::Array< value_type > &derivs) const
	Evaluate basis polynomials and their derivatives at given point `point`.

virtual void	setQuadZeroTol (value_type tol)
	Set tolerance for zero in quad point generation.

virtual ordinal_type	order () const
	Return order of basis (largest monomial degree ).

virtual ordinal_type	size () const
	Return total size of basis (given by order() + 1).

virtual const Teuchos::Array < value_type > &	norm_squared () const
	Return array storing norm-squared of each basis polynomial. More...

virtual const value_type &	norm_squared (ordinal_type i) const
	Return norm squared of basis polynomial `i`.

virtual Teuchos::RCP < Stokhos::Dense3Tensor < ordinal_type, value_type > >	computeTripleProductTensor () const
	Compute triple product tensor. More...

virtual Teuchos::RCP < Stokhos::Sparse3Tensor < ordinal_type, value_type > >	computeSparseTripleProductTensor (ordinal_type order) const
	Compute triple product tensor. More...

virtual Teuchos::RCP < Teuchos::SerialDenseMatrix < ordinal_type, value_type > >	computeDerivDoubleProductTensor () const
	Compute derivative double product tensor. More...

virtual void	evaluateBases (const value_type &point, Teuchos::Array< value_type > &basis_pts) const
	Evaluate each basis polynomial at given point `point`. More...

virtual value_type	evaluate (const value_type &point, ordinal_type order) const
	Evaluate basis polynomial given by order `order` at given point `point`.

virtual void	print (std::ostream &os) const
	Print basis to stream `os`.

virtual const std::string &	getName () const
	Return string name of basis.

virtual void	getQuadPoints (ordinal_type quad_order, Teuchos::Array< value_type > &points, Teuchos::Array< value_type > &weights, Teuchos::Array< Teuchos::Array< value_type > > &values) const
	Compute quadrature points, weights, and values of basis polynomials at given set of points `points`. More...

virtual ordinal_type	quadDegreeOfExactness (ordinal_type n) const

virtual ordinal_type	coefficientGrowth (ordinal_type n) const
	Evaluate coefficient growth rule for Smolyak-type bases.

virtual ordinal_type	pointGrowth (ordinal_type n) const
	Evaluate point growth rule for Smolyak-type bases.

virtual LevelToOrderFnPtr	getSparseGridGrowthRule () const
	Get sparse grid level_to_order mapping function. More...

virtual void	setSparseGridGrowthRule (LevelToOrderFnPtr ptr)
	Set sparse grid rule.

Public Member Functions inherited from Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >
	OneDOrthogPolyBasis ()
	Default constructor.

virtual	~OneDOrthogPolyBasis ()
	Destructor.

virtual void	setSparseGridGrowthRule (LevelToOrderFnPtr ptr)=0
	Set sparse grid rule.

Protected Member Functions
	JacobiBasis (ordinal_type p, const JacobiBasis &basis)
	Copy constructor with specified order.

Implementation of Stokhos::RecurrenceBasis methods
virtual bool	computeRecurrenceCoefficients (ordinal_type n, Teuchos::Array< value_type > &alpha, Teuchos::Array< value_type > &beta, Teuchos::Array< value_type > &delta, Teuchos::Array< value_type > &gamma) const
	Compute recurrence coefficients.

Protected Member Functions inherited from Stokhos::RecurrenceBasis< ordinal_type, value_type >
	RecurrenceBasis (const std::string &name, ordinal_type p, bool normalize, GrowthPolicy growth=SLOW_GROWTH)
	Constructor to be called by derived classes. More...

	RecurrenceBasis (ordinal_type p, const RecurrenceBasis &basis)
	Copy constructor with specified order.

virtual void	setup ()
	Setup basis after computing recurrence coefficients. More...

void	normalizeRecurrenceCoefficients (Teuchos::Array< value_type > &alpha, Teuchos::Array< value_type > &beta, Teuchos::Array< value_type > &delta, Teuchos::Array< value_type > &gamma) const
	Normalize coefficients.

Additional Inherited Members
Public Types inherited from Stokhos::RecurrenceBasis< ordinal_type, value_type >
typedef OneDOrthogPolyBasis < ordinal_type, value_type > ::LevelToOrderFnPtr	LevelToOrderFnPtr
	Function pointer needed for level_to_order mappings.

Public Types inherited from Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >
typedef int(*	LevelToOrderFnPtr )(int level, int growth)
	Function pointer needed for level_to_order mappings.

Protected Attributes inherited from Stokhos::RecurrenceBasis< ordinal_type, value_type >
std::string	name
	Name of basis.

ordinal_type	p
	Order of basis.

bool	normalize
	Normalize basis.

GrowthPolicy	growth
	Smolyak growth policy.

value_type	quad_zero_tol
	Tolerance for quadrature points near zero.

LevelToOrderFnPtr	sparse_grid_growth_rule
	Sparse grid growth rule (as determined by Pecos)

Teuchos::Array< value_type >	alpha
	Recurrence $\alpha$ coefficients.

Teuchos::Array< value_type >	beta
	Recurrence $\beta$ coefficients.

Teuchos::Array< value_type >	delta
	Recurrence $\delta$ coefficients.

Teuchos::Array< value_type >	gamma
	Recurrence $\gamma$ coefficients.

Teuchos::Array< value_type >	norms
	Norms.

Detailed Description

template<typename ordinal_type, typename value_type>
class Stokhos::JacobiBasis< ordinal_type, value_type >

Jacobi polynomial basis.

Jacobi polynomials are defined by the recurrence relationship

$A_k \psi_{k+1}(x) = \left(B_k-x C_k\right) \psi_k(x) - D_k \psi_{k-1}(x)\right)$

with $\psi_{-1}(x) = 0$ and $\psi_{0}(x) = 1$ where

$A_n = 2 (n+1)(n+\alpha+\beta+1)(2 n + \alpha+\beta)$

$B_n = -(2n+\alpha+\beta+1)(\alpha^2 - \beta^2)$

$C_n = (2n+\alpha+\beta)_3$

$D_n = 2(n+\alpha)(n+\beta)(2n+\alpha+\beta+2).$

In Stokhos notation we have

$\gamma_{k+1}=1/A_{k}$

$\alpha_k = B_k \] \f[ \delta_k = C_k \] \f[ \beta_k = D_k. \] The corresponding density function is \f[ \rho(x) = w_{\alpha_,\beta}(1-x)^\alpha (1+x)^\beta, \quad x\in[-1,1]$

with

$w_{\alpha,\beta}^{-1}=\frac{2^{\alpha+\beta+1}}{\alpha+\beta+1} \frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{\Gamma(\alpha+\beta+1)}.$

This class implements computeRecurrenceCoefficients() using the above formula.

Author: Kevin Long (kevin.nosp@m..lon.nosp@m.g@ttu.nosp@m..edu)

Constructor & Destructor Documentation

template<typename ordinal_type , typename value_type >

Stokhos::JacobiBasis< ordinal_type, value_type >::JacobiBasis	(	ordinal_type	p,
		value_type	alphaIndex,
		value_type	betaIndex,
		bool	normalize = `false`,
		Stokhos::GrowthPolicy	growth = `SLOW_GROWTH`
	)

Constructor.

Parameters

p	order of the basis
normalize	whether polynomials should be given unit norm

References Stokhos::RecurrenceBasis< ordinal_type, value_type >::setSparseGridGrowthRule(), and Stokhos::RecurrenceBasis< ordinal_type, value_type >::setup().

Member Function Documentation

template<typename ordinal_type , typename value_type >

Teuchos::RCP< Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type > > Stokhos::JacobiBasis< ordinal_type, value_type >::cloneWithOrder ( ordinal_type p ) const

virtual

Clone this object with the option of building a higher order basis.

This method is following the Prototype pattern (see Design Pattern's textbook). The slight variation is that it allows the order of the polynomial to be modified, otherwise an exact copy is formed. The use case for this is creating basis functions for column indices in a spatially varying adaptive refinement context.

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

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

Stokhos_JacobiBasis.hpp
Stokhos_JacobiBasisImp.hpp

Public Member Functions

Protected Member Functions

Additional Inherited Members

Detailed Description

template<typename ordinal_type, typename value_type> class Stokhos::JacobiBasis< ordinal_type, value_type >

Constructor & Destructor Documentation

Member Function Documentation

template<typename ordinal_type, typename value_type>
class Stokhos::JacobiBasis< ordinal_type, value_type >