Implementation of OneDOrthogPolyBasis based on the general three-term recurrence relationship:

$\gamma_{k+1}\psi_{k+1}(x) = (\delta_k x - \alpha_k)\psi_k(x) - \beta_k\psi_{k-1}(x)$

for $k=0,\dots,P$ where $\psi_{-1}(x) = 0$ , $\psi_{0}(x) = 1/\gamma_0$ , and $\beta_{0} = 1 = \int d\lambda$ . More...

#include <Stokhos_RecurrenceBasis.hpp>

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

Public Member Functions
virtual	~RecurrenceBasis ()
	Destructor. More...

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. More...

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`. More...

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

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

virtual	~OneDOrthogPolyBasis ()
	Destructor. More...

virtual Teuchos::RCP < OneDOrthogPolyBasis < ordinal_type, value_type > >	cloneWithOrder (ordinal_type p) const =0
	Clone this object with the option of building a higher order basis. More...

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

Protected Member Functions
	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. More...

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 =0
	Compute recurrence coefficients. More...

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. More...

Protected Attributes
std::string	name
	Name of basis. More...

ordinal_type	p
	Order of basis. More...

bool	normalize
	Normalize basis. More...

GrowthPolicy	growth
	Smolyak growth policy. More...

value_type	quad_zero_tol
	Tolerance for quadrature points near zero. More...

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

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

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

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

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

Teuchos::Array< value_type >	norms
	Norms. More...

Private Member Functions
	RecurrenceBasis (const RecurrenceBasis &)

RecurrenceBasis &	operator= (const RecurrenceBasis &b)

Implementation of Stokhos::OneDOrthogPolyBasis methods
typedef OneDOrthogPolyBasis < ordinal_type, value_type > ::LevelToOrderFnPtr	LevelToOrderFnPtr
	Function pointer needed for level_to_order mappings. More...

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

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

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`. More...

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`. More...

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

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

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. More...

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

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

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

Additional Inherited Members
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. More...

Detailed Description

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

Implementation of OneDOrthogPolyBasis based on the general three-term recurrence relationship:

$\gamma_{k+1}\psi_{k+1}(x) = (\delta_k x - \alpha_k)\psi_k(x) - \beta_k\psi_{k-1}(x)$

for $k=0,\dots,P$ where $\psi_{-1}(x) = 0$ , $\psi_{0}(x) = 1/\gamma_0$ , and $\beta_{0} = 1 = \int d\lambda$ .

Derived classes implement the recurrence relationship by implementing computeRecurrenceCoefficients(). If normalize = true in the constructor, then the recurrence relationship becomes:

$\sqrt{\frac{\gamma_{k+1}\beta_{k+1}}{\delta_{k+1}\delta_k}} \psi_{k+1}(x) = (x - \alpha_k/\delta_k)\psi_k(x) - \sqrt{\frac{\gamma_k\beta_k}{\delta_k\delta_{k-1}}} \psi_{k-1}(x)$

for $k=0,\dots,P$ where $\psi_{-1}(x) = 0$ , $\psi_{0}(x) = 1/\sqrt{\beta_0}$ , Note that a three term recurrence can always be defined with $\gamma_k = \delta_k = 1$ in which case the polynomials are monic. However typical normalizations of some polynomial families (see Stokhos::LegendreBasis) require the extra terms. Also, the quadrature rule (points and weights) is the same regardless if the polynomials are normalized. However the normalization can affect other algorithms.

Definition at line 53 of file Stokhos_RecurrenceBasis.hpp.

Member Typedef Documentation

template<typename ordinal_type, typename value_type>

typedef OneDOrthogPolyBasis<ordinal_type,value_type>::LevelToOrderFnPtr Stokhos::RecurrenceBasis< ordinal_type, value_type >::LevelToOrderFnPtr

Function pointer needed for level_to_order mappings.

Definition at line 177 of file Stokhos_RecurrenceBasis.hpp.

Constructor & Destructor Documentation

template<typename ordinal_type , typename value_type >

Stokhos::RecurrenceBasis< ordinal_type, value_type >::~RecurrenceBasis ( )

virtual

Destructor.

Definition at line 74 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type, typename value_type >

Stokhos::RecurrenceBasis< ordinal_type, value_type >::RecurrenceBasis	(	const std::string &	name,
		ordinal_type	p,
		bool	normalize,
		Stokhos::GrowthPolicy	growth_ = `SLOW_GROWTH`
	)

protected

Constructor to be called by derived classes.

name is the name for the basis that will be displayed when printing the basis, p is the order of the basis, normalize indicates whether the basis polynomials should have unit-norm, and quad_zero_tol is used to replace any quadrature point within this tolerance with zero (which can help with duplicate removal in sparse grid calculations).

Definition at line 15 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type, typename value_type >

Stokhos::RecurrenceBasis< ordinal_type, value_type >::RecurrenceBasis	(	ordinal_type	p,
		const RecurrenceBasis< ordinal_type, value_type > &	basis
	)

protected

Copy constructor with specified order.

Definition at line 37 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type, typename value_type>

Stokhos::RecurrenceBasis< ordinal_type, value_type >::RecurrenceBasis ( const RecurrenceBasis< ordinal_type, value_type > & )

private

Member Function Documentation

template<typename ordinal_type , typename value_type >

ordinal_type Stokhos::RecurrenceBasis< ordinal_type, value_type >::order ( ) const

virtual

Return order of basis (largest monomial degree $P$ ).

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

Definition at line 81 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type , typename value_type >

ordinal_type Stokhos::RecurrenceBasis< ordinal_type, value_type >::size ( ) const

virtual

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

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

Definition at line 89 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type , typename value_type >

const Teuchos::Array< value_type > & Stokhos::RecurrenceBasis< ordinal_type, value_type >::norm_squared ( ) const

virtual

Return array storing norm-squared of each basis polynomial.

Entry $l$ of returned array is given by $\langle\psi_l^2\rangle$ for $l=0,\dots,P$ where $P$ is given by order().

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

Definition at line 97 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type, typename value_type >

const value_type & Stokhos::RecurrenceBasis< ordinal_type, value_type >::norm_squared ( ordinal_type i ) const

virtual

Return norm squared of basis polynomial i.

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

Definition at line 105 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type , typename value_type >

Teuchos::RCP< Stokhos::Dense3Tensor< ordinal_type, value_type > > Stokhos::RecurrenceBasis< ordinal_type, value_type >::computeTripleProductTensor ( ) const

virtual

Compute triple product tensor.

The $(i,j,k)$ entry of the tensor $C_{ijk}$ is given by $C_{ijk} = \langle\Psi_i\Psi_j\Psi_k\rangle$ where $\Psi_l$ represents basis polynomial $l$ and $i,j=0,\dots,P$ where $P$ is size()-1 and $k=0,\dots,p$ where $p$ is the supplied order.

This method is implemented by computing $C_{ijk}$ using Gaussian quadrature.

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

Definition at line 113 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type, typename value_type >

Teuchos::RCP< Stokhos::Sparse3Tensor< ordinal_type, value_type > > Stokhos::RecurrenceBasis< ordinal_type, value_type >::computeSparseTripleProductTensor ( ordinal_type order ) const

virtual

Compute triple product tensor.

The $(i,j,k)$ entry of the tensor $C_{ijk}$ is given by $C_{ijk} = \langle\Psi_i\Psi_j\Psi_k\rangle$ where $\Psi_l$ represents basis polynomial $l$ and $i,j=0,\dots,P$ where $P$ is size()-1 and $k=0,\dots,p$ where $p$ is the supplied order.

This method is implemented by computing $C_{ijk}$ using Gaussian quadrature.

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

Definition at line 143 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type , typename value_type >

Teuchos::RCP< Teuchos::SerialDenseMatrix< ordinal_type, value_type > > Stokhos::RecurrenceBasis< ordinal_type, value_type >::computeDerivDoubleProductTensor ( ) const

virtual

Compute derivative double product tensor.

The $(i,j)$ entry of the tensor $B_{ij}$ is given by $B_{ij} = \langle\psi_i'\psi_j\rangle$ where $\psi_l$ represents basis polynomial $l$ and $i,j=0,\dots,P$ where $P$ is the order of the basis.

This method is implemented by computing $B_{ij}$ using Gaussian quadrature.

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

Definition at line 176 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type , typename value_type>

void Stokhos::RecurrenceBasis< ordinal_type, value_type >::evaluateBases	(	const value_type &	point,
		Teuchos::Array< value_type > &	basis_pts
	)		const

virtual

Evaluate each basis polynomial at given point point.

Size of returned array is given by size(), and coefficients are ordered from order 0 up to order order().

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

Definition at line 206 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type, typename value_type>

value_type Stokhos::RecurrenceBasis< ordinal_type, value_type >::evaluate	(	const value_type &	point,
		ordinal_type	order
	)		const

virtual

Evaluate basis polynomial given by order order at given point point.

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

Definition at line 240 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type , typename value_type >

void Stokhos::RecurrenceBasis< ordinal_type, value_type >::print ( std::ostream & os ) const

virtual

Print basis to stream os.

Reimplemented from Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

Definition at line 269 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type , typename value_type >

const std::string & Stokhos::RecurrenceBasis< ordinal_type, value_type >::getName ( ) const

virtual

Return string name of basis.

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

Definition at line 302 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type, typename value_type>

void Stokhos::RecurrenceBasis< ordinal_type, value_type >::getQuadPoints	(	ordinal_type	quad_order,
		Teuchos::Array< value_type > &	points,
		Teuchos::Array< value_type > &	weights,
		Teuchos::Array< Teuchos::Array< value_type > > &	values
	)		const

virtual

Compute quadrature points, weights, and values of basis polynomials at given set of points points.

quad_order specifies the order to which the quadrature should be accurate, not the number of quadrature points. The number of points is given by (quad_order + 1) / 2. Note however the passed arrays do NOT need to be sized correctly on input as they will be resized appropriately.

The quadrature points and weights are computed from the three-term recurrence by solving a tri-diagional symmetric eigenvalue problem (see Gene H. Golub and John H. Welsch, "Calculation of Gauss Quadrature Rules", Mathematics of Computation, Vol. 23, No. 106 (Apr., 1969), pp. 221-230).

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

Definition at line 310 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type, typename value_type >

ordinal_type Stokhos::RecurrenceBasis< ordinal_type, value_type >::quadDegreeOfExactness ( ordinal_type n ) const

virtual

Return polynomial degree of exactness for a given number of quadrature points.

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

Reimplemented in Stokhos::ClenshawCurtisLegendreBasis< ordinal_type, value_type >, and Stokhos::GaussPattersonLegendreBasis< ordinal_type, value_type >.

Definition at line 410 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type, typename value_type >

ordinal_type Stokhos::RecurrenceBasis< ordinal_type, value_type >::coefficientGrowth ( ordinal_type n ) const

virtual

Evaluate coefficient growth rule for Smolyak-type bases.

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

Reimplemented in Stokhos::ClenshawCurtisLegendreBasis< ordinal_type, value_type >, and Stokhos::GaussPattersonLegendreBasis< ordinal_type, value_type >.

Definition at line 418 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type, typename value_type >

ordinal_type Stokhos::RecurrenceBasis< ordinal_type, value_type >::pointGrowth ( ordinal_type n ) const

virtual

Evaluate point growth rule for Smolyak-type bases.

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

Reimplemented in Stokhos::ClenshawCurtisLegendreBasis< ordinal_type, value_type >, and Stokhos::GaussPattersonLegendreBasis< ordinal_type, value_type >.

Definition at line 430 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type, typename value_type>

virtual LevelToOrderFnPtr Stokhos::RecurrenceBasis< ordinal_type, value_type >::getSparseGridGrowthRule ( ) const

inlinevirtual

Get sparse grid level_to_order mapping function.

Predefined functions are: webbur::level_to_order_linear_wn Symmetric Gaussian linear growth webbur::level_to_order_linear_nn Asymmetric Gaussian linear growth webbur::level_to_order_exp_cc Clenshaw-Curtis exponential growth webbur::level_to_order_exp_gp Gauss-Patterson exponential growth webbur::level_to_order_exp_hgk Genz-Keister exponential growth webbur::level_to_order_exp_f2 Fejer-2 exponential growth

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

Definition at line 189 of file Stokhos_RecurrenceBasis.hpp.

template<typename ordinal_type, typename value_type>

virtual void Stokhos::RecurrenceBasis< ordinal_type, value_type >::setSparseGridGrowthRule ( LevelToOrderFnPtr ptr )

inlinevirtual

Set sparse grid rule.

Definition at line 193 of file Stokhos_RecurrenceBasis.hpp.

template<typename ordinal_type , typename value_type>

void Stokhos::RecurrenceBasis< ordinal_type, value_type >::getRecurrenceCoefficients	(	Teuchos::Array< value_type > &	alpha,
		Teuchos::Array< value_type > &	beta,
		Teuchos::Array< value_type > &	delta,
		Teuchos::Array< value_type > &	gamma
	)		const

virtual

Return recurrence coefficients defined by above formula.

Definition at line 446 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type , typename value_type>

void Stokhos::RecurrenceBasis< ordinal_type, value_type >::evaluateBasesAndDerivatives	(	const value_type &	point,
		Teuchos::Array< value_type > &	vals,
		Teuchos::Array< value_type > &	derivs
	)		const

virtual

Evaluate basis polynomials and their derivatives at given point point.

Definition at line 224 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type, typename value_type>

virtual void Stokhos::RecurrenceBasis< ordinal_type, value_type >::setQuadZeroTol ( value_type tol )

inlinevirtual

Set tolerance for zero in quad point generation.

Definition at line 210 of file Stokhos_RecurrenceBasis.hpp.

template<typename ordinal_type, typename value_type>

virtual bool Stokhos::RecurrenceBasis< ordinal_type, value_type >::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

protectedpure virtual

Compute recurrence coefficients.

Derived classes should implement this method to compute their recurrence coefficients. n is the number of coefficients to compute. Return value indicates whether coefficients correspond to normalized (i.e., orthonormal) polynomials.

Note: Owing to the description above, gamma should be an array of length n+1.

Implemented in Stokhos::LanczosPCEBasis< ordinal_type, value_type >, Stokhos::LanczosProjPCEBasis< ordinal_type, value_type >, Stokhos::LanczosPCEBasis< OrdinalType, ValueType >, Stokhos::LanczosProjPCEBasis< OrdinalType, ValueType >, Stokhos::JacobiBasis< ordinal_type, value_type >, Stokhos::StieltjesPCEBasis< ordinal_type, value_type >, Stokhos::StieltjesPCEBasis< OrdinalType, ValueType >, Stokhos::HouseTriDiagPCEBasis< ordinal_type, value_type >, Stokhos::HouseTriDiagPCEBasis< OrdinalType, ValueType >, Stokhos::MonoProjPCEBasis< ordinal_type, value_type >, Stokhos::MonoProjPCEBasis< OrdinalType, ValueType >, Stokhos::StieltjesBasis< ordinal_type, value_type, func_type >, Stokhos::DiscretizedStieltjesBasis< ordinal_type, value_type >, Stokhos::HermiteBasis< ordinal_type, value_type >, Stokhos::LegendreBasis< ordinal_type, value_type >, Stokhos::HermiteBasis< OrdinalType, ValueType >, and Stokhos::LegendreBasis< OrdinalType, ValueType >.

template<typename ordinal_type , typename value_type >

void Stokhos::RecurrenceBasis< ordinal_type, value_type >::setup ( )

protectedvirtual

Setup basis after computing recurrence coefficients.

Derived classes should call this method after computing their recurrence coefficients in their constructor to finish setting up the basis.

Reimplemented in Stokhos::LanczosPCEBasis< ordinal_type, value_type >, Stokhos::LanczosProjPCEBasis< ordinal_type, value_type >, Stokhos::LanczosPCEBasis< OrdinalType, ValueType >, Stokhos::LanczosProjPCEBasis< OrdinalType, ValueType >, Stokhos::StieltjesPCEBasis< ordinal_type, value_type >, Stokhos::StieltjesPCEBasis< OrdinalType, ValueType >, and Stokhos::StieltjesBasis< ordinal_type, value_type, func_type >.

Definition at line 55 of file Stokhos_RecurrenceBasisImp.hpp.

template<typename ordinal_type , typename value_type>

void Stokhos::RecurrenceBasis< ordinal_type, value_type >::normalizeRecurrenceCoefficients	(	Teuchos::Array< value_type > &	alpha,
		Teuchos::Array< value_type > &	beta,
		Teuchos::Array< value_type > &	delta,
		Teuchos::Array< value_type > &	gamma
	)		const