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

#include <Stokhos_RecurrenceBasis.hpp>

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

[legend]

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

Public Member Functions
virtual	~RecurrenceBasis ()
	Destructor.

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

virtual	~OneDOrthogPolyBasis ()
	Destructor.

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.

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

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

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.

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.

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.

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

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

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

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`.

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.

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

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.

	RecurrenceBasis (const std::string &name, ordinal_type p, bool normalize, GrowthPolicy growth=SLOW_GROWTH)
	Constructor to be called by derived classes.

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

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.

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

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

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.

Constructor & Destructor Documentation

◆ RecurrenceBasis()

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).

Member Function Documentation

◆ coefficientGrowth()

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 >.

◆ computeDerivDoubleProductTensor()

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

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

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

◆ computeRecurrenceCoefficients()

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.

◆ computeSparseTripleProductTensor()

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 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 and $i,j=0,\dots,P$ where is size()-1 and $k=0,\dots,p$ where is the supplied order.

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

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

◆ computeTripleProductTensor()

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 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 and $i,j=0,\dots,P$ where is size()-1 and $k=0,\dots,p$ where is the supplied order.

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

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

◆ evaluate()

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 >.

◆ evaluateBases()

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 >.

◆ getName()

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 >.

◆ getQuadPoints()

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 >.

Reimplemented in Stokhos::HouseTriDiagPCEBasis< ordinal_type, value_type >, Stokhos::MonoProjPCEBasis< ordinal_type, value_type >, Stokhos::ClenshawCurtisLegendreBasis< ordinal_type, value_type >, Stokhos::GaussPattersonLegendreBasis< ordinal_type, value_type >, Stokhos::LanczosPCEBasis< ordinal_type, value_type >, Stokhos::LanczosProjPCEBasis< ordinal_type, value_type >, Stokhos::StieltjesBasis< ordinal_type, value_type, func_type >, and Stokhos::StieltjesPCEBasis< ordinal_type, value_type >.

Referenced by Stokhos::HouseTriDiagPCEBasis< ordinal_type, value_type >::getQuadPoints(), Stokhos::MonoProjPCEBasis< ordinal_type, value_type >::getQuadPoints(), Stokhos::LanczosPCEBasis< ordinal_type, value_type >::getQuadPoints(), Stokhos::LanczosProjPCEBasis< ordinal_type, value_type >::getQuadPoints(), Stokhos::StieltjesBasis< ordinal_type, value_type, func_type >::getQuadPoints(), Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::getQuadPoints(), and Stokhos::LanczosPCEBasis< ordinal_type, value_type >::LanczosPCEBasis().

◆ getSparseGridGrowthRule()

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 >.

References Stokhos::RecurrenceBasis< ordinal_type, value_type >::sparse_grid_growth_rule.