Stokhos::DiscretizedStieltjesBasis< ordinal_type, value_type > Class Template Reference

Generates three-term recurrence using the Discretized Stieltjes procedure. More...

#include <Stokhos_DiscretizedStieltjesBasis.hpp>

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

[legend]Collaboration diagram for Stokhos::DiscretizedStieltjesBasis< ordinal_type, value_type >:

[legend]List of all members.

Public Member Functions
	DiscretizedStieltjesBasis (const std::string &name, const ordinal_type &p, const value_type(*weightFn)(const value_type), const value_type &leftEndPt, const value_type &rightEndPt, const bool normalize)
	Constructor.
	~DiscretizedStieltjesBasis ()
	Destructor.
value_type	eval_inner_product (const ordinal_type &order1, const ordinal_type &order2) const
	Evaluate inner product of two basis functions to test orthogonality.
Implementation of Stokhos::OneDOrthogPolyBasis methods
virtual ordinal_type	getRule () const
	Get sparse grid rule number as defined by Dakota's webbur package.
virtual value_type	getQuadWeightFactor () const
	Get quadrature weight factor as defined by Dakota's webbur package.
virtual value_type	getQuadPointFactor () const
	Get quadrature point factor as defined by Dakota's webbur package.
Protected Member Functions
value_type	evaluateRecurrence (const value_type &point, ordinal_type order, const Teuchos::Array< value_type > &alpha, const Teuchos::Array< value_type > &beta) const
	Evaluate 3-term recurrence formula using supplied coefficients.
value_type	evaluateWeight (const value_type &x) const
	Evaluates the scaled weight function.
value_type	expectedValue_tJ_nsquared (const ordinal_type &order, const Teuchos::Array< value_type > &alpha, const Teuchos::Array< value_type > &beta) const
	Approximates where = order.
value_type	expectedValue_J_nsquared (const ordinal_type &order, const Teuchos::Array< value_type > &alpha, const Teuchos::Array< value_type > &beta) const
	Approximates where = order.
Implementation of Stokhos::RecurrenceBasis methods
virtual void	computeRecurrenceCoefficients (ordinal_type n, Teuchos::Array< value_type > &alpha, Teuchos::Array< value_type > &beta, Teuchos::Array< value_type > &delta) const
	Compute recurrence coefficients.
Protected Attributes
value_type	scaleFactor
	Scale for the weight.
const value_type	leftEndPt_
	Left end point of domain.
const value_type	rightEndPt_
	Right end point of domain.
const value_type(*	weightFn_ )(const value_type)
	Weight function.
Teuchos::Array< value_type >	quad_points
	Quadrature points for discretized stieltjes procedure.
Teuchos::Array< value_type >	quad_weights
	Quadrature points for discretized stieltjes procedure.
Teuchos::Array< Teuchos::Array< value_type > >	quad_values
	Quadrature values for discretized stieltjes procedure.

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Detailed Description

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

Generates three-term recurrence using the Discretized Stieltjes procedure.

Class generates an orthogonal polynomial basis for a given weight function and interval. The Discretized Stiltjes procedure described in "On the Calculation of Rys Polynomials and Quadratures", Robin P. Sagar, Vedene H. Smith is used to generate the recurrence coefficients.

Please be aware that this method is not fullproof and that it appears to encounter trouble with strongly singular weights since Gaussan quadrature is used to compute the relevant integrals. For 'nice' weight functions the method seems relatively robust.

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Constructor & Destructor Documentation

template<typename ordinal_type, typename value_type>

Stokhos::DiscretizedStieltjesBasis< ordinal_type, value_type >::DiscretizedStieltjesBasis	(	const std::string &	name,
		const ordinal_type &	p,
		const value_type(*)(const value_type)	weightFn,
		const value_type &	leftEndPt,
		const value_type &	rightEndPt,
		const bool	normalize
	)

Constructor.

Parameters:

	name	name of the basis displayed when printing
	p	order of the basis
	weightFn	function pointer defining weight function
	leftEndPt	left end point of the domain of the weight function
	rightEndPt	right end point of the domain of the weight function
	normalize	whether polynomials should be given unit norm

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Member Function Documentation

template<typename ordinal_type, typename value_type>

virtual value_type Stokhos::DiscretizedStieltjesBasis< ordinal_type, value_type >::getQuadPointFactor

(

)

const [inline, virtual]

Get quadrature point factor as defined by Dakota's webbur package.

This method is needed for building Smolyak sparse grids out of this basis.

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

template<typename ordinal_type, typename value_type>

virtual value_type Stokhos::DiscretizedStieltjesBasis< ordinal_type, value_type >::getQuadWeightFactor

(

)

const [inline, virtual]

Get quadrature weight factor as defined by Dakota's webbur package.

This method is needed for building Smolyak sparse grids out of this basis.

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

template<typename ordinal_type, typename value_type>

virtual ordinal_type Stokhos::DiscretizedStieltjesBasis< ordinal_type, value_type >::getRule

(

)

const [inline, virtual]

Get sparse grid rule number as defined by Dakota's webbur package.

This method is needed for building Smolyak sparse grids out of this basis. A rule number of 10 is not defined by the webbur package, and this rule number is used internally by Stokhos::SparseGridQuadrature to pass an arbitrary one-dimensional basis to that package.

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

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

• Stokhos_DiscretizedStieltjesBasis.hpp
• Stokhos_DiscretizedStieltjesBasisImp.hpp

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