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

Abstract base class for 1-D orthogonal polynomials. More...

#include <Stokhos_OneDOrthogPolyBasis.hpp>

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


Public Member Functions
	OneDOrthogPolyBasis ()
	Default constructor.
virtual	~OneDOrthogPolyBasis ()
	Destructor.
virtual ordinal_type	order () const =0
	Return order of basis (largest monomial degree ).
virtual ordinal_type	size () const =0
	Return total size of basis (given by order() + 1).
virtual const Teuchos::Array< value_type > &	norm_squared () const =0
	Return array storing norm-squared of each basis polynomial.
virtual const value_type &	norm_squared (ordinal_type i) const =0
	Return norm squared of basis polynomial `i`.
virtual Teuchos::RCP< const Stokhos::Dense3Tensor< ordinal_type, value_type > >	getTripleProductTensor () const =0
	Compute triple product tensor.
virtual Teuchos::RCP< const Teuchos::SerialDenseMatrix< ordinal_type, value_type > >	getDerivDoubleProductTensor () const =0
	Compute derivative double product tensor.
virtual void	evaluateBases (const value_type &point, Teuchos::Array< value_type > &basis_pts) const =0
	Evaluate each basis polynomial at given point `point`.
virtual value_type	evaluate (const value_type &point, ordinal_type order) const =0
	Evaluate basis polynomial given by order `order` at given point `point`.
virtual void	print (std::ostream &os) const =0
	Print basis to stream `os`.
virtual const std::string &	getName () const =0
	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 =0
	Compute quadrature points, weights, and values of basis polynomials at given set of points `points`.
virtual ordinal_type	getRule () const =0
	Get sparse grid rule number as defined by Dakota's `webbur` package.
virtual value_type	getQuadWeightFactor () const =0
	Get quadrature weight factor as defined by Dakota's `webbur` package.
virtual value_type	getQuadPointFactor () const =0
	Get quadrature point factor as defined by Dakota's `webbur` package.

Detailed Description

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

Abstract base class for 1-D orthogonal polynomials.

This class provides an abstract interface for univariate orthogonal polynomials. Orthogonality is defined by the inner product

$(f,g) = \langle fg \rangle = \int_{-\infty}^{\infty} f(x)g(x) \rho(x) dx$

where $\rho$ is the density function of the measure associated with the orthogonal polynomials. See Stokhos::RecurrenceBasis for a general implementation of this interface based on the three-term recurrence satisfied by these polynomials. Multivariate polynomials can be formed from a collection of univariate polynomials through tensor products (see Stokhos::CompletePolynomialBasis).

Like most classes in Stokhos, the class is templated on the ordinal and value types. Typically ordinal_type = int and value_type = double.

Member Function Documentation

template<typename ordinal_type, typename value_type>

virtual void Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >::evaluateBases	(	const value_type &	point,
		Teuchos::Array< value_type > &	basis_pts
	)			const `[pure 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().

Implemented in Stokhos::RecurrenceBasis< ordinal_type, value_type >.

template<typename ordinal_type, typename value_type>

virtual Teuchos::RCP< const Teuchos::SerialDenseMatrix<ordinal_type, value_type> > Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >::getDerivDoubleProductTensor ( ) const [pure 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.

Implemented in Stokhos::RecurrenceBasis< ordinal_type, value_type >.

template<typename ordinal_type, typename value_type>

virtual value_type Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >::getQuadPointFactor ( ) const [pure 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.

Implemented in Stokhos::DiscretizedStieltjesBasis< ordinal_type, value_type >, Stokhos::HermiteBasis< ordinal_type, value_type >, and Stokhos::LegendreBasis< ordinal_type, value_type >.

template<typename ordinal_type, typename value_type>

virtual void Stokhos::OneDOrthogPolyBasis< 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 `[pure 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.

Implemented in Stokhos::RecurrenceBasis< ordinal_type, value_type >.

template<typename ordinal_type, typename value_type>

virtual value_type Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >::getQuadWeightFactor ( ) const [pure 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.

Implemented in Stokhos::DiscretizedStieltjesBasis< ordinal_type, value_type >, Stokhos::HermiteBasis< ordinal_type, value_type >, and Stokhos::LegendreBasis< ordinal_type, value_type >.

template<typename ordinal_type, typename value_type>

virtual ordinal_type Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >::getRule ( ) const [pure 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.

Implemented in Stokhos::DiscretizedStieltjesBasis< ordinal_type, value_type >, Stokhos::HermiteBasis< ordinal_type, value_type >, and Stokhos::LegendreBasis< ordinal_type, value_type >.

template<typename ordinal_type, typename value_type>

virtual Teuchos::RCP< const Stokhos::Dense3Tensor<ordinal_type, value_type> > Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >::getTripleProductTensor ( ) const [pure 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 $i,j,k=0,\dots,P$ where $P$ is the order of the basis.

Implemented in Stokhos::RecurrenceBasis< ordinal_type, value_type >.

template<typename ordinal_type, typename value_type>

virtual const Teuchos::Array<value_type>& Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >::norm_squared ( ) const [pure 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().

Implemented in Stokhos::RecurrenceBasis< ordinal_type, value_type >.

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

Stokhos_OneDOrthogPolyBasis.hpp

Generated on Wed May 12 21:25:04 2010 for Stokhos by

1.4.7

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

Public Member Functions

Detailed Description

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

Member Function Documentation

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