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 < Stokhos::Dense3Tensor < ordinal_type, value_type > >	computeTripleProductTensor () const =0
	Compute triple product tensor.
virtual Teuchos::RCP < Stokhos::Sparse3Tensor < ordinal_type, value_type > >	computeSparseTripleProductTensor (ordinal_type order) const =0
virtual Teuchos::RCP < Teuchos::SerialDenseMatrix < ordinal_type, value_type > >	computeDerivDoubleProductTensor () 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
	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 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 int	getSparseGridRule () const =0
	Get sparse grid rule number as defined by Dakota's `webbur` package.
virtual void	setSparseGridRule (int rule)=0
	Set sparse grid rule.
virtual int	getSparseGridGrowthRule () const =0
	Get sparse grid rule growth rule as defined by Dakota's `webbur` package.
virtual void	setSparseGridGrowthRule (int rule)=0
	Set sparse grid growth rule.

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 Teuchos::RCP<OneDOrthogPolyBasis<ordinal_type,value_type> > Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >::cloneWithOrder ( ordinal_type p ) const [pure 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.

template<typename ordinal_type , typename value_type >

virtual Teuchos::RCP< Teuchos::SerialDenseMatrix<ordinal_type, value_type> > Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >::computeDerivDoubleProductTensor ( ) 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::PecosOneDOrthogPolyBasis< ordinal_type, value_type >, and Stokhos::RecurrenceBasis< ordinal_type, value_type >.

template<typename ordinal_type , typename value_type >

virtual Teuchos::RCP< Stokhos::Dense3Tensor<ordinal_type, value_type> > Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >::computeTripleProductTensor ( ) 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 $i,j=0,\dots,P$ where $P$ is size()-1 and $k=0,\dots,p$ where $p$ is the supplied order.

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

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::PecosOneDOrthogPolyBasis< ordinal_type, value_type >, and Stokhos::RecurrenceBasis< 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::HouseTriDiagPCEBasis< ordinal_type, value_type >, Stokhos::MonoProjPCEBasis< ordinal_type, value_type >, Stokhos::LanczosPCEBasis< ordinal_type, value_type >, Stokhos::LanczosProjPCEBasis< ordinal_type, value_type >, Stokhos::PecosOneDOrthogPolyBasis< ordinal_type, value_type >, Stokhos::RecurrenceBasis< ordinal_type, value_type >, Stokhos::StieltjesBasis< ordinal_type, value_type, func_type >, and Stokhos::StieltjesPCEBasis< ordinal_type, value_type >.

template<typename ordinal_type , typename value_type >

virtual int Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >::getSparseGridGrowthRule ( ) const [pure virtual]

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

This method is needed for building Smolyak sparse grids out of this basis. The current rule definitions are: 1 Default growth 2 Slow linear 3 Slow linear odd 4 Moderate linear 5 Slow exponential 6 Moderate exponential 7 Full exponential

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

template<typename ordinal_type , typename value_type >

virtual int Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >::getSparseGridRule ( ) 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. The current rule definitions are: 1 Clenshaw-Curtis 2 Fejer Type 2 3 Gauss-Patterson 4 Gauss-Legendre 5 Gauss-Hermite 6 Generalized Gauss-Hermite 7 Gauss-Laguerre 8 Generalized Gauss-Laguerre 9 Gauss-Jacobi 10 Golub-Welsch (Gauss points for arbitrary weight function) 11 Genz-Keister (Gauss-Patterson-type Hermite) 12 Newton-Cotes

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

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

Stokhos_OneDOrthogPolyBasis.hpp

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 >