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 >:
Inheritance graph
[legend]

List of all members.

Public Member Functions

 OneDOrthogPolyBasis ()
 Default constructor.
virtual ~OneDOrthogPolyBasis ()
 Destructor.
virtual ordinal_type order () const =0
 Return order of basis (largest monomial degree $P$).
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
< 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 =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 int getSparseGridRule () const =0
 Get sparse grid rule number as defined by Dakota's webbur package.
virtual int getSparseGridGrowthRule () const =0
 Get sparse grid rule growth rule 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 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::PecosOneDOrthogPolyBasis< ordinal_type, value_type >, Stokhos::RecurrenceBasis< ordinal_type, value_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.

Implemented in Stokhos::ClenshawCurtisLegendreBasis< ordinal_type, value_type >, 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)

Implemented in Stokhos::ClenshawCurtisLegendreBasis< ordinal_type, value_type >, Stokhos::HermiteBasis< ordinal_type, value_type >, Stokhos::LegendreBasis< ordinal_type, value_type >, 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:
 All Classes Namespaces Functions Variables Typedefs Enumerations Enumerator
Generated on Wed Apr 13 09:58:15 2011 for Stokhos by  doxygen 1.6.3