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

$\psi_{k+1}(x) = \gamma_{k+1}\big( (\alpha_k - \delta_k x)\psi_k(x) - \beta_k\psi_{k-1}(x) \big)$

for $k=0,\dots,P$ where $\psi_{-1}(x) = 0$ , $\psi_{0}(x) = 1$ , and $\beta_{0} = 1$ . More...

#include <Stokhos_RecurrenceBasis.hpp>

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

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

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List of all members.

Public Member Functions
virtual	~RecurrenceBasis ()
	Destructor.
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`.
Implementation of Stokhos::OneDOrthogPolyBasis methods
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 < 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 int	getSparseGridRule () const
	Get sparse grid rule number as defined by Dakota's `webbur` package.
virtual int	getSparseGridGrowthRule () const
	Get sparse grid rule growth rule as defined by Dakota's `webbur` package.
Protected Member Functions
	RecurrenceBasis (const std::string &name, ordinal_type p, bool normalize)
	Constructor to be called by derived classes.
virtual void	computeRecurrenceCoefficients (ordinal_type n, Teuchos::Array< value_type > &alpha, Teuchos::Array< value_type > &beta, Teuchos::Array< value_type > &delta) const =0
	Compute recurrence coefficients.
void	setup ()
	Setup basis after computing recurrence coefficients.
Protected Attributes
std::string	name
	Name of basis.
ordinal_type	p
	Order of basis.
bool	normalize
	Normalize basis.
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.

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:

$\psi_{k+1}(x) = \gamma_{k+1}\big( (\alpha_k - \delta_k x)\psi_k(x) - \beta_k\psi_{k-1}(x) \big)$

for $k=0,\dots,P$ where $\psi_{-1}(x) = 0$ , $\psi_{0}(x) = 1$ , and $\beta_{0} = 1$ .

Derived classes implement the recurrence relationship by implementing computeRecurrenceCoefficients(). If normalize = true in the constructor, then $\gamma_k = 1/\sqrt{\langle\psi_k^2\rangle}$ , otherwise $\gamma_k = 1$ . Note that a three term recurrence can always be defined with $\delta_k = 1$ in which case the polynomials are monic. However typical normalizations of some polynomial families (see Stokhos::LegendreBasis) require the extra terms.

Constructor & Destructor Documentation

template<typename ordinal_type , typename value_type >

Stokhos::RecurrenceBasis< ordinal_type, value_type >::RecurrenceBasis	(	const std::string &	name,
		ordinal_type	p,
		bool	normalize
	)		`[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, and normalize indicates whether the basis polynomials should have unit-norm.

Member Function Documentation

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 $(i,j)$ 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.

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

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

template<typename ordinal_type , typename value_type >

virtual void 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
	)		const `[protected, pure virtual]`

Compute recurrence coefficients.

Derived classes should implement this method to compute their recurrence coefficients. n is the number of coefficients to compute. Derived classes should call this method in their constructor, followed be setup() to fully setup the basis. The method is also called by getQuadPoints() if a quadrature order greater than twice the polynomial order is requested.

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

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 $(i,j,k)$ 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.

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

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

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 >.

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::StieltjesPCEBasis< ordinal_type, value_type >.

template<typename ordinal_type , typename value_type >

virtual int Stokhos::RecurrenceBasis< ordinal_type, value_type >::getSparseGridGrowthRule ( ) const [inline, 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. Returns growth rule appropriate for Gaussian quadrature points.

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

Reimplemented in Stokhos::ClenshawCurtisLegendreBasis< ordinal_type, value_type >.

template<typename ordinal_type , typename value_type >

virtual int Stokhos::RecurrenceBasis< ordinal_type, value_type >::getSparseGridRule ( ) 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 >.

Reimplemented in Stokhos::ClenshawCurtisLegendreBasis< ordinal_type, value_type >, Stokhos::HermiteBasis< ordinal_type, value_type >, and Stokhos::LegendreBasis< ordinal_type, value_type >.

template<typename ordinal_type , typename value_type >

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

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

template<typename ordinal_type , typename value_type >

void Stokhos::RecurrenceBasis< ordinal_type, value_type >::setup ( ) [protected]

Setup basis after computing recurrence coefficients.

Derived classes should call this method after computing their recurrence coefficients in their constructor to finish setting up the basis.

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

Stokhos_RecurrenceBasis.hpp
Stokhos_RecurrenceBasisImp.hpp

Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

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

Constructor & Destructor Documentation

Member Function Documentation

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