Multivariate orthogonal polynomial basis generated from a total-order complete-polynomial tensor product of univariate polynomials. More...

#include <Stokhos_CompletePolynomialBasis.hpp>

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

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

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Public Member Functions
	CompletePolynomialBasis (const Teuchos::Array< Teuchos::RCP< const OneDOrthogPolyBasis< ordinal_type, value_type > > > &bases, const value_type &sparse_tol=1.0e-12, bool use_old_cijk_alg=false, const Teuchos::RCP< Teuchos::Array< value_type > > &deriv_coeffs=Teuchos::null)
	Constructor. More...

virtual	~CompletePolynomialBasis ()
	Destructor.

Implementation of Stokhos::OrthogPolyBasis methods
ordinal_type	order () const
	Return order of basis.

ordinal_type	dimension () const
	Return dimension of basis.

virtual ordinal_type	size () const
	Return total size of basis.

virtual const Teuchos::Array < value_type > &	norm_squared () const
	Return array storing norm-squared of each basis polynomial. More...

virtual const value_type &	norm_squared (ordinal_type i) const
	Return norm squared of basis polynomial `i`.

virtual Teuchos::RCP < Stokhos::Sparse3Tensor < ordinal_type, value_type > >	computeTripleProductTensor () const
	Compute triple product tensor. More...

virtual Teuchos::RCP < Stokhos::Sparse3Tensor < ordinal_type, value_type > >	computeLinearTripleProductTensor () const
	Compute linear triple product tensor where k = 0,1,..,d.

virtual value_type	evaluateZero (ordinal_type i) const
	Evaluate basis polynomial `i` at zero.

virtual void	evaluateBases (const Teuchos::ArrayView< const value_type > &point, Teuchos::Array< value_type > &basis_vals) const
	Evaluate basis polynomials at given point `point`. More...

virtual void	print (std::ostream &os) const
	Print basis to stream `os`.

virtual const std::string &	getName () const
	Return string name of basis.

Implementation of Stokhos::ProductBasis methods
virtual const MultiIndex < ordinal_type > &	term (ordinal_type i) const
	Get orders of each coordinate polynomial given an index `i`. More...

virtual ordinal_type	index (const MultiIndex< ordinal_type > &term) const
	Get index of the multivariate polynomial given orders of each coordinate. More...

Teuchos::Array< Teuchos::RCP < const OneDOrthogPolyBasis < ordinal_type, value_type > > >	getCoordinateBases () const
	Return coordinate bases. More...

virtual MultiIndex< ordinal_type >	getMaxOrders () const
	Return maximum order allowable for each coordinate basis.

Implementation of Stokhos::DerivBasis methods
virtual Teuchos::RCP < Stokhos::Dense3Tensor < ordinal_type, value_type > >	computeDerivTripleProductTensor (const Teuchos::RCP< const Teuchos::SerialDenseMatrix< ordinal_type, value_type > > &Bij, const Teuchos::RCP< const Stokhos::Sparse3Tensor< ordinal_type, value_type > > &Cijk) const
	Compute triple product tensor $D_{ijk} = \langle\Psi_i\Psi_j D_v\Psi_k\rangle$ where $D_v\Psi_k$ represents the derivative of $\Psi_k$ in the direction . More...

virtual Teuchos::RCP < Teuchos::SerialDenseMatrix < ordinal_type, value_type > >	computeDerivDoubleProductTensor () const
	Compute double product tensor $B_{ij} = \langle \Psi_i D_v\Psi_j\rangle$ where $D_v\Psi_j$ represents the derivative of $\Psi_j$ in the direction . More...

Public Member Functions inherited from Stokhos::ProductBasis< ordinal_type, value_type >
	ProductBasis ()
	Constructor.

virtual	~ProductBasis ()
	Destructor.

Public Member Functions inherited from Stokhos::OrthogPolyBasis< ordinal_type, value_type >
	OrthogPolyBasis ()
	Constructor.

virtual	~OrthogPolyBasis ()
	Destructor.

Public Member Functions inherited from Stokhos::DerivBasis< ordinal_type, value_type >
	DerivBasis ()
	Constructor.

virtual	~DerivBasis ()
	Destructor.

Protected Types
typedef Stokhos::CompletePolynomialBasisUtils < ordinal_type, value_type >	CPBUtils

typedef Stokhos::Sparse3Tensor < ordinal_type, value_type >	Cijk_type
	Short-hand for Cijk.

Protected Member Functions
virtual Teuchos::RCP < Stokhos::Sparse3Tensor < ordinal_type, value_type > >	computeTripleProductTensorOld (ordinal_type order) const
	Compute triple product tensor using old algorithm.

virtual Teuchos::RCP < Stokhos::Sparse3Tensor < ordinal_type, value_type > >	computeTripleProductTensorNew (ordinal_type order) const
	Compute triple product tensor using new algorithm.

Protected Attributes
std::string	name
	Name of basis.

ordinal_type	p
	Total order of basis.

ordinal_type	d
	Total dimension of basis.

ordinal_type	sz
	Total size of basis.

Teuchos::Array< Teuchos::RCP < const OneDOrthogPolyBasis < ordinal_type, value_type > > >	bases
	Array of bases.

Teuchos::Array< ordinal_type >	basis_orders
	Array storing order of each basis.

value_type	sparse_tol
	Tolerance for computing sparse Cijk.

bool	use_old_cijk_alg
	Use old algorithm for computing Cijk.

Teuchos::RCP< Teuchos::Array < value_type > >	deriv_coeffs
	Coefficients for derivative.

Teuchos::Array< value_type >	norms
	Norms.

Teuchos::Array< MultiIndex < ordinal_type > >	terms
	2-D array of basis terms

Teuchos::Array< ordinal_type >	num_terms
	Number of terms up to each order.

Teuchos::Array< Teuchos::Array < value_type > >	basis_eval_tmp
	Temporary array used in basis evaluation.

Detailed Description

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

Multivariate orthogonal polynomial basis generated from a total-order complete-polynomial tensor product of univariate polynomials.

The multivariate polynomials are given by

$\Psi_i(x) = \psi_{i_1}(x_1)\dots\psi_{i_d}(x_d)$

where $d$ is the dimension of the basis and $i_1+\dots+ i_d\leq p$ , where $p$ is the order of the basis. The size of the basis is given by $(d+p)!/(d!p!)$ .

NOTE: Currently all coordinate bases must be of the samer order $p$ .

Constructor & Destructor Documentation

template<typename ordinal_type , typename value_type >

Stokhos::CompletePolynomialBasis< ordinal_type, value_type >::CompletePolynomialBasis	(	const Teuchos::Array< Teuchos::RCP< const OneDOrthogPolyBasis< ordinal_type, value_type > > > &	bases,
		const value_type &	sparse_tol = `1.0e-12`,
		bool	use_old_cijk_alg = `false`,
		const Teuchos::RCP< Teuchos::Array< value_type > > &	deriv_coeffs = `Teuchos::null`
	)

Constructor.

Parameters

bases	array of 1-D coordinate bases
sparse_tol	tolerance used to drop terms in sparse triple-product tensors
use_old_cijk_alg	use old algorithm for computing the sparse triple product tensor (significantly slower, but simpler)
deriv_coeffs	direction used to define derivatives for derivative product tensors. Defaults to all one's if not supplied.

Member Function Documentation

template<typename ordinal_type , typename value_type >

Teuchos::RCP< Teuchos::SerialDenseMatrix< ordinal_type, value_type > > Stokhos::CompletePolynomialBasis< ordinal_type, value_type >::computeDerivDoubleProductTensor ( ) const

virtual

Compute double product tensor $B_{ij} = \langle \Psi_i D_v\Psi_j\rangle$ where $D_v\Psi_j$ represents the derivative of $\Psi_j$ in the direction $v$ .

The definition of $v$ is defined by the deriv_coeffs constructor argument.

Implements Stokhos::DerivBasis< ordinal_type, value_type >.

template<typename ordinal_type , typename value_type >

Teuchos::RCP< Stokhos::Dense3Tensor< ordinal_type, value_type > > Stokhos::CompletePolynomialBasis< ordinal_type, value_type >::computeDerivTripleProductTensor	(	const Teuchos::RCP< const Teuchos::SerialDenseMatrix< ordinal_type, value_type > > &	Bij,
		const Teuchos::RCP< const Stokhos::Sparse3Tensor< ordinal_type, value_type > > &	Cijk
	)		const

virtual

Compute triple product tensor $D_{ijk} = \langle\Psi_i\Psi_j D_v\Psi_k\rangle$ where $D_v\Psi_k$ represents the derivative of $\Psi_k$ in the direction $v$ .

The definition of $v$ is defined by the deriv_coeffs constructor argument.

Implements Stokhos::DerivBasis< ordinal_type, value_type >.

template<typename ordinal_type , typename value_type >

Teuchos::RCP< Stokhos::Sparse3Tensor< ordinal_type, value_type > > Stokhos::CompletePolynomialBasis< 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 $i,j,k=0,\dots,P$ where $P$ is size()-1.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

template<typename ordinal_type , typename value_type >

void Stokhos::CompletePolynomialBasis< ordinal_type, value_type >::evaluateBases	(	const Teuchos::ArrayView< const value_type > &	point,
		Teuchos::Array< value_type > &	basis_vals
	)		const

virtual

Evaluate basis polynomials at given point point.

Size of returned array is given by size(), and coefficients are ordered from order 0 up to size size()-1.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

template<typename ordinal_type , typename value_type >

Teuchos::Array< Teuchos::RCP< const Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type > > > Stokhos::CompletePolynomialBasis< ordinal_type, value_type >::getCoordinateBases ( ) const

virtual

Return coordinate bases.

Array is of size dimension().

Implements Stokhos::ProductBasis< ordinal_type, value_type >.

template<typename ordinal_type , typename value_type >

ordinal_type Stokhos::CompletePolynomialBasis< ordinal_type, value_type >::index ( const MultiIndex< ordinal_type > & term ) const

virtual

Get index of the multivariate polynomial given orders of each coordinate.

Given the array term storing $i_1,\dots,\i_d$ , returns the index $i$ such that $\Psi_i(x) = \psi_{i_1}(x_1)\dots\psi_{i_d}(x_d)$ .

Implements Stokhos::ProductBasis< ordinal_type, value_type >.

template<typename ordinal_type , typename value_type >

const Teuchos::Array< value_type > & Stokhos::CompletePolynomialBasis< 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 size()-1.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

Referenced by Stokhos::CompletePolynomialBasis< ordinal_type, value_type >::CompletePolynomialBasis().

template<typename ordinal_type , typename value_type >

const Stokhos::MultiIndex< ordinal_type > & Stokhos::CompletePolynomialBasis< ordinal_type, value_type >::term ( ordinal_type i ) const

virtual

Get orders of each coordinate polynomial given an index i.

The returned array is of size $d$ , where $d$ is the dimension of the basis, and entry $l$ is given by $i_l$ where $\Psi_i(x) = \psi_{i_1}(x_1)\dots\psi_{i_d}(x_d)$ .

Implements Stokhos::ProductBasis< ordinal_type, value_type >.

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

Stokhos_CompletePolynomialBasis.hpp
Stokhos_CompletePolynomialBasisImp.hpp

Public Member Functions

Protected Types

Protected Member Functions

Protected Attributes

Detailed Description

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

Constructor & Destructor Documentation

Member Function Documentation

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