Jacobi polynomial basis. More...

#include <Stokhos_JacobiBasis.hpp>

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

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

Public Member Functions
	JacobiBasis (ordinal_type p, value_type alphaIndex, value_type betaIndex, bool normalize=false)
	Constructor.
	~JacobiBasis ()
	Destructor.
Implementation of Stokhos::OneDOrthogPolyBasis methods
virtual ordinal_type	getRule () const
	Get sparse grid rule number as defined by Dakota's `webbur` package.
virtual value_type	getQuadWeightFactor () const
	Get quadrature weight factor as defined by Dakota's `webbur` package.
virtual value_type	getQuadPointFactor () const
	Get quadrature point factor as defined by Dakota's `webbur` package.
Protected Member Functions
Implementation of Stokhos::RecurrenceBasis methods
virtual void	computeRecurrenceCoefficients (ordinal_type n, Teuchos::Array< value_type > &alpha, Teuchos::Array< value_type > &beta, Teuchos::Array< value_type > &delta) const
	Compute recurrence coefficients.

Detailed Description

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

Jacobi polynomial basis.

Jacobi polynomials are defined by the recurrence relationship

$A_k \psi_{k+1}(x) = \left(B_k-x C_k\right) \psi_k(x) - D_k \psi_{k-1}(x)\right)$

with $\psi_{-1}(x) = 0$ and $\psi_{0}(x) = 1$ where

$A_n = 2 (n+1)(n+\alpha+\beta+1)(2 n + \alpha+\beta)$

$B_n = -(2n+\alpha+\beta+1)(\alpha^2 - \beta^2)$

$C_n = (2n+\alpha+\beta)_3$

$D_n = 2(n+\alpha)(n+\beta)(2n+\alpha+\beta+2).$

In Stokhos notation we have

$\gamma_{k+1}=1/A_{k}$

$\alpha_k = B_k \] \f[ \delta_k = C_k \] \f[ \beta_k = D_k. \] The corresponding density function is \f[ \rho(x) = w_{\alpha_,\beta}(1-x)^\alpha (1+x)^\beta, \quad x\in[-1,1]$

with

$w_{\alpha,\beta}^{-1}=\frac{2^{\alpha+\beta+1}}{\alpha+\beta+1} \frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{\Gamma(\alpha+\beta+1)}.$

This class implements computeRecurrenceCoefficients() using the above formula.

Author:: Kevin Long (kevin.long@ttu.edu)

Constructor & Destructor Documentation

template<typename ordinal_type , typename value_type >

Stokhos::JacobiBasis< ordinal_type, value_type >::JacobiBasis	(	ordinal_type	p,
		value_type	alphaIndex,
		value_type	betaIndex,
		bool	normalize = `false`
	)

Constructor.

Parameters:

p	order of the basis
normalize	whether polynomials should be given unit norm

Member Function Documentation

template<typename ordinal_type , typename value_type >

virtual value_type Stokhos::JacobiBasis< ordinal_type, value_type >::getQuadPointFactor ( ) const [inline, 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.

template<typename ordinal_type , typename value_type >

virtual value_type Stokhos::JacobiBasis< ordinal_type, value_type >::getQuadWeightFactor ( ) const [inline, 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.

template<typename ordinal_type , typename value_type >

virtual ordinal_type Stokhos::JacobiBasis< ordinal_type, value_type >::getRule ( ) 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.

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

Stokhos_JacobiBasis.hpp
Stokhos_JacobiBasisImp.hpp

Public Member Functions

Protected Member Functions

Detailed Description

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

Constructor & Destructor Documentation

Member Function Documentation

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