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Stokhos::JacobiBasis< ordinal_type, value_type > Class Template Reference

Jacobi polynomial basis. More...

#include <Stokhos_JacobiBasis.hpp>

Inheritance diagram for Stokhos::JacobiBasis< ordinal_type, value_type >:
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Public Member Functions

 JacobiBasis (ordinal_type p, value_type alphaIndex, value_type betaIndex, bool normalize=false, GrowthPolicy growth=SLOW_GROWTH)
 Constructor. More...
 
 ~JacobiBasis ()
 Destructor.
 
Implementation of Stokhos::OneDOrthogPolyBasis methods
virtual Teuchos::RCP
< OneDOrthogPolyBasis
< ordinal_type, value_type > > 
cloneWithOrder (ordinal_type p) const
 Clone this object with the option of building a higher order basis. More...
 
- Public Member Functions inherited from Stokhos::RecurrenceBasis< ordinal_type, value_type >
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.
 
virtual void setQuadZeroTol (value_type tol)
 Set tolerance for zero in quad point generation.
 
virtual ordinal_type order () const
 Return order of basis (largest monomial degree $P$).
 
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. More...
 
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. More...
 
virtual Teuchos::RCP
< Stokhos::Sparse3Tensor
< ordinal_type, value_type > > 
computeSparseTripleProductTensor (ordinal_type order) const
 Compute triple product tensor. More...
 
virtual Teuchos::RCP
< Teuchos::SerialDenseMatrix
< ordinal_type, value_type > > 
computeDerivDoubleProductTensor () const
 Compute derivative double product tensor. More...
 
virtual void evaluateBases (const value_type &point, Teuchos::Array< value_type > &basis_pts) const
 Evaluate each basis polynomial at given point point. More...
 
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. More...
 
virtual ordinal_type quadDegreeOfExactness (ordinal_type n) const
 
virtual ordinal_type coefficientGrowth (ordinal_type n) const
 Evaluate coefficient growth rule for Smolyak-type bases.
 
virtual ordinal_type pointGrowth (ordinal_type n) const
 Evaluate point growth rule for Smolyak-type bases.
 
virtual LevelToOrderFnPtr getSparseGridGrowthRule () const
 Get sparse grid level_to_order mapping function. More...
 
virtual void setSparseGridGrowthRule (LevelToOrderFnPtr ptr)
 Set sparse grid rule.
 
- Public Member Functions inherited from Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >
 OneDOrthogPolyBasis ()
 Default constructor.
 
virtual ~OneDOrthogPolyBasis ()
 Destructor.
 
virtual void setSparseGridGrowthRule (LevelToOrderFnPtr ptr)=0
 Set sparse grid rule.
 

Protected Member Functions

 JacobiBasis (ordinal_type p, const JacobiBasis &basis)
 Copy constructor with specified order.
 
Implementation of Stokhos::RecurrenceBasis methods
virtual bool computeRecurrenceCoefficients (ordinal_type n, Teuchos::Array< value_type > &alpha, Teuchos::Array< value_type > &beta, Teuchos::Array< value_type > &delta, Teuchos::Array< value_type > &gamma) const
 Compute recurrence coefficients.
 
- Protected Member Functions inherited from Stokhos::RecurrenceBasis< ordinal_type, value_type >
 RecurrenceBasis (const std::string &name, ordinal_type p, bool normalize, GrowthPolicy growth=SLOW_GROWTH)
 Constructor to be called by derived classes. More...
 
 RecurrenceBasis (ordinal_type p, const RecurrenceBasis &basis)
 Copy constructor with specified order.
 
virtual void setup ()
 Setup basis after computing recurrence coefficients. More...
 
void normalizeRecurrenceCoefficients (Teuchos::Array< value_type > &alpha, Teuchos::Array< value_type > &beta, Teuchos::Array< value_type > &delta, Teuchos::Array< value_type > &gamma) const
 Normalize coefficients.
 

Additional Inherited Members

- Public Types inherited from Stokhos::RecurrenceBasis< ordinal_type, value_type >
typedef OneDOrthogPolyBasis
< ordinal_type, value_type >
::LevelToOrderFnPtr 
LevelToOrderFnPtr
 Function pointer needed for level_to_order mappings.
 
- Public Types inherited from Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >
typedef int(* LevelToOrderFnPtr )(int level, int growth)
 Function pointer needed for level_to_order mappings.
 
- Protected Attributes inherited from Stokhos::RecurrenceBasis< ordinal_type, value_type >
std::string name
 Name of basis.
 
ordinal_type p
 Order of basis.
 
bool normalize
 Normalize basis.
 
GrowthPolicy growth
 Smolyak growth policy.
 
value_type quad_zero_tol
 Tolerance for quadrature points near zero.
 
LevelToOrderFnPtr sparse_grid_growth_rule
 Sparse grid growth rule (as determined by Pecos)
 
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::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.nosp@m..lon.nosp@m.g@ttu.nosp@m..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,
Stokhos::GrowthPolicy  growth = SLOW_GROWTH 
)

Constructor.

Parameters
porder of the basis
normalizewhether polynomials should be given unit norm

References Stokhos::RecurrenceBasis< ordinal_type, value_type >::setSparseGridGrowthRule(), and Stokhos::RecurrenceBasis< ordinal_type, value_type >::setup().

Member Function Documentation

template<typename ordinal_type , typename value_type >
Teuchos::RCP< Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type > > Stokhos::JacobiBasis< ordinal_type, value_type >::cloneWithOrder ( ordinal_type  p) const
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.

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.


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