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Stokhos::RecurrenceBasis< ordinal_type, value_type > Class Template Referenceabstract

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

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

for $k=0,\dots,P$ where $\psi_{-1}(x) = 0$, $\psi_{0}(x) = 1/\gamma_0$, and $\beta_{0} = 1 = \int d\lambda$. 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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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.
 
virtual void setQuadZeroTol (value_type tol)
 Set tolerance for zero in quad point generation.
 
- Public Member Functions inherited from Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >
 OneDOrthogPolyBasis ()
 Default constructor.
 
virtual ~OneDOrthogPolyBasis ()
 Destructor.
 
virtual Teuchos::RCP
< OneDOrthogPolyBasis
< ordinal_type, value_type > > 		cloneWithOrder (ordinal_type p) const =0
 Clone this object with the option of building a higher order basis. More...
 
virtual void setSparseGridGrowthRule (LevelToOrderFnPtr ptr)=0
 Set sparse grid rule.
 

Protected Member Functions

 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 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 =0
 Compute recurrence coefficients. More...
 
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.
 

Protected Attributes

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.
 

Implementation of Stokhos::OneDOrthogPolyBasis methods

typedef OneDOrthogPolyBasis
< ordinal_type, value_type >
::LevelToOrderFnPtr 		LevelToOrderFnPtr
 Function pointer needed for level_to_order mappings.
 
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.
 

Additional Inherited Members

- 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.
 

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:

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

for $k=0,\dots,P$ where $\psi_{-1}(x) = 0$, $\psi_{0}(x) = 1/\gamma_0$, and $\beta_{0} = 1 = \int d\lambda$.

Derived classes implement the recurrence relationship by implementing computeRecurrenceCoefficients(). If normalize = true in the constructor, then the recurrence relationship becomes:

\[ \sqrt{\frac{\gamma_{k+1}\beta_{k+1}}{\delta_{k+1}\delta_k}} \psi_{k+1}(x) = (x - \alpha_k/\delta_k)\psi_k(x) - \sqrt{\frac{\gamma_k\beta_k}{\delta_k\delta_{k-1}}} \psi_{k-1}(x) \]

for $k=0,\dots,P$ where $\psi_{-1}(x) = 0$, $\psi_{0}(x) = 1/\sqrt{\beta_0}$, Note that a three term recurrence can always be defined with $\gamma_k = \delta_k = 1$ in which case the polynomials are monic. However typical normalizations of some polynomial families (see Stokhos::LegendreBasis) require the extra terms. Also, the quadrature rule (points and weights) is the same regardless if the polynomials are normalized. However the normalization can affect other algorithms.

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,
Stokhos::GrowthPolicy  growth_ = SLOW_GROWTH 
)
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, normalize indicates whether the basis polynomials should have unit-norm, and quad_zero_tol is used to replace any quadrature point within this tolerance with zero (which can help with duplicate removal in sparse grid calculations).

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 bool 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,
Teuchos::Array< value_type > &  gamma 
) const
protectedpure virtual

Compute recurrence coefficients.

Derived classes should implement this method to compute their recurrence coefficients. n is the number of coefficients to compute. Return value indicates whether coefficients correspond to normalized (i.e., orthonormal) polynomials.

Note: Owing to the description above, gamma should be an array of length n+1.

Implemented in Stokhos::LanczosPCEBasis< ordinal_type, value_type >, Stokhos::LanczosProjPCEBasis< ordinal_type, value_type >, Stokhos::JacobiBasis< ordinal_type, value_type >, Stokhos::StieltjesPCEBasis< ordinal_type, value_type >, Stokhos::HouseTriDiagPCEBasis< ordinal_type, value_type >, Stokhos::MonoProjPCEBasis< ordinal_type, value_type >, Stokhos::StieltjesBasis< ordinal_type, value_type, func_type >, Stokhos::DiscretizedStieltjesBasis< ordinal_type, value_type >, Stokhos::HermiteBasis< ordinal_type, value_type >, and Stokhos::LegendreBasis< ordinal_type, value_type >.

template<typename ordinal_type , typename value_type >
Teuchos::RCP< Stokhos::Sparse3Tensor< ordinal_type, value_type > > Stokhos::RecurrenceBasis< ordinal_type, value_type >::computeSparseTripleProductTensor ( ordinal_type  order) 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 >
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::LanczosPCEBasis< ordinal_type, value_type >, Stokhos::LanczosProjPCEBasis< ordinal_type, value_type >, Stokhos::StieltjesPCEBasis< ordinal_type, value_type >, Stokhos::HouseTriDiagPCEBasis< ordinal_type, value_type >, Stokhos::MonoProjPCEBasis< ordinal_type, value_type >, Stokhos::StieltjesBasis< ordinal_type, value_type, func_type >, Stokhos::ClenshawCurtisLegendreBasis< ordinal_type, value_type >, and Stokhos::GaussPattersonLegendreBasis< ordinal_type, value_type >.

Referenced by Stokhos::HouseTriDiagPCEBasis< ordinal_type, value_type >::getQuadPoints(), Stokhos::MonoProjPCEBasis< ordinal_type, value_type >::getQuadPoints(), Stokhos::StieltjesBasis< ordinal_type, value_type, func_type >::getQuadPoints(), Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::getQuadPoints(), Stokhos::LanczosPCEBasis< ordinal_type, value_type >::getQuadPoints(), and Stokhos::LanczosProjPCEBasis< ordinal_type, value_type >::getQuadPoints().

template<typename ordinal_type , typename value_type >
virtual LevelToOrderFnPtr Stokhos::RecurrenceBasis< ordinal_type, value_type >::getSparseGridGrowthRule ( ) const
inlinevirtual

Get sparse grid level_to_order mapping function.

Predefined functions are: webbur::level_to_order_linear_wn Symmetric Gaussian linear growth webbur::level_to_order_linear_nn Asymmetric Gaussian linear growth webbur::level_to_order_exp_cc Clenshaw-Curtis exponential growth webbur::level_to_order_exp_gp Gauss-Patterson exponential growth webbur::level_to_order_exp_hgk Genz-Keister exponential growth webbur::level_to_order_exp_f2 Fejer-2 exponential growth

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

References Stokhos::RecurrenceBasis< ordinal_type, value_type >::sparse_grid_growth_rule.

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 >
ordinal_type Stokhos::RecurrenceBasis< ordinal_type, value_type >::quadDegreeOfExactness ( ordinal_type  n) const
virtual

Return polynomial degree of exactness for a given number of quadrature points.

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

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

template<typename ordinal_type , typename value_type >
void Stokhos::RecurrenceBasis< ordinal_type, value_type >::setup ( )
protectedvirtual

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.

Reimplemented in Stokhos::LanczosPCEBasis< ordinal_type, value_type >, Stokhos::LanczosProjPCEBasis< ordinal_type, value_type >, Stokhos::StieltjesPCEBasis< ordinal_type, value_type >, and Stokhos::StieltjesBasis< ordinal_type, value_type, func_type >.

Referenced by Stokhos::DiscretizedStieltjesBasis< ordinal_type, value_type >::DiscretizedStieltjesBasis(), Stokhos::HermiteBasis< ordinal_type, value_type >::HermiteBasis(), Stokhos::HouseTriDiagPCEBasis< ordinal_type, value_type >::HouseTriDiagPCEBasis(), Stokhos::JacobiBasis< ordinal_type, value_type >::JacobiBasis(), Stokhos::LegendreBasis< ordinal_type, value_type >::LegendreBasis(), Stokhos::MonoProjPCEBasis< ordinal_type, value_type >::MonoProjPCEBasis(), Stokhos::StieltjesBasis< ordinal_type, value_type, func_type >::setup(), Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::setup(), Stokhos::LanczosProjPCEBasis< ordinal_type, value_type >::setup(), and Stokhos::LanczosPCEBasis< ordinal_type, value_type >::setup().

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