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Public Member Functions | Protected Member Functions | Protected Attributes
Stokhos::RecurrenceBasis< ordinal_type, value_type > Class Template Reference

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 $\beta_{0} = (1,1) = \int d\lambda$. More...

#include <Stokhos_RecurrenceBasis.hpp>

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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.
virtual void setQuadZeroTol (value_type tol)
 Set tolerance for zero in quad point generation.

Protected Member Functions

 RecurrenceBasis (const std::string &name, ordinal_type p, bool normalize)
 Constructor to be called by derived classes.
 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.
virtual void setup ()
 Setup basis after computing recurrence coefficients.
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.
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.
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
< Stokhos::Sparse3Tensor
< ordinal_type, value_type > > 
computeSparseTripleProductTensor (ordinal_type order) const
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 LevelToOrderFnPtr getSparseGridGrowthRule () const
 Get sparse grid level_to_order mapping function.
virtual void setSparseGridGrowthRule (LevelToOrderFnPtr ptr)
 Set sparse grid rule.

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 $\beta_{0} = (1,1) = \int d\lambda$.

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 
) [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 [protected, pure virtual]
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::HouseTriDiagPCEBasis< ordinal_type, value_type >, Stokhos::MonoProjPCEBasis< ordinal_type, value_type >, Stokhos::LanczosPCEBasis< ordinal_type, value_type >, Stokhos::LanczosProjPCEBasis< ordinal_type, value_type >, Stokhos::StieltjesBasis< ordinal_type, value_type, func_type >, and Stokhos::StieltjesPCEBasis< ordinal_type, value_type >.

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

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 >.

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, virtual]

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::StieltjesBasis< ordinal_type, value_type, func_type >, and Stokhos::StieltjesPCEBasis< ordinal_type, value_type >.


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