A 1D orthogonal polynomial. More...

#include <OrthogonalPolynomial.h>

Inheritance diagram for muq::Approximation::OrthogonalPolynomial:

Detailed Description

A 1D orthogonal polynomial.

In general, we use an recursive formula to evaluate a \(d^{th}\) degree polynomial using a three term recurrence:

\begin{eqnarray*} p_0(x) &=& \phi_0(x) \\ p_1(x) &=& \phi_1(x) \\ p_{k}(x) & = & (a_k x + b_k) p_{k-1}(x) - c_k p_{k-2}(x) \end{eqnarray*}

Subclasses specialize for particular polynomials (e.g., Hermite) by implementing functions for \(a_k\), \(b_k\), \(c_k\), \(\phi_0(x)\), and \(\phi_1(x)\).

The BasisEvaluate function Uses the Clenshaw algorithm from: http://en.wikipedia.org/wiki/Clenshaw_algorithm.

Definition at line 31 of file OrthogonalPolynomial.h.

Public Member Functions
	OrthogonalPolynomial ()
	Create a polynomial. More...

virtual	~OrthogonalPolynomial ()=default

virtual double	BasisEvaluate (int const order, double const x) const override
	Evaluate the specific polynomial type (must be implemented by the child) More...

virtual Eigen::VectorXd	EvaluateAllTerms (int const maxOrder, double const x) const override
	Evaluates all basis functions with order <= maxOrder. More...

virtual double	Normalization (unsigned int polyOrder) const

Public Member Functions inherited from muq::Approximation::IndexedScalarBasis
	IndexedScalarBasis ()
	Create a basis function. More...

virtual	~IndexedScalarBasis ()=default

virtual double	DerivativeEvaluate (int const polyOrder, int const derivOrder, double const x) const =0

Public Member Functions inherited from muq::Modeling::WorkPiece
	WorkPiece ()
	Create a muq::Modeling::WorkPiece with no fixed number of inputs and outputs and variable input/output types. More...

	WorkPiece (int const num, WorkPiece::Fix const fix=WorkPiece::Fix::Inputs)
	Create a muq::Modeling::WorkPiece with either a fixed number of inputs or outputs and variable input/output types. More...

	WorkPiece (int const numIns, int const numOuts)
	Create a muq::Modeling::WorkPiece with a fixed number of inputs and outputs but variable input/output types. More...

	WorkPiece (std::vector< std::string > const &types, WorkPiece::Fix const fix=WorkPiece::Fix::Inputs)
	Create a muq::Modeling::WorkPiece with either a fixed number of inputs with specified types or a fixed number of outputs with specified types. More...

	WorkPiece (std::map< unsigned int, std::string > const &types, WorkPiece::Fix const fix=WorkPiece::Fix::Inputs)
	Create a muq::Modeling::WorkPiece where either some of the inputs have specified types or some of the outputs have specified types. More...

	WorkPiece (std::map< unsigned int, std::string > const &types, int const num, WorkPiece::Fix const fixTypes=WorkPiece::Fix::Inputs, WorkPiece::Fix const fixNum=WorkPiece::Fix::Inputs)
	Create a muq::Modeling::WorkPiece where either some of the inputs have specified types or some of the outputs have specified types and either the number of inputs or the number of outputs is fixed. More...

	WorkPiece (std::vector< std::string > const &types, int const num)
	Create a muq::Modeling::WorkPiece with a fixed number of inputs with specified types and a fixed number of outputs (of uknown type) More...

	WorkPiece (int const num, std::vector< std::string > const &types)
	Create a muq::Modeling::WorkPiece with a fixed number of outputs with specified types and a fixed number of inputs (of uknown type) More...

	WorkPiece (std::map< unsigned int, std::string > const &inTypes, int const numIns, int const numOuts)
	Create a muq::Modeling::WorkPiece where some of the inputs are known and we know the input and output numbers. More...

	WorkPiece (int const numIns, std::map< unsigned int, std::string > const &outTypes, int const numOuts)
	Create a muq::Modeling::WorkPiece where some of the outputs are known and we know the input and output numbers. More...

	WorkPiece (std::vector< std::string > const &inTypes, std::vector< std::string > const &outTypes)
	Create a muq::Modeling::WorkPiece with a fixed number of inputs and outputs with specified types. More...

	WorkPiece (std::map< unsigned int, std::string > const &inTypes, std::vector< std::string > const &outTypes)
	Create a muq::Modeling::WorkPiece where some of the inputs are known and all of the outputs have specified types. More...

	WorkPiece (std::map< unsigned int, std::string > const &inTypes, int const num, std::vector< std::string > const &outTypes)
	Create a muq::Modeling::WorkPiece where some of the inputs are known with a known number of inputs and all of the outputs have specified types. More...

	WorkPiece (std::vector< std::string > const &inTypes, std::map< unsigned int, std::string > const &outTypes)
	Create a muq::Modeling::WorkPiece where some of the outputs and all of the inputs have specified types. More...

	WorkPiece (std::vector< std::string > const &inTypes, std::map< unsigned int, std::string > const &outTypes, int const num)
	Create a muq::Modeling::WorkPiece where some of the outputs with a known number of outputs and all of the inputs have specified types. More...

	WorkPiece (std::map< unsigned int, std::string > const &inTypes, std::map< unsigned int, std::string > const &outTypes)
	Create a muq::Mdoeling::WorkPiece where some of the inputs and some of the outputs have specified types. More...

	WorkPiece (std::map< unsigned int, std::string > const &inTypes, int const numIn, std::map< unsigned int, std::string > const &outTypes)
	Create a muq::Mdoeling::WorkPiece where some of the inputs and some of the outputs have specified types with a fixed number of inputs. More...

	WorkPiece (std::map< unsigned int, std::string > const &inTypes, std::map< unsigned int, std::string > const &outTypes, int const numOut)
	Create a muq::Mdoeling::WorkPiece where some of the inputs and some of the outputs have specified types with a fixed number of outputs. More...

	WorkPiece (std::map< unsigned int, std::string > const &inTypes, int const numIn, std::map< unsigned int, std::string > const &outTypes, int const numOut)
	Create a muq::Mdoeling::WorkPiece where some of the inputs and some of the outputs have specified types with a fixed number of inputs and outputs. More...

virtual	~WorkPiece ()
	Default destructor. More...

std::vector< boost::any > const &	Evaluate (std::vector< boost::any > const &ins)
	Evaluate this muq::Modeling::WorkPiece. More...

std::vector< boost::any > const &	Evaluate (ref_vector< boost::any > const &ins)
	Evaluate this muq::Modeling::WorkPiece using references to the inputs. More...

std::vector< boost::any > const &	Evaluate ()
	Evaluate this muq::Modeling::WorkPiece in the case that there are no inputs. More...

template<typename... Args>
std::vector< boost::any > const &	Evaluate (Args... args)
	Evalaute this muq::Modeling::WorkPiece using multiple arguments. More...

std::string const &	Name ()
	Get the (unique) name of this work piece. More...

void	SetName (std::string const &newName)
	Set the name of this work piece. More...

std::string	InputType (unsigned int inputNum, bool const demangle=true) const
	Get the input type (if we know it) for a specific input. More...

int	InputSize (unsigned int inputNum) const
	Get the length of a vector valued input with fixed size. More...

void	SetInputSize (unsigned int inputNum, int newSize)

std::string	OutputType (unsigned int outputNum, bool const demangle=true) const
	Get the output type (if we know it) for a specific output. More...

std::map< unsigned int, std::string >	OutputTypes () const
	Get the output types. More...

std::map< unsigned int, std::string >	InputTypes () const
	Get the input types. More...

unsigned int	ID () const
	Get the unique ID number. More...

virtual double	GetRunTime (const std::string &method="Evaluate") const
	Get the average run time for one of the implemented methods. More...

virtual unsigned long int	GetNumCalls (const std::string &method="Evaluate") const
	get the number of times one of the implemented methods has been called. More...

virtual void	ResetCallTime ()
	Resets the number of call and times. More...

Static Public Member Functions
static std::shared_ptr< OrthogonalPolynomial >	Construct (std::string const &polyName)

Static Public Member Functions inherited from muq::Approximation::IndexedScalarBasis
static std::shared_ptr< ScalarBasisMapType >	GetScalarBasisMap ()

static std::shared_ptr< IndexedScalarBasis >	Construct (std::string const &polyName)

Static Public Member Functions inherited from muq::Modeling::WorkPiece
static ref_vector< const boost::any >	ToRefVector (std::vector< boost::any > const &anyVec)
	Create vector of references from a vector of boost::any's. More...

static ref_vector< const Eigen::VectorXd >	ToRefVector (std::vector< Eigen::VectorXd > const &anyVec)

Friends
class	GaussQuadrature

Additional Inherited Members
Public Types inherited from muq::Approximation::IndexedScalarBasis
typedef std::function< std::shared_ptr< IndexedScalarBasis >)>	ScalarBasisConstructorType

typedef std::map< std::string, ScalarBasisConstructorType >	ScalarBasisMapType

Public Attributes inherited from muq::Modeling::WorkPiece
int	numInputs
	The number of inputs. More...

int	numOutputs
	The number of outputs. More...

Constructor & Destructor Documentation

◆ OrthogonalPolynomial()

muq::Approximation::OrthogonalPolynomial::OrthogonalPolynomial ( )

inline

Create a polynomial.

Definition at line 38 of file OrthogonalPolynomial.h.

◆ ~OrthogonalPolynomial()

virtual muq::Approximation::OrthogonalPolynomial::~OrthogonalPolynomial ( )

virtualdefault

Member Function Documentation

◆ ak()

virtual double muq::Approximation::OrthogonalPolynomial::ak ( unsigned int k ) const

privatepure virtual

Implement \(a_k(x)\).

Parameters

[in] k The order of the polynomial

Implemented in muq::Approximation::ProbabilistHermite, muq::Approximation::PhysicistHermite, muq::Approximation::Legendre, muq::Approximation::Laguerre, and muq::Approximation::Jacobi.

◆ BasisEvaluate()

double OrthogonalPolynomial::BasisEvaluate	(	int const	order,
		double const	x
	)		const

overridevirtual

Evaluate the specific polynomial type (must be implemented by the child)

Inputs:

The order of the polynomial (unsigned int)
The point where we are evaluating the polynomial

Returns: The polynomial value

Implements muq::Approximation::IndexedScalarBasis.

Definition at line 39 of file OrthogonalPolynomial.cpp.

References nlohmann::detail::dtoa_impl::k.

Referenced by muq::Approximation::Legendre::DerivativeEvaluate(), muq::Approximation::PhysicistHermite::DerivativeEvaluate(), and muq::Approximation::ProbabilistHermite::DerivativeEvaluate().

◆ bk()

virtual double muq::Approximation::OrthogonalPolynomial::bk ( unsigned int k ) const

privatepure virtual

Implement \(b_k(x)\).

Parameters

[in] k The order of the polynomial

Implemented in muq::Approximation::ProbabilistHermite, muq::Approximation::PhysicistHermite, muq::Approximation::Legendre, muq::Approximation::Laguerre, and muq::Approximation::Jacobi.

◆ ck()

virtual double muq::Approximation::OrthogonalPolynomial::ck ( unsigned int k ) const

privatepure virtual

Implement \(c_k(x)\).

Parameters

[in] k The order of the polynomial

Implemented in muq::Approximation::ProbabilistHermite, muq::Approximation::PhysicistHermite, muq::Approximation::Legendre, muq::Approximation::Laguerre, and muq::Approximation::Jacobi.

◆ Construct()

std::shared_ptr< OrthogonalPolynomial > OrthogonalPolynomial::Construct ( std::string const & polyName )

static

Definition at line 14 of file OrthogonalPolynomial.cpp.

◆ EvaluateAllTerms()

Eigen::VectorXd OrthogonalPolynomial::EvaluateAllTerms	(	int const	maxOrder,
		double const	x
	)		const

overridevirtual

Evaluates all basis functions with order <= maxOrder.

Reimplemented from muq::Approximation::IndexedScalarBasis.

Definition at line 66 of file OrthogonalPolynomial.cpp.

◆ Normalization()

double OrthogonalPolynomial::Normalization ( unsigned int polyOrder ) const

virtual

     @brief Returns the normalization constant for the polynomial of order \f$p\f$.
     @details Many polynomial families are orthogonal with respect to some measure.  This means that for two polynomials \f$P_n(x)\f$ and \f$P_m(x)\f$ in the same family,

\[ \int P_n(x) P_m(x) w(x) dx = a_n \delta_{nm}, \]

where \(\delta_{mn}\) is the Kronecker delta function, \(w(x)\) is a weighting function tied to the polynomial family (e.g., the Askey scheme), and \(a_n\in\mathbb{R}\) is a scalar normalization constant. This function computes the normalization constant \(a_n\). Polynomial families that are not orthogonal do not have a normalizing constant \(a_n\) and will throw an exception if this function is called.

Parameters

[in] polyOrder The order of the polynomial

Returns: The normalization constant for a particular polynomial family and its weighting function \(w(x)\).

Reimplemented in muq::Approximation::ProbabilistHermite, muq::Approximation::PhysicistHermite, muq::Approximation::Legendre, muq::Approximation::Laguerre, and muq::Approximation::Jacobi.

Definition at line 20 of file OrthogonalPolynomial.cpp.

◆ phi0()

virtual double muq::Approximation::OrthogonalPolynomial::phi0 ( double x ) const

privatepure virtual

Implement \(\phi_0(x)\).

Parameters

[in] x The point where w are evaluating the polynomial

Implemented in muq::Approximation::ProbabilistHermite, muq::Approximation::PhysicistHermite, muq::Approximation::Legendre, muq::Approximation::Laguerre, and muq::Approximation::Jacobi.

◆ phi1()

virtual double muq::Approximation::OrthogonalPolynomial::phi1 ( double x ) const

privatepure virtual

Implement \(\phi_1(x)\).

Parameters

[in] x The point where w are evaluating the polynomial

Implemented in muq::Approximation::ProbabilistHermite, muq::Approximation::PhysicistHermite, muq::Approximation::Legendre, muq::Approximation::Laguerre, and muq::Approximation::Jacobi.

Friends And Related Function Documentation

◆ GaussQuadrature

friend class GaussQuadrature

friend

Definition at line 33 of file OrthogonalPolynomial.h.

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

doxygen_prep/MUQ/Approximation/Polynomials/OrthogonalPolynomial.h
doxygen_prep/modules/Approximation/src/Polynomials/OrthogonalPolynomial.cpp

Detailed Description

Public Member Functions

Static Public Member Functions

Friends

Additional Inherited Members

Constructor & Destructor Documentation

◆ OrthogonalPolynomial()

◆ ~OrthogonalPolynomial()

Member Function Documentation

◆ ak()

◆ BasisEvaluate()

◆ bk()

◆ ck()

◆ Construct()

◆ EvaluateAllTerms()

◆ Normalization()

◆ phi0()

◆ phi1()

Friends And Related Function Documentation

◆ GaussQuadrature