pfasst::quadrature::Polynomial< CoeffT > Class Template Reference

Representation of a polynomial including differentiation, integration and root finding. More...

#include <polynomial.hpp>

Collaboration diagram for pfasst::quadrature::Polynomial< CoeffT >:

Public Member Functions
	Polynomial (size_t n)
size_t	order () const
	Order of this polynomial. More...
CoeffT &	operator[] (const size_t i)
	Access coefficient i. More...
Polynomial< CoeffT >	differentiate () const
	Differentiate this polynomial. More...
Polynomial< CoeffT >	integrate () const
	Integrates this polynomial. More...
template<typename xtype >
xtype	evaluate (const xtype x) const
	Evaluate polynomial for given value. More...
Polynomial< CoeffT >	normalize () const
	Normalizes this polynomial with respect to \( c_0 \). More...
vector< CoeffT >	roots (size_t num_iterations=100, CoeffT ztol=1.0e-20) const
	Computes the roots of this polynomial. More...

Static Public Member Functions
static Polynomial< CoeffT >	legendre (const size_t order)
	Computes the Legendre polynomial of given order. More...

Protected Attributes
vector< CoeffT >	c
	Coefficients of the polynomial. More...

Detailed Description

template<typename CoeffT>
class pfasst::quadrature::Polynomial< CoeffT >

Representation of a polynomial including differentiation, integration and root finding.

Nothing more than a \( n \)-th order polynomial of the form \( P_n(x) = \sum_{i=0}^{n} c_i x^i \) with coefficients \( c_i \), \( i=0, \dots, n \).

Template Parameters

CoeffT

numerical precision of polynomial coefficients (e.g. double)

Since

v0.3.0

Definition at line 28 of file polynomial.hpp.

Constructor & Destructor Documentation

template<typename CoeffT >

pfasst::quadrature::Polynomial< CoeffT >::Polynomial

(

size_t

n

)

Todo:

Consider issuing a warning/assertion when n is zero.

Definition at line 18 of file polynomial_impl.hpp.

Member Function Documentation

template<typename CoeffT >

Polynomial< CoeffT > pfasst::quadrature::Polynomial< CoeffT >::differentiate

(

)

const

Differentiate this polynomial.

Computes standard differential of this polynomial.

Returns

differentiated polynomial

Definition at line 35 of file polynomial_impl.hpp.

template<typename CoeffT>

template<typename xtype >

xtype pfasst::quadrature::Polynomial< CoeffT >::evaluate ( const xtype  x ) const

inline

Evaluate polynomial for given value.

Template Parameters

xtype

numerical type of the value

Parameters

[in]

x

value to evaluate polynomial at

Returns

value of polynomial at x

Definition at line 95 of file polynomial.hpp.

Referenced by pfasst::quadrature::compute_q_matrix(), pfasst::quadrature::compute_q_vec(), and pfasst::quadrature::Polynomial< CoeffT >::roots().

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template<typename CoeffT >

Polynomial< CoeffT > pfasst::quadrature::Polynomial< CoeffT >::integrate

(

)

const

Integrates this polynomial.

Computes integral of this polynomial.

Returns

integrated polynomial

Definition at line 45 of file polynomial_impl.hpp.

Referenced by pfasst::quadrature::compute_q_matrix(), and pfasst::quadrature::compute_q_vec().

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template<typename CoeffT >

Polynomial< CoeffT > pfasst::quadrature::Polynomial< CoeffT >::legendre ( const size_t  order )

static

Computes the Legendre polynomial of given order.

Parameters

[in]

order

desired order of the Legendre polynomial

Returns

Legendre polynomial of order order

Definition at line 106 of file polynomial_impl.hpp.

Referenced by pfasst::quadrature::GaussLegendre< precision >::compute_nodes(), pfasst::quadrature::GaussRadau< precision >::compute_nodes(), pfasst::quadrature::GaussLobatto< precision >::compute_nodes(), pfasst::quadrature::ClenshawCurtis< precision >::compute_nodes(), and TEST().

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template<typename CoeffT >

Polynomial< CoeffT > pfasst::quadrature::Polynomial< CoeffT >::normalize

(

)

const

Normalizes this polynomial with respect to \( c_0 \).

Returns

normalized polynomial

Definition at line 55 of file polynomial_impl.hpp.

template<typename CoeffT >

CoeffT & pfasst::quadrature::Polynomial< CoeffT >::operator[]

(

const size_t

i

)

Access coefficient i.

Parameters

[in]

i

coefficient index (zero-based)

Returns

\( i+1 \)-th coefficient

Exceptions

std::out_of_range

if index is out of bounds, i.e. i >= Polynomial::order()

Note

The coefficients are stored in reversed order to the degree of the indeterminate, i.e. i=0 corresponds to \( c_n \) while i=n corresponds to \( c_0 \). See also Polynomial::c.

Definition at line 29 of file polynomial_impl.hpp.

template<typename CoeffT >

size_t pfasst::quadrature::Polynomial< CoeffT >::order

(

)

const

Order of this polynomial.

The order of this polynomial is one less the number of coefficients defined.

Returns

order of the polynomial

Definition at line 23 of file polynomial_impl.hpp.

template<typename CoeffT >

vector< CoeffT > pfasst::quadrature::Polynomial< CoeffT >::roots	(	size_t	num_iterations = 100,
		CoeffT	ztol = 1.0e-20
	)		const

Computes the roots of this polynomial.

Returns

roots sorted with respect to their value

Definition at line 70 of file polynomial_impl.hpp.

References pfasst::quadrature::Polynomial< CoeffT >::evaluate().

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Member Data Documentation

template<typename CoeffT>

vector<CoeffT> pfasst::quadrature::Polynomial< CoeffT >::c

protected

Coefficients of the polynomial.

The coefficient for the highest degree of \( x \) has index 0. The last coefficient is for \( x^0 \).

Definition at line 37 of file polynomial.hpp.

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

• include/pfasst/quadrature/polynomial.hpp
• src/pfasst/quadrature/polynomial_impl.hpp