#include <deal.II/base/polynomial.h>

Inheritance diagram for Polynomials::Lobatto:

Public Member Functions
	Lobatto (const unsigned int p=0)

Public Member Functions inherited from Polynomials::Polynomial< double >
	Polynomial (const std::vector< double > &coefficients)

	Polynomial (const unsigned int n)

	Polynomial (const std::vector< Point< 1 >> &lagrange_support_points, const unsigned int evaluation_point)

	Polynomial ()

double	value (const double x) const

void	value (const double x, std::vector< double > &values) const

void	value (const double x, const unsigned int n_derivatives, double *values) const

unsigned int	degree () const

void	scale (const double factor)

void	shift (const number2 offset)

Polynomial< double >	derivative () const

Polynomial< double >	primitive () const

Polynomial< double > &	operator*= (const double s)

Polynomial< double > &	operator*= (const Polynomial< double > &p)

Polynomial< double > &	operator+= (const Polynomial< double > &p)

Polynomial< double > &	operator-= (const Polynomial< double > &p)

bool	operator== (const Polynomial< double > &p) const

void	print (std::ostream &out) const

void	serialize (Archive &ar, const unsigned int version)

Public Member Functions inherited from Subscriptor
	Subscriptor ()

	Subscriptor (const Subscriptor &)

	Subscriptor (Subscriptor &&) noexcept

virtual	~Subscriptor ()

Subscriptor &	operator= (const Subscriptor &)

Subscriptor &	operator= (Subscriptor &&) noexcept

void	subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

void	unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

unsigned int	n_subscriptions () const

template<typename StreamType >
void	list_subscribers (StreamType &stream) const

void	list_subscribers () const

template<class Archive >
void	serialize (Archive &ar, const unsigned int version)

Static Public Member Functions
static std::vector< Polynomial< double > >	generate_complete_basis (const unsigned int p)

Static Public Member Functions inherited from Subscriptor
static ::ExceptionBase &	ExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBase &	ExcNoSubscriber (std::string arg1, std::string arg2)

Private Member Functions
std::vector< double >	compute_coefficients (const unsigned int p)

Additional Inherited Members
Protected Member Functions inherited from Polynomials::Polynomial< double >
void	transform_into_standard_form ()

Static Protected Member Functions inherited from Polynomials::Polynomial< double >
static void	scale (std::vector< double > &coefficients, const double factor)

static void	shift (std::vector< double > &coefficients, const number2 shift)

static void	multiply (std::vector< double > &coefficients, const double factor)

Protected Attributes inherited from Polynomials::Polynomial< double >
std::vector< double >	coefficients

bool	in_lagrange_product_form

std::vector< double >	lagrange_support_points

double	lagrange_weight

Detailed Description

Lobatto polynomials of arbitrary degree on [0,1].

These polynomials are the integrated Legendre polynomials on [0,1]. The first two polynomials are the standard linear shape functions given by \(l_0(x) = 1-x\) and \(l_1(x) = x\). For \(i\geq2\) we use the definition \(l_i(x) = \frac{1}{\Vert L_{i-1}\Vert_2}\int_0^x L_{i-1}(t)\,dt\), where \(L_i\) denotes the \(i\)-th Legendre polynomial on \([0,1]\). The Lobatto polynomials \(l_0,\ldots,l_k\) form a complete basis of the polynomials space of degree \(k\).

Calling the constructor with a given index k will generate the polynomial with index k. But only for \(k\geq 1\) the index equals the degree of the polynomial. For k==0 also a polynomial of degree 1 is generated.

These polynomials are used for the construction of the shape functions of Nédélec elements of arbitrary order.

Author: Markus Bürg, 2009

Definition at line 436 of file polynomial.h.

Constructor & Destructor Documentation

◆ Lobatto()

Polynomials::Lobatto::Lobatto ( const unsigned int p = 0 )

Constructor for polynomial of degree p. There is an exception for p==0, see the general documentation.

Definition at line 880 of file polynomial.cc.

Member Function Documentation

◆ generate_complete_basis()

std::vector< Polynomial< double > > Polynomials::Lobatto::generate_complete_basis ( const unsigned int p )

static

Return the polynomials with index 0 up to degree. There is an exception for p==0, see the general documentation.

Definition at line 971 of file polynomial.cc.

◆ compute_coefficients()

std::vector< double > Polynomials::Lobatto::compute_coefficients ( const unsigned int p )

private

Compute coefficients recursively.

Definition at line 885 of file polynomial.cc.

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

deal.II/base/polynomial.h
source/base/polynomial.cc

Public Member Functions

Static Public Member Functions