Polynomials::Lobatto Class Reference

#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 (const char *identifier=nullptr) const
void	unsubscribe (const char *identifier=nullptr) const
unsigned int	n_subscriptions () 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 429 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 888 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 960 of file polynomial.cc.

compute_coefficients()

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

private

Compute coefficients recursively.

Definition at line 892 of file polynomial.cc.

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

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