Reference documentation for deal.II version 8.1.0
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Polynomials::Lobatto Class Reference

#include <polynomial.h>

Inheritance diagram for Polynomials::Lobatto:
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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 doublex) const
 
void value (const doublex, std::vector< double > &values) const
 
unsigned int degree () const
 
void scale (const doublefactor)
 
void shift (const number2 offset)
 
Polynomial< doublederivative () const
 
Polynomial< doubleprimitive () 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 &)
 
virtual ~Subscriptor ()
 
Subscriptoroperator= (const Subscriptor &)
 
void subscribe (const char *identifier=0) const
 
void unsubscribe (const char *identifier=0) const
 
unsigned int n_subscriptions () const
 
void list_subscribers () const
 
 DeclException3 (ExcInUse, int, char *, std::string &,<< "Object of class "<< arg2<< " is still used by "<< arg1<< " other objects.\n"<< "(Additional information: "<< arg3<< ")\n"<< "Note the entry in the Frequently Asked Questions of "<< "deal.II (linked to from http://www.dealii.org/) for "<< "more information on what this error means.")
 
 DeclException2 (ExcNoSubscriber, char *, char *,<< "No subscriber with identifier \""<< arg2<< "\" did subscribe to this object of class "<< arg1)
 
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)
 

Private Member Functions

std::vector< doublecompute_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 doublefactor)
 
static void shift (std::vector< double > &coefficients, const number2 shift)
 
static void multiply (std::vector< double > &coefficients, const doublefactor)
 
- Protected Attributes inherited from Polynomials::Polynomial< double >
std::vector< doublecoefficients
 
bool in_lagrange_product_form
 
std::vector< doublelagrange_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 464 of file polynomial.h.

Constructor & Destructor Documentation

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.

Member Function Documentation

static 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.

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

Compute coefficients recursively.


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