Reference documentation for deal.II version 9.2.0
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#include <deal.II/base/polynomial.h>
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) |
virtual std::size_t | memory_consumption () const |
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 |
Lobatto polynomials of arbitrary degree on [0,1]
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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.
Definition at line 442 of file polynomial.h.
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 891 of file polynomial.cc.
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static |
Return the polynomials with index 0
up to degree
. There is an exception for p==0
, see the general documentation.
Definition at line 982 of file polynomial.cc.
Compute coefficients recursively.
Definition at line 896 of file polynomial.cc.