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Reference documentation for deal.II version 9.1.0-pre
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Reference documentation for deal.II version 9.1.0-pre
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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 doublex) const |
| void | value (const doublex, std::vector< double > &values) const |
| void | value (const doublex, const unsigned int n_derivatives, double *values) const |
| unsigned int | degree () const |
| void | scale (const doublefactor) |
| 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 |
| 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 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< double > | coefficients |
| bool | in_lagrange_product_form |
| std::vector< double > | lagrange_support_points |
| double | lagrange_weight |
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.
Definition at line 436 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 880 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 971 of file polynomial.cc.
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private |
Compute coefficients recursively.
Definition at line 885 of file polynomial.cc.
1.8.11
