Reference documentation for deal.II version Git bd58e18 2017-08-22 18:42:24 +0200
Public Member Functions | Static Public Member Functions | List of all members
Polynomials::HermiteInterpolation Class Reference

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

Inheritance diagram for Polynomials::HermiteInterpolation:
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Public Member Functions

 HermiteInterpolation (const unsigned int p)
 
- 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 &&)
 
virtual ~Subscriptor ()
 
Subscriptoroperator= (const Subscriptor &)
 
Subscriptoroperator= (Subscriptor &&)
 
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, char *arg2, std::string &arg3)
 
static::ExceptionBase & ExcNoSubscriber (char *arg1, char *arg2)
 

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
 

Detailed Description

Polynomials for Hermite interpolation condition.

This is the set of polynomials of degree at least three, such that the following interpolation conditions are met: the polynomials and their first derivatives vanish at the values x=0 and x=1, with the exceptions p0(0)=1, p1(1)=1, p'2(0)=1, p'3(1)=1.

For degree three, we obtain the standard four Hermitian interpolation polynomials, see for instance Wikipedia. For higher degrees, these are augmented first, by the polynomial of degree four with vanishing values and derivatives at x=0 and x=1, then by the product of this fourth order polynomial with Legendre polynomials of increasing order. The implementation is

\begin{align*} p_0(x) &= 2x^3-3x^2+1 \\ p_1(x) &= -2x^3+3x^2 \\ p_2(x) &= x^3-2x^2+x \\ p_3(x) &= x^3-x^2 \\ p_4(x) &= 16x^2(x-1)^2 \\ \ldots & \ldots \\ p_k(x) &= x^2(x-1)^2 L_{k-4}(x) \end{align*}

Author
Guido Kanschat
Date
2012

Definition at line 571 of file polynomial.h.

Constructor & Destructor Documentation

Polynomials::HermiteInterpolation::HermiteInterpolation ( const unsigned int  p)

Constructor for polynomial with index p. See the class documentation on the definition of the sequence of polynomials.

Definition at line 1168 of file polynomial.cc.

Member Function Documentation

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

Return the polynomials with index 0 up to p+1 in a space of degree up to p. Here, p has to be at least 3.

Definition at line 1221 of file polynomial.cc.


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