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

Inheritance diagram for Polynomials::HermiteInterpolation:

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 double x) const

void	value (const double x, std::vector< 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 &)

virtual	~Subscriptor ()

Subscriptor &	operator= (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."<< "\"<< "(Additional information: "<< arg3<< ")\"<< "See the entry in the Frequently Asked Questions of "<< "deal.II (linked to from http://www.dealii.org/) for "<< "a lot more information on what this error means and "<< "how to fix programs in which it happens.")

	DeclException2 (ExcNoSubscriber, char , char ,<< "No subscriber with identifier <"<< arg2<< "> subscribes to this object of class "<< arg1<< ". Consequently, it cannot be unsubscribed.")

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)

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

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

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

Author: Guido Kanschat

Date: 2012

Definition at line 576 of file polynomial.h.

Constructor & Destructor Documentation

§ HermiteInterpolation()

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 1302 of file polynomial.cc.

Member Function Documentation

§ generate_complete_basis()

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 1344 of file polynomial.cc.

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

deal.II/base/polynomial.h
/build/deal.ii-pdtiAo/deal.ii-8.4.2/source/base/polynomial.cc

Public Member Functions

Static Public Member Functions

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

§ HermiteInterpolation()

Member Function Documentation

§ generate_complete_basis()