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

Inheritance diagram for PolynomialsBernstein< number >:

Public Member Functions
	PolynomialsBernstein (const unsigned int index, const unsigned int degree)

number	value (const number x) const

void	value (const number x, std::vector< number > &values) const

template<typename Number2 >
void	value (const Number2 x, const unsigned int n_derivatives, Number2 *values) const

template<std::size_t n_entries, typename Number2 >
void	values_of_array (const std::array< Number2, n_entries > &points, const unsigned int n_derivatives, std::array< Number2, n_entries > *values) const

unsigned int	degree () const

void	scale (const number factor)

template<typename number2 >
void	shift (const number2 offset)

Polynomial< number >	derivative () const

Polynomial< number >	primitive () const

Polynomial< number > &	operator*= (const double s)

Polynomial< number > &	operator*= (const Polynomial< number > &p)

Polynomial< number > &	operator+= (const Polynomial< number > &p)

Polynomial< number > &	operator-= (const Polynomial< number > &p)

bool	operator== (const Polynomial< number > &p) const

void	print (std::ostream &out) const

template<class Archive >
void	serialize (Archive &ar, const unsigned int version)

virtual std::size_t	memory_consumption () const

Protected Member Functions
void	transform_into_standard_form ()

Static Protected Member Functions
static void	scale (std::vector< number > &coefficients, const number factor)

template<typename number2 >
static void	shift (std::vector< number > &coefficients, const number2 shift)

static void	multiply (std::vector< number > &coefficients, const number factor)

Protected Attributes
std::vector< number >	coefficients

bool	in_lagrange_product_form

std::vector< number >	lagrange_support_points

number	lagrange_weight

Subscriptor functionality
Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class.
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

using	map_value_type = decltype(counter_map)::value_type

using	map_iterator = decltype(counter_map)::iterator

std::atomic< unsigned int >	counter

std::map< std::string, unsigned int >	counter_map

std::vector< std::atomic< bool > * >	validity_pointers

const std::type_info *	object_info

static std::mutex	mutex

static ::ExceptionBase &	ExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBase &	ExcNoSubscriber (std::string arg1, std::string arg2)

void	check_no_subscribers () const noexcept

Detailed Description

template<typename number>
class PolynomialsBernstein< number >

This class implements Bernstein basis polynomials of desire degree as described in http://www.idav.ucdavis.edu/education/CAGDNotes/Bernstein-Polynomials.pdf in the paragraph "Converting from the Bernstein Basis to the Power Basis".

They are used to create the Bernstein finite element FE_Bernstein.

Definition at line 41 of file polynomials_bernstein.h.

Constructor & Destructor Documentation

◆ PolynomialsBernstein()

template<typename number >

PolynomialsBernstein< number >::PolynomialsBernstein	(	const unsigned int	index,
		const unsigned int	degree
	)

Construct the index -th Bernstein Polynomial of degree degree.

\begin{align*} B_{\text{index}, \text{degree}} (t) &= \text{binom}(\text{degree}, \text{index}) \cdot t^{\text{index}} \cdot (1 - t)^{\text{degree} - \text{index}} \\ &= \sum_{i = \text{index}}^\text{degree} \cdot (-1)^{i - \text{index}} \cdot \text{binom}(\text{degree}, i) \cdot \text{binom}(i, \text{index}) \cdot t^i \end{align*}

Parameters

index
degree

Definition at line 45 of file polynomials_bernstein.cc.

Member Function Documentation

◆ value() [1/3]

template<typename number >

number Polynomials::Polynomial< number >::value ( const number x ) const

inlineinherited

Return the value of this polynomial at the given point.

This function uses the most numerically stable evaluation algorithm for the provided form of the polynomial. If the polynomial is in the product form of roots, the evaluation is based on products of the form (x - x_i), whereas the Horner scheme is used for polynomials in the coefficient form.

Definition at line 816 of file polynomial.h.

◆ value() [2/3]

template<typename number >

void Polynomials::Polynomial< number >::value	(	const number	x,
		std::vector< number > &	values
	)		const

inherited

Return the values and the derivatives of the Polynomial at point x. values[i], i=0,...,values.size()-1 includes the ith derivative. The number of derivatives to be computed is thus determined by the size of the array passed.

This function uses the Horner scheme for numerical stability of the evaluation for polynomials in the coefficient form or the product of terms involving the roots if that representation is used.

Definition at line 101 of file polynomial.cc.

◆ value() [3/3]

template<typename number >

template<typename Number2 >

void Polynomials::Polynomial< number >::value	(	const Number2	x,
		const unsigned int	n_derivatives,
		Number2 *	values
	)		const

inlineinherited

Return the values and the derivatives of the Polynomial at point x. values[i], i=0,...,n_derivatives includes the ith derivative. The number of derivatives to be computed is determined by n_derivatives and values has to provide sufficient space for n_derivatives + 1 values.

This function uses the most numerically stable evaluation algorithm for the provided form of the polynomial. If the polynomial is in the product form of roots, the evaluation is based on products of the form (x - x_i), whereas the Horner scheme is used for polynomials in the coefficient form.

The template type Number2 must implement arithmetic operations such as additions or multiplication with the type number of the polynomial, and must be convertible from number by operator=.

Definition at line 846 of file polynomial.h.

◆ values_of_array()

template<typename number >

template<std::size_t n_entries, typename Number2 >

void Polynomials::Polynomial< number >::values_of_array	(	const std::array< Number2, n_entries > &	points,
		const unsigned int	n_derivatives,
		std::array< Number2, n_entries > *	values
	)		const

inlineinherited

Similar to the function above, but evaluate the polynomials on several positions at once, as described by the array argument points. This function is can be faster than the other function when the same polynomial should be evaluated on several positions at once, e.g., the x,y,z coordinates of a point for tensor-product polynomials.

The template type Number2 must implement arithmetic operations such as additions or multiplication with the type number of the polynomial, and must be convertible from number by operator=.

Definition at line 860 of file polynomial.h.

◆ degree()

template<typename number >

unsigned int Polynomials::Polynomial< number >::degree

inlineinherited

Degree of the polynomial. This is the degree reflected by the number of coefficients provided by the constructor. Leading non-zero coefficients are not treated separately.

Definition at line 799 of file polynomial.h.

◆ scale() [1/2]

template<typename number >

void Polynomials::Polynomial< number >::scale ( const number factor )

inherited

Scale the abscissa of the polynomial. Given the polynomial p(t) and the scaling t = ax, then the result of this operation is the polynomial q, such that q(x) = p(t).

The operation is performed in place.

Definition at line 166 of file polynomial.cc.

◆ scale() [2/2]

template<typename number >

void Polynomials::Polynomial< number >::scale	(	std::vector< number > &	coefficients,
		const number	factor
	)

staticprotectedinherited

This function performs the actual scaling.

Definition at line 149 of file polynomial.cc.

◆ shift() [1/2]

template<typename number >

template<typename number2 >

void Polynomials::Polynomial< number >::shift ( const number2 offset )

inherited

Shift the abscissa oft the polynomial. Given the polynomial p(t) and the shift t = x + a, then the result of this operation is the polynomial q, such that q(x) = p(t).

The template parameter allows to compute the new coefficients with higher accuracy, since all computations are performed with type number2. This may be necessary, since this operation involves a big number of additions. On a Sun Sparc Ultra with Solaris 2.8, the difference between double and long double was not significant, though.

The operation is performed in place, i.e. the coefficients of the present object are changed.

Definition at line 440 of file polynomial.cc.

◆ shift() [2/2]

template<typename number >

template<typename number2 >

void Polynomials::Polynomial< number >::shift	(	std::vector< number > &	coefficients,
		const number2	shift
	)

staticprotectedinherited

This function performs the actual shift

Definition at line 378 of file polynomial.cc.

◆ derivative()

template<typename number >

Polynomial< number > Polynomials::Polynomial< number >::derivative

inherited

Compute the derivative of a polynomial.

Definition at line 459 of file polynomial.cc.

◆ primitive()

template<typename number >

Polynomial< number > Polynomials::Polynomial< number >::primitive

inherited

Compute the primitive of a polynomial. the coefficient of the zero order term of the polynomial is zero.

Definition at line 488 of file polynomial.cc.

◆ operator*=() [1/2]

template<typename number >

Polynomial< number > & Polynomials::Polynomial< number >::operator*= ( const double s )

inherited

Multiply with a scalar.

Definition at line 204 of file polynomial.cc.

◆ operator*=() [2/2]

template<typename number >

Polynomial< number > & Polynomials::Polynomial< number >::operator*= ( const Polynomial< number > & p )

inherited

Multiply with another polynomial.

Definition at line 222 of file polynomial.cc.

◆ operator+=()

template<typename number >

Polynomial< number > & Polynomials::Polynomial< number >::operator+= ( const Polynomial< number > & p )

inherited

Add a second polynomial.

Definition at line 269 of file polynomial.cc.

◆ operator-=()

template<typename number >

Polynomial< number > & Polynomials::Polynomial< number >::operator-= ( const Polynomial< number > & p )

inherited

Subtract a second polynomial.

Definition at line 311 of file polynomial.cc.

◆ operator==()

template<typename number >

bool Polynomials::Polynomial< number >::operator== ( const Polynomial< number > & p ) const

inherited

Test for equality of two polynomials.

Definition at line 347 of file polynomial.cc.

◆ print()

template<typename number >

void Polynomials::Polynomial< number >::print ( std::ostream & out ) const

inherited

Print coefficients.

Definition at line 515 of file polynomial.cc.

◆ serialize()

template<typename number >

template<class Archive >

void Polynomials::Polynomial< number >::serialize	(	Archive &	ar,
		const unsigned int	version
	)

inlineinherited

Write or read the data of this object to or from a stream for the purpose of serialization using the BOOST serialization library.

Definition at line 1023 of file polynomial.h.

◆ memory_consumption()

template<typename number >

std::size_t Polynomials::Polynomial< number >::memory_consumption

virtualinherited

Return an estimate (in bytes) for the memory consumption of this object.

Definition at line 534 of file polynomial.cc.

◆ multiply()

template<typename number >

void Polynomials::Polynomial< number >::multiply	(	std::vector< number > &	coefficients,
		const number	factor
	)

staticprotectedinherited

Multiply polynomial by a factor.

Definition at line 191 of file polynomial.cc.

◆ transform_into_standard_form()

template<typename number >

void Polynomials::Polynomial< number >::transform_into_standard_form

protectedinherited

Transform polynomial form of product of linear factors into standard form, \(\sum_i a_i x^i\). Deletes all data structures related to the product form.

Definition at line 112 of file polynomial.cc.

Member Data Documentation

◆ coefficients

template<typename number >

std::vector<number> Polynomials::Polynomial< number >::coefficients

protectedinherited

Coefficients of the polynomial \(\sum_i a_i x^i\). This vector is filled by the constructor of this class and may be passed down by derived classes.

This vector cannot be constant since we want to allow copying of polynomials.

Definition at line 301 of file polynomial.h.

◆ in_lagrange_product_form

template<typename number >

bool Polynomials::Polynomial< number >::in_lagrange_product_form

protectedinherited

Stores whether the polynomial is in Lagrange product form, i.e., constructed as a product \((x-x_0) (x-x_1) \ldots (x-x_n)/c\), or not.

Definition at line 307 of file polynomial.h.

◆ lagrange_support_points

template<typename number >

std::vector<number> Polynomials::Polynomial< number >::lagrange_support_points

protectedinherited

If the polynomial is in Lagrange product form, i.e., constructed as a product \((x-x_0) (x-x_1) \ldots (x-x_n)/c\), store the shifts \(x_i\).

Definition at line 313 of file polynomial.h.

◆ lagrange_weight

template<typename number >

number Polynomials::Polynomial< number >::lagrange_weight

protectedinherited

If the polynomial is in Lagrange product form, i.e., constructed as a product \((x-x_0) (x-x_1) \ldots (x-x_n)/c\), store the weight c.

Definition at line 319 of file polynomial.h.

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

include/deal.II/base/polynomials_bernstein.h
source/base/polynomials_bernstein.cc

Public Member Functions

Protected Member Functions

Static Protected Member Functions

Protected Attributes

Subscriptor functionality

Detailed Description

Constructor & Destructor Documentation

◆ PolynomialsBernstein()

Member Function Documentation

◆ value() [1/3]

◆ value() [2/3]

◆ value() [3/3]

◆ values_of_array()

◆ degree()

◆ scale() [1/2]

◆ scale() [2/2]

◆ shift() [1/2]

◆ shift() [2/2]

◆ derivative()

◆ primitive()

◆ operator*=() [1/2]

◆ operator*=() [2/2]

◆ operator+=()

◆ operator-=()

◆ operator==()

◆ print()

◆ serialize()

◆ memory_consumption()

◆ multiply()

◆ transform_into_standard_form()

Member Data Documentation

◆ coefficients

◆ in_lagrange_product_form

◆ lagrange_support_points

◆ lagrange_weight