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

Detailed Description

template<typename number>
class Polynomials::Polynomial< number >

Base class for all 1d polynomials. A polynomial is represented in this class by its coefficients, which are set through the constructor or by derived classes.

There are two paths for evaluation of polynomials. One is based on the coefficients which are evaluated through the Horner scheme which is a robust general-purpose scheme. An alternative and more stable evaluation of high-degree polynomials with roots in the unit interval is provided by a product in terms of the roots. This form is available for special polynomials such as Lagrange polynomials or Legendre polynomials and used with the respective constructor. To obtain this more stable evaluation form, the constructor with the roots in form of a Lagrange polynomial must be used. In case a manipulation is done that changes the roots, the representation is switched to the coefficient form. The more stable Lagrange form inherits the stability properties from the related barycentric form, a popular form of polynomial interpolation formulas. However, deal.II does not have a dedicated class for an entire basis, so the overall O(n) efficiency characteristic for the barycentric form is not achieved.

This class is a typical example of a possible template argument for the TensorProductPolynomials class.

Definition at line 70 of file polynomial.h.

Inheritance diagram for Polynomials::Polynomial< number >:

[legend]

Public Member Functions
	Polynomial (const std::vector< number > &coefficients)

	Polynomial (const unsigned int n)

	Polynomial (const std::vector< Point< 1 > > &lagrange_support_points, const unsigned int j)

	Polynomial ()

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

Querying the observer pointers an object has.
unsigned int	n_subscriptions () const

template<typename StreamType >
void	list_subscribers (StreamType &stream) const

void	list_subscribers () const

Static Public Member Functions
static ::ExceptionBase &	ExcInUse (int arg1, std::string arg2, std::string arg3)

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

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

Private Member Functions
EnableObserverPointer functionality
Classes derived from EnableObserverPointer provide a facility to subscribe to this object. This is mostly used by the ObserverPointer 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

void	check_no_subscribers () const noexcept

Private Attributes
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 Private Attributes
static std::mutex	mutex

Constructor & Destructor Documentation

◆ Polynomial() [1/4]

template<typename number >

Polynomials::Polynomial< number >::Polynomial ( const std::vector< number > & coefficients )

Constructor creating a polynomial in monomial form, i.e. a polynomial that is represented in the form \(p(x) = a_0 + a_1x + a_2 x^2 + \cdots + a_n x^n = \sum_{i=0}^n a_i x^i\).

The coefficients \(a_i\) of the polynomial are passed as argument to this constructor, i.e. the first element of the array denotes the constant term, the second the linear one, and so on. The degree \(n\) of the polynomial represented by this object is thus the number of elements in the coefficient array minus one.

Parameters

coefficients The vector of coefficients \(a_i\) in the formula above.

Definition at line 36 of file polynomial.cc.

◆ Polynomial() [2/4]

template<typename number >

Polynomials::Polynomial< number >::Polynomial ( const unsigned int n )

Constructor creating a zero polynomial of degree n.

Definition at line 45 of file polynomial.cc.

◆ Polynomial() [3/4]

template<typename number >

Polynomials::Polynomial< number >::Polynomial	(	const std::vector< Point< 1 > > &	lagrange_support_points,
		const unsigned int	j
	)

Constructor for a poynomial in Lagrange form.

Lagrange polynomials are defined via their "support points" \(x_i\) as passed via the first argument to this constructor. For a given set of support points, there is a complete basis of polynomials that all have the form

\[ p_j(x) = \prod_{i\neq j} \frac{x-x_i}{x_j-x_i} \]

where \(j=0,\ldots,n-1\) where \(n\) is the number of support points and the degree of the polynomials is \(n-1\). The second argument of this constructor, \(j\), then selects which of these polynomials you want.

Definition at line 54 of file polynomial.cc.

◆ Polynomial() [4/4]

template<typename number >

Polynomials::Polynomial< number >::Polynomial ( )

inline

Default constructor creating an illegal object.

Definition at line 909 of file polynomial.h.

Member Function Documentation

◆ value() [1/3]

template<typename number >

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

inline

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 935 of file polynomial.h.

◆ value() [2/3]

template<typename number >

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

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 83 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

inline

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 965 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

inline

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 983 of file polynomial.h.

◆ degree()

template<typename number >

unsigned int Polynomials::Polynomial< number >::degree ( ) const

inline

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 918 of file polynomial.h.

◆ scale() [1/2]

template<typename number >

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

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 148 of file polynomial.cc.

◆ shift() [1/2]

template<typename number >

template<typename number2 >

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

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 422 of file polynomial.cc.

◆ derivative()

template<typename number >

Polynomial< number > Polynomials::Polynomial< number >::derivative ( ) const

Compute the derivative of a polynomial.

Definition at line 441 of file polynomial.cc.

◆ primitive()

template<typename number >

Polynomial< number > Polynomials::Polynomial< number >::primitive ( ) const

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

Definition at line 470 of file polynomial.cc.

◆ operator*=() [1/2]

template<typename number >

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

Multiply with a scalar.

Definition at line 186 of file polynomial.cc.

◆ operator*=() [2/2]

template<typename number >

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

Multiply with another polynomial.

Definition at line 204 of file polynomial.cc.

◆ operator+=()

template<typename number >

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

Add a second polynomial.

Definition at line 251 of file polynomial.cc.

◆ operator-=()

template<typename number >

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

Subtract a second polynomial.

Definition at line 293 of file polynomial.cc.

◆ operator==()

template<typename number >

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

Test for equality of two polynomials.

Definition at line 329 of file polynomial.cc.

◆ print()

template<typename number >

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

Print coefficients.

Definition at line 497 of file polynomial.cc.

◆ serialize()

template<typename number >

template<class Archive >

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

inline

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 1140 of file polynomial.h.

◆ memory_consumption()

template<typename number >

std::size_t Polynomials::Polynomial< number >::memory_consumption ( ) const

virtual

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

Definition at line 516 of file polynomial.cc.

◆ scale() [2/2]

template<typename number >

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

staticprotected

This function performs the actual scaling.

Definition at line 131 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
	)

staticprotected

This function performs the actual shift

Definition at line 360 of file polynomial.cc.

◆ multiply()

template<typename number >

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

staticprotected

Multiply polynomial by a factor.

Definition at line 173 of file polynomial.cc.

◆ transform_into_standard_form()

template<typename number >

void Polynomials::Polynomial< number >::transform_into_standard_form ( )

protected

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 94 of file polynomial.cc.

◆ n_subscriptions()

unsigned int EnableObserverPointer::n_subscriptions ( ) const

inlineinherited

Return the present number of subscriptions to this object. This allows to use this class for reference counted lifetime determination where the last one to unsubscribe also deletes the object.

Definition at line 318 of file enable_observer_pointer.h.

◆ list_subscribers() [1/2]

template<typename StreamType >

void EnableObserverPointer::list_subscribers ( StreamType & stream ) const

inlineinherited

List the subscribers to the input stream.

Definition at line 335 of file enable_observer_pointer.h.

◆ list_subscribers() [2/2]

void EnableObserverPointer::list_subscribers ( ) const

inherited

List the subscribers to deallog.

Definition at line 200 of file enable_observer_pointer.cc.

◆ subscribe()

void EnableObserverPointer::subscribe	(	std::atomic< bool > *const	validity,
		const std::string &	identifier = `""`
	)		const

privateinherited

Subscribes a user of the object by storing the pointer validity. The subscriber may be identified by text supplied as identifier.

Definition at line 136 of file enable_observer_pointer.cc.

◆ unsubscribe()

void EnableObserverPointer::unsubscribe	(	std::atomic< bool > *const	validity,
		const std::string &	identifier = `""`
	)		const

privateinherited

Unsubscribes a user from the object.

Note: The identifier and the validity pointer must be the same as the one supplied to subscribe().

Definition at line 154 of file enable_observer_pointer.cc.

◆ check_no_subscribers()

void EnableObserverPointer::check_no_subscribers ( ) const

privatenoexceptinherited

Check that there are no objects subscribing to this object. If this check passes then it is safe to destroy the current object. It this check fails then this function will either abort or print an error message to deallog (by using the AssertNothrow mechanism), but will not throw an exception.

Note: Since this function is just a consistency check it does nothing in release mode.; If this function is called when there is an uncaught exception then, rather than aborting, this function prints an error message to the standard error stream and returns.

Definition at line 58 of file enable_observer_pointer.cc.

Member Data Documentation

◆ coefficients

template<typename number >

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

protected

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 323 of file polynomial.h.

◆ in_lagrange_product_form

template<typename number >

bool Polynomials::Polynomial< number >::in_lagrange_product_form

protected

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 329 of file polynomial.h.

◆ lagrange_support_points

template<typename number >

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

protected

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 335 of file polynomial.h.

◆ lagrange_weight

template<typename number >

number Polynomials::Polynomial< number >::lagrange_weight

protected

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 341 of file polynomial.h.

◆ counter

std::atomic<unsigned int> EnableObserverPointer::counter

mutableprivateinherited

Store the number of objects which subscribed to this object. Initially, this number is zero, and upon destruction it shall be zero again (i.e. all objects which subscribed should have unsubscribed again).

The creator (and owner) of an object is counted in the map below if HE manages to supply identification.

We use the mutable keyword in order to allow subscription to constant objects also.

This counter may be read from and written to concurrently in multithreaded code: hence we use the std::atomic class template.

Definition at line 212 of file enable_observer_pointer.h.

◆ counter_map

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

mutableprivateinherited

In this map, we count subscriptions for each different identification string supplied to subscribe().

Definition at line 218 of file enable_observer_pointer.h.

◆ validity_pointers

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

mutableprivateinherited

In this vector, we store pointers to the validity bool in the ObserverPointer objects that subscribe to this class.

Definition at line 224 of file enable_observer_pointer.h.

◆ object_info

const std::type_info* EnableObserverPointer::object_info

mutableprivateinherited

Pointer to the typeinfo object of this object, from which we can later deduce the class name. Since this information on the derived class is neither available in the destructor, nor in the constructor, we obtain it in between and store it here.

Definition at line 232 of file enable_observer_pointer.h.

◆ mutex

std::mutex EnableObserverPointer::mutex

staticprivateinherited

A mutex used to ensure data consistency when accessing the mutable members of this class. This lock is used in the subscribe() and unsubscribe() functions, as well as in list_subscribers().

Definition at line 239 of file enable_observer_pointer.h.

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

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

Detailed Description

Public Member Functions

Static Public Member Functions

Protected Member Functions

Static Protected Member Functions

Protected Attributes

Private Member Functions

Private Attributes

Static Private Attributes

Constructor & Destructor Documentation

◆ Polynomial() [1/4]

◆ Polynomial() [2/4]

◆ Polynomial() [3/4]

◆ Polynomial() [4/4]

Member Function Documentation

◆ value() [1/3]

◆ value() [2/3]

◆ value() [3/3]

◆ values_of_array()

◆ degree()

◆ scale() [1/2]

◆ shift() [1/2]

◆ derivative()

◆ primitive()

◆ operator*=() [1/2]

◆ operator*=() [2/2]

◆ operator+=()

◆ operator-=()

◆ operator==()

◆ print()

◆ serialize()

◆ memory_consumption()

◆ scale() [2/2]

◆ shift() [2/2]

◆ multiply()

◆ transform_into_standard_form()

◆ n_subscriptions()

◆ list_subscribers() [1/2]

◆ list_subscribers() [2/2]

◆ subscribe()

◆ unsubscribe()

◆ check_no_subscribers()

Member Data Documentation

◆ coefficients

◆ in_lagrange_product_form

◆ lagrange_support_points

◆ lagrange_weight

◆ counter

◆ counter_map

◆ validity_pointers

◆ object_info

◆ mutex