Reference documentation for deal.II version 9.4.0
Polynomials::Polynomial< number > Class Template Reference

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

Inheritance diagram for Polynomials::Polynomial< number >:
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## 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 evaluation_point)

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

## 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 intcounter

std::map< std::string, unsigned intcounter_map

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

const std::type_info * object_info

static std::mutex mutex

static ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBaseExcNoSubscriber (std::string arg1, std::string arg2)

void check_no_subscribers () const noexcept

## 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.

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

Definition at line 64 of file polynomial.h.

## ◆ Polynomial() [1/4]

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

Constructor. The coefficients of the polynomial are passed as arguments, and denote the polynomial $$\sum_i a[i] x^i$$, i.e. the first element of the array denotes the constant term, the second the linear one, and so on. The degree of the polynomial represented by this object is thus the number of elements in the coefficient array minus one.

Definition at line 54 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 63 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 evaluation_point )

Constructor for a Lagrange polynomial and its point of evaluation. The idea is to construct $$\prod_{i\neq j} \frac{x-x_i}{x_j-x_i}$$, where j is the evaluation point specified as argument and the support points contain all points (including x_j, which will internally not be stored).

Definition at line 72 of file polynomial.cc.

## ◆ Polynomial() [4/4]

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

Default constructor creating an illegal object.

Definition at line 790 of file polynomial.h.

## ◆ 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 816 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 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
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 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
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 860 of file polynomial.h.

## ◆ degree()

template<typename number >
 unsigned int Polynomials::Polynomial< number >::degree
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 799 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 166 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 440 of file polynomial.cc.

## ◆ derivative()

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

Compute the derivative of a polynomial.

Definition at line 459 of file polynomial.cc.

## ◆ primitive()

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

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 )

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 )

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 )

Definition at line 269 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 311 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 347 of file polynomial.cc.

## ◆ print()

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

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 )
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 1023 of file polynomial.h.

## ◆ memory_consumption()

template<typename number >
 std::size_t Polynomials::Polynomial< number >::memory_consumption
virtual

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

Definition at line 534 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 149 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 378 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 191 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 112 of file polynomial.cc.

## ◆ coefficients

template<typename number >
 std::vector 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 301 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 307 of file polynomial.h.

## ◆ lagrange_support_points

template<typename number >
 std::vector 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 313 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 319 of file polynomial.h.

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