Reference documentation for deal.II version 9.3.3
Polynomials::Hierarchical Class Reference

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

Inheritance diagram for Polynomials::Hierarchical:
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## Public Member Functions

Hierarchical (const unsigned int p)

double value (const double x) const

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

void value (const Number2 x, const unsigned int n_derivatives, Number2 *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)

virtual std::size_t memory_consumption () const

## Static Public Member Functions

static std::vector< Polynomial< double > > generate_complete_basis (const unsigned int degree)

## Protected Member Functions

void transform_into_standard_form ()

## Static Protected Member Functions

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

std::vector< double > coefficients

bool in_lagrange_product_form

std::vector< double > lagrange_support_points

double lagrange_weight

## Static Private Member Functions

static void compute_coefficients (const unsigned int p)

static const std::vector< double > & get_coefficients (const unsigned int p)

## Static Private Attributes

static std::vector< std::unique_ptr< const std::vector< double > > > recursive_coefficients

## Subscriptor functionality

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

static std::mutex mutex

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

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

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

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

void check_no_subscribers () const noexcept

## Detailed Description

Hierarchical polynomials of arbitrary degree on [0,1].

When Constructing a Hierarchical polynomial of degree p, the coefficients will be computed by a recursion formula. The coefficients are stored in a static data vector to be available when needed next time.

These hierarchical polynomials are based on those of Demkowicz, Oden, Rachowicz, and Hardy (CMAME 77 (1989) 79-112, Sec. 4). The first two polynomials are the standard linear shape functions given by $$\phi_{0}(x) = 1 - x$$ and $$\phi_{1}(x) = x$$. For $$l \geq 2$$ we use the definitions $$\phi_{l}(x) = (2x-1)^l - 1, l = 2,4,6,...$$ and $$\phi_{l}(x) = (2x-1)^l - (2x-1), l = 3,5,7,...$$. These satisfy the recursion relations $$\phi_{l}(x) = (2x-1)\phi_{l-1}, l=3,5,7,...$$ and $$\phi_{l}(x) = (2x-1)\phi_{l-1} + \phi_{2}, l=4,6,8,...$$.

The degrees of freedom are the values at the vertices and the derivatives at the midpoint. Currently, we do not scale the polynomials in any way, although better conditioning of the element stiffness matrix could possibly be achieved with scaling.

Calling the constructor with a given index p will generate the following: if p==0, then the resulting polynomial is the linear function associated with the left vertex, if p==1 the one associated with the right vertex. For higher values of p, you get the polynomial of degree p that is orthogonal to all previous ones. Note that for p==0 you therefore do not get a polynomial of degree zero, but one of degree one. This is to allow generating a complete basis for polynomial spaces, by just iterating over the indices given to the constructor.

On the other hand, the function generate_complete_basis() creates a complete basis of given degree. In order to be consistent with the concept of a polynomial degree, if the given argument is zero, it does not return the linear polynomial described above, but rather a constant polynomial.

Definition at line 509 of file polynomial.h.

## ◆ Hierarchical()

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

Constructor for polynomial of degree p. There is an exception for p==0, see the general documentation.

Definition at line 871 of file polynomial.cc.

## ◆ generate_complete_basis()

 std::vector< Polynomial< double > > Polynomials::Hierarchical::generate_complete_basis ( const unsigned int degree )
static

Return a vector of Hierarchical polynomial objects of degrees zero through degree, which then spans the full space of polynomials up to the given degree. Note that there is an exception if the given degree equals zero, see the general documentation of this class.

This function may be used to initialize the TensorProductPolynomials, AnisotropicPolynomials, and PolynomialSpace classes.

Definition at line 1028 of file polynomial.cc.

## ◆ compute_coefficients()

 void Polynomials::Hierarchical::compute_coefficients ( const unsigned int p )
staticprivate

Compute coefficients recursively.

Definition at line 878 of file polynomial.cc.

## ◆ get_coefficients()

 const std::vector< double > & Polynomials::Hierarchical::get_coefficients ( const unsigned int p )
staticprivate

Get coefficients for constructor. This way, it can use the non- standard constructor of Polynomial.

Definition at line 1012 of file polynomial.cc.

## ◆ value() [1/3]

 double Polynomials::Polynomial< double >::value ( const double 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 106 of file polynomial.h.

## ◆ value() [2/3]

 void Polynomials::Polynomial< double >::value ( const double x, std::vector< double > & 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 119 of file polynomial.cc.

## ◆ value() [3/3]

 void Polynomials::Polynomial< double >::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 141 of file polynomial.h.

## ◆ degree()

 unsigned int Polynomials::Polynomial< double >::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 151 of file polynomial.h.

## ◆ scale() [1/2]

 void Polynomials::Polynomial< double >::scale ( const double 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 161 of file polynomial.cc.

## ◆ scale() [2/2]

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

This function performs the actual scaling.

Definition at line 251 of file polynomial.cc.

## ◆ shift() [1/2]

 void Polynomials::Polynomial< double >::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 180 of file polynomial.cc.

## ◆ shift() [2/2]

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

This function performs the actual shift

Definition at line 258 of file polynomial.cc.

## ◆ derivative()

 Polynomial< double > Polynomials::Polynomial< double >::derivative
inherited

Compute the derivative of a polynomial.

Definition at line 186 of file polynomial.cc.

## ◆ primitive()

 Polynomial< double > Polynomials::Polynomial< double >::primitive
inherited

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

Definition at line 193 of file polynomial.cc.

## ◆ operator*=() [1/2]

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

Multiply with a scalar.

Definition at line 199 of file polynomial.cc.

## ◆ operator*=() [2/2]

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

Multiply with another polynomial.

Definition at line 205 of file polynomial.cc.

## ◆ operator+=()

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

Definition at line 211 of file polynomial.cc.

## ◆ operator-=()

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

Subtract a second polynomial.

Definition at line 217 of file polynomial.cc.

## ◆ operator==()

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

Test for equality of two polynomials.

Definition at line 223 of file polynomial.cc.

## ◆ print()

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

Print coefficients.

Definition at line 229 of file polynomial.cc.

## ◆ serialize()

 void Polynomials::Polynomial< double >::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 238 of file polynomial.h.

## ◆ memory_consumption()

 std::size_t Polynomials::Polynomial< double >::memory_consumption
virtualinherited

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

Definition at line 244 of file polynomial.cc.

## ◆ multiply()

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

Multiply polynomial by a factor.

Definition at line 264 of file polynomial.cc.

## ◆ transform_into_standard_form()

 void Polynomials::Polynomial< double >::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 272 of file polynomial.cc.

## ◆ recursive_coefficients

 std::vector< std::unique_ptr< const std::vector< double > > > Polynomials::Hierarchical::recursive_coefficients
staticprivate

Vector with already computed coefficients. For each degree of the polynomial, we keep one pointer to the list of coefficients; we do so rather than keeping a vector of vectors in order to simplify programming multithread-safe. In order to avoid memory leak, we use a unique_ptr in order to correctly free the memory of the vectors when the global destructor is called.

Definition at line 554 of file polynomial.h.

## ◆ coefficients

 std::vector Polynomials::Polynomial< double >::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 282 of file polynomial.h.

## ◆ in_lagrange_product_form

 bool Polynomials::Polynomial< double >::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 288 of file polynomial.h.

## ◆ lagrange_support_points

 std::vector Polynomials::Polynomial< double >::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 294 of file polynomial.h.

## ◆ lagrange_weight

 double Polynomials::Polynomial< double >::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 300 of file polynomial.h.

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