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

Inheritance diagram for Polynomials::HermiteLikeInterpolation:

[legend]

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

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

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 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

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

Static Public Member Functions
static std::vector< Polynomial< double > >	generate_complete_basis (const unsigned int degree)

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< 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

Private Types
using	map_value_type = decltype(counter_map)::value_type

using	map_iterator = decltype(counter_map)::iterator

Private Member Functions
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

Detailed Description

Polynomials for a variant of Hermite polynomials with better condition number in the interpolation than the basis from HermiteInterpolation.

In analogy to the proper Hermite polynomials, this basis evaluates the first polynomial \(p_0\) to 1 at \(x=0\) and has both a zero value and zero derivative at \(x=1\). Likewise, the last polynomial \(p_n\) evaluates to 1 at \(x=1\) with a zero value and zero derivative at \(x=0\). The second polynomial \(p_1\) and the second to last polynomial \(p_{n-1}\) represent the derivative degree of freedom at \(x=0\) and \(x=1\), respectively. They are zero at both the end points \(x=0, x=1\) and have zero derivative at the opposite end, \(p_1'(1)=0\) and \(p_{n-1}'(0)=0\). As opposed to the original Hermite polynomials, \(p_0\) does not have zero derivative at \(x=0\). The additional degree of freedom is used to make \(p_0\) and \(p_1\) orthogonal, which for \(n=3\) results in a root at \(x=\frac{2}{7}\) for \(p_0\) and at \(x=\frac{5}{7}\) for \(p_n\), respectively. Furthermore, the extension of these polynomials to higher degrees \(n>3\) is constructed by adding additional nodes inside the unit interval, again ensuring better conditioning. The nodes are computed as the roots of the Jacobi polynomials for \(\alpha=\beta=4\), which are orthogonal against the square of the generating function \(x^2(1-x)^2\) with the Hermite property. Then, these polynomials are constructed in the usual way as Lagrange polynomials with double roots at \(x=0\) and \(x=1\). For example with \(n=4\), all of \(p_0, p_1, p_3, p_4\) get an additional root at \(x=0.5\) through the factor \((x-0.5)\). In summary, this basis is dominated by nodal contributions, but it is not a nodal one because the second and second to last polynomials that are non-nodal, and due to the presence of double nodes in \(x=0\) and \(x=1\). The weights of the basis functions are set such that the sum of all polynomials with unit weight represents the constant function 1, similarly to Lagrange polynomials.

The basis only contains Hermite information for degree>=3, but it is also implemented for degrees between 0 and two. For the linear case, the usual hat functions are implemented, whereas the polynomials for degree=2 are \(p_0(x)=(1-x)^2\), \(p_1(x)=2x(x-1)\), and \(p_2(x)=x^2\), in accordance with the construction principle for degree 3.

These two relaxations improve the condition number of the mass matrix (i.e., interpolation) significantly, as can be seen from the following table:

	Condition number mass matrix
degree	HermiteInterpolation	HermiteLikeInterpolation
n=3	1057	17.18
n=4	6580	16.83
n=5	1.875e+04	15.99
n=6	6.033e+04	16.34
n=10	9.756e+05	20.70
n=15	9.431e+06	27.91
n=25	2.220e+08	43.54
n=35	2.109e+09	59.51

This polynomial inherits the advantageous property of Hermite polynomials where only two functions have value and/or derivative nonzero on a face advantageous for discontinuous Galerkin methods but gives better condition numbers of interpolation, which improves the performance of some iterative schemes like conjugate gradients with point-Jacobi. This polynomial is used in FE_DGQHermite.

Definition at line 736 of file polynomial.h.

Member Typedef Documentation

◆ map_value_type

using Subscriptor::map_value_type = decltype(counter_map)::value_type

privateinherited

The data type used in counter_map.

Definition at line 229 of file subscriptor.h.

◆ map_iterator

using Subscriptor::map_iterator = decltype(counter_map)::iterator

privateinherited

The iterator type used in counter_map.

Definition at line 234 of file subscriptor.h.

Constructor & Destructor Documentation

◆ HermiteLikeInterpolation()

Polynomials::HermiteLikeInterpolation::HermiteLikeInterpolation	(	const unsigned int	degree,
		const unsigned int	index )

Constructor for the polynomial with index index within the set up polynomials of degree degree.

Definition at line 1166 of file polynomial.cc.

Member Function Documentation

◆ generate_complete_basis()

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

static

Return the polynomials with index 0 up to degree+1 in a space of degree up to degree.

Definition at line 1396 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 108 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 121 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 143 of file polynomial.h.

◆ values_of_array()

void Polynomials::Polynomial< double >::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 164 of file polynomial.h.

◆ degree()

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

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 174 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 184 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 274 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 203 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 281 of file polynomial.cc.

◆ derivative()

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

inherited

Compute the derivative of a polynomial.

Definition at line 209 of file polynomial.cc.

◆ primitive()

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

inherited

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

Definition at line 216 of file polynomial.cc.

◆ operator*=() [1/2]

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

inherited

Multiply with a scalar.

Definition at line 222 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 228 of file polynomial.cc.

◆ operator+=()

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

inherited

Add a second polynomial.

Definition at line 234 of file polynomial.cc.

◆ operator-=()

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

inherited

Subtract a second polynomial.

Definition at line 240 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 246 of file polynomial.cc.

◆ print()

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

inherited

Print coefficients.

Definition at line 252 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 261 of file polynomial.h.

◆ memory_consumption()

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

virtualinherited

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

Definition at line 267 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 287 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 295 of file polynomial.cc.

◆ subscribe()

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

inherited

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

Definition at line 135 of file subscriptor.cc.

◆ unsubscribe()

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

inherited

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 155 of file subscriptor.cc.

◆ n_subscriptions()

unsigned int Subscriptor::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 300 of file subscriptor.h.

◆ list_subscribers() [1/2]

template<typename StreamType >

void Subscriptor::list_subscribers ( StreamType & stream ) const

inlineinherited

List the subscribers to the input stream.

Definition at line 317 of file subscriptor.h.

◆ list_subscribers() [2/2]

void Subscriptor::list_subscribers ( ) const

inherited

List the subscribers to deallog.

Definition at line 203 of file subscriptor.cc.

◆ check_no_subscribers()

void Subscriptor::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 52 of file subscriptor.cc.

Member Data Documentation

◆ coefficients

std::vector<double> 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 305 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 311 of file polynomial.h.

◆ lagrange_support_points

std::vector<double> 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 317 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 323 of file polynomial.h.

◆ counter

std::atomic<unsigned int> Subscriptor::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 218 of file subscriptor.h.

◆ counter_map

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

mutableprivateinherited

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

Definition at line 224 of file subscriptor.h.

◆ validity_pointers

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

mutableprivateinherited

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

Definition at line 240 of file subscriptor.h.

◆ object_info

const std::type_info* Subscriptor::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 248 of file subscriptor.h.

◆ mutex

std::mutex Subscriptor::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 271 of file subscriptor.h.

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

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

Public Member Functions

Static Public Member Functions

Protected Member Functions

Static Protected Member Functions

Protected Attributes

Private Types

Private Member Functions

Private Attributes

Static Private Attributes

Detailed Description

Member Typedef Documentation

◆ map_value_type

◆ map_iterator

Constructor & Destructor Documentation

◆ HermiteLikeInterpolation()

Member Function Documentation

◆ generate_complete_basis()

◆ 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()

◆ subscribe()

◆ unsubscribe()

◆ n_subscriptions()

◆ list_subscribers() [1/2]

◆ list_subscribers() [2/2]

◆ check_no_subscribers()

Member Data Documentation

◆ coefficients

◆ in_lagrange_product_form

◆ lagrange_support_points

◆ lagrange_weight

◆ counter

◆ counter_map

◆ validity_pointers

◆ object_info

◆ mutex