Reference documentation for deal.II version 9.4.0
Polynomials::HermiteLikeInterpolation Class Reference

#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

## 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

## 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 ::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

static std::mutex mutex

void check_no_subscribers () const noexcept

## 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 726 of file polynomial.h.

## ◆ 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 1175 of file polynomial.cc.

## ◆ 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 1405 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 107 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 120 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 142 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 160 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 170 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 180 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 270 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 199 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 277 of file polynomial.cc.

## ◆ derivative()

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

Compute the derivative of a polynomial.

Definition at line 205 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 212 of file polynomial.cc.

## ◆ operator*=() [1/2]

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

Multiply with a scalar.

Definition at line 218 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 224 of file polynomial.cc.

## ◆ operator+=()

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

Definition at line 230 of file polynomial.cc.

## ◆ operator-=()

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

Subtract a second polynomial.

Definition at line 236 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 242 of file polynomial.cc.

## ◆ print()

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

Print coefficients.

Definition at line 248 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 257 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 263 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 283 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 291 of file polynomial.cc.

## ◆ 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 301 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 307 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 313 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 319 of file polynomial.h.

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