Reference documentation for deal.II version Git efdee58d95 2021-02-25 10:11:08 -0700
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Public Member Functions | Static Public Member Functions | Protected Member Functions | Static Protected Member Functions | Protected Attributes | List of all members
Polynomials::Legendre Class Reference

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

Inheritance diagram for Polynomials::Legendre:
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Public Member Functions

 Legendre (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< doublederivative () const
 
Polynomial< doubleprimitive () 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 ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)
 
static ::ExceptionBaseExcNoSubscriber (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< doublecoefficients
 
bool in_lagrange_product_form
 
std::vector< doublelagrange_support_points
 
double lagrange_weight
 

Detailed Description

Legendre polynomials of arbitrary degree. Constructing a Legendre polynomial of degree p, the roots will be computed by the Gauss formula of the respective number of points and a representation of the polynomial by its roots.

Note
The polynomials defined by this class differ in two aspects by what is usually referred to as Legendre polynomials: (i) This classes defines them on the reference interval \([0,1]\), rather than the commonly used interval \([-1,1]\). (ii) The polynomials have been scaled in such a way that they are orthonormal, not just orthogonal; consequently, the polynomials do not necessarily have boundary values equal to one.

Definition at line 409 of file polynomial.h.

Constructor & Destructor Documentation

◆ Legendre()

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

Constructor for polynomial of degree p.

Definition at line 717 of file polynomial.cc.

Member Function Documentation

◆ generate_complete_basis()

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

Return a vector of Legendre polynomial objects of degrees zero through degree, which then spans the full space of polynomials up to the given degree. This function may be used to initialize the TensorProductPolynomials and PolynomialSpace classes.

Definition at line 744 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 797 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 100 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 827 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 780 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 165 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 148 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 439 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 377 of file polynomial.cc.

◆ derivative()

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

Compute the derivative of a polynomial.

Definition at line 458 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 487 of file polynomial.cc.

◆ operator*=() [1/2]

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

Multiply with a scalar.

Definition at line 203 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 221 of file polynomial.cc.

◆ operator+=()

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

Add a second polynomial.

Definition at line 268 of file polynomial.cc.

◆ operator-=()

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

Subtract a second polynomial.

Definition at line 310 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 346 of file polynomial.cc.

◆ print()

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

Print coefficients.

Definition at line 514 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 962 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 533 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 190 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 111 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 136 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 156 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 301 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 318 of file subscriptor.h.

◆ list_subscribers() [2/2]

void Subscriptor::list_subscribers ( ) const
inherited

List the subscribers to deallog.

Definition at line 204 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 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<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 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: