Reference documentation for deal.II version Git 1ecc23629d 20210518 09:57:04 +0200

#include <deal.II/base/polynomial.h>
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 
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 
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 ::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< 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 
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 generalpurpose scheme. An alternative and more stable evaluation of highdegree 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 63 of file polynomial.h.
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 53 of file polynomial.cc.
Polynomials::Polynomial< number >::Polynomial  (  const unsigned int  n  ) 
Constructor creating a zero polynomial of degree n
.
Definition at line 62 of file polynomial.cc.
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 \(\prod_{i\neq j} \frac{xx_i}{x_jx_i}\), 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 71 of file polynomial.cc.

inline 
Default constructor creating an illegal object.
Definition at line 771 of file polynomial.h.

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 797 of file polynomial.h.
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 i
th 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.

inline 
Return the values and the derivatives of the Polynomial at point x
. values[i], i=0,...,n_derivatives
includes the i
th 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.

inline 
Degree of the polynomial. This is the degree reflected by the number of coefficients provided by the constructor. Leading nonzero coefficients are not treated separately.
Definition at line 780 of file polynomial.h.
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 165 of file polynomial.cc.
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 439 of file polynomial.cc.
Polynomial< number > Polynomials::Polynomial< number >::derivative  (  )  const 
Compute the derivative of a polynomial.
Definition at line 458 of file polynomial.cc.
Polynomial< number > Polynomials::Polynomial< number >::primitive  (  )  const 
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.
Polynomial< number > & Polynomials::Polynomial< number >::operator*=  (  const double  s  ) 
Multiply with a scalar.
Definition at line 203 of file polynomial.cc.
Polynomial< number > & Polynomials::Polynomial< number >::operator*=  (  const Polynomial< number > &  p  ) 
Multiply with another polynomial.
Definition at line 221 of file polynomial.cc.
Polynomial< number > & Polynomials::Polynomial< number >::operator+=  (  const Polynomial< number > &  p  ) 
Add a second polynomial.
Definition at line 268 of file polynomial.cc.
Polynomial< number > & Polynomials::Polynomial< number >::operator=  (  const Polynomial< number > &  p  ) 
Subtract a second polynomial.
Definition at line 310 of file polynomial.cc.
bool Polynomials::Polynomial< number >::operator==  (  const Polynomial< number > &  p  )  const 
Test for equality of two polynomials.
Definition at line 346 of file polynomial.cc.
void Polynomials::Polynomial< number >::print  (  std::ostream &  out  )  const 
Print coefficients.
Definition at line 514 of file polynomial.cc.

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 962 of file polynomial.h.

virtual 
Return an estimate (in bytes) for the memory consumption of this object.
Definition at line 533 of file polynomial.cc.

staticprotected 
This function performs the actual scaling.
Definition at line 148 of file polynomial.cc.

staticprotected 
This function performs the actual shift
Definition at line 377 of file polynomial.cc.

staticprotected 
Multiply polynomial by a factor.
Definition at line 190 of file polynomial.cc.

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 111 of file polynomial.cc.

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.

inherited 
Unsubscribes a user from the object.
identifier
and the validity
pointer must be the same as the one supplied to subscribe(). Definition at line 156 of file subscriptor.cc.

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.

inlineinherited 
List the subscribers to the input stream
.
Definition at line 318 of file subscriptor.h.

inherited 
List the subscribers to deallog
.
Definition at line 204 of file subscriptor.cc.

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 282 of file polynomial.h.

protected 
Stores whether the polynomial is in Lagrange product form, i.e., constructed as a product \((xx_0) (xx_1) \ldots (xx_n)/c\), or not.
Definition at line 288 of file polynomial.h.

protected 
If the polynomial is in Lagrange product form, i.e., constructed as a product \((xx_0) (xx_1) \ldots (xx_n)/c\), store the shifts \(x_i\).
Definition at line 294 of file polynomial.h.

protected 
If the polynomial is in Lagrange product form, i.e., constructed as a product \((xx_0) (xx_1) \ldots (xx_n)/c\), store the weight c.
Definition at line 300 of file polynomial.h.