Reference documentation for deal.II version 9.1.1

#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 
void  value (const number x, const unsigned int n_derivatives, number *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) 
Public Member Functions inherited from Subscriptor  
Subscriptor ()  
Subscriptor (const Subscriptor &)  
Subscriptor (Subscriptor &&) noexcept  
virtual  ~Subscriptor () 
Subscriptor &  operator= (const Subscriptor &) 
Subscriptor &  operator= (Subscriptor &&) noexcept 
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 
template<class Archive >  
void  serialize (Archive &ar, const unsigned int version) 
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 
Additional Inherited Members  
Static Public Member Functions inherited from Subscriptor  
static ::ExceptionBase &  ExcInUse (int arg1, std::string arg2, std::string arg3) 
static ::ExceptionBase &  ExcNoSubscriber (std::string arg1, std::string arg2) 
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.
Definition at line 62 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 52 of file polynomial.cc.
Polynomials::Polynomial< number >::Polynomial  (  const unsigned int  n  ) 
Constructor creating a zero polynomial of degree n
.
Definition at line 61 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 70 of file polynomial.cc.

inline 
Default constructor creating an illegal object.
Definition at line 766 of file polynomial.h.

inline 
Return the value of this polynomial at the given point.
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 792 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 99 of file polynomial.cc.
void Polynomials::Polynomial< number >::value  (  const number  x, 
const unsigned int  n_derivatives,  
number *  values  
)  const 
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 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 110 of file polynomial.cc.

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 775 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 297 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 571 of file polynomial.cc.
Polynomial< number > Polynomials::Polynomial< number >::derivative  (  )  const 
Compute the derivative of a polynomial.
Definition at line 590 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 619 of file polynomial.cc.
Polynomial< number > & Polynomials::Polynomial< number >::operator*=  (  const double  s  ) 
Multiply with a scalar.
Definition at line 335 of file polynomial.cc.
Polynomial< number > & Polynomials::Polynomial< number >::operator*=  (  const Polynomial< number > &  p  ) 
Multiply with another polynomial.
Definition at line 353 of file polynomial.cc.
Polynomial< number > & Polynomials::Polynomial< number >::operator+=  (  const Polynomial< number > &  p  ) 
Add a second polynomial.
Definition at line 400 of file polynomial.cc.
Polynomial< number > & Polynomials::Polynomial< number >::operator=  (  const Polynomial< number > &  p  ) 
Subtract a second polynomial.
Definition at line 442 of file polynomial.cc.
bool Polynomials::Polynomial< number >::operator==  (  const Polynomial< number > &  p  )  const 
Test for equality of two polynomials.
Definition at line 478 of file polynomial.cc.
void Polynomials::Polynomial< number >::print  (  std::ostream &  out  )  const 
Print coefficients.
Definition at line 646 of file polynomial.cc.

inline 
Write or read the data of this object to or from a stream for the purpose of serialization.
Definition at line 822 of file polynomial.h.

staticprotected 
This function performs the actual scaling.
Definition at line 280 of file polynomial.cc.

staticprotected 
This function performs the actual shift
Definition at line 509 of file polynomial.cc.

staticprotected 
Multiply polynomial by a factor.
Definition at line 322 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 243 of file polynomial.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 264 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 270 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 276 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 282 of file polynomial.h.