Reference documentation for deal.II version 9.4.1

#include <deal.II/base/polynomial.h>
Public Member Functions  
LagrangeEquidistant (const unsigned int n, const unsigned int support_point)  
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 
Static Private Member Functions  
static void  compute_coefficients (const unsigned int n, const unsigned int support_point, std::vector< double > &a) 
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 ::ExceptionBase &  ExcInUse (int arg1, std::string arg2, std::string arg3) 
static ::ExceptionBase &  ExcNoSubscriber (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 int >  counter 
std::map< std::string, unsigned int >  counter_map 
std::vector< std::atomic< bool > * >  validity_pointers 
const std::type_info *  object_info 
static std::mutex  mutex 
void  check_no_subscribers () const noexcept 
Lagrange polynomials with equidistant interpolation points in [0,1]. The polynomial of degree n
has got n+1
interpolation points. The interpolation points are sorted in ascending order. This order gives an index to each interpolation point. A Lagrangian polynomial equals to 1 at its ‘support point’, and 0 at all other interpolation points. For example, if the degree is 3, and the support point is 1, then the polynomial represented by this object is cubic and its value is 1 at the point x=1/3
, and zero at the point x=0
, x=2/3
, and x=1
. All the polynomials have polynomial degree equal to degree
, but together they span the entire space of polynomials of degree less than or equal degree
.
The Lagrange polynomials are implemented up to degree 10.
Definition at line 370 of file polynomial.h.
Polynomials::LagrangeEquidistant::LagrangeEquidistant  (  const unsigned int  n, 
const unsigned int  support_point  
) 
Constructor. Takes the degree n
of the Lagrangian polynomial and the index support_point
of the support point. Fills the coefficients
of the base class Polynomial.
Definition at line 597 of file polynomial.cc.

static 
Return a vector of polynomial objects of degree degree
, which then spans the full space of polynomials up to the given degree. The polynomials are generated by calling the constructor of this class with the same degree but support point running from zero to degree
. This function may be used to initialize the TensorProductPolynomials and PolynomialSpace classes.
Definition at line 679 of file polynomial.cc.

staticprivate 
Compute the coefficients
of the base class Polynomial. This function is static
to allow to be called in the constructor.
Definition at line 621 of file polynomial.cc.

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.

inherited 
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 120 of file polynomial.cc.

inlineinherited 
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 142 of file polynomial.h.

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 tensorproduct 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.

inlineinherited 
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 170 of file polynomial.h.

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.

staticprotectedinherited 
This function performs the actual scaling.
Definition at line 270 of file polynomial.cc.

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.

staticprotectedinherited 
This function performs the actual shift
Definition at line 277 of file polynomial.cc.

inherited 
Compute the derivative of a polynomial.
Definition at line 205 of file polynomial.cc.

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.

inherited 
Multiply with a scalar.
Definition at line 218 of file polynomial.cc.

inherited 
Multiply with another polynomial.
Definition at line 224 of file polynomial.cc.

inherited 
Add a second polynomial.
Definition at line 230 of file polynomial.cc.

inherited 
Subtract a second polynomial.
Definition at line 236 of file polynomial.cc.

inherited 
Test for equality of two polynomials.
Definition at line 242 of file polynomial.cc.

inherited 
Print coefficients.
Definition at line 248 of file polynomial.cc.

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.

virtualinherited 
Return an estimate (in bytes) for the memory consumption of this object.
Definition at line 263 of file polynomial.cc.

staticprotectedinherited 
Multiply polynomial by a factor.
Definition at line 283 of file polynomial.cc.

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.

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.

protectedinherited 
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 307 of file polynomial.h.

protectedinherited 
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 313 of file polynomial.h.

protectedinherited 
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 319 of file polynomial.h.