![]() |
Reference documentation for deal.II version 9.3.3
|
#include <deal.II/base/polynomials_bernstein.h>
Public Member Functions | |
PolynomialsBernstein (const unsigned int index, const unsigned int degree) | |
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 |
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 |
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 |
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 |
static ::ExceptionBase & | ExcInUse (int arg1, std::string arg2, std::string arg3) |
static ::ExceptionBase & | ExcNoSubscriber (std::string arg1, std::string arg2) |
void | check_no_subscribers () const noexcept |
This class implements Bernstein basis polynomials of desire degree as described in http://www.idav.ucdavis.edu/education/CAGDNotes/Bernstein-Polynomials.pdf in the paragraph "Converting from the Bernstein Basis to the Power Basis".
They are used to create the Bernstein finite element FE_Bernstein.
Definition at line 41 of file polynomials_bernstein.h.
PolynomialsBernstein< number >::PolynomialsBernstein | ( | const unsigned int | index, |
const unsigned int | degree | ||
) |
Construct the index
-th Bernstein Polynomial of degree degree
.
\begin{align*} B_{\text{index}, \text{degree}} (t) &= \text{binom}(\text{degree}, \text{index}) \cdot t^{\text{index}} \cdot (1 - t)^{\text{degree} - \text{index}} \\ &= \sum_{i = \text{index}}^\text{degree} \cdot (-1)^{i - \text{index}} \cdot \text{binom}(\text{degree}, i) \cdot \text{binom}(i, \text{index}) \cdot t^i \end{align*}
index | |
degree |
Definition at line 45 of file polynomials_bernstein.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 797 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 100 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 827 of file polynomial.h.
|
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.
|
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.
|
staticprotectedinherited |
This function performs the actual scaling.
Definition at line 148 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 439 of file polynomial.cc.
|
staticprotectedinherited |
This function performs the actual shift
Definition at line 377 of file polynomial.cc.
|
inherited |
Compute the derivative of a polynomial.
Definition at line 458 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 487 of file polynomial.cc.
|
inherited |
Multiply with a scalar.
Definition at line 203 of file polynomial.cc.
|
inherited |
Multiply with another polynomial.
Definition at line 221 of file polynomial.cc.
|
inherited |
Add a second polynomial.
Definition at line 268 of file polynomial.cc.
|
inherited |
Subtract a second polynomial.
Definition at line 310 of file polynomial.cc.
|
inherited |
Test for equality of two polynomials.
Definition at line 346 of file polynomial.cc.
|
inherited |
Print coefficients.
Definition at line 514 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 962 of file polynomial.h.
|
virtualinherited |
Return an estimate (in bytes) for the memory consumption of this object.
Definition at line 533 of file polynomial.cc.
|
staticprotectedinherited |
Multiply polynomial by a factor.
Definition at line 190 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 111 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 282 of file polynomial.h.
|
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.
|
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.
|
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.