Reference documentation for deal.II version GIT c545eda070 20230127 00:25:02+00:00

#include <deal.II/base/polynomial.h>
Public Member Functions  
HermiteLikeInterpolation (const unsigned int degree, const unsigned int index)  
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 
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 ::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< 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 
Private Types  
using  map_value_type = decltype(counter_map)::value_type 
using  map_iterator = decltype(counter_map)::iterator 
Private Member Functions  
void  check_no_subscribers () const noexcept 
Private Attributes  
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 Private Attributes  
static std::mutex  mutex 
Polynomials for a variant of Hermite polynomials with better condition number in the interpolation than the basis from HermiteInterpolation.
In analogy to the proper Hermite polynomials, this basis evaluates the first polynomial \(p_0\) to 1 at \(x=0\) and has both a zero value and zero derivative at \(x=1\). Likewise, the last polynomial \(p_n\) evaluates to 1 at \(x=1\) with a zero value and zero derivative at \(x=0\). The second polynomial \(p_1\) and the second to last polynomial \(p_{n1}\) represent the derivative degree of freedom at \(x=0\) and \(x=1\), respectively. They are zero at both the end points \(x=0, x=1\) and have zero derivative at the opposite end, \(p_1'(1)=0\) and \(p_{n1}'(0)=0\). As opposed to the original Hermite polynomials, \(p_0\) does not have zero derivative at \(x=0\). The additional degree of freedom is used to make \(p_0\) and \(p_1\) orthogonal, which for \(n=3\) results in a root at \(x=\frac{2}{7}\) for \(p_0\) and at \(x=\frac{5}{7}\) for \(p_n\), respectively. Furthermore, the extension of these polynomials to higher degrees \(n>3\) is constructed by adding additional nodes inside the unit interval, again ensuring better conditioning. The nodes are computed as the roots of the Jacobi polynomials for \(\alpha=\beta=4\), which are orthogonal against the square of the generating function \(x^2(1x)^2\) with the Hermite property. Then, these polynomials are constructed in the usual way as Lagrange polynomials with double roots at \(x=0\) and \(x=1\). For example with \(n=4\), all of \(p_0, p_1, p_3, p_4\) get an additional root at \(x=0.5\) through the factor \((x0.5)\). In summary, this basis is dominated by nodal contributions, but it is not a nodal one because the second and second to last polynomials that are nonnodal, and due to the presence of double nodes in \(x=0\) and \(x=1\). The weights of the basis functions are set such that the sum of all polynomials with unit weight represents the constant function 1, similarly to Lagrange polynomials.
The basis only contains Hermite information for degree>=3
, but it is also implemented for degrees between 0 and two. For the linear case, the usual hat functions are implemented, whereas the polynomials for degree=2
are \(p_0(x)=(1x)^2\), \(p_1(x)=2x(x1)\), and \(p_2(x)=x^2\), in accordance with the construction principle for degree 3.
These two relaxations improve the condition number of the mass matrix (i.e., interpolation) significantly, as can be seen from the following table:
Condition number mass matrix  

degree  HermiteInterpolation  HermiteLikeInterpolation 
n=3  1057  17.18 
n=4  6580  16.83 
n=5  1.875e+04  15.99 
n=6  6.033e+04  16.34 
n=10  9.756e+05  20.70 
n=15  9.431e+06  27.91 
n=25  2.220e+08  43.54 
n=35  2.109e+09  59.51 
This polynomial inherits the advantageous property of Hermite polynomials where only two functions have value and/or derivative nonzero on a face advantageous for discontinuous Galerkin methods but gives better condition numbers of interpolation, which improves the performance of some iterative schemes like conjugate gradients with pointJacobi. This polynomial is used in FE_DGQHermite.
Definition at line 727 of file polynomial.h.

privateinherited 
The data type used in counter_map.
Definition at line 230 of file subscriptor.h.

privateinherited 
The iterator type used in counter_map.
Definition at line 235 of file subscriptor.h.
Polynomials::HermiteLikeInterpolation::HermiteLikeInterpolation  (  const unsigned int  degree, 
const unsigned int  index  
) 
Constructor for the polynomial with index index
within the set up polynomials of degree degree
.
Definition at line 1175 of file polynomial.cc.

static 
Return the polynomials with index 0
up to degree+1
in a space of degree up to degree
.
Definition at line 1405 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 108 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 121 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 143 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 161 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 171 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 181 of file polynomial.cc.

staticprotectedinherited 
This function performs the actual scaling.
Definition at line 271 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 200 of file polynomial.cc.

staticprotectedinherited 
This function performs the actual shift
Definition at line 278 of file polynomial.cc.

inherited 
Compute the derivative of a polynomial.
Definition at line 206 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 213 of file polynomial.cc.

inherited 
Multiply with a scalar.
Definition at line 219 of file polynomial.cc.

inherited 
Multiply with another polynomial.
Definition at line 225 of file polynomial.cc.

inherited 
Add a second polynomial.
Definition at line 231 of file polynomial.cc.

inherited 
Subtract a second polynomial.
Definition at line 237 of file polynomial.cc.

inherited 
Test for equality of two polynomials.
Definition at line 243 of file polynomial.cc.

inherited 
Print coefficients.
Definition at line 249 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 258 of file polynomial.h.

virtualinherited 
Return an estimate (in bytes) for the memory consumption of this object.
Definition at line 264 of file polynomial.cc.

staticprotectedinherited 
Multiply polynomial by a factor.
Definition at line 284 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 292 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 300 of file subscriptor.h.

inlineinherited 
List the subscribers to the input stream
.
Definition at line 317 of file subscriptor.h.

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

privatenoexceptinherited 
Check that there are no objects subscribing to this object. If this check passes then it is safe to destroy the current object. It this check fails then this function will either abort or print an error message to deallog (by using the AssertNothrow mechanism), but will not throw an exception.
Definition at line 53 of file subscriptor.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 302 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 308 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 314 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 320 of file polynomial.h.

mutableprivateinherited 
Store the number of objects which subscribed to this object. Initially, this number is zero, and upon destruction it shall be zero again (i.e. all objects which subscribed should have unsubscribed again).
The creator (and owner) of an object is counted in the map below if HE manages to supply identification.
We use the mutable
keyword in order to allow subscription to constant objects also.
This counter may be read from and written to concurrently in multithreaded code: hence we use the std::atomic
class template.
Definition at line 219 of file subscriptor.h.

mutableprivateinherited 
In this map, we count subscriptions for each different identification string supplied to subscribe().
Definition at line 225 of file subscriptor.h.

mutableprivateinherited 
In this vector, we store pointers to the validity bool in the SmartPointer objects that subscribe to this class.
Definition at line 241 of file subscriptor.h.

mutableprivateinherited 
Pointer to the typeinfo object of this object, from which we can later deduce the class name. Since this information on the derived class is neither available in the destructor, nor in the constructor, we obtain it in between and store it here.
Definition at line 249 of file subscriptor.h.

staticprivateinherited 
A mutex used to ensure data consistency when printing out the list of subscribers.
Definition at line 271 of file subscriptor.h.