mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2024-09-05 12:56:44 -04:00
108 lines
3.2 KiB
Plaintext
108 lines
3.2 KiB
Plaintext
[section:hermite Hermite Polynomials]
|
|
|
|
[h4 Synopsis]
|
|
|
|
``
|
|
#include <boost/math/special_functions/hermite.hpp>
|
|
``
|
|
|
|
namespace boost{ namespace math{
|
|
|
|
template <class T>
|
|
``__sf_result`` hermite(unsigned n, T x);
|
|
|
|
template <class T, class ``__Policy``>
|
|
``__sf_result`` hermite(unsigned n, T x, const ``__Policy``&);
|
|
|
|
template <class T1, class T2, class T3>
|
|
``__sf_result`` hermite_next(unsigned n, T1 x, T2 Hn, T3 Hnm1);
|
|
|
|
}} // namespaces
|
|
|
|
[h4 Description]
|
|
|
|
The return type of these functions is computed using the __arg_promotion_rules:
|
|
note than when there is a single template argument the result is the same type
|
|
as that argument or `double` if the template argument is an integer type.
|
|
|
|
template <class T>
|
|
``__sf_result`` hermite(unsigned n, T x);
|
|
|
|
template <class T, class ``__Policy``>
|
|
``__sf_result`` hermite(unsigned n, T x, const ``__Policy``&);
|
|
|
|
Returns the value of the Hermite Polynomial of order /n/ at point /x/:
|
|
|
|
[equation hermite_0]
|
|
|
|
[optional_policy]
|
|
|
|
The following graph illustrates the behaviour of the first few
|
|
Hermite Polynomials:
|
|
|
|
[graph hermite]
|
|
|
|
template <class T1, class T2, class T3>
|
|
``__sf_result`` hermite_next(unsigned n, T1 x, T2 Hn, T3 Hnm1);
|
|
|
|
Implements the three term recurrence relation for the Hermite
|
|
polynomials, this function can be used to create a sequence of
|
|
values evaluated at the same /x/, and for rising /n/.
|
|
|
|
[equation hermite_1]
|
|
|
|
For example we could produce a vector of the first 10 polynomial
|
|
values using:
|
|
|
|
double x = 0.5; // Abscissa value
|
|
vector<double> v;
|
|
v.push_back(hermite(0, x)).push_back(hermite(1, x));
|
|
for(unsigned l = 1; l < 10; ++l)
|
|
v.push_back(hermite_next(l, x, v[l], v[l-1]));
|
|
|
|
Formally the arguments are:
|
|
|
|
[variablelist
|
|
[[n][The degree /n/ of the last polynomial calculated.]]
|
|
[[x][The abscissa value]]
|
|
[[Hn][The value of the polynomial evaluated at degree /n/.]]
|
|
[[Hnm1][The value of the polynomial evaluated at degree /n-1/.]]
|
|
]
|
|
|
|
[h4 Accuracy]
|
|
|
|
The following table shows peak errors (in units of epsilon)
|
|
for various domains of input arguments.
|
|
Note that only results for the widest floating point type on the system are
|
|
given as narrower types have __zero_error.
|
|
|
|
[table_hermite]
|
|
|
|
Note that the worst errors occur when the degree increases, values greater than
|
|
~120 are very unlikely to produce sensible results, especially in the associated
|
|
polynomial case when the order is also large. Further the relative errors
|
|
are likely to grow arbitrarily large when the function is very close to a root.
|
|
|
|
[h4 Testing]
|
|
|
|
A mixture of spot tests of values calculated using functions.wolfram.com,
|
|
and randomly generated test data are
|
|
used: the test data was computed using
|
|
[@http://shoup.net/ntl/doc/RR.txt NTL::RR] at 1000-bit precision.
|
|
|
|
[h4 Implementation]
|
|
|
|
These functions are implemented using the stable three term
|
|
recurrence relations. These relations guarantee low absolute error
|
|
but cannot guarantee low relative error near one of the roots of the
|
|
polynomials.
|
|
|
|
[endsect][/section:beta_function The Beta Function]
|
|
[/
|
|
Copyright 2006 John Maddock and Paul A. Bristow.
|
|
Distributed under the Boost Software License, Version 1.0.
|
|
(See accompanying file LICENSE_1_0.txt or copy at
|
|
http://www.boost.org/LICENSE_1_0.txt).
|
|
]
|
|
|