mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2024-11-07 09:44:16 -05:00
247 lines
6.6 KiB
Plaintext
247 lines
6.6 KiB
Plaintext
[/
|
|
Copyright (c) 2006 Xiaogang Zhang
|
|
Copyright (c) 2006 John Maddock
|
|
Use, modification and distribution are subject to 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)
|
|
]
|
|
|
|
[section:ellint_carlson Elliptic Integrals - Carlson Form]
|
|
|
|
[heading Synopsis]
|
|
|
|
``
|
|
#include <boost/math/special_functions/ellint_rf.hpp>
|
|
``
|
|
|
|
namespace boost { namespace math {
|
|
|
|
template <class T1, class T2, class T3>
|
|
``__sf_result`` ellint_rf(T1 x, T2 y, T3 z)
|
|
|
|
template <class T1, class T2, class T3, class ``__Policy``>
|
|
``__sf_result`` ellint_rf(T1 x, T2 y, T3 z, const ``__Policy``&)
|
|
|
|
}} // namespaces
|
|
|
|
|
|
``
|
|
#include <boost/math/special_functions/ellint_rd.hpp>
|
|
``
|
|
|
|
namespace boost { namespace math {
|
|
|
|
template <class T1, class T2, class T3>
|
|
``__sf_result`` ellint_rd(T1 x, T2 y, T3 z)
|
|
|
|
template <class T1, class T2, class T3, class ``__Policy``>
|
|
``__sf_result`` ellint_rd(T1 x, T2 y, T3 z, const ``__Policy``&)
|
|
|
|
}} // namespaces
|
|
|
|
|
|
``
|
|
#include <boost/math/special_functions/ellint_rj.hpp>
|
|
``
|
|
|
|
namespace boost { namespace math {
|
|
|
|
template <class T1, class T2, class T3, class T4>
|
|
``__sf_result`` ellint_rj(T1 x, T2 y, T3 z, T4 p)
|
|
|
|
template <class T1, class T2, class T3, class T4, class ``__Policy``>
|
|
``__sf_result`` ellint_rj(T1 x, T2 y, T3 z, T4 p, const ``__Policy``&)
|
|
|
|
}} // namespaces
|
|
|
|
|
|
``
|
|
#include <boost/math/special_functions/ellint_rc.hpp>
|
|
``
|
|
|
|
namespace boost { namespace math {
|
|
|
|
template <class T1, class T2>
|
|
``__sf_result`` ellint_rc(T1 x, T2 y)
|
|
|
|
template <class T1, class T2, class ``__Policy``>
|
|
``__sf_result`` ellint_rc(T1 x, T2 y, const ``__Policy``&)
|
|
|
|
}} // namespaces
|
|
|
|
``
|
|
#include <boost/math/special_functions/ellint_rg.hpp>
|
|
``
|
|
|
|
namespace boost { namespace math {
|
|
|
|
template <class T1, class T2, class T3>
|
|
``__sf_result`` ellint_rg(T1 x, T2 y, T3 z)
|
|
|
|
template <class T1, class T2, class T3, class ``__Policy``>
|
|
``__sf_result`` ellint_rg(T1 x, T2 y, T3 z, const ``__Policy``&)
|
|
|
|
}} // namespaces
|
|
|
|
|
|
|
|
[heading Description]
|
|
|
|
These functions return Carlson's symmetrical elliptic integrals, the functions
|
|
have complicated behavior over all their possible domains, but the following
|
|
graph gives an idea of their behavior:
|
|
|
|
[graph ellint_carlson]
|
|
|
|
The return type of these functions is computed using the __arg_promotion_rules
|
|
when the arguments are of different types: otherwise the return is the same type
|
|
as the arguments.
|
|
|
|
template <class T1, class T2, class T3>
|
|
``__sf_result`` ellint_rf(T1 x, T2 y, T3 z)
|
|
|
|
template <class T1, class T2, class T3, class ``__Policy``>
|
|
``__sf_result`` ellint_rf(T1 x, T2 y, T3 z, const ``__Policy``&)
|
|
|
|
Returns Carlson's Elliptic Integral R[sub F]:
|
|
|
|
[equation ellint9]
|
|
|
|
Requires that all of the arguments are non-negative, and at most
|
|
one may be zero. Otherwise returns the result of __domain_error.
|
|
|
|
[optional_policy]
|
|
|
|
template <class T1, class T2, class T3>
|
|
``__sf_result`` ellint_rd(T1 x, T2 y, T3 z)
|
|
|
|
template <class T1, class T2, class T3, class ``__Policy``>
|
|
``__sf_result`` ellint_rd(T1 x, T2 y, T3 z, const ``__Policy``&)
|
|
|
|
Returns Carlson's elliptic integral R[sub D]:
|
|
|
|
[equation ellint10]
|
|
|
|
Requires that x and y are non-negative, with at most one of them
|
|
zero, and that z >= 0. Otherwise returns the result of __domain_error.
|
|
|
|
[optional_policy]
|
|
|
|
template <class T1, class T2, class T3, class T4>
|
|
``__sf_result`` ellint_rj(T1 x, T2 y, T3 z, T4 p)
|
|
|
|
template <class T1, class T2, class T3, class T4, class ``__Policy``>
|
|
``__sf_result`` ellint_rj(T1 x, T2 y, T3 z, T4 p, const ``__Policy``&)
|
|
|
|
Returns Carlson's elliptic integral R[sub J]:
|
|
|
|
[equation ellint11]
|
|
|
|
Requires that x, y and z are non-negative, with at most one of them
|
|
zero, and that ['p != 0]. Otherwise returns the result of __domain_error.
|
|
|
|
[optional_policy]
|
|
|
|
When ['p < 0] the function returns the
|
|
[@http://en.wikipedia.org/wiki/Cauchy_principal_value Cauchy principal value]
|
|
using the relation:
|
|
|
|
[equation ellint17]
|
|
|
|
template <class T1, class T2>
|
|
``__sf_result`` ellint_rc(T1 x, T2 y)
|
|
|
|
template <class T1, class T2, class ``__Policy``>
|
|
``__sf_result`` ellint_rc(T1 x, T2 y, const ``__Policy``&)
|
|
|
|
Returns Carlson's elliptic integral R[sub C]:
|
|
|
|
[equation ellint12]
|
|
|
|
Requires that ['x > 0] and that ['y != 0].
|
|
Otherwise returns the result of __domain_error.
|
|
|
|
[optional_policy]
|
|
|
|
When ['y < 0] the function returns the
|
|
[@http://mathworld.wolfram.com/CauchyPrincipalValue.html Cauchy principal value]
|
|
using the relation:
|
|
|
|
[equation ellint18]
|
|
|
|
template <class T1, class T2, class T3>
|
|
``__sf_result`` ellint_rg(T1 x, T2 y, T3 z)
|
|
|
|
template <class T1, class T2, class T3, class ``__Policy``>
|
|
``__sf_result`` ellint_rg(T1 x, T2 y, T3 z, const ``__Policy``&)
|
|
|
|
Returns Carlson's elliptic integral R[sub G]:
|
|
|
|
[equation ellint27]
|
|
|
|
Requires that x and y are non-negative, otherwise returns the result of __domain_error.
|
|
|
|
[optional_policy]
|
|
|
|
[heading Testing]
|
|
|
|
There are two sets of tests.
|
|
|
|
Spot tests compare selected values with test data given in:
|
|
|
|
[:B. C. Carlson, ['[@http://arxiv.org/abs/math.CA/9409227
|
|
Numerical computation of real or complex elliptic integrals]]. Numerical Algorithms,
|
|
Volume 10, Number 1 / March, 1995, pp 13-26.]
|
|
|
|
Random test data generated using NTL::RR at 1000-bit precision and our
|
|
implementation checks for rounding-errors and/or regressions.
|
|
|
|
There are also sanity checks that use the inter-relations between the integrals
|
|
to verify their correctness: see the above Carlson paper for details.
|
|
|
|
[heading Accuracy]
|
|
|
|
These functions are computed using only basic arithmetic operations, so
|
|
there isn't much variation in accuracy over differing platforms.
|
|
Note that only results for the widest floating-point type on the
|
|
system are given as narrower types have __zero_error. All values
|
|
are relative errors in units of epsilon.
|
|
|
|
[table_ellint_rc]
|
|
|
|
[table_ellint_rd]
|
|
|
|
[table_ellint_rg]
|
|
|
|
[table_ellint_rf]
|
|
|
|
[table_ellint_rj]
|
|
|
|
|
|
[heading Implementation]
|
|
|
|
The key of Carlson's algorithm [[link ellint_ref_carlson79 Carlson79]] is the
|
|
duplication theorem:
|
|
|
|
[equation ellint13]
|
|
|
|
By applying it repeatedly, ['x], ['y], ['z] get
|
|
closer and closer. When they are nearly equal, the special case equation
|
|
|
|
[equation ellint16]
|
|
|
|
is used. More specifically, ['[R F]] is evaluated from a Taylor series
|
|
expansion to the fifth order. The calculations of the other three integrals
|
|
are analogous, except for R[sub C] which can be computed from elementary functions.
|
|
|
|
For ['p < 0] in ['R[sub J](x, y, z, p)] and ['y < 0] in ['R[sub C](x, y)],
|
|
the integrals are singular and their
|
|
[@http://mathworld.wolfram.com/CauchyPrincipalValue.html Cauchy principal values]
|
|
are returned via the relations:
|
|
|
|
[equation ellint17]
|
|
|
|
[equation ellint18]
|
|
|
|
[endsect]
|