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247 lines
6.6 KiB
Plaintext
247 lines
6.6 KiB
Plaintext
[/
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Copyright (c) 2006 Xiaogang Zhang
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Copyright (c) 2006 John Maddock
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Use, modification and distribution are subject to the
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Boost Software License, Version 1.0. (See accompanying file
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LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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]
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[section:ellint_carlson Elliptic Integrals - Carlson Form]
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[heading Synopsis]
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``
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#include <boost/math/special_functions/ellint_rf.hpp>
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``
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namespace boost { namespace math {
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template <class T1, class T2, class T3>
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``__sf_result`` ellint_rf(T1 x, T2 y, T3 z)
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template <class T1, class T2, class T3, class ``__Policy``>
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``__sf_result`` ellint_rf(T1 x, T2 y, T3 z, const ``__Policy``&)
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}} // namespaces
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``
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#include <boost/math/special_functions/ellint_rd.hpp>
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``
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namespace boost { namespace math {
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template <class T1, class T2, class T3>
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``__sf_result`` ellint_rd(T1 x, T2 y, T3 z)
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template <class T1, class T2, class T3, class ``__Policy``>
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``__sf_result`` ellint_rd(T1 x, T2 y, T3 z, const ``__Policy``&)
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}} // namespaces
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``
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#include <boost/math/special_functions/ellint_rj.hpp>
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``
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namespace boost { namespace math {
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template <class T1, class T2, class T3, class T4>
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``__sf_result`` ellint_rj(T1 x, T2 y, T3 z, T4 p)
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template <class T1, class T2, class T3, class T4, class ``__Policy``>
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``__sf_result`` ellint_rj(T1 x, T2 y, T3 z, T4 p, const ``__Policy``&)
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}} // namespaces
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``
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#include <boost/math/special_functions/ellint_rc.hpp>
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``
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namespace boost { namespace math {
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template <class T1, class T2>
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``__sf_result`` ellint_rc(T1 x, T2 y)
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` ellint_rc(T1 x, T2 y, const ``__Policy``&)
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}} // namespaces
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``
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#include <boost/math/special_functions/ellint_rg.hpp>
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``
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namespace boost { namespace math {
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template <class T1, class T2, class T3>
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``__sf_result`` ellint_rg(T1 x, T2 y, T3 z)
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template <class T1, class T2, class T3, class ``__Policy``>
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``__sf_result`` ellint_rg(T1 x, T2 y, T3 z, const ``__Policy``&)
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}} // namespaces
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[heading Description]
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These functions return Carlson's symmetrical elliptic integrals, the functions
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have complicated behavior over all their possible domains, but the following
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graph gives an idea of their behavior:
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[graph ellint_carlson]
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The return type of these functions is computed using the __arg_promotion_rules
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when the arguments are of different types: otherwise the return is the same type
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as the arguments.
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template <class T1, class T2, class T3>
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``__sf_result`` ellint_rf(T1 x, T2 y, T3 z)
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template <class T1, class T2, class T3, class ``__Policy``>
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``__sf_result`` ellint_rf(T1 x, T2 y, T3 z, const ``__Policy``&)
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Returns Carlson's Elliptic Integral R[sub F]:
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[equation ellint9]
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Requires that all of the arguments are non-negative, and at most
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one may be zero. Otherwise returns the result of __domain_error.
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[optional_policy]
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template <class T1, class T2, class T3>
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``__sf_result`` ellint_rd(T1 x, T2 y, T3 z)
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template <class T1, class T2, class T3, class ``__Policy``>
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``__sf_result`` ellint_rd(T1 x, T2 y, T3 z, const ``__Policy``&)
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Returns Carlson's elliptic integral R[sub D]:
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[equation ellint10]
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Requires that x and y are non-negative, with at most one of them
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zero, and that z >= 0. Otherwise returns the result of __domain_error.
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[optional_policy]
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template <class T1, class T2, class T3, class T4>
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``__sf_result`` ellint_rj(T1 x, T2 y, T3 z, T4 p)
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template <class T1, class T2, class T3, class T4, class ``__Policy``>
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``__sf_result`` ellint_rj(T1 x, T2 y, T3 z, T4 p, const ``__Policy``&)
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Returns Carlson's elliptic integral R[sub J]:
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[equation ellint11]
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Requires that x, y and z are non-negative, with at most one of them
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zero, and that ['p != 0]. Otherwise returns the result of __domain_error.
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[optional_policy]
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When ['p < 0] the function returns the
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[@http://en.wikipedia.org/wiki/Cauchy_principal_value Cauchy principal value]
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using the relation:
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[equation ellint17]
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template <class T1, class T2>
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``__sf_result`` ellint_rc(T1 x, T2 y)
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` ellint_rc(T1 x, T2 y, const ``__Policy``&)
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Returns Carlson's elliptic integral R[sub C]:
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[equation ellint12]
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Requires that ['x > 0] and that ['y != 0].
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Otherwise returns the result of __domain_error.
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[optional_policy]
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When ['y < 0] the function returns the
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[@http://mathworld.wolfram.com/CauchyPrincipalValue.html Cauchy principal value]
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using the relation:
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[equation ellint18]
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template <class T1, class T2, class T3>
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``__sf_result`` ellint_rg(T1 x, T2 y, T3 z)
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template <class T1, class T2, class T3, class ``__Policy``>
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``__sf_result`` ellint_rg(T1 x, T2 y, T3 z, const ``__Policy``&)
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Returns Carlson's elliptic integral R[sub G]:
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[equation ellint27]
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Requires that x and y are non-negative, otherwise returns the result of __domain_error.
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[optional_policy]
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[heading Testing]
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There are two sets of tests.
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Spot tests compare selected values with test data given in:
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[:B. C. Carlson, ['[@http://arxiv.org/abs/math.CA/9409227
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Numerical computation of real or complex elliptic integrals]]. Numerical Algorithms,
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Volume 10, Number 1 / March, 1995, pp 13-26.]
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Random test data generated using NTL::RR at 1000-bit precision and our
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implementation checks for rounding-errors and/or regressions.
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There are also sanity checks that use the inter-relations between the integrals
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to verify their correctness: see the above Carlson paper for details.
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[heading Accuracy]
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These functions are computed using only basic arithmetic operations, so
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there isn't much variation in accuracy over differing platforms.
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Note that only results for the widest floating-point type on the
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system are given as narrower types have __zero_error. All values
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are relative errors in units of epsilon.
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[table_ellint_rc]
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[table_ellint_rd]
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[table_ellint_rg]
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[table_ellint_rf]
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[table_ellint_rj]
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[heading Implementation]
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The key of Carlson's algorithm [[link ellint_ref_carlson79 Carlson79]] is the
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duplication theorem:
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[equation ellint13]
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By applying it repeatedly, ['x], ['y], ['z] get
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closer and closer. When they are nearly equal, the special case equation
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[equation ellint16]
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is used. More specifically, ['[R F]] is evaluated from a Taylor series
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expansion to the fifth order. The calculations of the other three integrals
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are analogous, except for R[sub C] which can be computed from elementary functions.
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For ['p < 0] in ['R[sub J](x, y, z, p)] and ['y < 0] in ['R[sub C](x, y)],
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the integrals are singular and their
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[@http://mathworld.wolfram.com/CauchyPrincipalValue.html Cauchy principal values]
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are returned via the relations:
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[equation ellint17]
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[equation ellint18]
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[endsect]
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