131 lines
3.8 KiB
C++
131 lines
3.8 KiB
C++
// Copyright (c) 2006 Xiaogang Zhang
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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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// History:
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// XZ wrote the original of this file as part of the Google
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// Summer of Code 2006. JM modified it slightly to fit into the
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// Boost.Math conceptual framework better.
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#ifndef BOOST_MATH_ELLINT_RD_HPP
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#define BOOST_MATH_ELLINT_RD_HPP
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#ifdef _MSC_VER
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#pragma once
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#endif
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#include <boost/math/special_functions/math_fwd.hpp>
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#include <boost/math/tools/config.hpp>
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#include <boost/math/policies/error_handling.hpp>
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// Carlson's elliptic integral of the second kind
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// R_D(x, y, z) = R_J(x, y, z, z) = 1.5 * \int_{0}^{\infty} [(t+x)(t+y)]^{-1/2} (t+z)^{-3/2} dt
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// Carlson, Numerische Mathematik, vol 33, 1 (1979)
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namespace boost { namespace math { namespace detail{
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template <typename T, typename Policy>
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T ellint_rd_imp(T x, T y, T z, const Policy& pol)
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{
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T value, u, lambda, sigma, factor, tolerance;
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T X, Y, Z, EA, EB, EC, ED, EE, S1, S2;
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unsigned long k;
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BOOST_MATH_STD_USING
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using namespace boost::math::tools;
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static const char* function = "boost::math::ellint_rd<%1%>(%1%,%1%,%1%)";
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if (x < 0)
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{
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return policies::raise_domain_error<T>(function,
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"Argument x must be >= 0, but got %1%", x, pol);
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}
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if (y < 0)
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{
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return policies::raise_domain_error<T>(function,
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"Argument y must be >= 0, but got %1%", y, pol);
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}
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if (z <= 0)
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{
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return policies::raise_domain_error<T>(function,
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"Argument z must be > 0, but got %1%", z, pol);
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}
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if (x + y == 0)
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{
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return policies::raise_domain_error<T>(function,
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"At most one argument can be zero, but got, x + y = %1%", x+y, pol);
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}
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// error scales as the 6th power of tolerance
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tolerance = pow(tools::epsilon<T>() / 3, T(1)/6);
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// duplication
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sigma = 0;
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factor = 1;
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k = 1;
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do
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{
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u = (x + y + z + z + z) / 5;
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X = (u - x) / u;
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Y = (u - y) / u;
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Z = (u - z) / u;
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if ((tools::max)(abs(X), abs(Y), abs(Z)) < tolerance)
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break;
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T sx = sqrt(x);
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T sy = sqrt(y);
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T sz = sqrt(z);
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lambda = sy * (sx + sz) + sz * sx; //sqrt(x * y) + sqrt(y * z) + sqrt(z * x);
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sigma += factor / (sz * (z + lambda));
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factor /= 4;
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x = (x + lambda) / 4;
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y = (y + lambda) / 4;
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z = (z + lambda) / 4;
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++k;
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}
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while(k < policies::get_max_series_iterations<Policy>());
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// Check to see if we gave up too soon:
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policies::check_series_iterations<T>(function, k, pol);
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// Taylor series expansion to the 5th order
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EA = X * Y;
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EB = Z * Z;
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EC = EA - EB;
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ED = EA - 6 * EB;
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EE = ED + EC + EC;
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S1 = ED * (ED * T(9) / 88 - Z * EE * T(9) / 52 - T(3) / 14);
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S2 = Z * (EE / 6 + Z * (-EC * T(9) / 22 + Z * EA * T(3) / 26));
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value = 3 * sigma + factor * (1 + S1 + S2) / (u * sqrt(u));
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return value;
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}
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} // namespace detail
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template <class T1, class T2, class T3, class Policy>
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inline typename tools::promote_args<T1, T2, T3>::type
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ellint_rd(T1 x, T2 y, T3 z, const Policy& pol)
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{
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typedef typename tools::promote_args<T1, T2, T3>::type result_type;
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typedef typename policies::evaluation<result_type, Policy>::type value_type;
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return policies::checked_narrowing_cast<result_type, Policy>(
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detail::ellint_rd_imp(
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static_cast<value_type>(x),
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static_cast<value_type>(y),
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static_cast<value_type>(z), pol), "boost::math::ellint_rd<%1%>(%1%,%1%,%1%)");
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}
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template <class T1, class T2, class T3>
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inline typename tools::promote_args<T1, T2, T3>::type
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ellint_rd(T1 x, T2 y, T3 z)
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{
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return ellint_rd(x, y, z, policies::policy<>());
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}
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}} // namespaces
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#endif // BOOST_MATH_ELLINT_RD_HPP
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