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