| // 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 to fit into the |
| // Boost.Math conceptual framework better, and to handle |
| // types longer than 80-bit reals. |
| // |
| #ifndef BOOST_MATH_ELLINT_RF_HPP |
| #define BOOST_MATH_ELLINT_RF_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 first kind |
| // R_F(x, y, z) = 0.5 * \int_{0}^{\infty} [(t+x)(t+y)(t+z)]^{-1/2} dt |
| // Carlson, Numerische Mathematik, vol 33, 1 (1979) |
| |
| namespace boost { namespace math { namespace detail{ |
| |
| template <typename T, typename Policy> |
| T ellint_rf_imp(T x, T y, T z, const Policy& pol) |
| { |
| T value, X, Y, Z, E2, E3, u, lambda, tolerance; |
| unsigned long k; |
| |
| BOOST_MATH_STD_USING |
| using namespace boost::math::tools; |
| |
| static const char* function = "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)"; |
| |
| if (x < 0 || y < 0 || z < 0) |
| { |
| return policies::raise_domain_error<T>(function, |
| "domain error, all arguments must be non-negative, " |
| "only sensible result is %1%.", |
| std::numeric_limits<T>::quiet_NaN(), pol); |
| } |
| if (x + y == 0 || y + z == 0 || z + x == 0) |
| { |
| return policies::raise_domain_error<T>(function, |
| "domain error, at most one argument can be zero, " |
| "only sensible result is %1%.", |
| std::numeric_limits<T>::quiet_NaN(), pol); |
| } |
| |
| // Carlson scales error as the 6th power of tolerance, |
| // but this seems not to work for types larger than |
| // 80-bit reals, this heuristic seems to work OK: |
| if(policies::digits<T, Policy>() > 64) |
| { |
| tolerance = pow(tools::epsilon<T>(), T(1)/4.25f); |
| BOOST_MATH_INSTRUMENT_VARIABLE(tolerance); |
| } |
| else |
| { |
| tolerance = pow(4*tools::epsilon<T>(), T(1)/6); |
| BOOST_MATH_INSTRUMENT_VARIABLE(tolerance); |
| } |
| |
| // duplication |
| k = 1; |
| do |
| { |
| u = (x + y + z) / 3; |
| X = (u - x) / u; |
| Y = (u - y) / u; |
| Z = (u - z) / u; |
| |
| // Termination condition: |
| 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; |
| 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); |
| BOOST_MATH_INSTRUMENT_VARIABLE(k); |
| |
| // Taylor series expansion to the 5th order |
| E2 = X * Y - Z * Z; |
| E3 = X * Y * Z; |
| value = (1 + E2*(E2/24 - E3*T(3)/44 - T(0.1)) + E3/14) / sqrt(u); |
| BOOST_MATH_INSTRUMENT_VARIABLE(value); |
| |
| return value; |
| } |
| |
| } // namespace detail |
| |
| template <class T1, class T2, class T3, class Policy> |
| inline typename tools::promote_args<T1, T2, T3>::type |
| ellint_rf(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_rf_imp( |
| static_cast<value_type>(x), |
| static_cast<value_type>(y), |
| static_cast<value_type>(z), pol), "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)"); |
| } |
| |
| template <class T1, class T2, class T3> |
| inline typename tools::promote_args<T1, T2, T3>::type |
| ellint_rf(T1 x, T2 y, T3 z) |
| { |
| return ellint_rf(x, y, z, policies::policy<>()); |
| } |
| |
| }} // namespaces |
| |
| #endif // BOOST_MATH_ELLINT_RF_HPP |
| |