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// Copyright (c) 2015 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) // #ifndef BOOST_MATH_ELLINT_RG_HPP #define BOOST_MATH_ELLINT_RG_HPP #ifdef _MSC_VER #pragma once #endif #include #include #include #include #include #include #include namespace boost { namespace math { namespace detail{ template BOOST_MATH_GPU_ENABLED T ellint_rg_imp(T x, T y, T z, const Policy& pol) { BOOST_MATH_STD_USING constexpr auto function = "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)"; if(x < 0 || y < 0 || z < 0) { return policies::raise_domain_error(function, "domain error, all arguments must be non-negative, only sensible result is %1%.", boost::math::numeric_limits::quiet_NaN(), pol); } // // Function is symmetric in x, y and z, but we require // (x - z)(y - z) >= 0 to avoid cancellation error in the result // which implies (for example) x >= z >= y // if(x < y) BOOST_MATH_GPU_SAFE_SWAP(x, y); if(x < z) BOOST_MATH_GPU_SAFE_SWAP(x, z); if(y > z) BOOST_MATH_GPU_SAFE_SWAP(y, z); BOOST_MATH_ASSERT(x >= z); BOOST_MATH_ASSERT(z >= y); // // Special cases from http://dlmf.nist.gov/19.20#ii // if(x == z) { if(y == z) { // x = y = z // This also works for x = y = z = 0 presumably. return sqrt(x); } else if(y == 0) { // x = y, z = 0 return constants::pi() * sqrt(x) / 4; } else { // x = z, y != 0 BOOST_MATH_GPU_SAFE_SWAP(x, y); return (x == 0) ? T(sqrt(z) / 2) : T((z * ellint_rc_imp(x, z, pol) + sqrt(x)) / 2); } } else if(y == z) { BOOST_MATH_ASSERT(x > 0); // Ordering of x,y,z above takes care of x == 0 case. return (y == 0) ? T(sqrt(x) / 2) : T((y * ellint_rc_imp(x, y, pol) + sqrt(x)) / 2); } else if(y == 0) { BOOST_MATH_GPU_SAFE_SWAP(y, z); // // Special handling for common case, from // Numerical Computation of Real or Complex Elliptic Integrals, eq.46 // T xn = sqrt(x); T yn = sqrt(y); T x0 = xn; T y0 = yn; T sum = 0; T sum_pow = 0.25f; while(fabs(xn - yn) >= T(2.7) * tools::root_epsilon() * fabs(xn)) { T t = sqrt(xn * yn); xn = (xn + yn) / 2; yn = t; sum_pow *= 2; sum += sum_pow * boost::math::pow<2>(xn - yn); } T RF = constants::pi() / (xn + yn); return ((boost::math::pow<2>((x0 + y0) / 2) - sum) * RF) / 2; } return (z * ellint_rf_imp(x, y, z, pol) - (x - z) * (y - z) * ellint_rd_imp(x, y, z, pol) / 3 + sqrt(x * y / z)) / 2; } } // namespace detail template BOOST_MATH_GPU_ENABLED inline typename tools::promote_args::type ellint_rg(T1 x, T2 y, T3 z, const Policy& pol) { typedef typename tools::promote_args::type result_type; typedef typename policies::evaluation::type value_type; return policies::checked_narrowing_cast( detail::ellint_rg_imp( static_cast(x), static_cast(y), static_cast(z), pol), "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)"); } template BOOST_MATH_GPU_ENABLED inline typename tools::promote_args::type ellint_rg(T1 x, T2 y, T3 z) { return ellint_rg(x, y, z, policies::policy<>()); } }} // namespaces #endif // BOOST_MATH_ELLINT_RG_HPP