// Copyright (c) 2006 Xiaogang Zhang // Copyright (c) 2006 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) // // 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 ensure // that the code continues to work no matter how many digits // type T has. #ifndef BOOST_MATH_ELLINT_D_HPP #define BOOST_MATH_ELLINT_D_HPP #ifdef _MSC_VER #pragma once #endif #include #include #include #include #include #include #include #include // Elliptic integrals (complete and incomplete) of the second kind // Carlson, Numerische Mathematik, vol 33, 1 (1979) namespace boost { namespace math { template typename tools::promote_args::type ellint_d(T1 k, T2 phi, const Policy& pol); namespace detail{ template T ellint_d_imp(T k, const Policy& pol); // Elliptic integral (Legendre form) of the second kind template T ellint_d_imp(T phi, T k, const Policy& pol) { BOOST_MATH_STD_USING using namespace boost::math::tools; using namespace boost::math::constants; bool invert = false; if(phi < 0) { phi = fabs(phi); invert = true; } T result; if(phi >= tools::max_value()) { // Need to handle infinity as a special case: result = policies::raise_overflow_error("boost::math::ellint_d<%1%>(%1%,%1%)", 0, pol); } else if(phi > 1 / tools::epsilon()) { // Phi is so large that phi%pi is necessarily zero (or garbage), // just return the second part of the duplication formula: result = 2 * phi * ellint_d_imp(k, pol) / constants::pi(); } else { // Carlson's algorithm works only for |phi| <= pi/2, // use the integrand's periodicity to normalize phi // T rphi = boost::math::tools::fmod_workaround(phi, T(constants::half_pi())); T m = boost::math::round((phi - rphi) / constants::half_pi()); int s = 1; if(boost::math::tools::fmod_workaround(m, T(2)) > 0.5) { m += 1; s = -1; rphi = constants::half_pi() - rphi; } T sinp = sin(rphi); T cosp = cos(rphi); T c = 1 / (sinp * sinp); T cm1 = cosp * cosp / (sinp * sinp); // c - 1 T k2 = k * k; if(k2 > 1) { return policies::raise_domain_error("boost::math::ellint_d<%1%>(%1%, %1%)", "The parameter k is out of range, got k = %1%", k, pol); } else if(rphi == 0) { result = 0; } else { // http://dlmf.nist.gov/19.25#E10 result = s * ellint_rd_imp(cm1, T(c - k2), c, pol) / 3; } if(m != 0) result += m * ellint_d_imp(k, pol); } return invert ? T(-result) : result; } // Complete elliptic integral (Legendre form) of the second kind template T ellint_d_imp(T k, const Policy& pol) { BOOST_MATH_STD_USING using namespace boost::math::tools; if (abs(k) >= 1) { return policies::raise_domain_error("boost::math::ellint_d<%1%>(%1%)", "Got k = %1%, function requires |k| <= 1", k, pol); } if(fabs(k) <= tools::root_epsilon()) return constants::pi() / 4; T x = 0; T t = k * k; T y = 1 - t; T z = 1; T value = ellint_rd_imp(x, y, z, pol) / 3; return value; } template inline typename tools::promote_args::type ellint_d(T k, const Policy& pol, const mpl::true_&) { typedef typename tools::promote_args::type result_type; typedef typename policies::evaluation::type value_type; return policies::checked_narrowing_cast(detail::ellint_d_imp(static_cast(k), pol), "boost::math::ellint_d<%1%>(%1%)"); } // Elliptic integral (Legendre form) of the second kind template inline typename tools::promote_args::type ellint_d(T1 k, T2 phi, const mpl::false_&) { return boost::math::ellint_d(k, phi, policies::policy<>()); } } // detail // Complete elliptic integral (Legendre form) of the second kind template inline typename tools::promote_args::type ellint_d(T k) { return ellint_d(k, policies::policy<>()); } // Elliptic integral (Legendre form) of the second kind template inline typename tools::promote_args::type ellint_d(T1 k, T2 phi) { typedef typename policies::is_policy::type tag_type; return detail::ellint_d(k, phi, tag_type()); } template inline typename tools::promote_args::type ellint_d(T1 k, T2 phi, 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_d_imp(static_cast(phi), static_cast(k), pol), "boost::math::ellint_2<%1%>(%1%,%1%)"); } }} // namespaces #endif // BOOST_MATH_ELLINT_D_HPP