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 ```1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 ``` ``````// 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_2_HPP #define BOOST_MATH_ELLINT_2_HPP #ifdef _MSC_VER #pragma once #endif #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_2(T1 k, T2 phi, const Policy& pol); namespace detail{ template T ellint_e_imp(T k, const Policy& pol); // Elliptic integral (Legendre form) of the second kind template T ellint_e_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_e<%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_e_imp(k, pol) / constants::pi(); } else { // Carlson's algorithm works only for |phi| <= pi/2, // use the integrand's periodicity to normalize phi // // Xiaogang's original code used a cast to long long here // but that fails if T has more digits than a long long, // so rewritten to use fmod instead: // 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 x = cosp * cosp; T t = k * k * sinp * sinp; T y = 1 - t; T z = 1; result = s * sinp * (ellint_rf_imp(x, y, z, pol) - t * ellint_rd_imp(x, y, z, pol) / 3); if(m != 0) result += m * ellint_e_imp(k, pol); } return invert ? T(-result) : result; } // Complete elliptic integral (Legendre form) of the second kind template T ellint_e_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_e<%1%>(%1%)", "Got k = %1%, function requires |k| <= 1", k, pol); } if (abs(k) == 1) { return static_cast(1); } T x = 0; T t = k * k; T y = 1 - t; T z = 1; T value = ellint_rf_imp(x, y, z, pol) - t * ellint_rd_imp(x, y, z, pol) / 3; return value; } template inline typename tools::promote_args::type ellint_2(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_e_imp(static_cast(k), pol), "boost::math::ellint_2<%1%>(%1%)"); } // Elliptic integral (Legendre form) of the second kind template inline typename tools::promote_args::type ellint_2(T1 k, T2 phi, const mpl::false_&) { return boost::math::ellint_2(k, phi, policies::policy<>()); } } // detail // Complete elliptic integral (Legendre form) of the second kind template inline typename tools::promote_args::type ellint_2(T k) { return ellint_2(k, policies::policy<>()); } // Elliptic integral (Legendre form) of the second kind template inline typename tools::promote_args::type ellint_2(T1 k, T2 phi) { typedef typename policies::is_policy::type tag_type; return detail::ellint_2(k, phi, tag_type()); } template inline typename tools::promote_args::type ellint_2(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_e_imp(static_cast(phi), static_cast(k), pol), "boost::math::ellint_2<%1%>(%1%,%1%)"); } }} // namespaces #endif // BOOST_MATH_ELLINT_2_HPP ``````