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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174  // Copyright (c) 2006 Xiaogang Zhang, 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) // // 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. // Updated 2015 to use Carlson's latest methods. // #ifndef BOOST_MATH_ELLINT_RF_HPP #define BOOST_MATH_ELLINT_RF_HPP #ifdef _MSC_VER #pragma once #endif #include #include #include #include #include // 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 T ellint_rf_imp(T x, T y, T z, const Policy& pol) { BOOST_MATH_STD_USING using namespace boost::math; using std::swap; static const char* 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%.", std::numeric_limits::quiet_NaN(), pol); } if(x + y == 0 || y + z == 0 || z + x == 0) { return policies::raise_domain_error(function, "domain error, at most one argument can be zero, " "only sensible result is %1%.", std::numeric_limits::quiet_NaN(), pol); } // // Special cases from http://dlmf.nist.gov/19.20#i // if(x == y) { if(x == z) { // x, y, z equal: return 1 / sqrt(x); } else { // 2 equal, x and y: if(z == 0) return constants::pi() / (2 * sqrt(x)); else return ellint_rc_imp(z, x, pol); } } if(x == z) { if(y == 0) return constants::pi() / (2 * sqrt(x)); else return ellint_rc_imp(y, x, pol); } if(y == z) { if(x == 0) return constants::pi() / (2 * sqrt(y)); else return ellint_rc_imp(x, y, pol); } if(x == 0) swap(x, z); else if(y == 0) swap(y, z); if(z == 0) { // // Special case for one value zero: // T xn = sqrt(x); T yn = sqrt(y); while(fabs(xn - yn) >= 2.7 * tools::root_epsilon() * fabs(xn)) { T t = sqrt(xn * yn); xn = (xn + yn) / 2; yn = t; } return constants::pi() / (xn + yn); } T xn = x; T yn = y; T zn = z; T An = (x + y + z) / 3; T A0 = An; T Q = pow(3 * boost::math::tools::epsilon(), T(-1) / 8) * (std::max)((std::max)(fabs(An - xn), fabs(An - yn)), fabs(An - zn)); T fn = 1; // duplication unsigned k = 1; for(; k < boost::math::policies::get_max_series_iterations(); ++k) { T root_x = sqrt(xn); T root_y = sqrt(yn); T root_z = sqrt(zn); T lambda = root_x * root_y + root_x * root_z + root_y * root_z; An = (An + lambda) / 4; xn = (xn + lambda) / 4; yn = (yn + lambda) / 4; zn = (zn + lambda) / 4; Q /= 4; fn *= 4; if(Q < fabs(An)) break; } // Check to see if we gave up too soon: policies::check_series_iterations(function, k, pol); BOOST_MATH_INSTRUMENT_VARIABLE(k); T X = (A0 - x) / (An * fn); T Y = (A0 - y) / (An * fn); T Z = -X - Y; // Taylor series expansion to the 7th order T E2 = X * Y - Z * Z; T E3 = X * Y * Z; return (1 + E3 * (T(1) / 14 + 3 * E3 / 104) + E2 * (T(-1) / 10 + E2 / 24 - (3 * E3) / 44 - 5 * E2 * E2 / 208 + E2 * E3 / 16)) / sqrt(An); } } // namespace detail template inline typename tools::promote_args::type ellint_rf(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_rf_imp( static_cast(x), static_cast(y), static_cast(z), pol), "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)"); } template inline typename tools::promote_args::type ellint_rf(T1 x, T2 y, T3 z) { return ellint_rf(x, y, z, policies::policy<>()); } }} // namespaces #endif // BOOST_MATH_ELLINT_RF_HPP