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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206  // 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 slightly to fit into the // Boost.Math conceptual framework better. // Updated 2015 to use Carlson's latest methods. #ifndef BOOST_MATH_ELLINT_RD_HPP #define BOOST_MATH_ELLINT_RD_HPP #ifdef _MSC_VER #pragma once #endif #include #include #include #include #include // Carlson's elliptic integral of the second kind // R_D(x, y, z) = R_J(x, y, z, z) = 1.5 * \int_{0}^{\infty} [(t+x)(t+y)]^{-1/2} (t+z)^{-3/2} dt // Carlson, Numerische Mathematik, vol 33, 1 (1979) namespace boost { namespace math { namespace detail{ template T ellint_rd_imp(T x, T y, T z, const Policy& pol) { BOOST_MATH_STD_USING using std::swap; static const char* function = "boost::math::ellint_rd<%1%>(%1%,%1%,%1%)"; if(x < 0) { return policies::raise_domain_error(function, "Argument x must be >= 0, but got %1%", x, pol); } if(y < 0) { return policies::raise_domain_error(function, "Argument y must be >= 0, but got %1%", y, pol); } if(z <= 0) { return policies::raise_domain_error(function, "Argument z must be > 0, but got %1%", z, pol); } if(x + y == 0) { return policies::raise_domain_error(function, "At most one argument can be zero, but got, x + y = %1%", x + y, pol); } // // Special cases from http://dlmf.nist.gov/19.20#iv // using std::swap; if(x == z) swap(x, y); if(y == z) { if(x == y) { return 1 / (x * sqrt(x)); } else if(x == 0) { return 3 * constants::pi() / (4 * y * sqrt(y)); } else { if((std::min)(x, y) / (std::max)(x, y) > 1.3) return 3 * (ellint_rc_imp(x, y, pol) - sqrt(x) / y) / (2 * (y - x)); // Otherwise fall through to avoid cancellation in the above (RC(x,y) -> 1/x^0.5 as x -> y) } } if(x == y) { if((std::min)(x, z) / (std::max)(x, z) > 1.3) return 3 * (ellint_rc_imp(z, x, pol) - 1 / sqrt(z)) / (z - x); // Otherwise fall through to avoid cancellation in the above (RC(x,y) -> 1/x^0.5 as x -> y) } if(y == 0) swap(x, y); if(x == 0) { // // Special handling for common case, from // Numerical Computation of Real or Complex Elliptic Integrals, eq.47 // T xn = sqrt(y); T yn = sqrt(z); T x0 = xn; T y0 = yn; T sum = 0; T sum_pow = 0.25f; while(fabs(xn - yn) >= 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); // // This following calculation suffers from serious cancellation when y ~ z // unless we combine terms. We have: // // ( ((x0 + y0)/2)^2 - z ) / (z(y-z)) // // Substituting y = x0^2 and z = y0^2 and simplifying we get the following: // T pt = (x0 + 3 * y0) / (4 * z * (x0 + y0)); // // Since we've moved the demoninator from eq.47 inside the expression, we // need to also scale "sum" by the same value: // pt -= sum / (z * (y - z)); return pt * RF * 3; } T xn = x; T yn = y; T zn = z; T An = (x + y + 3 * z) / 5; T A0 = An; // This has an extra 1.2 fudge factor which is really only needed when x, y and z are close in magnitude: T Q = pow(tools::epsilon() / 4, -T(1) / 8) * (std::max)((std::max)(An - x, An - y), An - z) * 1.2f; BOOST_MATH_INSTRUMENT_VARIABLE(Q); T lambda, rx, ry, rz; unsigned k = 0; T fn = 1; T RD_sum = 0; for(; k < policies::get_max_series_iterations(); ++k) { rx = sqrt(xn); ry = sqrt(yn); rz = sqrt(zn); lambda = rx * ry + rx * rz + ry * rz; RD_sum += fn / (rz * (zn + lambda)); An = (An + lambda) / 4; xn = (xn + lambda) / 4; yn = (yn + lambda) / 4; zn = (zn + lambda) / 4; fn /= 4; Q /= 4; BOOST_MATH_INSTRUMENT_VARIABLE(k); BOOST_MATH_INSTRUMENT_VARIABLE(RD_sum); BOOST_MATH_INSTRUMENT_VARIABLE(Q); if(Q < An) break; } policies::check_series_iterations(function, k, pol); T X = fn * (A0 - x) / An; T Y = fn * (A0 - y) / An; T Z = -(X + Y) / 3; T E2 = X * Y - 6 * Z * Z; T E3 = (3 * X * Y - 8 * Z * Z) * Z; T E4 = 3 * (X * Y - Z * Z) * Z * Z; T E5 = X * Y * Z * Z * Z; T result = fn * pow(An, T(-3) / 2) * (1 - 3 * E2 / 14 + E3 / 6 + 9 * E2 * E2 / 88 - 3 * E4 / 22 - 9 * E2 * E3 / 52 + 3 * E5 / 26 - E2 * E2 * E2 / 16 + 3 * E3 * E3 / 40 + 3 * E2 * E4 / 20 + 45 * E2 * E2 * E3 / 272 - 9 * (E3 * E4 + E2 * E5) / 68); BOOST_MATH_INSTRUMENT_VARIABLE(result); result += 3 * RD_sum; return result; } } // namespace detail template inline typename tools::promote_args::type ellint_rd(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_rd_imp( static_cast(x), static_cast(y), static_cast(z), pol), "boost::math::ellint_rd<%1%>(%1%,%1%,%1%)"); } template inline typename tools::promote_args::type ellint_rd(T1 x, T2 y, T3 z) { return ellint_rd(x, y, z, policies::policy<>()); } }} // namespaces #endif // BOOST_MATH_ELLINT_RD_HPP