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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  // Copyright (c) 2006 Xiaogang Zhang // 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 correctly // handle the p < 0 case. // #ifndef BOOST_MATH_ELLINT_RJ_HPP #define BOOST_MATH_ELLINT_RJ_HPP #ifdef _MSC_VER #pragma once #endif #include #include #include #include #include // Carlson's elliptic integral of the third kind // R_J(x, y, z, p) = 1.5 * \int_{0}^{\infty} (t+p)^{-1} [(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_rj_imp(T x, T y, T z, T p, const Policy& pol) { T value, u, lambda, alpha, beta, sigma, factor, tolerance; T X, Y, Z, P, EA, EB, EC, E2, E3, S1, S2, S3; unsigned long k; BOOST_MATH_STD_USING using namespace boost::math::tools; static const char* function = "boost::math::ellint_rj<%1%>(%1%,%1%,%1%)"; if (x < 0) { return policies::raise_domain_error(function, "Argument x must be non-negative, but got x = %1%", x, pol); } if(y < 0) { return policies::raise_domain_error(function, "Argument y must be non-negative, but got y = %1%", y, pol); } if(z < 0) { return policies::raise_domain_error(function, "Argument z must be non-negative, but got z = %1%", z, pol); } if(p == 0) { return policies::raise_domain_error(function, "Argument p must not be zero, but got p = %1%", p, pol); } if (x + y == 0 || y + z == 0 || z + x == 0) { return policies::raise_domain_error(function, "At most one argument can be zero, " "only possible result is %1%.", std::numeric_limits::quiet_NaN(), pol); } // error scales as the 6th power of tolerance tolerance = pow(T(1) * tools::epsilon() / 3, T(1) / 6); // for p < 0, the integral is singular, return Cauchy principal value if (p < 0) { // // We must ensure that (z - y) * (y - x) is positive. // Since the integral is symmetrical in x, y and z // we can just permute the values: // if(x > y) std::swap(x, y); if(y > z) std::swap(y, z); if(x > y) std::swap(x, y); T q = -p; T pmy = (z - y) * (y - x) / (y + q); // p - y BOOST_ASSERT(pmy >= 0); p = pmy + y; value = boost::math::ellint_rj(x, y, z, p, pol); value *= pmy; value -= 3 * boost::math::ellint_rf(x, y, z, pol); value += 3 * sqrt((x * y * z) / (x * z + p * q)) * boost::math::ellint_rc(x * z + p * q, p * q, pol); value /= (y + q); return value; } // duplication sigma = 0; factor = 1; k = 1; do { u = (x + y + z + p + p) / 5; X = (u - x) / u; Y = (u - y) / u; Z = (u - z) / u; P = (u - p) / u; if ((tools::max)(abs(X), abs(Y), abs(Z), abs(P)) < tolerance) break; T sx = sqrt(x); T sy = sqrt(y); T sz = sqrt(z); lambda = sy * (sx + sz) + sz * sx; alpha = p * (sx + sy + sz) + sx * sy * sz; alpha *= alpha; beta = p * (p + lambda) * (p + lambda); sigma += factor * boost::math::ellint_rc(alpha, beta, pol); factor /= 4; x = (x + lambda) / 4; y = (y + lambda) / 4; z = (z + lambda) / 4; p = (p + lambda) / 4; ++k; } while(k < policies::get_max_series_iterations()); // Check to see if we gave up too soon: policies::check_series_iterations(function, k, pol); // Taylor series expansion to the 5th order EA = X * Y + Y * Z + Z * X; EB = X * Y * Z; EC = P * P; E2 = EA - 3 * EC; E3 = EB + 2 * P * (EA - EC); S1 = 1 + E2 * (E2 * T(9) / 88 - E3 * T(9) / 52 - T(3) / 14); S2 = EB * (T(1) / 6 + P * (T(-6) / 22 + P * T(3) / 26)); S3 = P * ((EA - EC) / 3 - P * EA * T(3) / 22); value = 3 * sigma + factor * (S1 + S2 + S3) / (u * sqrt(u)); return value; } } // namespace detail template inline typename tools::promote_args::type ellint_rj(T1 x, T2 y, T3 z, T4 p, 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_rj_imp( static_cast(x), static_cast(y), static_cast(z), static_cast(p), pol), "boost::math::ellint_rj<%1%>(%1%,%1%,%1%,%1%)"); } template inline typename tools::promote_args::type ellint_rj(T1 x, T2 y, T3 z, T4 p) { return ellint_rj(x, y, z, p, policies::policy<>()); } }} // namespaces #endif // BOOST_MATH_ELLINT_RJ_HPP