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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  // 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 slightly to fit into the // Boost.Math conceptual framework better. #ifndef BOOST_MATH_ELLINT_RD_HPP #define BOOST_MATH_ELLINT_RD_HPP #ifdef _MSC_VER #pragma once #endif #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) { T value, u, lambda, sigma, factor, tolerance; T X, Y, Z, EA, EB, EC, ED, EE, S1, S2; unsigned long k; BOOST_MATH_STD_USING using namespace boost::math::tools; 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); } // error scales as the 6th power of tolerance tolerance = pow(tools::epsilon() / 3, T(1)/6); // duplication sigma = 0; factor = 1; k = 1; do { u = (x + y + z + z + z) / 5; X = (u - x) / u; Y = (u - y) / u; Z = (u - z) / u; if ((tools::max)(abs(X), abs(Y), abs(Z)) < tolerance) break; T sx = sqrt(x); T sy = sqrt(y); T sz = sqrt(z); lambda = sy * (sx + sz) + sz * sx; //sqrt(x * y) + sqrt(y * z) + sqrt(z * x); sigma += factor / (sz * (z + lambda)); factor /= 4; x = (x + lambda) / 4; y = (y + lambda) / 4; z = (z + 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; EB = Z * Z; EC = EA - EB; ED = EA - 6 * EB; EE = ED + EC + EC; S1 = ED * (ED * T(9) / 88 - Z * EE * T(9) / 52 - T(3) / 14); S2 = Z * (EE / 6 + Z * (-EC * T(9) / 22 + Z * EA * T(3) / 26)); value = 3 * sigma + factor * (1 + S1 + S2) / (u * sqrt(u)); return value; } } // 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