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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  // 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 handle // types longer than 80-bit reals. // #ifndef BOOST_MATH_ELLINT_RF_HPP #define BOOST_MATH_ELLINT_RF_HPP #ifdef _MSC_VER #pragma once #endif #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) { T value, X, Y, Z, E2, E3, u, lambda, tolerance; unsigned long k; BOOST_MATH_STD_USING using namespace boost::math::tools; 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); } // Carlson scales error as the 6th power of tolerance, // but this seems not to work for types larger than // 80-bit reals, this heuristic seems to work OK: if(policies::digits() > 64) { tolerance = pow(tools::epsilon(), T(1)/4.25f); BOOST_MATH_INSTRUMENT_VARIABLE(tolerance); } else { tolerance = pow(4*tools::epsilon(), T(1)/6); BOOST_MATH_INSTRUMENT_VARIABLE(tolerance); } // duplication k = 1; do { u = (x + y + z) / 3; X = (u - x) / u; Y = (u - y) / u; Z = (u - z) / u; // Termination condition: 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; 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); BOOST_MATH_INSTRUMENT_VARIABLE(k); // Taylor series expansion to the 5th order E2 = X * Y - Z * Z; E3 = X * Y * Z; value = (1 + E2*(E2/24 - E3*T(3)/44 - T(0.1)) + E3/14) / sqrt(u); BOOST_MATH_INSTRUMENT_VARIABLE(value); return value; } } // 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