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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  // 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 y < 0 case. // #ifndef BOOST_MATH_ELLINT_RC_HPP #define BOOST_MATH_ELLINT_RC_HPP #ifdef _MSC_VER #pragma once #endif #include #include #include // Carlson's degenerate elliptic integral // R_C(x, y) = R_F(x, y, y) = 0.5 * \int_{0}^{\infty} (t+x)^{-1/2} (t+y)^{-1} dt // Carlson, Numerische Mathematik, vol 33, 1 (1979) namespace boost { namespace math { namespace detail{ template T ellint_rc_imp(T x, T y, const Policy& pol) { T value, S, u, lambda, tolerance, prefix; unsigned long k; BOOST_MATH_STD_USING using namespace boost::math::tools; static const char* function = "boost::math::ellint_rc<%1%>(%1%,%1%)"; if(x < 0) { return policies::raise_domain_error(function, "Argument x must be non-negative but got %1%", x, pol); } if(y == 0) { return policies::raise_domain_error(function, "Argument y must not be zero but got %1%", y, pol); } // error scales as the 6th power of tolerance tolerance = pow(4 * tools::epsilon(), T(1) / 6); // for y < 0, the integral is singular, return Cauchy principal value if (y < 0) { prefix = sqrt(x / (x - y)); x = x - y; y = -y; } else prefix = 1; // duplication: k = 1; do { u = (x + y + y) / 3; S = y / u - 1; // 1 - x / u = 2 * S if (2 * abs(S) < tolerance) break; T sx = sqrt(x); T sy = sqrt(y); lambda = 2 * sx * sy + y; x = (x + lambda) / 4; y = (y + 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 value = (1 + S * S * (T(3) / 10 + S * (T(1) / 7 + S * (T(3) / 8 + S * T(9) / 22)))) / sqrt(u); return value * prefix; } } // namespace detail template inline typename tools::promote_args::type ellint_rc(T1 x, T2 y, 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_rc_imp( static_cast(x), static_cast(y), pol), "boost::math::ellint_rc<%1%>(%1%,%1%)"); } template inline typename tools::promote_args::type ellint_rc(T1 x, T2 y) { return ellint_rc(x, y, policies::policy<>()); } }} // namespaces #endif // BOOST_MATH_ELLINT_RC_HPP