boost/math/special_functions/ellint_rd.hpp
// 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 <boost/math/special_functions/math_fwd.hpp> #include <boost/math/special_functions/ellint_rc.hpp> #include <boost/math/special_functions/pow.hpp> #include <boost/math/tools/config.hpp> #include <boost/math/policies/error_handling.hpp> // 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 <typename T, typename Policy> 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<T>(function, "Argument x must be >= 0, but got %1%", x, pol); } if(y < 0) { return policies::raise_domain_error<T>(function, "Argument y must be >= 0, but got %1%", y, pol); } if(z <= 0) { return policies::raise_domain_error<T>(function, "Argument z must be > 0, but got %1%", z, pol); } if(x + y == 0) { return policies::raise_domain_error<T>(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<T>() / (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<T>() * 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<T>() / (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 denominator 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<T>() / 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<Policy>(); ++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<T, Policy>(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 <class T1, class T2, class T3, class Policy> inline typename tools::promote_args<T1, T2, T3>::type ellint_rd(T1 x, T2 y, T3 z, const Policy& pol) { typedef typename tools::promote_args<T1, T2, T3>::type result_type; typedef typename policies::evaluation<result_type, Policy>::type value_type; return policies::checked_narrowing_cast<result_type, Policy>( detail::ellint_rd_imp( static_cast<value_type>(x), static_cast<value_type>(y), static_cast<value_type>(z), pol), "boost::math::ellint_rd<%1%>(%1%,%1%,%1%)"); } template <class T1, class T2, class T3> inline typename tools::promote_args<T1, T2, T3>::type ellint_rd(T1 x, T2 y, T3 z) { return ellint_rd(x, y, z, policies::policy<>()); } }} // namespaces #endif // BOOST_MATH_ELLINT_RD_HPP