// Copyright (c) 2006 Xiaogang Zhang, 2015 John Maddock
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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//
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// History:
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// XZ wrote the original of this file as part of the Google
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// Summer of Code 2006. JM modified it to fit into the
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// Boost.Math conceptual framework better, and to handle
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// types longer than 80-bit reals.
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// Updated 2015 to use Carlson's latest methods.
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//
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#ifndef BOOST_MATH_ELLINT_RF_HPP
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#define BOOST_MATH_ELLINT_RF_HPP
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#ifdef _MSC_VER
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#pragma once
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#endif
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#include <boost/math/special_functions/math_fwd.hpp>
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#include <boost/math/tools/config.hpp>
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#include <boost/math/constants/constants.hpp>
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#include <boost/math/policies/error_handling.hpp>
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#include <boost/math/special_functions/ellint_rc.hpp>
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// Carlson's elliptic integral of the first kind
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// R_F(x, y, z) = 0.5 * \int_{0}^{\infty} [(t+x)(t+y)(t+z)]^{-1/2} dt
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// Carlson, Numerische Mathematik, vol 33, 1 (1979)
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namespace boost { namespace math { namespace detail{
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template <typename T, typename Policy>
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T ellint_rf_imp(T x, T y, T z, const Policy& pol)
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{
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BOOST_MATH_STD_USING
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using namespace boost::math;
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using std::swap;
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static const char* function = "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)";
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if(x < 0 || y < 0 || z < 0)
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{
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return policies::raise_domain_error<T>(function,
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"domain error, all arguments must be non-negative, "
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"only sensible result is %1%.",
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std::numeric_limits<T>::quiet_NaN(), pol);
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}
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if(x + y == 0 || y + z == 0 || z + x == 0)
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{
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return policies::raise_domain_error<T>(function,
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"domain error, at most one argument can be zero, "
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"only sensible result is %1%.",
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std::numeric_limits<T>::quiet_NaN(), pol);
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}
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//
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// Special cases from http://dlmf.nist.gov/19.20#i
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//
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if(x == y)
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{
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if(x == z)
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{
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// x, y, z equal:
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return 1 / sqrt(x);
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}
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else
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{
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// 2 equal, x and y:
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if(z == 0)
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return constants::pi<T>() / (2 * sqrt(x));
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else
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return ellint_rc_imp(z, x, pol);
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}
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}
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if(x == z)
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{
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if(y == 0)
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return constants::pi<T>() / (2 * sqrt(x));
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else
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return ellint_rc_imp(y, x, pol);
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}
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if(y == z)
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{
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if(x == 0)
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return constants::pi<T>() / (2 * sqrt(y));
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else
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return ellint_rc_imp(x, y, pol);
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}
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if(x == 0)
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swap(x, z);
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else if(y == 0)
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swap(y, z);
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if(z == 0)
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{
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//
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// Special case for one value zero:
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//
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T xn = sqrt(x);
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T yn = sqrt(y);
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while(fabs(xn - yn) >= 2.7 * tools::root_epsilon<T>() * fabs(xn))
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{
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T t = sqrt(xn * yn);
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xn = (xn + yn) / 2;
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yn = t;
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}
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return constants::pi<T>() / (xn + yn);
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}
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T xn = x;
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T yn = y;
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T zn = z;
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T An = (x + y + z) / 3;
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T A0 = An;
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T Q = pow(3 * boost::math::tools::epsilon<T>(), T(-1) / 8) * (std::max)((std::max)(fabs(An - xn), fabs(An - yn)), fabs(An - zn));
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T fn = 1;
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// duplication
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unsigned k = 1;
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for(; k < boost::math::policies::get_max_series_iterations<Policy>(); ++k)
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{
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T root_x = sqrt(xn);
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T root_y = sqrt(yn);
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T root_z = sqrt(zn);
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T lambda = root_x * root_y + root_x * root_z + root_y * root_z;
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An = (An + lambda) / 4;
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xn = (xn + lambda) / 4;
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yn = (yn + lambda) / 4;
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zn = (zn + lambda) / 4;
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Q /= 4;
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fn *= 4;
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if(Q < fabs(An))
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break;
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}
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// Check to see if we gave up too soon:
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policies::check_series_iterations<T>(function, k, pol);
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BOOST_MATH_INSTRUMENT_VARIABLE(k);
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T X = (A0 - x) / (An * fn);
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T Y = (A0 - y) / (An * fn);
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T Z = -X - Y;
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// Taylor series expansion to the 7th order
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T E2 = X * Y - Z * Z;
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T E3 = X * Y * Z;
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return (1 + E3 * (T(1) / 14 + 3 * E3 / 104) + E2 * (T(-1) / 10 + E2 / 24 - (3 * E3) / 44 - 5 * E2 * E2 / 208 + E2 * E3 / 16)) / sqrt(An);
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}
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} // namespace detail
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template <class T1, class T2, class T3, class Policy>
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inline typename tools::promote_args<T1, T2, T3>::type
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ellint_rf(T1 x, T2 y, T3 z, const Policy& pol)
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{
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typedef typename tools::promote_args<T1, T2, T3>::type result_type;
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typedef typename policies::evaluation<result_type, Policy>::type value_type;
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return policies::checked_narrowing_cast<result_type, Policy>(
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detail::ellint_rf_imp(
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static_cast<value_type>(x),
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static_cast<value_type>(y),
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static_cast<value_type>(z), pol), "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)");
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}
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template <class T1, class T2, class T3>
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inline typename tools::promote_args<T1, T2, T3>::type
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ellint_rf(T1 x, T2 y, T3 z)
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{
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return ellint_rf(x, y, z, policies::policy<>());
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}
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}} // namespaces
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#endif // BOOST_MATH_ELLINT_RF_HPP
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