// Copyright (c) 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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#ifndef BOOST_MATH_ELLINT_RG_HPP
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#define BOOST_MATH_ELLINT_RG_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_rd.hpp>
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#include <boost/math/special_functions/ellint_rf.hpp>
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#include <boost/math/special_functions/pow.hpp>
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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_rg_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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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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//
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// Function is symmetric in x, y and z, but we require
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// (x - z)(y - z) >= 0 to avoid cancellation error in the result
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// which implies (for example) x >= z >= y
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//
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using std::swap;
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if(x < y)
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swap(x, y);
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if(x < z)
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swap(x, z);
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if(y > z)
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swap(y, z);
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BOOST_ASSERT(x >= z);
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BOOST_ASSERT(z >= y);
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//
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// Special cases from http://dlmf.nist.gov/19.20#ii
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//
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if(x == z)
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{
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if(y == z)
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{
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// x = y = z
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// This also works for x = y = z = 0 presumably.
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return sqrt(x);
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}
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else if(y == 0)
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{
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// x = y, z = 0
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return constants::pi<T>() * sqrt(x) / 4;
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}
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else
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{
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// x = z, y != 0
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swap(x, y);
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return (x == 0) ? T(sqrt(z) / 2) : T((z * ellint_rc_imp(x, z, pol) + sqrt(x)) / 2);
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}
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}
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else if(y == z)
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{
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if(x == 0)
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return constants::pi<T>() * sqrt(y) / 4;
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else
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return (y == 0) ? T(sqrt(x) / 2) : T((y * ellint_rc_imp(x, y, pol) + sqrt(x)) / 2);
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}
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else if(y == 0)
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{
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swap(y, z);
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//
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// Special handling for common case, from
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// Numerical Computation of Real or Complex Elliptic Integrals, eq.46
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//
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T xn = sqrt(x);
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T yn = sqrt(y);
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T x0 = xn;
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T y0 = yn;
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T sum = 0;
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T sum_pow = 0.25f;
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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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sum_pow *= 2;
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sum += sum_pow * boost::math::pow<2>(xn - yn);
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}
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T RF = constants::pi<T>() / (xn + yn);
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return ((boost::math::pow<2>((x0 + y0) / 2) - sum) * RF) / 2;
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}
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return (z * ellint_rf_imp(x, y, z, pol)
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- (x - z) * (y - z) * ellint_rd_imp(x, y, z, pol) / 3
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+ sqrt(x * y / z)) / 2;
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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_rg(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_rg_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_rg(T1 x, T2 y, T3 z)
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{
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return ellint_rg(x, y, z, policies::policy<>());
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}
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}} // namespaces
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#endif // BOOST_MATH_ELLINT_RG_HPP
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