303 lines
8.9 KiB
Plaintext
303 lines
8.9 KiB
Plaintext
// 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 correctly
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// handle the p < 0 case.
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// Updated 2015 to use Carlson's latest methods.
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//
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#ifndef BOOST_MATH_ELLINT_RJ_HPP
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#define BOOST_MATH_ELLINT_RJ_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/policies/error_handling.hpp>
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#include <boost/math/special_functions/ellint_rc.hpp>
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#include <boost/math/special_functions/ellint_rf.hpp>
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#include <boost/math/special_functions/ellint_rd.hpp>
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// Carlson's elliptic integral of the third kind
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// R_J(x, y, z, p) = 1.5 * \int_{0}^{\infty} (t+p)^{-1} [(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_rc1p_imp(T y, const Policy& pol)
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{
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using namespace boost::math;
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// Calculate RC(1, 1 + x)
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BOOST_MATH_STD_USING
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static const char* function = "boost::math::ellint_rc<%1%>(%1%,%1%)";
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if(y == -1)
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{
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return policies::raise_domain_error<T>(function,
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"Argument y must not be zero but got %1%", y, pol);
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}
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// for 1 + y < 0, the integral is singular, return Cauchy principal value
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T result;
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if(y < -1)
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{
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result = sqrt(1 / -y) * detail::ellint_rc_imp(T(-y), T(-1 - y), pol);
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}
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else if(y == 0)
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{
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result = 1;
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}
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else if(y > 0)
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{
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result = atan(sqrt(y)) / sqrt(y);
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}
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else
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{
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if(y > -0.5)
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{
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T arg = sqrt(-y);
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result = (boost::math::log1p(arg) - boost::math::log1p(-arg)) / (2 * sqrt(-y));
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}
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else
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{
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result = log((1 + sqrt(-y)) / sqrt(1 + y)) / sqrt(-y);
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}
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}
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return result;
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}
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template <typename T, typename Policy>
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T ellint_rj_imp(T x, T y, T z, T p, 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_rj<%1%>(%1%,%1%,%1%)";
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if(x < 0)
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{
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return policies::raise_domain_error<T>(function,
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"Argument x must be non-negative, but got x = %1%", x, pol);
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}
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if(y < 0)
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{
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return policies::raise_domain_error<T>(function,
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"Argument y must be non-negative, but got y = %1%", y, pol);
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}
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if(z < 0)
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{
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return policies::raise_domain_error<T>(function,
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"Argument z must be non-negative, but got z = %1%", z, pol);
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}
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if(p == 0)
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{
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return policies::raise_domain_error<T>(function,
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"Argument p must not be zero, but got p = %1%", p, 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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"At most one argument can be zero, "
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"only possible result is %1%.", std::numeric_limits<T>::quiet_NaN(), pol);
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}
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// for p < 0, the integral is singular, return Cauchy principal value
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if(p < 0)
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{
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//
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// We must ensure that x < y < z.
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// Since the integral is symmetrical in x, y and z
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// we can just permute the values:
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//
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if(x > y)
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std::swap(x, y);
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if(y > z)
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std::swap(y, z);
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if(x > y)
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std::swap(x, y);
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BOOST_ASSERT(x <= y);
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BOOST_ASSERT(y <= z);
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T q = -p;
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p = (z * (x + y + q) - x * y) / (z + q);
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BOOST_ASSERT(p >= 0);
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T value = (p - z) * ellint_rj_imp(x, y, z, p, pol);
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value -= 3 * ellint_rf_imp(x, y, z, pol);
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value += 3 * sqrt((x * y * z) / (x * y + p * q)) * ellint_rc_imp(T(x * y + p * q), T(p * q), pol);
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value /= (z + q);
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return value;
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}
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//
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// Special cases from http://dlmf.nist.gov/19.20#iii
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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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if(x == p)
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{
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// All values equal:
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return 1 / (x * sqrt(x));
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}
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else
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{
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// x = y = z:
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return 3 * (ellint_rc_imp(x, p, pol) - 1 / sqrt(x)) / (x - p);
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}
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}
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else
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{
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// x = y only, permute so y = z:
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using std::swap;
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swap(x, z);
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if(y == p)
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{
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return ellint_rd_imp(x, y, y, pol);
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}
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else if((std::max)(y, p) / (std::min)(y, p) > 1.2)
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{
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return 3 * (ellint_rc_imp(x, y, pol) - ellint_rc_imp(x, p, pol)) / (p - y);
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}
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// Otherwise fall through to normal method, special case above will suffer too much cancellation...
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}
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}
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if(y == z)
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{
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if(y == p)
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{
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// y = z = p:
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return ellint_rd_imp(x, y, y, pol);
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}
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else if((std::max)(y, p) / (std::min)(y, p) > 1.2)
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{
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// y = z:
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return 3 * (ellint_rc_imp(x, y, pol) - ellint_rc_imp(x, p, pol)) / (p - y);
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}
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// Otherwise fall through to normal method, special case above will suffer too much cancellation...
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}
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if(z == p)
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{
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return ellint_rd_imp(x, y, z, pol);
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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 pn = p;
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T An = (x + y + z + 2 * p) / 5;
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T A0 = An;
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T delta = (p - x) * (p - y) * (p - z);
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T Q = pow(tools::epsilon<T>() / 5, -T(1) / 8) * (std::max)((std::max)(fabs(An - x), fabs(An - y)), (std::max)(fabs(An - z), fabs(An - p)));
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unsigned n;
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T lambda;
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T Dn;
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T En;
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T rx, ry, rz, rp;
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T fmn = 1; // 4^-n
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T RC_sum = 0;
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for(n = 0; n < policies::get_max_series_iterations<Policy>(); ++n)
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{
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rx = sqrt(xn);
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ry = sqrt(yn);
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rz = sqrt(zn);
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rp = sqrt(pn);
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Dn = (rp + rx) * (rp + ry) * (rp + rz);
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En = delta / Dn;
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En /= Dn;
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if((En < -0.5) && (En > -1.5))
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{
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//
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// Occationally En ~ -1, we then have no means of calculating
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// RC(1, 1+En) without terrible cancellation error, so we
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// need to get to 1+En directly. By substitution we have
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//
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// 1+E_0 = 1 + (p-x)*(p-y)*(p-z)/((sqrt(p) + sqrt(x))*(sqrt(p)+sqrt(y))*(sqrt(p)+sqrt(z)))^2
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// = 2*sqrt(p)*(p+sqrt(x) * (sqrt(y)+sqrt(z)) + sqrt(y)*sqrt(z)) / ((sqrt(p) + sqrt(x))*(sqrt(p) + sqrt(y)*(sqrt(p)+sqrt(z))))
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//
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// And since this is just an application of the duplication formula for RJ, the same
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// expression works for 1+En if we use x,y,z,p_n etc.
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// This branch is taken only once or twice at the start of iteration,
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// after than En reverts to it's usual very small values.
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//
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T b = 2 * rp * (pn + rx * (ry + rz) + ry * rz) / Dn;
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RC_sum += fmn / Dn * detail::ellint_rc_imp(T(1), b, pol);
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}
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else
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{
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RC_sum += fmn / Dn * ellint_rc1p_imp(En, pol);
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}
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lambda = rx * ry + rx * rz + ry * rz;
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// From here on we move to n+1:
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An = (An + lambda) / 4;
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fmn /= 4;
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if(fmn * Q < An)
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break;
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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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pn = (pn + lambda) / 4;
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delta /= 64;
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}
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T X = fmn * (A0 - x) / An;
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T Y = fmn * (A0 - y) / An;
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T Z = fmn * (A0 - z) / An;
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T P = (-X - Y - Z) / 2;
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T E2 = X * Y + X * Z + Y * Z - 3 * P * P;
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T E3 = X * Y * Z + 2 * E2 * P + 4 * P * P * P;
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T E4 = (2 * X * Y * Z + E2 * P + 3 * P * P * P) * P;
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T E5 = X * Y * Z * P * P;
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T result = fmn * pow(An, T(-3) / 2) *
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(1 - 3 * E2 / 14 + E3 / 6 + 9 * E2 * E2 / 88 - 3 * E4 / 22 - 9 * E2 * E3 / 52 + 3 * E5 / 26 - E2 * E2 * E2 / 16
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+ 3 * E3 * E3 / 40 + 3 * E2 * E4 / 20 + 45 * E2 * E2 * E3 / 272 - 9 * (E3 * E4 + E2 * E5) / 68);
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result += 6 * RC_sum;
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return result;
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}
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} // namespace detail
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template <class T1, class T2, class T3, class T4, class Policy>
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inline typename tools::promote_args<T1, T2, T3, T4>::type
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ellint_rj(T1 x, T2 y, T3 z, T4 p, const Policy& pol)
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{
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typedef typename tools::promote_args<T1, T2, T3, T4>::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_rj_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),
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static_cast<value_type>(p),
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pol), "boost::math::ellint_rj<%1%>(%1%,%1%,%1%,%1%)");
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}
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template <class T1, class T2, class T3, class T4>
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inline typename tools::promote_args<T1, T2, T3, T4>::type
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ellint_rj(T1 x, T2 y, T3 z, T4 p)
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{
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return ellint_rj(x, y, z, p, policies::policy<>());
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}
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}} // namespaces
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#endif // BOOST_MATH_ELLINT_RJ_HPP
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