--- /dev/null
+// Copyright (c) 2006 Xiaogang Zhang
+// Copyright (c) 2006 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 to fit into the
+// Boost.Math conceptual framework better, and to correctly
+// handle the various corner cases.
+//
+
+#ifndef BOOST_MATH_ELLINT_3_HPP
+#define BOOST_MATH_ELLINT_3_HPP
+
+#ifdef _MSC_VER
+#pragma once
+#endif
+
+#include <boost/math/special_functions/math_fwd.hpp>
+#include <boost/math/special_functions/ellint_rf.hpp>
+#include <boost/math/special_functions/ellint_rj.hpp>
+#include <boost/math/special_functions/ellint_1.hpp>
+#include <boost/math/special_functions/ellint_2.hpp>
+#include <boost/math/special_functions/log1p.hpp>
+#include <boost/math/special_functions/atanh.hpp>
+#include <boost/math/constants/constants.hpp>
+#include <boost/math/policies/error_handling.hpp>
+#include <boost/math/tools/workaround.hpp>
+#include <boost/math/special_functions/round.hpp>
+
+// Elliptic integrals (complete and incomplete) of the third kind
+// Carlson, Numerische Mathematik, vol 33, 1 (1979)
+
+namespace boost { namespace math {
+
+namespace detail{
+
+template <typename T, typename Policy>
+T ellint_pi_imp(T v, T k, T vc, const Policy& pol);
+
+// Elliptic integral (Legendre form) of the third kind
+template <typename T, typename Policy>
+T ellint_pi_imp(T v, T phi, T k, T vc, const Policy& pol)
+{
+ // Note vc = 1-v presumably without cancellation error.
+ BOOST_MATH_STD_USING
+
+ static const char* function = "boost::math::ellint_3<%1%>(%1%,%1%,%1%)";
+
+ if(abs(k) > 1)
+ {
+ return policies::raise_domain_error<T>(function,
+ "Got k = %1%, function requires |k| <= 1", k, pol);
+ }
+
+ T sphi = sin(fabs(phi));
+ T result = 0;
+
+ // Special cases first:
+ if(v == 0)
+ {
+ // A&S 17.7.18 & 19
+ return (k == 0) ? phi : ellint_f_imp(phi, k, pol);
+ }
+ if((v > 0) && (1 / v < (sphi * sphi)))
+ {
+ // Complex result is a domain error:
+ return policies::raise_domain_error<T>(function,
+ "Got v = %1%, but result is complex for v > 1 / sin^2(phi)", v, pol);
+ }
+
+ if(v == 1)
+ {
+ // http://functions.wolfram.com/08.06.03.0008.01
+ T m = k * k;
+ result = sqrt(1 - m * sphi * sphi) * tan(phi) - ellint_e_imp(phi, k, pol);
+ result /= 1 - m;
+ result += ellint_f_imp(phi, k, pol);
+ return result;
+ }
+ if(phi == constants::half_pi<T>())
+ {
+ // Have to filter this case out before the next
+ // special case, otherwise we might get an infinity from
+ // tan(phi).
+ // Also note that since we can't represent PI/2 exactly
+ // in a T, this is a bit of a guess as to the users true
+ // intent...
+ //
+ return ellint_pi_imp(v, k, vc, pol);
+ }
+ if((phi > constants::half_pi<T>()) || (phi < 0))
+ {
+ // Carlson's algorithm works only for |phi| <= pi/2,
+ // use the integrand's periodicity to normalize phi
+ //
+ // Xiaogang's original code used a cast to long long here
+ // but that fails if T has more digits than a long long,
+ // so rewritten to use fmod instead:
+ //
+ // See http://functions.wolfram.com/08.06.16.0002.01
+ //
+ if(fabs(phi) > 1 / tools::epsilon<T>())
+ {
+ if(v > 1)
+ return policies::raise_domain_error<T>(
+ function,
+ "Got v = %1%, but this is only supported for 0 <= phi <= pi/2", v, pol);
+ //
+ // Phi is so large that phi%pi is necessarily zero (or garbage),
+ // just return the second part of the duplication formula:
+ //
+ result = 2 * fabs(phi) * ellint_pi_imp(v, k, vc, pol) / constants::pi<T>();
+ }
+ else
+ {
+ T rphi = boost::math::tools::fmod_workaround(T(fabs(phi)), T(constants::half_pi<T>()));
+ T m = boost::math::round((fabs(phi) - rphi) / constants::half_pi<T>());
+ int sign = 1;
+ if((m != 0) && (k >= 1))
+ {
+ return policies::raise_domain_error<T>(function, "Got k=1 and phi=%1% but the result is complex in that domain", phi, pol);
+ }
+ if(boost::math::tools::fmod_workaround(m, T(2)) > 0.5)
+ {
+ m += 1;
+ sign = -1;
+ rphi = constants::half_pi<T>() - rphi;
+ }
+ result = sign * ellint_pi_imp(v, rphi, k, vc, pol);
+ if((m > 0) && (vc > 0))
+ result += m * ellint_pi_imp(v, k, vc, pol);
+ }
+ return phi < 0 ? T(-result) : result;
+ }
+ if(k == 0)
+ {
+ // A&S 17.7.20:
+ if(v < 1)
+ {
+ T vcr = sqrt(vc);
+ return atan(vcr * tan(phi)) / vcr;
+ }
+ else if(v == 1)
+ {
+ return tan(phi);
+ }
+ else
+ {
+ // v > 1:
+ T vcr = sqrt(-vc);
+ T arg = vcr * tan(phi);
+ return (boost::math::log1p(arg, pol) - boost::math::log1p(-arg, pol)) / (2 * vcr);
+ }
+ }
+ if(v < 0)
+ {
+ //
+ // If we don't shift to 0 <= v <= 1 we get
+ // cancellation errors later on. Use
+ // A&S 17.7.15/16 to shift to v > 0.
+ //
+ // Mathematica simplifies the expressions
+ // given in A&S as follows (with thanks to
+ // Rocco Romeo for figuring these out!):
+ //
+ // V = (k2 - n)/(1 - n)
+ // Assuming[(k2 >= 0 && k2 <= 1) && n < 0, FullSimplify[Sqrt[(1 - V)*(1 - k2 / V)] / Sqrt[((1 - n)*(1 - k2 / n))]]]
+ // Result: ((-1 + k2) n) / ((-1 + n) (-k2 + n))
+ //
+ // Assuming[(k2 >= 0 && k2 <= 1) && n < 0, FullSimplify[k2 / (Sqrt[-n*(k2 - n) / (1 - n)] * Sqrt[(1 - n)*(1 - k2 / n)])]]
+ // Result : k2 / (k2 - n)
+ //
+ // Assuming[(k2 >= 0 && k2 <= 1) && n < 0, FullSimplify[Sqrt[1 / ((1 - n)*(1 - k2 / n))]]]
+ // Result : Sqrt[n / ((k2 - n) (-1 + n))]
+ //
+ T k2 = k * k;
+ T N = (k2 - v) / (1 - v);
+ T Nm1 = (1 - k2) / (1 - v);
+ T p2 = -v * N;
+ T t;
+ if(p2 <= tools::min_value<T>())
+ p2 = sqrt(-v) * sqrt(N);
+ else
+ p2 = sqrt(p2);
+ T delta = sqrt(1 - k2 * sphi * sphi);
+ if(N > k2)
+ {
+ result = ellint_pi_imp(N, phi, k, Nm1, pol);
+ result *= v / (v - 1);
+ result *= (k2 - 1) / (v - k2);
+ }
+
+ if(k != 0)
+ {
+ t = ellint_f_imp(phi, k, pol);
+ t *= k2 / (k2 - v);
+ result += t;
+ }
+ t = v / ((k2 - v) * (v - 1));
+ if(t > tools::min_value<T>())
+ {
+ result += atan((p2 / 2) * sin(2 * phi) / delta) * sqrt(t);
+ }
+ else
+ {
+ result += atan((p2 / 2) * sin(2 * phi) / delta) * sqrt(fabs(1 / (k2 - v))) * sqrt(fabs(v / (v - 1)));
+ }
+ return result;
+ }
+ if(k == 1)
+ {
+ // See http://functions.wolfram.com/08.06.03.0013.01
+ result = sqrt(v) * atanh(sqrt(v) * sin(phi)) - log(1 / cos(phi) + tan(phi));
+ result /= v - 1;
+ return result;
+ }
+#if 0 // disabled but retained for future reference: see below.
+ if(v > 1)
+ {
+ //
+ // If v > 1 we can use the identity in A&S 17.7.7/8
+ // to shift to 0 <= v <= 1. In contrast to previous
+ // revisions of this header, this identity does now work
+ // but appears not to produce better error rates in
+ // practice. Archived here for future reference...
+ //
+ T k2 = k * k;
+ T N = k2 / v;
+ T Nm1 = (v - k2) / v;
+ T p1 = sqrt((-vc) * (1 - k2 / v));
+ T delta = sqrt(1 - k2 * sphi * sphi);
+ //
+ // These next two terms have a large amount of cancellation
+ // so it's not clear if this relation is useable even if
+ // the issues with phi > pi/2 can be fixed:
+ //
+ result = -ellint_pi_imp(N, phi, k, Nm1, pol);
+ result += ellint_f_imp(phi, k, pol);
+ //
+ // This log term gives the complex result when
+ // n > 1/sin^2(phi)
+ // However that case is dealt with as an error above,
+ // so we should always get a real result here:
+ //
+ result += log((delta + p1 * tan(phi)) / (delta - p1 * tan(phi))) / (2 * p1);
+ return result;
+ }
+#endif
+ //
+ // Carlson's algorithm works only for |phi| <= pi/2,
+ // by the time we get here phi should already have been
+ // normalised above.
+ //
+ BOOST_ASSERT(fabs(phi) < constants::half_pi<T>());
+ BOOST_ASSERT(phi >= 0);
+ T x, y, z, p, t;
+ T cosp = cos(phi);
+ x = cosp * cosp;
+ t = sphi * sphi;
+ y = 1 - k * k * t;
+ z = 1;
+ if(v * t < 0.5)
+ p = 1 - v * t;
+ else
+ p = x + vc * t;
+ result = sphi * (ellint_rf_imp(x, y, z, pol) + v * t * ellint_rj_imp(x, y, z, p, pol) / 3);
+
+ return result;
+}
+
+// Complete elliptic integral (Legendre form) of the third kind
+template <typename T, typename Policy>
+T ellint_pi_imp(T v, T k, T vc, const Policy& pol)
+{
+ // Note arg vc = 1-v, possibly without cancellation errors
+ BOOST_MATH_STD_USING
+ using namespace boost::math::tools;
+
+ static const char* function = "boost::math::ellint_pi<%1%>(%1%,%1%)";
+
+ if (abs(k) >= 1)
+ {
+ return policies::raise_domain_error<T>(function,
+ "Got k = %1%, function requires |k| <= 1", k, pol);
+ }
+ if(vc <= 0)
+ {
+ // Result is complex:
+ return policies::raise_domain_error<T>(function,
+ "Got v = %1%, function requires v < 1", v, pol);
+ }
+
+ if(v == 0)
+ {
+ return (k == 0) ? boost::math::constants::pi<T>() / 2 : ellint_k_imp(k, pol);
+ }
+
+ if(v < 0)
+ {
+ // Apply A&S 17.7.17:
+ T k2 = k * k;
+ T N = (k2 - v) / (1 - v);
+ T Nm1 = (1 - k2) / (1 - v);
+ T result = 0;
+ result = boost::math::detail::ellint_pi_imp(N, k, Nm1, pol);
+ // This next part is split in two to avoid spurious over/underflow:
+ result *= -v / (1 - v);
+ result *= (1 - k2) / (k2 - v);
+ result += ellint_k_imp(k, pol) * k2 / (k2 - v);
+ return result;
+ }
+
+ T x = 0;
+ T y = 1 - k * k;
+ T z = 1;
+ T p = vc;
+ T value = ellint_rf_imp(x, y, z, pol) + v * ellint_rj_imp(x, y, z, p, pol) / 3;
+
+ return value;
+}
+
+template <class T1, class T2, class T3>
+inline typename tools::promote_args<T1, T2, T3>::type ellint_3(T1 k, T2 v, T3 phi, const mpl::false_&)
+{
+ return boost::math::ellint_3(k, v, phi, policies::policy<>());
+}
+
+template <class T1, class T2, class Policy>
+inline typename tools::promote_args<T1, T2>::type ellint_3(T1 k, T2 v, const Policy& pol, const mpl::true_&)
+{
+ typedef typename tools::promote_args<T1, T2>::type result_type;
+ typedef typename policies::evaluation<result_type, Policy>::type value_type;
+ return policies::checked_narrowing_cast<result_type, Policy>(
+ detail::ellint_pi_imp(
+ static_cast<value_type>(v),
+ static_cast<value_type>(k),
+ static_cast<value_type>(1-v),
+ pol), "boost::math::ellint_3<%1%>(%1%,%1%)");
+}
+
+} // namespace detail
+
+template <class T1, class T2, class T3, class Policy>
+inline typename tools::promote_args<T1, T2, T3>::type ellint_3(T1 k, T2 v, T3 phi, 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_pi_imp(
+ static_cast<value_type>(v),
+ static_cast<value_type>(phi),
+ static_cast<value_type>(k),
+ static_cast<value_type>(1-v),
+ pol), "boost::math::ellint_3<%1%>(%1%,%1%,%1%)");
+}
+
+template <class T1, class T2, class T3>
+typename detail::ellint_3_result<T1, T2, T3>::type ellint_3(T1 k, T2 v, T3 phi)
+{
+ typedef typename policies::is_policy<T3>::type tag_type;
+ return detail::ellint_3(k, v, phi, tag_type());
+}
+
+template <class T1, class T2>
+inline typename tools::promote_args<T1, T2>::type ellint_3(T1 k, T2 v)
+{
+ return ellint_3(k, v, policies::policy<>());
+}
+
+}} // namespaces
+
+#endif // BOOST_MATH_ELLINT_3_HPP
+
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