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+// 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 ensure
+// that the code continues to work no matter how many digits
+// type T has.
+
+#ifndef BOOST_MATH_ELLINT_D_HPP
+#define BOOST_MATH_ELLINT_D_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_rd.hpp>
+#include <boost/math/special_functions/ellint_rg.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 second kind
+// Carlson, Numerische Mathematik, vol 33, 1 (1979)
+
+namespace boost { namespace math {
+
+template <class T1, class T2, class Policy>
+typename tools::promote_args<T1, T2>::type ellint_d(T1 k, T2 phi, const Policy& pol);
+
+namespace detail{
+
+template <typename T, typename Policy>
+T ellint_d_imp(T k, const Policy& pol);
+
+// Elliptic integral (Legendre form) of the second kind
+template <typename T, typename Policy>
+T ellint_d_imp(T phi, T k, const Policy& pol)
+{
+ BOOST_MATH_STD_USING
+ using namespace boost::math::tools;
+ using namespace boost::math::constants;
+
+ bool invert = false;
+ if(phi < 0)
+ {
+ phi = fabs(phi);
+ invert = true;
+ }
+
+ T result;
+
+ if(phi >= tools::max_value<T>())
+ {
+ // Need to handle infinity as a special case:
+ result = policies::raise_overflow_error<T>("boost::math::ellint_e<%1%>(%1%,%1%)", 0, pol);
+ }
+ else if(phi > 1 / tools::epsilon<T>())
+ {
+ // Phi is so large that phi%pi is necessarily zero (or garbage),
+ // just return the second part of the duplication formula:
+ result = 2 * phi * ellint_d_imp(k, pol) / constants::pi<T>();
+ }
+ else
+ {
+ // Carlson's algorithm works only for |phi| <= pi/2,
+ // use the integrand's periodicity to normalize phi
+ //
+ T rphi = boost::math::tools::fmod_workaround(phi, T(constants::half_pi<T>()));
+ T m = boost::math::round((phi - rphi) / constants::half_pi<T>());
+ int s = 1;
+ if(boost::math::tools::fmod_workaround(m, T(2)) > 0.5)
+ {
+ m += 1;
+ s = -1;
+ rphi = constants::half_pi<T>() - rphi;
+ }
+ T sinp = sin(rphi);
+ T cosp = cos(rphi);
+ T c = 1 / (sinp * sinp);
+ T cm1 = cosp * cosp / (sinp * sinp); // c - 1
+ T k2 = k * k;
+ if(k2 > 1)
+ {
+ return policies::raise_domain_error<T>("boost::math::ellint_d<%1%>(%1%, %1%)", "The parameter k is out of range, got k = %1%", k, pol);
+ }
+ else if(rphi == 0)
+ {
+ result = 0;
+ }
+ else
+ {
+ // http://dlmf.nist.gov/19.25#E10
+ result = s * ellint_rd_imp(cm1, T(c - k2), c, pol) / 3;
+ }
+ if(m != 0)
+ result += m * ellint_d_imp(k, pol);
+ }
+ return invert ? T(-result) : result;
+}
+
+// Complete elliptic integral (Legendre form) of the second kind
+template <typename T, typename Policy>
+T ellint_d_imp(T k, const Policy& pol)
+{
+ BOOST_MATH_STD_USING
+ using namespace boost::math::tools;
+
+ if (abs(k) > 1)
+ {
+ return policies::raise_domain_error<T>("boost::math::ellint_e<%1%>(%1%)",
+ "Got k = %1%, function requires |k| <= 1", k, pol);
+ }
+ if (abs(k) == 1)
+ {
+ return static_cast<T>(1);
+ }
+ if(fabs(k) <= tools::root_epsilon<T>())
+ return constants::pi<T>() / 4;
+
+ T x = 0;
+ T t = k * k;
+ T y = 1 - t;
+ T z = 1;
+ T value = ellint_rd_imp(x, y, z, pol) / 3;
+
+ return value;
+}
+
+template <typename T, typename Policy>
+inline typename tools::promote_args<T>::type ellint_d(T k, const Policy& pol, const mpl::true_&)
+{
+ typedef typename tools::promote_args<T>::type result_type;
+ typedef typename policies::evaluation<result_type, Policy>::type value_type;
+ return policies::checked_narrowing_cast<result_type, Policy>(detail::ellint_d_imp(static_cast<value_type>(k), pol), "boost::math::ellint_d<%1%>(%1%)");
+}
+
+// Elliptic integral (Legendre form) of the second kind
+template <class T1, class T2>
+inline typename tools::promote_args<T1, T2>::type ellint_d(T1 k, T2 phi, const mpl::false_&)
+{
+ return boost::math::ellint_d(k, phi, policies::policy<>());
+}
+
+} // detail
+
+// Complete elliptic integral (Legendre form) of the second kind
+template <typename T>
+inline typename tools::promote_args<T>::type ellint_d(T k)
+{
+ return ellint_d(k, policies::policy<>());
+}
+
+// Elliptic integral (Legendre form) of the second kind
+template <class T1, class T2>
+inline typename tools::promote_args<T1, T2>::type ellint_d(T1 k, T2 phi)
+{
+ typedef typename policies::is_policy<T2>::type tag_type;
+ return detail::ellint_d(k, phi, tag_type());
+}
+
+template <class T1, class T2, class Policy>
+inline typename tools::promote_args<T1, T2>::type ellint_d(T1 k, T2 phi, const Policy& pol)
+{
+ 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_d_imp(static_cast<value_type>(phi), static_cast<value_type>(k), pol), "boost::math::ellint_2<%1%>(%1%,%1%)");
+}
+
+}} // namespaces
+
+#endif // BOOST_MATH_ELLINT_D_HPP
+