Chris@16: //  Copyright (c) 2006 Xiaogang Zhang
Chris@16: //  Copyright (c) 2006 John Maddock
Chris@16: //  Use, modification and distribution are subject to the
Chris@16: //  Boost Software License, Version 1.0. (See accompanying file
Chris@16: //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@16: //
Chris@16: //  History:
Chris@16: //  XZ wrote the original of this file as part of the Google
Chris@16: //  Summer of Code 2006.  JM modified it to fit into the
Chris@16: //  Boost.Math conceptual framework better, and to ensure
Chris@16: //  that the code continues to work no matter how many digits
Chris@16: //  type T has.
Chris@16: 
Chris@16: #ifndef BOOST_MATH_ELLINT_2_HPP
Chris@16: #define BOOST_MATH_ELLINT_2_HPP
Chris@16: 
Chris@16: #ifdef _MSC_VER
Chris@16: #pragma once
Chris@16: #endif
Chris@16: 
Chris@101: #include <boost/math/special_functions/math_fwd.hpp>
Chris@16: #include <boost/math/special_functions/ellint_rf.hpp>
Chris@16: #include <boost/math/special_functions/ellint_rd.hpp>
Chris@101: #include <boost/math/special_functions/ellint_rg.hpp>
Chris@16: #include <boost/math/constants/constants.hpp>
Chris@16: #include <boost/math/policies/error_handling.hpp>
Chris@16: #include <boost/math/tools/workaround.hpp>
Chris@16: #include <boost/math/special_functions/round.hpp>
Chris@16: 
Chris@16: // Elliptic integrals (complete and incomplete) of the second kind
Chris@16: // Carlson, Numerische Mathematik, vol 33, 1 (1979)
Chris@16: 
Chris@16: namespace boost { namespace math { 
Chris@16:    
Chris@16: template <class T1, class T2, class Policy>
Chris@16: typename tools::promote_args<T1, T2>::type ellint_2(T1 k, T2 phi, const Policy& pol);
Chris@16:    
Chris@16: namespace detail{
Chris@16: 
Chris@16: template <typename T, typename Policy>
Chris@16: T ellint_e_imp(T k, const Policy& pol);
Chris@16: 
Chris@16: // Elliptic integral (Legendre form) of the second kind
Chris@16: template <typename T, typename Policy>
Chris@16: T ellint_e_imp(T phi, T k, const Policy& pol)
Chris@16: {
Chris@16:     BOOST_MATH_STD_USING
Chris@16:     using namespace boost::math::tools;
Chris@16:     using namespace boost::math::constants;
Chris@16: 
Chris@16:     bool invert = false;
Chris@16:     if(phi < 0)
Chris@16:     {
Chris@16:        phi = fabs(phi);
Chris@16:        invert = true;
Chris@16:     }
Chris@16: 
Chris@16:     T result;
Chris@16: 
Chris@16:     if(phi >= tools::max_value<T>())
Chris@16:     {
Chris@16:        // Need to handle infinity as a special case:
Chris@16:        result = policies::raise_overflow_error<T>("boost::math::ellint_e<%1%>(%1%,%1%)", 0, pol);
Chris@16:     }
Chris@16:     else if(phi > 1 / tools::epsilon<T>())
Chris@16:     {
Chris@16:        // Phi is so large that phi%pi is necessarily zero (or garbage),
Chris@16:        // just return the second part of the duplication formula:
Chris@16:        result = 2 * phi * ellint_e_imp(k, pol) / constants::pi<T>();
Chris@16:     }
Chris@101:     else if(k == 0)
Chris@101:     {
Chris@101:        return invert ? T(-phi) : phi;
Chris@101:     }
Chris@101:     else if(fabs(k) == 1)
Chris@101:     {
Chris@101:        return invert ? T(-sin(phi)) : sin(phi);
Chris@101:     }
Chris@16:     else
Chris@16:     {
Chris@16:        // Carlson's algorithm works only for |phi| <= pi/2,
Chris@16:        // use the integrand's periodicity to normalize phi
Chris@16:        //
Chris@16:        // Xiaogang's original code used a cast to long long here
Chris@16:        // but that fails if T has more digits than a long long,
Chris@16:        // so rewritten to use fmod instead:
Chris@16:        //
Chris@16:        T rphi = boost::math::tools::fmod_workaround(phi, T(constants::half_pi<T>()));
Chris@16:        T m = boost::math::round((phi - rphi) / constants::half_pi<T>());
Chris@16:        int s = 1;
Chris@16:        if(boost::math::tools::fmod_workaround(m, T(2)) > 0.5)
Chris@16:        {
Chris@16:           m += 1;
Chris@16:           s = -1;
Chris@16:           rphi = constants::half_pi<T>() - rphi;
Chris@16:        }
Chris@16:        T sinp = sin(rphi);
Chris@16:        T cosp = cos(rphi);
Chris@101:        T c = 1 / (sinp * sinp);
Chris@101:        T cm1 = cosp * cosp / (sinp * sinp);  // c - 1
Chris@101:        T k2 = k * k;
Chris@101:        if(k2 > 1)
Chris@101:        {
Chris@101:           return policies::raise_domain_error<T>("boost::math::ellint_2<%1%>(%1%, %1%)", "The parameter k is out of range, got k = %1%", k, pol);
Chris@101:        }
Chris@101:        else if(rphi == 0)
Chris@101:        {
Chris@101:           result = 0;
Chris@101:        }
Chris@101:        else if(sinp * sinp < tools::min_value<T>())
Chris@101:        {
Chris@101:           T x = cosp * cosp;
Chris@101:           T t = k * k * sinp * sinp;
Chris@101:           T y = 1 - t;
Chris@101:           T z = 1;
Chris@101:           result = s * sinp * (ellint_rf_imp(x, y, z, pol) - t * ellint_rd_imp(x, y, z, pol) / 3);
Chris@101:        }
Chris@101:        else
Chris@101:        {
Chris@101:           // http://dlmf.nist.gov/19.25#E10
Chris@101:           result = s * ((1 - k2) * ellint_rf_imp(cm1, T(c - k2), c, pol) + k2 * (1 - k2) * ellint_rd(cm1, c, T(c - k2), pol) / 3 + k2 * sqrt(cm1 / (c * (c - k2))));
Chris@101:        }
Chris@16:        if(m != 0)
Chris@16:           result += m * ellint_e_imp(k, pol);
Chris@16:     }
Chris@16:     return invert ? T(-result) : result;
Chris@16: }
Chris@16: 
Chris@16: // Complete elliptic integral (Legendre form) of the second kind
Chris@16: template <typename T, typename Policy>
Chris@16: T ellint_e_imp(T k, const Policy& pol)
Chris@16: {
Chris@16:     BOOST_MATH_STD_USING
Chris@16:     using namespace boost::math::tools;
Chris@16: 
Chris@16:     if (abs(k) > 1)
Chris@16:     {
Chris@16:        return policies::raise_domain_error<T>("boost::math::ellint_e<%1%>(%1%)",
Chris@16:             "Got k = %1%, function requires |k| <= 1", k, pol);
Chris@16:     }
Chris@16:     if (abs(k) == 1)
Chris@16:     {
Chris@16:         return static_cast<T>(1);
Chris@16:     }
Chris@16: 
Chris@16:     T x = 0;
Chris@16:     T t = k * k;
Chris@16:     T y = 1 - t;
Chris@16:     T z = 1;
Chris@101:     T value = 2 * ellint_rg_imp(x, y, z, pol);
Chris@16: 
Chris@16:     return value;
Chris@16: }
Chris@16: 
Chris@16: template <typename T, typename Policy>
Chris@16: inline typename tools::promote_args<T>::type ellint_2(T k, const Policy& pol, const mpl::true_&)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T>::type result_type;
Chris@16:    typedef typename policies::evaluation<result_type, Policy>::type value_type;
Chris@16:    return policies::checked_narrowing_cast<result_type, Policy>(detail::ellint_e_imp(static_cast<value_type>(k), pol), "boost::math::ellint_2<%1%>(%1%)");
Chris@16: }
Chris@16: 
Chris@16: // Elliptic integral (Legendre form) of the second kind
Chris@16: template <class T1, class T2>
Chris@16: inline typename tools::promote_args<T1, T2>::type ellint_2(T1 k, T2 phi, const mpl::false_&)
Chris@16: {
Chris@16:    return boost::math::ellint_2(k, phi, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: } // detail
Chris@16: 
Chris@16: // Complete elliptic integral (Legendre form) of the second kind
Chris@16: template <typename T>
Chris@16: inline typename tools::promote_args<T>::type ellint_2(T k)
Chris@16: {
Chris@16:    return ellint_2(k, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: // Elliptic integral (Legendre form) of the second kind
Chris@16: template <class T1, class T2>
Chris@16: inline typename tools::promote_args<T1, T2>::type ellint_2(T1 k, T2 phi)
Chris@16: {
Chris@16:    typedef typename policies::is_policy<T2>::type tag_type;
Chris@16:    return detail::ellint_2(k, phi, tag_type());
Chris@16: }
Chris@16: 
Chris@16: template <class T1, class T2, class Policy>
Chris@16: inline typename tools::promote_args<T1, T2>::type ellint_2(T1 k, T2 phi, const Policy& pol)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T1, T2>::type result_type;
Chris@16:    typedef typename policies::evaluation<result_type, Policy>::type value_type;
Chris@16:    return policies::checked_narrowing_cast<result_type, Policy>(detail::ellint_e_imp(static_cast<value_type>(phi), static_cast<value_type>(k), pol), "boost::math::ellint_2<%1%>(%1%,%1%)");
Chris@16: }
Chris@16: 
Chris@16: }} // namespaces
Chris@16: 
Chris@16: #endif // BOOST_MATH_ELLINT_2_HPP
Chris@16: