ceph/src/boost/boost/math/special_functions/ellint_d.hpp

   1 //  Copyright (c) 2006 Xiaogang Zhang
   2 //  Copyright (c) 2006 John Maddock
   3 //  Use, modification and distribution are subject to the
   4 //  Boost Software License, Version 1.0. (See accompanying file
   5 //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
   6 //
   7 //  History:
   8 //  XZ wrote the original of this file as part of the Google
   9 //  Summer of Code 2006.  JM modified it to fit into the
  10 //  Boost.Math conceptual framework better, and to ensure
  11 //  that the code continues to work no matter how many digits
  12 //  type T has.
  13
  14 #ifndef BOOST_MATH_ELLINT_D_HPP
  15 #define BOOST_MATH_ELLINT_D_HPP
  16
  17 #ifdef _MSC_VER
  18 #pragma once
  19 #endif
  20
  21 #include <boost/math/special_functions/math_fwd.hpp>
  22 #include <boost/math/special_functions/ellint_rf.hpp>
  23 #include <boost/math/special_functions/ellint_rd.hpp>
  24 #include <boost/math/special_functions/ellint_rg.hpp>
  25 #include <boost/math/constants/constants.hpp>
  26 #include <boost/math/policies/error_handling.hpp>
  27 #include <boost/math/tools/workaround.hpp>
  28 #include <boost/math/special_functions/round.hpp>
  29
  30 // Elliptic integrals (complete and incomplete) of the second kind
  31 // Carlson, Numerische Mathematik, vol 33, 1 (1979)
  32
  33 namespace boost { namespace math {
  34
  35 template <class T1, class T2, class Policy>
  36 typename tools::promote_args<T1, T2>::type ellint_d(T1 k, T2 phi, const Policy& pol);
  37
  38 namespace detail{
  39
  40 template <typename T, typename Policy>
  41 T ellint_d_imp(T k, const Policy& pol);
  42
  43 // Elliptic integral (Legendre form) of the second kind
  44 template <typename T, typename Policy>
  45 T ellint_d_imp(T phi, T k, const Policy& pol)
  46 {
  47     BOOST_MATH_STD_USING
  48     using namespace boost::math::tools;
  49     using namespace boost::math::constants;
  50
  51     bool invert = false;
  52     if(phi < 0)
  53     {
  54        phi = fabs(phi);
  55        invert = true;
  56     }
  57
  58     T result;
  59
  60     if(phi >= tools::max_value<T>())
  61     {
  62        // Need to handle infinity as a special case:
  63        result = policies::raise_overflow_error<T>("boost::math::ellint_d<%1%>(%1%,%1%)", 0, pol);
  64     }
  65     else if(phi > 1 / tools::epsilon<T>())
  66     {
  67        // Phi is so large that phi%pi is necessarily zero (or garbage),
  68        // just return the second part of the duplication formula:
  69        result = 2 * phi * ellint_d_imp(k, pol) / constants::pi<T>();
  70     }
  71     else
  72     {
  73        // Carlson's algorithm works only for |phi| <= pi/2,
  74        // use the integrand's periodicity to normalize phi
  75        //
  76        T rphi = boost::math::tools::fmod_workaround(phi, T(constants::half_pi<T>()));
  77        T m = boost::math::round((phi - rphi) / constants::half_pi<T>());
  78        int s = 1;
  79        if(boost::math::tools::fmod_workaround(m, T(2)) > 0.5)
  80        {
  81           m += 1;
  82           s = -1;
  83           rphi = constants::half_pi<T>() - rphi;
  84        }
  85        T sinp = sin(rphi);
  86        T cosp = cos(rphi);
  87        T c = 1 / (sinp * sinp);
  88        T cm1 = cosp * cosp / (sinp * sinp);  // c - 1
  89        T k2 = k * k;
  90        if(k2 > 1)
  91        {
  92           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);
  93        }
  94        else if(rphi == 0)
  95        {
  96           result = 0;
  97        }
  98        else
  99        {
 100           // http://dlmf.nist.gov/19.25#E10
 101           result = s * ellint_rd_imp(cm1, T(c - k2), c, pol) / 3;
 102        }
 103        if(m != 0)
 104           result += m * ellint_d_imp(k, pol);
 105     }
 106     return invert ? T(-result) : result;
 107 }
 108
 109 // Complete elliptic integral (Legendre form) of the second kind
 110 template <typename T, typename Policy>
 111 T ellint_d_imp(T k, const Policy& pol)
 112 {
 113     BOOST_MATH_STD_USING
 114     using namespace boost::math::tools;
 115
 116     if (abs(k) >= 1)
 117     {
 118        return policies::raise_domain_error<T>("boost::math::ellint_d<%1%>(%1%)",
 119             "Got k = %1%, function requires |k| <= 1", k, pol);
 120     }
 121     if(fabs(k) <= tools::root_epsilon<T>())
 122        return constants::pi<T>() / 4;
 123
 124     T x = 0;
 125     T t = k * k;
 126     T y = 1 - t;
 127     T z = 1;
 128     T value = ellint_rd_imp(x, y, z, pol) / 3;
 129
 130     return value;
 131 }
 132
 133 template <typename T, typename Policy>
 134 inline typename tools::promote_args<T>::type ellint_d(T k, const Policy& pol, const mpl::true_&)
 135 {
 136    typedef typename tools::promote_args<T>::type result_type;
 137    typedef typename policies::evaluation<result_type, Policy>::type value_type;
 138    return policies::checked_narrowing_cast<result_type, Policy>(detail::ellint_d_imp(static_cast<value_type>(k), pol), "boost::math::ellint_d<%1%>(%1%)");
 139 }
 140
 141 // Elliptic integral (Legendre form) of the second kind
 142 template <class T1, class T2>
 143 inline typename tools::promote_args<T1, T2>::type ellint_d(T1 k, T2 phi, const mpl::false_&)
 144 {
 145    return boost::math::ellint_d(k, phi, policies::policy<>());
 146 }
 147
 148 } // detail
 149
 150 // Complete elliptic integral (Legendre form) of the second kind
 151 template <typename T>
 152 inline typename tools::promote_args<T>::type ellint_d(T k)
 153 {
 154    return ellint_d(k, policies::policy<>());
 155 }
 156
 157 // Elliptic integral (Legendre form) of the second kind
 158 template <class T1, class T2>
 159 inline typename tools::promote_args<T1, T2>::type ellint_d(T1 k, T2 phi)
 160 {
 161    typedef typename policies::is_policy<T2>::type tag_type;
 162    return detail::ellint_d(k, phi, tag_type());
 163 }
 164
 165 template <class T1, class T2, class Policy>
 166 inline typename tools::promote_args<T1, T2>::type ellint_d(T1 k, T2 phi, const Policy& pol)
 167 {
 168    typedef typename tools::promote_args<T1, T2>::type result_type;
 169    typedef typename policies::evaluation<result_type, Policy>::type value_type;
 170    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%)");
 171 }
 172
 173 }} // namespaces
 174
 175 #endif // BOOST_MATH_ELLINT_D_HPP
 176