ceph/src/boost/boost/math/special_functions/ellint_2.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_2_HPP
  15 #define BOOST_MATH_ELLINT_2_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_2(T1 k, T2 phi, const Policy& pol);
  37
  38 namespace detail{
  39
  40 template <typename T, typename Policy>
  41 T ellint_e_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_e_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_e<%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_e_imp(k, pol) / constants::pi<T>();
  70     }
  71     else if(k == 0)
  72     {
  73        return invert ? T(-phi) : phi;
  74     }
  75     else if(fabs(k) == 1)
  76     {
  77        return invert ? T(-sin(phi)) : T(sin(phi));
  78     }
  79     else
  80     {
  81        // Carlson's algorithm works only for |phi| <= pi/2,
  82        // use the integrand's periodicity to normalize phi
  83        //
  84        // Xiaogang's original code used a cast to long long here
  85        // but that fails if T has more digits than a long long,
  86        // so rewritten to use fmod instead:
  87        //
  88        T rphi = boost::math::tools::fmod_workaround(phi, T(constants::half_pi<T>()));
  89        T m = boost::math::round((phi - rphi) / constants::half_pi<T>());
  90        int s = 1;
  91        if(boost::math::tools::fmod_workaround(m, T(2)) > 0.5)
  92        {
  93           m += 1;
  94           s = -1;
  95           rphi = constants::half_pi<T>() - rphi;
  96        }
  97        T k2 = k * k;
  98        if(k2 > 1)
  99        {
 100           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);
 101        }
 102        else if(rphi < tools::root_epsilon<T>())
 103        {
 104           // See http://functions.wolfram.com/EllipticIntegrals/EllipticE2/06/01/03/0001/
 105           result = s * rphi;
 106        }
 107        else
 108        {
 109           // http://dlmf.nist.gov/19.25#E10
 110           T sinp = sin(rphi);
 111           T cosp = cos(rphi);
 112           T c = 1 / (sinp * sinp);
 113           T cm1 = cosp * cosp / (sinp * sinp);  // c - 1
 114           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))));
 115        }
 116        if(m != 0)
 117           result += m * ellint_e_imp(k, pol);
 118     }
 119     return invert ? T(-result) : result;
 120 }
 121
 122 // Complete elliptic integral (Legendre form) of the second kind
 123 template <typename T, typename Policy>
 124 T ellint_e_imp(T k, const Policy& pol)
 125 {
 126     BOOST_MATH_STD_USING
 127     using namespace boost::math::tools;
 128
 129     if (abs(k) > 1)
 130     {
 131        return policies::raise_domain_error<T>("boost::math::ellint_e<%1%>(%1%)",
 132             "Got k = %1%, function requires |k| <= 1", k, pol);
 133     }
 134     if (abs(k) == 1)
 135     {
 136         return static_cast<T>(1);
 137     }
 138
 139     T x = 0;
 140     T t = k * k;
 141     T y = 1 - t;
 142     T z = 1;
 143     T value = 2 * ellint_rg_imp(x, y, z, pol);
 144
 145     return value;
 146 }
 147
 148 template <typename T, typename Policy>
 149 inline typename tools::promote_args<T>::type ellint_2(T k, const Policy& pol, const mpl::true_&)
 150 {
 151    typedef typename tools::promote_args<T>::type result_type;
 152    typedef typename policies::evaluation<result_type, Policy>::type value_type;
 153    return policies::checked_narrowing_cast<result_type, Policy>(detail::ellint_e_imp(static_cast<value_type>(k), pol), "boost::math::ellint_2<%1%>(%1%)");
 154 }
 155
 156 // Elliptic integral (Legendre form) of the second kind
 157 template <class T1, class T2>
 158 inline typename tools::promote_args<T1, T2>::type ellint_2(T1 k, T2 phi, const mpl::false_&)
 159 {
 160    return boost::math::ellint_2(k, phi, policies::policy<>());
 161 }
 162
 163 } // detail
 164
 165 // Complete elliptic integral (Legendre form) of the second kind
 166 template <typename T>
 167 inline typename tools::promote_args<T>::type ellint_2(T k)
 168 {
 169    return ellint_2(k, policies::policy<>());
 170 }
 171
 172 // Elliptic integral (Legendre form) of the second kind
 173 template <class T1, class T2>
 174 inline typename tools::promote_args<T1, T2>::type ellint_2(T1 k, T2 phi)
 175 {
 176    typedef typename policies::is_policy<T2>::type tag_type;
 177    return detail::ellint_2(k, phi, tag_type());
 178 }
 179
 180 template <class T1, class T2, class Policy>
 181 inline typename tools::promote_args<T1, T2>::type ellint_2(T1 k, T2 phi, const Policy& pol)
 182 {
 183    typedef typename tools::promote_args<T1, T2>::type result_type;
 184    typedef typename policies::evaluation<result_type, Policy>::type value_type;
 185    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%)");
 186 }
 187
 188 }} // namespaces
 189
 190 #endif // BOOST_MATH_ELLINT_2_HPP
 191