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        return 0;
  54
  55     if(phi < 0)
  56     {
  57        phi = fabs(phi);
  58        invert = true;
  59     }
  60
  61     T result;
  62
  63     if(phi >= tools::max_value<T>())
  64     {
  65        // Need to handle infinity as a special case:
  66        result = policies::raise_overflow_error<T>("boost::math::ellint_e<%1%>(%1%,%1%)", 0, pol);
  67     }
  68     else if(phi > 1 / tools::epsilon<T>())
  69     {
  70        // Phi is so large that phi%pi is necessarily zero (or garbage),
  71        // just return the second part of the duplication formula:
  72        result = 2 * phi * ellint_e_imp(k, pol) / constants::pi<T>();
  73     }
  74     else if(k == 0)
  75     {
  76        return invert ? T(-phi) : phi;
  77     }
  78     else if(fabs(k) == 1)
  79     {
  80        //
  81        // For k = 1 ellipse actually turns to a line and every pi/2 in phi is exactly 1 in arc length
  82        // Periodicity though is in pi, curve follows sin(pi) for 0 <= phi <= pi/2 and then
  83        // 2 - sin(pi- phi) = 2 + sin(phi - pi) for pi/2 <= phi <= pi, so general form is:
  84        //
  85        // 2n + sin(phi - n * pi) ; |phi - n * pi| <= pi / 2
  86        //
  87        T m = boost::math::round(phi / boost::math::constants::pi<T>());
  88        T remains = phi - m * boost::math::constants::pi<T>();
  89        T value = 2 * m + sin(remains);
  90
  91        // negative arc length for negative phi
  92        return invert ? -value : value;
  93     }
  94     else
  95     {
  96        // Carlson's algorithm works only for |phi| <= pi/2,
  97        // use the integrand's periodicity to normalize phi
  98        //
  99        // Xiaogang's original code used a cast to long long here
 100        // but that fails if T has more digits than a long long,
 101        // so rewritten to use fmod instead:
 102        //
 103        T rphi = boost::math::tools::fmod_workaround(phi, T(constants::half_pi<T>()));
 104        T m = boost::math::round((phi - rphi) / constants::half_pi<T>());
 105        int s = 1;
 106        if(boost::math::tools::fmod_workaround(m, T(2)) > 0.5)
 107        {
 108           m += 1;
 109           s = -1;
 110           rphi = constants::half_pi<T>() - rphi;
 111        }
 112        T k2 = k * k;
 113        if(boost::math::pow<3>(rphi) * k2 / 6 < tools::epsilon<T>() * fabs(rphi))
 114        {
 115           // See http://functions.wolfram.com/EllipticIntegrals/EllipticE2/06/01/03/0001/
 116           result = s * rphi;
 117        }
 118        else
 119        {
 120           // http://dlmf.nist.gov/19.25#E10
 121           T sinp = sin(rphi);
 122           if (k2 * sinp * sinp >= 1)
 123           {
 124              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);
 125           }
 126           T cosp = cos(rphi);
 127           T c = 1 / (sinp * sinp);
 128           T cm1 = cosp * cosp / (sinp * sinp);  // c - 1
 129           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))));
 130        }
 131        if(m != 0)
 132           result += m * ellint_e_imp(k, pol);
 133     }
 134     return invert ? T(-result) : result;
 135 }
 136
 137 // Complete elliptic integral (Legendre form) of the second kind
 138 template <typename T, typename Policy>
 139 T ellint_e_imp(T k, const Policy& pol)
 140 {
 141     BOOST_MATH_STD_USING
 142     using namespace boost::math::tools;
 143
 144     if (abs(k) > 1)
 145     {
 146        return policies::raise_domain_error<T>("boost::math::ellint_e<%1%>(%1%)",
 147             "Got k = %1%, function requires |k| <= 1", k, pol);
 148     }
 149     if (abs(k) == 1)
 150     {
 151         return static_cast<T>(1);
 152     }
 153
 154     T x = 0;
 155     T t = k * k;
 156     T y = 1 - t;
 157     T z = 1;
 158     T value = 2 * ellint_rg_imp(x, y, z, pol);
 159
 160     return value;
 161 }
 162
 163 template <typename T, typename Policy>
 164 inline typename tools::promote_args<T>::type ellint_2(T k, const Policy& pol, const std::true_type&)
 165 {
 166    typedef typename tools::promote_args<T>::type result_type;
 167    typedef typename policies::evaluation<result_type, Policy>::type value_type;
 168    return policies::checked_narrowing_cast<result_type, Policy>(detail::ellint_e_imp(static_cast<value_type>(k), pol), "boost::math::ellint_2<%1%>(%1%)");
 169 }
 170
 171 // Elliptic integral (Legendre form) of the second kind
 172 template <class T1, class T2>
 173 inline typename tools::promote_args<T1, T2>::type ellint_2(T1 k, T2 phi, const std::false_type&)
 174 {
 175    return boost::math::ellint_2(k, phi, policies::policy<>());
 176 }
 177
 178 } // detail
 179
 180 // Complete elliptic integral (Legendre form) of the second kind
 181 template <typename T>
 182 inline typename tools::promote_args<T>::type ellint_2(T k)
 183 {
 184    return ellint_2(k, policies::policy<>());
 185 }
 186
 187 // Elliptic integral (Legendre form) of the second kind
 188 template <class T1, class T2>
 189 inline typename tools::promote_args<T1, T2>::type ellint_2(T1 k, T2 phi)
 190 {
 191    typedef typename policies::is_policy<T2>::type tag_type;
 192    return detail::ellint_2(k, phi, tag_type());
 193 }
 194
 195 template <class T1, class T2, class Policy>
 196 inline typename tools::promote_args<T1, T2>::type ellint_2(T1 k, T2 phi, const Policy& pol)
 197 {
 198    typedef typename tools::promote_args<T1, T2>::type result_type;
 199    typedef typename policies::evaluation<result_type, Policy>::type value_type;
 200    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%)");
 201 }
 202
 203 }} // namespaces
 204
 205 #endif // BOOST_MATH_ELLINT_2_HPP
 206