ceph/src/boost/boost/math/special_functions/ellint_1.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_1_HPP
  15 #define BOOST_MATH_ELLINT_1_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/constants/constants.hpp>
  24 #include <boost/math/policies/error_handling.hpp>
  25 #include <boost/math/tools/workaround.hpp>
  26 #include <boost/math/special_functions/round.hpp>
  27
  28 // Elliptic integrals (complete and incomplete) of the first kind
  29 // Carlson, Numerische Mathematik, vol 33, 1 (1979)
  30
  31 namespace boost { namespace math {
  32
  33 template <class T1, class T2, class Policy>
  34 typename tools::promote_args<T1, T2>::type ellint_1(T1 k, T2 phi, const Policy& pol);
  35
  36 namespace detail{
  37
  38 template <typename T, typename Policy>
  39 T ellint_k_imp(T k, const Policy& pol);
  40
  41 // Elliptic integral (Legendre form) of the first kind
  42 template <typename T, typename Policy>
  43 T ellint_f_imp(T phi, T k, const Policy& pol)
  44 {
  45     BOOST_MATH_STD_USING
  46     using namespace boost::math::tools;
  47     using namespace boost::math::constants;
  48
  49     static const char* function = "boost::math::ellint_f<%1%>(%1%,%1%)";
  50     BOOST_MATH_INSTRUMENT_VARIABLE(phi);
  51     BOOST_MATH_INSTRUMENT_VARIABLE(k);
  52     BOOST_MATH_INSTRUMENT_VARIABLE(function);
  53
  54     bool invert = false;
  55     if(phi < 0)
  56     {
  57        BOOST_MATH_INSTRUMENT_VARIABLE(phi);
  58        phi = fabs(phi);
  59        invert = true;
  60     }
  61
  62     T result;
  63
  64     if(phi >= tools::max_value<T>())
  65     {
  66        // Need to handle infinity as a special case:
  67        result = policies::raise_overflow_error<T>(function, 0, pol);
  68        BOOST_MATH_INSTRUMENT_VARIABLE(result);
  69     }
  70     else if(phi > 1 / tools::epsilon<T>())
  71     {
  72        // Phi is so large that phi%pi is necessarily zero (or garbage),
  73        // just return the second part of the duplication formula:
  74        result = 2 * phi * ellint_k_imp(k, pol) / constants::pi<T>();
  75        BOOST_MATH_INSTRUMENT_VARIABLE(result);
  76     }
  77     else
  78     {
  79        // Carlson's algorithm works only for |phi| <= pi/2,
  80        // use the integrand's periodicity to normalize phi
  81        //
  82        // Xiaogang's original code used a cast to long long here
  83        // but that fails if T has more digits than a long long,
  84        // so rewritten to use fmod instead:
  85        //
  86        BOOST_MATH_INSTRUMENT_CODE("pi/2 = " << constants::pi<T>() / 2);
  87        T rphi = boost::math::tools::fmod_workaround(phi, T(constants::half_pi<T>()));
  88        BOOST_MATH_INSTRUMENT_VARIABLE(rphi);
  89        T m = boost::math::round((phi - rphi) / constants::half_pi<T>());
  90        BOOST_MATH_INSTRUMENT_VARIABLE(m);
  91        int s = 1;
  92        if(boost::math::tools::fmod_workaround(m, T(2)) > 0.5)
  93        {
  94           m += 1;
  95           s = -1;
  96           rphi = constants::half_pi<T>() - rphi;
  97           BOOST_MATH_INSTRUMENT_VARIABLE(rphi);
  98        }
  99        T sinp = sin(rphi);
 100        sinp *= sinp;
 101        if (sinp * k * k >= 1)
 102        {
 103           return policies::raise_domain_error<T>(function,
 104              "Got k^2 * sin^2(phi) = %1%, but the function requires this < 1", sinp * k * k, pol);
 105        }
 106        T cosp = cos(rphi);
 107        cosp *= cosp;
 108        BOOST_MATH_INSTRUMENT_VARIABLE(sinp);
 109        BOOST_MATH_INSTRUMENT_VARIABLE(cosp);
 110        if(sinp > tools::min_value<T>())
 111        {
 112           BOOST_ASSERT(rphi != 0); // precondition, can't be true if sin(rphi) != 0.
 113           //
 114           // Use http://dlmf.nist.gov/19.25#E5, note that
 115           // c-1 simplifies to cot^2(rphi) which avoid cancellation:
 116           //
 117           T c = 1 / sinp;
 118           result = static_cast<T>(s * ellint_rf_imp(T(cosp / sinp), T(c - k * k), c, pol));
 119        }
 120        else
 121           result = s * sin(rphi);
 122        BOOST_MATH_INSTRUMENT_VARIABLE(result);
 123        if(m != 0)
 124        {
 125           result += m * ellint_k_imp(k, pol);
 126           BOOST_MATH_INSTRUMENT_VARIABLE(result);
 127        }
 128     }
 129     return invert ? T(-result) : result;
 130 }
 131
 132 // Complete elliptic integral (Legendre form) of the first kind
 133 template <typename T, typename Policy>
 134 T ellint_k_imp(T k, const Policy& pol)
 135 {
 136     BOOST_MATH_STD_USING
 137     using namespace boost::math::tools;
 138
 139     static const char* function = "boost::math::ellint_k<%1%>(%1%)";
 140
 141     if (abs(k) > 1)
 142     {
 143        return policies::raise_domain_error<T>(function,
 144             "Got k = %1%, function requires |k| <= 1", k, pol);
 145     }
 146     if (abs(k) == 1)
 147     {
 148        return policies::raise_overflow_error<T>(function, 0, pol);
 149     }
 150
 151     T x = 0;
 152     T y = 1 - k * k;
 153     T z = 1;
 154     T value = ellint_rf_imp(x, y, z, pol);
 155
 156     return value;
 157 }
 158
 159 template <typename T, typename Policy>
 160 inline typename tools::promote_args<T>::type ellint_1(T k, const Policy& pol, const mpl::true_&)
 161 {
 162    typedef typename tools::promote_args<T>::type result_type;
 163    typedef typename policies::evaluation<result_type, Policy>::type value_type;
 164    return policies::checked_narrowing_cast<result_type, Policy>(detail::ellint_k_imp(static_cast<value_type>(k), pol), "boost::math::ellint_1<%1%>(%1%)");
 165 }
 166
 167 template <class T1, class T2>
 168 inline typename tools::promote_args<T1, T2>::type ellint_1(T1 k, T2 phi, const mpl::false_&)
 169 {
 170    return boost::math::ellint_1(k, phi, policies::policy<>());
 171 }
 172
 173 }
 174
 175 // Complete elliptic integral (Legendre form) of the first kind
 176 template <typename T>
 177 inline typename tools::promote_args<T>::type ellint_1(T k)
 178 {
 179    return ellint_1(k, policies::policy<>());
 180 }
 181
 182 // Elliptic integral (Legendre form) of the first kind
 183 template <class T1, class T2, class Policy>
 184 inline typename tools::promote_args<T1, T2>::type ellint_1(T1 k, T2 phi, const Policy& pol)
 185 {
 186    typedef typename tools::promote_args<T1, T2>::type result_type;
 187    typedef typename policies::evaluation<result_type, Policy>::type value_type;
 188    return policies::checked_narrowing_cast<result_type, Policy>(detail::ellint_f_imp(static_cast<value_type>(phi), static_cast<value_type>(k), pol), "boost::math::ellint_1<%1%>(%1%,%1%)");
 189 }
 190
 191 template <class T1, class T2>
 192 inline typename tools::promote_args<T1, T2>::type ellint_1(T1 k, T2 phi)
 193 {
 194    typedef typename policies::is_policy<T2>::type tag_type;
 195    return detail::ellint_1(k, phi, tag_type());
 196 }
 197
 198 }} // namespaces
 199
 200 #endif // BOOST_MATH_ELLINT_1_HPP
 201