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

   1 //  Copyright (c) 2006 Xiaogang Zhang, 2015 John Maddock
   2 //  Use, modification and distribution are subject to the
   3 //  Boost Software License, Version 1.0. (See accompanying file
   4 //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
   5 //
   6 //  History:
   7 //  XZ wrote the original of this file as part of the Google
   8 //  Summer of Code 2006.  JM modified it to fit into the
   9 //  Boost.Math conceptual framework better, and to handle
  10 //  types longer than 80-bit reals.
  11 //  Updated 2015 to use Carlson's latest methods.
  12 //
  13 #ifndef BOOST_MATH_ELLINT_RF_HPP
  14 #define BOOST_MATH_ELLINT_RF_HPP
  15
  16 #ifdef _MSC_VER
  17 #pragma once
  18 #endif
  19
  20 #include <boost/math/special_functions/math_fwd.hpp>
  21 #include <boost/math/tools/config.hpp>
  22 #include <boost/math/constants/constants.hpp>
  23 #include <boost/math/policies/error_handling.hpp>
  24 #include <boost/math/special_functions/ellint_rc.hpp>
  25
  26 // Carlson's elliptic integral of the first kind
  27 // R_F(x, y, z) = 0.5 * \int_{0}^{\infty} [(t+x)(t+y)(t+z)]^{-1/2} dt
  28 // Carlson, Numerische Mathematik, vol 33, 1 (1979)
  29
  30 namespace boost { namespace math { namespace detail{
  31
  32    template <typename T, typename Policy>
  33    T ellint_rf_imp(T x, T y, T z, const Policy& pol)
  34    {
  35       BOOST_MATH_STD_USING
  36       using namespace boost::math;
  37       using std::swap;
  38
  39       static const char* function = "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)";
  40
  41       if(x < 0 || y < 0 || z < 0)
  42       {
  43          return policies::raise_domain_error<T>(function,
  44             "domain error, all arguments must be non-negative, "
  45             "only sensible result is %1%.",
  46             std::numeric_limits<T>::quiet_NaN(), pol);
  47       }
  48       if(x + y == 0 || y + z == 0 || z + x == 0)
  49       {
  50          return policies::raise_domain_error<T>(function,
  51             "domain error, at most one argument can be zero, "
  52             "only sensible result is %1%.",
  53             std::numeric_limits<T>::quiet_NaN(), pol);
  54       }
  55       //
  56       // Special cases from http://dlmf.nist.gov/19.20#i
  57       //
  58       if(x == y)
  59       {
  60          if(x == z)
  61          {
  62             // x, y, z equal:
  63             return 1 / sqrt(x);
  64          }
  65          else
  66          {
  67             // 2 equal, x and y:
  68             if(z == 0)
  69                return constants::pi<T>() / (2 * sqrt(x));
  70             else
  71                return ellint_rc_imp(z, x, pol);
  72          }
  73       }
  74       if(x == z)
  75       {
  76          if(y == 0)
  77             return constants::pi<T>() / (2 * sqrt(x));
  78          else
  79             return ellint_rc_imp(y, x, pol);
  80       }
  81       if(y == z)
  82       {
  83          if(x == 0)
  84             return constants::pi<T>() / (2 * sqrt(y));
  85          else
  86             return ellint_rc_imp(x, y, pol);
  87       }
  88       if(x == 0)
  89          swap(x, z);
  90       else if(y == 0)
  91          swap(y, z);
  92       if(z == 0)
  93       {
  94          //
  95          // Special case for one value zero:
  96          //
  97          T xn = sqrt(x);
  98          T yn = sqrt(y);
  99
 100          while(fabs(xn - yn) >= 2.7 * tools::root_epsilon<T>() * fabs(xn))
 101          {
 102             T t = sqrt(xn * yn);
 103             xn = (xn + yn) / 2;
 104             yn = t;
 105          }
 106          return constants::pi<T>() / (xn + yn);
 107       }
 108
 109       T xn = x;
 110       T yn = y;
 111       T zn = z;
 112       T An = (x + y + z) / 3;
 113       T A0 = An;
 114       T Q = pow(3 * boost::math::tools::epsilon<T>(), T(-1) / 8) * (std::max)((std::max)(fabs(An - xn), fabs(An - yn)), fabs(An - zn));
 115       T fn = 1;
 116
 117
 118       // duplication
 119       unsigned k = 1;
 120       for(; k < boost::math::policies::get_max_series_iterations<Policy>(); ++k)
 121       {
 122          T root_x = sqrt(xn);
 123          T root_y = sqrt(yn);
 124          T root_z = sqrt(zn);
 125          T lambda = root_x * root_y + root_x * root_z + root_y * root_z;
 126          An = (An + lambda) / 4;
 127          xn = (xn + lambda) / 4;
 128          yn = (yn + lambda) / 4;
 129          zn = (zn + lambda) / 4;
 130          Q /= 4;
 131          fn *= 4;
 132          if(Q < fabs(An))
 133             break;
 134       }
 135       // Check to see if we gave up too soon:
 136       policies::check_series_iterations<T>(function, k, pol);
 137       BOOST_MATH_INSTRUMENT_VARIABLE(k);
 138
 139       T X = (A0 - x) / (An * fn);
 140       T Y = (A0 - y) / (An * fn);
 141       T Z = -X - Y;
 142
 143       // Taylor series expansion to the 7th order
 144       T E2 = X * Y - Z * Z;
 145       T E3 = X * Y * Z;
 146       return (1 + E3 * (T(1) / 14 + 3 * E3 / 104) + E2 * (T(-1) / 10 + E2 / 24 - (3 * E3) / 44 - 5 * E2 * E2 / 208 + E2 * E3 / 16)) / sqrt(An);
 147    }
 148
 149 } // namespace detail
 150
 151 template <class T1, class T2, class T3, class Policy>
 152 inline typename tools::promote_args<T1, T2, T3>::type
 153    ellint_rf(T1 x, T2 y, T3 z, const Policy& pol)
 154 {
 155    typedef typename tools::promote_args<T1, T2, T3>::type result_type;
 156    typedef typename policies::evaluation<result_type, Policy>::type value_type;
 157    return policies::checked_narrowing_cast<result_type, Policy>(
 158       detail::ellint_rf_imp(
 159          static_cast<value_type>(x),
 160          static_cast<value_type>(y),
 161          static_cast<value_type>(z), pol), "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)");
 162 }
 163
 164 template <class T1, class T2, class T3>
 165 inline typename tools::promote_args<T1, T2, T3>::type
 166    ellint_rf(T1 x, T2 y, T3 z)
 167 {
 168    return ellint_rf(x, y, z, policies::policy<>());
 169 }
 170
 171 }} // namespaces
 172
 173 #endif // BOOST_MATH_ELLINT_RF_HPP
 174