ceph/src/boost/libs/math/include/boost/math/special_functions/ellint_rd.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 slightly to fit into the
   9 //  Boost.Math conceptual framework better.
  10 //  Updated 2015 to use Carlson's latest methods.
  11
  12 #ifndef BOOST_MATH_ELLINT_RD_HPP
  13 #define BOOST_MATH_ELLINT_RD_HPP
  14
  15 #ifdef _MSC_VER
  16 #pragma once
  17 #endif
  18
  19 #include <boost/math/special_functions/math_fwd.hpp>
  20 #include <boost/math/special_functions/ellint_rc.hpp>
  21 #include <boost/math/special_functions/pow.hpp>
  22 #include <boost/math/tools/config.hpp>
  23 #include <boost/math/policies/error_handling.hpp>
  24
  25 // Carlson's elliptic integral of the second kind
  26 // R_D(x, y, z) = R_J(x, y, z, z) = 1.5 * \int_{0}^{\infty} [(t+x)(t+y)]^{-1/2} (t+z)^{-3/2} dt
  27 // Carlson, Numerische Mathematik, vol 33, 1 (1979)
  28
  29 namespace boost { namespace math { namespace detail{
  30
  31 template <typename T, typename Policy>
  32 T ellint_rd_imp(T x, T y, T z, const Policy& pol)
  33 {
  34    BOOST_MATH_STD_USING
  35    using std::swap;
  36
  37    static const char* function = "boost::math::ellint_rd<%1%>(%1%,%1%,%1%)";
  38
  39    if(x < 0)
  40    {
  41       return policies::raise_domain_error<T>(function,
  42          "Argument x must be >= 0, but got %1%", x, pol);
  43    }
  44    if(y < 0)
  45    {
  46       return policies::raise_domain_error<T>(function,
  47          "Argument y must be >= 0, but got %1%", y, pol);
  48    }
  49    if(z <= 0)
  50    {
  51       return policies::raise_domain_error<T>(function,
  52          "Argument z must be > 0, but got %1%", z, pol);
  53    }
  54    if(x + y == 0)
  55    {
  56       return policies::raise_domain_error<T>(function,
  57          "At most one argument can be zero, but got, x + y = %1%", x + y, pol);
  58    }
  59    //
  60    // Special cases from http://dlmf.nist.gov/19.20#iv
  61    //
  62    using std::swap;
  63    if(x == z)
  64       swap(x, y);
  65    if(y == z)
  66    {
  67       if(x == y)
  68       {
  69          return 1 / (x * sqrt(x));
  70       }
  71       else if(x == 0)
  72       {
  73          return 3 * constants::pi<T>() / (4 * y * sqrt(y));
  74       }
  75       else
  76       {
  77          if((std::min)(x, y) / (std::max)(x, y) > 1.3)
  78             return 3 * (ellint_rc_imp(x, y, pol) - sqrt(x) / y) / (2 * (y - x));
  79          // Otherwise fall through to avoid cancellation in the above (RC(x,y) -> 1/x^0.5 as x -> y)
  80       }
  81    }
  82    if(x == y)
  83    {
  84       if((std::min)(x, z) / (std::max)(x, z) > 1.3)
  85          return 3 * (ellint_rc_imp(z, x, pol) - 1 / sqrt(z)) / (z - x);
  86       // Otherwise fall through to avoid cancellation in the above (RC(x,y) -> 1/x^0.5 as x -> y)
  87    }
  88    if(y == 0)
  89       swap(x, y);
  90    if(x == 0)
  91    {
  92       //
  93       // Special handling for common case, from
  94       // Numerical Computation of Real or Complex Elliptic Integrals, eq.47
  95       //
  96       T xn = sqrt(y);
  97       T yn = sqrt(z);
  98       T x0 = xn;
  99       T y0 = yn;
 100       T sum = 0;
 101       T sum_pow = 0.25f;
 102
 103       while(fabs(xn - yn) >= 2.7 * tools::root_epsilon<T>() * fabs(xn))
 104       {
 105          T t = sqrt(xn * yn);
 106          xn = (xn + yn) / 2;
 107          yn = t;
 108          sum_pow *= 2;
 109          sum += sum_pow * boost::math::pow<2>(xn - yn);
 110       }
 111       T RF = constants::pi<T>() / (xn + yn);
 112       //
 113       // This following calculation suffers from serious cancellation when y ~ z
 114       // unless we combine terms.  We have:
 115       //
 116       // ( ((x0 + y0)/2)^2 - z ) / (z(y-z))
 117       //
 118       // Substituting y = x0^2 and z = y0^2 and simplifying we get the following:
 119       //
 120       T pt = (x0 + 3 * y0) / (4 * z * (x0 + y0));
 121       //
 122       // Since we've moved the demoninator from eq.47 inside the expression, we
 123       // need to also scale "sum" by the same value:
 124       //
 125       pt -= sum / (z * (y - z));
 126       return pt * RF * 3;
 127    }
 128
 129    T xn = x;
 130    T yn = y;
 131    T zn = z;
 132    T An = (x + y + 3 * z) / 5;
 133    T A0 = An;
 134    // This has an extra 1.2 fudge factor which is really only needed when x, y and z are close in magnitude:
 135    T Q = pow(tools::epsilon<T>() / 4, -T(1) / 8) * (std::max)((std::max)(An - x, An - y), An - z) * 1.2f;
 136    T lambda, rx, ry, rz;
 137    unsigned k = 0;
 138    T fn = 1;
 139    T RD_sum = 0;
 140
 141    for(; k < policies::get_max_series_iterations<Policy>(); ++k)
 142    {
 143       rx = sqrt(xn);
 144       ry = sqrt(yn);
 145       rz = sqrt(zn);
 146       lambda = rx * ry + rx * rz + ry * rz;
 147       RD_sum += fn / (rz * (zn + lambda));
 148       An = (An + lambda) / 4;
 149       xn = (xn + lambda) / 4;
 150       yn = (yn + lambda) / 4;
 151       zn = (zn + lambda) / 4;
 152       fn /= 4;
 153       Q /= 4;
 154       if(Q < An)
 155          break;
 156    }
 157
 158    policies::check_series_iterations<T, Policy>(function, k, pol);
 159
 160    T X = fn * (A0 - x) / An;
 161    T Y = fn * (A0 - y) / An;
 162    T Z = -(X + Y) / 3;
 163    T E2 = X * Y - 6 * Z * Z;
 164    T E3 = (3 * X * Y - 8 * Z * Z) * Z;
 165    T E4 = 3 * (X * Y - Z * Z) * Z * Z;
 166    T E5 = X * Y * Z * Z * Z;
 167
 168    T result = fn * pow(An, T(-3) / 2) *
 169       (1 - 3 * E2 / 14 + E3 / 6 + 9 * E2 * E2 / 88 - 3 * E4 / 22 - 9 * E2 * E3 / 52 + 3 * E5 / 26 - E2 * E2 * E2 / 16
 170       + 3 * E3 * E3 / 40 + 3 * E2 * E4 / 20 + 45 * E2 * E2 * E3 / 272 - 9 * (E3 * E4 + E2 * E5) / 68);
 171    result += 3 * RD_sum;
 172
 173    return result;
 174 }
 175
 176 } // namespace detail
 177
 178 template <class T1, class T2, class T3, class Policy>
 179 inline typename tools::promote_args<T1, T2, T3>::type
 180    ellint_rd(T1 x, T2 y, T3 z, const Policy& pol)
 181 {
 182    typedef typename tools::promote_args<T1, T2, T3>::type result_type;
 183    typedef typename policies::evaluation<result_type, Policy>::type value_type;
 184    return policies::checked_narrowing_cast<result_type, Policy>(
 185       detail::ellint_rd_imp(
 186          static_cast<value_type>(x),
 187          static_cast<value_type>(y),
 188          static_cast<value_type>(z), pol), "boost::math::ellint_rd<%1%>(%1%,%1%,%1%)");
 189 }
 190
 191 template <class T1, class T2, class T3>
 192 inline typename tools::promote_args<T1, T2, T3>::type
 193    ellint_rd(T1 x, T2 y, T3 z)
 194 {
 195    return ellint_rd(x, y, z, policies::policy<>());
 196 }
 197
 198 }} // namespaces
 199
 200 #endif // BOOST_MATH_ELLINT_RD_HPP
 201