Chris@101: //  Copyright (c) 2006 Xiaogang Zhang, 2015 John Maddock
Chris@16: //  Use, modification and distribution are subject to the
Chris@16: //  Boost Software License, Version 1.0. (See accompanying file
Chris@16: //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@16: //
Chris@16: //  History:
Chris@16: //  XZ wrote the original of this file as part of the Google
Chris@16: //  Summer of Code 2006.  JM modified it to fit into the
Chris@16: //  Boost.Math conceptual framework better, and to handle
Chris@16: //  types longer than 80-bit reals.
Chris@101: //  Updated 2015 to use Carlson's latest methods.
Chris@16: //
Chris@16: #ifndef BOOST_MATH_ELLINT_RF_HPP
Chris@16: #define BOOST_MATH_ELLINT_RF_HPP
Chris@16: 
Chris@16: #ifdef _MSC_VER
Chris@16: #pragma once
Chris@16: #endif
Chris@16: 
Chris@16: #include <boost/math/special_functions/math_fwd.hpp>
Chris@16: #include <boost/math/tools/config.hpp>
Chris@101: #include <boost/math/constants/constants.hpp>
Chris@16: #include <boost/math/policies/error_handling.hpp>
Chris@101: #include <boost/math/special_functions/ellint_rc.hpp>
Chris@16: 
Chris@16: // Carlson's elliptic integral of the first kind
Chris@16: // R_F(x, y, z) = 0.5 * \int_{0}^{\infty} [(t+x)(t+y)(t+z)]^{-1/2} dt
Chris@16: // Carlson, Numerische Mathematik, vol 33, 1 (1979)
Chris@16: 
Chris@16: namespace boost { namespace math { namespace detail{
Chris@16: 
Chris@101:    template <typename T, typename Policy>
Chris@101:    T ellint_rf_imp(T x, T y, T z, const Policy& pol)
Chris@101:    {
Chris@101:       BOOST_MATH_STD_USING
Chris@101:       using namespace boost::math;
Chris@101:       using std::swap;
Chris@16: 
Chris@101:       static const char* function = "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)";
Chris@16: 
Chris@101:       if(x < 0 || y < 0 || z < 0)
Chris@101:       {
Chris@101:          return policies::raise_domain_error<T>(function,
Chris@16:             "domain error, all arguments must be non-negative, "
Chris@16:             "only sensible result is %1%.",
Chris@16:             std::numeric_limits<T>::quiet_NaN(), pol);
Chris@101:       }
Chris@101:       if(x + y == 0 || y + z == 0 || z + x == 0)
Chris@101:       {
Chris@101:          return policies::raise_domain_error<T>(function,
Chris@16:             "domain error, at most one argument can be zero, "
Chris@16:             "only sensible result is %1%.",
Chris@16:             std::numeric_limits<T>::quiet_NaN(), pol);
Chris@101:       }
Chris@101:       //
Chris@101:       // Special cases from http://dlmf.nist.gov/19.20#i
Chris@101:       //
Chris@101:       if(x == y)
Chris@101:       {
Chris@101:          if(x == z)
Chris@101:          {
Chris@101:             // x, y, z equal:
Chris@101:             return 1 / sqrt(x);
Chris@101:          }
Chris@101:          else
Chris@101:          {
Chris@101:             // 2 equal, x and y:
Chris@101:             if(z == 0)
Chris@101:                return constants::pi<T>() / (2 * sqrt(x));
Chris@101:             else
Chris@101:                return ellint_rc_imp(z, x, pol);
Chris@101:          }
Chris@101:       }
Chris@101:       if(x == z)
Chris@101:       {
Chris@101:          if(y == 0)
Chris@101:             return constants::pi<T>() / (2 * sqrt(x));
Chris@101:          else
Chris@101:             return ellint_rc_imp(y, x, pol);
Chris@101:       }
Chris@101:       if(y == z)
Chris@101:       {
Chris@101:          if(x == 0)
Chris@101:             return constants::pi<T>() / (2 * sqrt(y));
Chris@101:          else
Chris@101:             return ellint_rc_imp(x, y, pol);
Chris@101:       }
Chris@101:       if(x == 0)
Chris@101:          swap(x, z);
Chris@101:       else if(y == 0)
Chris@101:          swap(y, z);
Chris@101:       if(z == 0)
Chris@101:       {
Chris@101:          //
Chris@101:          // Special case for one value zero:
Chris@101:          //
Chris@101:          T xn = sqrt(x);
Chris@101:          T yn = sqrt(y);
Chris@16: 
Chris@101:          while(fabs(xn - yn) >= 2.7 * tools::root_epsilon<T>() * fabs(xn))
Chris@101:          {
Chris@101:             T t = sqrt(xn * yn);
Chris@101:             xn = (xn + yn) / 2;
Chris@101:             yn = t;
Chris@101:          }
Chris@101:          return constants::pi<T>() / (xn + yn);
Chris@101:       }
Chris@16: 
Chris@101:       T xn = x;
Chris@101:       T yn = y;
Chris@101:       T zn = z;
Chris@101:       T An = (x + y + z) / 3;
Chris@101:       T A0 = An;
Chris@101:       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));
Chris@101:       T fn = 1;
Chris@16: 
Chris@16: 
Chris@101:       // duplication
Chris@101:       unsigned k = 1;
Chris@101:       for(; k < boost::math::policies::get_max_series_iterations<Policy>(); ++k)
Chris@101:       {
Chris@101:          T root_x = sqrt(xn);
Chris@101:          T root_y = sqrt(yn);
Chris@101:          T root_z = sqrt(zn);
Chris@101:          T lambda = root_x * root_y + root_x * root_z + root_y * root_z;
Chris@101:          An = (An + lambda) / 4;
Chris@101:          xn = (xn + lambda) / 4;
Chris@101:          yn = (yn + lambda) / 4;
Chris@101:          zn = (zn + lambda) / 4;
Chris@101:          Q /= 4;
Chris@101:          fn *= 4;
Chris@101:          if(Q < fabs(An))
Chris@101:             break;
Chris@101:       }
Chris@101:       // Check to see if we gave up too soon:
Chris@101:       policies::check_series_iterations<T>(function, k, pol);
Chris@101:       BOOST_MATH_INSTRUMENT_VARIABLE(k);
Chris@16: 
Chris@101:       T X = (A0 - x) / (An * fn);
Chris@101:       T Y = (A0 - y) / (An * fn);
Chris@101:       T Z = -X - Y;
Chris@16: 
Chris@101:       // Taylor series expansion to the 7th order
Chris@101:       T E2 = X * Y - Z * Z;
Chris@101:       T E3 = X * Y * Z;
Chris@101:       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);
Chris@101:    }
Chris@16: 
Chris@16: } // namespace detail
Chris@16: 
Chris@16: template <class T1, class T2, class T3, class Policy>
Chris@16: inline typename tools::promote_args<T1, T2, T3>::type 
Chris@16:    ellint_rf(T1 x, T2 y, T3 z, const Policy& pol)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T1, T2, T3>::type result_type;
Chris@16:    typedef typename policies::evaluation<result_type, Policy>::type value_type;
Chris@16:    return policies::checked_narrowing_cast<result_type, Policy>(
Chris@16:       detail::ellint_rf_imp(
Chris@16:          static_cast<value_type>(x),
Chris@16:          static_cast<value_type>(y),
Chris@16:          static_cast<value_type>(z), pol), "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)");
Chris@16: }
Chris@16: 
Chris@16: template <class T1, class T2, class T3>
Chris@16: inline typename tools::promote_args<T1, T2, T3>::type 
Chris@16:    ellint_rf(T1 x, T2 y, T3 z)
Chris@16: {
Chris@16:    return ellint_rf(x, y, z, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: }} // namespaces
Chris@16: 
Chris@16: #endif // BOOST_MATH_ELLINT_RF_HPP
Chris@16: