Chris@102: //  Copyright (c) 2015 John Maddock
Chris@102: //  Use, modification and distribution are subject to the
Chris@102: //  Boost Software License, Version 1.0. (See accompanying file
Chris@102: //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@102: //
Chris@102: #ifndef BOOST_MATH_ELLINT_RG_HPP
Chris@102: #define BOOST_MATH_ELLINT_RG_HPP
Chris@102: 
Chris@102: #ifdef _MSC_VER
Chris@102: #pragma once
Chris@102: #endif
Chris@102: 
Chris@102: #include <boost/math/special_functions/math_fwd.hpp>
Chris@102: #include <boost/math/tools/config.hpp>
Chris@102: #include <boost/math/constants/constants.hpp>
Chris@102: #include <boost/math/policies/error_handling.hpp>
Chris@102: #include <boost/math/special_functions/ellint_rd.hpp>
Chris@102: #include <boost/math/special_functions/ellint_rf.hpp>
Chris@102: #include <boost/math/special_functions/pow.hpp>
Chris@102: 
Chris@102: namespace boost { namespace math { namespace detail{
Chris@102: 
Chris@102:    template <typename T, typename Policy>
Chris@102:    T ellint_rg_imp(T x, T y, T z, const Policy& pol)
Chris@102:    {
Chris@102:       BOOST_MATH_STD_USING
Chris@102:       static const char* function = "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)";
Chris@102: 
Chris@102:       if(x < 0 || y < 0 || z < 0)
Chris@102:       {
Chris@102:          return policies::raise_domain_error<T>(function,
Chris@102:             "domain error, all arguments must be non-negative, "
Chris@102:             "only sensible result is %1%.",
Chris@102:             std::numeric_limits<T>::quiet_NaN(), pol);
Chris@102:       }
Chris@102:       //
Chris@102:       // Function is symmetric in x, y and z, but we require
Chris@102:       // (x - z)(y - z) >= 0 to avoid cancellation error in the result
Chris@102:       // which implies (for example) x >= z >= y
Chris@102:       //
Chris@102:       using std::swap;
Chris@102:       if(x < y)
Chris@102:          swap(x, y);
Chris@102:       if(x < z)
Chris@102:          swap(x, z);
Chris@102:       if(y > z)
Chris@102:          swap(y, z);
Chris@102:       
Chris@102:       BOOST_ASSERT(x >= z);
Chris@102:       BOOST_ASSERT(z >= y);
Chris@102:       //
Chris@102:       // Special cases from http://dlmf.nist.gov/19.20#ii
Chris@102:       //
Chris@102:       if(x == z)
Chris@102:       {
Chris@102:          if(y == z)
Chris@102:          {
Chris@102:             // x = y = z
Chris@102:             // This also works for x = y = z = 0 presumably.
Chris@102:             return sqrt(x);
Chris@102:          }
Chris@102:          else if(y == 0)
Chris@102:          {
Chris@102:             // x = y, z = 0
Chris@102:             return constants::pi<T>() * sqrt(x) / 4;
Chris@102:          }
Chris@102:          else
Chris@102:          {
Chris@102:             // x = z, y != 0
Chris@102:             swap(x, y);
Chris@102:             return (x == 0) ? T(sqrt(z) / 2) : T((z * ellint_rc_imp(x, z, pol) + sqrt(x)) / 2);
Chris@102:          }
Chris@102:       }
Chris@102:       else if(y == z)
Chris@102:       {
Chris@102:          if(x == 0)
Chris@102:             return constants::pi<T>() * sqrt(y) / 4;
Chris@102:          else
Chris@102:             return (y == 0) ? T(sqrt(x) / 2) : T((y * ellint_rc_imp(x, y, pol) + sqrt(x)) / 2);
Chris@102:       }
Chris@102:       else if(y == 0)
Chris@102:       {
Chris@102:          swap(y, z);
Chris@102:          //
Chris@102:          // Special handling for common case, from
Chris@102:          // Numerical Computation of Real or Complex Elliptic Integrals, eq.46
Chris@102:          //
Chris@102:          T xn = sqrt(x);
Chris@102:          T yn = sqrt(y);
Chris@102:          T x0 = xn;
Chris@102:          T y0 = yn;
Chris@102:          T sum = 0;
Chris@102:          T sum_pow = 0.25f;
Chris@102: 
Chris@102:          while(fabs(xn - yn) >= 2.7 * tools::root_epsilon<T>() * fabs(xn))
Chris@102:          {
Chris@102:             T t = sqrt(xn * yn);
Chris@102:             xn = (xn + yn) / 2;
Chris@102:             yn = t;
Chris@102:             sum_pow *= 2;
Chris@102:             sum += sum_pow * boost::math::pow<2>(xn - yn);
Chris@102:          }
Chris@102:          T RF = constants::pi<T>() / (xn + yn);
Chris@102:          return ((boost::math::pow<2>((x0 + y0) / 2) - sum) * RF) / 2;
Chris@102:       }
Chris@102:       return (z * ellint_rf_imp(x, y, z, pol)
Chris@102:          - (x - z) * (y - z) * ellint_rd_imp(x, y, z, pol) / 3
Chris@102:          + sqrt(x * y / z)) / 2;
Chris@102:    }
Chris@102: 
Chris@102: } // namespace detail
Chris@102: 
Chris@102: template <class T1, class T2, class T3, class Policy>
Chris@102: inline typename tools::promote_args<T1, T2, T3>::type 
Chris@102:    ellint_rg(T1 x, T2 y, T3 z, const Policy& pol)
Chris@102: {
Chris@102:    typedef typename tools::promote_args<T1, T2, T3>::type result_type;
Chris@102:    typedef typename policies::evaluation<result_type, Policy>::type value_type;
Chris@102:    return policies::checked_narrowing_cast<result_type, Policy>(
Chris@102:       detail::ellint_rg_imp(
Chris@102:          static_cast<value_type>(x),
Chris@102:          static_cast<value_type>(y),
Chris@102:          static_cast<value_type>(z), pol), "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)");
Chris@102: }
Chris@102: 
Chris@102: template <class T1, class T2, class T3>
Chris@102: inline typename tools::promote_args<T1, T2, T3>::type 
Chris@102:    ellint_rg(T1 x, T2 y, T3 z)
Chris@102: {
Chris@102:    return ellint_rg(x, y, z, policies::policy<>());
Chris@102: }
Chris@102: 
Chris@102: }} // namespaces
Chris@102: 
Chris@102: #endif // BOOST_MATH_ELLINT_RG_HPP
Chris@102: