math/special_functions/ellint_rg.hpp

0001 //  Copyright (c) 2015 John Maddock
0002 //  Use, modification and distribution are subject to the
0003 //  Boost Software License, Version 1.0. (See accompanying file
0004 //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
0005 //
0006 #ifndef BOOST_MATH_ELLINT_RG_HPP
0007 #define BOOST_MATH_ELLINT_RG_HPP
0008
0009 #ifdef _MSC_VER
0010 #pragma once
0011 #endif
0012
0013 #include <boost/math/tools/config.hpp>
0014 #include <boost/math/special_functions/math_fwd.hpp>
0015 #include <boost/math/constants/constants.hpp>
0016 #include <boost/math/policies/error_handling.hpp>
0017 #include <boost/math/special_functions/ellint_rd.hpp>
0018 #include <boost/math/special_functions/ellint_rf.hpp>
0019 #include <boost/math/special_functions/pow.hpp>
0020
0021 namespace boost { namespace math { namespace detail{
0022
0023    template <typename T, typename Policy>
0024    BOOST_MATH_GPU_ENABLED T ellint_rg_imp(T x, T y, T z, const Policy& pol)
0025    {
0026       BOOST_MATH_STD_USING
0027       constexpr auto function = "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)";
0028
0029       if(x < 0 || y < 0 || z < 0)
0030       {
0031          return policies::raise_domain_error<T>(function, "domain error, all arguments must be non-negative, only sensible result is %1%.", boost::math::numeric_limits<T>::quiet_NaN(), pol);
0032       }
0033       //
0034       // Function is symmetric in x, y and z, but we require
0035       // (x - z)(y - z) >= 0 to avoid cancellation error in the result
0036       // which implies (for example) x >= z >= y
0037       //
0038       if(x < y)
0039          BOOST_MATH_GPU_SAFE_SWAP(x, y);
0040       if(x < z)
0041          BOOST_MATH_GPU_SAFE_SWAP(x, z);
0042       if(y > z)
0043          BOOST_MATH_GPU_SAFE_SWAP(y, z);
0044
0045       BOOST_MATH_ASSERT(x >= z);
0046       BOOST_MATH_ASSERT(z >= y);
0047       //
0048       // Special cases from http://dlmf.nist.gov/19.20#ii
0049       //
0050       if(x == z)
0051       {
0052          if(y == z)
0053          {
0054             // x = y = z
0055             // This also works for x = y = z = 0 presumably.
0056             return sqrt(x);
0057          }
0058          else if(y == 0)
0059          {
0060             // x = y, z = 0
0061             return constants::pi<T>() * sqrt(x) / 4;
0062          }
0063          else
0064          {
0065             // x = z, y != 0
0066             BOOST_MATH_GPU_SAFE_SWAP(x, y);
0067             return (x == 0) ? T(sqrt(z) / 2) : T((z * ellint_rc_imp(x, z, pol) + sqrt(x)) / 2);
0068          }
0069       }
0070       else if(y == z)
0071       {
0072          BOOST_MATH_ASSERT(x > 0);  // Ordering of x,y,z above takes care of x == 0 case.
0073          return (y == 0) ? T(sqrt(x) / 2) : T((y * ellint_rc_imp(x, y, pol) + sqrt(x)) / 2);
0074       }
0075       else if(y == 0)
0076       {
0077          BOOST_MATH_GPU_SAFE_SWAP(y, z);
0078          //
0079          // Special handling for common case, from
0080          // Numerical Computation of Real or Complex Elliptic Integrals, eq.46
0081          //
0082          T xn = sqrt(x);
0083          T yn = sqrt(y);
0084          T x0 = xn;
0085          T y0 = yn;
0086          T sum = 0;
0087          T sum_pow = 0.25f;
0088
0089          while(fabs(xn - yn) >= T(2.7) * tools::root_epsilon<T>() * fabs(xn))
0090          {
0091             T t = sqrt(xn * yn);
0092             xn = (xn + yn) / 2;
0093             yn = t;
0094             sum_pow *= 2;
0095             sum += sum_pow * boost::math::pow<2>(xn - yn);
0096          }
0097          T RF = constants::pi<T>() / (xn + yn);
0098          return ((boost::math::pow<2>((x0 + y0) / 2) - sum) * RF) / 2;
0099       }
0100       return (z * ellint_rf_imp(x, y, z, pol)
0101          - (x - z) * (y - z) * ellint_rd_imp(x, y, z, pol) / 3
0102          + sqrt(x * y / z)) / 2;
0103    }
0104
0105 } // namespace detail
0106
0107 template <class T1, class T2, class T3, class Policy>
0108 BOOST_MATH_GPU_ENABLED inline typename tools::promote_args<T1, T2, T3>::type
0109    ellint_rg(T1 x, T2 y, T3 z, const Policy& pol)
0110 {
0111    typedef typename tools::promote_args<T1, T2, T3>::type result_type;
0112    typedef typename policies::evaluation<result_type, Policy>::type value_type;
0113    return policies::checked_narrowing_cast<result_type, Policy>(
0114       detail::ellint_rg_imp(
0115          static_cast<value_type>(x),
0116          static_cast<value_type>(y),
0117          static_cast<value_type>(z), pol), "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)");
0118 }
0119
0120 template <class T1, class T2, class T3>
0121 BOOST_MATH_GPU_ENABLED inline typename tools::promote_args<T1, T2, T3>::type
0122    ellint_rg(T1 x, T2 y, T3 z)
0123 {
0124    return ellint_rg(x, y, z, policies::policy<>());
0125 }
0126
0127 }} // namespaces
0128
0129 #endif // BOOST_MATH_ELLINT_RG_HPP
0130