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imagerie/boost/math/special_functions/ellint_rg.hpp

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