//  Copyright (c) 2006 Xiaogang Zhang

//  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)

//

//  History:

//  XZ wrote the original of this file as part of the Google

//  Summer of Code 2006.  JM modified it to fit into the

//  Boost.Math conceptual framework better, and to handle

//  types longer than 80-bit reals.

//

#ifndef BOOST_MATH_ELLINT_RF_HPP

#define BOOST_MATH_ELLINT_RF_HPP



#include <boost/math/special_functions/math_fwd.hpp>

#include <boost/math/tools/config.hpp>



#include <boost/math/policies/error_handling.hpp>



// Carlson's elliptic integral of the first kind

// R_F(x, y, z) = 0.5 * \int_{0}^{\infty} [(t+x)(t+y)(t+z)]^{-1/2} dt

// Carlson, Numerische Mathematik, vol 33, 1 (1979)



namespace boost { namespace math { namespace detail{



template <typename T, typename Policy>

T ellint_rf_imp(T x, T y, T z, const Policy& pol)

{

    T value, X, Y, Z, E2, E3, u, lambda, tolerance;

    unsigned long k;



    BOOST_MATH_STD_USING

    using namespace boost::math::tools;



    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);

    }

    if (x + y == 0 || y + z == 0 || z + x == 0)

    {

       return policies::raise_domain_error<T>(function,

            "domain error, at most one argument can be zero, "

            "only sensible result is %1%.",

            std::numeric_limits<T>::quiet_NaN(), pol);

    }



    // Carlson scales error as the 6th power of tolerance,

    // but this seems not to work for types larger than

    // 80-bit reals, this heuristic seems to work OK:

    if(policies::digits<T, Policy>() > 64)

    {

      tolerance = pow(tools::epsilon<T>(), T(1)/4.25f);

      BOOST_MATH_INSTRUMENT_VARIABLE(tolerance);

    }

    else

    {

      tolerance = pow(4*tools::epsilon<T>(), T(1)/6);

      BOOST_MATH_INSTRUMENT_VARIABLE(tolerance);

    }



    // duplication

    k = 1;

    do

    {

        u = (x + y + z) / 3;

        X = (u - x) / u;

        Y = (u - y) / u;

        Z = (u - z) / u;



        // Termination condition: 

        if ((tools::max)(abs(X), abs(Y), abs(Z)) < tolerance) 

           break; 



        T sx = sqrt(x);

        T sy = sqrt(y);

        T sz = sqrt(z);

        lambda = sy * (sx + sz) + sz * sx;

        x = (x + lambda) / 4;

        y = (y + lambda) / 4;

        z = (z + lambda) / 4;

        ++k;

    }

    while(k < policies::get_max_series_iterations<Policy>());



    // Check to see if we gave up too soon:

    policies::check_series_iterations(function, k, pol);

    BOOST_MATH_INSTRUMENT_VARIABLE(k);



    // Taylor series expansion to the 5th order

    E2 = X * Y - Z * Z;

    E3 = X * Y * Z;

    value = (1 + E2*(E2/24 - E3*T(3)/44 - T(0.1)) + E3/14) / sqrt(u);

    BOOST_MATH_INSTRUMENT_VARIABLE(value);



    return value;

}



} // namespace detail



template <class T1, class T2, class T3, class Policy>

inline typename tools::promote_args<T1, T2, T3>::type 

   ellint_rf(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_rf_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_rf(T1 x, T2 y, T3 z)

{

   return ellint_rf(x, y, z, policies::policy<>());

}



}} // namespaces



#endif // BOOST_MATH_ELLINT_RF_HPP

