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 slightly to fit into the
Chris@16: //  Boost.Math conceptual framework better.
Chris@101: //  Updated 2015 to use Carlson's latest methods.
Chris@16: 
Chris@16: #ifndef BOOST_MATH_ELLINT_RD_HPP
Chris@16: #define BOOST_MATH_ELLINT_RD_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@101: #include <boost/math/special_functions/ellint_rc.hpp>
Chris@101: #include <boost/math/special_functions/pow.hpp>
Chris@16: #include <boost/math/tools/config.hpp>
Chris@16: #include <boost/math/policies/error_handling.hpp>
Chris@16: 
Chris@16: // Carlson's elliptic integral of the second kind
Chris@16: // R_D(x, y, z) = R_J(x, y, z, z) = 1.5 * \int_{0}^{\infty} [(t+x)(t+y)]^{-1/2} (t+z)^{-3/2} dt
Chris@16: // Carlson, Numerische Mathematik, vol 33, 1 (1979)
Chris@16: 
Chris@16: namespace boost { namespace math { namespace detail{
Chris@16: 
Chris@16: template <typename T, typename Policy>
Chris@16: T ellint_rd_imp(T x, T y, T z, const Policy& pol)
Chris@16: {
Chris@101:    BOOST_MATH_STD_USING
Chris@101:    using std::swap;
Chris@16: 
Chris@101:    static const char* function = "boost::math::ellint_rd<%1%>(%1%,%1%,%1%)";
Chris@16: 
Chris@101:    if(x < 0)
Chris@101:    {
Chris@101:       return policies::raise_domain_error<T>(function,
Chris@101:          "Argument x must be >= 0, but got %1%", x, pol);
Chris@101:    }
Chris@101:    if(y < 0)
Chris@101:    {
Chris@101:       return policies::raise_domain_error<T>(function,
Chris@101:          "Argument y must be >= 0, but got %1%", y, pol);
Chris@101:    }
Chris@101:    if(z <= 0)
Chris@101:    {
Chris@101:       return policies::raise_domain_error<T>(function,
Chris@101:          "Argument z must be > 0, but got %1%", z, pol);
Chris@101:    }
Chris@101:    if(x + y == 0)
Chris@101:    {
Chris@101:       return policies::raise_domain_error<T>(function,
Chris@101:          "At most one argument can be zero, but got, x + y = %1%", x + y, pol);
Chris@101:    }
Chris@101:    //
Chris@101:    // Special cases from http://dlmf.nist.gov/19.20#iv
Chris@101:    //
Chris@101:    using std::swap;
Chris@101:    if(x == z)
Chris@101:       swap(x, y);
Chris@101:    if(y == z)
Chris@101:    {
Chris@101:       if(x == y)
Chris@101:       {
Chris@101:          return 1 / (x * sqrt(x));
Chris@101:       }
Chris@101:       else if(x == 0)
Chris@101:       {
Chris@101:          return 3 * constants::pi<T>() / (4 * y * sqrt(y));
Chris@101:       }
Chris@101:       else
Chris@101:       {
Chris@101:          if((std::min)(x, y) / (std::max)(x, y) > 1.3)
Chris@101:             return 3 * (ellint_rc_imp(x, y, pol) - sqrt(x) / y) / (2 * (y - x));
Chris@101:          // Otherwise fall through to avoid cancellation in the above (RC(x,y) -> 1/x^0.5 as x -> y)
Chris@101:       }
Chris@101:    }
Chris@101:    if(x == y)
Chris@101:    {
Chris@101:       if((std::min)(x, z) / (std::max)(x, z) > 1.3)
Chris@101:          return 3 * (ellint_rc_imp(z, x, pol) - 1 / sqrt(z)) / (z - x);
Chris@101:       // Otherwise fall through to avoid cancellation in the above (RC(x,y) -> 1/x^0.5 as x -> y)
Chris@101:    }
Chris@101:    if(y == 0)
Chris@101:       swap(x, y);
Chris@101:    if(x == 0)
Chris@101:    {
Chris@101:       //
Chris@101:       // Special handling for common case, from
Chris@101:       // Numerical Computation of Real or Complex Elliptic Integrals, eq.47
Chris@101:       //
Chris@101:       T xn = sqrt(y);
Chris@101:       T yn = sqrt(z);
Chris@101:       T x0 = xn;
Chris@101:       T y0 = yn;
Chris@101:       T sum = 0;
Chris@101:       T sum_pow = 0.25f;
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:          sum_pow *= 2;
Chris@101:          sum += sum_pow * boost::math::pow<2>(xn - yn);
Chris@101:       }
Chris@101:       T RF = constants::pi<T>() / (xn + yn);
Chris@101:       //
Chris@101:       // This following calculation suffers from serious cancellation when y ~ z
Chris@101:       // unless we combine terms.  We have:
Chris@101:       //
Chris@101:       // ( ((x0 + y0)/2)^2 - z ) / (z(y-z))
Chris@101:       //
Chris@101:       // Substituting y = x0^2 and z = y0^2 and simplifying we get the following:
Chris@101:       //
Chris@101:       T pt = (x0 + 3 * y0) / (4 * z * (x0 + y0));
Chris@101:       //
Chris@101:       // Since we've moved the demoninator from eq.47 inside the expression, we
Chris@101:       // need to also scale "sum" by the same value:
Chris@101:       //
Chris@101:       pt -= sum / (z * (y - z));
Chris@101:       return pt * RF * 3;
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 + 3 * z) / 5;
Chris@101:    T A0 = An;
Chris@101:    // This has an extra 1.2 fudge factor which is really only needed when x, y and z are close in magnitude:
Chris@101:    T Q = pow(tools::epsilon<T>() / 4, -T(1) / 8) * (std::max)((std::max)(An - x, An - y), An - z) * 1.2f;
Chris@101:    T lambda, rx, ry, rz;
Chris@101:    unsigned k = 0;
Chris@101:    T fn = 1;
Chris@101:    T RD_sum = 0;
Chris@16: 
Chris@101:    for(; k < policies::get_max_series_iterations<Policy>(); ++k)
Chris@101:    {
Chris@101:       rx = sqrt(xn);
Chris@101:       ry = sqrt(yn);
Chris@101:       rz = sqrt(zn);
Chris@101:       lambda = rx * ry + rx * rz + ry * rz;
Chris@101:       RD_sum += fn / (rz * (zn + lambda));
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:       fn /= 4;
Chris@101:       Q /= 4;
Chris@101:       if(Q < An)
Chris@101:          break;
Chris@101:    }
Chris@16: 
Chris@101:    policies::check_series_iterations<T, Policy>(function, k, pol);
Chris@16: 
Chris@101:    T X = fn * (A0 - x) / An;
Chris@101:    T Y = fn * (A0 - y) / An;
Chris@101:    T Z = -(X + Y) / 3;
Chris@101:    T E2 = X * Y - 6 * Z * Z;
Chris@101:    T E3 = (3 * X * Y - 8 * Z * Z) * Z;
Chris@101:    T E4 = 3 * (X * Y - Z * Z) * Z * Z;
Chris@101:    T E5 = X * Y * Z * Z * Z;
Chris@16: 
Chris@101:    T result = fn * pow(An, T(-3) / 2) *
Chris@101:       (1 - 3 * E2 / 14 + E3 / 6 + 9 * E2 * E2 / 88 - 3 * E4 / 22 - 9 * E2 * E3 / 52 + 3 * E5 / 26 - E2 * E2 * E2 / 16
Chris@101:       + 3 * E3 * E3 / 40 + 3 * E2 * E4 / 20 + 45 * E2 * E2 * E3 / 272 - 9 * (E3 * E4 + E2 * E5) / 68);
Chris@101:    result += 3 * RD_sum;
Chris@101: 
Chris@101:    return result;
Chris@16: }
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_rd(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_rd_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_rd<%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_rd(T1 x, T2 y, T3 z)
Chris@16: {
Chris@16:    return ellint_rd(x, y, z, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: }} // namespaces
Chris@16: 
Chris@16: #endif // BOOST_MATH_ELLINT_RD_HPP
Chris@16: