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 to fit into the
Chris@16: //  Boost.Math conceptual framework better, and to correctly
Chris@16: //  handle the p < 0 case.
Chris@101: //  Updated 2015 to use Carlson's latest methods.
Chris@16: //
Chris@16: 
Chris@16: #ifndef BOOST_MATH_ELLINT_RJ_HPP
Chris@16: #define BOOST_MATH_ELLINT_RJ_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@16: #include <boost/math/tools/config.hpp>
Chris@16: #include <boost/math/policies/error_handling.hpp>
Chris@16: #include <boost/math/special_functions/ellint_rc.hpp>
Chris@16: #include <boost/math/special_functions/ellint_rf.hpp>
Chris@101: #include <boost/math/special_functions/ellint_rd.hpp>
Chris@16: 
Chris@16: // Carlson's elliptic integral of the third kind
Chris@16: // R_J(x, y, z, p) = 1.5 * \int_{0}^{\infty} (t+p)^{-1} [(t+x)(t+y)(t+z)]^{-1/2} 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@101: T ellint_rc1p_imp(T y, const Policy& pol)
Chris@101: {
Chris@101:    using namespace boost::math;
Chris@101:    // Calculate RC(1, 1 + x)
Chris@101:    BOOST_MATH_STD_USING
Chris@101: 
Chris@101:   static const char* function = "boost::math::ellint_rc<%1%>(%1%,%1%)";
Chris@101: 
Chris@101:    if(y == -1)
Chris@101:    {
Chris@101:       return policies::raise_domain_error<T>(function,
Chris@101:          "Argument y must not be zero but got %1%", y, pol);
Chris@101:    }
Chris@101: 
Chris@101:    // for 1 + y < 0, the integral is singular, return Cauchy principal value
Chris@101:    T result;
Chris@101:    if(y < -1)
Chris@101:    {
Chris@101:       result = sqrt(1 / -y) * detail::ellint_rc_imp(T(-y), T(-1 - y), pol);
Chris@101:    }
Chris@101:    else if(y == 0)
Chris@101:    {
Chris@101:       result = 1;
Chris@101:    }
Chris@101:    else if(y > 0)
Chris@101:    {
Chris@101:       result = atan(sqrt(y)) / sqrt(y);
Chris@101:    }
Chris@101:    else
Chris@101:    {
Chris@101:       if(y > -0.5)
Chris@101:       {
Chris@101:          T arg = sqrt(-y);
Chris@101:          result = (boost::math::log1p(arg) - boost::math::log1p(-arg)) / (2 * sqrt(-y));
Chris@101:       }
Chris@101:       else
Chris@101:       {
Chris@101:          result = log((1 + sqrt(-y)) / sqrt(1 + y)) / sqrt(-y);
Chris@101:       }
Chris@101:    }
Chris@101:    return result;
Chris@101: }
Chris@101: 
Chris@101: template <typename T, typename Policy>
Chris@16: T ellint_rj_imp(T x, T y, T z, T p, const Policy& pol)
Chris@16: {
Chris@101:    BOOST_MATH_STD_USING
Chris@16: 
Chris@101:    static const char* function = "boost::math::ellint_rj<%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 non-negative, but got x = %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 non-negative, but got y = %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 non-negative, but got z = %1%", z, pol);
Chris@101:    }
Chris@101:    if(p == 0)
Chris@101:    {
Chris@101:       return policies::raise_domain_error<T>(function,
Chris@101:          "Argument p must not be zero, but got p = %1%", p, pol);
Chris@101:    }
Chris@101:    if(x + y == 0 || y + z == 0 || z + x == 0)
Chris@101:    {
Chris@101:       return policies::raise_domain_error<T>(function,
Chris@101:          "At most one argument can be zero, "
Chris@101:          "only possible result is %1%.", std::numeric_limits<T>::quiet_NaN(), pol);
Chris@101:    }
Chris@16: 
Chris@101:    // for p < 0, the integral is singular, return Cauchy principal value
Chris@101:    if(p < 0)
Chris@101:    {
Chris@101:       //
Chris@101:       // We must ensure that x < y < z.
Chris@101:       // Since the integral is symmetrical in x, y and z
Chris@101:       // we can just permute the values:
Chris@101:       //
Chris@101:       if(x > y)
Chris@101:          std::swap(x, y);
Chris@101:       if(y > z)
Chris@101:          std::swap(y, z);
Chris@101:       if(x > y)
Chris@101:          std::swap(x, y);
Chris@16: 
Chris@101:       BOOST_ASSERT(x <= y);
Chris@101:       BOOST_ASSERT(y <= z);
Chris@16: 
Chris@101:       T q = -p;
Chris@101:       p = (z * (x + y + q) - x * y) / (z + q);
Chris@16: 
Chris@101:       BOOST_ASSERT(p >= 0);
Chris@16: 
Chris@101:       T value = (p - z) * ellint_rj_imp(x, y, z, p, pol);
Chris@101:       value -= 3 * ellint_rf_imp(x, y, z, pol);
Chris@101:       value += 3 * sqrt((x * y * z) / (x * y + p * q)) * ellint_rc_imp(T(x * y + p * q), T(p * q), pol);
Chris@101:       value /= (z + q);
Chris@101:       return value;
Chris@101:    }
Chris@16: 
Chris@101:    //
Chris@101:    // Special cases from http://dlmf.nist.gov/19.20#iii
Chris@101:    //
Chris@101:    if(x == y)
Chris@101:    {
Chris@101:       if(x == z)
Chris@101:       {
Chris@101:          if(x == p)
Chris@101:          {
Chris@101:             // All values equal:
Chris@101:             return 1 / (x * sqrt(x));
Chris@101:          }
Chris@101:          else
Chris@101:          {
Chris@101:             // x = y = z:
Chris@101:             return 3 * (ellint_rc_imp(x, p, pol) - 1 / sqrt(x)) / (x - p);
Chris@101:          }
Chris@101:       }
Chris@101:       else
Chris@101:       {
Chris@101:          // x = y only, permute so y = z:
Chris@101:          using std::swap;
Chris@101:          swap(x, z);
Chris@101:          if(y == p)
Chris@101:          {
Chris@101:             return ellint_rd_imp(x, y, y, pol);
Chris@101:          }
Chris@101:          else if((std::max)(y, p) / (std::min)(y, p) > 1.2)
Chris@101:          {
Chris@101:             return 3 * (ellint_rc_imp(x, y, pol) - ellint_rc_imp(x, p, pol)) / (p - y);
Chris@101:          }
Chris@101:          // Otherwise fall through to normal method, special case above will suffer too much cancellation...
Chris@101:       }
Chris@101:    }
Chris@101:    if(y == z)
Chris@101:    {
Chris@101:       if(y == p)
Chris@101:       {
Chris@101:          // y = z = p:
Chris@101:          return ellint_rd_imp(x, y, y, pol);
Chris@101:       }
Chris@101:       else if((std::max)(y, p) / (std::min)(y, p) > 1.2)
Chris@101:       {
Chris@101:          // y = z:
Chris@101:          return 3 * (ellint_rc_imp(x, y, pol) - ellint_rc_imp(x, p, pol)) / (p - y);
Chris@101:       }
Chris@101:       // Otherwise fall through to normal method, special case above will suffer too much cancellation...
Chris@101:    }
Chris@101:    if(z == p)
Chris@101:    {
Chris@101:       return ellint_rd_imp(x, y, z, pol);
Chris@101:    }
Chris@16: 
Chris@101:    T xn = x;
Chris@101:    T yn = y;
Chris@101:    T zn = z;
Chris@101:    T pn = p;
Chris@101:    T An = (x + y + z + 2 * p) / 5;
Chris@101:    T A0 = An;
Chris@101:    T delta = (p - x) * (p - y) * (p - z);
Chris@101:    T Q = pow(tools::epsilon<T>() / 5, -T(1) / 8) * (std::max)((std::max)(fabs(An - x), fabs(An - y)), (std::max)(fabs(An - z), fabs(An - p)));
Chris@16: 
Chris@101:    unsigned n;
Chris@101:    T lambda;
Chris@101:    T Dn;
Chris@101:    T En;
Chris@101:    T rx, ry, rz, rp;
Chris@101:    T fmn = 1; // 4^-n
Chris@101:    T RC_sum = 0;
Chris@16: 
Chris@101:    for(n = 0; n < policies::get_max_series_iterations<Policy>(); ++n)
Chris@101:    {
Chris@101:       rx = sqrt(xn);
Chris@101:       ry = sqrt(yn);
Chris@101:       rz = sqrt(zn);
Chris@101:       rp = sqrt(pn);
Chris@101:       Dn = (rp + rx) * (rp + ry) * (rp + rz);
Chris@101:       En = delta / Dn;
Chris@101:       En /= Dn;
Chris@101:       if((En < -0.5) && (En > -1.5))
Chris@101:       {
Chris@101:          //
Chris@101:          // Occationally En ~ -1, we then have no means of calculating
Chris@101:          // RC(1, 1+En) without terrible cancellation error, so we
Chris@101:          // need to get to 1+En directly.  By substitution we have
Chris@101:          //
Chris@101:          // 1+E_0 = 1 + (p-x)*(p-y)*(p-z)/((sqrt(p) + sqrt(x))*(sqrt(p)+sqrt(y))*(sqrt(p)+sqrt(z)))^2
Chris@101:          //       = 2*sqrt(p)*(p+sqrt(x) * (sqrt(y)+sqrt(z)) + sqrt(y)*sqrt(z)) / ((sqrt(p) + sqrt(x))*(sqrt(p) + sqrt(y)*(sqrt(p)+sqrt(z))))
Chris@101:          //
Chris@101:          // And since this is just an application of the duplication formula for RJ, the same
Chris@101:          // expression works for 1+En if we use x,y,z,p_n etc.
Chris@101:          // This branch is taken only once or twice at the start of iteration,
Chris@101:          // after than En reverts to it's usual very small values.
Chris@101:          //
Chris@101:          T b = 2 * rp * (pn + rx * (ry + rz) + ry * rz) / Dn;
Chris@101:          RC_sum += fmn / Dn * detail::ellint_rc_imp(T(1), b, pol);
Chris@101:       }
Chris@101:       else
Chris@101:       {
Chris@101:          RC_sum += fmn / Dn * ellint_rc1p_imp(En, pol);
Chris@101:       }
Chris@101:       lambda = rx * ry + rx * rz + ry * rz;
Chris@16: 
Chris@101:       // From here on we move to n+1:
Chris@101:       An = (An + lambda) / 4;
Chris@101:       fmn /= 4;
Chris@16: 
Chris@101:       if(fmn * Q < An)
Chris@101:          break;
Chris@101: 
Chris@101:       xn = (xn + lambda) / 4;
Chris@101:       yn = (yn + lambda) / 4;
Chris@101:       zn = (zn + lambda) / 4;
Chris@101:       pn = (pn + lambda) / 4;
Chris@101:       delta /= 64;
Chris@101:    }
Chris@101: 
Chris@101:    T X = fmn * (A0 - x) / An;
Chris@101:    T Y = fmn * (A0 - y) / An;
Chris@101:    T Z = fmn * (A0 - z) / An;
Chris@101:    T P = (-X - Y - Z) / 2;
Chris@101:    T E2 = X * Y + X * Z + Y * Z - 3 * P * P;
Chris@101:    T E3 = X * Y * Z + 2 * E2 * P + 4 * P * P * P;
Chris@101:    T E4 = (2 * X * Y * Z + E2 * P + 3 * P * P * P) * P;
Chris@101:    T E5 = X * Y * Z * P * P;
Chris@101:    T result = fmn * 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: 
Chris@101:    result += 6 * RC_sum;
Chris@101:    return result;
Chris@16: }
Chris@16: 
Chris@16: } // namespace detail
Chris@16: 
Chris@16: template <class T1, class T2, class T3, class T4, class Policy>
Chris@16: inline typename tools::promote_args<T1, T2, T3, T4>::type 
Chris@16:    ellint_rj(T1 x, T2 y, T3 z, T4 p, const Policy& pol)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T1, T2, T3, T4>::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_rj_imp(
Chris@16:          static_cast<value_type>(x),
Chris@16:          static_cast<value_type>(y),
Chris@16:          static_cast<value_type>(z),
Chris@16:          static_cast<value_type>(p),
Chris@16:          pol), "boost::math::ellint_rj<%1%>(%1%,%1%,%1%,%1%)");
Chris@16: }
Chris@16: 
Chris@16: template <class T1, class T2, class T3, class T4>
Chris@16: inline typename tools::promote_args<T1, T2, T3, T4>::type 
Chris@16:    ellint_rj(T1 x, T2 y, T3 z, T4 p)
Chris@16: {
Chris@16:    return ellint_rj(x, y, z, p, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: }} // namespaces
Chris@16: 
Chris@16: #endif // BOOST_MATH_ELLINT_RJ_HPP
Chris@16: