math/special_functions/ellint_rj.hpp

0001 //  Copyright (c) 2006 Xiaogang Zhang, 2015 John Maddock
0002 //  Copyright (c) 2024 Matt Borland
0003 //  Use, modification and distribution are subject to the
0004 //  Boost Software License, Version 1.0. (See accompanying file
0005 //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
0006 //
0007 //  History:
0008 //  XZ wrote the original of this file as part of the Google
0009 //  Summer of Code 2006.  JM modified it to fit into the
0010 //  Boost.Math conceptual framework better, and to correctly
0011 //  handle the p < 0 case.
0012 //  Updated 2015 to use Carlson's latest methods.
0013 //
0014
0015 #ifndef BOOST_MATH_ELLINT_RJ_HPP
0016 #define BOOST_MATH_ELLINT_RJ_HPP
0017
0018 #ifdef _MSC_VER
0019 #pragma once
0020 #endif
0021
0022 #include <boost/math/tools/config.hpp>
0023 #include <boost/math/tools/numeric_limits.hpp>
0024 #include <boost/math/special_functions/math_fwd.hpp>
0025 #include <boost/math/policies/error_handling.hpp>
0026 #include <boost/math/special_functions/ellint_rc.hpp>
0027 #include <boost/math/special_functions/ellint_rf.hpp>
0028 #include <boost/math/special_functions/ellint_rd.hpp>
0029
0030 // Carlson's elliptic integral of the third kind
0031 // R_J(x, y, z, p) = 1.5 * \int_{0}^{\infty} (t+p)^{-1} [(t+x)(t+y)(t+z)]^{-1/2} dt
0032 // Carlson, Numerische Mathematik, vol 33, 1 (1979)
0033
0034 namespace boost { namespace math { namespace detail{
0035
0036 template <typename T, typename Policy>
0037 BOOST_MATH_GPU_ENABLED T ellint_rc1p_imp(T y, const Policy& pol)
0038 {
0039    using namespace boost::math;
0040    // Calculate RC(1, 1 + x)
0041    BOOST_MATH_STD_USING
0042
0043    BOOST_MATH_ASSERT(y != -1);
0044
0045    // for 1 + y < 0, the integral is singular, return Cauchy principal value
0046    T result;
0047    if(y < -1)
0048    {
0049       result = sqrt(1 / -y) * detail::ellint_rc_imp(T(-y), T(-1 - y), pol);
0050    }
0051    else if(y == 0)
0052    {
0053       result = 1;
0054    }
0055    else if(y > 0)
0056    {
0057       result = atan(sqrt(y)) / sqrt(y);
0058    }
0059    else
0060    {
0061       if(y > T(-0.5))
0062       {
0063          T arg = sqrt(-y);
0064          result = (boost::math::log1p(arg, pol) - boost::math::log1p(-arg, pol)) / (2 * sqrt(-y));
0065       }
0066       else
0067       {
0068          result = log((1 + sqrt(-y)) / sqrt(1 + y)) / sqrt(-y);
0069       }
0070    }
0071    return result;
0072 }
0073
0074 template <typename T, typename Policy>
0075 BOOST_MATH_GPU_ENABLED T ellint_rj_imp_final(T x, T y, T z, T p, const Policy& pol)
0076 {
0077    BOOST_MATH_STD_USING
0078
0079    constexpr auto function = "boost::math::ellint_rj<%1%>(%1%,%1%,%1%)";
0080
0081    if(x < 0)
0082    {
0083       return policies::raise_domain_error<T>(function, "Argument x must be non-negative, but got x = %1%", x, pol);
0084    }
0085    if(y < 0)
0086    {
0087       return policies::raise_domain_error<T>(function, "Argument y must be non-negative, but got y = %1%", y, pol);
0088    }
0089    if(z < 0)
0090    {
0091       return policies::raise_domain_error<T>(function, "Argument z must be non-negative, but got z = %1%", z, pol);
0092    }
0093    if(p == 0)
0094    {
0095       return policies::raise_domain_error<T>(function, "Argument p must not be zero, but got p = %1%", p, pol);
0096    }
0097    if(x + y == 0 || y + z == 0 || z + x == 0)
0098    {
0099       return policies::raise_domain_error<T>(function, "At most one argument can be zero, only possible result is %1%.", boost::math::numeric_limits<T>::quiet_NaN(), pol);
0100    }
0101
0102    //
0103    // Special cases from http://dlmf.nist.gov/19.20#iii
0104    //
0105    if(x == y)
0106    {
0107       if(x == z)
0108       {
0109          if(x == p)
0110          {
0111             // All values equal:
0112             return 1 / (x * sqrt(x));
0113          }
0114          else
0115          {
0116             // x = y = z:
0117             return 3 * (ellint_rc_imp(x, p, pol) - 1 / sqrt(x)) / (x - p);
0118          }
0119       }
0120       else
0121       {
0122          // x = y only, permute so y = z:
0123          BOOST_MATH_GPU_SAFE_SWAP(x, z);
0124          if(y == p)
0125          {
0126             return ellint_rd_imp(x, y, y, pol);
0127          }
0128          else if(BOOST_MATH_GPU_SAFE_MAX(y, p) / BOOST_MATH_GPU_SAFE_MIN(y, p) > T(1.2))
0129          {
0130             return 3 * (ellint_rc_imp(x, y, pol) - ellint_rc_imp(x, p, pol)) / (p - y);
0131          }
0132          // Otherwise fall through to normal method, special case above will suffer too much cancellation...
0133       }
0134    }
0135    if(y == z)
0136    {
0137       if(y == p)
0138       {
0139          // y = z = p:
0140          return ellint_rd_imp(x, y, y, pol);
0141       }
0142       else if(BOOST_MATH_GPU_SAFE_MAX(y, p) / BOOST_MATH_GPU_SAFE_MIN(y, p) > T(1.2))
0143       {
0144          // y = z:
0145          return 3 * (ellint_rc_imp(x, y, pol) - ellint_rc_imp(x, p, pol)) / (p - y);
0146       }
0147       // Otherwise fall through to normal method, special case above will suffer too much cancellation...
0148    }
0149    if(z == p)
0150    {
0151       return ellint_rd_imp(x, y, z, pol);
0152    }
0153
0154    T xn = x;
0155    T yn = y;
0156    T zn = z;
0157    T pn = p;
0158    T An = (x + y + z + 2 * p) / 5;
0159    T A0 = An;
0160    T delta = (p - x) * (p - y) * (p - z);
0161    T Q = pow(tools::epsilon<T>() / 5, -T(1) / 8) * BOOST_MATH_GPU_SAFE_MAX(BOOST_MATH_GPU_SAFE_MAX(fabs(An - x), fabs(An - y)), BOOST_MATH_GPU_SAFE_MAX(fabs(An - z), fabs(An - p)));
0162
0163    unsigned n;
0164    T lambda;
0165    T Dn;
0166    T En;
0167    T rx, ry, rz, rp;
0168    T fmn = 1; // 4^-n
0169    T RC_sum = 0;
0170
0171    for(n = 0; n < policies::get_max_series_iterations<Policy>(); ++n)
0172    {
0173       rx = sqrt(xn);
0174       ry = sqrt(yn);
0175       rz = sqrt(zn);
0176       rp = sqrt(pn);
0177       Dn = (rp + rx) * (rp + ry) * (rp + rz);
0178       En = delta / Dn;
0179       En /= Dn;
0180       if((En < T(-0.5)) && (En > T(-1.5)))
0181       {
0182          //
0183          // Occasionally En ~ -1, we then have no means of calculating
0184          // RC(1, 1+En) without terrible cancellation error, so we
0185          // need to get to 1+En directly.  By substitution we have
0186          //
0187          // 1+E_0 = 1 + (p-x)*(p-y)*(p-z)/((sqrt(p) + sqrt(x))*(sqrt(p)+sqrt(y))*(sqrt(p)+sqrt(z)))^2
0188          //       = 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))))
0189          //
0190          // And since this is just an application of the duplication formula for RJ, the same
0191          // expression works for 1+En if we use x,y,z,p_n etc.
0192          // This branch is taken only once or twice at the start of iteration,
0193          // after than En reverts to it's usual very small values.
0194          //
0195          T b = 2 * rp * (pn + rx * (ry + rz) + ry * rz) / Dn;
0196          RC_sum += fmn / Dn * detail::ellint_rc_imp(T(1), b, pol);
0197       }
0198       else
0199       {
0200          RC_sum += fmn / Dn * ellint_rc1p_imp(En, pol);
0201       }
0202       lambda = rx * ry + rx * rz + ry * rz;
0203
0204       // From here on we move to n+1:
0205       An = (An + lambda) / 4;
0206       fmn /= 4;
0207
0208       if(fmn * Q < An)
0209          break;
0210
0211       xn = (xn + lambda) / 4;
0212       yn = (yn + lambda) / 4;
0213       zn = (zn + lambda) / 4;
0214       pn = (pn + lambda) / 4;
0215       delta /= 64;
0216    }
0217
0218    T X = fmn * (A0 - x) / An;
0219    T Y = fmn * (A0 - y) / An;
0220    T Z = fmn * (A0 - z) / An;
0221    T P = (-X - Y - Z) / 2;
0222    T E2 = X * Y + X * Z + Y * Z - 3 * P * P;
0223    T E3 = X * Y * Z + 2 * E2 * P + 4 * P * P * P;
0224    T E4 = (2 * X * Y * Z + E2 * P + 3 * P * P * P) * P;
0225    T E5 = X * Y * Z * P * P;
0226    T result = fmn * pow(An, T(-3) / 2) *
0227       (1 - 3 * E2 / 14 + E3 / 6 + 9 * E2 * E2 / 88 - 3 * E4 / 22 - 9 * E2 * E3 / 52 + 3 * E5 / 26 - E2 * E2 * E2 / 16
0228       + 3 * E3 * E3 / 40 + 3 * E2 * E4 / 20 + 45 * E2 * E2 * E3 / 272 - 9 * (E3 * E4 + E2 * E5) / 68);
0229
0230    result += 6 * RC_sum;
0231    return result;
0232 }
0233
0234 template <typename T, typename Policy>
0235 BOOST_MATH_GPU_ENABLED T ellint_rj_imp(T x, T y, T z, T p, const Policy& pol)
0236 {
0237    BOOST_MATH_STD_USING
0238
0239    constexpr auto function = "boost::math::ellint_rj<%1%>(%1%,%1%,%1%)";
0240
0241    if(x < 0)
0242    {
0243       return policies::raise_domain_error<T>(function, "Argument x must be non-negative, but got x = %1%", x, pol);
0244    }
0245    if(y < 0)
0246    {
0247       return policies::raise_domain_error<T>(function, "Argument y must be non-negative, but got y = %1%", y, pol);
0248    }
0249    if(z < 0)
0250    {
0251       return policies::raise_domain_error<T>(function, "Argument z must be non-negative, but got z = %1%", z, pol);
0252    }
0253    if(p == 0)
0254    {
0255       return policies::raise_domain_error<T>(function, "Argument p must not be zero, but got p = %1%", p, pol);
0256    }
0257    if(x + y == 0 || y + z == 0 || z + x == 0)
0258    {
0259       return policies::raise_domain_error<T>(function, "At most one argument can be zero, only possible result is %1%.", boost::math::numeric_limits<T>::quiet_NaN(), pol);
0260    }
0261
0262    // for p < 0, the integral is singular, return Cauchy principal value
0263    if(p < 0)
0264    {
0265       //
0266       // We must ensure that x < y < z.
0267       // Since the integral is symmetrical in x, y and z
0268       // we can just permute the values:
0269       //
0270       if(x > y)
0271          BOOST_MATH_GPU_SAFE_SWAP(x, y);
0272       if(y > z)
0273          BOOST_MATH_GPU_SAFE_SWAP(y, z);
0274       if(x > y)
0275          BOOST_MATH_GPU_SAFE_SWAP(x, y);
0276
0277       BOOST_MATH_ASSERT(x <= y);
0278       BOOST_MATH_ASSERT(y <= z);
0279
0280       T q = -p;
0281       p = (z * (x + y + q) - x * y) / (z + q);
0282
0283       BOOST_MATH_ASSERT(p >= 0);
0284
0285       T value = (p - z) * ellint_rj_imp_final(x, y, z, p, pol);
0286       value -= 3 * ellint_rf_imp(x, y, z, pol);
0287       value += 3 * sqrt((x * y * z) / (x * y + p * q)) * ellint_rc_imp(T(x * y + p * q), T(p * q), pol);
0288       value /= (z + q);
0289       return value;
0290    }
0291
0292    return ellint_rj_imp_final(x, y, z, p, pol);
0293 }
0294
0295 } // namespace detail
0296
0297 template <class T1, class T2, class T3, class T4, class Policy>
0298 BOOST_MATH_GPU_ENABLED inline typename tools::promote_args<T1, T2, T3, T4>::type
0299    ellint_rj(T1 x, T2 y, T3 z, T4 p, const Policy& pol)
0300 {
0301    typedef typename tools::promote_args<T1, T2, T3, T4>::type result_type;
0302    typedef typename policies::evaluation<result_type, Policy>::type value_type;
0303    return policies::checked_narrowing_cast<result_type, Policy>(
0304       detail::ellint_rj_imp(
0305          static_cast<value_type>(x),
0306          static_cast<value_type>(y),
0307          static_cast<value_type>(z),
0308          static_cast<value_type>(p),
0309          pol), "boost::math::ellint_rj<%1%>(%1%,%1%,%1%,%1%)");
0310 }
0311
0312 template <class T1, class T2, class T3, class T4>
0313 BOOST_MATH_GPU_ENABLED inline typename tools::promote_args<T1, T2, T3, T4>::type
0314    ellint_rj(T1 x, T2 y, T3 z, T4 p)
0315 {
0316    return ellint_rj(x, y, z, p, policies::policy<>());
0317 }
0318
0319 }} // namespaces
0320
0321 #endif // BOOST_MATH_ELLINT_RJ_HPP
0322