math/special_functions/ellint_rj.hpp

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