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    BOOST_MATH_ASSERT(y != -1);
0042
0043    // for 1 + y < 0, the integral is singular, return Cauchy principal value
0044    T result;
0045    if(y < -1)
0046    {
0047       result = sqrt(1 / -y) * detail::ellint_rc_imp(T(-y), T(-1 - y), pol);
0048    }
0049    else if(y == 0)
0050    {
0051       result = 1;
0052    }
0053    else if(y > 0)
0054    {
0055       result = atan(sqrt(y)) / sqrt(y);
0056    }
0057    else
0058    {
0059       if(y > T(-0.5))
0060       {
0061          T arg = sqrt(-y);
0062          result = (boost::math::log1p(arg, pol) - boost::math::log1p(-arg, pol)) / (2 * sqrt(-y));
0063       }
0064       else
0065       {
0066          result = log((1 + sqrt(-y)) / sqrt(1 + y)) / sqrt(-y);
0067       }
0068    }
0069    return result;
0070 }
0071
0072 template <typename T, typename Policy>
0073 T ellint_rj_imp(T x, T y, T z, T p, const Policy& pol)
0074 {
0075    BOOST_MATH_STD_USING
0076
0077    static const char* function = "boost::math::ellint_rj<%1%>(%1%,%1%,%1%)";
0078
0079    if(x < 0)
0080    {
0081       return policies::raise_domain_error<T>(function, "Argument x must be non-negative, but got x = %1%", x, pol);
0082    }
0083    if(y < 0)
0084    {
0085       return policies::raise_domain_error<T>(function, "Argument y must be non-negative, but got y = %1%", y, pol);
0086    }
0087    if(z < 0)
0088    {
0089       return policies::raise_domain_error<T>(function, "Argument z must be non-negative, but got z = %1%", z, pol);
0090    }
0091    if(p == 0)
0092    {
0093       return policies::raise_domain_error<T>(function, "Argument p must not be zero, but got p = %1%", p, pol);
0094    }
0095    if(x + y == 0 || y + z == 0 || z + x == 0)
0096    {
0097       return policies::raise_domain_error<T>(function, "At most one argument can be zero, only possible result is %1%.", std::numeric_limits<T>::quiet_NaN(), pol);
0098    }
0099
0100    // for p < 0, the integral is singular, return Cauchy principal value
0101    if(p < 0)
0102    {
0103       //
0104       // We must ensure that x < y < z.
0105       // Since the integral is symmetrical in x, y and z
0106       // we can just permute the values:
0107       //
0108       if(x > y)
0109          std::swap(x, y);
0110       if(y > z)
0111          std::swap(y, z);
0112       if(x > y)
0113          std::swap(x, y);
0114
0115       BOOST_MATH_ASSERT(x <= y);
0116       BOOST_MATH_ASSERT(y <= z);
0117
0118       T q = -p;
0119       p = (z * (x + y + q) - x * y) / (z + q);
0120
0121       BOOST_MATH_ASSERT(p >= 0);
0122
0123       T value = (p - z) * ellint_rj_imp(x, y, z, p, pol);
0124       value -= 3 * ellint_rf_imp(x, y, z, pol);
0125       value += 3 * sqrt((x * y * z) / (x * y + p * q)) * ellint_rc_imp(T(x * y + p * q), T(p * q), pol);
0126       value /= (z + q);
0127       return value;
0128    }
0129
0130    //
0131    // Special cases from http://dlmf.nist.gov/19.20#iii
0132    //
0133    if(x == y)
0134    {
0135       if(x == z)
0136       {
0137          if(x == p)
0138          {
0139             // All values equal:
0140             return 1 / (x * sqrt(x));
0141          }
0142          else
0143          {
0144             // x = y = z:
0145             return 3 * (ellint_rc_imp(x, p, pol) - 1 / sqrt(x)) / (x - p);
0146          }
0147       }
0148       else
0149       {
0150          // x = y only, permute so y = z:
0151          using std::swap;
0152          swap(x, z);
0153          if(y == p)
0154          {
0155             return ellint_rd_imp(x, y, y, pol);
0156          }
0157          else if((std::max)(y, p) / (std::min)(y, p) > T(1.2))
0158          {
0159             return 3 * (ellint_rc_imp(x, y, pol) - ellint_rc_imp(x, p, pol)) / (p - y);
0160          }
0161          // Otherwise fall through to normal method, special case above will suffer too much cancellation...
0162       }
0163    }
0164    if(y == z)
0165    {
0166       if(y == p)
0167       {
0168          // y = z = p:
0169          return ellint_rd_imp(x, y, y, pol);
0170       }
0171       else if((std::max)(y, p) / (std::min)(y, p) > T(1.2))
0172       {
0173          // y = z:
0174          return 3 * (ellint_rc_imp(x, y, pol) - ellint_rc_imp(x, p, pol)) / (p - y);
0175       }
0176       // Otherwise fall through to normal method, special case above will suffer too much cancellation...
0177    }
0178    if(z == p)
0179    {
0180       return ellint_rd_imp(x, y, z, pol);
0181    }
0182
0183    T xn = x;
0184    T yn = y;
0185    T zn = z;
0186    T pn = p;
0187    T An = (x + y + z + 2 * p) / 5;
0188    T A0 = An;
0189    T delta = (p - x) * (p - y) * (p - z);
0190    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)));
0191
0192    unsigned n;
0193    T lambda;
0194    T Dn;
0195    T En;
0196    T rx, ry, rz, rp;
0197    T fmn = 1; // 4^-n
0198    T RC_sum = 0;
0199
0200    for(n = 0; n < policies::get_max_series_iterations<Policy>(); ++n)
0201    {
0202       rx = sqrt(xn);
0203       ry = sqrt(yn);
0204       rz = sqrt(zn);
0205       rp = sqrt(pn);
0206       Dn = (rp + rx) * (rp + ry) * (rp + rz);
0207       En = delta / Dn;
0208       En /= Dn;
0209       if((En < T(-0.5)) && (En > T(-1.5)))
0210       {
0211          //
0212          // Occasionally En ~ -1, we then have no means of calculating
0213          // RC(1, 1+En) without terrible cancellation error, so we
0214          // need to get to 1+En directly.  By substitution we have
0215          //
0216          // 1+E_0 = 1 + (p-x)*(p-y)*(p-z)/((sqrt(p) + sqrt(x))*(sqrt(p)+sqrt(y))*(sqrt(p)+sqrt(z)))^2
0217          //       = 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))))
0218          //
0219          // And since this is just an application of the duplication formula for RJ, the same
0220          // expression works for 1+En if we use x,y,z,p_n etc.
0221          // This branch is taken only once or twice at the start of iteration,
0222          // after than En reverts to it's usual very small values.
0223          //
0224          T b = 2 * rp * (pn + rx * (ry + rz) + ry * rz) / Dn;
0225          RC_sum += fmn / Dn * detail::ellint_rc_imp(T(1), b, pol);
0226       }
0227       else
0228       {
0229          RC_sum += fmn / Dn * ellint_rc1p_imp(En, pol);
0230       }
0231       lambda = rx * ry + rx * rz + ry * rz;
0232
0233       // From here on we move to n+1:
0234       An = (An + lambda) / 4;
0235       fmn /= 4;
0236
0237       if(fmn * Q < An)
0238          break;
0239
0240       xn = (xn + lambda) / 4;
0241       yn = (yn + lambda) / 4;
0242       zn = (zn + lambda) / 4;
0243       pn = (pn + lambda) / 4;
0244       delta /= 64;
0245    }
0246
0247    T X = fmn * (A0 - x) / An;
0248    T Y = fmn * (A0 - y) / An;
0249    T Z = fmn * (A0 - z) / An;
0250    T P = (-X - Y - Z) / 2;
0251    T E2 = X * Y + X * Z + Y * Z - 3 * P * P;
0252    T E3 = X * Y * Z + 2 * E2 * P + 4 * P * P * P;
0253    T E4 = (2 * X * Y * Z + E2 * P + 3 * P * P * P) * P;
0254    T E5 = X * Y * Z * P * P;
0255    T result = fmn * pow(An, T(-3) / 2) *
0256       (1 - 3 * E2 / 14 + E3 / 6 + 9 * E2 * E2 / 88 - 3 * E4 / 22 - 9 * E2 * E3 / 52 + 3 * E5 / 26 - E2 * E2 * E2 / 16
0257       + 3 * E3 * E3 / 40 + 3 * E2 * E4 / 20 + 45 * E2 * E2 * E3 / 272 - 9 * (E3 * E4 + E2 * E5) / 68);
0258
0259    result += 6 * RC_sum;
0260    return result;
0261 }
0262
0263 } // namespace detail
0264
0265 template <class T1, class T2, class T3, class T4, class Policy>
0266 inline typename tools::promote_args<T1, T2, T3, T4>::type
0267    ellint_rj(T1 x, T2 y, T3 z, T4 p, const Policy& pol)
0268 {
0269    typedef typename tools::promote_args<T1, T2, T3, T4>::type result_type;
0270    typedef typename policies::evaluation<result_type, Policy>::type value_type;
0271    return policies::checked_narrowing_cast<result_type, Policy>(
0272       detail::ellint_rj_imp(
0273          static_cast<value_type>(x),
0274          static_cast<value_type>(y),
0275          static_cast<value_type>(z),
0276          static_cast<value_type>(p),
0277          pol), "boost::math::ellint_rj<%1%>(%1%,%1%,%1%,%1%)");
0278 }
0279
0280 template <class T1, class T2, class T3, class T4>
0281 inline typename tools::promote_args<T1, T2, T3, T4>::type
0282    ellint_rj(T1 x, T2 y, T3 z, T4 p)
0283 {
0284    return ellint_rj(x, y, z, p, policies::policy<>());
0285 }
0286
0287 }} // namespaces
0288
0289 #endif // BOOST_MATH_ELLINT_RJ_HPP
0290