300 lines · plain
1// Copyright (c) 2006 Xiaogang Zhang, 2015 John Maddock2// Copyright (c) 2024 Matt Borland3// Use, modification and distribution are subject to the4// Boost Software License, Version 1.0. (See accompanying file5// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)6//7// History:8// XZ wrote the original of this file as part of the Google9// Summer of Code 2006. JM modified it to fit into the10// Boost.Math conceptual framework better, and to correctly11// handle the p < 0 case.12// Updated 2015 to use Carlson's latest methods.13//14 15#ifndef BOOST_MATH_ELLINT_RJ_HPP16#define BOOST_MATH_ELLINT_RJ_HPP17 18#ifdef _MSC_VER19#pragma once20#endif21 22#include <boost/math/tools/config.hpp>23#include <boost/math/tools/numeric_limits.hpp>24#include <boost/math/special_functions/math_fwd.hpp>25#include <boost/math/policies/error_handling.hpp>26#include <boost/math/special_functions/ellint_rc.hpp>27#include <boost/math/special_functions/ellint_rf.hpp>28#include <boost/math/special_functions/ellint_rd.hpp>29 30// Carlson's elliptic integral of the third kind31// R_J(x, y, z, p) = 1.5 * \int_{0}^{\infty} (t+p)^{-1} [(t+x)(t+y)(t+z)]^{-1/2} dt32// Carlson, Numerische Mathematik, vol 33, 1 (1979)33 34namespace boost { namespace math { namespace detail{35 36template <typename T, typename Policy>37BOOST_MATH_GPU_ENABLED T ellint_rc1p_imp(T y, const Policy& pol)38{39 using namespace boost::math;40 // Calculate RC(1, 1 + x)41 BOOST_MATH_STD_USING42 43 BOOST_MATH_ASSERT(y != -1);44 45 // for 1 + y < 0, the integral is singular, return Cauchy principal value46 T result;47 if(y < -1)48 {49 result = sqrt(1 / -y) * detail::ellint_rc_imp(T(-y), T(-1 - y), pol);50 }51 else if(y == 0)52 {53 result = 1;54 }55 else if(y > 0)56 {57 result = atan(sqrt(y)) / sqrt(y);58 }59 else60 {61 if(y > T(-0.5))62 {63 T arg = sqrt(-y);64 result = (boost::math::log1p(arg, pol) - boost::math::log1p(-arg, pol)) / (2 * sqrt(-y));65 }66 else67 {68 result = log((1 + sqrt(-y)) / sqrt(1 + y)) / sqrt(-y);69 }70 }71 return result;72}73 74template <typename T, typename Policy>75BOOST_MATH_GPU_ENABLED T ellint_rj_imp_final(T x, T y, T z, T p, const Policy& pol)76{77 BOOST_MATH_STD_USING78 79 //80 // Special cases from http://dlmf.nist.gov/19.20#iii81 //82 if(x == y)83 {84 if(x == z)85 {86 if(x == p)87 {88 // All values equal:89 return 1 / (x * sqrt(x));90 }91 else92 {93 // x = y = z:94 return 3 * (ellint_rc_imp(x, p, pol) - 1 / sqrt(x)) / (x - p);95 }96 }97 else98 {99 // x = y only, permute so y = z:100 BOOST_MATH_GPU_SAFE_SWAP(x, z);101 if(y == p)102 {103 return ellint_rd_imp(x, y, y, pol);104 }105 else if(BOOST_MATH_GPU_SAFE_MAX(y, p) / BOOST_MATH_GPU_SAFE_MIN(y, p) > T(1.2))106 {107 return 3 * (ellint_rc_imp(x, y, pol) - ellint_rc_imp(x, p, pol)) / (p - y);108 }109 // Otherwise fall through to normal method, special case above will suffer too much cancellation...110 }111 }112 if(y == z)113 {114 if(y == p)115 {116 // y = z = p:117 return ellint_rd_imp(x, y, y, pol);118 }119 else if(BOOST_MATH_GPU_SAFE_MAX(y, p) / BOOST_MATH_GPU_SAFE_MIN(y, p) > T(1.2))120 {121 // y = z:122 return 3 * (ellint_rc_imp(x, y, pol) - ellint_rc_imp(x, p, pol)) / (p - y);123 }124 // Otherwise fall through to normal method, special case above will suffer too much cancellation...125 }126 if(z == p)127 {128 return ellint_rd_imp(x, y, z, pol);129 }130 131 T xn = x;132 T yn = y;133 T zn = z;134 T pn = p;135 T An = (x + y + z + 2 * p) / 5;136 T A0 = An;137 T delta = (p - x) * (p - y) * (p - z);138 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)));139 140 unsigned n;141 T lambda;142 T Dn;143 T En;144 T rx, ry, rz, rp;145 T fmn = 1; // 4^-n146 T RC_sum = 0;147 148 for(n = 0; n < policies::get_max_series_iterations<Policy>(); ++n)149 {150 rx = sqrt(xn);151 ry = sqrt(yn);152 rz = sqrt(zn);153 rp = sqrt(pn);154 Dn = (rp + rx) * (rp + ry) * (rp + rz);155 En = delta / Dn;156 En /= Dn;157 if((En < T(-0.5)) && (En > T(-1.5)))158 {159 //160 // Occasionally En ~ -1, we then have no means of calculating161 // RC(1, 1+En) without terrible cancellation error, so we162 // need to get to 1+En directly. By substitution we have163 //164 // 1+E_0 = 1 + (p-x)*(p-y)*(p-z)/((sqrt(p) + sqrt(x))*(sqrt(p)+sqrt(y))*(sqrt(p)+sqrt(z)))^2165 // = 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))))166 //167 // And since this is just an application of the duplication formula for RJ, the same168 // expression works for 1+En if we use x,y,z,p_n etc.169 // This branch is taken only once or twice at the start of iteration,170 // after than En reverts to it's usual very small values.171 //172 T b = 2 * rp * (pn + rx * (ry + rz) + ry * rz) / Dn;173 RC_sum += fmn / Dn * detail::ellint_rc_imp(T(1), b, pol);174 }175 else176 {177 RC_sum += fmn / Dn * ellint_rc1p_imp(En, pol);178 }179 lambda = rx * ry + rx * rz + ry * rz;180 181 // From here on we move to n+1:182 An = (An + lambda) / 4;183 fmn /= 4;184 185 if(fmn * Q < An)186 break;187 188 xn = (xn + lambda) / 4;189 yn = (yn + lambda) / 4;190 zn = (zn + lambda) / 4;191 pn = (pn + lambda) / 4;192 delta /= 64;193 }194 195 T X = fmn * (A0 - x) / An;196 T Y = fmn * (A0 - y) / An;197 T Z = fmn * (A0 - z) / An;198 T P = (-X - Y - Z) / 2;199 T E2 = X * Y + X * Z + Y * Z - 3 * P * P;200 T E3 = X * Y * Z + 2 * E2 * P + 4 * P * P * P;201 T E4 = (2 * X * Y * Z + E2 * P + 3 * P * P * P) * P;202 T E5 = X * Y * Z * P * P;203 T result = fmn * pow(An, T(-3) / 2) *204 (1 - 3 * E2 / 14 + E3 / 6 + 9 * E2 * E2 / 88 - 3 * E4 / 22 - 9 * E2 * E3 / 52 + 3 * E5 / 26 - E2 * E2 * E2 / 16205 + 3 * E3 * E3 / 40 + 3 * E2 * E4 / 20 + 45 * E2 * E2 * E3 / 272 - 9 * (E3 * E4 + E2 * E5) / 68);206 207 result += 6 * RC_sum;208 return result;209}210 211template <typename T, typename Policy>212BOOST_MATH_GPU_ENABLED T ellint_rj_imp(T x, T y, T z, T p, const Policy& pol)213{214 BOOST_MATH_STD_USING215 216 constexpr auto function = "boost::math::ellint_rj<%1%>(%1%,%1%,%1%)";217 218 if(x < 0)219 {220 return policies::raise_domain_error<T>(function, "Argument x must be non-negative, but got x = %1%", x, pol);221 }222 if(y < 0)223 {224 return policies::raise_domain_error<T>(function, "Argument y must be non-negative, but got y = %1%", y, pol);225 }226 if(z < 0)227 {228 return policies::raise_domain_error<T>(function, "Argument z must be non-negative, but got z = %1%", z, pol);229 }230 if(p == 0)231 {232 return policies::raise_domain_error<T>(function, "Argument p must not be zero, but got p = %1%", p, pol);233 }234 if(x + y == 0 || y + z == 0 || z + x == 0)235 {236 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);237 }238 239 // for p < 0, the integral is singular, return Cauchy principal value240 if(p < 0)241 {242 //243 // We must ensure that x < y < z.244 // Since the integral is symmetrical in x, y and z245 // we can just permute the values:246 //247 if(x > y)248 BOOST_MATH_GPU_SAFE_SWAP(x, y);249 if(y > z)250 BOOST_MATH_GPU_SAFE_SWAP(y, z);251 if(x > y)252 BOOST_MATH_GPU_SAFE_SWAP(x, y);253 254 BOOST_MATH_ASSERT(x <= y);255 BOOST_MATH_ASSERT(y <= z);256 257 T q = -p;258 p = (z * (x + y + q) - x * y) / (z + q);259 260 BOOST_MATH_ASSERT(p >= 0);261 262 T value = (p - z) * ellint_rj_imp_final(x, y, z, p, pol);263 value -= 3 * ellint_rf_imp(x, y, z, pol);264 value += 3 * sqrt((x * y * z) / (x * y + p * q)) * ellint_rc_imp(T(x * y + p * q), T(p * q), pol);265 value /= (z + q);266 return value;267 }268 269 return ellint_rj_imp_final(x, y, z, p, pol);270}271 272} // namespace detail273 274template <class T1, class T2, class T3, class T4, class Policy>275BOOST_MATH_GPU_ENABLED inline typename tools::promote_args<T1, T2, T3, T4>::type 276 ellint_rj(T1 x, T2 y, T3 z, T4 p, const Policy& pol)277{278 typedef typename tools::promote_args<T1, T2, T3, T4>::type result_type;279 typedef typename policies::evaluation<result_type, Policy>::type value_type;280 return policies::checked_narrowing_cast<result_type, Policy>(281 detail::ellint_rj_imp(282 static_cast<value_type>(x),283 static_cast<value_type>(y),284 static_cast<value_type>(z),285 static_cast<value_type>(p),286 pol), "boost::math::ellint_rj<%1%>(%1%,%1%,%1%,%1%)");287}288 289template <class T1, class T2, class T3, class T4>290BOOST_MATH_GPU_ENABLED inline typename tools::promote_args<T1, T2, T3, T4>::type 291 ellint_rj(T1 x, T2 y, T3 z, T4 p)292{293 return ellint_rj(x, y, z, p, policies::policy<>());294}295 296}} // namespaces297 298#endif // BOOST_MATH_ELLINT_RJ_HPP299 300