1 // Copyright (c) 2006 Xiaogang Zhang, 2015 John Maddock 2 // Use, modification and distribution are subject to the 3 // Boost Software License, Version 1.0. (See accompanying file 4 // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) 5 // 6 // History: 7 // XZ wrote the original of this file as part of the Google 8 // Summer of Code 2006. JM modified it to fit into the 9 // Boost.Math conceptual framework better, and to correctly 10 // handle the p < 0 case. 11 // Updated 2015 to use Carlson's latest methods. 12 // 13 14 #ifndef BOOST_MATH_ELLINT_RJ_HPP 15 #define BOOST_MATH_ELLINT_RJ_HPP 16 17 #ifdef _MSC_VER 18 #pragma once 19 #endif 20 21 #include <boost/math/special_functions/math_fwd.hpp> 22 #include <boost/math/tools/config.hpp> 23 #include <boost/math/policies/error_handling.hpp> 24 #include <boost/math/special_functions/ellint_rc.hpp> 25 #include <boost/math/special_functions/ellint_rf.hpp> 26 #include <boost/math/special_functions/ellint_rd.hpp> 27 28 // Carlson's elliptic integral of the third kind 29 // R_J(x, y, z, p) = 1.5 * \int_{0}^{\infty} (t+p)^{-1} [(t+x)(t+y)(t+z)]^{-1/2} dt 30 // Carlson, Numerische Mathematik, vol 33, 1 (1979) 31 32 namespace boost { namespace math { namespace detail{ 33 34 template <typename T, typename Policy> 35 T ellint_rc1p_imp(T y, const Policy& pol) 36 { 37 using namespace boost::math; 38 // Calculate RC(1, 1 + x) 39 BOOST_MATH_STD_USING 40 41 static const char* function = "boost::math::ellint_rc<%1%>(%1%,%1%)"; 42 43 if(y == -1) 44 { 45 return policies::raise_domain_error<T>(function, 46 "Argument y must not be zero but got %1%", y, pol); 47 } 48 49 // for 1 + y < 0, the integral is singular, return Cauchy principal value 50 T result; 51 if(y < -1) 52 { 53 result = sqrt(1 / -y) * detail::ellint_rc_imp(T(-y), T(-1 - y), pol); 54 } 55 else if(y == 0) 56 { 57 result = 1; 58 } 59 else if(y > 0) 60 { 61 result = atan(sqrt(y)) / sqrt(y); 62 } 63 else 64 { 65 if(y > -0.5) 66 { 67 T arg = sqrt(-y); 68 result = (boost::math::log1p(arg) - boost::math::log1p(-arg)) / (2 * sqrt(-y)); 69 } 70 else 71 { 72 result = log((1 + sqrt(-y)) / sqrt(1 + y)) / sqrt(-y); 73 } 74 } 75 return result; 76 } 77 78 template <typename T, typename Policy> 79 T ellint_rj_imp(T x, T y, T z, T p, const Policy& pol) 80 { 81 BOOST_MATH_STD_USING 82 83 static const char* function = "boost::math::ellint_rj<%1%>(%1%,%1%,%1%)"; 84 85 if(x < 0) 86 { 87 return policies::raise_domain_error<T>(function, 88 "Argument x must be non-negative, but got x = %1%", x, pol); 89 } 90 if(y < 0) 91 { 92 return policies::raise_domain_error<T>(function, 93 "Argument y must be non-negative, but got y = %1%", y, pol); 94 } 95 if(z < 0) 96 { 97 return policies::raise_domain_error<T>(function, 98 "Argument z must be non-negative, but got z = %1%", z, pol); 99 } 100 if(p == 0) 101 { 102 return policies::raise_domain_error<T>(function, 103 "Argument p must not be zero, but got p = %1%", p, pol); 104 } 105 if(x + y == 0 || y + z == 0 || z + x == 0) 106 { 107 return policies::raise_domain_error<T>(function, 108 "At most one argument can be zero, " 109 "only possible result is %1%.", std::numeric_limits<T>::quiet_NaN(), pol); 110 } 111 112 // for p < 0, the integral is singular, return Cauchy principal value 113 if(p < 0) 114 { 115 // 116 // We must ensure that x < y < z. 117 // Since the integral is symmetrical in x, y and z 118 // we can just permute the values: 119 // 120 if(x > y) 121 std::swap(x, y); 122 if(y > z) 123 std::swap(y, z); 124 if(x > y) 125 std::swap(x, y); 126 127 BOOST_ASSERT(x <= y); 128 BOOST_ASSERT(y <= z); 129 130 T q = -p; 131 p = (z * (x + y + q) - x * y) / (z + q); 132 133 BOOST_ASSERT(p >= 0); 134 135 T value = (p - z) * ellint_rj_imp(x, y, z, p, pol); 136 value -= 3 * ellint_rf_imp(x, y, z, pol); 137 value += 3 * sqrt((x * y * z) / (x * y + p * q)) * ellint_rc_imp(T(x * y + p * q), T(p * q), pol); 138 value /= (z + q); 139 return value; 140 } 141 142 // 143 // Special cases from http://dlmf.nist.gov/19.20#iii 144 // 145 if(x == y) 146 { 147 if(x == z) 148 { 149 if(x == p) 150 { 151 // All values equal: 152 return 1 / (x * sqrt(x)); 153 } 154 else 155 { 156 // x = y = z: 157 return 3 * (ellint_rc_imp(x, p, pol) - 1 / sqrt(x)) / (x - p); 158 } 159 } 160 else 161 { 162 // x = y only, permute so y = z: 163 using std::swap; 164 swap(x, z); 165 if(y == p) 166 { 167 return ellint_rd_imp(x, y, y, pol); 168 } 169 else if((std::max)(y, p) / (std::min)(y, p) > 1.2) 170 { 171 return 3 * (ellint_rc_imp(x, y, pol) - ellint_rc_imp(x, p, pol)) / (p - y); 172 } 173 // Otherwise fall through to normal method, special case above will suffer too much cancellation... 174 } 175 } 176 if(y == z) 177 { 178 if(y == p) 179 { 180 // y = z = p: 181 return ellint_rd_imp(x, y, y, pol); 182 } 183 else if((std::max)(y, p) / (std::min)(y, p) > 1.2) 184 { 185 // y = z: 186 return 3 * (ellint_rc_imp(x, y, pol) - ellint_rc_imp(x, p, pol)) / (p - y); 187 } 188 // Otherwise fall through to normal method, special case above will suffer too much cancellation... 189 } 190 if(z == p) 191 { 192 return ellint_rd_imp(x, y, z, pol); 193 } 194 195 T xn = x; 196 T yn = y; 197 T zn = z; 198 T pn = p; 199 T An = (x + y + z + 2 * p) / 5; 200 T A0 = An; 201 T delta = (p - x) * (p - y) * (p - z); 202 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))); 203 204 unsigned n; 205 T lambda; 206 T Dn; 207 T En; 208 T rx, ry, rz, rp; 209 T fmn = 1; // 4^-n 210 T RC_sum = 0; 211 212 for(n = 0; n < policies::get_max_series_iterations<Policy>(); ++n) 213 { 214 rx = sqrt(xn); 215 ry = sqrt(yn); 216 rz = sqrt(zn); 217 rp = sqrt(pn); 218 Dn = (rp + rx) * (rp + ry) * (rp + rz); 219 En = delta / Dn; 220 En /= Dn; 221 if((En < -0.5) && (En > -1.5)) 222 { 223 // 224 // Occationally En ~ -1, we then have no means of calculating 225 // RC(1, 1+En) without terrible cancellation error, so we 226 // need to get to 1+En directly. By substitution we have 227 // 228 // 1+E_0 = 1 + (p-x)*(p-y)*(p-z)/((sqrt(p) + sqrt(x))*(sqrt(p)+sqrt(y))*(sqrt(p)+sqrt(z)))^2 229 // = 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)))) 230 // 231 // And since this is just an application of the duplication formula for RJ, the same 232 // expression works for 1+En if we use x,y,z,p_n etc. 233 // This branch is taken only once or twice at the start of iteration, 234 // after than En reverts to it's usual very small values. 235 // 236 T b = 2 * rp * (pn + rx * (ry + rz) + ry * rz) / Dn; 237 RC_sum += fmn / Dn * detail::ellint_rc_imp(T(1), b, pol); 238 } 239 else 240 { 241 RC_sum += fmn / Dn * ellint_rc1p_imp(En, pol); 242 } 243 lambda = rx * ry + rx * rz + ry * rz; 244 245 // From here on we move to n+1: 246 An = (An + lambda) / 4; 247 fmn /= 4; 248 249 if(fmn * Q < An) 250 break; 251 252 xn = (xn + lambda) / 4; 253 yn = (yn + lambda) / 4; 254 zn = (zn + lambda) / 4; 255 pn = (pn + lambda) / 4; 256 delta /= 64; 257 } 258 259 T X = fmn * (A0 - x) / An; 260 T Y = fmn * (A0 - y) / An; 261 T Z = fmn * (A0 - z) / An; 262 T P = (-X - Y - Z) / 2; 263 T E2 = X * Y + X * Z + Y * Z - 3 * P * P; 264 T E3 = X * Y * Z + 2 * E2 * P + 4 * P * P * P; 265 T E4 = (2 * X * Y * Z + E2 * P + 3 * P * P * P) * P; 266 T E5 = X * Y * Z * P * P; 267 T result = fmn * pow(An, T(-3) / 2) * 268 (1 - 3 * E2 / 14 + E3 / 6 + 9 * E2 * E2 / 88 - 3 * E4 / 22 - 9 * E2 * E3 / 52 + 3 * E5 / 26 - E2 * E2 * E2 / 16 269 + 3 * E3 * E3 / 40 + 3 * E2 * E4 / 20 + 45 * E2 * E2 * E3 / 272 - 9 * (E3 * E4 + E2 * E5) / 68); 270 271 result += 6 * RC_sum; 272 return result; 273 } 274 275 } // namespace detail 276 277 template <class T1, class T2, class T3, class T4, class Policy> 278 inline typename tools::promote_args<T1, T2, T3, T4>::type ellint_rj(T1 x,T2 y,T3 z,T4 p,const Policy & pol)279 ellint_rj(T1 x, T2 y, T3 z, T4 p, const Policy& pol) 280 { 281 typedef typename tools::promote_args<T1, T2, T3, T4>::type result_type; 282 typedef typename policies::evaluation<result_type, Policy>::type value_type; 283 return policies::checked_narrowing_cast<result_type, Policy>( 284 detail::ellint_rj_imp( 285 static_cast<value_type>(x), 286 static_cast<value_type>(y), 287 static_cast<value_type>(z), 288 static_cast<value_type>(p), 289 pol), "boost::math::ellint_rj<%1%>(%1%,%1%,%1%,%1%)"); 290 } 291 292 template <class T1, class T2, class T3, class T4> 293 inline typename tools::promote_args<T1, T2, T3, T4>::type ellint_rj(T1 x,T2 y,T3 z,T4 p)294 ellint_rj(T1 x, T2 y, T3 z, T4 p) 295 { 296 return ellint_rj(x, y, z, p, policies::policy<>()); 297 } 298 299 }} // namespaces 300 301 #endif // BOOST_MATH_ELLINT_RJ_HPP 302 303