1 // Special functions -*- C++ -*- 2 3 // Copyright (C) 2006-2014 Free Software Foundation, Inc. 4 // 5 // This file is part of the GNU ISO C++ Library. This library is free 6 // software; you can redistribute it and/or modify it under the 7 // terms of the GNU General Public License as published by the 8 // Free Software Foundation; either version 3, or (at your option) 9 // any later version. 10 // 11 // This library is distributed in the hope that it will be useful, 12 // but WITHOUT ANY WARRANTY; without even the implied warranty of 13 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 14 // GNU General Public License for more details. 15 // 16 // Under Section 7 of GPL version 3, you are granted additional 17 // permissions described in the GCC Runtime Library Exception, version 18 // 3.1, as published by the Free Software Foundation. 19 20 // You should have received a copy of the GNU General Public License and 21 // a copy of the GCC Runtime Library Exception along with this program; 22 // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see 23 // <http://www.gnu.org/licenses/>. 24 25 /** @file tr1/hypergeometric.tcc 26 * This is an internal header file, included by other library headers. 27 * Do not attempt to use it directly. @headername{tr1/cmath} 28 */ 29 30 // 31 // ISO C++ 14882 TR1: 5.2 Special functions 32 // 33 34 // Written by Edward Smith-Rowland based: 35 // (1) Handbook of Mathematical Functions, 36 // ed. Milton Abramowitz and Irene A. Stegun, 37 // Dover Publications, 38 // Section 6, pp. 555-566 39 // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl 40 41 #ifndef _GLIBCXX_TR1_HYPERGEOMETRIC_TCC 42 #define _GLIBCXX_TR1_HYPERGEOMETRIC_TCC 1 43 44 namespace std _GLIBCXX_VISIBILITY(default) 45 { 46 namespace tr1 47 { 48 // [5.2] Special functions 49 50 // Implementation-space details. 51 namespace __detail 52 { 53 _GLIBCXX_BEGIN_NAMESPACE_VERSION 54 55 /** 56 * @brief This routine returns the confluent hypergeometric function 57 * by series expansion. 58 * 59 * @f[ 60 * _1F_1(a;c;x) = \frac{\Gamma(c)}{\Gamma(a)} 61 * \sum_{n=0}^{\infty} 62 * \frac{\Gamma(a+n)}{\Gamma(c+n)} 63 * \frac{x^n}{n!} 64 * @f] 65 * 66 * If a and b are integers and a < 0 and either b > 0 or b < a 67 * then the series is a polynomial with a finite number of 68 * terms. If b is an integer and b <= 0 the confluent 69 * hypergeometric function is undefined. 70 * 71 * @param __a The "numerator" parameter. 72 * @param __c The "denominator" parameter. 73 * @param __x The argument of the confluent hypergeometric function. 74 * @return The confluent hypergeometric function. 75 */ 76 template<typename _Tp> 77 _Tp __conf_hyperg_series(_Tp __a,_Tp __c,_Tp __x)78 __conf_hyperg_series(_Tp __a, _Tp __c, _Tp __x) 79 { 80 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 81 82 _Tp __term = _Tp(1); 83 _Tp __Fac = _Tp(1); 84 const unsigned int __max_iter = 100000; 85 unsigned int __i; 86 for (__i = 0; __i < __max_iter; ++__i) 87 { 88 __term *= (__a + _Tp(__i)) * __x 89 / ((__c + _Tp(__i)) * _Tp(1 + __i)); 90 if (std::abs(__term) < __eps) 91 { 92 break; 93 } 94 __Fac += __term; 95 } 96 if (__i == __max_iter) 97 std::__throw_runtime_error(__N("Series failed to converge " 98 "in __conf_hyperg_series.")); 99 100 return __Fac; 101 } 102 103 104 /** 105 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ 106 * by an iterative procedure described in 107 * Luke, Algorithms for the Computation of Mathematical Functions. 108 * 109 * Like the case of the 2F1 rational approximations, these are 110 * probably guaranteed to converge for x < 0, barring gross 111 * numerical instability in the pre-asymptotic regime. 112 */ 113 template<typename _Tp> 114 _Tp __conf_hyperg_luke(_Tp __a,_Tp __c,_Tp __xin)115 __conf_hyperg_luke(_Tp __a, _Tp __c, _Tp __xin) 116 { 117 const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L)); 118 const int __nmax = 20000; 119 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 120 const _Tp __x = -__xin; 121 const _Tp __x3 = __x * __x * __x; 122 const _Tp __t0 = __a / __c; 123 const _Tp __t1 = (__a + _Tp(1)) / (_Tp(2) * __c); 124 const _Tp __t2 = (__a + _Tp(2)) / (_Tp(2) * (__c + _Tp(1))); 125 _Tp __F = _Tp(1); 126 _Tp __prec; 127 128 _Tp __Bnm3 = _Tp(1); 129 _Tp __Bnm2 = _Tp(1) + __t1 * __x; 130 _Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x); 131 132 _Tp __Anm3 = _Tp(1); 133 _Tp __Anm2 = __Bnm2 - __t0 * __x; 134 _Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x 135 + __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x; 136 137 int __n = 3; 138 while(1) 139 { 140 _Tp __npam1 = _Tp(__n - 1) + __a; 141 _Tp __npcm1 = _Tp(__n - 1) + __c; 142 _Tp __npam2 = _Tp(__n - 2) + __a; 143 _Tp __npcm2 = _Tp(__n - 2) + __c; 144 _Tp __tnm1 = _Tp(2 * __n - 1); 145 _Tp __tnm3 = _Tp(2 * __n - 3); 146 _Tp __tnm5 = _Tp(2 * __n - 5); 147 _Tp __F1 = (_Tp(__n - 2) - __a) / (_Tp(2) * __tnm3 * __npcm1); 148 _Tp __F2 = (_Tp(__n) + __a) * __npam1 149 / (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1); 150 _Tp __F3 = -__npam2 * __npam1 * (_Tp(__n - 2) - __a) 151 / (_Tp(8) * __tnm3 * __tnm3 * __tnm5 152 * (_Tp(__n - 3) + __c) * __npcm2 * __npcm1); 153 _Tp __E = -__npam1 * (_Tp(__n - 1) - __c) 154 / (_Tp(2) * __tnm3 * __npcm2 * __npcm1); 155 156 _Tp __An = (_Tp(1) + __F1 * __x) * __Anm1 157 + (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3; 158 _Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1 159 + (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3; 160 _Tp __r = __An / __Bn; 161 162 __prec = std::abs((__F - __r) / __F); 163 __F = __r; 164 165 if (__prec < __eps || __n > __nmax) 166 break; 167 168 if (std::abs(__An) > __big || std::abs(__Bn) > __big) 169 { 170 __An /= __big; 171 __Bn /= __big; 172 __Anm1 /= __big; 173 __Bnm1 /= __big; 174 __Anm2 /= __big; 175 __Bnm2 /= __big; 176 __Anm3 /= __big; 177 __Bnm3 /= __big; 178 } 179 else if (std::abs(__An) < _Tp(1) / __big 180 || std::abs(__Bn) < _Tp(1) / __big) 181 { 182 __An *= __big; 183 __Bn *= __big; 184 __Anm1 *= __big; 185 __Bnm1 *= __big; 186 __Anm2 *= __big; 187 __Bnm2 *= __big; 188 __Anm3 *= __big; 189 __Bnm3 *= __big; 190 } 191 192 ++__n; 193 __Bnm3 = __Bnm2; 194 __Bnm2 = __Bnm1; 195 __Bnm1 = __Bn; 196 __Anm3 = __Anm2; 197 __Anm2 = __Anm1; 198 __Anm1 = __An; 199 } 200 201 if (__n >= __nmax) 202 std::__throw_runtime_error(__N("Iteration failed to converge " 203 "in __conf_hyperg_luke.")); 204 205 return __F; 206 } 207 208 209 /** 210 * @brief Return the confluent hypogeometric function 211 * @f$ _1F_1(a;c;x) @f$. 212 * 213 * @todo Handle b == nonpositive integer blowup - return NaN. 214 * 215 * @param __a The @a numerator parameter. 216 * @param __c The @a denominator parameter. 217 * @param __x The argument of the confluent hypergeometric function. 218 * @return The confluent hypergeometric function. 219 */ 220 template<typename _Tp> 221 _Tp __conf_hyperg(_Tp __a,_Tp __c,_Tp __x)222 __conf_hyperg(_Tp __a, _Tp __c, _Tp __x) 223 { 224 #if _GLIBCXX_USE_C99_MATH_TR1 225 const _Tp __c_nint = std::tr1::nearbyint(__c); 226 #else 227 const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L)); 228 #endif 229 if (__isnan(__a) || __isnan(__c) || __isnan(__x)) 230 return std::numeric_limits<_Tp>::quiet_NaN(); 231 else if (__c_nint == __c && __c_nint <= 0) 232 return std::numeric_limits<_Tp>::infinity(); 233 else if (__a == _Tp(0)) 234 return _Tp(1); 235 else if (__c == __a) 236 return std::exp(__x); 237 else if (__x < _Tp(0)) 238 return __conf_hyperg_luke(__a, __c, __x); 239 else 240 return __conf_hyperg_series(__a, __c, __x); 241 } 242 243 244 /** 245 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ 246 * by series expansion. 247 * 248 * The hypogeometric function is defined by 249 * @f[ 250 * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} 251 * \sum_{n=0}^{\infty} 252 * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} 253 * \frac{x^n}{n!} 254 * @f] 255 * 256 * This works and it's pretty fast. 257 * 258 * @param __a The first @a numerator parameter. 259 * @param __a The second @a numerator parameter. 260 * @param __c The @a denominator parameter. 261 * @param __x The argument of the confluent hypergeometric function. 262 * @return The confluent hypergeometric function. 263 */ 264 template<typename _Tp> 265 _Tp __hyperg_series(_Tp __a,_Tp __b,_Tp __c,_Tp __x)266 __hyperg_series(_Tp __a, _Tp __b, _Tp __c, _Tp __x) 267 { 268 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 269 270 _Tp __term = _Tp(1); 271 _Tp __Fabc = _Tp(1); 272 const unsigned int __max_iter = 100000; 273 unsigned int __i; 274 for (__i = 0; __i < __max_iter; ++__i) 275 { 276 __term *= (__a + _Tp(__i)) * (__b + _Tp(__i)) * __x 277 / ((__c + _Tp(__i)) * _Tp(1 + __i)); 278 if (std::abs(__term) < __eps) 279 { 280 break; 281 } 282 __Fabc += __term; 283 } 284 if (__i == __max_iter) 285 std::__throw_runtime_error(__N("Series failed to converge " 286 "in __hyperg_series.")); 287 288 return __Fabc; 289 } 290 291 292 /** 293 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ 294 * by an iterative procedure described in 295 * Luke, Algorithms for the Computation of Mathematical Functions. 296 */ 297 template<typename _Tp> 298 _Tp __hyperg_luke(_Tp __a,_Tp __b,_Tp __c,_Tp __xin)299 __hyperg_luke(_Tp __a, _Tp __b, _Tp __c, _Tp __xin) 300 { 301 const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L)); 302 const int __nmax = 20000; 303 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 304 const _Tp __x = -__xin; 305 const _Tp __x3 = __x * __x * __x; 306 const _Tp __t0 = __a * __b / __c; 307 const _Tp __t1 = (__a + _Tp(1)) * (__b + _Tp(1)) / (_Tp(2) * __c); 308 const _Tp __t2 = (__a + _Tp(2)) * (__b + _Tp(2)) 309 / (_Tp(2) * (__c + _Tp(1))); 310 311 _Tp __F = _Tp(1); 312 313 _Tp __Bnm3 = _Tp(1); 314 _Tp __Bnm2 = _Tp(1) + __t1 * __x; 315 _Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x); 316 317 _Tp __Anm3 = _Tp(1); 318 _Tp __Anm2 = __Bnm2 - __t0 * __x; 319 _Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x 320 + __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x; 321 322 int __n = 3; 323 while (1) 324 { 325 const _Tp __npam1 = _Tp(__n - 1) + __a; 326 const _Tp __npbm1 = _Tp(__n - 1) + __b; 327 const _Tp __npcm1 = _Tp(__n - 1) + __c; 328 const _Tp __npam2 = _Tp(__n - 2) + __a; 329 const _Tp __npbm2 = _Tp(__n - 2) + __b; 330 const _Tp __npcm2 = _Tp(__n - 2) + __c; 331 const _Tp __tnm1 = _Tp(2 * __n - 1); 332 const _Tp __tnm3 = _Tp(2 * __n - 3); 333 const _Tp __tnm5 = _Tp(2 * __n - 5); 334 const _Tp __n2 = __n * __n; 335 const _Tp __F1 = (_Tp(3) * __n2 + (__a + __b - _Tp(6)) * __n 336 + _Tp(2) - __a * __b - _Tp(2) * (__a + __b)) 337 / (_Tp(2) * __tnm3 * __npcm1); 338 const _Tp __F2 = -(_Tp(3) * __n2 - (__a + __b + _Tp(6)) * __n 339 + _Tp(2) - __a * __b) * __npam1 * __npbm1 340 / (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1); 341 const _Tp __F3 = (__npam2 * __npam1 * __npbm2 * __npbm1 342 * (_Tp(__n - 2) - __a) * (_Tp(__n - 2) - __b)) 343 / (_Tp(8) * __tnm3 * __tnm3 * __tnm5 344 * (_Tp(__n - 3) + __c) * __npcm2 * __npcm1); 345 const _Tp __E = -__npam1 * __npbm1 * (_Tp(__n - 1) - __c) 346 / (_Tp(2) * __tnm3 * __npcm2 * __npcm1); 347 348 _Tp __An = (_Tp(1) + __F1 * __x) * __Anm1 349 + (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3; 350 _Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1 351 + (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3; 352 const _Tp __r = __An / __Bn; 353 354 const _Tp __prec = std::abs((__F - __r) / __F); 355 __F = __r; 356 357 if (__prec < __eps || __n > __nmax) 358 break; 359 360 if (std::abs(__An) > __big || std::abs(__Bn) > __big) 361 { 362 __An /= __big; 363 __Bn /= __big; 364 __Anm1 /= __big; 365 __Bnm1 /= __big; 366 __Anm2 /= __big; 367 __Bnm2 /= __big; 368 __Anm3 /= __big; 369 __Bnm3 /= __big; 370 } 371 else if (std::abs(__An) < _Tp(1) / __big 372 || std::abs(__Bn) < _Tp(1) / __big) 373 { 374 __An *= __big; 375 __Bn *= __big; 376 __Anm1 *= __big; 377 __Bnm1 *= __big; 378 __Anm2 *= __big; 379 __Bnm2 *= __big; 380 __Anm3 *= __big; 381 __Bnm3 *= __big; 382 } 383 384 ++__n; 385 __Bnm3 = __Bnm2; 386 __Bnm2 = __Bnm1; 387 __Bnm1 = __Bn; 388 __Anm3 = __Anm2; 389 __Anm2 = __Anm1; 390 __Anm1 = __An; 391 } 392 393 if (__n >= __nmax) 394 std::__throw_runtime_error(__N("Iteration failed to converge " 395 "in __hyperg_luke.")); 396 397 return __F; 398 } 399 400 401 /** 402 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ 403 * by the reflection formulae in Abramowitz & Stegun formula 404 * 15.3.6 for d = c - a - b not integral and formula 15.3.11 for 405 * d = c - a - b integral. This assumes a, b, c != negative 406 * integer. 407 * 408 * The hypogeometric function is defined by 409 * @f[ 410 * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} 411 * \sum_{n=0}^{\infty} 412 * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} 413 * \frac{x^n}{n!} 414 * @f] 415 * 416 * The reflection formula for nonintegral @f$ d = c - a - b @f$ is: 417 * @f[ 418 * _2F_1(a,b;c;x) = \frac{\Gamma(c)\Gamma(d)}{\Gamma(c-a)\Gamma(c-b)} 419 * _2F_1(a,b;1-d;1-x) 420 * + \frac{\Gamma(c)\Gamma(-d)}{\Gamma(a)\Gamma(b)} 421 * _2F_1(c-a,c-b;1+d;1-x) 422 * @f] 423 * 424 * The reflection formula for integral @f$ m = c - a - b @f$ is: 425 * @f[ 426 * _2F_1(a,b;a+b+m;x) = \frac{\Gamma(m)\Gamma(a+b+m)}{\Gamma(a+m)\Gamma(b+m)} 427 * \sum_{k=0}^{m-1} \frac{(m+a)_k(m+b)_k}{k!(1-m)_k} 428 * - 429 * @f] 430 */ 431 template<typename _Tp> 432 _Tp __hyperg_reflect(_Tp __a,_Tp __b,_Tp __c,_Tp __x)433 __hyperg_reflect(_Tp __a, _Tp __b, _Tp __c, _Tp __x) 434 { 435 const _Tp __d = __c - __a - __b; 436 const int __intd = std::floor(__d + _Tp(0.5L)); 437 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 438 const _Tp __toler = _Tp(1000) * __eps; 439 const _Tp __log_max = std::log(std::numeric_limits<_Tp>::max()); 440 const bool __d_integer = (std::abs(__d - __intd) < __toler); 441 442 if (__d_integer) 443 { 444 const _Tp __ln_omx = std::log(_Tp(1) - __x); 445 const _Tp __ad = std::abs(__d); 446 _Tp __F1, __F2; 447 448 _Tp __d1, __d2; 449 if (__d >= _Tp(0)) 450 { 451 __d1 = __d; 452 __d2 = _Tp(0); 453 } 454 else 455 { 456 __d1 = _Tp(0); 457 __d2 = __d; 458 } 459 460 const _Tp __lng_c = __log_gamma(__c); 461 462 // Evaluate F1. 463 if (__ad < __eps) 464 { 465 // d = c - a - b = 0. 466 __F1 = _Tp(0); 467 } 468 else 469 { 470 471 bool __ok_d1 = true; 472 _Tp __lng_ad, __lng_ad1, __lng_bd1; 473 __try 474 { 475 __lng_ad = __log_gamma(__ad); 476 __lng_ad1 = __log_gamma(__a + __d1); 477 __lng_bd1 = __log_gamma(__b + __d1); 478 } 479 __catch(...) 480 { 481 __ok_d1 = false; 482 } 483 484 if (__ok_d1) 485 { 486 /* Gamma functions in the denominator are ok. 487 * Proceed with evaluation. 488 */ 489 _Tp __sum1 = _Tp(1); 490 _Tp __term = _Tp(1); 491 _Tp __ln_pre1 = __lng_ad + __lng_c + __d2 * __ln_omx 492 - __lng_ad1 - __lng_bd1; 493 494 /* Do F1 sum. 495 */ 496 for (int __i = 1; __i < __ad; ++__i) 497 { 498 const int __j = __i - 1; 499 __term *= (__a + __d2 + __j) * (__b + __d2 + __j) 500 / (_Tp(1) + __d2 + __j) / __i * (_Tp(1) - __x); 501 __sum1 += __term; 502 } 503 504 if (__ln_pre1 > __log_max) 505 std::__throw_runtime_error(__N("Overflow of gamma functions" 506 " in __hyperg_luke.")); 507 else 508 __F1 = std::exp(__ln_pre1) * __sum1; 509 } 510 else 511 { 512 // Gamma functions in the denominator were not ok. 513 // So the F1 term is zero. 514 __F1 = _Tp(0); 515 } 516 } // end F1 evaluation 517 518 // Evaluate F2. 519 bool __ok_d2 = true; 520 _Tp __lng_ad2, __lng_bd2; 521 __try 522 { 523 __lng_ad2 = __log_gamma(__a + __d2); 524 __lng_bd2 = __log_gamma(__b + __d2); 525 } 526 __catch(...) 527 { 528 __ok_d2 = false; 529 } 530 531 if (__ok_d2) 532 { 533 // Gamma functions in the denominator are ok. 534 // Proceed with evaluation. 535 const int __maxiter = 2000; 536 const _Tp __psi_1 = -__numeric_constants<_Tp>::__gamma_e(); 537 const _Tp __psi_1pd = __psi(_Tp(1) + __ad); 538 const _Tp __psi_apd1 = __psi(__a + __d1); 539 const _Tp __psi_bpd1 = __psi(__b + __d1); 540 541 _Tp __psi_term = __psi_1 + __psi_1pd - __psi_apd1 542 - __psi_bpd1 - __ln_omx; 543 _Tp __fact = _Tp(1); 544 _Tp __sum2 = __psi_term; 545 _Tp __ln_pre2 = __lng_c + __d1 * __ln_omx 546 - __lng_ad2 - __lng_bd2; 547 548 // Do F2 sum. 549 int __j; 550 for (__j = 1; __j < __maxiter; ++__j) 551 { 552 // Values for psi functions use recurrence; 553 // Abramowitz & Stegun 6.3.5 554 const _Tp __term1 = _Tp(1) / _Tp(__j) 555 + _Tp(1) / (__ad + __j); 556 const _Tp __term2 = _Tp(1) / (__a + __d1 + _Tp(__j - 1)) 557 + _Tp(1) / (__b + __d1 + _Tp(__j - 1)); 558 __psi_term += __term1 - __term2; 559 __fact *= (__a + __d1 + _Tp(__j - 1)) 560 * (__b + __d1 + _Tp(__j - 1)) 561 / ((__ad + __j) * __j) * (_Tp(1) - __x); 562 const _Tp __delta = __fact * __psi_term; 563 __sum2 += __delta; 564 if (std::abs(__delta) < __eps * std::abs(__sum2)) 565 break; 566 } 567 if (__j == __maxiter) 568 std::__throw_runtime_error(__N("Sum F2 failed to converge " 569 "in __hyperg_reflect")); 570 571 if (__sum2 == _Tp(0)) 572 __F2 = _Tp(0); 573 else 574 __F2 = std::exp(__ln_pre2) * __sum2; 575 } 576 else 577 { 578 // Gamma functions in the denominator not ok. 579 // So the F2 term is zero. 580 __F2 = _Tp(0); 581 } // end F2 evaluation 582 583 const _Tp __sgn_2 = (__intd % 2 == 1 ? -_Tp(1) : _Tp(1)); 584 const _Tp __F = __F1 + __sgn_2 * __F2; 585 586 return __F; 587 } 588 else 589 { 590 // d = c - a - b not an integer. 591 592 // These gamma functions appear in the denominator, so we 593 // catch their harmless domain errors and set the terms to zero. 594 bool __ok1 = true; 595 _Tp __sgn_g1ca = _Tp(0), __ln_g1ca = _Tp(0); 596 _Tp __sgn_g1cb = _Tp(0), __ln_g1cb = _Tp(0); 597 __try 598 { 599 __sgn_g1ca = __log_gamma_sign(__c - __a); 600 __ln_g1ca = __log_gamma(__c - __a); 601 __sgn_g1cb = __log_gamma_sign(__c - __b); 602 __ln_g1cb = __log_gamma(__c - __b); 603 } 604 __catch(...) 605 { 606 __ok1 = false; 607 } 608 609 bool __ok2 = true; 610 _Tp __sgn_g2a = _Tp(0), __ln_g2a = _Tp(0); 611 _Tp __sgn_g2b = _Tp(0), __ln_g2b = _Tp(0); 612 __try 613 { 614 __sgn_g2a = __log_gamma_sign(__a); 615 __ln_g2a = __log_gamma(__a); 616 __sgn_g2b = __log_gamma_sign(__b); 617 __ln_g2b = __log_gamma(__b); 618 } 619 __catch(...) 620 { 621 __ok2 = false; 622 } 623 624 const _Tp __sgn_gc = __log_gamma_sign(__c); 625 const _Tp __ln_gc = __log_gamma(__c); 626 const _Tp __sgn_gd = __log_gamma_sign(__d); 627 const _Tp __ln_gd = __log_gamma(__d); 628 const _Tp __sgn_gmd = __log_gamma_sign(-__d); 629 const _Tp __ln_gmd = __log_gamma(-__d); 630 631 const _Tp __sgn1 = __sgn_gc * __sgn_gd * __sgn_g1ca * __sgn_g1cb; 632 const _Tp __sgn2 = __sgn_gc * __sgn_gmd * __sgn_g2a * __sgn_g2b; 633 634 _Tp __pre1, __pre2; 635 if (__ok1 && __ok2) 636 { 637 _Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb; 638 _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b 639 + __d * std::log(_Tp(1) - __x); 640 if (__ln_pre1 < __log_max && __ln_pre2 < __log_max) 641 { 642 __pre1 = std::exp(__ln_pre1); 643 __pre2 = std::exp(__ln_pre2); 644 __pre1 *= __sgn1; 645 __pre2 *= __sgn2; 646 } 647 else 648 { 649 std::__throw_runtime_error(__N("Overflow of gamma functions " 650 "in __hyperg_reflect")); 651 } 652 } 653 else if (__ok1 && !__ok2) 654 { 655 _Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb; 656 if (__ln_pre1 < __log_max) 657 { 658 __pre1 = std::exp(__ln_pre1); 659 __pre1 *= __sgn1; 660 __pre2 = _Tp(0); 661 } 662 else 663 { 664 std::__throw_runtime_error(__N("Overflow of gamma functions " 665 "in __hyperg_reflect")); 666 } 667 } 668 else if (!__ok1 && __ok2) 669 { 670 _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b 671 + __d * std::log(_Tp(1) - __x); 672 if (__ln_pre2 < __log_max) 673 { 674 __pre1 = _Tp(0); 675 __pre2 = std::exp(__ln_pre2); 676 __pre2 *= __sgn2; 677 } 678 else 679 { 680 std::__throw_runtime_error(__N("Overflow of gamma functions " 681 "in __hyperg_reflect")); 682 } 683 } 684 else 685 { 686 __pre1 = _Tp(0); 687 __pre2 = _Tp(0); 688 std::__throw_runtime_error(__N("Underflow of gamma functions " 689 "in __hyperg_reflect")); 690 } 691 692 const _Tp __F1 = __hyperg_series(__a, __b, _Tp(1) - __d, 693 _Tp(1) - __x); 694 const _Tp __F2 = __hyperg_series(__c - __a, __c - __b, _Tp(1) + __d, 695 _Tp(1) - __x); 696 697 const _Tp __F = __pre1 * __F1 + __pre2 * __F2; 698 699 return __F; 700 } 701 } 702 703 704 /** 705 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$. 706 * 707 * The hypogeometric function is defined by 708 * @f[ 709 * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} 710 * \sum_{n=0}^{\infty} 711 * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} 712 * \frac{x^n}{n!} 713 * @f] 714 * 715 * @param __a The first @a numerator parameter. 716 * @param __a The second @a numerator parameter. 717 * @param __c The @a denominator parameter. 718 * @param __x The argument of the confluent hypergeometric function. 719 * @return The confluent hypergeometric function. 720 */ 721 template<typename _Tp> 722 _Tp __hyperg(_Tp __a,_Tp __b,_Tp __c,_Tp __x)723 __hyperg(_Tp __a, _Tp __b, _Tp __c, _Tp __x) 724 { 725 #if _GLIBCXX_USE_C99_MATH_TR1 726 const _Tp __a_nint = std::tr1::nearbyint(__a); 727 const _Tp __b_nint = std::tr1::nearbyint(__b); 728 const _Tp __c_nint = std::tr1::nearbyint(__c); 729 #else 730 const _Tp __a_nint = static_cast<int>(__a + _Tp(0.5L)); 731 const _Tp __b_nint = static_cast<int>(__b + _Tp(0.5L)); 732 const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L)); 733 #endif 734 const _Tp __toler = _Tp(1000) * std::numeric_limits<_Tp>::epsilon(); 735 if (std::abs(__x) >= _Tp(1)) 736 std::__throw_domain_error(__N("Argument outside unit circle " 737 "in __hyperg.")); 738 else if (__isnan(__a) || __isnan(__b) 739 || __isnan(__c) || __isnan(__x)) 740 return std::numeric_limits<_Tp>::quiet_NaN(); 741 else if (__c_nint == __c && __c_nint <= _Tp(0)) 742 return std::numeric_limits<_Tp>::infinity(); 743 else if (std::abs(__c - __b) < __toler || std::abs(__c - __a) < __toler) 744 return std::pow(_Tp(1) - __x, __c - __a - __b); 745 else if (__a >= _Tp(0) && __b >= _Tp(0) && __c >= _Tp(0) 746 && __x >= _Tp(0) && __x < _Tp(0.995L)) 747 return __hyperg_series(__a, __b, __c, __x); 748 else if (std::abs(__a) < _Tp(10) && std::abs(__b) < _Tp(10)) 749 { 750 // For integer a and b the hypergeometric function is a 751 // finite polynomial. 752 if (__a < _Tp(0) && std::abs(__a - __a_nint) < __toler) 753 return __hyperg_series(__a_nint, __b, __c, __x); 754 else if (__b < _Tp(0) && std::abs(__b - __b_nint) < __toler) 755 return __hyperg_series(__a, __b_nint, __c, __x); 756 else if (__x < -_Tp(0.25L)) 757 return __hyperg_luke(__a, __b, __c, __x); 758 else if (__x < _Tp(0.5L)) 759 return __hyperg_series(__a, __b, __c, __x); 760 else 761 if (std::abs(__c) > _Tp(10)) 762 return __hyperg_series(__a, __b, __c, __x); 763 else 764 return __hyperg_reflect(__a, __b, __c, __x); 765 } 766 else 767 return __hyperg_luke(__a, __b, __c, __x); 768 } 769 770 _GLIBCXX_END_NAMESPACE_VERSION 771 } // namespace std::tr1::__detail 772 } 773 } 774 775 #endif // _GLIBCXX_TR1_HYPERGEOMETRIC_TCC 776