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$ cat -n /usr/include/c++/13/tr1/hypergeometric.tcc 1 // Special functions -*- C++ -*- 2 3 // Copyright (C) 2006-2023 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 _GLIBCXX_BEGIN_NAMESPACE_VERSION 47 48 #if _GLIBCXX_USE_STD_SPEC_FUNCS 49 # define _GLIBCXX_MATH_NS ::std 50 #elif defined(_GLIBCXX_TR1_CMATH) 51 namespace tr1 52 { 53 # define _GLIBCXX_MATH_NS ::std::tr1 54 #else 55 # error do not include this header directly, use <cmath> or <tr1/cmath> 56 #endif 57 // [5.2] Special functions 58 59 // Implementation-space details. 60 namespace __detail 61 { 62 /** 63 * @brief This routine returns the confluent hypergeometric function 64 * by series expansion. 65 * 66 * @f[ 67 * _1F_1(a;c;x) = \frac{\Gamma(c)}{\Gamma(a)} 68 * \sum_{n=0}^{\infty} 69 * \frac{\Gamma(a+n)}{\Gamma(c+n)} 70 * \frac{x^n}{n!} 71 * @f] 72 * 73 * If a and b are integers and a < 0 and either b > 0 or b < a 74 * then the series is a polynomial with a finite number of 75 * terms. If b is an integer and b <= 0 the confluent 76 * hypergeometric function is undefined. 77 * 78 * @param __a The "numerator" parameter. 79 * @param __c The "denominator" parameter. 80 * @param __x The argument of the confluent hypergeometric function. 81 * @return The confluent hypergeometric function. 82 */ 83 template<typename _Tp> 84 _Tp 85 __conf_hyperg_series(_Tp __a, _Tp __c, _Tp __x) 86 { 87 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 88 89 _Tp __term = _Tp(1); 90 _Tp __Fac = _Tp(1); 91 const unsigned int __max_iter = 100000; 92 unsigned int __i; 93 for (__i = 0; __i < __max_iter; ++__i) 94 { 95 __term *= (__a + _Tp(__i)) * __x 96 / ((__c + _Tp(__i)) * _Tp(1 + __i)); 97 if (std::abs(__term) < __eps) 98 { 99 break; 100 } 101 __Fac += __term; 102 } 103 if (__i == __max_iter) 104 std::__throw_runtime_error(__N("Series failed to converge " 105 "in __conf_hyperg_series.")); 106 107 return __Fac; 108 } 109 110 111 /** 112 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ 113 * by an iterative procedure described in 114 * Luke, Algorithms for the Computation of Mathematical Functions. 115 * 116 * Like the case of the 2F1 rational approximations, these are 117 * probably guaranteed to converge for x < 0, barring gross 118 * numerical instability in the pre-asymptotic regime. 119 */ 120 template<typename _Tp> 121 _Tp 122 __conf_hyperg_luke(_Tp __a, _Tp __c, _Tp __xin) 123 { 124 const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L)); 125 const int __nmax = 20000; 126 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 127 const _Tp __x = -__xin; 128 const _Tp __x3 = __x * __x * __x; 129 const _Tp __t0 = __a / __c; 130 const _Tp __t1 = (__a + _Tp(1)) / (_Tp(2) * __c); 131 const _Tp __t2 = (__a + _Tp(2)) / (_Tp(2) * (__c + _Tp(1))); 132 _Tp __F = _Tp(1); 133 _Tp __prec; 134 135 _Tp __Bnm3 = _Tp(1); 136 _Tp __Bnm2 = _Tp(1) + __t1 * __x; 137 _Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x); 138 139 _Tp __Anm3 = _Tp(1); 140 _Tp __Anm2 = __Bnm2 - __t0 * __x; 141 _Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x 142 + __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x; 143 144 int __n = 3; 145 while(1) 146 { 147 _Tp __npam1 = _Tp(__n - 1) + __a; 148 _Tp __npcm1 = _Tp(__n - 1) + __c; 149 _Tp __npam2 = _Tp(__n - 2) + __a; 150 _Tp __npcm2 = _Tp(__n - 2) + __c; 151 _Tp __tnm1 = _Tp(2 * __n - 1); 152 _Tp __tnm3 = _Tp(2 * __n - 3); 153 _Tp __tnm5 = _Tp(2 * __n - 5); 154 _Tp __F1 = (_Tp(__n - 2) - __a) / (_Tp(2) * __tnm3 * __npcm1); 155 _Tp __F2 = (_Tp(__n) + __a) * __npam1 156 / (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1); 157 _Tp __F3 = -__npam2 * __npam1 * (_Tp(__n - 2) - __a) 158 / (_Tp(8) * __tnm3 * __tnm3 * __tnm5 159 * (_Tp(__n - 3) + __c) * __npcm2 * __npcm1); 160 _Tp __E = -__npam1 * (_Tp(__n - 1) - __c) 161 / (_Tp(2) * __tnm3 * __npcm2 * __npcm1); 162 163 _Tp __An = (_Tp(1) + __F1 * __x) * __Anm1 164 + (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3; 165 _Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1 166 + (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3; 167 _Tp __r = __An / __Bn; 168 169 __prec = std::abs((__F - __r) / __F); 170 __F = __r; 171 172 if (__prec < __eps || __n > __nmax) 173 break; 174 175 if (std::abs(__An) > __big || std::abs(__Bn) > __big) 176 { 177 __An /= __big; 178 __Bn /= __big; 179 __Anm1 /= __big; 180 __Bnm1 /= __big; 181 __Anm2 /= __big; 182 __Bnm2 /= __big; 183 __Anm3 /= __big; 184 __Bnm3 /= __big; 185 } 186 else if (std::abs(__An) < _Tp(1) / __big 187 || std::abs(__Bn) < _Tp(1) / __big) 188 { 189 __An *= __big; 190 __Bn *= __big; 191 __Anm1 *= __big; 192 __Bnm1 *= __big; 193 __Anm2 *= __big; 194 __Bnm2 *= __big; 195 __Anm3 *= __big; 196 __Bnm3 *= __big; 197 } 198 199 ++__n; 200 __Bnm3 = __Bnm2; 201 __Bnm2 = __Bnm1; 202 __Bnm1 = __Bn; 203 __Anm3 = __Anm2; 204 __Anm2 = __Anm1; 205 __Anm1 = __An; 206 } 207 208 if (__n >= __nmax) 209 std::__throw_runtime_error(__N("Iteration failed to converge " 210 "in __conf_hyperg_luke.")); 211 212 return __F; 213 } 214 215 216 /** 217 * @brief Return the confluent hypogeometric function 218 * @f$ _1F_1(a;c;x) @f$. 219 * 220 * @todo Handle b == nonpositive integer blowup - return NaN. 221 * 222 * @param __a The @a numerator parameter. 223 * @param __c The @a denominator parameter. 224 * @param __x The argument of the confluent hypergeometric function. 225 * @return The confluent hypergeometric function. 226 */ 227 template<typename _Tp> 228 _Tp 229 __conf_hyperg(_Tp __a, _Tp __c, _Tp __x) 230 { 231 #if _GLIBCXX_USE_C99_MATH_TR1 232 const _Tp __c_nint = _GLIBCXX_MATH_NS::nearbyint(__c); 233 #else 234 const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L)); 235 #endif 236 if (__isnan(__a) || __isnan(__c) || __isnan(__x)) 237 return std::numeric_limits<_Tp>::quiet_NaN(); 238 else if (__c_nint == __c && __c_nint <= 0) 239 return std::numeric_limits<_Tp>::infinity(); 240 else if (__a == _Tp(0)) 241 return _Tp(1); 242 else if (__c == __a) 243 return std::exp(__x); 244 else if (__x < _Tp(0)) 245 return __conf_hyperg_luke(__a, __c, __x); 246 else 247 return __conf_hyperg_series(__a, __c, __x); 248 } 249 250 251 /** 252 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ 253 * by series expansion. 254 * 255 * The hypogeometric function is defined by 256 * @f[ 257 * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} 258 * \sum_{n=0}^{\infty} 259 * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} 260 * \frac{x^n}{n!} 261 * @f] 262 * 263 * This works and it's pretty fast. 264 * 265 * @param __a The first @a numerator parameter. 266 * @param __a The second @a numerator parameter. 267 * @param __c The @a denominator parameter. 268 * @param __x The argument of the confluent hypergeometric function. 269 * @return The confluent hypergeometric function. 270 */ 271 template<typename _Tp> 272 _Tp 273 __hyperg_series(_Tp __a, _Tp __b, _Tp __c, _Tp __x) 274 { 275 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 276 277 _Tp __term = _Tp(1); 278 _Tp __Fabc = _Tp(1); 279 const unsigned int __max_iter = 100000; 280 unsigned int __i; 281 for (__i = 0; __i < __max_iter; ++__i) 282 { 283 __term *= (__a + _Tp(__i)) * (__b + _Tp(__i)) * __x 284 / ((__c + _Tp(__i)) * _Tp(1 + __i)); 285 if (std::abs(__term) < __eps) 286 { 287 break; 288 } 289 __Fabc += __term; 290 } 291 if (__i == __max_iter) 292 std::__throw_runtime_error(__N("Series failed to converge " 293 "in __hyperg_series.")); 294 295 return __Fabc; 296 } 297 298 299 /** 300 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ 301 * by an iterative procedure described in 302 * Luke, Algorithms for the Computation of Mathematical Functions. 303 */ 304 template<typename _Tp> 305 _Tp 306 __hyperg_luke(_Tp __a, _Tp __b, _Tp __c, _Tp __xin) 307 { 308 const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L)); 309 const int __nmax = 20000; 310 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 311 const _Tp __x = -__xin; 312 const _Tp __x3 = __x * __x * __x; 313 const _Tp __t0 = __a * __b / __c; 314 const _Tp __t1 = (__a + _Tp(1)) * (__b + _Tp(1)) / (_Tp(2) * __c); 315 const _Tp __t2 = (__a + _Tp(2)) * (__b + _Tp(2)) 316 / (_Tp(2) * (__c + _Tp(1))); 317 318 _Tp __F = _Tp(1); 319 320 _Tp __Bnm3 = _Tp(1); 321 _Tp __Bnm2 = _Tp(1) + __t1 * __x; 322 _Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x); 323 324 _Tp __Anm3 = _Tp(1); 325 _Tp __Anm2 = __Bnm2 - __t0 * __x; 326 _Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x 327 + __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x; 328 329 int __n = 3; 330 while (1) 331 { 332 const _Tp __npam1 = _Tp(__n - 1) + __a; 333 const _Tp __npbm1 = _Tp(__n - 1) + __b; 334 const _Tp __npcm1 = _Tp(__n - 1) + __c; 335 const _Tp __npam2 = _Tp(__n - 2) + __a; 336 const _Tp __npbm2 = _Tp(__n - 2) + __b; 337 const _Tp __npcm2 = _Tp(__n - 2) + __c; 338 const _Tp __tnm1 = _Tp(2 * __n - 1); 339 const _Tp __tnm3 = _Tp(2 * __n - 3); 340 const _Tp __tnm5 = _Tp(2 * __n - 5); 341 const _Tp __n2 = __n * __n; 342 const _Tp __F1 = (_Tp(3) * __n2 + (__a + __b - _Tp(6)) * __n 343 + _Tp(2) - __a * __b - _Tp(2) * (__a + __b)) 344 / (_Tp(2) * __tnm3 * __npcm1); 345 const _Tp __F2 = -(_Tp(3) * __n2 - (__a + __b + _Tp(6)) * __n 346 + _Tp(2) - __a * __b) * __npam1 * __npbm1 347 / (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1); 348 const _Tp __F3 = (__npam2 * __npam1 * __npbm2 * __npbm1 349 * (_Tp(__n - 2) - __a) * (_Tp(__n - 2) - __b)) 350 / (_Tp(8) * __tnm3 * __tnm3 * __tnm5 351 * (_Tp(__n - 3) + __c) * __npcm2 * __npcm1); 352 const _Tp __E = -__npam1 * __npbm1 * (_Tp(__n - 1) - __c) 353 / (_Tp(2) * __tnm3 * __npcm2 * __npcm1); 354 355 _Tp __An = (_Tp(1) + __F1 * __x) * __Anm1 356 + (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3; 357 _Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1 358 + (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3; 359 const _Tp __r = __An / __Bn; 360 361 const _Tp __prec = std::abs((__F - __r) / __F); 362 __F = __r; 363 364 if (__prec < __eps || __n > __nmax) 365 break; 366 367 if (std::abs(__An) > __big || std::abs(__Bn) > __big) 368 { 369 __An /= __big; 370 __Bn /= __big; 371 __Anm1 /= __big; 372 __Bnm1 /= __big; 373 __Anm2 /= __big; 374 __Bnm2 /= __big; 375 __Anm3 /= __big; 376 __Bnm3 /= __big; 377 } 378 else if (std::abs(__An) < _Tp(1) / __big 379 || std::abs(__Bn) < _Tp(1) / __big) 380 { 381 __An *= __big; 382 __Bn *= __big; 383 __Anm1 *= __big; 384 __Bnm1 *= __big; 385 __Anm2 *= __big; 386 __Bnm2 *= __big; 387 __Anm3 *= __big; 388 __Bnm3 *= __big; 389 } 390 391 ++__n; 392 __Bnm3 = __Bnm2; 393 __Bnm2 = __Bnm1; 394 __Bnm1 = __Bn; 395 __Anm3 = __Anm2; 396 __Anm2 = __Anm1; 397 __Anm1 = __An; 398 } 399 400 if (__n >= __nmax) 401 std::__throw_runtime_error(__N("Iteration failed to converge " 402 "in __hyperg_luke.")); 403 404 return __F; 405 } 406 407 408 /** 409 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ 410 * by the reflection formulae in Abramowitz & Stegun formula 411 * 15.3.6 for d = c - a - b not integral and formula 15.3.11 for 412 * d = c - a - b integral. This assumes a, b, c != negative 413 * integer. 414 * 415 * The hypogeometric function is defined by 416 * @f[ 417 * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} 418 * \sum_{n=0}^{\infty} 419 * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} 420 * \frac{x^n}{n!} 421 * @f] 422 * 423 * The reflection formula for nonintegral @f$ d = c - a - b @f$ is: 424 * @f[ 425 * _2F_1(a,b;c;x) = \frac{\Gamma(c)\Gamma(d)}{\Gamma(c-a)\Gamma(c-b)} 426 * _2F_1(a,b;1-d;1-x) 427 * + \frac{\Gamma(c)\Gamma(-d)}{\Gamma(a)\Gamma(b)} 428 * _2F_1(c-a,c-b;1+d;1-x) 429 * @f] 430 * 431 * The reflection formula for integral @f$ m = c - a - b @f$ is: 432 * @f[ 433 * _2F_1(a,b;a+b+m;x) = \frac{\Gamma(m)\Gamma(a+b+m)}{\Gamma(a+m)\Gamma(b+m)} 434 * \sum_{k=0}^{m-1} \frac{(m+a)_k(m+b)_k}{k!(1-m)_k} 435 * - 436 * @f] 437 */ 438 template<typename _Tp> 439 _Tp 440 __hyperg_reflect(_Tp __a, _Tp __b, _Tp __c, _Tp __x) 441 { 442 const _Tp __d = __c - __a - __b; 443 const int __intd = std::floor(__d + _Tp(0.5L)); 444 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 445 const _Tp __toler = _Tp(1000) * __eps; 446 const _Tp __log_max = std::log(std::numeric_limits<_Tp>::max()); 447 const bool __d_integer = (std::abs(__d - __intd) < __toler); 448 449 if (__d_integer) 450 { 451 const _Tp __ln_omx = std::log(_Tp(1) - __x); 452 const _Tp __ad = std::abs(__d); 453 _Tp __F1, __F2; 454 455 _Tp __d1, __d2; 456 if (__d >= _Tp(0)) 457 { 458 __d1 = __d; 459 __d2 = _Tp(0); 460 } 461 else 462 { 463 __d1 = _Tp(0); 464 __d2 = __d; 465 } 466 467 const _Tp __lng_c = __log_gamma(__c); 468 469 // Evaluate F1. 470 if (__ad < __eps) 471 { 472 // d = c - a - b = 0. 473 __F1 = _Tp(0); 474 } 475 else 476 { 477 478 bool __ok_d1 = true; 479 _Tp __lng_ad, __lng_ad1, __lng_bd1; 480 __try 481 { 482 __lng_ad = __log_gamma(__ad); 483 __lng_ad1 = __log_gamma(__a + __d1); 484 __lng_bd1 = __log_gamma(__b + __d1); 485 } 486 __catch(...) 487 { 488 __ok_d1 = false; 489 } 490 491 if (__ok_d1) 492 { 493 /* Gamma functions in the denominator are ok. 494 * Proceed with evaluation. 495 */ 496 _Tp __sum1 = _Tp(1); 497 _Tp __term = _Tp(1); 498 _Tp __ln_pre1 = __lng_ad + __lng_c + __d2 * __ln_omx 499 - __lng_ad1 - __lng_bd1; 500 501 /* Do F1 sum. 502 */ 503 for (int __i = 1; __i < __ad; ++__i) 504 { 505 const int __j = __i - 1; 506 __term *= (__a + __d2 + __j) * (__b + __d2 + __j) 507 / (_Tp(1) + __d2 + __j) / __i * (_Tp(1) - __x); 508 __sum1 += __term; 509 } 510 511 if (__ln_pre1 > __log_max) 512 std::__throw_runtime_error(__N("Overflow of gamma functions" 513 " in __hyperg_luke.")); 514 else 515 __F1 = std::exp(__ln_pre1) * __sum1; 516 } 517 else 518 { 519 // Gamma functions in the denominator were not ok. 520 // So the F1 term is zero. 521 __F1 = _Tp(0); 522 } 523 } // end F1 evaluation 524 525 // Evaluate F2. 526 bool __ok_d2 = true; 527 _Tp __lng_ad2, __lng_bd2; 528 __try 529 { 530 __lng_ad2 = __log_gamma(__a + __d2); 531 __lng_bd2 = __log_gamma(__b + __d2); 532 } 533 __catch(...) 534 { 535 __ok_d2 = false; 536 } 537 538 if (__ok_d2) 539 { 540 // Gamma functions in the denominator are ok. 541 // Proceed with evaluation. 542 const int __maxiter = 2000; 543 const _Tp __psi_1 = -__numeric_constants<_Tp>::__gamma_e(); 544 const _Tp __psi_1pd = __psi(_Tp(1) + __ad); 545 const _Tp __psi_apd1 = __psi(__a + __d1); 546 const _Tp __psi_bpd1 = __psi(__b + __d1); 547 548 _Tp __psi_term = __psi_1 + __psi_1pd - __psi_apd1 549 - __psi_bpd1 - __ln_omx; 550 _Tp __fact = _Tp(1); 551 _Tp __sum2 = __psi_term; 552 _Tp __ln_pre2 = __lng_c + __d1 * __ln_omx 553 - __lng_ad2 - __lng_bd2; 554 555 // Do F2 sum. 556 int __j; 557 for (__j = 1; __j < __maxiter; ++__j) 558 { 559 // Values for psi functions use recurrence; 560 // Abramowitz & Stegun 6.3.5 561 const _Tp __term1 = _Tp(1) / _Tp(__j) 562 + _Tp(1) / (__ad + __j); 563 const _Tp __term2 = _Tp(1) / (__a + __d1 + _Tp(__j - 1)) 564 + _Tp(1) / (__b + __d1 + _Tp(__j - 1)); 565 __psi_term += __term1 - __term2; 566 __fact *= (__a + __d1 + _Tp(__j - 1)) 567 * (__b + __d1 + _Tp(__j - 1)) 568 / ((__ad + __j) * __j) * (_Tp(1) - __x); 569 const _Tp __delta = __fact * __psi_term; 570 __sum2 += __delta; 571 if (std::abs(__delta) < __eps * std::abs(__sum2)) 572 break; 573 } 574 if (__j == __maxiter) 575 std::__throw_runtime_error(__N("Sum F2 failed to converge " 576 "in __hyperg_reflect")); 577 578 if (__sum2 == _Tp(0)) 579 __F2 = _Tp(0); 580 else 581 __F2 = std::exp(__ln_pre2) * __sum2; 582 } 583 else 584 { 585 // Gamma functions in the denominator not ok. 586 // So the F2 term is zero. 587 __F2 = _Tp(0); 588 } // end F2 evaluation 589 590 const _Tp __sgn_2 = (__intd % 2 == 1 ? -_Tp(1) : _Tp(1)); 591 const _Tp __F = __F1 + __sgn_2 * __F2; 592 593 return __F; 594 } 595 else 596 { 597 // d = c - a - b not an integer. 598 599 // These gamma functions appear in the denominator, so we 600 // catch their harmless domain errors and set the terms to zero. 601 bool __ok1 = true; 602 _Tp __sgn_g1ca = _Tp(0), __ln_g1ca = _Tp(0); 603 _Tp __sgn_g1cb = _Tp(0), __ln_g1cb = _Tp(0); 604 __try 605 { 606 __sgn_g1ca = __log_gamma_sign(__c - __a); 607 __ln_g1ca = __log_gamma(__c - __a); 608 __sgn_g1cb = __log_gamma_sign(__c - __b); 609 __ln_g1cb = __log_gamma(__c - __b); 610 } 611 __catch(...) 612 { 613 __ok1 = false; 614 } 615 616 bool __ok2 = true; 617 _Tp __sgn_g2a = _Tp(0), __ln_g2a = _Tp(0); 618 _Tp __sgn_g2b = _Tp(0), __ln_g2b = _Tp(0); 619 __try 620 { 621 __sgn_g2a = __log_gamma_sign(__a); 622 __ln_g2a = __log_gamma(__a); 623 __sgn_g2b = __log_gamma_sign(__b); 624 __ln_g2b = __log_gamma(__b); 625 } 626 __catch(...) 627 { 628 __ok2 = false; 629 } 630 631 const _Tp __sgn_gc = __log_gamma_sign(__c); 632 const _Tp __ln_gc = __log_gamma(__c); 633 const _Tp __sgn_gd = __log_gamma_sign(__d); 634 const _Tp __ln_gd = __log_gamma(__d); 635 const _Tp __sgn_gmd = __log_gamma_sign(-__d); 636 const _Tp __ln_gmd = __log_gamma(-__d); 637 638 const _Tp __sgn1 = __sgn_gc * __sgn_gd * __sgn_g1ca * __sgn_g1cb; 639 const _Tp __sgn2 = __sgn_gc * __sgn_gmd * __sgn_g2a * __sgn_g2b; 640 641 _Tp __pre1, __pre2; 642 if (__ok1 && __ok2) 643 { 644 _Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb; 645 _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b 646 + __d * std::log(_Tp(1) - __x); 647 if (__ln_pre1 < __log_max && __ln_pre2 < __log_max) 648 { 649 __pre1 = std::exp(__ln_pre1); 650 __pre2 = std::exp(__ln_pre2); 651 __pre1 *= __sgn1; 652 __pre2 *= __sgn2; 653 } 654 else 655 { 656 std::__throw_runtime_error(__N("Overflow of gamma functions " 657 "in __hyperg_reflect")); 658 } 659 } 660 else if (__ok1 && !__ok2) 661 { 662 _Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb; 663 if (__ln_pre1 < __log_max) 664 { 665 __pre1 = std::exp(__ln_pre1); 666 __pre1 *= __sgn1; 667 __pre2 = _Tp(0); 668 } 669 else 670 { 671 std::__throw_runtime_error(__N("Overflow of gamma functions " 672 "in __hyperg_reflect")); 673 } 674 } 675 else if (!__ok1 && __ok2) 676 { 677 _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b 678 + __d * std::log(_Tp(1) - __x); 679 if (__ln_pre2 < __log_max) 680 { 681 __pre1 = _Tp(0); 682 __pre2 = std::exp(__ln_pre2); 683 __pre2 *= __sgn2; 684 } 685 else 686 { 687 std::__throw_runtime_error(__N("Overflow of gamma functions " 688 "in __hyperg_reflect")); 689 } 690 } 691 else 692 { 693 __pre1 = _Tp(0); 694 __pre2 = _Tp(0); 695 std::__throw_runtime_error(__N("Underflow of gamma functions " 696 "in __hyperg_reflect")); 697 } 698 699 const _Tp __F1 = __hyperg_series(__a, __b, _Tp(1) - __d, 700 _Tp(1) - __x); 701 const _Tp __F2 = __hyperg_series(__c - __a, __c - __b, _Tp(1) + __d, 702 _Tp(1) - __x); 703 704 const _Tp __F = __pre1 * __F1 + __pre2 * __F2; 705 706 return __F; 707 } 708 } 709 710 711 /** 712 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$. 713 * 714 * The hypogeometric function is defined by 715 * @f[ 716 * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} 717 * \sum_{n=0}^{\infty} 718 * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} 719 * \frac{x^n}{n!} 720 * @f] 721 * 722 * @param __a The first @a numerator parameter. 723 * @param __a The second @a numerator parameter. 724 * @param __c The @a denominator parameter. 725 * @param __x The argument of the confluent hypergeometric function. 726 * @return The confluent hypergeometric function. 727 */ 728 template<typename _Tp> 729 _Tp 730 __hyperg(_Tp __a, _Tp __b, _Tp __c, _Tp __x) 731 { 732 #if _GLIBCXX_USE_C99_MATH_TR1 733 const _Tp __a_nint = _GLIBCXX_MATH_NS::nearbyint(__a); 734 const _Tp __b_nint = _GLIBCXX_MATH_NS::nearbyint(__b); 735 const _Tp __c_nint = _GLIBCXX_MATH_NS::nearbyint(__c); 736 #else 737 const _Tp __a_nint = static_cast<int>(__a + _Tp(0.5L)); 738 const _Tp __b_nint = static_cast<int>(__b + _Tp(0.5L)); 739 const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L)); 740 #endif 741 const _Tp __toler = _Tp(1000) * std::numeric_limits<_Tp>::epsilon(); 742 if (std::abs(__x) >= _Tp(1)) 743 std::__throw_domain_error(__N("Argument outside unit circle " 744 "in __hyperg.")); 745 else if (__isnan(__a) || __isnan(__b) 746 || __isnan(__c) || __isnan(__x)) 747 return std::numeric_limits<_Tp>::quiet_NaN(); 748 else if (__c_nint == __c && __c_nint <= _Tp(0)) 749 return std::numeric_limits<_Tp>::infinity(); 750 else if (std::abs(__c - __b) < __toler || std::abs(__c - __a) < __toler) 751 return std::pow(_Tp(1) - __x, __c - __a - __b); 752 else if (__a >= _Tp(0) && __b >= _Tp(0) && __c >= _Tp(0) 753 && __x >= _Tp(0) && __x < _Tp(0.995L)) 754 return __hyperg_series(__a, __b, __c, __x); 755 else if (std::abs(__a) < _Tp(10) && std::abs(__b) < _Tp(10)) 756 { 757 // For integer a and b the hypergeometric function is a 758 // finite polynomial. 759 if (__a < _Tp(0) && std::abs(__a - __a_nint) < __toler) 760 return __hyperg_series(__a_nint, __b, __c, __x); 761 else if (__b < _Tp(0) && std::abs(__b - __b_nint) < __toler) 762 return __hyperg_series(__a, __b_nint, __c, __x); 763 else if (__x < -_Tp(0.25L)) 764 return __hyperg_luke(__a, __b, __c, __x); 765 else if (__x < _Tp(0.5L)) 766 return __hyperg_series(__a, __b, __c, __x); 767 else 768 if (std::abs(__c) > _Tp(10)) 769 return __hyperg_series(__a, __b, __c, __x); 770 else 771 return __hyperg_reflect(__a, __b, __c, __x); 772 } 773 else 774 return __hyperg_luke(__a, __b, __c, __x); 775 } 776 } // namespace __detail 777 #undef _GLIBCXX_MATH_NS 778 #if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) 779 } // namespace tr1 780 #endif 781 782 _GLIBCXX_END_NAMESPACE_VERSION 783 } 784 785 #endif // _GLIBCXX_TR1_HYPERGEOMETRIC_TCC
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