1 // Special functions -*- C++ -*- 2 3 // Copyright (C) 2006, 2007, 2008, 2009, 2010 4 // Free Software Foundation, Inc. 5 // 6 // This file is part of the GNU ISO C++ Library. This library is free 7 // software; you can redistribute it and/or modify it under the 8 // terms of the GNU General Public License as published by the 9 // Free Software Foundation; either version 3, or (at your option) 10 // any later version. 11 // 12 // This library is distributed in the hope that it will be useful, 13 // but WITHOUT ANY WARRANTY; without even the implied warranty of 14 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 15 // GNU General Public License for more details. 16 // 17 // Under Section 7 of GPL version 3, you are granted additional 18 // permissions described in the GCC Runtime Library Exception, version 19 // 3.1, as published by the Free Software Foundation. 20 21 // You should have received a copy of the GNU General Public License and 22 // a copy of the GCC Runtime Library Exception along with this program; 23 // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see 24 // <http://www.gnu.org/licenses/>. 25 26 /** @file tr1/hypergeometric.tcc 27 * This is an internal header file, included by other library headers. 28 * Do not attempt to use it directly. @headername{tr1/cmath} 29 */ 30 31 // 32 // ISO C++ 14882 TR1: 5.2 Special functions 33 // 34 35 // Written by Edward Smith-Rowland based: 36 // (1) Handbook of Mathematical Functions, 37 // ed. Milton Abramowitz and Irene A. Stegun, 38 // Dover Publications, 39 // Section 6, pp. 555-566 40 // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl 41 42 #ifndef _GLIBCXX_TR1_HYPERGEOMETRIC_TCC 43 #define _GLIBCXX_TR1_HYPERGEOMETRIC_TCC 1 44 45 namespace std _GLIBCXX_VISIBILITY(default) 46 { 47 namespace tr1 48 { 49 // [5.2] Special functions 50 51 // Implementation-space details. 52 namespace __detail 53 { 54 _GLIBCXX_BEGIN_NAMESPACE_VERSION 55 56 /** 57 * @brief This routine returns the confluent hypergeometric function 58 * by series expansion. 59 * 60 * @f[ 61 * _1F_1(a;c;x) = \frac{\Gamma(c)}{\Gamma(a)} 62 * \sum_{n=0}^{\infty} 63 * \frac{\Gamma(a+n)}{\Gamma(c+n)} 64 * \frac{x^n}{n!} 65 * @f] 66 * 67 * If a and b are integers and a < 0 and either b > 0 or b < a 68 * then the series is a polynomial with a finite number of 69 * terms. If b is an integer and b <= 0 the confluent 70 * hypergeometric function is undefined. 71 * 72 * @param __a The "numerator" parameter. 73 * @param __c The "denominator" parameter. 74 * @param __x The argument of the confluent hypergeometric function. 75 * @return The confluent hypergeometric function. 76 */ 77 template<typename _Tp> 78 _Tp 79 __conf_hyperg_series(const _Tp __a, const _Tp __c, const _Tp __x) 80 { 81 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 82 83 _Tp __term = _Tp(1); 84 _Tp __Fac = _Tp(1); 85 const unsigned int __max_iter = 100000; 86 unsigned int __i; 87 for (__i = 0; __i < __max_iter; ++__i) 88 { 89 __term *= (__a + _Tp(__i)) * __x 90 / ((__c + _Tp(__i)) * _Tp(1 + __i)); 91 if (std::abs(__term) < __eps) 92 { 93 break; 94 } 95 __Fac += __term; 96 } 97 if (__i == __max_iter) 98 std::__throw_runtime_error(__N("Series failed to converge " 99 "in __conf_hyperg_series.")); 100 101 return __Fac; 102 } 103 104 105 /** 106 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ 107 * by an iterative procedure described in 108 * Luke, Algorithms for the Computation of Mathematical Functions. 109 * 110 * Like the case of the 2F1 rational approximations, these are 111 * probably guaranteed to converge for x < 0, barring gross 112 * numerical instability in the pre-asymptotic regime. 113 */ 114 template<typename _Tp> 115 _Tp 116 __conf_hyperg_luke(const _Tp __a, const _Tp __c, const _Tp __xin) 117 { 118 const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L)); 119 const int __nmax = 20000; 120 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 121 const _Tp __x = -__xin; 122 const _Tp __x3 = __x * __x * __x; 123 const _Tp __t0 = __a / __c; 124 const _Tp __t1 = (__a + _Tp(1)) / (_Tp(2) * __c); 125 const _Tp __t2 = (__a + _Tp(2)) / (_Tp(2) * (__c + _Tp(1))); 126 _Tp __F = _Tp(1); 127 _Tp __prec; 128 129 _Tp __Bnm3 = _Tp(1); 130 _Tp __Bnm2 = _Tp(1) + __t1 * __x; 131 _Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x); 132 133 _Tp __Anm3 = _Tp(1); 134 _Tp __Anm2 = __Bnm2 - __t0 * __x; 135 _Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x 136 + __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x; 137 138 int __n = 3; 139 while(1) 140 { 141 _Tp __npam1 = _Tp(__n - 1) + __a; 142 _Tp __npcm1 = _Tp(__n - 1) + __c; 143 _Tp __npam2 = _Tp(__n - 2) + __a; 144 _Tp __npcm2 = _Tp(__n - 2) + __c; 145 _Tp __tnm1 = _Tp(2 * __n - 1); 146 _Tp __tnm3 = _Tp(2 * __n - 3); 147 _Tp __tnm5 = _Tp(2 * __n - 5); 148 _Tp __F1 = (_Tp(__n - 2) - __a) / (_Tp(2) * __tnm3 * __npcm1); 149 _Tp __F2 = (_Tp(__n) + __a) * __npam1 150 / (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1); 151 _Tp __F3 = -__npam2 * __npam1 * (_Tp(__n - 2) - __a) 152 / (_Tp(8) * __tnm3 * __tnm3 * __tnm5 153 * (_Tp(__n - 3) + __c) * __npcm2 * __npcm1); 154 _Tp __E = -__npam1 * (_Tp(__n - 1) - __c) 155 / (_Tp(2) * __tnm3 * __npcm2 * __npcm1); 156 157 _Tp __An = (_Tp(1) + __F1 * __x) * __Anm1 158 + (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3; 159 _Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1 160 + (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3; 161 _Tp __r = __An / __Bn; 162 163 __prec = std::abs((__F - __r) / __F); 164 __F = __r; 165 166 if (__prec < __eps || __n > __nmax) 167 break; 168 169 if (std::abs(__An) > __big || std::abs(__Bn) > __big) 170 { 171 __An /= __big; 172 __Bn /= __big; 173 __Anm1 /= __big; 174 __Bnm1 /= __big; 175 __Anm2 /= __big; 176 __Bnm2 /= __big; 177 __Anm3 /= __big; 178 __Bnm3 /= __big; 179 } 180 else if (std::abs(__An) < _Tp(1) / __big 181 || std::abs(__Bn) < _Tp(1) / __big) 182 { 183 __An *= __big; 184 __Bn *= __big; 185 __Anm1 *= __big; 186 __Bnm1 *= __big; 187 __Anm2 *= __big; 188 __Bnm2 *= __big; 189 __Anm3 *= __big; 190 __Bnm3 *= __big; 191 } 192 193 ++__n; 194 __Bnm3 = __Bnm2; 195 __Bnm2 = __Bnm1; 196 __Bnm1 = __Bn; 197 __Anm3 = __Anm2; 198 __Anm2 = __Anm1; 199 __Anm1 = __An; 200 } 201 202 if (__n >= __nmax) 203 std::__throw_runtime_error(__N("Iteration failed to converge " 204 "in __conf_hyperg_luke.")); 205 206 return __F; 207 } 208 209 210 /** 211 * @brief Return the confluent hypogeometric function 212 * @f$ _1F_1(a;c;x) @f$. 213 * 214 * @todo Handle b == nonpositive integer blowup - return NaN. 215 * 216 * @param __a The @a numerator parameter. 217 * @param __c The @a denominator parameter. 218 * @param __x The argument of the confluent hypergeometric function. 219 * @return The confluent hypergeometric function. 220 */ 221 template<typename _Tp> 222 inline _Tp 223 __conf_hyperg(const _Tp __a, const _Tp __c, const _Tp __x) 224 { 225 #if _GLIBCXX_USE_C99_MATH_TR1 226 const _Tp __c_nint = std::tr1::nearbyint(__c); 227 #else 228 const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L)); 229 #endif 230 if (__isnan(__a) || __isnan(__c) || __isnan(__x)) 231 return std::numeric_limits<_Tp>::quiet_NaN(); 232 else if (__c_nint == __c && __c_nint <= 0) 233 return std::numeric_limits<_Tp>::infinity(); 234 else if (__a == _Tp(0)) 235 return _Tp(1); 236 else if (__c == __a) 237 return std::exp(__x); 238 else if (__x < _Tp(0)) 239 return __conf_hyperg_luke(__a, __c, __x); 240 else 241 return __conf_hyperg_series(__a, __c, __x); 242 } 243 244 245 /** 246 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ 247 * by series expansion. 248 * 249 * The hypogeometric function is defined by 250 * @f[ 251 * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} 252 * \sum_{n=0}^{\infty} 253 * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} 254 * \frac{x^n}{n!} 255 * @f] 256 * 257 * This works and it's pretty fast. 258 * 259 * @param __a The first @a numerator parameter. 260 * @param __a The second @a numerator parameter. 261 * @param __c The @a denominator parameter. 262 * @param __x The argument of the confluent hypergeometric function. 263 * @return The confluent hypergeometric function. 264 */ 265 template<typename _Tp> 266 _Tp 267 __hyperg_series(const _Tp __a, const _Tp __b, 268 const _Tp __c, const _Tp __x) 269 { 270 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 271 272 _Tp __term = _Tp(1); 273 _Tp __Fabc = _Tp(1); 274 const unsigned int __max_iter = 100000; 275 unsigned int __i; 276 for (__i = 0; __i < __max_iter; ++__i) 277 { 278 __term *= (__a + _Tp(__i)) * (__b + _Tp(__i)) * __x 279 / ((__c + _Tp(__i)) * _Tp(1 + __i)); 280 if (std::abs(__term) < __eps) 281 { 282 break; 283 } 284 __Fabc += __term; 285 } 286 if (__i == __max_iter) 287 std::__throw_runtime_error(__N("Series failed to converge " 288 "in __hyperg_series.")); 289 290 return __Fabc; 291 } 292 293 294 /** 295 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ 296 * by an iterative procedure described in 297 * Luke, Algorithms for the Computation of Mathematical Functions. 298 */ 299 template<typename _Tp> 300 _Tp 301 __hyperg_luke(const _Tp __a, const _Tp __b, const _Tp __c, 302 const _Tp __xin) 303 { 304 const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L)); 305 const int __nmax = 20000; 306 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 307 const _Tp __x = -__xin; 308 const _Tp __x3 = __x * __x * __x; 309 const _Tp __t0 = __a * __b / __c; 310 const _Tp __t1 = (__a + _Tp(1)) * (__b + _Tp(1)) / (_Tp(2) * __c); 311 const _Tp __t2 = (__a + _Tp(2)) * (__b + _Tp(2)) 312 / (_Tp(2) * (__c + _Tp(1))); 313 314 _Tp __F = _Tp(1); 315 316 _Tp __Bnm3 = _Tp(1); 317 _Tp __Bnm2 = _Tp(1) + __t1 * __x; 318 _Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x); 319 320 _Tp __Anm3 = _Tp(1); 321 _Tp __Anm2 = __Bnm2 - __t0 * __x; 322 _Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x 323 + __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x; 324 325 int __n = 3; 326 while (1) 327 { 328 const _Tp __npam1 = _Tp(__n - 1) + __a; 329 const _Tp __npbm1 = _Tp(__n - 1) + __b; 330 const _Tp __npcm1 = _Tp(__n - 1) + __c; 331 const _Tp __npam2 = _Tp(__n - 2) + __a; 332 const _Tp __npbm2 = _Tp(__n - 2) + __b; 333 const _Tp __npcm2 = _Tp(__n - 2) + __c; 334 const _Tp __tnm1 = _Tp(2 * __n - 1); 335 const _Tp __tnm3 = _Tp(2 * __n - 3); 336 const _Tp __tnm5 = _Tp(2 * __n - 5); 337 const _Tp __n2 = __n * __n; 338 const _Tp __F1 = (_Tp(3) * __n2 + (__a + __b - _Tp(6)) * __n 339 + _Tp(2) - __a * __b - _Tp(2) * (__a + __b)) 340 / (_Tp(2) * __tnm3 * __npcm1); 341 const _Tp __F2 = -(_Tp(3) * __n2 - (__a + __b + _Tp(6)) * __n 342 + _Tp(2) - __a * __b) * __npam1 * __npbm1 343 / (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1); 344 const _Tp __F3 = (__npam2 * __npam1 * __npbm2 * __npbm1 345 * (_Tp(__n - 2) - __a) * (_Tp(__n - 2) - __b)) 346 / (_Tp(8) * __tnm3 * __tnm3 * __tnm5 347 * (_Tp(__n - 3) + __c) * __npcm2 * __npcm1); 348 const _Tp __E = -__npam1 * __npbm1 * (_Tp(__n - 1) - __c) 349 / (_Tp(2) * __tnm3 * __npcm2 * __npcm1); 350 351 _Tp __An = (_Tp(1) + __F1 * __x) * __Anm1 352 + (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3; 353 _Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1 354 + (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3; 355 const _Tp __r = __An / __Bn; 356 357 const _Tp __prec = std::abs((__F - __r) / __F); 358 __F = __r; 359 360 if (__prec < __eps || __n > __nmax) 361 break; 362 363 if (std::abs(__An) > __big || std::abs(__Bn) > __big) 364 { 365 __An /= __big; 366 __Bn /= __big; 367 __Anm1 /= __big; 368 __Bnm1 /= __big; 369 __Anm2 /= __big; 370 __Bnm2 /= __big; 371 __Anm3 /= __big; 372 __Bnm3 /= __big; 373 } 374 else if (std::abs(__An) < _Tp(1) / __big 375 || std::abs(__Bn) < _Tp(1) / __big) 376 { 377 __An *= __big; 378 __Bn *= __big; 379 __Anm1 *= __big; 380 __Bnm1 *= __big; 381 __Anm2 *= __big; 382 __Bnm2 *= __big; 383 __Anm3 *= __big; 384 __Bnm3 *= __big; 385 } 386 387 ++__n; 388 __Bnm3 = __Bnm2; 389 __Bnm2 = __Bnm1; 390 __Bnm1 = __Bn; 391 __Anm3 = __Anm2; 392 __Anm2 = __Anm1; 393 __Anm1 = __An; 394 } 395 396 if (__n >= __nmax) 397 std::__throw_runtime_error(__N("Iteration failed to converge " 398 "in __hyperg_luke.")); 399 400 return __F; 401 } 402 403 404 /** 405 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ 406 * by the reflection formulae in Abramowitz & Stegun formula 407 * 15.3.6 for d = c - a - b not integral and formula 15.3.11 for 408 * d = c - a - b integral. This assumes a, b, c != negative 409 * integer. 410 * 411 * The hypogeometric function is defined by 412 * @f[ 413 * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} 414 * \sum_{n=0}^{\infty} 415 * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} 416 * \frac{x^n}{n!} 417 * @f] 418 * 419 * The reflection formula for nonintegral @f$ d = c - a - b @f$ is: 420 * @f[ 421 * _2F_1(a,b;c;x) = \frac{\Gamma(c)\Gamma(d)}{\Gamma(c-a)\Gamma(c-b)} 422 * _2F_1(a,b;1-d;1-x) 423 * + \frac{\Gamma(c)\Gamma(-d)}{\Gamma(a)\Gamma(b)} 424 * _2F_1(c-a,c-b;1+d;1-x) 425 * @f] 426 * 427 * The reflection formula for integral @f$ m = c - a - b @f$ is: 428 * @f[ 429 * _2F_1(a,b;a+b+m;x) = \frac{\Gamma(m)\Gamma(a+b+m)}{\Gamma(a+m)\Gamma(b+m)} 430 * \sum_{k=0}^{m-1} \frac{(m+a)_k(m+b)_k}{k!(1-m)_k} 431 * - 432 * @f] 433 */ 434 template<typename _Tp> 435 _Tp 436 __hyperg_reflect(const _Tp __a, const _Tp __b, const _Tp __c, 437 const _Tp __x) 438 { 439 const _Tp __d = __c - __a - __b; 440 const int __intd = std::floor(__d + _Tp(0.5L)); 441 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 442 const _Tp __toler = _Tp(1000) * __eps; 443 const _Tp __log_max = std::log(std::numeric_limits<_Tp>::max()); 444 const bool __d_integer = (std::abs(__d - __intd) < __toler); 445 446 if (__d_integer) 447 { 448 const _Tp __ln_omx = std::log(_Tp(1) - __x); 449 const _Tp __ad = std::abs(__d); 450 _Tp __F1, __F2; 451 452 _Tp __d1, __d2; 453 if (__d >= _Tp(0)) 454 { 455 __d1 = __d; 456 __d2 = _Tp(0); 457 } 458 else 459 { 460 __d1 = _Tp(0); 461 __d2 = __d; 462 } 463 464 const _Tp __lng_c = __log_gamma(__c); 465 466 // Evaluate F1. 467 if (__ad < __eps) 468 { 469 // d = c - a - b = 0. 470 __F1 = _Tp(0); 471 } 472 else 473 { 474 475 bool __ok_d1 = true; 476 _Tp __lng_ad, __lng_ad1, __lng_bd1; 477 __try 478 { 479 __lng_ad = __log_gamma(__ad); 480 __lng_ad1 = __log_gamma(__a + __d1); 481 __lng_bd1 = __log_gamma(__b + __d1); 482 } 483 __catch(...) 484 { 485 __ok_d1 = false; 486 } 487 488 if (__ok_d1) 489 { 490 /* Gamma functions in the denominator are ok. 491 * Proceed with evaluation. 492 */ 493 _Tp __sum1 = _Tp(1); 494 _Tp __term = _Tp(1); 495 _Tp __ln_pre1 = __lng_ad + __lng_c + __d2 * __ln_omx 496 - __lng_ad1 - __lng_bd1; 497 498 /* Do F1 sum. 499 */ 500 for (int __i = 1; __i < __ad; ++__i) 501 { 502 const int __j = __i - 1; 503 __term *= (__a + __d2 + __j) * (__b + __d2 + __j) 504 / (_Tp(1) + __d2 + __j) / __i * (_Tp(1) - __x); 505 __sum1 += __term; 506 } 507 508 if (__ln_pre1 > __log_max) 509 std::__throw_runtime_error(__N("Overflow of gamma functions" 510 " in __hyperg_luke.")); 511 else 512 __F1 = std::exp(__ln_pre1) * __sum1; 513 } 514 else 515 { 516 // Gamma functions in the denominator were not ok. 517 // So the F1 term is zero. 518 __F1 = _Tp(0); 519 } 520 } // end F1 evaluation 521 522 // Evaluate F2. 523 bool __ok_d2 = true; 524 _Tp __lng_ad2, __lng_bd2; 525 __try 526 { 527 __lng_ad2 = __log_gamma(__a + __d2); 528 __lng_bd2 = __log_gamma(__b + __d2); 529 } 530 __catch(...) 531 { 532 __ok_d2 = false; 533 } 534 535 if (__ok_d2) 536 { 537 // Gamma functions in the denominator are ok. 538 // Proceed with evaluation. 539 const int __maxiter = 2000; 540 const _Tp __psi_1 = -__numeric_constants<_Tp>::__gamma_e(); 541 const _Tp __psi_1pd = __psi(_Tp(1) + __ad); 542 const _Tp __psi_apd1 = __psi(__a + __d1); 543 const _Tp __psi_bpd1 = __psi(__b + __d1); 544 545 _Tp __psi_term = __psi_1 + __psi_1pd - __psi_apd1 546 - __psi_bpd1 - __ln_omx; 547 _Tp __fact = _Tp(1); 548 _Tp __sum2 = __psi_term; 549 _Tp __ln_pre2 = __lng_c + __d1 * __ln_omx 550 - __lng_ad2 - __lng_bd2; 551 552 // Do F2 sum. 553 int __j; 554 for (__j = 1; __j < __maxiter; ++__j) 555 { 556 // Values for psi functions use recurrence; 557 // Abramowitz & Stegun 6.3.5 558 const _Tp __term1 = _Tp(1) / _Tp(__j) 559 + _Tp(1) / (__ad + __j); 560 const _Tp __term2 = _Tp(1) / (__a + __d1 + _Tp(__j - 1)) 561 + _Tp(1) / (__b + __d1 + _Tp(__j - 1)); 562 __psi_term += __term1 - __term2; 563 __fact *= (__a + __d1 + _Tp(__j - 1)) 564 * (__b + __d1 + _Tp(__j - 1)) 565 / ((__ad + __j) * __j) * (_Tp(1) - __x); 566 const _Tp __delta = __fact * __psi_term; 567 __sum2 += __delta; 568 if (std::abs(__delta) < __eps * std::abs(__sum2)) 569 break; 570 } 571 if (__j == __maxiter) 572 std::__throw_runtime_error(__N("Sum F2 failed to converge " 573 "in __hyperg_reflect")); 574 575 if (__sum2 == _Tp(0)) 576 __F2 = _Tp(0); 577 else 578 __F2 = std::exp(__ln_pre2) * __sum2; 579 } 580 else 581 { 582 // Gamma functions in the denominator not ok. 583 // So the F2 term is zero. 584 __F2 = _Tp(0); 585 } // end F2 evaluation 586 587 const _Tp __sgn_2 = (__intd % 2 == 1 ? -_Tp(1) : _Tp(1)); 588 const _Tp __F = __F1 + __sgn_2 * __F2; 589 590 return __F; 591 } 592 else 593 { 594 // d = c - a - b not an integer. 595 596 // These gamma functions appear in the denominator, so we 597 // catch their harmless domain errors and set the terms to zero. 598 bool __ok1 = true; 599 _Tp __sgn_g1ca = _Tp(0), __ln_g1ca = _Tp(0); 600 _Tp __sgn_g1cb = _Tp(0), __ln_g1cb = _Tp(0); 601 __try 602 { 603 __sgn_g1ca = __log_gamma_sign(__c - __a); 604 __ln_g1ca = __log_gamma(__c - __a); 605 __sgn_g1cb = __log_gamma_sign(__c - __b); 606 __ln_g1cb = __log_gamma(__c - __b); 607 } 608 __catch(...) 609 { 610 __ok1 = false; 611 } 612 613 bool __ok2 = true; 614 _Tp __sgn_g2a = _Tp(0), __ln_g2a = _Tp(0); 615 _Tp __sgn_g2b = _Tp(0), __ln_g2b = _Tp(0); 616 __try 617 { 618 __sgn_g2a = __log_gamma_sign(__a); 619 __ln_g2a = __log_gamma(__a); 620 __sgn_g2b = __log_gamma_sign(__b); 621 __ln_g2b = __log_gamma(__b); 622 } 623 __catch(...) 624 { 625 __ok2 = false; 626 } 627 628 const _Tp __sgn_gc = __log_gamma_sign(__c); 629 const _Tp __ln_gc = __log_gamma(__c); 630 const _Tp __sgn_gd = __log_gamma_sign(__d); 631 const _Tp __ln_gd = __log_gamma(__d); 632 const _Tp __sgn_gmd = __log_gamma_sign(-__d); 633 const _Tp __ln_gmd = __log_gamma(-__d); 634 635 const _Tp __sgn1 = __sgn_gc * __sgn_gd * __sgn_g1ca * __sgn_g1cb; 636 const _Tp __sgn2 = __sgn_gc * __sgn_gmd * __sgn_g2a * __sgn_g2b; 637 638 _Tp __pre1, __pre2; 639 if (__ok1 && __ok2) 640 { 641 _Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb; 642 _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b 643 + __d * std::log(_Tp(1) - __x); 644 if (__ln_pre1 < __log_max && __ln_pre2 < __log_max) 645 { 646 __pre1 = std::exp(__ln_pre1); 647 __pre2 = std::exp(__ln_pre2); 648 __pre1 *= __sgn1; 649 __pre2 *= __sgn2; 650 } 651 else 652 { 653 std::__throw_runtime_error(__N("Overflow of gamma functions " 654 "in __hyperg_reflect")); 655 } 656 } 657 else if (__ok1 && !__ok2) 658 { 659 _Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb; 660 if (__ln_pre1 < __log_max) 661 { 662 __pre1 = std::exp(__ln_pre1); 663 __pre1 *= __sgn1; 664 __pre2 = _Tp(0); 665 } 666 else 667 { 668 std::__throw_runtime_error(__N("Overflow of gamma functions " 669 "in __hyperg_reflect")); 670 } 671 } 672 else if (!__ok1 && __ok2) 673 { 674 _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b 675 + __d * std::log(_Tp(1) - __x); 676 if (__ln_pre2 < __log_max) 677 { 678 __pre1 = _Tp(0); 679 __pre2 = std::exp(__ln_pre2); 680 __pre2 *= __sgn2; 681 } 682 else 683 { 684 std::__throw_runtime_error(__N("Overflow of gamma functions " 685 "in __hyperg_reflect")); 686 } 687 } 688 else 689 { 690 __pre1 = _Tp(0); 691 __pre2 = _Tp(0); 692 std::__throw_runtime_error(__N("Underflow of gamma functions " 693 "in __hyperg_reflect")); 694 } 695 696 const _Tp __F1 = __hyperg_series(__a, __b, _Tp(1) - __d, 697 _Tp(1) - __x); 698 const _Tp __F2 = __hyperg_series(__c - __a, __c - __b, _Tp(1) + __d, 699 _Tp(1) - __x); 700 701 const _Tp __F = __pre1 * __F1 + __pre2 * __F2; 702 703 return __F; 704 } 705 } 706 707 708 /** 709 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$. 710 * 711 * The hypogeometric function is defined by 712 * @f[ 713 * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} 714 * \sum_{n=0}^{\infty} 715 * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} 716 * \frac{x^n}{n!} 717 * @f] 718 * 719 * @param __a The first @a numerator parameter. 720 * @param __a The second @a numerator parameter. 721 * @param __c The @a denominator parameter. 722 * @param __x The argument of the confluent hypergeometric function. 723 * @return The confluent hypergeometric function. 724 */ 725 template<typename _Tp> 726 inline _Tp 727 __hyperg(const _Tp __a, const _Tp __b, const _Tp __c, const _Tp __x) 728 { 729 #if _GLIBCXX_USE_C99_MATH_TR1 730 const _Tp __a_nint = std::tr1::nearbyint(__a); 731 const _Tp __b_nint = std::tr1::nearbyint(__b); 732 const _Tp __c_nint = std::tr1::nearbyint(__c); 733 #else 734 const _Tp __a_nint = static_cast<int>(__a + _Tp(0.5L)); 735 const _Tp __b_nint = static_cast<int>(__b + _Tp(0.5L)); 736 const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L)); 737 #endif 738 const _Tp __toler = _Tp(1000) * std::numeric_limits<_Tp>::epsilon(); 739 if (std::abs(__x) >= _Tp(1)) 740 std::__throw_domain_error(__N("Argument outside unit circle " 741 "in __hyperg.")); 742 else if (__isnan(__a) || __isnan(__b) 743 || __isnan(__c) || __isnan(__x)) 744 return std::numeric_limits<_Tp>::quiet_NaN(); 745 else if (__c_nint == __c && __c_nint <= _Tp(0)) 746 return std::numeric_limits<_Tp>::infinity(); 747 else if (std::abs(__c - __b) < __toler || std::abs(__c - __a) < __toler) 748 return std::pow(_Tp(1) - __x, __c - __a - __b); 749 else if (__a >= _Tp(0) && __b >= _Tp(0) && __c >= _Tp(0) 750 && __x >= _Tp(0) && __x < _Tp(0.995L)) 751 return __hyperg_series(__a, __b, __c, __x); 752 else if (std::abs(__a) < _Tp(10) && std::abs(__b) < _Tp(10)) 753 { 754 // For integer a and b the hypergeometric function is a 755 // finite polynomial. 756 if (__a < _Tp(0) && std::abs(__a - __a_nint) < __toler) 757 return __hyperg_series(__a_nint, __b, __c, __x); 758 else if (__b < _Tp(0) && std::abs(__b - __b_nint) < __toler) 759 return __hyperg_series(__a, __b_nint, __c, __x); 760 else if (__x < -_Tp(0.25L)) 761 return __hyperg_luke(__a, __b, __c, __x); 762 else if (__x < _Tp(0.5L)) 763 return __hyperg_series(__a, __b, __c, __x); 764 else 765 if (std::abs(__c) > _Tp(10)) 766 return __hyperg_series(__a, __b, __c, __x); 767 else 768 return __hyperg_reflect(__a, __b, __c, __x); 769 } 770 else 771 return __hyperg_luke(__a, __b, __c, __x); 772 } 773 774 _GLIBCXX_END_NAMESPACE_VERSION 775 } // namespace std::tr1::__detail 776 } 777 } 778 779 #endif // _GLIBCXX_TR1_HYPERGEOMETRIC_TCC 780
