1 // Special functions -*- C++ -*- 2 3 // Copyright (C) 2006, 2007, 2008, 2009 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 * You should not attempt to use it directly. 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 46 { 47 namespace tr1 48 { 49 50 // [5.2] Special functions 51 52 // Implementation-space details. 53 namespace __detail 54 { 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 then the 68 * series is a polynomial with a finite number of terms. If b is an integer 69 * and b <= 0 the confluent 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 78 __conf_hyperg_series(const _Tp __a, const _Tp __c, const _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 115 __conf_hyperg_luke(const _Tp __a, const _Tp __c, const _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$. 212 * 213 * @todo Handle b == nonpositive integer blowup - return NaN. 214 * 215 * @param __a The "numerator" parameter. 216 * @param __c The "denominator" parameter. 217 * @param __x The argument of the confluent hypergeometric function. 218 * @return The confluent hypergeometric function. 219 */ 220 template<typename _Tp> 221 inline _Tp 222 __conf_hyperg(const _Tp __a, const _Tp __c, const _Tp __x) 223 { 224 #if _GLIBCXX_USE_C99_MATH_TR1 225 const _Tp __c_nint = std::tr1::nearbyint(__c); 226 #else 227 const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L)); 228 #endif 229 if (__isnan(__a) || __isnan(__c) || __isnan(__x)) 230 return std::numeric_limits<_Tp>::quiet_NaN(); 231 else if (__c_nint == __c && __c_nint <= 0) 232 return std::numeric_limits<_Tp>::infinity(); 233 else if (__a == _Tp(0)) 234 return _Tp(1); 235 else if (__c == __a) 236 return std::exp(__x); 237 else if (__x < _Tp(0)) 238 return __conf_hyperg_luke(__a, __c, __x); 239 else 240 return __conf_hyperg_series(__a, __c, __x); 241 } 242 243 244 /** 245 * @brief Return the hypogeometric function @f$ _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 "numerator" parameter. 259 * @param __a The second "numerator" parameter. 260 * @param __c The "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 266 __hyperg_series(const _Tp __a, const _Tp __b, 267 const _Tp __c, const _Tp __x) 268 { 269 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 270 271 _Tp __term = _Tp(1); 272 _Tp __Fabc = _Tp(1); 273 const unsigned int __max_iter = 100000; 274 unsigned int __i; 275 for (__i = 0; __i < __max_iter; ++__i) 276 { 277 __term *= (__a + _Tp(__i)) * (__b + _Tp(__i)) * __x 278 / ((__c + _Tp(__i)) * _Tp(1 + __i)); 279 if (std::abs(__term) < __eps) 280 { 281 break; 282 } 283 __Fabc += __term; 284 } 285 if (__i == __max_iter) 286 std::__throw_runtime_error(__N("Series failed to converge " 287 "in __hyperg_series.")); 288 289 return __Fabc; 290 } 291 292 293 /** 294 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ 295 * by an iterative procedure described in 296 * Luke, Algorithms for the Computation of Mathematical Functions. 297 */ 298 template<typename _Tp> 299 _Tp 300 __hyperg_luke(const _Tp __a, const _Tp __b, const _Tp __c, 301 const _Tp __xin) 302 { 303 const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L)); 304 const int __nmax = 20000; 305 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 306 const _Tp __x = -__xin; 307 const _Tp __x3 = __x * __x * __x; 308 const _Tp __t0 = __a * __b / __c; 309 const _Tp __t1 = (__a + _Tp(1)) * (__b + _Tp(1)) / (_Tp(2) * __c); 310 const _Tp __t2 = (__a + _Tp(2)) * (__b + _Tp(2)) 311 / (_Tp(2) * (__c + _Tp(1))); 312 313 _Tp __F = _Tp(1); 314 315 _Tp __Bnm3 = _Tp(1); 316 _Tp __Bnm2 = _Tp(1) + __t1 * __x; 317 _Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x); 318 319 _Tp __Anm3 = _Tp(1); 320 _Tp __Anm2 = __Bnm2 - __t0 * __x; 321 _Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x 322 + __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x; 323 324 int __n = 3; 325 while (1) 326 { 327 const _Tp __npam1 = _Tp(__n - 1) + __a; 328 const _Tp __npbm1 = _Tp(__n - 1) + __b; 329 const _Tp __npcm1 = _Tp(__n - 1) + __c; 330 const _Tp __npam2 = _Tp(__n - 2) + __a; 331 const _Tp __npbm2 = _Tp(__n - 2) + __b; 332 const _Tp __npcm2 = _Tp(__n - 2) + __c; 333 const _Tp __tnm1 = _Tp(2 * __n - 1); 334 const _Tp __tnm3 = _Tp(2 * __n - 3); 335 const _Tp __tnm5 = _Tp(2 * __n - 5); 336 const _Tp __n2 = __n * __n; 337 const _Tp __F1 = (_Tp(3) * __n2 + (__a + __b - _Tp(6)) * __n 338 + _Tp(2) - __a * __b - _Tp(2) * (__a + __b)) 339 / (_Tp(2) * __tnm3 * __npcm1); 340 const _Tp __F2 = -(_Tp(3) * __n2 - (__a + __b + _Tp(6)) * __n 341 + _Tp(2) - __a * __b) * __npam1 * __npbm1 342 / (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1); 343 const _Tp __F3 = (__npam2 * __npam1 * __npbm2 * __npbm1 344 * (_Tp(__n - 2) - __a) * (_Tp(__n - 2) - __b)) 345 / (_Tp(8) * __tnm3 * __tnm3 * __tnm5 346 * (_Tp(__n - 3) + __c) * __npcm2 * __npcm1); 347 const _Tp __E = -__npam1 * __npbm1 * (_Tp(__n - 1) - __c) 348 / (_Tp(2) * __tnm3 * __npcm2 * __npcm1); 349 350 _Tp __An = (_Tp(1) + __F1 * __x) * __Anm1 351 + (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3; 352 _Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1 353 + (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3; 354 const _Tp __r = __An / __Bn; 355 356 const _Tp __prec = std::abs((__F - __r) / __F); 357 __F = __r; 358 359 if (__prec < __eps || __n > __nmax) 360 break; 361 362 if (std::abs(__An) > __big || std::abs(__Bn) > __big) 363 { 364 __An /= __big; 365 __Bn /= __big; 366 __Anm1 /= __big; 367 __Bnm1 /= __big; 368 __Anm2 /= __big; 369 __Bnm2 /= __big; 370 __Anm3 /= __big; 371 __Bnm3 /= __big; 372 } 373 else if (std::abs(__An) < _Tp(1) / __big 374 || std::abs(__Bn) < _Tp(1) / __big) 375 { 376 __An *= __big; 377 __Bn *= __big; 378 __Anm1 *= __big; 379 __Bnm1 *= __big; 380 __Anm2 *= __big; 381 __Bnm2 *= __big; 382 __Anm3 *= __big; 383 __Bnm3 *= __big; 384 } 385 386 ++__n; 387 __Bnm3 = __Bnm2; 388 __Bnm2 = __Bnm1; 389 __Bnm1 = __Bn; 390 __Anm3 = __Anm2; 391 __Anm2 = __Anm1; 392 __Anm1 = __An; 393 } 394 395 if (__n >= __nmax) 396 std::__throw_runtime_error(__N("Iteration failed to converge " 397 "in __hyperg_luke.")); 398 399 return __F; 400 } 401 402 403 /** 404 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ by the reflection 405 * formulae in Abramowitz & Stegun formula 15.3.6 for d = c - a - b not integral 406 * and formula 15.3.11 for d = c - a - b integral. 407 * This assumes a, b, c != negative integer. 408 * 409 * The hypogeometric function is defined by 410 * @f[ 411 * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} 412 * \sum_{n=0}^{\infty} 413 * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} 414 * \frac{x^n}{n!} 415 * @f] 416 * 417 * The reflection formula for nonintegral @f$ d = c - a - b @f$ is: 418 * @f[ 419 * _2F_1(a,b;c;x) = \frac{\Gamma(c)\Gamma(d)}{\Gamma(c-a)\Gamma(c-b)} 420 * _2F_1(a,b;1-d;1-x) 421 * + \frac{\Gamma(c)\Gamma(-d)}{\Gamma(a)\Gamma(b)} 422 * _2F_1(c-a,c-b;1+d;1-x) 423 * @f] 424 * 425 * The reflection formula for integral @f$ m = c - a - b @f$ is: 426 * @f[ 427 * _2F_1(a,b;a+b+m;x) = \frac{\Gamma(m)\Gamma(a+b+m)}{\Gamma(a+m)\Gamma(b+m)} 428 * \sum_{k=0}^{m-1} \frac{(m+a)_k(m+b)_k}{k!(1-m)_k} 429 * - 430 * @f] 431 */ 432 template<typename _Tp> 433 _Tp 434 __hyperg_reflect(const _Tp __a, const _Tp __b, const _Tp __c, 435 const _Tp __x) 436 { 437 const _Tp __d = __c - __a - __b; 438 const int __intd = std::floor(__d + _Tp(0.5L)); 439 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 440 const _Tp __toler = _Tp(1000) * __eps; 441 const _Tp __log_max = std::log(std::numeric_limits<_Tp>::max()); 442 const bool __d_integer = (std::abs(__d - __intd) < __toler); 443 444 if (__d_integer) 445 { 446 const _Tp __ln_omx = std::log(_Tp(1) - __x); 447 const _Tp __ad = std::abs(__d); 448 _Tp __F1, __F2; 449 450 _Tp __d1, __d2; 451 if (__d >= _Tp(0)) 452 { 453 __d1 = __d; 454 __d2 = _Tp(0); 455 } 456 else 457 { 458 __d1 = _Tp(0); 459 __d2 = __d; 460 } 461 462 const _Tp __lng_c = __log_gamma(__c); 463 464 // Evaluate F1. 465 if (__ad < __eps) 466 { 467 // d = c - a - b = 0. 468 __F1 = _Tp(0); 469 } 470 else 471 { 472 473 bool __ok_d1 = true; 474 _Tp __lng_ad, __lng_ad1, __lng_bd1; 475 __try 476 { 477 __lng_ad = __log_gamma(__ad); 478 __lng_ad1 = __log_gamma(__a + __d1); 479 __lng_bd1 = __log_gamma(__b + __d1); 480 } 481 __catch(...) 482 { 483 __ok_d1 = false; 484 } 485 486 if (__ok_d1) 487 { 488 /* Gamma functions in the denominator are ok. 489 * Proceed with evaluation. 490 */ 491 _Tp __sum1 = _Tp(1); 492 _Tp __term = _Tp(1); 493 _Tp __ln_pre1 = __lng_ad + __lng_c + __d2 * __ln_omx 494 - __lng_ad1 - __lng_bd1; 495 496 /* Do F1 sum. 497 */ 498 for (int __i = 1; __i < __ad; ++__i) 499 { 500 const int __j = __i - 1; 501 __term *= (__a + __d2 + __j) * (__b + __d2 + __j) 502 / (_Tp(1) + __d2 + __j) / __i * (_Tp(1) - __x); 503 __sum1 += __term; 504 } 505 506 if (__ln_pre1 > __log_max) 507 std::__throw_runtime_error(__N("Overflow of gamma functions " 508 "in __hyperg_luke.")); 509 else 510 __F1 = std::exp(__ln_pre1) * __sum1; 511 } 512 else 513 { 514 // Gamma functions in the denominator were not ok. 515 // So the F1 term is zero. 516 __F1 = _Tp(0); 517 } 518 } // end F1 evaluation 519 520 // Evaluate F2. 521 bool __ok_d2 = true; 522 _Tp __lng_ad2, __lng_bd2; 523 __try 524 { 525 __lng_ad2 = __log_gamma(__a + __d2); 526 __lng_bd2 = __log_gamma(__b + __d2); 527 } 528 __catch(...) 529 { 530 __ok_d2 = false; 531 } 532 533 if (__ok_d2) 534 { 535 // Gamma functions in the denominator are ok. 536 // Proceed with evaluation. 537 const int __maxiter = 2000; 538 const _Tp __psi_1 = -__numeric_constants<_Tp>::__gamma_e(); 539 const _Tp __psi_1pd = __psi(_Tp(1) + __ad); 540 const _Tp __psi_apd1 = __psi(__a + __d1); 541 const _Tp __psi_bpd1 = __psi(__b + __d1); 542 543 _Tp __psi_term = __psi_1 + __psi_1pd - __psi_apd1 544 - __psi_bpd1 - __ln_omx; 545 _Tp __fact = _Tp(1); 546 _Tp __sum2 = __psi_term; 547 _Tp __ln_pre2 = __lng_c + __d1 * __ln_omx 548 - __lng_ad2 - __lng_bd2; 549 550 // Do F2 sum. 551 int __j; 552 for (__j = 1; __j < __maxiter; ++__j) 553 { 554 // Values for psi functions use recurrence; Abramowitz & Stegun 6.3.5 555 const _Tp __term1 = _Tp(1) / _Tp(__j) 556 + _Tp(1) / (__ad + __j); 557 const _Tp __term2 = _Tp(1) / (__a + __d1 + _Tp(__j - 1)) 558 + _Tp(1) / (__b + __d1 + _Tp(__j - 1)); 559 __psi_term += __term1 - __term2; 560 __fact *= (__a + __d1 + _Tp(__j - 1)) 561 * (__b + __d1 + _Tp(__j - 1)) 562 / ((__ad + __j) * __j) * (_Tp(1) - __x); 563 const _Tp __delta = __fact * __psi_term; 564 __sum2 += __delta; 565 if (std::abs(__delta) < __eps * std::abs(__sum2)) 566 break; 567 } 568 if (__j == __maxiter) 569 std::__throw_runtime_error(__N("Sum F2 failed to converge " 570 "in __hyperg_reflect")); 571 572 if (__sum2 == _Tp(0)) 573 __F2 = _Tp(0); 574 else 575 __F2 = std::exp(__ln_pre2) * __sum2; 576 } 577 else 578 { 579 // Gamma functions in the denominator not ok. 580 // So the F2 term is zero. 581 __F2 = _Tp(0); 582 } // end F2 evaluation 583 584 const _Tp __sgn_2 = (__intd % 2 == 1 ? -_Tp(1) : _Tp(1)); 585 const _Tp __F = __F1 + __sgn_2 * __F2; 586 587 return __F; 588 } 589 else 590 { 591 // d = c - a - b not an integer. 592 593 // These gamma functions appear in the denominator, so we 594 // catch their harmless domain errors and set the terms to zero. 595 bool __ok1 = true; 596 _Tp __sgn_g1ca = _Tp(0), __ln_g1ca = _Tp(0); 597 _Tp __sgn_g1cb = _Tp(0), __ln_g1cb = _Tp(0); 598 __try 599 { 600 __sgn_g1ca = __log_gamma_sign(__c - __a); 601 __ln_g1ca = __log_gamma(__c - __a); 602 __sgn_g1cb = __log_gamma_sign(__c - __b); 603 __ln_g1cb = __log_gamma(__c - __b); 604 } 605 __catch(...) 606 { 607 __ok1 = false; 608 } 609 610 bool __ok2 = true; 611 _Tp __sgn_g2a = _Tp(0), __ln_g2a = _Tp(0); 612 _Tp __sgn_g2b = _Tp(0), __ln_g2b = _Tp(0); 613 __try 614 { 615 __sgn_g2a = __log_gamma_sign(__a); 616 __ln_g2a = __log_gamma(__a); 617 __sgn_g2b = __log_gamma_sign(__b); 618 __ln_g2b = __log_gamma(__b); 619 } 620 __catch(...) 621 { 622 __ok2 = false; 623 } 624 625 const _Tp __sgn_gc = __log_gamma_sign(__c); 626 const _Tp __ln_gc = __log_gamma(__c); 627 const _Tp __sgn_gd = __log_gamma_sign(__d); 628 const _Tp __ln_gd = __log_gamma(__d); 629 const _Tp __sgn_gmd = __log_gamma_sign(-__d); 630 const _Tp __ln_gmd = __log_gamma(-__d); 631 632 const _Tp __sgn1 = __sgn_gc * __sgn_gd * __sgn_g1ca * __sgn_g1cb; 633 const _Tp __sgn2 = __sgn_gc * __sgn_gmd * __sgn_g2a * __sgn_g2b; 634 635 _Tp __pre1, __pre2; 636 if (__ok1 && __ok2) 637 { 638 _Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb; 639 _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b 640 + __d * std::log(_Tp(1) - __x); 641 if (__ln_pre1 < __log_max && __ln_pre2 < __log_max) 642 { 643 __pre1 = std::exp(__ln_pre1); 644 __pre2 = std::exp(__ln_pre2); 645 __pre1 *= __sgn1; 646 __pre2 *= __sgn2; 647 } 648 else 649 { 650 std::__throw_runtime_error(__N("Overflow of gamma functions " 651 "in __hyperg_reflect")); 652 } 653 } 654 else if (__ok1 && !__ok2) 655 { 656 _Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb; 657 if (__ln_pre1 < __log_max) 658 { 659 __pre1 = std::exp(__ln_pre1); 660 __pre1 *= __sgn1; 661 __pre2 = _Tp(0); 662 } 663 else 664 { 665 std::__throw_runtime_error(__N("Overflow of gamma functions " 666 "in __hyperg_reflect")); 667 } 668 } 669 else if (!__ok1 && __ok2) 670 { 671 _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b 672 + __d * std::log(_Tp(1) - __x); 673 if (__ln_pre2 < __log_max) 674 { 675 __pre1 = _Tp(0); 676 __pre2 = std::exp(__ln_pre2); 677 __pre2 *= __sgn2; 678 } 679 else 680 { 681 std::__throw_runtime_error(__N("Overflow of gamma functions " 682 "in __hyperg_reflect")); 683 } 684 } 685 else 686 { 687 __pre1 = _Tp(0); 688 __pre2 = _Tp(0); 689 std::__throw_runtime_error(__N("Underflow of gamma functions " 690 "in __hyperg_reflect")); 691 } 692 693 const _Tp __F1 = __hyperg_series(__a, __b, _Tp(1) - __d, 694 _Tp(1) - __x); 695 const _Tp __F2 = __hyperg_series(__c - __a, __c - __b, _Tp(1) + __d, 696 _Tp(1) - __x); 697 698 const _Tp __F = __pre1 * __F1 + __pre2 * __F2; 699 700 return __F; 701 } 702 } 703 704 705 /** 706 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$. 707 * 708 * The hypogeometric function is defined by 709 * @f[ 710 * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} 711 * \sum_{n=0}^{\infty} 712 * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} 713 * \frac{x^n}{n!} 714 * @f] 715 * 716 * @param __a The first "numerator" parameter. 717 * @param __a The second "numerator" parameter. 718 * @param __c The "denominator" parameter. 719 * @param __x The argument of the confluent hypergeometric function. 720 * @return The confluent hypergeometric function. 721 */ 722 template<typename _Tp> 723 inline _Tp 724 __hyperg(const _Tp __a, const _Tp __b, const _Tp __c, const _Tp __x) 725 { 726 #if _GLIBCXX_USE_C99_MATH_TR1 727 const _Tp __a_nint = std::tr1::nearbyint(__a); 728 const _Tp __b_nint = std::tr1::nearbyint(__b); 729 const _Tp __c_nint = std::tr1::nearbyint(__c); 730 #else 731 const _Tp __a_nint = static_cast<int>(__a + _Tp(0.5L)); 732 const _Tp __b_nint = static_cast<int>(__b + _Tp(0.5L)); 733 const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L)); 734 #endif 735 const _Tp __toler = _Tp(1000) * std::numeric_limits<_Tp>::epsilon(); 736 if (std::abs(__x) >= _Tp(1)) 737 std::__throw_domain_error(__N("Argument outside unit circle " 738 "in __hyperg.")); 739 else if (__isnan(__a) || __isnan(__b) 740 || __isnan(__c) || __isnan(__x)) 741 return std::numeric_limits<_Tp>::quiet_NaN(); 742 else if (__c_nint == __c && __c_nint <= _Tp(0)) 743 return std::numeric_limits<_Tp>::infinity(); 744 else if (std::abs(__c - __b) < __toler || std::abs(__c - __a) < __toler) 745 return std::pow(_Tp(1) - __x, __c - __a - __b); 746 else if (__a >= _Tp(0) && __b >= _Tp(0) && __c >= _Tp(0) 747 && __x >= _Tp(0) && __x < _Tp(0.995L)) 748 return __hyperg_series(__a, __b, __c, __x); 749 else if (std::abs(__a) < _Tp(10) && std::abs(__b) < _Tp(10)) 750 { 751 // For integer a and b the hypergeometric function is a 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 } // namespace std::tr1::__detail 771 } 772 } 773 774 #endif // _GLIBCXX_TR1_HYPERGEOMETRIC_TCC 775
