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1 // Special functions -*- C++ -*-
3 // Copyright (C) 2006, 2007, 2008, 2009, 2010, 2011
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.
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/>.
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 */
31 //
32 // ISO C++ 14882 TR1: 5.2 Special functions
33 //
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
42 #ifndef _GLIBCXX_TR1_HYPERGEOMETRIC_TCC
43 #define _GLIBCXX_TR1_HYPERGEOMETRIC_TCC 1
45 namespace std _GLIBCXX_VISIBILITY(default)
46 {
47 namespace tr1
48 {
49 // [5.2] Special functions
51 // Implementation-space details.
52 namespace __detail
53 {
54 _GLIBCXX_BEGIN_NAMESPACE_VERSION
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();
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."));
101 return __Fac;
102 }
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;
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);
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;
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);
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;
163 __prec = std::abs((__F - __r) / __F);
164 __F = __r;
166 if (__prec < __eps || __n > __nmax)
167 break;
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 }
193 ++__n;
194 __Bnm3 = __Bnm2;
195 __Bnm2 = __Bnm1;
196 __Bnm1 = __Bn;
197 __Anm3 = __Anm2;
198 __Anm2 = __Anm1;
199 __Anm1 = __An;
200 }
202 if (__n >= __nmax)
203 std::__throw_runtime_error(__N("Iteration failed to converge "
204 "in __conf_hyperg_luke."));
206 return __F;
207 }
210 /**
211 * @brief Return the confluent hypogeometric function
212 * @f$ _1F_1(a;c;x) @f$.
213 *
214 * @todo Handle b == nonpositive integer blowup - return NaN.
215 *
216 * @param __a The @a numerator parameter.
217 * @param __c The @a denominator parameter.
218 * @param __x The argument of the confluent hypergeometric function.
219 * @return The confluent hypergeometric function.
220 */
221 template<typename _Tp>
222 inline _Tp
223 __conf_hyperg(const _Tp __a, const _Tp __c, const _Tp __x)
224 {
225 #if _GLIBCXX_USE_C99_MATH_TR1
226 const _Tp __c_nint = std::tr1::nearbyint(__c);
227 #else
228 const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L));
229 #endif
230 if (__isnan(__a) || __isnan(__c) || __isnan(__x))
231 return std::numeric_limits<_Tp>::quiet_NaN();
232 else if (__c_nint == __c && __c_nint <= 0)
233 return std::numeric_limits<_Tp>::infinity();
234 else if (__a == _Tp(0))
235 return _Tp(1);
236 else if (__c == __a)
237 return std::exp(__x);
238 else if (__x < _Tp(0))
239 return __conf_hyperg_luke(__a, __c, __x);
240 else
241 return __conf_hyperg_series(__a, __c, __x);
242 }
245 /**
246 * @brief Return the hypogeometric function @f$ _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();
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."));
290 return __Fabc;
291 }
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)));
314 _Tp __F = _Tp(1);
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);
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;
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);
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;
357 const _Tp __prec = std::abs((__F - __r) / __F);
358 __F = __r;
360 if (__prec < __eps || __n > __nmax)
361 break;
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 }
387 ++__n;
388 __Bnm3 = __Bnm2;
389 __Bnm2 = __Bnm1;
390 __Bnm1 = __Bn;
391 __Anm3 = __Anm2;
392 __Anm2 = __Anm1;
393 __Anm1 = __An;
394 }
396 if (__n >= __nmax)
397 std::__throw_runtime_error(__N("Iteration failed to converge "
398 "in __hyperg_luke."));
400 return __F;
401 }
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);
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;
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 }
464 const _Tp __lng_c = __log_gamma(__c);
466 // Evaluate F1.
467 if (__ad < __eps)
468 {
469 // d = c - a - b = 0.
470 __F1 = _Tp(0);
471 }
472 else
473 {
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 }
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;
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 }
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
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 }
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);
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;
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"));
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
587 const _Tp __sgn_2 = (__intd % 2 == 1 ? -_Tp(1) : _Tp(1));
588 const _Tp __F = __F1 + __sgn_2 * __F2;
590 return __F;
591 }
592 else
593 {
594 // d = c - a - b not an integer.
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 }
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 }
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);
635 const _Tp __sgn1 = __sgn_gc * __sgn_gd * __sgn_g1ca * __sgn_g1cb;
636 const _Tp __sgn2 = __sgn_gc * __sgn_gmd * __sgn_g2a * __sgn_g2b;
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 }
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);
701 const _Tp __F = __pre1 * __F1 + __pre2 * __F2;
703 return __F;
704 }
705 }
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 }
774 _GLIBCXX_END_NAMESPACE_VERSION
775 } // namespace std::tr1::__detail
776 }
777 }
779 #endif // _GLIBCXX_TR1_HYPERGEOMETRIC_TCC