1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
|
// Special functions -*- C++ -*-
// Copyright (C) 2006-2015 Free Software Foundation, Inc.
//
// This file is part of the GNU ISO C++ Library. This library is free
// software; you can redistribute it and/or modify it under the
// terms of the GNU General Public License as published by the
// Free Software Foundation; either version 3, or (at your option)
// any later version.
//
// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
// GNU General Public License for more details.
//
// Under Section 7 of GPL version 3, you are granted additional
// permissions described in the GCC Runtime Library Exception, version
// 3.1, as published by the Free Software Foundation.
// You should have received a copy of the GNU General Public License and
// a copy of the GCC Runtime Library Exception along with this program;
// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
// <http://www.gnu.org/licenses/>.
/** @file tr1/hypergeometric.tcc
* This is an internal header file, included by other library headers.
* Do not attempt to use it directly. @headername{tr1/cmath}
*/
//
// ISO C++ 14882 TR1: 5.2 Special functions
//
// Written by Edward Smith-Rowland based:
// (1) Handbook of Mathematical Functions,
// ed. Milton Abramowitz and Irene A. Stegun,
// Dover Publications,
// Section 6, pp. 555-566
// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
#ifndef _GLIBCXX_TR1_HYPERGEOMETRIC_TCC
#define _GLIBCXX_TR1_HYPERGEOMETRIC_TCC 1
namespace std _GLIBCXX_VISIBILITY(default)
{
namespace tr1
{
// [5.2] Special functions
// Implementation-space details.
namespace __detail
{
_GLIBCXX_BEGIN_NAMESPACE_VERSION
/**
* @brief This routine returns the confluent hypergeometric function
* by series expansion.
*
* @f[
* _1F_1(a;c;x) = \frac{\Gamma(c)}{\Gamma(a)}
* \sum_{n=0}^{\infty}
* \frac{\Gamma(a+n)}{\Gamma(c+n)}
* \frac{x^n}{n!}
* @f]
*
* If a and b are integers and a < 0 and either b > 0 or b < a
* then the series is a polynomial with a finite number of
* terms. If b is an integer and b <= 0 the confluent
* hypergeometric function is undefined.
*
* @param __a The "numerator" parameter.
* @param __c The "denominator" parameter.
* @param __x The argument of the confluent hypergeometric function.
* @return The confluent hypergeometric function.
*/
template<typename _Tp>
_Tp
__conf_hyperg_series(_Tp __a, _Tp __c, _Tp __x)
{
const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
_Tp __term = _Tp(1);
_Tp __Fac = _Tp(1);
const unsigned int __max_iter = 100000;
unsigned int __i;
for (__i = 0; __i < __max_iter; ++__i)
{
__term *= (__a + _Tp(__i)) * __x
/ ((__c + _Tp(__i)) * _Tp(1 + __i));
if (std::abs(__term) < __eps)
{
break;
}
__Fac += __term;
}
if (__i == __max_iter)
std::__throw_runtime_error(__N("Series failed to converge "
"in __conf_hyperg_series."));
return __Fac;
}
/**
* @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$
* by an iterative procedure described in
* Luke, Algorithms for the Computation of Mathematical Functions.
*
* Like the case of the 2F1 rational approximations, these are
* probably guaranteed to converge for x < 0, barring gross
* numerical instability in the pre-asymptotic regime.
*/
template<typename _Tp>
_Tp
__conf_hyperg_luke(_Tp __a, _Tp __c, _Tp __xin)
{
const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L));
const int __nmax = 20000;
const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
const _Tp __x = -__xin;
const _Tp __x3 = __x * __x * __x;
const _Tp __t0 = __a / __c;
const _Tp __t1 = (__a + _Tp(1)) / (_Tp(2) * __c);
const _Tp __t2 = (__a + _Tp(2)) / (_Tp(2) * (__c + _Tp(1)));
_Tp __F = _Tp(1);
_Tp __prec;
_Tp __Bnm3 = _Tp(1);
_Tp __Bnm2 = _Tp(1) + __t1 * __x;
_Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x);
_Tp __Anm3 = _Tp(1);
_Tp __Anm2 = __Bnm2 - __t0 * __x;
_Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x
+ __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x;
int __n = 3;
while(1)
{
_Tp __npam1 = _Tp(__n - 1) + __a;
_Tp __npcm1 = _Tp(__n - 1) + __c;
_Tp __npam2 = _Tp(__n - 2) + __a;
_Tp __npcm2 = _Tp(__n - 2) + __c;
_Tp __tnm1 = _Tp(2 * __n - 1);
_Tp __tnm3 = _Tp(2 * __n - 3);
_Tp __tnm5 = _Tp(2 * __n - 5);
_Tp __F1 = (_Tp(__n - 2) - __a) / (_Tp(2) * __tnm3 * __npcm1);
_Tp __F2 = (_Tp(__n) + __a) * __npam1
/ (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1);
_Tp __F3 = -__npam2 * __npam1 * (_Tp(__n - 2) - __a)
/ (_Tp(8) * __tnm3 * __tnm3 * __tnm5
* (_Tp(__n - 3) + __c) * __npcm2 * __npcm1);
_Tp __E = -__npam1 * (_Tp(__n - 1) - __c)
/ (_Tp(2) * __tnm3 * __npcm2 * __npcm1);
_Tp __An = (_Tp(1) + __F1 * __x) * __Anm1
+ (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3;
_Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1
+ (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3;
_Tp __r = __An / __Bn;
__prec = std::abs((__F - __r) / __F);
__F = __r;
if (__prec < __eps || __n > __nmax)
break;
if (std::abs(__An) > __big || std::abs(__Bn) > __big)
{
__An /= __big;
__Bn /= __big;
__Anm1 /= __big;
__Bnm1 /= __big;
__Anm2 /= __big;
__Bnm2 /= __big;
__Anm3 /= __big;
__Bnm3 /= __big;
}
else if (std::abs(__An) < _Tp(1) / __big
|| std::abs(__Bn) < _Tp(1) / __big)
{
__An *= __big;
__Bn *= __big;
__Anm1 *= __big;
__Bnm1 *= __big;
__Anm2 *= __big;
__Bnm2 *= __big;
__Anm3 *= __big;
__Bnm3 *= __big;
}
++__n;
__Bnm3 = __Bnm2;
__Bnm2 = __Bnm1;
__Bnm1 = __Bn;
__Anm3 = __Anm2;
__Anm2 = __Anm1;
__Anm1 = __An;
}
if (__n >= __nmax)
std::__throw_runtime_error(__N("Iteration failed to converge "
"in __conf_hyperg_luke."));
return __F;
}
/**
* @brief Return the confluent hypogeometric function
* @f$ _1F_1(a;c;x) @f$.
*
* @todo Handle b == nonpositive integer blowup - return NaN.
*
* @param __a The @a numerator parameter.
* @param __c The @a denominator parameter.
* @param __x The argument of the confluent hypergeometric function.
* @return The confluent hypergeometric function.
*/
template<typename _Tp>
_Tp
__conf_hyperg(_Tp __a, _Tp __c, _Tp __x)
{
#if _GLIBCXX_USE_C99_MATH_TR1
const _Tp __c_nint = std::tr1::nearbyint(__c);
#else
const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L));
#endif
if (__isnan(__a) || __isnan(__c) || __isnan(__x))
return std::numeric_limits<_Tp>::quiet_NaN();
else if (__c_nint == __c && __c_nint <= 0)
return std::numeric_limits<_Tp>::infinity();
else if (__a == _Tp(0))
return _Tp(1);
else if (__c == __a)
return std::exp(__x);
else if (__x < _Tp(0))
return __conf_hyperg_luke(__a, __c, __x);
else
return __conf_hyperg_series(__a, __c, __x);
}
/**
* @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$
* by series expansion.
*
* The hypogeometric function is defined by
* @f[
* _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}
* \sum_{n=0}^{\infty}
* \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}
* \frac{x^n}{n!}
* @f]
*
* This works and it's pretty fast.
*
* @param __a The first @a numerator parameter.
* @param __a The second @a numerator parameter.
* @param __c The @a denominator parameter.
* @param __x The argument of the confluent hypergeometric function.
* @return The confluent hypergeometric function.
*/
template<typename _Tp>
_Tp
__hyperg_series(_Tp __a, _Tp __b, _Tp __c, _Tp __x)
{
const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
_Tp __term = _Tp(1);
_Tp __Fabc = _Tp(1);
const unsigned int __max_iter = 100000;
unsigned int __i;
for (__i = 0; __i < __max_iter; ++__i)
{
__term *= (__a + _Tp(__i)) * (__b + _Tp(__i)) * __x
/ ((__c + _Tp(__i)) * _Tp(1 + __i));
if (std::abs(__term) < __eps)
{
break;
}
__Fabc += __term;
}
if (__i == __max_iter)
std::__throw_runtime_error(__N("Series failed to converge "
"in __hyperg_series."));
return __Fabc;
}
/**
* @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$
* by an iterative procedure described in
* Luke, Algorithms for the Computation of Mathematical Functions.
*/
template<typename _Tp>
_Tp
__hyperg_luke(_Tp __a, _Tp __b, _Tp __c, _Tp __xin)
{
const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L));
const int __nmax = 20000;
const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
const _Tp __x = -__xin;
const _Tp __x3 = __x * __x * __x;
const _Tp __t0 = __a * __b / __c;
const _Tp __t1 = (__a + _Tp(1)) * (__b + _Tp(1)) / (_Tp(2) * __c);
const _Tp __t2 = (__a + _Tp(2)) * (__b + _Tp(2))
/ (_Tp(2) * (__c + _Tp(1)));
_Tp __F = _Tp(1);
_Tp __Bnm3 = _Tp(1);
_Tp __Bnm2 = _Tp(1) + __t1 * __x;
_Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x);
_Tp __Anm3 = _Tp(1);
_Tp __Anm2 = __Bnm2 - __t0 * __x;
_Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x
+ __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x;
int __n = 3;
while (1)
{
const _Tp __npam1 = _Tp(__n - 1) + __a;
const _Tp __npbm1 = _Tp(__n - 1) + __b;
const _Tp __npcm1 = _Tp(__n - 1) + __c;
const _Tp __npam2 = _Tp(__n - 2) + __a;
const _Tp __npbm2 = _Tp(__n - 2) + __b;
const _Tp __npcm2 = _Tp(__n - 2) + __c;
const _Tp __tnm1 = _Tp(2 * __n - 1);
const _Tp __tnm3 = _Tp(2 * __n - 3);
const _Tp __tnm5 = _Tp(2 * __n - 5);
const _Tp __n2 = __n * __n;
const _Tp __F1 = (_Tp(3) * __n2 + (__a + __b - _Tp(6)) * __n
+ _Tp(2) - __a * __b - _Tp(2) * (__a + __b))
/ (_Tp(2) * __tnm3 * __npcm1);
const _Tp __F2 = -(_Tp(3) * __n2 - (__a + __b + _Tp(6)) * __n
+ _Tp(2) - __a * __b) * __npam1 * __npbm1
/ (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1);
const _Tp __F3 = (__npam2 * __npam1 * __npbm2 * __npbm1
* (_Tp(__n - 2) - __a) * (_Tp(__n - 2) - __b))
/ (_Tp(8) * __tnm3 * __tnm3 * __tnm5
* (_Tp(__n - 3) + __c) * __npcm2 * __npcm1);
const _Tp __E = -__npam1 * __npbm1 * (_Tp(__n - 1) - __c)
/ (_Tp(2) * __tnm3 * __npcm2 * __npcm1);
_Tp __An = (_Tp(1) + __F1 * __x) * __Anm1
+ (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3;
_Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1
+ (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3;
const _Tp __r = __An / __Bn;
const _Tp __prec = std::abs((__F - __r) / __F);
__F = __r;
if (__prec < __eps || __n > __nmax)
break;
if (std::abs(__An) > __big || std::abs(__Bn) > __big)
{
__An /= __big;
__Bn /= __big;
__Anm1 /= __big;
__Bnm1 /= __big;
__Anm2 /= __big;
__Bnm2 /= __big;
__Anm3 /= __big;
__Bnm3 /= __big;
}
else if (std::abs(__An) < _Tp(1) / __big
|| std::abs(__Bn) < _Tp(1) / __big)
{
__An *= __big;
__Bn *= __big;
__Anm1 *= __big;
__Bnm1 *= __big;
__Anm2 *= __big;
__Bnm2 *= __big;
__Anm3 *= __big;
__Bnm3 *= __big;
}
++__n;
__Bnm3 = __Bnm2;
__Bnm2 = __Bnm1;
__Bnm1 = __Bn;
__Anm3 = __Anm2;
__Anm2 = __Anm1;
__Anm1 = __An;
}
if (__n >= __nmax)
std::__throw_runtime_error(__N("Iteration failed to converge "
"in __hyperg_luke."));
return __F;
}
/**
* @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$
* by the reflection formulae in Abramowitz & Stegun formula
* 15.3.6 for d = c - a - b not integral and formula 15.3.11 for
* d = c - a - b integral. This assumes a, b, c != negative
* integer.
*
* The hypogeometric function is defined by
* @f[
* _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}
* \sum_{n=0}^{\infty}
* \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}
* \frac{x^n}{n!}
* @f]
*
* The reflection formula for nonintegral @f$ d = c - a - b @f$ is:
* @f[
* _2F_1(a,b;c;x) = \frac{\Gamma(c)\Gamma(d)}{\Gamma(c-a)\Gamma(c-b)}
* _2F_1(a,b;1-d;1-x)
* + \frac{\Gamma(c)\Gamma(-d)}{\Gamma(a)\Gamma(b)}
* _2F_1(c-a,c-b;1+d;1-x)
* @f]
*
* The reflection formula for integral @f$ m = c - a - b @f$ is:
* @f[
* _2F_1(a,b;a+b+m;x) = \frac{\Gamma(m)\Gamma(a+b+m)}{\Gamma(a+m)\Gamma(b+m)}
* \sum_{k=0}^{m-1} \frac{(m+a)_k(m+b)_k}{k!(1-m)_k}
* -
* @f]
*/
template<typename _Tp>
_Tp
__hyperg_reflect(_Tp __a, _Tp __b, _Tp __c, _Tp __x)
{
const _Tp __d = __c - __a - __b;
const int __intd = std::floor(__d + _Tp(0.5L));
const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
const _Tp __toler = _Tp(1000) * __eps;
const _Tp __log_max = std::log(std::numeric_limits<_Tp>::max());
const bool __d_integer = (std::abs(__d - __intd) < __toler);
if (__d_integer)
{
const _Tp __ln_omx = std::log(_Tp(1) - __x);
const _Tp __ad = std::abs(__d);
_Tp __F1, __F2;
_Tp __d1, __d2;
if (__d >= _Tp(0))
{
__d1 = __d;
__d2 = _Tp(0);
}
else
{
__d1 = _Tp(0);
__d2 = __d;
}
const _Tp __lng_c = __log_gamma(__c);
// Evaluate F1.
if (__ad < __eps)
{
// d = c - a - b = 0.
__F1 = _Tp(0);
}
else
{
bool __ok_d1 = true;
_Tp __lng_ad, __lng_ad1, __lng_bd1;
__try
{
__lng_ad = __log_gamma(__ad);
__lng_ad1 = __log_gamma(__a + __d1);
__lng_bd1 = __log_gamma(__b + __d1);
}
__catch(...)
{
__ok_d1 = false;
}
if (__ok_d1)
{
/* Gamma functions in the denominator are ok.
* Proceed with evaluation.
*/
_Tp __sum1 = _Tp(1);
_Tp __term = _Tp(1);
_Tp __ln_pre1 = __lng_ad + __lng_c + __d2 * __ln_omx
- __lng_ad1 - __lng_bd1;
/* Do F1 sum.
*/
for (int __i = 1; __i < __ad; ++__i)
{
const int __j = __i - 1;
__term *= (__a + __d2 + __j) * (__b + __d2 + __j)
/ (_Tp(1) + __d2 + __j) / __i * (_Tp(1) - __x);
__sum1 += __term;
}
if (__ln_pre1 > __log_max)
std::__throw_runtime_error(__N("Overflow of gamma functions"
" in __hyperg_luke."));
else
__F1 = std::exp(__ln_pre1) * __sum1;
}
else
{
// Gamma functions in the denominator were not ok.
// So the F1 term is zero.
__F1 = _Tp(0);
}
} // end F1 evaluation
// Evaluate F2.
bool __ok_d2 = true;
_Tp __lng_ad2, __lng_bd2;
__try
{
__lng_ad2 = __log_gamma(__a + __d2);
__lng_bd2 = __log_gamma(__b + __d2);
}
__catch(...)
{
__ok_d2 = false;
}
if (__ok_d2)
{
// Gamma functions in the denominator are ok.
// Proceed with evaluation.
const int __maxiter = 2000;
const _Tp __psi_1 = -__numeric_constants<_Tp>::__gamma_e();
const _Tp __psi_1pd = __psi(_Tp(1) + __ad);
const _Tp __psi_apd1 = __psi(__a + __d1);
const _Tp __psi_bpd1 = __psi(__b + __d1);
_Tp __psi_term = __psi_1 + __psi_1pd - __psi_apd1
- __psi_bpd1 - __ln_omx;
_Tp __fact = _Tp(1);
_Tp __sum2 = __psi_term;
_Tp __ln_pre2 = __lng_c + __d1 * __ln_omx
- __lng_ad2 - __lng_bd2;
// Do F2 sum.
int __j;
for (__j = 1; __j < __maxiter; ++__j)
{
// Values for psi functions use recurrence;
// Abramowitz & Stegun 6.3.5
const _Tp __term1 = _Tp(1) / _Tp(__j)
+ _Tp(1) / (__ad + __j);
const _Tp __term2 = _Tp(1) / (__a + __d1 + _Tp(__j - 1))
+ _Tp(1) / (__b + __d1 + _Tp(__j - 1));
__psi_term += __term1 - __term2;
__fact *= (__a + __d1 + _Tp(__j - 1))
* (__b + __d1 + _Tp(__j - 1))
/ ((__ad + __j) * __j) * (_Tp(1) - __x);
const _Tp __delta = __fact * __psi_term;
__sum2 += __delta;
if (std::abs(__delta) < __eps * std::abs(__sum2))
break;
}
if (__j == __maxiter)
std::__throw_runtime_error(__N("Sum F2 failed to converge "
"in __hyperg_reflect"));
if (__sum2 == _Tp(0))
__F2 = _Tp(0);
else
__F2 = std::exp(__ln_pre2) * __sum2;
}
else
{
// Gamma functions in the denominator not ok.
// So the F2 term is zero.
__F2 = _Tp(0);
} // end F2 evaluation
const _Tp __sgn_2 = (__intd % 2 == 1 ? -_Tp(1) : _Tp(1));
const _Tp __F = __F1 + __sgn_2 * __F2;
return __F;
}
else
{
// d = c - a - b not an integer.
// These gamma functions appear in the denominator, so we
// catch their harmless domain errors and set the terms to zero.
bool __ok1 = true;
_Tp __sgn_g1ca = _Tp(0), __ln_g1ca = _Tp(0);
_Tp __sgn_g1cb = _Tp(0), __ln_g1cb = _Tp(0);
__try
{
__sgn_g1ca = __log_gamma_sign(__c - __a);
__ln_g1ca = __log_gamma(__c - __a);
__sgn_g1cb = __log_gamma_sign(__c - __b);
__ln_g1cb = __log_gamma(__c - __b);
}
__catch(...)
{
__ok1 = false;
}
bool __ok2 = true;
_Tp __sgn_g2a = _Tp(0), __ln_g2a = _Tp(0);
_Tp __sgn_g2b = _Tp(0), __ln_g2b = _Tp(0);
__try
{
__sgn_g2a = __log_gamma_sign(__a);
__ln_g2a = __log_gamma(__a);
__sgn_g2b = __log_gamma_sign(__b);
__ln_g2b = __log_gamma(__b);
}
__catch(...)
{
__ok2 = false;
}
const _Tp __sgn_gc = __log_gamma_sign(__c);
const _Tp __ln_gc = __log_gamma(__c);
const _Tp __sgn_gd = __log_gamma_sign(__d);
const _Tp __ln_gd = __log_gamma(__d);
const _Tp __sgn_gmd = __log_gamma_sign(-__d);
const _Tp __ln_gmd = __log_gamma(-__d);
const _Tp __sgn1 = __sgn_gc * __sgn_gd * __sgn_g1ca * __sgn_g1cb;
const _Tp __sgn2 = __sgn_gc * __sgn_gmd * __sgn_g2a * __sgn_g2b;
_Tp __pre1, __pre2;
if (__ok1 && __ok2)
{
_Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb;
_Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b
+ __d * std::log(_Tp(1) - __x);
if (__ln_pre1 < __log_max && __ln_pre2 < __log_max)
{
__pre1 = std::exp(__ln_pre1);
__pre2 = std::exp(__ln_pre2);
__pre1 *= __sgn1;
__pre2 *= __sgn2;
}
else
{
std::__throw_runtime_error(__N("Overflow of gamma functions "
"in __hyperg_reflect"));
}
}
else if (__ok1 && !__ok2)
{
_Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb;
if (__ln_pre1 < __log_max)
{
__pre1 = std::exp(__ln_pre1);
__pre1 *= __sgn1;
__pre2 = _Tp(0);
}
else
{
std::__throw_runtime_error(__N("Overflow of gamma functions "
"in __hyperg_reflect"));
}
}
else if (!__ok1 && __ok2)
{
_Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b
+ __d * std::log(_Tp(1) - __x);
if (__ln_pre2 < __log_max)
{
__pre1 = _Tp(0);
__pre2 = std::exp(__ln_pre2);
__pre2 *= __sgn2;
}
else
{
std::__throw_runtime_error(__N("Overflow of gamma functions "
"in __hyperg_reflect"));
}
}
else
{
__pre1 = _Tp(0);
__pre2 = _Tp(0);
std::__throw_runtime_error(__N("Underflow of gamma functions "
"in __hyperg_reflect"));
}
const _Tp __F1 = __hyperg_series(__a, __b, _Tp(1) - __d,
_Tp(1) - __x);
const _Tp __F2 = __hyperg_series(__c - __a, __c - __b, _Tp(1) + __d,
_Tp(1) - __x);
const _Tp __F = __pre1 * __F1 + __pre2 * __F2;
return __F;
}
}
/**
* @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$.
*
* The hypogeometric function is defined by
* @f[
* _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}
* \sum_{n=0}^{\infty}
* \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}
* \frac{x^n}{n!}
* @f]
*
* @param __a The first @a numerator parameter.
* @param __a The second @a numerator parameter.
* @param __c The @a denominator parameter.
* @param __x The argument of the confluent hypergeometric function.
* @return The confluent hypergeometric function.
*/
template<typename _Tp>
_Tp
__hyperg(_Tp __a, _Tp __b, _Tp __c, _Tp __x)
{
#if _GLIBCXX_USE_C99_MATH_TR1
const _Tp __a_nint = std::tr1::nearbyint(__a);
const _Tp __b_nint = std::tr1::nearbyint(__b);
const _Tp __c_nint = std::tr1::nearbyint(__c);
#else
const _Tp __a_nint = static_cast<int>(__a + _Tp(0.5L));
const _Tp __b_nint = static_cast<int>(__b + _Tp(0.5L));
const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L));
#endif
const _Tp __toler = _Tp(1000) * std::numeric_limits<_Tp>::epsilon();
if (std::abs(__x) >= _Tp(1))
std::__throw_domain_error(__N("Argument outside unit circle "
"in __hyperg."));
else if (__isnan(__a) || __isnan(__b)
|| __isnan(__c) || __isnan(__x))
return std::numeric_limits<_Tp>::quiet_NaN();
else if (__c_nint == __c && __c_nint <= _Tp(0))
return std::numeric_limits<_Tp>::infinity();
else if (std::abs(__c - __b) < __toler || std::abs(__c - __a) < __toler)
return std::pow(_Tp(1) - __x, __c - __a - __b);
else if (__a >= _Tp(0) && __b >= _Tp(0) && __c >= _Tp(0)
&& __x >= _Tp(0) && __x < _Tp(0.995L))
return __hyperg_series(__a, __b, __c, __x);
else if (std::abs(__a) < _Tp(10) && std::abs(__b) < _Tp(10))
{
// For integer a and b the hypergeometric function is a
// finite polynomial.
if (__a < _Tp(0) && std::abs(__a - __a_nint) < __toler)
return __hyperg_series(__a_nint, __b, __c, __x);
else if (__b < _Tp(0) && std::abs(__b - __b_nint) < __toler)
return __hyperg_series(__a, __b_nint, __c, __x);
else if (__x < -_Tp(0.25L))
return __hyperg_luke(__a, __b, __c, __x);
else if (__x < _Tp(0.5L))
return __hyperg_series(__a, __b, __c, __x);
else
if (std::abs(__c) > _Tp(10))
return __hyperg_series(__a, __b, __c, __x);
else
return __hyperg_reflect(__a, __b, __c, __x);
}
else
return __hyperg_luke(__a, __b, __c, __x);
}
_GLIBCXX_END_NAMESPACE_VERSION
} // namespace std::tr1::__detail
}
}
#endif // _GLIBCXX_TR1_HYPERGEOMETRIC_TCC
|
