1// Special functions -*- C++ -*- 
2 
3// Copyright (C) 2006-2019 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 
44namespace 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) 
51namespace 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 
786