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1 // Special functions -*- C++ -*-
3 // Copyright (C) 2006-2018 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.
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/>.
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 */
30 //
31 // ISO C++ 14882 TR1: 5.2 Special functions
32 //
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
41 #ifndef _GLIBCXX_TR1_HYPERGEOMETRIC_TCC
42 #define _GLIBCXX_TR1_HYPERGEOMETRIC_TCC 1
44 namespace std _GLIBCXX_VISIBILITY(default)
45 {
46 _GLIBCXX_BEGIN_NAMESPACE_VERSION
48 #if _GLIBCXX_USE_STD_SPEC_FUNCS
49 # define _GLIBCXX_MATH_NS ::std
50 #elif defined(_GLIBCXX_TR1_CMATH)
51 namespace 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
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();
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."));
107 return __Fac;
108 }
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;
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);
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;
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);
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;
169 __prec = std::abs((__F - __r) / __F);
170 __F = __r;
172 if (__prec < __eps || __n > __nmax)
173 break;
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 }
199 ++__n;
200 __Bnm3 = __Bnm2;
201 __Bnm2 = __Bnm1;
202 __Bnm1 = __Bn;
203 __Anm3 = __Anm2;
204 __Anm2 = __Anm1;
205 __Anm1 = __An;
206 }
208 if (__n >= __nmax)
209 std::__throw_runtime_error(__N("Iteration failed to converge "
210 "in __conf_hyperg_luke."));
212 return __F;
213 }
216 /**
217 * @brief Return the confluent hypogeometric function
218 * @f$ _1F_1(a;c;x) @f$.
219 *
220 * @todo Handle b == nonpositive integer blowup - return NaN.
221 *
222 * @param __a The @a numerator parameter.
223 * @param __c The @a denominator parameter.
224 * @param __x The argument of the confluent hypergeometric function.
225 * @return The confluent hypergeometric function.
226 */
227 template<typename _Tp>
228 _Tp
229 __conf_hyperg(_Tp __a, _Tp __c, _Tp __x)
230 {
231 #if _GLIBCXX_USE_C99_MATH_TR1
232 const _Tp __c_nint = _GLIBCXX_MATH_NS::nearbyint(__c);
233 #else
234 const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L));
235 #endif
236 if (__isnan(__a) || __isnan(__c) || __isnan(__x))
237 return std::numeric_limits<_Tp>::quiet_NaN();
238 else if (__c_nint == __c && __c_nint <= 0)
239 return std::numeric_limits<_Tp>::infinity();
240 else if (__a == _Tp(0))
241 return _Tp(1);
242 else if (__c == __a)
243 return std::exp(__x);
244 else if (__x < _Tp(0))
245 return __conf_hyperg_luke(__a, __c, __x);
246 else
247 return __conf_hyperg_series(__a, __c, __x);
248 }
251 /**
252 * @brief Return the hypogeometric function @f$ _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();
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."));
295 return __Fabc;
296 }
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)));
318 _Tp __F = _Tp(1);
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);
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;
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);
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;
361 const _Tp __prec = std::abs((__F - __r) / __F);
362 __F = __r;
364 if (__prec < __eps || __n > __nmax)
365 break;
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 }
391 ++__n;
392 __Bnm3 = __Bnm2;
393 __Bnm2 = __Bnm1;
394 __Bnm1 = __Bn;
395 __Anm3 = __Anm2;
396 __Anm2 = __Anm1;
397 __Anm1 = __An;
398 }
400 if (__n >= __nmax)
401 std::__throw_runtime_error(__N("Iteration failed to converge "
402 "in __hyperg_luke."));
404 return __F;
405 }
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);
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;
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 }
467 const _Tp __lng_c = __log_gamma(__c);
469 // Evaluate F1.
470 if (__ad < __eps)
471 {
472 // d = c - a - b = 0.
473 __F1 = _Tp(0);
474 }
475 else
476 {
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 }
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;
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 }
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
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 }
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);
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;
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"));
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
590 const _Tp __sgn_2 = (__intd % 2 == 1 ? -_Tp(1) : _Tp(1));
591 const _Tp __F = __F1 + __sgn_2 * __F2;
593 return __F;
594 }
595 else
596 {
597 // d = c - a - b not an integer.
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 }
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 }
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);
638 const _Tp __sgn1 = __sgn_gc * __sgn_gd * __sgn_g1ca * __sgn_g1cb;
639 const _Tp __sgn2 = __sgn_gc * __sgn_gmd * __sgn_g2a * __sgn_g2b;
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 }
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);
704 const _Tp __F = __pre1 * __F1 + __pre2 * __F2;
706 return __F;
707 }
708 }
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
782 _GLIBCXX_END_NAMESPACE_VERSION
783 }
785 #endif // _GLIBCXX_TR1_HYPERGEOMETRIC_TCC