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b34f60ac 1// Special functions -*- C++ -*-
2
f1717362 3// Copyright (C) 2006-2016 Free Software Foundation, Inc.
b34f60ac 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
6bc9506f 8// Free Software Foundation; either version 3, or (at your option)
b34f60ac 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//
6bc9506f 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/>.
b34f60ac 24
25/** @file tr1/hypergeometric.tcc
26 * This is an internal header file, included by other library headers.
5846aeac 27 * Do not attempt to use it directly. @headername{tr1/cmath}
b34f60ac 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
c17b0a1c 41#ifndef _GLIBCXX_TR1_HYPERGEOMETRIC_TCC
42#define _GLIBCXX_TR1_HYPERGEOMETRIC_TCC 1
b34f60ac 43
2948dd21 44namespace std _GLIBCXX_VISIBILITY(default)
b34f60ac 45{
c17b0a1c 46namespace tr1
47{
b34f60ac 48 // [5.2] Special functions
49
b34f60ac 50 // Implementation-space details.
b34f60ac 51 namespace __detail
52 {
2948dd21 53 _GLIBCXX_BEGIN_NAMESPACE_VERSION
b34f60ac 54
55 /**
9fc1117c 56 * @brief This routine returns the confluent hypergeometric function
b34f60ac 57 * by series expansion.
58 *
59 * @f[
60 * _1F_1(a;c;x) = \frac{\Gamma(c)}{\Gamma(a)}
61 * \sum_{n=0}^{\infty}
62 * \frac{\Gamma(a+n)}{\Gamma(c+n)}
63 * \frac{x^n}{n!}
64 * @f]
65 *
72117d76 66 * If a and b are integers and a < 0 and either b > 0 or b < a
67 * then the series is a polynomial with a finite number of
68 * terms. If b is an integer and b <= 0 the confluent
69 * hypergeometric function is undefined.
b34f60ac 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
cd7f5f45 78 __conf_hyperg_series(_Tp __a, _Tp __c, _Tp __x)
b34f60ac 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
cd7f5f45 115 __conf_hyperg_luke(_Tp __a, _Tp __c, _Tp __xin)
b34f60ac 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$.
212 *
213 * @todo Handle b == nonpositive integer blowup - return NaN.
214 *
72117d76 215 * @param __a The @a numerator parameter.
216 * @param __c The @a denominator parameter.
b34f60ac 217 * @param __x The argument of the confluent hypergeometric function.
218 * @return The confluent hypergeometric function.
219 */
220 template<typename _Tp>
cd7f5f45 221 _Tp
222 __conf_hyperg(_Tp __a, _Tp __c, _Tp __x)
b34f60ac 223 {
224#if _GLIBCXX_USE_C99_MATH_TR1
c17b0a1c 225 const _Tp __c_nint = std::tr1::nearbyint(__c);
b34f60ac 226#else
227 const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L));
228#endif
229 if (__isnan(__a) || __isnan(__c) || __isnan(__x))
230 return std::numeric_limits<_Tp>::quiet_NaN();
231 else if (__c_nint == __c && __c_nint <= 0)
232 return std::numeric_limits<_Tp>::infinity();
233 else if (__a == _Tp(0))
234 return _Tp(1);
235 else if (__c == __a)
236 return std::exp(__x);
237 else if (__x < _Tp(0))
238 return __conf_hyperg_luke(__a, __c, __x);
239 else
240 return __conf_hyperg_series(__a, __c, __x);
241 }
242
243
244 /**
245 * @brief Return the hypogeometric function @f$ _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 *
72117d76 258 * @param __a The first @a numerator parameter.
259 * @param __a The second @a numerator parameter.
260 * @param __c The @a denominator parameter.
b34f60ac 261 * @param __x The argument of the confluent hypergeometric function.
262 * @return The confluent hypergeometric function.
263 */
264 template<typename _Tp>
265 _Tp
cd7f5f45 266 __hyperg_series(_Tp __a, _Tp __b, _Tp __c, _Tp __x)
b34f60ac 267 {
268 const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
269
270 _Tp __term = _Tp(1);
271 _Tp __Fabc = _Tp(1);
272 const unsigned int __max_iter = 100000;
273 unsigned int __i;
274 for (__i = 0; __i < __max_iter; ++__i)
275 {
276 __term *= (__a + _Tp(__i)) * (__b + _Tp(__i)) * __x
277 / ((__c + _Tp(__i)) * _Tp(1 + __i));
278 if (std::abs(__term) < __eps)
279 {
280 break;
281 }
282 __Fabc += __term;
283 }
284 if (__i == __max_iter)
285 std::__throw_runtime_error(__N("Series failed to converge "
286 "in __hyperg_series."));
287
288 return __Fabc;
289 }
290
291
292 /**
293 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$
294 * by an iterative procedure described in
295 * Luke, Algorithms for the Computation of Mathematical Functions.
296 */
297 template<typename _Tp>
298 _Tp
cd7f5f45 299 __hyperg_luke(_Tp __a, _Tp __b, _Tp __c, _Tp __xin)
b34f60ac 300 {
301 const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L));
302 const int __nmax = 20000;
303 const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
304 const _Tp __x = -__xin;
305 const _Tp __x3 = __x * __x * __x;
306 const _Tp __t0 = __a * __b / __c;
307 const _Tp __t1 = (__a + _Tp(1)) * (__b + _Tp(1)) / (_Tp(2) * __c);
308 const _Tp __t2 = (__a + _Tp(2)) * (__b + _Tp(2))
309 / (_Tp(2) * (__c + _Tp(1)));
310
311 _Tp __F = _Tp(1);
312
313 _Tp __Bnm3 = _Tp(1);
314 _Tp __Bnm2 = _Tp(1) + __t1 * __x;
315 _Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x);
316
317 _Tp __Anm3 = _Tp(1);
318 _Tp __Anm2 = __Bnm2 - __t0 * __x;
319 _Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x
320 + __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x;
321
322 int __n = 3;
323 while (1)
324 {
325 const _Tp __npam1 = _Tp(__n - 1) + __a;
326 const _Tp __npbm1 = _Tp(__n - 1) + __b;
327 const _Tp __npcm1 = _Tp(__n - 1) + __c;
328 const _Tp __npam2 = _Tp(__n - 2) + __a;
329 const _Tp __npbm2 = _Tp(__n - 2) + __b;
330 const _Tp __npcm2 = _Tp(__n - 2) + __c;
331 const _Tp __tnm1 = _Tp(2 * __n - 1);
332 const _Tp __tnm3 = _Tp(2 * __n - 3);
333 const _Tp __tnm5 = _Tp(2 * __n - 5);
334 const _Tp __n2 = __n * __n;
335 const _Tp __F1 = (_Tp(3) * __n2 + (__a + __b - _Tp(6)) * __n
336 + _Tp(2) - __a * __b - _Tp(2) * (__a + __b))
337 / (_Tp(2) * __tnm3 * __npcm1);
338 const _Tp __F2 = -(_Tp(3) * __n2 - (__a + __b + _Tp(6)) * __n
339 + _Tp(2) - __a * __b) * __npam1 * __npbm1
340 / (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1);
341 const _Tp __F3 = (__npam2 * __npam1 * __npbm2 * __npbm1
342 * (_Tp(__n - 2) - __a) * (_Tp(__n - 2) - __b))
343 / (_Tp(8) * __tnm3 * __tnm3 * __tnm5
344 * (_Tp(__n - 3) + __c) * __npcm2 * __npcm1);
345 const _Tp __E = -__npam1 * __npbm1 * (_Tp(__n - 1) - __c)
346 / (_Tp(2) * __tnm3 * __npcm2 * __npcm1);
347
348 _Tp __An = (_Tp(1) + __F1 * __x) * __Anm1
349 + (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3;
350 _Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1
351 + (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3;
352 const _Tp __r = __An / __Bn;
353
354 const _Tp __prec = std::abs((__F - __r) / __F);
355 __F = __r;
356
357 if (__prec < __eps || __n > __nmax)
358 break;
359
360 if (std::abs(__An) > __big || std::abs(__Bn) > __big)
361 {
362 __An /= __big;
363 __Bn /= __big;
364 __Anm1 /= __big;
365 __Bnm1 /= __big;
366 __Anm2 /= __big;
367 __Bnm2 /= __big;
368 __Anm3 /= __big;
369 __Bnm3 /= __big;
370 }
371 else if (std::abs(__An) < _Tp(1) / __big
372 || std::abs(__Bn) < _Tp(1) / __big)
373 {
374 __An *= __big;
375 __Bn *= __big;
376 __Anm1 *= __big;
377 __Bnm1 *= __big;
378 __Anm2 *= __big;
379 __Bnm2 *= __big;
380 __Anm3 *= __big;
381 __Bnm3 *= __big;
382 }
383
384 ++__n;
385 __Bnm3 = __Bnm2;
386 __Bnm2 = __Bnm1;
387 __Bnm1 = __Bn;
388 __Anm3 = __Anm2;
389 __Anm2 = __Anm1;
390 __Anm1 = __An;
391 }
392
393 if (__n >= __nmax)
394 std::__throw_runtime_error(__N("Iteration failed to converge "
395 "in __hyperg_luke."));
396
397 return __F;
398 }
399
400
401 /**
72117d76 402 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$
403 * by the reflection formulae in Abramowitz & Stegun formula
404 * 15.3.6 for d = c - a - b not integral and formula 15.3.11 for
405 * d = c - a - b integral. This assumes a, b, c != negative
406 * integer.
b34f60ac 407 *
408 * The hypogeometric function is defined by
409 * @f[
410 * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}
411 * \sum_{n=0}^{\infty}
412 * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}
413 * \frac{x^n}{n!}
414 * @f]
415 *
416 * The reflection formula for nonintegral @f$ d = c - a - b @f$ is:
417 * @f[
418 * _2F_1(a,b;c;x) = \frac{\Gamma(c)\Gamma(d)}{\Gamma(c-a)\Gamma(c-b)}
419 * _2F_1(a,b;1-d;1-x)
420 * + \frac{\Gamma(c)\Gamma(-d)}{\Gamma(a)\Gamma(b)}
421 * _2F_1(c-a,c-b;1+d;1-x)
422 * @f]
423 *
424 * The reflection formula for integral @f$ m = c - a - b @f$ is:
425 * @f[
426 * _2F_1(a,b;a+b+m;x) = \frac{\Gamma(m)\Gamma(a+b+m)}{\Gamma(a+m)\Gamma(b+m)}
427 * \sum_{k=0}^{m-1} \frac{(m+a)_k(m+b)_k}{k!(1-m)_k}
428 * -
429 * @f]
430 */
431 template<typename _Tp>
432 _Tp
cd7f5f45 433 __hyperg_reflect(_Tp __a, _Tp __b, _Tp __c, _Tp __x)
b34f60ac 434 {
435 const _Tp __d = __c - __a - __b;
436 const int __intd = std::floor(__d + _Tp(0.5L));
437 const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
438 const _Tp __toler = _Tp(1000) * __eps;
439 const _Tp __log_max = std::log(std::numeric_limits<_Tp>::max());
440 const bool __d_integer = (std::abs(__d - __intd) < __toler);
441
442 if (__d_integer)
443 {
444 const _Tp __ln_omx = std::log(_Tp(1) - __x);
445 const _Tp __ad = std::abs(__d);
446 _Tp __F1, __F2;
447
448 _Tp __d1, __d2;
449 if (__d >= _Tp(0))
450 {
451 __d1 = __d;
452 __d2 = _Tp(0);
453 }
454 else
455 {
456 __d1 = _Tp(0);
457 __d2 = __d;
458 }
459
460 const _Tp __lng_c = __log_gamma(__c);
461
462 // Evaluate F1.
463 if (__ad < __eps)
464 {
465 // d = c - a - b = 0.
466 __F1 = _Tp(0);
467 }
468 else
469 {
470
471 bool __ok_d1 = true;
472 _Tp __lng_ad, __lng_ad1, __lng_bd1;
b1a09c64 473 __try
b34f60ac 474 {
475 __lng_ad = __log_gamma(__ad);
476 __lng_ad1 = __log_gamma(__a + __d1);
477 __lng_bd1 = __log_gamma(__b + __d1);
478 }
b1a09c64 479 __catch(...)
b34f60ac 480 {
481 __ok_d1 = false;
482 }
483
484 if (__ok_d1)
485 {
486 /* Gamma functions in the denominator are ok.
487 * Proceed with evaluation.
488 */
489 _Tp __sum1 = _Tp(1);
490 _Tp __term = _Tp(1);
491 _Tp __ln_pre1 = __lng_ad + __lng_c + __d2 * __ln_omx
492 - __lng_ad1 - __lng_bd1;
493
494 /* Do F1 sum.
495 */
496 for (int __i = 1; __i < __ad; ++__i)
497 {
498 const int __j = __i - 1;
499 __term *= (__a + __d2 + __j) * (__b + __d2 + __j)
500 / (_Tp(1) + __d2 + __j) / __i * (_Tp(1) - __x);
501 __sum1 += __term;
502 }
503
504 if (__ln_pre1 > __log_max)
72117d76 505 std::__throw_runtime_error(__N("Overflow of gamma functions"
506 " in __hyperg_luke."));
b34f60ac 507 else
508 __F1 = std::exp(__ln_pre1) * __sum1;
509 }
510 else
511 {
512 // Gamma functions in the denominator were not ok.
513 // So the F1 term is zero.
514 __F1 = _Tp(0);
515 }
516 } // end F1 evaluation
517
518 // Evaluate F2.
519 bool __ok_d2 = true;
520 _Tp __lng_ad2, __lng_bd2;
b1a09c64 521 __try
b34f60ac 522 {
523 __lng_ad2 = __log_gamma(__a + __d2);
524 __lng_bd2 = __log_gamma(__b + __d2);
525 }
b1a09c64 526 __catch(...)
b34f60ac 527 {
528 __ok_d2 = false;
529 }
530
531 if (__ok_d2)
532 {
533 // Gamma functions in the denominator are ok.
534 // Proceed with evaluation.
535 const int __maxiter = 2000;
536 const _Tp __psi_1 = -__numeric_constants<_Tp>::__gamma_e();
537 const _Tp __psi_1pd = __psi(_Tp(1) + __ad);
538 const _Tp __psi_apd1 = __psi(__a + __d1);
539 const _Tp __psi_bpd1 = __psi(__b + __d1);
540
541 _Tp __psi_term = __psi_1 + __psi_1pd - __psi_apd1
542 - __psi_bpd1 - __ln_omx;
543 _Tp __fact = _Tp(1);
544 _Tp __sum2 = __psi_term;
545 _Tp __ln_pre2 = __lng_c + __d1 * __ln_omx
546 - __lng_ad2 - __lng_bd2;
547
548 // Do F2 sum.
549 int __j;
550 for (__j = 1; __j < __maxiter; ++__j)
551 {
72117d76 552 // Values for psi functions use recurrence;
553 // Abramowitz & Stegun 6.3.5
b34f60ac 554 const _Tp __term1 = _Tp(1) / _Tp(__j)
555 + _Tp(1) / (__ad + __j);
556 const _Tp __term2 = _Tp(1) / (__a + __d1 + _Tp(__j - 1))
557 + _Tp(1) / (__b + __d1 + _Tp(__j - 1));
558 __psi_term += __term1 - __term2;
559 __fact *= (__a + __d1 + _Tp(__j - 1))
560 * (__b + __d1 + _Tp(__j - 1))
561 / ((__ad + __j) * __j) * (_Tp(1) - __x);
562 const _Tp __delta = __fact * __psi_term;
563 __sum2 += __delta;
564 if (std::abs(__delta) < __eps * std::abs(__sum2))
565 break;
566 }
567 if (__j == __maxiter)
568 std::__throw_runtime_error(__N("Sum F2 failed to converge "
569 "in __hyperg_reflect"));
570
571 if (__sum2 == _Tp(0))
572 __F2 = _Tp(0);
573 else
574 __F2 = std::exp(__ln_pre2) * __sum2;
575 }
576 else
577 {
578 // Gamma functions in the denominator not ok.
579 // So the F2 term is zero.
580 __F2 = _Tp(0);
581 } // end F2 evaluation
582
583 const _Tp __sgn_2 = (__intd % 2 == 1 ? -_Tp(1) : _Tp(1));
584 const _Tp __F = __F1 + __sgn_2 * __F2;
585
586 return __F;
587 }
588 else
589 {
590 // d = c - a - b not an integer.
591
592 // These gamma functions appear in the denominator, so we
593 // catch their harmless domain errors and set the terms to zero.
594 bool __ok1 = true;
595 _Tp __sgn_g1ca = _Tp(0), __ln_g1ca = _Tp(0);
596 _Tp __sgn_g1cb = _Tp(0), __ln_g1cb = _Tp(0);
b1a09c64 597 __try
b34f60ac 598 {
599 __sgn_g1ca = __log_gamma_sign(__c - __a);
600 __ln_g1ca = __log_gamma(__c - __a);
601 __sgn_g1cb = __log_gamma_sign(__c - __b);
602 __ln_g1cb = __log_gamma(__c - __b);
603 }
b1a09c64 604 __catch(...)
b34f60ac 605 {
606 __ok1 = false;
607 }
608
609 bool __ok2 = true;
610 _Tp __sgn_g2a = _Tp(0), __ln_g2a = _Tp(0);
611 _Tp __sgn_g2b = _Tp(0), __ln_g2b = _Tp(0);
b1a09c64 612 __try
b34f60ac 613 {
614 __sgn_g2a = __log_gamma_sign(__a);
615 __ln_g2a = __log_gamma(__a);
616 __sgn_g2b = __log_gamma_sign(__b);
617 __ln_g2b = __log_gamma(__b);
618 }
b1a09c64 619 __catch(...)
b34f60ac 620 {
621 __ok2 = false;
622 }
623
624 const _Tp __sgn_gc = __log_gamma_sign(__c);
625 const _Tp __ln_gc = __log_gamma(__c);
626 const _Tp __sgn_gd = __log_gamma_sign(__d);
627 const _Tp __ln_gd = __log_gamma(__d);
628 const _Tp __sgn_gmd = __log_gamma_sign(-__d);
629 const _Tp __ln_gmd = __log_gamma(-__d);
630
631 const _Tp __sgn1 = __sgn_gc * __sgn_gd * __sgn_g1ca * __sgn_g1cb;
632 const _Tp __sgn2 = __sgn_gc * __sgn_gmd * __sgn_g2a * __sgn_g2b;
633
634 _Tp __pre1, __pre2;
635 if (__ok1 && __ok2)
636 {
637 _Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb;
638 _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b
639 + __d * std::log(_Tp(1) - __x);
640 if (__ln_pre1 < __log_max && __ln_pre2 < __log_max)
641 {
642 __pre1 = std::exp(__ln_pre1);
643 __pre2 = std::exp(__ln_pre2);
644 __pre1 *= __sgn1;
645 __pre2 *= __sgn2;
646 }
647 else
648 {
649 std::__throw_runtime_error(__N("Overflow of gamma functions "
650 "in __hyperg_reflect"));
651 }
652 }
653 else if (__ok1 && !__ok2)
654 {
655 _Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb;
656 if (__ln_pre1 < __log_max)
657 {
658 __pre1 = std::exp(__ln_pre1);
659 __pre1 *= __sgn1;
660 __pre2 = _Tp(0);
661 }
662 else
663 {
664 std::__throw_runtime_error(__N("Overflow of gamma functions "
665 "in __hyperg_reflect"));
666 }
667 }
668 else if (!__ok1 && __ok2)
669 {
670 _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b
671 + __d * std::log(_Tp(1) - __x);
672 if (__ln_pre2 < __log_max)
673 {
674 __pre1 = _Tp(0);
675 __pre2 = std::exp(__ln_pre2);
676 __pre2 *= __sgn2;
677 }
678 else
679 {
680 std::__throw_runtime_error(__N("Overflow of gamma functions "
681 "in __hyperg_reflect"));
682 }
683 }
684 else
685 {
686 __pre1 = _Tp(0);
687 __pre2 = _Tp(0);
688 std::__throw_runtime_error(__N("Underflow of gamma functions "
689 "in __hyperg_reflect"));
690 }
691
692 const _Tp __F1 = __hyperg_series(__a, __b, _Tp(1) - __d,
693 _Tp(1) - __x);
694 const _Tp __F2 = __hyperg_series(__c - __a, __c - __b, _Tp(1) + __d,
695 _Tp(1) - __x);
696
697 const _Tp __F = __pre1 * __F1 + __pre2 * __F2;
698
699 return __F;
700 }
701 }
702
703
048ff85f 704 /**
b34f60ac 705 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$.
706 *
707 * The hypogeometric function is defined by
708 * @f[
709 * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}
710 * \sum_{n=0}^{\infty}
711 * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}
712 * \frac{x^n}{n!}
713 * @f]
714 *
72117d76 715 * @param __a The first @a numerator parameter.
716 * @param __a The second @a numerator parameter.
717 * @param __c The @a denominator parameter.
b34f60ac 718 * @param __x The argument of the confluent hypergeometric function.
719 * @return The confluent hypergeometric function.
720 */
721 template<typename _Tp>
cd7f5f45 722 _Tp
723 __hyperg(_Tp __a, _Tp __b, _Tp __c, _Tp __x)
b34f60ac 724 {
725#if _GLIBCXX_USE_C99_MATH_TR1
c17b0a1c 726 const _Tp __a_nint = std::tr1::nearbyint(__a);
727 const _Tp __b_nint = std::tr1::nearbyint(__b);
728 const _Tp __c_nint = std::tr1::nearbyint(__c);
b34f60ac 729#else
730 const _Tp __a_nint = static_cast<int>(__a + _Tp(0.5L));
731 const _Tp __b_nint = static_cast<int>(__b + _Tp(0.5L));
732 const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L));
733#endif
734 const _Tp __toler = _Tp(1000) * std::numeric_limits<_Tp>::epsilon();
735 if (std::abs(__x) >= _Tp(1))
736 std::__throw_domain_error(__N("Argument outside unit circle "
737 "in __hyperg."));
738 else if (__isnan(__a) || __isnan(__b)
739 || __isnan(__c) || __isnan(__x))
740 return std::numeric_limits<_Tp>::quiet_NaN();
741 else if (__c_nint == __c && __c_nint <= _Tp(0))
742 return std::numeric_limits<_Tp>::infinity();
743 else if (std::abs(__c - __b) < __toler || std::abs(__c - __a) < __toler)
744 return std::pow(_Tp(1) - __x, __c - __a - __b);
745 else if (__a >= _Tp(0) && __b >= _Tp(0) && __c >= _Tp(0)
746 && __x >= _Tp(0) && __x < _Tp(0.995L))
747 return __hyperg_series(__a, __b, __c, __x);
748 else if (std::abs(__a) < _Tp(10) && std::abs(__b) < _Tp(10))
749 {
72117d76 750 // For integer a and b the hypergeometric function is a
751 // finite polynomial.
b34f60ac 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
2948dd21 770 _GLIBCXX_END_NAMESPACE_VERSION
b34f60ac 771 } // namespace std::tr1::__detail
c17b0a1c 772}
b34f60ac 773}
774
c17b0a1c 775#endif // _GLIBCXX_TR1_HYPERGEOMETRIC_TCC
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