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[thirdparty/gcc.git] / libstdc++-v3 / include / tr1 / hypergeometric.tcc
CommitLineData
7c62b943
BK		 1// Special functions -*- C++ -*-
2
bc2631e0 3// Copyright (C) 2006, 2007, 2008, 2009
7c62b943
BK		 4// Free Software Foundation, Inc.
5//
6// This file is part of the GNU ISO C++ Library. This library is free
7// software; you can redistribute it and/or modify it under the
8// terms of the GNU General Public License as published by the
748086b7 9// Free Software Foundation; either version 3, or (at your option)
7c62b943
BK		 10// any later version.
11//
12// This library is distributed in the hope that it will be useful,
13// but WITHOUT ANY WARRANTY; without even the implied warranty of
14// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
15// GNU General Public License for more details.
16//
748086b7
JJ		 17// Under Section 7 of GPL version 3, you are granted additional
18// permissions described in the GCC Runtime Library Exception, version
19// 3.1, as published by the Free Software Foundation.
20
21// You should have received a copy of the GNU General Public License and
22// a copy of the GCC Runtime Library Exception along with this program;
23// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
24// <http://www.gnu.org/licenses/>.
7c62b943
BK		 25
26/** @file tr1/hypergeometric.tcc
27 * This is an internal header file, included by other library headers.
28 * You should not attempt to use it directly.
29 */
30
31//
32// ISO C++ 14882 TR1: 5.2 Special functions
33//
34
35// Written by Edward Smith-Rowland based:
36// (1) Handbook of Mathematical Functions,
37// ed. Milton Abramowitz and Irene A. Stegun,
38// Dover Publications,
39// Section 6, pp. 555-566
40// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
41
e133ace8
PC		 42#ifndef _GLIBCXX_TR1_HYPERGEOMETRIC_TCC
43#define _GLIBCXX_TR1_HYPERGEOMETRIC_TCC 1
7c62b943
BK		 44
45namespace std
46{
e133ace8
PC		 47namespace tr1
48{
7c62b943
BK		 49
50 // [5.2] Special functions
51
7c62b943 52 // Implementation-space details.
7c62b943
BK		 53 namespace __detail
54 {
55
56 /**
28dac70a 57 * @brief This routine returns the confluent hypergeometric function
7c62b943
BK		 58 * by series expansion.
59 *
60 * @f[
61 * _1F_1(a;c;x) = \frac{\Gamma(c)}{\Gamma(a)}
62 * \sum_{n=0}^{\infty}
63 * \frac{\Gamma(a+n)}{\Gamma(c+n)}
64 * \frac{x^n}{n!}
65 * @f]
66 *
67 * If a and b are integers and a < 0 and either b > 0 or b < a then the
68 * series is a polynomial with a finite number of terms. If b is an integer
28dac70a 69 * and b <= 0 the confluent hypergeometric function is undefined.
7c62b943
BK		 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
78 __conf_hyperg_series(const _Tp __a, const _Tp __c, const _Tp __x)
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
115 __conf_hyperg_luke(const _Tp __a, const _Tp __c, const _Tp __xin)
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$.
212 *
213 * @todo Handle b == nonpositive integer blowup - return NaN.
214 *
215 * @param __a The "numerator" parameter.
216 * @param __c The "denominator" parameter.
217 * @param __x The argument of the confluent hypergeometric function.
218 * @return The confluent hypergeometric function.
219 */
220 template<typename _Tp>
221 inline _Tp
222 __conf_hyperg(const _Tp __a, const _Tp __c, const _Tp __x)
223 {
224#if _GLIBCXX_USE_C99_MATH_TR1
e133ace8 225 const _Tp __c_nint = std::tr1::nearbyint(__c);
7c62b943
BK		 226#else
227 const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L));
228#endif
229 if (__isnan(__a) || __isnan(__c) || __isnan(__x))
230 return std::numeric_limits<_Tp>::quiet_NaN();
231 else if (__c_nint == __c && __c_nint <= 0)
232 return std::numeric_limits<_Tp>::infinity();
233 else if (__a == _Tp(0))
234 return _Tp(1);
235 else if (__c == __a)
236 return std::exp(__x);
237 else if (__x < _Tp(0))
238 return __conf_hyperg_luke(__a, __c, __x);
239 else
240 return __conf_hyperg_series(__a, __c, __x);
241 }
242
243
244 /**
245 * @brief Return the hypogeometric function @f$ _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 *
258 * @param __a The first "numerator" parameter.
259 * @param __a The second "numerator" parameter.
260 * @param __c The "denominator" parameter.
261 * @param __x The argument of the confluent hypergeometric function.
262 * @return The confluent hypergeometric function.
263 */
264 template<typename _Tp>
265 _Tp
266 __hyperg_series(const _Tp __a, const _Tp __b,
267 const _Tp __c, const _Tp __x)
268 {
269 const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
270
271 _Tp __term = _Tp(1);
272 _Tp __Fabc = _Tp(1);
273 const unsigned int __max_iter = 100000;
274 unsigned int __i;
275 for (__i = 0; __i < __max_iter; ++__i)
276 {
277 __term *= (__a + _Tp(__i)) * (__b + _Tp(__i)) * __x
278 / ((__c + _Tp(__i)) * _Tp(1 + __i));
279 if (std::abs(__term) < __eps)
280 {
281 break;
282 }
283 __Fabc += __term;
284 }
285 if (__i == __max_iter)
286 std::__throw_runtime_error(__N("Series failed to converge "
287 "in __hyperg_series."));
288
289 return __Fabc;
290 }
291
292
293 /**
294 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$
295 * by an iterative procedure described in
296 * Luke, Algorithms for the Computation of Mathematical Functions.
297 */
298 template<typename _Tp>
299 _Tp
300 __hyperg_luke(const _Tp __a, const _Tp __b, const _Tp __c,
301 const _Tp __xin)
302 {
303 const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L));
304 const int __nmax = 20000;
305 const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
306 const _Tp __x = -__xin;
307 const _Tp __x3 = __x * __x * __x;
308 const _Tp __t0 = __a * __b / __c;
309 const _Tp __t1 = (__a + _Tp(1)) * (__b + _Tp(1)) / (_Tp(2) * __c);
310 const _Tp __t2 = (__a + _Tp(2)) * (__b + _Tp(2))
311 / (_Tp(2) * (__c + _Tp(1)));
312
313 _Tp __F = _Tp(1);
314
315 _Tp __Bnm3 = _Tp(1);
316 _Tp __Bnm2 = _Tp(1) + __t1 * __x;
317 _Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x);
318
319 _Tp __Anm3 = _Tp(1);
320 _Tp __Anm2 = __Bnm2 - __t0 * __x;
321 _Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x
322 + __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x;
323
324 int __n = 3;
325 while (1)
326 {
327 const _Tp __npam1 = _Tp(__n - 1) + __a;
328 const _Tp __npbm1 = _Tp(__n - 1) + __b;
329 const _Tp __npcm1 = _Tp(__n - 1) + __c;
330 const _Tp __npam2 = _Tp(__n - 2) + __a;
331 const _Tp __npbm2 = _Tp(__n - 2) + __b;
332 const _Tp __npcm2 = _Tp(__n - 2) + __c;
333 const _Tp __tnm1 = _Tp(2 * __n - 1);
334 const _Tp __tnm3 = _Tp(2 * __n - 3);
335 const _Tp __tnm5 = _Tp(2 * __n - 5);
336 const _Tp __n2 = __n * __n;
337 const _Tp __F1 = (_Tp(3) * __n2 + (__a + __b - _Tp(6)) * __n
338 + _Tp(2) - __a * __b - _Tp(2) * (__a + __b))
339 / (_Tp(2) * __tnm3 * __npcm1);
340 const _Tp __F2 = -(_Tp(3) * __n2 - (__a + __b + _Tp(6)) * __n
341 + _Tp(2) - __a * __b) * __npam1 * __npbm1
342 / (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1);
343 const _Tp __F3 = (__npam2 * __npam1 * __npbm2 * __npbm1
344 * (_Tp(__n - 2) - __a) * (_Tp(__n - 2) - __b))
345 / (_Tp(8) * __tnm3 * __tnm3 * __tnm5
346 * (_Tp(__n - 3) + __c) * __npcm2 * __npcm1);
347 const _Tp __E = -__npam1 * __npbm1 * (_Tp(__n - 1) - __c)
348 / (_Tp(2) * __tnm3 * __npcm2 * __npcm1);
349
350 _Tp __An = (_Tp(1) + __F1 * __x) * __Anm1
351 + (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3;
352 _Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1
353 + (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3;
354 const _Tp __r = __An / __Bn;
355
356 const _Tp __prec = std::abs((__F - __r) / __F);
357 __F = __r;
358
359 if (__prec < __eps || __n > __nmax)
360 break;
361
362 if (std::abs(__An) > __big || std::abs(__Bn) > __big)
363 {
364 __An /= __big;
365 __Bn /= __big;
366 __Anm1 /= __big;
367 __Bnm1 /= __big;
368 __Anm2 /= __big;
369 __Bnm2 /= __big;
370 __Anm3 /= __big;
371 __Bnm3 /= __big;
372 }
373 else if (std::abs(__An) < _Tp(1) / __big
374 || std::abs(__Bn) < _Tp(1) / __big)
375 {
376 __An *= __big;
377 __Bn *= __big;
378 __Anm1 *= __big;
379 __Bnm1 *= __big;
380 __Anm2 *= __big;
381 __Bnm2 *= __big;
382 __Anm3 *= __big;
383 __Bnm3 *= __big;
384 }
385
386 ++__n;
387 __Bnm3 = __Bnm2;
388 __Bnm2 = __Bnm1;
389 __Bnm1 = __Bn;
390 __Anm3 = __Anm2;
391 __Anm2 = __Anm1;
392 __Anm1 = __An;
393 }
394
395 if (__n >= __nmax)
396 std::__throw_runtime_error(__N("Iteration failed to converge "
397 "in __hyperg_luke."));
398
399 return __F;
400 }
401
402
403 /**
404 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ by the reflection
405 * formulae in Abramowitz & Stegun formula 15.3.6 for d = c - a - b not integral
406 * and formula 15.3.11 for d = c - a - b integral.
407 * This assumes a, b, c != negative integer.
408 *
409 * The hypogeometric function is defined by
410 * @f[
411 * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}
412 * \sum_{n=0}^{\infty}
413 * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}
414 * \frac{x^n}{n!}
415 * @f]
416 *
417 * The reflection formula for nonintegral @f$ d = c - a - b @f$ is:
418 * @f[
419 * _2F_1(a,b;c;x) = \frac{\Gamma(c)\Gamma(d)}{\Gamma(c-a)\Gamma(c-b)}
420 * _2F_1(a,b;1-d;1-x)
421 * + \frac{\Gamma(c)\Gamma(-d)}{\Gamma(a)\Gamma(b)}
422 * _2F_1(c-a,c-b;1+d;1-x)
423 * @f]
424 *
425 * The reflection formula for integral @f$ m = c - a - b @f$ is:
426 * @f[
427 * _2F_1(a,b;a+b+m;x) = \frac{\Gamma(m)\Gamma(a+b+m)}{\Gamma(a+m)\Gamma(b+m)}
428 * \sum_{k=0}^{m-1} \frac{(m+a)_k(m+b)_k}{k!(1-m)_k}
429 * -
430 * @f]
431 */
432 template<typename _Tp>
433 _Tp
434 __hyperg_reflect(const _Tp __a, const _Tp __b, const _Tp __c,
435 const _Tp __x)
436 {
437 const _Tp __d = __c - __a - __b;
438 const int __intd = std::floor(__d + _Tp(0.5L));
439 const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
440 const _Tp __toler = _Tp(1000) * __eps;
441 const _Tp __log_max = std::log(std::numeric_limits<_Tp>::max());
442 const bool __d_integer = (std::abs(__d - __intd) < __toler);
443
444 if (__d_integer)
445 {
446 const _Tp __ln_omx = std::log(_Tp(1) - __x);
447 const _Tp __ad = std::abs(__d);
448 _Tp __F1, __F2;
449
450 _Tp __d1, __d2;
451 if (__d >= _Tp(0))
452 {
453 __d1 = __d;
454 __d2 = _Tp(0);
455 }
456 else
457 {
458 __d1 = _Tp(0);
459 __d2 = __d;
460 }
461
462 const _Tp __lng_c = __log_gamma(__c);
463
464 // Evaluate F1.
465 if (__ad < __eps)
466 {
467 // d = c - a - b = 0.
468 __F1 = _Tp(0);
469 }
470 else
471 {
472
473 bool __ok_d1 = true;
474 _Tp __lng_ad, __lng_ad1, __lng_bd1;
bc2631e0 475 __try
7c62b943
BK		 476 {
477 __lng_ad = __log_gamma(__ad);
478 __lng_ad1 = __log_gamma(__a + __d1);
479 __lng_bd1 = __log_gamma(__b + __d1);
480 }
bc2631e0 481 __catch(...)
7c62b943
BK		 482 {
483 __ok_d1 = false;
484 }
485
486 if (__ok_d1)
487 {
488 /* Gamma functions in the denominator are ok.
489 * Proceed with evaluation.
490 */
491 _Tp __sum1 = _Tp(1);
492 _Tp __term = _Tp(1);
493 _Tp __ln_pre1 = __lng_ad + __lng_c + __d2 * __ln_omx
494 - __lng_ad1 - __lng_bd1;
495
496 /* Do F1 sum.
497 */
498 for (int __i = 1; __i < __ad; ++__i)
499 {
500 const int __j = __i - 1;
501 __term *= (__a + __d2 + __j) * (__b + __d2 + __j)
502 / (_Tp(1) + __d2 + __j) / __i * (_Tp(1) - __x);
503 __sum1 += __term;
504 }
505
506 if (__ln_pre1 > __log_max)
507 std::__throw_runtime_error(__N("Overflow of gamma functions "
508 "in __hyperg_luke."));
509 else
510 __F1 = std::exp(__ln_pre1) * __sum1;
511 }
512 else
513 {
514 // Gamma functions in the denominator were not ok.
515 // So the F1 term is zero.
516 __F1 = _Tp(0);
517 }
518 } // end F1 evaluation
519
520 // Evaluate F2.
521 bool __ok_d2 = true;
522 _Tp __lng_ad2, __lng_bd2;
bc2631e0 523 __try
7c62b943
BK		 524 {
525 __lng_ad2 = __log_gamma(__a + __d2);
526 __lng_bd2 = __log_gamma(__b + __d2);
527 }
bc2631e0 528 __catch(...)
7c62b943
BK		 529 {
530 __ok_d2 = false;
531 }
532
533 if (__ok_d2)
534 {
535 // Gamma functions in the denominator are ok.
536 // Proceed with evaluation.
537 const int __maxiter = 2000;
538 const _Tp __psi_1 = -__numeric_constants<_Tp>::__gamma_e();
539 const _Tp __psi_1pd = __psi(_Tp(1) + __ad);
540 const _Tp __psi_apd1 = __psi(__a + __d1);
541 const _Tp __psi_bpd1 = __psi(__b + __d1);
542
543 _Tp __psi_term = __psi_1 + __psi_1pd - __psi_apd1
544 - __psi_bpd1 - __ln_omx;
545 _Tp __fact = _Tp(1);
546 _Tp __sum2 = __psi_term;
547 _Tp __ln_pre2 = __lng_c + __d1 * __ln_omx
548 - __lng_ad2 - __lng_bd2;
549
550 // Do F2 sum.
551 int __j;
552 for (__j = 1; __j < __maxiter; ++__j)
553 {
554 // Values for psi functions use recurrence; Abramowitz & Stegun 6.3.5
555 const _Tp __term1 = _Tp(1) / _Tp(__j)
556 + _Tp(1) / (__ad + __j);
557 const _Tp __term2 = _Tp(1) / (__a + __d1 + _Tp(__j - 1))
558 + _Tp(1) / (__b + __d1 + _Tp(__j - 1));
559 __psi_term += __term1 - __term2;
560 __fact *= (__a + __d1 + _Tp(__j - 1))
561 * (__b + __d1 + _Tp(__j - 1))
562 / ((__ad + __j) * __j) * (_Tp(1) - __x);
563 const _Tp __delta = __fact * __psi_term;
564 __sum2 += __delta;
565 if (std::abs(__delta) < __eps * std::abs(__sum2))
566 break;
567 }
568 if (__j == __maxiter)
569 std::__throw_runtime_error(__N("Sum F2 failed to converge "
570 "in __hyperg_reflect"));
571
572 if (__sum2 == _Tp(0))
573 __F2 = _Tp(0);
574 else
575 __F2 = std::exp(__ln_pre2) * __sum2;
576 }
577 else
578 {
579 // Gamma functions in the denominator not ok.
580 // So the F2 term is zero.
581 __F2 = _Tp(0);
582 } // end F2 evaluation
583
584 const _Tp __sgn_2 = (__intd % 2 == 1 ? -_Tp(1) : _Tp(1));
585 const _Tp __F = __F1 + __sgn_2 * __F2;
586
587 return __F;
588 }
589 else
590 {
591 // d = c - a - b not an integer.
592
593 // These gamma functions appear in the denominator, so we
594 // catch their harmless domain errors and set the terms to zero.
595 bool __ok1 = true;
596 _Tp __sgn_g1ca = _Tp(0), __ln_g1ca = _Tp(0);
597 _Tp __sgn_g1cb = _Tp(0), __ln_g1cb = _Tp(0);
bc2631e0 598 __try
7c62b943
BK		 599 {
600 __sgn_g1ca = __log_gamma_sign(__c - __a);
601 __ln_g1ca = __log_gamma(__c - __a);
602 __sgn_g1cb = __log_gamma_sign(__c - __b);
603 __ln_g1cb = __log_gamma(__c - __b);
604 }
bc2631e0 605 __catch(...)
7c62b943
BK		 606 {
607 __ok1 = false;
608 }
609
610 bool __ok2 = true;
611 _Tp __sgn_g2a = _Tp(0), __ln_g2a = _Tp(0);
612 _Tp __sgn_g2b = _Tp(0), __ln_g2b = _Tp(0);
bc2631e0 613 __try
7c62b943
BK		 614 {
615 __sgn_g2a = __log_gamma_sign(__a);
616 __ln_g2a = __log_gamma(__a);
617 __sgn_g2b = __log_gamma_sign(__b);
618 __ln_g2b = __log_gamma(__b);
619 }
bc2631e0 620 __catch(...)
7c62b943
BK		 621 {
622 __ok2 = false;
623 }
624
625 const _Tp __sgn_gc = __log_gamma_sign(__c);
626 const _Tp __ln_gc = __log_gamma(__c);
627 const _Tp __sgn_gd = __log_gamma_sign(__d);
628 const _Tp __ln_gd = __log_gamma(__d);
629 const _Tp __sgn_gmd = __log_gamma_sign(-__d);
630 const _Tp __ln_gmd = __log_gamma(-__d);
631
632 const _Tp __sgn1 = __sgn_gc * __sgn_gd * __sgn_g1ca * __sgn_g1cb;
633 const _Tp __sgn2 = __sgn_gc * __sgn_gmd * __sgn_g2a * __sgn_g2b;
634
635 _Tp __pre1, __pre2;
636 if (__ok1 && __ok2)
637 {
638 _Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb;
639 _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b
640 + __d * std::log(_Tp(1) - __x);
641 if (__ln_pre1 < __log_max && __ln_pre2 < __log_max)
642 {
643 __pre1 = std::exp(__ln_pre1);
644 __pre2 = std::exp(__ln_pre2);
645 __pre1 *= __sgn1;
646 __pre2 *= __sgn2;
647 }
648 else
649 {
650 std::__throw_runtime_error(__N("Overflow of gamma functions "
651 "in __hyperg_reflect"));
652 }
653 }
654 else if (__ok1 && !__ok2)
655 {
656 _Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb;
657 if (__ln_pre1 < __log_max)
658 {
659 __pre1 = std::exp(__ln_pre1);
660 __pre1 *= __sgn1;
661 __pre2 = _Tp(0);
662 }
663 else
664 {
665 std::__throw_runtime_error(__N("Overflow of gamma functions "
666 "in __hyperg_reflect"));
667 }
668 }
669 else if (!__ok1 && __ok2)
670 {
671 _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b
672 + __d * std::log(_Tp(1) - __x);
673 if (__ln_pre2 < __log_max)
674 {
675 __pre1 = _Tp(0);
676 __pre2 = std::exp(__ln_pre2);
677 __pre2 *= __sgn2;
678 }
679 else
680 {
681 std::__throw_runtime_error(__N("Overflow of gamma functions "
682 "in __hyperg_reflect"));
683 }
684 }
685 else
686 {
687 __pre1 = _Tp(0);
688 __pre2 = _Tp(0);
689 std::__throw_runtime_error(__N("Underflow of gamma functions "
690 "in __hyperg_reflect"));
691 }
692
693 const _Tp __F1 = __hyperg_series(__a, __b, _Tp(1) - __d,
694 _Tp(1) - __x);
695 const _Tp __F2 = __hyperg_series(__c - __a, __c - __b, _Tp(1) + __d,
696 _Tp(1) - __x);
697
698 const _Tp __F = __pre1 * __F1 + __pre2 * __F2;
699
700 return __F;
701 }
702 }
703
704
6165bbdd 705 /**
7c62b943
BK		 706 * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$.
707 *
708 * The hypogeometric function is defined by
709 * @f[
710 * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}
711 * \sum_{n=0}^{\infty}
712 * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}
713 * \frac{x^n}{n!}
714 * @f]
715 *
716 * @param __a The first "numerator" parameter.
717 * @param __a The second "numerator" parameter.
718 * @param __c The "denominator" parameter.
719 * @param __x The argument of the confluent hypergeometric function.
720 * @return The confluent hypergeometric function.
721 */
722 template<typename _Tp>
723 inline _Tp
724 __hyperg(const _Tp __a, const _Tp __b, const _Tp __c, const _Tp __x)
725 {
726#if _GLIBCXX_USE_C99_MATH_TR1
e133ace8
PC		 727 const _Tp __a_nint = std::tr1::nearbyint(__a);
728 const _Tp __b_nint = std::tr1::nearbyint(__b);
729 const _Tp __c_nint = std::tr1::nearbyint(__c);
7c62b943
BK		 730#else
731 const _Tp __a_nint = static_cast<int>(__a + _Tp(0.5L));
732 const _Tp __b_nint = static_cast<int>(__b + _Tp(0.5L));
733 const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L));
734#endif
735 const _Tp __toler = _Tp(1000) * std::numeric_limits<_Tp>::epsilon();
736 if (std::abs(__x) >= _Tp(1))
737 std::__throw_domain_error(__N("Argument outside unit circle "
738 "in __hyperg."));
739 else if (__isnan(__a) || __isnan(__b)
740 || __isnan(__c) || __isnan(__x))
741 return std::numeric_limits<_Tp>::quiet_NaN();
742 else if (__c_nint == __c && __c_nint <= _Tp(0))
743 return std::numeric_limits<_Tp>::infinity();
744 else if (std::abs(__c - __b) < __toler || std::abs(__c - __a) < __toler)
745 return std::pow(_Tp(1) - __x, __c - __a - __b);
746 else if (__a >= _Tp(0) && __b >= _Tp(0) && __c >= _Tp(0)
747 && __x >= _Tp(0) && __x < _Tp(0.995L))
748 return __hyperg_series(__a, __b, __c, __x);
749 else if (std::abs(__a) < _Tp(10) && std::abs(__b) < _Tp(10))
750 {
751 // For integer a and b the hypergeometric function is a finite polynomial.
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
770 } // namespace std::tr1::__detail
e133ace8 771}
7c62b943
BK		 772}
773
e133ace8 774#endif // _GLIBCXX_TR1_HYPERGEOMETRIC_TCC
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