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/riemann_zeta.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 on: 
35// (1) Handbook of Mathematical Functions, 
36// Ed. by Milton Abramowitz and Irene A. Stegun, 
37// Dover Publications, New-York, Section 5, pp. 807-808. 
38// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl 
39// (3) Gamma, Exploring Euler's Constant, Julian Havil, 
40// Princeton, 2003. 
41 
42#ifndef _GLIBCXX_TR1_RIEMANN_ZETA_TCC 
43#define _GLIBCXX_TR1_RIEMANN_ZETA_TCC 1 
44 
45#include <tr1/special_function_util.h> 
46 
47namespace std _GLIBCXX_VISIBILITY(default) 
48{ 
49_GLIBCXX_BEGIN_NAMESPACE_VERSION 
50 
51#if _GLIBCXX_USE_STD_SPEC_FUNCS 
52# define _GLIBCXX_MATH_NS ::std 
53#elif defined(_GLIBCXX_TR1_CMATH) 
54namespace tr1 
55{ 
56# define _GLIBCXX_MATH_NS ::std::tr1 
57#else 
58# error do not include this header directly, use <cmath> or <tr1/cmath> 
59#endif 
60 // [5.2] Special functions 
61 
62 // Implementation-space details. 
63 namespace __detail 
64 { 
65 /** 
66 * @brief Compute the Riemann zeta function @f$ \zeta(s) @f$ 
67 * by summation for s > 1. 
68 *  
69 * The Riemann zeta function is defined by: 
70 * \f[ 
71 * \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 
72 * \f] 
73 * For s < 1 use the reflection formula: 
74 * \f[ 
75 * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) 
76 * \f] 
77 */ 
78 template<typename _Tp> 
79 _Tp 
80 __riemann_zeta_sum(_Tp __s) 
81 { 
82 // A user shouldn't get to this. 
83 if (__s < _Tp(1)) 
84 std::__throw_domain_error(__N("Bad argument in zeta sum.")); 
85 
86 const unsigned int max_iter = 10000; 
87 _Tp __zeta = _Tp(0); 
88 for (unsigned int __k = 1; __k < max_iter; ++__k) 
89 { 
90 _Tp __term = std::pow(static_cast<_Tp>(__k), -__s); 
91 if (__term < std::numeric_limits<_Tp>::epsilon()) 
92 { 
93 break; 
94 } 
95 __zeta += __term; 
96 } 
97 
98 return __zeta; 
99 } 
100 
101 
102 /** 
103 * @brief Evaluate the Riemann zeta function @f$ \zeta(s) @f$ 
104 * by an alternate series for s > 0. 
105 *  
106 * The Riemann zeta function is defined by: 
107 * \f[ 
108 * \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 
109 * \f] 
110 * For s < 1 use the reflection formula: 
111 * \f[ 
112 * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) 
113 * \f] 
114 */ 
115 template<typename _Tp> 
116 _Tp 
117 __riemann_zeta_alt(_Tp __s) 
118 { 
119 _Tp __sgn = _Tp(1); 
120 _Tp __zeta = _Tp(0); 
121 for (unsigned int __i = 1; __i < 10000000; ++__i) 
122 { 
123 _Tp __term = __sgn / std::pow(__i, __s); 
124 if (std::abs(__term) < std::numeric_limits<_Tp>::epsilon()) 
125 break; 
126 __zeta += __term; 
127 __sgn *= _Tp(-1); 
128 } 
129 __zeta /= _Tp(1) - std::pow(_Tp(2), _Tp(1) - __s); 
130 
131 return __zeta; 
132 } 
133 
134 
135 /** 
136 * @brief Evaluate the Riemann zeta function by series for all s != 1. 
137 * Convergence is great until largish negative numbers. 
138 * Then the convergence of the > 0 sum gets better. 
139 * 
140 * The series is: 
141 * \f[ 
142 * \zeta(s) = \frac{1}{1-2^{1-s}} 
143 * \sum_{n=0}^{\infty} \frac{1}{2^{n+1}} 
144 * \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (k+1)^{-s} 
145 * \f] 
146 * Havil 2003, p. 206. 
147 * 
148 * The Riemann zeta function is defined by: 
149 * \f[ 
150 * \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 
151 * \f] 
152 * For s < 1 use the reflection formula: 
153 * \f[ 
154 * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) 
155 * \f] 
156 */ 
157 template<typename _Tp> 
158 _Tp 
159 __riemann_zeta_glob(_Tp __s) 
160 { 
161 _Tp __zeta = _Tp(0); 
162 
163 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 
164 // Max e exponent before overflow. 
165 const _Tp __max_bincoeff = std::numeric_limits<_Tp>::max_exponent10 
166 * std::log(_Tp(10)) - _Tp(1); 
167 
168 // This series works until the binomial coefficient blows up 
169 // so use reflection. 
170 if (__s < _Tp(0)) 
171 { 
172#if _GLIBCXX_USE_C99_MATH_TR1 
173 if (_GLIBCXX_MATH_NS::fmod(__s,_Tp(2)) == _Tp(0)) 
174 return _Tp(0); 
175 else 
176#endif 
177 { 
178 _Tp __zeta = __riemann_zeta_glob(_Tp(1) - __s); 
179 __zeta *= std::pow(_Tp(2) 
180 * __numeric_constants<_Tp>::__pi(), __s) 
181 * std::sin(__numeric_constants<_Tp>::__pi_2() * __s) 
182#if _GLIBCXX_USE_C99_MATH_TR1 
183 * std::exp(_GLIBCXX_MATH_NS::lgamma(_Tp(1) - __s)) 
184#else 
185 * std::exp(__log_gamma(_Tp(1) - __s)) 
186#endif 
187 / __numeric_constants<_Tp>::__pi(); 
188 return __zeta; 
189 } 
190 } 
191 
192 _Tp __num = _Tp(0.5L); 
193 const unsigned int __maxit = 10000; 
194 for (unsigned int __i = 0; __i < __maxit; ++__i) 
195 { 
196 bool __punt = false; 
197 _Tp __sgn = _Tp(1); 
198 _Tp __term = _Tp(0); 
199 for (unsigned int __j = 0; __j <= __i; ++__j) 
200 { 
201#if _GLIBCXX_USE_C99_MATH_TR1 
202 _Tp __bincoeff = _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __i)) 
203 - _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __j)) 
204 - _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __i - __j)); 
205#else 
206 _Tp __bincoeff = __log_gamma(_Tp(1 + __i)) 
207 - __log_gamma(_Tp(1 + __j)) 
208 - __log_gamma(_Tp(1 + __i - __j)); 
209#endif 
210 if (__bincoeff > __max_bincoeff) 
211 { 
212 // This only gets hit for x << 0. 
213 __punt = true; 
214 break; 
215 } 
216 __bincoeff = std::exp(__bincoeff); 
217 __term += __sgn * __bincoeff * std::pow(_Tp(1 + __j), -__s); 
218 __sgn *= _Tp(-1); 
219 } 
220 if (__punt) 
221 break; 
222 __term *= __num; 
223 __zeta += __term; 
224 if (std::abs(__term/__zeta) < __eps) 
225 break; 
226 __num *= _Tp(0.5L); 
227 } 
228 
229 __zeta /= _Tp(1) - std::pow(_Tp(2), _Tp(1) - __s); 
230 
231 return __zeta; 
232 } 
233 
234 
235 /** 
236 * @brief Compute the Riemann zeta function @f$ \zeta(s) @f$ 
237 * using the product over prime factors. 
238 * \f[ 
239 * \zeta(s) = \Pi_{i=1}^\infty \frac{1}{1 - p_i^{-s}} 
240 * \f] 
241 * where @f$ {p_i} @f$ are the prime numbers. 
242 *  
243 * The Riemann zeta function is defined by: 
244 * \f[ 
245 * \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 
246 * \f] 
247 * For s < 1 use the reflection formula: 
248 * \f[ 
249 * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) 
250 * \f] 
251 */ 
252 template<typename _Tp> 
253 _Tp 
254 __riemann_zeta_product(_Tp __s) 
255 { 
256 static const _Tp __prime[] = { 
257 _Tp(2), _Tp(3), _Tp(5), _Tp(7), _Tp(11), _Tp(13), _Tp(17), _Tp(19), 
258 _Tp(23), _Tp(29), _Tp(31), _Tp(37), _Tp(41), _Tp(43), _Tp(47), 
259 _Tp(53), _Tp(59), _Tp(61), _Tp(67), _Tp(71), _Tp(73), _Tp(79), 
260 _Tp(83), _Tp(89), _Tp(97), _Tp(101), _Tp(103), _Tp(107), _Tp(109) 
261 }; 
262 static const unsigned int __num_primes = sizeof(__prime) / sizeof(_Tp); 
263 
264 _Tp __zeta = _Tp(1); 
265 for (unsigned int __i = 0; __i < __num_primes; ++__i) 
266 { 
267 const _Tp __fact = _Tp(1) - std::pow(__prime[__i], -__s); 
268 __zeta *= __fact; 
269 if (_Tp(1) - __fact < std::numeric_limits<_Tp>::epsilon()) 
270 break; 
271 } 
272 
273 __zeta = _Tp(1) / __zeta; 
274 
275 return __zeta; 
276 } 
277 
278 
279 /** 
280 * @brief Return the Riemann zeta function @f$ \zeta(s) @f$. 
281 *  
282 * The Riemann zeta function is defined by: 
283 * \f[ 
284 * \zeta(s) = \sum_{k=1}^{\infty} k^{-s} for s > 1 
285 * \frac{(2\pi)^s}{pi} sin(\frac{\pi s}{2}) 
286 * \Gamma (1 - s) \zeta (1 - s) for s < 1 
287 * \f] 
288 * For s < 1 use the reflection formula: 
289 * \f[ 
290 * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) 
291 * \f] 
292 */ 
293 template<typename _Tp> 
294 _Tp 
295 __riemann_zeta(_Tp __s) 
296 { 
297 if (__isnan(__s)) 
298 return std::numeric_limits<_Tp>::quiet_NaN(); 
299 else if (__s == _Tp(1)) 
300 return std::numeric_limits<_Tp>::infinity(); 
301 else if (__s < -_Tp(19)) 
302 { 
303 _Tp __zeta = __riemann_zeta_product(_Tp(1) - __s); 
304 __zeta *= std::pow(_Tp(2) * __numeric_constants<_Tp>::__pi(), __s) 
305 * std::sin(__numeric_constants<_Tp>::__pi_2() * __s) 
306#if _GLIBCXX_USE_C99_MATH_TR1 
307 * std::exp(_GLIBCXX_MATH_NS::lgamma(_Tp(1) - __s)) 
308#else 
309 * std::exp(__log_gamma(_Tp(1) - __s)) 
310#endif 
311 / __numeric_constants<_Tp>::__pi(); 
312 return __zeta; 
313 } 
314 else if (__s < _Tp(20)) 
315 { 
316 // Global double sum or McLaurin? 
317 bool __glob = true; 
318 if (__glob) 
319 return __riemann_zeta_glob(__s); 
320 else 
321 { 
322 if (__s > _Tp(1)) 
323 return __riemann_zeta_sum(__s); 
324 else 
325 { 
326 _Tp __zeta = std::pow(_Tp(2) 
327 * __numeric_constants<_Tp>::__pi(), __s) 
328 * std::sin(__numeric_constants<_Tp>::__pi_2() * __s) 
329#if _GLIBCXX_USE_C99_MATH_TR1 
330 * _GLIBCXX_MATH_NS::tgamma(_Tp(1) - __s) 
331#else 
332 * std::exp(__log_gamma(_Tp(1) - __s)) 
333#endif 
334 * __riemann_zeta_sum(_Tp(1) - __s); 
335 return __zeta; 
336 } 
337 } 
338 } 
339 else 
340 return __riemann_zeta_product(__s); 
341 } 
342 
343 
344 /** 
345 * @brief Return the Hurwitz zeta function @f$ \zeta(x,s) @f$ 
346 * for all s != 1 and x > -1. 
347 *  
348 * The Hurwitz zeta function is defined by: 
349 * @f[ 
350 * \zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s} 
351 * @f] 
352 * The Riemann zeta function is a special case: 
353 * @f[ 
354 * \zeta(s) = \zeta(1,s) 
355 * @f] 
356 *  
357 * This functions uses the double sum that converges for s != 1 
358 * and x > -1: 
359 * @f[ 
360 * \zeta(x,s) = \frac{1}{s-1} 
361 * \sum_{n=0}^{\infty} \frac{1}{n + 1} 
362 * \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (x+k)^{-s} 
363 * @f] 
364 */ 
365 template<typename _Tp> 
366 _Tp 
367 __hurwitz_zeta_glob(_Tp __a, _Tp __s) 
368 { 
369 _Tp __zeta = _Tp(0); 
370 
371 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 
372 // Max e exponent before overflow. 
373 const _Tp __max_bincoeff = std::numeric_limits<_Tp>::max_exponent10 
374 * std::log(_Tp(10)) - _Tp(1); 
375 
376 const unsigned int __maxit = 10000; 
377 for (unsigned int __i = 0; __i < __maxit; ++__i) 
378 { 
379 bool __punt = false; 
380 _Tp __sgn = _Tp(1); 
381 _Tp __term = _Tp(0); 
382 for (unsigned int __j = 0; __j <= __i; ++__j) 
383 { 
384#if _GLIBCXX_USE_C99_MATH_TR1 
385 _Tp __bincoeff = _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __i)) 
386 - _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __j)) 
387 - _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __i - __j)); 
388#else 
389 _Tp __bincoeff = __log_gamma(_Tp(1 + __i)) 
390 - __log_gamma(_Tp(1 + __j)) 
391 - __log_gamma(_Tp(1 + __i - __j)); 
392#endif 
393 if (__bincoeff > __max_bincoeff) 
394 { 
395 // This only gets hit for x << 0. 
396 __punt = true; 
397 break; 
398 } 
399 __bincoeff = std::exp(__bincoeff); 
400 __term += __sgn * __bincoeff * std::pow(_Tp(__a + __j), -__s); 
401 __sgn *= _Tp(-1); 
402 } 
403 if (__punt) 
404 break; 
405 __term /= _Tp(__i + 1); 
406 if (std::abs(__term / __zeta) < __eps) 
407 break; 
408 __zeta += __term; 
409 } 
410 
411 __zeta /= __s - _Tp(1); 
412 
413 return __zeta; 
414 } 
415 
416 
417 /** 
418 * @brief Return the Hurwitz zeta function @f$ \zeta(x,s) @f$ 
419 * for all s != 1 and x > -1. 
420 *  
421 * The Hurwitz zeta function is defined by: 
422 * @f[ 
423 * \zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s} 
424 * @f] 
425 * The Riemann zeta function is a special case: 
426 * @f[ 
427 * \zeta(s) = \zeta(1,s) 
428 * @f] 
429 */ 
430 template<typename _Tp> 
431 inline _Tp 
432 __hurwitz_zeta(_Tp __a, _Tp __s) 
433 { return __hurwitz_zeta_glob(__a, __s); } 
434 } // namespace __detail 
435#undef _GLIBCXX_MATH_NS 
436#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) 
437} // namespace tr1 
438#endif 
439 
440_GLIBCXX_END_NAMESPACE_VERSION 
441} 
442 
443#endif // _GLIBCXX_TR1_RIEMANN_ZETA_TCC 
444