1// Special functions -*- C++ -*-
2
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.
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 "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