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 47 namespace 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) 54 namespace 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