1 // Special functions -*- C++ -*- 2 3 // Copyright (C) 2006, 2007, 2008, 2009 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 9 // Free Software Foundation; either version 3, or (at your option) 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 // 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/>. 25 26 /** @file tr1/riemann_zeta.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 on: 36 // (1) Handbook of Mathematical Functions, 37 // Ed. by Milton Abramowitz and Irene A. Stegun, 38 // Dover Publications, New-York, Section 5, pp. 807-808. 39 // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl 40 // (3) Gamma, Exploring Euler's Constant, Julian Havil, 41 // Princeton, 2003. 42 43 #ifndef _GLIBCXX_TR1_RIEMANN_ZETA_TCC 44 #define _GLIBCXX_TR1_RIEMANN_ZETA_TCC 1 45 46 #include "special_function_util.h" 47 48 namespace std 49 { 50 namespace tr1 51 { 52 53 // [5.2] Special functions 54 55 // Implementation-space details. 56 namespace __detail 57 { 58 59 /** 60 * @brief Compute the Riemann zeta function @f$ \zeta(s) @f$ 61 * by summation for s > 1. 62 * 63 * The Riemann zeta function is defined by: 64 * \f[ 65 * \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 66 * \f] 67 * For s < 1 use the reflection formula: 68 * \f[ 69 * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) 70 * \f] 71 */ 72 template<typename _Tp> 73 _Tp 74 __riemann_zeta_sum(const _Tp __s) 75 { 76 // A user shouldn't get to this. 77 if (__s < _Tp(1)) 78 std::__throw_domain_error(__N("Bad argument in zeta sum.")); 79 80 const unsigned int max_iter = 10000; 81 _Tp __zeta = _Tp(0); 82 for (unsigned int __k = 1; __k < max_iter; ++__k) 83 { 84 _Tp __term = std::pow(static_cast<_Tp>(__k), -__s); 85 if (__term < std::numeric_limits<_Tp>::epsilon()) 86 { 87 break; 88 } 89 __zeta += __term; 90 } 91 92 return __zeta; 93 } 94 95 96 /** 97 * @brief Evaluate the Riemann zeta function @f$ \zeta(s) @f$ 98 * by an alternate series for s > 0. 99 * 100 * The Riemann zeta function is defined by: 101 * \f[ 102 * \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 103 * \f] 104 * For s < 1 use the reflection formula: 105 * \f[ 106 * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) 107 * \f] 108 */ 109 template<typename _Tp> 110 _Tp 111 __riemann_zeta_alt(const _Tp __s) 112 { 113 _Tp __sgn = _Tp(1); 114 _Tp __zeta = _Tp(0); 115 for (unsigned int __i = 1; __i < 10000000; ++__i) 116 { 117 _Tp __term = __sgn / std::pow(__i, __s); 118 if (std::abs(__term) < std::numeric_limits<_Tp>::epsilon()) 119 break; 120 __zeta += __term; 121 __sgn *= _Tp(-1); 122 } 123 __zeta /= _Tp(1) - std::pow(_Tp(2), _Tp(1) - __s); 124 125 return __zeta; 126 } 127 128 129 /** 130 * @brief Evaluate the Riemann zeta function by series for all s != 1. 131 * Convergence is great until largish negative numbers. 132 * Then the convergence of the > 0 sum gets better. 133 * 134 * The series is: 135 * \f[ 136 * \zeta(s) = \frac{1}{1-2^{1-s}} 137 * \sum_{n=0}^{\infty} \frac{1}{2^{n+1}} 138 * \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (k+1)^{-s} 139 * \f] 140 * Havil 2003, p. 206. 141 * 142 * The Riemann zeta function is defined by: 143 * \f[ 144 * \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 145 * \f] 146 * For s < 1 use the reflection formula: 147 * \f[ 148 * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) 149 * \f] 150 */ 151 template<typename _Tp> 152 _Tp 153 __riemann_zeta_glob(const _Tp __s) 154 { 155 _Tp __zeta = _Tp(0); 156 157 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 158 // Max e exponent before overflow. 159 const _Tp __max_bincoeff = std::numeric_limits<_Tp>::max_exponent10 160 * std::log(_Tp(10)) - _Tp(1); 161 162 // This series works until the binomial coefficient blows up 163 // so use reflection. 164 if (__s < _Tp(0)) 165 { 166 #if _GLIBCXX_USE_C99_MATH_TR1 167 if (std::tr1::fmod(__s,_Tp(2)) == _Tp(0)) 168 return _Tp(0); 169 else 170 #endif 171 { 172 _Tp __zeta = __riemann_zeta_glob(_Tp(1) - __s); 173 __zeta *= std::pow(_Tp(2) 174 * __numeric_constants<_Tp>::__pi(), __s) 175 * std::sin(__numeric_constants<_Tp>::__pi_2() * __s) 176 #if _GLIBCXX_USE_C99_MATH_TR1 177 * std::exp(std::tr1::lgamma(_Tp(1) - __s)) 178 #else 179 * std::exp(__log_gamma(_Tp(1) - __s)) 180 #endif 181 / __numeric_constants<_Tp>::__pi(); 182 return __zeta; 183 } 184 } 185 186 _Tp __num = _Tp(0.5L); 187 const unsigned int __maxit = 10000; 188 for (unsigned int __i = 0; __i < __maxit; ++__i) 189 { 190 bool __punt = false; 191 _Tp __sgn = _Tp(1); 192 _Tp __term = _Tp(0); 193 for (unsigned int __j = 0; __j <= __i; ++__j) 194 { 195 #if _GLIBCXX_USE_C99_MATH_TR1 196 _Tp __bincoeff = std::tr1::lgamma(_Tp(1 + __i)) 197 - std::tr1::lgamma(_Tp(1 + __j)) 198 - std::tr1::lgamma(_Tp(1 + __i - __j)); 199 #else 200 _Tp __bincoeff = __log_gamma(_Tp(1 + __i)) 201 - __log_gamma(_Tp(1 + __j)) 202 - __log_gamma(_Tp(1 + __i - __j)); 203 #endif 204 if (__bincoeff > __max_bincoeff) 205 { 206 // This only gets hit for x << 0. 207 __punt = true; 208 break; 209 } 210 __bincoeff = std::exp(__bincoeff); 211 __term += __sgn * __bincoeff * std::pow(_Tp(1 + __j), -__s); 212 __sgn *= _Tp(-1); 213 } 214 if (__punt) 215 break; 216 __term *= __num; 217 __zeta += __term; 218 if (std::abs(__term/__zeta) < __eps) 219 break; 220 __num *= _Tp(0.5L); 221 } 222 223 __zeta /= _Tp(1) - std::pow(_Tp(2), _Tp(1) - __s); 224 225 return __zeta; 226 } 227 228 229 /** 230 * @brief Compute the Riemann zeta function @f$ \zeta(s) @f$ 231 * using the product over prime factors. 232 * \f[ 233 * \zeta(s) = \Pi_{i=1}^\infty \frac{1}{1 - p_i^{-s}} 234 * \f] 235 * where @f$ {p_i} @f$ are the prime numbers. 236 * 237 * The Riemann zeta function is defined by: 238 * \f[ 239 * \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 240 * \f] 241 * For s < 1 use the reflection formula: 242 * \f[ 243 * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) 244 * \f] 245 */ 246 template<typename _Tp> 247 _Tp 248 __riemann_zeta_product(const _Tp __s) 249 { 250 static const _Tp __prime[] = { 251 _Tp(2), _Tp(3), _Tp(5), _Tp(7), _Tp(11), _Tp(13), _Tp(17), _Tp(19), 252 _Tp(23), _Tp(29), _Tp(31), _Tp(37), _Tp(41), _Tp(43), _Tp(47), 253 _Tp(53), _Tp(59), _Tp(61), _Tp(67), _Tp(71), _Tp(73), _Tp(79), 254 _Tp(83), _Tp(89), _Tp(97), _Tp(101), _Tp(103), _Tp(107), _Tp(109) 255 }; 256 static const unsigned int __num_primes = sizeof(__prime) / sizeof(_Tp); 257 258 _Tp __zeta = _Tp(1); 259 for (unsigned int __i = 0; __i < __num_primes; ++__i) 260 { 261 const _Tp __fact = _Tp(1) - std::pow(__prime[__i], -__s); 262 __zeta *= __fact; 263 if (_Tp(1) - __fact < std::numeric_limits<_Tp>::epsilon()) 264 break; 265 } 266 267 __zeta = _Tp(1) / __zeta; 268 269 return __zeta; 270 } 271 272 273 /** 274 * @brief Return the Riemann zeta function @f$ \zeta(s) @f$. 275 * 276 * The Riemann zeta function is defined by: 277 * \f[ 278 * \zeta(s) = \sum_{k=1}^{\infty} k^{-s} for s > 1 279 * \frac{(2\pi)^s}{pi} sin(\frac{\pi s}{2}) 280 * \Gamma (1 - s) \zeta (1 - s) for s < 1 281 * \f] 282 * For s < 1 use the reflection formula: 283 * \f[ 284 * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) 285 * \f] 286 */ 287 template<typename _Tp> 288 _Tp 289 __riemann_zeta(const _Tp __s) 290 { 291 if (__isnan(__s)) 292 return std::numeric_limits<_Tp>::quiet_NaN(); 293 else if (__s == _Tp(1)) 294 return std::numeric_limits<_Tp>::infinity(); 295 else if (__s < -_Tp(19)) 296 { 297 _Tp __zeta = __riemann_zeta_product(_Tp(1) - __s); 298 __zeta *= std::pow(_Tp(2) * __numeric_constants<_Tp>::__pi(), __s) 299 * std::sin(__numeric_constants<_Tp>::__pi_2() * __s) 300 #if _GLIBCXX_USE_C99_MATH_TR1 301 * std::exp(std::tr1::lgamma(_Tp(1) - __s)) 302 #else 303 * std::exp(__log_gamma(_Tp(1) - __s)) 304 #endif 305 / __numeric_constants<_Tp>::__pi(); 306 return __zeta; 307 } 308 else if (__s < _Tp(20)) 309 { 310 // Global double sum or McLaurin? 311 bool __glob = true; 312 if (__glob) 313 return __riemann_zeta_glob(__s); 314 else 315 { 316 if (__s > _Tp(1)) 317 return __riemann_zeta_sum(__s); 318 else 319 { 320 _Tp __zeta = std::pow(_Tp(2) 321 * __numeric_constants<_Tp>::__pi(), __s) 322 * std::sin(__numeric_constants<_Tp>::__pi_2() * __s) 323 #if _GLIBCXX_USE_C99_MATH_TR1 324 * std::tr1::tgamma(_Tp(1) - __s) 325 #else 326 * std::exp(__log_gamma(_Tp(1) - __s)) 327 #endif 328 * __riemann_zeta_sum(_Tp(1) - __s); 329 return __zeta; 330 } 331 } 332 } 333 else 334 return __riemann_zeta_product(__s); 335 } 336 337 338 /** 339 * @brief Return the Hurwitz zeta function @f$ \zeta(x,s) @f$ 340 * for all s != 1 and x > -1. 341 * 342 * The Hurwitz zeta function is defined by: 343 * @f[ 344 * \zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s} 345 * @f] 346 * The Riemann zeta function is a special case: 347 * @f[ 348 * \zeta(s) = \zeta(1,s) 349 * @f] 350 * 351 * This functions uses the double sum that converges for s != 1 352 * and x > -1: 353 * @f[ 354 * \zeta(x,s) = \frac{1}{s-1} 355 * \sum_{n=0}^{\infty} \frac{1}{n + 1} 356 * \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (x+k)^{-s} 357 * @f] 358 */ 359 template<typename _Tp> 360 _Tp 361 __hurwitz_zeta_glob(const _Tp __a, const _Tp __s) 362 { 363 _Tp __zeta = _Tp(0); 364 365 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 366 // Max e exponent before overflow. 367 const _Tp __max_bincoeff = std::numeric_limits<_Tp>::max_exponent10 368 * std::log(_Tp(10)) - _Tp(1); 369 370 const unsigned int __maxit = 10000; 371 for (unsigned int __i = 0; __i < __maxit; ++__i) 372 { 373 bool __punt = false; 374 _Tp __sgn = _Tp(1); 375 _Tp __term = _Tp(0); 376 for (unsigned int __j = 0; __j <= __i; ++__j) 377 { 378 #if _GLIBCXX_USE_C99_MATH_TR1 379 _Tp __bincoeff = std::tr1::lgamma(_Tp(1 + __i)) 380 - std::tr1::lgamma(_Tp(1 + __j)) 381 - std::tr1::lgamma(_Tp(1 + __i - __j)); 382 #else 383 _Tp __bincoeff = __log_gamma(_Tp(1 + __i)) 384 - __log_gamma(_Tp(1 + __j)) 385 - __log_gamma(_Tp(1 + __i - __j)); 386 #endif 387 if (__bincoeff > __max_bincoeff) 388 { 389 // This only gets hit for x << 0. 390 __punt = true; 391 break; 392 } 393 __bincoeff = std::exp(__bincoeff); 394 __term += __sgn * __bincoeff * std::pow(_Tp(__a + __j), -__s); 395 __sgn *= _Tp(-1); 396 } 397 if (__punt) 398 break; 399 __term /= _Tp(__i + 1); 400 if (std::abs(__term / __zeta) < __eps) 401 break; 402 __zeta += __term; 403 } 404 405 __zeta /= __s - _Tp(1); 406 407 return __zeta; 408 } 409 410 411 /** 412 * @brief Return the Hurwitz zeta function @f$ \zeta(x,s) @f$ 413 * for all s != 1 and x > -1. 414 * 415 * The Hurwitz zeta function is defined by: 416 * @f[ 417 * \zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s} 418 * @f] 419 * The Riemann zeta function is a special case: 420 * @f[ 421 * \zeta(s) = \zeta(1,s) 422 * @f] 423 */ 424 template<typename _Tp> 425 inline _Tp 426 __hurwitz_zeta(const _Tp __a, const _Tp __s) 427 { 428 return __hurwitz_zeta_glob(__a, __s); 429 } 430 431 } // namespace std::tr1::__detail 432 } 433 } 434 435 #endif // _GLIBCXX_TR1_RIEMANN_ZETA_TCC 436
