libstdc++
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00001 // Special functions -*- C++ -*- 00002 00003 // Copyright (C) 2006, 2007, 2008, 2009 00004 // Free Software Foundation, Inc. 00005 // 00006 // This file is part of the GNU ISO C++ Library. This library is free 00007 // software; you can redistribute it and/or modify it under the 00008 // terms of the GNU General Public License as published by the 00009 // Free Software Foundation; either version 3, or (at your option) 00010 // any later version. 00011 // 00012 // This library is distributed in the hope that it will be useful, 00013 // but WITHOUT ANY WARRANTY; without even the implied warranty of 00014 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 00015 // GNU General Public License for more details. 00016 // 00017 // Under Section 7 of GPL version 3, you are granted additional 00018 // permissions described in the GCC Runtime Library Exception, version 00019 // 3.1, as published by the Free Software Foundation. 00020 00021 // You should have received a copy of the GNU General Public License and 00022 // a copy of the GCC Runtime Library Exception along with this program; 00023 // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see 00024 // <http://www.gnu.org/licenses/>. 00025 00026 /** @file tr1/riemann_zeta.tcc 00027 * This is an internal header file, included by other library headers. 00028 * You should not attempt to use it directly. 00029 */ 00030 00031 // 00032 // ISO C++ 14882 TR1: 5.2 Special functions 00033 // 00034 00035 // Written by Edward Smith-Rowland based on: 00036 // (1) Handbook of Mathematical Functions, 00037 // Ed. by Milton Abramowitz and Irene A. Stegun, 00038 // Dover Publications, New-York, Section 5, pp. 807-808. 00039 // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl 00040 // (3) Gamma, Exploring Euler's Constant, Julian Havil, 00041 // Princeton, 2003. 00042 00043 #ifndef _GLIBCXX_TR1_RIEMANN_ZETA_TCC 00044 #define _GLIBCXX_TR1_RIEMANN_ZETA_TCC 1 00045 00046 #include "special_function_util.h" 00047 00048 namespace std 00049 { 00050 namespace tr1 00051 { 00052 00053 // [5.2] Special functions 00054 00055 // Implementation-space details. 00056 namespace __detail 00057 { 00058 00059 /** 00060 * @brief Compute the Riemann zeta function @f$ \zeta(s) @f$ 00061 * by summation for s > 1. 00062 * 00063 * The Riemann zeta function is defined by: 00064 * \f[ 00065 * \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 00066 * \f] 00067 * For s < 1 use the reflection formula: 00068 * \f[ 00069 * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) 00070 * \f] 00071 */ 00072 template<typename _Tp> 00073 _Tp 00074 __riemann_zeta_sum(const _Tp __s) 00075 { 00076 // A user shouldn't get to this. 00077 if (__s < _Tp(1)) 00078 std::__throw_domain_error(__N("Bad argument in zeta sum.")); 00079 00080 const unsigned int max_iter = 10000; 00081 _Tp __zeta = _Tp(0); 00082 for (unsigned int __k = 1; __k < max_iter; ++__k) 00083 { 00084 _Tp __term = std::pow(static_cast<_Tp>(__k), -__s); 00085 if (__term < std::numeric_limits<_Tp>::epsilon()) 00086 { 00087 break; 00088 } 00089 __zeta += __term; 00090 } 00091 00092 return __zeta; 00093 } 00094 00095 00096 /** 00097 * @brief Evaluate the Riemann zeta function @f$ \zeta(s) @f$ 00098 * by an alternate series for s > 0. 00099 * 00100 * The Riemann zeta function is defined by: 00101 * \f[ 00102 * \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 00103 * \f] 00104 * For s < 1 use the reflection formula: 00105 * \f[ 00106 * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) 00107 * \f] 00108 */ 00109 template<typename _Tp> 00110 _Tp 00111 __riemann_zeta_alt(const _Tp __s) 00112 { 00113 _Tp __sgn = _Tp(1); 00114 _Tp __zeta = _Tp(0); 00115 for (unsigned int __i = 1; __i < 10000000; ++__i) 00116 { 00117 _Tp __term = __sgn / std::pow(__i, __s); 00118 if (std::abs(__term) < std::numeric_limits<_Tp>::epsilon()) 00119 break; 00120 __zeta += __term; 00121 __sgn *= _Tp(-1); 00122 } 00123 __zeta /= _Tp(1) - std::pow(_Tp(2), _Tp(1) - __s); 00124 00125 return __zeta; 00126 } 00127 00128 00129 /** 00130 * @brief Evaluate the Riemann zeta function by series for all s != 1. 00131 * Convergence is great until largish negative numbers. 00132 * Then the convergence of the > 0 sum gets better. 00133 * 00134 * The series is: 00135 * \f[ 00136 * \zeta(s) = \frac{1}{1-2^{1-s}} 00137 * \sum_{n=0}^{\infty} \frac{1}{2^{n+1}} 00138 * \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (k+1)^{-s} 00139 * \f] 00140 * Havil 2003, p. 206. 00141 * 00142 * The Riemann zeta function is defined by: 00143 * \f[ 00144 * \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 00145 * \f] 00146 * For s < 1 use the reflection formula: 00147 * \f[ 00148 * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) 00149 * \f] 00150 */ 00151 template<typename _Tp> 00152 _Tp 00153 __riemann_zeta_glob(const _Tp __s) 00154 { 00155 _Tp __zeta = _Tp(0); 00156 00157 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 00158 // Max e exponent before overflow. 00159 const _Tp __max_bincoeff = std::numeric_limits<_Tp>::max_exponent10 00160 * std::log(_Tp(10)) - _Tp(1); 00161 00162 // This series works until the binomial coefficient blows up 00163 // so use reflection. 00164 if (__s < _Tp(0)) 00165 { 00166 #if _GLIBCXX_USE_C99_MATH_TR1 00167 if (std::tr1::fmod(__s,_Tp(2)) == _Tp(0)) 00168 return _Tp(0); 00169 else 00170 #endif 00171 { 00172 _Tp __zeta = __riemann_zeta_glob(_Tp(1) - __s); 00173 __zeta *= std::pow(_Tp(2) 00174 * __numeric_constants<_Tp>::__pi(), __s) 00175 * std::sin(__numeric_constants<_Tp>::__pi_2() * __s) 00176 #if _GLIBCXX_USE_C99_MATH_TR1 00177 * std::exp(std::tr1::lgamma(_Tp(1) - __s)) 00178 #else 00179 * std::exp(__log_gamma(_Tp(1) - __s)) 00180 #endif 00181 / __numeric_constants<_Tp>::__pi(); 00182 return __zeta; 00183 } 00184 } 00185 00186 _Tp __num = _Tp(0.5L); 00187 const unsigned int __maxit = 10000; 00188 for (unsigned int __i = 0; __i < __maxit; ++__i) 00189 { 00190 bool __punt = false; 00191 _Tp __sgn = _Tp(1); 00192 _Tp __term = _Tp(0); 00193 for (unsigned int __j = 0; __j <= __i; ++__j) 00194 { 00195 #if _GLIBCXX_USE_C99_MATH_TR1 00196 _Tp __bincoeff = std::tr1::lgamma(_Tp(1 + __i)) 00197 - std::tr1::lgamma(_Tp(1 + __j)) 00198 - std::tr1::lgamma(_Tp(1 + __i - __j)); 00199 #else 00200 _Tp __bincoeff = __log_gamma(_Tp(1 + __i)) 00201 - __log_gamma(_Tp(1 + __j)) 00202 - __log_gamma(_Tp(1 + __i - __j)); 00203 #endif 00204 if (__bincoeff > __max_bincoeff) 00205 { 00206 // This only gets hit for x << 0. 00207 __punt = true; 00208 break; 00209 } 00210 __bincoeff = std::exp(__bincoeff); 00211 __term += __sgn * __bincoeff * std::pow(_Tp(1 + __j), -__s); 00212 __sgn *= _Tp(-1); 00213 } 00214 if (__punt) 00215 break; 00216 __term *= __num; 00217 __zeta += __term; 00218 if (std::abs(__term/__zeta) < __eps) 00219 break; 00220 __num *= _Tp(0.5L); 00221 } 00222 00223 __zeta /= _Tp(1) - std::pow(_Tp(2), _Tp(1) - __s); 00224 00225 return __zeta; 00226 } 00227 00228 00229 /** 00230 * @brief Compute the Riemann zeta function @f$ \zeta(s) @f$ 00231 * using the product over prime factors. 00232 * \f[ 00233 * \zeta(s) = \Pi_{i=1}^\infty \frac{1}{1 - p_i^{-s}} 00234 * \f] 00235 * where @f$ {p_i} @f$ are the prime numbers. 00236 * 00237 * The Riemann zeta function is defined by: 00238 * \f[ 00239 * \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 00240 * \f] 00241 * For s < 1 use the reflection formula: 00242 * \f[ 00243 * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) 00244 * \f] 00245 */ 00246 template<typename _Tp> 00247 _Tp 00248 __riemann_zeta_product(const _Tp __s) 00249 { 00250 static const _Tp __prime[] = { 00251 _Tp(2), _Tp(3), _Tp(5), _Tp(7), _Tp(11), _Tp(13), _Tp(17), _Tp(19), 00252 _Tp(23), _Tp(29), _Tp(31), _Tp(37), _Tp(41), _Tp(43), _Tp(47), 00253 _Tp(53), _Tp(59), _Tp(61), _Tp(67), _Tp(71), _Tp(73), _Tp(79), 00254 _Tp(83), _Tp(89), _Tp(97), _Tp(101), _Tp(103), _Tp(107), _Tp(109) 00255 }; 00256 static const unsigned int __num_primes = sizeof(__prime) / sizeof(_Tp); 00257 00258 _Tp __zeta = _Tp(1); 00259 for (unsigned int __i = 0; __i < __num_primes; ++__i) 00260 { 00261 const _Tp __fact = _Tp(1) - std::pow(__prime[__i], -__s); 00262 __zeta *= __fact; 00263 if (_Tp(1) - __fact < std::numeric_limits<_Tp>::epsilon()) 00264 break; 00265 } 00266 00267 __zeta = _Tp(1) / __zeta; 00268 00269 return __zeta; 00270 } 00271 00272 00273 /** 00274 * @brief Return the Riemann zeta function @f$ \zeta(s) @f$. 00275 * 00276 * The Riemann zeta function is defined by: 00277 * \f[ 00278 * \zeta(s) = \sum_{k=1}^{\infty} k^{-s} for s > 1 00279 * \frac{(2\pi)^s}{pi} sin(\frac{\pi s}{2}) 00280 * \Gamma (1 - s) \zeta (1 - s) for s < 1 00281 * \f] 00282 * For s < 1 use the reflection formula: 00283 * \f[ 00284 * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) 00285 * \f] 00286 */ 00287 template<typename _Tp> 00288 _Tp 00289 __riemann_zeta(const _Tp __s) 00290 { 00291 if (__isnan(__s)) 00292 return std::numeric_limits<_Tp>::quiet_NaN(); 00293 else if (__s == _Tp(1)) 00294 return std::numeric_limits<_Tp>::infinity(); 00295 else if (__s < -_Tp(19)) 00296 { 00297 _Tp __zeta = __riemann_zeta_product(_Tp(1) - __s); 00298 __zeta *= std::pow(_Tp(2) * __numeric_constants<_Tp>::__pi(), __s) 00299 * std::sin(__numeric_constants<_Tp>::__pi_2() * __s) 00300 #if _GLIBCXX_USE_C99_MATH_TR1 00301 * std::exp(std::tr1::lgamma(_Tp(1) - __s)) 00302 #else 00303 * std::exp(__log_gamma(_Tp(1) - __s)) 00304 #endif 00305 / __numeric_constants<_Tp>::__pi(); 00306 return __zeta; 00307 } 00308 else if (__s < _Tp(20)) 00309 { 00310 // Global double sum or McLaurin? 00311 bool __glob = true; 00312 if (__glob) 00313 return __riemann_zeta_glob(__s); 00314 else 00315 { 00316 if (__s > _Tp(1)) 00317 return __riemann_zeta_sum(__s); 00318 else 00319 { 00320 _Tp __zeta = std::pow(_Tp(2) 00321 * __numeric_constants<_Tp>::__pi(), __s) 00322 * std::sin(__numeric_constants<_Tp>::__pi_2() * __s) 00323 #if _GLIBCXX_USE_C99_MATH_TR1 00324 * std::tr1::tgamma(_Tp(1) - __s) 00325 #else 00326 * std::exp(__log_gamma(_Tp(1) - __s)) 00327 #endif 00328 * __riemann_zeta_sum(_Tp(1) - __s); 00329 return __zeta; 00330 } 00331 } 00332 } 00333 else 00334 return __riemann_zeta_product(__s); 00335 } 00336 00337 00338 /** 00339 * @brief Return the Hurwitz zeta function @f$ \zeta(x,s) @f$ 00340 * for all s != 1 and x > -1. 00341 * 00342 * The Hurwitz zeta function is defined by: 00343 * @f[ 00344 * \zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s} 00345 * @f] 00346 * The Riemann zeta function is a special case: 00347 * @f[ 00348 * \zeta(s) = \zeta(1,s) 00349 * @f] 00350 * 00351 * This functions uses the double sum that converges for s != 1 00352 * and x > -1: 00353 * @f[ 00354 * \zeta(x,s) = \frac{1}{s-1} 00355 * \sum_{n=0}^{\infty} \frac{1}{n + 1} 00356 * \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (x+k)^{-s} 00357 * @f] 00358 */ 00359 template<typename _Tp> 00360 _Tp 00361 __hurwitz_zeta_glob(const _Tp __a, const _Tp __s) 00362 { 00363 _Tp __zeta = _Tp(0); 00364 00365 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 00366 // Max e exponent before overflow. 00367 const _Tp __max_bincoeff = std::numeric_limits<_Tp>::max_exponent10 00368 * std::log(_Tp(10)) - _Tp(1); 00369 00370 const unsigned int __maxit = 10000; 00371 for (unsigned int __i = 0; __i < __maxit; ++__i) 00372 { 00373 bool __punt = false; 00374 _Tp __sgn = _Tp(1); 00375 _Tp __term = _Tp(0); 00376 for (unsigned int __j = 0; __j <= __i; ++__j) 00377 { 00378 #if _GLIBCXX_USE_C99_MATH_TR1 00379 _Tp __bincoeff = std::tr1::lgamma(_Tp(1 + __i)) 00380 - std::tr1::lgamma(_Tp(1 + __j)) 00381 - std::tr1::lgamma(_Tp(1 + __i - __j)); 00382 #else 00383 _Tp __bincoeff = __log_gamma(_Tp(1 + __i)) 00384 - __log_gamma(_Tp(1 + __j)) 00385 - __log_gamma(_Tp(1 + __i - __j)); 00386 #endif 00387 if (__bincoeff > __max_bincoeff) 00388 { 00389 // This only gets hit for x << 0. 00390 __punt = true; 00391 break; 00392 } 00393 __bincoeff = std::exp(__bincoeff); 00394 __term += __sgn * __bincoeff * std::pow(_Tp(__a + __j), -__s); 00395 __sgn *= _Tp(-1); 00396 } 00397 if (__punt) 00398 break; 00399 __term /= _Tp(__i + 1); 00400 if (std::abs(__term / __zeta) < __eps) 00401 break; 00402 __zeta += __term; 00403 } 00404 00405 __zeta /= __s - _Tp(1); 00406 00407 return __zeta; 00408 } 00409 00410 00411 /** 00412 * @brief Return the Hurwitz zeta function @f$ \zeta(x,s) @f$ 00413 * for all s != 1 and x > -1. 00414 * 00415 * The Hurwitz zeta function is defined by: 00416 * @f[ 00417 * \zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s} 00418 * @f] 00419 * The Riemann zeta function is a special case: 00420 * @f[ 00421 * \zeta(s) = \zeta(1,s) 00422 * @f] 00423 */ 00424 template<typename _Tp> 00425 inline _Tp 00426 __hurwitz_zeta(const _Tp __a, const _Tp __s) 00427 { 00428 return __hurwitz_zeta_glob(__a, __s); 00429 } 00430 00431 } // namespace std::tr1::__detail 00432 } 00433 } 00434 00435 #endif // _GLIBCXX_TR1_RIEMANN_ZETA_TCC