# riemann_zeta.tcc

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    00001 // Special functions -*- C++ -*-
    00002
    00003 // Copyright (C) 2006, 2007, 2008
    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 2, 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 // You should have received a copy of the GNU General Public License along
    00018 // with this library; see the file COPYING.  If not, write to the Free
    00019 // Software Foundation, 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301,
    00020 // USA.
    00021 //
    00022 // As a special exception, you may use this file as part of a free software
    00023 // library without restriction.  Specifically, if other files instantiate
    00024 // templates or use macros or inline functions from this file, or you compile
    00025 // this file and link it with other files to produce an executable, this
    00026 // file does not by itself cause the resulting executable to be covered by
    00027 // the GNU General Public License.  This exception does not however
    00028 // invalidate any other reasons why the executable file might be covered by
    00029 // the GNU General Public License.
    00030
    00031 /** @file tr1/riemann_zeta.tcc
    00032  *  This is an internal header file, included by other library headers.
    00033  *  You should not attempt to use it directly.
    00034  */
    00035
    00036 //
    00037 // ISO C++ 14882 TR1: 5.2  Special functions
    00038 //
    00039
    00040 // Written by Edward Smith-Rowland based on:
    00041 //   (1) Handbook of Mathematical Functions,
    00042 //       Ed. by Milton Abramowitz and Irene A. Stegun,
    00043 //       Dover Publications, New-York, Section 5, pp. 807-808.
    00044 //   (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
    00045 //   (3) Gamma, Exploring Euler's Constant, Julian Havil,
    00046 //       Princeton, 2003.
    00047
    00048 #ifndef _GLIBCXX_TR1_RIEMANN_ZETA_TCC
    00049 #define _GLIBCXX_TR1_RIEMANN_ZETA_TCC 1
    00050
    00051 #include "special_function_util.h"
    00052
    00053 namespace std
    00054 {
    00055 namespace tr1
    00056 {
    00057
    00058   // [5.2] Special functions
    00059
    00060   // Implementation-space details.
    00061   namespace __detail
    00062   {
    00063
    00064     /**
    00065      *   @brief  Compute the Riemann zeta function @f$ \zeta(s) @f$
    00066      *           by summation for s > 1.
    00067      *
    00068      *   The Riemann zeta function is defined by:
    00069      *    \f[
    00070      *      \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1
    00071      *    \f]
    00072      *   For s < 1 use the reflection formula:
    00073      *    \f[
    00074      *      \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s)
    00075      *    \f]
    00076      */
    00077     template<typename _Tp>
    00078     _Tp
    00079     __riemann_zeta_sum(const _Tp __s)
    00080     {
    00081       //  A user shouldn't get to this.
    00082       if (__s < _Tp(1))
    00083         std::__throw_domain_error(__N("Bad argument in zeta sum."));
    00084
    00085       const unsigned int max_iter = 10000;
    00086       _Tp __zeta = _Tp(0);
    00087       for (unsigned int __k = 1; __k < max_iter; ++__k)
    00088         {
    00089           _Tp __term = std::pow(static_cast<_Tp>(__k), -__s);
    00090           if (__term < std::numeric_limits<_Tp>::epsilon())
    00091             {
    00092               break;
    00093             }
    00094           __zeta += __term;
    00095         }
    00096
    00097       return __zeta;
    00098     }
    00099
    00100
    00101     /**
    00102      *   @brief  Evaluate the Riemann zeta function @f$ \zeta(s) @f$
    00103      *           by an alternate series for s > 0.
    00104      *
    00105      *   The Riemann zeta function is defined by:
    00106      *    \f[
    00107      *      \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1
    00108      *    \f]
    00109      *   For s < 1 use the reflection formula:
    00110      *    \f[
    00111      *      \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s)
    00112      *    \f]
    00113      */
    00114     template<typename _Tp>
    00115     _Tp
    00116     __riemann_zeta_alt(const _Tp __s)
    00117     {
    00118       _Tp __sgn = _Tp(1);
    00119       _Tp __zeta = _Tp(0);
    00120       for (unsigned int __i = 1; __i < 10000000; ++__i)
    00121         {
    00122           _Tp __term = __sgn / std::pow(__i, __s);
    00123           if (std::abs(__term) < std::numeric_limits<_Tp>::epsilon())
    00124             break;
    00125           __zeta += __term;
    00126           __sgn *= _Tp(-1);
    00127         }
    00128       __zeta /= _Tp(1) - std::pow(_Tp(2), _Tp(1) - __s);
    00129
    00130       return __zeta;
    00131     }
    00132
    00133
    00134     /**
    00135      *   @brief  Evaluate the Riemann zeta function by series for all s != 1.
    00136      *           Convergence is great until largish negative numbers.
    00137      *           Then the convergence of the > 0 sum gets better.
    00138      *
    00139      *   The series is:
    00140      *    \f[
    00141      *      \zeta(s) = \frac{1}{1-2^{1-s}}
    00142      *                 \sum_{n=0}^{\infty} \frac{1}{2^{n+1}}
    00143      *                 \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (k+1)^{-s}
    00144      *    \f]
    00145      *   Havil 2003, p. 206.
    00146      *
    00147      *   The Riemann zeta function is defined by:
    00148      *    \f[
    00149      *      \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1
    00150      *    \f]
    00151      *   For s < 1 use the reflection formula:
    00152      *    \f[
    00153      *      \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s)
    00154      *    \f]
    00155      */
    00156     template<typename _Tp>
    00157     _Tp
    00158     __riemann_zeta_glob(const _Tp __s)
    00159     {
    00160       _Tp __zeta = _Tp(0);
    00161
    00162       const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
    00163       //  Max e exponent before overflow.
    00164       const _Tp __max_bincoeff = std::numeric_limits<_Tp>::max_exponent10
    00165                                * std::log(_Tp(10)) - _Tp(1);
    00166
    00167       //  This series works until the binomial coefficient blows up
    00168       //  so use reflection.
    00169       if (__s < _Tp(0))
    00170         {
    00171 #if _GLIBCXX_USE_C99_MATH_TR1
    00172           if (std::tr1::fmod(__s,_Tp(2)) == _Tp(0))
    00173             return _Tp(0);
    00174           else
    00175 #endif
    00176             {
    00177               _Tp __zeta = __riemann_zeta_glob(_Tp(1) - __s);
    00178               __zeta *= std::pow(_Tp(2)
    00179                      * __numeric_constants<_Tp>::__pi(), __s)
    00180                      * std::sin(__numeric_constants<_Tp>::__pi_2() * __s)
    00181 #if _GLIBCXX_USE_C99_MATH_TR1
    00182                      * std::exp(std::tr1::lgamma(_Tp(1) - __s))
    00183 #else
    00184                      * std::exp(__log_gamma(_Tp(1) - __s))
    00185 #endif
    00186                      / __numeric_constants<_Tp>::__pi();
    00187               return __zeta;
    00188             }
    00189         }
    00190
    00191       _Tp __num = _Tp(0.5L);
    00192       const unsigned int __maxit = 10000;
    00193       for (unsigned int __i = 0; __i < __maxit; ++__i)
    00194         {
    00195           bool __punt = false;
    00196           _Tp __sgn = _Tp(1);
    00197           _Tp __term = _Tp(0);
    00198           for (unsigned int __j = 0; __j <= __i; ++__j)
    00199             {
    00200 #if _GLIBCXX_USE_C99_MATH_TR1
    00201               _Tp __bincoeff =  std::tr1::lgamma(_Tp(1 + __i))
    00202                               - std::tr1::lgamma(_Tp(1 + __j))
    00203                               - std::tr1::lgamma(_Tp(1 + __i - __j));
    00204 #else
    00205               _Tp __bincoeff =  __log_gamma(_Tp(1 + __i))
    00206                               - __log_gamma(_Tp(1 + __j))
    00207                               - __log_gamma(_Tp(1 + __i - __j));
    00208 #endif
    00209               if (__bincoeff > __max_bincoeff)
    00210                 {
    00211                   //  This only gets hit for x << 0.
    00212                   __punt = true;
    00213                   break;
    00214                 }
    00215               __bincoeff = std::exp(__bincoeff);
    00216               __term += __sgn * __bincoeff * std::pow(_Tp(1 + __j), -__s);
    00217               __sgn *= _Tp(-1);
    00218             }
    00219           if (__punt)
    00220             break;
    00221           __term *= __num;
    00222           __zeta += __term;
    00223           if (std::abs(__term/__zeta) < __eps)
    00224             break;
    00225           __num *= _Tp(0.5L);
    00226         }
    00227
    00228       __zeta /= _Tp(1) - std::pow(_Tp(2), _Tp(1) - __s);
    00229
    00230       return __zeta;
    00231     }
    00232
    00233
    00234     /**
    00235      *   @brief  Compute the Riemann zeta function @f$ \zeta(s) @f$
    00236      *           using the product over prime factors.
    00237      *    \f[
    00238      *      \zeta(s) = \Pi_{i=1}^\infty \frac{1}{1 - p_i^{-s}}
    00239      *    \f]
    00240      *    where @f$ {p_i} @f$ are the prime numbers.
    00241      *
    00242      *   The Riemann zeta function is defined by:
    00243      *    \f[
    00244      *      \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1
    00245      *    \f]
    00246      *   For s < 1 use the reflection formula:
    00247      *    \f[
    00248      *      \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s)
    00249      *    \f]
    00250      */
    00251     template<typename _Tp>
    00252     _Tp
    00253     __riemann_zeta_product(const _Tp __s)
    00254     {
    00255       static const _Tp __prime[] = {
    00256         _Tp(2), _Tp(3), _Tp(5), _Tp(7), _Tp(11), _Tp(13), _Tp(17), _Tp(19),
    00257         _Tp(23), _Tp(29), _Tp(31), _Tp(37), _Tp(41), _Tp(43), _Tp(47),
    00258         _Tp(53), _Tp(59), _Tp(61), _Tp(67), _Tp(71), _Tp(73), _Tp(79),
    00259         _Tp(83), _Tp(89), _Tp(97), _Tp(101), _Tp(103), _Tp(107), _Tp(109)
    00260       };
    00261       static const unsigned int __num_primes = sizeof(__prime) / sizeof(_Tp);
    00262
    00263       _Tp __zeta = _Tp(1);
    00264       for (unsigned int __i = 0; __i < __num_primes; ++__i)
    00265         {
    00266           const _Tp __fact = _Tp(1) - std::pow(__prime[__i], -__s);
    00267           __zeta *= __fact;
    00268           if (_Tp(1) - __fact < std::numeric_limits<_Tp>::epsilon())
    00269             break;
    00270         }
    00271
    00272       __zeta = _Tp(1) / __zeta;
    00273
    00274       return __zeta;
    00275     }
    00276
    00277
    00278     /**
    00279      *   @brief  Return the Riemann zeta function @f$ \zeta(s) @f$.
    00280      *
    00281      *   The Riemann zeta function is defined by:
    00282      *    \f[
    00283      *      \zeta(s) = \sum_{k=1}^{\infty} k^{-s} for s > 1
    00284      *                 \frac{(2\pi)^s}{pi} sin(\frac{\pi s}{2})
    00285      *                 \Gamma (1 - s) \zeta (1 - s) for s < 1
    00286      *    \f]
    00287      *   For s < 1 use the reflection formula:
    00288      *    \f[
    00289      *      \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s)
    00290      *    \f]
    00291      */
    00292     template<typename _Tp>
    00293     _Tp
    00294     __riemann_zeta(const _Tp __s)
    00295     {
    00296       if (__isnan(__s))
    00297         return std::numeric_limits<_Tp>::quiet_NaN();
    00298       else if (__s == _Tp(1))
    00299         return std::numeric_limits<_Tp>::infinity();
    00300       else if (__s < -_Tp(19))
    00301         {
    00302           _Tp __zeta = __riemann_zeta_product(_Tp(1) - __s);
    00303           __zeta *= std::pow(_Tp(2) * __numeric_constants<_Tp>::__pi(), __s)
    00304                  * std::sin(__numeric_constants<_Tp>::__pi_2() * __s)
    00305 #if _GLIBCXX_USE_C99_MATH_TR1
    00306                  * std::exp(std::tr1::lgamma(_Tp(1) - __s))
    00307 #else
    00308                  * std::exp(__log_gamma(_Tp(1) - __s))
    00309 #endif
    00310                  / __numeric_constants<_Tp>::__pi();
    00311           return __zeta;
    00312         }
    00313       else if (__s < _Tp(20))
    00314         {
    00315           //  Global double sum or McLaurin?
    00316           bool __glob = true;
    00317           if (__glob)
    00318             return __riemann_zeta_glob(__s);
    00319           else
    00320             {
    00321               if (__s > _Tp(1))
    00322                 return __riemann_zeta_sum(__s);
    00323               else
    00324                 {
    00325                   _Tp __zeta = std::pow(_Tp(2)
    00326                                 * __numeric_constants<_Tp>::__pi(), __s)
    00327                          * std::sin(__numeric_constants<_Tp>::__pi_2() * __s)
    00328 #if _GLIBCXX_USE_C99_MATH_TR1
    00329                              * std::tr1::tgamma(_Tp(1) - __s)
    00330 #else
    00331                              * std::exp(__log_gamma(_Tp(1) - __s))
    00332 #endif
    00333                              * __riemann_zeta_sum(_Tp(1) - __s);
    00334                   return __zeta;
    00335                 }
    00336             }
    00337         }
    00338       else
    00339         return __riemann_zeta_product(__s);
    00340     }
    00341
    00342
    00343     /**
    00344      *   @brief  Return the Hurwitz zeta function @f$ \zeta(x,s) @f$
    00345      *           for all s != 1 and x > -1.
    00346      *
    00347      *   The Hurwitz zeta function is defined by:
    00348      *   @f[
    00349      *     \zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s}
    00350      *   @f]
    00351      *   The Riemann zeta function is a special case:
    00352      *   @f[
    00353      *     \zeta(s) = \zeta(1,s)
    00354      *   @f]
    00355      *
    00356      *   This functions uses the double sum that converges for s != 1
    00357      *   and x > -1:
    00358      *   @f[
    00359      *     \zeta(x,s) = \frac{1}{s-1}
    00360      *                \sum_{n=0}^{\infty} \frac{1}{n + 1}
    00361      *                \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (x+k)^{-s}
    00362      *   @f]
    00363      */
    00364     template<typename _Tp>
    00365     _Tp
    00366     __hurwitz_zeta_glob(const _Tp __a, const _Tp __s)
    00367     {
    00368       _Tp __zeta = _Tp(0);
    00369
    00370       const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
    00371       //  Max e exponent before overflow.
    00372       const _Tp __max_bincoeff = std::numeric_limits<_Tp>::max_exponent10
    00373                                * std::log(_Tp(10)) - _Tp(1);
    00374
    00375       const unsigned int __maxit = 10000;
    00376       for (unsigned int __i = 0; __i < __maxit; ++__i)
    00377         {
    00378           bool __punt = false;
    00379           _Tp __sgn = _Tp(1);
    00380           _Tp __term = _Tp(0);
    00381           for (unsigned int __j = 0; __j <= __i; ++__j)
    00382             {
    00383 #if _GLIBCXX_USE_C99_MATH_TR1
    00384               _Tp __bincoeff =  std::tr1::lgamma(_Tp(1 + __i))
    00385                               - std::tr1::lgamma(_Tp(1 + __j))
    00386                               - std::tr1::lgamma(_Tp(1 + __i - __j));
    00387 #else
    00388               _Tp __bincoeff =  __log_gamma(_Tp(1 + __i))
    00389                               - __log_gamma(_Tp(1 + __j))
    00390                               - __log_gamma(_Tp(1 + __i - __j));
    00391 #endif
    00392               if (__bincoeff > __max_bincoeff)
    00393                 {
    00394                   //  This only gets hit for x << 0.
    00395                   __punt = true;
    00396                   break;
    00397                 }
    00398               __bincoeff = std::exp(__bincoeff);
    00399               __term += __sgn * __bincoeff * std::pow(_Tp(__a + __j), -__s);
    00400               __sgn *= _Tp(-1);
    00401             }
    00402           if (__punt)
    00403             break;
    00404           __term /= _Tp(__i + 1);
    00405           if (std::abs(__term / __zeta) < __eps)
    00406             break;
    00407           __zeta += __term;
    00408         }
    00409
    00410       __zeta /= __s - _Tp(1);
    00411
    00412       return __zeta;
    00413     }
    00414
    00415
    00416     /**
    00417      *   @brief  Return the Hurwitz zeta function @f$ \zeta(x,s) @f$
    00418      *           for all s != 1 and x > -1.
    00419      *
    00420      *   The Hurwitz zeta function is defined by:
    00421      *   @f[
    00422      *     \zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s}
    00423      *   @f]
    00424      *   The Riemann zeta function is a special case:
    00425      *   @f[
    00426      *     \zeta(s) = \zeta(1,s)
    00427      *   @f]
    00428      */
    00429     template<typename _Tp>
    00430     inline _Tp
    00431     __hurwitz_zeta(const _Tp __a, const _Tp __s)
    00432     {
    00433       return __hurwitz_zeta_glob(__a, __s);
    00434     }
    00435
    00436   } // namespace std::tr1::__detail
    00437 }
    00438 }
    00439
    00440 #endif // _GLIBCXX_TR1_RIEMANN_ZETA_TCC