# riemann_zeta.tcc

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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