#include "special_function_util.h"

Macros
#define	_GLIBCXX_TR1_RIEMANN_ZETA_TCC 1

Functions
namespace std	_GLIBCXX_VISIBILITY (default)

Detailed Description

This is an internal header file, included by other library headers. Do not attempt to use it directly. {tr1/cmath}

Macro Definition Documentation

#define _GLIBCXX_TR1_RIEMANN_ZETA_TCC 1

Function Documentation

namespace std _GLIBCXX_VISIBILITY ( default )

Compute the Riemann zeta function $\zeta(s)$ by summation for s > 1.

The Riemann zeta function is defined by:

$\zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1$

For s < 1 use the reflection formula:

$\zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s)$

Evaluate the Riemann zeta function $\zeta(s)$ by an alternate series for s > 0.

The Riemann zeta function is defined by:

$\zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1$

For s < 1 use the reflection formula:

$\zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s)$

Evaluate the Riemann zeta function by series for all s != 1. Convergence is great until largish negative numbers. Then the convergence of the > 0 sum gets better.

The series is:

$\zeta(s) = \frac{1}{1-2^{1-s}} \sum_{n=0}^{\infty} \frac{1}{2^{n+1}} \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (k+1)^{-s}$

Havil 2003, p. 206.

The Riemann zeta function is defined by:

$\zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1$

For s < 1 use the reflection formula:

$\zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s)$

Compute the Riemann zeta function $\zeta(s)$ using the product over prime factors.

$\zeta(s) = \Pi_{i=1}^\infty \frac{1}{1 - p_i^{-s}}$

where ${p_i}$ are the prime numbers.

The Riemann zeta function is defined by:

$\zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1$

For s < 1 use the reflection formula:

$\zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s)$

Return the Riemann zeta function $\zeta(s)$ .

The Riemann zeta function is defined by:

$\zeta(s) = \sum_{k=1}^{\infty} k^{-s} for s > 1 \frac{(2\pi)^s}{pi} sin(\frac{\pi s}{2}) \Gamma (1 - s) \zeta (1 - s) for s < 1$

For s < 1 use the reflection formula:

$\zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s)$

Return the Hurwitz zeta function $\zeta(x,s)$ for all s != 1 and x > -1.

The Hurwitz zeta function is defined by:

$\zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s}$

The Riemann zeta function is a special case:

$\zeta(s) = \zeta(1,s)$

This functions uses the double sum that converges for s != 1 and x > -1:

$\zeta(x,s) = \frac{1}{s-1} \sum_{n=0}^{\infty} \frac{1}{n + 1} \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (x+k)^{-s}$

Return the Hurwitz zeta function $\zeta(x,s)$ for all s != 1 and x > -1.

The Hurwitz zeta function is defined by:

$\zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s}$

The Riemann zeta function is a special case:

$\zeta(s) = \zeta(1,s)$

 {
 namespace tr1
 {
   // [5.2] Special functions
 
   // Implementation-space details.
   namespace __detail
   {
   _GLIBCXX_BEGIN_NAMESPACE_VERSION
 
     template<typename _Tp>
     _Tp
     __riemann_zeta_sum(_Tp __s)
     {
       //  A user shouldn't get to this.
       if (__s < _Tp(1))
         std::__throw_domain_error(__N("Bad argument in zeta sum."));
 
       const unsigned int max_iter = 10000;
       _Tp __zeta = _Tp(0);
       for (unsigned int __k = 1; __k < max_iter; ++__k)
         {
           _Tp __term = std::pow(static_cast<_Tp>(__k), -__s);
           if (__term < std::numeric_limits<_Tp>::epsilon())
             {
               break;
             }
           __zeta += __term;
         }
 
       return __zeta;
     }
 
 
     template<typename _Tp>
     _Tp
     __riemann_zeta_alt(_Tp __s)
     {
       _Tp __sgn = _Tp(1);
       _Tp __zeta = _Tp(0);
       for (unsigned int __i = 1; __i < 10000000; ++__i)
         {
           _Tp __term = __sgn / std::pow(__i, __s);
           if (std::abs(__term) < std::numeric_limits<_Tp>::epsilon())
             break;
           __zeta += __term;
           __sgn *= _Tp(-1);
         }
       __zeta /= _Tp(1) - std::pow(_Tp(2), _Tp(1) - __s);
 
       return __zeta;
     }
 
 
     template<typename _Tp>
     _Tp
     __riemann_zeta_glob(_Tp __s)
     {
       _Tp __zeta = _Tp(0);
 
       const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
       //  Max e exponent before overflow.
       const _Tp __max_bincoeff = std::numeric_limits<_Tp>::max_exponent10
                                * std::log(_Tp(10)) - _Tp(1);
 
       //  This series works until the binomial coefficient blows up
       //  so use reflection.
       if (__s < _Tp(0))
         {
 #if _GLIBCXX_USE_C99_MATH_TR1
           if (std::tr1::fmod(__s,_Tp(2)) == _Tp(0))
             return _Tp(0);
           else
 #endif
             {
               _Tp __zeta = __riemann_zeta_glob(_Tp(1) - __s);
               __zeta *= std::pow(_Tp(2)
                      * __numeric_constants<_Tp>::__pi(), __s)
                      * std::sin(__numeric_constants<_Tp>::__pi_2() * __s)
 #if _GLIBCXX_USE_C99_MATH_TR1
                      * std::exp(std::tr1::lgamma(_Tp(1) - __s))
 #else
                      * std::exp(__log_gamma(_Tp(1) - __s))
 #endif
                      / __numeric_constants<_Tp>::__pi();
               return __zeta;
             }
         }
 
       _Tp __num = _Tp(0.5L);
       const unsigned int __maxit = 10000;
       for (unsigned int __i = 0; __i < __maxit; ++__i)
         {
           bool __punt = false;
           _Tp __sgn = _Tp(1);
           _Tp __term = _Tp(0);
           for (unsigned int __j = 0; __j <= __i; ++__j)
             {
 #if _GLIBCXX_USE_C99_MATH_TR1
               _Tp __bincoeff =  std::tr1::lgamma(_Tp(1 + __i))
                               - std::tr1::lgamma(_Tp(1 + __j))
                               - std::tr1::lgamma(_Tp(1 + __i - __j));
 #else
               _Tp __bincoeff =  __log_gamma(_Tp(1 + __i))
                               - __log_gamma(_Tp(1 + __j))
                               - __log_gamma(_Tp(1 + __i - __j));
 #endif
               if (__bincoeff > __max_bincoeff)
                 {
                   //  This only gets hit for x << 0.
                   __punt = true;
                   break;
                 }
               __bincoeff = std::exp(__bincoeff);
               __term += __sgn * __bincoeff * std::pow(_Tp(1 + __j), -__s);
               __sgn *= _Tp(-1);
             }
           if (__punt)
             break;
           __term *= __num;
           __zeta += __term;
           if (std::abs(__term/__zeta) < __eps)
             break;
           __num *= _Tp(0.5L);
         }
 
       __zeta /= _Tp(1) - std::pow(_Tp(2), _Tp(1) - __s);
 
       return __zeta;
     }
 
 
     template<typename _Tp>
     _Tp
     __riemann_zeta_product(_Tp __s)
     {
       static const _Tp __prime[] = {
         _Tp(2), _Tp(3), _Tp(5), _Tp(7), _Tp(11), _Tp(13), _Tp(17), _Tp(19),
         _Tp(23), _Tp(29), _Tp(31), _Tp(37), _Tp(41), _Tp(43), _Tp(47),
         _Tp(53), _Tp(59), _Tp(61), _Tp(67), _Tp(71), _Tp(73), _Tp(79),
         _Tp(83), _Tp(89), _Tp(97), _Tp(101), _Tp(103), _Tp(107), _Tp(109)
       };
       static const unsigned int __num_primes = sizeof(__prime) / sizeof(_Tp);
 
       _Tp __zeta = _Tp(1);
       for (unsigned int __i = 0; __i < __num_primes; ++__i)
         {
           const _Tp __fact = _Tp(1) - std::pow(__prime[__i], -__s);
           __zeta *= __fact;
           if (_Tp(1) - __fact < std::numeric_limits<_Tp>::epsilon())
             break;
         }
 
       __zeta = _Tp(1) / __zeta;
 
       return __zeta;
     }
 
 
     template<typename _Tp>
     _Tp
     __riemann_zeta(_Tp __s)
     {
       if (__isnan(__s))
         return std::numeric_limits<_Tp>::quiet_NaN();
       else if (__s == _Tp(1))
         return std::numeric_limits<_Tp>::infinity();
       else if (__s < -_Tp(19))
         {
           _Tp __zeta = __riemann_zeta_product(_Tp(1) - __s);
           __zeta *= std::pow(_Tp(2) * __numeric_constants<_Tp>::__pi(), __s)
                  * std::sin(__numeric_constants<_Tp>::__pi_2() * __s)
 #if _GLIBCXX_USE_C99_MATH_TR1
                  * std::exp(std::tr1::lgamma(_Tp(1) - __s))
 #else
                  * std::exp(__log_gamma(_Tp(1) - __s))
 #endif
                  / __numeric_constants<_Tp>::__pi();
           return __zeta;
         }
       else if (__s < _Tp(20))
         {
           //  Global double sum or McLaurin?
           bool __glob = true;
           if (__glob)
             return __riemann_zeta_glob(__s);
           else
             {
               if (__s > _Tp(1))
                 return __riemann_zeta_sum(__s);
               else
                 {
                   _Tp __zeta = std::pow(_Tp(2)
                                 * __numeric_constants<_Tp>::__pi(), __s)
                          * std::sin(__numeric_constants<_Tp>::__pi_2() * __s)
 #if _GLIBCXX_USE_C99_MATH_TR1
                              * std::tr1::tgamma(_Tp(1) - __s)
 #else
                              * std::exp(__log_gamma(_Tp(1) - __s))
 #endif
                              * __riemann_zeta_sum(_Tp(1) - __s);
                   return __zeta;
                 }
             }
         }
       else
         return __riemann_zeta_product(__s);
     }
 
 
     template<typename _Tp>
     _Tp
     __hurwitz_zeta_glob(_Tp __a, _Tp __s)
     {
       _Tp __zeta = _Tp(0);
 
       const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
       //  Max e exponent before overflow.
       const _Tp __max_bincoeff = std::numeric_limits<_Tp>::max_exponent10
                                * std::log(_Tp(10)) - _Tp(1);
 
       const unsigned int __maxit = 10000;
       for (unsigned int __i = 0; __i < __maxit; ++__i)
         {
           bool __punt = false;
           _Tp __sgn = _Tp(1);
           _Tp __term = _Tp(0);
           for (unsigned int __j = 0; __j <= __i; ++__j)
             {
 #if _GLIBCXX_USE_C99_MATH_TR1
               _Tp __bincoeff =  std::tr1::lgamma(_Tp(1 + __i))
                               - std::tr1::lgamma(_Tp(1 + __j))
                               - std::tr1::lgamma(_Tp(1 + __i - __j));
 #else
               _Tp __bincoeff =  __log_gamma(_Tp(1 + __i))
                               - __log_gamma(_Tp(1 + __j))
                               - __log_gamma(_Tp(1 + __i - __j));
 #endif
               if (__bincoeff > __max_bincoeff)
                 {
                   //  This only gets hit for x << 0.
                   __punt = true;
                   break;
                 }
               __bincoeff = std::exp(__bincoeff);
               __term += __sgn * __bincoeff * std::pow(_Tp(__a + __j), -__s);
               __sgn *= _Tp(-1);
             }
           if (__punt)
             break;
           __term /= _Tp(__i + 1);
           if (std::abs(__term / __zeta) < __eps)
             break;
           __zeta += __term;
         }
 
       __zeta /= __s - _Tp(1);
 
       return __zeta;
     }
 
 
     template<typename _Tp>
     inline _Tp
     __hurwitz_zeta(_Tp __a, _Tp __s)
     { return __hurwitz_zeta_glob(__a, __s); }
 
   _GLIBCXX_END_NAMESPACE_VERSION
   } // namespace std::tr1::__detail
 }
 }

Macros

Functions

Detailed Description

Macro Definition Documentation

Function Documentation