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// Special functions -*- C++ -*-

// Copyright (C) 2006, 2007, 2008, 2009, 2010

// Free Software Foundation, Inc.

//

// This file is part of the GNU ISO C++ Library.  This library is free

// software; you can redistribute it and/or modify it under the

// terms of the GNU General Public License as published by the

// Free Software Foundation; either version 3, or (at your option)

// any later version.

//

// This library is distributed in the hope that it will be useful,

// but WITHOUT ANY WARRANTY; without even the implied warranty of

// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the

// GNU General Public License for more details.

//

// Under Section 7 of GPL version 3, you are granted additional

// permissions described in the GCC Runtime Library Exception, version

// 3.1, as published by the Free Software Foundation.

// You should have received a copy of the GNU General Public License and

// a copy of the GCC Runtime Library Exception along with this program;

// see the files COPYING3 and COPYING.RUNTIME respectively.  If not, see

// .

/** @file tr1/riemann_zeta.tcc

 *  This is an internal header file, included by other library headers.

 *  Do not attempt to use it directly. @headername{tr1/cmath}

 */

//

// ISO C++ 14882 TR1: 5.2  Special functions

//

// Written by Edward Smith-Rowland based on:

//   (1) Handbook of Mathematical Functions,

//       Ed. by Milton Abramowitz and Irene A. Stegun,

//       Dover Publications, New-York, Section 5, pp. 807-808.

//   (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl

//   (3) Gamma, Exploring Euler's Constant, Julian Havil,

//       Princeton, 2003.

#ifndef _GLIBCXX_TR1_RIEMANN_ZETA_TCC

#define _GLIBCXX_TR1_RIEMANN_ZETA_TCC 1

#include "special_function_util.h"

namespace std _GLIBCXX_VISIBILITY(default)

{

namespace tr1

{

  // [5.2] Special functions

  // Implementation-space details.

  namespace __detail

  {

  _GLIBCXX_BEGIN_NAMESPACE_VERSION

    /**

     *   @brief  Compute the Riemann zeta function @f$ \zeta(s) @f$

     *           by summation for s > 1.

     *

     *   The Riemann zeta function is defined by:

     *    \f[

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

     *    \f]

     *   For s < 1 use the reflection formula:

     *    \f[

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

     *    \f]

     */

    template

    _Tp

    __riemann_zeta_sum(const _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;

    }

    /**

     *   @brief  Evaluate the Riemann zeta function @f$ \zeta(s) @f$

     *           by an alternate series for s > 0.

     *

     *   The Riemann zeta function is defined by:

     *    \f[

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

     *    \f]

     *   For s < 1 use the reflection formula:

     *    \f[

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

     *    \f]

     */

    template

    _Tp

    __riemann_zeta_alt(const _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;

    }

    /**

     *   @brief  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:

     *    \f[

     *      \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}

     *    \f]

     *   Havil 2003, p. 206.

     *

     *   The Riemann zeta function is defined by:

     *    \f[

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

     *    \f]

     *   For s < 1 use the reflection formula:

     *    \f[

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

     *    \f]

     */

    template

    _Tp

    __riemann_zeta_glob(const _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;

    }

    /**

     *   @brief  Compute the Riemann zeta function @f$ \zeta(s) @f$

     *           using the product over prime factors.

     *    \f[

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

     *    \f]

     *    where @f$ {p_i} @f$ are the prime numbers.

     *

     *   The Riemann zeta function is defined by:

     *    \f[

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

     *    \f]

     *   For s < 1 use the reflection formula:

     *    \f[

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

     *    \f]

     */

    template

    _Tp

    __riemann_zeta_product(const _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;

    }

    /**

     *   @brief  Return the Riemann zeta function @f$ \zeta(s) @f$.

     *

     *   The Riemann zeta function is defined by:

     *    \f[

     *      \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

     *    \f]

     *   For s < 1 use the reflection formula:

     *    \f[

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

     *    \f]

     */

    template

    _Tp

    __riemann_zeta(const _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);

    }

    /**

     *   @brief  Return the Hurwitz zeta function @f$ \zeta(x,s) @f$

     *           for all s != 1 and x > -1.

     *

     *   The Hurwitz zeta function is defined by:

     *   @f[

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

     *   @f]

     *   The Riemann zeta function is a special case:

     *   @f[

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

     *   @f]

     *

     *   This functions uses the double sum that converges for s != 1

     *   and x > -1:

     *   @f[

     *     \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}

     *   @f]

     */

    template

    _Tp

    __hurwitz_zeta_glob(const _Tp __a, const _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;

    }

    /**

     *   @brief  Return the Hurwitz zeta function @f$ \zeta(x,s) @f$

     *           for all s != 1 and x > -1.

     *

     *   The Hurwitz zeta function is defined by:

     *   @f[

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

     *   @f]

     *   The Riemann zeta function is a special case:

     *   @f[

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

     *   @f]

     */

    template

    inline _Tp

    __hurwitz_zeta(const _Tp __a, const _Tp __s)

    {

      return __hurwitz_zeta_glob(__a, __s);

    }

  _GLIBCXX_END_NAMESPACE_VERSION

  } // namespace std::tr1::__detail

}

}

#endif // _GLIBCXX_TR1_RIEMANN_ZETA_TCC