--- /dev/null
+// 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
+// <http://www.gnu.org/licenses/>.
+
+/** @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<typename _Tp>
+ _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<typename _Tp>
+ _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<typename _Tp>
+ _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<typename _Tp>
+ _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$.
+ *
+ * The Riemann zeta function is defined by:
+ * \f[
+ * \zeta(s) = \sum_{k=1}^{\infty} k^{-s} for s > 1
+ * \frac{(2\pi)^s}{pi} sin(\frac{\pi s}{2})
+ * \Gamma (1 - s) \zeta (1 - s) for s < 1
+ * \f]
+ * For s < 1 use the reflection formula:
+ * \f[
+ * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s)
+ * \f]
+ */
+ template<typename _Tp>
+ _Tp
+ __riemann_zeta(const _Tp __s)
+ {
+ if (__isnan(__s))
+ return std::numeric_limits<_Tp>::quiet_NaN();
+ else if (__s == _Tp(1))
+ return std::numeric_limits<_Tp>::infinity();
+ else if (__s < -_Tp(19))
+ {
+ _Tp __zeta = __riemann_zeta_product(_Tp(1) - __s);
+ __zeta *= std::pow(_Tp(2) * __numeric_constants<_Tp>::__pi(), __s)
+ * std::sin(__numeric_constants<_Tp>::__pi_2() * __s)
+#if _GLIBCXX_USE_C99_MATH_TR1
+ * std::exp(std::tr1::lgamma(_Tp(1) - __s))
+#else
+ * std::exp(__log_gamma(_Tp(1) - __s))
+#endif
+ / __numeric_constants<_Tp>::__pi();
+ return __zeta;
+ }
+ else if (__s < _Tp(20))
+ {
+ // Global double sum or McLaurin?
+ bool __glob = true;
+ if (__glob)
+ return __riemann_zeta_glob(__s);
+ else
+ {
+ if (__s > _Tp(1))
+ return __riemann_zeta_sum(__s);
+ else
+ {
+ _Tp __zeta = std::pow(_Tp(2)
+ * __numeric_constants<_Tp>::__pi(), __s)
+ * std::sin(__numeric_constants<_Tp>::__pi_2() * __s)
+#if _GLIBCXX_USE_C99_MATH_TR1
+ * std::tr1::tgamma(_Tp(1) - __s)
+#else
+ * std::exp(__log_gamma(_Tp(1) - __s))
+#endif
+ * __riemann_zeta_sum(_Tp(1) - __s);
+ return __zeta;
+ }
+ }
+ }
+ else
+ return __riemann_zeta_product(__s);
+ }
+
+
+ /**
+ * @brief Return the Hurwitz zeta function @f$ \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<typename _Tp>
+ _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<typename _Tp>
+ 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
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