// Special functions -*- C++ -*- // Copyright (C) 2006-2020 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/gamma.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. Milton Abramowitz and Irene A. Stegun, // Dover Publications, // Section 6, pp. 253-266 // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl // (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky, // W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992), // 2nd ed, pp. 213-216 // (4) Gamma, Exploring Euler's Constant, Julian Havil, // Princeton, 2003. #ifndef _GLIBCXX_TR1_GAMMA_TCC #define _GLIBCXX_TR1_GAMMA_TCC 1 #include <tr1/special_function_util.h> namespace std _GLIBCXX_VISIBILITY(default) { _GLIBCXX_BEGIN_NAMESPACE_VERSION #if _GLIBCXX_USE_STD_SPEC_FUNCS # define _GLIBCXX_MATH_NS ::std #elif defined(_GLIBCXX_TR1_CMATH) namespace tr1 { # define _GLIBCXX_MATH_NS ::std::tr1 #else # error do not include this header directly, use <cmath> or <tr1/cmath> #endif // Implementation-space details. namespace __detail { /** * @brief This returns Bernoulli numbers from a table or by summation * for larger values. * * Recursion is unstable. * * @param __n the order n of the Bernoulli number. * @return The Bernoulli number of order n. */ template <typename _Tp> _Tp __bernoulli_series(unsigned int __n) { static const _Tp __num[28] = { _Tp(1UL), -_Tp(1UL) / _Tp(2UL), _Tp(1UL) / _Tp(6UL), _Tp(0UL), -_Tp(1UL) / _Tp(30UL), _Tp(0UL), _Tp(1UL) / _Tp(42UL), _Tp(0UL), -_Tp(1UL) / _Tp(30UL), _Tp(0UL), _Tp(5UL) / _Tp(66UL), _Tp(0UL), -_Tp(691UL) / _Tp(2730UL), _Tp(0UL), _Tp(7UL) / _Tp(6UL), _Tp(0UL), -_Tp(3617UL) / _Tp(510UL), _Tp(0UL), _Tp(43867UL) / _Tp(798UL), _Tp(0UL), -_Tp(174611) / _Tp(330UL), _Tp(0UL), _Tp(854513UL) / _Tp(138UL), _Tp(0UL), -_Tp(236364091UL) / _Tp(2730UL), _Tp(0UL), _Tp(8553103UL) / _Tp(6UL), _Tp(0UL) }; if (__n == 0) return _Tp(1); if (__n == 1) return -_Tp(1) / _Tp(2); // Take care of the rest of the odd ones. if (__n % 2 == 1) return _Tp(0); // Take care of some small evens that are painful for the series. if (__n < 28) return __num[__n]; _Tp __fact = _Tp(1); if ((__n / 2) % 2 == 0) __fact *= _Tp(-1); for (unsigned int __k = 1; __k <= __n; ++__k) __fact *= __k / (_Tp(2) * __numeric_constants<_Tp>::__pi()); __fact *= _Tp(2); _Tp __sum = _Tp(0); for (unsigned int __i = 1; __i < 1000; ++__i) { _Tp __term = std::pow(_Tp(__i), -_Tp(__n)); if (__term < std::numeric_limits<_Tp>::epsilon()) break; __sum += __term; } return __fact * __sum; } /** * @brief This returns Bernoulli number \f$B_n\f$. * * @param __n the order n of the Bernoulli number. * @return The Bernoulli number of order n. */ template<typename _Tp> inline _Tp __bernoulli(int __n) { return __bernoulli_series<_Tp>(__n); } /** * @brief Return \f$log(\Gamma(x))\f$ by asymptotic expansion * with Bernoulli number coefficients. This is like * Sterling's approximation. * * @param __x The argument of the log of the gamma function. * @return The logarithm of the gamma function. */ template<typename _Tp> _Tp __log_gamma_bernoulli(_Tp __x) { _Tp __lg = (__x - _Tp(0.5L)) * std::log(__x) - __x + _Tp(0.5L) * std::log(_Tp(2) * __numeric_constants<_Tp>::__pi()); const _Tp __xx = __x * __x; _Tp __help = _Tp(1) / __x; for ( unsigned int __i = 1; __i < 20; ++__i ) { const _Tp __2i = _Tp(2 * __i); __help /= __2i * (__2i - _Tp(1)) * __xx; __lg += __bernoulli<_Tp>(2 * __i) * __help; } return __lg; } /** * @brief Return \f$log(\Gamma(x))\f$ by the Lanczos method. * This method dominates all others on the positive axis I think. * * @param __x The argument of the log of the gamma function. * @return The logarithm of the gamma function. */ template<typename _Tp> _Tp __log_gamma_lanczos(_Tp __x) { const _Tp __xm1 = __x - _Tp(1); static const _Tp __lanczos_cheb_7[9] = { _Tp( 0.99999999999980993227684700473478L), _Tp( 676.520368121885098567009190444019L), _Tp(-1259.13921672240287047156078755283L), _Tp( 771.3234287776530788486528258894L), _Tp(-176.61502916214059906584551354L), _Tp( 12.507343278686904814458936853L), _Tp(-0.13857109526572011689554707L), _Tp( 9.984369578019570859563e-6L), _Tp( 1.50563273514931155834e-7L) }; static const _Tp __LOGROOT2PI = _Tp(0.9189385332046727417803297364056176L); _Tp __sum = __lanczos_cheb_7[0]; for(unsigned int __k = 1; __k < 9; ++__k) __sum += __lanczos_cheb_7[__k] / (__xm1 + __k); const _Tp __term1 = (__xm1 + _Tp(0.5L)) * std::log((__xm1 + _Tp(7.5L)) / __numeric_constants<_Tp>::__euler()); const _Tp __term2 = __LOGROOT2PI + std::log(__sum); const _Tp __result = __term1 + (__term2 - _Tp(7)); return __result; } /** * @brief Return \f$ log(|\Gamma(x)|) \f$. * This will return values even for \f$ x < 0 \f$. * To recover the sign of \f$ \Gamma(x) \f$ for * any argument use @a __log_gamma_sign. * * @param __x The argument of the log of the gamma function. * @return The logarithm of the gamma function. */ template<typename _Tp> _Tp __log_gamma(_Tp __x) { if (__x > _Tp(0.5L)) return __log_gamma_lanczos(__x); else { const _Tp __sin_fact = std::abs(std::sin(__numeric_constants<_Tp>::__pi() * __x)); if (__sin_fact == _Tp(0)) std::__throw_domain_error(__N("Argument is nonpositive integer " "in __log_gamma")); return __numeric_constants<_Tp>::__lnpi() - std::log(__sin_fact) - __log_gamma_lanczos(_Tp(1) - __x); } } /** * @brief Return the sign of \f$ \Gamma(x) \f$. * At nonpositive integers zero is returned. * * @param __x The argument of the gamma function. * @return The sign of the gamma function. */ template<typename _Tp> _Tp __log_gamma_sign(_Tp __x) { if (__x > _Tp(0)) return _Tp(1); else { const _Tp __sin_fact = std::sin(__numeric_constants<_Tp>::__pi() * __x); if (__sin_fact > _Tp(0)) return (1); else if (__sin_fact < _Tp(0)) return -_Tp(1); else return _Tp(0); } } /** * @brief Return the logarithm of the binomial coefficient. * The binomial coefficient is given by: * @f[ * \left( \right) = \frac{n!}{(n-k)! k!} * @f] * * @param __n The first argument of the binomial coefficient. * @param __k The second argument of the binomial coefficient. * @return The binomial coefficient. */ template<typename _Tp> _Tp __log_bincoef(unsigned int __n, unsigned int __k) { // Max e exponent before overflow. static const _Tp __max_bincoeff = std::numeric_limits<_Tp>::max_exponent10 * std::log(_Tp(10)) - _Tp(1); #if _GLIBCXX_USE_C99_MATH_TR1 _Tp __coeff = _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __n)) - _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __k)) - _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __n - __k)); #else _Tp __coeff = __log_gamma(_Tp(1 + __n)) - __log_gamma(_Tp(1 + __k)) - __log_gamma(_Tp(1 + __n - __k)); #endif } /** * @brief Return the binomial coefficient. * The binomial coefficient is given by: * @f[ * \left( \right) = \frac{n!}{(n-k)! k!} * @f] * * @param __n The first argument of the binomial coefficient. * @param __k The second argument of the binomial coefficient. * @return The binomial coefficient. */ template<typename _Tp> _Tp __bincoef(unsigned int __n, unsigned int __k) { // Max e exponent before overflow. static const _Tp __max_bincoeff = std::numeric_limits<_Tp>::max_exponent10 * std::log(_Tp(10)) - _Tp(1); const _Tp __log_coeff = __log_bincoef<_Tp>(__n, __k); if (__log_coeff > __max_bincoeff) return std::numeric_limits<_Tp>::quiet_NaN(); else return std::exp(__log_coeff); } /** * @brief Return \f$ \Gamma(x) \f$. * * @param __x The argument of the gamma function. * @return The gamma function. */ template<typename _Tp> inline _Tp __gamma(_Tp __x) { return std::exp(__log_gamma(__x)); } /** * @brief Return the digamma function by series expansion. * The digamma or @f$ \psi(x) @f$ function is defined by * @f[ * \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)} * @f] * * The series is given by: * @f[ * \psi(x) = -\gamma_E - \frac{1}{x} * \sum_{k=1}^{\infty} \frac{x}{k(x + k)} * @f] */ template<typename _Tp> _Tp __psi_series(_Tp __x) { _Tp __sum = -__numeric_constants<_Tp>::__gamma_e() - _Tp(1) / __x; const unsigned int __max_iter = 100000; for (unsigned int __k = 1; __k < __max_iter; ++__k) { const _Tp __term = __x / (__k * (__k + __x)); __sum += __term; if (std::abs(__term / __sum) < std::numeric_limits<_Tp>::epsilon()) break; } return __sum; } /** * @brief Return the digamma function for large argument. * The digamma or @f$ \psi(x) @f$ function is defined by * @f[ * \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)} * @f] * * The asymptotic series is given by: * @f[ * \psi(x) = \ln(x) - \frac{1}{2x} * - \sum_{n=1}^{\infty} \frac{B_{2n}}{2 n x^{2n}} * @f] */ template<typename _Tp> _Tp __psi_asymp(_Tp __x) { _Tp __sum = std::log(__x) - _Tp(0.5L) / __x; const _Tp __xx = __x * __x; _Tp __xp = __xx; const unsigned int __max_iter = 100; for (unsigned int __k = 1; __k < __max_iter; ++__k) { const _Tp __term = __bernoulli<_Tp>(2 * __k) / (2 * __k * __xp); __sum -= __term; if (std::abs(__term / __sum) < std::numeric_limits<_Tp>::epsilon()) break; __xp *= __xx; } return __sum; } /** * @brief Return the digamma function. * The digamma or @f$ \psi(x) @f$ function is defined by * @f[ * \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)} * @f] * For negative argument the reflection formula is used: * @f[ * \psi(x) = \psi(1-x) - \pi \cot(\pi x) * @f] */ template<typename _Tp> _Tp __psi(_Tp __x) { const int __n = static_cast<int>(__x + 0.5L); const _Tp __eps = _Tp(4) * std::numeric_limits<_Tp>::epsilon(); if (__n <= 0 && std::abs(__x - _Tp(__n)) < __eps) return std::numeric_limits<_Tp>::quiet_NaN(); else if (__x < _Tp(0)) { const _Tp __pi = __numeric_constants<_Tp>::__pi(); return __psi(_Tp(1) - __x) - __pi * std::cos(__pi * __x) / std::sin(__pi * __x); } else if (__x > _Tp(100)) return __psi_asymp(__x); else return __psi_series(__x); } /** * @brief Return the polygamma function @f$ \psi^{(n)}(x) @f$. * * The polygamma function is related to the Hurwitz zeta function: * @f[ * \psi^{(n)}(x) = (-1)^{n+1} m! \zeta(m+1,x) * @f] */ template<typename _Tp> _Tp __psi(unsigned int __n, _Tp __x) { if (__x <= _Tp(0)) std::__throw_domain_error(__N("Argument out of range " "in __psi")); else if (__n == 0) return __psi(__x); else { const _Tp __hzeta = __hurwitz_zeta(_Tp(__n + 1), __x); #if _GLIBCXX_USE_C99_MATH_TR1 const _Tp __ln_nfact = _GLIBCXX_MATH_NS::lgamma(_Tp(__n + 1)); #else const _Tp __ln_nfact = __log_gamma(_Tp(__n + 1)); #endif _Tp __result = std::exp(__ln_nfact) * __hzeta; if (__n % 2 == 1) __result = -__result; return __result; } } } // namespace __detail #undef _GLIBCXX_MATH_NS #if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) } // namespace tr1 #endif _GLIBCXX_END_NAMESPACE_VERSION } // namespace std #endif // _GLIBCXX_TR1_GAMMA_TCC