// Special functions -*- C++ -*- // Copyright (C) 2006-2013 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/exp_integral.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. 228-251. // (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. 222-225. // #ifndef _GLIBCXX_TR1_EXP_INTEGRAL_TCC #define _GLIBCXX_TR1_EXP_INTEGRAL_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 template<typename _Tp> _Tp __expint_E1(_Tp); /** * @brief Return the exponential integral @f$ E_1(x) @f$ * by series summation. This should be good * for @f$ x < 1 @f$. * * The exponential integral is given by * \f[ * E_1(x) = \int_{1}^{\infty} \frac{e^{-xt}}{t} dt * \f] * * @param __x The argument of the exponential integral function. * @return The exponential integral. */ template<typename _Tp> _Tp __expint_E1_series(_Tp __x) { const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); _Tp __term = _Tp(1); _Tp __esum = _Tp(0); _Tp __osum = _Tp(0); const unsigned int __max_iter = 100; for (unsigned int __i = 1; __i < __max_iter; ++__i) { __term *= - __x / __i; if (std::abs(__term) < __eps) break; if (__term >= _Tp(0)) __esum += __term / __i; else __osum += __term / __i; } return - __esum - __osum - __numeric_constants<_Tp>::__gamma_e() - std::log(__x); } /** * @brief Return the exponential integral @f$ E_1(x) @f$ * by asymptotic expansion. * * The exponential integral is given by * \f[ * E_1(x) = \int_{1}^\infty \frac{e^{-xt}}{t} dt * \f] * * @param __x The argument of the exponential integral function. * @return The exponential integral. */ template<typename _Tp> _Tp __expint_E1_asymp(_Tp __x) { _Tp __term = _Tp(1); _Tp __esum = _Tp(1); _Tp __osum = _Tp(0); const unsigned int __max_iter = 1000; for (unsigned int __i = 1; __i < __max_iter; ++__i) { _Tp __prev = __term; __term *= - __i / __x; if (std::abs(__term) > std::abs(__prev)) break; if (__term >= _Tp(0)) __esum += __term; else __osum += __term; } return std::exp(- __x) * (__esum + __osum) / __x; } /** * @brief Return the exponential integral @f$ E_n(x) @f$ * by series summation. * * The exponential integral is given by * \f[ * E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt * \f] * * @param __n The order of the exponential integral function. * @param __x The argument of the exponential integral function. * @return The exponential integral. */ template<typename _Tp> _Tp __expint_En_series(unsigned int __n, _Tp __x) { const unsigned int __max_iter = 100; const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); const int __nm1 = __n - 1; _Tp __ans = (__nm1 != 0 ? _Tp(1) / __nm1 : -std::log(__x) - __numeric_constants<_Tp>::__gamma_e()); _Tp __fact = _Tp(1); for (int __i = 1; __i <= __max_iter; ++__i) { __fact *= -__x / _Tp(__i); _Tp __del; if ( __i != __nm1 ) __del = -__fact / _Tp(__i - __nm1); else { _Tp __psi = -__numeric_constants<_Tp>::gamma_e(); for (int __ii = 1; __ii <= __nm1; ++__ii) __psi += _Tp(1) / _Tp(__ii); __del = __fact * (__psi - std::log(__x)); } __ans += __del; if (std::abs(__del) < __eps * std::abs(__ans)) return __ans; } std::__throw_runtime_error(__N("Series summation failed " "in __expint_En_series.")); } /** * @brief Return the exponential integral @f$ E_n(x) @f$ * by continued fractions. * * The exponential integral is given by * \f[ * E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt * \f] * * @param __n The order of the exponential integral function. * @param __x The argument of the exponential integral function. * @return The exponential integral. */ template<typename _Tp> _Tp __expint_En_cont_frac(unsigned int __n, _Tp __x) { const unsigned int __max_iter = 100; const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); const _Tp __fp_min = std::numeric_limits<_Tp>::min(); const int __nm1 = __n - 1; _Tp __b = __x + _Tp(__n); _Tp __c = _Tp(1) / __fp_min; _Tp __d = _Tp(1) / __b; _Tp __h = __d; for ( unsigned int __i = 1; __i <= __max_iter; ++__i ) { _Tp __a = -_Tp(__i * (__nm1 + __i)); __b += _Tp(2); __d = _Tp(1) / (__a * __d + __b); __c = __b + __a / __c; const _Tp __del = __c * __d; __h *= __del; if (std::abs(__del - _Tp(1)) < __eps) { const _Tp __ans = __h * std::exp(-__x); return __ans; } } std::__throw_runtime_error(__N("Continued fraction failed " "in __expint_En_cont_frac.")); } /** * @brief Return the exponential integral @f$ E_n(x) @f$ * by recursion. Use upward recursion for @f$ x < n @f$ * and downward recursion (Miller's algorithm) otherwise. * * The exponential integral is given by * \f[ * E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt * \f] * * @param __n The order of the exponential integral function. * @param __x The argument of the exponential integral function. * @return The exponential integral. */ template<typename _Tp> _Tp __expint_En_recursion(unsigned int __n, _Tp __x) { _Tp __En; _Tp __E1 = __expint_E1(__x); if (__x < _Tp(__n)) { // Forward recursion is stable only for n < x. __En = __E1; for (unsigned int __j = 2; __j < __n; ++__j) __En = (std::exp(-__x) - __x * __En) / _Tp(__j - 1); } else { // Backward recursion is stable only for n >= x. __En = _Tp(1); const int __N = __n + 20; // TODO: Check this starting number. _Tp __save = _Tp(0); for (int __j = __N; __j > 0; --__j) { __En = (std::exp(-__x) - __j * __En) / __x; if (__j == __n) __save = __En; } _Tp __norm = __En / __E1; __En /= __norm; } return __En; } /** * @brief Return the exponential integral @f$ Ei(x) @f$ * by series summation. * * The exponential integral is given by * \f[ * Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt * \f] * * @param __x The argument of the exponential integral function. * @return The exponential integral. */ template<typename _Tp> _Tp __expint_Ei_series(_Tp __x) { _Tp __term = _Tp(1); _Tp __sum = _Tp(0); const unsigned int __max_iter = 1000; for (unsigned int __i = 1; __i < __max_iter; ++__i) { __term *= __x / __i; __sum += __term / __i; if (__term < std::numeric_limits<_Tp>::epsilon() * __sum) break; } return __numeric_constants<_Tp>::__gamma_e() + __sum + std::log(__x); } /** * @brief Return the exponential integral @f$ Ei(x) @f$ * by asymptotic expansion. * * The exponential integral is given by * \f[ * Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt * \f] * * @param __x The argument of the exponential integral function. * @return The exponential integral. */ template<typename _Tp> _Tp __expint_Ei_asymp(_Tp __x) { _Tp __term = _Tp(1); _Tp __sum = _Tp(1); const unsigned int __max_iter = 1000; for (unsigned int __i = 1; __i < __max_iter; ++__i) { _Tp __prev = __term; __term *= __i / __x; if (__term < std::numeric_limits<_Tp>::epsilon()) break; if (__term >= __prev) break; __sum += __term; } return std::exp(__x) * __sum / __x; } /** * @brief Return the exponential integral @f$ Ei(x) @f$. * * The exponential integral is given by * \f[ * Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt * \f] * * @param __x The argument of the exponential integral function. * @return The exponential integral. */ template<typename _Tp> _Tp __expint_Ei(_Tp __x) { if (__x < _Tp(0)) return -__expint_E1(-__x); else if (__x < -std::log(std::numeric_limits<_Tp>::epsilon())) return __expint_Ei_series(__x); else return __expint_Ei_asymp(__x); } /** * @brief Return the exponential integral @f$ E_1(x) @f$. * * The exponential integral is given by * \f[ * E_1(x) = \int_{1}^\infty \frac{e^{-xt}}{t} dt * \f] * * @param __x The argument of the exponential integral function. * @return The exponential integral. */ template<typename _Tp> _Tp __expint_E1(_Tp __x) { if (__x < _Tp(0)) return -__expint_Ei(-__x); else if (__x < _Tp(1)) return __expint_E1_series(__x); else if (__x < _Tp(100)) // TODO: Find a good asymptotic switch point. return __expint_En_cont_frac(1, __x); else return __expint_E1_asymp(__x); } /** * @brief Return the exponential integral @f$ E_n(x) @f$ * for large argument. * * The exponential integral is given by * \f[ * E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt * \f] * * This is something of an extension. * * @param __n The order of the exponential integral function. * @param __x The argument of the exponential integral function. * @return The exponential integral. */ template<typename _Tp> _Tp __expint_asymp(unsigned int __n, _Tp __x) { _Tp __term = _Tp(1); _Tp __sum = _Tp(1); for (unsigned int __i = 1; __i <= __n; ++__i) { _Tp __prev = __term; __term *= -(__n - __i + 1) / __x; if (std::abs(__term) > std::abs(__prev)) break; __sum += __term; } return std::exp(-__x) * __sum / __x; } /** * @brief Return the exponential integral @f$ E_n(x) @f$ * for large order. * * The exponential integral is given by * \f[ * E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt * \f] * * This is something of an extension. * * @param __n The order of the exponential integral function. * @param __x The argument of the exponential integral function. * @return The exponential integral. */ template<typename _Tp> _Tp __expint_large_n(unsigned int __n, _Tp __x) { const _Tp __xpn = __x + __n; const _Tp __xpn2 = __xpn * __xpn; _Tp __term = _Tp(1); _Tp __sum = _Tp(1); for (unsigned int __i = 1; __i <= __n; ++__i) { _Tp __prev = __term; __term *= (__n - 2 * (__i - 1) * __x) / __xpn2; if (std::abs(__term) < std::numeric_limits<_Tp>::epsilon()) break; __sum += __term; } return std::exp(-__x) * __sum / __xpn; } /** * @brief Return the exponential integral @f$ E_n(x) @f$. * * The exponential integral is given by * \f[ * E_n(x) = \int_{1}^\infty \frac{e^{-xt}}{t^n} dt * \f] * This is something of an extension. * * @param __n The order of the exponential integral function. * @param __x The argument of the exponential integral function. * @return The exponential integral. */ template<typename _Tp> _Tp __expint(unsigned int __n, _Tp __x) { // Return NaN on NaN input. if (__isnan(__x)) return std::numeric_limits<_Tp>::quiet_NaN(); else if (__n <= 1 && __x == _Tp(0)) return std::numeric_limits<_Tp>::infinity(); else { _Tp __E0 = std::exp(__x) / __x; if (__n == 0) return __E0; _Tp __E1 = __expint_E1(__x); if (__n == 1) return __E1; if (__x == _Tp(0)) return _Tp(1) / static_cast<_Tp>(__n - 1); _Tp __En = __expint_En_recursion(__n, __x); return __En; } } /** * @brief Return the exponential integral @f$ Ei(x) @f$. * * The exponential integral is given by * \f[ * Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt * \f] * * @param __x The argument of the exponential integral function. * @return The exponential integral. */ template<typename _Tp> inline _Tp __expint(_Tp __x) { if (__isnan(__x)) return std::numeric_limits<_Tp>::quiet_NaN(); else return __expint_Ei(__x); } _GLIBCXX_END_NAMESPACE_VERSION } // namespace std::tr1::__detail } } #endif // _GLIBCXX_TR1_EXP_INTEGRAL_TCC

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