exp_integral.tcc

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

// Copyright (C) 2006, 2007, 2008, 2009
// 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.
 *  You should not attempt to use it directly.
 */

//
// 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
{
namespace tr1
{

  // [5.2] Special functions

  // Implementation-space details.
  namespace __detail
  {

    /**
     *   @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(const _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(const _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(const unsigned int __n, const _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 = -_TR1_GAMMA_TCC;
              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(const unsigned int __n, const _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(const unsigned int __n, const _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(const _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(const _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(const _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(const _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(const unsigned int __n, const _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(const unsigned int __n, const _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(const unsigned int __n, const _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(const _Tp __x)
    {
      if (__isnan(__x))
        return std::numeric_limits<_Tp>::quiet_NaN();
      else
        return __expint_Ei(__x);
    }

  } // namespace std::tr1::__detail
}
}

#endif // _GLIBCXX_TR1_EXP_INTEGRAL_TCC