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// Special functions -*- C++ -*-
// Copyright (C) 2006-2014 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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