// Special functions -*- C++ -*-
// Copyright (C) 2006-2015 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/modified_bessel_func.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.
//
// References:
// (1) Handbook of Mathematical Functions,
// Ed. Milton Abramowitz and Irene A. Stegun,
// Dover Publications,
// Section 9, pp. 355-434, Section 10 pp. 435-478
// (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. 246-249.
#ifndef _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC
#define _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_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 modified Bessel functions @f$ I_\nu(x) @f$ and
* @f$ K_\nu(x) @f$ and their first derivatives
* @f$ I'_\nu(x) @f$ and @f$ K'_\nu(x) @f$ respectively.
* These four functions are computed together for numerical
* stability.
*
* @param __nu The order of the Bessel functions.
* @param __x The argument of the Bessel functions.
* @param __Inu The output regular modified Bessel function.
* @param __Knu The output irregular modified Bessel function.
* @param __Ipnu The output derivative of the regular
* modified Bessel function.
* @param __Kpnu The output derivative of the irregular
* modified Bessel function.
*/
template <typename _Tp>
void
__bessel_ik(_Tp __nu, _Tp __x,
_Tp & __Inu, _Tp & __Knu, _Tp & __Ipnu, _Tp & __Kpnu)
{
if (__x == _Tp(0))
{
if (__nu == _Tp(0))
{
__Inu = _Tp(1);
__Ipnu = _Tp(0);
}
else if (__nu == _Tp(1))
{
__Inu = _Tp(0);
__Ipnu = _Tp(0.5L);
}
else
{
__Inu = _Tp(0);
__Ipnu = _Tp(0);
}
__Knu = std::numeric_limits<_Tp>::infinity();
__Kpnu = -std::numeric_limits<_Tp>::infinity();
return;
}
const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
const _Tp __fp_min = _Tp(10) * std::numeric_limits<_Tp>::epsilon();
const int __max_iter = 15000;
const _Tp __x_min = _Tp(2);
const int __nl = static_cast<int>(__nu + _Tp(0.5L));
const _Tp __mu = __nu - __nl;
const _Tp __mu2 = __mu * __mu;
const _Tp __xi = _Tp(1) / __x;
const _Tp __xi2 = _Tp(2) * __xi;
_Tp __h = __nu * __xi;
if ( __h < __fp_min )
__h = __fp_min;
_Tp __b = __xi2 * __nu;
_Tp __d = _Tp(0);
_Tp __c = __h;
int __i;
for ( __i = 1; __i <= __max_iter; ++__i )
{
__b += __xi2;
__d = _Tp(1) / (__b + __d);
__c = __b + _Tp(1) / __c;
const _Tp __del = __c * __d;
__h *= __del;
if (std::abs(__del - _Tp(1)) < __eps)
break;
}
if (__i > __max_iter)
std::__throw_runtime_error(__N("Argument x too large "
"in __bessel_ik; "
"try asymptotic expansion."));
_Tp __Inul = __fp_min;
_Tp __Ipnul = __h * __Inul;
_Tp __Inul1 = __Inul;
_Tp __Ipnu1 = __Ipnul;
_Tp __fact = __nu * __xi;
for (int __l = __nl; __l >= 1; --__l)
{
const _Tp __Inutemp = __fact * __Inul + __Ipnul;
__fact -= __xi;
__Ipnul = __fact * __Inutemp + __Inul;
__Inul = __Inutemp;
}
_Tp __f = __Ipnul / __Inul;
_Tp __Kmu, __Knu1;
if (__x < __x_min)
{
const _Tp __x2 = __x / _Tp(2);
const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu;
const _Tp __fact = (std::abs(__pimu) < __eps
? _Tp(1) : __pimu / std::sin(__pimu));
_Tp __d = -std::log(__x2);
_Tp __e = __mu * __d;
const _Tp __fact2 = (std::abs(__e) < __eps
? _Tp(1) : std::sinh(__e) / __e);
_Tp __gam1, __gam2, __gampl, __gammi;
__gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi);
_Tp __ff = __fact
* (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d);
_Tp __sum = __ff;
__e = std::exp(__e);
_Tp __p = __e / (_Tp(2) * __gampl);
_Tp __q = _Tp(1) / (_Tp(2) * __e * __gammi);
_Tp __c = _Tp(1);
__d = __x2 * __x2;
_Tp __sum1 = __p;
int __i;
for (__i = 1; __i <= __max_iter; ++__i)
{
__ff = (__i * __ff + __p + __q) / (__i * __i - __mu2);
__c *= __d / __i;
__p /= __i - __mu;
__q /= __i + __mu;
const _Tp __del = __c * __ff;
__sum += __del;
const _Tp __del1 = __c * (__p - __i * __ff);
__sum1 += __del1;
if (std::abs(__del) < __eps * std::abs(__sum))
break;
}
if (__i > __max_iter)
std::__throw_runtime_error(__N("Bessel k series failed to converge "
"in __bessel_ik."));
__Kmu = __sum;
__Knu1 = __sum1 * __xi2;
}
else
{
_Tp __b = _Tp(2) * (_Tp(1) + __x);
_Tp __d = _Tp(1) / __b;
_Tp __delh = __d;
_Tp __h = __delh;
_Tp __q1 = _Tp(0);
_Tp __q2 = _Tp(1);
_Tp __a1 = _Tp(0.25L) - __mu2;
_Tp __q = __c = __a1;
_Tp __a = -__a1;
_Tp __s = _Tp(1) + __q * __delh;
int __i;
for (__i = 2; __i <= __max_iter; ++__i)
{
__a -= 2 * (__i - 1);
__c = -__a * __c / __i;
const _Tp __qnew = (__q1 - __b * __q2) / __a;
__q1 = __q2;
__q2 = __qnew;
__q += __c * __qnew;
__b += _Tp(2);
__d = _Tp(1) / (__b + __a * __d);
__delh = (__b * __d - _Tp(1)) * __delh;
__h += __delh;
const _Tp __dels = __q * __delh;
__s += __dels;
if ( std::abs(__dels / __s) < __eps )
break;
}
if (__i > __max_iter)
std::__throw_runtime_error(__N("Steed's method failed "
"in __bessel_ik."));
__h = __a1 * __h;
__Kmu = std::sqrt(__numeric_constants<_Tp>::__pi() / (_Tp(2) * __x))
* std::exp(-__x) / __s;
__Knu1 = __Kmu * (__mu + __x + _Tp(0.5L) - __h) * __xi;
}
_Tp __Kpmu = __mu * __xi * __Kmu - __Knu1;
_Tp __Inumu = __xi / (__f * __Kmu - __Kpmu);
__Inu = __Inumu * __Inul1 / __Inul;
__Ipnu = __Inumu * __Ipnu1 / __Inul;
for ( __i = 1; __i <= __nl; ++__i )
{
const _Tp __Knutemp = (__mu + __i) * __xi2 * __Knu1 + __Kmu;
__Kmu = __Knu1;
__Knu1 = __Knutemp;
}
__Knu = __Kmu;
__Kpnu = __nu * __xi * __Kmu - __Knu1;
return;
}
/**
* @brief Return the regular modified Bessel function of order
* \f$ \nu \f$: \f$ I_{\nu}(x) \f$.
*
* The regular modified cylindrical Bessel function is:
* @f[
* I_{\nu}(x) = \sum_{k=0}^{\infty}
* \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
* @f]
*
* @param __nu The order of the regular modified Bessel function.
* @param __x The argument of the regular modified Bessel function.
* @return The output regular modified Bessel function.
*/
template<typename _Tp>
_Tp
__cyl_bessel_i(_Tp __nu, _Tp __x)
{
if (__nu < _Tp(0) || __x < _Tp(0))
std::__throw_domain_error(__N("Bad argument "
"in __cyl_bessel_i."));
else if (__isnan(__nu) || __isnan(__x))
return std::numeric_limits<_Tp>::quiet_NaN();
else if (__x * __x < _Tp(10) * (__nu + _Tp(1)))
return __cyl_bessel_ij_series(__nu, __x, +_Tp(1), 200);
else
{
_Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu;
__bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
return __I_nu;
}
}
/**
* @brief Return the irregular modified Bessel function
* \f$ K_{\nu}(x) \f$ of order \f$ \nu \f$.
*
* The irregular modified Bessel function is defined by:
* @f[
* K_{\nu}(x) = \frac{\pi}{2}
* \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi}
* @f]
* where for integral \f$ \nu = n \f$ a limit is taken:
* \f$ lim_{\nu \to n} \f$.
*
* @param __nu The order of the irregular modified Bessel function.
* @param __x The argument of the irregular modified Bessel function.
* @return The output irregular modified Bessel function.
*/
template<typename _Tp>
_Tp
__cyl_bessel_k(_Tp __nu, _Tp __x)
{
if (__nu < _Tp(0) || __x < _Tp(0))
std::__throw_domain_error(__N("Bad argument "
"in __cyl_bessel_k."));
else if (__isnan(__nu) || __isnan(__x))
return std::numeric_limits<_Tp>::quiet_NaN();
else
{
_Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu;
__bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
return __K_nu;
}
}
/**
* @brief Compute the spherical modified Bessel functions
* @f$ i_n(x) @f$ and @f$ k_n(x) @f$ and their first
* derivatives @f$ i'_n(x) @f$ and @f$ k'_n(x) @f$
* respectively.
*
* @param __n The order of the modified spherical Bessel function.
* @param __x The argument of the modified spherical Bessel function.
* @param __i_n The output regular modified spherical Bessel function.
* @param __k_n The output irregular modified spherical
* Bessel function.
* @param __ip_n The output derivative of the regular modified
* spherical Bessel function.
* @param __kp_n The output derivative of the irregular modified
* spherical Bessel function.
*/
template <typename _Tp>
void
__sph_bessel_ik(unsigned int __n, _Tp __x,
_Tp & __i_n, _Tp & __k_n, _Tp & __ip_n, _Tp & __kp_n)
{
const _Tp __nu = _Tp(__n) + _Tp(0.5L);
_Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu;
__bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2()
/ std::sqrt(__x);
__i_n = __factor * __I_nu;
__k_n = __factor * __K_nu;
__ip_n = __factor * __Ip_nu - __i_n / (_Tp(2) * __x);
__kp_n = __factor * __Kp_nu - __k_n / (_Tp(2) * __x);
return;
}
/**
* @brief Compute the Airy functions
* @f$ Ai(x) @f$ and @f$ Bi(x) @f$ and their first
* derivatives @f$ Ai'(x) @f$ and @f$ Bi(x) @f$
* respectively.
*
* @param __x The argument of the Airy functions.
* @param __Ai The output Airy function of the first kind.
* @param __Bi The output Airy function of the second kind.
* @param __Aip The output derivative of the Airy function
* of the first kind.
* @param __Bip The output derivative of the Airy function
* of the second kind.
*/
template <typename _Tp>
void
__airy(_Tp __x, _Tp & __Ai, _Tp & __Bi, _Tp & __Aip, _Tp & __Bip)
{
const _Tp __absx = std::abs(__x);
const _Tp __rootx = std::sqrt(__absx);
const _Tp __z = _Tp(2) * __absx * __rootx / _Tp(3);
if (__x > _Tp(0))
{
_Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu;
__bessel_ik(_Tp(1) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
__Ai = __rootx * __K_nu
/ (__numeric_constants<_Tp>::__sqrt3()
* __numeric_constants<_Tp>::__pi());
__Bi = __rootx * (__K_nu / __numeric_constants<_Tp>::__pi()
+ _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3());
__bessel_ik(_Tp(2) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
__Aip = -__x * __K_nu
/ (__numeric_constants<_Tp>::__sqrt3()
* __numeric_constants<_Tp>::__pi());
__Bip = __x * (__K_nu / __numeric_constants<_Tp>::__pi()
+ _Tp(2) * __I_nu
/ __numeric_constants<_Tp>::__sqrt3());
}
else if (__x < _Tp(0))
{
_Tp __J_nu, __Jp_nu, __N_nu, __Np_nu;
__bessel_jn(_Tp(1) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu);
__Ai = __rootx * (__J_nu
- __N_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2);
__Bi = -__rootx * (__N_nu
+ __J_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2);
__bessel_jn(_Tp(2) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu);
__Aip = __absx * (__N_nu / __numeric_constants<_Tp>::__sqrt3()
+ __J_nu) / _Tp(2);
__Bip = __absx * (__J_nu / __numeric_constants<_Tp>::__sqrt3()
- __N_nu) / _Tp(2);
}
else
{
// Reference:
// Abramowitz & Stegun, page 446 section 10.4.4 on Airy functions.
// The number is Ai(0) = 3^{-2/3}/\Gamma(2/3).
__Ai = _Tp(0.35502805388781723926L);
__Bi = __Ai * __numeric_constants<_Tp>::__sqrt3();
// Reference:
// Abramowitz & Stegun, page 446 section 10.4.5 on Airy functions.
// The number is Ai'(0) = -3^{-1/3}/\Gamma(1/3).
__Aip = -_Tp(0.25881940379280679840L);
__Bip = -__Aip * __numeric_constants<_Tp>::__sqrt3();
}
return;
}
_GLIBCXX_END_NAMESPACE_VERSION
} // namespace std::tr1::__detail
}
}
#endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC