// Special functions -*- C++ -*-
// Copyright (C) 2006-2017 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/bessel_function.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. 240-245
#ifndef _GLIBCXX_TR1_BESSEL_FUNCTION_TCC
#define _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 1
#include "special_function_util.h"
namespace std _GLIBCXX_VISIBILITY(default)
{
#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
// [5.2] Special functions
// Implementation-space details.
namespace __detail
{
_GLIBCXX_BEGIN_NAMESPACE_VERSION
/**
* @brief Compute the gamma functions required by the Temme series
* expansions of @f$ N_\nu(x) @f$ and @f$ K_\nu(x) @f$.
* @f[
* \Gamma_1 = \frac{1}{2\mu}
* [\frac{1}{\Gamma(1 - \mu)} - \frac{1}{\Gamma(1 + \mu)}]
* @f]
* and
* @f[
* \Gamma_2 = \frac{1}{2}
* [\frac{1}{\Gamma(1 - \mu)} + \frac{1}{\Gamma(1 + \mu)}]
* @f]
* where @f$ -1/2 <= \mu <= 1/2 @f$ is @f$ \mu = \nu - N @f$ and @f$ N @f$.
* is the nearest integer to @f$ \nu @f$.
* The values of \f$ \Gamma(1 + \mu) \f$ and \f$ \Gamma(1 - \mu) \f$
* are returned as well.
*
* The accuracy requirements on this are exquisite.
*
* @param __mu The input parameter of the gamma functions.
* @param __gam1 The output function \f$ \Gamma_1(\mu) \f$
* @param __gam2 The output function \f$ \Gamma_2(\mu) \f$
* @param __gampl The output function \f$ \Gamma(1 + \mu) \f$
* @param __gammi The output function \f$ \Gamma(1 - \mu) \f$
*/
template <typename _Tp>
void
__gamma_temme(_Tp __mu,
_Tp & __gam1, _Tp & __gam2, _Tp & __gampl, _Tp & __gammi)
{
#if _GLIBCXX_USE_C99_MATH_TR1
__gampl = _Tp(1) / _GLIBCXX_MATH_NS::tgamma(_Tp(1) + __mu);
__gammi = _Tp(1) / _GLIBCXX_MATH_NS::tgamma(_Tp(1) - __mu);
#else
__gampl = _Tp(1) / __gamma(_Tp(1) + __mu);
__gammi = _Tp(1) / __gamma(_Tp(1) - __mu);
#endif
if (std::abs(__mu) < std::numeric_limits<_Tp>::epsilon())
__gam1 = -_Tp(__numeric_constants<_Tp>::__gamma_e());
else
__gam1 = (__gammi - __gampl) / (_Tp(2) * __mu);
__gam2 = (__gammi + __gampl) / (_Tp(2));
return;
}
/**
* @brief Compute the Bessel @f$ J_\nu(x) @f$ and Neumann
* @f$ N_\nu(x) @f$ functions and their first derivatives
* @f$ J'_\nu(x) @f$ and @f$ N'_\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 __Jnu The output Bessel function of the first kind.
* @param __Nnu The output Neumann function (Bessel function of the second kind).
* @param __Jpnu The output derivative of the Bessel function of the first kind.
* @param __Npnu The output derivative of the Neumann function.
*/
template <typename _Tp>
void
__bessel_jn(_Tp __nu, _Tp __x,
_Tp & __Jnu, _Tp & __Nnu, _Tp & __Jpnu, _Tp & __Npnu)
{
if (__x == _Tp(0))
{
if (__nu == _Tp(0))
{
__Jnu = _Tp(1);
__Jpnu = _Tp(0);
}
else if (__nu == _Tp(1))
{
__Jnu = _Tp(0);
__Jpnu = _Tp(0.5L);
}
else
{
__Jnu = _Tp(0);
__Jpnu = _Tp(0);
}
__Nnu = -std::numeric_limits<_Tp>::infinity();
__Npnu = std::numeric_limits<_Tp>::infinity();
return;
}
const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
// When the multiplier is N i.e.
// fp_min = N * min()
// Then J_0 and N_0 tank at x = 8 * N (J_0 = 0 and N_0 = nan)!
//const _Tp __fp_min = _Tp(20) * std::numeric_limits<_Tp>::min();
const _Tp __fp_min = std::sqrt(std::numeric_limits<_Tp>::min());
const int __max_iter = 15000;
const _Tp __x_min = _Tp(2);
const int __nl = (__x < __x_min
? static_cast<int>(__nu + _Tp(0.5L))
: std::max(0, static_cast<int>(__nu - __x + _Tp(1.5L))));
const _Tp __mu = __nu - __nl;
const _Tp __mu2 = __mu * __mu;
const _Tp __xi = _Tp(1) / __x;
const _Tp __xi2 = _Tp(2) * __xi;
_Tp __w = __xi2 / __numeric_constants<_Tp>::__pi();
int __isign = 1;
_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 = __b - __d;
if (std::abs(__d) < __fp_min)
__d = __fp_min;
__c = __b - _Tp(1) / __c;
if (std::abs(__c) < __fp_min)
__c = __fp_min;
__d = _Tp(1) / __d;
const _Tp __del = __c * __d;
__h *= __del;
if (__d < _Tp(0))
__isign = -__isign;
if (std::abs(__del - _Tp(1)) < __eps)
break;
}
if (__i > __max_iter)
std::__throw_runtime_error(__N("Argument x too large in __bessel_jn; "
"try asymptotic expansion."));
_Tp __Jnul = __isign * __fp_min;
_Tp __Jpnul = __h * __Jnul;
_Tp __Jnul1 = __Jnul;
_Tp __Jpnu1 = __Jpnul;
_Tp __fact = __nu * __xi;
for ( int __l = __nl; __l >= 1; --__l )
{
const _Tp __Jnutemp = __fact * __Jnul + __Jpnul;
__fact -= __xi;
__Jpnul = __fact * __Jnutemp - __Jnul;
__Jnul = __Jnutemp;
}
if (__Jnul == _Tp(0))
__Jnul = __eps;
_Tp __f= __Jpnul / __Jnul;
_Tp __Nmu, __Nnu1, __Npmu, __Jmu;
if (__x < __x_min)
{
const _Tp __x2 = __x / _Tp(2);
const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu;
_Tp __fact = (std::abs(__pimu) < __eps
? _Tp(1) : __pimu / std::sin(__pimu));
_Tp __d = -std::log(__x2);
_Tp __e = __mu * __d;
_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 = (_Tp(2) / __numeric_constants<_Tp>::__pi())
* __fact * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d);
__e = std::exp(__e);
_Tp __p = __e / (__numeric_constants<_Tp>::__pi() * __gampl);
_Tp __q = _Tp(1) / (__e * __numeric_constants<_Tp>::__pi() * __gammi);
const _Tp __pimu2 = __pimu / _Tp(2);
_Tp __fact3 = (std::abs(__pimu2) < __eps
? _Tp(1) : std::sin(__pimu2) / __pimu2 );
_Tp __r = __numeric_constants<_Tp>::__pi() * __pimu2 * __fact3 * __fact3;
_Tp __c = _Tp(1);
__d = -__x2 * __x2;
_Tp __sum = __ff + __r * __q;
_Tp __sum1 = __p;
for (__i = 1; __i <= __max_iter; ++__i)
{
__ff = (__i * __ff + __p + __q) / (__i * __i - __mu2);
__c *= __d / _Tp(__i);
__p /= _Tp(__i) - __mu;
__q /= _Tp(__i) + __mu;
const _Tp __del = __c * (__ff + __r * __q);
__sum += __del;
const _Tp __del1 = __c * __p - __i * __del;
__sum1 += __del1;
if ( std::abs(__del) < __eps * (_Tp(1) + std::abs(__sum)) )
break;
}
if ( __i > __max_iter )
std::__throw_runtime_error(__N("Bessel y series failed to converge "
"in __bessel_jn."));
__Nmu = -__sum;
__Nnu1 = -__sum1 * __xi2;
__Npmu = __mu * __xi * __Nmu - __Nnu1;
__Jmu = __w / (__Npmu - __f * __Nmu);
}
else
{
_Tp __a = _Tp(0.25L) - __mu2;
_Tp __q = _Tp(1);
_Tp __p = -__xi / _Tp(2);
_Tp __br = _Tp(2) * __x;
_Tp __bi = _Tp(2);
_Tp __fact = __a * __xi / (__p * __p + __q * __q);
_Tp __cr = __br + __q * __fact;
_Tp __ci = __bi + __p * __fact;
_Tp __den = __br * __br + __bi * __bi;
_Tp __dr = __br / __den;
_Tp __di = -__bi / __den;
_Tp __dlr = __cr * __dr - __ci * __di;
_Tp __dli = __cr * __di + __ci * __dr;
_Tp __temp = __p * __dlr - __q * __dli;
__q = __p * __dli + __q * __dlr;
__p = __temp;
int __i;
for (__i = 2; __i <= __max_iter; ++__i)
{
__a += _Tp(2 * (__i - 1));
__bi += _Tp(2);
__dr = __a * __dr + __br;
__di = __a * __di + __bi;
if (std::abs(__dr) + std::abs(__di) < __fp_min)
__dr = __fp_min;
__fact = __a / (__cr * __cr + __ci * __ci);
__cr = __br + __cr * __fact;
__ci = __bi - __ci * __fact;
if (std::abs(__cr) + std::abs(__ci) < __fp_min)
__cr = __fp_min;
__den = __dr * __dr + __di * __di;
__dr /= __den;
__di /= -__den;
__dlr = __cr * __dr - __ci * __di;
__dli = __cr * __di + __ci * __dr;
__temp = __p * __dlr - __q * __dli;
__q = __p * __dli + __q * __dlr;
__p = __temp;
if (std::abs(__dlr - _Tp(1)) + std::abs(__dli) < __eps)
break;
}
if (__i > __max_iter)
std::__throw_runtime_error(__N("Lentz's method failed "
"in __bessel_jn."));
const _Tp __gam = (__p - __f) / __q;
__Jmu = std::sqrt(__w / ((__p - __f) * __gam + __q));
#if _GLIBCXX_USE_C99_MATH_TR1
__Jmu = _GLIBCXX_MATH_NS::copysign(__Jmu, __Jnul);
#else
if (__Jmu * __Jnul < _Tp(0))
__Jmu = -__Jmu;
#endif
__Nmu = __gam * __Jmu;
__Npmu = (__p + __q / __gam) * __Nmu;
__Nnu1 = __mu * __xi * __Nmu - __Npmu;
}
__fact = __Jmu / __Jnul;
__Jnu = __fact * __Jnul1;
__Jpnu = __fact * __Jpnu1;
for (__i = 1; __i <= __nl; ++__i)
{
const _Tp __Nnutemp = (__mu + __i) * __xi2 * __Nnu1 - __Nmu;
__Nmu = __Nnu1;
__Nnu1 = __Nnutemp;
}
__Nnu = __Nmu;
__Npnu = __nu * __xi * __Nmu - __Nnu1;
return;
}
/**
* @brief This routine computes the asymptotic cylindrical Bessel
* and Neumann functions of order nu: \f$ J_{\nu} \f$,
* \f$ N_{\nu} \f$.
*
* References:
* (1) Handbook of Mathematical Functions,
* ed. Milton Abramowitz and Irene A. Stegun,
* Dover Publications,
* Section 9 p. 364, Equations 9.2.5-9.2.10
*
* @param __nu The order of the Bessel functions.
* @param __x The argument of the Bessel functions.
* @param __Jnu The output Bessel function of the first kind.
* @param __Nnu The output Neumann function (Bessel function of the second kind).
*/
template <typename _Tp>
void
__cyl_bessel_jn_asymp(_Tp __nu, _Tp __x, _Tp & __Jnu, _Tp & __Nnu)
{
const _Tp __mu = _Tp(4) * __nu * __nu;
const _Tp __mum1 = __mu - _Tp(1);
const _Tp __mum9 = __mu - _Tp(9);
const _Tp __mum25 = __mu - _Tp(25);
const _Tp __mum49 = __mu - _Tp(49);
const _Tp __xx = _Tp(64) * __x * __x;
const _Tp __P = _Tp(1) - __mum1 * __mum9 / (_Tp(2) * __xx)
* (_Tp(1) - __mum25 * __mum49 / (_Tp(12) * __xx));
const _Tp __Q = __mum1 / (_Tp(8) * __x)
* (_Tp(1) - __mum9 * __mum25 / (_Tp(6) * __xx));
const _Tp __chi = __x - (__nu + _Tp(0.5L))
* __numeric_constants<_Tp>::__pi_2();
const _Tp __c = std::cos(__chi);
const _Tp __s = std::sin(__chi);
const _Tp __coef = std::sqrt(_Tp(2)
/ (__numeric_constants<_Tp>::__pi() * __x));
__Jnu = __coef * (__c * __P - __s * __Q);
__Nnu = __coef * (__s * __P + __c * __Q);
return;
}
/**
* @brief This routine returns the cylindrical Bessel functions
* of order \f$ \nu \f$: \f$ J_{\nu} \f$ or \f$ I_{\nu} \f$
* by series expansion.
*
* The modified cylindrical Bessel function is:
* @f[
* Z_{\nu}(x) = \sum_{k=0}^{\infty}
* \frac{\sigma^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
* @f]
* where \f$ \sigma = +1 \f$ or\f$ -1 \f$ for
* \f$ Z = I \f$ or \f$ J \f$ respectively.
*
* See Abramowitz & Stegun, 9.1.10
* Abramowitz & Stegun, 9.6.7
* (1) Handbook of Mathematical Functions,
* ed. Milton Abramowitz and Irene A. Stegun,
* Dover Publications,
* Equation 9.1.10 p. 360 and Equation 9.6.10 p. 375
*
* @param __nu The order of the Bessel function.
* @param __x The argument of the Bessel function.
* @param __sgn The sign of the alternate terms
* -1 for the Bessel function of the first kind.
* +1 for the modified Bessel function of the first kind.
* @return The output Bessel function.
*/
template <typename _Tp>
_Tp
__cyl_bessel_ij_series(_Tp __nu, _Tp __x, _Tp __sgn,
unsigned int __max_iter)
{
if (__x == _Tp(0))
return __nu == _Tp(0) ? _Tp(1) : _Tp(0);
const _Tp __x2 = __x / _Tp(2);
_Tp __fact = __nu * std::log(__x2);
#if _GLIBCXX_USE_C99_MATH_TR1
__fact -= _GLIBCXX_MATH_NS::lgamma(__nu + _Tp(1));
#else
__fact -= __log_gamma(__nu + _Tp(1));
#endif
__fact = std::exp(__fact);
const _Tp __xx4 = __sgn * __x2 * __x2;
_Tp __Jn = _Tp(1);
_Tp __term = _Tp(1);
for (unsigned int __i = 1; __i < __max_iter; ++__i)
{
__term *= __xx4 / (_Tp(__i) * (__nu + _Tp(__i)));
__Jn += __term;
if (std::abs(__term / __Jn) < std::numeric_limits<_Tp>::epsilon())
break;
}
return __fact * __Jn;
}
/**
* @brief Return the Bessel function of order \f$ \nu \f$:
* \f$ J_{\nu}(x) \f$.
*
* The cylindrical Bessel function is:
* @f[
* J_{\nu}(x) = \sum_{k=0}^{\infty}
* \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
* @f]
*
* @param __nu The order of the Bessel function.
* @param __x The argument of the Bessel function.
* @return The output Bessel function.
*/
template<typename _Tp>
_Tp
__cyl_bessel_j(_Tp __nu, _Tp __x)
{
if (__nu < _Tp(0) || __x < _Tp(0))
std::__throw_domain_error(__N("Bad argument "
"in __cyl_bessel_j."));
else if (__isnan(__nu) || __isnan(__x))
return std::numeric_limits<_Tp>::quiet_NaN();
else if (__x * __x < _Tp(10) * (__nu + _Tp(1)))
return __cyl_bessel_ij_series(__nu, __x, -_Tp(1), 200);
else if (__x > _Tp(1000))
{
_Tp __J_nu, __N_nu;
__cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu);
return __J_nu;
}
else
{
_Tp __J_nu, __N_nu, __Jp_nu, __Np_nu;
__bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu);
return __J_nu;
}
}
/**
* @brief Return the Neumann function of order \f$ \nu \f$:
* \f$ N_{\nu}(x) \f$.
*
* The Neumann function is defined by:
* @f[
* N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)}
* {\sin \nu\pi}
* @f]
* where for integral \f$ \nu = n \f$ a limit is taken:
* \f$ lim_{\nu \to n} \f$.
*
* @param __nu The order of the Neumann function.
* @param __x The argument of the Neumann function.
* @return The output Neumann function.
*/
template<typename _Tp>
_Tp
__cyl_neumann_n(_Tp __nu, _Tp __x)
{
if (__nu < _Tp(0) || __x < _Tp(0))
std::__throw_domain_error(__N("Bad argument "
"in __cyl_neumann_n."));
else if (__isnan(__nu) || __isnan(__x))
return std::numeric_limits<_Tp>::quiet_NaN();
else if (__x > _Tp(1000))
{
_Tp __J_nu, __N_nu;
__cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu);
return __N_nu;
}
else
{
_Tp __J_nu, __N_nu, __Jp_nu, __Np_nu;
__bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu);
return __N_nu;
}
}
/**
* @brief Compute the spherical Bessel @f$ j_n(x) @f$
* and Neumann @f$ n_n(x) @f$ functions and their first
* derivatives @f$ j'_n(x) @f$ and @f$ n'_n(x) @f$
* respectively.
*
* @param __n The order of the spherical Bessel function.
* @param __x The argument of the spherical Bessel function.
* @param __j_n The output spherical Bessel function.
* @param __n_n The output spherical Neumann function.
* @param __jp_n The output derivative of the spherical Bessel function.
* @param __np_n The output derivative of the spherical Neumann function.
*/
template <typename _Tp>
void
__sph_bessel_jn(unsigned int __n, _Tp __x,
_Tp & __j_n, _Tp & __n_n, _Tp & __jp_n, _Tp & __np_n)
{
const _Tp __nu = _Tp(__n) + _Tp(0.5L);
_Tp __J_nu, __N_nu, __Jp_nu, __Np_nu;
__bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu);
const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2()
/ std::sqrt(__x);
__j_n = __factor * __J_nu;
__n_n = __factor * __N_nu;
__jp_n = __factor * __Jp_nu - __j_n / (_Tp(2) * __x);
__np_n = __factor * __Np_nu - __n_n / (_Tp(2) * __x);
return;
}
/**
* @brief Return the spherical Bessel function
* @f$ j_n(x) @f$ of order n.
*
* The spherical Bessel function is defined by:
* @f[
* j_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x)
* @f]
*
* @param __n The order of the spherical Bessel function.
* @param __x The argument of the spherical Bessel function.
* @return The output spherical Bessel function.
*/
template <typename _Tp>
_Tp
__sph_bessel(unsigned int __n, _Tp __x)
{
if (__x < _Tp(0))
std::__throw_domain_error(__N("Bad argument "
"in __sph_bessel."));
else if (__isnan(__x))
return std::numeric_limits<_Tp>::quiet_NaN();
else if (__x == _Tp(0))
{
if (__n == 0)
return _Tp(1);
else
return _Tp(0);
}
else
{
_Tp __j_n, __n_n, __jp_n, __np_n;
__sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n);
return __j_n;
}
}
/**
* @brief Return the spherical Neumann function
* @f$ n_n(x) @f$.
*
* The spherical Neumann function is defined by:
* @f[
* n_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x)
* @f]
*
* @param __n The order of the spherical Neumann function.
* @param __x The argument of the spherical Neumann function.
* @return The output spherical Neumann function.
*/
template <typename _Tp>
_Tp
__sph_neumann(unsigned int __n, _Tp __x)
{
if (__x < _Tp(0))
std::__throw_domain_error(__N("Bad argument "
"in __sph_neumann."));
else if (__isnan(__x))
return std::numeric_limits<_Tp>::quiet_NaN();
else if (__x == _Tp(0))
return -std::numeric_limits<_Tp>::infinity();
else
{
_Tp __j_n, __n_n, __jp_n, __np_n;
__sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n);
return __n_n;
}
}
_GLIBCXX_END_NAMESPACE_VERSION
} // namespace __detail
#undef _GLIBCXX_MATH_NS
#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
} // namespace tr1
#endif
}
#endif // _GLIBCXX_TR1_BESSEL_FUNCTION_TCC