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// Special functions -*- C++ -*- |
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|
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// Copyright (C) 2006-2021 Free Software Foundation, Inc. |
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// |
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// This file is part of the GNU ISO C++ Library. This library is free |
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// software; you can redistribute it and/or modify it under the |
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// terms of the GNU General Public License as published by the |
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// Free Software Foundation; either version 3, or (at your option) |
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// any later version. |
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// |
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// This library is distributed in the hope that it will be useful, |
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// but WITHOUT ANY WARRANTY; without even the implied warranty of |
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// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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// GNU General Public License for more details. |
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// |
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// Under Section 7 of GPL version 3, you are granted additional |
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// permissions described in the GCC Runtime Library Exception, version |
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// 3.1, as published by the Free Software Foundation. |
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|
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// You should have received a copy of the GNU General Public License and |
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// a copy of the GCC Runtime Library Exception along with this program; |
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// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see |
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// <http://www.gnu.org/licenses/>. |
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|
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/** @file tr1/modified_bessel_func.tcc |
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* This is an internal header file, included by other library headers. |
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* Do not attempt to use it directly. @headername{tr1/cmath} |
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*/ |
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|
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// |
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// ISO C++ 14882 TR1: 5.2 Special functions |
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// |
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|
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// Written by Edward Smith-Rowland. |
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// |
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// References: |
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// (1) Handbook of Mathematical Functions, |
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// Ed. Milton Abramowitz and Irene A. Stegun, |
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// Dover Publications, |
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// Section 9, pp. 355-434, Section 10 pp. 435-478 |
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// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl |
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// (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky, |
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// W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992), |
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// 2nd ed, pp. 246-249. |
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|
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#ifndef _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC |
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#define _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 1 |
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|
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#include <tr1/special_function_util.h> |
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|
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namespace std _GLIBCXX_VISIBILITY(default) |
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{ |
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_GLIBCXX_BEGIN_NAMESPACE_VERSION |
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|
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#if _GLIBCXX_USE_STD_SPEC_FUNCS |
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#elif defined(_GLIBCXX_TR1_CMATH) |
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namespace tr1 |
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{ |
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#else |
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# error do not include this header directly, use <cmath> or <tr1/cmath> |
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#endif |
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// [5.2] Special functions |
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|
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// Implementation-space details. |
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namespace __detail |
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{ |
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/** |
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* @brief Compute the modified Bessel functions @f$ I_\nu(x) @f$ and |
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* @f$ K_\nu(x) @f$ and their first derivatives |
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* @f$ I'_\nu(x) @f$ and @f$ K'_\nu(x) @f$ respectively. |
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* These four functions are computed together for numerical |
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* stability. |
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* |
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* @param __nu The order of the Bessel functions. |
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* @param __x The argument of the Bessel functions. |
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* @param __Inu The output regular modified Bessel function. |
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* @param __Knu The output irregular modified Bessel function. |
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* @param __Ipnu The output derivative of the regular |
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* modified Bessel function. |
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* @param __Kpnu The output derivative of the irregular |
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* modified Bessel function. |
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*/ |
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template <typename _Tp> |
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void |
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__bessel_ik(_Tp __nu, _Tp __x, |
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_Tp & __Inu, _Tp & __Knu, _Tp & __Ipnu, _Tp & __Kpnu) |
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{ |
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if (__x == _Tp(0)) |
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{ |
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if (__nu == _Tp(0)) |
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{ |
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__Inu = _Tp(1); |
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__Ipnu = _Tp(0); |
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} |
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else if (__nu == _Tp(1)) |
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{ |
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__Inu = _Tp(0); |
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__Ipnu = _Tp(0.5L); |
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} |
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else |
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{ |
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__Inu = _Tp(0); |
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__Ipnu = _Tp(0); |
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} |
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__Knu = std::numeric_limits<_Tp>::infinity(); |
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__Kpnu = -std::numeric_limits<_Tp>::infinity(); |
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return; |
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} |
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|
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const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); |
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const _Tp __fp_min = _Tp(10) * std::numeric_limits<_Tp>::epsilon(); |
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const int __max_iter = 15000; |
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const _Tp __x_min = _Tp(2); |
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|
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const int __nl = static_cast<int>(__nu + _Tp(0.5L)); |
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|
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const _Tp __mu = __nu - __nl; |
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const _Tp __mu2 = __mu * __mu; |
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const _Tp __xi = _Tp(1) / __x; |
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const _Tp __xi2 = _Tp(2) * __xi; |
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_Tp __h = __nu * __xi; |
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if ( __h < __fp_min ) |
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__h = __fp_min; |
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_Tp __b = __xi2 * __nu; |
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_Tp __d = _Tp(0); |
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_Tp __c = __h; |
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int __i; |
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for ( __i = 1; __i <= __max_iter; ++__i ) |
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{ |
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__b += __xi2; |
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__d = _Tp(1) / (__b + __d); |
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__c = __b + _Tp(1) / __c; |
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const _Tp __del = __c * __d; |
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__h *= __del; |
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if (std::abs(__del - _Tp(1)) < __eps) |
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break; |
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} |
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if (__i > __max_iter) |
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std::__throw_runtime_error(__N("Argument x too large " |
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"in __bessel_ik; " |
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"try asymptotic expansion.")); |
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_Tp __Inul = __fp_min; |
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_Tp __Ipnul = __h * __Inul; |
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_Tp __Inul1 = __Inul; |
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_Tp __Ipnu1 = __Ipnul; |
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_Tp __fact = __nu * __xi; |
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for (int __l = __nl; __l >= 1; --__l) |
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{ |
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const _Tp __Inutemp = __fact * __Inul + __Ipnul; |
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__fact -= __xi; |
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__Ipnul = __fact * __Inutemp + __Inul; |
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__Inul = __Inutemp; |
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} |
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_Tp __f = __Ipnul / __Inul; |
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_Tp __Kmu, __Knu1; |
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if (__x < __x_min) |
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{ |
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const _Tp __x2 = __x / _Tp(2); |
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const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu; |
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const _Tp __fact = (std::abs(__pimu) < __eps |
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? _Tp(1) : __pimu / std::sin(__pimu)); |
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_Tp __d = -std::log(__x2); |
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_Tp __e = __mu * __d; |
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const _Tp __fact2 = (std::abs(__e) < __eps |
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? _Tp(1) : std::sinh(__e) / __e); |
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_Tp __gam1, __gam2, __gampl, __gammi; |
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__gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi); |
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_Tp __ff = __fact |
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* (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d); |
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_Tp __sum = __ff; |
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__e = std::exp(__e); |
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_Tp __p = __e / (_Tp(2) * __gampl); |
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_Tp __q = _Tp(1) / (_Tp(2) * __e * __gammi); |
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_Tp __c = _Tp(1); |
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__d = __x2 * __x2; |
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_Tp __sum1 = __p; |
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int __i; |
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for (__i = 1; __i <= __max_iter; ++__i) |
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{ |
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__ff = (__i * __ff + __p + __q) / (__i * __i - __mu2); |
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__c *= __d / __i; |
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__p /= __i - __mu; |
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__q /= __i + __mu; |
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const _Tp __del = __c * __ff; |
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__sum += __del; |
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const _Tp __del1 = __c * (__p - __i * __ff); |
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__sum1 += __del1; |
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if (std::abs(__del) < __eps * std::abs(__sum)) |
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break; |
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} |
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if (__i > __max_iter) |
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std::__throw_runtime_error(__N("Bessel k series failed to converge " |
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"in __bessel_ik.")); |
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__Kmu = __sum; |
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__Knu1 = __sum1 * __xi2; |
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} |
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else |
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{ |
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_Tp __b = _Tp(2) * (_Tp(1) + __x); |
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_Tp __d = _Tp(1) / __b; |
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_Tp __delh = __d; |
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_Tp __h = __delh; |
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_Tp __q1 = _Tp(0); |
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_Tp __q2 = _Tp(1); |
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_Tp __a1 = _Tp(0.25L) - __mu2; |
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_Tp __q = __c = __a1; |
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_Tp __a = -__a1; |
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_Tp __s = _Tp(1) + __q * __delh; |
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int __i; |
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for (__i = 2; __i <= __max_iter; ++__i) |
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{ |
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__a -= 2 * (__i - 1); |
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__c = -__a * __c / __i; |
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const _Tp __qnew = (__q1 - __b * __q2) / __a; |
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__q1 = __q2; |
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__q2 = __qnew; |
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__q += __c * __qnew; |
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__b += _Tp(2); |
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__d = _Tp(1) / (__b + __a * __d); |
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__delh = (__b * __d - _Tp(1)) * __delh; |
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__h += __delh; |
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const _Tp __dels = __q * __delh; |
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__s += __dels; |
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if ( std::abs(__dels / __s) < __eps ) |
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break; |
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} |
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if (__i > __max_iter) |
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std::__throw_runtime_error(__N("Steed's method failed " |
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"in __bessel_ik.")); |
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__h = __a1 * __h; |
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__Kmu = std::sqrt(__numeric_constants<_Tp>::__pi() / (_Tp(2) * __x)) |
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* std::exp(-__x) / __s; |
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__Knu1 = __Kmu * (__mu + __x + _Tp(0.5L) - __h) * __xi; |
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} |
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|
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_Tp __Kpmu = __mu * __xi * __Kmu - __Knu1; |
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_Tp __Inumu = __xi / (__f * __Kmu - __Kpmu); |
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__Inu = __Inumu * __Inul1 / __Inul; |
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__Ipnu = __Inumu * __Ipnu1 / __Inul; |
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for ( __i = 1; __i <= __nl; ++__i ) |
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{ |
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const _Tp __Knutemp = (__mu + __i) * __xi2 * __Knu1 + __Kmu; |
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__Kmu = __Knu1; |
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__Knu1 = __Knutemp; |
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} |
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__Knu = __Kmu; |
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__Kpnu = __nu * __xi * __Kmu - __Knu1; |
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|
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return; |
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} |
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|
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|
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/** |
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* @brief Return the regular modified Bessel function of order |
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* \f$ \nu \f$: \f$ I_{\nu}(x) \f$. |
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* |
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* The regular modified cylindrical Bessel function is: |
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* @f[ |
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* I_{\nu}(x) = \sum_{k=0}^{\infty} |
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* \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} |
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* @f] |
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* |
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* @param __nu The order of the regular modified Bessel function. |
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* @param __x The argument of the regular modified Bessel function. |
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* @return The output regular modified Bessel function. |
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*/ |
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template<typename _Tp> |
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_Tp |
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__cyl_bessel_i(_Tp __nu, _Tp __x) |
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{ |
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if (__nu < _Tp(0) || __x < _Tp(0)) |
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std::__throw_domain_error(__N("Bad argument " |
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"in __cyl_bessel_i.")); |
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else if (__isnan(__nu) || __isnan(__x)) |
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return std::numeric_limits<_Tp>::quiet_NaN(); |
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else if (__x * __x < _Tp(10) * (__nu + _Tp(1))) |
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return __cyl_bessel_ij_series(__nu, __x, +_Tp(1), 200); |
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else |
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{ |
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_Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu; |
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__bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); |
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return __I_nu; |
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} |
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} |
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|
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|
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/** |
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* @brief Return the irregular modified Bessel function |
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* \f$ K_{\nu}(x) \f$ of order \f$ \nu \f$. |
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* |
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* The irregular modified Bessel function is defined by: |
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* @f[ |
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* K_{\nu}(x) = \frac{\pi}{2} |
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* \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi} |
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* @f] |
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* where for integral \f$ \nu = n \f$ a limit is taken: |
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* \f$ lim_{\nu \to n} \f$. |
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* |
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* @param __nu The order of the irregular modified Bessel function. |
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* @param __x The argument of the irregular modified Bessel function. |
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* @return The output irregular modified Bessel function. |
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*/ |
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template<typename _Tp> |
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_Tp |
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__cyl_bessel_k(_Tp __nu, _Tp __x) |
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{ |
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if (__nu < _Tp(0) || __x < _Tp(0)) |
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std::__throw_domain_error(__N("Bad argument " |
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"in __cyl_bessel_k.")); |
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else if (__isnan(__nu) || __isnan(__x)) |
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return std::numeric_limits<_Tp>::quiet_NaN(); |
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else |
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{ |
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_Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu; |
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__bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); |
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return __K_nu; |
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} |
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} |
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|
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|
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/** |
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* @brief Compute the spherical modified Bessel functions |
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* @f$ i_n(x) @f$ and @f$ k_n(x) @f$ and their first |
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* derivatives @f$ i'_n(x) @f$ and @f$ k'_n(x) @f$ |
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* respectively. |
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* |
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* @param __n The order of the modified spherical Bessel function. |
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* @param __x The argument of the modified spherical Bessel function. |
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* @param __i_n The output regular modified spherical Bessel function. |
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* @param __k_n The output irregular modified spherical |
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* Bessel function. |
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* @param __ip_n The output derivative of the regular modified |
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* spherical Bessel function. |
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* @param __kp_n The output derivative of the irregular modified |
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* spherical Bessel function. |
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*/ |
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template <typename _Tp> |
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void |
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__sph_bessel_ik(unsigned int __n, _Tp __x, |
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_Tp & __i_n, _Tp & __k_n, _Tp & __ip_n, _Tp & __kp_n) |
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{ |
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const _Tp __nu = _Tp(__n) + _Tp(0.5L); |
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|
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_Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu; |
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__bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); |
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|
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const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2() |
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/ std::sqrt(__x); |
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|
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__i_n = __factor * __I_nu; |
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__k_n = __factor * __K_nu; |
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__ip_n = __factor * __Ip_nu - __i_n / (_Tp(2) * __x); |
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__kp_n = __factor * __Kp_nu - __k_n / (_Tp(2) * __x); |
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|
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return; |
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} |
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|
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|
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/** |
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* @brief Compute the Airy functions |
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* @f$ Ai(x) @f$ and @f$ Bi(x) @f$ and their first |
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* derivatives @f$ Ai'(x) @f$ and @f$ Bi(x) @f$ |
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* respectively. |
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* |
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* @param __x The argument of the Airy functions. |
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* @param __Ai The output Airy function of the first kind. |
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* @param __Bi The output Airy function of the second kind. |
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* @param __Aip The output derivative of the Airy function |
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* of the first kind. |
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* @param __Bip The output derivative of the Airy function |
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* of the second kind. |
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*/ |
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template <typename _Tp> |
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void |
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__airy(_Tp __x, _Tp & __Ai, _Tp & __Bi, _Tp & __Aip, _Tp & __Bip) |
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{ |
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const _Tp __absx = std::abs(__x); |
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const _Tp __rootx = std::sqrt(__absx); |
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const _Tp __z = _Tp(2) * __absx * __rootx / _Tp(3); |
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const _Tp _S_inf = std::numeric_limits<_Tp>::infinity(); |
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|
| 382 |
if (__isnan(__x)) |
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__Bip = __Aip = __Bi = __Ai = std::numeric_limits<_Tp>::quiet_NaN(); |
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else if (__z == _S_inf) |
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{ |
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__Aip = __Ai = _Tp(0); |
| 387 |
__Bip = __Bi = _S_inf; |
| 388 |
} |
| 389 |
else if (__z == -_S_inf) |
| 390 |
__Bip = __Aip = __Bi = __Ai = _Tp(0); |
| 391 |
else if (__x > _Tp(0)) |
| 392 |
{ |
| 393 |
_Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu; |
| 394 |
|
| 395 |
__bessel_ik(_Tp(1) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); |
| 396 |
__Ai = __rootx * __K_nu |
| 397 |
/ (__numeric_constants<_Tp>::__sqrt3() |
| 398 |
* __numeric_constants<_Tp>::__pi()); |
| 399 |
__Bi = __rootx * (__K_nu / __numeric_constants<_Tp>::__pi() |
| 400 |
+ _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3()); |
| 401 |
|
| 402 |
__bessel_ik(_Tp(2) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); |
| 403 |
__Aip = -__x * __K_nu |
| 404 |
/ (__numeric_constants<_Tp>::__sqrt3() |
| 405 |
* __numeric_constants<_Tp>::__pi()); |
| 406 |
__Bip = __x * (__K_nu / __numeric_constants<_Tp>::__pi() |
| 407 |
+ _Tp(2) * __I_nu |
| 408 |
/ __numeric_constants<_Tp>::__sqrt3()); |
| 409 |
} |
| 410 |
else if (__x < _Tp(0)) |
| 411 |
{ |
| 412 |
_Tp __J_nu, __Jp_nu, __N_nu, __Np_nu; |
| 413 |
|
| 414 |
__bessel_jn(_Tp(1) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); |
| 415 |
__Ai = __rootx * (__J_nu |
| 416 |
- __N_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); |
| 417 |
__Bi = -__rootx * (__N_nu |
| 418 |
+ __J_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); |
| 419 |
|
| 420 |
__bessel_jn(_Tp(2) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); |
| 421 |
__Aip = __absx * (__N_nu / __numeric_constants<_Tp>::__sqrt3() |
| 422 |
+ __J_nu) / _Tp(2); |
| 423 |
__Bip = __absx * (__J_nu / __numeric_constants<_Tp>::__sqrt3() |
| 424 |
- __N_nu) / _Tp(2); |
| 425 |
} |
| 426 |
else |
| 427 |
{ |
| 428 |
// Reference: |
| 429 |
// Abramowitz & Stegun, page 446 section 10.4.4 on Airy functions. |
| 430 |
// The number is Ai(0) = 3^{-2/3}/\Gamma(2/3). |
| 431 |
__Ai = _Tp(0.35502805388781723926L); |
| 432 |
__Bi = __Ai * __numeric_constants<_Tp>::__sqrt3(); |
| 433 |
|
| 434 |
// Reference: |
| 435 |
// Abramowitz & Stegun, page 446 section 10.4.5 on Airy functions. |
| 436 |
// The number is Ai'(0) = -3^{-1/3}/\Gamma(1/3). |
| 437 |
__Aip = -_Tp(0.25881940379280679840L); |
| 438 |
__Bip = -__Aip * __numeric_constants<_Tp>::__sqrt3(); |
| 439 |
} |
| 440 |
|
| 441 |
return; |
| 442 |
} |
| 443 |
} // namespace __detail |
| 444 |
#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) |
| 445 |
} // namespace tr1 |
| 446 |
#endif |
| 447 |
|
| 448 |
_GLIBCXX_END_NAMESPACE_VERSION |
| 449 |
} |
| 450 |
|
| 451 |
#endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC |
