442 lines
16 KiB
C++
442 lines
16 KiB
C++
// Special functions -*- C++ -*-
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// Copyright (C) 2006-2017 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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// 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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/** @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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// ISO C++ 14882 TR1: 5.2 Special functions
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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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#ifndef _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC
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#define _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 1
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#include "special_function_util.h"
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namespace std _GLIBCXX_VISIBILITY(default)
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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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// Implementation-space details.
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namespace __detail
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{
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_GLIBCXX_BEGIN_NAMESPACE_VERSION
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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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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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const int __nl = static_cast<int>(__nu + _Tp(0.5L));
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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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_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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return;
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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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* @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$.
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*
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* @param __nu The order of the irregular modified Bessel function.
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* @param __x The argument of the irregular modified Bessel function.
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* @return The output irregular modified Bessel function.
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*/
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template<typename _Tp>
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_Tp
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__cyl_bessel_k(_Tp __nu, _Tp __x)
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{
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if (__nu < _Tp(0) || __x < _Tp(0))
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std::__throw_domain_error(__N("Bad argument "
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"in __cyl_bessel_k."));
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else if (__isnan(__nu) || __isnan(__x))
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return std::numeric_limits<_Tp>::quiet_NaN();
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else
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{
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_Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu;
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__bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
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return __K_nu;
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}
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}
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/**
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* @brief Compute the spherical modified Bessel functions
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* @f$ 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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_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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const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2()
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/ std::sqrt(__x);
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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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return;
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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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if (__x > _Tp(0))
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{
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_Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu;
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__bessel_ik(_Tp(1) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
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__Ai = __rootx * __K_nu
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/ (__numeric_constants<_Tp>::__sqrt3()
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* __numeric_constants<_Tp>::__pi());
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__Bi = __rootx * (__K_nu / __numeric_constants<_Tp>::__pi()
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+ _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3());
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__bessel_ik(_Tp(2) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
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__Aip = -__x * __K_nu
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/ (__numeric_constants<_Tp>::__sqrt3()
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* __numeric_constants<_Tp>::__pi());
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__Bip = __x * (__K_nu / __numeric_constants<_Tp>::__pi()
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+ _Tp(2) * __I_nu
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/ __numeric_constants<_Tp>::__sqrt3());
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}
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else if (__x < _Tp(0))
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{
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_Tp __J_nu, __Jp_nu, __N_nu, __Np_nu;
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__bessel_jn(_Tp(1) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu);
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__Ai = __rootx * (__J_nu
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- __N_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2);
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__Bi = -__rootx * (__N_nu
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+ __J_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2);
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__bessel_jn(_Tp(2) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu);
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__Aip = __absx * (__N_nu / __numeric_constants<_Tp>::__sqrt3()
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+ __J_nu) / _Tp(2);
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__Bip = __absx * (__J_nu / __numeric_constants<_Tp>::__sqrt3()
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- __N_nu) / _Tp(2);
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}
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else
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{
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// Reference:
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// Abramowitz & Stegun, page 446 section 10.4.4 on Airy functions.
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// The number is Ai(0) = 3^{-2/3}/\Gamma(2/3).
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__Ai = _Tp(0.35502805388781723926L);
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__Bi = __Ai * __numeric_constants<_Tp>::__sqrt3();
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// Reference:
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// Abramowitz & Stegun, page 446 section 10.4.5 on Airy functions.
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// The number is Ai'(0) = -3^{-1/3}/\Gamma(1/3).
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__Aip = -_Tp(0.25881940379280679840L);
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__Bip = -__Aip * __numeric_constants<_Tp>::__sqrt3();
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}
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return;
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}
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_GLIBCXX_END_NAMESPACE_VERSION
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} // namespace __detail
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#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
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} // namespace tr1
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#endif
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}
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#endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC
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