670 lines
22 KiB
C++
670 lines
22 KiB
C++
// Special functions -*- C++ -*-
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// Copyright (C) 2006-2020 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/bessel_function.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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/* __cyl_bessel_jn_asymp adapted from GNU GSL version 2.4 specfunc/bessel_j.c
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* Copyright (C) 1996-2003 Gerard Jungman
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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. 240-245
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#ifndef _GLIBCXX_TR1_BESSEL_FUNCTION_TCC
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#define _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 1
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#include <tr1/special_function_util.h>
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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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#if _GLIBCXX_USE_STD_SPEC_FUNCS
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# define _GLIBCXX_MATH_NS ::std
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#elif defined(_GLIBCXX_TR1_CMATH)
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namespace tr1
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{
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# define _GLIBCXX_MATH_NS ::std::tr1
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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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/**
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* @brief Compute the gamma functions required by the Temme series
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* expansions of @f$ N_\nu(x) @f$ and @f$ K_\nu(x) @f$.
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* @f[
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* \Gamma_1 = \frac{1}{2\mu}
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* [\frac{1}{\Gamma(1 - \mu)} - \frac{1}{\Gamma(1 + \mu)}]
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* @f]
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* and
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* @f[
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* \Gamma_2 = \frac{1}{2}
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* [\frac{1}{\Gamma(1 - \mu)} + \frac{1}{\Gamma(1 + \mu)}]
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* @f]
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* where @f$ -1/2 <= \mu <= 1/2 @f$ is @f$ \mu = \nu - N @f$ and @f$ N @f$.
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* is the nearest integer to @f$ \nu @f$.
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* The values of \f$ \Gamma(1 + \mu) \f$ and \f$ \Gamma(1 - \mu) \f$
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* are returned as well.
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*
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* The accuracy requirements on this are exquisite.
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*
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* @param __mu The input parameter of the gamma functions.
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* @param __gam1 The output function \f$ \Gamma_1(\mu) \f$
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* @param __gam2 The output function \f$ \Gamma_2(\mu) \f$
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* @param __gampl The output function \f$ \Gamma(1 + \mu) \f$
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* @param __gammi The output function \f$ \Gamma(1 - \mu) \f$
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*/
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template <typename _Tp>
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void
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__gamma_temme(_Tp __mu,
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_Tp & __gam1, _Tp & __gam2, _Tp & __gampl, _Tp & __gammi)
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{
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#if _GLIBCXX_USE_C99_MATH_TR1
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__gampl = _Tp(1) / _GLIBCXX_MATH_NS::tgamma(_Tp(1) + __mu);
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__gammi = _Tp(1) / _GLIBCXX_MATH_NS::tgamma(_Tp(1) - __mu);
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#else
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__gampl = _Tp(1) / __gamma(_Tp(1) + __mu);
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__gammi = _Tp(1) / __gamma(_Tp(1) - __mu);
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#endif
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if (std::abs(__mu) < std::numeric_limits<_Tp>::epsilon())
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__gam1 = -_Tp(__numeric_constants<_Tp>::__gamma_e());
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else
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__gam1 = (__gammi - __gampl) / (_Tp(2) * __mu);
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__gam2 = (__gammi + __gampl) / (_Tp(2));
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return;
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}
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/**
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* @brief Compute the Bessel @f$ J_\nu(x) @f$ and Neumann
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* @f$ N_\nu(x) @f$ functions and their first derivatives
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* @f$ J'_\nu(x) @f$ and @f$ N'_\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 __Jnu The output Bessel function of the first kind.
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* @param __Nnu The output Neumann function (Bessel function of the second kind).
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* @param __Jpnu The output derivative of the Bessel function of the first kind.
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* @param __Npnu The output derivative of the Neumann function.
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*/
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template <typename _Tp>
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void
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__bessel_jn(_Tp __nu, _Tp __x,
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_Tp & __Jnu, _Tp & __Nnu, _Tp & __Jpnu, _Tp & __Npnu)
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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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__Jnu = _Tp(1);
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__Jpnu = _Tp(0);
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}
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else if (__nu == _Tp(1))
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{
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__Jnu = _Tp(0);
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__Jpnu = _Tp(0.5L);
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}
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else
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{
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__Jnu = _Tp(0);
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__Jpnu = _Tp(0);
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}
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__Nnu = -std::numeric_limits<_Tp>::infinity();
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__Npnu = 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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// When the multiplier is N i.e.
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// fp_min = N * min()
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// Then J_0 and N_0 tank at x = 8 * N (J_0 = 0 and N_0 = nan)!
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//const _Tp __fp_min = _Tp(20) * std::numeric_limits<_Tp>::min();
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const _Tp __fp_min = std::sqrt(std::numeric_limits<_Tp>::min());
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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 = (__x < __x_min
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? static_cast<int>(__nu + _Tp(0.5L))
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: std::max(0, static_cast<int>(__nu - __x + _Tp(1.5L))));
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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 __w = __xi2 / __numeric_constants<_Tp>::__pi();
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int __isign = 1;
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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 = __b - __d;
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if (std::abs(__d) < __fp_min)
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__d = __fp_min;
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__c = __b - _Tp(1) / __c;
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if (std::abs(__c) < __fp_min)
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__c = __fp_min;
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__d = _Tp(1) / __d;
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const _Tp __del = __c * __d;
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__h *= __del;
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if (__d < _Tp(0))
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__isign = -__isign;
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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 in __bessel_jn; "
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"try asymptotic expansion."));
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_Tp __Jnul = __isign * __fp_min;
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_Tp __Jpnul = __h * __Jnul;
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_Tp __Jnul1 = __Jnul;
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_Tp __Jpnu1 = __Jpnul;
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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 __Jnutemp = __fact * __Jnul + __Jpnul;
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__fact -= __xi;
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__Jpnul = __fact * __Jnutemp - __Jnul;
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__Jnul = __Jnutemp;
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}
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if (__Jnul == _Tp(0))
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__Jnul = __eps;
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_Tp __f= __Jpnul / __Jnul;
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_Tp __Nmu, __Nnu1, __Npmu, __Jmu;
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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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_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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_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 = (_Tp(2) / __numeric_constants<_Tp>::__pi())
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* __fact * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d);
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__e = std::exp(__e);
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_Tp __p = __e / (__numeric_constants<_Tp>::__pi() * __gampl);
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_Tp __q = _Tp(1) / (__e * __numeric_constants<_Tp>::__pi() * __gammi);
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const _Tp __pimu2 = __pimu / _Tp(2);
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_Tp __fact3 = (std::abs(__pimu2) < __eps
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? _Tp(1) : std::sin(__pimu2) / __pimu2 );
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_Tp __r = __numeric_constants<_Tp>::__pi() * __pimu2 * __fact3 * __fact3;
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_Tp __c = _Tp(1);
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__d = -__x2 * __x2;
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_Tp __sum = __ff + __r * __q;
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_Tp __sum1 = __p;
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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 / _Tp(__i);
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__p /= _Tp(__i) - __mu;
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__q /= _Tp(__i) + __mu;
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const _Tp __del = __c * (__ff + __r * __q);
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__sum += __del;
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const _Tp __del1 = __c * __p - __i * __del;
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__sum1 += __del1;
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if ( std::abs(__del) < __eps * (_Tp(1) + 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 y series failed to converge "
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"in __bessel_jn."));
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__Nmu = -__sum;
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__Nnu1 = -__sum1 * __xi2;
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__Npmu = __mu * __xi * __Nmu - __Nnu1;
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__Jmu = __w / (__Npmu - __f * __Nmu);
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}
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else
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{
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_Tp __a = _Tp(0.25L) - __mu2;
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_Tp __q = _Tp(1);
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_Tp __p = -__xi / _Tp(2);
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_Tp __br = _Tp(2) * __x;
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_Tp __bi = _Tp(2);
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_Tp __fact = __a * __xi / (__p * __p + __q * __q);
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_Tp __cr = __br + __q * __fact;
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_Tp __ci = __bi + __p * __fact;
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_Tp __den = __br * __br + __bi * __bi;
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_Tp __dr = __br / __den;
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_Tp __di = -__bi / __den;
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_Tp __dlr = __cr * __dr - __ci * __di;
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_Tp __dli = __cr * __di + __ci * __dr;
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_Tp __temp = __p * __dlr - __q * __dli;
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__q = __p * __dli + __q * __dlr;
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__p = __temp;
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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 += _Tp(2 * (__i - 1));
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__bi += _Tp(2);
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__dr = __a * __dr + __br;
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__di = __a * __di + __bi;
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if (std::abs(__dr) + std::abs(__di) < __fp_min)
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__dr = __fp_min;
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__fact = __a / (__cr * __cr + __ci * __ci);
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__cr = __br + __cr * __fact;
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__ci = __bi - __ci * __fact;
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if (std::abs(__cr) + std::abs(__ci) < __fp_min)
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__cr = __fp_min;
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__den = __dr * __dr + __di * __di;
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__dr /= __den;
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__di /= -__den;
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__dlr = __cr * __dr - __ci * __di;
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__dli = __cr * __di + __ci * __dr;
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__temp = __p * __dlr - __q * __dli;
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__q = __p * __dli + __q * __dlr;
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__p = __temp;
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if (std::abs(__dlr - _Tp(1)) + std::abs(__dli) < __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("Lentz's method failed "
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"in __bessel_jn."));
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const _Tp __gam = (__p - __f) / __q;
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__Jmu = std::sqrt(__w / ((__p - __f) * __gam + __q));
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#if _GLIBCXX_USE_C99_MATH_TR1
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__Jmu = _GLIBCXX_MATH_NS::copysign(__Jmu, __Jnul);
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#else
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if (__Jmu * __Jnul < _Tp(0))
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__Jmu = -__Jmu;
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#endif
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__Nmu = __gam * __Jmu;
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__Npmu = (__p + __q / __gam) * __Nmu;
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__Nnu1 = __mu * __xi * __Nmu - __Npmu;
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}
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__fact = __Jmu / __Jnul;
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__Jnu = __fact * __Jnul1;
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__Jpnu = __fact * __Jpnu1;
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for (__i = 1; __i <= __nl; ++__i)
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{
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const _Tp __Nnutemp = (__mu + __i) * __xi2 * __Nnu1 - __Nmu;
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__Nmu = __Nnu1;
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__Nnu1 = __Nnutemp;
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}
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__Nnu = __Nmu;
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__Npnu = __nu * __xi * __Nmu - __Nnu1;
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return;
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}
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/**
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* @brief This routine computes the asymptotic cylindrical Bessel
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* and Neumann functions of order nu: \f$ J_{\nu} \f$,
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* \f$ N_{\nu} \f$.
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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 p. 364, Equations 9.2.5-9.2.10
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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 __Jnu The output Bessel function of the first kind.
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* @param __Nnu The output Neumann function (Bessel function of the second kind).
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*/
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template <typename _Tp>
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void
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__cyl_bessel_jn_asymp(_Tp __nu, _Tp __x, _Tp & __Jnu, _Tp & __Nnu)
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{
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const _Tp __mu = _Tp(4) * __nu * __nu;
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const _Tp __8x = _Tp(8) * __x;
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_Tp __P = _Tp(0);
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_Tp __Q = _Tp(0);
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_Tp __k = _Tp(0);
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_Tp __term = _Tp(1);
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int __epsP = 0;
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int __epsQ = 0;
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_Tp __eps = std::numeric_limits<_Tp>::epsilon();
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do
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{
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__term *= (__k == 0
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? _Tp(1)
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: -(__mu - (2 * __k - 1) * (2 * __k - 1)) / (__k * __8x));
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__epsP = std::abs(__term) < __eps * std::abs(__P);
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__P += __term;
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__k++;
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__term *= (__mu - (2 * __k - 1) * (2 * __k - 1)) / (__k * __8x);
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__epsQ = std::abs(__term) < __eps * std::abs(__Q);
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__Q += __term;
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if (__epsP && __epsQ && __k > (__nu / 2.))
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break;
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__k++;
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}
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while (__k < 1000);
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const _Tp __chi = __x - (__nu + _Tp(0.5L))
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* __numeric_constants<_Tp>::__pi_2();
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const _Tp __c = std::cos(__chi);
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const _Tp __s = std::sin(__chi);
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const _Tp __coef = std::sqrt(_Tp(2)
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/ (__numeric_constants<_Tp>::__pi() * __x));
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__Jnu = __coef * (__c * __P - __s * __Q);
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__Nnu = __coef * (__s * __P + __c * __Q);
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return;
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}
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/**
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* @brief This routine returns the cylindrical Bessel functions
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* of order \f$ \nu \f$: \f$ J_{\nu} \f$ or \f$ I_{\nu} \f$
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* by series expansion.
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*
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* The modified cylindrical Bessel function is:
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* @f[
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* Z_{\nu}(x) = \sum_{k=0}^{\infty}
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* \frac{\sigma^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
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* @f]
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* where \f$ \sigma = +1 \f$ or\f$ -1 \f$ for
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* \f$ Z = I \f$ or \f$ J \f$ respectively.
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*
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* See Abramowitz & Stegun, 9.1.10
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* Abramowitz & Stegun, 9.6.7
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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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* 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;
|
|
}
|
|
}
|
|
} // namespace __detail
|
|
#undef _GLIBCXX_MATH_NS
|
|
#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
|
|
} // namespace tr1
|
|
#endif
|
|
|
|
_GLIBCXX_END_NAMESPACE_VERSION
|
|
}
|
|
|
|
#endif // _GLIBCXX_TR1_BESSEL_FUNCTION_TCC
|