--- /dev/null
+// Special functions -*- C++ -*-
+
+// Copyright (C) 2006, 2007, 2008, 2009
+// Free Software Foundation, Inc.
+//
+// This file is part of the GNU ISO C++ Library. This library is free
+// software; you can redistribute it and/or modify it under the
+// terms of the GNU General Public License as published by the
+// Free Software Foundation; either version 3, or (at your option)
+// any later version.
+//
+// This library is distributed in the hope that it will be useful,
+// but WITHOUT ANY WARRANTY; without even the implied warranty of
+// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+// GNU General Public License for more details.
+//
+// Under Section 7 of GPL version 3, you are granted additional
+// permissions described in the GCC Runtime Library Exception, version
+// 3.1, as published by the Free Software Foundation.
+
+// You should have received a copy of the GNU General Public License and
+// a copy of the GCC Runtime Library Exception along with this program;
+// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
+// <http://www.gnu.org/licenses/>.
+
+/** @file tr1/modified_bessel_func.tcc
+ * This is an internal header file, included by other library headers.
+ * You should not attempt to use it directly.
+ */
+
+//
+// ISO C++ 14882 TR1: 5.2 Special functions
+//
+
+// Written by Edward Smith-Rowland.
+//
+// References:
+// (1) Handbook of Mathematical Functions,
+// Ed. Milton Abramowitz and Irene A. Stegun,
+// Dover Publications,
+// Section 9, pp. 355-434, Section 10 pp. 435-478
+// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
+// (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,
+// W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),
+// 2nd ed, pp. 246-249.
+
+#ifndef _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC
+#define _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 1
+
+#include "special_function_util.h"
+
+namespace std
+{
+namespace tr1
+{
+
+ // [5.2] Special functions
+
+ // Implementation-space details.
+ namespace __detail
+ {
+
+ /**
+ * @brief Compute the modified Bessel functions @f$ I_\nu(x) @f$ and
+ * @f$ K_\nu(x) @f$ and their first derivatives
+ * @f$ I'_\nu(x) @f$ and @f$ K'_\nu(x) @f$ respectively.
+ * These four functions are computed together for numerical
+ * stability.
+ *
+ * @param __nu The order of the Bessel functions.
+ * @param __x The argument of the Bessel functions.
+ * @param __Inu The output regular modified Bessel function.
+ * @param __Knu The output irregular modified Bessel function.
+ * @param __Ipnu The output derivative of the regular
+ * modified Bessel function.
+ * @param __Kpnu The output derivative of the irregular
+ * modified Bessel function.
+ */
+ template <typename _Tp>
+ void
+ __bessel_ik(const _Tp __nu, const _Tp __x,
+ _Tp & __Inu, _Tp & __Knu, _Tp & __Ipnu, _Tp & __Kpnu)
+ {
+ if (__x == _Tp(0))
+ {
+ if (__nu == _Tp(0))
+ {
+ __Inu = _Tp(1);
+ __Ipnu = _Tp(0);
+ }
+ else if (__nu == _Tp(1))
+ {
+ __Inu = _Tp(0);
+ __Ipnu = _Tp(0.5L);
+ }
+ else
+ {
+ __Inu = _Tp(0);
+ __Ipnu = _Tp(0);
+ }
+ __Knu = std::numeric_limits<_Tp>::infinity();
+ __Kpnu = -std::numeric_limits<_Tp>::infinity();
+ return;
+ }
+
+ const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
+ const _Tp __fp_min = _Tp(10) * std::numeric_limits<_Tp>::epsilon();
+ const int __max_iter = 15000;
+ const _Tp __x_min = _Tp(2);
+
+ const int __nl = static_cast<int>(__nu + _Tp(0.5L));
+
+ const _Tp __mu = __nu - __nl;
+ const _Tp __mu2 = __mu * __mu;
+ const _Tp __xi = _Tp(1) / __x;
+ const _Tp __xi2 = _Tp(2) * __xi;
+ _Tp __h = __nu * __xi;
+ if ( __h < __fp_min )
+ __h = __fp_min;
+ _Tp __b = __xi2 * __nu;
+ _Tp __d = _Tp(0);
+ _Tp __c = __h;
+ int __i;
+ for ( __i = 1; __i <= __max_iter; ++__i )
+ {
+ __b += __xi2;
+ __d = _Tp(1) / (__b + __d);
+ __c = __b + _Tp(1) / __c;
+ const _Tp __del = __c * __d;
+ __h *= __del;
+ if (std::abs(__del - _Tp(1)) < __eps)
+ break;
+ }
+ if (__i > __max_iter)
+ std::__throw_runtime_error(__N("Argument x too large "
+ "in __bessel_jn; "
+ "try asymptotic expansion."));
+ _Tp __Inul = __fp_min;
+ _Tp __Ipnul = __h * __Inul;
+ _Tp __Inul1 = __Inul;
+ _Tp __Ipnu1 = __Ipnul;
+ _Tp __fact = __nu * __xi;
+ for (int __l = __nl; __l >= 1; --__l)
+ {
+ const _Tp __Inutemp = __fact * __Inul + __Ipnul;
+ __fact -= __xi;
+ __Ipnul = __fact * __Inutemp + __Inul;
+ __Inul = __Inutemp;
+ }
+ _Tp __f = __Ipnul / __Inul;
+ _Tp __Kmu, __Knu1;
+ if (__x < __x_min)
+ {
+ const _Tp __x2 = __x / _Tp(2);
+ const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu;
+ const _Tp __fact = (std::abs(__pimu) < __eps
+ ? _Tp(1) : __pimu / std::sin(__pimu));
+ _Tp __d = -std::log(__x2);
+ _Tp __e = __mu * __d;
+ const _Tp __fact2 = (std::abs(__e) < __eps
+ ? _Tp(1) : std::sinh(__e) / __e);
+ _Tp __gam1, __gam2, __gampl, __gammi;
+ __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi);
+ _Tp __ff = __fact
+ * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d);
+ _Tp __sum = __ff;
+ __e = std::exp(__e);
+ _Tp __p = __e / (_Tp(2) * __gampl);
+ _Tp __q = _Tp(1) / (_Tp(2) * __e * __gammi);
+ _Tp __c = _Tp(1);
+ __d = __x2 * __x2;
+ _Tp __sum1 = __p;
+ int __i;
+ for (__i = 1; __i <= __max_iter; ++__i)
+ {
+ __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2);
+ __c *= __d / __i;
+ __p /= __i - __mu;
+ __q /= __i + __mu;
+ const _Tp __del = __c * __ff;
+ __sum += __del;
+ const _Tp __del1 = __c * (__p - __i * __ff);
+ __sum1 += __del1;
+ if (std::abs(__del) < __eps * std::abs(__sum))
+ break;
+ }
+ if (__i > __max_iter)
+ std::__throw_runtime_error(__N("Bessel k series failed to converge "
+ "in __bessel_jn."));
+ __Kmu = __sum;
+ __Knu1 = __sum1 * __xi2;
+ }
+ else
+ {
+ _Tp __b = _Tp(2) * (_Tp(1) + __x);
+ _Tp __d = _Tp(1) / __b;
+ _Tp __delh = __d;
+ _Tp __h = __delh;
+ _Tp __q1 = _Tp(0);
+ _Tp __q2 = _Tp(1);
+ _Tp __a1 = _Tp(0.25L) - __mu2;
+ _Tp __q = __c = __a1;
+ _Tp __a = -__a1;
+ _Tp __s = _Tp(1) + __q * __delh;
+ int __i;
+ for (__i = 2; __i <= __max_iter; ++__i)
+ {
+ __a -= 2 * (__i - 1);
+ __c = -__a * __c / __i;
+ const _Tp __qnew = (__q1 - __b * __q2) / __a;
+ __q1 = __q2;
+ __q2 = __qnew;
+ __q += __c * __qnew;
+ __b += _Tp(2);
+ __d = _Tp(1) / (__b + __a * __d);
+ __delh = (__b * __d - _Tp(1)) * __delh;
+ __h += __delh;
+ const _Tp __dels = __q * __delh;
+ __s += __dels;
+ if ( std::abs(__dels / __s) < __eps )
+ break;
+ }
+ if (__i > __max_iter)
+ std::__throw_runtime_error(__N("Steed's method failed "
+ "in __bessel_jn."));
+ __h = __a1 * __h;
+ __Kmu = std::sqrt(__numeric_constants<_Tp>::__pi() / (_Tp(2) * __x))
+ * std::exp(-__x) / __s;
+ __Knu1 = __Kmu * (__mu + __x + _Tp(0.5L) - __h) * __xi;
+ }
+
+ _Tp __Kpmu = __mu * __xi * __Kmu - __Knu1;
+ _Tp __Inumu = __xi / (__f * __Kmu - __Kpmu);
+ __Inu = __Inumu * __Inul1 / __Inul;
+ __Ipnu = __Inumu * __Ipnu1 / __Inul;
+ for ( __i = 1; __i <= __nl; ++__i )
+ {
+ const _Tp __Knutemp = (__mu + __i) * __xi2 * __Knu1 + __Kmu;
+ __Kmu = __Knu1;
+ __Knu1 = __Knutemp;
+ }
+ __Knu = __Kmu;
+ __Kpnu = __nu * __xi * __Kmu - __Knu1;
+
+ return;
+ }
+
+
+ /**
+ * @brief Return the regular modified Bessel function of order
+ * \f$ \nu \f$: \f$ I_{\nu}(x) \f$.
+ *
+ * The regular modified cylindrical Bessel function is:
+ * @f[
+ * I_{\nu}(x) = \sum_{k=0}^{\infty}
+ * \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
+ * @f]
+ *
+ * @param __nu The order of the regular modified Bessel function.
+ * @param __x The argument of the regular modified Bessel function.
+ * @return The output regular modified Bessel function.
+ */
+ template<typename _Tp>
+ _Tp
+ __cyl_bessel_i(const _Tp __nu, const _Tp __x)
+ {
+ if (__nu < _Tp(0) || __x < _Tp(0))
+ std::__throw_domain_error(__N("Bad argument "
+ "in __cyl_bessel_i."));
+ else if (__isnan(__nu) || __isnan(__x))
+ return std::numeric_limits<_Tp>::quiet_NaN();
+ else if (__x * __x < _Tp(10) * (__nu + _Tp(1)))
+ return __cyl_bessel_ij_series(__nu, __x, +_Tp(1), 200);
+ else
+ {
+ _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu;
+ __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
+ return __I_nu;
+ }
+ }
+
+
+ /**
+ * @brief Return the irregular modified Bessel function
+ * \f$ K_{\nu}(x) \f$ of order \f$ \nu \f$.
+ *
+ * The irregular modified Bessel function is defined by:
+ * @f[
+ * K_{\nu}(x) = \frac{\pi}{2}
+ * \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi}
+ * @f]
+ * where for integral \f$ \nu = n \f$ a limit is taken:
+ * \f$ lim_{\nu \to n} \f$.
+ *
+ * @param __nu The order of the irregular modified Bessel function.
+ * @param __x The argument of the irregular modified Bessel function.
+ * @return The output irregular modified Bessel function.
+ */
+ template<typename _Tp>
+ _Tp
+ __cyl_bessel_k(const _Tp __nu, const _Tp __x)
+ {
+ if (__nu < _Tp(0) || __x < _Tp(0))
+ std::__throw_domain_error(__N("Bad argument "
+ "in __cyl_bessel_k."));
+ else if (__isnan(__nu) || __isnan(__x))
+ return std::numeric_limits<_Tp>::quiet_NaN();
+ else
+ {
+ _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu;
+ __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
+ return __K_nu;
+ }
+ }
+
+
+ /**
+ * @brief Compute the spherical modified Bessel functions
+ * @f$ i_n(x) @f$ and @f$ k_n(x) @f$ and their first
+ * derivatives @f$ i'_n(x) @f$ and @f$ k'_n(x) @f$
+ * respectively.
+ *
+ * @param __n The order of the modified spherical Bessel function.
+ * @param __x The argument of the modified spherical Bessel function.
+ * @param __i_n The output regular modified spherical Bessel function.
+ * @param __k_n The output irregular modified spherical
+ * Bessel function.
+ * @param __ip_n The output derivative of the regular modified
+ * spherical Bessel function.
+ * @param __kp_n The output derivative of the irregular modified
+ * spherical Bessel function.
+ */
+ template <typename _Tp>
+ void
+ __sph_bessel_ik(const unsigned int __n, const _Tp __x,
+ _Tp & __i_n, _Tp & __k_n, _Tp & __ip_n, _Tp & __kp_n)
+ {
+ const _Tp __nu = _Tp(__n) + _Tp(0.5L);
+
+ _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu;
+ __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
+
+ const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2()
+ / std::sqrt(__x);
+
+ __i_n = __factor * __I_nu;
+ __k_n = __factor * __K_nu;
+ __ip_n = __factor * __Ip_nu - __i_n / (_Tp(2) * __x);
+ __kp_n = __factor * __Kp_nu - __k_n / (_Tp(2) * __x);
+
+ return;
+ }
+
+
+ /**
+ * @brief Compute the Airy functions
+ * @f$ Ai(x) @f$ and @f$ Bi(x) @f$ and their first
+ * derivatives @f$ Ai'(x) @f$ and @f$ Bi(x) @f$
+ * respectively.
+ *
+ * @param __n The order of the Airy functions.
+ * @param __x The argument of the Airy functions.
+ * @param __i_n The output Airy function.
+ * @param __k_n The output Airy function.
+ * @param __ip_n The output derivative of the Airy function.
+ * @param __kp_n The output derivative of the Airy function.
+ */
+ template <typename _Tp>
+ void
+ __airy(const _Tp __x,
+ _Tp & __Ai, _Tp & __Bi, _Tp & __Aip, _Tp & __Bip)
+ {
+ const _Tp __absx = std::abs(__x);
+ const _Tp __rootx = std::sqrt(__absx);
+ const _Tp __z = _Tp(2) * __absx * __rootx / _Tp(3);
+
+ if (__isnan(__x))
+ return std::numeric_limits<_Tp>::quiet_NaN();
+ else if (__x > _Tp(0))
+ {
+ _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu;
+
+ __bessel_ik(_Tp(1) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
+ __Ai = __rootx * __K_nu
+ / (__numeric_constants<_Tp>::__sqrt3()
+ * __numeric_constants<_Tp>::__pi());
+ __Bi = __rootx * (__K_nu / __numeric_constants<_Tp>::__pi()
+ + _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3());
+
+ __bessel_ik(_Tp(2) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
+ __Aip = -__x * __K_nu
+ / (__numeric_constants<_Tp>::__sqrt3()
+ * __numeric_constants<_Tp>::__pi());
+ __Bip = __x * (__K_nu / __numeric_constants<_Tp>::__pi()
+ + _Tp(2) * __I_nu
+ / __numeric_constants<_Tp>::__sqrt3());
+ }
+ else if (__x < _Tp(0))
+ {
+ _Tp __J_nu, __Jp_nu, __N_nu, __Np_nu;
+
+ __bessel_jn(_Tp(1) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu);
+ __Ai = __rootx * (__J_nu
+ - __N_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2);
+ __Bi = -__rootx * (__N_nu
+ + __J_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2);
+
+ __bessel_jn(_Tp(2) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu);
+ __Aip = __absx * (__N_nu / __numeric_constants<_Tp>::__sqrt3()
+ + __J_nu) / _Tp(2);
+ __Bip = __absx * (__J_nu / __numeric_constants<_Tp>::__sqrt3()
+ - __N_nu) / _Tp(2);
+ }
+ else
+ {
+ // Reference:
+ // Abramowitz & Stegun, page 446 section 10.4.4 on Airy functions.
+ // The number is Ai(0) = 3^{-2/3}/\Gamma(2/3).
+ __Ai = _Tp(0.35502805388781723926L);
+ __Bi = __Ai * __numeric_constants<_Tp>::__sqrt3();
+
+ // Reference:
+ // Abramowitz & Stegun, page 446 section 10.4.5 on Airy functions.
+ // The number is Ai'(0) = -3^{-1/3}/\Gamma(1/3).
+ __Aip = -_Tp(0.25881940379280679840L);
+ __Bip = -__Aip * __numeric_constants<_Tp>::__sqrt3();
+ }
+
+ return;
+ }
+
+ } // namespace std::tr1::__detail
+}
+}
+
+#endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC
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