X-Git-Url: https://oss.titaniummirror.com/gitweb?a=blobdiff_plain;f=libstdc%2B%2B-v3%2Finclude%2Ftr1%2Fbessel_function.tcc;fp=libstdc%2B%2B-v3%2Finclude%2Ftr1%2Fbessel_function.tcc;h=25102151b56d4f54095196e89747ff5c0530e397;hb=6fed43773c9b0ce596dca5686f37ac3fc0fa11c0;hp=0000000000000000000000000000000000000000;hpb=27b11d56b743098deb193d510b337ba22dc52e5c;p=msp430-gcc.git
diff --git a/libstdc++-v3/include/tr1/bessel_function.tcc b/libstdc++-v3/include/tr1/bessel_function.tcc
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+// 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
+// .
+
+/** @file tr1/bessel_function.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. 240-245
+
+#ifndef _GLIBCXX_TR1_BESSEL_FUNCTION_TCC
+#define _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 1
+
+#include "special_function_util.h"
+
+namespace std
+{
+namespace tr1
+{
+
+ // [5.2] Special functions
+
+ // Implementation-space details.
+ namespace __detail
+ {
+
+ /**
+ * @brief Compute the gamma functions required by the Temme series
+ * expansions of @f$ N_\nu(x) @f$ and @f$ K_\nu(x) @f$.
+ * @f[
+ * \Gamma_1 = \frac{1}{2\mu}
+ * [\frac{1}{\Gamma(1 - \mu)} - \frac{1}{\Gamma(1 + \mu)}]
+ * @f]
+ * and
+ * @f[
+ * \Gamma_2 = \frac{1}{2}
+ * [\frac{1}{\Gamma(1 - \mu)} + \frac{1}{\Gamma(1 + \mu)}]
+ * @f]
+ * where @f$ -1/2 <= \mu <= 1/2 @f$ is @f$ \mu = \nu - N @f$ and @f$ N @f$.
+ * is the nearest integer to @f$ \nu @f$.
+ * The values of \f$ \Gamma(1 + \mu) \f$ and \f$ \Gamma(1 - \mu) \f$
+ * are returned as well.
+ *
+ * The accuracy requirements on this are exquisite.
+ *
+ * @param __mu The input parameter of the gamma functions.
+ * @param __gam1 The output function \f$ \Gamma_1(\mu) \f$
+ * @param __gam2 The output function \f$ \Gamma_2(\mu) \f$
+ * @param __gampl The output function \f$ \Gamma(1 + \mu) \f$
+ * @param __gammi The output function \f$ \Gamma(1 - \mu) \f$
+ */
+ template
+ void
+ __gamma_temme(const _Tp __mu,
+ _Tp & __gam1, _Tp & __gam2, _Tp & __gampl, _Tp & __gammi)
+ {
+#if _GLIBCXX_USE_C99_MATH_TR1
+ __gampl = _Tp(1) / std::tr1::tgamma(_Tp(1) + __mu);
+ __gammi = _Tp(1) / std::tr1::tgamma(_Tp(1) - __mu);
+#else
+ __gampl = _Tp(1) / __gamma(_Tp(1) + __mu);
+ __gammi = _Tp(1) / __gamma(_Tp(1) - __mu);
+#endif
+
+ if (std::abs(__mu) < std::numeric_limits<_Tp>::epsilon())
+ __gam1 = -_Tp(__numeric_constants<_Tp>::__gamma_e());
+ else
+ __gam1 = (__gammi - __gampl) / (_Tp(2) * __mu);
+
+ __gam2 = (__gammi + __gampl) / (_Tp(2));
+
+ return;
+ }
+
+
+ /**
+ * @brief Compute the Bessel @f$ J_\nu(x) @f$ and Neumann
+ * @f$ N_\nu(x) @f$ functions and their first derivatives
+ * @f$ J'_\nu(x) @f$ and @f$ N'_\nu(x) @f$ respectively.
+ * These four functions are computed together for numerical
+ * stability.
+ *
+ * @param __nu The order of the Bessel functions.
+ * @param __x The argument of the Bessel functions.
+ * @param __Jnu The output Bessel function of the first kind.
+ * @param __Nnu The output Neumann function (Bessel function of the second kind).
+ * @param __Jpnu The output derivative of the Bessel function of the first kind.
+ * @param __Npnu The output derivative of the Neumann function.
+ */
+ template
+ void
+ __bessel_jn(const _Tp __nu, const _Tp __x,
+ _Tp & __Jnu, _Tp & __Nnu, _Tp & __Jpnu, _Tp & __Npnu)
+ {
+ if (__x == _Tp(0))
+ {
+ if (__nu == _Tp(0))
+ {
+ __Jnu = _Tp(1);
+ __Jpnu = _Tp(0);
+ }
+ else if (__nu == _Tp(1))
+ {
+ __Jnu = _Tp(0);
+ __Jpnu = _Tp(0.5L);
+ }
+ else
+ {
+ __Jnu = _Tp(0);
+ __Jpnu = _Tp(0);
+ }
+ __Nnu = -std::numeric_limits<_Tp>::infinity();
+ __Npnu = std::numeric_limits<_Tp>::infinity();
+ return;
+ }
+
+ const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
+ // When the multiplier is N i.e.
+ // fp_min = N * min()
+ // Then J_0 and N_0 tank at x = 8 * N (J_0 = 0 and N_0 = nan)!
+ //const _Tp __fp_min = _Tp(20) * std::numeric_limits<_Tp>::min();
+ const _Tp __fp_min = std::sqrt(std::numeric_limits<_Tp>::min());
+ const int __max_iter = 15000;
+ const _Tp __x_min = _Tp(2);
+
+ const int __nl = (__x < __x_min
+ ? static_cast(__nu + _Tp(0.5L))
+ : std::max(0, static_cast(__nu - __x + _Tp(1.5L))));
+
+ const _Tp __mu = __nu - __nl;
+ const _Tp __mu2 = __mu * __mu;
+ const _Tp __xi = _Tp(1) / __x;
+ const _Tp __xi2 = _Tp(2) * __xi;
+ _Tp __w = __xi2 / __numeric_constants<_Tp>::__pi();
+ int __isign = 1;
+ _Tp __h = __nu * __xi;
+ if (__h < __fp_min)
+ __h = __fp_min;
+ _Tp __b = __xi2 * __nu;
+ _Tp __d = _Tp(0);
+ _Tp __c = __h;
+ int __i;
+ for (__i = 1; __i <= __max_iter; ++__i)
+ {
+ __b += __xi2;
+ __d = __b - __d;
+ if (std::abs(__d) < __fp_min)
+ __d = __fp_min;
+ __c = __b - _Tp(1) / __c;
+ if (std::abs(__c) < __fp_min)
+ __c = __fp_min;
+ __d = _Tp(1) / __d;
+ const _Tp __del = __c * __d;
+ __h *= __del;
+ if (__d < _Tp(0))
+ __isign = -__isign;
+ if (std::abs(__del - _Tp(1)) < __eps)
+ break;
+ }
+ if (__i > __max_iter)
+ std::__throw_runtime_error(__N("Argument x too large in __bessel_jn; "
+ "try asymptotic expansion."));
+ _Tp __Jnul = __isign * __fp_min;
+ _Tp __Jpnul = __h * __Jnul;
+ _Tp __Jnul1 = __Jnul;
+ _Tp __Jpnu1 = __Jpnul;
+ _Tp __fact = __nu * __xi;
+ for ( int __l = __nl; __l >= 1; --__l )
+ {
+ const _Tp __Jnutemp = __fact * __Jnul + __Jpnul;
+ __fact -= __xi;
+ __Jpnul = __fact * __Jnutemp - __Jnul;
+ __Jnul = __Jnutemp;
+ }
+ if (__Jnul == _Tp(0))
+ __Jnul = __eps;
+ _Tp __f= __Jpnul / __Jnul;
+ _Tp __Nmu, __Nnu1, __Npmu, __Jmu;
+ if (__x < __x_min)
+ {
+ const _Tp __x2 = __x / _Tp(2);
+ const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu;
+ _Tp __fact = (std::abs(__pimu) < __eps
+ ? _Tp(1) : __pimu / std::sin(__pimu));
+ _Tp __d = -std::log(__x2);
+ _Tp __e = __mu * __d;
+ _Tp __fact2 = (std::abs(__e) < __eps
+ ? _Tp(1) : std::sinh(__e) / __e);
+ _Tp __gam1, __gam2, __gampl, __gammi;
+ __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi);
+ _Tp __ff = (_Tp(2) / __numeric_constants<_Tp>::__pi())
+ * __fact * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d);
+ __e = std::exp(__e);
+ _Tp __p = __e / (__numeric_constants<_Tp>::__pi() * __gampl);
+ _Tp __q = _Tp(1) / (__e * __numeric_constants<_Tp>::__pi() * __gammi);
+ const _Tp __pimu2 = __pimu / _Tp(2);
+ _Tp __fact3 = (std::abs(__pimu2) < __eps
+ ? _Tp(1) : std::sin(__pimu2) / __pimu2 );
+ _Tp __r = __numeric_constants<_Tp>::__pi() * __pimu2 * __fact3 * __fact3;
+ _Tp __c = _Tp(1);
+ __d = -__x2 * __x2;
+ _Tp __sum = __ff + __r * __q;
+ _Tp __sum1 = __p;
+ for (__i = 1; __i <= __max_iter; ++__i)
+ {
+ __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2);
+ __c *= __d / _Tp(__i);
+ __p /= _Tp(__i) - __mu;
+ __q /= _Tp(__i) + __mu;
+ const _Tp __del = __c * (__ff + __r * __q);
+ __sum += __del;
+ const _Tp __del1 = __c * __p - __i * __del;
+ __sum1 += __del1;
+ if ( std::abs(__del) < __eps * (_Tp(1) + std::abs(__sum)) )
+ break;
+ }
+ if ( __i > __max_iter )
+ std::__throw_runtime_error(__N("Bessel y series failed to converge "
+ "in __bessel_jn."));
+ __Nmu = -__sum;
+ __Nnu1 = -__sum1 * __xi2;
+ __Npmu = __mu * __xi * __Nmu - __Nnu1;
+ __Jmu = __w / (__Npmu - __f * __Nmu);
+ }
+ else
+ {
+ _Tp __a = _Tp(0.25L) - __mu2;
+ _Tp __q = _Tp(1);
+ _Tp __p = -__xi / _Tp(2);
+ _Tp __br = _Tp(2) * __x;
+ _Tp __bi = _Tp(2);
+ _Tp __fact = __a * __xi / (__p * __p + __q * __q);
+ _Tp __cr = __br + __q * __fact;
+ _Tp __ci = __bi + __p * __fact;
+ _Tp __den = __br * __br + __bi * __bi;
+ _Tp __dr = __br / __den;
+ _Tp __di = -__bi / __den;
+ _Tp __dlr = __cr * __dr - __ci * __di;
+ _Tp __dli = __cr * __di + __ci * __dr;
+ _Tp __temp = __p * __dlr - __q * __dli;
+ __q = __p * __dli + __q * __dlr;
+ __p = __temp;
+ int __i;
+ for (__i = 2; __i <= __max_iter; ++__i)
+ {
+ __a += _Tp(2 * (__i - 1));
+ __bi += _Tp(2);
+ __dr = __a * __dr + __br;
+ __di = __a * __di + __bi;
+ if (std::abs(__dr) + std::abs(__di) < __fp_min)
+ __dr = __fp_min;
+ __fact = __a / (__cr * __cr + __ci * __ci);
+ __cr = __br + __cr * __fact;
+ __ci = __bi - __ci * __fact;
+ if (std::abs(__cr) + std::abs(__ci) < __fp_min)
+ __cr = __fp_min;
+ __den = __dr * __dr + __di * __di;
+ __dr /= __den;
+ __di /= -__den;
+ __dlr = __cr * __dr - __ci * __di;
+ __dli = __cr * __di + __ci * __dr;
+ __temp = __p * __dlr - __q * __dli;
+ __q = __p * __dli + __q * __dlr;
+ __p = __temp;
+ if (std::abs(__dlr - _Tp(1)) + std::abs(__dli) < __eps)
+ break;
+ }
+ if (__i > __max_iter)
+ std::__throw_runtime_error(__N("Lentz's method failed "
+ "in __bessel_jn."));
+ const _Tp __gam = (__p - __f) / __q;
+ __Jmu = std::sqrt(__w / ((__p - __f) * __gam + __q));
+#if _GLIBCXX_USE_C99_MATH_TR1
+ __Jmu = std::tr1::copysign(__Jmu, __Jnul);
+#else
+ if (__Jmu * __Jnul < _Tp(0))
+ __Jmu = -__Jmu;
+#endif
+ __Nmu = __gam * __Jmu;
+ __Npmu = (__p + __q / __gam) * __Nmu;
+ __Nnu1 = __mu * __xi * __Nmu - __Npmu;
+ }
+ __fact = __Jmu / __Jnul;
+ __Jnu = __fact * __Jnul1;
+ __Jpnu = __fact * __Jpnu1;
+ for (__i = 1; __i <= __nl; ++__i)
+ {
+ const _Tp __Nnutemp = (__mu + __i) * __xi2 * __Nnu1 - __Nmu;
+ __Nmu = __Nnu1;
+ __Nnu1 = __Nnutemp;
+ }
+ __Nnu = __Nmu;
+ __Npnu = __nu * __xi * __Nmu - __Nnu1;
+
+ return;
+ }
+
+
+ /**
+ * @brief This routine computes the asymptotic cylindrical Bessel
+ * and Neumann functions of order nu: \f$ J_{\nu} \f$,
+ * \f$ N_{\nu} \f$.
+ *
+ * References:
+ * (1) Handbook of Mathematical Functions,
+ * ed. Milton Abramowitz and Irene A. Stegun,
+ * Dover Publications,
+ * Section 9 p. 364, Equations 9.2.5-9.2.10
+ *
+ * @param __nu The order of the Bessel functions.
+ * @param __x The argument of the Bessel functions.
+ * @param __Jnu The output Bessel function of the first kind.
+ * @param __Nnu The output Neumann function (Bessel function of the second kind).
+ */
+ template
+ void
+ __cyl_bessel_jn_asymp(const _Tp __nu, const _Tp __x,
+ _Tp & __Jnu, _Tp & __Nnu)
+ {
+ const _Tp __coef = std::sqrt(_Tp(2)
+ / (__numeric_constants<_Tp>::__pi() * __x));
+ const _Tp __mu = _Tp(4) * __nu * __nu;
+ const _Tp __mum1 = __mu - _Tp(1);
+ const _Tp __mum9 = __mu - _Tp(9);
+ const _Tp __mum25 = __mu - _Tp(25);
+ const _Tp __mum49 = __mu - _Tp(49);
+ const _Tp __xx = _Tp(64) * __x * __x;
+ const _Tp __P = _Tp(1) - __mum1 * __mum9 / (_Tp(2) * __xx)
+ * (_Tp(1) - __mum25 * __mum49 / (_Tp(12) * __xx));
+ const _Tp __Q = __mum1 / (_Tp(8) * __x)
+ * (_Tp(1) - __mum9 * __mum25 / (_Tp(6) * __xx));
+
+ const _Tp __chi = __x - (__nu + _Tp(0.5L))
+ * __numeric_constants<_Tp>::__pi_2();
+ const _Tp __c = std::cos(__chi);
+ const _Tp __s = std::sin(__chi);
+
+ __Jnu = __coef * (__c * __P - __s * __Q);
+ __Nnu = __coef * (__s * __P + __c * __Q);
+
+ return;
+ }
+
+
+ /**
+ * @brief This routine returns the cylindrical Bessel functions
+ * of order \f$ \nu \f$: \f$ J_{\nu} \f$ or \f$ I_{\nu} \f$
+ * by series expansion.
+ *
+ * The modified cylindrical Bessel function is:
+ * @f[
+ * Z_{\nu}(x) = \sum_{k=0}^{\infty}
+ * \frac{\sigma^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
+ * @f]
+ * where \f$ \sigma = +1 \f$ or\f$ -1 \f$ for
+ * \f$ Z = I \f$ or \f$ J \f$ respectively.
+ *
+ * See Abramowitz & Stegun, 9.1.10
+ * Abramowitz & Stegun, 9.6.7
+ * (1) Handbook of Mathematical Functions,
+ * ed. Milton Abramowitz and Irene A. Stegun,
+ * Dover Publications,
+ * Equation 9.1.10 p. 360 and Equation 9.6.10 p. 375
+ *
+ * @param __nu The order of the Bessel function.
+ * @param __x The argument of the Bessel function.
+ * @param __sgn The sign of the alternate terms
+ * -1 for the Bessel function of the first kind.
+ * +1 for the modified Bessel function of the first kind.
+ * @return The output Bessel function.
+ */
+ template
+ _Tp
+ __cyl_bessel_ij_series(const _Tp __nu, const _Tp __x, const _Tp __sgn,
+ const unsigned int __max_iter)
+ {
+
+ const _Tp __x2 = __x / _Tp(2);
+ _Tp __fact = __nu * std::log(__x2);
+#if _GLIBCXX_USE_C99_MATH_TR1
+ __fact -= std::tr1::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$.
+ *
+ * The cylindrical Bessel function is:
+ * @f[
+ * J_{\nu}(x) = \sum_{k=0}^{\infty}
+ * \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
+ * @f]
+ *
+ * @param __nu The order of the Bessel function.
+ * @param __x The argument of the Bessel function.
+ * @return The output Bessel function.
+ */
+ template
+ _Tp
+ __cyl_bessel_j(const _Tp __nu, const _Tp __x)
+ {
+ if (__nu < _Tp(0) || __x < _Tp(0))
+ std::__throw_domain_error(__N("Bad argument "
+ "in __cyl_bessel_j."));
+ else if (__isnan(__nu) || __isnan(__x))
+ return std::numeric_limits<_Tp>::quiet_NaN();
+ else if (__x * __x < _Tp(10) * (__nu + _Tp(1)))
+ return __cyl_bessel_ij_series(__nu, __x, -_Tp(1), 200);
+ else if (__x > _Tp(1000))
+ {
+ _Tp __J_nu, __N_nu;
+ __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu);
+ return __J_nu;
+ }
+ else
+ {
+ _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu;
+ __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu);
+ return __J_nu;
+ }
+ }
+
+
+ /**
+ * @brief Return the Neumann function of order \f$ \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$.
+ *
+ * @param __nu The order of the Neumann function.
+ * @param __x The argument of the Neumann function.
+ * @return The output Neumann function.
+ */
+ template
+ _Tp
+ __cyl_neumann_n(const _Tp __nu, const _Tp __x)
+ {
+ if (__nu < _Tp(0) || __x < _Tp(0))
+ std::__throw_domain_error(__N("Bad argument "
+ "in __cyl_neumann_n."));
+ else if (__isnan(__nu) || __isnan(__x))
+ return std::numeric_limits<_Tp>::quiet_NaN();
+ else if (__x > _Tp(1000))
+ {
+ _Tp __J_nu, __N_nu;
+ __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu);
+ return __N_nu;
+ }
+ else
+ {
+ _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu;
+ __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu);
+ return __N_nu;
+ }
+ }
+
+
+ /**
+ * @brief Compute the spherical Bessel @f$ 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
+ void
+ __sph_bessel_jn(const unsigned int __n, const _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
+ _Tp
+ __sph_bessel(const unsigned int __n, const _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
+ _Tp
+ __sph_neumann(const unsigned int __n, const _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 std::tr1::__detail
+}
+}
+
+#endif // _GLIBCXX_TR1_BESSEL_FUNCTION_TCC