libstdc++
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00001 // Special functions -*- C++ -*- 00002 00003 // Copyright (C) 2006, 2007, 2008, 2009 00004 // Free Software Foundation, Inc. 00005 // 00006 // This file is part of the GNU ISO C++ Library. This library is free 00007 // software; you can redistribute it and/or modify it under the 00008 // terms of the GNU General Public License as published by the 00009 // Free Software Foundation; either version 3, or (at your option) 00010 // any later version. 00011 // 00012 // This library is distributed in the hope that it will be useful, 00013 // but WITHOUT ANY WARRANTY; without even the implied warranty of 00014 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 00015 // GNU General Public License for more details. 00016 // 00017 // Under Section 7 of GPL version 3, you are granted additional 00018 // permissions described in the GCC Runtime Library Exception, version 00019 // 3.1, as published by the Free Software Foundation. 00020 00021 // You should have received a copy of the GNU General Public License and 00022 // a copy of the GCC Runtime Library Exception along with this program; 00023 // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see 00024 // <http://www.gnu.org/licenses/>. 00025 00026 /** @file tr1/bessel_function.tcc 00027 * This is an internal header file, included by other library headers. 00028 * You should not attempt to use it directly. 00029 */ 00030 00031 // 00032 // ISO C++ 14882 TR1: 5.2 Special functions 00033 // 00034 00035 // Written by Edward Smith-Rowland. 00036 // 00037 // References: 00038 // (1) Handbook of Mathematical Functions, 00039 // ed. Milton Abramowitz and Irene A. Stegun, 00040 // Dover Publications, 00041 // Section 9, pp. 355-434, Section 10 pp. 435-478 00042 // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl 00043 // (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky, 00044 // W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992), 00045 // 2nd ed, pp. 240-245 00046 00047 #ifndef _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 00048 #define _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 1 00049 00050 #include "special_function_util.h" 00051 00052 namespace std 00053 { 00054 namespace tr1 00055 { 00056 00057 // [5.2] Special functions 00058 00059 // Implementation-space details. 00060 namespace __detail 00061 { 00062 00063 /** 00064 * @brief Compute the gamma functions required by the Temme series 00065 * expansions of @f$ N_\nu(x) @f$ and @f$ K_\nu(x) @f$. 00066 * @f[ 00067 * \Gamma_1 = \frac{1}{2\mu} 00068 * [\frac{1}{\Gamma(1 - \mu)} - \frac{1}{\Gamma(1 + \mu)}] 00069 * @f] 00070 * and 00071 * @f[ 00072 * \Gamma_2 = \frac{1}{2} 00073 * [\frac{1}{\Gamma(1 - \mu)} + \frac{1}{\Gamma(1 + \mu)}] 00074 * @f] 00075 * where @f$ -1/2 <= \mu <= 1/2 @f$ is @f$ \mu = \nu - N @f$ and @f$ N @f$. 00076 * is the nearest integer to @f$ \nu @f$. 00077 * The values of \f$ \Gamma(1 + \mu) \f$ and \f$ \Gamma(1 - \mu) \f$ 00078 * are returned as well. 00079 * 00080 * The accuracy requirements on this are exquisite. 00081 * 00082 * @param __mu The input parameter of the gamma functions. 00083 * @param __gam1 The output function \f$ \Gamma_1(\mu) \f$ 00084 * @param __gam2 The output function \f$ \Gamma_2(\mu) \f$ 00085 * @param __gampl The output function \f$ \Gamma(1 + \mu) \f$ 00086 * @param __gammi The output function \f$ \Gamma(1 - \mu) \f$ 00087 */ 00088 template <typename _Tp> 00089 void 00090 __gamma_temme(const _Tp __mu, 00091 _Tp & __gam1, _Tp & __gam2, _Tp & __gampl, _Tp & __gammi) 00092 { 00093 #if _GLIBCXX_USE_C99_MATH_TR1 00094 __gampl = _Tp(1) / std::tr1::tgamma(_Tp(1) + __mu); 00095 __gammi = _Tp(1) / std::tr1::tgamma(_Tp(1) - __mu); 00096 #else 00097 __gampl = _Tp(1) / __gamma(_Tp(1) + __mu); 00098 __gammi = _Tp(1) / __gamma(_Tp(1) - __mu); 00099 #endif 00100 00101 if (std::abs(__mu) < std::numeric_limits<_Tp>::epsilon()) 00102 __gam1 = -_Tp(__numeric_constants<_Tp>::__gamma_e()); 00103 else 00104 __gam1 = (__gammi - __gampl) / (_Tp(2) * __mu); 00105 00106 __gam2 = (__gammi + __gampl) / (_Tp(2)); 00107 00108 return; 00109 } 00110 00111 00112 /** 00113 * @brief Compute the Bessel @f$ J_\nu(x) @f$ and Neumann 00114 * @f$ N_\nu(x) @f$ functions and their first derivatives 00115 * @f$ J'_\nu(x) @f$ and @f$ N'_\nu(x) @f$ respectively. 00116 * These four functions are computed together for numerical 00117 * stability. 00118 * 00119 * @param __nu The order of the Bessel functions. 00120 * @param __x The argument of the Bessel functions. 00121 * @param __Jnu The output Bessel function of the first kind. 00122 * @param __Nnu The output Neumann function (Bessel function of the second kind). 00123 * @param __Jpnu The output derivative of the Bessel function of the first kind. 00124 * @param __Npnu The output derivative of the Neumann function. 00125 */ 00126 template <typename _Tp> 00127 void 00128 __bessel_jn(const _Tp __nu, const _Tp __x, 00129 _Tp & __Jnu, _Tp & __Nnu, _Tp & __Jpnu, _Tp & __Npnu) 00130 { 00131 if (__x == _Tp(0)) 00132 { 00133 if (__nu == _Tp(0)) 00134 { 00135 __Jnu = _Tp(1); 00136 __Jpnu = _Tp(0); 00137 } 00138 else if (__nu == _Tp(1)) 00139 { 00140 __Jnu = _Tp(0); 00141 __Jpnu = _Tp(0.5L); 00142 } 00143 else 00144 { 00145 __Jnu = _Tp(0); 00146 __Jpnu = _Tp(0); 00147 } 00148 __Nnu = -std::numeric_limits<_Tp>::infinity(); 00149 __Npnu = std::numeric_limits<_Tp>::infinity(); 00150 return; 00151 } 00152 00153 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 00154 // When the multiplier is N i.e. 00155 // fp_min = N * min() 00156 // Then J_0 and N_0 tank at x = 8 * N (J_0 = 0 and N_0 = nan)! 00157 //const _Tp __fp_min = _Tp(20) * std::numeric_limits<_Tp>::min(); 00158 const _Tp __fp_min = std::sqrt(std::numeric_limits<_Tp>::min()); 00159 const int __max_iter = 15000; 00160 const _Tp __x_min = _Tp(2); 00161 00162 const int __nl = (__x < __x_min 00163 ? static_cast<int>(__nu + _Tp(0.5L)) 00164 : std::max(0, static_cast<int>(__nu - __x + _Tp(1.5L)))); 00165 00166 const _Tp __mu = __nu - __nl; 00167 const _Tp __mu2 = __mu * __mu; 00168 const _Tp __xi = _Tp(1) / __x; 00169 const _Tp __xi2 = _Tp(2) * __xi; 00170 _Tp __w = __xi2 / __numeric_constants<_Tp>::__pi(); 00171 int __isign = 1; 00172 _Tp __h = __nu * __xi; 00173 if (__h < __fp_min) 00174 __h = __fp_min; 00175 _Tp __b = __xi2 * __nu; 00176 _Tp __d = _Tp(0); 00177 _Tp __c = __h; 00178 int __i; 00179 for (__i = 1; __i <= __max_iter; ++__i) 00180 { 00181 __b += __xi2; 00182 __d = __b - __d; 00183 if (std::abs(__d) < __fp_min) 00184 __d = __fp_min; 00185 __c = __b - _Tp(1) / __c; 00186 if (std::abs(__c) < __fp_min) 00187 __c = __fp_min; 00188 __d = _Tp(1) / __d; 00189 const _Tp __del = __c * __d; 00190 __h *= __del; 00191 if (__d < _Tp(0)) 00192 __isign = -__isign; 00193 if (std::abs(__del - _Tp(1)) < __eps) 00194 break; 00195 } 00196 if (__i > __max_iter) 00197 std::__throw_runtime_error(__N("Argument x too large in __bessel_jn; " 00198 "try asymptotic expansion.")); 00199 _Tp __Jnul = __isign * __fp_min; 00200 _Tp __Jpnul = __h * __Jnul; 00201 _Tp __Jnul1 = __Jnul; 00202 _Tp __Jpnu1 = __Jpnul; 00203 _Tp __fact = __nu * __xi; 00204 for ( int __l = __nl; __l >= 1; --__l ) 00205 { 00206 const _Tp __Jnutemp = __fact * __Jnul + __Jpnul; 00207 __fact -= __xi; 00208 __Jpnul = __fact * __Jnutemp - __Jnul; 00209 __Jnul = __Jnutemp; 00210 } 00211 if (__Jnul == _Tp(0)) 00212 __Jnul = __eps; 00213 _Tp __f= __Jpnul / __Jnul; 00214 _Tp __Nmu, __Nnu1, __Npmu, __Jmu; 00215 if (__x < __x_min) 00216 { 00217 const _Tp __x2 = __x / _Tp(2); 00218 const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu; 00219 _Tp __fact = (std::abs(__pimu) < __eps 00220 ? _Tp(1) : __pimu / std::sin(__pimu)); 00221 _Tp __d = -std::log(__x2); 00222 _Tp __e = __mu * __d; 00223 _Tp __fact2 = (std::abs(__e) < __eps 00224 ? _Tp(1) : std::sinh(__e) / __e); 00225 _Tp __gam1, __gam2, __gampl, __gammi; 00226 __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi); 00227 _Tp __ff = (_Tp(2) / __numeric_constants<_Tp>::__pi()) 00228 * __fact * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d); 00229 __e = std::exp(__e); 00230 _Tp __p = __e / (__numeric_constants<_Tp>::__pi() * __gampl); 00231 _Tp __q = _Tp(1) / (__e * __numeric_constants<_Tp>::__pi() * __gammi); 00232 const _Tp __pimu2 = __pimu / _Tp(2); 00233 _Tp __fact3 = (std::abs(__pimu2) < __eps 00234 ? _Tp(1) : std::sin(__pimu2) / __pimu2 ); 00235 _Tp __r = __numeric_constants<_Tp>::__pi() * __pimu2 * __fact3 * __fact3; 00236 _Tp __c = _Tp(1); 00237 __d = -__x2 * __x2; 00238 _Tp __sum = __ff + __r * __q; 00239 _Tp __sum1 = __p; 00240 for (__i = 1; __i <= __max_iter; ++__i) 00241 { 00242 __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2); 00243 __c *= __d / _Tp(__i); 00244 __p /= _Tp(__i) - __mu; 00245 __q /= _Tp(__i) + __mu; 00246 const _Tp __del = __c * (__ff + __r * __q); 00247 __sum += __del; 00248 const _Tp __del1 = __c * __p - __i * __del; 00249 __sum1 += __del1; 00250 if ( std::abs(__del) < __eps * (_Tp(1) + std::abs(__sum)) ) 00251 break; 00252 } 00253 if ( __i > __max_iter ) 00254 std::__throw_runtime_error(__N("Bessel y series failed to converge " 00255 "in __bessel_jn.")); 00256 __Nmu = -__sum; 00257 __Nnu1 = -__sum1 * __xi2; 00258 __Npmu = __mu * __xi * __Nmu - __Nnu1; 00259 __Jmu = __w / (__Npmu - __f * __Nmu); 00260 } 00261 else 00262 { 00263 _Tp __a = _Tp(0.25L) - __mu2; 00264 _Tp __q = _Tp(1); 00265 _Tp __p = -__xi / _Tp(2); 00266 _Tp __br = _Tp(2) * __x; 00267 _Tp __bi = _Tp(2); 00268 _Tp __fact = __a * __xi / (__p * __p + __q * __q); 00269 _Tp __cr = __br + __q * __fact; 00270 _Tp __ci = __bi + __p * __fact; 00271 _Tp __den = __br * __br + __bi * __bi; 00272 _Tp __dr = __br / __den; 00273 _Tp __di = -__bi / __den; 00274 _Tp __dlr = __cr * __dr - __ci * __di; 00275 _Tp __dli = __cr * __di + __ci * __dr; 00276 _Tp __temp = __p * __dlr - __q * __dli; 00277 __q = __p * __dli + __q * __dlr; 00278 __p = __temp; 00279 int __i; 00280 for (__i = 2; __i <= __max_iter; ++__i) 00281 { 00282 __a += _Tp(2 * (__i - 1)); 00283 __bi += _Tp(2); 00284 __dr = __a * __dr + __br; 00285 __di = __a * __di + __bi; 00286 if (std::abs(__dr) + std::abs(__di) < __fp_min) 00287 __dr = __fp_min; 00288 __fact = __a / (__cr * __cr + __ci * __ci); 00289 __cr = __br + __cr * __fact; 00290 __ci = __bi - __ci * __fact; 00291 if (std::abs(__cr) + std::abs(__ci) < __fp_min) 00292 __cr = __fp_min; 00293 __den = __dr * __dr + __di * __di; 00294 __dr /= __den; 00295 __di /= -__den; 00296 __dlr = __cr * __dr - __ci * __di; 00297 __dli = __cr * __di + __ci * __dr; 00298 __temp = __p * __dlr - __q * __dli; 00299 __q = __p * __dli + __q * __dlr; 00300 __p = __temp; 00301 if (std::abs(__dlr - _Tp(1)) + std::abs(__dli) < __eps) 00302 break; 00303 } 00304 if (__i > __max_iter) 00305 std::__throw_runtime_error(__N("Lentz's method failed " 00306 "in __bessel_jn.")); 00307 const _Tp __gam = (__p - __f) / __q; 00308 __Jmu = std::sqrt(__w / ((__p - __f) * __gam + __q)); 00309 #if _GLIBCXX_USE_C99_MATH_TR1 00310 __Jmu = std::tr1::copysign(__Jmu, __Jnul); 00311 #else 00312 if (__Jmu * __Jnul < _Tp(0)) 00313 __Jmu = -__Jmu; 00314 #endif 00315 __Nmu = __gam * __Jmu; 00316 __Npmu = (__p + __q / __gam) * __Nmu; 00317 __Nnu1 = __mu * __xi * __Nmu - __Npmu; 00318 } 00319 __fact = __Jmu / __Jnul; 00320 __Jnu = __fact * __Jnul1; 00321 __Jpnu = __fact * __Jpnu1; 00322 for (__i = 1; __i <= __nl; ++__i) 00323 { 00324 const _Tp __Nnutemp = (__mu + __i) * __xi2 * __Nnu1 - __Nmu; 00325 __Nmu = __Nnu1; 00326 __Nnu1 = __Nnutemp; 00327 } 00328 __Nnu = __Nmu; 00329 __Npnu = __nu * __xi * __Nmu - __Nnu1; 00330 00331 return; 00332 } 00333 00334 00335 /** 00336 * @brief This routine computes the asymptotic cylindrical Bessel 00337 * and Neumann functions of order nu: \f$ J_{\nu} \f$, 00338 * \f$ N_{\nu} \f$. 00339 * 00340 * References: 00341 * (1) Handbook of Mathematical Functions, 00342 * ed. Milton Abramowitz and Irene A. Stegun, 00343 * Dover Publications, 00344 * Section 9 p. 364, Equations 9.2.5-9.2.10 00345 * 00346 * @param __nu The order of the Bessel functions. 00347 * @param __x The argument of the Bessel functions. 00348 * @param __Jnu The output Bessel function of the first kind. 00349 * @param __Nnu The output Neumann function (Bessel function of the second kind). 00350 */ 00351 template <typename _Tp> 00352 void 00353 __cyl_bessel_jn_asymp(const _Tp __nu, const _Tp __x, 00354 _Tp & __Jnu, _Tp & __Nnu) 00355 { 00356 const _Tp __coef = std::sqrt(_Tp(2) 00357 / (__numeric_constants<_Tp>::__pi() * __x)); 00358 const _Tp __mu = _Tp(4) * __nu * __nu; 00359 const _Tp __mum1 = __mu - _Tp(1); 00360 const _Tp __mum9 = __mu - _Tp(9); 00361 const _Tp __mum25 = __mu - _Tp(25); 00362 const _Tp __mum49 = __mu - _Tp(49); 00363 const _Tp __xx = _Tp(64) * __x * __x; 00364 const _Tp __P = _Tp(1) - __mum1 * __mum9 / (_Tp(2) * __xx) 00365 * (_Tp(1) - __mum25 * __mum49 / (_Tp(12) * __xx)); 00366 const _Tp __Q = __mum1 / (_Tp(8) * __x) 00367 * (_Tp(1) - __mum9 * __mum25 / (_Tp(6) * __xx)); 00368 00369 const _Tp __chi = __x - (__nu + _Tp(0.5L)) 00370 * __numeric_constants<_Tp>::__pi_2(); 00371 const _Tp __c = std::cos(__chi); 00372 const _Tp __s = std::sin(__chi); 00373 00374 __Jnu = __coef * (__c * __P - __s * __Q); 00375 __Nnu = __coef * (__s * __P + __c * __Q); 00376 00377 return; 00378 } 00379 00380 00381 /** 00382 * @brief This routine returns the cylindrical Bessel functions 00383 * of order \f$ \nu \f$: \f$ J_{\nu} \f$ or \f$ I_{\nu} \f$ 00384 * by series expansion. 00385 * 00386 * The modified cylindrical Bessel function is: 00387 * @f[ 00388 * Z_{\nu}(x) = \sum_{k=0}^{\infty} 00389 * \frac{\sigma^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} 00390 * @f] 00391 * where \f$ \sigma = +1 \f$ or\f$ -1 \f$ for 00392 * \f$ Z = I \f$ or \f$ J \f$ respectively. 00393 * 00394 * See Abramowitz & Stegun, 9.1.10 00395 * Abramowitz & Stegun, 9.6.7 00396 * (1) Handbook of Mathematical Functions, 00397 * ed. Milton Abramowitz and Irene A. Stegun, 00398 * Dover Publications, 00399 * Equation 9.1.10 p. 360 and Equation 9.6.10 p. 375 00400 * 00401 * @param __nu The order of the Bessel function. 00402 * @param __x The argument of the Bessel function. 00403 * @param __sgn The sign of the alternate terms 00404 * -1 for the Bessel function of the first kind. 00405 * +1 for the modified Bessel function of the first kind. 00406 * @return The output Bessel function. 00407 */ 00408 template <typename _Tp> 00409 _Tp 00410 __cyl_bessel_ij_series(const _Tp __nu, const _Tp __x, const _Tp __sgn, 00411 const unsigned int __max_iter) 00412 { 00413 00414 const _Tp __x2 = __x / _Tp(2); 00415 _Tp __fact = __nu * std::log(__x2); 00416 #if _GLIBCXX_USE_C99_MATH_TR1 00417 __fact -= std::tr1::lgamma(__nu + _Tp(1)); 00418 #else 00419 __fact -= __log_gamma(__nu + _Tp(1)); 00420 #endif 00421 __fact = std::exp(__fact); 00422 const _Tp __xx4 = __sgn * __x2 * __x2; 00423 _Tp __Jn = _Tp(1); 00424 _Tp __term = _Tp(1); 00425 00426 for (unsigned int __i = 1; __i < __max_iter; ++__i) 00427 { 00428 __term *= __xx4 / (_Tp(__i) * (__nu + _Tp(__i))); 00429 __Jn += __term; 00430 if (std::abs(__term / __Jn) < std::numeric_limits<_Tp>::epsilon()) 00431 break; 00432 } 00433 00434 return __fact * __Jn; 00435 } 00436 00437 00438 /** 00439 * @brief Return the Bessel function of order \f$ \nu \f$: 00440 * \f$ J_{\nu}(x) \f$. 00441 * 00442 * The cylindrical Bessel function is: 00443 * @f[ 00444 * J_{\nu}(x) = \sum_{k=0}^{\infty} 00445 * \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} 00446 * @f] 00447 * 00448 * @param __nu The order of the Bessel function. 00449 * @param __x The argument of the Bessel function. 00450 * @return The output Bessel function. 00451 */ 00452 template<typename _Tp> 00453 _Tp 00454 __cyl_bessel_j(const _Tp __nu, const _Tp __x) 00455 { 00456 if (__nu < _Tp(0) || __x < _Tp(0)) 00457 std::__throw_domain_error(__N("Bad argument " 00458 "in __cyl_bessel_j.")); 00459 else if (__isnan(__nu) || __isnan(__x)) 00460 return std::numeric_limits<_Tp>::quiet_NaN(); 00461 else if (__x * __x < _Tp(10) * (__nu + _Tp(1))) 00462 return __cyl_bessel_ij_series(__nu, __x, -_Tp(1), 200); 00463 else if (__x > _Tp(1000)) 00464 { 00465 _Tp __J_nu, __N_nu; 00466 __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu); 00467 return __J_nu; 00468 } 00469 else 00470 { 00471 _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; 00472 __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); 00473 return __J_nu; 00474 } 00475 } 00476 00477 00478 /** 00479 * @brief Return the Neumann function of order \f$ \nu \f$: 00480 * \f$ N_{\nu}(x) \f$. 00481 * 00482 * The Neumann function is defined by: 00483 * @f[ 00484 * N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)} 00485 * {\sin \nu\pi} 00486 * @f] 00487 * where for integral \f$ \nu = n \f$ a limit is taken: 00488 * \f$ lim_{\nu \to n} \f$. 00489 * 00490 * @param __nu The order of the Neumann function. 00491 * @param __x The argument of the Neumann function. 00492 * @return The output Neumann function. 00493 */ 00494 template<typename _Tp> 00495 _Tp 00496 __cyl_neumann_n(const _Tp __nu, const _Tp __x) 00497 { 00498 if (__nu < _Tp(0) || __x < _Tp(0)) 00499 std::__throw_domain_error(__N("Bad argument " 00500 "in __cyl_neumann_n.")); 00501 else if (__isnan(__nu) || __isnan(__x)) 00502 return std::numeric_limits<_Tp>::quiet_NaN(); 00503 else if (__x > _Tp(1000)) 00504 { 00505 _Tp __J_nu, __N_nu; 00506 __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu); 00507 return __N_nu; 00508 } 00509 else 00510 { 00511 _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; 00512 __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); 00513 return __N_nu; 00514 } 00515 } 00516 00517 00518 /** 00519 * @brief Compute the spherical Bessel @f$ j_n(x) @f$ 00520 * and Neumann @f$ n_n(x) @f$ functions and their first 00521 * derivatives @f$ j'_n(x) @f$ and @f$ n'_n(x) @f$ 00522 * respectively. 00523 * 00524 * @param __n The order of the spherical Bessel function. 00525 * @param __x The argument of the spherical Bessel function. 00526 * @param __j_n The output spherical Bessel function. 00527 * @param __n_n The output spherical Neumann function. 00528 * @param __jp_n The output derivative of the spherical Bessel function. 00529 * @param __np_n The output derivative of the spherical Neumann function. 00530 */ 00531 template <typename _Tp> 00532 void 00533 __sph_bessel_jn(const unsigned int __n, const _Tp __x, 00534 _Tp & __j_n, _Tp & __n_n, _Tp & __jp_n, _Tp & __np_n) 00535 { 00536 const _Tp __nu = _Tp(__n) + _Tp(0.5L); 00537 00538 _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; 00539 __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); 00540 00541 const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2() 00542 / std::sqrt(__x); 00543 00544 __j_n = __factor * __J_nu; 00545 __n_n = __factor * __N_nu; 00546 __jp_n = __factor * __Jp_nu - __j_n / (_Tp(2) * __x); 00547 __np_n = __factor * __Np_nu - __n_n / (_Tp(2) * __x); 00548 00549 return; 00550 } 00551 00552 00553 /** 00554 * @brief Return the spherical Bessel function 00555 * @f$ j_n(x) @f$ of order n. 00556 * 00557 * The spherical Bessel function is defined by: 00558 * @f[ 00559 * j_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x) 00560 * @f] 00561 * 00562 * @param __n The order of the spherical Bessel function. 00563 * @param __x The argument of the spherical Bessel function. 00564 * @return The output spherical Bessel function. 00565 */ 00566 template <typename _Tp> 00567 _Tp 00568 __sph_bessel(const unsigned int __n, const _Tp __x) 00569 { 00570 if (__x < _Tp(0)) 00571 std::__throw_domain_error(__N("Bad argument " 00572 "in __sph_bessel.")); 00573 else if (__isnan(__x)) 00574 return std::numeric_limits<_Tp>::quiet_NaN(); 00575 else if (__x == _Tp(0)) 00576 { 00577 if (__n == 0) 00578 return _Tp(1); 00579 else 00580 return _Tp(0); 00581 } 00582 else 00583 { 00584 _Tp __j_n, __n_n, __jp_n, __np_n; 00585 __sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n); 00586 return __j_n; 00587 } 00588 } 00589 00590 00591 /** 00592 * @brief Return the spherical Neumann function 00593 * @f$ n_n(x) @f$. 00594 * 00595 * The spherical Neumann function is defined by: 00596 * @f[ 00597 * n_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x) 00598 * @f] 00599 * 00600 * @param __n The order of the spherical Neumann function. 00601 * @param __x The argument of the spherical Neumann function. 00602 * @return The output spherical Neumann function. 00603 */ 00604 template <typename _Tp> 00605 _Tp 00606 __sph_neumann(const unsigned int __n, const _Tp __x) 00607 { 00608 if (__x < _Tp(0)) 00609 std::__throw_domain_error(__N("Bad argument " 00610 "in __sph_neumann.")); 00611 else if (__isnan(__x)) 00612 return std::numeric_limits<_Tp>::quiet_NaN(); 00613 else if (__x == _Tp(0)) 00614 return -std::numeric_limits<_Tp>::infinity(); 00615 else 00616 { 00617 _Tp __j_n, __n_n, __jp_n, __np_n; 00618 __sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n); 00619 return __n_n; 00620 } 00621 } 00622 00623 } // namespace std::tr1::__detail 00624 } 00625 } 00626 00627 #endif // _GLIBCXX_TR1_BESSEL_FUNCTION_TCC