1 // Special functions -*- C++ -*- 2 3 // Copyright (C) 2006, 2007, 2008, 2009 4 // Free Software Foundation, Inc. 5 // 6 // This file is part of the GNU ISO C++ Library. This library is free 7 // software; you can redistribute it and/or modify it under the 8 // terms of the GNU General Public License as published by the 9 // Free Software Foundation; either version 3, or (at your option) 10 // any later version. 11 // 12 // This library is distributed in the hope that it will be useful, 13 // but WITHOUT ANY WARRANTY; without even the implied warranty of 14 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 15 // GNU General Public License for more details. 16 // 17 // Under Section 7 of GPL version 3, you are granted additional 18 // permissions described in the GCC Runtime Library Exception, version 19 // 3.1, as published by the Free Software Foundation. 20 21 // You should have received a copy of the GNU General Public License and 22 // a copy of the GCC Runtime Library Exception along with this program; 23 // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see 24 // <http://www.gnu.org/licenses/>. 25 26 /** @file tr1/bessel_function.tcc 27 * This is an internal header file, included by other library headers. 28 * You should not attempt to use it directly. 29 */ 30 31 // 32 // ISO C++ 14882 TR1: 5.2 Special functions 33 // 34 35 // Written by Edward Smith-Rowland. 36 // 37 // References: 38 // (1) Handbook of Mathematical Functions, 39 // ed. Milton Abramowitz and Irene A. Stegun, 40 // Dover Publications, 41 // Section 9, pp. 355-434, Section 10 pp. 435-478 42 // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl 43 // (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky, 44 // W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992), 45 // 2nd ed, pp. 240-245 46 47 #ifndef _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 48 #define _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 1 49 50 #include "special_function_util.h" 51 52 namespace std 53 { 54 namespace tr1 55 { 56 57 // [5.2] Special functions 58 59 // Implementation-space details. 60 namespace __detail 61 { 62 63 /** 64 * @brief Compute the gamma functions required by the Temme series 65 * expansions of @f$ N_\nu(x) @f$ and @f$ K_\nu(x) @f$. 66 * @f[ 67 * \Gamma_1 = \frac{1}{2\mu} 68 * [\frac{1}{\Gamma(1 - \mu)} - \frac{1}{\Gamma(1 + \mu)}] 69 * @f] 70 * and 71 * @f[ 72 * \Gamma_2 = \frac{1}{2} 73 * [\frac{1}{\Gamma(1 - \mu)} + \frac{1}{\Gamma(1 + \mu)}] 74 * @f] 75 * where @f$ -1/2 <= \mu <= 1/2 @f$ is @f$ \mu = \nu - N @f$ and @f$ N @f$. 76 * is the nearest integer to @f$ \nu @f$. 77 * The values of \f$ \Gamma(1 + \mu) \f$ and \f$ \Gamma(1 - \mu) \f$ 78 * are returned as well. 79 * 80 * The accuracy requirements on this are exquisite. 81 * 82 * @param __mu The input parameter of the gamma functions. 83 * @param __gam1 The output function \f$ \Gamma_1(\mu) \f$ 84 * @param __gam2 The output function \f$ \Gamma_2(\mu) \f$ 85 * @param __gampl The output function \f$ \Gamma(1 + \mu) \f$ 86 * @param __gammi The output function \f$ \Gamma(1 - \mu) \f$ 87 */ 88 template <typename _Tp> 89 void 90 __gamma_temme(const _Tp __mu, 91 _Tp & __gam1, _Tp & __gam2, _Tp & __gampl, _Tp & __gammi) 92 { 93 #if _GLIBCXX_USE_C99_MATH_TR1 94 __gampl = _Tp(1) / std::tr1::tgamma(_Tp(1) + __mu); 95 __gammi = _Tp(1) / std::tr1::tgamma(_Tp(1) - __mu); 96 #else 97 __gampl = _Tp(1) / __gamma(_Tp(1) + __mu); 98 __gammi = _Tp(1) / __gamma(_Tp(1) - __mu); 99 #endif 100 101 if (std::abs(__mu) < std::numeric_limits<_Tp>::epsilon()) 102 __gam1 = -_Tp(__numeric_constants<_Tp>::__gamma_e()); 103 else 104 __gam1 = (__gammi - __gampl) / (_Tp(2) * __mu); 105 106 __gam2 = (__gammi + __gampl) / (_Tp(2)); 107 108 return; 109 } 110 111 112 /** 113 * @brief Compute the Bessel @f$ J_\nu(x) @f$ and Neumann 114 * @f$ N_\nu(x) @f$ functions and their first derivatives 115 * @f$ J'_\nu(x) @f$ and @f$ N'_\nu(x) @f$ respectively. 116 * These four functions are computed together for numerical 117 * stability. 118 * 119 * @param __nu The order of the Bessel functions. 120 * @param __x The argument of the Bessel functions. 121 * @param __Jnu The output Bessel function of the first kind. 122 * @param __Nnu The output Neumann function (Bessel function of the second kind). 123 * @param __Jpnu The output derivative of the Bessel function of the first kind. 124 * @param __Npnu The output derivative of the Neumann function. 125 */ 126 template <typename _Tp> 127 void 128 __bessel_jn(const _Tp __nu, const _Tp __x, 129 _Tp & __Jnu, _Tp & __Nnu, _Tp & __Jpnu, _Tp & __Npnu) 130 { 131 if (__x == _Tp(0)) 132 { 133 if (__nu == _Tp(0)) 134 { 135 __Jnu = _Tp(1); 136 __Jpnu = _Tp(0); 137 } 138 else if (__nu == _Tp(1)) 139 { 140 __Jnu = _Tp(0); 141 __Jpnu = _Tp(0.5L); 142 } 143 else 144 { 145 __Jnu = _Tp(0); 146 __Jpnu = _Tp(0); 147 } 148 __Nnu = -std::numeric_limits<_Tp>::infinity(); 149 __Npnu = std::numeric_limits<_Tp>::infinity(); 150 return; 151 } 152 153 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 154 // When the multiplier is N i.e. 155 // fp_min = N * min() 156 // Then J_0 and N_0 tank at x = 8 * N (J_0 = 0 and N_0 = nan)! 157 //const _Tp __fp_min = _Tp(20) * std::numeric_limits<_Tp>::min(); 158 const _Tp __fp_min = std::sqrt(std::numeric_limits<_Tp>::min()); 159 const int __max_iter = 15000; 160 const _Tp __x_min = _Tp(2); 161 162 const int __nl = (__x < __x_min 163 ? static_cast<int>(__nu + _Tp(0.5L)) 164 : std::max(0, static_cast<int>(__nu - __x + _Tp(1.5L)))); 165 166 const _Tp __mu = __nu - __nl; 167 const _Tp __mu2 = __mu * __mu; 168 const _Tp __xi = _Tp(1) / __x; 169 const _Tp __xi2 = _Tp(2) * __xi; 170 _Tp __w = __xi2 / __numeric_constants<_Tp>::__pi(); 171 int __isign = 1; 172 _Tp __h = __nu * __xi; 173 if (__h < __fp_min) 174 __h = __fp_min; 175 _Tp __b = __xi2 * __nu; 176 _Tp __d = _Tp(0); 177 _Tp __c = __h; 178 int __i; 179 for (__i = 1; __i <= __max_iter; ++__i) 180 { 181 __b += __xi2; 182 __d = __b - __d; 183 if (std::abs(__d) < __fp_min) 184 __d = __fp_min; 185 __c = __b - _Tp(1) / __c; 186 if (std::abs(__c) < __fp_min) 187 __c = __fp_min; 188 __d = _Tp(1) / __d; 189 const _Tp __del = __c * __d; 190 __h *= __del; 191 if (__d < _Tp(0)) 192 __isign = -__isign; 193 if (std::abs(__del - _Tp(1)) < __eps) 194 break; 195 } 196 if (__i > __max_iter) 197 std::__throw_runtime_error(__N("Argument x too large in __bessel_jn; " 198 "try asymptotic expansion.")); 199 _Tp __Jnul = __isign * __fp_min; 200 _Tp __Jpnul = __h * __Jnul; 201 _Tp __Jnul1 = __Jnul; 202 _Tp __Jpnu1 = __Jpnul; 203 _Tp __fact = __nu * __xi; 204 for ( int __l = __nl; __l >= 1; --__l ) 205 { 206 const _Tp __Jnutemp = __fact * __Jnul + __Jpnul; 207 __fact -= __xi; 208 __Jpnul = __fact * __Jnutemp - __Jnul; 209 __Jnul = __Jnutemp; 210 } 211 if (__Jnul == _Tp(0)) 212 __Jnul = __eps; 213 _Tp __f= __Jpnul / __Jnul; 214 _Tp __Nmu, __Nnu1, __Npmu, __Jmu; 215 if (__x < __x_min) 216 { 217 const _Tp __x2 = __x / _Tp(2); 218 const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu; 219 _Tp __fact = (std::abs(__pimu) < __eps 220 ? _Tp(1) : __pimu / std::sin(__pimu)); 221 _Tp __d = -std::log(__x2); 222 _Tp __e = __mu * __d; 223 _Tp __fact2 = (std::abs(__e) < __eps 224 ? _Tp(1) : std::sinh(__e) / __e); 225 _Tp __gam1, __gam2, __gampl, __gammi; 226 __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi); 227 _Tp __ff = (_Tp(2) / __numeric_constants<_Tp>::__pi()) 228 * __fact * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d); 229 __e = std::exp(__e); 230 _Tp __p = __e / (__numeric_constants<_Tp>::__pi() * __gampl); 231 _Tp __q = _Tp(1) / (__e * __numeric_constants<_Tp>::__pi() * __gammi); 232 const _Tp __pimu2 = __pimu / _Tp(2); 233 _Tp __fact3 = (std::abs(__pimu2) < __eps 234 ? _Tp(1) : std::sin(__pimu2) / __pimu2 ); 235 _Tp __r = __numeric_constants<_Tp>::__pi() * __pimu2 * __fact3 * __fact3; 236 _Tp __c = _Tp(1); 237 __d = -__x2 * __x2; 238 _Tp __sum = __ff + __r * __q; 239 _Tp __sum1 = __p; 240 for (__i = 1; __i <= __max_iter; ++__i) 241 { 242 __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2); 243 __c *= __d / _Tp(__i); 244 __p /= _Tp(__i) - __mu; 245 __q /= _Tp(__i) + __mu; 246 const _Tp __del = __c * (__ff + __r * __q); 247 __sum += __del; 248 const _Tp __del1 = __c * __p - __i * __del; 249 __sum1 += __del1; 250 if ( std::abs(__del) < __eps * (_Tp(1) + std::abs(__sum)) ) 251 break; 252 } 253 if ( __i > __max_iter ) 254 std::__throw_runtime_error(__N("Bessel y series failed to converge " 255 "in __bessel_jn.")); 256 __Nmu = -__sum; 257 __Nnu1 = -__sum1 * __xi2; 258 __Npmu = __mu * __xi * __Nmu - __Nnu1; 259 __Jmu = __w / (__Npmu - __f * __Nmu); 260 } 261 else 262 { 263 _Tp __a = _Tp(0.25L) - __mu2; 264 _Tp __q = _Tp(1); 265 _Tp __p = -__xi / _Tp(2); 266 _Tp __br = _Tp(2) * __x; 267 _Tp __bi = _Tp(2); 268 _Tp __fact = __a * __xi / (__p * __p + __q * __q); 269 _Tp __cr = __br + __q * __fact; 270 _Tp __ci = __bi + __p * __fact; 271 _Tp __den = __br * __br + __bi * __bi; 272 _Tp __dr = __br / __den; 273 _Tp __di = -__bi / __den; 274 _Tp __dlr = __cr * __dr - __ci * __di; 275 _Tp __dli = __cr * __di + __ci * __dr; 276 _Tp __temp = __p * __dlr - __q * __dli; 277 __q = __p * __dli + __q * __dlr; 278 __p = __temp; 279 int __i; 280 for (__i = 2; __i <= __max_iter; ++__i) 281 { 282 __a += _Tp(2 * (__i - 1)); 283 __bi += _Tp(2); 284 __dr = __a * __dr + __br; 285 __di = __a * __di + __bi; 286 if (std::abs(__dr) + std::abs(__di) < __fp_min) 287 __dr = __fp_min; 288 __fact = __a / (__cr * __cr + __ci * __ci); 289 __cr = __br + __cr * __fact; 290 __ci = __bi - __ci * __fact; 291 if (std::abs(__cr) + std::abs(__ci) < __fp_min) 292 __cr = __fp_min; 293 __den = __dr * __dr + __di * __di; 294 __dr /= __den; 295 __di /= -__den; 296 __dlr = __cr * __dr - __ci * __di; 297 __dli = __cr * __di + __ci * __dr; 298 __temp = __p * __dlr - __q * __dli; 299 __q = __p * __dli + __q * __dlr; 300 __p = __temp; 301 if (std::abs(__dlr - _Tp(1)) + std::abs(__dli) < __eps) 302 break; 303 } 304 if (__i > __max_iter) 305 std::__throw_runtime_error(__N("Lentz's method failed " 306 "in __bessel_jn.")); 307 const _Tp __gam = (__p - __f) / __q; 308 __Jmu = std::sqrt(__w / ((__p - __f) * __gam + __q)); 309 #if _GLIBCXX_USE_C99_MATH_TR1 310 __Jmu = std::tr1::copysign(__Jmu, __Jnul); 311 #else 312 if (__Jmu * __Jnul < _Tp(0)) 313 __Jmu = -__Jmu; 314 #endif 315 __Nmu = __gam * __Jmu; 316 __Npmu = (__p + __q / __gam) * __Nmu; 317 __Nnu1 = __mu * __xi * __Nmu - __Npmu; 318 } 319 __fact = __Jmu / __Jnul; 320 __Jnu = __fact * __Jnul1; 321 __Jpnu = __fact * __Jpnu1; 322 for (__i = 1; __i <= __nl; ++__i) 323 { 324 const _Tp __Nnutemp = (__mu + __i) * __xi2 * __Nnu1 - __Nmu; 325 __Nmu = __Nnu1; 326 __Nnu1 = __Nnutemp; 327 } 328 __Nnu = __Nmu; 329 __Npnu = __nu * __xi * __Nmu - __Nnu1; 330 331 return; 332 } 333 334 335 /** 336 * @brief This routine computes the asymptotic cylindrical Bessel 337 * and Neumann functions of order nu: \f$ J_{\nu} \f$, 338 * \f$ N_{\nu} \f$. 339 * 340 * References: 341 * (1) Handbook of Mathematical Functions, 342 * ed. Milton Abramowitz and Irene A. Stegun, 343 * Dover Publications, 344 * Section 9 p. 364, Equations 9.2.5-9.2.10 345 * 346 * @param __nu The order of the Bessel functions. 347 * @param __x The argument of the Bessel functions. 348 * @param __Jnu The output Bessel function of the first kind. 349 * @param __Nnu The output Neumann function (Bessel function of the second kind). 350 */ 351 template <typename _Tp> 352 void 353 __cyl_bessel_jn_asymp(const _Tp __nu, const _Tp __x, 354 _Tp & __Jnu, _Tp & __Nnu) 355 { 356 const _Tp __coef = std::sqrt(_Tp(2) 357 / (__numeric_constants<_Tp>::__pi() * __x)); 358 const _Tp __mu = _Tp(4) * __nu * __nu; 359 const _Tp __mum1 = __mu - _Tp(1); 360 const _Tp __mum9 = __mu - _Tp(9); 361 const _Tp __mum25 = __mu - _Tp(25); 362 const _Tp __mum49 = __mu - _Tp(49); 363 const _Tp __xx = _Tp(64) * __x * __x; 364 const _Tp __P = _Tp(1) - __mum1 * __mum9 / (_Tp(2) * __xx) 365 * (_Tp(1) - __mum25 * __mum49 / (_Tp(12) * __xx)); 366 const _Tp __Q = __mum1 / (_Tp(8) * __x) 367 * (_Tp(1) - __mum9 * __mum25 / (_Tp(6) * __xx)); 368 369 const _Tp __chi = __x - (__nu + _Tp(0.5L)) 370 * __numeric_constants<_Tp>::__pi_2(); 371 const _Tp __c = std::cos(__chi); 372 const _Tp __s = std::sin(__chi); 373 374 __Jnu = __coef * (__c * __P - __s * __Q); 375 __Nnu = __coef * (__s * __P + __c * __Q); 376 377 return; 378 } 379 380 381 /** 382 * @brief This routine returns the cylindrical Bessel functions 383 * of order \f$ \nu \f$: \f$ J_{\nu} \f$ or \f$ I_{\nu} \f$ 384 * by series expansion. 385 * 386 * The modified cylindrical Bessel function is: 387 * @f[ 388 * Z_{\nu}(x) = \sum_{k=0}^{\infty} 389 * \frac{\sigma^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} 390 * @f] 391 * where \f$ \sigma = +1 \f$ or\f$ -1 \f$ for 392 * \f$ Z = I \f$ or \f$ J \f$ respectively. 393 * 394 * See Abramowitz & Stegun, 9.1.10 395 * Abramowitz & Stegun, 9.6.7 396 * (1) Handbook of Mathematical Functions, 397 * ed. Milton Abramowitz and Irene A. Stegun, 398 * Dover Publications, 399 * Equation 9.1.10 p. 360 and Equation 9.6.10 p. 375 400 * 401 * @param __nu The order of the Bessel function. 402 * @param __x The argument of the Bessel function. 403 * @param __sgn The sign of the alternate terms 404 * -1 for the Bessel function of the first kind. 405 * +1 for the modified Bessel function of the first kind. 406 * @return The output Bessel function. 407 */ 408 template <typename _Tp> 409 _Tp 410 __cyl_bessel_ij_series(const _Tp __nu, const _Tp __x, const _Tp __sgn, 411 const unsigned int __max_iter) 412 { 413 414 const _Tp __x2 = __x / _Tp(2); 415 _Tp __fact = __nu * std::log(__x2); 416 #if _GLIBCXX_USE_C99_MATH_TR1 417 __fact -= std::tr1::lgamma(__nu + _Tp(1)); 418 #else 419 __fact -= __log_gamma(__nu + _Tp(1)); 420 #endif 421 __fact = std::exp(__fact); 422 const _Tp __xx4 = __sgn * __x2 * __x2; 423 _Tp __Jn = _Tp(1); 424 _Tp __term = _Tp(1); 425 426 for (unsigned int __i = 1; __i < __max_iter; ++__i) 427 { 428 __term *= __xx4 / (_Tp(__i) * (__nu + _Tp(__i))); 429 __Jn += __term; 430 if (std::abs(__term / __Jn) < std::numeric_limits<_Tp>::epsilon()) 431 break; 432 } 433 434 return __fact * __Jn; 435 } 436 437 438 /** 439 * @brief Return the Bessel function of order \f$ \nu \f$: 440 * \f$ J_{\nu}(x) \f$. 441 * 442 * The cylindrical Bessel function is: 443 * @f[ 444 * J_{\nu}(x) = \sum_{k=0}^{\infty} 445 * \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} 446 * @f] 447 * 448 * @param __nu The order of the Bessel function. 449 * @param __x The argument of the Bessel function. 450 * @return The output Bessel function. 451 */ 452 template<typename _Tp> 453 _Tp 454 __cyl_bessel_j(const _Tp __nu, const _Tp __x) 455 { 456 if (__nu < _Tp(0) || __x < _Tp(0)) 457 std::__throw_domain_error(__N("Bad argument " 458 "in __cyl_bessel_j.")); 459 else if (__isnan(__nu) || __isnan(__x)) 460 return std::numeric_limits<_Tp>::quiet_NaN(); 461 else if (__x * __x < _Tp(10) * (__nu + _Tp(1))) 462 return __cyl_bessel_ij_series(__nu, __x, -_Tp(1), 200); 463 else if (__x > _Tp(1000)) 464 { 465 _Tp __J_nu, __N_nu; 466 __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu); 467 return __J_nu; 468 } 469 else 470 { 471 _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; 472 __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); 473 return __J_nu; 474 } 475 } 476 477 478 /** 479 * @brief Return the Neumann function of order \f$ \nu \f$: 480 * \f$ N_{\nu}(x) \f$. 481 * 482 * The Neumann function is defined by: 483 * @f[ 484 * N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)} 485 * {\sin \nu\pi} 486 * @f] 487 * where for integral \f$ \nu = n \f$ a limit is taken: 488 * \f$ lim_{\nu \to n} \f$. 489 * 490 * @param __nu The order of the Neumann function. 491 * @param __x The argument of the Neumann function. 492 * @return The output Neumann function. 493 */ 494 template<typename _Tp> 495 _Tp 496 __cyl_neumann_n(const _Tp __nu, const _Tp __x) 497 { 498 if (__nu < _Tp(0) || __x < _Tp(0)) 499 std::__throw_domain_error(__N("Bad argument " 500 "in __cyl_neumann_n.")); 501 else if (__isnan(__nu) || __isnan(__x)) 502 return std::numeric_limits<_Tp>::quiet_NaN(); 503 else if (__x > _Tp(1000)) 504 { 505 _Tp __J_nu, __N_nu; 506 __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu); 507 return __N_nu; 508 } 509 else 510 { 511 _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; 512 __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); 513 return __N_nu; 514 } 515 } 516 517 518 /** 519 * @brief Compute the spherical Bessel @f$ j_n(x) @f$ 520 * and Neumann @f$ n_n(x) @f$ functions and their first 521 * derivatives @f$ j'_n(x) @f$ and @f$ n'_n(x) @f$ 522 * respectively. 523 * 524 * @param __n The order of the spherical Bessel function. 525 * @param __x The argument of the spherical Bessel function. 526 * @param __j_n The output spherical Bessel function. 527 * @param __n_n The output spherical Neumann function. 528 * @param __jp_n The output derivative of the spherical Bessel function. 529 * @param __np_n The output derivative of the spherical Neumann function. 530 */ 531 template <typename _Tp> 532 void 533 __sph_bessel_jn(const unsigned int __n, const _Tp __x, 534 _Tp & __j_n, _Tp & __n_n, _Tp & __jp_n, _Tp & __np_n) 535 { 536 const _Tp __nu = _Tp(__n) + _Tp(0.5L); 537 538 _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; 539 __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); 540 541 const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2() 542 / std::sqrt(__x); 543 544 __j_n = __factor * __J_nu; 545 __n_n = __factor * __N_nu; 546 __jp_n = __factor * __Jp_nu - __j_n / (_Tp(2) * __x); 547 __np_n = __factor * __Np_nu - __n_n / (_Tp(2) * __x); 548 549 return; 550 } 551 552 553 /** 554 * @brief Return the spherical Bessel function 555 * @f$ j_n(x) @f$ of order n. 556 * 557 * The spherical Bessel function is defined by: 558 * @f[ 559 * j_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x) 560 * @f] 561 * 562 * @param __n The order of the spherical Bessel function. 563 * @param __x The argument of the spherical Bessel function. 564 * @return The output spherical Bessel function. 565 */ 566 template <typename _Tp> 567 _Tp 568 __sph_bessel(const unsigned int __n, const _Tp __x) 569 { 570 if (__x < _Tp(0)) 571 std::__throw_domain_error(__N("Bad argument " 572 "in __sph_bessel.")); 573 else if (__isnan(__x)) 574 return std::numeric_limits<_Tp>::quiet_NaN(); 575 else if (__x == _Tp(0)) 576 { 577 if (__n == 0) 578 return _Tp(1); 579 else 580 return _Tp(0); 581 } 582 else 583 { 584 _Tp __j_n, __n_n, __jp_n, __np_n; 585 __sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n); 586 return __j_n; 587 } 588 } 589 590 591 /** 592 * @brief Return the spherical Neumann function 593 * @f$ n_n(x) @f$. 594 * 595 * The spherical Neumann function is defined by: 596 * @f[ 597 * n_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x) 598 * @f] 599 * 600 * @param __n The order of the spherical Neumann function. 601 * @param __x The argument of the spherical Neumann function. 602 * @return The output spherical Neumann function. 603 */ 604 template <typename _Tp> 605 _Tp 606 __sph_neumann(const unsigned int __n, const _Tp __x) 607 { 608 if (__x < _Tp(0)) 609 std::__throw_domain_error(__N("Bad argument " 610 "in __sph_neumann.")); 611 else if (__isnan(__x)) 612 return std::numeric_limits<_Tp>::quiet_NaN(); 613 else if (__x == _Tp(0)) 614 return -std::numeric_limits<_Tp>::infinity(); 615 else 616 { 617 _Tp __j_n, __n_n, __jp_n, __np_n; 618 __sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n); 619 return __n_n; 620 } 621 } 622 623 } // namespace std::tr1::__detail 624 } 625 } 626 627 #endif // _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 628
