1 // Special functions -*- C++ -*- 2 3 // Copyright (C) 2006-2013 Free Software Foundation, Inc. 4 // 5 // This file is part of the GNU ISO C++ Library. This library is free 6 // software; you can redistribute it and/or modify it under the 7 // terms of the GNU General Public License as published by the 8 // Free Software Foundation; either version 3, or (at your option) 9 // any later version. 10 // 11 // This library is distributed in the hope that it will be useful, 12 // but WITHOUT ANY WARRANTY; without even the implied warranty of 13 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 14 // GNU General Public License for more details. 15 // 16 // Under Section 7 of GPL version 3, you are granted additional 17 // permissions described in the GCC Runtime Library Exception, version 18 // 3.1, as published by the Free Software Foundation. 19 20 // You should have received a copy of the GNU General Public License and 21 // a copy of the GCC Runtime Library Exception along with this program; 22 // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see 23 // <http://www.gnu.org/licenses/>. 24 25 /** @file tr1/bessel_function.tcc 26 * This is an internal header file, included by other library headers. 27 * Do not attempt to use it directly. @headername{tr1/cmath} 28 */ 29 30 // 31 // ISO C++ 14882 TR1: 5.2 Special functions 32 // 33 34 // Written by Edward Smith-Rowland. 35 // 36 // References: 37 // (1) Handbook of Mathematical Functions, 38 // ed. Milton Abramowitz and Irene A. Stegun, 39 // Dover Publications, 40 // Section 9, pp. 355-434, Section 10 pp. 435-478 41 // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl 42 // (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky, 43 // W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992), 44 // 2nd ed, pp. 240-245 45 46 #ifndef _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 47 #define _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 1 48 49 #include "special_function_util.h" 50 51 namespace std _GLIBCXX_VISIBILITY(default) 52 { 53 namespace tr1 54 { 55 // [5.2] Special functions 56 57 // Implementation-space details. 58 namespace __detail 59 { 60 _GLIBCXX_BEGIN_NAMESPACE_VERSION 61 62 /** 63 * @brief Compute the gamma functions required by the Temme series 64 * expansions of @f$ N_\nu(x) @f$ and @f$ K_\nu(x) @f$. 65 * @f[ 66 * \Gamma_1 = \frac{1}{2\mu} 67 * [\frac{1}{\Gamma(1 - \mu)} - \frac{1}{\Gamma(1 + \mu)}] 68 * @f] 69 * and 70 * @f[ 71 * \Gamma_2 = \frac{1}{2} 72 * [\frac{1}{\Gamma(1 - \mu)} + \frac{1}{\Gamma(1 + \mu)}] 73 * @f] 74 * where @f$ -1/2 <= \mu <= 1/2 @f$ is @f$ \mu = \nu - N @f$ and @f$ N @f$. 75 * is the nearest integer to @f$ \nu @f$. 76 * The values of \f$ \Gamma(1 + \mu) \f$ and \f$ \Gamma(1 - \mu) \f$ 77 * are returned as well. 78 * 79 * The accuracy requirements on this are exquisite. 80 * 81 * @param __mu The input parameter of the gamma functions. 82 * @param __gam1 The output function \f$ \Gamma_1(\mu) \f$ 83 * @param __gam2 The output function \f$ \Gamma_2(\mu) \f$ 84 * @param __gampl The output function \f$ \Gamma(1 + \mu) \f$ 85 * @param __gammi The output function \f$ \Gamma(1 - \mu) \f$ 86 */ 87 template <typename _Tp> 88 void 89 __gamma_temme(_Tp __mu, 90 _Tp & __gam1, _Tp & __gam2, _Tp & __gampl, _Tp & __gammi) 91 { 92 #if _GLIBCXX_USE_C99_MATH_TR1 93 __gampl = _Tp(1) / std::tr1::tgamma(_Tp(1) + __mu); 94 __gammi = _Tp(1) / std::tr1::tgamma(_Tp(1) - __mu); 95 #else 96 __gampl = _Tp(1) / __gamma(_Tp(1) + __mu); 97 __gammi = _Tp(1) / __gamma(_Tp(1) - __mu); 98 #endif 99 100 if (std::abs(__mu) < std::numeric_limits<_Tp>::epsilon()) 101 __gam1 = -_Tp(__numeric_constants<_Tp>::__gamma_e()); 102 else 103 __gam1 = (__gammi - __gampl) / (_Tp(2) * __mu); 104 105 __gam2 = (__gammi + __gampl) / (_Tp(2)); 106 107 return; 108 } 109 110 111 /** 112 * @brief Compute the Bessel @f$ J_\nu(x) @f$ and Neumann 113 * @f$ N_\nu(x) @f$ functions and their first derivatives 114 * @f$ J'_\nu(x) @f$ and @f$ N'_\nu(x) @f$ respectively. 115 * These four functions are computed together for numerical 116 * stability. 117 * 118 * @param __nu The order of the Bessel functions. 119 * @param __x The argument of the Bessel functions. 120 * @param __Jnu The output Bessel function of the first kind. 121 * @param __Nnu The output Neumann function (Bessel function of the second kind). 122 * @param __Jpnu The output derivative of the Bessel function of the first kind. 123 * @param __Npnu The output derivative of the Neumann function. 124 */ 125 template <typename _Tp> 126 void 127 __bessel_jn(_Tp __nu, _Tp __x, 128 _Tp & __Jnu, _Tp & __Nnu, _Tp & __Jpnu, _Tp & __Npnu) 129 { 130 if (__x == _Tp(0)) 131 { 132 if (__nu == _Tp(0)) 133 { 134 __Jnu = _Tp(1); 135 __Jpnu = _Tp(0); 136 } 137 else if (__nu == _Tp(1)) 138 { 139 __Jnu = _Tp(0); 140 __Jpnu = _Tp(0.5L); 141 } 142 else 143 { 144 __Jnu = _Tp(0); 145 __Jpnu = _Tp(0); 146 } 147 __Nnu = -std::numeric_limits<_Tp>::infinity(); 148 __Npnu = std::numeric_limits<_Tp>::infinity(); 149 return; 150 } 151 152 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 153 // When the multiplier is N i.e. 154 // fp_min = N * min() 155 // Then J_0 and N_0 tank at x = 8 * N (J_0 = 0 and N_0 = nan)! 156 //const _Tp __fp_min = _Tp(20) * std::numeric_limits<_Tp>::min(); 157 const _Tp __fp_min = std::sqrt(std::numeric_limits<_Tp>::min()); 158 const int __max_iter = 15000; 159 const _Tp __x_min = _Tp(2); 160 161 const int __nl = (__x < __x_min 162 ? static_cast<int>(__nu + _Tp(0.5L)) 163 : std::max(0, static_cast<int>(__nu - __x + _Tp(1.5L)))); 164 165 const _Tp __mu = __nu - __nl; 166 const _Tp __mu2 = __mu * __mu; 167 const _Tp __xi = _Tp(1) / __x; 168 const _Tp __xi2 = _Tp(2) * __xi; 169 _Tp __w = __xi2 / __numeric_constants<_Tp>::__pi(); 170 int __isign = 1; 171 _Tp __h = __nu * __xi; 172 if (__h < __fp_min) 173 __h = __fp_min; 174 _Tp __b = __xi2 * __nu; 175 _Tp __d = _Tp(0); 176 _Tp __c = __h; 177 int __i; 178 for (__i = 1; __i <= __max_iter; ++__i) 179 { 180 __b += __xi2; 181 __d = __b - __d; 182 if (std::abs(__d) < __fp_min) 183 __d = __fp_min; 184 __c = __b - _Tp(1) / __c; 185 if (std::abs(__c) < __fp_min) 186 __c = __fp_min; 187 __d = _Tp(1) / __d; 188 const _Tp __del = __c * __d; 189 __h *= __del; 190 if (__d < _Tp(0)) 191 __isign = -__isign; 192 if (std::abs(__del - _Tp(1)) < __eps) 193 break; 194 } 195 if (__i > __max_iter) 196 std::__throw_runtime_error(__N("Argument x too large in __bessel_jn; " 197 "try asymptotic expansion.")); 198 _Tp __Jnul = __isign * __fp_min; 199 _Tp __Jpnul = __h * __Jnul; 200 _Tp __Jnul1 = __Jnul; 201 _Tp __Jpnu1 = __Jpnul; 202 _Tp __fact = __nu * __xi; 203 for ( int __l = __nl; __l >= 1; --__l ) 204 { 205 const _Tp __Jnutemp = __fact * __Jnul + __Jpnul; 206 __fact -= __xi; 207 __Jpnul = __fact * __Jnutemp - __Jnul; 208 __Jnul = __Jnutemp; 209 } 210 if (__Jnul == _Tp(0)) 211 __Jnul = __eps; 212 _Tp __f= __Jpnul / __Jnul; 213 _Tp __Nmu, __Nnu1, __Npmu, __Jmu; 214 if (__x < __x_min) 215 { 216 const _Tp __x2 = __x / _Tp(2); 217 const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu; 218 _Tp __fact = (std::abs(__pimu) < __eps 219 ? _Tp(1) : __pimu / std::sin(__pimu)); 220 _Tp __d = -std::log(__x2); 221 _Tp __e = __mu * __d; 222 _Tp __fact2 = (std::abs(__e) < __eps 223 ? _Tp(1) : std::sinh(__e) / __e); 224 _Tp __gam1, __gam2, __gampl, __gammi; 225 __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi); 226 _Tp __ff = (_Tp(2) / __numeric_constants<_Tp>::__pi()) 227 * __fact * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d); 228 __e = std::exp(__e); 229 _Tp __p = __e / (__numeric_constants<_Tp>::__pi() * __gampl); 230 _Tp __q = _Tp(1) / (__e * __numeric_constants<_Tp>::__pi() * __gammi); 231 const _Tp __pimu2 = __pimu / _Tp(2); 232 _Tp __fact3 = (std::abs(__pimu2) < __eps 233 ? _Tp(1) : std::sin(__pimu2) / __pimu2 ); 234 _Tp __r = __numeric_constants<_Tp>::__pi() * __pimu2 * __fact3 * __fact3; 235 _Tp __c = _Tp(1); 236 __d = -__x2 * __x2; 237 _Tp __sum = __ff + __r * __q; 238 _Tp __sum1 = __p; 239 for (__i = 1; __i <= __max_iter; ++__i) 240 { 241 __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2); 242 __c *= __d / _Tp(__i); 243 __p /= _Tp(__i) - __mu; 244 __q /= _Tp(__i) + __mu; 245 const _Tp __del = __c * (__ff + __r * __q); 246 __sum += __del; 247 const _Tp __del1 = __c * __p - __i * __del; 248 __sum1 += __del1; 249 if ( std::abs(__del) < __eps * (_Tp(1) + std::abs(__sum)) ) 250 break; 251 } 252 if ( __i > __max_iter ) 253 std::__throw_runtime_error(__N("Bessel y series failed to converge " 254 "in __bessel_jn.")); 255 __Nmu = -__sum; 256 __Nnu1 = -__sum1 * __xi2; 257 __Npmu = __mu * __xi * __Nmu - __Nnu1; 258 __Jmu = __w / (__Npmu - __f * __Nmu); 259 } 260 else 261 { 262 _Tp __a = _Tp(0.25L) - __mu2; 263 _Tp __q = _Tp(1); 264 _Tp __p = -__xi / _Tp(2); 265 _Tp __br = _Tp(2) * __x; 266 _Tp __bi = _Tp(2); 267 _Tp __fact = __a * __xi / (__p * __p + __q * __q); 268 _Tp __cr = __br + __q * __fact; 269 _Tp __ci = __bi + __p * __fact; 270 _Tp __den = __br * __br + __bi * __bi; 271 _Tp __dr = __br / __den; 272 _Tp __di = -__bi / __den; 273 _Tp __dlr = __cr * __dr - __ci * __di; 274 _Tp __dli = __cr * __di + __ci * __dr; 275 _Tp __temp = __p * __dlr - __q * __dli; 276 __q = __p * __dli + __q * __dlr; 277 __p = __temp; 278 int __i; 279 for (__i = 2; __i <= __max_iter; ++__i) 280 { 281 __a += _Tp(2 * (__i - 1)); 282 __bi += _Tp(2); 283 __dr = __a * __dr + __br; 284 __di = __a * __di + __bi; 285 if (std::abs(__dr) + std::abs(__di) < __fp_min) 286 __dr = __fp_min; 287 __fact = __a / (__cr * __cr + __ci * __ci); 288 __cr = __br + __cr * __fact; 289 __ci = __bi - __ci * __fact; 290 if (std::abs(__cr) + std::abs(__ci) < __fp_min) 291 __cr = __fp_min; 292 __den = __dr * __dr + __di * __di; 293 __dr /= __den; 294 __di /= -__den; 295 __dlr = __cr * __dr - __ci * __di; 296 __dli = __cr * __di + __ci * __dr; 297 __temp = __p * __dlr - __q * __dli; 298 __q = __p * __dli + __q * __dlr; 299 __p = __temp; 300 if (std::abs(__dlr - _Tp(1)) + std::abs(__dli) < __eps) 301 break; 302 } 303 if (__i > __max_iter) 304 std::__throw_runtime_error(__N("Lentz's method failed " 305 "in __bessel_jn.")); 306 const _Tp __gam = (__p - __f) / __q; 307 __Jmu = std::sqrt(__w / ((__p - __f) * __gam + __q)); 308 #if _GLIBCXX_USE_C99_MATH_TR1 309 __Jmu = std::tr1::copysign(__Jmu, __Jnul); 310 #else 311 if (__Jmu * __Jnul < _Tp(0)) 312 __Jmu = -__Jmu; 313 #endif 314 __Nmu = __gam * __Jmu; 315 __Npmu = (__p + __q / __gam) * __Nmu; 316 __Nnu1 = __mu * __xi * __Nmu - __Npmu; 317 } 318 __fact = __Jmu / __Jnul; 319 __Jnu = __fact * __Jnul1; 320 __Jpnu = __fact * __Jpnu1; 321 for (__i = 1; __i <= __nl; ++__i) 322 { 323 const _Tp __Nnutemp = (__mu + __i) * __xi2 * __Nnu1 - __Nmu; 324 __Nmu = __Nnu1; 325 __Nnu1 = __Nnutemp; 326 } 327 __Nnu = __Nmu; 328 __Npnu = __nu * __xi * __Nmu - __Nnu1; 329 330 return; 331 } 332 333 334 /** 335 * @brief This routine computes the asymptotic cylindrical Bessel 336 * and Neumann functions of order nu: \f$ J_{\nu} \f$, 337 * \f$ N_{\nu} \f$. 338 * 339 * References: 340 * (1) Handbook of Mathematical Functions, 341 * ed. Milton Abramowitz and Irene A. Stegun, 342 * Dover Publications, 343 * Section 9 p. 364, Equations 9.2.5-9.2.10 344 * 345 * @param __nu The order of the Bessel functions. 346 * @param __x The argument of the Bessel functions. 347 * @param __Jnu The output Bessel function of the first kind. 348 * @param __Nnu The output Neumann function (Bessel function of the second kind). 349 */ 350 template <typename _Tp> 351 void 352 __cyl_bessel_jn_asymp(_Tp __nu, _Tp __x, _Tp & __Jnu, _Tp & __Nnu) 353 { 354 const _Tp __mu = _Tp(4) * __nu * __nu; 355 const _Tp __mum1 = __mu - _Tp(1); 356 const _Tp __mum9 = __mu - _Tp(9); 357 const _Tp __mum25 = __mu - _Tp(25); 358 const _Tp __mum49 = __mu - _Tp(49); 359 const _Tp __xx = _Tp(64) * __x * __x; 360 const _Tp __P = _Tp(1) - __mum1 * __mum9 / (_Tp(2) * __xx) 361 * (_Tp(1) - __mum25 * __mum49 / (_Tp(12) * __xx)); 362 const _Tp __Q = __mum1 / (_Tp(8) * __x) 363 * (_Tp(1) - __mum9 * __mum25 / (_Tp(6) * __xx)); 364 365 const _Tp __chi = __x - (__nu + _Tp(0.5L)) 366 * __numeric_constants<_Tp>::__pi_2(); 367 const _Tp __c = std::cos(__chi); 368 const _Tp __s = std::sin(__chi); 369 370 const _Tp __coef = std::sqrt(_Tp(2) 371 / (__numeric_constants<_Tp>::__pi() * __x)); 372 __Jnu = __coef * (__c * __P - __s * __Q); 373 __Nnu = __coef * (__s * __P + __c * __Q); 374 375 return; 376 } 377 378 379 /** 380 * @brief This routine returns the cylindrical Bessel functions 381 * of order \f$ \nu \f$: \f$ J_{\nu} \f$ or \f$ I_{\nu} \f$ 382 * by series expansion. 383 * 384 * The modified cylindrical Bessel function is: 385 * @f[ 386 * Z_{\nu}(x) = \sum_{k=0}^{\infty} 387 * \frac{\sigma^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} 388 * @f] 389 * where \f$ \sigma = +1 \f$ or\f$ -1 \f$ for 390 * \f$ Z = I \f$ or \f$ J \f$ respectively. 391 * 392 * See Abramowitz & Stegun, 9.1.10 393 * Abramowitz & Stegun, 9.6.7 394 * (1) Handbook of Mathematical Functions, 395 * ed. Milton Abramowitz and Irene A. Stegun, 396 * Dover Publications, 397 * Equation 9.1.10 p. 360 and Equation 9.6.10 p. 375 398 * 399 * @param __nu The order of the Bessel function. 400 * @param __x The argument of the Bessel function. 401 * @param __sgn The sign of the alternate terms 402 * -1 for the Bessel function of the first kind. 403 * +1 for the modified Bessel function of the first kind. 404 * @return The output Bessel function. 405 */ 406 template <typename _Tp> 407 _Tp 408 __cyl_bessel_ij_series(_Tp __nu, _Tp __x, _Tp __sgn, 409 unsigned int __max_iter) 410 { 411 if (__x == _Tp(0)) 412 return __nu == _Tp(0) ? _Tp(1) : _Tp(0); 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$. 441 * 442 * The cylindrical Bessel function is: 443 * @f[ 444 * J_{\nu}(x) = \sum_{k=0}^{\infty} 445 * \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} 446 * @f] 447 * 448 * @param __nu The order of the Bessel function. 449 * @param __x The argument of the Bessel function. 450 * @return The output Bessel function. 451 */ 452 template<typename _Tp> 453 _Tp 454 __cyl_bessel_j(_Tp __nu, _Tp __x) 455 { 456 if (__nu < _Tp(0) || __x < _Tp(0)) 457 std::__throw_domain_error(__N("Bad argument " 458 "in __cyl_bessel_j.")); 459 else if (__isnan(__nu) || __isnan(__x)) 460 return std::numeric_limits<_Tp>::quiet_NaN(); 461 else if (__x * __x < _Tp(10) * (__nu + _Tp(1))) 462 return __cyl_bessel_ij_series(__nu, __x, -_Tp(1), 200); 463 else if (__x > _Tp(1000)) 464 { 465 _Tp __J_nu, __N_nu; 466 __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu); 467 return __J_nu; 468 } 469 else 470 { 471 _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; 472 __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); 473 return __J_nu; 474 } 475 } 476 477 478 /** 479 * @brief Return the Neumann function of order \f$ \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$. 489 * 490 * @param __nu The order of the Neumann function. 491 * @param __x The argument of the Neumann function. 492 * @return The output Neumann function. 493 */ 494 template<typename _Tp> 495 _Tp 496 __cyl_neumann_n(_Tp __nu, _Tp __x) 497 { 498 if (__nu < _Tp(0) || __x < _Tp(0)) 499 std::__throw_domain_error(__N("Bad argument " 500 "in __cyl_neumann_n.")); 501 else if (__isnan(__nu) || __isnan(__x)) 502 return std::numeric_limits<_Tp>::quiet_NaN(); 503 else if (__x > _Tp(1000)) 504 { 505 _Tp __J_nu, __N_nu; 506 __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu); 507 return __N_nu; 508 } 509 else 510 { 511 _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; 512 __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); 513 return __N_nu; 514 } 515 } 516 517 518 /** 519 * @brief Compute the spherical Bessel @f$ 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(unsigned int __n, _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(unsigned int __n, _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(unsigned int __n, _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 _GLIBCXX_END_NAMESPACE_VERSION 624 } // namespace std::tr1::__detail 625 } 626 } 627 628 #endif // _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 629
