1// Special functions -*- C++ -*- 2 3// Copyright (C) 2006-2020 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/* __cyl_bessel_jn_asymp adapted from GNU GSL version 2.4 specfunc/bessel_j.c 31 * Copyright (C) 1996-2003 Gerard Jungman 32 */ 33 34// 35// ISO C++ 14882 TR1: 5.2 Special functions 36// 37 38// Written by Edward Smith-Rowland. 39// 40// References: 41// (1) Handbook of Mathematical Functions, 42// ed. Milton Abramowitz and Irene A. Stegun, 43// Dover Publications, 44// Section 9, pp. 355-434, Section 10 pp. 435-478 45// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl 46// (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky, 47// W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992), 48// 2nd ed, pp. 240-245 49 50#ifndef _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 51#define _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 1 52 53#include <tr1/special_function_util.h> 54 55namespace std _GLIBCXX_VISIBILITY(default) 56{ 57_GLIBCXX_BEGIN_NAMESPACE_VERSION 58 59#if _GLIBCXX_USE_STD_SPEC_FUNCS 60# define _GLIBCXX_MATH_NS ::std 61#elif defined(_GLIBCXX_TR1_CMATH) 62namespace tr1 63{ 64# define _GLIBCXX_MATH_NS ::std::tr1 65#else 66# error do not include this header directly, use <cmath> or <tr1/cmath> 67#endif 68 // [5.2] Special functions 69 70 // Implementation-space details. 71 namespace __detail 72 { 73 /** 74 * @brief Compute the gamma functions required by the Temme series 75 * expansions of @f$ N_\nu(x) @f$ and @f$ K_\nu(x) @f$. 76 * @f[ 77 * \Gamma_1 = \frac{1}{2\mu} 78 * [\frac{1}{\Gamma(1 - \mu)} - \frac{1}{\Gamma(1 + \mu)}] 79 * @f] 80 * and 81 * @f[ 82 * \Gamma_2 = \frac{1}{2} 83 * [\frac{1}{\Gamma(1 - \mu)} + \frac{1}{\Gamma(1 + \mu)}] 84 * @f] 85 * where @f$ -1/2 <= \mu <= 1/2 @f$ is @f$ \mu = \nu - N @f$ and @f$ N @f$. 86 * is the nearest integer to @f$ \nu @f$. 87 * The values of \f$ \Gamma(1 + \mu) \f$ and \f$ \Gamma(1 - \mu) \f$ 88 * are returned as well. 89 * 90 * The accuracy requirements on this are exquisite. 91 * 92 * @param __mu The input parameter of the gamma functions. 93 * @param __gam1 The output function \f$ \Gamma_1(\mu) \f$ 94 * @param __gam2 The output function \f$ \Gamma_2(\mu) \f$ 95 * @param __gampl The output function \f$ \Gamma(1 + \mu) \f$ 96 * @param __gammi The output function \f$ \Gamma(1 - \mu) \f$ 97 */ 98 template <typename _Tp> 99 void 100 __gamma_temme(_Tp __mu, 101 _Tp & __gam1, _Tp & __gam2, _Tp & __gampl, _Tp & __gammi) 102 { 103#if _GLIBCXX_USE_C99_MATH_TR1 104 __gampl = _Tp(1) / _GLIBCXX_MATH_NS::tgamma(_Tp(1) + __mu); 105 __gammi = _Tp(1) / _GLIBCXX_MATH_NS::tgamma(_Tp(1) - __mu); 106#else 107 __gampl = _Tp(1) / __gamma(_Tp(1) + __mu); 108 __gammi = _Tp(1) / __gamma(_Tp(1) - __mu); 109#endif 110 111 if (std::abs(__mu) < std::numeric_limits<_Tp>::epsilon()) 112 __gam1 = -_Tp(__numeric_constants<_Tp>::__gamma_e()); 113 else 114 __gam1 = (__gammi - __gampl) / (_Tp(2) * __mu); 115 116 __gam2 = (__gammi + __gampl) / (_Tp(2)); 117 118 return; 119 } 120 121 122 /** 123 * @brief Compute the Bessel @f$ J_\nu(x) @f$ and Neumann 124 * @f$ N_\nu(x) @f$ functions and their first derivatives 125 * @f$ J'_\nu(x) @f$ and @f$ N'_\nu(x) @f$ respectively. 126 * These four functions are computed together for numerical 127 * stability. 128 * 129 * @param __nu The order of the Bessel functions. 130 * @param __x The argument of the Bessel functions. 131 * @param __Jnu The output Bessel function of the first kind. 132 * @param __Nnu The output Neumann function (Bessel function of the second kind). 133 * @param __Jpnu The output derivative of the Bessel function of the first kind. 134 * @param __Npnu The output derivative of the Neumann function. 135 */ 136 template <typename _Tp> 137 void 138 __bessel_jn(_Tp __nu, _Tp __x, 139 _Tp & __Jnu, _Tp & __Nnu, _Tp & __Jpnu, _Tp & __Npnu) 140 { 141 if (__x == _Tp(0)) 142 { 143 if (__nu == _Tp(0)) 144 { 145 __Jnu = _Tp(1); 146 __Jpnu = _Tp(0); 147 } 148 else if (__nu == _Tp(1)) 149 { 150 __Jnu = _Tp(0); 151 __Jpnu = _Tp(0.5L); 152 } 153 else 154 { 155 __Jnu = _Tp(0); 156 __Jpnu = _Tp(0); 157 } 158 __Nnu = -std::numeric_limits<_Tp>::infinity(); 159 __Npnu = std::numeric_limits<_Tp>::infinity(); 160 return; 161 } 162 163 const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 164 // When the multiplier is N i.e. 165 // fp_min = N * min() 166 // Then J_0 and N_0 tank at x = 8 * N (J_0 = 0 and N_0 = nan)! 167 //const _Tp __fp_min = _Tp(20) * std::numeric_limits<_Tp>::min(); 168 const _Tp __fp_min = std::sqrt(std::numeric_limits<_Tp>::min()); 169 const int __max_iter = 15000; 170 const _Tp __x_min = _Tp(2); 171 172 const int __nl = (__x < __x_min 173 ? static_cast<int>(__nu + _Tp(0.5L)) 174 : std::max(0, static_cast<int>(__nu - __x + _Tp(1.5L)))); 175 176 const _Tp __mu = __nu - __nl; 177 const _Tp __mu2 = __mu * __mu; 178 const _Tp __xi = _Tp(1) / __x; 179 const _Tp __xi2 = _Tp(2) * __xi; 180 _Tp __w = __xi2 / __numeric_constants<_Tp>::__pi(); 181 int __isign = 1; 182 _Tp __h = __nu * __xi; 183 if (__h < __fp_min) 184 __h = __fp_min; 185 _Tp __b = __xi2 * __nu; 186 _Tp __d = _Tp(0); 187 _Tp __c = __h; 188 int __i; 189 for (__i = 1; __i <= __max_iter; ++__i) 190 { 191 __b += __xi2; 192 __d = __b - __d; 193 if (std::abs(__d) < __fp_min) 194 __d = __fp_min; 195 __c = __b - _Tp(1) / __c; 196 if (std::abs(__c) < __fp_min) 197 __c = __fp_min; 198 __d = _Tp(1) / __d; 199 const _Tp __del = __c * __d; 200 __h *= __del; 201 if (__d < _Tp(0)) 202 __isign = -__isign; 203 if (std::abs(__del - _Tp(1)) < __eps) 204 break; 205 } 206 if (__i > __max_iter) 207 std::__throw_runtime_error(__N("Argument x too large in __bessel_jn; " 208 "try asymptotic expansion.")); 209 _Tp __Jnul = __isign * __fp_min; 210 _Tp __Jpnul = __h * __Jnul; 211 _Tp __Jnul1 = __Jnul; 212 _Tp __Jpnu1 = __Jpnul; 213 _Tp __fact = __nu * __xi; 214 for ( int __l = __nl; __l >= 1; --__l ) 215 { 216 const _Tp __Jnutemp = __fact * __Jnul + __Jpnul; 217 __fact -= __xi; 218 __Jpnul = __fact * __Jnutemp - __Jnul; 219 __Jnul = __Jnutemp; 220 } 221 if (__Jnul == _Tp(0)) 222 __Jnul = __eps; 223 _Tp __f= __Jpnul / __Jnul; 224 _Tp __Nmu, __Nnu1, __Npmu, __Jmu; 225 if (__x < __x_min) 226 { 227 const _Tp __x2 = __x / _Tp(2); 228 const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu; 229 _Tp __fact = (std::abs(__pimu) < __eps 230 ? _Tp(1) : __pimu / std::sin(__pimu)); 231 _Tp __d = -std::log(__x2); 232 _Tp __e = __mu * __d; 233 _Tp __fact2 = (std::abs(__e) < __eps 234 ? _Tp(1) : std::sinh(__e) / __e); 235 _Tp __gam1, __gam2, __gampl, __gammi; 236 __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi); 237 _Tp __ff = (_Tp(2) / __numeric_constants<_Tp>::__pi()) 238 * __fact * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d); 239 __e = std::exp(__e); 240 _Tp __p = __e / (__numeric_constants<_Tp>::__pi() * __gampl); 241 _Tp __q = _Tp(1) / (__e * __numeric_constants<_Tp>::__pi() * __gammi); 242 const _Tp __pimu2 = __pimu / _Tp(2); 243 _Tp __fact3 = (std::abs(__pimu2) < __eps 244 ? _Tp(1) : std::sin(__pimu2) / __pimu2 ); 245 _Tp __r = __numeric_constants<_Tp>::__pi() * __pimu2 * __fact3 * __fact3; 246 _Tp __c = _Tp(1); 247 __d = -__x2 * __x2; 248 _Tp __sum = __ff + __r * __q; 249 _Tp __sum1 = __p; 250 for (__i = 1; __i <= __max_iter; ++__i) 251 { 252 __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2); 253 __c *= __d / _Tp(__i); 254 __p /= _Tp(__i) - __mu; 255 __q /= _Tp(__i) + __mu; 256 const _Tp __del = __c * (__ff + __r * __q); 257 __sum += __del; 258 const _Tp __del1 = __c * __p - __i * __del; 259 __sum1 += __del1; 260 if ( std::abs(__del) < __eps * (_Tp(1) + std::abs(__sum)) ) 261 break; 262 } 263 if ( __i > __max_iter ) 264 std::__throw_runtime_error(__N("Bessel y series failed to converge " 265 "in __bessel_jn.")); 266 __Nmu = -__sum; 267 __Nnu1 = -__sum1 * __xi2; 268 __Npmu = __mu * __xi * __Nmu - __Nnu1; 269 __Jmu = __w / (__Npmu - __f * __Nmu); 270 } 271 else 272 { 273 _Tp __a = _Tp(0.25L) - __mu2; 274 _Tp __q = _Tp(1); 275 _Tp __p = -__xi / _Tp(2); 276 _Tp __br = _Tp(2) * __x; 277 _Tp __bi = _Tp(2); 278 _Tp __fact = __a * __xi / (__p * __p + __q * __q); 279 _Tp __cr = __br + __q * __fact; 280 _Tp __ci = __bi + __p * __fact; 281 _Tp __den = __br * __br + __bi * __bi; 282 _Tp __dr = __br / __den; 283 _Tp __di = -__bi / __den; 284 _Tp __dlr = __cr * __dr - __ci * __di; 285 _Tp __dli = __cr * __di + __ci * __dr; 286 _Tp __temp = __p * __dlr - __q * __dli; 287 __q = __p * __dli + __q * __dlr; 288 __p = __temp; 289 int __i; 290 for (__i = 2; __i <= __max_iter; ++__i) 291 { 292 __a += _Tp(2 * (__i - 1)); 293 __bi += _Tp(2); 294 __dr = __a * __dr + __br; 295 __di = __a * __di + __bi; 296 if (std::abs(__dr) + std::abs(__di) < __fp_min) 297 __dr = __fp_min; 298 __fact = __a / (__cr * __cr + __ci * __ci); 299 __cr = __br + __cr * __fact; 300 __ci = __bi - __ci * __fact; 301 if (std::abs(__cr) + std::abs(__ci) < __fp_min) 302 __cr = __fp_min; 303 __den = __dr * __dr + __di * __di; 304 __dr /= __den; 305 __di /= -__den; 306 __dlr = __cr * __dr - __ci * __di; 307 __dli = __cr * __di + __ci * __dr; 308 __temp = __p * __dlr - __q * __dli; 309 __q = __p * __dli + __q * __dlr; 310 __p = __temp; 311 if (std::abs(__dlr - _Tp(1)) + std::abs(__dli) < __eps) 312 break; 313 } 314 if (__i > __max_iter) 315 std::__throw_runtime_error(__N("Lentz's method failed " 316 "in __bessel_jn.")); 317 const _Tp __gam = (__p - __f) / __q; 318 __Jmu = std::sqrt(__w / ((__p - __f) * __gam + __q)); 319#if _GLIBCXX_USE_C99_MATH_TR1 320 __Jmu = _GLIBCXX_MATH_NS::copysign(__Jmu, __Jnul); 321#else 322 if (__Jmu * __Jnul < _Tp(0)) 323 __Jmu = -__Jmu; 324#endif 325 __Nmu = __gam * __Jmu; 326 __Npmu = (__p + __q / __gam) * __Nmu; 327 __Nnu1 = __mu * __xi * __Nmu - __Npmu; 328 } 329 __fact = __Jmu / __Jnul; 330 __Jnu = __fact * __Jnul1; 331 __Jpnu = __fact * __Jpnu1; 332 for (__i = 1; __i <= __nl; ++__i) 333 { 334 const _Tp __Nnutemp = (__mu + __i) * __xi2 * __Nnu1 - __Nmu; 335 __Nmu = __Nnu1; 336 __Nnu1 = __Nnutemp; 337 } 338 __Nnu = __Nmu; 339 __Npnu = __nu * __xi * __Nmu - __Nnu1; 340 341 return; 342 } 343 344 345 /** 346 * @brief This routine computes the asymptotic cylindrical Bessel 347 * and Neumann functions of order nu: \f$ J_{\nu} \f$, 348 * \f$ N_{\nu} \f$. 349 * 350 * References: 351 * (1) Handbook of Mathematical Functions, 352 * ed. Milton Abramowitz and Irene A. Stegun, 353 * Dover Publications, 354 * Section 9 p. 364, Equations 9.2.5-9.2.10 355 * 356 * @param __nu The order of the Bessel functions. 357 * @param __x The argument of the Bessel functions. 358 * @param __Jnu The output Bessel function of the first kind. 359 * @param __Nnu The output Neumann function (Bessel function of the second kind). 360 */ 361 template <typename _Tp> 362 void 363 __cyl_bessel_jn_asymp(_Tp __nu, _Tp __x, _Tp & __Jnu, _Tp & __Nnu) 364 { 365 const _Tp __mu = _Tp(4) * __nu * __nu; 366 const _Tp __8x = _Tp(8) * __x; 367 368 _Tp __P = _Tp(0); 369 _Tp __Q = _Tp(0); 370 371 _Tp __k = _Tp(0); 372 _Tp __term = _Tp(1); 373 374 int __epsP = 0; 375 int __epsQ = 0; 376 377 _Tp __eps = std::numeric_limits<_Tp>::epsilon(); 378 379 do 380 { 381 __term *= (__k == 0 382 ? _Tp(1) 383 : -(__mu - (2 * __k - 1) * (2 * __k - 1)) / (__k * __8x)); 384 385 __epsP = std::abs(__term) < __eps * std::abs(__P); 386 __P += __term; 387 388 __k++; 389 390 __term *= (__mu - (2 * __k - 1) * (2 * __k - 1)) / (__k * __8x); 391 __epsQ = std::abs(__term) < __eps * std::abs(__Q); 392 __Q += __term; 393 394 if (__epsP && __epsQ && __k > (__nu / 2.)) 395 break; 396 397 __k++; 398 } 399 while (__k < 1000); 400 401 const _Tp __chi = __x - (__nu + _Tp(0.5L)) 402 * __numeric_constants<_Tp>::__pi_2(); 403 404 const _Tp __c = std::cos(__chi); 405 const _Tp __s = std::sin(__chi); 406 407 const _Tp __coef = std::sqrt(_Tp(2) 408 / (__numeric_constants<_Tp>::__pi() * __x)); 409 410 __Jnu = __coef * (__c * __P - __s * __Q); 411 __Nnu = __coef * (__s * __P + __c * __Q); 412 413 return; 414 } 415 416 417 /** 418 * @brief This routine returns the cylindrical Bessel functions 419 * of order \f$ \nu \f$: \f$ J_{\nu} \f$ or \f$ I_{\nu} \f$ 420 * by series expansion. 421 * 422 * The modified cylindrical Bessel function is: 423 * @f[ 424 * Z_{\nu}(x) = \sum_{k=0}^{\infty} 425 * \frac{\sigma^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} 426 * @f] 427 * where \f$ \sigma = +1 \f$ or\f$ -1 \f$ for 428 * \f$ Z = I \f$ or \f$ J \f$ respectively. 429 * 430 * See Abramowitz & Stegun, 9.1.10 431 * Abramowitz & Stegun, 9.6.7 432 * (1) Handbook of Mathematical Functions, 433 * ed. Milton Abramowitz and Irene A. Stegun, 434 * Dover Publications, 435 * Equation 9.1.10 p. 360 and Equation 9.6.10 p. 375 436 * 437 * @param __nu The order of the Bessel function. 438 * @param __x The argument of the Bessel function. 439 * @param __sgn The sign of the alternate terms 440 * -1 for the Bessel function of the first kind. 441 * +1 for the modified Bessel function of the first kind. 442 * @return The output Bessel function. 443 */ 444 template <typename _Tp> 445 _Tp 446 __cyl_bessel_ij_series(_Tp __nu, _Tp __x, _Tp __sgn, 447 unsigned int __max_iter) 448 { 449 if (__x == _Tp(0)) 450 return __nu == _Tp(0) ? _Tp(1) : _Tp(0); 451 452 const _Tp __x2 = __x / _Tp(2); 453 _Tp __fact = __nu * std::log(__x2); 454#if _GLIBCXX_USE_C99_MATH_TR1 455 __fact -= _GLIBCXX_MATH_NS::lgamma(__nu + _Tp(1)); 456#else 457 __fact -= __log_gamma(__nu + _Tp(1)); 458#endif 459 __fact = std::exp(__fact); 460 const _Tp __xx4 = __sgn * __x2 * __x2; 461 _Tp __Jn = _Tp(1); 462 _Tp __term = _Tp(1); 463 464 for (unsigned int __i = 1; __i < __max_iter; ++__i) 465 { 466 __term *= __xx4 / (_Tp(__i) * (__nu + _Tp(__i))); 467 __Jn += __term; 468 if (std::abs(__term / __Jn) < std::numeric_limits<_Tp>::epsilon()) 469 break; 470 } 471 472 return __fact * __Jn; 473 } 474 475 476 /** 477 * @brief Return the Bessel function of order \f$ \nu \f$: 478 * \f$ J_{\nu}(x) \f$. 479 * 480 * The cylindrical Bessel function is: 481 * @f[ 482 * J_{\nu}(x) = \sum_{k=0}^{\infty} 483 * \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} 484 * @f] 485 * 486 * @param __nu The order of the Bessel function. 487 * @param __x The argument of the Bessel function. 488 * @return The output Bessel function. 489 */ 490 template<typename _Tp> 491 _Tp 492 __cyl_bessel_j(_Tp __nu, _Tp __x) 493 { 494 if (__nu < _Tp(0) || __x < _Tp(0)) 495 std::__throw_domain_error(__N("Bad argument " 496 "in __cyl_bessel_j.")); 497 else if (__isnan(__nu) || __isnan(__x)) 498 return std::numeric_limits<_Tp>::quiet_NaN(); 499 else if (__x * __x < _Tp(10) * (__nu + _Tp(1))) 500 return __cyl_bessel_ij_series(__nu, __x, -_Tp(1), 200); 501 else if (__x > _Tp(1000)) 502 { 503 _Tp __J_nu, __N_nu; 504 __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu); 505 return __J_nu; 506 } 507 else 508 { 509 _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; 510 __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); 511 return __J_nu; 512 } 513 } 514 515 516 /** 517 * @brief Return the Neumann function of order \f$ \nu \f$: 518 * \f$ N_{\nu}(x) \f$. 519 * 520 * The Neumann function is defined by: 521 * @f[ 522 * N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)} 523 * {\sin \nu\pi} 524 * @f] 525 * where for integral \f$ \nu = n \f$ a limit is taken: 526 * \f$ lim_{\nu \to n} \f$. 527 * 528 * @param __nu The order of the Neumann function. 529 * @param __x The argument of the Neumann function. 530 * @return The output Neumann function. 531 */ 532 template<typename _Tp> 533 _Tp 534 __cyl_neumann_n(_Tp __nu, _Tp __x) 535 { 536 if (__nu < _Tp(0) || __x < _Tp(0)) 537 std::__throw_domain_error(__N("Bad argument " 538 "in __cyl_neumann_n.")); 539 else if (__isnan(__nu) || __isnan(__x)) 540 return std::numeric_limits<_Tp>::quiet_NaN(); 541 else if (__x > _Tp(1000)) 542 { 543 _Tp __J_nu, __N_nu; 544 __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu); 545 return __N_nu; 546 } 547 else 548 { 549 _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; 550 __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); 551 return __N_nu; 552 } 553 } 554 555 556 /** 557 * @brief Compute the spherical Bessel @f$ j_n(x) @f$ 558 * and Neumann @f$ n_n(x) @f$ functions and their first 559 * derivatives @f$ j'_n(x) @f$ and @f$ n'_n(x) @f$ 560 * respectively. 561 * 562 * @param __n The order of the spherical Bessel function. 563 * @param __x The argument of the spherical Bessel function. 564 * @param __j_n The output spherical Bessel function. 565 * @param __n_n The output spherical Neumann function. 566 * @param __jp_n The output derivative of the spherical Bessel function. 567 * @param __np_n The output derivative of the spherical Neumann function. 568 */ 569 template <typename _Tp> 570 void 571 __sph_bessel_jn(unsigned int __n, _Tp __x, 572 _Tp & __j_n, _Tp & __n_n, _Tp & __jp_n, _Tp & __np_n) 573 { 574 const _Tp __nu = _Tp(__n) + _Tp(0.5L); 575 576 _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; 577 __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); 578 579 const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2() 580 / std::sqrt(__x); 581 582 __j_n = __factor * __J_nu; 583 __n_n = __factor * __N_nu; 584 __jp_n = __factor * __Jp_nu - __j_n / (_Tp(2) * __x); 585 __np_n = __factor * __Np_nu - __n_n / (_Tp(2) * __x); 586 587 return; 588 } 589 590 591 /** 592 * @brief Return the spherical Bessel function 593 * @f$ j_n(x) @f$ of order n. 594 * 595 * The spherical Bessel function is defined by: 596 * @f[ 597 * j_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x) 598 * @f] 599 * 600 * @param __n The order of the spherical Bessel function. 601 * @param __x The argument of the spherical Bessel function. 602 * @return The output spherical Bessel function. 603 */ 604 template <typename _Tp> 605 _Tp 606 __sph_bessel(unsigned int __n, _Tp __x) 607 { 608 if (__x < _Tp(0)) 609 std::__throw_domain_error(__N("Bad argument " 610 "in __sph_bessel.")); 611 else if (__isnan(__x)) 612 return std::numeric_limits<_Tp>::quiet_NaN(); 613 else if (__x == _Tp(0)) 614 { 615 if (__n == 0) 616 return _Tp(1); 617 else 618 return _Tp(0); 619 } 620 else 621 { 622 _Tp __j_n, __n_n, __jp_n, __np_n; 623 __sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n); 624 return __j_n; 625 } 626 } 627 628 629 /** 630 * @brief Return the spherical Neumann function 631 * @f$ n_n(x) @f$. 632 * 633 * The spherical Neumann function is defined by: 634 * @f[ 635 * n_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x) 636 * @f] 637 * 638 * @param __n The order of the spherical Neumann function. 639 * @param __x The argument of the spherical Neumann function. 640 * @return The output spherical Neumann function. 641 */ 642 template <typename _Tp> 643 _Tp 644 __sph_neumann(unsigned int __n, _Tp __x) 645 { 646 if (__x < _Tp(0)) 647 std::__throw_domain_error(__N("Bad argument " 648 "in __sph_neumann.")); 649 else if (__isnan(__x)) 650 return std::numeric_limits<_Tp>::quiet_NaN(); 651 else if (__x == _Tp(0)) 652 return -std::numeric_limits<_Tp>::infinity(); 653 else 654 { 655 _Tp __j_n, __n_n, __jp_n, __np_n; 656 __sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n); 657 return __n_n; 658 } 659 } 660 } // namespace __detail 661#undef _GLIBCXX_MATH_NS 662#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) 663} // namespace tr1 664#endif 665 666_GLIBCXX_END_NAMESPACE_VERSION 667} 668 669#endif // _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 670