| 1 | // Special functions -*- C++ -*-
|
|---|
| 2 |
|
|---|
| 3 | // Copyright (C) 2006-2021 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 |
|
|---|
| 55 | namespace 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)
|
|---|
| 62 | namespace 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
|
|---|
