-
Notifications
You must be signed in to change notification settings - Fork 1
/
Copy pathbessel_j_standalone.c
589 lines (532 loc) · 22.4 KB
/
bessel_j_standalone.c
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
/*
* Mathlib : A C Library of Special Functions
* Copyright (C) 1998-2015 Ross Ihaka and the R Core team.
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, a copy is available at
* https://www.R-project.org/Licenses/
*/
/* DESCRIPTION --> see below */
/* From http://www.netlib.org/specfun/rjbesl Fortran translated by f2c,...
* ------------------------------=#---- Martin Maechler, ETH Zurich
* Additional code for nu == alpha < 0 MM
*/
#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <float.h>
#define IEEE_754
#define min0(x, y)(((x) <= (y)) ? (x) : (y))
#define ME_RANGE "range error"
#define ML_WARNING(x, y) fprintf(stderr, "%s %s\n", x, y)
#define MATHLIB_ERROR(x, y) fprintf(stderr, x, y);
#define MATHLIB_WARNING4(a, b, c, d, e) fprintf(stderr, a, b, c, d, e)
#define MATHLIB_WARNING2(a, b, c) fprintf(stderr, a, b, c)
#define MATHLIB_WARNING(a, b) fprintf(stderr, a, b)
#define thresh_BESS_Y 16.
#define M_eps_sinc 2.149e-8
#define M_SQRT_2dPI 0.797884560802865355879892119869 // sqrt(2/pi)
#define xlrg_BESS_Y 1e8
#define ME_RANGE_NAN 0x7FF8000000000001
#define ensig_BESS 1e16
#define enten_BESS 1e308
#define enmten_BESS 8.9e-308
#define rtnsig_BESS 1e-4
#define xlrg_BESS_IJ 1e5
double Rf_gamma_cody(double x);
double cospi(double x);
double sinpi(double x);
double bessel_y(double x, double alpha);
static void J_bessel(double * x, double * alpha, int * nb,
double * b, int * ncalc);
// unused now from R
double bessel_j(double x, double alpha) {
int nb, ncalc;
double na, * bj;
/* NaNs propagated correctly */
if (isnan(x) || isnan(alpha)) {
return x + alpha;
}
if (x < 0) {
ML_WARNING(ME_RANGE, "bessel_j");
return NAN;
}
na = floor(alpha);
if (alpha < 0) {
/* Using Abramowitz & Stegun 9.1.2
* this may not be quite optimal (CPU and accuracy wise) */
return (((alpha - na == 0.5) ? 0 : bessel_j(x, -alpha) * cospi(alpha)) +
((alpha == na) ? 0 : bessel_y(x, -alpha) * sinpi(alpha)));
} else if (alpha > 1e7) {
MATHLIB_WARNING("besselJ(x, nu): nu=%g too large for bessel_j() algorithm", alpha);
return NAN; //ME_RANGE_NAN;
}
nb = 1 + (int) na; /* nb-1 <= alpha < nb */
alpha -= (double)(nb - 1); // ==> alpha' in [0, 1)
bj = (double * ) calloc(nb, sizeof(double));
if (!bj) MATHLIB_ERROR("%s", "bessel_j allocation error");
J_bessel( & x, & alpha, & nb, bj, & ncalc);
if (ncalc != nb) {
/* error input */
if (ncalc < 0)
MATHLIB_WARNING4("bessel_j(%g): ncalc (=%d) != nb (=%d); alpha=%g. Arg. out of range?\n",
x, ncalc, nb, alpha);
else
MATHLIB_WARNING2("bessel_j(%g,nu=%g): precision lost in result\n",
x, alpha + (double) nb - 1);
}
x = bj[nb - 1];
free(bj);
return x;
}
static void J_bessel(double * x, double * alpha, int * nb,
double * b, int * ncalc) {
/*
Calculates Bessel functions J_{n+alpha} (x)
for non-negative argument x, and non-negative order n+alpha, n = 0,1,..,nb-1.
Explanation of variables in the calling sequence.
X - Non-negative argument for which J's are to be calculated.
ALPHA - Fractional part of order for which
J's are to be calculated. 0 <= ALPHA < 1.
NB - Number of functions to be calculated, NB >= 1.
The first function calculated is of order ALPHA, and the
last is of order (NB - 1 + ALPHA).
B - Output vector of length NB. If RJBESL
terminates normally (NCALC=NB), the vector B contains the
functions J/ALPHA/(X) through J/NB-1+ALPHA/(X).
NCALC - Output variable indicating possible errors.
Before using the vector B, the user should check that
NCALC=NB, i.e., all orders have been calculated to
the desired accuracy. See the following
****************************************************************
Error return codes
In case of an error, NCALC != NB, and not all J's are
calculated to the desired accuracy.
NCALC < 0: An argument is out of range. For example,
NBES <= 0, ALPHA < 0 or > 1, or X is too large.
In this case, b[1] is set to zero, the remainder of the
B-vector is not calculated, and NCALC is set to
MIN(NB,0)-1 so that NCALC != NB.
NB > NCALC > 0: Not all requested function values could
be calculated accurately. This usually occurs because NB is
much larger than ABS(X). In this case, b[N] is calculated
to the desired accuracy for N <= NCALC, but precision
is lost for NCALC < N <= NB. If b[N] does not vanish
for N > NCALC (because it is too small to be represented),
and b[N]/b[NCALC] = 10^(-K), then only the first NSIG - K
significant figures of b[N] can be trusted.
Acknowledgement
This program is based on a program written by David J. Sookne
(2) that computes values of the Bessel functions J or I of float
argument and long order. Modifications include the restriction
of the computation to the J Bessel function of non-negative float
argument, the extension of the computation to arbitrary positive
order, and the elimination of most underflow.
References:
Olver, F.W.J., and Sookne, D.J. (1972)
"A Note on Backward Recurrence Algorithms";
Math. Comp. 26, 941-947.
Sookne, D.J. (1973)
"Bessel Functions of Real Argument and Integer Order";
NBS Jour. of Res. B. 77B, 125-132.
Latest modification: March 19, 1990
Author: W. J. Cody
Applied Mathematics Division
Argonne National Laboratory
Argonne, IL 60439
*******************************************************************
*/
/* ---------------------------------------------------------------------
Mathematical constants
PI2 = 2 / PI
TWOPI1 = first few significant digits of 2 * PI
TWOPI2 = (2*PI - TWOPI1) to working precision, i.e.,
TWOPI1 + TWOPI2 = 2 * PI to extra precision.
--------------------------------------------------------------------- */
const static double pi2 = .636619772367581343075535;
const static double twopi1 = 6.28125;
const static double twopi2 = .001935307179586476925286767;
/*---------------------------------------------------------------------
* Factorial(N)
*--------------------------------------------------------------------- */
const static double fact[25] = {
1.,
1.,
2.,
6.,
24.,
120.,
720.,
5040.,
40320.,
362880.,
3628800.,
39916800.,
479001600.,
6227020800.,
87178291200.,
1.307674368e12,
2.0922789888e13,
3.55687428096e14,
6.402373705728e15,
1.21645100408832e17,
2.43290200817664e18,
5.109094217170944e19,
1.12400072777760768e21,
2.585201673888497664e22,
6.2044840173323943936e23
};
/* Local variables */
int nend, intx, nbmx, i, j, k, l, m, n, nstart;
double nu, twonu, capp, capq, pold, vcos, test, vsin;
double p, s, t, z, alpem, halfx, aa, bb, cc, psave, plast;
double tover, t1, alp2em, em, en, xc, xk, xm, psavel, gnu, xin, sum;
/* Parameter adjustment */
--b;
nu = * alpha;
twonu = nu + nu;
/*-------------------------------------------------------------------
Check for out of range arguments.
-------------------------------------------------------------------*/
if ( * nb > 0 && * x >= 0. && 0. <= nu && nu < 1.) {
* ncalc = * nb;
if ( * x > xlrg_BESS_IJ) {
ML_WARNING(ME_RANGE, "J_bessel");
/* indeed, the limit is 0,
* but the cutoff happens too early */
for (i = 1; i <= * nb; i++)
b[i] = 0.; /*was ML_POSINF (really nonsense) */
return;
}
intx = (int)( * x);
/* Initialize result array to zero. */
for (i = 1; i <= * nb; ++i)
b[i] = 0.;
/*===================================================================
Branch into 3 cases :
1) use 2-term ascending series for small X
2) use asymptotic form for large X when NB is not too large
3) use recursion otherwise
===================================================================*/
if ( * x < rtnsig_BESS) {
/* ---------------------------------------------------------------
Two-term ascending series for small X.
--------------------------------------------------------------- */
alpem = 1. + nu;
halfx = ( * x > enmten_BESS) ? .5 * * x : 0.;
aa = (nu != 0.) ? pow(halfx, nu) / (nu * Rf_gamma_cody(nu)) : 1.;
bb = ( * x + 1. > 1.) ? -halfx * halfx : 0.;
b[1] = aa + aa * bb / alpem;
if ( * x != 0. && b[1] == 0.) {
* ncalc = 0;
}
if ( * nb != 1) {
if ( * x <= 0.) {
for (n = 2; n <= * nb; ++n) {
b[n] = 0.;
}
} else {
/* ----------------------------------------------
Calculate higher order functions.
---------------------------------------------- */
if (bb == 0.) {
tover = (enmten_BESS + enmten_BESS) / * x;
} else {
tover = enmten_BESS / bb;
}
cc = halfx;
for (n = 2; n <= * nb; ++n) {
aa /= alpem;
alpem += 1.;
aa *= cc;
if (aa <= tover * alpem) {
aa = 0.;
}
b[n] = aa + aa * bb / alpem;
if (b[n] == 0. && * ncalc > n) {
* ncalc = n - 1;
}
}
}
}
} else if ( * x > 25. && * nb <= intx + 1) {
/* ------------------------------------------------------------
Asymptotic series for X > 25 (and not too large nb)
------------------------------------------------------------ */
xc = sqrt(pi2 / * x);
xin = 1 / (64 * * x * * x);
if ( * x >= 130.) m = 4;
else if ( * x >= 35.) m = 8;
else m = 11;
xm = 4. * (double) m;
/* ------------------------------------------------
Argument reduction for SIN and COS routines.
------------------------------------------------ */
t = trunc( * x / (twopi1 + twopi2) + .5);
z = ( * x - t * twopi1) - t * twopi2 - (nu + .5) / pi2;
vsin = sin(z);
vcos = cos(z);
gnu = twonu;
for (i = 1; i <= 2; ++i) {
s = (xm - 1. - gnu) * (xm - 1. + gnu) * xin * .5;
t = (gnu - (xm - 3.)) * (gnu + (xm - 3.));
t1 = (gnu - (xm + 1.)) * (gnu + (xm + 1.));
k = m + m;
capp = s * t / fact[k];
capq = s * t1 / fact[k + 1];
xk = xm;
for (; k >= 4; k -= 2) {
/* k + 2(j-2) == 2m */
xk -= 4.;
s = (xk - 1. - gnu) * (xk - 1. + gnu);
t1 = t;
t = (gnu - (xk - 3.)) * (gnu + (xk - 3.));
capp = (capp + 1. / fact[k - 2]) * s * t * xin;
capq = (capq + 1. / fact[k - 1]) * s * t1 * xin;
}
capp += 1.;
capq = (capq + 1.) * (gnu * gnu - 1.) * (.125 / * x);
b[i] = xc * (capp * vcos - capq * vsin);
if ( * nb == 1)
return;
/* vsin <--> vcos */
t = vsin;
vsin = -vcos;
vcos = t;
gnu += 2.;
}
/* -----------------------------------------------
If NB > 2, compute J(X,ORDER+I) for I = 2, NB-1
----------------------------------------------- */
if ( * nb > 2)
for (gnu = twonu + 2., j = 3; j <= * nb; j++, gnu += 2.)
b[j] = gnu * b[j - 1] / * x - b[j - 2];
} else {
/* rtnsig_BESS <= x && ( x <= 25 || intx+1 < *nb ) :
--------------------------------------------------------
Use recurrence to generate results.
First initialize the calculation of P*S.
-------------------------------------------------------- */
nbmx = * nb - intx;
n = intx + 1;
en = (double)(n + n) + twonu;
plast = 1.;
p = en / * x;
/* ---------------------------------------------------
Calculate general significance test.
--------------------------------------------------- */
test = ensig_BESS + ensig_BESS;
if (nbmx >= 3) {
/* ------------------------------------------------------------
Calculate P*S until N = NB-1. Check for possible overflow.
---------------------------------------------------------- */
tover = enten_BESS / ensig_BESS;
nstart = intx + 2;
nend = * nb - 1;
en = (double)(nstart + nstart) - 2. + twonu;
for (k = nstart; k <= nend; ++k) {
n = k;
en += 2.;
pold = plast;
plast = p;
p = en * plast / * x - pold;
if (p > tover) {
/* -------------------------------------------
To avoid overflow, divide P*S by TOVER.
Calculate P*S until ABS(P) > 1.
-------------------------------------------*/
tover = enten_BESS;
p /= tover;
plast /= tover;
psave = p;
psavel = plast;
nstart = n + 1;
do {
++n;
en += 2.;
pold = plast;
plast = p;
p = en * plast / * x - pold;
} while (p <= 1.);
bb = en / * x;
/* -----------------------------------------------
Calculate backward test and find NCALC,
the highest N such that the test is passed.
----------------------------------------------- */
test = pold * plast * (.5 - .5 / (bb * bb));
test /= ensig_BESS;
p = plast * tover;
--n;
en -= 2.;
nend = min0( * nb, n);
for (l = nstart; l <= nend; ++l) {
pold = psavel;
psavel = psave;
psave = en * psavel / * x - pold;
if (psave * psavel > test) {
* ncalc = l - 1;
goto L190;
}
} //for
* ncalc = nend;
goto L190;
} // if
} //for
n = nend;
en = (double)(n + n) + twonu;
/* -----------------------------------------------------
Calculate special significance test for NBMX > 2.
-----------------------------------------------------*/
test = fmax(test, sqrt(plast * ensig_BESS) * sqrt(p + p));
}
/* ------------------------------------------------
Calculate P*S until significance test passes. */
do {
++n;
en += 2.;
pold = plast;
plast = p;
p = en * plast / * x - pold;
} while (p < test);
L190:
/*---------------------------------------------------------------
Initialize the backward recursion and the normalization sum.
--------------------------------------------------------------- */
++n;
en += 2.;
bb = 0.;
aa = 1. / p;
m = n / 2;
em = (double) m;
m = (n << 1) - (m << 2);
/* = 2 n - 4 (n/2)
= 0 for even, 2 for odd n */
if (m == 0) {
sum = 0.;
} else {
alpem = em - 1. + nu;
alp2em = em + em + nu;
sum = aa * alpem * alp2em / em;
}
nend = n - * nb;
/* if (nend > 0) */
/* --------------------------------------------------------
Recur backward via difference equation, calculating
(but not storing) b[N], until N = NB.
-------------------------------------------------------- */
for (l = 1; l <= nend; ++l) {
--n;
en -= 2.;
cc = bb;
bb = aa;
aa = en * bb / * x - cc;
m = m ? 0 : 2; /* m = 2 - m failed on gcc4-20041019 */
if (m != 0) {
em -= 1.;
alp2em = em + em + nu;
if (n == 1)
break;
alpem = em - 1. + nu;
if (alpem == 0.)
alpem = 1.;
sum = (sum + aa * alp2em) * alpem / em;
}
}
/*--------------------------------------------------
Store b[NB].
--------------------------------------------------*/
b[n] = aa;
if (nend >= 0) {
if ( * nb <= 1) {
if (nu + 1. == 1.)
alp2em = 1.;
else
alp2em = nu;
sum += b[1] * alp2em;
goto L250;
} else {
/*-- nb >= 2 : ---------------------------
Calculate and store b[NB-1].
----------------------------------------*/
--n;
en -= 2.;
b[n] = en * aa / * x - bb;
if (n == 1) {
goto L240;
}
m = m ? 0 : 2; /* m = 2 - m failed on gcc4-20041019 */
if (m != 0) {
em -= 1.;
alp2em = em + em + nu;
alpem = em - 1. + nu;
if (alpem == 0.) {
alpem = 1.;
}
sum = (sum + b[n] * alp2em) * alpem / em;
}
}
}
/* if (n - 2 != 0) */
/* --------------------------------------------------------
Calculate via difference equation and store b[N],
until N = 2.
-------------------------------------------------------- */
for (n = n - 1; n >= 2; n--) {
en -= 2.;
b[n] = en * b[n + 1] / * x - b[n + 2];
m = m ? 0 : 2; /* m = 2 - m failed on gcc4-20041019 */
if (m != 0) {
em -= 1.;
alp2em = em + em + nu;
alpem = em - 1. + nu;
if (alpem == 0.) {
alpem = 1.;
}
sum = (sum + b[n] * alp2em) * alpem / em;
}
}
/* ---------------------------------------
Calculate b[1].
-----------------------------------------*/
b[1] = 2. * (nu + 1.) * b[2] / * x - b[3];
L240:
em -= 1.;
alp2em = em + em + nu;
if (alp2em == 0.)
alp2em = 1.;
sum += b[1] * alp2em;
L250:
/* ---------------------------------------------------
Normalize. Divide all b[N] by sum.
---------------------------------------------------*/
/* if (nu + 1. != 1.) poor test */
if (fabs(nu) > 1e-15)
sum *= (Rf_gamma_cody(nu) * pow(.5 * * x, -nu));
aa = enmten_BESS;
if (sum > 1.) {
aa *= sum;
}
for (n = 1; n <= * nb; ++n) {
if (fabs(b[n]) < aa) {
b[n] = 0.;
} else {
b[n] /= sum;
}
}
}
} else {
/* Error return -- X, NB, or ALPHA is out of range : */
b[1] = 0.;
* ncalc = min0( * nb, 0) - 1;
}
}