Source

enzo-dev / src / enzo / fft66.F

Full commit
  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
#include "error.def"


* ======================================================================
* NIST Guide to Available Math Software.
* Source for module FFT from package GO.
* Retrieved from NETLIB on Wed Jul  5 11:50:07 1995.
* ======================================================================

      subroutine fft66(a,b,ntot,n,nspan,isn)

c  multivariate complex fourier transform, computed in place
c    using mixed-radix fast fourier transform algorithm.
c  by r. c. singleton, stanford research institute, sept. 1968
c  arrays a and b originally hold the real and imaginary
c    components of the data, and return the real and
c    imaginary components of the resulting fourier coefficients.
c  multivariate data is indexed according to the fortran
c    array element successor function, without limit
c    on the number of implied multiple subscripts.
c    the subroutine is called once for each variate.
c    the calls for a multivariate transform may be in any order.
c  ntot is the total number of complex data values.
c  n is the dimension of the current variable.
c  nspan/n is the spacing of consecutive data values
c    while indexing the current variable.
c  the sign of isn determines the sign of the complex
c    exponential, and the magnitude of isn is normally one.
c  a tri-variate transform with a(n1,n2,n3), b(n1,n2,n3)
c    is computed by
c      call fft(a,b,n1*n2*n3,n1,n1,1)
c      call fft(a,b,n1*n2*n3,n2,n1*n2,1)
c      call fft(a,b,n1*n2*n3,n3,n1*n2*n3,1)
c  for a single-variate transform,
c    ntot = n = nspan = (number of complex data values), e.g.
c      call fft(a,b,n,n,n,1)
c  the data can alternatively be stored in a single complex array c
c    in standard fortran fashion, i.e. alternating real and imaginary
c    parts. then with most fortran compilers, the complex array c can
c    be equivalenced to a real array a, the magnitude of isn changed
c    to two to give correct indexing increment, and a(1) and a(2) used
c    to pass the initial addresses for the sequences of real and
c    imaginary values, e.g.
c       complex c(ntot)
c       real    a(2*ntot)
c       equivalence (c(1),a(1))
c       call fft(a(1),a(2),ntot,n,nspan,2)
c  arrays at(maxf), ck(maxf), bt(maxf), sk(maxf), and np(maxp)
c    are used for temporary storage.  if the available storage
c    is insufficient, the program is terminated by a stop.
c    maxf must be .ge. the maximum prime factor of n.
c    maxp must be .gt. the number of prime factors of n.
c    in addition, if the square-free portion k of n has two or
c    more prime factors, then maxp must be .ge. k-1.
      dimension a(1),b(1)
c  array storage in nfac for a maximum of 15 prime factors of n.
c  if n has more than one square-free factor, the product of the
c    square-free factors must be .le. 210
      dimension nfac(11),np(209)
c  array storage for maximum prime factor of 23
      dimension at(23),ck(23),bt(23),sk(23)
      equivalence (i,ii)
c  the following two constants should agree with the array dimensions.
      maxp=209
      maxf=23
      if(n .lt. 2) return
      inc=isn
      c72=0.30901699437494742
      s72=0.95105651629515357
      s120=0.86602540378443865
      rad=6.2831853071796
      if(isn .ge. 0) go to 10
      s72=-s72
      s120=-s120
      rad=-rad
      inc=-inc
   10 nt=inc*ntot
      ks=inc*nspan
      kspan=ks
      nn=nt-inc
      jc=ks/n
      radf=rad*float(jc)*0.5
      i=0
      jf=0
c  determine the factors of n
      m=0
      k=n
      go to 20
   15 m=m+1
      nfac(m)=4
      k=k/16
   20 if(k-(k/16)*16 .eq. 0) go to 15
      j=3
      jj=9
      go to 30
   25 m=m+1
      nfac(m)=j
      k=k/jj
   30 if(mod(k,jj) .eq. 0) go to 25
      j=j+2
      jj=j**2
      if(jj .le. k) go to 30
      if(k .gt. 4) go to 40
      kt=m
      nfac(m+1)=k
      if(k .ne. 1) m=m+1
      go to 80
   40 if(k-(k/4)*4 .ne. 0) go to 50
      m=m+1
      nfac(m)=2
      k=k/4
   50 kt=m
      j=2
   60 if(mod(k,j) .ne. 0) go to 70
      m=m+1
      nfac(m)=j
      k=k/j
   70 j=((j+1)/2)*2+1
      if(j .le. k) go to 60
   80 if(kt .eq. 0) go to 100
      j=kt
   90 m=m+1
      nfac(m)=nfac(j)
      j=j-1
      if(j .ne. 0) go to 90
c  compute fourier transform
  100 sd=radf/float(kspan)
      cd=2.0*sin(sd)**2
      sd=sin(sd+sd)
      kk=1
      i=i+1
      if(nfac(i) .ne. 2) go to 400
c  transform for factor of 2 (including rotation factor)
      kspan=kspan/2
      k1=kspan+2
  210 k2=kk+kspan
      ak=a(k2)
      bk=b(k2)
      a(k2)=a(kk)-ak
      b(k2)=b(kk)-bk
      a(kk)=a(kk)+ak
      b(kk)=b(kk)+bk
      kk=k2+kspan
      if(kk .le. nn) go to 210
      kk=kk-nn
      if(kk .le. jc) go to 210
      if(kk .gt. kspan) go to 800
  220 c1=1.0-cd
      s1=sd
  230 k2=kk+kspan
      ak=a(kk)-a(k2)
      bk=b(kk)-b(k2)
      a(kk)=a(kk)+a(k2)
      b(kk)=b(kk)+b(k2)
      a(k2)=c1*ak-s1*bk
      b(k2)=s1*ak+c1*bk
      kk=k2+kspan
      if(kk .lt. nt) go to 230
      k2=kk-nt
      c1=-c1
      kk=k1-k2
      if(kk .gt. k2) go to 230
      ak=c1-(cd*c1+sd*s1)
      s1=(sd*c1-cd*s1)+s1
      c1=2.0-(ak**2+s1**2)
      s1=c1*s1
      c1=c1*ak
      kk=kk+jc
      if(kk .lt. k2) go to 230
      k1=k1+inc+inc
      kk=(k1-kspan)/2+jc
      if(kk .le. jc+jc) go to 220
      go to 100
c  transform for factor of 3 (optional code)
  320 k1=kk+kspan
      k2=k1+kspan
      ak=a(kk)
      bk=b(kk)
      aj=a(k1)+a(k2)
      bj=b(k1)+b(k2)
      a(kk)=ak+aj
      b(kk)=bk+bj
      ak=-0.5*aj+ak
      bk=-0.5*bj+bk
      aj=(a(k1)-a(k2))*s120
      bj=(b(k1)-b(k2))*s120
      a(k1)=ak-bj
      b(k1)=bk+aj
      a(k2)=ak+bj
      b(k2)=bk-aj
      kk=k2+kspan
      if(kk .lt. nn) go to 320
      kk=kk-nn
      if(kk .le. kspan) go to 320
      go to 700
c  transform for factor of 4
  400 if(nfac(i) .ne. 4) go to 600
      kspnn=kspan
      kspan=kspan/4
  410 c1=1.0
      s1=0
  420 k1=kk+kspan
      k2=k1+kspan
      k3=k2+kspan
      akp=a(kk)+a(k2)
      akm=a(kk)-a(k2)
      ajp=a(k1)+a(k3)
      ajm=a(k1)-a(k3)
      a(kk)=akp+ajp
      ajp=akp-ajp
      bkp=b(kk)+b(k2)
      bkm=b(kk)-b(k2)
      bjp=b(k1)+b(k3)
      bjm=b(k1)-b(k3)
      b(kk)=bkp+bjp
      bjp=bkp-bjp
      if(isn .lt. 0) go to 450
      akp=akm-bjm
      akm=akm+bjm
      bkp=bkm+ajm
      bkm=bkm-ajm
      if(s1 .eq. 0) go to 460
  430 a(k1)=akp*c1-bkp*s1
      b(k1)=akp*s1+bkp*c1
      a(k2)=ajp*c2-bjp*s2
      b(k2)=ajp*s2+bjp*c2
      a(k3)=akm*c3-bkm*s3
      b(k3)=akm*s3+bkm*c3
      kk=k3+kspan
      if(kk .le. nt) go to 420
  440 c2=c1-(cd*c1+sd*s1)
      s1=(sd*c1-cd*s1)+s1
      c1=2.0-(c2**2+s1**2)
      s1=c1*s1
      c1=c1*c2
      c2=c1**2-s1**2
      s2=2.0*c1*s1
      c3=c2*c1-s2*s1
      s3=c2*s1+s2*c1
      kk=kk-nt+jc
      if(kk .le. kspan) go to 420
      kk=kk-kspan+inc
      if(kk .le. jc) go to 410
      if(kspan .eq. jc) go to 800
      go to 100
  450 akp=akm+bjm
      akm=akm-bjm
      bkp=bkm-ajm
      bkm=bkm+ajm
      if(s1 .ne. 0) go to 430
  460 a(k1)=akp
      b(k1)=bkp
      a(k2)=ajp
      b(k2)=bjp
      a(k3)=akm
      b(k3)=bkm
      kk=k3+kspan
      if(kk .le. nt) go to 420
      go to 440
c  transform for factor of 5 (optional code)
  510 c2=c72**2-s72**2
      s2=2.0*c72*s72
  520 k1=kk+kspan
      k2=k1+kspan
      k3=k2+kspan
      k4=k3+kspan
      akp=a(k1)+a(k4)
      akm=a(k1)-a(k4)
      bkp=b(k1)+b(k4)
      bkm=b(k1)-b(k4)
      ajp=a(k2)+a(k3)
      ajm=a(k2)-a(k3)
      bjp=b(k2)+b(k3)
      bjm=b(k2)-b(k3)
      aa=a(kk)
      bb=b(kk)
      a(kk)=aa+akp+ajp
      b(kk)=bb+bkp+bjp
      ak=akp*c72+ajp*c2+aa
      bk=bkp*c72+bjp*c2+bb
      aj=akm*s72+ajm*s2
      bj=bkm*s72+bjm*s2
      a(k1)=ak-bj
      a(k4)=ak+bj
      b(k1)=bk+aj
      b(k4)=bk-aj
      ak=akp*c2+ajp*c72+aa
      bk=bkp*c2+bjp*c72+bb
      aj=akm*s2-ajm*s72
      bj=bkm*s2-bjm*s72
      a(k2)=ak-bj
      a(k3)=ak+bj
      b(k2)=bk+aj
      b(k3)=bk-aj
      kk=k4+kspan
      if(kk .lt. nn) go to 520
      kk=kk-nn
      if(kk .le. kspan) go to 520
      go to 700
c  transform for odd factors
  600 k=nfac(i)
      kspnn=kspan
      kspan=kspan/k
      if(k .eq. 3) go to 320
      if(k .eq. 5) go to 510
      if(k .eq. jf) go to 640
      jf=k
      s1=rad/float(k)
      c1=cos(s1)
      s1=sin(s1)
      if(jf .gt. maxf) go to 998
      ck(jf)=1.0
      sk(jf)=0.0
      j=1
  630 ck(j)=ck(k)*c1+sk(k)*s1
      sk(j)=ck(k)*s1-sk(k)*c1
      k=k-1
      ck(k)=ck(j)
      sk(k)=-sk(j)
      j=j+1
      if(j .lt. k) go to 630
  640 k1=kk
      k2=kk+kspnn
      aa=a(kk)
      bb=b(kk)
      ak=aa
      bk=bb
      j=1
      k1=k1+kspan
  650 k2=k2-kspan
      j=j+1
      at(j)=a(k1)+a(k2)
      ak=at(j)+ak
      bt(j)=b(k1)+b(k2)
      bk=bt(j)+bk
      j=j+1
      at(j)=a(k1)-a(k2)
      bt(j)=b(k1)-b(k2)
      k1=k1+kspan
      if(k1 .lt. k2) go to 650
      a(kk)=ak
      b(kk)=bk
      k1=kk
      k2=kk+kspnn
      j=1
  660 k1=k1+kspan
      k2=k2-kspan
      jj=j
      ak=aa
      bk=bb
      aj=0.0
      bj=0.0
      k=1
  670 k=k+1
      ak=at(k)*ck(jj)+ak
      bk=bt(k)*ck(jj)+bk
      k=k+1
      aj=at(k)*sk(jj)+aj
      bj=bt(k)*sk(jj)+bj
      jj=jj+j
      if(jj .gt. jf) jj=jj-jf
      if(k .lt. jf) go to 670
      k=jf-j
      a(k1)=ak-bj
      b(k1)=bk+aj
      a(k2)=ak+bj
      b(k2)=bk-aj
      j=j+1
      if(j .lt. k) go to 660
      kk=kk+kspnn
      if(kk .le. nn) go to 640
      kk=kk-nn
      if(kk .le. kspan) go to 640
c  multiply by rotation factor (except for factors of 2 and 4)
  700 if(i .eq. m) go to 800
      kk=jc+1
  710 c2=1.0-cd
      s1=sd
  720 c1=c2
      s2=s1
      kk=kk+kspan
  730 ak=a(kk)
      a(kk)=c2*ak-s2*b(kk)
      b(kk)=s2*ak+c2*b(kk)
      kk=kk+kspnn
      if(kk .le. nt) go to 730
      ak=s1*s2
      s2=s1*c2+c1*s2
      c2=c1*c2-ak
      kk=kk-nt+kspan
      if(kk .le. kspnn) go to 730
      c2=c1-(cd*c1+sd*s1)
      s1=s1+(sd*c1-cd*s1)
      c1=2.0-(c2**2+s1**2)
      s1=c1*s1
      c2=c1*c2
      kk=kk-kspnn+jc
      if(kk .le. kspan) go to 720
      kk=kk-kspan+jc+inc
      if(kk .le. jc+jc) go to 710
      go to 100
c  permute the results to normal order---done in two stages
c  permutation for square factors of n
  800 np(1)=ks
      if(kt .eq. 0) go to 890
      k=kt+kt+1
      if(m .lt. k) k=k-1
      j=1
      np(k+1)=jc
  810 np(j+1)=np(j)/nfac(j)
      np(k)=np(k+1)*nfac(j)
      j=j+1
      k=k-1
      if(j .lt. k) go to 810
      k3=np(k+1)
      kspan=np(2)
      kk=jc+1
      k2=kspan+1
      j=1
      if(n .ne. ntot) go to 850
c  permutation for single-variate transform (optional code)
  820 ak=a(kk)
      a(kk)=a(k2)
      a(k2)=ak
      bk=b(kk)
      b(kk)=b(k2)
      b(k2)=bk
      kk=kk+inc
      k2=kspan+k2
      if(k2 .lt. ks) go to 820
  830 k2=k2-np(j)
      j=j+1
      k2=np(j+1)+k2
      if(k2 .gt. np(j)) go to 830
      j=1
  840 if(kk .lt. k2) go to 820
      kk=kk+inc
      k2=kspan+k2
      if(k2 .lt. ks) go to 840
      if(kk .lt. ks) go to 830
      jc=k3
      go to 890
c  permutation for multivariate transform
  850 k=kk+jc
  860 ak=a(kk)
      a(kk)=a(k2)
      a(k2)=ak
      bk=b(kk)
      b(kk)=b(k2)
      b(k2)=bk
      kk=kk+inc
      k2=k2+inc
      if(kk .lt. k) go to 860
      kk=kk+ks-jc
      k2=k2+ks-jc
      if(kk .lt. nt) go to 850
      k2=k2-nt+kspan
      kk=kk-nt+jc
      if(k2 .lt. ks) go to 850
  870 k2=k2-np(j)
      j=j+1
      k2=np(j+1)+k2
      if(k2 .gt. np(j)) go to 870
      j=1
  880 if(kk .lt. k2) go to 850
      kk=kk+jc
      k2=kspan+k2
      if(k2 .lt. ks) go to 880
      if(kk .lt. ks) go to 870
      jc=k3
  890 if(2*kt+1 .ge. m) return
      kspnn=np(kt+1)
c  permutation for square-free factors of n
      j=m-kt
      nfac(j+1)=1
  900 nfac(j)=nfac(j)*nfac(j+1)
      j=j-1
      if(j .ne. kt) go to 900
      kt=kt+1
      nn=nfac(kt)-1
      if(nn .gt. maxp) go to 998
      jj=0
      j=0
      go to 906
  902 jj=jj-k2
      k2=kk
      k=k+1
      kk=nfac(k)
  904 jj=kk+jj
      if(jj .ge. k2) go to 902
      np(j)=jj
  906 k2=nfac(kt)
      k=kt+1
      kk=nfac(k)
      j=j+1
      if(j .le. nn) go to 904
c  determine the permutation cycles of length greater than 1
      j=0
      go to 914
  910 k=kk
      kk=np(k)
      np(k)=-kk
      if(kk .ne. j) go to 910
      k3=kk
  914 j=j+1
      kk=np(j)
      if(kk .lt. 0) go to 914
      if(kk .ne. j) go to 910
      np(j)=-j
      if(j .ne. nn) go to 914
      maxf=inc*maxf
c  reorder a and b, following the permutation cycles
      go to 950
  924 j=j-1
      if(np(j) .lt. 0) go to 924
      jj=jc
  926 kspan=jj
      if(jj .gt. maxf) kspan=maxf
      jj=jj-kspan
      k=np(j)
      kk=jc*k+ii+jj
      k1=kk+kspan
      k2=0
  928 k2=k2+1
      at(k2)=a(k1)
      bt(k2)=b(k1)
      k1=k1-inc
      if(k1 .ne. kk) go to 928
  932 k1=kk+kspan
      k2=k1-jc*(k+np(k))
      k=-np(k)
  936 a(k1)=a(k2)
      b(k1)=b(k2)
      k1=k1-inc
      k2=k2-inc
      if(k1 .ne. kk) go to 936
      kk=k2
      if(k .ne. j) go to 932
      k1=kk+kspan
      k2=0
  940 k2=k2+1
      a(k1)=at(k2)
      b(k1)=bt(k2)
      k1=k1-inc
      if(k1 .ne. kk) go to 940
      if(jj .ne. 0) go to 926
      if(j .ne. 1) go to 924
  950 j=k3+1
      nt=nt-kspnn
      ii=nt-inc+1
      if(nt .ge. 0) go to 924
      return
c  error finish, insufficient array storage
  998 isn=0
      print 999
      ERROR_MESSAGE
  999 format('array bounds exceeded within subroutine fft')
      end