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
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
|
SUBROUTINE DLATMS( M, N, DIST, ISEED, SYM, D, MODE, COND, DMAX,
$ KL, KU, PACK, A, LDA, WORK, INFO )
*
* -- LAPACK test routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
CHARACTER DIST, PACK, SYM
INTEGER INFO, KL, KU, LDA, M, MODE, N
DOUBLE PRECISION COND, DMAX
* ..
* .. Array Arguments ..
INTEGER ISEED( 4 )
DOUBLE PRECISION A( LDA, * ), D( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* DLATMS generates random matrices with specified singular values
* (or symmetric/hermitian with specified eigenvalues)
* for testing LAPACK programs.
*
* DLATMS operates by applying the following sequence of
* operations:
*
* Set the diagonal to D, where D may be input or
* computed according to MODE, COND, DMAX, and SYM
* as described below.
*
* Generate a matrix with the appropriate band structure, by one
* of two methods:
*
* Method A:
* Generate a dense M x N matrix by multiplying D on the left
* and the right by random unitary matrices, then:
*
* Reduce the bandwidth according to KL and KU, using
* Householder transformations.
*
* Method B:
* Convert the bandwidth-0 (i.e., diagonal) matrix to a
* bandwidth-1 matrix using Givens rotations, "chasing"
* out-of-band elements back, much as in QR; then
* convert the bandwidth-1 to a bandwidth-2 matrix, etc.
* Note that for reasonably small bandwidths (relative to
* M and N) this requires less storage, as a dense matrix
* is not generated. Also, for symmetric matrices, only
* one triangle is generated.
*
* Method A is chosen if the bandwidth is a large fraction of the
* order of the matrix, and LDA is at least M (so a dense
* matrix can be stored.) Method B is chosen if the bandwidth
* is small (< 1/2 N for symmetric, < .3 N+M for
* non-symmetric), or LDA is less than M and not less than the
* bandwidth.
*
* Pack the matrix if desired. Options specified by PACK are:
* no packing
* zero out upper half (if symmetric)
* zero out lower half (if symmetric)
* store the upper half columnwise (if symmetric or upper
* triangular)
* store the lower half columnwise (if symmetric or lower
* triangular)
* store the lower triangle in banded format (if symmetric
* or lower triangular)
* store the upper triangle in banded format (if symmetric
* or upper triangular)
* store the entire matrix in banded format
* If Method B is chosen, and band format is specified, then the
* matrix will be generated in the band format, so no repacking
* will be necessary.
*
* Arguments
* =========
*
* M - INTEGER
* The number of rows of A. Not modified.
*
* N - INTEGER
* The number of columns of A. Not modified.
*
* DIST - CHARACTER*1
* On entry, DIST specifies the type of distribution to be used
* to generate the random eigen-/singular values.
* 'U' => UNIFORM( 0, 1 ) ( 'U' for uniform )
* 'S' => UNIFORM( -1, 1 ) ( 'S' for symmetric )
* 'N' => NORMAL( 0, 1 ) ( 'N' for normal )
* Not modified.
*
* ISEED - INTEGER array, dimension ( 4 )
* On entry ISEED specifies the seed of the random number
* generator. They should lie between 0 and 4095 inclusive,
* and ISEED(4) should be odd. The random number generator
* uses a linear congruential sequence limited to small
* integers, and so should produce machine independent
* random numbers. The values of ISEED are changed on
* exit, and can be used in the next call to DLATMS
* to continue the same random number sequence.
* Changed on exit.
*
* SYM - CHARACTER*1
* If SYM='S' or 'H', the generated matrix is symmetric, with
* eigenvalues specified by D, COND, MODE, and DMAX; they
* may be positive, negative, or zero.
* If SYM='P', the generated matrix is symmetric, with
* eigenvalues (= singular values) specified by D, COND,
* MODE, and DMAX; they will not be negative.
* If SYM='N', the generated matrix is nonsymmetric, with
* singular values specified by D, COND, MODE, and DMAX;
* they will not be negative.
* Not modified.
*
* D - DOUBLE PRECISION array, dimension ( MIN( M , N ) )
* This array is used to specify the singular values or
* eigenvalues of A (see SYM, above.) If MODE=0, then D is
* assumed to contain the singular/eigenvalues, otherwise
* they will be computed according to MODE, COND, and DMAX,
* and placed in D.
* Modified if MODE is nonzero.
*
* MODE - INTEGER
* On entry this describes how the singular/eigenvalues are to
* be specified:
* MODE = 0 means use D as input
* MODE = 1 sets D(1)=1 and D(2:N)=1.0/COND
* MODE = 2 sets D(1:N-1)=1 and D(N)=1.0/COND
* MODE = 3 sets D(I)=COND**(-(I-1)/(N-1))
* MODE = 4 sets D(i)=1 - (i-1)/(N-1)*(1 - 1/COND)
* MODE = 5 sets D to random numbers in the range
* ( 1/COND , 1 ) such that their logarithms
* are uniformly distributed.
* MODE = 6 set D to random numbers from same distribution
* as the rest of the matrix.
* MODE < 0 has the same meaning as ABS(MODE), except that
* the order of the elements of D is reversed.
* Thus if MODE is positive, D has entries ranging from
* 1 to 1/COND, if negative, from 1/COND to 1,
* If SYM='S' or 'H', and MODE is neither 0, 6, nor -6, then
* the elements of D will also be multiplied by a random
* sign (i.e., +1 or -1.)
* Not modified.
*
* COND - DOUBLE PRECISION
* On entry, this is used as described under MODE above.
* If used, it must be >= 1. Not modified.
*
* DMAX - DOUBLE PRECISION
* If MODE is neither -6, 0 nor 6, the contents of D, as
* computed according to MODE and COND, will be scaled by
* DMAX / max(abs(D(i))); thus, the maximum absolute eigen- or
* singular value (which is to say the norm) will be abs(DMAX).
* Note that DMAX need not be positive: if DMAX is negative
* (or zero), D will be scaled by a negative number (or zero).
* Not modified.
*
* KL - INTEGER
* This specifies the lower bandwidth of the matrix. For
* example, KL=0 implies upper triangular, KL=1 implies upper
* Hessenberg, and KL being at least M-1 means that the matrix
* has full lower bandwidth. KL must equal KU if the matrix
* is symmetric.
* Not modified.
*
* KU - INTEGER
* This specifies the upper bandwidth of the matrix. For
* example, KU=0 implies lower triangular, KU=1 implies lower
* Hessenberg, and KU being at least N-1 means that the matrix
* has full upper bandwidth. KL must equal KU if the matrix
* is symmetric.
* Not modified.
*
* PACK - CHARACTER*1
* This specifies packing of matrix as follows:
* 'N' => no packing
* 'U' => zero out all subdiagonal entries (if symmetric)
* 'L' => zero out all superdiagonal entries (if symmetric)
* 'C' => store the upper triangle columnwise
* (only if the matrix is symmetric or upper triangular)
* 'R' => store the lower triangle columnwise
* (only if the matrix is symmetric or lower triangular)
* 'B' => store the lower triangle in band storage scheme
* (only if matrix symmetric or lower triangular)
* 'Q' => store the upper triangle in band storage scheme
* (only if matrix symmetric or upper triangular)
* 'Z' => store the entire matrix in band storage scheme
* (pivoting can be provided for by using this
* option to store A in the trailing rows of
* the allocated storage)
*
* Using these options, the various LAPACK packed and banded
* storage schemes can be obtained:
* GB - use 'Z'
* PB, SB or TB - use 'B' or 'Q'
* PP, SP or TP - use 'C' or 'R'
*
* If two calls to DLATMS differ only in the PACK parameter,
* they will generate mathematically equivalent matrices.
* Not modified.
*
* A - DOUBLE PRECISION array, dimension ( LDA, N )
* On exit A is the desired test matrix. A is first generated
* in full (unpacked) form, and then packed, if so specified
* by PACK. Thus, the first M elements of the first N
* columns will always be modified. If PACK specifies a
* packed or banded storage scheme, all LDA elements of the
* first N columns will be modified; the elements of the
* array which do not correspond to elements of the generated
* matrix are set to zero.
* Modified.
*
* LDA - INTEGER
* LDA specifies the first dimension of A as declared in the
* calling program. If PACK='N', 'U', 'L', 'C', or 'R', then
* LDA must be at least M. If PACK='B' or 'Q', then LDA must
* be at least MIN( KL, M-1) (which is equal to MIN(KU,N-1)).
* If PACK='Z', LDA must be large enough to hold the packed
* array: MIN( KU, N-1) + MIN( KL, M-1) + 1.
* Not modified.
*
* WORK - DOUBLE PRECISION array, dimension ( 3*MAX( N , M ) )
* Workspace.
* Modified.
*
* INFO - INTEGER
* Error code. On exit, INFO will be set to one of the
* following values:
* 0 => normal return
* -1 => M negative or unequal to N and SYM='S', 'H', or 'P'
* -2 => N negative
* -3 => DIST illegal string
* -5 => SYM illegal string
* -7 => MODE not in range -6 to 6
* -8 => COND less than 1.0, and MODE neither -6, 0 nor 6
* -10 => KL negative
* -11 => KU negative, or SYM='S' or 'H' and KU not equal to KL
* -12 => PACK illegal string, or PACK='U' or 'L', and SYM='N';
* or PACK='C' or 'Q' and SYM='N' and KL is not zero;
* or PACK='R' or 'B' and SYM='N' and KU is not zero;
* or PACK='U', 'L', 'C', 'R', 'B', or 'Q', and M is not
* N.
* -14 => LDA is less than M, or PACK='Z' and LDA is less than
* MIN(KU,N-1) + MIN(KL,M-1) + 1.
* 1 => Error return from DLATM1
* 2 => Cannot scale to DMAX (max. sing. value is 0)
* 3 => Error return from DLAGGE or SLAGSY
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
DOUBLE PRECISION TWOPI
PARAMETER ( TWOPI = 6.2831853071795864769252867663D+0 )
* ..
* .. Local Scalars ..
LOGICAL GIVENS, ILEXTR, ILTEMP, TOPDWN
INTEGER I, IC, ICOL, IDIST, IENDCH, IINFO, IL, ILDA,
$ IOFFG, IOFFST, IPACK, IPACKG, IR, IR1, IR2,
$ IROW, IRSIGN, ISKEW, ISYM, ISYMPK, J, JC, JCH,
$ JKL, JKU, JR, K, LLB, MINLDA, MNMIN, MR, NC,
$ UUB
DOUBLE PRECISION ALPHA, ANGLE, C, DUMMY, EXTRA, S, TEMP
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLARND
EXTERNAL LSAME, DLARND
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLAGGE, DLAGSY, DLAROT, DLARTG, DLASET,
$ DLATM1, DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, COS, DBLE, MAX, MIN, MOD, SIN
* ..
* .. Executable Statements ..
*
* 1) Decode and Test the input parameters.
* Initialize flags & seed.
*
INFO = 0
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 )
$ RETURN
*
* Decode DIST
*
IF( LSAME( DIST, 'U' ) ) THEN
IDIST = 1
ELSE IF( LSAME( DIST, 'S' ) ) THEN
IDIST = 2
ELSE IF( LSAME( DIST, 'N' ) ) THEN
IDIST = 3
ELSE
IDIST = -1
END IF
*
* Decode SYM
*
IF( LSAME( SYM, 'N' ) ) THEN
ISYM = 1
IRSIGN = 0
ELSE IF( LSAME( SYM, 'P' ) ) THEN
ISYM = 2
IRSIGN = 0
ELSE IF( LSAME( SYM, 'S' ) ) THEN
ISYM = 2
IRSIGN = 1
ELSE IF( LSAME( SYM, 'H' ) ) THEN
ISYM = 2
IRSIGN = 1
ELSE
ISYM = -1
END IF
*
* Decode PACK
*
ISYMPK = 0
IF( LSAME( PACK, 'N' ) ) THEN
IPACK = 0
ELSE IF( LSAME( PACK, 'U' ) ) THEN
IPACK = 1
ISYMPK = 1
ELSE IF( LSAME( PACK, 'L' ) ) THEN
IPACK = 2
ISYMPK = 1
ELSE IF( LSAME( PACK, 'C' ) ) THEN
IPACK = 3
ISYMPK = 2
ELSE IF( LSAME( PACK, 'R' ) ) THEN
IPACK = 4
ISYMPK = 3
ELSE IF( LSAME( PACK, 'B' ) ) THEN
IPACK = 5
ISYMPK = 3
ELSE IF( LSAME( PACK, 'Q' ) ) THEN
IPACK = 6
ISYMPK = 2
ELSE IF( LSAME( PACK, 'Z' ) ) THEN
IPACK = 7
ELSE
IPACK = -1
END IF
*
* Set certain internal parameters
*
MNMIN = MIN( M, N )
LLB = MIN( KL, M-1 )
UUB = MIN( KU, N-1 )
MR = MIN( M, N+LLB )
NC = MIN( N, M+UUB )
*
IF( IPACK.EQ.5 .OR. IPACK.EQ.6 ) THEN
MINLDA = UUB + 1
ELSE IF( IPACK.EQ.7 ) THEN
MINLDA = LLB + UUB + 1
ELSE
MINLDA = M
END IF
*
* Use Givens rotation method if bandwidth small enough,
* or if LDA is too small to store the matrix unpacked.
*
GIVENS = .FALSE.
IF( ISYM.EQ.1 ) THEN
IF( DBLE( LLB+UUB ).LT.0.3D0*DBLE( MAX( 1, MR+NC ) ) )
$ GIVENS = .TRUE.
ELSE
IF( 2*LLB.LT.M )
$ GIVENS = .TRUE.
END IF
IF( LDA.LT.M .AND. LDA.GE.MINLDA )
$ GIVENS = .TRUE.
*
* Set INFO if an error
*
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( M.NE.N .AND. ISYM.NE.1 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( IDIST.EQ.-1 ) THEN
INFO = -3
ELSE IF( ISYM.EQ.-1 ) THEN
INFO = -5
ELSE IF( ABS( MODE ).GT.6 ) THEN
INFO = -7
ELSE IF( ( MODE.NE.0 .AND. ABS( MODE ).NE.6 ) .AND. COND.LT.ONE )
$ THEN
INFO = -8
ELSE IF( KL.LT.0 ) THEN
INFO = -10
ELSE IF( KU.LT.0 .OR. ( ISYM.NE.1 .AND. KL.NE.KU ) ) THEN
INFO = -11
ELSE IF( IPACK.EQ.-1 .OR. ( ISYMPK.EQ.1 .AND. ISYM.EQ.1 ) .OR.
$ ( ISYMPK.EQ.2 .AND. ISYM.EQ.1 .AND. KL.GT.0 ) .OR.
$ ( ISYMPK.EQ.3 .AND. ISYM.EQ.1 .AND. KU.GT.0 ) .OR.
$ ( ISYMPK.NE.0 .AND. M.NE.N ) ) THEN
INFO = -12
ELSE IF( LDA.LT.MAX( 1, MINLDA ) ) THEN
INFO = -14
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLATMS', -INFO )
RETURN
END IF
*
* Initialize random number generator
*
DO 10 I = 1, 4
ISEED( I ) = MOD( ABS( ISEED( I ) ), 4096 )
10 CONTINUE
*
IF( MOD( ISEED( 4 ), 2 ).NE.1 )
$ ISEED( 4 ) = ISEED( 4 ) + 1
*
* 2) Set up D if indicated.
*
* Compute D according to COND and MODE
*
CALL DLATM1( MODE, COND, IRSIGN, IDIST, ISEED, D, MNMIN, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = 1
RETURN
END IF
*
* Choose Top-Down if D is (apparently) increasing,
* Bottom-Up if D is (apparently) decreasing.
*
IF( ABS( D( 1 ) ).LE.ABS( D( MNMIN ) ) ) THEN
TOPDWN = .TRUE.
ELSE
TOPDWN = .FALSE.
END IF
*
IF( MODE.NE.0 .AND. ABS( MODE ).NE.6 ) THEN
*
* Scale by DMAX
*
TEMP = ABS( D( 1 ) )
DO 20 I = 2, MNMIN
TEMP = MAX( TEMP, ABS( D( I ) ) )
20 CONTINUE
*
IF( TEMP.GT.ZERO ) THEN
ALPHA = DMAX / TEMP
ELSE
INFO = 2
RETURN
END IF
*
CALL DSCAL( MNMIN, ALPHA, D, 1 )
*
END IF
*
* 3) Generate Banded Matrix using Givens rotations.
* Also the special case of UUB=LLB=0
*
* Compute Addressing constants to cover all
* storage formats. Whether GE, SY, GB, or SB,
* upper or lower triangle or both,
* the (i,j)-th element is in
* A( i - ISKEW*j + IOFFST, j )
*
IF( IPACK.GT.4 ) THEN
ILDA = LDA - 1
ISKEW = 1
IF( IPACK.GT.5 ) THEN
IOFFST = UUB + 1
ELSE
IOFFST = 1
END IF
ELSE
ILDA = LDA
ISKEW = 0
IOFFST = 0
END IF
*
* IPACKG is the format that the matrix is generated in. If this is
* different from IPACK, then the matrix must be repacked at the
* end. It also signals how to compute the norm, for scaling.
*
IPACKG = 0
CALL DLASET( 'Full', LDA, N, ZERO, ZERO, A, LDA )
*
* Diagonal Matrix -- We are done, unless it
* is to be stored SP/PP/TP (PACK='R' or 'C')
*
IF( LLB.EQ.0 .AND. UUB.EQ.0 ) THEN
CALL DCOPY( MNMIN, D, 1, A( 1-ISKEW+IOFFST, 1 ), ILDA+1 )
IF( IPACK.LE.2 .OR. IPACK.GE.5 )
$ IPACKG = IPACK
*
ELSE IF( GIVENS ) THEN
*
* Check whether to use Givens rotations,
* Householder transformations, or nothing.
*
IF( ISYM.EQ.1 ) THEN
*
* Non-symmetric -- A = U D V
*
IF( IPACK.GT.4 ) THEN
IPACKG = IPACK
ELSE
IPACKG = 0
END IF
*
CALL DCOPY( MNMIN, D, 1, A( 1-ISKEW+IOFFST, 1 ), ILDA+1 )
*
IF( TOPDWN ) THEN
JKL = 0
DO 50 JKU = 1, UUB
*
* Transform from bandwidth JKL, JKU-1 to JKL, JKU
*
* Last row actually rotated is M
* Last column actually rotated is MIN( M+JKU, N )
*
DO 40 JR = 1, MIN( M+JKU, N ) + JKL - 1
EXTRA = ZERO
ANGLE = TWOPI*DLARND( 1, ISEED )
C = COS( ANGLE )
S = SIN( ANGLE )
ICOL = MAX( 1, JR-JKL )
IF( JR.LT.M ) THEN
IL = MIN( N, JR+JKU ) + 1 - ICOL
CALL DLAROT( .TRUE., JR.GT.JKL, .FALSE., IL, C,
$ S, A( JR-ISKEW*ICOL+IOFFST, ICOL ),
$ ILDA, EXTRA, DUMMY )
END IF
*
* Chase "EXTRA" back up
*
IR = JR
IC = ICOL
DO 30 JCH = JR - JKL, 1, -JKL - JKU
IF( IR.LT.M ) THEN
CALL DLARTG( A( IR+1-ISKEW*( IC+1 )+IOFFST,
$ IC+1 ), EXTRA, C, S, DUMMY )
END IF
IROW = MAX( 1, JCH-JKU )
IL = IR + 2 - IROW
TEMP = ZERO
ILTEMP = JCH.GT.JKU
CALL DLAROT( .FALSE., ILTEMP, .TRUE., IL, C, -S,
$ A( IROW-ISKEW*IC+IOFFST, IC ),
$ ILDA, TEMP, EXTRA )
IF( ILTEMP ) THEN
CALL DLARTG( A( IROW+1-ISKEW*( IC+1 )+IOFFST,
$ IC+1 ), TEMP, C, S, DUMMY )
ICOL = MAX( 1, JCH-JKU-JKL )
IL = IC + 2 - ICOL
EXTRA = ZERO
CALL DLAROT( .TRUE., JCH.GT.JKU+JKL, .TRUE.,
$ IL, C, -S, A( IROW-ISKEW*ICOL+
$ IOFFST, ICOL ), ILDA, EXTRA,
$ TEMP )
IC = ICOL
IR = IROW
END IF
30 CONTINUE
40 CONTINUE
50 CONTINUE
*
JKU = UUB
DO 80 JKL = 1, LLB
*
* Transform from bandwidth JKL-1, JKU to JKL, JKU
*
DO 70 JC = 1, MIN( N+JKL, M ) + JKU - 1
EXTRA = ZERO
ANGLE = TWOPI*DLARND( 1, ISEED )
C = COS( ANGLE )
S = SIN( ANGLE )
IROW = MAX( 1, JC-JKU )
IF( JC.LT.N ) THEN
IL = MIN( M, JC+JKL ) + 1 - IROW
CALL DLAROT( .FALSE., JC.GT.JKU, .FALSE., IL, C,
$ S, A( IROW-ISKEW*JC+IOFFST, JC ),
$ ILDA, EXTRA, DUMMY )
END IF
*
* Chase "EXTRA" back up
*
IC = JC
IR = IROW
DO 60 JCH = JC - JKU, 1, -JKL - JKU
IF( IC.LT.N ) THEN
CALL DLARTG( A( IR+1-ISKEW*( IC+1 )+IOFFST,
$ IC+1 ), EXTRA, C, S, DUMMY )
END IF
ICOL = MAX( 1, JCH-JKL )
IL = IC + 2 - ICOL
TEMP = ZERO
ILTEMP = JCH.GT.JKL
CALL DLAROT( .TRUE., ILTEMP, .TRUE., IL, C, -S,
$ A( IR-ISKEW*ICOL+IOFFST, ICOL ),
$ ILDA, TEMP, EXTRA )
IF( ILTEMP ) THEN
CALL DLARTG( A( IR+1-ISKEW*( ICOL+1 )+IOFFST,
$ ICOL+1 ), TEMP, C, S, DUMMY )
IROW = MAX( 1, JCH-JKL-JKU )
IL = IR + 2 - IROW
EXTRA = ZERO
CALL DLAROT( .FALSE., JCH.GT.JKL+JKU, .TRUE.,
$ IL, C, -S, A( IROW-ISKEW*ICOL+
$ IOFFST, ICOL ), ILDA, EXTRA,
$ TEMP )
IC = ICOL
IR = IROW
END IF
60 CONTINUE
70 CONTINUE
80 CONTINUE
*
ELSE
*
* Bottom-Up -- Start at the bottom right.
*
JKL = 0
DO 110 JKU = 1, UUB
*
* Transform from bandwidth JKL, JKU-1 to JKL, JKU
*
* First row actually rotated is M
* First column actually rotated is MIN( M+JKU, N )
*
IENDCH = MIN( M, N+JKL ) - 1
DO 100 JC = MIN( M+JKU, N ) - 1, 1 - JKL, -1
EXTRA = ZERO
ANGLE = TWOPI*DLARND( 1, ISEED )
C = COS( ANGLE )
S = SIN( ANGLE )
IROW = MAX( 1, JC-JKU+1 )
IF( JC.GT.0 ) THEN
IL = MIN( M, JC+JKL+1 ) + 1 - IROW
CALL DLAROT( .FALSE., .FALSE., JC+JKL.LT.M, IL,
$ C, S, A( IROW-ISKEW*JC+IOFFST,
$ JC ), ILDA, DUMMY, EXTRA )
END IF
*
* Chase "EXTRA" back down
*
IC = JC
DO 90 JCH = JC + JKL, IENDCH, JKL + JKU
ILEXTR = IC.GT.0
IF( ILEXTR ) THEN
CALL DLARTG( A( JCH-ISKEW*IC+IOFFST, IC ),
$ EXTRA, C, S, DUMMY )
END IF
IC = MAX( 1, IC )
ICOL = MIN( N-1, JCH+JKU )
ILTEMP = JCH + JKU.LT.N
TEMP = ZERO
CALL DLAROT( .TRUE., ILEXTR, ILTEMP, ICOL+2-IC,
$ C, S, A( JCH-ISKEW*IC+IOFFST, IC ),
$ ILDA, EXTRA, TEMP )
IF( ILTEMP ) THEN
CALL DLARTG( A( JCH-ISKEW*ICOL+IOFFST,
$ ICOL ), TEMP, C, S, DUMMY )
IL = MIN( IENDCH, JCH+JKL+JKU ) + 2 - JCH
EXTRA = ZERO
CALL DLAROT( .FALSE., .TRUE.,
$ JCH+JKL+JKU.LE.IENDCH, IL, C, S,
$ A( JCH-ISKEW*ICOL+IOFFST,
$ ICOL ), ILDA, TEMP, EXTRA )
IC = ICOL
END IF
90 CONTINUE
100 CONTINUE
110 CONTINUE
*
JKU = UUB
DO 140 JKL = 1, LLB
*
* Transform from bandwidth JKL-1, JKU to JKL, JKU
*
* First row actually rotated is MIN( N+JKL, M )
* First column actually rotated is N
*
IENDCH = MIN( N, M+JKU ) - 1
DO 130 JR = MIN( N+JKL, M ) - 1, 1 - JKU, -1
EXTRA = ZERO
ANGLE = TWOPI*DLARND( 1, ISEED )
C = COS( ANGLE )
S = SIN( ANGLE )
ICOL = MAX( 1, JR-JKL+1 )
IF( JR.GT.0 ) THEN
IL = MIN( N, JR+JKU+1 ) + 1 - ICOL
CALL DLAROT( .TRUE., .FALSE., JR+JKU.LT.N, IL,
$ C, S, A( JR-ISKEW*ICOL+IOFFST,
$ ICOL ), ILDA, DUMMY, EXTRA )
END IF
*
* Chase "EXTRA" back down
*
IR = JR
DO 120 JCH = JR + JKU, IENDCH, JKL + JKU
ILEXTR = IR.GT.0
IF( ILEXTR ) THEN
CALL DLARTG( A( IR-ISKEW*JCH+IOFFST, JCH ),
$ EXTRA, C, S, DUMMY )
END IF
IR = MAX( 1, IR )
IROW = MIN( M-1, JCH+JKL )
ILTEMP = JCH + JKL.LT.M
TEMP = ZERO
CALL DLAROT( .FALSE., ILEXTR, ILTEMP, IROW+2-IR,
$ C, S, A( IR-ISKEW*JCH+IOFFST,
$ JCH ), ILDA, EXTRA, TEMP )
IF( ILTEMP ) THEN
CALL DLARTG( A( IROW-ISKEW*JCH+IOFFST, JCH ),
$ TEMP, C, S, DUMMY )
IL = MIN( IENDCH, JCH+JKL+JKU ) + 2 - JCH
EXTRA = ZERO
CALL DLAROT( .TRUE., .TRUE.,
$ JCH+JKL+JKU.LE.IENDCH, IL, C, S,
$ A( IROW-ISKEW*JCH+IOFFST, JCH ),
$ ILDA, TEMP, EXTRA )
IR = IROW
END IF
120 CONTINUE
130 CONTINUE
140 CONTINUE
END IF
*
ELSE
*
* Symmetric -- A = U D U'
*
IPACKG = IPACK
IOFFG = IOFFST
*
IF( TOPDWN ) THEN
*
* Top-Down -- Generate Upper triangle only
*
IF( IPACK.GE.5 ) THEN
IPACKG = 6
IOFFG = UUB + 1
ELSE
IPACKG = 1
END IF
CALL DCOPY( MNMIN, D, 1, A( 1-ISKEW+IOFFG, 1 ), ILDA+1 )
*
DO 170 K = 1, UUB
DO 160 JC = 1, N - 1
IROW = MAX( 1, JC-K )
IL = MIN( JC+1, K+2 )
EXTRA = ZERO
TEMP = A( JC-ISKEW*( JC+1 )+IOFFG, JC+1 )
ANGLE = TWOPI*DLARND( 1, ISEED )
C = COS( ANGLE )
S = SIN( ANGLE )
CALL DLAROT( .FALSE., JC.GT.K, .TRUE., IL, C, S,
$ A( IROW-ISKEW*JC+IOFFG, JC ), ILDA,
$ EXTRA, TEMP )
CALL DLAROT( .TRUE., .TRUE., .FALSE.,
$ MIN( K, N-JC )+1, C, S,
$ A( ( 1-ISKEW )*JC+IOFFG, JC ), ILDA,
$ TEMP, DUMMY )
*
* Chase EXTRA back up the matrix
*
ICOL = JC
DO 150 JCH = JC - K, 1, -K
CALL DLARTG( A( JCH+1-ISKEW*( ICOL+1 )+IOFFG,
$ ICOL+1 ), EXTRA, C, S, DUMMY )
TEMP = A( JCH-ISKEW*( JCH+1 )+IOFFG, JCH+1 )
CALL DLAROT( .TRUE., .TRUE., .TRUE., K+2, C, -S,
$ A( ( 1-ISKEW )*JCH+IOFFG, JCH ),
$ ILDA, TEMP, EXTRA )
IROW = MAX( 1, JCH-K )
IL = MIN( JCH+1, K+2 )
EXTRA = ZERO
CALL DLAROT( .FALSE., JCH.GT.K, .TRUE., IL, C,
$ -S, A( IROW-ISKEW*JCH+IOFFG, JCH ),
$ ILDA, EXTRA, TEMP )
ICOL = JCH
150 CONTINUE
160 CONTINUE
170 CONTINUE
*
* If we need lower triangle, copy from upper. Note that
* the order of copying is chosen to work for 'q' -> 'b'
*
IF( IPACK.NE.IPACKG .AND. IPACK.NE.3 ) THEN
DO 190 JC = 1, N
IROW = IOFFST - ISKEW*JC
DO 180 JR = JC, MIN( N, JC+UUB )
A( JR+IROW, JC ) = A( JC-ISKEW*JR+IOFFG, JR )
180 CONTINUE
190 CONTINUE
IF( IPACK.EQ.5 ) THEN
DO 210 JC = N - UUB + 1, N
DO 200 JR = N + 2 - JC, UUB + 1
A( JR, JC ) = ZERO
200 CONTINUE
210 CONTINUE
END IF
IF( IPACKG.EQ.6 ) THEN
IPACKG = IPACK
ELSE
IPACKG = 0
END IF
END IF
ELSE
*
* Bottom-Up -- Generate Lower triangle only
*
IF( IPACK.GE.5 ) THEN
IPACKG = 5
IF( IPACK.EQ.6 )
$ IOFFG = 1
ELSE
IPACKG = 2
END IF
CALL DCOPY( MNMIN, D, 1, A( 1-ISKEW+IOFFG, 1 ), ILDA+1 )
*
DO 240 K = 1, UUB
DO 230 JC = N - 1, 1, -1
IL = MIN( N+1-JC, K+2 )
EXTRA = ZERO
TEMP = A( 1+( 1-ISKEW )*JC+IOFFG, JC )
ANGLE = TWOPI*DLARND( 1, ISEED )
C = COS( ANGLE )
S = -SIN( ANGLE )
CALL DLAROT( .FALSE., .TRUE., N-JC.GT.K, IL, C, S,
$ A( ( 1-ISKEW )*JC+IOFFG, JC ), ILDA,
$ TEMP, EXTRA )
ICOL = MAX( 1, JC-K+1 )
CALL DLAROT( .TRUE., .FALSE., .TRUE., JC+2-ICOL, C,
$ S, A( JC-ISKEW*ICOL+IOFFG, ICOL ),
$ ILDA, DUMMY, TEMP )
*
* Chase EXTRA back down the matrix
*
ICOL = JC
DO 220 JCH = JC + K, N - 1, K
CALL DLARTG( A( JCH-ISKEW*ICOL+IOFFG, ICOL ),
$ EXTRA, C, S, DUMMY )
TEMP = A( 1+( 1-ISKEW )*JCH+IOFFG, JCH )
CALL DLAROT( .TRUE., .TRUE., .TRUE., K+2, C, S,
$ A( JCH-ISKEW*ICOL+IOFFG, ICOL ),
$ ILDA, EXTRA, TEMP )
IL = MIN( N+1-JCH, K+2 )
EXTRA = ZERO
CALL DLAROT( .FALSE., .TRUE., N-JCH.GT.K, IL, C,
$ S, A( ( 1-ISKEW )*JCH+IOFFG, JCH ),
$ ILDA, TEMP, EXTRA )
ICOL = JCH
220 CONTINUE
230 CONTINUE
240 CONTINUE
*
* If we need upper triangle, copy from lower. Note that
* the order of copying is chosen to work for 'b' -> 'q'
*
IF( IPACK.NE.IPACKG .AND. IPACK.NE.4 ) THEN
DO 260 JC = N, 1, -1
IROW = IOFFST - ISKEW*JC
DO 250 JR = JC, MAX( 1, JC-UUB ), -1
A( JR+IROW, JC ) = A( JC-ISKEW*JR+IOFFG, JR )
250 CONTINUE
260 CONTINUE
IF( IPACK.EQ.6 ) THEN
DO 280 JC = 1, UUB
DO 270 JR = 1, UUB + 1 - JC
A( JR, JC ) = ZERO
270 CONTINUE
280 CONTINUE
END IF
IF( IPACKG.EQ.5 ) THEN
IPACKG = IPACK
ELSE
IPACKG = 0
END IF
END IF
END IF
END IF
*
ELSE
*
* 4) Generate Banded Matrix by first
* Rotating by random Unitary matrices,
* then reducing the bandwidth using Householder
* transformations.
*
* Note: we should get here only if LDA .ge. N
*
IF( ISYM.EQ.1 ) THEN
*
* Non-symmetric -- A = U D V
*
CALL DLAGGE( MR, NC, LLB, UUB, D, A, LDA, ISEED, WORK,
$ IINFO )
ELSE
*
* Symmetric -- A = U D U'
*
CALL DLAGSY( M, LLB, D, A, LDA, ISEED, WORK, IINFO )
*
END IF
IF( IINFO.NE.0 ) THEN
INFO = 3
RETURN
END IF
END IF
*
* 5) Pack the matrix
*
IF( IPACK.NE.IPACKG ) THEN
IF( IPACK.EQ.1 ) THEN
*
* 'U' -- Upper triangular, not packed
*
DO 300 J = 1, M
DO 290 I = J + 1, M
A( I, J ) = ZERO
290 CONTINUE
300 CONTINUE
*
ELSE IF( IPACK.EQ.2 ) THEN
*
* 'L' -- Lower triangular, not packed
*
DO 320 J = 2, M
DO 310 I = 1, J - 1
A( I, J ) = ZERO
310 CONTINUE
320 CONTINUE
*
ELSE IF( IPACK.EQ.3 ) THEN
*
* 'C' -- Upper triangle packed Columnwise.
*
ICOL = 1
IROW = 0
DO 340 J = 1, M
DO 330 I = 1, J
IROW = IROW + 1
IF( IROW.GT.LDA ) THEN
IROW = 1
ICOL = ICOL + 1
END IF
A( IROW, ICOL ) = A( I, J )
330 CONTINUE
340 CONTINUE
*
ELSE IF( IPACK.EQ.4 ) THEN
*
* 'R' -- Lower triangle packed Columnwise.
*
ICOL = 1
IROW = 0
DO 360 J = 1, M
DO 350 I = J, M
IROW = IROW + 1
IF( IROW.GT.LDA ) THEN
IROW = 1
ICOL = ICOL + 1
END IF
A( IROW, ICOL ) = A( I, J )
350 CONTINUE
360 CONTINUE
*
ELSE IF( IPACK.GE.5 ) THEN
*
* 'B' -- The lower triangle is packed as a band matrix.
* 'Q' -- The upper triangle is packed as a band matrix.
* 'Z' -- The whole matrix is packed as a band matrix.
*
IF( IPACK.EQ.5 )
$ UUB = 0
IF( IPACK.EQ.6 )
$ LLB = 0
*
DO 380 J = 1, UUB
DO 370 I = MIN( J+LLB, M ), 1, -1
A( I-J+UUB+1, J ) = A( I, J )
370 CONTINUE
380 CONTINUE
*
DO 400 J = UUB + 2, N
DO 390 I = J - UUB, MIN( J+LLB, M )
A( I-J+UUB+1, J ) = A( I, J )
390 CONTINUE
400 CONTINUE
END IF
*
* If packed, zero out extraneous elements.
*
* Symmetric/Triangular Packed --
* zero out everything after A(IROW,ICOL)
*
IF( IPACK.EQ.3 .OR. IPACK.EQ.4 ) THEN
DO 420 JC = ICOL, M
DO 410 JR = IROW + 1, LDA
A( JR, JC ) = ZERO
410 CONTINUE
IROW = 0
420 CONTINUE
*
ELSE IF( IPACK.GE.5 ) THEN
*
* Packed Band --
* 1st row is now in A( UUB+2-j, j), zero above it
* m-th row is now in A( M+UUB-j,j), zero below it
* last non-zero diagonal is now in A( UUB+LLB+1,j ),
* zero below it, too.
*
IR1 = UUB + LLB + 2
IR2 = UUB + M + 2
DO 450 JC = 1, N
DO 430 JR = 1, UUB + 1 - JC
A( JR, JC ) = ZERO
430 CONTINUE
DO 440 JR = MAX( 1, MIN( IR1, IR2-JC ) ), LDA
A( JR, JC ) = ZERO
440 CONTINUE
450 CONTINUE
END IF
END IF
*
RETURN
*
* End of DLATMS
*
END
|
