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
|
SUBROUTINE CLATME( N, DIST, ISEED, D, MODE, COND, DMAX, EI, RSIGN,
$ UPPER, SIM, DS, MODES, CONDS, KL, KU, ANORM, 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, EI, RSIGN, SIM, UPPER
INTEGER INFO, KL, KU, LDA, MODE, MODES, N
REAL ANORM, COND, CONDS
COMPLEX DMAX
* ..
* .. Array Arguments ..
INTEGER ISEED( 4 )
REAL DS( * )
COMPLEX A( LDA, * ), D( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* CLATME generates random non-symmetric square matrices with
* specified eigenvalues for testing LAPACK programs.
*
* CLATME operates by applying the following sequence of
* operations:
*
* 1. Set the diagonal to D, where D may be input or
* computed according to MODE, COND, DMAX, and RSIGN
* as described below.
*
* 2. If UPPER='T', the upper triangle of A is set to random values
* out of distribution DIST.
*
* 3. If SIM='T', A is multiplied on the left by a random matrix
* X, whose singular values are specified by DS, MODES, and
* CONDS, and on the right by X inverse.
*
* 4. If KL < N-1, the lower bandwidth is reduced to KL using
* Householder transformations. If KU < N-1, the upper
* bandwidth is reduced to KU.
*
* 5. If ANORM is not negative, the matrix is scaled to have
* maximum-element-norm ANORM.
*
* (Note: since the matrix cannot be reduced beyond Hessenberg form,
* no packing options are available.)
*
* Arguments
* =========
*
* N - INTEGER
* The number of columns (or rows) 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, and on the
* upper triangle (see UPPER).
* 'U' => UNIFORM( 0, 1 ) ( 'U' for uniform )
* 'S' => UNIFORM( -1, 1 ) ( 'S' for symmetric )
* 'N' => NORMAL( 0, 1 ) ( 'N' for normal )
* 'D' => uniform on the complex disc |z| < 1.
* 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 CLATME
* to continue the same random number sequence.
* Changed on exit.
*
* D - COMPLEX array, dimension ( N )
* This array is used to specify the eigenvalues of A. If
* MODE=0, then D is assumed to contain the eigenvalues
* otherwise they will be computed according to MODE, COND,
* DMAX, and RSIGN and placed in D.
* Modified if MODE is nonzero.
*
* MODE - INTEGER
* On entry this describes how the 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 between 1 and 4, D has entries ranging
* from 1 to 1/COND, if between -1 and -4, D has entries
* ranging from 1/COND to 1,
* Not modified.
*
* COND - REAL
* On entry, this is used as described under MODE above.
* If used, it must be >= 1. Not modified.
*
* DMAX - COMPLEX
* 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))). Note that DMAX need not be
* positive or real: if DMAX is negative or complex (or zero),
* D will be scaled by a negative or complex number (or zero).
* If RSIGN='F' then the largest (absolute) eigenvalue will be
* equal to DMAX.
* Not modified.
*
* EI - CHARACTER*1 (ignored)
* Not modified.
*
* RSIGN - CHARACTER*1
* If MODE is not 0, 6, or -6, and RSIGN='T', then the
* elements of D, as computed according to MODE and COND, will
* be multiplied by a random complex number from the unit
* circle |z| = 1. If RSIGN='F', they will not be. RSIGN may
* only have the values 'T' or 'F'.
* Not modified.
*
* UPPER - CHARACTER*1
* If UPPER='T', then the elements of A above the diagonal
* will be set to random numbers out of DIST. If UPPER='F',
* they will not. UPPER may only have the values 'T' or 'F'.
* Not modified.
*
* SIM - CHARACTER*1
* If SIM='T', then A will be operated on by a "similarity
* transform", i.e., multiplied on the left by a matrix X and
* on the right by X inverse. X = U S V, where U and V are
* random unitary matrices and S is a (diagonal) matrix of
* singular values specified by DS, MODES, and CONDS. If
* SIM='F', then A will not be transformed.
* Not modified.
*
* DS - REAL array, dimension ( N )
* This array is used to specify the singular values of X,
* in the same way that D specifies the eigenvalues of A.
* If MODE=0, the DS contains the singular values, which
* may not be zero.
* Modified if MODE is nonzero.
*
* MODES - INTEGER
* CONDS - REAL
* Similar to MODE and COND, but for specifying the diagonal
* of S. MODES=-6 and +6 are not allowed (since they would
* result in randomly ill-conditioned eigenvalues.)
*
* KL - INTEGER
* This specifies the lower bandwidth of the matrix. KL=1
* specifies upper Hessenberg form. If KL is at least N-1,
* then A will have full lower bandwidth.
* Not modified.
*
* KU - INTEGER
* This specifies the upper bandwidth of the matrix. KU=1
* specifies lower Hessenberg form. If KU is at least N-1,
* then A will have full upper bandwidth; if KU and KL
* are both at least N-1, then A will be dense. Only one of
* KU and KL may be less than N-1.
* Not modified.
*
* ANORM - REAL
* If ANORM is not negative, then A will be scaled by a non-
* negative real number to make the maximum-element-norm of A
* to be ANORM.
* Not modified.
*
* A - COMPLEX array, dimension ( LDA, N )
* On exit A is the desired test matrix.
* Modified.
*
* LDA - INTEGER
* LDA specifies the first dimension of A as declared in the
* calling program. LDA must be at least M.
* Not modified.
*
* WORK - COMPLEX array, dimension ( 3*N )
* Workspace.
* Modified.
*
* INFO - INTEGER
* Error code. On exit, INFO will be set to one of the
* following values:
* 0 => normal return
* -1 => N negative
* -2 => DIST illegal string
* -5 => MODE not in range -6 to 6
* -6 => COND less than 1.0, and MODE neither -6, 0 nor 6
* -9 => RSIGN is not 'T' or 'F'
* -10 => UPPER is not 'T' or 'F'
* -11 => SIM is not 'T' or 'F'
* -12 => MODES=0 and DS has a zero singular value.
* -13 => MODES is not in the range -5 to 5.
* -14 => MODES is nonzero and CONDS is less than 1.
* -15 => KL is less than 1.
* -16 => KU is less than 1, or KL and KU are both less than
* N-1.
* -19 => LDA is less than M.
* 1 => Error return from CLATM1 (computing D)
* 2 => Cannot scale to DMAX (max. eigenvalue is 0)
* 3 => Error return from SLATM1 (computing DS)
* 4 => Error return from CLARGE
* 5 => Zero singular value from SLATM1.
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO
PARAMETER ( ZERO = 0.0E+0 )
REAL ONE
PARAMETER ( ONE = 1.0E+0 )
COMPLEX CZERO
PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ) )
COMPLEX CONE
PARAMETER ( CONE = ( 1.0E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
LOGICAL BADS
INTEGER I, IC, ICOLS, IDIST, IINFO, IR, IROWS, IRSIGN,
$ ISIM, IUPPER, J, JC, JCR
REAL RALPHA, TEMP
COMPLEX ALPHA, TAU, XNORMS
* ..
* .. Local Arrays ..
REAL TEMPA( 1 )
* ..
* .. External Functions ..
LOGICAL LSAME
REAL CLANGE
COMPLEX CLARND
EXTERNAL LSAME, CLANGE, CLARND
* ..
* .. External Subroutines ..
EXTERNAL CCOPY, CGEMV, CGERC, CLACGV, CLARFG, CLARGE,
$ CLARNV, CLATM1, CLASET, CSCAL, CSSCAL, SLATM1,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, CONJG, MAX, MOD
* ..
* .. Executable Statements ..
*
* 1) Decode and Test the input parameters.
* Initialize flags & seed.
*
INFO = 0
*
* Quick return if possible
*
IF( 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 IF( LSAME( DIST, 'D' ) ) THEN
IDIST = 4
ELSE
IDIST = -1
END IF
*
* Decode RSIGN
*
IF( LSAME( RSIGN, 'T' ) ) THEN
IRSIGN = 1
ELSE IF( LSAME( RSIGN, 'F' ) ) THEN
IRSIGN = 0
ELSE
IRSIGN = -1
END IF
*
* Decode UPPER
*
IF( LSAME( UPPER, 'T' ) ) THEN
IUPPER = 1
ELSE IF( LSAME( UPPER, 'F' ) ) THEN
IUPPER = 0
ELSE
IUPPER = -1
END IF
*
* Decode SIM
*
IF( LSAME( SIM, 'T' ) ) THEN
ISIM = 1
ELSE IF( LSAME( SIM, 'F' ) ) THEN
ISIM = 0
ELSE
ISIM = -1
END IF
*
* Check DS, if MODES=0 and ISIM=1
*
BADS = .FALSE.
IF( MODES.EQ.0 .AND. ISIM.EQ.1 ) THEN
DO 10 J = 1, N
IF( DS( J ).EQ.ZERO )
$ BADS = .TRUE.
10 CONTINUE
END IF
*
* Set INFO if an error
*
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( IDIST.EQ.-1 ) THEN
INFO = -2
ELSE IF( ABS( MODE ).GT.6 ) THEN
INFO = -5
ELSE IF( ( MODE.NE.0 .AND. ABS( MODE ).NE.6 ) .AND. COND.LT.ONE )
$ THEN
INFO = -6
ELSE IF( IRSIGN.EQ.-1 ) THEN
INFO = -9
ELSE IF( IUPPER.EQ.-1 ) THEN
INFO = -10
ELSE IF( ISIM.EQ.-1 ) THEN
INFO = -11
ELSE IF( BADS ) THEN
INFO = -12
ELSE IF( ISIM.EQ.1 .AND. ABS( MODES ).GT.5 ) THEN
INFO = -13
ELSE IF( ISIM.EQ.1 .AND. MODES.NE.0 .AND. CONDS.LT.ONE ) THEN
INFO = -14
ELSE IF( KL.LT.1 ) THEN
INFO = -15
ELSE IF( KU.LT.1 .OR. ( KU.LT.N-1 .AND. KL.LT.N-1 ) ) THEN
INFO = -16
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -19
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CLATME', -INFO )
RETURN
END IF
*
* Initialize random number generator
*
DO 20 I = 1, 4
ISEED( I ) = MOD( ABS( ISEED( I ) ), 4096 )
20 CONTINUE
*
IF( MOD( ISEED( 4 ), 2 ).NE.1 )
$ ISEED( 4 ) = ISEED( 4 ) + 1
*
* 2) Set up diagonal of A
*
* Compute D according to COND and MODE
*
CALL CLATM1( MODE, COND, IRSIGN, IDIST, ISEED, D, N, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = 1
RETURN
END IF
IF( MODE.NE.0 .AND. ABS( MODE ).NE.6 ) THEN
*
* Scale by DMAX
*
TEMP = ABS( D( 1 ) )
DO 30 I = 2, N
TEMP = MAX( TEMP, ABS( D( I ) ) )
30 CONTINUE
*
IF( TEMP.GT.ZERO ) THEN
ALPHA = DMAX / TEMP
ELSE
INFO = 2
RETURN
END IF
*
CALL CSCAL( N, ALPHA, D, 1 )
*
END IF
*
CALL CLASET( 'Full', N, N, CZERO, CZERO, A, LDA )
CALL CCOPY( N, D, 1, A, LDA+1 )
*
* 3) If UPPER='T', set upper triangle of A to random numbers.
*
IF( IUPPER.NE.0 ) THEN
DO 40 JC = 2, N
CALL CLARNV( IDIST, ISEED, JC-1, A( 1, JC ) )
40 CONTINUE
END IF
*
* 4) If SIM='T', apply similarity transformation.
*
* -1
* Transform is X A X , where X = U S V, thus
*
* it is U S V A V' (1/S) U'
*
IF( ISIM.NE.0 ) THEN
*
* Compute S (singular values of the eigenvector matrix)
* according to CONDS and MODES
*
CALL SLATM1( MODES, CONDS, 0, 0, ISEED, DS, N, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = 3
RETURN
END IF
*
* Multiply by V and V'
*
CALL CLARGE( N, A, LDA, ISEED, WORK, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = 4
RETURN
END IF
*
* Multiply by S and (1/S)
*
DO 50 J = 1, N
CALL CSSCAL( N, DS( J ), A( J, 1 ), LDA )
IF( DS( J ).NE.ZERO ) THEN
CALL CSSCAL( N, ONE / DS( J ), A( 1, J ), 1 )
ELSE
INFO = 5
RETURN
END IF
50 CONTINUE
*
* Multiply by U and U'
*
CALL CLARGE( N, A, LDA, ISEED, WORK, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = 4
RETURN
END IF
END IF
*
* 5) Reduce the bandwidth.
*
IF( KL.LT.N-1 ) THEN
*
* Reduce bandwidth -- kill column
*
DO 60 JCR = KL + 1, N - 1
IC = JCR - KL
IROWS = N + 1 - JCR
ICOLS = N + KL - JCR
*
CALL CCOPY( IROWS, A( JCR, IC ), 1, WORK, 1 )
XNORMS = WORK( 1 )
CALL CLARFG( IROWS, XNORMS, WORK( 2 ), 1, TAU )
TAU = CONJG( TAU )
WORK( 1 ) = CONE
ALPHA = CLARND( 5, ISEED )
*
CALL CGEMV( 'C', IROWS, ICOLS, CONE, A( JCR, IC+1 ), LDA,
$ WORK, 1, CZERO, WORK( IROWS+1 ), 1 )
CALL CGERC( IROWS, ICOLS, -TAU, WORK, 1, WORK( IROWS+1 ), 1,
$ A( JCR, IC+1 ), LDA )
*
CALL CGEMV( 'N', N, IROWS, CONE, A( 1, JCR ), LDA, WORK, 1,
$ CZERO, WORK( IROWS+1 ), 1 )
CALL CGERC( N, IROWS, -CONJG( TAU ), WORK( IROWS+1 ), 1,
$ WORK, 1, A( 1, JCR ), LDA )
*
A( JCR, IC ) = XNORMS
CALL CLASET( 'Full', IROWS-1, 1, CZERO, CZERO,
$ A( JCR+1, IC ), LDA )
*
CALL CSCAL( ICOLS+1, ALPHA, A( JCR, IC ), LDA )
CALL CSCAL( N, CONJG( ALPHA ), A( 1, JCR ), 1 )
60 CONTINUE
ELSE IF( KU.LT.N-1 ) THEN
*
* Reduce upper bandwidth -- kill a row at a time.
*
DO 70 JCR = KU + 1, N - 1
IR = JCR - KU
IROWS = N + KU - JCR
ICOLS = N + 1 - JCR
*
CALL CCOPY( ICOLS, A( IR, JCR ), LDA, WORK, 1 )
XNORMS = WORK( 1 )
CALL CLARFG( ICOLS, XNORMS, WORK( 2 ), 1, TAU )
TAU = CONJG( TAU )
WORK( 1 ) = CONE
CALL CLACGV( ICOLS-1, WORK( 2 ), 1 )
ALPHA = CLARND( 5, ISEED )
*
CALL CGEMV( 'N', IROWS, ICOLS, CONE, A( IR+1, JCR ), LDA,
$ WORK, 1, CZERO, WORK( ICOLS+1 ), 1 )
CALL CGERC( IROWS, ICOLS, -TAU, WORK( ICOLS+1 ), 1, WORK, 1,
$ A( IR+1, JCR ), LDA )
*
CALL CGEMV( 'C', ICOLS, N, CONE, A( JCR, 1 ), LDA, WORK, 1,
$ CZERO, WORK( ICOLS+1 ), 1 )
CALL CGERC( ICOLS, N, -CONJG( TAU ), WORK, 1,
$ WORK( ICOLS+1 ), 1, A( JCR, 1 ), LDA )
*
A( IR, JCR ) = XNORMS
CALL CLASET( 'Full', 1, ICOLS-1, CZERO, CZERO,
$ A( IR, JCR+1 ), LDA )
*
CALL CSCAL( IROWS+1, ALPHA, A( IR, JCR ), 1 )
CALL CSCAL( N, CONJG( ALPHA ), A( JCR, 1 ), LDA )
70 CONTINUE
END IF
*
* Scale the matrix to have norm ANORM
*
IF( ANORM.GE.ZERO ) THEN
TEMP = CLANGE( 'M', N, N, A, LDA, TEMPA )
IF( TEMP.GT.ZERO ) THEN
RALPHA = ANORM / TEMP
DO 80 J = 1, N
CALL CSSCAL( N, RALPHA, A( 1, J ), 1 )
80 CONTINUE
END IF
END IF
*
RETURN
*
* End of CLATME
*
END
|
