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
|
*> \brief \b ZLAVSY_ROOK
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE ZLAVSY_ROOK( UPLO, TRANS, DIAG, N, NRHS, A, LDA, IPIV, B,
* LDB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, TRANS, UPLO
* INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* COMPLEX*16 A( LDA, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZLAVSY_ROOK performs one of the matrix-vector operations
*> x := A*x or x := A'*x,
*> where x is an N element vector and A is one of the factors
*> from the block U*D*U' or L*D*L' factorization computed by ZSYTRF_ROOK.
*>
*> If TRANS = 'N', multiplies by U or U * D (or L or L * D)
*> If TRANS = 'T', multiplies by U' or D * U' (or L' or D * L')
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the factor stored in A is upper or lower
*> triangular.
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the operation to be performed:
*> = 'N': x := A*x
*> = 'T': x := A'*x
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> Specifies whether or not the diagonal blocks are unit
*> matrices. If the diagonal blocks are assumed to be unit,
*> then A = U or A = L, otherwise A = U*D or A = L*D.
*> = 'U': Diagonal blocks are assumed to be unit matrices.
*> = 'N': Diagonal blocks are assumed to be non-unit matrices.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows and columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of vectors
*> x to be multiplied by A. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is COMPLEX*16 array, dimension (LDA,N)
*> The block diagonal matrix D and the multipliers used to
*> obtain the factor U or L as computed by ZSYTRF_ROOK.
*> Stored as a 2-D triangular matrix.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D,
*> as determined by ZSYTRF_ROOK.
*>
*> If UPLO = 'U':
*> If IPIV(k) > 0, then rows and columns k and IPIV(k)
*> were interchanged and D(k,k) is a 1-by-1 diagonal block.
*> (If IPIV( k ) = k, no interchange was done).
*>
*> If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
*> columns k and -IPIV(k) were interchanged and rows and
*> columns k-1 and -IPIV(k-1) were inerchaged,
*> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
*>
*> If UPLO = 'L':
*> If IPIV(k) > 0, then rows and columns k and IPIV(k)
*> were interchanged and D(k,k) is a 1-by-1 diagonal block.
*> (If IPIV( k ) = k, no interchange was done).
*>
*> If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
*> columns k and -IPIV(k) were interchanged and rows and
*> columns k+1 and -IPIV(k+1) were inerchaged,
*> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is COMPLEX*16 array, dimension (LDB,NRHS)
*> On entry, B contains NRHS vectors of length N.
*> On exit, B is overwritten with the product A * B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -k, the k-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2013
*
*> \ingroup complex16_lin
*
* =====================================================================
SUBROUTINE ZLAVSY_ROOK( UPLO, TRANS, DIAG, N, NRHS, A, LDA, IPIV,
$ B, LDB, INFO )
*
* -- LAPACK test routine (version 3.5.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2013
*
* .. Scalar Arguments ..
CHARACTER DIAG, TRANS, UPLO
INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
COMPLEX*16 A( LDA, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
COMPLEX*16 CONE
PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
LOGICAL NOUNIT
INTEGER J, K, KP
COMPLEX*16 D11, D12, D21, D22, T1, T2
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZGEMV, ZGERU, ZSCAL, ZSWAP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( TRANS, 'N' ) .AND. .NOT.LSAME( TRANS, 'T' ) )
$ THEN
INFO = -2
ELSE IF( .NOT.LSAME( DIAG, 'U' ) .AND. .NOT.LSAME( DIAG, 'N' ) )
$ THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZLAVSY_ROOK ', -INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( N.EQ.0 )
$ RETURN
*
NOUNIT = LSAME( DIAG, 'N' )
*------------------------------------------
*
* Compute B := A * B (No transpose)
*
*------------------------------------------
IF( LSAME( TRANS, 'N' ) ) THEN
*
* Compute B := U*B
* where U = P(m)*inv(U(m))* ... *P(1)*inv(U(1))
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* Loop forward applying the transformations.
*
K = 1
10 CONTINUE
IF( K.GT.N )
$ GO TO 30
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 pivot block
*
* Multiply by the diagonal element if forming U * D.
*
IF( NOUNIT )
$ CALL ZSCAL( NRHS, A( K, K ), B( K, 1 ), LDB )
*
* Multiply by P(K) * inv(U(K)) if K > 1.
*
IF( K.GT.1 ) THEN
*
* Apply the transformation.
*
CALL ZGERU( K-1, NRHS, CONE, A( 1, K ), 1, B( K, 1 ),
$ LDB, B( 1, 1 ), LDB )
*
* Interchange if P(K) != I.
*
KP = IPIV( K )
IF( KP.NE.K )
$ CALL ZSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
END IF
K = K + 1
ELSE
*
* 2 x 2 pivot block
*
* Multiply by the diagonal block if forming U * D.
*
IF( NOUNIT ) THEN
D11 = A( K, K )
D22 = A( K+1, K+1 )
D12 = A( K, K+1 )
D21 = D12
DO 20 J = 1, NRHS
T1 = B( K, J )
T2 = B( K+1, J )
B( K, J ) = D11*T1 + D12*T2
B( K+1, J ) = D21*T1 + D22*T2
20 CONTINUE
END IF
*
* Multiply by P(K) * inv(U(K)) if K > 1.
*
IF( K.GT.1 ) THEN
*
* Apply the transformations.
*
CALL ZGERU( K-1, NRHS, CONE, A( 1, K ), 1, B( K, 1 ),
$ LDB, B( 1, 1 ), LDB )
CALL ZGERU( K-1, NRHS, CONE, A( 1, K+1 ), 1,
$ B( K+1, 1 ), LDB, B( 1, 1 ), LDB )
*
* Interchange if a permutation was applied at the
* K-th step of the factorization.
*
* Swap the first of pair with IMAXth
*
KP = ABS( IPIV( K ) )
IF( KP.NE.K )
$ CALL ZSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
*
* NOW swap the first of pair with Pth
*
KP = ABS( IPIV( K+1 ) )
IF( KP.NE.K+1 )
$ CALL ZSWAP( NRHS, B( K+1, 1 ), LDB, B( KP, 1 ),
$ LDB )
END IF
K = K + 2
END IF
GO TO 10
30 CONTINUE
*
* Compute B := L*B
* where L = P(1)*inv(L(1))* ... *P(m)*inv(L(m)) .
*
ELSE
*
* Loop backward applying the transformations to B.
*
K = N
40 CONTINUE
IF( K.LT.1 )
$ GO TO 60
*
* Test the pivot index. If greater than zero, a 1 x 1
* pivot was used, otherwise a 2 x 2 pivot was used.
*
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 pivot block:
*
* Multiply by the diagonal element if forming L * D.
*
IF( NOUNIT )
$ CALL ZSCAL( NRHS, A( K, K ), B( K, 1 ), LDB )
*
* Multiply by P(K) * inv(L(K)) if K < N.
*
IF( K.NE.N ) THEN
KP = IPIV( K )
*
* Apply the transformation.
*
CALL ZGERU( N-K, NRHS, CONE, A( K+1, K ), 1,
$ B( K, 1 ), LDB, B( K+1, 1 ), LDB )
*
* Interchange if a permutation was applied at the
* K-th step of the factorization.
*
IF( KP.NE.K )
$ CALL ZSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
END IF
K = K - 1
*
ELSE
*
* 2 x 2 pivot block:
*
* Multiply by the diagonal block if forming L * D.
*
IF( NOUNIT ) THEN
D11 = A( K-1, K-1 )
D22 = A( K, K )
D21 = A( K, K-1 )
D12 = D21
DO 50 J = 1, NRHS
T1 = B( K-1, J )
T2 = B( K, J )
B( K-1, J ) = D11*T1 + D12*T2
B( K, J ) = D21*T1 + D22*T2
50 CONTINUE
END IF
*
* Multiply by P(K) * inv(L(K)) if K < N.
*
IF( K.NE.N ) THEN
*
* Apply the transformation.
*
CALL ZGERU( N-K, NRHS, CONE, A( K+1, K ), 1,
$ B( K, 1 ), LDB, B( K+1, 1 ), LDB )
CALL ZGERU( N-K, NRHS, CONE, A( K+1, K-1 ), 1,
$ B( K-1, 1 ), LDB, B( K+1, 1 ), LDB )
*
* Interchange if a permutation was applied at the
* K-th step of the factorization.
*
* Swap the second of pair with IMAXth
*
KP = ABS( IPIV( K ) )
IF( KP.NE.K )
$ CALL ZSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
*
* NOW swap the first of pair with Pth
*
KP = ABS( IPIV( K-1 ) )
IF( KP.NE.K-1 )
$ CALL ZSWAP( NRHS, B( K-1, 1 ), LDB, B( KP, 1 ),
$ LDB )
END IF
K = K - 2
END IF
GO TO 40
60 CONTINUE
END IF
*----------------------------------------
*
* Compute B := A' * B (transpose)
*
*----------------------------------------
ELSE IF( LSAME( TRANS, 'T' ) ) THEN
*
* Form B := U'*B
* where U = P(m)*inv(U(m))* ... *P(1)*inv(U(1))
* and U' = inv(U'(1))*P(1)* ... *inv(U'(m))*P(m)
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* Loop backward applying the transformations.
*
K = N
70 CONTINUE
IF( K.LT.1 )
$ GO TO 90
*
* 1 x 1 pivot block.
*
IF( IPIV( K ).GT.0 ) THEN
IF( K.GT.1 ) THEN
*
* Interchange if P(K) != I.
*
KP = IPIV( K )
IF( KP.NE.K )
$ CALL ZSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
*
* Apply the transformation
*
CALL ZGEMV( 'Transpose', K-1, NRHS, CONE, B, LDB,
$ A( 1, K ), 1, CONE, B( K, 1 ), LDB )
END IF
IF( NOUNIT )
$ CALL ZSCAL( NRHS, A( K, K ), B( K, 1 ), LDB )
K = K - 1
*
* 2 x 2 pivot block.
*
ELSE
IF( K.GT.2 ) THEN
*
* Swap the second of pair with Pth
*
KP = ABS( IPIV( K ) )
IF( KP.NE.K )
$ CALL ZSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
*
* Now swap the first of pair with IMAX(r)th
*
KP = ABS( IPIV( K-1 ) )
IF( KP.NE.K-1 )
$ CALL ZSWAP( NRHS, B( K-1, 1 ), LDB, B( KP, 1 ),
$ LDB )
*
* Apply the transformations
*
CALL ZGEMV( 'Transpose', K-2, NRHS, CONE, B, LDB,
$ A( 1, K ), 1, CONE, B( K, 1 ), LDB )
CALL ZGEMV( 'Transpose', K-2, NRHS, CONE, B, LDB,
$ A( 1, K-1 ), 1, CONE, B( K-1, 1 ), LDB )
END IF
*
* Multiply by the diagonal block if non-unit.
*
IF( NOUNIT ) THEN
D11 = A( K-1, K-1 )
D22 = A( K, K )
D12 = A( K-1, K )
D21 = D12
DO 80 J = 1, NRHS
T1 = B( K-1, J )
T2 = B( K, J )
B( K-1, J ) = D11*T1 + D12*T2
B( K, J ) = D21*T1 + D22*T2
80 CONTINUE
END IF
K = K - 2
END IF
GO TO 70
90 CONTINUE
*
* Form B := L'*B
* where L = P(1)*inv(L(1))* ... *P(m)*inv(L(m))
* and L' = inv(L'(m))*P(m)* ... *inv(L'(1))*P(1)
*
ELSE
*
* Loop forward applying the L-transformations.
*
K = 1
100 CONTINUE
IF( K.GT.N )
$ GO TO 120
*
* 1 x 1 pivot block
*
IF( IPIV( K ).GT.0 ) THEN
IF( K.LT.N ) THEN
*
* Interchange if P(K) != I.
*
KP = IPIV( K )
IF( KP.NE.K )
$ CALL ZSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
*
* Apply the transformation
*
CALL ZGEMV( 'Transpose', N-K, NRHS, CONE, B( K+1, 1 ),
$ LDB, A( K+1, K ), 1, CONE, B( K, 1 ), LDB )
END IF
IF( NOUNIT )
$ CALL ZSCAL( NRHS, A( K, K ), B( K, 1 ), LDB )
K = K + 1
*
* 2 x 2 pivot block.
*
ELSE
IF( K.LT.N-1 ) THEN
*
* Swap the first of pair with Pth
*
KP = ABS( IPIV( K ) )
IF( KP.NE.K )
$ CALL ZSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
*
* Now swap the second of pair with IMAX(r)th
*
KP = ABS( IPIV( K+1 ) )
IF( KP.NE.K+1 )
$ CALL ZSWAP( NRHS, B( K+1, 1 ), LDB, B( KP, 1 ),
$ LDB )
*
* Apply the transformation
*
CALL ZGEMV( 'Transpose', N-K-1, NRHS, CONE,
$ B( K+2, 1 ), LDB, A( K+2, K+1 ), 1, CONE,
$ B( K+1, 1 ), LDB )
CALL ZGEMV( 'Transpose', N-K-1, NRHS, CONE,
$ B( K+2, 1 ), LDB, A( K+2, K ), 1, CONE,
$ B( K, 1 ), LDB )
END IF
*
* Multiply by the diagonal block if non-unit.
*
IF( NOUNIT ) THEN
D11 = A( K, K )
D22 = A( K+1, K+1 )
D21 = A( K+1, K )
D12 = D21
DO 110 J = 1, NRHS
T1 = B( K, J )
T2 = B( K+1, J )
B( K, J ) = D11*T1 + D12*T2
B( K+1, J ) = D21*T1 + D22*T2
110 CONTINUE
END IF
K = K + 2
END IF
GO TO 100
120 CONTINUE
END IF
END IF
RETURN
*
* End of ZLAVSY_ROOK
*
END
|
