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
|
SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
$ ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
$ LWORK, INFO )
*
* -- LAPACK driver routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER JOBVSL, JOBVSR
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
$ VSR( LDVSR, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* This routine is deprecated and has been replaced by routine DGGES.
*
* DGEGS computes the eigenvalues, real Schur form, and, optionally,
* left and or/right Schur vectors of a real matrix pair (A,B).
* Given two square matrices A and B, the generalized real Schur
* factorization has the form
*
* A = Q*S*Z**T, B = Q*T*Z**T
*
* where Q and Z are orthogonal matrices, T is upper triangular, and S
* is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal
* blocks, the 2-by-2 blocks corresponding to complex conjugate pairs
* of eigenvalues of (A,B). The columns of Q are the left Schur vectors
* and the columns of Z are the right Schur vectors.
*
* If only the eigenvalues of (A,B) are needed, the driver routine
* DGEGV should be used instead. See DGEGV for a description of the
* eigenvalues of the generalized nonsymmetric eigenvalue problem
* (GNEP).
*
* Arguments
* =========
*
* JOBVSL (input) CHARACTER*1
* = 'N': do not compute the left Schur vectors;
* = 'V': compute the left Schur vectors (returned in VSL).
*
* JOBVSR (input) CHARACTER*1
* = 'N': do not compute the right Schur vectors;
* = 'V': compute the right Schur vectors (returned in VSR).
*
* N (input) INTEGER
* The order of the matrices A, B, VSL, and VSR. N >= 0.
*
* A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
* On entry, the matrix A.
* On exit, the upper quasi-triangular matrix S from the
* generalized real Schur factorization.
*
* LDA (input) INTEGER
* The leading dimension of A. LDA >= max(1,N).
*
* B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
* On entry, the matrix B.
* On exit, the upper triangular matrix T from the generalized
* real Schur factorization.
*
* LDB (input) INTEGER
* The leading dimension of B. LDB >= max(1,N).
*
* ALPHAR (output) DOUBLE PRECISION array, dimension (N)
* The real parts of each scalar alpha defining an eigenvalue
* of GNEP.
*
* ALPHAI (output) DOUBLE PRECISION array, dimension (N)
* The imaginary parts of each scalar alpha defining an
* eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th
* eigenvalue is real; if positive, then the j-th and (j+1)-st
* eigenvalues are a complex conjugate pair, with
* ALPHAI(j+1) = -ALPHAI(j).
*
* BETA (output) DOUBLE PRECISION array, dimension (N)
* The scalars beta that define the eigenvalues of GNEP.
* Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
* beta = BETA(j) represent the j-th eigenvalue of the matrix
* pair (A,B), in one of the forms lambda = alpha/beta or
* mu = beta/alpha. Since either lambda or mu may overflow,
* they should not, in general, be computed.
*
* VSL (output) DOUBLE PRECISION array, dimension (LDVSL,N)
* If JOBVSL = 'V', the matrix of left Schur vectors Q.
* Not referenced if JOBVSL = 'N'.
*
* LDVSL (input) INTEGER
* The leading dimension of the matrix VSL. LDVSL >=1, and
* if JOBVSL = 'V', LDVSL >= N.
*
* VSR (output) DOUBLE PRECISION array, dimension (LDVSR,N)
* If JOBVSR = 'V', the matrix of right Schur vectors Z.
* Not referenced if JOBVSR = 'N'.
*
* LDVSR (input) INTEGER
* The leading dimension of the matrix VSR. LDVSR >= 1, and
* if JOBVSR = 'V', LDVSR >= N.
*
* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,4*N).
* For good performance, LWORK must generally be larger.
* To compute the optimal value of LWORK, call ILAENV to get
* blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute:
* NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR
* The optimal LWORK is 2*N + N*(NB+1).
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
* = 1,...,N:
* The QZ iteration failed. (A,B) are not in Schur
* form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
* be correct for j=INFO+1,...,N.
* > N: errors that usually indicate LAPACK problems:
* =N+1: error return from DGGBAL
* =N+2: error return from DGEQRF
* =N+3: error return from DORMQR
* =N+4: error return from DORGQR
* =N+5: error return from DGGHRD
* =N+6: error return from DHGEQZ (other than failed
* iteration)
* =N+7: error return from DGGBAK (computing VSL)
* =N+8: error return from DGGBAK (computing VSR)
* =N+9: error return from DLASCL (various places)
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL ILASCL, ILBSCL, ILVSL, ILVSR, LQUERY
INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
$ IRIGHT, IROWS, ITAU, IWORK, LOPT, LWKMIN,
$ LWKOPT, NB, NB1, NB2, NB3
DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
$ SAFMIN, SMLNUM
* ..
* .. External Subroutines ..
EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLACPY,
$ DLASCL, DLASET, DORGQR, DORMQR, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX
* ..
* .. Executable Statements ..
*
* Decode the input arguments
*
IF( LSAME( JOBVSL, 'N' ) ) THEN
IJOBVL = 1
ILVSL = .FALSE.
ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
IJOBVL = 2
ILVSL = .TRUE.
ELSE
IJOBVL = -1
ILVSL = .FALSE.
END IF
*
IF( LSAME( JOBVSR, 'N' ) ) THEN
IJOBVR = 1
ILVSR = .FALSE.
ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
IJOBVR = 2
ILVSR = .TRUE.
ELSE
IJOBVR = -1
ILVSR = .FALSE.
END IF
*
* Test the input arguments
*
LWKMIN = MAX( 4*N, 1 )
LWKOPT = LWKMIN
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
INFO = 0
IF( IJOBVL.LE.0 ) THEN
INFO = -1
ELSE IF( IJOBVR.LE.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
INFO = -12
ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
INFO = -14
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -16
END IF
*
IF( INFO.EQ.0 ) THEN
NB1 = ILAENV( 1, 'DGEQRF', ' ', N, N, -1, -1 )
NB2 = ILAENV( 1, 'DORMQR', ' ', N, N, N, -1 )
NB3 = ILAENV( 1, 'DORGQR', ' ', N, N, N, -1 )
NB = MAX( NB1, NB2, NB3 )
LOPT = 2*N + N*( NB+1 )
WORK( 1 ) = LOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEGS ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
SAFMIN = DLAMCH( 'S' )
SMLNUM = N*SAFMIN / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
ILASCL = .FALSE.
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
ANRMTO = SMLNUM
ILASCL = .TRUE.
ELSE IF( ANRM.GT.BIGNUM ) THEN
ANRMTO = BIGNUM
ILASCL = .TRUE.
END IF
*
IF( ILASCL ) THEN
CALL DLASCL( 'G', -1, -1, ANRM, ANRMTO, N, N, A, LDA, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
* Scale B if max element outside range [SMLNUM,BIGNUM]
*
BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
ILBSCL = .FALSE.
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
BNRMTO = SMLNUM
ILBSCL = .TRUE.
ELSE IF( BNRM.GT.BIGNUM ) THEN
BNRMTO = BIGNUM
ILBSCL = .TRUE.
END IF
*
IF( ILBSCL ) THEN
CALL DLASCL( 'G', -1, -1, BNRM, BNRMTO, N, N, B, LDB, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
* Permute the matrix to make it more nearly triangular
* Workspace layout: (2*N words -- "work..." not actually used)
* left_permutation, right_permutation, work...
*
ILEFT = 1
IRIGHT = N + 1
IWORK = IRIGHT + N
CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), WORK( IWORK ), IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 1
GO TO 10
END IF
*
* Reduce B to triangular form, and initialize VSL and/or VSR
* Workspace layout: ("work..." must have at least N words)
* left_permutation, right_permutation, tau, work...
*
IROWS = IHI + 1 - ILO
ICOLS = N + 1 - ILO
ITAU = IWORK
IWORK = ITAU + IROWS
CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 2
GO TO 10
END IF
*
CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
$ LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 3
GO TO 10
END IF
*
IF( ILVSL ) THEN
CALL DLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL )
CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VSL( ILO+1, ILO ), LDVSL )
CALL DORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
$ IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 4
GO TO 10
END IF
END IF
*
IF( ILVSR )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR )
*
* Reduce to generalized Hessenberg form
*
CALL DGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
$ LDVSL, VSR, LDVSR, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 5
GO TO 10
END IF
*
* Perform QZ algorithm, computing Schur vectors if desired
* Workspace layout: ("work..." must have at least 1 word)
* left_permutation, right_permutation, work...
*
IWORK = ITAU
CALL DHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
INFO = IINFO
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
INFO = IINFO - N
ELSE
INFO = N + 6
END IF
GO TO 10
END IF
*
* Apply permutation to VSL and VSR
*
IF( ILVSL ) THEN
CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VSL, LDVSL, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 7
GO TO 10
END IF
END IF
IF( ILVSR ) THEN
CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VSR, LDVSR, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 8
GO TO 10
END IF
END IF
*
* Undo scaling
*
IF( ILASCL ) THEN
CALL DLASCL( 'H', -1, -1, ANRMTO, ANRM, N, N, A, LDA, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
CALL DLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAR, N,
$ IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
CALL DLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAI, N,
$ IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
IF( ILBSCL ) THEN
CALL DLASCL( 'U', -1, -1, BNRMTO, BNRM, N, N, B, LDB, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
CALL DLASCL( 'G', -1, -1, BNRMTO, BNRM, N, 1, BETA, N, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
10 CONTINUE
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of DGEGS
*
END
|
