1 SUBROUTINE DGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
2 $ SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR,
3 $ LDVSR, WORK, LWORK, BWORK, INFO )
4 *
5 * -- LAPACK driver routine (version 3.2) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * November 2006
9 *
10 * .. Scalar Arguments ..
11 CHARACTER JOBVSL, JOBVSR, SORT
12 INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
13 * ..
14 * .. Array Arguments ..
15 LOGICAL BWORK( * )
16 DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
17 $ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
18 $ VSR( LDVSR, * ), WORK( * )
19 * ..
20 * .. Function Arguments ..
21 LOGICAL SELCTG
22 EXTERNAL SELCTG
23 * ..
24 *
25 * Purpose
26 * =======
27 *
28 * DGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B),
29 * the generalized eigenvalues, the generalized real Schur form (S,T),
30 * optionally, the left and/or right matrices of Schur vectors (VSL and
31 * VSR). This gives the generalized Schur factorization
32 *
33 * (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )
34 *
35 * Optionally, it also orders the eigenvalues so that a selected cluster
36 * of eigenvalues appears in the leading diagonal blocks of the upper
37 * quasi-triangular matrix S and the upper triangular matrix T.The
38 * leading columns of VSL and VSR then form an orthonormal basis for the
39 * corresponding left and right eigenspaces (deflating subspaces).
40 *
41 * (If only the generalized eigenvalues are needed, use the driver
42 * DGGEV instead, which is faster.)
43 *
44 * A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
45 * or a ratio alpha/beta = w, such that A - w*B is singular. It is
46 * usually represented as the pair (alpha,beta), as there is a
47 * reasonable interpretation for beta=0 or both being zero.
48 *
49 * A pair of matrices (S,T) is in generalized real Schur form if T is
50 * upper triangular with non-negative diagonal and S is block upper
51 * triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond
52 * to real generalized eigenvalues, while 2-by-2 blocks of S will be
53 * "standardized" by making the corresponding elements of T have the
54 * form:
55 * [ a 0 ]
56 * [ 0 b ]
57 *
58 * and the pair of corresponding 2-by-2 blocks in S and T will have a
59 * complex conjugate pair of generalized eigenvalues.
60 *
61 *
62 * Arguments
63 * =========
64 *
65 * JOBVSL (input) CHARACTER*1
66 * = 'N': do not compute the left Schur vectors;
67 * = 'V': compute the left Schur vectors.
68 *
69 * JOBVSR (input) CHARACTER*1
70 * = 'N': do not compute the right Schur vectors;
71 * = 'V': compute the right Schur vectors.
72 *
73 * SORT (input) CHARACTER*1
74 * Specifies whether or not to order the eigenvalues on the
75 * diagonal of the generalized Schur form.
76 * = 'N': Eigenvalues are not ordered;
77 * = 'S': Eigenvalues are ordered (see SELCTG);
78 *
79 * SELCTG (external procedure) LOGICAL FUNCTION of three DOUBLE PRECISION arguments
80 * SELCTG must be declared EXTERNAL in the calling subroutine.
81 * If SORT = 'N', SELCTG is not referenced.
82 * If SORT = 'S', SELCTG is used to select eigenvalues to sort
83 * to the top left of the Schur form.
84 * An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
85 * SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
86 * one of a complex conjugate pair of eigenvalues is selected,
87 * then both complex eigenvalues are selected.
88 *
89 * Note that in the ill-conditioned case, a selected complex
90 * eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
91 * BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
92 * in this case.
93 *
94 * N (input) INTEGER
95 * The order of the matrices A, B, VSL, and VSR. N >= 0.
96 *
97 * A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
98 * On entry, the first of the pair of matrices.
99 * On exit, A has been overwritten by its generalized Schur
100 * form S.
101 *
102 * LDA (input) INTEGER
103 * The leading dimension of A. LDA >= max(1,N).
104 *
105 * B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
106 * On entry, the second of the pair of matrices.
107 * On exit, B has been overwritten by its generalized Schur
108 * form T.
109 *
110 * LDB (input) INTEGER
111 * The leading dimension of B. LDB >= max(1,N).
112 *
113 * SDIM (output) INTEGER
114 * If SORT = 'N', SDIM = 0.
115 * If SORT = 'S', SDIM = number of eigenvalues (after sorting)
116 * for which SELCTG is true. (Complex conjugate pairs for which
117 * SELCTG is true for either eigenvalue count as 2.)
118 *
119 * ALPHAR (output) DOUBLE PRECISION array, dimension (N)
120 * ALPHAI (output) DOUBLE PRECISION array, dimension (N)
121 * BETA (output) DOUBLE PRECISION array, dimension (N)
122 * On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
123 * be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i,
124 * and BETA(j),j=1,...,N are the diagonals of the complex Schur
125 * form (S,T) that would result if the 2-by-2 diagonal blocks of
126 * the real Schur form of (A,B) were further reduced to
127 * triangular form using 2-by-2 complex unitary transformations.
128 * If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
129 * positive, then the j-th and (j+1)-st eigenvalues are a
130 * complex conjugate pair, with ALPHAI(j+1) negative.
131 *
132 * Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
133 * may easily over- or underflow, and BETA(j) may even be zero.
134 * Thus, the user should avoid naively computing the ratio.
135 * However, ALPHAR and ALPHAI will be always less than and
136 * usually comparable with norm(A) in magnitude, and BETA always
137 * less than and usually comparable with norm(B).
138 *
139 * VSL (output) DOUBLE PRECISION array, dimension (LDVSL,N)
140 * If JOBVSL = 'V', VSL will contain the left Schur vectors.
141 * Not referenced if JOBVSL = 'N'.
142 *
143 * LDVSL (input) INTEGER
144 * The leading dimension of the matrix VSL. LDVSL >=1, and
145 * if JOBVSL = 'V', LDVSL >= N.
146 *
147 * VSR (output) DOUBLE PRECISION array, dimension (LDVSR,N)
148 * If JOBVSR = 'V', VSR will contain the right Schur vectors.
149 * Not referenced if JOBVSR = 'N'.
150 *
151 * LDVSR (input) INTEGER
152 * The leading dimension of the matrix VSR. LDVSR >= 1, and
153 * if JOBVSR = 'V', LDVSR >= N.
154 *
155 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
156 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
157 *
158 * LWORK (input) INTEGER
159 * The dimension of the array WORK.
160 * If N = 0, LWORK >= 1, else LWORK >= 8*N+16.
161 * For good performance , LWORK must generally be larger.
162 *
163 * If LWORK = -1, then a workspace query is assumed; the routine
164 * only calculates the optimal size of the WORK array, returns
165 * this value as the first entry of the WORK array, and no error
166 * message related to LWORK is issued by XERBLA.
167 *
168 * BWORK (workspace) LOGICAL array, dimension (N)
169 * Not referenced if SORT = 'N'.
170 *
171 * INFO (output) INTEGER
172 * = 0: successful exit
173 * < 0: if INFO = -i, the i-th argument had an illegal value.
174 * = 1,...,N:
175 * The QZ iteration failed. (A,B) are not in Schur
176 * form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
177 * be correct for j=INFO+1,...,N.
178 * > N: =N+1: other than QZ iteration failed in DHGEQZ.
179 * =N+2: after reordering, roundoff changed values of
180 * some complex eigenvalues so that leading
181 * eigenvalues in the Generalized Schur form no
182 * longer satisfy SELCTG=.TRUE. This could also
183 * be caused due to scaling.
184 * =N+3: reordering failed in DTGSEN.
185 *
186 * =====================================================================
187 *
188 * .. Parameters ..
189 DOUBLE PRECISION ZERO, ONE
190 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
191 * ..
192 * .. Local Scalars ..
193 LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
194 $ LQUERY, LST2SL, WANTST
195 INTEGER I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT,
196 $ ILO, IP, IRIGHT, IROWS, ITAU, IWRK, MAXWRK,
197 $ MINWRK
198 DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
199 $ PVSR, SAFMAX, SAFMIN, SMLNUM
200 * ..
201 * .. Local Arrays ..
202 INTEGER IDUM( 1 )
203 DOUBLE PRECISION DIF( 2 )
204 * ..
205 * .. External Subroutines ..
206 EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLABAD,
207 $ DLACPY, DLASCL, DLASET, DORGQR, DORMQR, DTGSEN,
208 $ XERBLA
209 * ..
210 * .. External Functions ..
211 LOGICAL LSAME
212 INTEGER ILAENV
213 DOUBLE PRECISION DLAMCH, DLANGE
214 EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
215 * ..
216 * .. Intrinsic Functions ..
217 INTRINSIC ABS, MAX, SQRT
218 * ..
219 * .. Executable Statements ..
220 *
221 * Decode the input arguments
222 *
223 IF( LSAME( JOBVSL, 'N' ) ) THEN
224 IJOBVL = 1
225 ILVSL = .FALSE.
226 ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
227 IJOBVL = 2
228 ILVSL = .TRUE.
229 ELSE
230 IJOBVL = -1
231 ILVSL = .FALSE.
232 END IF
233 *
234 IF( LSAME( JOBVSR, 'N' ) ) THEN
235 IJOBVR = 1
236 ILVSR = .FALSE.
237 ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
238 IJOBVR = 2
239 ILVSR = .TRUE.
240 ELSE
241 IJOBVR = -1
242 ILVSR = .FALSE.
243 END IF
244 *
245 WANTST = LSAME( SORT, 'S' )
246 *
247 * Test the input arguments
248 *
249 INFO = 0
250 LQUERY = ( LWORK.EQ.-1 )
251 IF( IJOBVL.LE.0 ) THEN
252 INFO = -1
253 ELSE IF( IJOBVR.LE.0 ) THEN
254 INFO = -2
255 ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
256 INFO = -3
257 ELSE IF( N.LT.0 ) THEN
258 INFO = -5
259 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
260 INFO = -7
261 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
262 INFO = -9
263 ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
264 INFO = -15
265 ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
266 INFO = -17
267 END IF
268 *
269 * Compute workspace
270 * (Note: Comments in the code beginning "Workspace:" describe the
271 * minimal amount of workspace needed at that point in the code,
272 * as well as the preferred amount for good performance.
273 * NB refers to the optimal block size for the immediately
274 * following subroutine, as returned by ILAENV.)
275 *
276 IF( INFO.EQ.0 ) THEN
277 IF( N.GT.0 )THEN
278 MINWRK = MAX( 8*N, 6*N + 16 )
279 MAXWRK = MINWRK - N +
280 $ N*ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 )
281 MAXWRK = MAX( MAXWRK, MINWRK - N +
282 $ N*ILAENV( 1, 'DORMQR', ' ', N, 1, N, -1 ) )
283 IF( ILVSL ) THEN
284 MAXWRK = MAX( MAXWRK, MINWRK - N +
285 $ N*ILAENV( 1, 'DORGQR', ' ', N, 1, N, -1 ) )
286 END IF
287 ELSE
288 MINWRK = 1
289 MAXWRK = 1
290 END IF
291 WORK( 1 ) = MAXWRK
292 *
293 IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY )
294 $ INFO = -19
295 END IF
296 *
297 IF( INFO.NE.0 ) THEN
298 CALL XERBLA( 'DGGES ', -INFO )
299 RETURN
300 ELSE IF( LQUERY ) THEN
301 RETURN
302 END IF
303 *
304 * Quick return if possible
305 *
306 IF( N.EQ.0 ) THEN
307 SDIM = 0
308 RETURN
309 END IF
310 *
311 * Get machine constants
312 *
313 EPS = DLAMCH( 'P' )
314 SAFMIN = DLAMCH( 'S' )
315 SAFMAX = ONE / SAFMIN
316 CALL DLABAD( SAFMIN, SAFMAX )
317 SMLNUM = SQRT( SAFMIN ) / EPS
318 BIGNUM = ONE / SMLNUM
319 *
320 * Scale A if max element outside range [SMLNUM,BIGNUM]
321 *
322 ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
323 ILASCL = .FALSE.
324 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
325 ANRMTO = SMLNUM
326 ILASCL = .TRUE.
327 ELSE IF( ANRM.GT.BIGNUM ) THEN
328 ANRMTO = BIGNUM
329 ILASCL = .TRUE.
330 END IF
331 IF( ILASCL )
332 $ CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
333 *
334 * Scale B if max element outside range [SMLNUM,BIGNUM]
335 *
336 BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
337 ILBSCL = .FALSE.
338 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
339 BNRMTO = SMLNUM
340 ILBSCL = .TRUE.
341 ELSE IF( BNRM.GT.BIGNUM ) THEN
342 BNRMTO = BIGNUM
343 ILBSCL = .TRUE.
344 END IF
345 IF( ILBSCL )
346 $ CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
347 *
348 * Permute the matrix to make it more nearly triangular
349 * (Workspace: need 6*N + 2*N space for storing balancing factors)
350 *
351 ILEFT = 1
352 IRIGHT = N + 1
353 IWRK = IRIGHT + N
354 CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
355 $ WORK( IRIGHT ), WORK( IWRK ), IERR )
356 *
357 * Reduce B to triangular form (QR decomposition of B)
358 * (Workspace: need N, prefer N*NB)
359 *
360 IROWS = IHI + 1 - ILO
361 ICOLS = N + 1 - ILO
362 ITAU = IWRK
363 IWRK = ITAU + IROWS
364 CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
365 $ WORK( IWRK ), LWORK+1-IWRK, IERR )
366 *
367 * Apply the orthogonal transformation to matrix A
368 * (Workspace: need N, prefer N*NB)
369 *
370 CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
371 $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
372 $ LWORK+1-IWRK, IERR )
373 *
374 * Initialize VSL
375 * (Workspace: need N, prefer N*NB)
376 *
377 IF( ILVSL ) THEN
378 CALL DLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL )
379 IF( IROWS.GT.1 ) THEN
380 CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
381 $ VSL( ILO+1, ILO ), LDVSL )
382 END IF
383 CALL DORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
384 $ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
385 END IF
386 *
387 * Initialize VSR
388 *
389 IF( ILVSR )
390 $ CALL DLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR )
391 *
392 * Reduce to generalized Hessenberg form
393 * (Workspace: none needed)
394 *
395 CALL DGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
396 $ LDVSL, VSR, LDVSR, IERR )
397 *
398 * Perform QZ algorithm, computing Schur vectors if desired
399 * (Workspace: need N)
400 *
401 IWRK = ITAU
402 CALL DHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
403 $ ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
404 $ WORK( IWRK ), LWORK+1-IWRK, IERR )
405 IF( IERR.NE.0 ) THEN
406 IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
407 INFO = IERR
408 ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
409 INFO = IERR - N
410 ELSE
411 INFO = N + 1
412 END IF
413 GO TO 50
414 END IF
415 *
416 * Sort eigenvalues ALPHA/BETA if desired
417 * (Workspace: need 4*N+16 )
418 *
419 SDIM = 0
420 IF( WANTST ) THEN
421 *
422 * Undo scaling on eigenvalues before SELCTGing
423 *
424 IF( ILASCL ) THEN
425 CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N,
426 $ IERR )
427 CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N,
428 $ IERR )
429 END IF
430 IF( ILBSCL )
431 $ CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
432 *
433 * Select eigenvalues
434 *
435 DO 10 I = 1, N
436 BWORK( I ) = SELCTG( ALPHAR( I ), ALPHAI( I ), BETA( I ) )
437 10 CONTINUE
438 *
439 CALL DTGSEN( 0, ILVSL, ILVSR, BWORK, N, A, LDA, B, LDB, ALPHAR,
440 $ ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, SDIM, PVSL,
441 $ PVSR, DIF, WORK( IWRK ), LWORK-IWRK+1, IDUM, 1,
442 $ IERR )
443 IF( IERR.EQ.1 )
444 $ INFO = N + 3
445 *
446 END IF
447 *
448 * Apply back-permutation to VSL and VSR
449 * (Workspace: none needed)
450 *
451 IF( ILVSL )
452 $ CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
453 $ WORK( IRIGHT ), N, VSL, LDVSL, IERR )
454 *
455 IF( ILVSR )
456 $ CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
457 $ WORK( IRIGHT ), N, VSR, LDVSR, IERR )
458 *
459 * Check if unscaling would cause over/underflow, if so, rescale
460 * (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
461 * B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
462 *
463 IF( ILASCL ) THEN
464 DO 20 I = 1, N
465 IF( ALPHAI( I ).NE.ZERO ) THEN
466 IF( ( ALPHAR( I ) / SAFMAX ).GT.( ANRMTO / ANRM ) .OR.
467 $ ( SAFMIN / ALPHAR( I ) ).GT.( ANRM / ANRMTO ) ) THEN
468 WORK( 1 ) = ABS( A( I, I ) / ALPHAR( I ) )
469 BETA( I ) = BETA( I )*WORK( 1 )
470 ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
471 ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
472 ELSE IF( ( ALPHAI( I ) / SAFMAX ).GT.
473 $ ( ANRMTO / ANRM ) .OR.
474 $ ( SAFMIN / ALPHAI( I ) ).GT.( ANRM / ANRMTO ) )
475 $ THEN
476 WORK( 1 ) = ABS( A( I, I+1 ) / ALPHAI( I ) )
477 BETA( I ) = BETA( I )*WORK( 1 )
478 ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
479 ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
480 END IF
481 END IF
482 20 CONTINUE
483 END IF
484 *
485 IF( ILBSCL ) THEN
486 DO 30 I = 1, N
487 IF( ALPHAI( I ).NE.ZERO ) THEN
488 IF( ( BETA( I ) / SAFMAX ).GT.( BNRMTO / BNRM ) .OR.
489 $ ( SAFMIN / BETA( I ) ).GT.( BNRM / BNRMTO ) ) THEN
490 WORK( 1 ) = ABS( B( I, I ) / BETA( I ) )
491 BETA( I ) = BETA( I )*WORK( 1 )
492 ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
493 ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
494 END IF
495 END IF
496 30 CONTINUE
497 END IF
498 *
499 * Undo scaling
500 *
501 IF( ILASCL ) THEN
502 CALL DLASCL( 'H', 0, 0, ANRMTO, ANRM, N, N, A, LDA, IERR )
503 CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR )
504 CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR )
505 END IF
506 *
507 IF( ILBSCL ) THEN
508 CALL DLASCL( 'U', 0, 0, BNRMTO, BNRM, N, N, B, LDB, IERR )
509 CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
510 END IF
511 *
512 IF( WANTST ) THEN
513 *
514 * Check if reordering is correct
515 *
516 LASTSL = .TRUE.
517 LST2SL = .TRUE.
518 SDIM = 0
519 IP = 0
520 DO 40 I = 1, N
521 CURSL = SELCTG( ALPHAR( I ), ALPHAI( I ), BETA( I ) )
522 IF( ALPHAI( I ).EQ.ZERO ) THEN
523 IF( CURSL )
524 $ SDIM = SDIM + 1
525 IP = 0
526 IF( CURSL .AND. .NOT.LASTSL )
527 $ INFO = N + 2
528 ELSE
529 IF( IP.EQ.1 ) THEN
530 *
531 * Last eigenvalue of conjugate pair
532 *
533 CURSL = CURSL .OR. LASTSL
534 LASTSL = CURSL
535 IF( CURSL )
536 $ SDIM = SDIM + 2
537 IP = -1
538 IF( CURSL .AND. .NOT.LST2SL )
539 $ INFO = N + 2
540 ELSE
541 *
542 * First eigenvalue of conjugate pair
543 *
544 IP = 1
545 END IF
546 END IF
547 LST2SL = LASTSL
548 LASTSL = CURSL
549 40 CONTINUE
550 *
551 END IF
552 *
553 50 CONTINUE
554 *
555 WORK( 1 ) = MAXWRK
556 *
557 RETURN
558 *
559 * End of DGGES
560 *
561 END
2 $ SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR,
3 $ LDVSR, WORK, LWORK, BWORK, INFO )
4 *
5 * -- LAPACK driver routine (version 3.2) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * November 2006
9 *
10 * .. Scalar Arguments ..
11 CHARACTER JOBVSL, JOBVSR, SORT
12 INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
13 * ..
14 * .. Array Arguments ..
15 LOGICAL BWORK( * )
16 DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
17 $ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
18 $ VSR( LDVSR, * ), WORK( * )
19 * ..
20 * .. Function Arguments ..
21 LOGICAL SELCTG
22 EXTERNAL SELCTG
23 * ..
24 *
25 * Purpose
26 * =======
27 *
28 * DGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B),
29 * the generalized eigenvalues, the generalized real Schur form (S,T),
30 * optionally, the left and/or right matrices of Schur vectors (VSL and
31 * VSR). This gives the generalized Schur factorization
32 *
33 * (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )
34 *
35 * Optionally, it also orders the eigenvalues so that a selected cluster
36 * of eigenvalues appears in the leading diagonal blocks of the upper
37 * quasi-triangular matrix S and the upper triangular matrix T.The
38 * leading columns of VSL and VSR then form an orthonormal basis for the
39 * corresponding left and right eigenspaces (deflating subspaces).
40 *
41 * (If only the generalized eigenvalues are needed, use the driver
42 * DGGEV instead, which is faster.)
43 *
44 * A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
45 * or a ratio alpha/beta = w, such that A - w*B is singular. It is
46 * usually represented as the pair (alpha,beta), as there is a
47 * reasonable interpretation for beta=0 or both being zero.
48 *
49 * A pair of matrices (S,T) is in generalized real Schur form if T is
50 * upper triangular with non-negative diagonal and S is block upper
51 * triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond
52 * to real generalized eigenvalues, while 2-by-2 blocks of S will be
53 * "standardized" by making the corresponding elements of T have the
54 * form:
55 * [ a 0 ]
56 * [ 0 b ]
57 *
58 * and the pair of corresponding 2-by-2 blocks in S and T will have a
59 * complex conjugate pair of generalized eigenvalues.
60 *
61 *
62 * Arguments
63 * =========
64 *
65 * JOBVSL (input) CHARACTER*1
66 * = 'N': do not compute the left Schur vectors;
67 * = 'V': compute the left Schur vectors.
68 *
69 * JOBVSR (input) CHARACTER*1
70 * = 'N': do not compute the right Schur vectors;
71 * = 'V': compute the right Schur vectors.
72 *
73 * SORT (input) CHARACTER*1
74 * Specifies whether or not to order the eigenvalues on the
75 * diagonal of the generalized Schur form.
76 * = 'N': Eigenvalues are not ordered;
77 * = 'S': Eigenvalues are ordered (see SELCTG);
78 *
79 * SELCTG (external procedure) LOGICAL FUNCTION of three DOUBLE PRECISION arguments
80 * SELCTG must be declared EXTERNAL in the calling subroutine.
81 * If SORT = 'N', SELCTG is not referenced.
82 * If SORT = 'S', SELCTG is used to select eigenvalues to sort
83 * to the top left of the Schur form.
84 * An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
85 * SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
86 * one of a complex conjugate pair of eigenvalues is selected,
87 * then both complex eigenvalues are selected.
88 *
89 * Note that in the ill-conditioned case, a selected complex
90 * eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
91 * BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
92 * in this case.
93 *
94 * N (input) INTEGER
95 * The order of the matrices A, B, VSL, and VSR. N >= 0.
96 *
97 * A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
98 * On entry, the first of the pair of matrices.
99 * On exit, A has been overwritten by its generalized Schur
100 * form S.
101 *
102 * LDA (input) INTEGER
103 * The leading dimension of A. LDA >= max(1,N).
104 *
105 * B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
106 * On entry, the second of the pair of matrices.
107 * On exit, B has been overwritten by its generalized Schur
108 * form T.
109 *
110 * LDB (input) INTEGER
111 * The leading dimension of B. LDB >= max(1,N).
112 *
113 * SDIM (output) INTEGER
114 * If SORT = 'N', SDIM = 0.
115 * If SORT = 'S', SDIM = number of eigenvalues (after sorting)
116 * for which SELCTG is true. (Complex conjugate pairs for which
117 * SELCTG is true for either eigenvalue count as 2.)
118 *
119 * ALPHAR (output) DOUBLE PRECISION array, dimension (N)
120 * ALPHAI (output) DOUBLE PRECISION array, dimension (N)
121 * BETA (output) DOUBLE PRECISION array, dimension (N)
122 * On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
123 * be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i,
124 * and BETA(j),j=1,...,N are the diagonals of the complex Schur
125 * form (S,T) that would result if the 2-by-2 diagonal blocks of
126 * the real Schur form of (A,B) were further reduced to
127 * triangular form using 2-by-2 complex unitary transformations.
128 * If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
129 * positive, then the j-th and (j+1)-st eigenvalues are a
130 * complex conjugate pair, with ALPHAI(j+1) negative.
131 *
132 * Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
133 * may easily over- or underflow, and BETA(j) may even be zero.
134 * Thus, the user should avoid naively computing the ratio.
135 * However, ALPHAR and ALPHAI will be always less than and
136 * usually comparable with norm(A) in magnitude, and BETA always
137 * less than and usually comparable with norm(B).
138 *
139 * VSL (output) DOUBLE PRECISION array, dimension (LDVSL,N)
140 * If JOBVSL = 'V', VSL will contain the left Schur vectors.
141 * Not referenced if JOBVSL = 'N'.
142 *
143 * LDVSL (input) INTEGER
144 * The leading dimension of the matrix VSL. LDVSL >=1, and
145 * if JOBVSL = 'V', LDVSL >= N.
146 *
147 * VSR (output) DOUBLE PRECISION array, dimension (LDVSR,N)
148 * If JOBVSR = 'V', VSR will contain the right Schur vectors.
149 * Not referenced if JOBVSR = 'N'.
150 *
151 * LDVSR (input) INTEGER
152 * The leading dimension of the matrix VSR. LDVSR >= 1, and
153 * if JOBVSR = 'V', LDVSR >= N.
154 *
155 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
156 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
157 *
158 * LWORK (input) INTEGER
159 * The dimension of the array WORK.
160 * If N = 0, LWORK >= 1, else LWORK >= 8*N+16.
161 * For good performance , LWORK must generally be larger.
162 *
163 * If LWORK = -1, then a workspace query is assumed; the routine
164 * only calculates the optimal size of the WORK array, returns
165 * this value as the first entry of the WORK array, and no error
166 * message related to LWORK is issued by XERBLA.
167 *
168 * BWORK (workspace) LOGICAL array, dimension (N)
169 * Not referenced if SORT = 'N'.
170 *
171 * INFO (output) INTEGER
172 * = 0: successful exit
173 * < 0: if INFO = -i, the i-th argument had an illegal value.
174 * = 1,...,N:
175 * The QZ iteration failed. (A,B) are not in Schur
176 * form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
177 * be correct for j=INFO+1,...,N.
178 * > N: =N+1: other than QZ iteration failed in DHGEQZ.
179 * =N+2: after reordering, roundoff changed values of
180 * some complex eigenvalues so that leading
181 * eigenvalues in the Generalized Schur form no
182 * longer satisfy SELCTG=.TRUE. This could also
183 * be caused due to scaling.
184 * =N+3: reordering failed in DTGSEN.
185 *
186 * =====================================================================
187 *
188 * .. Parameters ..
189 DOUBLE PRECISION ZERO, ONE
190 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
191 * ..
192 * .. Local Scalars ..
193 LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
194 $ LQUERY, LST2SL, WANTST
195 INTEGER I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT,
196 $ ILO, IP, IRIGHT, IROWS, ITAU, IWRK, MAXWRK,
197 $ MINWRK
198 DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
199 $ PVSR, SAFMAX, SAFMIN, SMLNUM
200 * ..
201 * .. Local Arrays ..
202 INTEGER IDUM( 1 )
203 DOUBLE PRECISION DIF( 2 )
204 * ..
205 * .. External Subroutines ..
206 EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLABAD,
207 $ DLACPY, DLASCL, DLASET, DORGQR, DORMQR, DTGSEN,
208 $ XERBLA
209 * ..
210 * .. External Functions ..
211 LOGICAL LSAME
212 INTEGER ILAENV
213 DOUBLE PRECISION DLAMCH, DLANGE
214 EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
215 * ..
216 * .. Intrinsic Functions ..
217 INTRINSIC ABS, MAX, SQRT
218 * ..
219 * .. Executable Statements ..
220 *
221 * Decode the input arguments
222 *
223 IF( LSAME( JOBVSL, 'N' ) ) THEN
224 IJOBVL = 1
225 ILVSL = .FALSE.
226 ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
227 IJOBVL = 2
228 ILVSL = .TRUE.
229 ELSE
230 IJOBVL = -1
231 ILVSL = .FALSE.
232 END IF
233 *
234 IF( LSAME( JOBVSR, 'N' ) ) THEN
235 IJOBVR = 1
236 ILVSR = .FALSE.
237 ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
238 IJOBVR = 2
239 ILVSR = .TRUE.
240 ELSE
241 IJOBVR = -1
242 ILVSR = .FALSE.
243 END IF
244 *
245 WANTST = LSAME( SORT, 'S' )
246 *
247 * Test the input arguments
248 *
249 INFO = 0
250 LQUERY = ( LWORK.EQ.-1 )
251 IF( IJOBVL.LE.0 ) THEN
252 INFO = -1
253 ELSE IF( IJOBVR.LE.0 ) THEN
254 INFO = -2
255 ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
256 INFO = -3
257 ELSE IF( N.LT.0 ) THEN
258 INFO = -5
259 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
260 INFO = -7
261 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
262 INFO = -9
263 ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
264 INFO = -15
265 ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
266 INFO = -17
267 END IF
268 *
269 * Compute workspace
270 * (Note: Comments in the code beginning "Workspace:" describe the
271 * minimal amount of workspace needed at that point in the code,
272 * as well as the preferred amount for good performance.
273 * NB refers to the optimal block size for the immediately
274 * following subroutine, as returned by ILAENV.)
275 *
276 IF( INFO.EQ.0 ) THEN
277 IF( N.GT.0 )THEN
278 MINWRK = MAX( 8*N, 6*N + 16 )
279 MAXWRK = MINWRK - N +
280 $ N*ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 )
281 MAXWRK = MAX( MAXWRK, MINWRK - N +
282 $ N*ILAENV( 1, 'DORMQR', ' ', N, 1, N, -1 ) )
283 IF( ILVSL ) THEN
284 MAXWRK = MAX( MAXWRK, MINWRK - N +
285 $ N*ILAENV( 1, 'DORGQR', ' ', N, 1, N, -1 ) )
286 END IF
287 ELSE
288 MINWRK = 1
289 MAXWRK = 1
290 END IF
291 WORK( 1 ) = MAXWRK
292 *
293 IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY )
294 $ INFO = -19
295 END IF
296 *
297 IF( INFO.NE.0 ) THEN
298 CALL XERBLA( 'DGGES ', -INFO )
299 RETURN
300 ELSE IF( LQUERY ) THEN
301 RETURN
302 END IF
303 *
304 * Quick return if possible
305 *
306 IF( N.EQ.0 ) THEN
307 SDIM = 0
308 RETURN
309 END IF
310 *
311 * Get machine constants
312 *
313 EPS = DLAMCH( 'P' )
314 SAFMIN = DLAMCH( 'S' )
315 SAFMAX = ONE / SAFMIN
316 CALL DLABAD( SAFMIN, SAFMAX )
317 SMLNUM = SQRT( SAFMIN ) / EPS
318 BIGNUM = ONE / SMLNUM
319 *
320 * Scale A if max element outside range [SMLNUM,BIGNUM]
321 *
322 ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
323 ILASCL = .FALSE.
324 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
325 ANRMTO = SMLNUM
326 ILASCL = .TRUE.
327 ELSE IF( ANRM.GT.BIGNUM ) THEN
328 ANRMTO = BIGNUM
329 ILASCL = .TRUE.
330 END IF
331 IF( ILASCL )
332 $ CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
333 *
334 * Scale B if max element outside range [SMLNUM,BIGNUM]
335 *
336 BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
337 ILBSCL = .FALSE.
338 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
339 BNRMTO = SMLNUM
340 ILBSCL = .TRUE.
341 ELSE IF( BNRM.GT.BIGNUM ) THEN
342 BNRMTO = BIGNUM
343 ILBSCL = .TRUE.
344 END IF
345 IF( ILBSCL )
346 $ CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
347 *
348 * Permute the matrix to make it more nearly triangular
349 * (Workspace: need 6*N + 2*N space for storing balancing factors)
350 *
351 ILEFT = 1
352 IRIGHT = N + 1
353 IWRK = IRIGHT + N
354 CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
355 $ WORK( IRIGHT ), WORK( IWRK ), IERR )
356 *
357 * Reduce B to triangular form (QR decomposition of B)
358 * (Workspace: need N, prefer N*NB)
359 *
360 IROWS = IHI + 1 - ILO
361 ICOLS = N + 1 - ILO
362 ITAU = IWRK
363 IWRK = ITAU + IROWS
364 CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
365 $ WORK( IWRK ), LWORK+1-IWRK, IERR )
366 *
367 * Apply the orthogonal transformation to matrix A
368 * (Workspace: need N, prefer N*NB)
369 *
370 CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
371 $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
372 $ LWORK+1-IWRK, IERR )
373 *
374 * Initialize VSL
375 * (Workspace: need N, prefer N*NB)
376 *
377 IF( ILVSL ) THEN
378 CALL DLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL )
379 IF( IROWS.GT.1 ) THEN
380 CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
381 $ VSL( ILO+1, ILO ), LDVSL )
382 END IF
383 CALL DORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
384 $ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
385 END IF
386 *
387 * Initialize VSR
388 *
389 IF( ILVSR )
390 $ CALL DLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR )
391 *
392 * Reduce to generalized Hessenberg form
393 * (Workspace: none needed)
394 *
395 CALL DGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
396 $ LDVSL, VSR, LDVSR, IERR )
397 *
398 * Perform QZ algorithm, computing Schur vectors if desired
399 * (Workspace: need N)
400 *
401 IWRK = ITAU
402 CALL DHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
403 $ ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
404 $ WORK( IWRK ), LWORK+1-IWRK, IERR )
405 IF( IERR.NE.0 ) THEN
406 IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
407 INFO = IERR
408 ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
409 INFO = IERR - N
410 ELSE
411 INFO = N + 1
412 END IF
413 GO TO 50
414 END IF
415 *
416 * Sort eigenvalues ALPHA/BETA if desired
417 * (Workspace: need 4*N+16 )
418 *
419 SDIM = 0
420 IF( WANTST ) THEN
421 *
422 * Undo scaling on eigenvalues before SELCTGing
423 *
424 IF( ILASCL ) THEN
425 CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N,
426 $ IERR )
427 CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N,
428 $ IERR )
429 END IF
430 IF( ILBSCL )
431 $ CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
432 *
433 * Select eigenvalues
434 *
435 DO 10 I = 1, N
436 BWORK( I ) = SELCTG( ALPHAR( I ), ALPHAI( I ), BETA( I ) )
437 10 CONTINUE
438 *
439 CALL DTGSEN( 0, ILVSL, ILVSR, BWORK, N, A, LDA, B, LDB, ALPHAR,
440 $ ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, SDIM, PVSL,
441 $ PVSR, DIF, WORK( IWRK ), LWORK-IWRK+1, IDUM, 1,
442 $ IERR )
443 IF( IERR.EQ.1 )
444 $ INFO = N + 3
445 *
446 END IF
447 *
448 * Apply back-permutation to VSL and VSR
449 * (Workspace: none needed)
450 *
451 IF( ILVSL )
452 $ CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
453 $ WORK( IRIGHT ), N, VSL, LDVSL, IERR )
454 *
455 IF( ILVSR )
456 $ CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
457 $ WORK( IRIGHT ), N, VSR, LDVSR, IERR )
458 *
459 * Check if unscaling would cause over/underflow, if so, rescale
460 * (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
461 * B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
462 *
463 IF( ILASCL ) THEN
464 DO 20 I = 1, N
465 IF( ALPHAI( I ).NE.ZERO ) THEN
466 IF( ( ALPHAR( I ) / SAFMAX ).GT.( ANRMTO / ANRM ) .OR.
467 $ ( SAFMIN / ALPHAR( I ) ).GT.( ANRM / ANRMTO ) ) THEN
468 WORK( 1 ) = ABS( A( I, I ) / ALPHAR( I ) )
469 BETA( I ) = BETA( I )*WORK( 1 )
470 ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
471 ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
472 ELSE IF( ( ALPHAI( I ) / SAFMAX ).GT.
473 $ ( ANRMTO / ANRM ) .OR.
474 $ ( SAFMIN / ALPHAI( I ) ).GT.( ANRM / ANRMTO ) )
475 $ THEN
476 WORK( 1 ) = ABS( A( I, I+1 ) / ALPHAI( I ) )
477 BETA( I ) = BETA( I )*WORK( 1 )
478 ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
479 ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
480 END IF
481 END IF
482 20 CONTINUE
483 END IF
484 *
485 IF( ILBSCL ) THEN
486 DO 30 I = 1, N
487 IF( ALPHAI( I ).NE.ZERO ) THEN
488 IF( ( BETA( I ) / SAFMAX ).GT.( BNRMTO / BNRM ) .OR.
489 $ ( SAFMIN / BETA( I ) ).GT.( BNRM / BNRMTO ) ) THEN
490 WORK( 1 ) = ABS( B( I, I ) / BETA( I ) )
491 BETA( I ) = BETA( I )*WORK( 1 )
492 ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
493 ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
494 END IF
495 END IF
496 30 CONTINUE
497 END IF
498 *
499 * Undo scaling
500 *
501 IF( ILASCL ) THEN
502 CALL DLASCL( 'H', 0, 0, ANRMTO, ANRM, N, N, A, LDA, IERR )
503 CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR )
504 CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR )
505 END IF
506 *
507 IF( ILBSCL ) THEN
508 CALL DLASCL( 'U', 0, 0, BNRMTO, BNRM, N, N, B, LDB, IERR )
509 CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
510 END IF
511 *
512 IF( WANTST ) THEN
513 *
514 * Check if reordering is correct
515 *
516 LASTSL = .TRUE.
517 LST2SL = .TRUE.
518 SDIM = 0
519 IP = 0
520 DO 40 I = 1, N
521 CURSL = SELCTG( ALPHAR( I ), ALPHAI( I ), BETA( I ) )
522 IF( ALPHAI( I ).EQ.ZERO ) THEN
523 IF( CURSL )
524 $ SDIM = SDIM + 1
525 IP = 0
526 IF( CURSL .AND. .NOT.LASTSL )
527 $ INFO = N + 2
528 ELSE
529 IF( IP.EQ.1 ) THEN
530 *
531 * Last eigenvalue of conjugate pair
532 *
533 CURSL = CURSL .OR. LASTSL
534 LASTSL = CURSL
535 IF( CURSL )
536 $ SDIM = SDIM + 2
537 IP = -1
538 IF( CURSL .AND. .NOT.LST2SL )
539 $ INFO = N + 2
540 ELSE
541 *
542 * First eigenvalue of conjugate pair
543 *
544 IP = 1
545 END IF
546 END IF
547 LST2SL = LASTSL
548 LASTSL = CURSL
549 40 CONTINUE
550 *
551 END IF
552 *
553 50 CONTINUE
554 *
555 WORK( 1 ) = MAXWRK
556 *
557 RETURN
558 *
559 * End of DGGES
560 *
561 END