LAPACK  3.10.1
LAPACK: Linear Algebra PACKage
zgges3.f
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1 *> \brief <b> ZGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)</b>
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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13 *> [ZIP]</a>
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZGGES3( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B,
22 * $ LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR,
23 * $ WORK, LWORK, RWORK, BWORK, INFO )
24 *
25 * .. Scalar Arguments ..
26 * CHARACTER JOBVSL, JOBVSR, SORT
27 * INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
28 * ..
29 * .. Array Arguments ..
30 * LOGICAL BWORK( * )
31 * DOUBLE PRECISION RWORK( * )
32 * COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
33 * $ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ),
34 * $ WORK( * )
35 * ..
36 * .. Function Arguments ..
37 * LOGICAL SELCTG
38 * EXTERNAL SELCTG
39 * ..
40 *
41 *
42 *> \par Purpose:
43 * =============
44 *>
45 *> \verbatim
46 *>
47 *> ZGGES3 computes for a pair of N-by-N complex nonsymmetric matrices
48 *> (A,B), the generalized eigenvalues, the generalized complex Schur
49 *> form (S, T), and optionally left and/or right Schur vectors (VSL
50 *> and VSR). This gives the generalized Schur factorization
51 *>
52 *> (A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H )
53 *>
54 *> where (VSR)**H is the conjugate-transpose of VSR.
55 *>
56 *> Optionally, it also orders the eigenvalues so that a selected cluster
57 *> of eigenvalues appears in the leading diagonal blocks of the upper
58 *> triangular matrix S and the upper triangular matrix T. The leading
59 *> columns of VSL and VSR then form an unitary basis for the
60 *> corresponding left and right eigenspaces (deflating subspaces).
61 *>
62 *> (If only the generalized eigenvalues are needed, use the driver
63 *> ZGGEV instead, which is faster.)
64 *>
65 *> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
66 *> or a ratio alpha/beta = w, such that A - w*B is singular. It is
67 *> usually represented as the pair (alpha,beta), as there is a
68 *> reasonable interpretation for beta=0, and even for both being zero.
69 *>
70 *> A pair of matrices (S,T) is in generalized complex Schur form if S
71 *> and T are upper triangular and, in addition, the diagonal elements
72 *> of T are non-negative real numbers.
73 *> \endverbatim
74 *
75 * Arguments:
76 * ==========
77 *
78 *> \param[in] JOBVSL
79 *> \verbatim
80 *> JOBVSL is CHARACTER*1
81 *> = 'N': do not compute the left Schur vectors;
82 *> = 'V': compute the left Schur vectors.
83 *> \endverbatim
84 *>
85 *> \param[in] JOBVSR
86 *> \verbatim
87 *> JOBVSR is CHARACTER*1
88 *> = 'N': do not compute the right Schur vectors;
89 *> = 'V': compute the right Schur vectors.
90 *> \endverbatim
91 *>
92 *> \param[in] SORT
93 *> \verbatim
94 *> SORT is CHARACTER*1
95 *> Specifies whether or not to order the eigenvalues on the
96 *> diagonal of the generalized Schur form.
97 *> = 'N': Eigenvalues are not ordered;
98 *> = 'S': Eigenvalues are ordered (see SELCTG).
99 *> \endverbatim
100 *>
101 *> \param[in] SELCTG
102 *> \verbatim
103 *> SELCTG is a LOGICAL FUNCTION of two COMPLEX*16 arguments
104 *> SELCTG must be declared EXTERNAL in the calling subroutine.
105 *> If SORT = 'N', SELCTG is not referenced.
106 *> If SORT = 'S', SELCTG is used to select eigenvalues to sort
107 *> to the top left of the Schur form.
108 *> An eigenvalue ALPHA(j)/BETA(j) is selected if
109 *> SELCTG(ALPHA(j),BETA(j)) is true.
110 *>
111 *> Note that a selected complex eigenvalue may no longer satisfy
112 *> SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since
113 *> ordering may change the value of complex eigenvalues
114 *> (especially if the eigenvalue is ill-conditioned), in this
115 *> case INFO is set to N+2 (See INFO below).
116 *> \endverbatim
117 *>
118 *> \param[in] N
119 *> \verbatim
120 *> N is INTEGER
121 *> The order of the matrices A, B, VSL, and VSR. N >= 0.
122 *> \endverbatim
123 *>
124 *> \param[in,out] A
125 *> \verbatim
126 *> A is COMPLEX*16 array, dimension (LDA, N)
127 *> On entry, the first of the pair of matrices.
128 *> On exit, A has been overwritten by its generalized Schur
129 *> form S.
130 *> \endverbatim
131 *>
132 *> \param[in] LDA
133 *> \verbatim
134 *> LDA is INTEGER
135 *> The leading dimension of A. LDA >= max(1,N).
136 *> \endverbatim
137 *>
138 *> \param[in,out] B
139 *> \verbatim
140 *> B is COMPLEX*16 array, dimension (LDB, N)
141 *> On entry, the second of the pair of matrices.
142 *> On exit, B has been overwritten by its generalized Schur
143 *> form T.
144 *> \endverbatim
145 *>
146 *> \param[in] LDB
147 *> \verbatim
148 *> LDB is INTEGER
149 *> The leading dimension of B. LDB >= max(1,N).
150 *> \endverbatim
151 *>
152 *> \param[out] SDIM
153 *> \verbatim
154 *> SDIM is INTEGER
155 *> If SORT = 'N', SDIM = 0.
156 *> If SORT = 'S', SDIM = number of eigenvalues (after sorting)
157 *> for which SELCTG is true.
158 *> \endverbatim
159 *>
160 *> \param[out] ALPHA
161 *> \verbatim
162 *> ALPHA is COMPLEX*16 array, dimension (N)
163 *> \endverbatim
164 *>
165 *> \param[out] BETA
166 *> \verbatim
167 *> BETA is COMPLEX*16 array, dimension (N)
168 *> On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
169 *> generalized eigenvalues. ALPHA(j), j=1,...,N and BETA(j),
170 *> j=1,...,N are the diagonals of the complex Schur form (A,B)
171 *> output by ZGGES3. The BETA(j) will be non-negative real.
172 *>
173 *> Note: the quotients ALPHA(j)/BETA(j) may easily over- or
174 *> underflow, and BETA(j) may even be zero. Thus, the user
175 *> should avoid naively computing the ratio alpha/beta.
176 *> However, ALPHA will be always less than and usually
177 *> comparable with norm(A) in magnitude, and BETA always less
178 *> than and usually comparable with norm(B).
179 *> \endverbatim
180 *>
181 *> \param[out] VSL
182 *> \verbatim
183 *> VSL is COMPLEX*16 array, dimension (LDVSL,N)
184 *> If JOBVSL = 'V', VSL will contain the left Schur vectors.
185 *> Not referenced if JOBVSL = 'N'.
186 *> \endverbatim
187 *>
188 *> \param[in] LDVSL
189 *> \verbatim
190 *> LDVSL is INTEGER
191 *> The leading dimension of the matrix VSL. LDVSL >= 1, and
192 *> if JOBVSL = 'V', LDVSL >= N.
193 *> \endverbatim
194 *>
195 *> \param[out] VSR
196 *> \verbatim
197 *> VSR is COMPLEX*16 array, dimension (LDVSR,N)
198 *> If JOBVSR = 'V', VSR will contain the right Schur vectors.
199 *> Not referenced if JOBVSR = 'N'.
200 *> \endverbatim
201 *>
202 *> \param[in] LDVSR
203 *> \verbatim
204 *> LDVSR is INTEGER
205 *> The leading dimension of the matrix VSR. LDVSR >= 1, and
206 *> if JOBVSR = 'V', LDVSR >= N.
207 *> \endverbatim
208 *>
209 *> \param[out] WORK
210 *> \verbatim
211 *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
212 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
213 *> \endverbatim
214 *>
215 *> \param[in] LWORK
216 *> \verbatim
217 *> LWORK is INTEGER
218 *> The dimension of the array WORK.
219 *>
220 *> If LWORK = -1, then a workspace query is assumed; the routine
221 *> only calculates the optimal size of the WORK array, returns
222 *> this value as the first entry of the WORK array, and no error
223 *> message related to LWORK is issued by XERBLA.
224 *> \endverbatim
225 *>
226 *> \param[out] RWORK
227 *> \verbatim
228 *> RWORK is DOUBLE PRECISION array, dimension (8*N)
229 *> \endverbatim
230 *>
231 *> \param[out] BWORK
232 *> \verbatim
233 *> BWORK is LOGICAL array, dimension (N)
234 *> Not referenced if SORT = 'N'.
235 *> \endverbatim
236 *>
237 *> \param[out] INFO
238 *> \verbatim
239 *> INFO is INTEGER
240 *> = 0: successful exit
241 *> < 0: if INFO = -i, the i-th argument had an illegal value.
242 *> =1,...,N:
243 *> The QZ iteration failed. (A,B) are not in Schur
244 *> form, but ALPHA(j) and BETA(j) should be correct for
245 *> j=INFO+1,...,N.
246 *> > N: =N+1: other than QZ iteration failed in ZLAQZ0
247 *> =N+2: after reordering, roundoff changed values of
248 *> some complex eigenvalues so that leading
249 *> eigenvalues in the Generalized Schur form no
250 *> longer satisfy SELCTG=.TRUE. This could also
251 *> be caused due to scaling.
252 *> =N+3: reordering failed in ZTGSEN.
253 *> \endverbatim
254 *
255 * Authors:
256 * ========
257 *
258 *> \author Univ. of Tennessee
259 *> \author Univ. of California Berkeley
260 *> \author Univ. of Colorado Denver
261 *> \author NAG Ltd.
262 *
263 *> \ingroup complex16GEeigen
264 *
265 * =====================================================================
266  SUBROUTINE zgges3( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B,
267  $ LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR,
268  $ WORK, LWORK, RWORK, BWORK, INFO )
269 *
270 * -- LAPACK driver routine --
271 * -- LAPACK is a software package provided by Univ. of Tennessee, --
272 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
273 *
274 * .. Scalar Arguments ..
275  CHARACTER JOBVSL, JOBVSR, SORT
276  INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
277 * ..
278 * .. Array Arguments ..
279  LOGICAL BWORK( * )
280  DOUBLE PRECISION RWORK( * )
281  COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
282  $ beta( * ), vsl( ldvsl, * ), vsr( ldvsr, * ),
283  $ work( * )
284 * ..
285 * .. Function Arguments ..
286  LOGICAL SELCTG
287  EXTERNAL SELCTG
288 * ..
289 *
290 * =====================================================================
291 *
292 * .. Parameters ..
293  DOUBLE PRECISION ZERO, ONE
294  PARAMETER ( ZERO = 0.0d0, one = 1.0d0 )
295  COMPLEX*16 CZERO, CONE
296  parameter( czero = ( 0.0d0, 0.0d0 ),
297  $ cone = ( 1.0d0, 0.0d0 ) )
298 * ..
299 * .. Local Scalars ..
300  LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
301  $ LQUERY, WANTST
302  INTEGER I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT,
303  $ ILO, IRIGHT, IROWS, IRWRK, ITAU, IWRK, LWKOPT
304  DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
305  $ PVSR, SMLNUM
306 * ..
307 * .. Local Arrays ..
308  INTEGER IDUM( 1 )
309  DOUBLE PRECISION DIF( 2 )
310 * ..
311 * .. External Subroutines ..
312  EXTERNAL dlabad, xerbla, zgeqrf, zggbak, zggbal, zgghd3,
314  $ zunmqr
315 * ..
316 * .. External Functions ..
317  LOGICAL LSAME
318  DOUBLE PRECISION DLAMCH, ZLANGE
319  EXTERNAL lsame, dlamch, zlange
320 * ..
321 * .. Intrinsic Functions ..
322  INTRINSIC max, sqrt
323 * ..
324 * .. Executable Statements ..
325 *
326 * Decode the input arguments
327 *
328  IF( lsame( jobvsl, 'N' ) ) THEN
329  ijobvl = 1
330  ilvsl = .false.
331  ELSE IF( lsame( jobvsl, 'V' ) ) THEN
332  ijobvl = 2
333  ilvsl = .true.
334  ELSE
335  ijobvl = -1
336  ilvsl = .false.
337  END IF
338 *
339  IF( lsame( jobvsr, 'N' ) ) THEN
340  ijobvr = 1
341  ilvsr = .false.
342  ELSE IF( lsame( jobvsr, 'V' ) ) THEN
343  ijobvr = 2
344  ilvsr = .true.
345  ELSE
346  ijobvr = -1
347  ilvsr = .false.
348  END IF
349 *
350  wantst = lsame( sort, 'S' )
351 *
352 * Test the input arguments
353 *
354  info = 0
355  lquery = ( lwork.EQ.-1 )
356  IF( ijobvl.LE.0 ) THEN
357  info = -1
358  ELSE IF( ijobvr.LE.0 ) THEN
359  info = -2
360  ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
361  info = -3
362  ELSE IF( n.LT.0 ) THEN
363  info = -5
364  ELSE IF( lda.LT.max( 1, n ) ) THEN
365  info = -7
366  ELSE IF( ldb.LT.max( 1, n ) ) THEN
367  info = -9
368  ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
369  info = -14
370  ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
371  info = -16
372  ELSE IF( lwork.LT.max( 1, 2*n ) .AND. .NOT.lquery ) THEN
373  info = -18
374  END IF
375 *
376 * Compute workspace
377 *
378  IF( info.EQ.0 ) THEN
379  CALL zgeqrf( n, n, b, ldb, work, work, -1, ierr )
380  lwkopt = max( 1, n + int( work( 1 ) ) )
381  CALL zunmqr( 'L', 'C', n, n, n, b, ldb, work, a, lda, work,
382  $ -1, ierr )
383  lwkopt = max( lwkopt, n + int( work( 1 ) ) )
384  IF( ilvsl ) THEN
385  CALL zungqr( n, n, n, vsl, ldvsl, work, work, -1, ierr )
386  lwkopt = max( lwkopt, n + int( work( 1 ) ) )
387  END IF
388  CALL zgghd3( jobvsl, jobvsr, n, 1, n, a, lda, b, ldb, vsl,
389  $ ldvsl, vsr, ldvsr, work, -1, ierr )
390  lwkopt = max( lwkopt, n + int( work( 1 ) ) )
391  CALL zlaqz0( 'S', jobvsl, jobvsr, n, 1, n, a, lda, b, ldb,
392  $ alpha, beta, vsl, ldvsl, vsr, ldvsr, work, -1,
393  $ rwork, 0, ierr )
394  lwkopt = max( lwkopt, int( work( 1 ) ) )
395  IF( wantst ) THEN
396  CALL ztgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb,
397  $ alpha, beta, vsl, ldvsl, vsr, ldvsr, sdim,
398  $ pvsl, pvsr, dif, work, -1, idum, 1, ierr )
399  lwkopt = max( lwkopt, int( work( 1 ) ) )
400  END IF
401  work( 1 ) = dcmplx( lwkopt )
402  END IF
403 *
404  IF( info.NE.0 ) THEN
405  CALL xerbla( 'ZGGES3 ', -info )
406  RETURN
407  ELSE IF( lquery ) THEN
408  RETURN
409  END IF
410 *
411 * Quick return if possible
412 *
413  IF( n.EQ.0 ) THEN
414  sdim = 0
415  RETURN
416  END IF
417 *
418 * Get machine constants
419 *
420  eps = dlamch( 'P' )
421  smlnum = dlamch( 'S' )
422  bignum = one / smlnum
423  CALL dlabad( smlnum, bignum )
424  smlnum = sqrt( smlnum ) / eps
425  bignum = one / smlnum
426 *
427 * Scale A if max element outside range [SMLNUM,BIGNUM]
428 *
429  anrm = zlange( 'M', n, n, a, lda, rwork )
430  ilascl = .false.
431  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
432  anrmto = smlnum
433  ilascl = .true.
434  ELSE IF( anrm.GT.bignum ) THEN
435  anrmto = bignum
436  ilascl = .true.
437  END IF
438 *
439  IF( ilascl )
440  $ CALL zlascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
441 *
442 * Scale B if max element outside range [SMLNUM,BIGNUM]
443 *
444  bnrm = zlange( 'M', n, n, b, ldb, rwork )
445  ilbscl = .false.
446  IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
447  bnrmto = smlnum
448  ilbscl = .true.
449  ELSE IF( bnrm.GT.bignum ) THEN
450  bnrmto = bignum
451  ilbscl = .true.
452  END IF
453 *
454  IF( ilbscl )
455  $ CALL zlascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
456 *
457 * Permute the matrix to make it more nearly triangular
458 *
459  ileft = 1
460  iright = n + 1
461  irwrk = iright + n
462  CALL zggbal( 'P', n, a, lda, b, ldb, ilo, ihi, rwork( ileft ),
463  $ rwork( iright ), rwork( irwrk ), ierr )
464 *
465 * Reduce B to triangular form (QR decomposition of B)
466 *
467  irows = ihi + 1 - ilo
468  icols = n + 1 - ilo
469  itau = 1
470  iwrk = itau + irows
471  CALL zgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
472  $ work( iwrk ), lwork+1-iwrk, ierr )
473 *
474 * Apply the orthogonal transformation to matrix A
475 *
476  CALL zunmqr( 'L', 'C', irows, icols, irows, b( ilo, ilo ), ldb,
477  $ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
478  $ lwork+1-iwrk, ierr )
479 *
480 * Initialize VSL
481 *
482  IF( ilvsl ) THEN
483  CALL zlaset( 'Full', n, n, czero, cone, vsl, ldvsl )
484  IF( irows.GT.1 ) THEN
485  CALL zlacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
486  $ vsl( ilo+1, ilo ), ldvsl )
487  END IF
488  CALL zungqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
489  $ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
490  END IF
491 *
492 * Initialize VSR
493 *
494  IF( ilvsr )
495  $ CALL zlaset( 'Full', n, n, czero, cone, vsr, ldvsr )
496 *
497 * Reduce to generalized Hessenberg form
498 *
499  CALL zgghd3( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
500  $ ldvsl, vsr, ldvsr, work( iwrk ), lwork+1-iwrk, ierr )
501 *
502  sdim = 0
503 *
504 * Perform QZ algorithm, computing Schur vectors if desired
505 *
506  iwrk = itau
507  CALL zlaqz0( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
508  $ alpha, beta, vsl, ldvsl, vsr, ldvsr, work( iwrk ),
509  $ lwork+1-iwrk, rwork( irwrk ), 0, ierr )
510  IF( ierr.NE.0 ) THEN
511  IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
512  info = ierr
513  ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
514  info = ierr - n
515  ELSE
516  info = n + 1
517  END IF
518  GO TO 30
519  END IF
520 *
521 * Sort eigenvalues ALPHA/BETA if desired
522 *
523  IF( wantst ) THEN
524 *
525 * Undo scaling on eigenvalues before selecting
526 *
527  IF( ilascl )
528  $ CALL zlascl( 'G', 0, 0, anrm, anrmto, n, 1, alpha, n, ierr )
529  IF( ilbscl )
530  $ CALL zlascl( 'G', 0, 0, bnrm, bnrmto, n, 1, beta, n, ierr )
531 *
532 * Select eigenvalues
533 *
534  DO 10 i = 1, n
535  bwork( i ) = selctg( alpha( i ), beta( i ) )
536  10 CONTINUE
537 *
538  CALL ztgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb, alpha,
539  $ beta, vsl, ldvsl, vsr, ldvsr, sdim, pvsl, pvsr,
540  $ dif, work( iwrk ), lwork-iwrk+1, idum, 1, ierr )
541  IF( ierr.EQ.1 )
542  $ info = n + 3
543 *
544  END IF
545 *
546 * Apply back-permutation to VSL and VSR
547 *
548  IF( ilvsl )
549  $ CALL zggbak( 'P', 'L', n, ilo, ihi, rwork( ileft ),
550  $ rwork( iright ), n, vsl, ldvsl, ierr )
551  IF( ilvsr )
552  $ CALL zggbak( 'P', 'R', n, ilo, ihi, rwork( ileft ),
553  $ rwork( iright ), n, vsr, ldvsr, ierr )
554 *
555 * Undo scaling
556 *
557  IF( ilascl ) THEN
558  CALL zlascl( 'U', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
559  CALL zlascl( 'G', 0, 0, anrmto, anrm, n, 1, alpha, n, ierr )
560  END IF
561 *
562  IF( ilbscl ) THEN
563  CALL zlascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
564  CALL zlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
565  END IF
566 *
567  IF( wantst ) THEN
568 *
569 * Check if reordering is correct
570 *
571  lastsl = .true.
572  sdim = 0
573  DO 20 i = 1, n
574  cursl = selctg( alpha( i ), beta( i ) )
575  IF( cursl )
576  $ sdim = sdim + 1
577  IF( cursl .AND. .NOT.lastsl )
578  $ info = n + 2
579  lastsl = cursl
580  20 CONTINUE
581 *
582  END IF
583 *
584  30 CONTINUE
585 *
586  work( 1 ) = dcmplx( lwkopt )
587 *
588  RETURN
589 *
590 * End of ZGGES3
591 *
592  END
subroutine dlabad(SMALL, LARGE)
DLABAD
Definition: dlabad.f:74
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zggbal(JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
ZGGBAL
Definition: zggbal.f:177
subroutine zggbak(JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV, INFO)
ZGGBAK
Definition: zggbak.f:148
recursive subroutine zlaqz0(WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, REC, INFO)
ZLAQZ0
Definition: zlaqz0.f:284
subroutine zgges3(JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, BWORK, INFO)
ZGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE ...
Definition: zgges3.f:269
subroutine zlascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
ZLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: zlascl.f:143
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
subroutine zlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: zlaset.f:106
subroutine ztgsen(IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, ALPHA, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO)
ZTGSEN
Definition: ztgsen.f:433
subroutine zungqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
ZUNGQR
Definition: zungqr.f:128
subroutine zgghd3(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
ZGGHD3
Definition: zgghd3.f:227
subroutine zunmqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
ZUNMQR
Definition: zunmqr.f:167
subroutine zgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
ZGEQRF VARIANT: left-looking Level 3 BLAS of the algorithm.
Definition: zgeqrf.f:151