LAPACK  3.6.0
LAPACK: Linear Algebra PACKage
cgges.f
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1 *> \brief <b> CGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices</b>
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download CGGES + dependencies
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11 *> [TGZ]</a>
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13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgges.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
22 * SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
23 * 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 * REAL RWORK( * )
32 * COMPLEX 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 *> CGGES 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 *> CGGEV 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 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 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 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 array, dimension (N)
163 *> \endverbatim
164 *>
165 *> \param[out] BETA
166 *> \verbatim
167 *> BETA is COMPLEX 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 CGGES. 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 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 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 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. LWORK >= max(1,2*N).
219 *> For good performance, LWORK must generally be larger.
220 *>
221 *> If LWORK = -1, then a workspace query is assumed; the routine
222 *> only calculates the optimal size of the WORK array, returns
223 *> this value as the first entry of the WORK array, and no error
224 *> message related to LWORK is issued by XERBLA.
225 *> \endverbatim
226 *>
227 *> \param[out] RWORK
228 *> \verbatim
229 *> RWORK is REAL array, dimension (8*N)
230 *> \endverbatim
231 *>
232 *> \param[out] BWORK
233 *> \verbatim
234 *> BWORK is LOGICAL array, dimension (N)
235 *> Not referenced if SORT = 'N'.
236 *> \endverbatim
237 *>
238 *> \param[out] INFO
239 *> \verbatim
240 *> INFO is INTEGER
241 *> = 0: successful exit
242 *> < 0: if INFO = -i, the i-th argument had an illegal value.
243 *> =1,...,N:
244 *> The QZ iteration failed. (A,B) are not in Schur
245 *> form, but ALPHA(j) and BETA(j) should be correct for
246 *> j=INFO+1,...,N.
247 *> > N: =N+1: other than QZ iteration failed in CHGEQZ
248 *> =N+2: after reordering, roundoff changed values of
249 *> some complex eigenvalues so that leading
250 *> eigenvalues in the Generalized Schur form no
251 *> longer satisfy SELCTG=.TRUE. This could also
252 *> be caused due to scaling.
253 *> =N+3: reordering failed in CTGSEN.
254 *> \endverbatim
255 *
256 * Authors:
257 * ========
258 *
259 *> \author Univ. of Tennessee
260 *> \author Univ. of California Berkeley
261 *> \author Univ. of Colorado Denver
262 *> \author NAG Ltd.
263 *
264 *> \date November 2015
265 *
266 *> \ingroup complexGEeigen
267 *
268 * =====================================================================
269  SUBROUTINE cgges( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
270  $ sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, work,
271  $ lwork, rwork, bwork, info )
272 *
273 * -- LAPACK driver routine (version 3.6.0) --
274 * -- LAPACK is a software package provided by Univ. of Tennessee, --
275 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
276 * November 2015
277 *
278 * .. Scalar Arguments ..
279  CHARACTER JOBVSL, JOBVSR, SORT
280  INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
281 * ..
282 * .. Array Arguments ..
283  LOGICAL BWORK( * )
284  REAL RWORK( * )
285  COMPLEX A( lda, * ), ALPHA( * ), B( ldb, * ),
286  $ beta( * ), vsl( ldvsl, * ), vsr( ldvsr, * ),
287  $ work( * )
288 * ..
289 * .. Function Arguments ..
290  LOGICAL SELCTG
291  EXTERNAL selctg
292 * ..
293 *
294 * =====================================================================
295 *
296 * .. Parameters ..
297  REAL ZERO, ONE
298  parameter( zero = 0.0e0, one = 1.0e0 )
299  COMPLEX CZERO, CONE
300  parameter( czero = ( 0.0e0, 0.0e0 ),
301  $ cone = ( 1.0e0, 0.0e0 ) )
302 * ..
303 * .. Local Scalars ..
304  LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
305  $ lquery, wantst
306  INTEGER I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT,
307  $ ilo, iright, irows, irwrk, itau, iwrk, lwkmin,
308  $ lwkopt
309  REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
310  $ pvsr, smlnum
311 * ..
312 * .. Local Arrays ..
313  INTEGER IDUM( 1 )
314  REAL DIF( 2 )
315 * ..
316 * .. External Subroutines ..
317  EXTERNAL cgeqrf, cggbak, cggbal, cgghrd, chgeqz, clacpy,
319  $ xerbla
320 * ..
321 * .. External Functions ..
322  LOGICAL LSAME
323  INTEGER ILAENV
324  REAL CLANGE, SLAMCH
325  EXTERNAL lsame, ilaenv, clange, slamch
326 * ..
327 * .. Intrinsic Functions ..
328  INTRINSIC max, sqrt
329 * ..
330 * .. Executable Statements ..
331 *
332 * Decode the input arguments
333 *
334  IF( lsame( jobvsl, 'N' ) ) THEN
335  ijobvl = 1
336  ilvsl = .false.
337  ELSE IF( lsame( jobvsl, 'V' ) ) THEN
338  ijobvl = 2
339  ilvsl = .true.
340  ELSE
341  ijobvl = -1
342  ilvsl = .false.
343  END IF
344 *
345  IF( lsame( jobvsr, 'N' ) ) THEN
346  ijobvr = 1
347  ilvsr = .false.
348  ELSE IF( lsame( jobvsr, 'V' ) ) THEN
349  ijobvr = 2
350  ilvsr = .true.
351  ELSE
352  ijobvr = -1
353  ilvsr = .false.
354  END IF
355 *
356  wantst = lsame( sort, 'S' )
357 *
358 * Test the input arguments
359 *
360  info = 0
361  lquery = ( lwork.EQ.-1 )
362  IF( ijobvl.LE.0 ) THEN
363  info = -1
364  ELSE IF( ijobvr.LE.0 ) THEN
365  info = -2
366  ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
367  info = -3
368  ELSE IF( n.LT.0 ) THEN
369  info = -5
370  ELSE IF( lda.LT.max( 1, n ) ) THEN
371  info = -7
372  ELSE IF( ldb.LT.max( 1, n ) ) THEN
373  info = -9
374  ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
375  info = -14
376  ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
377  info = -16
378  END IF
379 *
380 * Compute workspace
381 * (Note: Comments in the code beginning "Workspace:" describe the
382 * minimal amount of workspace needed at that point in the code,
383 * as well as the preferred amount for good performance.
384 * NB refers to the optimal block size for the immediately
385 * following subroutine, as returned by ILAENV.)
386 *
387  IF( info.EQ.0 ) THEN
388  lwkmin = max( 1, 2*n )
389  lwkopt = max( 1, n + n*ilaenv( 1, 'CGEQRF', ' ', n, 1, n, 0 ) )
390  lwkopt = max( lwkopt, n +
391  $ n*ilaenv( 1, 'CUNMQR', ' ', n, 1, n, -1 ) )
392  IF( ilvsl ) THEN
393  lwkopt = max( lwkopt, n +
394  $ n*ilaenv( 1, 'CUNGQR', ' ', n, 1, n, -1 ) )
395  END IF
396  work( 1 ) = lwkopt
397 *
398  IF( lwork.LT.lwkmin .AND. .NOT.lquery )
399  $ info = -18
400  END IF
401 *
402  IF( info.NE.0 ) THEN
403  CALL xerbla( 'CGGES ', -info )
404  RETURN
405  ELSE IF( lquery ) THEN
406  RETURN
407  END IF
408 *
409 * Quick return if possible
410 *
411  IF( n.EQ.0 ) THEN
412  sdim = 0
413  RETURN
414  END IF
415 *
416 * Get machine constants
417 *
418  eps = slamch( 'P' )
419  smlnum = slamch( 'S' )
420  bignum = one / smlnum
421  CALL slabad( smlnum, bignum )
422  smlnum = sqrt( smlnum ) / eps
423  bignum = one / smlnum
424 *
425 * Scale A if max element outside range [SMLNUM,BIGNUM]
426 *
427  anrm = clange( 'M', n, n, a, lda, rwork )
428  ilascl = .false.
429  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
430  anrmto = smlnum
431  ilascl = .true.
432  ELSE IF( anrm.GT.bignum ) THEN
433  anrmto = bignum
434  ilascl = .true.
435  END IF
436 *
437  IF( ilascl )
438  $ CALL clascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
439 *
440 * Scale B if max element outside range [SMLNUM,BIGNUM]
441 *
442  bnrm = clange( 'M', n, n, b, ldb, rwork )
443  ilbscl = .false.
444  IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
445  bnrmto = smlnum
446  ilbscl = .true.
447  ELSE IF( bnrm.GT.bignum ) THEN
448  bnrmto = bignum
449  ilbscl = .true.
450  END IF
451 *
452  IF( ilbscl )
453  $ CALL clascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
454 *
455 * Permute the matrix to make it more nearly triangular
456 * (Real Workspace: need 6*N)
457 *
458  ileft = 1
459  iright = n + 1
460  irwrk = iright + n
461  CALL cggbal( 'P', n, a, lda, b, ldb, ilo, ihi, rwork( ileft ),
462  $ rwork( iright ), rwork( irwrk ), ierr )
463 *
464 * Reduce B to triangular form (QR decomposition of B)
465 * (Complex Workspace: need N, prefer N*NB)
466 *
467  irows = ihi + 1 - ilo
468  icols = n + 1 - ilo
469  itau = 1
470  iwrk = itau + irows
471  CALL cgeqrf( 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 * (Complex Workspace: need N, prefer N*NB)
476 *
477  CALL cunmqr( 'L', 'C', irows, icols, irows, b( ilo, ilo ), ldb,
478  $ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
479  $ lwork+1-iwrk, ierr )
480 *
481 * Initialize VSL
482 * (Complex Workspace: need N, prefer N*NB)
483 *
484  IF( ilvsl ) THEN
485  CALL claset( 'Full', n, n, czero, cone, vsl, ldvsl )
486  IF( irows.GT.1 ) THEN
487  CALL clacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
488  $ vsl( ilo+1, ilo ), ldvsl )
489  END IF
490  CALL cungqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
491  $ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
492  END IF
493 *
494 * Initialize VSR
495 *
496  IF( ilvsr )
497  $ CALL claset( 'Full', n, n, czero, cone, vsr, ldvsr )
498 *
499 * Reduce to generalized Hessenberg form
500 * (Workspace: none needed)
501 *
502  CALL cgghrd( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
503  $ ldvsl, vsr, ldvsr, ierr )
504 *
505  sdim = 0
506 *
507 * Perform QZ algorithm, computing Schur vectors if desired
508 * (Complex Workspace: need N)
509 * (Real Workspace: need N)
510 *
511  iwrk = itau
512  CALL chgeqz( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
513  $ alpha, beta, vsl, ldvsl, vsr, ldvsr, work( iwrk ),
514  $ lwork+1-iwrk, rwork( irwrk ), ierr )
515  IF( ierr.NE.0 ) THEN
516  IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
517  info = ierr
518  ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
519  info = ierr - n
520  ELSE
521  info = n + 1
522  END IF
523  GO TO 30
524  END IF
525 *
526 * Sort eigenvalues ALPHA/BETA if desired
527 * (Workspace: none needed)
528 *
529  IF( wantst ) THEN
530 *
531 * Undo scaling on eigenvalues before selecting
532 *
533  IF( ilascl )
534  $ CALL clascl( 'G', 0, 0, anrm, anrmto, n, 1, alpha, n, ierr )
535  IF( ilbscl )
536  $ CALL clascl( 'G', 0, 0, bnrm, bnrmto, n, 1, beta, n, ierr )
537 *
538 * Select eigenvalues
539 *
540  DO 10 i = 1, n
541  bwork( i ) = selctg( alpha( i ), beta( i ) )
542  10 CONTINUE
543 *
544  CALL ctgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb, alpha,
545  $ beta, vsl, ldvsl, vsr, ldvsr, sdim, pvsl, pvsr,
546  $ dif, work( iwrk ), lwork-iwrk+1, idum, 1, ierr )
547  IF( ierr.EQ.1 )
548  $ info = n + 3
549 *
550  END IF
551 *
552 * Apply back-permutation to VSL and VSR
553 * (Workspace: none needed)
554 *
555  IF( ilvsl )
556  $ CALL cggbak( 'P', 'L', n, ilo, ihi, rwork( ileft ),
557  $ rwork( iright ), n, vsl, ldvsl, ierr )
558  IF( ilvsr )
559  $ CALL cggbak( 'P', 'R', n, ilo, ihi, rwork( ileft ),
560  $ rwork( iright ), n, vsr, ldvsr, ierr )
561 *
562 * Undo scaling
563 *
564  IF( ilascl ) THEN
565  CALL clascl( 'U', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
566  CALL clascl( 'G', 0, 0, anrmto, anrm, n, 1, alpha, n, ierr )
567  END IF
568 *
569  IF( ilbscl ) THEN
570  CALL clascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
571  CALL clascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
572  END IF
573 *
574  IF( wantst ) THEN
575 *
576 * Check if reordering is correct
577 *
578  lastsl = .true.
579  sdim = 0
580  DO 20 i = 1, n
581  cursl = selctg( alpha( i ), beta( i ) )
582  IF( cursl )
583  $ sdim = sdim + 1
584  IF( cursl .AND. .NOT.lastsl )
585  $ info = n + 2
586  lastsl = cursl
587  20 CONTINUE
588 *
589  END IF
590 *
591  30 CONTINUE
592 *
593  work( 1 ) = lwkopt
594 *
595  RETURN
596 *
597 * End of CGGES
598 *
599  END
subroutine claset(UPLO, M, N, ALPHA, BETA, A, LDA)
CLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values...
Definition: claset.f:108
subroutine cgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
CGEQRF
Definition: cgeqrf.f:138
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slabad(SMALL, LARGE)
SLABAD
Definition: slabad.f:76
subroutine cunmqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMQR
Definition: cunmqr.f:170
subroutine cgges(JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, BWORK, INFO)
CGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE ...
Definition: cgges.f:272
subroutine cggbal(JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
CGGBAL
Definition: cggbal.f:179
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:105
subroutine cggbak(JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV, INFO)
CGGBAK
Definition: cggbak.f:150
subroutine cgghrd(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
CGGHRD
Definition: cgghrd.f:206
subroutine clascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
CLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: clascl.f:141
subroutine ctgsen(IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, ALPHA, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO)
CTGSEN
Definition: ctgsen.f:435
subroutine chgeqz(JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO)
CHGEQZ
Definition: chgeqz.f:286
subroutine cungqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
CUNGQR
Definition: cungqr.f:130