LAPACK  3.10.1
LAPACK: Linear Algebra PACKage
sgges3.f
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1 *> \brief <b> SGGES3 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 *
8 *> \htmlonly
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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/sgges3.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SGGES3( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B,
22 * $ LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL,
23 * $ VSR, LDVSR, WORK, LWORK, 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 A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
32 * $ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
33 * $ VSR( LDVSR, * ), WORK( * )
34 * ..
35 * .. Function Arguments ..
36 * LOGICAL SELCTG
37 * EXTERNAL SELCTG
38 * ..
39 *
40 *
41 *> \par Purpose:
42 * =============
43 *>
44 *> \verbatim
45 *>
46 *> SGGES3 computes for a pair of N-by-N real nonsymmetric matrices (A,B),
47 *> the generalized eigenvalues, the generalized real Schur form (S,T),
48 *> optionally, the left and/or right matrices of Schur vectors (VSL and
49 *> VSR). This gives the generalized Schur factorization
50 *>
51 *> (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )
52 *>
53 *> Optionally, it also orders the eigenvalues so that a selected cluster
54 *> of eigenvalues appears in the leading diagonal blocks of the upper
55 *> quasi-triangular matrix S and the upper triangular matrix T.The
56 *> leading columns of VSL and VSR then form an orthonormal basis for the
57 *> corresponding left and right eigenspaces (deflating subspaces).
58 *>
59 *> (If only the generalized eigenvalues are needed, use the driver
60 *> SGGEV instead, which is faster.)
61 *>
62 *> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
63 *> or a ratio alpha/beta = w, such that A - w*B is singular. It is
64 *> usually represented as the pair (alpha,beta), as there is a
65 *> reasonable interpretation for beta=0 or both being zero.
66 *>
67 *> A pair of matrices (S,T) is in generalized real Schur form if T is
68 *> upper triangular with non-negative diagonal and S is block upper
69 *> triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond
70 *> to real generalized eigenvalues, while 2-by-2 blocks of S will be
71 *> "standardized" by making the corresponding elements of T have the
72 *> form:
73 *> [ a 0 ]
74 *> [ 0 b ]
75 *>
76 *> and the pair of corresponding 2-by-2 blocks in S and T will have a
77 *> complex conjugate pair of generalized eigenvalues.
78 *>
79 *> \endverbatim
80 *
81 * Arguments:
82 * ==========
83 *
84 *> \param[in] JOBVSL
85 *> \verbatim
86 *> JOBVSL is CHARACTER*1
87 *> = 'N': do not compute the left Schur vectors;
88 *> = 'V': compute the left Schur vectors.
89 *> \endverbatim
90 *>
91 *> \param[in] JOBVSR
92 *> \verbatim
93 *> JOBVSR is CHARACTER*1
94 *> = 'N': do not compute the right Schur vectors;
95 *> = 'V': compute the right Schur vectors.
96 *> \endverbatim
97 *>
98 *> \param[in] SORT
99 *> \verbatim
100 *> SORT is CHARACTER*1
101 *> Specifies whether or not to order the eigenvalues on the
102 *> diagonal of the generalized Schur form.
103 *> = 'N': Eigenvalues are not ordered;
104 *> = 'S': Eigenvalues are ordered (see SELCTG);
105 *> \endverbatim
106 *>
107 *> \param[in] SELCTG
108 *> \verbatim
109 *> SELCTG is a LOGICAL FUNCTION of three REAL arguments
110 *> SELCTG must be declared EXTERNAL in the calling subroutine.
111 *> If SORT = 'N', SELCTG is not referenced.
112 *> If SORT = 'S', SELCTG is used to select eigenvalues to sort
113 *> to the top left of the Schur form.
114 *> An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
115 *> SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
116 *> one of a complex conjugate pair of eigenvalues is selected,
117 *> then both complex eigenvalues are selected.
118 *>
119 *> Note that in the ill-conditioned case, a selected complex
120 *> eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
121 *> BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
122 *> in this case.
123 *> \endverbatim
124 *>
125 *> \param[in] N
126 *> \verbatim
127 *> N is INTEGER
128 *> The order of the matrices A, B, VSL, and VSR. N >= 0.
129 *> \endverbatim
130 *>
131 *> \param[in,out] A
132 *> \verbatim
133 *> A is REAL array, dimension (LDA, N)
134 *> On entry, the first of the pair of matrices.
135 *> On exit, A has been overwritten by its generalized Schur
136 *> form S.
137 *> \endverbatim
138 *>
139 *> \param[in] LDA
140 *> \verbatim
141 *> LDA is INTEGER
142 *> The leading dimension of A. LDA >= max(1,N).
143 *> \endverbatim
144 *>
145 *> \param[in,out] B
146 *> \verbatim
147 *> B is REAL array, dimension (LDB, N)
148 *> On entry, the second of the pair of matrices.
149 *> On exit, B has been overwritten by its generalized Schur
150 *> form T.
151 *> \endverbatim
152 *>
153 *> \param[in] LDB
154 *> \verbatim
155 *> LDB is INTEGER
156 *> The leading dimension of B. LDB >= max(1,N).
157 *> \endverbatim
158 *>
159 *> \param[out] SDIM
160 *> \verbatim
161 *> SDIM is INTEGER
162 *> If SORT = 'N', SDIM = 0.
163 *> If SORT = 'S', SDIM = number of eigenvalues (after sorting)
164 *> for which SELCTG is true. (Complex conjugate pairs for which
165 *> SELCTG is true for either eigenvalue count as 2.)
166 *> \endverbatim
167 *>
168 *> \param[out] ALPHAR
169 *> \verbatim
170 *> ALPHAR is REAL array, dimension (N)
171 *> \endverbatim
172 *>
173 *> \param[out] ALPHAI
174 *> \verbatim
175 *> ALPHAI is REAL array, dimension (N)
176 *> \endverbatim
177 *>
178 *> \param[out] BETA
179 *> \verbatim
180 *> BETA is REAL array, dimension (N)
181 *> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
182 *> be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i,
183 *> and BETA(j),j=1,...,N are the diagonals of the complex Schur
184 *> form (S,T) that would result if the 2-by-2 diagonal blocks of
185 *> the real Schur form of (A,B) were further reduced to
186 *> triangular form using 2-by-2 complex unitary transformations.
187 *> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
188 *> positive, then the j-th and (j+1)-st eigenvalues are a
189 *> complex conjugate pair, with ALPHAI(j+1) negative.
190 *>
191 *> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
192 *> may easily over- or underflow, and BETA(j) may even be zero.
193 *> Thus, the user should avoid naively computing the ratio.
194 *> However, ALPHAR and ALPHAI will be always less than and
195 *> usually comparable with norm(A) in magnitude, and BETA always
196 *> less than and usually comparable with norm(B).
197 *> \endverbatim
198 *>
199 *> \param[out] VSL
200 *> \verbatim
201 *> VSL is REAL array, dimension (LDVSL,N)
202 *> If JOBVSL = 'V', VSL will contain the left Schur vectors.
203 *> Not referenced if JOBVSL = 'N'.
204 *> \endverbatim
205 *>
206 *> \param[in] LDVSL
207 *> \verbatim
208 *> LDVSL is INTEGER
209 *> The leading dimension of the matrix VSL. LDVSL >=1, and
210 *> if JOBVSL = 'V', LDVSL >= N.
211 *> \endverbatim
212 *>
213 *> \param[out] VSR
214 *> \verbatim
215 *> VSR is REAL array, dimension (LDVSR,N)
216 *> If JOBVSR = 'V', VSR will contain the right Schur vectors.
217 *> Not referenced if JOBVSR = 'N'.
218 *> \endverbatim
219 *>
220 *> \param[in] LDVSR
221 *> \verbatim
222 *> LDVSR is INTEGER
223 *> The leading dimension of the matrix VSR. LDVSR >= 1, and
224 *> if JOBVSR = 'V', LDVSR >= N.
225 *> \endverbatim
226 *>
227 *> \param[out] WORK
228 *> \verbatim
229 *> WORK is REAL array, dimension (MAX(1,LWORK))
230 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
231 *> \endverbatim
232 *>
233 *> \param[in] LWORK
234 *> \verbatim
235 *> LWORK is INTEGER
236 *> The dimension of the array WORK.
237 *>
238 *> If LWORK = -1, then a workspace query is assumed; the routine
239 *> only calculates the optimal size of the WORK array, returns
240 *> this value as the first entry of the WORK array, and no error
241 *> message related to LWORK is issued by XERBLA.
242 *> \endverbatim
243 *>
244 *> \param[out] BWORK
245 *> \verbatim
246 *> BWORK is LOGICAL array, dimension (N)
247 *> Not referenced if SORT = 'N'.
248 *> \endverbatim
249 *>
250 *> \param[out] INFO
251 *> \verbatim
252 *> INFO is INTEGER
253 *> = 0: successful exit
254 *> < 0: if INFO = -i, the i-th argument had an illegal value.
255 *> = 1,...,N:
256 *> The QZ iteration failed. (A,B) are not in Schur
257 *> form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
258 *> be correct for j=INFO+1,...,N.
259 *> > N: =N+1: other than QZ iteration failed in SLAQZ0.
260 *> =N+2: after reordering, roundoff changed values of
261 *> some complex eigenvalues so that leading
262 *> eigenvalues in the Generalized Schur form no
263 *> longer satisfy SELCTG=.TRUE. This could also
264 *> be caused due to scaling.
265 *> =N+3: reordering failed in STGSEN.
266 *> \endverbatim
267 *
268 * Authors:
269 * ========
270 *
271 *> \author Univ. of Tennessee
272 *> \author Univ. of California Berkeley
273 *> \author Univ. of Colorado Denver
274 *> \author NAG Ltd.
275 *
276 *> \ingroup realGEeigen
277 *
278 * =====================================================================
279  SUBROUTINE sgges3( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B,
280  $ LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL,
281  $ VSR, LDVSR, WORK, LWORK, BWORK, INFO )
282 *
283 * -- LAPACK driver routine --
284 * -- LAPACK is a software package provided by Univ. of Tennessee, --
285 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
286 *
287 * .. Scalar Arguments ..
288  CHARACTER JOBVSL, JOBVSR, SORT
289  INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
290 * ..
291 * .. Array Arguments ..
292  LOGICAL BWORK( * )
293  REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
294  $ b( ldb, * ), beta( * ), vsl( ldvsl, * ),
295  $ vsr( ldvsr, * ), work( * )
296 * ..
297 * .. Function Arguments ..
298  LOGICAL SELCTG
299  EXTERNAL SELCTG
300 * ..
301 *
302 * =====================================================================
303 *
304 * .. Parameters ..
305  REAL ZERO, ONE
306  PARAMETER ( ZERO = 0.0e+0, one = 1.0e+0 )
307 * ..
308 * .. Local Scalars ..
309  LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
310  $ LQUERY, LST2SL, WANTST
311  INTEGER I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT,
312  $ ILO, IP, IRIGHT, IROWS, ITAU, IWRK, LWKOPT
313  REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
314  $ PVSR, SAFMAX, SAFMIN, SMLNUM
315 * ..
316 * .. Local Arrays ..
317  INTEGER IDUM( 1 )
318  REAL DIF( 2 )
319 * ..
320 * .. External Subroutines ..
321  EXTERNAL sgeqrf, sggbak, sggbal, sgghd3, slaqz0, slabad,
323  $ xerbla
324 * ..
325 * .. External Functions ..
326  LOGICAL LSAME
327  REAL SLAMCH, SLANGE
328  EXTERNAL lsame, slamch, slange
329 * ..
330 * .. Intrinsic Functions ..
331  INTRINSIC abs, max, sqrt
332 * ..
333 * .. Executable Statements ..
334 *
335 * Decode the input arguments
336 *
337  IF( lsame( jobvsl, 'N' ) ) THEN
338  ijobvl = 1
339  ilvsl = .false.
340  ELSE IF( lsame( jobvsl, 'V' ) ) THEN
341  ijobvl = 2
342  ilvsl = .true.
343  ELSE
344  ijobvl = -1
345  ilvsl = .false.
346  END IF
347 *
348  IF( lsame( jobvsr, 'N' ) ) THEN
349  ijobvr = 1
350  ilvsr = .false.
351  ELSE IF( lsame( jobvsr, 'V' ) ) THEN
352  ijobvr = 2
353  ilvsr = .true.
354  ELSE
355  ijobvr = -1
356  ilvsr = .false.
357  END IF
358 *
359  wantst = lsame( sort, 'S' )
360 *
361 * Test the input arguments
362 *
363  info = 0
364  lquery = ( lwork.EQ.-1 )
365  IF( ijobvl.LE.0 ) THEN
366  info = -1
367  ELSE IF( ijobvr.LE.0 ) THEN
368  info = -2
369  ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
370  info = -3
371  ELSE IF( n.LT.0 ) THEN
372  info = -5
373  ELSE IF( lda.LT.max( 1, n ) ) THEN
374  info = -7
375  ELSE IF( ldb.LT.max( 1, n ) ) THEN
376  info = -9
377  ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
378  info = -15
379  ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
380  info = -17
381  ELSE IF( lwork.LT.6*n+16 .AND. .NOT.lquery ) THEN
382  info = -19
383  END IF
384 *
385 * Compute workspace
386 *
387  IF( info.EQ.0 ) THEN
388  CALL sgeqrf( n, n, b, ldb, work, work, -1, ierr )
389  lwkopt = max( 6*n+16, 3*n+int( work( 1 ) ) )
390  CALL sormqr( 'L', 'T', n, n, n, b, ldb, work, a, lda, work,
391  $ -1, ierr )
392  lwkopt = max( lwkopt, 3*n+int( work( 1 ) ) )
393  IF( ilvsl ) THEN
394  CALL sorgqr( n, n, n, vsl, ldvsl, work, work, -1, ierr )
395  lwkopt = max( lwkopt, 3*n+int( work( 1 ) ) )
396  END IF
397  CALL sgghd3( jobvsl, jobvsr, n, 1, n, a, lda, b, ldb, vsl,
398  $ ldvsl, vsr, ldvsr, work, -1, ierr )
399  lwkopt = max( lwkopt, 3*n+int( work( 1 ) ) )
400  CALL slaqz0( 'S', jobvsl, jobvsr, n, 1, n, a, lda, b, ldb,
401  $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
402  $ work, -1, 0, ierr )
403  lwkopt = max( lwkopt, 2*n+int( work( 1 ) ) )
404  IF( wantst ) THEN
405  CALL stgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb,
406  $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
407  $ sdim, pvsl, pvsr, dif, work, -1, idum, 1,
408  $ ierr )
409  lwkopt = max( lwkopt, 2*n+int( work( 1 ) ) )
410  END IF
411  work( 1 ) = lwkopt
412  END IF
413 *
414  IF( info.NE.0 ) THEN
415  CALL xerbla( 'SGGES3 ', -info )
416  RETURN
417  ELSE IF( lquery ) THEN
418  RETURN
419  END IF
420 *
421 * Quick return if possible
422 *
423  IF( n.EQ.0 ) THEN
424  sdim = 0
425  RETURN
426  END IF
427 *
428 * Get machine constants
429 *
430  eps = slamch( 'P' )
431  safmin = slamch( 'S' )
432  safmax = one / safmin
433  CALL slabad( safmin, safmax )
434  smlnum = sqrt( safmin ) / eps
435  bignum = one / smlnum
436 *
437 * Scale A if max element outside range [SMLNUM,BIGNUM]
438 *
439  anrm = slange( 'M', n, n, a, lda, work )
440  ilascl = .false.
441  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
442  anrmto = smlnum
443  ilascl = .true.
444  ELSE IF( anrm.GT.bignum ) THEN
445  anrmto = bignum
446  ilascl = .true.
447  END IF
448  IF( ilascl )
449  $ CALL slascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
450 *
451 * Scale B if max element outside range [SMLNUM,BIGNUM]
452 *
453  bnrm = slange( 'M', n, n, b, ldb, work )
454  ilbscl = .false.
455  IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
456  bnrmto = smlnum
457  ilbscl = .true.
458  ELSE IF( bnrm.GT.bignum ) THEN
459  bnrmto = bignum
460  ilbscl = .true.
461  END IF
462  IF( ilbscl )
463  $ CALL slascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
464 *
465 * Permute the matrix to make it more nearly triangular
466 *
467  ileft = 1
468  iright = n + 1
469  iwrk = iright + n
470  CALL sggbal( 'P', n, a, lda, b, ldb, ilo, ihi, work( ileft ),
471  $ work( iright ), work( iwrk ), ierr )
472 *
473 * Reduce B to triangular form (QR decomposition of B)
474 *
475  irows = ihi + 1 - ilo
476  icols = n + 1 - ilo
477  itau = iwrk
478  iwrk = itau + irows
479  CALL sgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
480  $ work( iwrk ), lwork+1-iwrk, ierr )
481 *
482 * Apply the orthogonal transformation to matrix A
483 *
484  CALL sormqr( 'L', 'T', irows, icols, irows, b( ilo, ilo ), ldb,
485  $ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
486  $ lwork+1-iwrk, ierr )
487 *
488 * Initialize VSL
489 *
490  IF( ilvsl ) THEN
491  CALL slaset( 'Full', n, n, zero, one, vsl, ldvsl )
492  IF( irows.GT.1 ) THEN
493  CALL slacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
494  $ vsl( ilo+1, ilo ), ldvsl )
495  END IF
496  CALL sorgqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
497  $ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
498  END IF
499 *
500 * Initialize VSR
501 *
502  IF( ilvsr )
503  $ CALL slaset( 'Full', n, n, zero, one, vsr, ldvsr )
504 *
505 * Reduce to generalized Hessenberg form
506 *
507  CALL sgghd3( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
508  $ ldvsl, vsr, ldvsr, work( iwrk ), lwork+1-iwrk, ierr )
509 *
510 * Perform QZ algorithm, computing Schur vectors if desired
511 *
512  iwrk = itau
513  CALL slaqz0( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
514  $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
515  $ work( iwrk ), lwork+1-iwrk, 0, ierr )
516  IF( ierr.NE.0 ) THEN
517  IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
518  info = ierr
519  ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
520  info = ierr - n
521  ELSE
522  info = n + 1
523  END IF
524  GO TO 40
525  END IF
526 *
527 * Sort eigenvalues ALPHA/BETA if desired
528 *
529  sdim = 0
530  IF( wantst ) THEN
531 *
532 * Undo scaling on eigenvalues before SELCTGing
533 *
534  IF( ilascl ) THEN
535  CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n,
536  $ ierr )
537  CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n,
538  $ ierr )
539  END IF
540  IF( ilbscl )
541  $ CALL slascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
542 *
543 * Select eigenvalues
544 *
545  DO 10 i = 1, n
546  bwork( i ) = selctg( alphar( i ), alphai( i ), beta( i ) )
547  10 CONTINUE
548 *
549  CALL stgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb, alphar,
550  $ alphai, beta, vsl, ldvsl, vsr, ldvsr, sdim, pvsl,
551  $ pvsr, dif, work( iwrk ), lwork-iwrk+1, idum, 1,
552  $ ierr )
553  IF( ierr.EQ.1 )
554  $ info = n + 3
555 *
556  END IF
557 *
558 * Apply back-permutation to VSL and VSR
559 *
560  IF( ilvsl )
561  $ CALL sggbak( 'P', 'L', n, ilo, ihi, work( ileft ),
562  $ work( iright ), n, vsl, ldvsl, ierr )
563 *
564  IF( ilvsr )
565  $ CALL sggbak( 'P', 'R', n, ilo, ihi, work( ileft ),
566  $ work( iright ), n, vsr, ldvsr, ierr )
567 *
568 * Check if unscaling would cause over/underflow, if so, rescale
569 * (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
570 * B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
571 *
572  IF( ilascl )THEN
573  DO 50 i = 1, n
574  IF( alphai( i ).NE.zero ) THEN
575  IF( ( alphar( i )/safmax ).GT.( anrmto/anrm ) .OR.
576  $ ( safmin/alphar( i ) ).GT.( anrm/anrmto ) ) THEN
577  work( 1 ) = abs( a( i, i )/alphar( i ) )
578  beta( i ) = beta( i )*work( 1 )
579  alphar( i ) = alphar( i )*work( 1 )
580  alphai( i ) = alphai( i )*work( 1 )
581  ELSE IF( ( alphai( i )/safmax ).GT.( anrmto/anrm ) .OR.
582  $ ( safmin/alphai( i ) ).GT.( anrm/anrmto ) ) THEN
583  work( 1 ) = abs( a( i, i+1 )/alphai( i ) )
584  beta( i ) = beta( i )*work( 1 )
585  alphar( i ) = alphar( i )*work( 1 )
586  alphai( i ) = alphai( i )*work( 1 )
587  END IF
588  END IF
589  50 CONTINUE
590  END IF
591 *
592  IF( ilbscl )THEN
593  DO 60 i = 1, n
594  IF( alphai( i ).NE.zero ) THEN
595  IF( ( beta( i )/safmax ).GT.( bnrmto/bnrm ) .OR.
596  $ ( safmin/beta( i ) ).GT.( bnrm/bnrmto ) ) THEN
597  work( 1 ) = abs(b( i, i )/beta( i ))
598  beta( i ) = beta( i )*work( 1 )
599  alphar( i ) = alphar( i )*work( 1 )
600  alphai( i ) = alphai( i )*work( 1 )
601  END IF
602  END IF
603  60 CONTINUE
604  END IF
605 *
606 * Undo scaling
607 *
608  IF( ilascl ) THEN
609  CALL slascl( 'H', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
610  CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n, ierr )
611  CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n, ierr )
612  END IF
613 *
614  IF( ilbscl ) THEN
615  CALL slascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
616  CALL slascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
617  END IF
618 *
619  IF( wantst ) THEN
620 *
621 * Check if reordering is correct
622 *
623  lastsl = .true.
624  lst2sl = .true.
625  sdim = 0
626  ip = 0
627  DO 30 i = 1, n
628  cursl = selctg( alphar( i ), alphai( i ), beta( i ) )
629  IF( alphai( i ).EQ.zero ) THEN
630  IF( cursl )
631  $ sdim = sdim + 1
632  ip = 0
633  IF( cursl .AND. .NOT.lastsl )
634  $ info = n + 2
635  ELSE
636  IF( ip.EQ.1 ) THEN
637 *
638 * Last eigenvalue of conjugate pair
639 *
640  cursl = cursl .OR. lastsl
641  lastsl = cursl
642  IF( cursl )
643  $ sdim = sdim + 2
644  ip = -1
645  IF( cursl .AND. .NOT.lst2sl )
646  $ info = n + 2
647  ELSE
648 *
649 * First eigenvalue of conjugate pair
650 *
651  ip = 1
652  END IF
653  END IF
654  lst2sl = lastsl
655  lastsl = cursl
656  30 CONTINUE
657 *
658  END IF
659 *
660  40 CONTINUE
661 *
662  work( 1 ) = lwkopt
663 *
664  RETURN
665 *
666 * End of SGGES3
667 *
668  END
subroutine slabad(SMALL, LARGE)
SLABAD
Definition: slabad.f:74
subroutine slascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
SLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: slascl.f:143
subroutine slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: slaset.f:110
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
recursive subroutine slaqz0(WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, REC, INFO)
SLAQZ0
Definition: slaqz0.f:304
subroutine sggbak(JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV, INFO)
SGGBAK
Definition: sggbak.f:147
subroutine sggbal(JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
SGGBAL
Definition: sggbal.f:177
subroutine sgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGEQRF
Definition: sgeqrf.f:146
subroutine sgges3(JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, BWORK, INFO)
SGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE ...
Definition: sgges3.f:282
subroutine stgsen(IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO)
STGSEN
Definition: stgsen.f:451
subroutine sorgqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
SORGQR
Definition: sorgqr.f:128
subroutine sormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMQR
Definition: sormqr.f:168
subroutine sgghd3(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
SGGHD3
Definition: sgghd3.f:230