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slaqr3.f
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1 *> \brief \b SLAQR3 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download SLAQR3 + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slaqr3.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slaqr3.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slaqr3.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
22 * IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T,
23 * LDT, NV, WV, LDWV, WORK, LWORK )
24 *
25 * .. Scalar Arguments ..
26 * INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
27 * $ LDZ, LWORK, N, ND, NH, NS, NV, NW
28 * LOGICAL WANTT, WANTZ
29 * ..
30 * .. Array Arguments ..
31 * REAL H( LDH, * ), SI( * ), SR( * ), T( LDT, * ),
32 * $ V( LDV, * ), WORK( * ), WV( LDWV, * ),
33 * $ Z( LDZ, * )
34 * ..
35 *
36 *
37 *> \par Purpose:
38 * =============
39 *>
40 *> \verbatim
41 *>
42 *> Aggressive early deflation:
43 *>
44 *> SLAQR3 accepts as input an upper Hessenberg matrix
45 *> H and performs an orthogonal similarity transformation
46 *> designed to detect and deflate fully converged eigenvalues from
47 *> a trailing principal submatrix. On output H has been over-
48 *> written by a new Hessenberg matrix that is a perturbation of
49 *> an orthogonal similarity transformation of H. It is to be
50 *> hoped that the final version of H has many zero subdiagonal
51 *> entries.
52 *> \endverbatim
53 *
54 * Arguments:
55 * ==========
56 *
57 *> \param[in] WANTT
58 *> \verbatim
59 *> WANTT is LOGICAL
60 *> If .TRUE., then the Hessenberg matrix H is fully updated
61 *> so that the quasi-triangular Schur factor may be
62 *> computed (in cooperation with the calling subroutine).
63 *> If .FALSE., then only enough of H is updated to preserve
64 *> the eigenvalues.
65 *> \endverbatim
66 *>
67 *> \param[in] WANTZ
68 *> \verbatim
69 *> WANTZ is LOGICAL
70 *> If .TRUE., then the orthogonal matrix Z is updated so
71 *> so that the orthogonal Schur factor may be computed
72 *> (in cooperation with the calling subroutine).
73 *> If .FALSE., then Z is not referenced.
74 *> \endverbatim
75 *>
76 *> \param[in] N
77 *> \verbatim
78 *> N is INTEGER
79 *> The order of the matrix H and (if WANTZ is .TRUE.) the
80 *> order of the orthogonal matrix Z.
81 *> \endverbatim
82 *>
83 *> \param[in] KTOP
84 *> \verbatim
85 *> KTOP is INTEGER
86 *> It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
87 *> KBOT and KTOP together determine an isolated block
88 *> along the diagonal of the Hessenberg matrix.
89 *> \endverbatim
90 *>
91 *> \param[in] KBOT
92 *> \verbatim
93 *> KBOT is INTEGER
94 *> It is assumed without a check that either
95 *> KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together
96 *> determine an isolated block along the diagonal of the
97 *> Hessenberg matrix.
98 *> \endverbatim
99 *>
100 *> \param[in] NW
101 *> \verbatim
102 *> NW is INTEGER
103 *> Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1).
104 *> \endverbatim
105 *>
106 *> \param[in,out] H
107 *> \verbatim
108 *> H is REAL array, dimension (LDH,N)
109 *> On input the initial N-by-N section of H stores the
110 *> Hessenberg matrix undergoing aggressive early deflation.
111 *> On output H has been transformed by an orthogonal
112 *> similarity transformation, perturbed, and the returned
113 *> to Hessenberg form that (it is to be hoped) has some
114 *> zero subdiagonal entries.
115 *> \endverbatim
116 *>
117 *> \param[in] LDH
118 *> \verbatim
119 *> LDH is integer
120 *> Leading dimension of H just as declared in the calling
121 *> subroutine. N .LE. LDH
122 *> \endverbatim
123 *>
124 *> \param[in] ILOZ
125 *> \verbatim
126 *> ILOZ is INTEGER
127 *> \endverbatim
128 *>
129 *> \param[in] IHIZ
130 *> \verbatim
131 *> IHIZ is INTEGER
132 *> Specify the rows of Z to which transformations must be
133 *> applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.
134 *> \endverbatim
135 *>
136 *> \param[in,out] Z
137 *> \verbatim
138 *> Z is REAL array, dimension (LDZ,N)
139 *> IF WANTZ is .TRUE., then on output, the orthogonal
140 *> similarity transformation mentioned above has been
141 *> accumulated into Z(ILOZ:IHIZ,ILO:IHI) from the right.
142 *> If WANTZ is .FALSE., then Z is unreferenced.
143 *> \endverbatim
144 *>
145 *> \param[in] LDZ
146 *> \verbatim
147 *> LDZ is integer
148 *> The leading dimension of Z just as declared in the
149 *> calling subroutine. 1 .LE. LDZ.
150 *> \endverbatim
151 *>
152 *> \param[out] NS
153 *> \verbatim
154 *> NS is integer
155 *> The number of unconverged (ie approximate) eigenvalues
156 *> returned in SR and SI that may be used as shifts by the
157 *> calling subroutine.
158 *> \endverbatim
159 *>
160 *> \param[out] ND
161 *> \verbatim
162 *> ND is integer
163 *> The number of converged eigenvalues uncovered by this
164 *> subroutine.
165 *> \endverbatim
166 *>
167 *> \param[out] SR
168 *> \verbatim
169 *> SR is REAL array, dimension KBOT
170 *> \endverbatim
171 *>
172 *> \param[out] SI
173 *> \verbatim
174 *> SI is REAL array, dimension KBOT
175 *> On output, the real and imaginary parts of approximate
176 *> eigenvalues that may be used for shifts are stored in
177 *> SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
178 *> SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
179 *> The real and imaginary parts of converged eigenvalues
180 *> are stored in SR(KBOT-ND+1) through SR(KBOT) and
181 *> SI(KBOT-ND+1) through SI(KBOT), respectively.
182 *> \endverbatim
183 *>
184 *> \param[out] V
185 *> \verbatim
186 *> V is REAL array, dimension (LDV,NW)
187 *> An NW-by-NW work array.
188 *> \endverbatim
189 *>
190 *> \param[in] LDV
191 *> \verbatim
192 *> LDV is integer scalar
193 *> The leading dimension of V just as declared in the
194 *> calling subroutine. NW .LE. LDV
195 *> \endverbatim
196 *>
197 *> \param[in] NH
198 *> \verbatim
199 *> NH is integer scalar
200 *> The number of columns of T. NH.GE.NW.
201 *> \endverbatim
202 *>
203 *> \param[out] T
204 *> \verbatim
205 *> T is REAL array, dimension (LDT,NW)
206 *> \endverbatim
207 *>
208 *> \param[in] LDT
209 *> \verbatim
210 *> LDT is integer
211 *> The leading dimension of T just as declared in the
212 *> calling subroutine. NW .LE. LDT
213 *> \endverbatim
214 *>
215 *> \param[in] NV
216 *> \verbatim
217 *> NV is integer
218 *> The number of rows of work array WV available for
219 *> workspace. NV.GE.NW.
220 *> \endverbatim
221 *>
222 *> \param[out] WV
223 *> \verbatim
224 *> WV is REAL array, dimension (LDWV,NW)
225 *> \endverbatim
226 *>
227 *> \param[in] LDWV
228 *> \verbatim
229 *> LDWV is integer
230 *> The leading dimension of W just as declared in the
231 *> calling subroutine. NW .LE. LDV
232 *> \endverbatim
233 *>
234 *> \param[out] WORK
235 *> \verbatim
236 *> WORK is REAL array, dimension LWORK.
237 *> On exit, WORK(1) is set to an estimate of the optimal value
238 *> of LWORK for the given values of N, NW, KTOP and KBOT.
239 *> \endverbatim
240 *>
241 *> \param[in] LWORK
242 *> \verbatim
243 *> LWORK is integer
244 *> The dimension of the work array WORK. LWORK = 2*NW
245 *> suffices, but greater efficiency may result from larger
246 *> values of LWORK.
247 *>
248 *> If LWORK = -1, then a workspace query is assumed; SLAQR3
249 *> only estimates the optimal workspace size for the given
250 *> values of N, NW, KTOP and KBOT. The estimate is returned
251 *> in WORK(1). No error message related to LWORK is issued
252 *> by XERBLA. Neither H nor Z are accessed.
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 *> \date September 2012
264 *
265 *> \ingroup realOTHERauxiliary
266 *
267 *> \par Contributors:
268 * ==================
269 *>
270 *> Karen Braman and Ralph Byers, Department of Mathematics,
271 *> University of Kansas, USA
272 *>
273 * =====================================================================
274  SUBROUTINE slaqr3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
275  $ ihiz, z, ldz, ns, nd, sr, si, v, ldv, nh, t,
276  $ ldt, nv, wv, ldwv, work, lwork )
277 *
278 * -- LAPACK auxiliary routine (version 3.4.2) --
279 * -- LAPACK is a software package provided by Univ. of Tennessee, --
280 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
281 * September 2012
282 *
283 * .. Scalar Arguments ..
284  INTEGER ihiz, iloz, kbot, ktop, ldh, ldt, ldv, ldwv,
285  $ ldz, lwork, n, nd, nh, ns, nv, nw
286  LOGICAL wantt, wantz
287 * ..
288 * .. Array Arguments ..
289  REAL h( ldh, * ), si( * ), sr( * ), t( ldt, * ),
290  $ v( ldv, * ), work( * ), wv( ldwv, * ),
291  $ z( ldz, * )
292 * ..
293 *
294 * ================================================================
295 * .. Parameters ..
296  REAL zero, one
297  parameter( zero = 0.0e0, one = 1.0e0 )
298 * ..
299 * .. Local Scalars ..
300  REAL aa, bb, beta, cc, cs, dd, evi, evk, foo, s,
301  $ safmax, safmin, smlnum, sn, tau, ulp
302  INTEGER i, ifst, ilst, info, infqr, j, jw, k, kcol,
303  $ kend, kln, krow, kwtop, ltop, lwk1, lwk2, lwk3,
304  $ lwkopt, nmin
305  LOGICAL bulge, sorted
306 * ..
307 * .. External Functions ..
308  REAL slamch
309  INTEGER ilaenv
310  EXTERNAL slamch, ilaenv
311 * ..
312 * .. External Subroutines ..
313  EXTERNAL scopy, sgehrd, sgemm, slabad, slacpy, slahqr,
315  $ strexc
316 * ..
317 * .. Intrinsic Functions ..
318  INTRINSIC abs, int, max, min, REAL, sqrt
319 * ..
320 * .. Executable Statements ..
321 *
322 * ==== Estimate optimal workspace. ====
323 *
324  jw = min( nw, kbot-ktop+1 )
325  IF( jw.LE.2 ) THEN
326  lwkopt = 1
327  ELSE
328 *
329 * ==== Workspace query call to SGEHRD ====
330 *
331  CALL sgehrd( jw, 1, jw-1, t, ldt, work, work, -1, info )
332  lwk1 = int( work( 1 ) )
333 *
334 * ==== Workspace query call to SORMHR ====
335 *
336  CALL sormhr( 'R', 'N', jw, jw, 1, jw-1, t, ldt, work, v, ldv,
337  $ work, -1, info )
338  lwk2 = int( work( 1 ) )
339 *
340 * ==== Workspace query call to SLAQR4 ====
341 *
342  CALL slaqr4( .true., .true., jw, 1, jw, t, ldt, sr, si, 1, jw,
343  $ v, ldv, work, -1, infqr )
344  lwk3 = int( work( 1 ) )
345 *
346 * ==== Optimal workspace ====
347 *
348  lwkopt = max( jw+max( lwk1, lwk2 ), lwk3 )
349  END IF
350 *
351 * ==== Quick return in case of workspace query. ====
352 *
353  IF( lwork.EQ.-1 ) THEN
354  work( 1 ) = REAL( lwkopt )
355  return
356  END IF
357 *
358 * ==== Nothing to do ...
359 * ... for an empty active block ... ====
360  ns = 0
361  nd = 0
362  work( 1 ) = one
363  IF( ktop.GT.kbot )
364  $ return
365 * ... nor for an empty deflation window. ====
366  IF( nw.LT.1 )
367  $ return
368 *
369 * ==== Machine constants ====
370 *
371  safmin = slamch( 'SAFE MINIMUM' )
372  safmax = one / safmin
373  CALL slabad( safmin, safmax )
374  ulp = slamch( 'PRECISION' )
375  smlnum = safmin*( REAL( N ) / ulp )
376 *
377 * ==== Setup deflation window ====
378 *
379  jw = min( nw, kbot-ktop+1 )
380  kwtop = kbot - jw + 1
381  IF( kwtop.EQ.ktop ) THEN
382  s = zero
383  ELSE
384  s = h( kwtop, kwtop-1 )
385  END IF
386 *
387  IF( kbot.EQ.kwtop ) THEN
388 *
389 * ==== 1-by-1 deflation window: not much to do ====
390 *
391  sr( kwtop ) = h( kwtop, kwtop )
392  si( kwtop ) = zero
393  ns = 1
394  nd = 0
395  IF( abs( s ).LE.max( smlnum, ulp*abs( h( kwtop, kwtop ) ) ) )
396  $ THEN
397  ns = 0
398  nd = 1
399  IF( kwtop.GT.ktop )
400  $ h( kwtop, kwtop-1 ) = zero
401  END IF
402  work( 1 ) = one
403  return
404  END IF
405 *
406 * ==== Convert to spike-triangular form. (In case of a
407 * . rare QR failure, this routine continues to do
408 * . aggressive early deflation using that part of
409 * . the deflation window that converged using INFQR
410 * . here and there to keep track.) ====
411 *
412  CALL slacpy( 'U', jw, jw, h( kwtop, kwtop ), ldh, t, ldt )
413  CALL scopy( jw-1, h( kwtop+1, kwtop ), ldh+1, t( 2, 1 ), ldt+1 )
414 *
415  CALL slaset( 'A', jw, jw, zero, one, v, ldv )
416  nmin = ilaenv( 12, 'SLAQR3', 'SV', jw, 1, jw, lwork )
417  IF( jw.GT.nmin ) THEN
418  CALL slaqr4( .true., .true., jw, 1, jw, t, ldt, sr( kwtop ),
419  $ si( kwtop ), 1, jw, v, ldv, work, lwork, infqr )
420  ELSE
421  CALL slahqr( .true., .true., jw, 1, jw, t, ldt, sr( kwtop ),
422  $ si( kwtop ), 1, jw, v, ldv, infqr )
423  END IF
424 *
425 * ==== STREXC needs a clean margin near the diagonal ====
426 *
427  DO 10 j = 1, jw - 3
428  t( j+2, j ) = zero
429  t( j+3, j ) = zero
430  10 continue
431  IF( jw.GT.2 )
432  $ t( jw, jw-2 ) = zero
433 *
434 * ==== Deflation detection loop ====
435 *
436  ns = jw
437  ilst = infqr + 1
438  20 continue
439  IF( ilst.LE.ns ) THEN
440  IF( ns.EQ.1 ) THEN
441  bulge = .false.
442  ELSE
443  bulge = t( ns, ns-1 ).NE.zero
444  END IF
445 *
446 * ==== Small spike tip test for deflation ====
447 *
448  IF( .NOT. bulge ) THEN
449 *
450 * ==== Real eigenvalue ====
451 *
452  foo = abs( t( ns, ns ) )
453  IF( foo.EQ.zero )
454  $ foo = abs( s )
455  IF( abs( s*v( 1, ns ) ).LE.max( smlnum, ulp*foo ) ) THEN
456 *
457 * ==== Deflatable ====
458 *
459  ns = ns - 1
460  ELSE
461 *
462 * ==== Undeflatable. Move it up out of the way.
463 * . (STREXC can not fail in this case.) ====
464 *
465  ifst = ns
466  CALL strexc( 'V', jw, t, ldt, v, ldv, ifst, ilst, work,
467  $ info )
468  ilst = ilst + 1
469  END IF
470  ELSE
471 *
472 * ==== Complex conjugate pair ====
473 *
474  foo = abs( t( ns, ns ) ) + sqrt( abs( t( ns, ns-1 ) ) )*
475  $ sqrt( abs( t( ns-1, ns ) ) )
476  IF( foo.EQ.zero )
477  $ foo = abs( s )
478  IF( max( abs( s*v( 1, ns ) ), abs( s*v( 1, ns-1 ) ) ).LE.
479  $ max( smlnum, ulp*foo ) ) THEN
480 *
481 * ==== Deflatable ====
482 *
483  ns = ns - 2
484  ELSE
485 *
486 * ==== Undeflatable. Move them up out of the way.
487 * . Fortunately, STREXC does the right thing with
488 * . ILST in case of a rare exchange failure. ====
489 *
490  ifst = ns
491  CALL strexc( 'V', jw, t, ldt, v, ldv, ifst, ilst, work,
492  $ info )
493  ilst = ilst + 2
494  END IF
495  END IF
496 *
497 * ==== End deflation detection loop ====
498 *
499  go to 20
500  END IF
501 *
502 * ==== Return to Hessenberg form ====
503 *
504  IF( ns.EQ.0 )
505  $ s = zero
506 *
507  IF( ns.LT.jw ) THEN
508 *
509 * ==== sorting diagonal blocks of T improves accuracy for
510 * . graded matrices. Bubble sort deals well with
511 * . exchange failures. ====
512 *
513  sorted = .false.
514  i = ns + 1
515  30 continue
516  IF( sorted )
517  $ go to 50
518  sorted = .true.
519 *
520  kend = i - 1
521  i = infqr + 1
522  IF( i.EQ.ns ) THEN
523  k = i + 1
524  ELSE IF( t( i+1, i ).EQ.zero ) THEN
525  k = i + 1
526  ELSE
527  k = i + 2
528  END IF
529  40 continue
530  IF( k.LE.kend ) THEN
531  IF( k.EQ.i+1 ) THEN
532  evi = abs( t( i, i ) )
533  ELSE
534  evi = abs( t( i, i ) ) + sqrt( abs( t( i+1, i ) ) )*
535  $ sqrt( abs( t( i, i+1 ) ) )
536  END IF
537 *
538  IF( k.EQ.kend ) THEN
539  evk = abs( t( k, k ) )
540  ELSE IF( t( k+1, k ).EQ.zero ) THEN
541  evk = abs( t( k, k ) )
542  ELSE
543  evk = abs( t( k, k ) ) + sqrt( abs( t( k+1, k ) ) )*
544  $ sqrt( abs( t( k, k+1 ) ) )
545  END IF
546 *
547  IF( evi.GE.evk ) THEN
548  i = k
549  ELSE
550  sorted = .false.
551  ifst = i
552  ilst = k
553  CALL strexc( 'V', jw, t, ldt, v, ldv, ifst, ilst, work,
554  $ info )
555  IF( info.EQ.0 ) THEN
556  i = ilst
557  ELSE
558  i = k
559  END IF
560  END IF
561  IF( i.EQ.kend ) THEN
562  k = i + 1
563  ELSE IF( t( i+1, i ).EQ.zero ) THEN
564  k = i + 1
565  ELSE
566  k = i + 2
567  END IF
568  go to 40
569  END IF
570  go to 30
571  50 continue
572  END IF
573 *
574 * ==== Restore shift/eigenvalue array from T ====
575 *
576  i = jw
577  60 continue
578  IF( i.GE.infqr+1 ) THEN
579  IF( i.EQ.infqr+1 ) THEN
580  sr( kwtop+i-1 ) = t( i, i )
581  si( kwtop+i-1 ) = zero
582  i = i - 1
583  ELSE IF( t( i, i-1 ).EQ.zero ) THEN
584  sr( kwtop+i-1 ) = t( i, i )
585  si( kwtop+i-1 ) = zero
586  i = i - 1
587  ELSE
588  aa = t( i-1, i-1 )
589  cc = t( i, i-1 )
590  bb = t( i-1, i )
591  dd = t( i, i )
592  CALL slanv2( aa, bb, cc, dd, sr( kwtop+i-2 ),
593  $ si( kwtop+i-2 ), sr( kwtop+i-1 ),
594  $ si( kwtop+i-1 ), cs, sn )
595  i = i - 2
596  END IF
597  go to 60
598  END IF
599 *
600  IF( ns.LT.jw .OR. s.EQ.zero ) THEN
601  IF( ns.GT.1 .AND. s.NE.zero ) THEN
602 *
603 * ==== Reflect spike back into lower triangle ====
604 *
605  CALL scopy( ns, v, ldv, work, 1 )
606  beta = work( 1 )
607  CALL slarfg( ns, beta, work( 2 ), 1, tau )
608  work( 1 ) = one
609 *
610  CALL slaset( 'L', jw-2, jw-2, zero, zero, t( 3, 1 ), ldt )
611 *
612  CALL slarf( 'L', ns, jw, work, 1, tau, t, ldt,
613  $ work( jw+1 ) )
614  CALL slarf( 'R', ns, ns, work, 1, tau, t, ldt,
615  $ work( jw+1 ) )
616  CALL slarf( 'R', jw, ns, work, 1, tau, v, ldv,
617  $ work( jw+1 ) )
618 *
619  CALL sgehrd( jw, 1, ns, t, ldt, work, work( jw+1 ),
620  $ lwork-jw, info )
621  END IF
622 *
623 * ==== Copy updated reduced window into place ====
624 *
625  IF( kwtop.GT.1 )
626  $ h( kwtop, kwtop-1 ) = s*v( 1, 1 )
627  CALL slacpy( 'U', jw, jw, t, ldt, h( kwtop, kwtop ), ldh )
628  CALL scopy( jw-1, t( 2, 1 ), ldt+1, h( kwtop+1, kwtop ),
629  $ ldh+1 )
630 *
631 * ==== Accumulate orthogonal matrix in order update
632 * . H and Z, if requested. ====
633 *
634  IF( ns.GT.1 .AND. s.NE.zero )
635  $ CALL sormhr( 'R', 'N', jw, ns, 1, ns, t, ldt, work, v, ldv,
636  $ work( jw+1 ), lwork-jw, info )
637 *
638 * ==== Update vertical slab in H ====
639 *
640  IF( wantt ) THEN
641  ltop = 1
642  ELSE
643  ltop = ktop
644  END IF
645  DO 70 krow = ltop, kwtop - 1, nv
646  kln = min( nv, kwtop-krow )
647  CALL sgemm( 'N', 'N', kln, jw, jw, one, h( krow, kwtop ),
648  $ ldh, v, ldv, zero, wv, ldwv )
649  CALL slacpy( 'A', kln, jw, wv, ldwv, h( krow, kwtop ), ldh )
650  70 continue
651 *
652 * ==== Update horizontal slab in H ====
653 *
654  IF( wantt ) THEN
655  DO 80 kcol = kbot + 1, n, nh
656  kln = min( nh, n-kcol+1 )
657  CALL sgemm( 'C', 'N', jw, kln, jw, one, v, ldv,
658  $ h( kwtop, kcol ), ldh, zero, t, ldt )
659  CALL slacpy( 'A', jw, kln, t, ldt, h( kwtop, kcol ),
660  $ ldh )
661  80 continue
662  END IF
663 *
664 * ==== Update vertical slab in Z ====
665 *
666  IF( wantz ) THEN
667  DO 90 krow = iloz, ihiz, nv
668  kln = min( nv, ihiz-krow+1 )
669  CALL sgemm( 'N', 'N', kln, jw, jw, one, z( krow, kwtop ),
670  $ ldz, v, ldv, zero, wv, ldwv )
671  CALL slacpy( 'A', kln, jw, wv, ldwv, z( krow, kwtop ),
672  $ ldz )
673  90 continue
674  END IF
675  END IF
676 *
677 * ==== Return the number of deflations ... ====
678 *
679  nd = jw - ns
680 *
681 * ==== ... and the number of shifts. (Subtracting
682 * . INFQR from the spike length takes care
683 * . of the case of a rare QR failure while
684 * . calculating eigenvalues of the deflation
685 * . window.) ====
686 *
687  ns = ns - infqr
688 *
689 * ==== Return optimal workspace. ====
690 *
691  work( 1 ) = REAL( lwkopt )
692 *
693 * ==== End of SLAQR3 ====
694 *
695  END