1*> \brief \b SLAQR2 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 SLAQR2 + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slaqr2.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slaqr2.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slaqr2.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE SLAQR2( 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*>    SLAQR2 is identical to SLAQR3 except that it avoids
43*>    recursion by calling SLAHQR instead of SLAQR4.
44*>
45*>    Aggressive early deflation:
46*>
47*>    This subroutine accepts as input an upper Hessenberg matrix
48*>    H and performs an orthogonal similarity transformation
49*>    designed to detect and deflate fully converged eigenvalues from
50*>    a trailing principal submatrix.  On output H has been over-
51*>    written by a new Hessenberg matrix that is a perturbation of
52*>    an orthogonal similarity transformation of H.  It is to be
53*>    hoped that the final version of H has many zero subdiagonal
54*>    entries.
55*> \endverbatim
56*
57*  Arguments:
58*  ==========
59*
60*> \param[in] WANTT
61*> \verbatim
62*>          WANTT is LOGICAL
63*>          If .TRUE., then the Hessenberg matrix H is fully updated
64*>          so that the quasi-triangular Schur factor may be
65*>          computed (in cooperation with the calling subroutine).
66*>          If .FALSE., then only enough of H is updated to preserve
67*>          the eigenvalues.
68*> \endverbatim
69*>
70*> \param[in] WANTZ
71*> \verbatim
72*>          WANTZ is LOGICAL
73*>          If .TRUE., then the orthogonal matrix Z is updated so
74*>          so that the orthogonal Schur factor may be computed
75*>          (in cooperation with the calling subroutine).
76*>          If .FALSE., then Z is not referenced.
77*> \endverbatim
78*>
79*> \param[in] N
80*> \verbatim
81*>          N is INTEGER
82*>          The order of the matrix H and (if WANTZ is .TRUE.) the
83*>          order of the orthogonal matrix Z.
84*> \endverbatim
85*>
86*> \param[in] KTOP
87*> \verbatim
88*>          KTOP is INTEGER
89*>          It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
90*>          KBOT and KTOP together determine an isolated block
91*>          along the diagonal of the Hessenberg matrix.
92*> \endverbatim
93*>
94*> \param[in] KBOT
95*> \verbatim
96*>          KBOT is INTEGER
97*>          It is assumed without a check that either
98*>          KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together
99*>          determine an isolated block along the diagonal of the
100*>          Hessenberg matrix.
101*> \endverbatim
102*>
103*> \param[in] NW
104*> \verbatim
105*>          NW is INTEGER
106*>          Deflation window size.  1 .LE. NW .LE. (KBOT-KTOP+1).
107*> \endverbatim
108*>
109*> \param[in,out] H
110*> \verbatim
111*>          H is REAL array, dimension (LDH,N)
112*>          On input the initial N-by-N section of H stores the
113*>          Hessenberg matrix undergoing aggressive early deflation.
114*>          On output H has been transformed by an orthogonal
115*>          similarity transformation, perturbed, and the returned
116*>          to Hessenberg form that (it is to be hoped) has some
117*>          zero subdiagonal entries.
118*> \endverbatim
119*>
120*> \param[in] LDH
121*> \verbatim
122*>          LDH is integer
123*>          Leading dimension of H just as declared in the calling
124*>          subroutine.  N .LE. LDH
125*> \endverbatim
126*>
127*> \param[in] ILOZ
128*> \verbatim
129*>          ILOZ is INTEGER
130*> \endverbatim
131*>
132*> \param[in] IHIZ
133*> \verbatim
134*>          IHIZ is INTEGER
135*>          Specify the rows of Z to which transformations must be
136*>          applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.
137*> \endverbatim
138*>
139*> \param[in,out] Z
140*> \verbatim
141*>          Z is REAL array, dimension (LDZ,N)
142*>          IF WANTZ is .TRUE., then on output, the orthogonal
143*>          similarity transformation mentioned above has been
144*>          accumulated into Z(ILOZ:IHIZ,ILO:IHI) from the right.
145*>          If WANTZ is .FALSE., then Z is unreferenced.
146*> \endverbatim
147*>
148*> \param[in] LDZ
149*> \verbatim
150*>          LDZ is integer
151*>          The leading dimension of Z just as declared in the
152*>          calling subroutine.  1 .LE. LDZ.
153*> \endverbatim
154*>
155*> \param[out] NS
156*> \verbatim
157*>          NS is integer
158*>          The number of unconverged (ie approximate) eigenvalues
159*>          returned in SR and SI that may be used as shifts by the
160*>          calling subroutine.
161*> \endverbatim
162*>
163*> \param[out] ND
164*> \verbatim
165*>          ND is integer
166*>          The number of converged eigenvalues uncovered by this
167*>          subroutine.
168*> \endverbatim
169*>
170*> \param[out] SR
171*> \verbatim
172*>          SR is REAL array, dimension KBOT
173*> \endverbatim
174*>
175*> \param[out] SI
176*> \verbatim
177*>          SI is REAL array, dimension KBOT
178*>          On output, the real and imaginary parts of approximate
179*>          eigenvalues that may be used for shifts are stored in
180*>          SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
181*>          SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
182*>          The real and imaginary parts of converged eigenvalues
183*>          are stored in SR(KBOT-ND+1) through SR(KBOT) and
184*>          SI(KBOT-ND+1) through SI(KBOT), respectively.
185*> \endverbatim
186*>
187*> \param[out] V
188*> \verbatim
189*>          V is REAL array, dimension (LDV,NW)
190*>          An NW-by-NW work array.
191*> \endverbatim
192*>
193*> \param[in] LDV
194*> \verbatim
195*>          LDV is integer scalar
196*>          The leading dimension of V just as declared in the
197*>          calling subroutine.  NW .LE. LDV
198*> \endverbatim
199*>
200*> \param[in] NH
201*> \verbatim
202*>          NH is integer scalar
203*>          The number of columns of T.  NH.GE.NW.
204*> \endverbatim
205*>
206*> \param[out] T
207*> \verbatim
208*>          T is REAL array, dimension (LDT,NW)
209*> \endverbatim
210*>
211*> \param[in] LDT
212*> \verbatim
213*>          LDT is integer
214*>          The leading dimension of T just as declared in the
215*>          calling subroutine.  NW .LE. LDT
216*> \endverbatim
217*>
218*> \param[in] NV
219*> \verbatim
220*>          NV is integer
221*>          The number of rows of work array WV available for
222*>          workspace.  NV.GE.NW.
223*> \endverbatim
224*>
225*> \param[out] WV
226*> \verbatim
227*>          WV is REAL array, dimension (LDWV,NW)
228*> \endverbatim
229*>
230*> \param[in] LDWV
231*> \verbatim
232*>          LDWV is integer
233*>          The leading dimension of W just as declared in the
234*>          calling subroutine.  NW .LE. LDV
235*> \endverbatim
236*>
237*> \param[out] WORK
238*> \verbatim
239*>          WORK is REAL array, dimension LWORK.
240*>          On exit, WORK(1) is set to an estimate of the optimal value
241*>          of LWORK for the given values of N, NW, KTOP and KBOT.
242*> \endverbatim
243*>
244*> \param[in] LWORK
245*> \verbatim
246*>          LWORK is integer
247*>          The dimension of the work array WORK.  LWORK = 2*NW
248*>          suffices, but greater efficiency may result from larger
249*>          values of LWORK.
250*>
251*>          If LWORK = -1, then a workspace query is assumed; SLAQR2
252*>          only estimates the optimal workspace size for the given
253*>          values of N, NW, KTOP and KBOT.  The estimate is returned
254*>          in WORK(1).  No error message related to LWORK is issued
255*>          by XERBLA.  Neither H nor Z are accessed.
256*> \endverbatim
257*
258*  Authors:
259*  ========
260*
261*> \author Univ. of Tennessee
262*> \author Univ. of California Berkeley
263*> \author Univ. of Colorado Denver
264*> \author NAG Ltd.
265*
266*> \date September 2012
267*
268*> \ingroup realOTHERauxiliary
269*
270*> \par Contributors:
271*  ==================
272*>
273*>       Karen Braman and Ralph Byers, Department of Mathematics,
274*>       University of Kansas, USA
275*>
276*  =====================================================================
277      SUBROUTINE SLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
278     $                   IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T,
279     $                   LDT, NV, WV, LDWV, WORK, LWORK )
280*
281*  -- LAPACK auxiliary routine (version 3.4.2) --
282*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
283*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
284*     September 2012
285*
286*     .. Scalar Arguments ..
287      INTEGER            IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
288     $                   LDZ, LWORK, N, ND, NH, NS, NV, NW
289      LOGICAL            WANTT, WANTZ
290*     ..
291*     .. Array Arguments ..
292      REAL               H( LDH, * ), SI( * ), SR( * ), T( LDT, * ),
293     $                   V( LDV, * ), WORK( * ), WV( LDWV, * ),
294     $                   Z( LDZ, * )
295*     ..
296*
297*  ================================================================
298*     .. Parameters ..
299      REAL               ZERO, ONE
300      PARAMETER          ( ZERO = 0.0e0, ONE = 1.0e0 )
301*     ..
302*     .. Local Scalars ..
303      REAL               AA, BB, BETA, CC, CS, DD, EVI, EVK, FOO, S,
304     $                   SAFMAX, SAFMIN, SMLNUM, SN, TAU, ULP
305      INTEGER            I, IFST, ILST, INFO, INFQR, J, JW, K, KCOL,
306     $                   KEND, KLN, KROW, KWTOP, LTOP, LWK1, LWK2,
307     $                   LWKOPT
308      LOGICAL            BULGE, SORTED
309*     ..
310*     .. External Functions ..
311      REAL               SLAMCH
312      EXTERNAL           SLAMCH
313*     ..
314*     .. External Subroutines ..
315      EXTERNAL           SCOPY, SGEHRD, SGEMM, SLABAD, SLACPY, SLAHQR,
316     $                   SLANV2, SLARF, SLARFG, SLASET, SORMHR, STREXC
317*     ..
318*     .. Intrinsic Functions ..
319      INTRINSIC          ABS, INT, MAX, MIN, REAL, SQRT
320*     ..
321*     .. Executable Statements ..
322*
323*     ==== Estimate optimal workspace. ====
324*
325      JW = MIN( NW, KBOT-KTOP+1 )
326      IF( JW.LE.2 ) THEN
327         LWKOPT = 1
328      ELSE
329*
330*        ==== Workspace query call to SGEHRD ====
331*
332         CALL SGEHRD( JW, 1, JW-1, T, LDT, WORK, WORK, -1, INFO )
333         LWK1 = INT( WORK( 1 ) )
334*
335*        ==== Workspace query call to SORMHR ====
336*
337         CALL SORMHR( 'R', 'N', JW, JW, 1, JW-1, T, LDT, WORK, V, LDV,
338     $                WORK, -1, INFO )
339         LWK2 = INT( WORK( 1 ) )
340*
341*        ==== Optimal workspace ====
342*
343         LWKOPT = JW + MAX( LWK1, LWK2 )
344      END IF
345*
346*     ==== Quick return in case of workspace query. ====
347*
348      IF( LWORK.EQ.-1 ) THEN
349         WORK( 1 ) = REAL( LWKOPT )
350         RETURN
351      END IF
352*
353*     ==== Nothing to do ...
354*     ... for an empty active block ... ====
355      NS = 0
356      ND = 0
357      WORK( 1 ) = ONE
358      IF( KTOP.GT.KBOT )
359     $   RETURN
360*     ... nor for an empty deflation window. ====
361      IF( NW.LT.1 )
362     $   RETURN
363*
364*     ==== Machine constants ====
365*
366      SAFMIN = SLAMCH( 'SAFE MINIMUM' )
367      SAFMAX = ONE / SAFMIN
368      CALL SLABAD( SAFMIN, SAFMAX )
369      ULP = SLAMCH( 'PRECISION' )
370      SMLNUM = SAFMIN*( REAL( N ) / ULP )
371*
372*     ==== Setup deflation window ====
373*
374      JW = MIN( NW, KBOT-KTOP+1 )
375      KWTOP = KBOT - JW + 1
376      IF( KWTOP.EQ.KTOP ) THEN
377         S = ZERO
378      ELSE
379         S = H( KWTOP, KWTOP-1 )
380      END IF
381*
382      IF( KBOT.EQ.KWTOP ) THEN
383*
384*        ==== 1-by-1 deflation window: not much to do ====
385*
386         SR( KWTOP ) = H( KWTOP, KWTOP )
387         SI( KWTOP ) = ZERO
388         NS = 1
389         ND = 0
390         IF( ABS( S ).LE.MAX( SMLNUM, ULP*ABS( H( KWTOP, KWTOP ) ) ) )
391     $        THEN
392            NS = 0
393            ND = 1
394            IF( KWTOP.GT.KTOP )
395     $         H( KWTOP, KWTOP-1 ) = ZERO
396         END IF
397         WORK( 1 ) = ONE
398         RETURN
399      END IF
400*
401*     ==== Convert to spike-triangular form.  (In case of a
402*     .    rare QR failure, this routine continues to do
403*     .    aggressive early deflation using that part of
404*     .    the deflation window that converged using INFQR
405*     .    here and there to keep track.) ====
406*
407      CALL SLACPY( 'U', JW, JW, H( KWTOP, KWTOP ), LDH, T, LDT )
408      CALL SCOPY( JW-1, H( KWTOP+1, KWTOP ), LDH+1, T( 2, 1 ), LDT+1 )
409*
410      CALL SLASET( 'A', JW, JW, ZERO, ONE, V, LDV )
411      CALL SLAHQR( .true., .true., JW, 1, JW, T, LDT, SR( KWTOP ),
412     $             SI( KWTOP ), 1, JW, V, LDV, INFQR )
413*
414*     ==== STREXC needs a clean margin near the diagonal ====
415*
416      DO 10 J = 1, JW - 3
417         T( J+2, J ) = ZERO
418         T( J+3, J ) = ZERO
419   10 CONTINUE
420      IF( JW.GT.2 )
421     $   T( JW, JW-2 ) = ZERO
422*
423*     ==== Deflation detection loop ====
424*
425      NS = JW
426      ILST = INFQR + 1
427   20 CONTINUE
428      IF( ILST.LE.NS ) THEN
429         IF( NS.EQ.1 ) THEN
430            BULGE = .FALSE.
431         ELSE
432            BULGE = T( NS, NS-1 ).NE.ZERO
433         END IF
434*
435*        ==== Small spike tip test for deflation ====
436*
437         IF( .NOT.BULGE ) THEN
438*
439*           ==== Real eigenvalue ====
440*
441            FOO = ABS( T( NS, NS ) )
442            IF( FOO.EQ.ZERO )
443     $         FOO = ABS( S )
444            IF( ABS( S*V( 1, NS ) ).LE.MAX( SMLNUM, ULP*FOO ) ) THEN
445*
446*              ==== Deflatable ====
447*
448               NS = NS - 1
449            ELSE
450*
451*              ==== Undeflatable.   Move it up out of the way.
452*              .    (STREXC can not fail in this case.) ====
453*
454               IFST = NS
455               CALL STREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, WORK,
456     $                      INFO )
457               ILST = ILST + 1
458            END IF
459         ELSE
460*
461*           ==== Complex conjugate pair ====
462*
463            FOO = ABS( T( NS, NS ) ) + SQRT( ABS( T( NS, NS-1 ) ) )*
464     $            SQRT( ABS( T( NS-1, NS ) ) )
465            IF( FOO.EQ.ZERO )
466     $         FOO = ABS( S )
467            IF( MAX( ABS( S*V( 1, NS ) ), ABS( S*V( 1, NS-1 ) ) ).LE.
468     $          MAX( SMLNUM, ULP*FOO ) ) THEN
469*
470*              ==== Deflatable ====
471*
472               NS = NS - 2
473            ELSE
474*
475*              ==== Undeflatable. Move them up out of the way.
476*              .    Fortunately, STREXC does the right thing with
477*              .    ILST in case of a rare exchange failure. ====
478*
479               IFST = NS
480               CALL STREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, WORK,
481     $                      INFO )
482               ILST = ILST + 2
483            END IF
484         END IF
485*
486*        ==== End deflation detection loop ====
487*
488         GO TO 20
489      END IF
490*
491*        ==== Return to Hessenberg form ====
492*
493      IF( NS.EQ.0 )
494     $   S = ZERO
495*
496      IF( NS.LT.JW ) THEN
497*
498*        ==== sorting diagonal blocks of T improves accuracy for
499*        .    graded matrices.  Bubble sort deals well with
500*        .    exchange failures. ====
501*
502         SORTED = .false.
503         I = NS + 1
504   30    CONTINUE
505         IF( SORTED )
506     $      GO TO 50
507         SORTED = .true.
508*
509         KEND = I - 1
510         I = INFQR + 1
511         IF( I.EQ.NS ) THEN
512            K = I + 1
513         ELSE IF( T( I+1, I ).EQ.ZERO ) THEN
514            K = I + 1
515         ELSE
516            K = I + 2
517         END IF
518   40    CONTINUE
519         IF( K.LE.KEND ) THEN
520            IF( K.EQ.I+1 ) THEN
521               EVI = ABS( T( I, I ) )
522            ELSE
523               EVI = ABS( T( I, I ) ) + SQRT( ABS( T( I+1, I ) ) )*
524     $               SQRT( ABS( T( I, I+1 ) ) )
525            END IF
526*
527            IF( K.EQ.KEND ) THEN
528               EVK = ABS( T( K, K ) )
529            ELSE IF( T( K+1, K ).EQ.ZERO ) THEN
530               EVK = ABS( T( K, K ) )
531            ELSE
532               EVK = ABS( T( K, K ) ) + SQRT( ABS( T( K+1, K ) ) )*
533     $               SQRT( ABS( T( K, K+1 ) ) )
534            END IF
535*
536            IF( EVI.GE.EVK ) THEN
537               I = K
538            ELSE
539               SORTED = .false.
540               IFST = I
541               ILST = K
542               CALL STREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, WORK,
543     $                      INFO )
544               IF( INFO.EQ.0 ) THEN
545                  I = ILST
546               ELSE
547                  I = K
548               END IF
549            END IF
550            IF( I.EQ.KEND ) THEN
551               K = I + 1
552            ELSE IF( T( I+1, I ).EQ.ZERO ) THEN
553               K = I + 1
554            ELSE
555               K = I + 2
556            END IF
557            GO TO 40
558         END IF
559         GO TO 30
560   50    CONTINUE
561      END IF
562*
563*     ==== Restore shift/eigenvalue array from T ====
564*
565      I = JW
566   60 CONTINUE
567      IF( I.GE.INFQR+1 ) THEN
568         IF( I.EQ.INFQR+1 ) THEN
569            SR( KWTOP+I-1 ) = T( I, I )
570            SI( KWTOP+I-1 ) = ZERO
571            I = I - 1
572         ELSE IF( T( I, I-1 ).EQ.ZERO ) THEN
573            SR( KWTOP+I-1 ) = T( I, I )
574            SI( KWTOP+I-1 ) = ZERO
575            I = I - 1
576         ELSE
577            AA = T( I-1, I-1 )
578            CC = T( I, I-1 )
579            BB = T( I-1, I )
580            DD = T( I, I )
581            CALL SLANV2( AA, BB, CC, DD, SR( KWTOP+I-2 ),
582     $                   SI( KWTOP+I-2 ), SR( KWTOP+I-1 ),
583     $                   SI( KWTOP+I-1 ), CS, SN )
584            I = I - 2
585         END IF
586         GO TO 60
587      END IF
588*
589      IF( NS.LT.JW .OR. S.EQ.ZERO ) THEN
590         IF( NS.GT.1 .AND. S.NE.ZERO ) THEN
591*
592*           ==== Reflect spike back into lower triangle ====
593*
594            CALL SCOPY( NS, V, LDV, WORK, 1 )
595            BETA = WORK( 1 )
596            CALL SLARFG( NS, BETA, WORK( 2 ), 1, TAU )
597            WORK( 1 ) = ONE
598*
599            CALL SLASET( 'L', JW-2, JW-2, ZERO, ZERO, T( 3, 1 ), LDT )
600*
601            CALL SLARF( 'L', NS, JW, WORK, 1, TAU, T, LDT,
602     $                  WORK( JW+1 ) )
603            CALL SLARF( 'R', NS, NS, WORK, 1, TAU, T, LDT,
604     $                  WORK( JW+1 ) )
605            CALL SLARF( 'R', JW, NS, WORK, 1, TAU, V, LDV,
606     $                  WORK( JW+1 ) )
607*
608            CALL SGEHRD( JW, 1, NS, T, LDT, WORK, WORK( JW+1 ),
609     $                   LWORK-JW, INFO )
610         END IF
611*
612*        ==== Copy updated reduced window into place ====
613*
614         IF( KWTOP.GT.1 )
615     $      H( KWTOP, KWTOP-1 ) = S*V( 1, 1 )
616         CALL SLACPY( 'U', JW, JW, T, LDT, H( KWTOP, KWTOP ), LDH )
617         CALL SCOPY( JW-1, T( 2, 1 ), LDT+1, H( KWTOP+1, KWTOP ),
618     $               LDH+1 )
619*
620*        ==== Accumulate orthogonal matrix in order update
621*        .    H and Z, if requested.  ====
622*
623         IF( NS.GT.1 .AND. S.NE.ZERO )
624     $      CALL SORMHR( 'R', 'N', JW, NS, 1, NS, T, LDT, WORK, V, LDV,
625     $                   WORK( JW+1 ), LWORK-JW, INFO )
626*
627*        ==== Update vertical slab in H ====
628*
629         IF( WANTT ) THEN
630            LTOP = 1
631         ELSE
632            LTOP = KTOP
633         END IF
634         DO 70 KROW = LTOP, KWTOP - 1, NV
635            KLN = MIN( NV, KWTOP-KROW )
636            CALL SGEMM( 'N', 'N', KLN, JW, JW, ONE, H( KROW, KWTOP ),
637     $                  LDH, V, LDV, ZERO, WV, LDWV )
638            CALL SLACPY( 'A', KLN, JW, WV, LDWV, H( KROW, KWTOP ), LDH )
639   70    CONTINUE
640*
641*        ==== Update horizontal slab in H ====
642*
643         IF( WANTT ) THEN
644            DO 80 KCOL = KBOT + 1, N, NH
645               KLN = MIN( NH, N-KCOL+1 )
646               CALL SGEMM( 'C', 'N', JW, KLN, JW, ONE, V, LDV,
647     $                     H( KWTOP, KCOL ), LDH, ZERO, T, LDT )
648               CALL SLACPY( 'A', JW, KLN, T, LDT, H( KWTOP, KCOL ),
649     $                      LDH )
650   80       CONTINUE
651         END IF
652*
653*        ==== Update vertical slab in Z ====
654*
655         IF( WANTZ ) THEN
656            DO 90 KROW = ILOZ, IHIZ, NV
657               KLN = MIN( NV, IHIZ-KROW+1 )
658               CALL SGEMM( 'N', 'N', KLN, JW, JW, ONE, Z( KROW, KWTOP ),
659     $                     LDZ, V, LDV, ZERO, WV, LDWV )
660               CALL SLACPY( 'A', KLN, JW, WV, LDWV, Z( KROW, KWTOP ),
661     $                      LDZ )
662   90       CONTINUE
663         END IF
664      END IF
665*
666*     ==== Return the number of deflations ... ====
667*
668      ND = JW - NS
669*
670*     ==== ... and the number of shifts. (Subtracting
671*     .    INFQR from the spike length takes care
672*     .    of the case of a rare QR failure while
673*     .    calculating eigenvalues of the deflation
674*     .    window.)  ====
675*
676      NS = NS - INFQR
677*
678*      ==== Return optimal workspace. ====
679*
680      WORK( 1 ) = REAL( LWKOPT )
681*
682*     ==== End of SLAQR2 ====
683*
684      END
685c $Id$
686