1*> \brief \b DLAQR3 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 DLAQR3 + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr3.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqr3.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqr3.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE DLAQR3( 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*       DOUBLE PRECISION   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*>    DLAQR3 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (KBOT)
170*> \endverbatim
171*>
172*> \param[out] SI
173*> \verbatim
174*>          SI is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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; DLAQR3
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 doubleOTHERauxiliary
266*
267*> \par Contributors:
268*  ==================
269*>
270*>       Karen Braman and Ralph Byers, Department of Mathematics,
271*>       University of Kansas, USA
272*>
273*  =====================================================================
274      SUBROUTINE DLAQR3( 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      DOUBLE PRECISION   H( LDH, * ), SI( * ), SR( * ), T( LDT, * ),
290     $                   V( LDV, * ), WORK( * ), WV( LDWV, * ),
291     $                   Z( LDZ, * )
292*     ..
293*
294*  ================================================================
295*     .. Parameters ..
296      DOUBLE PRECISION   ZERO, ONE
297      PARAMETER          ( ZERO = 0.0d0, ONE = 1.0d0 )
298*     ..
299*     .. Local Scalars ..
300      DOUBLE PRECISION   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      DOUBLE PRECISION   DLAMCH
309      INTEGER            ILAENV
310      EXTERNAL           DLAMCH, ILAENV
311*     ..
312*     .. External Subroutines ..
313      EXTERNAL           DCOPY, DGEHRD, DGEMM, DLABAD, DLACPY, DLAHQR,
314     $                   DLANV2, DLAQR4, DLARF, DLARFG, DLASET, DORMHR,
315     $                   DTREXC
316*     ..
317*     .. Intrinsic Functions ..
318      INTRINSIC          ABS, DBLE, INT, MAX, MIN, 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 DGEHRD ====
330*
331         CALL DGEHRD( JW, 1, JW-1, T, LDT, WORK, WORK, -1, INFO )
332         LWK1 = INT( WORK( 1 ) )
333*
334*        ==== Workspace query call to DORMHR ====
335*
336         CALL DORMHR( '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 DLAQR4 ====
341*
342         CALL DLAQR4( .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 ) = DBLE( 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 = DLAMCH( 'SAFE MINIMUM' )
372      SAFMAX = ONE / SAFMIN
373      CALL DLABAD( SAFMIN, SAFMAX )
374      ULP = DLAMCH( 'PRECISION' )
375      SMLNUM = SAFMIN*( DBLE( 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 DLACPY( 'U', JW, JW, H( KWTOP, KWTOP ), LDH, T, LDT )
413      CALL DCOPY( JW-1, H( KWTOP+1, KWTOP ), LDH+1, T( 2, 1 ), LDT+1 )
414*
415      CALL DLASET( 'A', JW, JW, ZERO, ONE, V, LDV )
416      NMIN = ILAENV( 12, 'DLAQR3', 'SV', JW, 1, JW, LWORK )
417      IF( JW.GT.NMIN ) THEN
418         CALL DLAQR4( .true., .true., JW, 1, JW, T, LDT, SR( KWTOP ),
419     $                SI( KWTOP ), 1, JW, V, LDV, WORK, LWORK, INFQR )
420      ELSE
421         CALL DLAHQR( .true., .true., JW, 1, JW, T, LDT, SR( KWTOP ),
422     $                SI( KWTOP ), 1, JW, V, LDV, INFQR )
423      END IF
424*
425*     ==== DTREXC 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*              .    (DTREXC can not fail in this case.) ====
464*
465               IFST = NS
466               CALL DTREXC( '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, DTREXC does the right thing with
488*              .    ILST in case of a rare exchange failure. ====
489*
490               IFST = NS
491               CALL DTREXC( '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 DTREXC( '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 DLANV2( 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 DCOPY( NS, V, LDV, WORK, 1 )
606            BETA = WORK( 1 )
607            CALL DLARFG( NS, BETA, WORK( 2 ), 1, TAU )
608            WORK( 1 ) = ONE
609*
610            CALL DLASET( 'L', JW-2, JW-2, ZERO, ZERO, T( 3, 1 ), LDT )
611*
612            CALL DLARF( 'L', NS, JW, WORK, 1, TAU, T, LDT,
613     $                  WORK( JW+1 ) )
614            CALL DLARF( 'R', NS, NS, WORK, 1, TAU, T, LDT,
615     $                  WORK( JW+1 ) )
616            CALL DLARF( 'R', JW, NS, WORK, 1, TAU, V, LDV,
617     $                  WORK( JW+1 ) )
618*
619            CALL DGEHRD( 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 DLACPY( 'U', JW, JW, T, LDT, H( KWTOP, KWTOP ), LDH )
628         CALL DCOPY( 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 DORMHR( '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 DGEMM( 'N', 'N', KLN, JW, JW, ONE, H( KROW, KWTOP ),
648     $                  LDH, V, LDV, ZERO, WV, LDWV )
649            CALL DLACPY( '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 DGEMM( 'C', 'N', JW, KLN, JW, ONE, V, LDV,
658     $                     H( KWTOP, KCOL ), LDH, ZERO, T, LDT )
659               CALL DLACPY( '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 DGEMM( 'N', 'N', KLN, JW, JW, ONE, Z( KROW, KWTOP ),
670     $                     LDZ, V, LDV, ZERO, WV, LDWV )
671               CALL DLACPY( '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 ) = DBLE( LWKOPT )
692*
693*     ==== End of DLAQR3 ====
694*
695      END
696c $Id$
697