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dlaqr4.f
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1 *> \brief \b DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download DLAQR4 + 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/dlaqr4.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
22 * ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
23 *
24 * .. Scalar Arguments ..
25 * INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
26 * LOGICAL WANTT, WANTZ
27 * ..
28 * .. Array Arguments ..
29 * DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
30 * $ Z( LDZ, * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *>
39 *> DLAQR4 implements one level of recursion for DLAQR0.
40 *> It is a complete implementation of the small bulge multi-shift
41 *> QR algorithm. It may be called by DLAQR0 and, for large enough
42 *> deflation window size, it may be called by DLAQR3. This
43 *> subroutine is identical to DLAQR0 except that it calls DLAQR2
44 *> instead of DLAQR3.
45 *>
46 *> DLAQR4 computes the eigenvalues of a Hessenberg matrix H
47 *> and, optionally, the matrices T and Z from the Schur decomposition
48 *> H = Z T Z**T, where T is an upper quasi-triangular matrix (the
49 *> Schur form), and Z is the orthogonal matrix of Schur vectors.
50 *>
51 *> Optionally Z may be postmultiplied into an input orthogonal
52 *> matrix Q so that this routine can give the Schur factorization
53 *> of a matrix A which has been reduced to the Hessenberg form H
54 *> by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
55 *> \endverbatim
56 *
57 * Arguments:
58 * ==========
59 *
60 *> \param[in] WANTT
61 *> \verbatim
62 *> WANTT is LOGICAL
63 *> = .TRUE. : the full Schur form T is required;
64 *> = .FALSE.: only eigenvalues are required.
65 *> \endverbatim
66 *>
67 *> \param[in] WANTZ
68 *> \verbatim
69 *> WANTZ is LOGICAL
70 *> = .TRUE. : the matrix of Schur vectors Z is required;
71 *> = .FALSE.: Schur vectors are not required.
72 *> \endverbatim
73 *>
74 *> \param[in] N
75 *> \verbatim
76 *> N is INTEGER
77 *> The order of the matrix H. N .GE. 0.
78 *> \endverbatim
79 *>
80 *> \param[in] ILO
81 *> \verbatim
82 *> ILO is INTEGER
83 *> \endverbatim
84 *>
85 *> \param[in] IHI
86 *> \verbatim
87 *> IHI is INTEGER
88 *> It is assumed that H is already upper triangular in rows
89 *> and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
90 *> H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
91 *> previous call to DGEBAL, and then passed to DGEHRD when the
92 *> matrix output by DGEBAL is reduced to Hessenberg form.
93 *> Otherwise, ILO and IHI should be set to 1 and N,
94 *> respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
95 *> If N = 0, then ILO = 1 and IHI = 0.
96 *> \endverbatim
97 *>
98 *> \param[in,out] H
99 *> \verbatim
100 *> H is DOUBLE PRECISION array, dimension (LDH,N)
101 *> On entry, the upper Hessenberg matrix H.
102 *> On exit, if INFO = 0 and WANTT is .TRUE., then H contains
103 *> the upper quasi-triangular matrix T from the Schur
104 *> decomposition (the Schur form); 2-by-2 diagonal blocks
105 *> (corresponding to complex conjugate pairs of eigenvalues)
106 *> are returned in standard form, with H(i,i) = H(i+1,i+1)
107 *> and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
108 *> .FALSE., then the contents of H are unspecified on exit.
109 *> (The output value of H when INFO.GT.0 is given under the
110 *> description of INFO below.)
111 *>
112 *> This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
113 *> j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
114 *> \endverbatim
115 *>
116 *> \param[in] LDH
117 *> \verbatim
118 *> LDH is INTEGER
119 *> The leading dimension of the array H. LDH .GE. max(1,N).
120 *> \endverbatim
121 *>
122 *> \param[out] WR
123 *> \verbatim
124 *> WR is DOUBLE PRECISION array, dimension (IHI)
125 *> \endverbatim
126 *>
127 *> \param[out] WI
128 *> \verbatim
129 *> WI is DOUBLE PRECISION array, dimension (IHI)
130 *> The real and imaginary parts, respectively, of the computed
131 *> eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
132 *> and WI(ILO:IHI). If two eigenvalues are computed as a
133 *> complex conjugate pair, they are stored in consecutive
134 *> elements of WR and WI, say the i-th and (i+1)th, with
135 *> WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
136 *> the eigenvalues are stored in the same order as on the
137 *> diagonal of the Schur form returned in H, with
138 *> WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
139 *> block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
140 *> WI(i+1) = -WI(i).
141 *> \endverbatim
142 *>
143 *> \param[in] ILOZ
144 *> \verbatim
145 *> ILOZ is INTEGER
146 *> \endverbatim
147 *>
148 *> \param[in] IHIZ
149 *> \verbatim
150 *> IHIZ is INTEGER
151 *> Specify the rows of Z to which transformations must be
152 *> applied if WANTZ is .TRUE..
153 *> 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
154 *> \endverbatim
155 *>
156 *> \param[in,out] Z
157 *> \verbatim
158 *> Z is DOUBLE PRECISION array, dimension (LDZ,IHI)
159 *> If WANTZ is .FALSE., then Z is not referenced.
160 *> If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
161 *> replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
162 *> orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
163 *> (The output value of Z when INFO.GT.0 is given under
164 *> the description of INFO below.)
165 *> \endverbatim
166 *>
167 *> \param[in] LDZ
168 *> \verbatim
169 *> LDZ is INTEGER
170 *> The leading dimension of the array Z. if WANTZ is .TRUE.
171 *> then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.
172 *> \endverbatim
173 *>
174 *> \param[out] WORK
175 *> \verbatim
176 *> WORK is DOUBLE PRECISION array, dimension LWORK
177 *> On exit, if LWORK = -1, WORK(1) returns an estimate of
178 *> the optimal value for LWORK.
179 *> \endverbatim
180 *>
181 *> \param[in] LWORK
182 *> \verbatim
183 *> LWORK is INTEGER
184 *> The dimension of the array WORK. LWORK .GE. max(1,N)
185 *> is sufficient, but LWORK typically as large as 6*N may
186 *> be required for optimal performance. A workspace query
187 *> to determine the optimal workspace size is recommended.
188 *>
189 *> If LWORK = -1, then DLAQR4 does a workspace query.
190 *> In this case, DLAQR4 checks the input parameters and
191 *> estimates the optimal workspace size for the given
192 *> values of N, ILO and IHI. The estimate is returned
193 *> in WORK(1). No error message related to LWORK is
194 *> issued by XERBLA. Neither H nor Z are accessed.
195 *> \endverbatim
196 *>
197 *> \param[out] INFO
198 *> \verbatim
199 *> INFO is INTEGER
200 *> = 0: successful exit
201 *> .GT. 0: if INFO = i, DLAQR4 failed to compute all of
202 *> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
203 *> and WI contain those eigenvalues which have been
204 *> successfully computed. (Failures are rare.)
205 *>
206 *> If INFO .GT. 0 and WANT is .FALSE., then on exit,
207 *> the remaining unconverged eigenvalues are the eigen-
208 *> values of the upper Hessenberg matrix rows and
209 *> columns ILO through INFO of the final, output
210 *> value of H.
211 *>
212 *> If INFO .GT. 0 and WANTT is .TRUE., then on exit
213 *>
214 *> (*) (initial value of H)*U = U*(final value of H)
215 *>
216 *> where U is a orthogonal matrix. The final
217 *> value of H is upper Hessenberg and triangular in
218 *> rows and columns INFO+1 through IHI.
219 *>
220 *> If INFO .GT. 0 and WANTZ is .TRUE., then on exit
221 *>
222 *> (final value of Z(ILO:IHI,ILOZ:IHIZ)
223 *> = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
224 *>
225 *> where U is the orthogonal matrix in (*) (regard-
226 *> less of the value of WANTT.)
227 *>
228 *> If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
229 *> accessed.
230 *> \endverbatim
231 *
232 * Authors:
233 * ========
234 *
235 *> \author Univ. of Tennessee
236 *> \author Univ. of California Berkeley
237 *> \author Univ. of Colorado Denver
238 *> \author NAG Ltd.
239 *
240 *> \date September 2012
241 *
242 *> \ingroup doubleOTHERauxiliary
243 *
244 *> \par Contributors:
245 * ==================
246 *>
247 *> Karen Braman and Ralph Byers, Department of Mathematics,
248 *> University of Kansas, USA
249 *
250 *> \par References:
251 * ================
252 *>
253 *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
254 *> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
255 *> Performance, SIAM Journal of Matrix Analysis, volume 23, pages
256 *> 929--947, 2002.
257 *> \n
258 *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
259 *> Algorithm Part II: Aggressive Early Deflation, SIAM Journal
260 *> of Matrix Analysis, volume 23, pages 948--973, 2002.
261 *>
262 * =====================================================================
263  SUBROUTINE dlaqr4( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
264  $ iloz, ihiz, z, ldz, work, lwork, info )
265 *
266 * -- LAPACK auxiliary routine (version 3.4.2) --
267 * -- LAPACK is a software package provided by Univ. of Tennessee, --
268 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
269 * September 2012
270 *
271 * .. Scalar Arguments ..
272  INTEGER ihi, ihiz, ilo, iloz, info, ldh, ldz, lwork, n
273  LOGICAL wantt, wantz
274 * ..
275 * .. Array Arguments ..
276  DOUBLE PRECISION h( ldh, * ), wi( * ), work( * ), wr( * ),
277  $ z( ldz, * )
278 * ..
279 *
280 * ================================================================
281 * .. Parameters ..
282 *
283 * ==== Matrices of order NTINY or smaller must be processed by
284 * . DLAHQR because of insufficient subdiagonal scratch space.
285 * . (This is a hard limit.) ====
286  INTEGER ntiny
287  parameter( ntiny = 11 )
288 *
289 * ==== Exceptional deflation windows: try to cure rare
290 * . slow convergence by varying the size of the
291 * . deflation window after KEXNW iterations. ====
292  INTEGER kexnw
293  parameter( kexnw = 5 )
294 *
295 * ==== Exceptional shifts: try to cure rare slow convergence
296 * . with ad-hoc exceptional shifts every KEXSH iterations.
297 * . ====
298  INTEGER kexsh
299  parameter( kexsh = 6 )
300 *
301 * ==== The constants WILK1 and WILK2 are used to form the
302 * . exceptional shifts. ====
303  DOUBLE PRECISION wilk1, wilk2
304  parameter( wilk1 = 0.75d0, wilk2 = -0.4375d0 )
305  DOUBLE PRECISION zero, one
306  parameter( zero = 0.0d0, one = 1.0d0 )
307 * ..
308 * .. Local Scalars ..
309  DOUBLE PRECISION aa, bb, cc, cs, dd, sn, ss, swap
310  INTEGER i, inf, it, itmax, k, kacc22, kbot, kdu, ks,
311  $ kt, ktop, ku, kv, kwh, kwtop, kwv, ld, ls,
312  $ lwkopt, ndec, ndfl, nh, nho, nibble, nmin, ns,
313  $ nsmax, nsr, nve, nw, nwmax, nwr, nwupbd
314  LOGICAL sorted
315  CHARACTER jbcmpz*2
316 * ..
317 * .. External Functions ..
318  INTEGER ilaenv
319  EXTERNAL ilaenv
320 * ..
321 * .. Local Arrays ..
322  DOUBLE PRECISION zdum( 1, 1 )
323 * ..
324 * .. External Subroutines ..
325  EXTERNAL dlacpy, dlahqr, dlanv2, dlaqr2, dlaqr5
326 * ..
327 * .. Intrinsic Functions ..
328  INTRINSIC abs, dble, int, max, min, mod
329 * ..
330 * .. Executable Statements ..
331  info = 0
332 *
333 * ==== Quick return for N = 0: nothing to do. ====
334 *
335  IF( n.EQ.0 ) THEN
336  work( 1 ) = one
337  return
338  END IF
339 *
340  IF( n.LE.ntiny ) THEN
341 *
342 * ==== Tiny matrices must use DLAHQR. ====
343 *
344  lwkopt = 1
345  IF( lwork.NE.-1 )
346  $ CALL dlahqr( wantt, wantz, n, ilo, ihi, h, ldh, wr, wi,
347  $ iloz, ihiz, z, ldz, info )
348  ELSE
349 *
350 * ==== Use small bulge multi-shift QR with aggressive early
351 * . deflation on larger-than-tiny matrices. ====
352 *
353 * ==== Hope for the best. ====
354 *
355  info = 0
356 *
357 * ==== Set up job flags for ILAENV. ====
358 *
359  IF( wantt ) THEN
360  jbcmpz( 1: 1 ) = 'S'
361  ELSE
362  jbcmpz( 1: 1 ) = 'E'
363  END IF
364  IF( wantz ) THEN
365  jbcmpz( 2: 2 ) = 'V'
366  ELSE
367  jbcmpz( 2: 2 ) = 'N'
368  END IF
369 *
370 * ==== NWR = recommended deflation window size. At this
371 * . point, N .GT. NTINY = 11, so there is enough
372 * . subdiagonal workspace for NWR.GE.2 as required.
373 * . (In fact, there is enough subdiagonal space for
374 * . NWR.GE.3.) ====
375 *
376  nwr = ilaenv( 13, 'DLAQR4', jbcmpz, n, ilo, ihi, lwork )
377  nwr = max( 2, nwr )
378  nwr = min( ihi-ilo+1, ( n-1 ) / 3, nwr )
379 *
380 * ==== NSR = recommended number of simultaneous shifts.
381 * . At this point N .GT. NTINY = 11, so there is at
382 * . enough subdiagonal workspace for NSR to be even
383 * . and greater than or equal to two as required. ====
384 *
385  nsr = ilaenv( 15, 'DLAQR4', jbcmpz, n, ilo, ihi, lwork )
386  nsr = min( nsr, ( n+6 ) / 9, ihi-ilo )
387  nsr = max( 2, nsr-mod( nsr, 2 ) )
388 *
389 * ==== Estimate optimal workspace ====
390 *
391 * ==== Workspace query call to DLAQR2 ====
392 *
393  CALL dlaqr2( wantt, wantz, n, ilo, ihi, nwr+1, h, ldh, iloz,
394  $ ihiz, z, ldz, ls, ld, wr, wi, h, ldh, n, h, ldh,
395  $ n, h, ldh, work, -1 )
396 *
397 * ==== Optimal workspace = MAX(DLAQR5, DLAQR2) ====
398 *
399  lwkopt = max( 3*nsr / 2, int( work( 1 ) ) )
400 *
401 * ==== Quick return in case of workspace query. ====
402 *
403  IF( lwork.EQ.-1 ) THEN
404  work( 1 ) = dble( lwkopt )
405  return
406  END IF
407 *
408 * ==== DLAHQR/DLAQR0 crossover point ====
409 *
410  nmin = ilaenv( 12, 'DLAQR4', jbcmpz, n, ilo, ihi, lwork )
411  nmin = max( ntiny, nmin )
412 *
413 * ==== Nibble crossover point ====
414 *
415  nibble = ilaenv( 14, 'DLAQR4', jbcmpz, n, ilo, ihi, lwork )
416  nibble = max( 0, nibble )
417 *
418 * ==== Accumulate reflections during ttswp? Use block
419 * . 2-by-2 structure during matrix-matrix multiply? ====
420 *
421  kacc22 = ilaenv( 16, 'DLAQR4', jbcmpz, n, ilo, ihi, lwork )
422  kacc22 = max( 0, kacc22 )
423  kacc22 = min( 2, kacc22 )
424 *
425 * ==== NWMAX = the largest possible deflation window for
426 * . which there is sufficient workspace. ====
427 *
428  nwmax = min( ( n-1 ) / 3, lwork / 2 )
429  nw = nwmax
430 *
431 * ==== NSMAX = the Largest number of simultaneous shifts
432 * . for which there is sufficient workspace. ====
433 *
434  nsmax = min( ( n+6 ) / 9, 2*lwork / 3 )
435  nsmax = nsmax - mod( nsmax, 2 )
436 *
437 * ==== NDFL: an iteration count restarted at deflation. ====
438 *
439  ndfl = 1
440 *
441 * ==== ITMAX = iteration limit ====
442 *
443  itmax = max( 30, 2*kexsh )*max( 10, ( ihi-ilo+1 ) )
444 *
445 * ==== Last row and column in the active block ====
446 *
447  kbot = ihi
448 *
449 * ==== Main Loop ====
450 *
451  DO 80 it = 1, itmax
452 *
453 * ==== Done when KBOT falls below ILO ====
454 *
455  IF( kbot.LT.ilo )
456  $ go to 90
457 *
458 * ==== Locate active block ====
459 *
460  DO 10 k = kbot, ilo + 1, -1
461  IF( h( k, k-1 ).EQ.zero )
462  $ go to 20
463  10 continue
464  k = ilo
465  20 continue
466  ktop = k
467 *
468 * ==== Select deflation window size:
469 * . Typical Case:
470 * . If possible and advisable, nibble the entire
471 * . active block. If not, use size MIN(NWR,NWMAX)
472 * . or MIN(NWR+1,NWMAX) depending upon which has
473 * . the smaller corresponding subdiagonal entry
474 * . (a heuristic).
475 * .
476 * . Exceptional Case:
477 * . If there have been no deflations in KEXNW or
478 * . more iterations, then vary the deflation window
479 * . size. At first, because, larger windows are,
480 * . in general, more powerful than smaller ones,
481 * . rapidly increase the window to the maximum possible.
482 * . Then, gradually reduce the window size. ====
483 *
484  nh = kbot - ktop + 1
485  nwupbd = min( nh, nwmax )
486  IF( ndfl.LT.kexnw ) THEN
487  nw = min( nwupbd, nwr )
488  ELSE
489  nw = min( nwupbd, 2*nw )
490  END IF
491  IF( nw.LT.nwmax ) THEN
492  IF( nw.GE.nh-1 ) THEN
493  nw = nh
494  ELSE
495  kwtop = kbot - nw + 1
496  IF( abs( h( kwtop, kwtop-1 ) ).GT.
497  $ abs( h( kwtop-1, kwtop-2 ) ) )nw = nw + 1
498  END IF
499  END IF
500  IF( ndfl.LT.kexnw ) THEN
501  ndec = -1
502  ELSE IF( ndec.GE.0 .OR. nw.GE.nwupbd ) THEN
503  ndec = ndec + 1
504  IF( nw-ndec.LT.2 )
505  $ ndec = 0
506  nw = nw - ndec
507  END IF
508 *
509 * ==== Aggressive early deflation:
510 * . split workspace under the subdiagonal into
511 * . - an nw-by-nw work array V in the lower
512 * . left-hand-corner,
513 * . - an NW-by-at-least-NW-but-more-is-better
514 * . (NW-by-NHO) horizontal work array along
515 * . the bottom edge,
516 * . - an at-least-NW-but-more-is-better (NHV-by-NW)
517 * . vertical work array along the left-hand-edge.
518 * . ====
519 *
520  kv = n - nw + 1
521  kt = nw + 1
522  nho = ( n-nw-1 ) - kt + 1
523  kwv = nw + 2
524  nve = ( n-nw ) - kwv + 1
525 *
526 * ==== Aggressive early deflation ====
527 *
528  CALL dlaqr2( wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz,
529  $ ihiz, z, ldz, ls, ld, wr, wi, h( kv, 1 ), ldh,
530  $ nho, h( kv, kt ), ldh, nve, h( kwv, 1 ), ldh,
531  $ work, lwork )
532 *
533 * ==== Adjust KBOT accounting for new deflations. ====
534 *
535  kbot = kbot - ld
536 *
537 * ==== KS points to the shifts. ====
538 *
539  ks = kbot - ls + 1
540 *
541 * ==== Skip an expensive QR sweep if there is a (partly
542 * . heuristic) reason to expect that many eigenvalues
543 * . will deflate without it. Here, the QR sweep is
544 * . skipped if many eigenvalues have just been deflated
545 * . or if the remaining active block is small.
546 *
547  IF( ( ld.EQ.0 ) .OR. ( ( 100*ld.LE.nw*nibble ) .AND. ( kbot-
548  $ ktop+1.GT.min( nmin, nwmax ) ) ) ) THEN
549 *
550 * ==== NS = nominal number of simultaneous shifts.
551 * . This may be lowered (slightly) if DLAQR2
552 * . did not provide that many shifts. ====
553 *
554  ns = min( nsmax, nsr, max( 2, kbot-ktop ) )
555  ns = ns - mod( ns, 2 )
556 *
557 * ==== If there have been no deflations
558 * . in a multiple of KEXSH iterations,
559 * . then try exceptional shifts.
560 * . Otherwise use shifts provided by
561 * . DLAQR2 above or from the eigenvalues
562 * . of a trailing principal submatrix. ====
563 *
564  IF( mod( ndfl, kexsh ).EQ.0 ) THEN
565  ks = kbot - ns + 1
566  DO 30 i = kbot, max( ks+1, ktop+2 ), -2
567  ss = abs( h( i, i-1 ) ) + abs( h( i-1, i-2 ) )
568  aa = wilk1*ss + h( i, i )
569  bb = ss
570  cc = wilk2*ss
571  dd = aa
572  CALL dlanv2( aa, bb, cc, dd, wr( i-1 ), wi( i-1 ),
573  $ wr( i ), wi( i ), cs, sn )
574  30 continue
575  IF( ks.EQ.ktop ) THEN
576  wr( ks+1 ) = h( ks+1, ks+1 )
577  wi( ks+1 ) = zero
578  wr( ks ) = wr( ks+1 )
579  wi( ks ) = wi( ks+1 )
580  END IF
581  ELSE
582 *
583 * ==== Got NS/2 or fewer shifts? Use DLAHQR
584 * . on a trailing principal submatrix to
585 * . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
586 * . there is enough space below the subdiagonal
587 * . to fit an NS-by-NS scratch array.) ====
588 *
589  IF( kbot-ks+1.LE.ns / 2 ) THEN
590  ks = kbot - ns + 1
591  kt = n - ns + 1
592  CALL dlacpy( 'A', ns, ns, h( ks, ks ), ldh,
593  $ h( kt, 1 ), ldh )
594  CALL dlahqr( .false., .false., ns, 1, ns,
595  $ h( kt, 1 ), ldh, wr( ks ), wi( ks ),
596  $ 1, 1, zdum, 1, inf )
597  ks = ks + inf
598 *
599 * ==== In case of a rare QR failure use
600 * . eigenvalues of the trailing 2-by-2
601 * . principal submatrix. ====
602 *
603  IF( ks.GE.kbot ) THEN
604  aa = h( kbot-1, kbot-1 )
605  cc = h( kbot, kbot-1 )
606  bb = h( kbot-1, kbot )
607  dd = h( kbot, kbot )
608  CALL dlanv2( aa, bb, cc, dd, wr( kbot-1 ),
609  $ wi( kbot-1 ), wr( kbot ),
610  $ wi( kbot ), cs, sn )
611  ks = kbot - 1
612  END IF
613  END IF
614 *
615  IF( kbot-ks+1.GT.ns ) THEN
616 *
617 * ==== Sort the shifts (Helps a little)
618 * . Bubble sort keeps complex conjugate
619 * . pairs together. ====
620 *
621  sorted = .false.
622  DO 50 k = kbot, ks + 1, -1
623  IF( sorted )
624  $ go to 60
625  sorted = .true.
626  DO 40 i = ks, k - 1
627  IF( abs( wr( i ) )+abs( wi( i ) ).LT.
628  $ abs( wr( i+1 ) )+abs( wi( i+1 ) ) ) THEN
629  sorted = .false.
630 *
631  swap = wr( i )
632  wr( i ) = wr( i+1 )
633  wr( i+1 ) = swap
634 *
635  swap = wi( i )
636  wi( i ) = wi( i+1 )
637  wi( i+1 ) = swap
638  END IF
639  40 continue
640  50 continue
641  60 continue
642  END IF
643 *
644 * ==== Shuffle shifts into pairs of real shifts
645 * . and pairs of complex conjugate shifts
646 * . assuming complex conjugate shifts are
647 * . already adjacent to one another. (Yes,
648 * . they are.) ====
649 *
650  DO 70 i = kbot, ks + 2, -2
651  IF( wi( i ).NE.-wi( i-1 ) ) THEN
652 *
653  swap = wr( i )
654  wr( i ) = wr( i-1 )
655  wr( i-1 ) = wr( i-2 )
656  wr( i-2 ) = swap
657 *
658  swap = wi( i )
659  wi( i ) = wi( i-1 )
660  wi( i-1 ) = wi( i-2 )
661  wi( i-2 ) = swap
662  END IF
663  70 continue
664  END IF
665 *
666 * ==== If there are only two shifts and both are
667 * . real, then use only one. ====
668 *
669  IF( kbot-ks+1.EQ.2 ) THEN
670  IF( wi( kbot ).EQ.zero ) THEN
671  IF( abs( wr( kbot )-h( kbot, kbot ) ).LT.
672  $ abs( wr( kbot-1 )-h( kbot, kbot ) ) ) THEN
673  wr( kbot-1 ) = wr( kbot )
674  ELSE
675  wr( kbot ) = wr( kbot-1 )
676  END IF
677  END IF
678  END IF
679 *
680 * ==== Use up to NS of the the smallest magnatiude
681 * . shifts. If there aren't NS shifts available,
682 * . then use them all, possibly dropping one to
683 * . make the number of shifts even. ====
684 *
685  ns = min( ns, kbot-ks+1 )
686  ns = ns - mod( ns, 2 )
687  ks = kbot - ns + 1
688 *
689 * ==== Small-bulge multi-shift QR sweep:
690 * . split workspace under the subdiagonal into
691 * . - a KDU-by-KDU work array U in the lower
692 * . left-hand-corner,
693 * . - a KDU-by-at-least-KDU-but-more-is-better
694 * . (KDU-by-NHo) horizontal work array WH along
695 * . the bottom edge,
696 * . - and an at-least-KDU-but-more-is-better-by-KDU
697 * . (NVE-by-KDU) vertical work WV arrow along
698 * . the left-hand-edge. ====
699 *
700  kdu = 3*ns - 3
701  ku = n - kdu + 1
702  kwh = kdu + 1
703  nho = ( n-kdu+1-4 ) - ( kdu+1 ) + 1
704  kwv = kdu + 4
705  nve = n - kdu - kwv + 1
706 *
707 * ==== Small-bulge multi-shift QR sweep ====
708 *
709  CALL dlaqr5( wantt, wantz, kacc22, n, ktop, kbot, ns,
710  $ wr( ks ), wi( ks ), h, ldh, iloz, ihiz, z,
711  $ ldz, work, 3, h( ku, 1 ), ldh, nve,
712  $ h( kwv, 1 ), ldh, nho, h( ku, kwh ), ldh )
713  END IF
714 *
715 * ==== Note progress (or the lack of it). ====
716 *
717  IF( ld.GT.0 ) THEN
718  ndfl = 1
719  ELSE
720  ndfl = ndfl + 1
721  END IF
722 *
723 * ==== End of main loop ====
724  80 continue
725 *
726 * ==== Iteration limit exceeded. Set INFO to show where
727 * . the problem occurred and exit. ====
728 *
729  info = kbot
730  90 continue
731  END IF
732 *
733 * ==== Return the optimal value of LWORK. ====
734 *
735  work( 1 ) = dble( lwkopt )
736 *
737 * ==== End of DLAQR4 ====
738 *
739  END