001:       SUBROUTINE DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
002:      $                   ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
003: *
004: *  -- LAPACK auxiliary routine (version 3.2) --
005: *     Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..
006: *     November 2006
007: *
008: *     .. Scalar Arguments ..
009:       INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
010:       LOGICAL            WANTT, WANTZ
011: *     ..
012: *     .. Array Arguments ..
013:       DOUBLE PRECISION   H( LDH, * ), WI( * ), WORK( * ), WR( * ),
014:      $                   Z( LDZ, * )
015: *     ..
016: *
017: *     Purpose
018: *     =======
019: *
020: *     DLAQR0 computes the eigenvalues of a Hessenberg matrix H
021: *     and, optionally, the matrices T and Z from the Schur decomposition
022: *     H = Z T Z**T, where T is an upper quasi-triangular matrix (the
023: *     Schur form), and Z is the orthogonal matrix of Schur vectors.
024: *
025: *     Optionally Z may be postmultiplied into an input orthogonal
026: *     matrix Q so that this routine can give the Schur factorization
027: *     of a matrix A which has been reduced to the Hessenberg form H
028: *     by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.
029: *
030: *     Arguments
031: *     =========
032: *
033: *     WANTT   (input) LOGICAL
034: *          = .TRUE. : the full Schur form T is required;
035: *          = .FALSE.: only eigenvalues are required.
036: *
037: *     WANTZ   (input) LOGICAL
038: *          = .TRUE. : the matrix of Schur vectors Z is required;
039: *          = .FALSE.: Schur vectors are not required.
040: *
041: *     N     (input) INTEGER
042: *           The order of the matrix H.  N .GE. 0.
043: *
044: *     ILO   (input) INTEGER
045: *     IHI   (input) INTEGER
046: *           It is assumed that H is already upper triangular in rows
047: *           and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
048: *           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
049: *           previous call to DGEBAL, and then passed to DGEHRD when the
050: *           matrix output by DGEBAL is reduced to Hessenberg form.
051: *           Otherwise, ILO and IHI should be set to 1 and N,
052: *           respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
053: *           If N = 0, then ILO = 1 and IHI = 0.
054: *
055: *     H     (input/output) DOUBLE PRECISION array, dimension (LDH,N)
056: *           On entry, the upper Hessenberg matrix H.
057: *           On exit, if INFO = 0 and WANTT is .TRUE., then H contains
058: *           the upper quasi-triangular matrix T from the Schur
059: *           decomposition (the Schur form); 2-by-2 diagonal blocks
060: *           (corresponding to complex conjugate pairs of eigenvalues)
061: *           are returned in standard form, with H(i,i) = H(i+1,i+1)
062: *           and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
063: *           .FALSE., then the contents of H are unspecified on exit.
064: *           (The output value of H when INFO.GT.0 is given under the
065: *           description of INFO below.)
066: *
067: *           This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
068: *           j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
069: *
070: *     LDH   (input) INTEGER
071: *           The leading dimension of the array H. LDH .GE. max(1,N).
072: *
073: *     WR    (output) DOUBLE PRECISION array, dimension (IHI)
074: *     WI    (output) DOUBLE PRECISION array, dimension (IHI)
075: *           The real and imaginary parts, respectively, of the computed
076: *           eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
077: *           and WI(ILO:IHI). If two eigenvalues are computed as a
078: *           complex conjugate pair, they are stored in consecutive
079: *           elements of WR and WI, say the i-th and (i+1)th, with
080: *           WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
081: *           the eigenvalues are stored in the same order as on the
082: *           diagonal of the Schur form returned in H, with
083: *           WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
084: *           block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
085: *           WI(i+1) = -WI(i).
086: *
087: *     ILOZ     (input) INTEGER
088: *     IHIZ     (input) INTEGER
089: *           Specify the rows of Z to which transformations must be
090: *           applied if WANTZ is .TRUE..
091: *           1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
092: *
093: *     Z     (input/output) DOUBLE PRECISION array, dimension (LDZ,IHI)
094: *           If WANTZ is .FALSE., then Z is not referenced.
095: *           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
096: *           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
097: *           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
098: *           (The output value of Z when INFO.GT.0 is given under
099: *           the description of INFO below.)
100: *
101: *     LDZ   (input) INTEGER
102: *           The leading dimension of the array Z.  if WANTZ is .TRUE.
103: *           then LDZ.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.
104: *
105: *     WORK  (workspace/output) DOUBLE PRECISION array, dimension LWORK
106: *           On exit, if LWORK = -1, WORK(1) returns an estimate of
107: *           the optimal value for LWORK.
108: *
109: *     LWORK (input) INTEGER
110: *           The dimension of the array WORK.  LWORK .GE. max(1,N)
111: *           is sufficient, but LWORK typically as large as 6*N may
112: *           be required for optimal performance.  A workspace query
113: *           to determine the optimal workspace size is recommended.
114: *
115: *           If LWORK = -1, then DLAQR0 does a workspace query.
116: *           In this case, DLAQR0 checks the input parameters and
117: *           estimates the optimal workspace size for the given
118: *           values of N, ILO and IHI.  The estimate is returned
119: *           in WORK(1).  No error message related to LWORK is
120: *           issued by XERBLA.  Neither H nor Z are accessed.
121: *
122: *
123: *     INFO  (output) INTEGER
124: *             =  0:  successful exit
125: *           .GT. 0:  if INFO = i, DLAQR0 failed to compute all of
126: *                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
127: *                and WI contain those eigenvalues which have been
128: *                successfully computed.  (Failures are rare.)
129: *
130: *                If INFO .GT. 0 and WANT is .FALSE., then on exit,
131: *                the remaining unconverged eigenvalues are the eigen-
132: *                values of the upper Hessenberg matrix rows and
133: *                columns ILO through INFO of the final, output
134: *                value of H.
135: *
136: *                If INFO .GT. 0 and WANTT is .TRUE., then on exit
137: *
138: *           (*)  (initial value of H)*U  = U*(final value of H)
139: *
140: *                where U is an orthogonal matrix.  The final
141: *                value of H is upper Hessenberg and quasi-triangular
142: *                in rows and columns INFO+1 through IHI.
143: *
144: *                If INFO .GT. 0 and WANTZ is .TRUE., then on exit
145: *
146: *                  (final value of Z(ILO:IHI,ILOZ:IHIZ)
147: *                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
148: *
149: *                where U is the orthogonal matrix in (*) (regard-
150: *                less of the value of WANTT.)
151: *
152: *                If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
153: *                accessed.
154: *
155: *     ================================================================
156: *     Based on contributions by
157: *        Karen Braman and Ralph Byers, Department of Mathematics,
158: *        University of Kansas, USA
159: *
160: *     ================================================================
161: *     References:
162: *       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
163: *       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
164: *       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
165: *       929--947, 2002.
166: *
167: *       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
168: *       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
169: *       of Matrix Analysis, volume 23, pages 948--973, 2002.
170: *
171: *     ================================================================
172: *     .. Parameters ..
173: *
174: *     ==== Matrices of order NTINY or smaller must be processed by
175: *     .    DLAHQR because of insufficient subdiagonal scratch space.
176: *     .    (This is a hard limit.) ====
177:       INTEGER            NTINY
178:       PARAMETER          ( NTINY = 11 )
179: *
180: *     ==== Exceptional deflation windows:  try to cure rare
181: *     .    slow convergence by varying the size of the
182: *     .    deflation window after KEXNW iterations. ====
183:       INTEGER            KEXNW
184:       PARAMETER          ( KEXNW = 5 )
185: *
186: *     ==== Exceptional shifts: try to cure rare slow convergence
187: *     .    with ad-hoc exceptional shifts every KEXSH iterations.
188: *     .    ====
189:       INTEGER            KEXSH
190:       PARAMETER          ( KEXSH = 6 )
191: *
192: *     ==== The constants WILK1 and WILK2 are used to form the
193: *     .    exceptional shifts. ====
194:       DOUBLE PRECISION   WILK1, WILK2
195:       PARAMETER          ( WILK1 = 0.75d0, WILK2 = -0.4375d0 )
196:       DOUBLE PRECISION   ZERO, ONE
197:       PARAMETER          ( ZERO = 0.0d0, ONE = 1.0d0 )
198: *     ..
199: *     .. Local Scalars ..
200:       DOUBLE PRECISION   AA, BB, CC, CS, DD, SN, SS, SWAP
201:       INTEGER            I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
202:      $                   KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
203:      $                   LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
204:      $                   NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
205:       LOGICAL            SORTED
206:       CHARACTER          JBCMPZ*2
207: *     ..
208: *     .. External Functions ..
209:       INTEGER            ILAENV
210:       EXTERNAL           ILAENV
211: *     ..
212: *     .. Local Arrays ..
213:       DOUBLE PRECISION   ZDUM( 1, 1 )
214: *     ..
215: *     .. External Subroutines ..
216:       EXTERNAL           DLACPY, DLAHQR, DLANV2, DLAQR3, DLAQR4, DLAQR5
217: *     ..
218: *     .. Intrinsic Functions ..
219:       INTRINSIC          ABS, DBLE, INT, MAX, MIN, MOD
220: *     ..
221: *     .. Executable Statements ..
222:       INFO = 0
223: *
224: *     ==== Quick return for N = 0: nothing to do. ====
225: *
226:       IF( N.EQ.0 ) THEN
227:          WORK( 1 ) = ONE
228:          RETURN
229:       END IF
230: *
231:       IF( N.LE.NTINY ) THEN
232: *
233: *        ==== Tiny matrices must use DLAHQR. ====
234: *
235:          LWKOPT = 1
236:          IF( LWORK.NE.-1 )
237:      $      CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
238:      $                   ILOZ, IHIZ, Z, LDZ, INFO )
239:       ELSE
240: *
241: *        ==== Use small bulge multi-shift QR with aggressive early
242: *        .    deflation on larger-than-tiny matrices. ====
243: *
244: *        ==== Hope for the best. ====
245: *
246:          INFO = 0
247: *
248: *        ==== Set up job flags for ILAENV. ====
249: *
250:          IF( WANTT ) THEN
251:             JBCMPZ( 1: 1 ) = 'S'
252:          ELSE
253:             JBCMPZ( 1: 1 ) = 'E'
254:          END IF
255:          IF( WANTZ ) THEN
256:             JBCMPZ( 2: 2 ) = 'V'
257:          ELSE
258:             JBCMPZ( 2: 2 ) = 'N'
259:          END IF
260: *
261: *        ==== NWR = recommended deflation window size.  At this
262: *        .    point,  N .GT. NTINY = 11, so there is enough
263: *        .    subdiagonal workspace for NWR.GE.2 as required.
264: *        .    (In fact, there is enough subdiagonal space for
265: *        .    NWR.GE.3.) ====
266: *
267:          NWR = ILAENV( 13, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
268:          NWR = MAX( 2, NWR )
269:          NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
270: *
271: *        ==== NSR = recommended number of simultaneous shifts.
272: *        .    At this point N .GT. NTINY = 11, so there is at
273: *        .    enough subdiagonal workspace for NSR to be even
274: *        .    and greater than or equal to two as required. ====
275: *
276:          NSR = ILAENV( 15, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
277:          NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
278:          NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
279: *
280: *        ==== Estimate optimal workspace ====
281: *
282: *        ==== Workspace query call to DLAQR3 ====
283: *
284:          CALL DLAQR3( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
285:      $                IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH,
286:      $                N, H, LDH, WORK, -1 )
287: *
288: *        ==== Optimal workspace = MAX(DLAQR5, DLAQR3) ====
289: *
290:          LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
291: *
292: *        ==== Quick return in case of workspace query. ====
293: *
294:          IF( LWORK.EQ.-1 ) THEN
295:             WORK( 1 ) = DBLE( LWKOPT )
296:             RETURN
297:          END IF
298: *
299: *        ==== DLAHQR/DLAQR0 crossover point ====
300: *
301:          NMIN = ILAENV( 12, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
302:          NMIN = MAX( NTINY, NMIN )
303: *
304: *        ==== Nibble crossover point ====
305: *
306:          NIBBLE = ILAENV( 14, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
307:          NIBBLE = MAX( 0, NIBBLE )
308: *
309: *        ==== Accumulate reflections during ttswp?  Use block
310: *        .    2-by-2 structure during matrix-matrix multiply? ====
311: *
312:          KACC22 = ILAENV( 16, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
313:          KACC22 = MAX( 0, KACC22 )
314:          KACC22 = MIN( 2, KACC22 )
315: *
316: *        ==== NWMAX = the largest possible deflation window for
317: *        .    which there is sufficient workspace. ====
318: *
319:          NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
320:          NW = NWMAX
321: *
322: *        ==== NSMAX = the Largest number of simultaneous shifts
323: *        .    for which there is sufficient workspace. ====
324: *
325:          NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
326:          NSMAX = NSMAX - MOD( NSMAX, 2 )
327: *
328: *        ==== NDFL: an iteration count restarted at deflation. ====
329: *
330:          NDFL = 1
331: *
332: *        ==== ITMAX = iteration limit ====
333: *
334:          ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
335: *
336: *        ==== Last row and column in the active block ====
337: *
338:          KBOT = IHI
339: *
340: *        ==== Main Loop ====
341: *
342:          DO 80 IT = 1, ITMAX
343: *
344: *           ==== Done when KBOT falls below ILO ====
345: *
346:             IF( KBOT.LT.ILO )
347:      $         GO TO 90
348: *
349: *           ==== Locate active block ====
350: *
351:             DO 10 K = KBOT, ILO + 1, -1
352:                IF( H( K, K-1 ).EQ.ZERO )
353:      $            GO TO 20
354:    10       CONTINUE
355:             K = ILO
356:    20       CONTINUE
357:             KTOP = K
358: *
359: *           ==== Select deflation window size:
360: *           .    Typical Case:
361: *           .      If possible and advisable, nibble the entire
362: *           .      active block.  If not, use size MIN(NWR,NWMAX)
363: *           .      or MIN(NWR+1,NWMAX) depending upon which has
364: *           .      the smaller corresponding subdiagonal entry
365: *           .      (a heuristic).
366: *           .
367: *           .    Exceptional Case:
368: *           .      If there have been no deflations in KEXNW or
369: *           .      more iterations, then vary the deflation window
370: *           .      size.   At first, because, larger windows are,
371: *           .      in general, more powerful than smaller ones,
372: *           .      rapidly increase the window to the maximum possible.
373: *           .      Then, gradually reduce the window size. ====
374: *
375:             NH = KBOT - KTOP + 1
376:             NWUPBD = MIN( NH, NWMAX )
377:             IF( NDFL.LT.KEXNW ) THEN
378:                NW = MIN( NWUPBD, NWR )
379:             ELSE
380:                NW = MIN( NWUPBD, 2*NW )
381:             END IF
382:             IF( NW.LT.NWMAX ) THEN
383:                IF( NW.GE.NH-1 ) THEN
384:                   NW = NH
385:                ELSE
386:                   KWTOP = KBOT - NW + 1
387:                   IF( ABS( H( KWTOP, KWTOP-1 ) ).GT.
388:      $                ABS( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
389:                END IF
390:             END IF
391:             IF( NDFL.LT.KEXNW ) THEN
392:                NDEC = -1
393:             ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
394:                NDEC = NDEC + 1
395:                IF( NW-NDEC.LT.2 )
396:      $            NDEC = 0
397:                NW = NW - NDEC
398:             END IF
399: *
400: *           ==== Aggressive early deflation:
401: *           .    split workspace under the subdiagonal into
402: *           .      - an nw-by-nw work array V in the lower
403: *           .        left-hand-corner,
404: *           .      - an NW-by-at-least-NW-but-more-is-better
405: *           .        (NW-by-NHO) horizontal work array along
406: *           .        the bottom edge,
407: *           .      - an at-least-NW-but-more-is-better (NHV-by-NW)
408: *           .        vertical work array along the left-hand-edge.
409: *           .        ====
410: *
411:             KV = N - NW + 1
412:             KT = NW + 1
413:             NHO = ( N-NW-1 ) - KT + 1
414:             KWV = NW + 2
415:             NVE = ( N-NW ) - KWV + 1
416: *
417: *           ==== Aggressive early deflation ====
418: *
419:             CALL DLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
420:      $                   IHIZ, Z, LDZ, LS, LD, WR, WI, H( KV, 1 ), LDH,
421:      $                   NHO, H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH,
422:      $                   WORK, LWORK )
423: *
424: *           ==== Adjust KBOT accounting for new deflations. ====
425: *
426:             KBOT = KBOT - LD
427: *
428: *           ==== KS points to the shifts. ====
429: *
430:             KS = KBOT - LS + 1
431: *
432: *           ==== Skip an expensive QR sweep if there is a (partly
433: *           .    heuristic) reason to expect that many eigenvalues
434: *           .    will deflate without it.  Here, the QR sweep is
435: *           .    skipped if many eigenvalues have just been deflated
436: *           .    or if the remaining active block is small.
437: *
438:             IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
439:      $          KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
440: *
441: *              ==== NS = nominal number of simultaneous shifts.
442: *              .    This may be lowered (slightly) if DLAQR3
443: *              .    did not provide that many shifts. ====
444: *
445:                NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
446:                NS = NS - MOD( NS, 2 )
447: *
448: *              ==== If there have been no deflations
449: *              .    in a multiple of KEXSH iterations,
450: *              .    then try exceptional shifts.
451: *              .    Otherwise use shifts provided by
452: *              .    DLAQR3 above or from the eigenvalues
453: *              .    of a trailing principal submatrix. ====
454: *
455:                IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
456:                   KS = KBOT - NS + 1
457:                   DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2
458:                      SS = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
459:                      AA = WILK1*SS + H( I, I )
460:                      BB = SS
461:                      CC = WILK2*SS
462:                      DD = AA
463:                      CALL DLANV2( AA, BB, CC, DD, WR( I-1 ), WI( I-1 ),
464:      $                            WR( I ), WI( I ), CS, SN )
465:    30             CONTINUE
466:                   IF( KS.EQ.KTOP ) THEN
467:                      WR( KS+1 ) = H( KS+1, KS+1 )
468:                      WI( KS+1 ) = ZERO
469:                      WR( KS ) = WR( KS+1 )
470:                      WI( KS ) = WI( KS+1 )
471:                   END IF
472:                ELSE
473: *
474: *                 ==== Got NS/2 or fewer shifts? Use DLAQR4 or
475: *                 .    DLAHQR on a trailing principal submatrix to
476: *                 .    get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
477: *                 .    there is enough space below the subdiagonal
478: *                 .    to fit an NS-by-NS scratch array.) ====
479: *
480:                   IF( KBOT-KS+1.LE.NS / 2 ) THEN
481:                      KS = KBOT - NS + 1
482:                      KT = N - NS + 1
483:                      CALL DLACPY( 'A', NS, NS, H( KS, KS ), LDH,
484:      $                            H( KT, 1 ), LDH )
485:                      IF( NS.GT.NMIN ) THEN
486:                         CALL DLAQR4( .false., .false., NS, 1, NS,
487:      $                               H( KT, 1 ), LDH, WR( KS ),
488:      $                               WI( KS ), 1, 1, ZDUM, 1, WORK,
489:      $                               LWORK, INF )
490:                      ELSE
491:                         CALL DLAHQR( .false., .false., NS, 1, NS,
492:      $                               H( KT, 1 ), LDH, WR( KS ),
493:      $                               WI( KS ), 1, 1, ZDUM, 1, INF )
494:                      END IF
495:                      KS = KS + INF
496: *
497: *                    ==== In case of a rare QR failure use
498: *                    .    eigenvalues of the trailing 2-by-2
499: *                    .    principal submatrix.  ====
500: *
501:                      IF( KS.GE.KBOT ) THEN
502:                         AA = H( KBOT-1, KBOT-1 )
503:                         CC = H( KBOT, KBOT-1 )
504:                         BB = H( KBOT-1, KBOT )
505:                         DD = H( KBOT, KBOT )
506:                         CALL DLANV2( AA, BB, CC, DD, WR( KBOT-1 ),
507:      $                               WI( KBOT-1 ), WR( KBOT ),
508:      $                               WI( KBOT ), CS, SN )
509:                         KS = KBOT - 1
510:                      END IF
511:                   END IF
512: *
513:                   IF( KBOT-KS+1.GT.NS ) THEN
514: *
515: *                    ==== Sort the shifts (Helps a little)
516: *                    .    Bubble sort keeps complex conjugate
517: *                    .    pairs together. ====
518: *
519:                      SORTED = .false.
520:                      DO 50 K = KBOT, KS + 1, -1
521:                         IF( SORTED )
522:      $                     GO TO 60
523:                         SORTED = .true.
524:                         DO 40 I = KS, K - 1
525:                            IF( ABS( WR( I ) )+ABS( WI( I ) ).LT.
526:      $                         ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN
527:                               SORTED = .false.
528: *
529:                               SWAP = WR( I )
530:                               WR( I ) = WR( I+1 )
531:                               WR( I+1 ) = SWAP
532: *
533:                               SWAP = WI( I )
534:                               WI( I ) = WI( I+1 )
535:                               WI( I+1 ) = SWAP
536:                            END IF
537:    40                   CONTINUE
538:    50                CONTINUE
539:    60                CONTINUE
540:                   END IF
541: *
542: *                 ==== Shuffle shifts into pairs of real shifts
543: *                 .    and pairs of complex conjugate shifts
544: *                 .    assuming complex conjugate shifts are
545: *                 .    already adjacent to one another. (Yes,
546: *                 .    they are.)  ====
547: *
548:                   DO 70 I = KBOT, KS + 2, -2
549:                      IF( WI( I ).NE.-WI( I-1 ) ) THEN
550: *
551:                         SWAP = WR( I )
552:                         WR( I ) = WR( I-1 )
553:                         WR( I-1 ) = WR( I-2 )
554:                         WR( I-2 ) = SWAP
555: *
556:                         SWAP = WI( I )
557:                         WI( I ) = WI( I-1 )
558:                         WI( I-1 ) = WI( I-2 )
559:                         WI( I-2 ) = SWAP
560:                      END IF
561:    70             CONTINUE
562:                END IF
563: *
564: *              ==== If there are only two shifts and both are
565: *              .    real, then use only one.  ====
566: *
567:                IF( KBOT-KS+1.EQ.2 ) THEN
568:                   IF( WI( KBOT ).EQ.ZERO ) THEN
569:                      IF( ABS( WR( KBOT )-H( KBOT, KBOT ) ).LT.
570:      $                   ABS( WR( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
571:                         WR( KBOT-1 ) = WR( KBOT )
572:                      ELSE
573:                         WR( KBOT ) = WR( KBOT-1 )
574:                      END IF
575:                   END IF
576:                END IF
577: *
578: *              ==== Use up to NS of the the smallest magnatiude
579: *              .    shifts.  If there aren't NS shifts available,
580: *              .    then use them all, possibly dropping one to
581: *              .    make the number of shifts even. ====
582: *
583:                NS = MIN( NS, KBOT-KS+1 )
584:                NS = NS - MOD( NS, 2 )
585:                KS = KBOT - NS + 1
586: *
587: *              ==== Small-bulge multi-shift QR sweep:
588: *              .    split workspace under the subdiagonal into
589: *              .    - a KDU-by-KDU work array U in the lower
590: *              .      left-hand-corner,
591: *              .    - a KDU-by-at-least-KDU-but-more-is-better
592: *              .      (KDU-by-NHo) horizontal work array WH along
593: *              .      the bottom edge,
594: *              .    - and an at-least-KDU-but-more-is-better-by-KDU
595: *              .      (NVE-by-KDU) vertical work WV arrow along
596: *              .      the left-hand-edge. ====
597: *
598:                KDU = 3*NS - 3
599:                KU = N - KDU + 1
600:                KWH = KDU + 1
601:                NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
602:                KWV = KDU + 4
603:                NVE = N - KDU - KWV + 1
604: *
605: *              ==== Small-bulge multi-shift QR sweep ====
606: *
607:                CALL DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
608:      $                      WR( KS ), WI( KS ), H, LDH, ILOZ, IHIZ, Z,
609:      $                      LDZ, WORK, 3, H( KU, 1 ), LDH, NVE,
610:      $                      H( KWV, 1 ), LDH, NHO, H( KU, KWH ), LDH )
611:             END IF
612: *
613: *           ==== Note progress (or the lack of it). ====
614: *
615:             IF( LD.GT.0 ) THEN
616:                NDFL = 1
617:             ELSE
618:                NDFL = NDFL + 1
619:             END IF
620: *
621: *           ==== End of main loop ====
622:    80    CONTINUE
623: *
624: *        ==== Iteration limit exceeded.  Set INFO to show where
625: *        .    the problem occurred and exit. ====
626: *
627:          INFO = KBOT
628:    90    CONTINUE
629:       END IF
630: *
631: *     ==== Return the optimal value of LWORK. ====
632: *
633:       WORK( 1 ) = DBLE( LWKOPT )
634: *
635: *     ==== End of DLAQR0 ====
636: *
637:       END
638: