1: SUBROUTINE ZLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
2: $ IHIZ, Z, LDZ, WORK, LWORK, INFO )
3: *
4: * -- LAPACK auxiliary routine (version 3.2) --
5: * Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..
6: * November 2006
7: *
8: * .. Scalar Arguments ..
9: INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
10: LOGICAL WANTT, WANTZ
11: * ..
12: * .. Array Arguments ..
13: COMPLEX*16 H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
14: * ..
15: *
16: * This subroutine implements one level of recursion for ZLAQR0.
17: * It is a complete implementation of the small bulge multi-shift
18: * QR algorithm. It may be called by ZLAQR0 and, for large enough
19: * deflation window size, it may be called by ZLAQR3. This
20: * subroutine is identical to ZLAQR0 except that it calls ZLAQR2
21: * instead of ZLAQR3.
22: *
23: * Purpose
24: * =======
25: *
26: * ZLAQR4 computes the eigenvalues of a Hessenberg matrix H
27: * and, optionally, the matrices T and Z from the Schur decomposition
28: * H = Z T Z**H, where T is an upper triangular matrix (the
29: * Schur form), and Z is the unitary matrix of Schur vectors.
30: *
31: * Optionally Z may be postmultiplied into an input unitary
32: * matrix Q so that this routine can give the Schur factorization
33: * of a matrix A which has been reduced to the Hessenberg form H
34: * by the unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H.
35: *
36: * Arguments
37: * =========
38: *
39: * WANTT (input) LOGICAL
40: * = .TRUE. : the full Schur form T is required;
41: * = .FALSE.: only eigenvalues are required.
42: *
43: * WANTZ (input) LOGICAL
44: * = .TRUE. : the matrix of Schur vectors Z is required;
45: * = .FALSE.: Schur vectors are not required.
46: *
47: * N (input) INTEGER
48: * The order of the matrix H. N .GE. 0.
49: *
50: * ILO (input) INTEGER
51: * IHI (input) INTEGER
52: * It is assumed that H is already upper triangular in rows
53: * and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
54: * H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
55: * previous call to ZGEBAL, and then passed to ZGEHRD when the
56: * matrix output by ZGEBAL is reduced to Hessenberg form.
57: * Otherwise, ILO and IHI should be set to 1 and N,
58: * respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
59: * If N = 0, then ILO = 1 and IHI = 0.
60: *
61: * H (input/output) COMPLEX*16 array, dimension (LDH,N)
62: * On entry, the upper Hessenberg matrix H.
63: * On exit, if INFO = 0 and WANTT is .TRUE., then H
64: * contains the upper triangular matrix T from the Schur
65: * decomposition (the Schur form). If INFO = 0 and WANT is
66: * .FALSE., then the contents of H are unspecified on exit.
67: * (The output value of H when INFO.GT.0 is given under the
68: * description of INFO below.)
69: *
70: * This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
71: * j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
72: *
73: * LDH (input) INTEGER
74: * The leading dimension of the array H. LDH .GE. max(1,N).
75: *
76: * W (output) COMPLEX*16 array, dimension (N)
77: * The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored
78: * in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are
79: * stored in the same order as on the diagonal of the Schur
80: * form returned in H, with W(i) = H(i,i).
81: *
82: * Z (input/output) COMPLEX*16 array, dimension (LDZ,IHI)
83: * If WANTZ is .FALSE., then Z is not referenced.
84: * If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
85: * replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
86: * orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
87: * (The output value of Z when INFO.GT.0 is given under
88: * the description of INFO below.)
89: *
90: * LDZ (input) INTEGER
91: * The leading dimension of the array Z. if WANTZ is .TRUE.
92: * then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.
93: *
94: * WORK (workspace/output) COMPLEX*16 array, dimension LWORK
95: * On exit, if LWORK = -1, WORK(1) returns an estimate of
96: * the optimal value for LWORK.
97: *
98: * LWORK (input) INTEGER
99: * The dimension of the array WORK. LWORK .GE. max(1,N)
100: * is sufficient, but LWORK typically as large as 6*N may
101: * be required for optimal performance. A workspace query
102: * to determine the optimal workspace size is recommended.
103: *
104: * If LWORK = -1, then ZLAQR4 does a workspace query.
105: * In this case, ZLAQR4 checks the input parameters and
106: * estimates the optimal workspace size for the given
107: * values of N, ILO and IHI. The estimate is returned
108: * in WORK(1). No error message related to LWORK is
109: * issued by XERBLA. Neither H nor Z are accessed.
110: *
111: *
112: * INFO (output) INTEGER
113: * = 0: successful exit
114: * .GT. 0: if INFO = i, ZLAQR4 failed to compute all of
115: * the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
116: * and WI contain those eigenvalues which have been
117: * successfully computed. (Failures are rare.)
118: *
119: * If INFO .GT. 0 and WANT is .FALSE., then on exit,
120: * the remaining unconverged eigenvalues are the eigen-
121: * values of the upper Hessenberg matrix rows and
122: * columns ILO through INFO of the final, output
123: * value of H.
124: *
125: * If INFO .GT. 0 and WANTT is .TRUE., then on exit
126: *
127: * (*) (initial value of H)*U = U*(final value of H)
128: *
129: * where U is a unitary matrix. The final
130: * value of H is upper Hessenberg and triangular in
131: * rows and columns INFO+1 through IHI.
132: *
133: * If INFO .GT. 0 and WANTZ is .TRUE., then on exit
134: *
135: * (final value of Z(ILO:IHI,ILOZ:IHIZ)
136: * = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
137: *
138: * where U is the unitary matrix in (*) (regard-
139: * less of the value of WANTT.)
140: *
141: * If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
142: * accessed.
143: *
144: * ================================================================
145: * Based on contributions by
146: * Karen Braman and Ralph Byers, Department of Mathematics,
147: * University of Kansas, USA
148: *
149: * ================================================================
150: * References:
151: * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
152: * Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
153: * Performance, SIAM Journal of Matrix Analysis, volume 23, pages
154: * 929--947, 2002.
155: *
156: * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
157: * Algorithm Part II: Aggressive Early Deflation, SIAM Journal
158: * of Matrix Analysis, volume 23, pages 948--973, 2002.
159: *
160: * ================================================================
161: * .. Parameters ..
162: *
163: * ==== Matrices of order NTINY or smaller must be processed by
164: * . ZLAHQR because of insufficient subdiagonal scratch space.
165: * . (This is a hard limit.) ====
166: INTEGER NTINY
167: PARAMETER ( NTINY = 11 )
168: *
169: * ==== Exceptional deflation windows: try to cure rare
170: * . slow convergence by varying the size of the
171: * . deflation window after KEXNW iterations. ====
172: INTEGER KEXNW
173: PARAMETER ( KEXNW = 5 )
174: *
175: * ==== Exceptional shifts: try to cure rare slow convergence
176: * . with ad-hoc exceptional shifts every KEXSH iterations.
177: * . ====
178: INTEGER KEXSH
179: PARAMETER ( KEXSH = 6 )
180: *
181: * ==== The constant WILK1 is used to form the exceptional
182: * . shifts. ====
183: DOUBLE PRECISION WILK1
184: PARAMETER ( WILK1 = 0.75d0 )
185: COMPLEX*16 ZERO, ONE
186: PARAMETER ( ZERO = ( 0.0d0, 0.0d0 ),
187: $ ONE = ( 1.0d0, 0.0d0 ) )
188: DOUBLE PRECISION TWO
189: PARAMETER ( TWO = 2.0d0 )
190: * ..
191: * .. Local Scalars ..
192: COMPLEX*16 AA, BB, CC, CDUM, DD, DET, RTDISC, SWAP, TR2
193: DOUBLE PRECISION S
194: INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
195: $ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
196: $ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
197: $ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
198: LOGICAL SORTED
199: CHARACTER JBCMPZ*2
200: * ..
201: * .. External Functions ..
202: INTEGER ILAENV
203: EXTERNAL ILAENV
204: * ..
205: * .. Local Arrays ..
206: COMPLEX*16 ZDUM( 1, 1 )
207: * ..
208: * .. External Subroutines ..
209: EXTERNAL ZLACPY, ZLAHQR, ZLAQR2, ZLAQR5
210: * ..
211: * .. Intrinsic Functions ..
212: INTRINSIC ABS, DBLE, DCMPLX, DIMAG, INT, MAX, MIN, MOD,
213: $ SQRT
214: * ..
215: * .. Statement Functions ..
216: DOUBLE PRECISION CABS1
217: * ..
218: * .. Statement Function definitions ..
219: CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
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 ZLAHQR. ====
234: *
235: LWKOPT = 1
236: IF( LWORK.NE.-1 )
237: $ CALL ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
238: $ 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, 'ZLAQR4', 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, 'ZLAQR4', 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 ZLAQR2 ====
283: *
284: CALL ZLAQR2( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
285: $ IHIZ, Z, LDZ, LS, LD, W, H, LDH, N, H, LDH, N, H,
286: $ LDH, WORK, -1 )
287: *
288: * ==== Optimal workspace = MAX(ZLAQR5, ZLAQR2) ====
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 ) = DCMPLX( LWKOPT, 0 )
296: RETURN
297: END IF
298: *
299: * ==== ZLAHQR/ZLAQR0 crossover point ====
300: *
301: NMIN = ILAENV( 12, 'ZLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
302: NMIN = MAX( NTINY, NMIN )
303: *
304: * ==== Nibble crossover point ====
305: *
306: NIBBLE = ILAENV( 14, 'ZLAQR4', 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, 'ZLAQR4', 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 70 IT = 1, ITMAX
343: *
344: * ==== Done when KBOT falls below ILO ====
345: *
346: IF( KBOT.LT.ILO )
347: $ GO TO 80
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( CABS1( H( KWTOP, KWTOP-1 ) ).GT.
388: $ CABS1( 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 ZLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
420: $ IHIZ, Z, LDZ, LS, LD, W, H( KV, 1 ), LDH, NHO,
421: $ H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH, WORK,
422: $ 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 ZLAQR2
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: * . ZLAQR2 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, KS + 1, -2
458: W( I ) = H( I, I ) + WILK1*CABS1( H( I, I-1 ) )
459: W( I-1 ) = W( I )
460: 30 CONTINUE
461: ELSE
462: *
463: * ==== Got NS/2 or fewer shifts? Use ZLAHQR
464: * . on a trailing principal submatrix to
465: * . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
466: * . there is enough space below the subdiagonal
467: * . to fit an NS-by-NS scratch array.) ====
468: *
469: IF( KBOT-KS+1.LE.NS / 2 ) THEN
470: KS = KBOT - NS + 1
471: KT = N - NS + 1
472: CALL ZLACPY( 'A', NS, NS, H( KS, KS ), LDH,
473: $ H( KT, 1 ), LDH )
474: CALL ZLAHQR( .false., .false., NS, 1, NS,
475: $ H( KT, 1 ), LDH, W( KS ), 1, 1, ZDUM,
476: $ 1, INF )
477: KS = KS + INF
478: *
479: * ==== In case of a rare QR failure use
480: * . eigenvalues of the trailing 2-by-2
481: * . principal submatrix. Scale to avoid
482: * . overflows, underflows and subnormals.
483: * . (The scale factor S can not be zero,
484: * . because H(KBOT,KBOT-1) is nonzero.) ====
485: *
486: IF( KS.GE.KBOT ) THEN
487: S = CABS1( H( KBOT-1, KBOT-1 ) ) +
488: $ CABS1( H( KBOT, KBOT-1 ) ) +
489: $ CABS1( H( KBOT-1, KBOT ) ) +
490: $ CABS1( H( KBOT, KBOT ) )
491: AA = H( KBOT-1, KBOT-1 ) / S
492: CC = H( KBOT, KBOT-1 ) / S
493: BB = H( KBOT-1, KBOT ) / S
494: DD = H( KBOT, KBOT ) / S
495: TR2 = ( AA+DD ) / TWO
496: DET = ( AA-TR2 )*( DD-TR2 ) - BB*CC
497: RTDISC = SQRT( -DET )
498: W( KBOT-1 ) = ( TR2+RTDISC )*S
499: W( KBOT ) = ( TR2-RTDISC )*S
500: *
501: KS = KBOT - 1
502: END IF
503: END IF
504: *
505: IF( KBOT-KS+1.GT.NS ) THEN
506: *
507: * ==== Sort the shifts (Helps a little) ====
508: *
509: SORTED = .false.
510: DO 50 K = KBOT, KS + 1, -1
511: IF( SORTED )
512: $ GO TO 60
513: SORTED = .true.
514: DO 40 I = KS, K - 1
515: IF( CABS1( W( I ) ).LT.CABS1( W( I+1 ) ) )
516: $ THEN
517: SORTED = .false.
518: SWAP = W( I )
519: W( I ) = W( I+1 )
520: W( I+1 ) = SWAP
521: END IF
522: 40 CONTINUE
523: 50 CONTINUE
524: 60 CONTINUE
525: END IF
526: END IF
527: *
528: * ==== If there are only two shifts, then use
529: * . only one. ====
530: *
531: IF( KBOT-KS+1.EQ.2 ) THEN
532: IF( CABS1( W( KBOT )-H( KBOT, KBOT ) ).LT.
533: $ CABS1( W( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
534: W( KBOT-1 ) = W( KBOT )
535: ELSE
536: W( KBOT ) = W( KBOT-1 )
537: END IF
538: END IF
539: *
540: * ==== Use up to NS of the the smallest magnatiude
541: * . shifts. If there aren't NS shifts available,
542: * . then use them all, possibly dropping one to
543: * . make the number of shifts even. ====
544: *
545: NS = MIN( NS, KBOT-KS+1 )
546: NS = NS - MOD( NS, 2 )
547: KS = KBOT - NS + 1
548: *
549: * ==== Small-bulge multi-shift QR sweep:
550: * . split workspace under the subdiagonal into
551: * . - a KDU-by-KDU work array U in the lower
552: * . left-hand-corner,
553: * . - a KDU-by-at-least-KDU-but-more-is-better
554: * . (KDU-by-NHo) horizontal work array WH along
555: * . the bottom edge,
556: * . - and an at-least-KDU-but-more-is-better-by-KDU
557: * . (NVE-by-KDU) vertical work WV arrow along
558: * . the left-hand-edge. ====
559: *
560: KDU = 3*NS - 3
561: KU = N - KDU + 1
562: KWH = KDU + 1
563: NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
564: KWV = KDU + 4
565: NVE = N - KDU - KWV + 1
566: *
567: * ==== Small-bulge multi-shift QR sweep ====
568: *
569: CALL ZLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
570: $ W( KS ), H, LDH, ILOZ, IHIZ, Z, LDZ, WORK,
571: $ 3, H( KU, 1 ), LDH, NVE, H( KWV, 1 ), LDH,
572: $ NHO, H( KU, KWH ), LDH )
573: END IF
574: *
575: * ==== Note progress (or the lack of it). ====
576: *
577: IF( LD.GT.0 ) THEN
578: NDFL = 1
579: ELSE
580: NDFL = NDFL + 1
581: END IF
582: *
583: * ==== End of main loop ====
584: 70 CONTINUE
585: *
586: * ==== Iteration limit exceeded. Set INFO to show where
587: * . the problem occurred and exit. ====
588: *
589: INFO = KBOT
590: 80 CONTINUE
591: END IF
592: *
593: * ==== Return the optimal value of LWORK. ====
594: *
595: WORK( 1 ) = DCMPLX( LWKOPT, 0 )
596: *
597: * ==== End of ZLAQR4 ====
598: *
599: END
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