Annotation of rpl/lapack/lapack/zlaqr0.f, revision 1.2
1.1 bertrand 1: SUBROUTINE ZLAQR0( 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: * Purpose
17: * =======
18: *
19: * ZLAQR0 computes the eigenvalues of a Hessenberg matrix H
20: * and, optionally, the matrices T and Z from the Schur decomposition
21: * H = Z T Z**H, where T is an upper triangular matrix (the
22: * Schur form), and Z is the unitary matrix of Schur vectors.
23: *
24: * Optionally Z may be postmultiplied into an input unitary
25: * matrix Q so that this routine can give the Schur factorization
26: * of a matrix A which has been reduced to the Hessenberg form H
27: * by the unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H.
28: *
29: * Arguments
30: * =========
31: *
32: * WANTT (input) LOGICAL
33: * = .TRUE. : the full Schur form T is required;
34: * = .FALSE.: only eigenvalues are required.
35: *
36: * WANTZ (input) LOGICAL
37: * = .TRUE. : the matrix of Schur vectors Z is required;
38: * = .FALSE.: Schur vectors are not required.
39: *
40: * N (input) INTEGER
41: * The order of the matrix H. N .GE. 0.
42: *
43: * ILO (input) INTEGER
44: * IHI (input) INTEGER
45: * It is assumed that H is already upper triangular in rows
46: * and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
47: * H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
48: * previous call to ZGEBAL, and then passed to ZGEHRD when the
49: * matrix output by ZGEBAL is reduced to Hessenberg form.
50: * Otherwise, ILO and IHI should be set to 1 and N,
51: * respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
52: * If N = 0, then ILO = 1 and IHI = 0.
53: *
54: * H (input/output) COMPLEX*16 array, dimension (LDH,N)
55: * On entry, the upper Hessenberg matrix H.
56: * On exit, if INFO = 0 and WANTT is .TRUE., then H
57: * contains the upper triangular matrix T from the Schur
58: * decomposition (the Schur form). If INFO = 0 and WANT is
59: * .FALSE., then the contents of H are unspecified on exit.
60: * (The output value of H when INFO.GT.0 is given under the
61: * description of INFO below.)
62: *
63: * This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
64: * j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
65: *
66: * LDH (input) INTEGER
67: * The leading dimension of the array H. LDH .GE. max(1,N).
68: *
69: * W (output) COMPLEX*16 array, dimension (N)
70: * The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored
71: * in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are
72: * stored in the same order as on the diagonal of the Schur
73: * form returned in H, with W(i) = H(i,i).
74: *
75: * Z (input/output) COMPLEX*16 array, dimension (LDZ,IHI)
76: * If WANTZ is .FALSE., then Z is not referenced.
77: * If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
78: * replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
79: * orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
80: * (The output value of Z when INFO.GT.0 is given under
81: * the description of INFO below.)
82: *
83: * LDZ (input) INTEGER
84: * The leading dimension of the array Z. if WANTZ is .TRUE.
85: * then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.
86: *
87: * WORK (workspace/output) COMPLEX*16 array, dimension LWORK
88: * On exit, if LWORK = -1, WORK(1) returns an estimate of
89: * the optimal value for LWORK.
90: *
91: * LWORK (input) INTEGER
92: * The dimension of the array WORK. LWORK .GE. max(1,N)
93: * is sufficient, but LWORK typically as large as 6*N may
94: * be required for optimal performance. A workspace query
95: * to determine the optimal workspace size is recommended.
96: *
97: * If LWORK = -1, then ZLAQR0 does a workspace query.
98: * In this case, ZLAQR0 checks the input parameters and
99: * estimates the optimal workspace size for the given
100: * values of N, ILO and IHI. The estimate is returned
101: * in WORK(1). No error message related to LWORK is
102: * issued by XERBLA. Neither H nor Z are accessed.
103: *
104: *
105: * INFO (output) INTEGER
106: * = 0: successful exit
107: * .GT. 0: if INFO = i, ZLAQR0 failed to compute all of
108: * the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
109: * and WI contain those eigenvalues which have been
110: * successfully computed. (Failures are rare.)
111: *
112: * If INFO .GT. 0 and WANT is .FALSE., then on exit,
113: * the remaining unconverged eigenvalues are the eigen-
114: * values of the upper Hessenberg matrix rows and
115: * columns ILO through INFO of the final, output
116: * value of H.
117: *
118: * If INFO .GT. 0 and WANTT is .TRUE., then on exit
119: *
120: * (*) (initial value of H)*U = U*(final value of H)
121: *
122: * where U is a unitary matrix. The final
123: * value of H is upper Hessenberg and triangular in
124: * rows and columns INFO+1 through IHI.
125: *
126: * If INFO .GT. 0 and WANTZ is .TRUE., then on exit
127: *
128: * (final value of Z(ILO:IHI,ILOZ:IHIZ)
129: * = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
130: *
131: * where U is the unitary matrix in (*) (regard-
132: * less of the value of WANTT.)
133: *
134: * If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
135: * accessed.
136: *
137: * ================================================================
138: * Based on contributions by
139: * Karen Braman and Ralph Byers, Department of Mathematics,
140: * University of Kansas, USA
141: *
142: * ================================================================
143: * References:
144: * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
145: * Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
146: * Performance, SIAM Journal of Matrix Analysis, volume 23, pages
147: * 929--947, 2002.
148: *
149: * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
150: * Algorithm Part II: Aggressive Early Deflation, SIAM Journal
151: * of Matrix Analysis, volume 23, pages 948--973, 2002.
152: *
153: * ================================================================
154: * .. Parameters ..
155: *
156: * ==== Matrices of order NTINY or smaller must be processed by
157: * . ZLAHQR because of insufficient subdiagonal scratch space.
158: * . (This is a hard limit.) ====
159: INTEGER NTINY
160: PARAMETER ( NTINY = 11 )
161: *
162: * ==== Exceptional deflation windows: try to cure rare
163: * . slow convergence by varying the size of the
164: * . deflation window after KEXNW iterations. ====
165: INTEGER KEXNW
166: PARAMETER ( KEXNW = 5 )
167: *
168: * ==== Exceptional shifts: try to cure rare slow convergence
169: * . with ad-hoc exceptional shifts every KEXSH iterations.
170: * . ====
171: INTEGER KEXSH
172: PARAMETER ( KEXSH = 6 )
173: *
174: * ==== The constant WILK1 is used to form the exceptional
175: * . shifts. ====
176: DOUBLE PRECISION WILK1
177: PARAMETER ( WILK1 = 0.75d0 )
178: COMPLEX*16 ZERO, ONE
179: PARAMETER ( ZERO = ( 0.0d0, 0.0d0 ),
180: $ ONE = ( 1.0d0, 0.0d0 ) )
181: DOUBLE PRECISION TWO
182: PARAMETER ( TWO = 2.0d0 )
183: * ..
184: * .. Local Scalars ..
185: COMPLEX*16 AA, BB, CC, CDUM, DD, DET, RTDISC, SWAP, TR2
186: DOUBLE PRECISION S
187: INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
188: $ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
189: $ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
190: $ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
191: LOGICAL SORTED
192: CHARACTER JBCMPZ*2
193: * ..
194: * .. External Functions ..
195: INTEGER ILAENV
196: EXTERNAL ILAENV
197: * ..
198: * .. Local Arrays ..
199: COMPLEX*16 ZDUM( 1, 1 )
200: * ..
201: * .. External Subroutines ..
202: EXTERNAL ZLACPY, ZLAHQR, ZLAQR3, ZLAQR4, ZLAQR5
203: * ..
204: * .. Intrinsic Functions ..
205: INTRINSIC ABS, DBLE, DCMPLX, DIMAG, INT, MAX, MIN, MOD,
206: $ SQRT
207: * ..
208: * .. Statement Functions ..
209: DOUBLE PRECISION CABS1
210: * ..
211: * .. Statement Function definitions ..
212: CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
213: * ..
214: * .. Executable Statements ..
215: INFO = 0
216: *
217: * ==== Quick return for N = 0: nothing to do. ====
218: *
219: IF( N.EQ.0 ) THEN
220: WORK( 1 ) = ONE
221: RETURN
222: END IF
223: *
224: IF( N.LE.NTINY ) THEN
225: *
226: * ==== Tiny matrices must use ZLAHQR. ====
227: *
228: LWKOPT = 1
229: IF( LWORK.NE.-1 )
230: $ CALL ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
231: $ IHIZ, Z, LDZ, INFO )
232: ELSE
233: *
234: * ==== Use small bulge multi-shift QR with aggressive early
235: * . deflation on larger-than-tiny matrices. ====
236: *
237: * ==== Hope for the best. ====
238: *
239: INFO = 0
240: *
241: * ==== Set up job flags for ILAENV. ====
242: *
243: IF( WANTT ) THEN
244: JBCMPZ( 1: 1 ) = 'S'
245: ELSE
246: JBCMPZ( 1: 1 ) = 'E'
247: END IF
248: IF( WANTZ ) THEN
249: JBCMPZ( 2: 2 ) = 'V'
250: ELSE
251: JBCMPZ( 2: 2 ) = 'N'
252: END IF
253: *
254: * ==== NWR = recommended deflation window size. At this
255: * . point, N .GT. NTINY = 11, so there is enough
256: * . subdiagonal workspace for NWR.GE.2 as required.
257: * . (In fact, there is enough subdiagonal space for
258: * . NWR.GE.3.) ====
259: *
260: NWR = ILAENV( 13, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
261: NWR = MAX( 2, NWR )
262: NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
263: *
264: * ==== NSR = recommended number of simultaneous shifts.
265: * . At this point N .GT. NTINY = 11, so there is at
266: * . enough subdiagonal workspace for NSR to be even
267: * . and greater than or equal to two as required. ====
268: *
269: NSR = ILAENV( 15, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
270: NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
271: NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
272: *
273: * ==== Estimate optimal workspace ====
274: *
275: * ==== Workspace query call to ZLAQR3 ====
276: *
277: CALL ZLAQR3( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
278: $ IHIZ, Z, LDZ, LS, LD, W, H, LDH, N, H, LDH, N, H,
279: $ LDH, WORK, -1 )
280: *
281: * ==== Optimal workspace = MAX(ZLAQR5, ZLAQR3) ====
282: *
283: LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
284: *
285: * ==== Quick return in case of workspace query. ====
286: *
287: IF( LWORK.EQ.-1 ) THEN
288: WORK( 1 ) = DCMPLX( LWKOPT, 0 )
289: RETURN
290: END IF
291: *
292: * ==== ZLAHQR/ZLAQR0 crossover point ====
293: *
294: NMIN = ILAENV( 12, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
295: NMIN = MAX( NTINY, NMIN )
296: *
297: * ==== Nibble crossover point ====
298: *
299: NIBBLE = ILAENV( 14, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
300: NIBBLE = MAX( 0, NIBBLE )
301: *
302: * ==== Accumulate reflections during ttswp? Use block
303: * . 2-by-2 structure during matrix-matrix multiply? ====
304: *
305: KACC22 = ILAENV( 16, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
306: KACC22 = MAX( 0, KACC22 )
307: KACC22 = MIN( 2, KACC22 )
308: *
309: * ==== NWMAX = the largest possible deflation window for
310: * . which there is sufficient workspace. ====
311: *
312: NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
313: NW = NWMAX
314: *
315: * ==== NSMAX = the Largest number of simultaneous shifts
316: * . for which there is sufficient workspace. ====
317: *
318: NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
319: NSMAX = NSMAX - MOD( NSMAX, 2 )
320: *
321: * ==== NDFL: an iteration count restarted at deflation. ====
322: *
323: NDFL = 1
324: *
325: * ==== ITMAX = iteration limit ====
326: *
327: ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
328: *
329: * ==== Last row and column in the active block ====
330: *
331: KBOT = IHI
332: *
333: * ==== Main Loop ====
334: *
335: DO 70 IT = 1, ITMAX
336: *
337: * ==== Done when KBOT falls below ILO ====
338: *
339: IF( KBOT.LT.ILO )
340: $ GO TO 80
341: *
342: * ==== Locate active block ====
343: *
344: DO 10 K = KBOT, ILO + 1, -1
345: IF( H( K, K-1 ).EQ.ZERO )
346: $ GO TO 20
347: 10 CONTINUE
348: K = ILO
349: 20 CONTINUE
350: KTOP = K
351: *
352: * ==== Select deflation window size:
353: * . Typical Case:
354: * . If possible and advisable, nibble the entire
355: * . active block. If not, use size MIN(NWR,NWMAX)
356: * . or MIN(NWR+1,NWMAX) depending upon which has
357: * . the smaller corresponding subdiagonal entry
358: * . (a heuristic).
359: * .
360: * . Exceptional Case:
361: * . If there have been no deflations in KEXNW or
362: * . more iterations, then vary the deflation window
363: * . size. At first, because, larger windows are,
364: * . in general, more powerful than smaller ones,
365: * . rapidly increase the window to the maximum possible.
366: * . Then, gradually reduce the window size. ====
367: *
368: NH = KBOT - KTOP + 1
369: NWUPBD = MIN( NH, NWMAX )
370: IF( NDFL.LT.KEXNW ) THEN
371: NW = MIN( NWUPBD, NWR )
372: ELSE
373: NW = MIN( NWUPBD, 2*NW )
374: END IF
375: IF( NW.LT.NWMAX ) THEN
376: IF( NW.GE.NH-1 ) THEN
377: NW = NH
378: ELSE
379: KWTOP = KBOT - NW + 1
380: IF( CABS1( H( KWTOP, KWTOP-1 ) ).GT.
381: $ CABS1( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
382: END IF
383: END IF
384: IF( NDFL.LT.KEXNW ) THEN
385: NDEC = -1
386: ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
387: NDEC = NDEC + 1
388: IF( NW-NDEC.LT.2 )
389: $ NDEC = 0
390: NW = NW - NDEC
391: END IF
392: *
393: * ==== Aggressive early deflation:
394: * . split workspace under the subdiagonal into
395: * . - an nw-by-nw work array V in the lower
396: * . left-hand-corner,
397: * . - an NW-by-at-least-NW-but-more-is-better
398: * . (NW-by-NHO) horizontal work array along
399: * . the bottom edge,
400: * . - an at-least-NW-but-more-is-better (NHV-by-NW)
401: * . vertical work array along the left-hand-edge.
402: * . ====
403: *
404: KV = N - NW + 1
405: KT = NW + 1
406: NHO = ( N-NW-1 ) - KT + 1
407: KWV = NW + 2
408: NVE = ( N-NW ) - KWV + 1
409: *
410: * ==== Aggressive early deflation ====
411: *
412: CALL ZLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
413: $ IHIZ, Z, LDZ, LS, LD, W, H( KV, 1 ), LDH, NHO,
414: $ H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH, WORK,
415: $ LWORK )
416: *
417: * ==== Adjust KBOT accounting for new deflations. ====
418: *
419: KBOT = KBOT - LD
420: *
421: * ==== KS points to the shifts. ====
422: *
423: KS = KBOT - LS + 1
424: *
425: * ==== Skip an expensive QR sweep if there is a (partly
426: * . heuristic) reason to expect that many eigenvalues
427: * . will deflate without it. Here, the QR sweep is
428: * . skipped if many eigenvalues have just been deflated
429: * . or if the remaining active block is small.
430: *
431: IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
432: $ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
433: *
434: * ==== NS = nominal number of simultaneous shifts.
435: * . This may be lowered (slightly) if ZLAQR3
436: * . did not provide that many shifts. ====
437: *
438: NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
439: NS = NS - MOD( NS, 2 )
440: *
441: * ==== If there have been no deflations
442: * . in a multiple of KEXSH iterations,
443: * . then try exceptional shifts.
444: * . Otherwise use shifts provided by
445: * . ZLAQR3 above or from the eigenvalues
446: * . of a trailing principal submatrix. ====
447: *
448: IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
449: KS = KBOT - NS + 1
450: DO 30 I = KBOT, KS + 1, -2
451: W( I ) = H( I, I ) + WILK1*CABS1( H( I, I-1 ) )
452: W( I-1 ) = W( I )
453: 30 CONTINUE
454: ELSE
455: *
456: * ==== Got NS/2 or fewer shifts? Use ZLAQR4 or
457: * . ZLAHQR on a trailing principal submatrix to
458: * . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
459: * . there is enough space below the subdiagonal
460: * . to fit an NS-by-NS scratch array.) ====
461: *
462: IF( KBOT-KS+1.LE.NS / 2 ) THEN
463: KS = KBOT - NS + 1
464: KT = N - NS + 1
465: CALL ZLACPY( 'A', NS, NS, H( KS, KS ), LDH,
466: $ H( KT, 1 ), LDH )
467: IF( NS.GT.NMIN ) THEN
468: CALL ZLAQR4( .false., .false., NS, 1, NS,
469: $ H( KT, 1 ), LDH, W( KS ), 1, 1,
470: $ ZDUM, 1, WORK, LWORK, INF )
471: ELSE
472: CALL ZLAHQR( .false., .false., NS, 1, NS,
473: $ H( KT, 1 ), LDH, W( KS ), 1, 1,
474: $ ZDUM, 1, INF )
475: END IF
476: KS = KS + INF
477: *
478: * ==== In case of a rare QR failure use
479: * . eigenvalues of the trailing 2-by-2
480: * . principal submatrix. Scale to avoid
481: * . overflows, underflows and subnormals.
482: * . (The scale factor S can not be zero,
483: * . because H(KBOT,KBOT-1) is nonzero.) ====
484: *
485: IF( KS.GE.KBOT ) THEN
486: S = CABS1( H( KBOT-1, KBOT-1 ) ) +
487: $ CABS1( H( KBOT, KBOT-1 ) ) +
488: $ CABS1( H( KBOT-1, KBOT ) ) +
489: $ CABS1( H( KBOT, KBOT ) )
490: AA = H( KBOT-1, KBOT-1 ) / S
491: CC = H( KBOT, KBOT-1 ) / S
492: BB = H( KBOT-1, KBOT ) / S
493: DD = H( KBOT, KBOT ) / S
494: TR2 = ( AA+DD ) / TWO
495: DET = ( AA-TR2 )*( DD-TR2 ) - BB*CC
496: RTDISC = SQRT( -DET )
497: W( KBOT-1 ) = ( TR2+RTDISC )*S
498: W( KBOT ) = ( TR2-RTDISC )*S
499: *
500: KS = KBOT - 1
501: END IF
502: END IF
503: *
504: IF( KBOT-KS+1.GT.NS ) THEN
505: *
506: * ==== Sort the shifts (Helps a little) ====
507: *
508: SORTED = .false.
509: DO 50 K = KBOT, KS + 1, -1
510: IF( SORTED )
511: $ GO TO 60
512: SORTED = .true.
513: DO 40 I = KS, K - 1
514: IF( CABS1( W( I ) ).LT.CABS1( W( I+1 ) ) )
515: $ THEN
516: SORTED = .false.
517: SWAP = W( I )
518: W( I ) = W( I+1 )
519: W( I+1 ) = SWAP
520: END IF
521: 40 CONTINUE
522: 50 CONTINUE
523: 60 CONTINUE
524: END IF
525: END IF
526: *
527: * ==== If there are only two shifts, then use
528: * . only one. ====
529: *
530: IF( KBOT-KS+1.EQ.2 ) THEN
531: IF( CABS1( W( KBOT )-H( KBOT, KBOT ) ).LT.
532: $ CABS1( W( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
533: W( KBOT-1 ) = W( KBOT )
534: ELSE
535: W( KBOT ) = W( KBOT-1 )
536: END IF
537: END IF
538: *
539: * ==== Use up to NS of the the smallest magnatiude
540: * . shifts. If there aren't NS shifts available,
541: * . then use them all, possibly dropping one to
542: * . make the number of shifts even. ====
543: *
544: NS = MIN( NS, KBOT-KS+1 )
545: NS = NS - MOD( NS, 2 )
546: KS = KBOT - NS + 1
547: *
548: * ==== Small-bulge multi-shift QR sweep:
549: * . split workspace under the subdiagonal into
550: * . - a KDU-by-KDU work array U in the lower
551: * . left-hand-corner,
552: * . - a KDU-by-at-least-KDU-but-more-is-better
553: * . (KDU-by-NHo) horizontal work array WH along
554: * . the bottom edge,
555: * . - and an at-least-KDU-but-more-is-better-by-KDU
556: * . (NVE-by-KDU) vertical work WV arrow along
557: * . the left-hand-edge. ====
558: *
559: KDU = 3*NS - 3
560: KU = N - KDU + 1
561: KWH = KDU + 1
562: NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
563: KWV = KDU + 4
564: NVE = N - KDU - KWV + 1
565: *
566: * ==== Small-bulge multi-shift QR sweep ====
567: *
568: CALL ZLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
569: $ W( KS ), H, LDH, ILOZ, IHIZ, Z, LDZ, WORK,
570: $ 3, H( KU, 1 ), LDH, NVE, H( KWV, 1 ), LDH,
571: $ NHO, H( KU, KWH ), LDH )
572: END IF
573: *
574: * ==== Note progress (or the lack of it). ====
575: *
576: IF( LD.GT.0 ) THEN
577: NDFL = 1
578: ELSE
579: NDFL = NDFL + 1
580: END IF
581: *
582: * ==== End of main loop ====
583: 70 CONTINUE
584: *
585: * ==== Iteration limit exceeded. Set INFO to show where
586: * . the problem occurred and exit. ====
587: *
588: INFO = KBOT
589: 80 CONTINUE
590: END IF
591: *
592: * ==== Return the optimal value of LWORK. ====
593: *
594: WORK( 1 ) = DCMPLX( LWKOPT, 0 )
595: *
596: * ==== End of ZLAQR0 ====
597: *
598: END
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