1: *> \brief \b ZLAQR4 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 ZLAQR4 + dependencies
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11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlaqr4.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlaqr4.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
22: * 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: * COMPLEX*16 H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> ZLAQR4 implements one level of recursion for ZLAQR0.
39: *> It is a complete implementation of the small bulge multi-shift
40: *> QR algorithm. It may be called by ZLAQR0 and, for large enough
41: *> deflation window size, it may be called by ZLAQR3. This
42: *> subroutine is identical to ZLAQR0 except that it calls ZLAQR2
43: *> instead of ZLAQR3.
44: *>
45: *> ZLAQR4 computes the eigenvalues of a Hessenberg matrix H
46: *> and, optionally, the matrices T and Z from the Schur decomposition
47: *> H = Z T Z**H, where T is an upper triangular matrix (the
48: *> Schur form), and Z is the unitary matrix of Schur vectors.
49: *>
50: *> Optionally Z may be postmultiplied into an input unitary
51: *> matrix Q so that this routine can give the Schur factorization
52: *> of a matrix A which has been reduced to the Hessenberg form H
53: *> by the unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H.
54: *> \endverbatim
55: *
56: * Arguments:
57: * ==========
58: *
59: *> \param[in] WANTT
60: *> \verbatim
61: *> WANTT is LOGICAL
62: *> = .TRUE. : the full Schur form T is required;
63: *> = .FALSE.: only eigenvalues are required.
64: *> \endverbatim
65: *>
66: *> \param[in] WANTZ
67: *> \verbatim
68: *> WANTZ is LOGICAL
69: *> = .TRUE. : the matrix of Schur vectors Z is required;
70: *> = .FALSE.: Schur vectors are not required.
71: *> \endverbatim
72: *>
73: *> \param[in] N
74: *> \verbatim
75: *> N is INTEGER
76: *> The order of the matrix H. N >= 0.
77: *> \endverbatim
78: *>
79: *> \param[in] ILO
80: *> \verbatim
81: *> ILO is INTEGER
82: *> \endverbatim
83: *>
84: *> \param[in] IHI
85: *> \verbatim
86: *> IHI is INTEGER
87: *> It is assumed that H is already upper triangular in rows
88: *> and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,
89: *> H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
90: *> previous call to ZGEBAL, and then passed to ZGEHRD when the
91: *> matrix output by ZGEBAL is reduced to Hessenberg form.
92: *> Otherwise, ILO and IHI should be set to 1 and N,
93: *> respectively. If N > 0, then 1 <= ILO <= IHI <= N.
94: *> If N = 0, then ILO = 1 and IHI = 0.
95: *> \endverbatim
96: *>
97: *> \param[in,out] H
98: *> \verbatim
99: *> H is COMPLEX*16 array, dimension (LDH,N)
100: *> On entry, the upper Hessenberg matrix H.
101: *> On exit, if INFO = 0 and WANTT is .TRUE., then H
102: *> contains the upper triangular matrix T from the Schur
103: *> decomposition (the Schur form). If INFO = 0 and WANT is
104: *> .FALSE., then the contents of H are unspecified on exit.
105: *> (The output value of H when INFO > 0 is given under the
106: *> description of INFO below.)
107: *>
108: *> This subroutine may explicitly set H(i,j) = 0 for i > j and
109: *> j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
110: *> \endverbatim
111: *>
112: *> \param[in] LDH
113: *> \verbatim
114: *> LDH is INTEGER
115: *> The leading dimension of the array H. LDH >= max(1,N).
116: *> \endverbatim
117: *>
118: *> \param[out] W
119: *> \verbatim
120: *> W is COMPLEX*16 array, dimension (N)
121: *> The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored
122: *> in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are
123: *> stored in the same order as on the diagonal of the Schur
124: *> form returned in H, with W(i) = H(i,i).
125: *> \endverbatim
126: *>
127: *> \param[in] ILOZ
128: *> \verbatim
129: *> ILOZ is INTEGER
130: *> \endverbatim
131: *>
132: *> \param[in] IHIZ
133: *> \verbatim
134: *> IHIZ is INTEGER
135: *> Specify the rows of Z to which transformations must be
136: *> applied if WANTZ is .TRUE..
137: *> 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
138: *> \endverbatim
139: *>
140: *> \param[in,out] Z
141: *> \verbatim
142: *> Z is COMPLEX*16 array, dimension (LDZ,IHI)
143: *> If WANTZ is .FALSE., then Z is not referenced.
144: *> If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
145: *> replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
146: *> orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
147: *> (The output value of Z when INFO > 0 is given under
148: *> the description of INFO below.)
149: *> \endverbatim
150: *>
151: *> \param[in] LDZ
152: *> \verbatim
153: *> LDZ is INTEGER
154: *> The leading dimension of the array Z. if WANTZ is .TRUE.
155: *> then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1.
156: *> \endverbatim
157: *>
158: *> \param[out] WORK
159: *> \verbatim
160: *> WORK is COMPLEX*16 array, dimension LWORK
161: *> On exit, if LWORK = -1, WORK(1) returns an estimate of
162: *> the optimal value for LWORK.
163: *> \endverbatim
164: *>
165: *> \param[in] LWORK
166: *> \verbatim
167: *> LWORK is INTEGER
168: *> The dimension of the array WORK. LWORK >= max(1,N)
169: *> is sufficient, but LWORK typically as large as 6*N may
170: *> be required for optimal performance. A workspace query
171: *> to determine the optimal workspace size is recommended.
172: *>
173: *> If LWORK = -1, then ZLAQR4 does a workspace query.
174: *> In this case, ZLAQR4 checks the input parameters and
175: *> estimates the optimal workspace size for the given
176: *> values of N, ILO and IHI. The estimate is returned
177: *> in WORK(1). No error message related to LWORK is
178: *> issued by XERBLA. Neither H nor Z are accessed.
179: *> \endverbatim
180: *>
181: *> \param[out] INFO
182: *> \verbatim
183: *> INFO is INTEGER
184: *> = 0: successful exit
185: *> > 0: if INFO = i, ZLAQR4 failed to compute all of
186: *> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
187: *> and WI contain those eigenvalues which have been
188: *> successfully computed. (Failures are rare.)
189: *>
190: *> If INFO > 0 and WANT is .FALSE., then on exit,
191: *> the remaining unconverged eigenvalues are the eigen-
192: *> values of the upper Hessenberg matrix rows and
193: *> columns ILO through INFO of the final, output
194: *> value of H.
195: *>
196: *> If INFO > 0 and WANTT is .TRUE., then on exit
197: *>
198: *> (*) (initial value of H)*U = U*(final value of H)
199: *>
200: *> where U is a unitary matrix. The final
201: *> value of H is upper Hessenberg and triangular in
202: *> rows and columns INFO+1 through IHI.
203: *>
204: *> If INFO > 0 and WANTZ is .TRUE., then on exit
205: *>
206: *> (final value of Z(ILO:IHI,ILOZ:IHIZ)
207: *> = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
208: *>
209: *> where U is the unitary matrix in (*) (regard-
210: *> less of the value of WANTT.)
211: *>
212: *> If INFO > 0 and WANTZ is .FALSE., then Z is not
213: *> accessed.
214: *> \endverbatim
215: *
216: * Authors:
217: * ========
218: *
219: *> \author Univ. of Tennessee
220: *> \author Univ. of California Berkeley
221: *> \author Univ. of Colorado Denver
222: *> \author NAG Ltd.
223: *
224: *> \ingroup complex16OTHERauxiliary
225: *
226: *> \par Contributors:
227: * ==================
228: *>
229: *> Karen Braman and Ralph Byers, Department of Mathematics,
230: *> University of Kansas, USA
231: *
232: *> \par References:
233: * ================
234: *>
235: *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
236: *> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
237: *> Performance, SIAM Journal of Matrix Analysis, volume 23, pages
238: *> 929--947, 2002.
239: *> \n
240: *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
241: *> Algorithm Part II: Aggressive Early Deflation, SIAM Journal
242: *> of Matrix Analysis, volume 23, pages 948--973, 2002.
243: *>
244: * =====================================================================
245: SUBROUTINE ZLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
246: $ IHIZ, Z, LDZ, WORK, LWORK, INFO )
247: *
248: * -- LAPACK auxiliary routine --
249: * -- LAPACK is a software package provided by Univ. of Tennessee, --
250: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
251: *
252: * .. Scalar Arguments ..
253: INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
254: LOGICAL WANTT, WANTZ
255: * ..
256: * .. Array Arguments ..
257: COMPLEX*16 H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
258: * ..
259: *
260: * ================================================================
261: *
262: * .. Parameters ..
263: *
264: * ==== Matrices of order NTINY or smaller must be processed by
265: * . ZLAHQR because of insufficient subdiagonal scratch space.
266: * . (This is a hard limit.) ====
267: INTEGER NTINY
268: PARAMETER ( NTINY = 15 )
269: *
270: * ==== Exceptional deflation windows: try to cure rare
271: * . slow convergence by varying the size of the
272: * . deflation window after KEXNW iterations. ====
273: INTEGER KEXNW
274: PARAMETER ( KEXNW = 5 )
275: *
276: * ==== Exceptional shifts: try to cure rare slow convergence
277: * . with ad-hoc exceptional shifts every KEXSH iterations.
278: * . ====
279: INTEGER KEXSH
280: PARAMETER ( KEXSH = 6 )
281: *
282: * ==== The constant WILK1 is used to form the exceptional
283: * . shifts. ====
284: DOUBLE PRECISION WILK1
285: PARAMETER ( WILK1 = 0.75d0 )
286: COMPLEX*16 ZERO, ONE
287: PARAMETER ( ZERO = ( 0.0d0, 0.0d0 ),
288: $ ONE = ( 1.0d0, 0.0d0 ) )
289: DOUBLE PRECISION TWO
290: PARAMETER ( TWO = 2.0d0 )
291: * ..
292: * .. Local Scalars ..
293: COMPLEX*16 AA, BB, CC, CDUM, DD, DET, RTDISC, SWAP, TR2
294: DOUBLE PRECISION S
295: INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
296: $ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
297: $ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
298: $ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
299: LOGICAL SORTED
300: CHARACTER JBCMPZ*2
301: * ..
302: * .. External Functions ..
303: INTEGER ILAENV
304: EXTERNAL ILAENV
305: * ..
306: * .. Local Arrays ..
307: COMPLEX*16 ZDUM( 1, 1 )
308: * ..
309: * .. External Subroutines ..
310: EXTERNAL ZLACPY, ZLAHQR, ZLAQR2, ZLAQR5
311: * ..
312: * .. Intrinsic Functions ..
313: INTRINSIC ABS, DBLE, DCMPLX, DIMAG, INT, MAX, MIN, MOD,
314: $ SQRT
315: * ..
316: * .. Statement Functions ..
317: DOUBLE PRECISION CABS1
318: * ..
319: * .. Statement Function definitions ..
320: CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
321: * ..
322: * .. Executable Statements ..
323: INFO = 0
324: *
325: * ==== Quick return for N = 0: nothing to do. ====
326: *
327: IF( N.EQ.0 ) THEN
328: WORK( 1 ) = ONE
329: RETURN
330: END IF
331: *
332: IF( N.LE.NTINY ) THEN
333: *
334: * ==== Tiny matrices must use ZLAHQR. ====
335: *
336: LWKOPT = 1
337: IF( LWORK.NE.-1 )
338: $ CALL ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
339: $ IHIZ, Z, LDZ, INFO )
340: ELSE
341: *
342: * ==== Use small bulge multi-shift QR with aggressive early
343: * . deflation on larger-than-tiny matrices. ====
344: *
345: * ==== Hope for the best. ====
346: *
347: INFO = 0
348: *
349: * ==== Set up job flags for ILAENV. ====
350: *
351: IF( WANTT ) THEN
352: JBCMPZ( 1: 1 ) = 'S'
353: ELSE
354: JBCMPZ( 1: 1 ) = 'E'
355: END IF
356: IF( WANTZ ) THEN
357: JBCMPZ( 2: 2 ) = 'V'
358: ELSE
359: JBCMPZ( 2: 2 ) = 'N'
360: END IF
361: *
362: * ==== NWR = recommended deflation window size. At this
363: * . point, N .GT. NTINY = 15, so there is enough
364: * . subdiagonal workspace for NWR.GE.2 as required.
365: * . (In fact, there is enough subdiagonal space for
366: * . NWR.GE.4.) ====
367: *
368: NWR = ILAENV( 13, 'ZLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
369: NWR = MAX( 2, NWR )
370: NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
371: *
372: * ==== NSR = recommended number of simultaneous shifts.
373: * . At this point N .GT. NTINY = 15, so there is at
374: * . enough subdiagonal workspace for NSR to be even
375: * . and greater than or equal to two as required. ====
376: *
377: NSR = ILAENV( 15, 'ZLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
378: NSR = MIN( NSR, ( N-3 ) / 6, IHI-ILO )
379: NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
380: *
381: * ==== Estimate optimal workspace ====
382: *
383: * ==== Workspace query call to ZLAQR2 ====
384: *
385: CALL ZLAQR2( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
386: $ IHIZ, Z, LDZ, LS, LD, W, H, LDH, N, H, LDH, N, H,
387: $ LDH, WORK, -1 )
388: *
389: * ==== Optimal workspace = MAX(ZLAQR5, ZLAQR2) ====
390: *
391: LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
392: *
393: * ==== Quick return in case of workspace query. ====
394: *
395: IF( LWORK.EQ.-1 ) THEN
396: WORK( 1 ) = DCMPLX( LWKOPT, 0 )
397: RETURN
398: END IF
399: *
400: * ==== ZLAHQR/ZLAQR0 crossover point ====
401: *
402: NMIN = ILAENV( 12, 'ZLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
403: NMIN = MAX( NTINY, NMIN )
404: *
405: * ==== Nibble crossover point ====
406: *
407: NIBBLE = ILAENV( 14, 'ZLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
408: NIBBLE = MAX( 0, NIBBLE )
409: *
410: * ==== Accumulate reflections during ttswp? Use block
411: * . 2-by-2 structure during matrix-matrix multiply? ====
412: *
413: KACC22 = ILAENV( 16, 'ZLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
414: KACC22 = MAX( 0, KACC22 )
415: KACC22 = MIN( 2, KACC22 )
416: *
417: * ==== NWMAX = the largest possible deflation window for
418: * . which there is sufficient workspace. ====
419: *
420: NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
421: NW = NWMAX
422: *
423: * ==== NSMAX = the Largest number of simultaneous shifts
424: * . for which there is sufficient workspace. ====
425: *
426: NSMAX = MIN( ( N-3 ) / 6, 2*LWORK / 3 )
427: NSMAX = NSMAX - MOD( NSMAX, 2 )
428: *
429: * ==== NDFL: an iteration count restarted at deflation. ====
430: *
431: NDFL = 1
432: *
433: * ==== ITMAX = iteration limit ====
434: *
435: ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
436: *
437: * ==== Last row and column in the active block ====
438: *
439: KBOT = IHI
440: *
441: * ==== Main Loop ====
442: *
443: DO 70 IT = 1, ITMAX
444: *
445: * ==== Done when KBOT falls below ILO ====
446: *
447: IF( KBOT.LT.ILO )
448: $ GO TO 80
449: *
450: * ==== Locate active block ====
451: *
452: DO 10 K = KBOT, ILO + 1, -1
453: IF( H( K, K-1 ).EQ.ZERO )
454: $ GO TO 20
455: 10 CONTINUE
456: K = ILO
457: 20 CONTINUE
458: KTOP = K
459: *
460: * ==== Select deflation window size:
461: * . Typical Case:
462: * . If possible and advisable, nibble the entire
463: * . active block. If not, use size MIN(NWR,NWMAX)
464: * . or MIN(NWR+1,NWMAX) depending upon which has
465: * . the smaller corresponding subdiagonal entry
466: * . (a heuristic).
467: * .
468: * . Exceptional Case:
469: * . If there have been no deflations in KEXNW or
470: * . more iterations, then vary the deflation window
471: * . size. At first, because, larger windows are,
472: * . in general, more powerful than smaller ones,
473: * . rapidly increase the window to the maximum possible.
474: * . Then, gradually reduce the window size. ====
475: *
476: NH = KBOT - KTOP + 1
477: NWUPBD = MIN( NH, NWMAX )
478: IF( NDFL.LT.KEXNW ) THEN
479: NW = MIN( NWUPBD, NWR )
480: ELSE
481: NW = MIN( NWUPBD, 2*NW )
482: END IF
483: IF( NW.LT.NWMAX ) THEN
484: IF( NW.GE.NH-1 ) THEN
485: NW = NH
486: ELSE
487: KWTOP = KBOT - NW + 1
488: IF( CABS1( H( KWTOP, KWTOP-1 ) ).GT.
489: $ CABS1( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
490: END IF
491: END IF
492: IF( NDFL.LT.KEXNW ) THEN
493: NDEC = -1
494: ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
495: NDEC = NDEC + 1
496: IF( NW-NDEC.LT.2 )
497: $ NDEC = 0
498: NW = NW - NDEC
499: END IF
500: *
501: * ==== Aggressive early deflation:
502: * . split workspace under the subdiagonal into
503: * . - an nw-by-nw work array V in the lower
504: * . left-hand-corner,
505: * . - an NW-by-at-least-NW-but-more-is-better
506: * . (NW-by-NHO) horizontal work array along
507: * . the bottom edge,
508: * . - an at-least-NW-but-more-is-better (NHV-by-NW)
509: * . vertical work array along the left-hand-edge.
510: * . ====
511: *
512: KV = N - NW + 1
513: KT = NW + 1
514: NHO = ( N-NW-1 ) - KT + 1
515: KWV = NW + 2
516: NVE = ( N-NW ) - KWV + 1
517: *
518: * ==== Aggressive early deflation ====
519: *
520: CALL ZLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
521: $ IHIZ, Z, LDZ, LS, LD, W, H( KV, 1 ), LDH, NHO,
522: $ H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH, WORK,
523: $ LWORK )
524: *
525: * ==== Adjust KBOT accounting for new deflations. ====
526: *
527: KBOT = KBOT - LD
528: *
529: * ==== KS points to the shifts. ====
530: *
531: KS = KBOT - LS + 1
532: *
533: * ==== Skip an expensive QR sweep if there is a (partly
534: * . heuristic) reason to expect that many eigenvalues
535: * . will deflate without it. Here, the QR sweep is
536: * . skipped if many eigenvalues have just been deflated
537: * . or if the remaining active block is small.
538: *
539: IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
540: $ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
541: *
542: * ==== NS = nominal number of simultaneous shifts.
543: * . This may be lowered (slightly) if ZLAQR2
544: * . did not provide that many shifts. ====
545: *
546: NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
547: NS = NS - MOD( NS, 2 )
548: *
549: * ==== If there have been no deflations
550: * . in a multiple of KEXSH iterations,
551: * . then try exceptional shifts.
552: * . Otherwise use shifts provided by
553: * . ZLAQR2 above or from the eigenvalues
554: * . of a trailing principal submatrix. ====
555: *
556: IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
557: KS = KBOT - NS + 1
558: DO 30 I = KBOT, KS + 1, -2
559: W( I ) = H( I, I ) + WILK1*CABS1( H( I, I-1 ) )
560: W( I-1 ) = W( I )
561: 30 CONTINUE
562: ELSE
563: *
564: * ==== Got NS/2 or fewer shifts? Use ZLAHQR
565: * . on a trailing principal submatrix to
566: * . get more. (Since NS.LE.NSMAX.LE.(N-3)/6,
567: * . there is enough space below the subdiagonal
568: * . to fit an NS-by-NS scratch array.) ====
569: *
570: IF( KBOT-KS+1.LE.NS / 2 ) THEN
571: KS = KBOT - NS + 1
572: KT = N - NS + 1
573: CALL ZLACPY( 'A', NS, NS, H( KS, KS ), LDH,
574: $ H( KT, 1 ), LDH )
575: CALL ZLAHQR( .false., .false., NS, 1, NS,
576: $ H( KT, 1 ), LDH, W( KS ), 1, 1, ZDUM,
577: $ 1, INF )
578: KS = KS + INF
579: *
580: * ==== In case of a rare QR failure use
581: * . eigenvalues of the trailing 2-by-2
582: * . principal submatrix. Scale to avoid
583: * . overflows, underflows and subnormals.
584: * . (The scale factor S can not be zero,
585: * . because H(KBOT,KBOT-1) is nonzero.) ====
586: *
587: IF( KS.GE.KBOT ) THEN
588: S = CABS1( H( KBOT-1, KBOT-1 ) ) +
589: $ CABS1( H( KBOT, KBOT-1 ) ) +
590: $ CABS1( H( KBOT-1, KBOT ) ) +
591: $ CABS1( H( KBOT, KBOT ) )
592: AA = H( KBOT-1, KBOT-1 ) / S
593: CC = H( KBOT, KBOT-1 ) / S
594: BB = H( KBOT-1, KBOT ) / S
595: DD = H( KBOT, KBOT ) / S
596: TR2 = ( AA+DD ) / TWO
597: DET = ( AA-TR2 )*( DD-TR2 ) - BB*CC
598: RTDISC = SQRT( -DET )
599: W( KBOT-1 ) = ( TR2+RTDISC )*S
600: W( KBOT ) = ( TR2-RTDISC )*S
601: *
602: KS = KBOT - 1
603: END IF
604: END IF
605: *
606: IF( KBOT-KS+1.GT.NS ) THEN
607: *
608: * ==== Sort the shifts (Helps a little) ====
609: *
610: SORTED = .false.
611: DO 50 K = KBOT, KS + 1, -1
612: IF( SORTED )
613: $ GO TO 60
614: SORTED = .true.
615: DO 40 I = KS, K - 1
616: IF( CABS1( W( I ) ).LT.CABS1( W( I+1 ) ) )
617: $ THEN
618: SORTED = .false.
619: SWAP = W( I )
620: W( I ) = W( I+1 )
621: W( I+1 ) = SWAP
622: END IF
623: 40 CONTINUE
624: 50 CONTINUE
625: 60 CONTINUE
626: END IF
627: END IF
628: *
629: * ==== If there are only two shifts, then use
630: * . only one. ====
631: *
632: IF( KBOT-KS+1.EQ.2 ) THEN
633: IF( CABS1( W( KBOT )-H( KBOT, KBOT ) ).LT.
634: $ CABS1( W( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
635: W( KBOT-1 ) = W( KBOT )
636: ELSE
637: W( KBOT ) = W( KBOT-1 )
638: END IF
639: END IF
640: *
641: * ==== Use up to NS of the the smallest magnitude
642: * . shifts. If there aren't NS shifts available,
643: * . then use them all, possibly dropping one to
644: * . make the number of shifts even. ====
645: *
646: NS = MIN( NS, KBOT-KS+1 )
647: NS = NS - MOD( NS, 2 )
648: KS = KBOT - NS + 1
649: *
650: * ==== Small-bulge multi-shift QR sweep:
651: * . split workspace under the subdiagonal into
652: * . - a KDU-by-KDU work array U in the lower
653: * . left-hand-corner,
654: * . - a KDU-by-at-least-KDU-but-more-is-better
655: * . (KDU-by-NHo) horizontal work array WH along
656: * . the bottom edge,
657: * . - and an at-least-KDU-but-more-is-better-by-KDU
658: * . (NVE-by-KDU) vertical work WV arrow along
659: * . the left-hand-edge. ====
660: *
661: KDU = 2*NS
662: KU = N - KDU + 1
663: KWH = KDU + 1
664: NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
665: KWV = KDU + 4
666: NVE = N - KDU - KWV + 1
667: *
668: * ==== Small-bulge multi-shift QR sweep ====
669: *
670: CALL ZLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
671: $ W( KS ), H, LDH, ILOZ, IHIZ, Z, LDZ, WORK,
672: $ 3, H( KU, 1 ), LDH, NVE, H( KWV, 1 ), LDH,
673: $ NHO, H( KU, KWH ), LDH )
674: END IF
675: *
676: * ==== Note progress (or the lack of it). ====
677: *
678: IF( LD.GT.0 ) THEN
679: NDFL = 1
680: ELSE
681: NDFL = NDFL + 1
682: END IF
683: *
684: * ==== End of main loop ====
685: 70 CONTINUE
686: *
687: * ==== Iteration limit exceeded. Set INFO to show where
688: * . the problem occurred and exit. ====
689: *
690: INFO = KBOT
691: 80 CONTINUE
692: END IF
693: *
694: * ==== Return the optimal value of LWORK. ====
695: *
696: WORK( 1 ) = DCMPLX( LWKOPT, 0 )
697: *
698: * ==== End of ZLAQR4 ====
699: *
700: END
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