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 .GE. 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.GT.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.GT.0, then 1.LE.ILO.LE.IHI.LE.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.GT.0 is given under the
106: *> description of INFO below.)
107: *>
108: *> This subroutine may explicitly set H(i,j) = 0 for i.GT.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 .GE. 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 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. 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.GT.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.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.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 .GE. 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: *> .GT. 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 .GT. 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 .GT. 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 .GT. 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 .GT. 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: *> \date September 2012
225: *
226: *> \ingroup complex16OTHERauxiliary
227: *
228: *> \par Contributors:
229: * ==================
230: *>
231: *> Karen Braman and Ralph Byers, Department of Mathematics,
232: *> University of Kansas, USA
233: *
234: *> \par References:
235: * ================
236: *>
237: *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
238: *> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
239: *> Performance, SIAM Journal of Matrix Analysis, volume 23, pages
240: *> 929--947, 2002.
241: *> \n
242: *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
243: *> Algorithm Part II: Aggressive Early Deflation, SIAM Journal
244: *> of Matrix Analysis, volume 23, pages 948--973, 2002.
245: *>
246: * =====================================================================
247: SUBROUTINE ZLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
248: $ IHIZ, Z, LDZ, WORK, LWORK, INFO )
249: *
250: * -- LAPACK auxiliary routine (version 3.4.2) --
251: * -- LAPACK is a software package provided by Univ. of Tennessee, --
252: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
253: * September 2012
254: *
255: * .. Scalar Arguments ..
256: INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
257: LOGICAL WANTT, WANTZ
258: * ..
259: * .. Array Arguments ..
260: COMPLEX*16 H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
261: * ..
262: *
263: * ================================================================
264: *
265: * .. Parameters ..
266: *
267: * ==== Matrices of order NTINY or smaller must be processed by
268: * . ZLAHQR because of insufficient subdiagonal scratch space.
269: * . (This is a hard limit.) ====
270: INTEGER NTINY
271: PARAMETER ( NTINY = 11 )
272: *
273: * ==== Exceptional deflation windows: try to cure rare
274: * . slow convergence by varying the size of the
275: * . deflation window after KEXNW iterations. ====
276: INTEGER KEXNW
277: PARAMETER ( KEXNW = 5 )
278: *
279: * ==== Exceptional shifts: try to cure rare slow convergence
280: * . with ad-hoc exceptional shifts every KEXSH iterations.
281: * . ====
282: INTEGER KEXSH
283: PARAMETER ( KEXSH = 6 )
284: *
285: * ==== The constant WILK1 is used to form the exceptional
286: * . shifts. ====
287: DOUBLE PRECISION WILK1
288: PARAMETER ( WILK1 = 0.75d0 )
289: COMPLEX*16 ZERO, ONE
290: PARAMETER ( ZERO = ( 0.0d0, 0.0d0 ),
291: $ ONE = ( 1.0d0, 0.0d0 ) )
292: DOUBLE PRECISION TWO
293: PARAMETER ( TWO = 2.0d0 )
294: * ..
295: * .. Local Scalars ..
296: COMPLEX*16 AA, BB, CC, CDUM, DD, DET, RTDISC, SWAP, TR2
297: DOUBLE PRECISION S
298: INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
299: $ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
300: $ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
301: $ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
302: LOGICAL SORTED
303: CHARACTER JBCMPZ*2
304: * ..
305: * .. External Functions ..
306: INTEGER ILAENV
307: EXTERNAL ILAENV
308: * ..
309: * .. Local Arrays ..
310: COMPLEX*16 ZDUM( 1, 1 )
311: * ..
312: * .. External Subroutines ..
313: EXTERNAL ZLACPY, ZLAHQR, ZLAQR2, ZLAQR5
314: * ..
315: * .. Intrinsic Functions ..
316: INTRINSIC ABS, DBLE, DCMPLX, DIMAG, INT, MAX, MIN, MOD,
317: $ SQRT
318: * ..
319: * .. Statement Functions ..
320: DOUBLE PRECISION CABS1
321: * ..
322: * .. Statement Function definitions ..
323: CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
324: * ..
325: * .. Executable Statements ..
326: INFO = 0
327: *
328: * ==== Quick return for N = 0: nothing to do. ====
329: *
330: IF( N.EQ.0 ) THEN
331: WORK( 1 ) = ONE
332: RETURN
333: END IF
334: *
335: IF( N.LE.NTINY ) THEN
336: *
337: * ==== Tiny matrices must use ZLAHQR. ====
338: *
339: LWKOPT = 1
340: IF( LWORK.NE.-1 )
341: $ CALL ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
342: $ IHIZ, Z, LDZ, INFO )
343: ELSE
344: *
345: * ==== Use small bulge multi-shift QR with aggressive early
346: * . deflation on larger-than-tiny matrices. ====
347: *
348: * ==== Hope for the best. ====
349: *
350: INFO = 0
351: *
352: * ==== Set up job flags for ILAENV. ====
353: *
354: IF( WANTT ) THEN
355: JBCMPZ( 1: 1 ) = 'S'
356: ELSE
357: JBCMPZ( 1: 1 ) = 'E'
358: END IF
359: IF( WANTZ ) THEN
360: JBCMPZ( 2: 2 ) = 'V'
361: ELSE
362: JBCMPZ( 2: 2 ) = 'N'
363: END IF
364: *
365: * ==== NWR = recommended deflation window size. At this
366: * . point, N .GT. NTINY = 11, so there is enough
367: * . subdiagonal workspace for NWR.GE.2 as required.
368: * . (In fact, there is enough subdiagonal space for
369: * . NWR.GE.3.) ====
370: *
371: NWR = ILAENV( 13, 'ZLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
372: NWR = MAX( 2, NWR )
373: NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
374: *
375: * ==== NSR = recommended number of simultaneous shifts.
376: * . At this point N .GT. NTINY = 11, so there is at
377: * . enough subdiagonal workspace for NSR to be even
378: * . and greater than or equal to two as required. ====
379: *
380: NSR = ILAENV( 15, 'ZLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
381: NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
382: NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
383: *
384: * ==== Estimate optimal workspace ====
385: *
386: * ==== Workspace query call to ZLAQR2 ====
387: *
388: CALL ZLAQR2( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
389: $ IHIZ, Z, LDZ, LS, LD, W, H, LDH, N, H, LDH, N, H,
390: $ LDH, WORK, -1 )
391: *
392: * ==== Optimal workspace = MAX(ZLAQR5, ZLAQR2) ====
393: *
394: LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
395: *
396: * ==== Quick return in case of workspace query. ====
397: *
398: IF( LWORK.EQ.-1 ) THEN
399: WORK( 1 ) = DCMPLX( LWKOPT, 0 )
400: RETURN
401: END IF
402: *
403: * ==== ZLAHQR/ZLAQR0 crossover point ====
404: *
405: NMIN = ILAENV( 12, 'ZLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
406: NMIN = MAX( NTINY, NMIN )
407: *
408: * ==== Nibble crossover point ====
409: *
410: NIBBLE = ILAENV( 14, 'ZLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
411: NIBBLE = MAX( 0, NIBBLE )
412: *
413: * ==== Accumulate reflections during ttswp? Use block
414: * . 2-by-2 structure during matrix-matrix multiply? ====
415: *
416: KACC22 = ILAENV( 16, 'ZLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
417: KACC22 = MAX( 0, KACC22 )
418: KACC22 = MIN( 2, KACC22 )
419: *
420: * ==== NWMAX = the largest possible deflation window for
421: * . which there is sufficient workspace. ====
422: *
423: NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
424: NW = NWMAX
425: *
426: * ==== NSMAX = the Largest number of simultaneous shifts
427: * . for which there is sufficient workspace. ====
428: *
429: NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
430: NSMAX = NSMAX - MOD( NSMAX, 2 )
431: *
432: * ==== NDFL: an iteration count restarted at deflation. ====
433: *
434: NDFL = 1
435: *
436: * ==== ITMAX = iteration limit ====
437: *
438: ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
439: *
440: * ==== Last row and column in the active block ====
441: *
442: KBOT = IHI
443: *
444: * ==== Main Loop ====
445: *
446: DO 70 IT = 1, ITMAX
447: *
448: * ==== Done when KBOT falls below ILO ====
449: *
450: IF( KBOT.LT.ILO )
451: $ GO TO 80
452: *
453: * ==== Locate active block ====
454: *
455: DO 10 K = KBOT, ILO + 1, -1
456: IF( H( K, K-1 ).EQ.ZERO )
457: $ GO TO 20
458: 10 CONTINUE
459: K = ILO
460: 20 CONTINUE
461: KTOP = K
462: *
463: * ==== Select deflation window size:
464: * . Typical Case:
465: * . If possible and advisable, nibble the entire
466: * . active block. If not, use size MIN(NWR,NWMAX)
467: * . or MIN(NWR+1,NWMAX) depending upon which has
468: * . the smaller corresponding subdiagonal entry
469: * . (a heuristic).
470: * .
471: * . Exceptional Case:
472: * . If there have been no deflations in KEXNW or
473: * . more iterations, then vary the deflation window
474: * . size. At first, because, larger windows are,
475: * . in general, more powerful than smaller ones,
476: * . rapidly increase the window to the maximum possible.
477: * . Then, gradually reduce the window size. ====
478: *
479: NH = KBOT - KTOP + 1
480: NWUPBD = MIN( NH, NWMAX )
481: IF( NDFL.LT.KEXNW ) THEN
482: NW = MIN( NWUPBD, NWR )
483: ELSE
484: NW = MIN( NWUPBD, 2*NW )
485: END IF
486: IF( NW.LT.NWMAX ) THEN
487: IF( NW.GE.NH-1 ) THEN
488: NW = NH
489: ELSE
490: KWTOP = KBOT - NW + 1
491: IF( CABS1( H( KWTOP, KWTOP-1 ) ).GT.
492: $ CABS1( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
493: END IF
494: END IF
495: IF( NDFL.LT.KEXNW ) THEN
496: NDEC = -1
497: ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
498: NDEC = NDEC + 1
499: IF( NW-NDEC.LT.2 )
500: $ NDEC = 0
501: NW = NW - NDEC
502: END IF
503: *
504: * ==== Aggressive early deflation:
505: * . split workspace under the subdiagonal into
506: * . - an nw-by-nw work array V in the lower
507: * . left-hand-corner,
508: * . - an NW-by-at-least-NW-but-more-is-better
509: * . (NW-by-NHO) horizontal work array along
510: * . the bottom edge,
511: * . - an at-least-NW-but-more-is-better (NHV-by-NW)
512: * . vertical work array along the left-hand-edge.
513: * . ====
514: *
515: KV = N - NW + 1
516: KT = NW + 1
517: NHO = ( N-NW-1 ) - KT + 1
518: KWV = NW + 2
519: NVE = ( N-NW ) - KWV + 1
520: *
521: * ==== Aggressive early deflation ====
522: *
523: CALL ZLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
524: $ IHIZ, Z, LDZ, LS, LD, W, H( KV, 1 ), LDH, NHO,
525: $ H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH, WORK,
526: $ LWORK )
527: *
528: * ==== Adjust KBOT accounting for new deflations. ====
529: *
530: KBOT = KBOT - LD
531: *
532: * ==== KS points to the shifts. ====
533: *
534: KS = KBOT - LS + 1
535: *
536: * ==== Skip an expensive QR sweep if there is a (partly
537: * . heuristic) reason to expect that many eigenvalues
538: * . will deflate without it. Here, the QR sweep is
539: * . skipped if many eigenvalues have just been deflated
540: * . or if the remaining active block is small.
541: *
542: IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
543: $ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
544: *
545: * ==== NS = nominal number of simultaneous shifts.
546: * . This may be lowered (slightly) if ZLAQR2
547: * . did not provide that many shifts. ====
548: *
549: NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
550: NS = NS - MOD( NS, 2 )
551: *
552: * ==== If there have been no deflations
553: * . in a multiple of KEXSH iterations,
554: * . then try exceptional shifts.
555: * . Otherwise use shifts provided by
556: * . ZLAQR2 above or from the eigenvalues
557: * . of a trailing principal submatrix. ====
558: *
559: IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
560: KS = KBOT - NS + 1
561: DO 30 I = KBOT, KS + 1, -2
562: W( I ) = H( I, I ) + WILK1*CABS1( H( I, I-1 ) )
563: W( I-1 ) = W( I )
564: 30 CONTINUE
565: ELSE
566: *
567: * ==== Got NS/2 or fewer shifts? Use ZLAHQR
568: * . on a trailing principal submatrix to
569: * . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
570: * . there is enough space below the subdiagonal
571: * . to fit an NS-by-NS scratch array.) ====
572: *
573: IF( KBOT-KS+1.LE.NS / 2 ) THEN
574: KS = KBOT - NS + 1
575: KT = N - NS + 1
576: CALL ZLACPY( 'A', NS, NS, H( KS, KS ), LDH,
577: $ H( KT, 1 ), LDH )
578: CALL ZLAHQR( .false., .false., NS, 1, NS,
579: $ H( KT, 1 ), LDH, W( KS ), 1, 1, ZDUM,
580: $ 1, INF )
581: KS = KS + INF
582: *
583: * ==== In case of a rare QR failure use
584: * . eigenvalues of the trailing 2-by-2
585: * . principal submatrix. Scale to avoid
586: * . overflows, underflows and subnormals.
587: * . (The scale factor S can not be zero,
588: * . because H(KBOT,KBOT-1) is nonzero.) ====
589: *
590: IF( KS.GE.KBOT ) THEN
591: S = CABS1( H( KBOT-1, KBOT-1 ) ) +
592: $ CABS1( H( KBOT, KBOT-1 ) ) +
593: $ CABS1( H( KBOT-1, KBOT ) ) +
594: $ CABS1( H( KBOT, KBOT ) )
595: AA = H( KBOT-1, KBOT-1 ) / S
596: CC = H( KBOT, KBOT-1 ) / S
597: BB = H( KBOT-1, KBOT ) / S
598: DD = H( KBOT, KBOT ) / S
599: TR2 = ( AA+DD ) / TWO
600: DET = ( AA-TR2 )*( DD-TR2 ) - BB*CC
601: RTDISC = SQRT( -DET )
602: W( KBOT-1 ) = ( TR2+RTDISC )*S
603: W( KBOT ) = ( TR2-RTDISC )*S
604: *
605: KS = KBOT - 1
606: END IF
607: END IF
608: *
609: IF( KBOT-KS+1.GT.NS ) THEN
610: *
611: * ==== Sort the shifts (Helps a little) ====
612: *
613: SORTED = .false.
614: DO 50 K = KBOT, KS + 1, -1
615: IF( SORTED )
616: $ GO TO 60
617: SORTED = .true.
618: DO 40 I = KS, K - 1
619: IF( CABS1( W( I ) ).LT.CABS1( W( I+1 ) ) )
620: $ THEN
621: SORTED = .false.
622: SWAP = W( I )
623: W( I ) = W( I+1 )
624: W( I+1 ) = SWAP
625: END IF
626: 40 CONTINUE
627: 50 CONTINUE
628: 60 CONTINUE
629: END IF
630: END IF
631: *
632: * ==== If there are only two shifts, then use
633: * . only one. ====
634: *
635: IF( KBOT-KS+1.EQ.2 ) THEN
636: IF( CABS1( W( KBOT )-H( KBOT, KBOT ) ).LT.
637: $ CABS1( W( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
638: W( KBOT-1 ) = W( KBOT )
639: ELSE
640: W( KBOT ) = W( KBOT-1 )
641: END IF
642: END IF
643: *
644: * ==== Use up to NS of the the smallest magnatiude
645: * . shifts. If there aren't NS shifts available,
646: * . then use them all, possibly dropping one to
647: * . make the number of shifts even. ====
648: *
649: NS = MIN( NS, KBOT-KS+1 )
650: NS = NS - MOD( NS, 2 )
651: KS = KBOT - NS + 1
652: *
653: * ==== Small-bulge multi-shift QR sweep:
654: * . split workspace under the subdiagonal into
655: * . - a KDU-by-KDU work array U in the lower
656: * . left-hand-corner,
657: * . - a KDU-by-at-least-KDU-but-more-is-better
658: * . (KDU-by-NHo) horizontal work array WH along
659: * . the bottom edge,
660: * . - and an at-least-KDU-but-more-is-better-by-KDU
661: * . (NVE-by-KDU) vertical work WV arrow along
662: * . the left-hand-edge. ====
663: *
664: KDU = 3*NS - 3
665: KU = N - KDU + 1
666: KWH = KDU + 1
667: NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
668: KWV = KDU + 4
669: NVE = N - KDU - KWV + 1
670: *
671: * ==== Small-bulge multi-shift QR sweep ====
672: *
673: CALL ZLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
674: $ W( KS ), H, LDH, ILOZ, IHIZ, Z, LDZ, WORK,
675: $ 3, H( KU, 1 ), LDH, NVE, H( KWV, 1 ), LDH,
676: $ NHO, H( KU, KWH ), LDH )
677: END IF
678: *
679: * ==== Note progress (or the lack of it). ====
680: *
681: IF( LD.GT.0 ) THEN
682: NDFL = 1
683: ELSE
684: NDFL = NDFL + 1
685: END IF
686: *
687: * ==== End of main loop ====
688: 70 CONTINUE
689: *
690: * ==== Iteration limit exceeded. Set INFO to show where
691: * . the problem occurred and exit. ====
692: *
693: INFO = KBOT
694: 80 CONTINUE
695: END IF
696: *
697: * ==== Return the optimal value of LWORK. ====
698: *
699: WORK( 1 ) = DCMPLX( LWKOPT, 0 )
700: *
701: * ==== End of ZLAQR4 ====
702: *
703: END
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