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
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