1 SUBROUTINE DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
2 $ ILOZ, 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 DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
14 $ Z( LDZ, * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * DLAQR0 computes the eigenvalues of a Hessenberg matrix H
21 * and, optionally, the matrices T and Z from the Schur decomposition
22 * H = Z T Z**T, where T is an upper quasi-triangular matrix (the
23 * Schur form), and Z is the orthogonal matrix of Schur vectors.
24 *
25 * Optionally Z may be postmultiplied into an input orthogonal
26 * matrix Q so that this routine can give the Schur factorization
27 * of a matrix A which has been reduced to the Hessenberg form H
28 * by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
29 *
30 * Arguments
31 * =========
32 *
33 * WANTT (input) LOGICAL
34 * = .TRUE. : the full Schur form T is required;
35 * = .FALSE.: only eigenvalues are required.
36 *
37 * WANTZ (input) LOGICAL
38 * = .TRUE. : the matrix of Schur vectors Z is required;
39 * = .FALSE.: Schur vectors are not required.
40 *
41 * N (input) INTEGER
42 * The order of the matrix H. N .GE. 0.
43 *
44 * ILO (input) INTEGER
45 * IHI (input) INTEGER
46 * It is assumed that H is already upper triangular in rows
47 * and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
48 * H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
49 * previous call to DGEBAL, and then passed to DGEHRD when the
50 * matrix output by DGEBAL is reduced to Hessenberg form.
51 * Otherwise, ILO and IHI should be set to 1 and N,
52 * respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
53 * If N = 0, then ILO = 1 and IHI = 0.
54 *
55 * H (input/output) DOUBLE PRECISION array, dimension (LDH,N)
56 * On entry, the upper Hessenberg matrix H.
57 * On exit, if INFO = 0 and WANTT is .TRUE., then H contains
58 * the upper quasi-triangular matrix T from the Schur
59 * decomposition (the Schur form); 2-by-2 diagonal blocks
60 * (corresponding to complex conjugate pairs of eigenvalues)
61 * are returned in standard form, with H(i,i) = H(i+1,i+1)
62 * and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
63 * .FALSE., then the contents of H are unspecified on exit.
64 * (The output value of H when INFO.GT.0 is given under the
65 * description of INFO below.)
66 *
67 * This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
68 * j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
69 *
70 * LDH (input) INTEGER
71 * The leading dimension of the array H. LDH .GE. max(1,N).
72 *
73 * WR (output) DOUBLE PRECISION array, dimension (IHI)
74 * WI (output) DOUBLE PRECISION array, dimension (IHI)
75 * The real and imaginary parts, respectively, of the computed
76 * eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
77 * and WI(ILO:IHI). If two eigenvalues are computed as a
78 * complex conjugate pair, they are stored in consecutive
79 * elements of WR and WI, say the i-th and (i+1)th, with
80 * WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
81 * the eigenvalues are stored in the same order as on the
82 * diagonal of the Schur form returned in H, with
83 * WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
84 * block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
85 * WI(i+1) = -WI(i).
86 *
87 * ILOZ (input) INTEGER
88 * IHIZ (input) INTEGER
89 * Specify the rows of Z to which transformations must be
90 * applied if WANTZ is .TRUE..
91 * 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
92 *
93 * Z (input/output) DOUBLE PRECISION array, dimension (LDZ,IHI)
94 * If WANTZ is .FALSE., then Z is not referenced.
95 * If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
96 * replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
97 * orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
98 * (The output value of Z when INFO.GT.0 is given under
99 * the description of INFO below.)
100 *
101 * LDZ (input) INTEGER
102 * The leading dimension of the array Z. if WANTZ is .TRUE.
103 * then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.
104 *
105 * WORK (workspace/output) DOUBLE PRECISION array, dimension LWORK
106 * On exit, if LWORK = -1, WORK(1) returns an estimate of
107 * the optimal value for LWORK.
108 *
109 * LWORK (input) INTEGER
110 * The dimension of the array WORK. LWORK .GE. max(1,N)
111 * is sufficient, but LWORK typically as large as 6*N may
112 * be required for optimal performance. A workspace query
113 * to determine the optimal workspace size is recommended.
114 *
115 * If LWORK = -1, then DLAQR0 does a workspace query.
116 * In this case, DLAQR0 checks the input parameters and
117 * estimates the optimal workspace size for the given
118 * values of N, ILO and IHI. The estimate is returned
119 * in WORK(1). No error message related to LWORK is
120 * issued by XERBLA. Neither H nor Z are accessed.
121 *
122 *
123 * INFO (output) INTEGER
124 * = 0: successful exit
125 * .GT. 0: if INFO = i, DLAQR0 failed to compute all of
126 * the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
127 * and WI contain those eigenvalues which have been
128 * successfully computed. (Failures are rare.)
129 *
130 * If INFO .GT. 0 and WANT is .FALSE., then on exit,
131 * the remaining unconverged eigenvalues are the eigen-
132 * values of the upper Hessenberg matrix rows and
133 * columns ILO through INFO of the final, output
134 * value of H.
135 *
136 * If INFO .GT. 0 and WANTT is .TRUE., then on exit
137 *
138 * (*) (initial value of H)*U = U*(final value of H)
139 *
140 * where U is an orthogonal matrix. The final
141 * value of H is upper Hessenberg and quasi-triangular
142 * in rows and columns INFO+1 through IHI.
143 *
144 * If INFO .GT. 0 and WANTZ is .TRUE., then on exit
145 *
146 * (final value of Z(ILO:IHI,ILOZ:IHIZ)
147 * = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
148 *
149 * where U is the orthogonal matrix in (*) (regard-
150 * less of the value of WANTT.)
151 *
152 * If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
153 * accessed.
154 *
155 * ================================================================
156 * Based on contributions by
157 * Karen Braman and Ralph Byers, Department of Mathematics,
158 * University of Kansas, USA
159 *
160 * ================================================================
161 * References:
162 * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
163 * Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
164 * Performance, SIAM Journal of Matrix Analysis, volume 23, pages
165 * 929--947, 2002.
166 *
167 * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
168 * Algorithm Part II: Aggressive Early Deflation, SIAM Journal
169 * of Matrix Analysis, volume 23, pages 948--973, 2002.
170 *
171 * ================================================================
172 * .. Parameters ..
173 *
174 * ==== Matrices of order NTINY or smaller must be processed by
175 * . DLAHQR because of insufficient subdiagonal scratch space.
176 * . (This is a hard limit.) ====
177 INTEGER NTINY
178 PARAMETER ( NTINY = 11 )
179 *
180 * ==== Exceptional deflation windows: try to cure rare
181 * . slow convergence by varying the size of the
182 * . deflation window after KEXNW iterations. ====
183 INTEGER KEXNW
184 PARAMETER ( KEXNW = 5 )
185 *
186 * ==== Exceptional shifts: try to cure rare slow convergence
187 * . with ad-hoc exceptional shifts every KEXSH iterations.
188 * . ====
189 INTEGER KEXSH
190 PARAMETER ( KEXSH = 6 )
191 *
192 * ==== The constants WILK1 and WILK2 are used to form the
193 * . exceptional shifts. ====
194 DOUBLE PRECISION WILK1, WILK2
195 PARAMETER ( WILK1 = 0.75d0, WILK2 = -0.4375d0 )
196 DOUBLE PRECISION ZERO, ONE
197 PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
198 * ..
199 * .. Local Scalars ..
200 DOUBLE PRECISION AA, BB, CC, CS, DD, SN, SS, SWAP
201 INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
202 $ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
203 $ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
204 $ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
205 LOGICAL SORTED
206 CHARACTER JBCMPZ*2
207 * ..
208 * .. External Functions ..
209 INTEGER ILAENV
210 EXTERNAL ILAENV
211 * ..
212 * .. Local Arrays ..
213 DOUBLE PRECISION ZDUM( 1, 1 )
214 * ..
215 * .. External Subroutines ..
216 EXTERNAL DLACPY, DLAHQR, DLANV2, DLAQR3, DLAQR4, DLAQR5
217 * ..
218 * .. Intrinsic Functions ..
219 INTRINSIC ABS, DBLE, INT, MAX, MIN, MOD
220 * ..
221 * .. Executable Statements ..
222 INFO = 0
223 *
224 * ==== Quick return for N = 0: nothing to do. ====
225 *
226 IF( N.EQ.0 ) THEN
227 WORK( 1 ) = ONE
228 RETURN
229 END IF
230 *
231 IF( N.LE.NTINY ) THEN
232 *
233 * ==== Tiny matrices must use DLAHQR. ====
234 *
235 LWKOPT = 1
236 IF( LWORK.NE.-1 )
237 $ CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
238 $ ILOZ, IHIZ, Z, LDZ, INFO )
239 ELSE
240 *
241 * ==== Use small bulge multi-shift QR with aggressive early
242 * . deflation on larger-than-tiny matrices. ====
243 *
244 * ==== Hope for the best. ====
245 *
246 INFO = 0
247 *
248 * ==== Set up job flags for ILAENV. ====
249 *
250 IF( WANTT ) THEN
251 JBCMPZ( 1: 1 ) = 'S'
252 ELSE
253 JBCMPZ( 1: 1 ) = 'E'
254 END IF
255 IF( WANTZ ) THEN
256 JBCMPZ( 2: 2 ) = 'V'
257 ELSE
258 JBCMPZ( 2: 2 ) = 'N'
259 END IF
260 *
261 * ==== NWR = recommended deflation window size. At this
262 * . point, N .GT. NTINY = 11, so there is enough
263 * . subdiagonal workspace for NWR.GE.2 as required.
264 * . (In fact, there is enough subdiagonal space for
265 * . NWR.GE.3.) ====
266 *
267 NWR = ILAENV( 13, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
268 NWR = MAX( 2, NWR )
269 NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
270 *
271 * ==== NSR = recommended number of simultaneous shifts.
272 * . At this point N .GT. NTINY = 11, so there is at
273 * . enough subdiagonal workspace for NSR to be even
274 * . and greater than or equal to two as required. ====
275 *
276 NSR = ILAENV( 15, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
277 NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
278 NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
279 *
280 * ==== Estimate optimal workspace ====
281 *
282 * ==== Workspace query call to DLAQR3 ====
283 *
284 CALL DLAQR3( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
285 $ IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH,
286 $ N, H, LDH, WORK, -1 )
287 *
288 * ==== Optimal workspace = MAX(DLAQR5, DLAQR3) ====
289 *
290 LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
291 *
292 * ==== Quick return in case of workspace query. ====
293 *
294 IF( LWORK.EQ.-1 ) THEN
295 WORK( 1 ) = DBLE( LWKOPT )
296 RETURN
297 END IF
298 *
299 * ==== DLAHQR/DLAQR0 crossover point ====
300 *
301 NMIN = ILAENV( 12, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
302 NMIN = MAX( NTINY, NMIN )
303 *
304 * ==== Nibble crossover point ====
305 *
306 NIBBLE = ILAENV( 14, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
307 NIBBLE = MAX( 0, NIBBLE )
308 *
309 * ==== Accumulate reflections during ttswp? Use block
310 * . 2-by-2 structure during matrix-matrix multiply? ====
311 *
312 KACC22 = ILAENV( 16, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
313 KACC22 = MAX( 0, KACC22 )
314 KACC22 = MIN( 2, KACC22 )
315 *
316 * ==== NWMAX = the largest possible deflation window for
317 * . which there is sufficient workspace. ====
318 *
319 NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
320 NW = NWMAX
321 *
322 * ==== NSMAX = the Largest number of simultaneous shifts
323 * . for which there is sufficient workspace. ====
324 *
325 NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
326 NSMAX = NSMAX - MOD( NSMAX, 2 )
327 *
328 * ==== NDFL: an iteration count restarted at deflation. ====
329 *
330 NDFL = 1
331 *
332 * ==== ITMAX = iteration limit ====
333 *
334 ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
335 *
336 * ==== Last row and column in the active block ====
337 *
338 KBOT = IHI
339 *
340 * ==== Main Loop ====
341 *
342 DO 80 IT = 1, ITMAX
343 *
344 * ==== Done when KBOT falls below ILO ====
345 *
346 IF( KBOT.LT.ILO )
347 $ GO TO 90
348 *
349 * ==== Locate active block ====
350 *
351 DO 10 K = KBOT, ILO + 1, -1
352 IF( H( K, K-1 ).EQ.ZERO )
353 $ GO TO 20
354 10 CONTINUE
355 K = ILO
356 20 CONTINUE
357 KTOP = K
358 *
359 * ==== Select deflation window size:
360 * . Typical Case:
361 * . If possible and advisable, nibble the entire
362 * . active block. If not, use size MIN(NWR,NWMAX)
363 * . or MIN(NWR+1,NWMAX) depending upon which has
364 * . the smaller corresponding subdiagonal entry
365 * . (a heuristic).
366 * .
367 * . Exceptional Case:
368 * . If there have been no deflations in KEXNW or
369 * . more iterations, then vary the deflation window
370 * . size. At first, because, larger windows are,
371 * . in general, more powerful than smaller ones,
372 * . rapidly increase the window to the maximum possible.
373 * . Then, gradually reduce the window size. ====
374 *
375 NH = KBOT - KTOP + 1
376 NWUPBD = MIN( NH, NWMAX )
377 IF( NDFL.LT.KEXNW ) THEN
378 NW = MIN( NWUPBD, NWR )
379 ELSE
380 NW = MIN( NWUPBD, 2*NW )
381 END IF
382 IF( NW.LT.NWMAX ) THEN
383 IF( NW.GE.NH-1 ) THEN
384 NW = NH
385 ELSE
386 KWTOP = KBOT - NW + 1
387 IF( ABS( H( KWTOP, KWTOP-1 ) ).GT.
388 $ ABS( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
389 END IF
390 END IF
391 IF( NDFL.LT.KEXNW ) THEN
392 NDEC = -1
393 ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
394 NDEC = NDEC + 1
395 IF( NW-NDEC.LT.2 )
396 $ NDEC = 0
397 NW = NW - NDEC
398 END IF
399 *
400 * ==== Aggressive early deflation:
401 * . split workspace under the subdiagonal into
402 * . - an nw-by-nw work array V in the lower
403 * . left-hand-corner,
404 * . - an NW-by-at-least-NW-but-more-is-better
405 * . (NW-by-NHO) horizontal work array along
406 * . the bottom edge,
407 * . - an at-least-NW-but-more-is-better (NHV-by-NW)
408 * . vertical work array along the left-hand-edge.
409 * . ====
410 *
411 KV = N - NW + 1
412 KT = NW + 1
413 NHO = ( N-NW-1 ) - KT + 1
414 KWV = NW + 2
415 NVE = ( N-NW ) - KWV + 1
416 *
417 * ==== Aggressive early deflation ====
418 *
419 CALL DLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
420 $ IHIZ, Z, LDZ, LS, LD, WR, WI, H( KV, 1 ), LDH,
421 $ NHO, H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH,
422 $ WORK, LWORK )
423 *
424 * ==== Adjust KBOT accounting for new deflations. ====
425 *
426 KBOT = KBOT - LD
427 *
428 * ==== KS points to the shifts. ====
429 *
430 KS = KBOT - LS + 1
431 *
432 * ==== Skip an expensive QR sweep if there is a (partly
433 * . heuristic) reason to expect that many eigenvalues
434 * . will deflate without it. Here, the QR sweep is
435 * . skipped if many eigenvalues have just been deflated
436 * . or if the remaining active block is small.
437 *
438 IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
439 $ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
440 *
441 * ==== NS = nominal number of simultaneous shifts.
442 * . This may be lowered (slightly) if DLAQR3
443 * . did not provide that many shifts. ====
444 *
445 NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
446 NS = NS - MOD( NS, 2 )
447 *
448 * ==== If there have been no deflations
449 * . in a multiple of KEXSH iterations,
450 * . then try exceptional shifts.
451 * . Otherwise use shifts provided by
452 * . DLAQR3 above or from the eigenvalues
453 * . of a trailing principal submatrix. ====
454 *
455 IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
456 KS = KBOT - NS + 1
457 DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2
458 SS = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
459 AA = WILK1*SS + H( I, I )
460 BB = SS
461 CC = WILK2*SS
462 DD = AA
463 CALL DLANV2( AA, BB, CC, DD, WR( I-1 ), WI( I-1 ),
464 $ WR( I ), WI( I ), CS, SN )
465 30 CONTINUE
466 IF( KS.EQ.KTOP ) THEN
467 WR( KS+1 ) = H( KS+1, KS+1 )
468 WI( KS+1 ) = ZERO
469 WR( KS ) = WR( KS+1 )
470 WI( KS ) = WI( KS+1 )
471 END IF
472 ELSE
473 *
474 * ==== Got NS/2 or fewer shifts? Use DLAQR4 or
475 * . DLAHQR on a trailing principal submatrix to
476 * . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
477 * . there is enough space below the subdiagonal
478 * . to fit an NS-by-NS scratch array.) ====
479 *
480 IF( KBOT-KS+1.LE.NS / 2 ) THEN
481 KS = KBOT - NS + 1
482 KT = N - NS + 1
483 CALL DLACPY( 'A', NS, NS, H( KS, KS ), LDH,
484 $ H( KT, 1 ), LDH )
485 IF( NS.GT.NMIN ) THEN
486 CALL DLAQR4( .false., .false., NS, 1, NS,
487 $ H( KT, 1 ), LDH, WR( KS ),
488 $ WI( KS ), 1, 1, ZDUM, 1, WORK,
489 $ LWORK, INF )
490 ELSE
491 CALL DLAHQR( .false., .false., NS, 1, NS,
492 $ H( KT, 1 ), LDH, WR( KS ),
493 $ WI( KS ), 1, 1, ZDUM, 1, INF )
494 END IF
495 KS = KS + INF
496 *
497 * ==== In case of a rare QR failure use
498 * . eigenvalues of the trailing 2-by-2
499 * . principal submatrix. ====
500 *
501 IF( KS.GE.KBOT ) THEN
502 AA = H( KBOT-1, KBOT-1 )
503 CC = H( KBOT, KBOT-1 )
504 BB = H( KBOT-1, KBOT )
505 DD = H( KBOT, KBOT )
506 CALL DLANV2( AA, BB, CC, DD, WR( KBOT-1 ),
507 $ WI( KBOT-1 ), WR( KBOT ),
508 $ WI( KBOT ), CS, SN )
509 KS = KBOT - 1
510 END IF
511 END IF
512 *
513 IF( KBOT-KS+1.GT.NS ) THEN
514 *
515 * ==== Sort the shifts (Helps a little)
516 * . Bubble sort keeps complex conjugate
517 * . pairs together. ====
518 *
519 SORTED = .false.
520 DO 50 K = KBOT, KS + 1, -1
521 IF( SORTED )
522 $ GO TO 60
523 SORTED = .true.
524 DO 40 I = KS, K - 1
525 IF( ABS( WR( I ) )+ABS( WI( I ) ).LT.
526 $ ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN
527 SORTED = .false.
528 *
529 SWAP = WR( I )
530 WR( I ) = WR( I+1 )
531 WR( I+1 ) = SWAP
532 *
533 SWAP = WI( I )
534 WI( I ) = WI( I+1 )
535 WI( I+1 ) = SWAP
536 END IF
537 40 CONTINUE
538 50 CONTINUE
539 60 CONTINUE
540 END IF
541 *
542 * ==== Shuffle shifts into pairs of real shifts
543 * . and pairs of complex conjugate shifts
544 * . assuming complex conjugate shifts are
545 * . already adjacent to one another. (Yes,
546 * . they are.) ====
547 *
548 DO 70 I = KBOT, KS + 2, -2
549 IF( WI( I ).NE.-WI( I-1 ) ) THEN
550 *
551 SWAP = WR( I )
552 WR( I ) = WR( I-1 )
553 WR( I-1 ) = WR( I-2 )
554 WR( I-2 ) = SWAP
555 *
556 SWAP = WI( I )
557 WI( I ) = WI( I-1 )
558 WI( I-1 ) = WI( I-2 )
559 WI( I-2 ) = SWAP
560 END IF
561 70 CONTINUE
562 END IF
563 *
564 * ==== If there are only two shifts and both are
565 * . real, then use only one. ====
566 *
567 IF( KBOT-KS+1.EQ.2 ) THEN
568 IF( WI( KBOT ).EQ.ZERO ) THEN
569 IF( ABS( WR( KBOT )-H( KBOT, KBOT ) ).LT.
570 $ ABS( WR( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
571 WR( KBOT-1 ) = WR( KBOT )
572 ELSE
573 WR( KBOT ) = WR( KBOT-1 )
574 END IF
575 END IF
576 END IF
577 *
578 * ==== Use up to NS of the the smallest magnatiude
579 * . shifts. If there aren't NS shifts available,
580 * . then use them all, possibly dropping one to
581 * . make the number of shifts even. ====
582 *
583 NS = MIN( NS, KBOT-KS+1 )
584 NS = NS - MOD( NS, 2 )
585 KS = KBOT - NS + 1
586 *
587 * ==== Small-bulge multi-shift QR sweep:
588 * . split workspace under the subdiagonal into
589 * . - a KDU-by-KDU work array U in the lower
590 * . left-hand-corner,
591 * . - a KDU-by-at-least-KDU-but-more-is-better
592 * . (KDU-by-NHo) horizontal work array WH along
593 * . the bottom edge,
594 * . - and an at-least-KDU-but-more-is-better-by-KDU
595 * . (NVE-by-KDU) vertical work WV arrow along
596 * . the left-hand-edge. ====
597 *
598 KDU = 3*NS - 3
599 KU = N - KDU + 1
600 KWH = KDU + 1
601 NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
602 KWV = KDU + 4
603 NVE = N - KDU - KWV + 1
604 *
605 * ==== Small-bulge multi-shift QR sweep ====
606 *
607 CALL DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
608 $ WR( KS ), WI( KS ), H, LDH, ILOZ, IHIZ, Z,
609 $ LDZ, WORK, 3, H( KU, 1 ), LDH, NVE,
610 $ H( KWV, 1 ), LDH, NHO, H( KU, KWH ), LDH )
611 END IF
612 *
613 * ==== Note progress (or the lack of it). ====
614 *
615 IF( LD.GT.0 ) THEN
616 NDFL = 1
617 ELSE
618 NDFL = NDFL + 1
619 END IF
620 *
621 * ==== End of main loop ====
622 80 CONTINUE
623 *
624 * ==== Iteration limit exceeded. Set INFO to show where
625 * . the problem occurred and exit. ====
626 *
627 INFO = KBOT
628 90 CONTINUE
629 END IF
630 *
631 * ==== Return the optimal value of LWORK. ====
632 *
633 WORK( 1 ) = DBLE( LWKOPT )
634 *
635 * ==== End of DLAQR0 ====
636 *
637 END
2 $ ILOZ, 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 DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
14 $ Z( LDZ, * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * DLAQR0 computes the eigenvalues of a Hessenberg matrix H
21 * and, optionally, the matrices T and Z from the Schur decomposition
22 * H = Z T Z**T, where T is an upper quasi-triangular matrix (the
23 * Schur form), and Z is the orthogonal matrix of Schur vectors.
24 *
25 * Optionally Z may be postmultiplied into an input orthogonal
26 * matrix Q so that this routine can give the Schur factorization
27 * of a matrix A which has been reduced to the Hessenberg form H
28 * by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
29 *
30 * Arguments
31 * =========
32 *
33 * WANTT (input) LOGICAL
34 * = .TRUE. : the full Schur form T is required;
35 * = .FALSE.: only eigenvalues are required.
36 *
37 * WANTZ (input) LOGICAL
38 * = .TRUE. : the matrix of Schur vectors Z is required;
39 * = .FALSE.: Schur vectors are not required.
40 *
41 * N (input) INTEGER
42 * The order of the matrix H. N .GE. 0.
43 *
44 * ILO (input) INTEGER
45 * IHI (input) INTEGER
46 * It is assumed that H is already upper triangular in rows
47 * and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
48 * H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
49 * previous call to DGEBAL, and then passed to DGEHRD when the
50 * matrix output by DGEBAL is reduced to Hessenberg form.
51 * Otherwise, ILO and IHI should be set to 1 and N,
52 * respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
53 * If N = 0, then ILO = 1 and IHI = 0.
54 *
55 * H (input/output) DOUBLE PRECISION array, dimension (LDH,N)
56 * On entry, the upper Hessenberg matrix H.
57 * On exit, if INFO = 0 and WANTT is .TRUE., then H contains
58 * the upper quasi-triangular matrix T from the Schur
59 * decomposition (the Schur form); 2-by-2 diagonal blocks
60 * (corresponding to complex conjugate pairs of eigenvalues)
61 * are returned in standard form, with H(i,i) = H(i+1,i+1)
62 * and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
63 * .FALSE., then the contents of H are unspecified on exit.
64 * (The output value of H when INFO.GT.0 is given under the
65 * description of INFO below.)
66 *
67 * This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
68 * j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
69 *
70 * LDH (input) INTEGER
71 * The leading dimension of the array H. LDH .GE. max(1,N).
72 *
73 * WR (output) DOUBLE PRECISION array, dimension (IHI)
74 * WI (output) DOUBLE PRECISION array, dimension (IHI)
75 * The real and imaginary parts, respectively, of the computed
76 * eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
77 * and WI(ILO:IHI). If two eigenvalues are computed as a
78 * complex conjugate pair, they are stored in consecutive
79 * elements of WR and WI, say the i-th and (i+1)th, with
80 * WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
81 * the eigenvalues are stored in the same order as on the
82 * diagonal of the Schur form returned in H, with
83 * WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
84 * block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
85 * WI(i+1) = -WI(i).
86 *
87 * ILOZ (input) INTEGER
88 * IHIZ (input) INTEGER
89 * Specify the rows of Z to which transformations must be
90 * applied if WANTZ is .TRUE..
91 * 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
92 *
93 * Z (input/output) DOUBLE PRECISION array, dimension (LDZ,IHI)
94 * If WANTZ is .FALSE., then Z is not referenced.
95 * If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
96 * replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
97 * orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
98 * (The output value of Z when INFO.GT.0 is given under
99 * the description of INFO below.)
100 *
101 * LDZ (input) INTEGER
102 * The leading dimension of the array Z. if WANTZ is .TRUE.
103 * then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.
104 *
105 * WORK (workspace/output) DOUBLE PRECISION array, dimension LWORK
106 * On exit, if LWORK = -1, WORK(1) returns an estimate of
107 * the optimal value for LWORK.
108 *
109 * LWORK (input) INTEGER
110 * The dimension of the array WORK. LWORK .GE. max(1,N)
111 * is sufficient, but LWORK typically as large as 6*N may
112 * be required for optimal performance. A workspace query
113 * to determine the optimal workspace size is recommended.
114 *
115 * If LWORK = -1, then DLAQR0 does a workspace query.
116 * In this case, DLAQR0 checks the input parameters and
117 * estimates the optimal workspace size for the given
118 * values of N, ILO and IHI. The estimate is returned
119 * in WORK(1). No error message related to LWORK is
120 * issued by XERBLA. Neither H nor Z are accessed.
121 *
122 *
123 * INFO (output) INTEGER
124 * = 0: successful exit
125 * .GT. 0: if INFO = i, DLAQR0 failed to compute all of
126 * the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
127 * and WI contain those eigenvalues which have been
128 * successfully computed. (Failures are rare.)
129 *
130 * If INFO .GT. 0 and WANT is .FALSE., then on exit,
131 * the remaining unconverged eigenvalues are the eigen-
132 * values of the upper Hessenberg matrix rows and
133 * columns ILO through INFO of the final, output
134 * value of H.
135 *
136 * If INFO .GT. 0 and WANTT is .TRUE., then on exit
137 *
138 * (*) (initial value of H)*U = U*(final value of H)
139 *
140 * where U is an orthogonal matrix. The final
141 * value of H is upper Hessenberg and quasi-triangular
142 * in rows and columns INFO+1 through IHI.
143 *
144 * If INFO .GT. 0 and WANTZ is .TRUE., then on exit
145 *
146 * (final value of Z(ILO:IHI,ILOZ:IHIZ)
147 * = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
148 *
149 * where U is the orthogonal matrix in (*) (regard-
150 * less of the value of WANTT.)
151 *
152 * If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
153 * accessed.
154 *
155 * ================================================================
156 * Based on contributions by
157 * Karen Braman and Ralph Byers, Department of Mathematics,
158 * University of Kansas, USA
159 *
160 * ================================================================
161 * References:
162 * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
163 * Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
164 * Performance, SIAM Journal of Matrix Analysis, volume 23, pages
165 * 929--947, 2002.
166 *
167 * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
168 * Algorithm Part II: Aggressive Early Deflation, SIAM Journal
169 * of Matrix Analysis, volume 23, pages 948--973, 2002.
170 *
171 * ================================================================
172 * .. Parameters ..
173 *
174 * ==== Matrices of order NTINY or smaller must be processed by
175 * . DLAHQR because of insufficient subdiagonal scratch space.
176 * . (This is a hard limit.) ====
177 INTEGER NTINY
178 PARAMETER ( NTINY = 11 )
179 *
180 * ==== Exceptional deflation windows: try to cure rare
181 * . slow convergence by varying the size of the
182 * . deflation window after KEXNW iterations. ====
183 INTEGER KEXNW
184 PARAMETER ( KEXNW = 5 )
185 *
186 * ==== Exceptional shifts: try to cure rare slow convergence
187 * . with ad-hoc exceptional shifts every KEXSH iterations.
188 * . ====
189 INTEGER KEXSH
190 PARAMETER ( KEXSH = 6 )
191 *
192 * ==== The constants WILK1 and WILK2 are used to form the
193 * . exceptional shifts. ====
194 DOUBLE PRECISION WILK1, WILK2
195 PARAMETER ( WILK1 = 0.75d0, WILK2 = -0.4375d0 )
196 DOUBLE PRECISION ZERO, ONE
197 PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
198 * ..
199 * .. Local Scalars ..
200 DOUBLE PRECISION AA, BB, CC, CS, DD, SN, SS, SWAP
201 INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
202 $ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
203 $ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
204 $ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
205 LOGICAL SORTED
206 CHARACTER JBCMPZ*2
207 * ..
208 * .. External Functions ..
209 INTEGER ILAENV
210 EXTERNAL ILAENV
211 * ..
212 * .. Local Arrays ..
213 DOUBLE PRECISION ZDUM( 1, 1 )
214 * ..
215 * .. External Subroutines ..
216 EXTERNAL DLACPY, DLAHQR, DLANV2, DLAQR3, DLAQR4, DLAQR5
217 * ..
218 * .. Intrinsic Functions ..
219 INTRINSIC ABS, DBLE, INT, MAX, MIN, MOD
220 * ..
221 * .. Executable Statements ..
222 INFO = 0
223 *
224 * ==== Quick return for N = 0: nothing to do. ====
225 *
226 IF( N.EQ.0 ) THEN
227 WORK( 1 ) = ONE
228 RETURN
229 END IF
230 *
231 IF( N.LE.NTINY ) THEN
232 *
233 * ==== Tiny matrices must use DLAHQR. ====
234 *
235 LWKOPT = 1
236 IF( LWORK.NE.-1 )
237 $ CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
238 $ ILOZ, IHIZ, Z, LDZ, INFO )
239 ELSE
240 *
241 * ==== Use small bulge multi-shift QR with aggressive early
242 * . deflation on larger-than-tiny matrices. ====
243 *
244 * ==== Hope for the best. ====
245 *
246 INFO = 0
247 *
248 * ==== Set up job flags for ILAENV. ====
249 *
250 IF( WANTT ) THEN
251 JBCMPZ( 1: 1 ) = 'S'
252 ELSE
253 JBCMPZ( 1: 1 ) = 'E'
254 END IF
255 IF( WANTZ ) THEN
256 JBCMPZ( 2: 2 ) = 'V'
257 ELSE
258 JBCMPZ( 2: 2 ) = 'N'
259 END IF
260 *
261 * ==== NWR = recommended deflation window size. At this
262 * . point, N .GT. NTINY = 11, so there is enough
263 * . subdiagonal workspace for NWR.GE.2 as required.
264 * . (In fact, there is enough subdiagonal space for
265 * . NWR.GE.3.) ====
266 *
267 NWR = ILAENV( 13, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
268 NWR = MAX( 2, NWR )
269 NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
270 *
271 * ==== NSR = recommended number of simultaneous shifts.
272 * . At this point N .GT. NTINY = 11, so there is at
273 * . enough subdiagonal workspace for NSR to be even
274 * . and greater than or equal to two as required. ====
275 *
276 NSR = ILAENV( 15, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
277 NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
278 NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
279 *
280 * ==== Estimate optimal workspace ====
281 *
282 * ==== Workspace query call to DLAQR3 ====
283 *
284 CALL DLAQR3( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
285 $ IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH,
286 $ N, H, LDH, WORK, -1 )
287 *
288 * ==== Optimal workspace = MAX(DLAQR5, DLAQR3) ====
289 *
290 LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
291 *
292 * ==== Quick return in case of workspace query. ====
293 *
294 IF( LWORK.EQ.-1 ) THEN
295 WORK( 1 ) = DBLE( LWKOPT )
296 RETURN
297 END IF
298 *
299 * ==== DLAHQR/DLAQR0 crossover point ====
300 *
301 NMIN = ILAENV( 12, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
302 NMIN = MAX( NTINY, NMIN )
303 *
304 * ==== Nibble crossover point ====
305 *
306 NIBBLE = ILAENV( 14, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
307 NIBBLE = MAX( 0, NIBBLE )
308 *
309 * ==== Accumulate reflections during ttswp? Use block
310 * . 2-by-2 structure during matrix-matrix multiply? ====
311 *
312 KACC22 = ILAENV( 16, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
313 KACC22 = MAX( 0, KACC22 )
314 KACC22 = MIN( 2, KACC22 )
315 *
316 * ==== NWMAX = the largest possible deflation window for
317 * . which there is sufficient workspace. ====
318 *
319 NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
320 NW = NWMAX
321 *
322 * ==== NSMAX = the Largest number of simultaneous shifts
323 * . for which there is sufficient workspace. ====
324 *
325 NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
326 NSMAX = NSMAX - MOD( NSMAX, 2 )
327 *
328 * ==== NDFL: an iteration count restarted at deflation. ====
329 *
330 NDFL = 1
331 *
332 * ==== ITMAX = iteration limit ====
333 *
334 ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
335 *
336 * ==== Last row and column in the active block ====
337 *
338 KBOT = IHI
339 *
340 * ==== Main Loop ====
341 *
342 DO 80 IT = 1, ITMAX
343 *
344 * ==== Done when KBOT falls below ILO ====
345 *
346 IF( KBOT.LT.ILO )
347 $ GO TO 90
348 *
349 * ==== Locate active block ====
350 *
351 DO 10 K = KBOT, ILO + 1, -1
352 IF( H( K, K-1 ).EQ.ZERO )
353 $ GO TO 20
354 10 CONTINUE
355 K = ILO
356 20 CONTINUE
357 KTOP = K
358 *
359 * ==== Select deflation window size:
360 * . Typical Case:
361 * . If possible and advisable, nibble the entire
362 * . active block. If not, use size MIN(NWR,NWMAX)
363 * . or MIN(NWR+1,NWMAX) depending upon which has
364 * . the smaller corresponding subdiagonal entry
365 * . (a heuristic).
366 * .
367 * . Exceptional Case:
368 * . If there have been no deflations in KEXNW or
369 * . more iterations, then vary the deflation window
370 * . size. At first, because, larger windows are,
371 * . in general, more powerful than smaller ones,
372 * . rapidly increase the window to the maximum possible.
373 * . Then, gradually reduce the window size. ====
374 *
375 NH = KBOT - KTOP + 1
376 NWUPBD = MIN( NH, NWMAX )
377 IF( NDFL.LT.KEXNW ) THEN
378 NW = MIN( NWUPBD, NWR )
379 ELSE
380 NW = MIN( NWUPBD, 2*NW )
381 END IF
382 IF( NW.LT.NWMAX ) THEN
383 IF( NW.GE.NH-1 ) THEN
384 NW = NH
385 ELSE
386 KWTOP = KBOT - NW + 1
387 IF( ABS( H( KWTOP, KWTOP-1 ) ).GT.
388 $ ABS( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
389 END IF
390 END IF
391 IF( NDFL.LT.KEXNW ) THEN
392 NDEC = -1
393 ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
394 NDEC = NDEC + 1
395 IF( NW-NDEC.LT.2 )
396 $ NDEC = 0
397 NW = NW - NDEC
398 END IF
399 *
400 * ==== Aggressive early deflation:
401 * . split workspace under the subdiagonal into
402 * . - an nw-by-nw work array V in the lower
403 * . left-hand-corner,
404 * . - an NW-by-at-least-NW-but-more-is-better
405 * . (NW-by-NHO) horizontal work array along
406 * . the bottom edge,
407 * . - an at-least-NW-but-more-is-better (NHV-by-NW)
408 * . vertical work array along the left-hand-edge.
409 * . ====
410 *
411 KV = N - NW + 1
412 KT = NW + 1
413 NHO = ( N-NW-1 ) - KT + 1
414 KWV = NW + 2
415 NVE = ( N-NW ) - KWV + 1
416 *
417 * ==== Aggressive early deflation ====
418 *
419 CALL DLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
420 $ IHIZ, Z, LDZ, LS, LD, WR, WI, H( KV, 1 ), LDH,
421 $ NHO, H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH,
422 $ WORK, LWORK )
423 *
424 * ==== Adjust KBOT accounting for new deflations. ====
425 *
426 KBOT = KBOT - LD
427 *
428 * ==== KS points to the shifts. ====
429 *
430 KS = KBOT - LS + 1
431 *
432 * ==== Skip an expensive QR sweep if there is a (partly
433 * . heuristic) reason to expect that many eigenvalues
434 * . will deflate without it. Here, the QR sweep is
435 * . skipped if many eigenvalues have just been deflated
436 * . or if the remaining active block is small.
437 *
438 IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
439 $ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
440 *
441 * ==== NS = nominal number of simultaneous shifts.
442 * . This may be lowered (slightly) if DLAQR3
443 * . did not provide that many shifts. ====
444 *
445 NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
446 NS = NS - MOD( NS, 2 )
447 *
448 * ==== If there have been no deflations
449 * . in a multiple of KEXSH iterations,
450 * . then try exceptional shifts.
451 * . Otherwise use shifts provided by
452 * . DLAQR3 above or from the eigenvalues
453 * . of a trailing principal submatrix. ====
454 *
455 IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
456 KS = KBOT - NS + 1
457 DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2
458 SS = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
459 AA = WILK1*SS + H( I, I )
460 BB = SS
461 CC = WILK2*SS
462 DD = AA
463 CALL DLANV2( AA, BB, CC, DD, WR( I-1 ), WI( I-1 ),
464 $ WR( I ), WI( I ), CS, SN )
465 30 CONTINUE
466 IF( KS.EQ.KTOP ) THEN
467 WR( KS+1 ) = H( KS+1, KS+1 )
468 WI( KS+1 ) = ZERO
469 WR( KS ) = WR( KS+1 )
470 WI( KS ) = WI( KS+1 )
471 END IF
472 ELSE
473 *
474 * ==== Got NS/2 or fewer shifts? Use DLAQR4 or
475 * . DLAHQR on a trailing principal submatrix to
476 * . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
477 * . there is enough space below the subdiagonal
478 * . to fit an NS-by-NS scratch array.) ====
479 *
480 IF( KBOT-KS+1.LE.NS / 2 ) THEN
481 KS = KBOT - NS + 1
482 KT = N - NS + 1
483 CALL DLACPY( 'A', NS, NS, H( KS, KS ), LDH,
484 $ H( KT, 1 ), LDH )
485 IF( NS.GT.NMIN ) THEN
486 CALL DLAQR4( .false., .false., NS, 1, NS,
487 $ H( KT, 1 ), LDH, WR( KS ),
488 $ WI( KS ), 1, 1, ZDUM, 1, WORK,
489 $ LWORK, INF )
490 ELSE
491 CALL DLAHQR( .false., .false., NS, 1, NS,
492 $ H( KT, 1 ), LDH, WR( KS ),
493 $ WI( KS ), 1, 1, ZDUM, 1, INF )
494 END IF
495 KS = KS + INF
496 *
497 * ==== In case of a rare QR failure use
498 * . eigenvalues of the trailing 2-by-2
499 * . principal submatrix. ====
500 *
501 IF( KS.GE.KBOT ) THEN
502 AA = H( KBOT-1, KBOT-1 )
503 CC = H( KBOT, KBOT-1 )
504 BB = H( KBOT-1, KBOT )
505 DD = H( KBOT, KBOT )
506 CALL DLANV2( AA, BB, CC, DD, WR( KBOT-1 ),
507 $ WI( KBOT-1 ), WR( KBOT ),
508 $ WI( KBOT ), CS, SN )
509 KS = KBOT - 1
510 END IF
511 END IF
512 *
513 IF( KBOT-KS+1.GT.NS ) THEN
514 *
515 * ==== Sort the shifts (Helps a little)
516 * . Bubble sort keeps complex conjugate
517 * . pairs together. ====
518 *
519 SORTED = .false.
520 DO 50 K = KBOT, KS + 1, -1
521 IF( SORTED )
522 $ GO TO 60
523 SORTED = .true.
524 DO 40 I = KS, K - 1
525 IF( ABS( WR( I ) )+ABS( WI( I ) ).LT.
526 $ ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN
527 SORTED = .false.
528 *
529 SWAP = WR( I )
530 WR( I ) = WR( I+1 )
531 WR( I+1 ) = SWAP
532 *
533 SWAP = WI( I )
534 WI( I ) = WI( I+1 )
535 WI( I+1 ) = SWAP
536 END IF
537 40 CONTINUE
538 50 CONTINUE
539 60 CONTINUE
540 END IF
541 *
542 * ==== Shuffle shifts into pairs of real shifts
543 * . and pairs of complex conjugate shifts
544 * . assuming complex conjugate shifts are
545 * . already adjacent to one another. (Yes,
546 * . they are.) ====
547 *
548 DO 70 I = KBOT, KS + 2, -2
549 IF( WI( I ).NE.-WI( I-1 ) ) THEN
550 *
551 SWAP = WR( I )
552 WR( I ) = WR( I-1 )
553 WR( I-1 ) = WR( I-2 )
554 WR( I-2 ) = SWAP
555 *
556 SWAP = WI( I )
557 WI( I ) = WI( I-1 )
558 WI( I-1 ) = WI( I-2 )
559 WI( I-2 ) = SWAP
560 END IF
561 70 CONTINUE
562 END IF
563 *
564 * ==== If there are only two shifts and both are
565 * . real, then use only one. ====
566 *
567 IF( KBOT-KS+1.EQ.2 ) THEN
568 IF( WI( KBOT ).EQ.ZERO ) THEN
569 IF( ABS( WR( KBOT )-H( KBOT, KBOT ) ).LT.
570 $ ABS( WR( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
571 WR( KBOT-1 ) = WR( KBOT )
572 ELSE
573 WR( KBOT ) = WR( KBOT-1 )
574 END IF
575 END IF
576 END IF
577 *
578 * ==== Use up to NS of the the smallest magnatiude
579 * . shifts. If there aren't NS shifts available,
580 * . then use them all, possibly dropping one to
581 * . make the number of shifts even. ====
582 *
583 NS = MIN( NS, KBOT-KS+1 )
584 NS = NS - MOD( NS, 2 )
585 KS = KBOT - NS + 1
586 *
587 * ==== Small-bulge multi-shift QR sweep:
588 * . split workspace under the subdiagonal into
589 * . - a KDU-by-KDU work array U in the lower
590 * . left-hand-corner,
591 * . - a KDU-by-at-least-KDU-but-more-is-better
592 * . (KDU-by-NHo) horizontal work array WH along
593 * . the bottom edge,
594 * . - and an at-least-KDU-but-more-is-better-by-KDU
595 * . (NVE-by-KDU) vertical work WV arrow along
596 * . the left-hand-edge. ====
597 *
598 KDU = 3*NS - 3
599 KU = N - KDU + 1
600 KWH = KDU + 1
601 NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
602 KWV = KDU + 4
603 NVE = N - KDU - KWV + 1
604 *
605 * ==== Small-bulge multi-shift QR sweep ====
606 *
607 CALL DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
608 $ WR( KS ), WI( KS ), H, LDH, ILOZ, IHIZ, Z,
609 $ LDZ, WORK, 3, H( KU, 1 ), LDH, NVE,
610 $ H( KWV, 1 ), LDH, NHO, H( KU, KWH ), LDH )
611 END IF
612 *
613 * ==== Note progress (or the lack of it). ====
614 *
615 IF( LD.GT.0 ) THEN
616 NDFL = 1
617 ELSE
618 NDFL = NDFL + 1
619 END IF
620 *
621 * ==== End of main loop ====
622 80 CONTINUE
623 *
624 * ==== Iteration limit exceeded. Set INFO to show where
625 * . the problem occurred and exit. ====
626 *
627 INFO = KBOT
628 90 CONTINUE
629 END IF
630 *
631 * ==== Return the optimal value of LWORK. ====
632 *
633 WORK( 1 ) = DBLE( LWKOPT )
634 *
635 * ==== End of DLAQR0 ====
636 *
637 END