1 SUBROUTINE DLAQR4( 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 * This subroutine implements one level of recursion for DLAQR0.
18 * It is a complete implementation of the small bulge multi-shift
19 * QR algorithm. It may be called by DLAQR0 and, for large enough
20 * deflation window size, it may be called by DLAQR3. This
21 * subroutine is identical to DLAQR0 except that it calls DLAQR2
22 * instead of DLAQR3.
23 *
24 * Purpose
25 * =======
26 *
27 * DLAQR4 computes the eigenvalues of a Hessenberg matrix H
28 * and, optionally, the matrices T and Z from the Schur decomposition
29 * H = Z T Z**T, where T is an upper quasi-triangular matrix (the
30 * Schur form), and Z is the orthogonal matrix of Schur vectors.
31 *
32 * Optionally Z may be postmultiplied into an input orthogonal
33 * matrix Q so that this routine can give the Schur factorization
34 * of a matrix A which has been reduced to the Hessenberg form H
35 * by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
36 *
37 * Arguments
38 * =========
39 *
40 * WANTT (input) LOGICAL
41 * = .TRUE. : the full Schur form T is required;
42 * = .FALSE.: only eigenvalues are required.
43 *
44 * WANTZ (input) LOGICAL
45 * = .TRUE. : the matrix of Schur vectors Z is required;
46 * = .FALSE.: Schur vectors are not required.
47 *
48 * N (input) INTEGER
49 * The order of the matrix H. N .GE. 0.
50 *
51 * ILO (input) INTEGER
52 * IHI (input) INTEGER
53 * It is assumed that H is already upper triangular in rows
54 * and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
55 * H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
56 * previous call to DGEBAL, and then passed to DGEHRD when the
57 * matrix output by DGEBAL is reduced to Hessenberg form.
58 * Otherwise, ILO and IHI should be set to 1 and N,
59 * respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
60 * If N = 0, then ILO = 1 and IHI = 0.
61 *
62 * H (input/output) DOUBLE PRECISION array, dimension (LDH,N)
63 * On entry, the upper Hessenberg matrix H.
64 * On exit, if INFO = 0 and WANTT is .TRUE., then H contains
65 * the upper quasi-triangular matrix T from the Schur
66 * decomposition (the Schur form); 2-by-2 diagonal blocks
67 * (corresponding to complex conjugate pairs of eigenvalues)
68 * are returned in standard form, with H(i,i) = H(i+1,i+1)
69 * and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
70 * .FALSE., then the contents of H are unspecified on exit.
71 * (The output value of H when INFO.GT.0 is given under the
72 * description of INFO below.)
73 *
74 * This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
75 * j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
76 *
77 * LDH (input) INTEGER
78 * The leading dimension of the array H. LDH .GE. max(1,N).
79 *
80 * WR (output) DOUBLE PRECISION array, dimension (IHI)
81 * WI (output) DOUBLE PRECISION array, dimension (IHI)
82 * The real and imaginary parts, respectively, of the computed
83 * eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
84 * and WI(ILO:IHI). If two eigenvalues are computed as a
85 * complex conjugate pair, they are stored in consecutive
86 * elements of WR and WI, say the i-th and (i+1)th, with
87 * WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
88 * the eigenvalues are stored in the same order as on the
89 * diagonal of the Schur form returned in H, with
90 * WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
91 * block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
92 * WI(i+1) = -WI(i).
93 *
94 * ILOZ (input) INTEGER
95 * IHIZ (input) INTEGER
96 * Specify the rows of Z to which transformations must be
97 * applied if WANTZ is .TRUE..
98 * 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
99 *
100 * Z (input/output) DOUBLE PRECISION array, dimension (LDZ,IHI)
101 * If WANTZ is .FALSE., then Z is not referenced.
102 * If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
103 * replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
104 * orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
105 * (The output value of Z when INFO.GT.0 is given under
106 * the description of INFO below.)
107 *
108 * LDZ (input) INTEGER
109 * The leading dimension of the array Z. if WANTZ is .TRUE.
110 * then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.
111 *
112 * WORK (workspace/output) DOUBLE PRECISION array, dimension LWORK
113 * On exit, if LWORK = -1, WORK(1) returns an estimate of
114 * the optimal value for LWORK.
115 *
116 * LWORK (input) INTEGER
117 * The dimension of the array WORK. LWORK .GE. max(1,N)
118 * is sufficient, but LWORK typically as large as 6*N may
119 * be required for optimal performance. A workspace query
120 * to determine the optimal workspace size is recommended.
121 *
122 * If LWORK = -1, then DLAQR4 does a workspace query.
123 * In this case, DLAQR4 checks the input parameters and
124 * estimates the optimal workspace size for the given
125 * values of N, ILO and IHI. The estimate is returned
126 * in WORK(1). No error message related to LWORK is
127 * issued by XERBLA. Neither H nor Z are accessed.
128 *
129 *
130 * INFO (output) INTEGER
131 * = 0: successful exit
132 * .GT. 0: if INFO = i, DLAQR4 failed to compute all of
133 * the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
134 * and WI contain those eigenvalues which have been
135 * successfully computed. (Failures are rare.)
136 *
137 * If INFO .GT. 0 and WANT is .FALSE., then on exit,
138 * the remaining unconverged eigenvalues are the eigen-
139 * values of the upper Hessenberg matrix rows and
140 * columns ILO through INFO of the final, output
141 * value of H.
142 *
143 * If INFO .GT. 0 and WANTT is .TRUE., then on exit
144 *
145 * (*) (initial value of H)*U = U*(final value of H)
146 *
147 * where U is an orthogonal matrix. The final
148 * value of H is upper Hessenberg and quasi-triangular
149 * in rows and columns INFO+1 through IHI.
150 *
151 * If INFO .GT. 0 and WANTZ is .TRUE., then on exit
152 *
153 * (final value of Z(ILO:IHI,ILOZ:IHIZ)
154 * = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
155 *
156 * where U is the orthogonal matrix in (*) (regard-
157 * less of the value of WANTT.)
158 *
159 * If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
160 * accessed.
161 *
162 * ================================================================
163 * Based on contributions by
164 * Karen Braman and Ralph Byers, Department of Mathematics,
165 * University of Kansas, USA
166 *
167 * ================================================================
168 * References:
169 * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
170 * Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
171 * Performance, SIAM Journal of Matrix Analysis, volume 23, pages
172 * 929--947, 2002.
173 *
174 * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
175 * Algorithm Part II: Aggressive Early Deflation, SIAM Journal
176 * of Matrix Analysis, volume 23, pages 948--973, 2002.
177 *
178 * ================================================================
179 * .. Parameters ..
180 *
181 * ==== Matrices of order NTINY or smaller must be processed by
182 * . DLAHQR because of insufficient subdiagonal scratch space.
183 * . (This is a hard limit.) ====
184 INTEGER NTINY
185 PARAMETER ( NTINY = 11 )
186 *
187 * ==== Exceptional deflation windows: try to cure rare
188 * . slow convergence by varying the size of the
189 * . deflation window after KEXNW iterations. ====
190 INTEGER KEXNW
191 PARAMETER ( KEXNW = 5 )
192 *
193 * ==== Exceptional shifts: try to cure rare slow convergence
194 * . with ad-hoc exceptional shifts every KEXSH iterations.
195 * . ====
196 INTEGER KEXSH
197 PARAMETER ( KEXSH = 6 )
198 *
199 * ==== The constants WILK1 and WILK2 are used to form the
200 * . exceptional shifts. ====
201 DOUBLE PRECISION WILK1, WILK2
202 PARAMETER ( WILK1 = 0.75d0, WILK2 = -0.4375d0 )
203 DOUBLE PRECISION ZERO, ONE
204 PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
205 * ..
206 * .. Local Scalars ..
207 DOUBLE PRECISION AA, BB, CC, CS, DD, SN, SS, SWAP
208 INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
209 $ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
210 $ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
211 $ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
212 LOGICAL SORTED
213 CHARACTER JBCMPZ*2
214 * ..
215 * .. External Functions ..
216 INTEGER ILAENV
217 EXTERNAL ILAENV
218 * ..
219 * .. Local Arrays ..
220 DOUBLE PRECISION ZDUM( 1, 1 )
221 * ..
222 * .. External Subroutines ..
223 EXTERNAL DLACPY, DLAHQR, DLANV2, DLAQR2, DLAQR5
224 * ..
225 * .. Intrinsic Functions ..
226 INTRINSIC ABS, DBLE, INT, MAX, MIN, MOD
227 * ..
228 * .. Executable Statements ..
229 INFO = 0
230
231 *
232 * ==== Quick return for N = 0: nothing to do. ====
233 *
234 IF( N.EQ.0 ) THEN
235 WORK( 1 ) = ONE
236 RETURN
237 END IF
238 *
239 IF( N.LE.NTINY ) THEN
240 *
241 * ==== Tiny matrices must use DLAHQR. ====
242 *
243 LWKOPT = 1
244 IF( LWORK.NE.-1 )
245 $ CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
246 $ ILOZ, IHIZ, Z, LDZ, INFO )
247 ELSE
248 *
249 * ==== Use small bulge multi-shift QR with aggressive early
250 * . deflation on larger-than-tiny matrices. ====
251 *
252 * ==== Hope for the best. ====
253 *
254 INFO = 0
255 *
256 * ==== Set up job flags for ILAENV. ====
257 *
258 IF( WANTT ) THEN
259 JBCMPZ( 1: 1 ) = 'S'
260 ELSE
261 JBCMPZ( 1: 1 ) = 'E'
262 END IF
263 IF( WANTZ ) THEN
264 JBCMPZ( 2: 2 ) = 'V'
265 ELSE
266 JBCMPZ( 2: 2 ) = 'N'
267 END IF
268 *
269 * ==== NWR = recommended deflation window size. At this
270 * . point, N .GT. NTINY = 11, so there is enough
271 * . subdiagonal workspace for NWR.GE.2 as required.
272 * . (In fact, there is enough subdiagonal space for
273 * . NWR.GE.3.) ====
274 *
275 NWR = ILAENV( 13, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
276 NWR = MAX( 2, NWR )
277 NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
278 *
279 * ==== NSR = recommended number of simultaneous shifts.
280 * . At this point N .GT. NTINY = 11, so there is at
281 * . enough subdiagonal workspace for NSR to be even
282 * . and greater than or equal to two as required. ====
283 *
284 NSR = ILAENV( 15, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
285 NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
286 NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
287 *
288 * ==== Estimate optimal workspace ====
289 *
290 * ==== Workspace query call to DLAQR2 ====
291 *
292 CALL DLAQR2( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
293 $ IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH,
294 $ N, H, LDH, WORK, -1 )
295 *
296 * ==== Optimal workspace = MAX(DLAQR5, DLAQR2) ====
297 *
298 LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
299 *
300 * ==== Quick return in case of workspace query. ====
301 *
302 IF( LWORK.EQ.-1 ) THEN
303 WORK( 1 ) = DBLE( LWKOPT )
304 RETURN
305 END IF
306 *
307 * ==== DLAHQR/DLAQR0 crossover point ====
308 *
309 NMIN = ILAENV( 12, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
310 NMIN = MAX( NTINY, NMIN )
311 *
312 * ==== Nibble crossover point ====
313 *
314 NIBBLE = ILAENV( 14, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
315 NIBBLE = MAX( 0, NIBBLE )
316 *
317 * ==== Accumulate reflections during ttswp? Use block
318 * . 2-by-2 structure during matrix-matrix multiply? ====
319 *
320 KACC22 = ILAENV( 16, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
321 KACC22 = MAX( 0, KACC22 )
322 KACC22 = MIN( 2, KACC22 )
323 *
324 * ==== NWMAX = the largest possible deflation window for
325 * . which there is sufficient workspace. ====
326 *
327 NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
328 NW = NWMAX
329 *
330 * ==== NSMAX = the Largest number of simultaneous shifts
331 * . for which there is sufficient workspace. ====
332 *
333 NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
334 NSMAX = NSMAX - MOD( NSMAX, 2 )
335 *
336 * ==== NDFL: an iteration count restarted at deflation. ====
337 *
338 NDFL = 1
339 *
340 * ==== ITMAX = iteration limit ====
341 *
342 ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
343 *
344 * ==== Last row and column in the active block ====
345 *
346 KBOT = IHI
347 *
348 * ==== Main Loop ====
349 *
350 DO 80 IT = 1, ITMAX
351 *
352 * ==== Done when KBOT falls below ILO ====
353 *
354 IF( KBOT.LT.ILO )
355 $ GO TO 90
356 *
357 * ==== Locate active block ====
358 *
359 DO 10 K = KBOT, ILO + 1, -1
360 IF( H( K, K-1 ).EQ.ZERO )
361 $ GO TO 20
362 10 CONTINUE
363 K = ILO
364 20 CONTINUE
365 KTOP = K
366 *
367 * ==== Select deflation window size:
368 * . Typical Case:
369 * . If possible and advisable, nibble the entire
370 * . active block. If not, use size MIN(NWR,NWMAX)
371 * . or MIN(NWR+1,NWMAX) depending upon which has
372 * . the smaller corresponding subdiagonal entry
373 * . (a heuristic).
374 * .
375 * . Exceptional Case:
376 * . If there have been no deflations in KEXNW or
377 * . more iterations, then vary the deflation window
378 * . size. At first, because, larger windows are,
379 * . in general, more powerful than smaller ones,
380 * . rapidly increase the window to the maximum possible.
381 * . Then, gradually reduce the window size. ====
382 *
383 NH = KBOT - KTOP + 1
384 NWUPBD = MIN( NH, NWMAX )
385 IF( NDFL.LT.KEXNW ) THEN
386 NW = MIN( NWUPBD, NWR )
387 ELSE
388 NW = MIN( NWUPBD, 2*NW )
389 END IF
390 IF( NW.LT.NWMAX ) THEN
391 IF( NW.GE.NH-1 ) THEN
392 NW = NH
393 ELSE
394 KWTOP = KBOT - NW + 1
395 IF( ABS( H( KWTOP, KWTOP-1 ) ).GT.
396 $ ABS( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
397 END IF
398 END IF
399 IF( NDFL.LT.KEXNW ) THEN
400 NDEC = -1
401 ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
402 NDEC = NDEC + 1
403 IF( NW-NDEC.LT.2 )
404 $ NDEC = 0
405 NW = NW - NDEC
406 END IF
407 *
408 * ==== Aggressive early deflation:
409 * . split workspace under the subdiagonal into
410 * . - an nw-by-nw work array V in the lower
411 * . left-hand-corner,
412 * . - an NW-by-at-least-NW-but-more-is-better
413 * . (NW-by-NHO) horizontal work array along
414 * . the bottom edge,
415 * . - an at-least-NW-but-more-is-better (NHV-by-NW)
416 * . vertical work array along the left-hand-edge.
417 * . ====
418 *
419 KV = N - NW + 1
420 KT = NW + 1
421 NHO = ( N-NW-1 ) - KT + 1
422 KWV = NW + 2
423 NVE = ( N-NW ) - KWV + 1
424 *
425 * ==== Aggressive early deflation ====
426 *
427 CALL DLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
428 $ IHIZ, Z, LDZ, LS, LD, WR, WI, H( KV, 1 ), LDH,
429 $ NHO, H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH,
430 $ WORK, LWORK )
431 *
432 * ==== Adjust KBOT accounting for new deflations. ====
433 *
434 KBOT = KBOT - LD
435 *
436 * ==== KS points to the shifts. ====
437 *
438 KS = KBOT - LS + 1
439 *
440 * ==== Skip an expensive QR sweep if there is a (partly
441 * . heuristic) reason to expect that many eigenvalues
442 * . will deflate without it. Here, the QR sweep is
443 * . skipped if many eigenvalues have just been deflated
444 * . or if the remaining active block is small.
445 *
446 IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
447 $ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
448 *
449 * ==== NS = nominal number of simultaneous shifts.
450 * . This may be lowered (slightly) if DLAQR2
451 * . did not provide that many shifts. ====
452 *
453 NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
454 NS = NS - MOD( NS, 2 )
455 *
456 * ==== If there have been no deflations
457 * . in a multiple of KEXSH iterations,
458 * . then try exceptional shifts.
459 * . Otherwise use shifts provided by
460 * . DLAQR2 above or from the eigenvalues
461 * . of a trailing principal submatrix. ====
462 *
463 IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
464 KS = KBOT - NS + 1
465 DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2
466 SS = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
467 AA = WILK1*SS + H( I, I )
468 BB = SS
469 CC = WILK2*SS
470 DD = AA
471 CALL DLANV2( AA, BB, CC, DD, WR( I-1 ), WI( I-1 ),
472 $ WR( I ), WI( I ), CS, SN )
473 30 CONTINUE
474 IF( KS.EQ.KTOP ) THEN
475 WR( KS+1 ) = H( KS+1, KS+1 )
476 WI( KS+1 ) = ZERO
477 WR( KS ) = WR( KS+1 )
478 WI( KS ) = WI( KS+1 )
479 END IF
480 ELSE
481 *
482 * ==== Got NS/2 or fewer shifts? Use DLAHQR
483 * . on a trailing principal submatrix to
484 * . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
485 * . there is enough space below the subdiagonal
486 * . to fit an NS-by-NS scratch array.) ====
487 *
488 IF( KBOT-KS+1.LE.NS / 2 ) THEN
489 KS = KBOT - NS + 1
490 KT = N - NS + 1
491 CALL DLACPY( 'A', NS, NS, H( KS, KS ), LDH,
492 $ H( KT, 1 ), LDH )
493 CALL DLAHQR( .false., .false., NS, 1, NS,
494 $ H( KT, 1 ), LDH, WR( KS ), WI( KS ),
495 $ 1, 1, ZDUM, 1, INF )
496 KS = KS + INF
497 *
498 * ==== In case of a rare QR failure use
499 * . eigenvalues of the trailing 2-by-2
500 * . principal submatrix. ====
501 *
502 IF( KS.GE.KBOT ) THEN
503 AA = H( KBOT-1, KBOT-1 )
504 CC = H( KBOT, KBOT-1 )
505 BB = H( KBOT-1, KBOT )
506 DD = H( KBOT, KBOT )
507 CALL DLANV2( AA, BB, CC, DD, WR( KBOT-1 ),
508 $ WI( KBOT-1 ), WR( KBOT ),
509 $ WI( KBOT ), CS, SN )
510 KS = KBOT - 1
511 END IF
512 END IF
513 *
514 IF( KBOT-KS+1.GT.NS ) THEN
515 *
516 * ==== Sort the shifts (Helps a little)
517 * . Bubble sort keeps complex conjugate
518 * . pairs together. ====
519 *
520 SORTED = .false.
521 DO 50 K = KBOT, KS + 1, -1
522 IF( SORTED )
523 $ GO TO 60
524 SORTED = .true.
525 DO 40 I = KS, K - 1
526 IF( ABS( WR( I ) )+ABS( WI( I ) ).LT.
527 $ ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN
528 SORTED = .false.
529 *
530 SWAP = WR( I )
531 WR( I ) = WR( I+1 )
532 WR( I+1 ) = SWAP
533 *
534 SWAP = WI( I )
535 WI( I ) = WI( I+1 )
536 WI( I+1 ) = SWAP
537 END IF
538 40 CONTINUE
539 50 CONTINUE
540 60 CONTINUE
541 END IF
542 *
543 * ==== Shuffle shifts into pairs of real shifts
544 * . and pairs of complex conjugate shifts
545 * . assuming complex conjugate shifts are
546 * . already adjacent to one another. (Yes,
547 * . they are.) ====
548 *
549 DO 70 I = KBOT, KS + 2, -2
550 IF( WI( I ).NE.-WI( I-1 ) ) THEN
551 *
552 SWAP = WR( I )
553 WR( I ) = WR( I-1 )
554 WR( I-1 ) = WR( I-2 )
555 WR( I-2 ) = SWAP
556 *
557 SWAP = WI( I )
558 WI( I ) = WI( I-1 )
559 WI( I-1 ) = WI( I-2 )
560 WI( I-2 ) = SWAP
561 END IF
562 70 CONTINUE
563 END IF
564 *
565 * ==== If there are only two shifts and both are
566 * . real, then use only one. ====
567 *
568 IF( KBOT-KS+1.EQ.2 ) THEN
569 IF( WI( KBOT ).EQ.ZERO ) THEN
570 IF( ABS( WR( KBOT )-H( KBOT, KBOT ) ).LT.
571 $ ABS( WR( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
572 WR( KBOT-1 ) = WR( KBOT )
573 ELSE
574 WR( KBOT ) = WR( KBOT-1 )
575 END IF
576 END IF
577 END IF
578 *
579 * ==== Use up to NS of the the smallest magnatiude
580 * . shifts. If there aren't NS shifts available,
581 * . then use them all, possibly dropping one to
582 * . make the number of shifts even. ====
583 *
584 NS = MIN( NS, KBOT-KS+1 )
585 NS = NS - MOD( NS, 2 )
586 KS = KBOT - NS + 1
587 *
588 * ==== Small-bulge multi-shift QR sweep:
589 * . split workspace under the subdiagonal into
590 * . - a KDU-by-KDU work array U in the lower
591 * . left-hand-corner,
592 * . - a KDU-by-at-least-KDU-but-more-is-better
593 * . (KDU-by-NHo) horizontal work array WH along
594 * . the bottom edge,
595 * . - and an at-least-KDU-but-more-is-better-by-KDU
596 * . (NVE-by-KDU) vertical work WV arrow along
597 * . the left-hand-edge. ====
598 *
599 KDU = 3*NS - 3
600 KU = N - KDU + 1
601 KWH = KDU + 1
602 NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
603 KWV = KDU + 4
604 NVE = N - KDU - KWV + 1
605 *
606 * ==== Small-bulge multi-shift QR sweep ====
607 *
608 CALL DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
609 $ WR( KS ), WI( KS ), H, LDH, ILOZ, IHIZ, Z,
610 $ LDZ, WORK, 3, H( KU, 1 ), LDH, NVE,
611 $ H( KWV, 1 ), LDH, NHO, H( KU, KWH ), LDH )
612 END IF
613 *
614 * ==== Note progress (or the lack of it). ====
615 *
616 IF( LD.GT.0 ) THEN
617 NDFL = 1
618 ELSE
619 NDFL = NDFL + 1
620 END IF
621 *
622 * ==== End of main loop ====
623 80 CONTINUE
624 *
625 * ==== Iteration limit exceeded. Set INFO to show where
626 * . the problem occurred and exit. ====
627 *
628 INFO = KBOT
629 90 CONTINUE
630 END IF
631 *
632 * ==== Return the optimal value of LWORK. ====
633 *
634 WORK( 1 ) = DBLE( LWKOPT )
635 *
636 * ==== End of DLAQR4 ====
637 *
638 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 * This subroutine implements one level of recursion for DLAQR0.
18 * It is a complete implementation of the small bulge multi-shift
19 * QR algorithm. It may be called by DLAQR0 and, for large enough
20 * deflation window size, it may be called by DLAQR3. This
21 * subroutine is identical to DLAQR0 except that it calls DLAQR2
22 * instead of DLAQR3.
23 *
24 * Purpose
25 * =======
26 *
27 * DLAQR4 computes the eigenvalues of a Hessenberg matrix H
28 * and, optionally, the matrices T and Z from the Schur decomposition
29 * H = Z T Z**T, where T is an upper quasi-triangular matrix (the
30 * Schur form), and Z is the orthogonal matrix of Schur vectors.
31 *
32 * Optionally Z may be postmultiplied into an input orthogonal
33 * matrix Q so that this routine can give the Schur factorization
34 * of a matrix A which has been reduced to the Hessenberg form H
35 * by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
36 *
37 * Arguments
38 * =========
39 *
40 * WANTT (input) LOGICAL
41 * = .TRUE. : the full Schur form T is required;
42 * = .FALSE.: only eigenvalues are required.
43 *
44 * WANTZ (input) LOGICAL
45 * = .TRUE. : the matrix of Schur vectors Z is required;
46 * = .FALSE.: Schur vectors are not required.
47 *
48 * N (input) INTEGER
49 * The order of the matrix H. N .GE. 0.
50 *
51 * ILO (input) INTEGER
52 * IHI (input) INTEGER
53 * It is assumed that H is already upper triangular in rows
54 * and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
55 * H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
56 * previous call to DGEBAL, and then passed to DGEHRD when the
57 * matrix output by DGEBAL is reduced to Hessenberg form.
58 * Otherwise, ILO and IHI should be set to 1 and N,
59 * respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
60 * If N = 0, then ILO = 1 and IHI = 0.
61 *
62 * H (input/output) DOUBLE PRECISION array, dimension (LDH,N)
63 * On entry, the upper Hessenberg matrix H.
64 * On exit, if INFO = 0 and WANTT is .TRUE., then H contains
65 * the upper quasi-triangular matrix T from the Schur
66 * decomposition (the Schur form); 2-by-2 diagonal blocks
67 * (corresponding to complex conjugate pairs of eigenvalues)
68 * are returned in standard form, with H(i,i) = H(i+1,i+1)
69 * and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
70 * .FALSE., then the contents of H are unspecified on exit.
71 * (The output value of H when INFO.GT.0 is given under the
72 * description of INFO below.)
73 *
74 * This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
75 * j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
76 *
77 * LDH (input) INTEGER
78 * The leading dimension of the array H. LDH .GE. max(1,N).
79 *
80 * WR (output) DOUBLE PRECISION array, dimension (IHI)
81 * WI (output) DOUBLE PRECISION array, dimension (IHI)
82 * The real and imaginary parts, respectively, of the computed
83 * eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
84 * and WI(ILO:IHI). If two eigenvalues are computed as a
85 * complex conjugate pair, they are stored in consecutive
86 * elements of WR and WI, say the i-th and (i+1)th, with
87 * WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
88 * the eigenvalues are stored in the same order as on the
89 * diagonal of the Schur form returned in H, with
90 * WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
91 * block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
92 * WI(i+1) = -WI(i).
93 *
94 * ILOZ (input) INTEGER
95 * IHIZ (input) INTEGER
96 * Specify the rows of Z to which transformations must be
97 * applied if WANTZ is .TRUE..
98 * 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
99 *
100 * Z (input/output) DOUBLE PRECISION array, dimension (LDZ,IHI)
101 * If WANTZ is .FALSE., then Z is not referenced.
102 * If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
103 * replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
104 * orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
105 * (The output value of Z when INFO.GT.0 is given under
106 * the description of INFO below.)
107 *
108 * LDZ (input) INTEGER
109 * The leading dimension of the array Z. if WANTZ is .TRUE.
110 * then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.
111 *
112 * WORK (workspace/output) DOUBLE PRECISION array, dimension LWORK
113 * On exit, if LWORK = -1, WORK(1) returns an estimate of
114 * the optimal value for LWORK.
115 *
116 * LWORK (input) INTEGER
117 * The dimension of the array WORK. LWORK .GE. max(1,N)
118 * is sufficient, but LWORK typically as large as 6*N may
119 * be required for optimal performance. A workspace query
120 * to determine the optimal workspace size is recommended.
121 *
122 * If LWORK = -1, then DLAQR4 does a workspace query.
123 * In this case, DLAQR4 checks the input parameters and
124 * estimates the optimal workspace size for the given
125 * values of N, ILO and IHI. The estimate is returned
126 * in WORK(1). No error message related to LWORK is
127 * issued by XERBLA. Neither H nor Z are accessed.
128 *
129 *
130 * INFO (output) INTEGER
131 * = 0: successful exit
132 * .GT. 0: if INFO = i, DLAQR4 failed to compute all of
133 * the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
134 * and WI contain those eigenvalues which have been
135 * successfully computed. (Failures are rare.)
136 *
137 * If INFO .GT. 0 and WANT is .FALSE., then on exit,
138 * the remaining unconverged eigenvalues are the eigen-
139 * values of the upper Hessenberg matrix rows and
140 * columns ILO through INFO of the final, output
141 * value of H.
142 *
143 * If INFO .GT. 0 and WANTT is .TRUE., then on exit
144 *
145 * (*) (initial value of H)*U = U*(final value of H)
146 *
147 * where U is an orthogonal matrix. The final
148 * value of H is upper Hessenberg and quasi-triangular
149 * in rows and columns INFO+1 through IHI.
150 *
151 * If INFO .GT. 0 and WANTZ is .TRUE., then on exit
152 *
153 * (final value of Z(ILO:IHI,ILOZ:IHIZ)
154 * = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
155 *
156 * where U is the orthogonal matrix in (*) (regard-
157 * less of the value of WANTT.)
158 *
159 * If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
160 * accessed.
161 *
162 * ================================================================
163 * Based on contributions by
164 * Karen Braman and Ralph Byers, Department of Mathematics,
165 * University of Kansas, USA
166 *
167 * ================================================================
168 * References:
169 * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
170 * Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
171 * Performance, SIAM Journal of Matrix Analysis, volume 23, pages
172 * 929--947, 2002.
173 *
174 * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
175 * Algorithm Part II: Aggressive Early Deflation, SIAM Journal
176 * of Matrix Analysis, volume 23, pages 948--973, 2002.
177 *
178 * ================================================================
179 * .. Parameters ..
180 *
181 * ==== Matrices of order NTINY or smaller must be processed by
182 * . DLAHQR because of insufficient subdiagonal scratch space.
183 * . (This is a hard limit.) ====
184 INTEGER NTINY
185 PARAMETER ( NTINY = 11 )
186 *
187 * ==== Exceptional deflation windows: try to cure rare
188 * . slow convergence by varying the size of the
189 * . deflation window after KEXNW iterations. ====
190 INTEGER KEXNW
191 PARAMETER ( KEXNW = 5 )
192 *
193 * ==== Exceptional shifts: try to cure rare slow convergence
194 * . with ad-hoc exceptional shifts every KEXSH iterations.
195 * . ====
196 INTEGER KEXSH
197 PARAMETER ( KEXSH = 6 )
198 *
199 * ==== The constants WILK1 and WILK2 are used to form the
200 * . exceptional shifts. ====
201 DOUBLE PRECISION WILK1, WILK2
202 PARAMETER ( WILK1 = 0.75d0, WILK2 = -0.4375d0 )
203 DOUBLE PRECISION ZERO, ONE
204 PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
205 * ..
206 * .. Local Scalars ..
207 DOUBLE PRECISION AA, BB, CC, CS, DD, SN, SS, SWAP
208 INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
209 $ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
210 $ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
211 $ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
212 LOGICAL SORTED
213 CHARACTER JBCMPZ*2
214 * ..
215 * .. External Functions ..
216 INTEGER ILAENV
217 EXTERNAL ILAENV
218 * ..
219 * .. Local Arrays ..
220 DOUBLE PRECISION ZDUM( 1, 1 )
221 * ..
222 * .. External Subroutines ..
223 EXTERNAL DLACPY, DLAHQR, DLANV2, DLAQR2, DLAQR5
224 * ..
225 * .. Intrinsic Functions ..
226 INTRINSIC ABS, DBLE, INT, MAX, MIN, MOD
227 * ..
228 * .. Executable Statements ..
229 INFO = 0
230
231 *
232 * ==== Quick return for N = 0: nothing to do. ====
233 *
234 IF( N.EQ.0 ) THEN
235 WORK( 1 ) = ONE
236 RETURN
237 END IF
238 *
239 IF( N.LE.NTINY ) THEN
240 *
241 * ==== Tiny matrices must use DLAHQR. ====
242 *
243 LWKOPT = 1
244 IF( LWORK.NE.-1 )
245 $ CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
246 $ ILOZ, IHIZ, Z, LDZ, INFO )
247 ELSE
248 *
249 * ==== Use small bulge multi-shift QR with aggressive early
250 * . deflation on larger-than-tiny matrices. ====
251 *
252 * ==== Hope for the best. ====
253 *
254 INFO = 0
255 *
256 * ==== Set up job flags for ILAENV. ====
257 *
258 IF( WANTT ) THEN
259 JBCMPZ( 1: 1 ) = 'S'
260 ELSE
261 JBCMPZ( 1: 1 ) = 'E'
262 END IF
263 IF( WANTZ ) THEN
264 JBCMPZ( 2: 2 ) = 'V'
265 ELSE
266 JBCMPZ( 2: 2 ) = 'N'
267 END IF
268 *
269 * ==== NWR = recommended deflation window size. At this
270 * . point, N .GT. NTINY = 11, so there is enough
271 * . subdiagonal workspace for NWR.GE.2 as required.
272 * . (In fact, there is enough subdiagonal space for
273 * . NWR.GE.3.) ====
274 *
275 NWR = ILAENV( 13, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
276 NWR = MAX( 2, NWR )
277 NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
278 *
279 * ==== NSR = recommended number of simultaneous shifts.
280 * . At this point N .GT. NTINY = 11, so there is at
281 * . enough subdiagonal workspace for NSR to be even
282 * . and greater than or equal to two as required. ====
283 *
284 NSR = ILAENV( 15, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
285 NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
286 NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
287 *
288 * ==== Estimate optimal workspace ====
289 *
290 * ==== Workspace query call to DLAQR2 ====
291 *
292 CALL DLAQR2( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
293 $ IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH,
294 $ N, H, LDH, WORK, -1 )
295 *
296 * ==== Optimal workspace = MAX(DLAQR5, DLAQR2) ====
297 *
298 LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
299 *
300 * ==== Quick return in case of workspace query. ====
301 *
302 IF( LWORK.EQ.-1 ) THEN
303 WORK( 1 ) = DBLE( LWKOPT )
304 RETURN
305 END IF
306 *
307 * ==== DLAHQR/DLAQR0 crossover point ====
308 *
309 NMIN = ILAENV( 12, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
310 NMIN = MAX( NTINY, NMIN )
311 *
312 * ==== Nibble crossover point ====
313 *
314 NIBBLE = ILAENV( 14, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
315 NIBBLE = MAX( 0, NIBBLE )
316 *
317 * ==== Accumulate reflections during ttswp? Use block
318 * . 2-by-2 structure during matrix-matrix multiply? ====
319 *
320 KACC22 = ILAENV( 16, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
321 KACC22 = MAX( 0, KACC22 )
322 KACC22 = MIN( 2, KACC22 )
323 *
324 * ==== NWMAX = the largest possible deflation window for
325 * . which there is sufficient workspace. ====
326 *
327 NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
328 NW = NWMAX
329 *
330 * ==== NSMAX = the Largest number of simultaneous shifts
331 * . for which there is sufficient workspace. ====
332 *
333 NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
334 NSMAX = NSMAX - MOD( NSMAX, 2 )
335 *
336 * ==== NDFL: an iteration count restarted at deflation. ====
337 *
338 NDFL = 1
339 *
340 * ==== ITMAX = iteration limit ====
341 *
342 ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
343 *
344 * ==== Last row and column in the active block ====
345 *
346 KBOT = IHI
347 *
348 * ==== Main Loop ====
349 *
350 DO 80 IT = 1, ITMAX
351 *
352 * ==== Done when KBOT falls below ILO ====
353 *
354 IF( KBOT.LT.ILO )
355 $ GO TO 90
356 *
357 * ==== Locate active block ====
358 *
359 DO 10 K = KBOT, ILO + 1, -1
360 IF( H( K, K-1 ).EQ.ZERO )
361 $ GO TO 20
362 10 CONTINUE
363 K = ILO
364 20 CONTINUE
365 KTOP = K
366 *
367 * ==== Select deflation window size:
368 * . Typical Case:
369 * . If possible and advisable, nibble the entire
370 * . active block. If not, use size MIN(NWR,NWMAX)
371 * . or MIN(NWR+1,NWMAX) depending upon which has
372 * . the smaller corresponding subdiagonal entry
373 * . (a heuristic).
374 * .
375 * . Exceptional Case:
376 * . If there have been no deflations in KEXNW or
377 * . more iterations, then vary the deflation window
378 * . size. At first, because, larger windows are,
379 * . in general, more powerful than smaller ones,
380 * . rapidly increase the window to the maximum possible.
381 * . Then, gradually reduce the window size. ====
382 *
383 NH = KBOT - KTOP + 1
384 NWUPBD = MIN( NH, NWMAX )
385 IF( NDFL.LT.KEXNW ) THEN
386 NW = MIN( NWUPBD, NWR )
387 ELSE
388 NW = MIN( NWUPBD, 2*NW )
389 END IF
390 IF( NW.LT.NWMAX ) THEN
391 IF( NW.GE.NH-1 ) THEN
392 NW = NH
393 ELSE
394 KWTOP = KBOT - NW + 1
395 IF( ABS( H( KWTOP, KWTOP-1 ) ).GT.
396 $ ABS( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
397 END IF
398 END IF
399 IF( NDFL.LT.KEXNW ) THEN
400 NDEC = -1
401 ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
402 NDEC = NDEC + 1
403 IF( NW-NDEC.LT.2 )
404 $ NDEC = 0
405 NW = NW - NDEC
406 END IF
407 *
408 * ==== Aggressive early deflation:
409 * . split workspace under the subdiagonal into
410 * . - an nw-by-nw work array V in the lower
411 * . left-hand-corner,
412 * . - an NW-by-at-least-NW-but-more-is-better
413 * . (NW-by-NHO) horizontal work array along
414 * . the bottom edge,
415 * . - an at-least-NW-but-more-is-better (NHV-by-NW)
416 * . vertical work array along the left-hand-edge.
417 * . ====
418 *
419 KV = N - NW + 1
420 KT = NW + 1
421 NHO = ( N-NW-1 ) - KT + 1
422 KWV = NW + 2
423 NVE = ( N-NW ) - KWV + 1
424 *
425 * ==== Aggressive early deflation ====
426 *
427 CALL DLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
428 $ IHIZ, Z, LDZ, LS, LD, WR, WI, H( KV, 1 ), LDH,
429 $ NHO, H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH,
430 $ WORK, LWORK )
431 *
432 * ==== Adjust KBOT accounting for new deflations. ====
433 *
434 KBOT = KBOT - LD
435 *
436 * ==== KS points to the shifts. ====
437 *
438 KS = KBOT - LS + 1
439 *
440 * ==== Skip an expensive QR sweep if there is a (partly
441 * . heuristic) reason to expect that many eigenvalues
442 * . will deflate without it. Here, the QR sweep is
443 * . skipped if many eigenvalues have just been deflated
444 * . or if the remaining active block is small.
445 *
446 IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
447 $ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
448 *
449 * ==== NS = nominal number of simultaneous shifts.
450 * . This may be lowered (slightly) if DLAQR2
451 * . did not provide that many shifts. ====
452 *
453 NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
454 NS = NS - MOD( NS, 2 )
455 *
456 * ==== If there have been no deflations
457 * . in a multiple of KEXSH iterations,
458 * . then try exceptional shifts.
459 * . Otherwise use shifts provided by
460 * . DLAQR2 above or from the eigenvalues
461 * . of a trailing principal submatrix. ====
462 *
463 IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
464 KS = KBOT - NS + 1
465 DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2
466 SS = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
467 AA = WILK1*SS + H( I, I )
468 BB = SS
469 CC = WILK2*SS
470 DD = AA
471 CALL DLANV2( AA, BB, CC, DD, WR( I-1 ), WI( I-1 ),
472 $ WR( I ), WI( I ), CS, SN )
473 30 CONTINUE
474 IF( KS.EQ.KTOP ) THEN
475 WR( KS+1 ) = H( KS+1, KS+1 )
476 WI( KS+1 ) = ZERO
477 WR( KS ) = WR( KS+1 )
478 WI( KS ) = WI( KS+1 )
479 END IF
480 ELSE
481 *
482 * ==== Got NS/2 or fewer shifts? Use DLAHQR
483 * . on a trailing principal submatrix to
484 * . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
485 * . there is enough space below the subdiagonal
486 * . to fit an NS-by-NS scratch array.) ====
487 *
488 IF( KBOT-KS+1.LE.NS / 2 ) THEN
489 KS = KBOT - NS + 1
490 KT = N - NS + 1
491 CALL DLACPY( 'A', NS, NS, H( KS, KS ), LDH,
492 $ H( KT, 1 ), LDH )
493 CALL DLAHQR( .false., .false., NS, 1, NS,
494 $ H( KT, 1 ), LDH, WR( KS ), WI( KS ),
495 $ 1, 1, ZDUM, 1, INF )
496 KS = KS + INF
497 *
498 * ==== In case of a rare QR failure use
499 * . eigenvalues of the trailing 2-by-2
500 * . principal submatrix. ====
501 *
502 IF( KS.GE.KBOT ) THEN
503 AA = H( KBOT-1, KBOT-1 )
504 CC = H( KBOT, KBOT-1 )
505 BB = H( KBOT-1, KBOT )
506 DD = H( KBOT, KBOT )
507 CALL DLANV2( AA, BB, CC, DD, WR( KBOT-1 ),
508 $ WI( KBOT-1 ), WR( KBOT ),
509 $ WI( KBOT ), CS, SN )
510 KS = KBOT - 1
511 END IF
512 END IF
513 *
514 IF( KBOT-KS+1.GT.NS ) THEN
515 *
516 * ==== Sort the shifts (Helps a little)
517 * . Bubble sort keeps complex conjugate
518 * . pairs together. ====
519 *
520 SORTED = .false.
521 DO 50 K = KBOT, KS + 1, -1
522 IF( SORTED )
523 $ GO TO 60
524 SORTED = .true.
525 DO 40 I = KS, K - 1
526 IF( ABS( WR( I ) )+ABS( WI( I ) ).LT.
527 $ ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN
528 SORTED = .false.
529 *
530 SWAP = WR( I )
531 WR( I ) = WR( I+1 )
532 WR( I+1 ) = SWAP
533 *
534 SWAP = WI( I )
535 WI( I ) = WI( I+1 )
536 WI( I+1 ) = SWAP
537 END IF
538 40 CONTINUE
539 50 CONTINUE
540 60 CONTINUE
541 END IF
542 *
543 * ==== Shuffle shifts into pairs of real shifts
544 * . and pairs of complex conjugate shifts
545 * . assuming complex conjugate shifts are
546 * . already adjacent to one another. (Yes,
547 * . they are.) ====
548 *
549 DO 70 I = KBOT, KS + 2, -2
550 IF( WI( I ).NE.-WI( I-1 ) ) THEN
551 *
552 SWAP = WR( I )
553 WR( I ) = WR( I-1 )
554 WR( I-1 ) = WR( I-2 )
555 WR( I-2 ) = SWAP
556 *
557 SWAP = WI( I )
558 WI( I ) = WI( I-1 )
559 WI( I-1 ) = WI( I-2 )
560 WI( I-2 ) = SWAP
561 END IF
562 70 CONTINUE
563 END IF
564 *
565 * ==== If there are only two shifts and both are
566 * . real, then use only one. ====
567 *
568 IF( KBOT-KS+1.EQ.2 ) THEN
569 IF( WI( KBOT ).EQ.ZERO ) THEN
570 IF( ABS( WR( KBOT )-H( KBOT, KBOT ) ).LT.
571 $ ABS( WR( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
572 WR( KBOT-1 ) = WR( KBOT )
573 ELSE
574 WR( KBOT ) = WR( KBOT-1 )
575 END IF
576 END IF
577 END IF
578 *
579 * ==== Use up to NS of the the smallest magnatiude
580 * . shifts. If there aren't NS shifts available,
581 * . then use them all, possibly dropping one to
582 * . make the number of shifts even. ====
583 *
584 NS = MIN( NS, KBOT-KS+1 )
585 NS = NS - MOD( NS, 2 )
586 KS = KBOT - NS + 1
587 *
588 * ==== Small-bulge multi-shift QR sweep:
589 * . split workspace under the subdiagonal into
590 * . - a KDU-by-KDU work array U in the lower
591 * . left-hand-corner,
592 * . - a KDU-by-at-least-KDU-but-more-is-better
593 * . (KDU-by-NHo) horizontal work array WH along
594 * . the bottom edge,
595 * . - and an at-least-KDU-but-more-is-better-by-KDU
596 * . (NVE-by-KDU) vertical work WV arrow along
597 * . the left-hand-edge. ====
598 *
599 KDU = 3*NS - 3
600 KU = N - KDU + 1
601 KWH = KDU + 1
602 NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
603 KWV = KDU + 4
604 NVE = N - KDU - KWV + 1
605 *
606 * ==== Small-bulge multi-shift QR sweep ====
607 *
608 CALL DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
609 $ WR( KS ), WI( KS ), H, LDH, ILOZ, IHIZ, Z,
610 $ LDZ, WORK, 3, H( KU, 1 ), LDH, NVE,
611 $ H( KWV, 1 ), LDH, NHO, H( KU, KWH ), LDH )
612 END IF
613 *
614 * ==== Note progress (or the lack of it). ====
615 *
616 IF( LD.GT.0 ) THEN
617 NDFL = 1
618 ELSE
619 NDFL = NDFL + 1
620 END IF
621 *
622 * ==== End of main loop ====
623 80 CONTINUE
624 *
625 * ==== Iteration limit exceeded. Set INFO to show where
626 * . the problem occurred and exit. ====
627 *
628 INFO = KBOT
629 90 CONTINUE
630 END IF
631 *
632 * ==== Return the optimal value of LWORK. ====
633 *
634 WORK( 1 ) = DBLE( LWKOPT )
635 *
636 * ==== End of DLAQR4 ====
637 *
638 END