Annotation of rpl/lapack/lapack/dlaqr0.f, revision 1.5
1.1 bertrand 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
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