Annotation of rpl/lapack/lapack/dlaqr4.f, revision 1.14
1.11 bertrand 1: *> \brief \b DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
1.8 bertrand 2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLAQR4 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr4.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqr4.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqr4.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
22: * ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
26: * LOGICAL WANTT, WANTZ
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
30: * $ Z( LDZ, * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *>
39: *> DLAQR4 implements one level of recursion for DLAQR0.
40: *> It is a complete implementation of the small bulge multi-shift
41: *> QR algorithm. It may be called by DLAQR0 and, for large enough
42: *> deflation window size, it may be called by DLAQR3. This
43: *> subroutine is identical to DLAQR0 except that it calls DLAQR2
44: *> instead of DLAQR3.
45: *>
46: *> DLAQR4 computes the eigenvalues of a Hessenberg matrix H
47: *> and, optionally, the matrices T and Z from the Schur decomposition
48: *> H = Z T Z**T, where T is an upper quasi-triangular matrix (the
49: *> Schur form), and Z is the orthogonal matrix of Schur vectors.
50: *>
51: *> Optionally Z may be postmultiplied into an input orthogonal
52: *> matrix Q so that this routine can give the Schur factorization
53: *> of a matrix A which has been reduced to the Hessenberg form H
54: *> by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
55: *> \endverbatim
56: *
57: * Arguments:
58: * ==========
59: *
60: *> \param[in] WANTT
61: *> \verbatim
62: *> WANTT is LOGICAL
63: *> = .TRUE. : the full Schur form T is required;
64: *> = .FALSE.: only eigenvalues are required.
65: *> \endverbatim
66: *>
67: *> \param[in] WANTZ
68: *> \verbatim
69: *> WANTZ is LOGICAL
70: *> = .TRUE. : the matrix of Schur vectors Z is required;
71: *> = .FALSE.: Schur vectors are not required.
72: *> \endverbatim
73: *>
74: *> \param[in] N
75: *> \verbatim
76: *> N is INTEGER
77: *> The order of the matrix H. N .GE. 0.
78: *> \endverbatim
79: *>
80: *> \param[in] ILO
81: *> \verbatim
82: *> ILO is INTEGER
83: *> \endverbatim
84: *>
85: *> \param[in] IHI
86: *> \verbatim
87: *> IHI is INTEGER
88: *> It is assumed that H is already upper triangular in rows
89: *> and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
90: *> H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
91: *> previous call to DGEBAL, and then passed to DGEHRD when the
92: *> matrix output by DGEBAL is reduced to Hessenberg form.
93: *> Otherwise, ILO and IHI should be set to 1 and N,
94: *> respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
95: *> If N = 0, then ILO = 1 and IHI = 0.
96: *> \endverbatim
97: *>
98: *> \param[in,out] H
99: *> \verbatim
100: *> H is DOUBLE PRECISION array, dimension (LDH,N)
101: *> On entry, the upper Hessenberg matrix H.
102: *> On exit, if INFO = 0 and WANTT is .TRUE., then H contains
103: *> the upper quasi-triangular matrix T from the Schur
104: *> decomposition (the Schur form); 2-by-2 diagonal blocks
105: *> (corresponding to complex conjugate pairs of eigenvalues)
106: *> are returned in standard form, with H(i,i) = H(i+1,i+1)
107: *> and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
108: *> .FALSE., then the contents of H are unspecified on exit.
109: *> (The output value of H when INFO.GT.0 is given under the
110: *> description of INFO below.)
111: *>
112: *> This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
113: *> j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
114: *> \endverbatim
115: *>
116: *> \param[in] LDH
117: *> \verbatim
118: *> LDH is INTEGER
119: *> The leading dimension of the array H. LDH .GE. max(1,N).
120: *> \endverbatim
121: *>
122: *> \param[out] WR
123: *> \verbatim
124: *> WR is DOUBLE PRECISION array, dimension (IHI)
125: *> \endverbatim
126: *>
127: *> \param[out] WI
128: *> \verbatim
129: *> WI is DOUBLE PRECISION array, dimension (IHI)
130: *> The real and imaginary parts, respectively, of the computed
131: *> eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
132: *> and WI(ILO:IHI). If two eigenvalues are computed as a
133: *> complex conjugate pair, they are stored in consecutive
134: *> elements of WR and WI, say the i-th and (i+1)th, with
135: *> WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
136: *> the eigenvalues are stored in the same order as on the
137: *> diagonal of the Schur form returned in H, with
138: *> WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
139: *> block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
140: *> WI(i+1) = -WI(i).
141: *> \endverbatim
142: *>
143: *> \param[in] ILOZ
144: *> \verbatim
145: *> ILOZ is INTEGER
146: *> \endverbatim
147: *>
148: *> \param[in] IHIZ
149: *> \verbatim
150: *> IHIZ is INTEGER
151: *> Specify the rows of Z to which transformations must be
152: *> applied if WANTZ is .TRUE..
153: *> 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
154: *> \endverbatim
155: *>
156: *> \param[in,out] Z
157: *> \verbatim
158: *> Z is DOUBLE PRECISION array, dimension (LDZ,IHI)
159: *> If WANTZ is .FALSE., then Z is not referenced.
160: *> If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
161: *> replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
162: *> orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
163: *> (The output value of Z when INFO.GT.0 is given under
164: *> the description of INFO below.)
165: *> \endverbatim
166: *>
167: *> \param[in] LDZ
168: *> \verbatim
169: *> LDZ is INTEGER
170: *> The leading dimension of the array Z. if WANTZ is .TRUE.
171: *> then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.
172: *> \endverbatim
173: *>
174: *> \param[out] WORK
175: *> \verbatim
176: *> WORK is DOUBLE PRECISION array, dimension LWORK
177: *> On exit, if LWORK = -1, WORK(1) returns an estimate of
178: *> the optimal value for LWORK.
179: *> \endverbatim
180: *>
181: *> \param[in] LWORK
182: *> \verbatim
183: *> LWORK is INTEGER
184: *> The dimension of the array WORK. LWORK .GE. max(1,N)
185: *> is sufficient, but LWORK typically as large as 6*N may
186: *> be required for optimal performance. A workspace query
187: *> to determine the optimal workspace size is recommended.
188: *>
189: *> If LWORK = -1, then DLAQR4 does a workspace query.
190: *> In this case, DLAQR4 checks the input parameters and
191: *> estimates the optimal workspace size for the given
192: *> values of N, ILO and IHI. The estimate is returned
193: *> in WORK(1). No error message related to LWORK is
194: *> issued by XERBLA. Neither H nor Z are accessed.
195: *> \endverbatim
196: *>
197: *> \param[out] INFO
198: *> \verbatim
199: *> INFO is INTEGER
200: *> = 0: successful exit
201: *> .GT. 0: if INFO = i, DLAQR4 failed to compute all of
202: *> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
203: *> and WI contain those eigenvalues which have been
204: *> successfully computed. (Failures are rare.)
205: *>
206: *> If INFO .GT. 0 and WANT is .FALSE., then on exit,
207: *> the remaining unconverged eigenvalues are the eigen-
208: *> values of the upper Hessenberg matrix rows and
209: *> columns ILO through INFO of the final, output
210: *> value of H.
211: *>
212: *> If INFO .GT. 0 and WANTT is .TRUE., then on exit
213: *>
214: *> (*) (initial value of H)*U = U*(final value of H)
215: *>
216: *> where U is a orthogonal matrix. The final
217: *> value of H is upper Hessenberg and triangular in
218: *> rows and columns INFO+1 through IHI.
219: *>
220: *> If INFO .GT. 0 and WANTZ is .TRUE., then on exit
221: *>
222: *> (final value of Z(ILO:IHI,ILOZ:IHIZ)
223: *> = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
224: *>
225: *> where U is the orthogonal matrix in (*) (regard-
226: *> less of the value of WANTT.)
227: *>
228: *> If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
229: *> accessed.
230: *> \endverbatim
231: *
232: * Authors:
233: * ========
234: *
235: *> \author Univ. of Tennessee
236: *> \author Univ. of California Berkeley
237: *> \author Univ. of Colorado Denver
238: *> \author NAG Ltd.
239: *
1.11 bertrand 240: *> \date September 2012
1.8 bertrand 241: *
242: *> \ingroup doubleOTHERauxiliary
243: *
244: *> \par Contributors:
245: * ==================
246: *>
247: *> Karen Braman and Ralph Byers, Department of Mathematics,
248: *> University of Kansas, USA
249: *
250: *> \par References:
251: * ================
252: *>
253: *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
254: *> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
255: *> Performance, SIAM Journal of Matrix Analysis, volume 23, pages
256: *> 929--947, 2002.
257: *> \n
258: *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
259: *> Algorithm Part II: Aggressive Early Deflation, SIAM Journal
260: *> of Matrix Analysis, volume 23, pages 948--973, 2002.
261: *>
262: * =====================================================================
1.1 bertrand 263: SUBROUTINE DLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
264: $ ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
265: *
1.11 bertrand 266: * -- LAPACK auxiliary routine (version 3.4.2) --
1.8 bertrand 267: * -- LAPACK is a software package provided by Univ. of Tennessee, --
268: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.11 bertrand 269: * September 2012
1.1 bertrand 270: *
271: * .. Scalar Arguments ..
272: INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
273: LOGICAL WANTT, WANTZ
274: * ..
275: * .. Array Arguments ..
276: DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
277: $ Z( LDZ, * )
278: * ..
279: *
1.8 bertrand 280: * ================================================================
1.1 bertrand 281: * .. Parameters ..
282: *
283: * ==== Matrices of order NTINY or smaller must be processed by
284: * . DLAHQR because of insufficient subdiagonal scratch space.
285: * . (This is a hard limit.) ====
286: INTEGER NTINY
287: PARAMETER ( NTINY = 11 )
288: *
289: * ==== Exceptional deflation windows: try to cure rare
290: * . slow convergence by varying the size of the
291: * . deflation window after KEXNW iterations. ====
292: INTEGER KEXNW
293: PARAMETER ( KEXNW = 5 )
294: *
295: * ==== Exceptional shifts: try to cure rare slow convergence
296: * . with ad-hoc exceptional shifts every KEXSH iterations.
297: * . ====
298: INTEGER KEXSH
299: PARAMETER ( KEXSH = 6 )
300: *
301: * ==== The constants WILK1 and WILK2 are used to form the
302: * . exceptional shifts. ====
303: DOUBLE PRECISION WILK1, WILK2
304: PARAMETER ( WILK1 = 0.75d0, WILK2 = -0.4375d0 )
305: DOUBLE PRECISION ZERO, ONE
306: PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
307: * ..
308: * .. Local Scalars ..
309: DOUBLE PRECISION AA, BB, CC, CS, DD, SN, SS, SWAP
310: INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
311: $ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
312: $ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
313: $ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
314: LOGICAL SORTED
315: CHARACTER JBCMPZ*2
316: * ..
317: * .. External Functions ..
318: INTEGER ILAENV
319: EXTERNAL ILAENV
320: * ..
321: * .. Local Arrays ..
322: DOUBLE PRECISION ZDUM( 1, 1 )
323: * ..
324: * .. External Subroutines ..
325: EXTERNAL DLACPY, DLAHQR, DLANV2, DLAQR2, DLAQR5
326: * ..
327: * .. Intrinsic Functions ..
328: INTRINSIC ABS, DBLE, INT, MAX, MIN, MOD
329: * ..
330: * .. Executable Statements ..
331: INFO = 0
332: *
333: * ==== Quick return for N = 0: nothing to do. ====
334: *
335: IF( N.EQ.0 ) THEN
336: WORK( 1 ) = ONE
337: RETURN
338: END IF
339: *
340: IF( N.LE.NTINY ) THEN
341: *
342: * ==== Tiny matrices must use DLAHQR. ====
343: *
344: LWKOPT = 1
345: IF( LWORK.NE.-1 )
346: $ CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
347: $ ILOZ, IHIZ, Z, LDZ, INFO )
348: ELSE
349: *
350: * ==== Use small bulge multi-shift QR with aggressive early
351: * . deflation on larger-than-tiny matrices. ====
352: *
353: * ==== Hope for the best. ====
354: *
355: INFO = 0
356: *
357: * ==== Set up job flags for ILAENV. ====
358: *
359: IF( WANTT ) THEN
360: JBCMPZ( 1: 1 ) = 'S'
361: ELSE
362: JBCMPZ( 1: 1 ) = 'E'
363: END IF
364: IF( WANTZ ) THEN
365: JBCMPZ( 2: 2 ) = 'V'
366: ELSE
367: JBCMPZ( 2: 2 ) = 'N'
368: END IF
369: *
370: * ==== NWR = recommended deflation window size. At this
371: * . point, N .GT. NTINY = 11, so there is enough
372: * . subdiagonal workspace for NWR.GE.2 as required.
373: * . (In fact, there is enough subdiagonal space for
374: * . NWR.GE.3.) ====
375: *
376: NWR = ILAENV( 13, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
377: NWR = MAX( 2, NWR )
378: NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
379: *
380: * ==== NSR = recommended number of simultaneous shifts.
381: * . At this point N .GT. NTINY = 11, so there is at
382: * . enough subdiagonal workspace for NSR to be even
383: * . and greater than or equal to two as required. ====
384: *
385: NSR = ILAENV( 15, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
386: NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
387: NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
388: *
389: * ==== Estimate optimal workspace ====
390: *
391: * ==== Workspace query call to DLAQR2 ====
392: *
393: CALL DLAQR2( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
394: $ IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH,
395: $ N, H, LDH, WORK, -1 )
396: *
397: * ==== Optimal workspace = MAX(DLAQR5, DLAQR2) ====
398: *
399: LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
400: *
401: * ==== Quick return in case of workspace query. ====
402: *
403: IF( LWORK.EQ.-1 ) THEN
404: WORK( 1 ) = DBLE( LWKOPT )
405: RETURN
406: END IF
407: *
408: * ==== DLAHQR/DLAQR0 crossover point ====
409: *
410: NMIN = ILAENV( 12, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
411: NMIN = MAX( NTINY, NMIN )
412: *
413: * ==== Nibble crossover point ====
414: *
415: NIBBLE = ILAENV( 14, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
416: NIBBLE = MAX( 0, NIBBLE )
417: *
418: * ==== Accumulate reflections during ttswp? Use block
419: * . 2-by-2 structure during matrix-matrix multiply? ====
420: *
421: KACC22 = ILAENV( 16, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
422: KACC22 = MAX( 0, KACC22 )
423: KACC22 = MIN( 2, KACC22 )
424: *
425: * ==== NWMAX = the largest possible deflation window for
426: * . which there is sufficient workspace. ====
427: *
428: NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
429: NW = NWMAX
430: *
431: * ==== NSMAX = the Largest number of simultaneous shifts
432: * . for which there is sufficient workspace. ====
433: *
434: NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
435: NSMAX = NSMAX - MOD( NSMAX, 2 )
436: *
437: * ==== NDFL: an iteration count restarted at deflation. ====
438: *
439: NDFL = 1
440: *
441: * ==== ITMAX = iteration limit ====
442: *
443: ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
444: *
445: * ==== Last row and column in the active block ====
446: *
447: KBOT = IHI
448: *
449: * ==== Main Loop ====
450: *
451: DO 80 IT = 1, ITMAX
452: *
453: * ==== Done when KBOT falls below ILO ====
454: *
455: IF( KBOT.LT.ILO )
456: $ GO TO 90
457: *
458: * ==== Locate active block ====
459: *
460: DO 10 K = KBOT, ILO + 1, -1
461: IF( H( K, K-1 ).EQ.ZERO )
462: $ GO TO 20
463: 10 CONTINUE
464: K = ILO
465: 20 CONTINUE
466: KTOP = K
467: *
468: * ==== Select deflation window size:
469: * . Typical Case:
470: * . If possible and advisable, nibble the entire
471: * . active block. If not, use size MIN(NWR,NWMAX)
472: * . or MIN(NWR+1,NWMAX) depending upon which has
473: * . the smaller corresponding subdiagonal entry
474: * . (a heuristic).
475: * .
476: * . Exceptional Case:
477: * . If there have been no deflations in KEXNW or
478: * . more iterations, then vary the deflation window
479: * . size. At first, because, larger windows are,
480: * . in general, more powerful than smaller ones,
481: * . rapidly increase the window to the maximum possible.
482: * . Then, gradually reduce the window size. ====
483: *
484: NH = KBOT - KTOP + 1
485: NWUPBD = MIN( NH, NWMAX )
486: IF( NDFL.LT.KEXNW ) THEN
487: NW = MIN( NWUPBD, NWR )
488: ELSE
489: NW = MIN( NWUPBD, 2*NW )
490: END IF
491: IF( NW.LT.NWMAX ) THEN
492: IF( NW.GE.NH-1 ) THEN
493: NW = NH
494: ELSE
495: KWTOP = KBOT - NW + 1
496: IF( ABS( H( KWTOP, KWTOP-1 ) ).GT.
497: $ ABS( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
498: END IF
499: END IF
500: IF( NDFL.LT.KEXNW ) THEN
501: NDEC = -1
502: ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
503: NDEC = NDEC + 1
504: IF( NW-NDEC.LT.2 )
505: $ NDEC = 0
506: NW = NW - NDEC
507: END IF
508: *
509: * ==== Aggressive early deflation:
510: * . split workspace under the subdiagonal into
511: * . - an nw-by-nw work array V in the lower
512: * . left-hand-corner,
513: * . - an NW-by-at-least-NW-but-more-is-better
514: * . (NW-by-NHO) horizontal work array along
515: * . the bottom edge,
516: * . - an at-least-NW-but-more-is-better (NHV-by-NW)
517: * . vertical work array along the left-hand-edge.
518: * . ====
519: *
520: KV = N - NW + 1
521: KT = NW + 1
522: NHO = ( N-NW-1 ) - KT + 1
523: KWV = NW + 2
524: NVE = ( N-NW ) - KWV + 1
525: *
526: * ==== Aggressive early deflation ====
527: *
528: CALL DLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
529: $ IHIZ, Z, LDZ, LS, LD, WR, WI, H( KV, 1 ), LDH,
530: $ NHO, H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH,
531: $ WORK, LWORK )
532: *
533: * ==== Adjust KBOT accounting for new deflations. ====
534: *
535: KBOT = KBOT - LD
536: *
537: * ==== KS points to the shifts. ====
538: *
539: KS = KBOT - LS + 1
540: *
541: * ==== Skip an expensive QR sweep if there is a (partly
542: * . heuristic) reason to expect that many eigenvalues
543: * . will deflate without it. Here, the QR sweep is
544: * . skipped if many eigenvalues have just been deflated
545: * . or if the remaining active block is small.
546: *
547: IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
548: $ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
549: *
550: * ==== NS = nominal number of simultaneous shifts.
551: * . This may be lowered (slightly) if DLAQR2
552: * . did not provide that many shifts. ====
553: *
554: NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
555: NS = NS - MOD( NS, 2 )
556: *
557: * ==== If there have been no deflations
558: * . in a multiple of KEXSH iterations,
559: * . then try exceptional shifts.
560: * . Otherwise use shifts provided by
561: * . DLAQR2 above or from the eigenvalues
562: * . of a trailing principal submatrix. ====
563: *
564: IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
565: KS = KBOT - NS + 1
566: DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2
567: SS = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
568: AA = WILK1*SS + H( I, I )
569: BB = SS
570: CC = WILK2*SS
571: DD = AA
572: CALL DLANV2( AA, BB, CC, DD, WR( I-1 ), WI( I-1 ),
573: $ WR( I ), WI( I ), CS, SN )
574: 30 CONTINUE
575: IF( KS.EQ.KTOP ) THEN
576: WR( KS+1 ) = H( KS+1, KS+1 )
577: WI( KS+1 ) = ZERO
578: WR( KS ) = WR( KS+1 )
579: WI( KS ) = WI( KS+1 )
580: END IF
581: ELSE
582: *
583: * ==== Got NS/2 or fewer shifts? Use DLAHQR
584: * . on a trailing principal submatrix to
585: * . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
586: * . there is enough space below the subdiagonal
587: * . to fit an NS-by-NS scratch array.) ====
588: *
589: IF( KBOT-KS+1.LE.NS / 2 ) THEN
590: KS = KBOT - NS + 1
591: KT = N - NS + 1
592: CALL DLACPY( 'A', NS, NS, H( KS, KS ), LDH,
593: $ H( KT, 1 ), LDH )
594: CALL DLAHQR( .false., .false., NS, 1, NS,
595: $ H( KT, 1 ), LDH, WR( KS ), WI( KS ),
596: $ 1, 1, ZDUM, 1, INF )
597: KS = KS + INF
598: *
599: * ==== In case of a rare QR failure use
600: * . eigenvalues of the trailing 2-by-2
601: * . principal submatrix. ====
602: *
603: IF( KS.GE.KBOT ) THEN
604: AA = H( KBOT-1, KBOT-1 )
605: CC = H( KBOT, KBOT-1 )
606: BB = H( KBOT-1, KBOT )
607: DD = H( KBOT, KBOT )
608: CALL DLANV2( AA, BB, CC, DD, WR( KBOT-1 ),
609: $ WI( KBOT-1 ), WR( KBOT ),
610: $ WI( KBOT ), CS, SN )
611: KS = KBOT - 1
612: END IF
613: END IF
614: *
615: IF( KBOT-KS+1.GT.NS ) THEN
616: *
617: * ==== Sort the shifts (Helps a little)
618: * . Bubble sort keeps complex conjugate
619: * . pairs together. ====
620: *
621: SORTED = .false.
622: DO 50 K = KBOT, KS + 1, -1
623: IF( SORTED )
624: $ GO TO 60
625: SORTED = .true.
626: DO 40 I = KS, K - 1
627: IF( ABS( WR( I ) )+ABS( WI( I ) ).LT.
628: $ ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN
629: SORTED = .false.
630: *
631: SWAP = WR( I )
632: WR( I ) = WR( I+1 )
633: WR( I+1 ) = SWAP
634: *
635: SWAP = WI( I )
636: WI( I ) = WI( I+1 )
637: WI( I+1 ) = SWAP
638: END IF
639: 40 CONTINUE
640: 50 CONTINUE
641: 60 CONTINUE
642: END IF
643: *
644: * ==== Shuffle shifts into pairs of real shifts
645: * . and pairs of complex conjugate shifts
646: * . assuming complex conjugate shifts are
647: * . already adjacent to one another. (Yes,
648: * . they are.) ====
649: *
650: DO 70 I = KBOT, KS + 2, -2
651: IF( WI( I ).NE.-WI( I-1 ) ) THEN
652: *
653: SWAP = WR( I )
654: WR( I ) = WR( I-1 )
655: WR( I-1 ) = WR( I-2 )
656: WR( I-2 ) = SWAP
657: *
658: SWAP = WI( I )
659: WI( I ) = WI( I-1 )
660: WI( I-1 ) = WI( I-2 )
661: WI( I-2 ) = SWAP
662: END IF
663: 70 CONTINUE
664: END IF
665: *
666: * ==== If there are only two shifts and both are
667: * . real, then use only one. ====
668: *
669: IF( KBOT-KS+1.EQ.2 ) THEN
670: IF( WI( KBOT ).EQ.ZERO ) THEN
671: IF( ABS( WR( KBOT )-H( KBOT, KBOT ) ).LT.
672: $ ABS( WR( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
673: WR( KBOT-1 ) = WR( KBOT )
674: ELSE
675: WR( KBOT ) = WR( KBOT-1 )
676: END IF
677: END IF
678: END IF
679: *
680: * ==== Use up to NS of the the smallest magnatiude
681: * . shifts. If there aren't NS shifts available,
682: * . then use them all, possibly dropping one to
683: * . make the number of shifts even. ====
684: *
685: NS = MIN( NS, KBOT-KS+1 )
686: NS = NS - MOD( NS, 2 )
687: KS = KBOT - NS + 1
688: *
689: * ==== Small-bulge multi-shift QR sweep:
690: * . split workspace under the subdiagonal into
691: * . - a KDU-by-KDU work array U in the lower
692: * . left-hand-corner,
693: * . - a KDU-by-at-least-KDU-but-more-is-better
694: * . (KDU-by-NHo) horizontal work array WH along
695: * . the bottom edge,
696: * . - and an at-least-KDU-but-more-is-better-by-KDU
697: * . (NVE-by-KDU) vertical work WV arrow along
698: * . the left-hand-edge. ====
699: *
700: KDU = 3*NS - 3
701: KU = N - KDU + 1
702: KWH = KDU + 1
703: NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
704: KWV = KDU + 4
705: NVE = N - KDU - KWV + 1
706: *
707: * ==== Small-bulge multi-shift QR sweep ====
708: *
709: CALL DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
710: $ WR( KS ), WI( KS ), H, LDH, ILOZ, IHIZ, Z,
711: $ LDZ, WORK, 3, H( KU, 1 ), LDH, NVE,
712: $ H( KWV, 1 ), LDH, NHO, H( KU, KWH ), LDH )
713: END IF
714: *
715: * ==== Note progress (or the lack of it). ====
716: *
717: IF( LD.GT.0 ) THEN
718: NDFL = 1
719: ELSE
720: NDFL = NDFL + 1
721: END IF
722: *
723: * ==== End of main loop ====
724: 80 CONTINUE
725: *
726: * ==== Iteration limit exceeded. Set INFO to show where
727: * . the problem occurred and exit. ====
728: *
729: INFO = KBOT
730: 90 CONTINUE
731: END IF
732: *
733: * ==== Return the optimal value of LWORK. ====
734: *
735: WORK( 1 ) = DBLE( LWKOPT )
736: *
737: * ==== End of DLAQR4 ====
738: *
739: END
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