1: *> \brief \b ZLAQR2 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZLAQR2 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlaqr2.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlaqr2.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlaqr2.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
22: * IHIZ, Z, LDZ, NS, ND, SH, V, LDV, NH, T, LDT,
23: * NV, WV, LDWV, WORK, LWORK )
24: *
25: * .. Scalar Arguments ..
26: * INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
27: * $ LDZ, LWORK, N, ND, NH, NS, NV, NW
28: * LOGICAL WANTT, WANTZ
29: * ..
30: * .. Array Arguments ..
31: * COMPLEX*16 H( LDH, * ), SH( * ), T( LDT, * ), V( LDV, * ),
32: * $ WORK( * ), WV( LDWV, * ), Z( LDZ, * )
33: * ..
34: *
35: *
36: *> \par Purpose:
37: * =============
38: *>
39: *> \verbatim
40: *>
41: *> ZLAQR2 is identical to ZLAQR3 except that it avoids
42: *> recursion by calling ZLAHQR instead of ZLAQR4.
43: *>
44: *> Aggressive early deflation:
45: *>
46: *> ZLAQR2 accepts as input an upper Hessenberg matrix
47: *> H and performs an unitary similarity transformation
48: *> designed to detect and deflate fully converged eigenvalues from
49: *> a trailing principal submatrix. On output H has been over-
50: *> written by a new Hessenberg matrix that is a perturbation of
51: *> an unitary similarity transformation of H. It is to be
52: *> hoped that the final version of H has many zero subdiagonal
53: *> entries.
54: *>
55: *> \endverbatim
56: *
57: * Arguments:
58: * ==========
59: *
60: *> \param[in] WANTT
61: *> \verbatim
62: *> WANTT is LOGICAL
63: *> If .TRUE., then the Hessenberg matrix H is fully updated
64: *> so that the triangular Schur factor may be
65: *> computed (in cooperation with the calling subroutine).
66: *> If .FALSE., then only enough of H is updated to preserve
67: *> the eigenvalues.
68: *> \endverbatim
69: *>
70: *> \param[in] WANTZ
71: *> \verbatim
72: *> WANTZ is LOGICAL
73: *> If .TRUE., then the unitary matrix Z is updated so
74: *> so that the unitary Schur factor may be computed
75: *> (in cooperation with the calling subroutine).
76: *> If .FALSE., then Z is not referenced.
77: *> \endverbatim
78: *>
79: *> \param[in] N
80: *> \verbatim
81: *> N is INTEGER
82: *> The order of the matrix H and (if WANTZ is .TRUE.) the
83: *> order of the unitary matrix Z.
84: *> \endverbatim
85: *>
86: *> \param[in] KTOP
87: *> \verbatim
88: *> KTOP is INTEGER
89: *> It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
90: *> KBOT and KTOP together determine an isolated block
91: *> along the diagonal of the Hessenberg matrix.
92: *> \endverbatim
93: *>
94: *> \param[in] KBOT
95: *> \verbatim
96: *> KBOT is INTEGER
97: *> It is assumed without a check that either
98: *> KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together
99: *> determine an isolated block along the diagonal of the
100: *> Hessenberg matrix.
101: *> \endverbatim
102: *>
103: *> \param[in] NW
104: *> \verbatim
105: *> NW is INTEGER
106: *> Deflation window size. 1 <= NW <= (KBOT-KTOP+1).
107: *> \endverbatim
108: *>
109: *> \param[in,out] H
110: *> \verbatim
111: *> H is COMPLEX*16 array, dimension (LDH,N)
112: *> On input the initial N-by-N section of H stores the
113: *> Hessenberg matrix undergoing aggressive early deflation.
114: *> On output H has been transformed by a unitary
115: *> similarity transformation, perturbed, and the returned
116: *> to Hessenberg form that (it is to be hoped) has some
117: *> zero subdiagonal entries.
118: *> \endverbatim
119: *>
120: *> \param[in] LDH
121: *> \verbatim
122: *> LDH is INTEGER
123: *> Leading dimension of H just as declared in the calling
124: *> subroutine. N <= LDH
125: *> \endverbatim
126: *>
127: *> \param[in] ILOZ
128: *> \verbatim
129: *> ILOZ is INTEGER
130: *> \endverbatim
131: *>
132: *> \param[in] IHIZ
133: *> \verbatim
134: *> IHIZ is INTEGER
135: *> Specify the rows of Z to which transformations must be
136: *> applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
137: *> \endverbatim
138: *>
139: *> \param[in,out] Z
140: *> \verbatim
141: *> Z is COMPLEX*16 array, dimension (LDZ,N)
142: *> IF WANTZ is .TRUE., then on output, the unitary
143: *> similarity transformation mentioned above has been
144: *> accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
145: *> If WANTZ is .FALSE., then Z is unreferenced.
146: *> \endverbatim
147: *>
148: *> \param[in] LDZ
149: *> \verbatim
150: *> LDZ is INTEGER
151: *> The leading dimension of Z just as declared in the
152: *> calling subroutine. 1 <= LDZ.
153: *> \endverbatim
154: *>
155: *> \param[out] NS
156: *> \verbatim
157: *> NS is INTEGER
158: *> The number of unconverged (ie approximate) eigenvalues
159: *> returned in SR and SI that may be used as shifts by the
160: *> calling subroutine.
161: *> \endverbatim
162: *>
163: *> \param[out] ND
164: *> \verbatim
165: *> ND is INTEGER
166: *> The number of converged eigenvalues uncovered by this
167: *> subroutine.
168: *> \endverbatim
169: *>
170: *> \param[out] SH
171: *> \verbatim
172: *> SH is COMPLEX*16 array, dimension (KBOT)
173: *> On output, approximate eigenvalues that may
174: *> be used for shifts are stored in SH(KBOT-ND-NS+1)
175: *> through SR(KBOT-ND). Converged eigenvalues are
176: *> stored in SH(KBOT-ND+1) through SH(KBOT).
177: *> \endverbatim
178: *>
179: *> \param[out] V
180: *> \verbatim
181: *> V is COMPLEX*16 array, dimension (LDV,NW)
182: *> An NW-by-NW work array.
183: *> \endverbatim
184: *>
185: *> \param[in] LDV
186: *> \verbatim
187: *> LDV is INTEGER
188: *> The leading dimension of V just as declared in the
189: *> calling subroutine. NW <= LDV
190: *> \endverbatim
191: *>
192: *> \param[in] NH
193: *> \verbatim
194: *> NH is INTEGER
195: *> The number of columns of T. NH >= NW.
196: *> \endverbatim
197: *>
198: *> \param[out] T
199: *> \verbatim
200: *> T is COMPLEX*16 array, dimension (LDT,NW)
201: *> \endverbatim
202: *>
203: *> \param[in] LDT
204: *> \verbatim
205: *> LDT is INTEGER
206: *> The leading dimension of T just as declared in the
207: *> calling subroutine. NW <= LDT
208: *> \endverbatim
209: *>
210: *> \param[in] NV
211: *> \verbatim
212: *> NV is INTEGER
213: *> The number of rows of work array WV available for
214: *> workspace. NV >= NW.
215: *> \endverbatim
216: *>
217: *> \param[out] WV
218: *> \verbatim
219: *> WV is COMPLEX*16 array, dimension (LDWV,NW)
220: *> \endverbatim
221: *>
222: *> \param[in] LDWV
223: *> \verbatim
224: *> LDWV is INTEGER
225: *> The leading dimension of W just as declared in the
226: *> calling subroutine. NW <= LDV
227: *> \endverbatim
228: *>
229: *> \param[out] WORK
230: *> \verbatim
231: *> WORK is COMPLEX*16 array, dimension (LWORK)
232: *> On exit, WORK(1) is set to an estimate of the optimal value
233: *> of LWORK for the given values of N, NW, KTOP and KBOT.
234: *> \endverbatim
235: *>
236: *> \param[in] LWORK
237: *> \verbatim
238: *> LWORK is INTEGER
239: *> The dimension of the work array WORK. LWORK = 2*NW
240: *> suffices, but greater efficiency may result from larger
241: *> values of LWORK.
242: *>
243: *> If LWORK = -1, then a workspace query is assumed; ZLAQR2
244: *> only estimates the optimal workspace size for the given
245: *> values of N, NW, KTOP and KBOT. The estimate is returned
246: *> in WORK(1). No error message related to LWORK is issued
247: *> by XERBLA. Neither H nor Z are accessed.
248: *> \endverbatim
249: *
250: * Authors:
251: * ========
252: *
253: *> \author Univ. of Tennessee
254: *> \author Univ. of California Berkeley
255: *> \author Univ. of Colorado Denver
256: *> \author NAG Ltd.
257: *
258: *> \ingroup complex16OTHERauxiliary
259: *
260: *> \par Contributors:
261: * ==================
262: *>
263: *> Karen Braman and Ralph Byers, Department of Mathematics,
264: *> University of Kansas, USA
265: *>
266: * =====================================================================
267: SUBROUTINE ZLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
268: $ IHIZ, Z, LDZ, NS, ND, SH, V, LDV, NH, T, LDT,
269: $ NV, WV, LDWV, WORK, LWORK )
270: *
271: * -- LAPACK auxiliary routine --
272: * -- LAPACK is a software package provided by Univ. of Tennessee, --
273: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
274: *
275: * .. Scalar Arguments ..
276: INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
277: $ LDZ, LWORK, N, ND, NH, NS, NV, NW
278: LOGICAL WANTT, WANTZ
279: * ..
280: * .. Array Arguments ..
281: COMPLEX*16 H( LDH, * ), SH( * ), T( LDT, * ), V( LDV, * ),
282: $ WORK( * ), WV( LDWV, * ), Z( LDZ, * )
283: * ..
284: *
285: * ================================================================
286: *
287: * .. Parameters ..
288: COMPLEX*16 ZERO, ONE
289: PARAMETER ( ZERO = ( 0.0d0, 0.0d0 ),
290: $ ONE = ( 1.0d0, 0.0d0 ) )
291: DOUBLE PRECISION RZERO, RONE
292: PARAMETER ( RZERO = 0.0d0, RONE = 1.0d0 )
293: * ..
294: * .. Local Scalars ..
295: COMPLEX*16 BETA, CDUM, S, TAU
296: DOUBLE PRECISION FOO, SAFMAX, SAFMIN, SMLNUM, ULP
297: INTEGER I, IFST, ILST, INFO, INFQR, J, JW, KCOL, KLN,
298: $ KNT, KROW, KWTOP, LTOP, LWK1, LWK2, LWKOPT
299: * ..
300: * .. External Functions ..
301: DOUBLE PRECISION DLAMCH
302: EXTERNAL DLAMCH
303: * ..
304: * .. External Subroutines ..
305: EXTERNAL DLABAD, ZCOPY, ZGEHRD, ZGEMM, ZLACPY, ZLAHQR,
306: $ ZLARF, ZLARFG, ZLASET, ZTREXC, ZUNMHR
307: * ..
308: * .. Intrinsic Functions ..
309: INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, INT, MAX, MIN
310: * ..
311: * .. Statement Functions ..
312: DOUBLE PRECISION CABS1
313: * ..
314: * .. Statement Function definitions ..
315: CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
316: * ..
317: * .. Executable Statements ..
318: *
319: * ==== Estimate optimal workspace. ====
320: *
321: JW = MIN( NW, KBOT-KTOP+1 )
322: IF( JW.LE.2 ) THEN
323: LWKOPT = 1
324: ELSE
325: *
326: * ==== Workspace query call to ZGEHRD ====
327: *
328: CALL ZGEHRD( JW, 1, JW-1, T, LDT, WORK, WORK, -1, INFO )
329: LWK1 = INT( WORK( 1 ) )
330: *
331: * ==== Workspace query call to ZUNMHR ====
332: *
333: CALL ZUNMHR( 'R', 'N', JW, JW, 1, JW-1, T, LDT, WORK, V, LDV,
334: $ WORK, -1, INFO )
335: LWK2 = INT( WORK( 1 ) )
336: *
337: * ==== Optimal workspace ====
338: *
339: LWKOPT = JW + MAX( LWK1, LWK2 )
340: END IF
341: *
342: * ==== Quick return in case of workspace query. ====
343: *
344: IF( LWORK.EQ.-1 ) THEN
345: WORK( 1 ) = DCMPLX( LWKOPT, 0 )
346: RETURN
347: END IF
348: *
349: * ==== Nothing to do ...
350: * ... for an empty active block ... ====
351: NS = 0
352: ND = 0
353: WORK( 1 ) = ONE
354: IF( KTOP.GT.KBOT )
355: $ RETURN
356: * ... nor for an empty deflation window. ====
357: IF( NW.LT.1 )
358: $ RETURN
359: *
360: * ==== Machine constants ====
361: *
362: SAFMIN = DLAMCH( 'SAFE MINIMUM' )
363: SAFMAX = RONE / SAFMIN
364: CALL DLABAD( SAFMIN, SAFMAX )
365: ULP = DLAMCH( 'PRECISION' )
366: SMLNUM = SAFMIN*( DBLE( N ) / ULP )
367: *
368: * ==== Setup deflation window ====
369: *
370: JW = MIN( NW, KBOT-KTOP+1 )
371: KWTOP = KBOT - JW + 1
372: IF( KWTOP.EQ.KTOP ) THEN
373: S = ZERO
374: ELSE
375: S = H( KWTOP, KWTOP-1 )
376: END IF
377: *
378: IF( KBOT.EQ.KWTOP ) THEN
379: *
380: * ==== 1-by-1 deflation window: not much to do ====
381: *
382: SH( KWTOP ) = H( KWTOP, KWTOP )
383: NS = 1
384: ND = 0
385: IF( CABS1( S ).LE.MAX( SMLNUM, ULP*CABS1( H( KWTOP,
386: $ KWTOP ) ) ) ) THEN
387: NS = 0
388: ND = 1
389: IF( KWTOP.GT.KTOP )
390: $ H( KWTOP, KWTOP-1 ) = ZERO
391: END IF
392: WORK( 1 ) = ONE
393: RETURN
394: END IF
395: *
396: * ==== Convert to spike-triangular form. (In case of a
397: * . rare QR failure, this routine continues to do
398: * . aggressive early deflation using that part of
399: * . the deflation window that converged using INFQR
400: * . here and there to keep track.) ====
401: *
402: CALL ZLACPY( 'U', JW, JW, H( KWTOP, KWTOP ), LDH, T, LDT )
403: CALL ZCOPY( JW-1, H( KWTOP+1, KWTOP ), LDH+1, T( 2, 1 ), LDT+1 )
404: *
405: CALL ZLASET( 'A', JW, JW, ZERO, ONE, V, LDV )
406: CALL ZLAHQR( .true., .true., JW, 1, JW, T, LDT, SH( KWTOP ), 1,
407: $ JW, V, LDV, INFQR )
408: *
409: * ==== Deflation detection loop ====
410: *
411: NS = JW
412: ILST = INFQR + 1
413: DO 10 KNT = INFQR + 1, JW
414: *
415: * ==== Small spike tip deflation test ====
416: *
417: FOO = CABS1( T( NS, NS ) )
418: IF( FOO.EQ.RZERO )
419: $ FOO = CABS1( S )
420: IF( CABS1( S )*CABS1( V( 1, NS ) ).LE.MAX( SMLNUM, ULP*FOO ) )
421: $ THEN
422: *
423: * ==== One more converged eigenvalue ====
424: *
425: NS = NS - 1
426: ELSE
427: *
428: * ==== One undeflatable eigenvalue. Move it up out of the
429: * . way. (ZTREXC can not fail in this case.) ====
430: *
431: IFST = NS
432: CALL ZTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, INFO )
433: ILST = ILST + 1
434: END IF
435: 10 CONTINUE
436: *
437: * ==== Return to Hessenberg form ====
438: *
439: IF( NS.EQ.0 )
440: $ S = ZERO
441: *
442: IF( NS.LT.JW ) THEN
443: *
444: * ==== sorting the diagonal of T improves accuracy for
445: * . graded matrices. ====
446: *
447: DO 30 I = INFQR + 1, NS
448: IFST = I
449: DO 20 J = I + 1, NS
450: IF( CABS1( T( J, J ) ).GT.CABS1( T( IFST, IFST ) ) )
451: $ IFST = J
452: 20 CONTINUE
453: ILST = I
454: IF( IFST.NE.ILST )
455: $ CALL ZTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, INFO )
456: 30 CONTINUE
457: END IF
458: *
459: * ==== Restore shift/eigenvalue array from T ====
460: *
461: DO 40 I = INFQR + 1, JW
462: SH( KWTOP+I-1 ) = T( I, I )
463: 40 CONTINUE
464: *
465: *
466: IF( NS.LT.JW .OR. S.EQ.ZERO ) THEN
467: IF( NS.GT.1 .AND. S.NE.ZERO ) THEN
468: *
469: * ==== Reflect spike back into lower triangle ====
470: *
471: CALL ZCOPY( NS, V, LDV, WORK, 1 )
472: DO 50 I = 1, NS
473: WORK( I ) = DCONJG( WORK( I ) )
474: 50 CONTINUE
475: BETA = WORK( 1 )
476: CALL ZLARFG( NS, BETA, WORK( 2 ), 1, TAU )
477: WORK( 1 ) = ONE
478: *
479: CALL ZLASET( 'L', JW-2, JW-2, ZERO, ZERO, T( 3, 1 ), LDT )
480: *
481: CALL ZLARF( 'L', NS, JW, WORK, 1, DCONJG( TAU ), T, LDT,
482: $ WORK( JW+1 ) )
483: CALL ZLARF( 'R', NS, NS, WORK, 1, TAU, T, LDT,
484: $ WORK( JW+1 ) )
485: CALL ZLARF( 'R', JW, NS, WORK, 1, TAU, V, LDV,
486: $ WORK( JW+1 ) )
487: *
488: CALL ZGEHRD( JW, 1, NS, T, LDT, WORK, WORK( JW+1 ),
489: $ LWORK-JW, INFO )
490: END IF
491: *
492: * ==== Copy updated reduced window into place ====
493: *
494: IF( KWTOP.GT.1 )
495: $ H( KWTOP, KWTOP-1 ) = S*DCONJG( V( 1, 1 ) )
496: CALL ZLACPY( 'U', JW, JW, T, LDT, H( KWTOP, KWTOP ), LDH )
497: CALL ZCOPY( JW-1, T( 2, 1 ), LDT+1, H( KWTOP+1, KWTOP ),
498: $ LDH+1 )
499: *
500: * ==== Accumulate orthogonal matrix in order update
501: * . H and Z, if requested. ====
502: *
503: IF( NS.GT.1 .AND. S.NE.ZERO )
504: $ CALL ZUNMHR( 'R', 'N', JW, NS, 1, NS, T, LDT, WORK, V, LDV,
505: $ WORK( JW+1 ), LWORK-JW, INFO )
506: *
507: * ==== Update vertical slab in H ====
508: *
509: IF( WANTT ) THEN
510: LTOP = 1
511: ELSE
512: LTOP = KTOP
513: END IF
514: DO 60 KROW = LTOP, KWTOP - 1, NV
515: KLN = MIN( NV, KWTOP-KROW )
516: CALL ZGEMM( 'N', 'N', KLN, JW, JW, ONE, H( KROW, KWTOP ),
517: $ LDH, V, LDV, ZERO, WV, LDWV )
518: CALL ZLACPY( 'A', KLN, JW, WV, LDWV, H( KROW, KWTOP ), LDH )
519: 60 CONTINUE
520: *
521: * ==== Update horizontal slab in H ====
522: *
523: IF( WANTT ) THEN
524: DO 70 KCOL = KBOT + 1, N, NH
525: KLN = MIN( NH, N-KCOL+1 )
526: CALL ZGEMM( 'C', 'N', JW, KLN, JW, ONE, V, LDV,
527: $ H( KWTOP, KCOL ), LDH, ZERO, T, LDT )
528: CALL ZLACPY( 'A', JW, KLN, T, LDT, H( KWTOP, KCOL ),
529: $ LDH )
530: 70 CONTINUE
531: END IF
532: *
533: * ==== Update vertical slab in Z ====
534: *
535: IF( WANTZ ) THEN
536: DO 80 KROW = ILOZ, IHIZ, NV
537: KLN = MIN( NV, IHIZ-KROW+1 )
538: CALL ZGEMM( 'N', 'N', KLN, JW, JW, ONE, Z( KROW, KWTOP ),
539: $ LDZ, V, LDV, ZERO, WV, LDWV )
540: CALL ZLACPY( 'A', KLN, JW, WV, LDWV, Z( KROW, KWTOP ),
541: $ LDZ )
542: 80 CONTINUE
543: END IF
544: END IF
545: *
546: * ==== Return the number of deflations ... ====
547: *
548: ND = JW - NS
549: *
550: * ==== ... and the number of shifts. (Subtracting
551: * . INFQR from the spike length takes care
552: * . of the case of a rare QR failure while
553: * . calculating eigenvalues of the deflation
554: * . window.) ====
555: *
556: NS = NS - INFQR
557: *
558: * ==== Return optimal workspace. ====
559: *
560: WORK( 1 ) = DCMPLX( LWKOPT, 0 )
561: *
562: * ==== End of ZLAQR2 ====
563: *
564: END
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