LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
claqr2.f
Go to the documentation of this file.
1 *> \brief \b CLAQR2 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 CLAQR2 + dependencies
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11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/claqr2.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/claqr2.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CLAQR2( 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 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 *> CLAQR2 is identical to CLAQR3 except that it avoids
42 *> recursion by calling CLAHQR instead of CLAQR4.
43 *>
44 *> Aggressive early deflation:
45 *>
46 *> This subroutine 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 *> \endverbatim
55 *
56 * Arguments:
57 * ==========
58 *
59 *> \param[in] WANTT
60 *> \verbatim
61 *> WANTT is LOGICAL
62 *> If .TRUE., then the Hessenberg matrix H is fully updated
63 *> so that the triangular Schur factor may be
64 *> computed (in cooperation with the calling subroutine).
65 *> If .FALSE., then only enough of H is updated to preserve
66 *> the eigenvalues.
67 *> \endverbatim
68 *>
69 *> \param[in] WANTZ
70 *> \verbatim
71 *> WANTZ is LOGICAL
72 *> If .TRUE., then the unitary matrix Z is updated so
73 *> so that the unitary Schur factor may be computed
74 *> (in cooperation with the calling subroutine).
75 *> If .FALSE., then Z is not referenced.
76 *> \endverbatim
77 *>
78 *> \param[in] N
79 *> \verbatim
80 *> N is INTEGER
81 *> The order of the matrix H and (if WANTZ is .TRUE.) the
82 *> order of the unitary matrix Z.
83 *> \endverbatim
84 *>
85 *> \param[in] KTOP
86 *> \verbatim
87 *> KTOP is INTEGER
88 *> It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
89 *> KBOT and KTOP together determine an isolated block
90 *> along the diagonal of the Hessenberg matrix.
91 *> \endverbatim
92 *>
93 *> \param[in] KBOT
94 *> \verbatim
95 *> KBOT is INTEGER
96 *> It is assumed without a check that either
97 *> KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together
98 *> determine an isolated block along the diagonal of the
99 *> Hessenberg matrix.
100 *> \endverbatim
101 *>
102 *> \param[in] NW
103 *> \verbatim
104 *> NW is INTEGER
105 *> Deflation window size. 1 <= NW <= (KBOT-KTOP+1).
106 *> \endverbatim
107 *>
108 *> \param[in,out] H
109 *> \verbatim
110 *> H is COMPLEX array, dimension (LDH,N)
111 *> On input the initial N-by-N section of H stores the
112 *> Hessenberg matrix undergoing aggressive early deflation.
113 *> On output H has been transformed by a unitary
114 *> similarity transformation, perturbed, and the returned
115 *> to Hessenberg form that (it is to be hoped) has some
116 *> zero subdiagonal entries.
117 *> \endverbatim
118 *>
119 *> \param[in] LDH
120 *> \verbatim
121 *> LDH is INTEGER
122 *> Leading dimension of H just as declared in the calling
123 *> subroutine. N <= LDH
124 *> \endverbatim
125 *>
126 *> \param[in] ILOZ
127 *> \verbatim
128 *> ILOZ is INTEGER
129 *> \endverbatim
130 *>
131 *> \param[in] IHIZ
132 *> \verbatim
133 *> IHIZ is INTEGER
134 *> Specify the rows of Z to which transformations must be
135 *> applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
136 *> \endverbatim
137 *>
138 *> \param[in,out] Z
139 *> \verbatim
140 *> Z is COMPLEX array, dimension (LDZ,N)
141 *> IF WANTZ is .TRUE., then on output, the unitary
142 *> similarity transformation mentioned above has been
143 *> accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
144 *> If WANTZ is .FALSE., then Z is unreferenced.
145 *> \endverbatim
146 *>
147 *> \param[in] LDZ
148 *> \verbatim
149 *> LDZ is INTEGER
150 *> The leading dimension of Z just as declared in the
151 *> calling subroutine. 1 <= LDZ.
152 *> \endverbatim
153 *>
154 *> \param[out] NS
155 *> \verbatim
156 *> NS is INTEGER
157 *> The number of unconverged (ie approximate) eigenvalues
158 *> returned in SR and SI that may be used as shifts by the
159 *> calling subroutine.
160 *> \endverbatim
161 *>
162 *> \param[out] ND
163 *> \verbatim
164 *> ND is INTEGER
165 *> The number of converged eigenvalues uncovered by this
166 *> subroutine.
167 *> \endverbatim
168 *>
169 *> \param[out] SH
170 *> \verbatim
171 *> SH is COMPLEX array, dimension (KBOT)
172 *> On output, approximate eigenvalues that may
173 *> be used for shifts are stored in SH(KBOT-ND-NS+1)
174 *> through SR(KBOT-ND). Converged eigenvalues are
175 *> stored in SH(KBOT-ND+1) through SH(KBOT).
176 *> \endverbatim
177 *>
178 *> \param[out] V
179 *> \verbatim
180 *> V is COMPLEX array, dimension (LDV,NW)
181 *> An NW-by-NW work array.
182 *> \endverbatim
183 *>
184 *> \param[in] LDV
185 *> \verbatim
186 *> LDV is INTEGER
187 *> The leading dimension of V just as declared in the
188 *> calling subroutine. NW <= LDV
189 *> \endverbatim
190 *>
191 *> \param[in] NH
192 *> \verbatim
193 *> NH is INTEGER
194 *> The number of columns of T. NH >= NW.
195 *> \endverbatim
196 *>
197 *> \param[out] T
198 *> \verbatim
199 *> T is COMPLEX array, dimension (LDT,NW)
200 *> \endverbatim
201 *>
202 *> \param[in] LDT
203 *> \verbatim
204 *> LDT is INTEGER
205 *> The leading dimension of T just as declared in the
206 *> calling subroutine. NW <= LDT
207 *> \endverbatim
208 *>
209 *> \param[in] NV
210 *> \verbatim
211 *> NV is INTEGER
212 *> The number of rows of work array WV available for
213 *> workspace. NV >= NW.
214 *> \endverbatim
215 *>
216 *> \param[out] WV
217 *> \verbatim
218 *> WV is COMPLEX array, dimension (LDWV,NW)
219 *> \endverbatim
220 *>
221 *> \param[in] LDWV
222 *> \verbatim
223 *> LDWV is INTEGER
224 *> The leading dimension of W just as declared in the
225 *> calling subroutine. NW <= LDV
226 *> \endverbatim
227 *>
228 *> \param[out] WORK
229 *> \verbatim
230 *> WORK is COMPLEX array, dimension (LWORK)
231 *> On exit, WORK(1) is set to an estimate of the optimal value
232 *> of LWORK for the given values of N, NW, KTOP and KBOT.
233 *> \endverbatim
234 *>
235 *> \param[in] LWORK
236 *> \verbatim
237 *> LWORK is INTEGER
238 *> The dimension of the work array WORK. LWORK = 2*NW
239 *> suffices, but greater efficiency may result from larger
240 *> values of LWORK.
241 *>
242 *> If LWORK = -1, then a workspace query is assumed; CLAQR2
243 *> only estimates the optimal workspace size for the given
244 *> values of N, NW, KTOP and KBOT. The estimate is returned
245 *> in WORK(1). No error message related to LWORK is issued
246 *> by XERBLA. Neither H nor Z are accessed.
247 *> \endverbatim
248 *
249 * Authors:
250 * ========
251 *
252 *> \author Univ. of Tennessee
253 *> \author Univ. of California Berkeley
254 *> \author Univ. of Colorado Denver
255 *> \author NAG Ltd.
256 *
257 *> \ingroup complexOTHERauxiliary
258 *
259 *> \par Contributors:
260 * ==================
261 *>
262 *> Karen Braman and Ralph Byers, Department of Mathematics,
263 *> University of Kansas, USA
264 *>
265 * =====================================================================
266  SUBROUTINE claqr2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
267  $ IHIZ, Z, LDZ, NS, ND, SH, V, LDV, NH, T, LDT,
268  $ NV, WV, LDWV, WORK, LWORK )
269 *
270 * -- LAPACK auxiliary routine --
271 * -- LAPACK is a software package provided by Univ. of Tennessee, --
272 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
273 *
274 * .. Scalar Arguments ..
275  INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
276  $ LDZ, LWORK, N, ND, NH, NS, NV, NW
277  LOGICAL WANTT, WANTZ
278 * ..
279 * .. Array Arguments ..
280  COMPLEX H( LDH, * ), SH( * ), T( LDT, * ), V( LDV, * ),
281  $ WORK( * ), WV( LDWV, * ), Z( LDZ, * )
282 * ..
283 *
284 * ================================================================
285 *
286 * .. Parameters ..
287  COMPLEX ZERO, ONE
288  PARAMETER ( ZERO = ( 0.0e0, 0.0e0 ),
289  $ one = ( 1.0e0, 0.0e0 ) )
290  REAL RZERO, RONE
291  PARAMETER ( RZERO = 0.0e0, rone = 1.0e0 )
292 * ..
293 * .. Local Scalars ..
294  COMPLEX BETA, CDUM, S, TAU
295  REAL FOO, SAFMAX, SAFMIN, SMLNUM, ULP
296  INTEGER I, IFST, ILST, INFO, INFQR, J, JW, KCOL, KLN,
297  $ knt, krow, kwtop, ltop, lwk1, lwk2, lwkopt
298 * ..
299 * .. External Functions ..
300  REAL SLAMCH
301  EXTERNAL SLAMCH
302 * ..
303 * .. External Subroutines ..
304  EXTERNAL ccopy, cgehrd, cgemm, clacpy, clahqr, clarf,
306 * ..
307 * .. Intrinsic Functions ..
308  INTRINSIC abs, aimag, cmplx, conjg, int, max, min, real
309 * ..
310 * .. Statement Functions ..
311  REAL CABS1
312 * ..
313 * .. Statement Function definitions ..
314  cabs1( cdum ) = abs( real( cdum ) ) + abs( aimag( cdum ) )
315 * ..
316 * .. Executable Statements ..
317 *
318 * ==== Estimate optimal workspace. ====
319 *
320  jw = min( nw, kbot-ktop+1 )
321  IF( jw.LE.2 ) THEN
322  lwkopt = 1
323  ELSE
324 *
325 * ==== Workspace query call to CGEHRD ====
326 *
327  CALL cgehrd( jw, 1, jw-1, t, ldt, work, work, -1, info )
328  lwk1 = int( work( 1 ) )
329 *
330 * ==== Workspace query call to CUNMHR ====
331 *
332  CALL cunmhr( 'R', 'N', jw, jw, 1, jw-1, t, ldt, work, v, ldv,
333  $ work, -1, info )
334  lwk2 = int( work( 1 ) )
335 *
336 * ==== Optimal workspace ====
337 *
338  lwkopt = jw + max( lwk1, lwk2 )
339  END IF
340 *
341 * ==== Quick return in case of workspace query. ====
342 *
343  IF( lwork.EQ.-1 ) THEN
344  work( 1 ) = cmplx( lwkopt, 0 )
345  RETURN
346  END IF
347 *
348 * ==== Nothing to do ...
349 * ... for an empty active block ... ====
350  ns = 0
351  nd = 0
352  work( 1 ) = one
353  IF( ktop.GT.kbot )
354  $ RETURN
355 * ... nor for an empty deflation window. ====
356  IF( nw.LT.1 )
357  $ RETURN
358 *
359 * ==== Machine constants ====
360 *
361  safmin = slamch( 'SAFE MINIMUM' )
362  safmax = rone / safmin
363  CALL slabad( safmin, safmax )
364  ulp = slamch( 'PRECISION' )
365  smlnum = safmin*( real( n ) / ulp )
366 *
367 * ==== Setup deflation window ====
368 *
369  jw = min( nw, kbot-ktop+1 )
370  kwtop = kbot - jw + 1
371  IF( kwtop.EQ.ktop ) THEN
372  s = zero
373  ELSE
374  s = h( kwtop, kwtop-1 )
375  END IF
376 *
377  IF( kbot.EQ.kwtop ) THEN
378 *
379 * ==== 1-by-1 deflation window: not much to do ====
380 *
381  sh( kwtop ) = h( kwtop, kwtop )
382  ns = 1
383  nd = 0
384  IF( cabs1( s ).LE.max( smlnum, ulp*cabs1( h( kwtop,
385  $ kwtop ) ) ) ) THEN
386  ns = 0
387  nd = 1
388  IF( kwtop.GT.ktop )
389  $ h( kwtop, kwtop-1 ) = zero
390  END IF
391  work( 1 ) = one
392  RETURN
393  END IF
394 *
395 * ==== Convert to spike-triangular form. (In case of a
396 * . rare QR failure, this routine continues to do
397 * . aggressive early deflation using that part of
398 * . the deflation window that converged using INFQR
399 * . here and there to keep track.) ====
400 *
401  CALL clacpy( 'U', jw, jw, h( kwtop, kwtop ), ldh, t, ldt )
402  CALL ccopy( jw-1, h( kwtop+1, kwtop ), ldh+1, t( 2, 1 ), ldt+1 )
403 *
404  CALL claset( 'A', jw, jw, zero, one, v, ldv )
405  CALL clahqr( .true., .true., jw, 1, jw, t, ldt, sh( kwtop ), 1,
406  $ jw, v, ldv, infqr )
407 *
408 * ==== Deflation detection loop ====
409 *
410  ns = jw
411  ilst = infqr + 1
412  DO 10 knt = infqr + 1, jw
413 *
414 * ==== Small spike tip deflation test ====
415 *
416  foo = cabs1( t( ns, ns ) )
417  IF( foo.EQ.rzero )
418  $ foo = cabs1( s )
419  IF( cabs1( s )*cabs1( v( 1, ns ) ).LE.max( smlnum, ulp*foo ) )
420  $ THEN
421 *
422 * ==== One more converged eigenvalue ====
423 *
424  ns = ns - 1
425  ELSE
426 *
427 * ==== One undeflatable eigenvalue. Move it up out of the
428 * . way. (CTREXC can not fail in this case.) ====
429 *
430  ifst = ns
431  CALL ctrexc( 'V', jw, t, ldt, v, ldv, ifst, ilst, info )
432  ilst = ilst + 1
433  END IF
434  10 CONTINUE
435 *
436 * ==== Return to Hessenberg form ====
437 *
438  IF( ns.EQ.0 )
439  $ s = zero
440 *
441  IF( ns.LT.jw ) THEN
442 *
443 * ==== sorting the diagonal of T improves accuracy for
444 * . graded matrices. ====
445 *
446  DO 30 i = infqr + 1, ns
447  ifst = i
448  DO 20 j = i + 1, ns
449  IF( cabs1( t( j, j ) ).GT.cabs1( t( ifst, ifst ) ) )
450  $ ifst = j
451  20 CONTINUE
452  ilst = i
453  IF( ifst.NE.ilst )
454  $ CALL ctrexc( 'V', jw, t, ldt, v, ldv, ifst, ilst, info )
455  30 CONTINUE
456  END IF
457 *
458 * ==== Restore shift/eigenvalue array from T ====
459 *
460  DO 40 i = infqr + 1, jw
461  sh( kwtop+i-1 ) = t( i, i )
462  40 CONTINUE
463 *
464 *
465  IF( ns.LT.jw .OR. s.EQ.zero ) THEN
466  IF( ns.GT.1 .AND. s.NE.zero ) THEN
467 *
468 * ==== Reflect spike back into lower triangle ====
469 *
470  CALL ccopy( ns, v, ldv, work, 1 )
471  DO 50 i = 1, ns
472  work( i ) = conjg( work( i ) )
473  50 CONTINUE
474  beta = work( 1 )
475  CALL clarfg( ns, beta, work( 2 ), 1, tau )
476  work( 1 ) = one
477 *
478  CALL claset( 'L', jw-2, jw-2, zero, zero, t( 3, 1 ), ldt )
479 *
480  CALL clarf( 'L', ns, jw, work, 1, conjg( tau ), t, ldt,
481  $ work( jw+1 ) )
482  CALL clarf( 'R', ns, ns, work, 1, tau, t, ldt,
483  $ work( jw+1 ) )
484  CALL clarf( 'R', jw, ns, work, 1, tau, v, ldv,
485  $ work( jw+1 ) )
486 *
487  CALL cgehrd( jw, 1, ns, t, ldt, work, work( jw+1 ),
488  $ lwork-jw, info )
489  END IF
490 *
491 * ==== Copy updated reduced window into place ====
492 *
493  IF( kwtop.GT.1 )
494  $ h( kwtop, kwtop-1 ) = s*conjg( v( 1, 1 ) )
495  CALL clacpy( 'U', jw, jw, t, ldt, h( kwtop, kwtop ), ldh )
496  CALL ccopy( jw-1, t( 2, 1 ), ldt+1, h( kwtop+1, kwtop ),
497  $ ldh+1 )
498 *
499 * ==== Accumulate orthogonal matrix in order update
500 * . H and Z, if requested. ====
501 *
502  IF( ns.GT.1 .AND. s.NE.zero )
503  $ CALL cunmhr( 'R', 'N', jw, ns, 1, ns, t, ldt, work, v, ldv,
504  $ work( jw+1 ), lwork-jw, info )
505 *
506 * ==== Update vertical slab in H ====
507 *
508  IF( wantt ) THEN
509  ltop = 1
510  ELSE
511  ltop = ktop
512  END IF
513  DO 60 krow = ltop, kwtop - 1, nv
514  kln = min( nv, kwtop-krow )
515  CALL cgemm( 'N', 'N', kln, jw, jw, one, h( krow, kwtop ),
516  $ ldh, v, ldv, zero, wv, ldwv )
517  CALL clacpy( 'A', kln, jw, wv, ldwv, h( krow, kwtop ), ldh )
518  60 CONTINUE
519 *
520 * ==== Update horizontal slab in H ====
521 *
522  IF( wantt ) THEN
523  DO 70 kcol = kbot + 1, n, nh
524  kln = min( nh, n-kcol+1 )
525  CALL cgemm( 'C', 'N', jw, kln, jw, one, v, ldv,
526  $ h( kwtop, kcol ), ldh, zero, t, ldt )
527  CALL clacpy( 'A', jw, kln, t, ldt, h( kwtop, kcol ),
528  $ ldh )
529  70 CONTINUE
530  END IF
531 *
532 * ==== Update vertical slab in Z ====
533 *
534  IF( wantz ) THEN
535  DO 80 krow = iloz, ihiz, nv
536  kln = min( nv, ihiz-krow+1 )
537  CALL cgemm( 'N', 'N', kln, jw, jw, one, z( krow, kwtop ),
538  $ ldz, v, ldv, zero, wv, ldwv )
539  CALL clacpy( 'A', kln, jw, wv, ldwv, z( krow, kwtop ),
540  $ ldz )
541  80 CONTINUE
542  END IF
543  END IF
544 *
545 * ==== Return the number of deflations ... ====
546 *
547  nd = jw - ns
548 *
549 * ==== ... and the number of shifts. (Subtracting
550 * . INFQR from the spike length takes care
551 * . of the case of a rare QR failure while
552 * . calculating eigenvalues of the deflation
553 * . window.) ====
554 *
555  ns = ns - infqr
556 *
557 * ==== Return optimal workspace. ====
558 *
559  work( 1 ) = cmplx( lwkopt, 0 )
560 *
561 * ==== End of CLAQR2 ====
562 *
563  END
subroutine slabad(SMALL, LARGE)
SLABAD
Definition: slabad.f:74
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:81
subroutine cgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
CGEMM
Definition: cgemm.f:187
subroutine cgehrd(N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
CGEHRD
Definition: cgehrd.f:167
subroutine claset(UPLO, M, N, ALPHA, BETA, A, LDA)
CLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: claset.f:106
subroutine clarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
CLARF applies an elementary reflector to a general rectangular matrix.
Definition: clarf.f:128
subroutine claqr2(WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND, SH, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)
CLAQR2 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fu...
Definition: claqr2.f:269
subroutine clarfg(N, ALPHA, X, INCX, TAU)
CLARFG generates an elementary reflector (Householder matrix).
Definition: clarfg.f:106
subroutine clahqr(WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, INFO)
CLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix,...
Definition: clahqr.f:195
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:103
subroutine cunmhr(SIDE, TRANS, M, N, ILO, IHI, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMHR
Definition: cunmhr.f:179
subroutine ctrexc(COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, INFO)
CTREXC
Definition: ctrexc.f:126