LAPACK  3.8.0
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 .LE. NW .LE. (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 .LE. 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 .LE. ILOZ .LE. IHIZ .LE. 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 .LE. 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 .LE. LDV
189 *> \endverbatim
190 *>
191 *> \param[in] NH
192 *> \verbatim
193 *> NH is INTEGER
194 *> The number of columns of T. NH.GE.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 .LE. 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.GE.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 .LE. 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 *> \date June 2017
258 *
259 *> \ingroup complexOTHERauxiliary
260 *
261 *> \par Contributors:
262 * ==================
263 *>
264 *> Karen Braman and Ralph Byers, Department of Mathematics,
265 *> University of Kansas, USA
266 *>
267 * =====================================================================
268  SUBROUTINE claqr2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
269  $ IHIZ, Z, LDZ, NS, ND, SH, V, LDV, NH, T, LDT,
270  $ NV, WV, LDWV, WORK, LWORK )
271 *
272 * -- LAPACK auxiliary routine (version 3.7.1) --
273 * -- LAPACK is a software package provided by Univ. of Tennessee, --
274 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
275 * June 2017
276 *
277 * .. Scalar Arguments ..
278  INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
279  $ ldz, lwork, n, nd, nh, ns, nv, nw
280  LOGICAL WANTT, WANTZ
281 * ..
282 * .. Array Arguments ..
283  COMPLEX H( ldh, * ), SH( * ), T( ldt, * ), V( ldv, * ),
284  $ work( * ), wv( ldwv, * ), z( ldz, * )
285 * ..
286 *
287 * ================================================================
288 *
289 * .. Parameters ..
290  COMPLEX ZERO, ONE
291  parameter( zero = ( 0.0e0, 0.0e0 ),
292  $ one = ( 1.0e0, 0.0e0 ) )
293  REAL RZERO, RONE
294  parameter( rzero = 0.0e0, rone = 1.0e0 )
295 * ..
296 * .. Local Scalars ..
297  COMPLEX BETA, CDUM, S, TAU
298  REAL FOO, SAFMAX, SAFMIN, SMLNUM, ULP
299  INTEGER I, IFST, ILST, INFO, INFQR, J, JW, KCOL, KLN,
300  $ knt, krow, kwtop, ltop, lwk1, lwk2, lwkopt
301 * ..
302 * .. External Functions ..
303  REAL SLAMCH
304  EXTERNAL slamch
305 * ..
306 * .. External Subroutines ..
307  EXTERNAL ccopy, cgehrd, cgemm, clacpy, clahqr, clarf,
309 * ..
310 * .. Intrinsic Functions ..
311  INTRINSIC abs, aimag, cmplx, conjg, int, max, min, real
312 * ..
313 * .. Statement Functions ..
314  REAL CABS1
315 * ..
316 * .. Statement Function definitions ..
317  cabs1( cdum ) = abs( REAL( CDUM ) ) + abs( AIMAG( cdum ) )
318 * ..
319 * .. Executable Statements ..
320 *
321 * ==== Estimate optimal workspace. ====
322 *
323  jw = min( nw, kbot-ktop+1 )
324  IF( jw.LE.2 ) THEN
325  lwkopt = 1
326  ELSE
327 *
328 * ==== Workspace query call to CGEHRD ====
329 *
330  CALL cgehrd( jw, 1, jw-1, t, ldt, work, work, -1, info )
331  lwk1 = int( work( 1 ) )
332 *
333 * ==== Workspace query call to CUNMHR ====
334 *
335  CALL cunmhr( 'R', 'N', jw, jw, 1, jw-1, t, ldt, work, v, ldv,
336  $ work, -1, info )
337  lwk2 = int( work( 1 ) )
338 *
339 * ==== Optimal workspace ====
340 *
341  lwkopt = jw + max( lwk1, lwk2 )
342  END IF
343 *
344 * ==== Quick return in case of workspace query. ====
345 *
346  IF( lwork.EQ.-1 ) THEN
347  work( 1 ) = cmplx( lwkopt, 0 )
348  RETURN
349  END IF
350 *
351 * ==== Nothing to do ...
352 * ... for an empty active block ... ====
353  ns = 0
354  nd = 0
355  work( 1 ) = one
356  IF( ktop.GT.kbot )
357  $ RETURN
358 * ... nor for an empty deflation window. ====
359  IF( nw.LT.1 )
360  $ RETURN
361 *
362 * ==== Machine constants ====
363 *
364  safmin = slamch( 'SAFE MINIMUM' )
365  safmax = rone / safmin
366  CALL slabad( safmin, safmax )
367  ulp = slamch( 'PRECISION' )
368  smlnum = safmin*( REAL( N ) / ULP )
369 *
370 * ==== Setup deflation window ====
371 *
372  jw = min( nw, kbot-ktop+1 )
373  kwtop = kbot - jw + 1
374  IF( kwtop.EQ.ktop ) THEN
375  s = zero
376  ELSE
377  s = h( kwtop, kwtop-1 )
378  END IF
379 *
380  IF( kbot.EQ.kwtop ) THEN
381 *
382 * ==== 1-by-1 deflation window: not much to do ====
383 *
384  sh( kwtop ) = h( kwtop, kwtop )
385  ns = 1
386  nd = 0
387  IF( cabs1( s ).LE.max( smlnum, ulp*cabs1( h( kwtop,
388  $ kwtop ) ) ) ) THEN
389  ns = 0
390  nd = 1
391  IF( kwtop.GT.ktop )
392  $ h( kwtop, kwtop-1 ) = zero
393  END IF
394  work( 1 ) = one
395  RETURN
396  END IF
397 *
398 * ==== Convert to spike-triangular form. (In case of a
399 * . rare QR failure, this routine continues to do
400 * . aggressive early deflation using that part of
401 * . the deflation window that converged using INFQR
402 * . here and there to keep track.) ====
403 *
404  CALL clacpy( 'U', jw, jw, h( kwtop, kwtop ), ldh, t, ldt )
405  CALL ccopy( jw-1, h( kwtop+1, kwtop ), ldh+1, t( 2, 1 ), ldt+1 )
406 *
407  CALL claset( 'A', jw, jw, zero, one, v, ldv )
408  CALL clahqr( .true., .true., jw, 1, jw, t, ldt, sh( kwtop ), 1,
409  $ jw, v, ldv, infqr )
410 *
411 * ==== Deflation detection loop ====
412 *
413  ns = jw
414  ilst = infqr + 1
415  DO 10 knt = infqr + 1, jw
416 *
417 * ==== Small spike tip deflation test ====
418 *
419  foo = cabs1( t( ns, ns ) )
420  IF( foo.EQ.rzero )
421  $ foo = cabs1( s )
422  IF( cabs1( s )*cabs1( v( 1, ns ) ).LE.max( smlnum, ulp*foo ) )
423  $ THEN
424 *
425 * ==== One more converged eigenvalue ====
426 *
427  ns = ns - 1
428  ELSE
429 *
430 * ==== One undeflatable eigenvalue. Move it up out of the
431 * . way. (CTREXC can not fail in this case.) ====
432 *
433  ifst = ns
434  CALL ctrexc( 'V', jw, t, ldt, v, ldv, ifst, ilst, info )
435  ilst = ilst + 1
436  END IF
437  10 CONTINUE
438 *
439 * ==== Return to Hessenberg form ====
440 *
441  IF( ns.EQ.0 )
442  $ s = zero
443 *
444  IF( ns.LT.jw ) THEN
445 *
446 * ==== sorting the diagonal of T improves accuracy for
447 * . graded matrices. ====
448 *
449  DO 30 i = infqr + 1, ns
450  ifst = i
451  DO 20 j = i + 1, ns
452  IF( cabs1( t( j, j ) ).GT.cabs1( t( ifst, ifst ) ) )
453  $ ifst = j
454  20 CONTINUE
455  ilst = i
456  IF( ifst.NE.ilst )
457  $ CALL ctrexc( 'V', jw, t, ldt, v, ldv, ifst, ilst, info )
458  30 CONTINUE
459  END IF
460 *
461 * ==== Restore shift/eigenvalue array from T ====
462 *
463  DO 40 i = infqr + 1, jw
464  sh( kwtop+i-1 ) = t( i, i )
465  40 CONTINUE
466 *
467 *
468  IF( ns.LT.jw .OR. s.EQ.zero ) THEN
469  IF( ns.GT.1 .AND. s.NE.zero ) THEN
470 *
471 * ==== Reflect spike back into lower triangle ====
472 *
473  CALL ccopy( ns, v, ldv, work, 1 )
474  DO 50 i = 1, ns
475  work( i ) = conjg( work( i ) )
476  50 CONTINUE
477  beta = work( 1 )
478  CALL clarfg( ns, beta, work( 2 ), 1, tau )
479  work( 1 ) = one
480 *
481  CALL claset( 'L', jw-2, jw-2, zero, zero, t( 3, 1 ), ldt )
482 *
483  CALL clarf( 'L', ns, jw, work, 1, conjg( tau ), t, ldt,
484  $ work( jw+1 ) )
485  CALL clarf( 'R', ns, ns, work, 1, tau, t, ldt,
486  $ work( jw+1 ) )
487  CALL clarf( 'R', jw, ns, work, 1, tau, v, ldv,
488  $ work( jw+1 ) )
489 *
490  CALL cgehrd( jw, 1, ns, t, ldt, work, work( jw+1 ),
491  $ lwork-jw, info )
492  END IF
493 *
494 * ==== Copy updated reduced window into place ====
495 *
496  IF( kwtop.GT.1 )
497  $ h( kwtop, kwtop-1 ) = s*conjg( v( 1, 1 ) )
498  CALL clacpy( 'U', jw, jw, t, ldt, h( kwtop, kwtop ), ldh )
499  CALL ccopy( jw-1, t( 2, 1 ), ldt+1, h( kwtop+1, kwtop ),
500  $ ldh+1 )
501 *
502 * ==== Accumulate orthogonal matrix in order update
503 * . H and Z, if requested. ====
504 *
505  IF( ns.GT.1 .AND. s.NE.zero )
506  $ CALL cunmhr( 'R', 'N', jw, ns, 1, ns, t, ldt, work, v, ldv,
507  $ work( jw+1 ), lwork-jw, info )
508 *
509 * ==== Update vertical slab in H ====
510 *
511  IF( wantt ) THEN
512  ltop = 1
513  ELSE
514  ltop = ktop
515  END IF
516  DO 60 krow = ltop, kwtop - 1, nv
517  kln = min( nv, kwtop-krow )
518  CALL cgemm( 'N', 'N', kln, jw, jw, one, h( krow, kwtop ),
519  $ ldh, v, ldv, zero, wv, ldwv )
520  CALL clacpy( 'A', kln, jw, wv, ldwv, h( krow, kwtop ), ldh )
521  60 CONTINUE
522 *
523 * ==== Update horizontal slab in H ====
524 *
525  IF( wantt ) THEN
526  DO 70 kcol = kbot + 1, n, nh
527  kln = min( nh, n-kcol+1 )
528  CALL cgemm( 'C', 'N', jw, kln, jw, one, v, ldv,
529  $ h( kwtop, kcol ), ldh, zero, t, ldt )
530  CALL clacpy( 'A', jw, kln, t, ldt, h( kwtop, kcol ),
531  $ ldh )
532  70 CONTINUE
533  END IF
534 *
535 * ==== Update vertical slab in Z ====
536 *
537  IF( wantz ) THEN
538  DO 80 krow = iloz, ihiz, nv
539  kln = min( nv, ihiz-krow+1 )
540  CALL cgemm( 'N', 'N', kln, jw, jw, one, z( krow, kwtop ),
541  $ ldz, v, ldv, zero, wv, ldwv )
542  CALL clacpy( 'A', kln, jw, wv, ldwv, z( krow, kwtop ),
543  $ ldz )
544  80 CONTINUE
545  END IF
546  END IF
547 *
548 * ==== Return the number of deflations ... ====
549 *
550  nd = jw - ns
551 *
552 * ==== ... and the number of shifts. (Subtracting
553 * . INFQR from the spike length takes care
554 * . of the case of a rare QR failure while
555 * . calculating eigenvalues of the deflation
556 * . window.) ====
557 *
558  ns = ns - infqr
559 *
560 * ==== Return optimal workspace. ====
561 *
562  work( 1 ) = cmplx( lwkopt, 0 )
563 *
564 * ==== End of CLAQR2 ====
565 *
566  END
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, using the double-shift/single-shift QR algorithm.
Definition: clahqr.f:197
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:271
subroutine clarfg(N, ALPHA, X, INCX, TAU)
CLARFG generates an elementary reflector (Householder matrix).
Definition: clarfg.f:108
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:108
subroutine cunmhr(SIDE, TRANS, M, N, ILO, IHI, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMHR
Definition: cunmhr.f:181
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:105
subroutine slabad(SMALL, LARGE)
SLABAD
Definition: slabad.f:76
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:83
subroutine ctrexc(COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, INFO)
CTREXC
Definition: ctrexc.f:128
subroutine clarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
CLARF applies an elementary reflector to a general rectangular matrix.
Definition: clarf.f:130
subroutine cgehrd(N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
CGEHRD
Definition: cgehrd.f:169
subroutine cgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
CGEMM
Definition: cgemm.f:189