LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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claqz3.f
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1*> \brief \b CLAQZ3
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download CLAQZ3 + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/CLAQZ3.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/CLAQZ3.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/CLAQZ3.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE CLAQZ3( ILSCHUR, ILQ, ILZ, N, ILO, IHI, NSHIFTS,
22* $ NBLOCK_DESIRED, ALPHA, BETA, A, LDA, B, LDB, Q, LDQ, Z, LDZ,
23* $ QC, LDQC, ZC, LDZC, WORK, LWORK, INFO )
24* IMPLICIT NONE
25*
26* Function arguments
27* LOGICAL, INTENT( IN ) :: ILSCHUR, ILQ, ILZ
28* INTEGER, INTENT( IN ) :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK,
29* $ NSHIFTS, NBLOCK_DESIRED, LDQC, LDZC
30*
31* COMPLEX, INTENT( INOUT ) :: A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
32* $ Z( LDZ, * ), QC( LDQC, * ), ZC( LDZC, * ), WORK( * ),
33* $ ALPHA( * ), BETA( * )
34*
35* INTEGER, INTENT( OUT ) :: INFO
36* ..
37*
38*
39*> \par Purpose:
40* =============
41*>
42*> \verbatim
43*>
44*> CLAQZ3 Executes a single multishift QZ sweep
45*> \endverbatim
46*
47* Arguments:
48* ==========
49*
50*> \param[in] ILSCHUR
51*> \verbatim
52*> ILSCHUR is LOGICAL
53*> Determines whether or not to update the full Schur form
54*> \endverbatim
55*> \param[in] ILQ
56*> \verbatim
57*> ILQ is LOGICAL
58*> Determines whether or not to update the matrix Q
59*> \endverbatim
60*>
61*> \param[in] ILZ
62*> \verbatim
63*> ILZ is LOGICAL
64*> Determines whether or not to update the matrix Z
65*> \endverbatim
66*>
67*> \param[in] N
68*> \verbatim
69*> N is INTEGER
70*> The order of the matrices A, B, Q, and Z. N >= 0.
71*> \endverbatim
72*>
73*> \param[in] ILO
74*> \verbatim
75*> ILO is INTEGER
76*> \endverbatim
77*>
78*> \param[in] IHI
79*> \verbatim
80*> IHI is INTEGER
81*> \endverbatim
82*>
83*> \param[in] NSHIFTS
84*> \verbatim
85*> NSHIFTS is INTEGER
86*> The desired number of shifts to use
87*> \endverbatim
88*>
89*> \param[in] NBLOCK_DESIRED
90*> \verbatim
91*> NBLOCK_DESIRED is INTEGER
92*> The desired size of the computational windows
93*> \endverbatim
94*>
95*> \param[in] ALPHA
96*> \verbatim
97*> ALPHA is COMPLEX array. SR contains
98*> the alpha parts of the shifts to use.
99*> \endverbatim
100*>
101*> \param[in] BETA
102*> \verbatim
103*> BETA is COMPLEX array. SS contains
104*> the scale of the shifts to use.
105*> \endverbatim
106*>
107*> \param[in,out] A
108*> \verbatim
109*> A is COMPLEX array, dimension (LDA, N)
110*> \endverbatim
111*>
112*> \param[in] LDA
113*> \verbatim
114*> LDA is INTEGER
115*> The leading dimension of the array A. LDA >= max( 1, N ).
116*> \endverbatim
117*>
118*> \param[in,out] B
119*> \verbatim
120*> B is COMPLEX array, dimension (LDB, N)
121*> \endverbatim
122*>
123*> \param[in] LDB
124*> \verbatim
125*> LDB is INTEGER
126*> The leading dimension of the array B. LDB >= max( 1, N ).
127*> \endverbatim
128*>
129*> \param[in,out] Q
130*> \verbatim
131*> Q is COMPLEX array, dimension (LDQ, N)
132*> \endverbatim
133*>
134*> \param[in] LDQ
135*> \verbatim
136*> LDQ is INTEGER
137*> \endverbatim
138*>
139*> \param[in,out] Z
140*> \verbatim
141*> Z is COMPLEX array, dimension (LDZ, N)
142*> \endverbatim
143*>
144*> \param[in] LDZ
145*> \verbatim
146*> LDZ is INTEGER
147*> \endverbatim
148*>
149*> \param[in,out] QC
150*> \verbatim
151*> QC is COMPLEX array, dimension (LDQC, NBLOCK_DESIRED)
152*> \endverbatim
153*>
154*> \param[in] LDQC
155*> \verbatim
156*> LDQC is INTEGER
157*> \endverbatim
158*>
159*> \param[in,out] ZC
160*> \verbatim
161*> ZC is COMPLEX array, dimension (LDZC, NBLOCK_DESIRED)
162*> \endverbatim
163*>
164*> \param[in] LDZC
165*> \verbatim
166*> LDZ is INTEGER
167*> \endverbatim
168*>
169*> \param[out] WORK
170*> \verbatim
171*> WORK is COMPLEX array, dimension (MAX(1,LWORK))
172*> On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
173*> \endverbatim
174*>
175*> \param[in] LWORK
176*> \verbatim
177*> LWORK is INTEGER
178*> The dimension of the array WORK. LWORK >= max(1,N).
179*>
180*> If LWORK = -1, then a workspace query is assumed; the routine
181*> only calculates the optimal size of the WORK array, returns
182*> this value as the first entry of the WORK array, and no error
183*> message related to LWORK is issued by XERBLA.
184*> \endverbatim
185*>
186*> \param[out] INFO
187*> \verbatim
188*> INFO is INTEGER
189*> = 0: successful exit
190*> < 0: if INFO = -i, the i-th argument had an illegal value
191*> \endverbatim
192*
193* Authors:
194* ========
195*
196*> \author Thijs Steel, KU Leuven
197*
198*> \date May 2020
199*
200*> \ingroup laqz3
201*>
202* =====================================================================
203 SUBROUTINE claqz3( ILSCHUR, ILQ, ILZ, N, ILO, IHI, NSHIFTS,
204 $ NBLOCK_DESIRED, ALPHA, BETA, A, LDA, B, LDB,
205 $ Q, LDQ, Z, LDZ, QC, LDQC, ZC, LDZC, WORK,
206 $ LWORK, INFO )
207 IMPLICIT NONE
208
209* Function arguments
210 LOGICAL, INTENT( IN ) :: ILSCHUR, ILQ, ILZ
211 INTEGER, INTENT( IN ) :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK,
212 $ NSHIFTS, NBLOCK_DESIRED, LDQC, LDZC
213
214 COMPLEX, INTENT( INOUT ) :: A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
215 $ Z( LDZ, * ), QC( LDQC, * ), ZC( LDZC, * ), WORK( * ),
216 $ ALPHA( * ), BETA( * )
217
218 INTEGER, INTENT( OUT ) :: INFO
219
220* Parameters
221 COMPLEX CZERO, CONE
222 PARAMETER ( CZERO = ( 0.0, 0.0 ), cone = ( 1.0, 0.0 ) )
223 REAL :: ZERO, ONE, HALF
224 parameter( zero = 0.0, one = 1.0, half = 0.5 )
225
226* Local scalars
227 INTEGER :: I, J, NS, ISTARTM, ISTOPM, SHEIGHT, SWIDTH, K, NP,
228 $ ISTARTB, ISTOPB, ISHIFT, NBLOCK, NPOS
229 REAL :: SAFMIN, SAFMAX, C, SCALE
230 COMPLEX :: TEMP, TEMP2, TEMP3, S
231
232* External Functions
233 EXTERNAL :: xerbla, claset, clartg, crot, claqz1, cgemm, clacpy
234 REAL, EXTERNAL :: SLAMCH
235
236 info = 0
237 IF ( nblock_desired .LT. nshifts+1 ) THEN
238 info = -8
239 END IF
240 IF ( lwork .EQ.-1 ) THEN
241* workspace query, quick return
242 work( 1 ) = n*nblock_desired
243 RETURN
244 ELSE IF ( lwork .LT. n*nblock_desired ) THEN
245 info = -25
246 END IF
247
248 IF( info.NE.0 ) THEN
249 CALL xerbla( 'CLAQZ3', -info )
250 RETURN
251 END IF
252
253*
254* Executable statements
255*
256
257* Get machine constants
258 safmin = slamch( 'SAFE MINIMUM' )
259 safmax = one/safmin
260
261 IF ( ilo .GE. ihi ) THEN
262 RETURN
263 END IF
264
265 IF ( ilschur ) THEN
266 istartm = 1
267 istopm = n
268 ELSE
269 istartm = ilo
270 istopm = ihi
271 END IF
272
273 ns = nshifts
274 npos = max( nblock_desired-ns, 1 )
275
276
277* The following block introduces the shifts and chases
278* them down one by one just enough to make space for
279* the other shifts. The near-the-diagonal block is
280* of size (ns+1) x ns.
281
282 CALL claset( 'FULL', ns+1, ns+1, czero, cone, qc, ldqc )
283 CALL claset( 'FULL', ns, ns, czero, cone, zc, ldzc )
284
285 DO i = 1, ns
286* Introduce the shift
287 scale = sqrt( abs( alpha( i ) ) ) * sqrt( abs( beta( i ) ) )
288 IF( scale .GE. safmin .AND. scale .LE. safmax ) THEN
289 alpha( i ) = alpha( i )/scale
290 beta( i ) = beta( i )/scale
291 END IF
292
293 temp2 = beta( i )*a( ilo, ilo )-alpha( i )*b( ilo, ilo )
294 temp3 = beta( i )*a( ilo+1, ilo )
295
296 IF ( abs( temp2 ) .GT. safmax .OR.
297 $ abs( temp3 ) .GT. safmax ) THEN
298 temp2 = cone
299 temp3 = czero
300 END IF
301
302 CALL clartg( temp2, temp3, c, s, temp )
303 CALL crot( ns, a( ilo, ilo ), lda, a( ilo+1, ilo ), lda, c,
304 $ s )
305 CALL crot( ns, b( ilo, ilo ), ldb, b( ilo+1, ilo ), ldb, c,
306 $ s )
307 CALL crot( ns+1, qc( 1, 1 ), 1, qc( 1, 2 ), 1, c, conjg( s ) )
308
309* Chase the shift down
310 DO j = 1, ns-i
311
312 CALL claqz1( .true., .true., j, 1, ns, ihi-ilo+1, a( ilo,
313 $ ilo ), lda, b( ilo, ilo ), ldb, ns+1, 1, qc,
314 $ ldqc, ns, 1, zc, ldzc )
315
316 END DO
317
318 END DO
319
320* Update the rest of the pencil
321
322* Update A(ilo:ilo+ns,ilo+ns:istopm) and B(ilo:ilo+ns,ilo+ns:istopm)
323* from the left with Qc(1:ns+1,1:ns+1)'
324 sheight = ns+1
325 swidth = istopm-( ilo+ns )+1
326 IF ( swidth > 0 ) THEN
327 CALL cgemm( 'C', 'N', sheight, swidth, sheight, cone, qc, ldqc,
328 $ a( ilo, ilo+ns ), lda, czero, work, sheight )
329 CALL clacpy( 'ALL', sheight, swidth, work, sheight, a( ilo,
330 $ ilo+ns ), lda )
331 CALL cgemm( 'C', 'N', sheight, swidth, sheight, cone, qc, ldqc,
332 $ b( ilo, ilo+ns ), ldb, czero, work, sheight )
333 CALL clacpy( 'ALL', sheight, swidth, work, sheight, b( ilo,
334 $ ilo+ns ), ldb )
335 END IF
336 IF ( ilq ) THEN
337 CALL cgemm( 'N', 'N', n, sheight, sheight, cone, q( 1, ilo ),
338 $ ldq, qc, ldqc, czero, work, n )
339 CALL clacpy( 'ALL', n, sheight, work, n, q( 1, ilo ), ldq )
340 END IF
341
342* Update A(istartm:ilo-1,ilo:ilo+ns-1) and B(istartm:ilo-1,ilo:ilo+ns-1)
343* from the right with Zc(1:ns,1:ns)
344 sheight = ilo-1-istartm+1
345 swidth = ns
346 IF ( sheight > 0 ) THEN
347 CALL cgemm( 'N', 'N', sheight, swidth, swidth, cone,
348 $ a( istartm, ilo ), lda, zc, ldzc, czero, work,
349 $ sheight )
350 CALL clacpy( 'ALL', sheight, swidth, work, sheight, a( istartm,
351 $ ilo ), lda )
352 CALL cgemm( 'N', 'N', sheight, swidth, swidth, cone,
353 $ b( istartm, ilo ), ldb, zc, ldzc, czero, work,
354 $ sheight )
355 CALL clacpy( 'ALL', sheight, swidth, work, sheight, b( istartm,
356 $ ilo ), ldb )
357 END IF
358 IF ( ilz ) THEN
359 CALL cgemm( 'N', 'N', n, swidth, swidth, cone, z( 1, ilo ),
360 $ ldz, zc, ldzc, czero, work, n )
361 CALL clacpy( 'ALL', n, swidth, work, n, z( 1, ilo ), ldz )
362 END IF
363
364* The following block chases the shifts down to the bottom
365* right block. If possible, a shift is moved down npos
366* positions at a time
367
368 k = ilo
369 DO WHILE ( k < ihi-ns )
370 np = min( ihi-ns-k, npos )
371* Size of the near-the-diagonal block
372 nblock = ns+np
373* istartb points to the first row we will be updating
374 istartb = k+1
375* istopb points to the last column we will be updating
376 istopb = k+nblock-1
377
378 CALL claset( 'FULL', ns+np, ns+np, czero, cone, qc, ldqc )
379 CALL claset( 'FULL', ns+np, ns+np, czero, cone, zc, ldzc )
380
381* Near the diagonal shift chase
382 DO i = ns-1, 0, -1
383 DO j = 0, np-1
384* Move down the block with index k+i+j, updating
385* the (ns+np x ns+np) block:
386* (k:k+ns+np,k:k+ns+np-1)
387 CALL claqz1( .true., .true., k+i+j, istartb, istopb, ihi,
388 $ a, lda, b, ldb, nblock, k+1, qc, ldqc,
389 $ nblock, k, zc, ldzc )
390 END DO
391 END DO
392
393* Update rest of the pencil
394
395* Update A(k+1:k+ns+np, k+ns+np:istopm) and
396* B(k+1:k+ns+np, k+ns+np:istopm)
397* from the left with Qc(1:ns+np,1:ns+np)'
398 sheight = ns+np
399 swidth = istopm-( k+ns+np )+1
400 IF ( swidth > 0 ) THEN
401 CALL cgemm( 'C', 'N', sheight, swidth, sheight, cone, qc,
402 $ ldqc, a( k+1, k+ns+np ), lda, czero, work,
403 $ sheight )
404 CALL clacpy( 'ALL', sheight, swidth, work, sheight, a( k+1,
405 $ k+ns+np ), lda )
406 CALL cgemm( 'C', 'N', sheight, swidth, sheight, cone, qc,
407 $ ldqc, b( k+1, k+ns+np ), ldb, czero, work,
408 $ sheight )
409 CALL clacpy( 'ALL', sheight, swidth, work, sheight, b( k+1,
410 $ k+ns+np ), ldb )
411 END IF
412 IF ( ilq ) THEN
413 CALL cgemm( 'N', 'N', n, nblock, nblock, cone, q( 1, k+1 ),
414 $ ldq, qc, ldqc, czero, work, n )
415 CALL clacpy( 'ALL', n, nblock, work, n, q( 1, k+1 ), ldq )
416 END IF
417
418* Update A(istartm:k,k:k+ns+npos-1) and B(istartm:k,k:k+ns+npos-1)
419* from the right with Zc(1:ns+np,1:ns+np)
420 sheight = k-istartm+1
421 swidth = nblock
422 IF ( sheight > 0 ) THEN
423 CALL cgemm( 'N', 'N', sheight, swidth, swidth, cone,
424 $ a( istartm, k ), lda, zc, ldzc, czero, work,
425 $ sheight )
426 CALL clacpy( 'ALL', sheight, swidth, work, sheight,
427 $ a( istartm, k ), lda )
428 CALL cgemm( 'N', 'N', sheight, swidth, swidth, cone,
429 $ b( istartm, k ), ldb, zc, ldzc, czero, work,
430 $ sheight )
431 CALL clacpy( 'ALL', sheight, swidth, work, sheight,
432 $ b( istartm, k ), ldb )
433 END IF
434 IF ( ilz ) THEN
435 CALL cgemm( 'N', 'N', n, nblock, nblock, cone, z( 1, k ),
436 $ ldz, zc, ldzc, czero, work, n )
437 CALL clacpy( 'ALL', n, nblock, work, n, z( 1, k ), ldz )
438 END IF
439
440 k = k+np
441
442 END DO
443
444* The following block removes the shifts from the bottom right corner
445* one by one. Updates are initially applied to A(ihi-ns+1:ihi,ihi-ns:ihi).
446
447 CALL claset( 'FULL', ns, ns, czero, cone, qc, ldqc )
448 CALL claset( 'FULL', ns+1, ns+1, czero, cone, zc, ldzc )
449
450* istartb points to the first row we will be updating
451 istartb = ihi-ns+1
452* istopb points to the last column we will be updating
453 istopb = ihi
454
455 DO i = 1, ns
456* Chase the shift down to the bottom right corner
457 DO ishift = ihi-i, ihi-1
458 CALL claqz1( .true., .true., ishift, istartb, istopb, ihi,
459 $ a, lda, b, ldb, ns, ihi-ns+1, qc, ldqc, ns+1,
460 $ ihi-ns, zc, ldzc )
461 END DO
462
463 END DO
464
465* Update rest of the pencil
466
467* Update A(ihi-ns+1:ihi, ihi+1:istopm)
468* from the left with Qc(1:ns,1:ns)'
469 sheight = ns
470 swidth = istopm-( ihi+1 )+1
471 IF ( swidth > 0 ) THEN
472 CALL cgemm( 'C', 'N', sheight, swidth, sheight, cone, qc, ldqc,
473 $ a( ihi-ns+1, ihi+1 ), lda, czero, work, sheight )
474 CALL clacpy( 'ALL', sheight, swidth, work, sheight,
475 $ a( ihi-ns+1, ihi+1 ), lda )
476 CALL cgemm( 'C', 'N', sheight, swidth, sheight, cone, qc, ldqc,
477 $ b( ihi-ns+1, ihi+1 ), ldb, czero, work, sheight )
478 CALL clacpy( 'ALL', sheight, swidth, work, sheight,
479 $ b( ihi-ns+1, ihi+1 ), ldb )
480 END IF
481 IF ( ilq ) THEN
482 CALL cgemm( 'N', 'N', n, ns, ns, cone, q( 1, ihi-ns+1 ), ldq,
483 $ qc, ldqc, czero, work, n )
484 CALL clacpy( 'ALL', n, ns, work, n, q( 1, ihi-ns+1 ), ldq )
485 END IF
486
487* Update A(istartm:ihi-ns,ihi-ns:ihi)
488* from the right with Zc(1:ns+1,1:ns+1)
489 sheight = ihi-ns-istartm+1
490 swidth = ns+1
491 IF ( sheight > 0 ) THEN
492 CALL cgemm( 'N', 'N', sheight, swidth, swidth, cone,
493 $ a( istartm, ihi-ns ), lda, zc, ldzc, czero, work,
494 $ sheight )
495 CALL clacpy( 'ALL', sheight, swidth, work, sheight, a( istartm,
496 $ ihi-ns ), lda )
497 CALL cgemm( 'N', 'N', sheight, swidth, swidth, cone,
498 $ b( istartm, ihi-ns ), ldb, zc, ldzc, czero, work,
499 $ sheight )
500 CALL clacpy( 'ALL', sheight, swidth, work, sheight, b( istartm,
501 $ ihi-ns ), ldb )
502 END IF
503 IF ( ilz ) THEN
504 CALL cgemm( 'N', 'N', n, ns+1, ns+1, cone, z( 1, ihi-ns ), ldz,
505 $ zc, ldzc, czero, work, n )
506 CALL clacpy( 'ALL', n, ns+1, work, n, z( 1, ihi-ns ), ldz )
507 END IF
508
509 END SUBROUTINE
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
cgemm
subroutine cgemm(transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc)
CGEMM
Definition cgemm.f:188
clacpy
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
claqz1
subroutine claqz1(ilq, ilz, k, istartm, istopm, ihi, a, lda, b, ldb, nq, qstart, q, ldq, nz, zstart, z, ldz)
CLAQZ1
Definition claqz1.f:173
claqz3
subroutine claqz3(ilschur, ilq, ilz, n, ilo, ihi, nshifts, nblock_desired, alpha, beta, a, lda, b, ldb, q, ldq, z, ldz, qc, ldqc, zc, ldzc, work, lwork, info)
CLAQZ3
Definition claqz3.f:207
clartg
subroutine clartg(f, g, c, s, r)
CLARTG generates a plane rotation with real cosine and complex sine.
Definition clartg.f90:116
claset
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
crot
subroutine crot(n, cx, incx, cy, incy, c, s)
CROT applies a plane rotation with real cosine and complex sine to a pair of complex vectors.
Definition crot.f:103