LAPACK  3.8.0
LAPACK: Linear Algebra PACKage
chseqr.f
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1 *> \brief \b CHSEQR
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
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7 *
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ,
22 * WORK, LWORK, INFO )
23 *
24 * .. Scalar Arguments ..
25 * INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
26 * CHARACTER COMPZ, JOB
27 * ..
28 * .. Array Arguments ..
29 * COMPLEX H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> CHSEQR computes the eigenvalues of a Hessenberg matrix H
39 *> and, optionally, the matrices T and Z from the Schur decomposition
40 *> H = Z T Z**H, where T is an upper triangular matrix (the
41 *> Schur form), and Z is the unitary matrix of Schur vectors.
42 *>
43 *> Optionally Z may be postmultiplied into an input unitary
44 *> matrix Q so that this routine can give the Schur factorization
45 *> of a matrix A which has been reduced to the Hessenberg form H
46 *> by the unitary matrix Q: A = Q*H*Q**H = (QZ)*T*(QZ)**H.
47 *> \endverbatim
48 *
49 * Arguments:
50 * ==========
51 *
52 *> \param[in] JOB
53 *> \verbatim
54 *> JOB is CHARACTER*1
55 *> = 'E': compute eigenvalues only;
56 *> = 'S': compute eigenvalues and the Schur form T.
57 *> \endverbatim
58 *>
59 *> \param[in] COMPZ
60 *> \verbatim
61 *> COMPZ is CHARACTER*1
62 *> = 'N': no Schur vectors are computed;
63 *> = 'I': Z is initialized to the unit matrix and the matrix Z
64 *> of Schur vectors of H is returned;
65 *> = 'V': Z must contain an unitary matrix Q on entry, and
66 *> the product Q*Z is returned.
67 *> \endverbatim
68 *>
69 *> \param[in] N
70 *> \verbatim
71 *> N is INTEGER
72 *> The order of the matrix H. N .GE. 0.
73 *> \endverbatim
74 *>
75 *> \param[in] ILO
76 *> \verbatim
77 *> ILO is INTEGER
78 *> \endverbatim
79 *>
80 *> \param[in] IHI
81 *> \verbatim
82 *> IHI is INTEGER
83 *>
84 *> It is assumed that H is already upper triangular in rows
85 *> and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
86 *> set by a previous call to CGEBAL, and then passed to ZGEHRD
87 *> when the matrix output by CGEBAL is reduced to Hessenberg
88 *> form. Otherwise ILO and IHI should be set to 1 and N
89 *> respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
90 *> If N = 0, then ILO = 1 and IHI = 0.
91 *> \endverbatim
92 *>
93 *> \param[in,out] H
94 *> \verbatim
95 *> H is COMPLEX array, dimension (LDH,N)
96 *> On entry, the upper Hessenberg matrix H.
97 *> On exit, if INFO = 0 and JOB = 'S', H contains the upper
98 *> triangular matrix T from the Schur decomposition (the
99 *> Schur form). If INFO = 0 and JOB = 'E', the contents of
100 *> H are unspecified on exit. (The output value of H when
101 *> INFO.GT.0 is given under the description of INFO below.)
102 *>
103 *> Unlike earlier versions of CHSEQR, this subroutine may
104 *> explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1
105 *> or j = IHI+1, IHI+2, ... N.
106 *> \endverbatim
107 *>
108 *> \param[in] LDH
109 *> \verbatim
110 *> LDH is INTEGER
111 *> The leading dimension of the array H. LDH .GE. max(1,N).
112 *> \endverbatim
113 *>
114 *> \param[out] W
115 *> \verbatim
116 *> W is COMPLEX array, dimension (N)
117 *> The computed eigenvalues. If JOB = 'S', the eigenvalues are
118 *> stored in the same order as on the diagonal of the Schur
119 *> form returned in H, with W(i) = H(i,i).
120 *> \endverbatim
121 *>
122 *> \param[in,out] Z
123 *> \verbatim
124 *> Z is COMPLEX array, dimension (LDZ,N)
125 *> If COMPZ = 'N', Z is not referenced.
126 *> If COMPZ = 'I', on entry Z need not be set and on exit,
127 *> if INFO = 0, Z contains the unitary matrix Z of the Schur
128 *> vectors of H. If COMPZ = 'V', on entry Z must contain an
129 *> N-by-N matrix Q, which is assumed to be equal to the unit
130 *> matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
131 *> if INFO = 0, Z contains Q*Z.
132 *> Normally Q is the unitary matrix generated by CUNGHR
133 *> after the call to CGEHRD which formed the Hessenberg matrix
134 *> H. (The output value of Z when INFO.GT.0 is given under
135 *> the description of INFO below.)
136 *> \endverbatim
137 *>
138 *> \param[in] LDZ
139 *> \verbatim
140 *> LDZ is INTEGER
141 *> The leading dimension of the array Z. if COMPZ = 'I' or
142 *> COMPZ = 'V', then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.1.
143 *> \endverbatim
144 *>
145 *> \param[out] WORK
146 *> \verbatim
147 *> WORK is COMPLEX array, dimension (LWORK)
148 *> On exit, if INFO = 0, WORK(1) returns an estimate of
149 *> the optimal value for LWORK.
150 *> \endverbatim
151 *>
152 *> \param[in] LWORK
153 *> \verbatim
154 *> LWORK is INTEGER
155 *> The dimension of the array WORK. LWORK .GE. max(1,N)
156 *> is sufficient and delivers very good and sometimes
157 *> optimal performance. However, LWORK as large as 11*N
158 *> may be required for optimal performance. A workspace
159 *> query is recommended to determine the optimal workspace
160 *> size.
161 *>
162 *> If LWORK = -1, then CHSEQR does a workspace query.
163 *> In this case, CHSEQR checks the input parameters and
164 *> estimates the optimal workspace size for the given
165 *> values of N, ILO and IHI. The estimate is returned
166 *> in WORK(1). No error message related to LWORK is
167 *> issued by XERBLA. Neither H nor Z are accessed.
168 *> \endverbatim
169 *>
170 *> \param[out] INFO
171 *> \verbatim
172 *> INFO is INTEGER
173 *> = 0: successful exit
174 *> .LT. 0: if INFO = -i, the i-th argument had an illegal
175 *> value
176 *> .GT. 0: if INFO = i, CHSEQR failed to compute all of
177 *> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
178 *> and WI contain those eigenvalues which have been
179 *> successfully computed. (Failures are rare.)
180 *>
181 *> If INFO .GT. 0 and JOB = 'E', then on exit, the
182 *> remaining unconverged eigenvalues are the eigen-
183 *> values of the upper Hessenberg matrix rows and
184 *> columns ILO through INFO of the final, output
185 *> value of H.
186 *>
187 *> If INFO .GT. 0 and JOB = 'S', then on exit
188 *>
189 *> (*) (initial value of H)*U = U*(final value of H)
190 *>
191 *> where U is a unitary matrix. The final
192 *> value of H is upper Hessenberg and triangular in
193 *> rows and columns INFO+1 through IHI.
194 *>
195 *> If INFO .GT. 0 and COMPZ = 'V', then on exit
196 *>
197 *> (final value of Z) = (initial value of Z)*U
198 *>
199 *> where U is the unitary matrix in (*) (regard-
200 *> less of the value of JOB.)
201 *>
202 *> If INFO .GT. 0 and COMPZ = 'I', then on exit
203 *> (final value of Z) = U
204 *> where U is the unitary matrix in (*) (regard-
205 *> less of the value of JOB.)
206 *>
207 *> If INFO .GT. 0 and COMPZ = 'N', then Z is not
208 *> accessed.
209 *> \endverbatim
210 *
211 * Authors:
212 * ========
213 *
214 *> \author Univ. of Tennessee
215 *> \author Univ. of California Berkeley
216 *> \author Univ. of Colorado Denver
217 *> \author NAG Ltd.
218 *
219 *> \date December 2016
220 *
221 *> \ingroup complexOTHERcomputational
222 *
223 *> \par Contributors:
224 * ==================
225 *>
226 *> Karen Braman and Ralph Byers, Department of Mathematics,
227 *> University of Kansas, USA
228 *
229 *> \par Further Details:
230 * =====================
231 *>
232 *> \verbatim
233 *>
234 *> Default values supplied by
235 *> ILAENV(ISPEC,'CHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
236 *> It is suggested that these defaults be adjusted in order
237 *> to attain best performance in each particular
238 *> computational environment.
239 *>
240 *> ISPEC=12: The CLAHQR vs CLAQR0 crossover point.
241 *> Default: 75. (Must be at least 11.)
242 *>
243 *> ISPEC=13: Recommended deflation window size.
244 *> This depends on ILO, IHI and NS. NS is the
245 *> number of simultaneous shifts returned
246 *> by ILAENV(ISPEC=15). (See ISPEC=15 below.)
247 *> The default for (IHI-ILO+1).LE.500 is NS.
248 *> The default for (IHI-ILO+1).GT.500 is 3*NS/2.
249 *>
250 *> ISPEC=14: Nibble crossover point. (See IPARMQ for
251 *> details.) Default: 14% of deflation window
252 *> size.
253 *>
254 *> ISPEC=15: Number of simultaneous shifts in a multishift
255 *> QR iteration.
256 *>
257 *> If IHI-ILO+1 is ...
258 *>
259 *> greater than ...but less ... the
260 *> or equal to ... than default is
261 *>
262 *> 1 30 NS = 2(+)
263 *> 30 60 NS = 4(+)
264 *> 60 150 NS = 10(+)
265 *> 150 590 NS = **
266 *> 590 3000 NS = 64
267 *> 3000 6000 NS = 128
268 *> 6000 infinity NS = 256
269 *>
270 *> (+) By default some or all matrices of this order
271 *> are passed to the implicit double shift routine
272 *> CLAHQR and this parameter is ignored. See
273 *> ISPEC=12 above and comments in IPARMQ for
274 *> details.
275 *>
276 *> (**) The asterisks (**) indicate an ad-hoc
277 *> function of N increasing from 10 to 64.
278 *>
279 *> ISPEC=16: Select structured matrix multiply.
280 *> If the number of simultaneous shifts (specified
281 *> by ISPEC=15) is less than 14, then the default
282 *> for ISPEC=16 is 0. Otherwise the default for
283 *> ISPEC=16 is 2.
284 *> \endverbatim
285 *
286 *> \par References:
287 * ================
288 *>
289 *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
290 *> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
291 *> Performance, SIAM Journal of Matrix Analysis, volume 23, pages
292 *> 929--947, 2002.
293 *> \n
294 *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
295 *> Algorithm Part II: Aggressive Early Deflation, SIAM Journal
296 *> of Matrix Analysis, volume 23, pages 948--973, 2002.
297 *
298 * =====================================================================
299  SUBROUTINE chseqr( JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ,
300  $ WORK, LWORK, INFO )
301 *
302 * -- LAPACK computational routine (version 3.7.0) --
303 * -- LAPACK is a software package provided by Univ. of Tennessee, --
304 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
305 * December 2016
306 *
307 * .. Scalar Arguments ..
308  INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
309  CHARACTER COMPZ, JOB
310 * ..
311 * .. Array Arguments ..
312  COMPLEX H( ldh, * ), W( * ), WORK( * ), Z( ldz, * )
313 * ..
314 *
315 * =====================================================================
316 *
317 * .. Parameters ..
318 *
319 * ==== Matrices of order NTINY or smaller must be processed by
320 * . CLAHQR because of insufficient subdiagonal scratch space.
321 * . (This is a hard limit.) ====
322  INTEGER NTINY
323  parameter( ntiny = 11 )
324 *
325 * ==== NL allocates some local workspace to help small matrices
326 * . through a rare CLAHQR failure. NL .GT. NTINY = 11 is
327 * . required and NL .LE. NMIN = ILAENV(ISPEC=12,...) is recom-
328 * . mended. (The default value of NMIN is 75.) Using NL = 49
329 * . allows up to six simultaneous shifts and a 16-by-16
330 * . deflation window. ====
331  INTEGER NL
332  parameter( nl = 49 )
333  COMPLEX ZERO, ONE
334  parameter( zero = ( 0.0e0, 0.0e0 ),
335  $ one = ( 1.0e0, 0.0e0 ) )
336  REAL RZERO
337  parameter( rzero = 0.0e0 )
338 * ..
339 * .. Local Arrays ..
340  COMPLEX HL( nl, nl ), WORKL( nl )
341 * ..
342 * .. Local Scalars ..
343  INTEGER KBOT, NMIN
344  LOGICAL INITZ, LQUERY, WANTT, WANTZ
345 * ..
346 * .. External Functions ..
347  INTEGER ILAENV
348  LOGICAL LSAME
349  EXTERNAL ilaenv, lsame
350 * ..
351 * .. External Subroutines ..
352  EXTERNAL ccopy, clacpy, clahqr, claqr0, claset, xerbla
353 * ..
354 * .. Intrinsic Functions ..
355  INTRINSIC cmplx, max, min, real
356 * ..
357 * .. Executable Statements ..
358 *
359 * ==== Decode and check the input parameters. ====
360 *
361  wantt = lsame( job, 'S' )
362  initz = lsame( compz, 'I' )
363  wantz = initz .OR. lsame( compz, 'V' )
364  work( 1 ) = cmplx( REAL( MAX( 1, N ) ), RZERO )
365  lquery = lwork.EQ.-1
366 *
367  info = 0
368  IF( .NOT.lsame( job, 'E' ) .AND. .NOT.wantt ) THEN
369  info = -1
370  ELSE IF( .NOT.lsame( compz, 'N' ) .AND. .NOT.wantz ) THEN
371  info = -2
372  ELSE IF( n.LT.0 ) THEN
373  info = -3
374  ELSE IF( ilo.LT.1 .OR. ilo.GT.max( 1, n ) ) THEN
375  info = -4
376  ELSE IF( ihi.LT.min( ilo, n ) .OR. ihi.GT.n ) THEN
377  info = -5
378  ELSE IF( ldh.LT.max( 1, n ) ) THEN
379  info = -7
380  ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.max( 1, n ) ) ) THEN
381  info = -10
382  ELSE IF( lwork.LT.max( 1, n ) .AND. .NOT.lquery ) THEN
383  info = -12
384  END IF
385 *
386  IF( info.NE.0 ) THEN
387 *
388 * ==== Quick return in case of invalid argument. ====
389 *
390  CALL xerbla( 'CHSEQR', -info )
391  RETURN
392 *
393  ELSE IF( n.EQ.0 ) THEN
394 *
395 * ==== Quick return in case N = 0; nothing to do. ====
396 *
397  RETURN
398 *
399  ELSE IF( lquery ) THEN
400 *
401 * ==== Quick return in case of a workspace query ====
402 *
403  CALL claqr0( wantt, wantz, n, ilo, ihi, h, ldh, w, ilo, ihi, z,
404  $ ldz, work, lwork, info )
405 * ==== Ensure reported workspace size is backward-compatible with
406 * . previous LAPACK versions. ====
407  work( 1 ) = cmplx( max( REAL( WORK( 1 ) ), REAL( MAX( 1, $ N ) ) ), rzero )
408  RETURN
409 *
410  ELSE
411 *
412 * ==== copy eigenvalues isolated by CGEBAL ====
413 *
414  IF( ilo.GT.1 )
415  $ CALL ccopy( ilo-1, h, ldh+1, w, 1 )
416  IF( ihi.LT.n )
417  $ CALL ccopy( n-ihi, h( ihi+1, ihi+1 ), ldh+1, w( ihi+1 ), 1 )
418 *
419 * ==== Initialize Z, if requested ====
420 *
421  IF( initz )
422  $ CALL claset( 'A', n, n, zero, one, z, ldz )
423 *
424 * ==== Quick return if possible ====
425 *
426  IF( ilo.EQ.ihi ) THEN
427  w( ilo ) = h( ilo, ilo )
428  RETURN
429  END IF
430 *
431 * ==== CLAHQR/CLAQR0 crossover point ====
432 *
433  nmin = ilaenv( 12, 'CHSEQR', job( : 1 ) // compz( : 1 ), n,
434  $ ilo, ihi, lwork )
435  nmin = max( ntiny, nmin )
436 *
437 * ==== CLAQR0 for big matrices; CLAHQR for small ones ====
438 *
439  IF( n.GT.nmin ) THEN
440  CALL claqr0( wantt, wantz, n, ilo, ihi, h, ldh, w, ilo, ihi,
441  $ z, ldz, work, lwork, info )
442  ELSE
443 *
444 * ==== Small matrix ====
445 *
446  CALL clahqr( wantt, wantz, n, ilo, ihi, h, ldh, w, ilo, ihi,
447  $ z, ldz, info )
448 *
449  IF( info.GT.0 ) THEN
450 *
451 * ==== A rare CLAHQR failure! CLAQR0 sometimes succeeds
452 * . when CLAHQR fails. ====
453 *
454  kbot = info
455 *
456  IF( n.GE.nl ) THEN
457 *
458 * ==== Larger matrices have enough subdiagonal scratch
459 * . space to call CLAQR0 directly. ====
460 *
461  CALL claqr0( wantt, wantz, n, ilo, kbot, h, ldh, w,
462  $ ilo, ihi, z, ldz, work, lwork, info )
463 *
464  ELSE
465 *
466 * ==== Tiny matrices don't have enough subdiagonal
467 * . scratch space to benefit from CLAQR0. Hence,
468 * . tiny matrices must be copied into a larger
469 * . array before calling CLAQR0. ====
470 *
471  CALL clacpy( 'A', n, n, h, ldh, hl, nl )
472  hl( n+1, n ) = zero
473  CALL claset( 'A', nl, nl-n, zero, zero, hl( 1, n+1 ),
474  $ nl )
475  CALL claqr0( wantt, wantz, nl, ilo, kbot, hl, nl, w,
476  $ ilo, ihi, z, ldz, workl, nl, info )
477  IF( wantt .OR. info.NE.0 )
478  $ CALL clacpy( 'A', n, n, hl, nl, h, ldh )
479  END IF
480  END IF
481  END IF
482 *
483 * ==== Clear out the trash, if necessary. ====
484 *
485  IF( ( wantt .OR. info.NE.0 ) .AND. n.GT.2 )
486  $ CALL claset( 'L', n-2, n-2, zero, zero, h( 3, 1 ), ldh )
487 *
488 * ==== Ensure reported workspace size is backward-compatible with
489 * . previous LAPACK versions. ====
490 *
491  work( 1 ) = cmplx( max( REAL( MAX( 1, N ) ),
492  $ REAL( WORK( 1 ) ) ), rzero )
493  END IF
494 *
495 * ==== End of CHSEQR ====
496 *
497  END
498 
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 claqr0(WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO)
CLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur de...
Definition: claqr0.f:242
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 xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
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 ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:83
subroutine chseqr(JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, WORK, LWORK, INFO)
CHSEQR
Definition: chseqr.f:301