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cheevx.f
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1 *> \brief <b> CHEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices</b>
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download CHEEVX + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cheevx.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cheevx.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cheevx.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
22 * ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK,
23 * IWORK, IFAIL, INFO )
24 *
25 * .. Scalar Arguments ..
26 * CHARACTER JOBZ, RANGE, UPLO
27 * INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
28 * REAL ABSTOL, VL, VU
29 * ..
30 * .. Array Arguments ..
31 * INTEGER IFAIL( * ), IWORK( * )
32 * REAL RWORK( * ), W( * )
33 * COMPLEX A( LDA, * ), WORK( * ), Z( LDZ, * )
34 * ..
35 *
36 *
37 *> \par Purpose:
38 * =============
39 *>
40 *> \verbatim
41 *>
42 *> CHEEVX computes selected eigenvalues and, optionally, eigenvectors
43 *> of a complex Hermitian matrix A. Eigenvalues and eigenvectors can
44 *> be selected by specifying either a range of values or a range of
45 *> indices for the desired eigenvalues.
46 *> \endverbatim
47 *
48 * Arguments:
49 * ==========
50 *
51 *> \param[in] JOBZ
52 *> \verbatim
53 *> JOBZ is CHARACTER*1
54 *> = 'N': Compute eigenvalues only;
55 *> = 'V': Compute eigenvalues and eigenvectors.
56 *> \endverbatim
57 *>
58 *> \param[in] RANGE
59 *> \verbatim
60 *> RANGE is CHARACTER*1
61 *> = 'A': all eigenvalues will be found.
62 *> = 'V': all eigenvalues in the half-open interval (VL,VU]
63 *> will be found.
64 *> = 'I': the IL-th through IU-th eigenvalues will be found.
65 *> \endverbatim
66 *>
67 *> \param[in] UPLO
68 *> \verbatim
69 *> UPLO is CHARACTER*1
70 *> = 'U': Upper triangle of A is stored;
71 *> = 'L': Lower triangle of A is stored.
72 *> \endverbatim
73 *>
74 *> \param[in] N
75 *> \verbatim
76 *> N is INTEGER
77 *> The order of the matrix A. N >= 0.
78 *> \endverbatim
79 *>
80 *> \param[in,out] A
81 *> \verbatim
82 *> A is COMPLEX array, dimension (LDA, N)
83 *> On entry, the Hermitian matrix A. If UPLO = 'U', the
84 *> leading N-by-N upper triangular part of A contains the
85 *> upper triangular part of the matrix A. If UPLO = 'L',
86 *> the leading N-by-N lower triangular part of A contains
87 *> the lower triangular part of the matrix A.
88 *> On exit, the lower triangle (if UPLO='L') or the upper
89 *> triangle (if UPLO='U') of A, including the diagonal, is
90 *> destroyed.
91 *> \endverbatim
92 *>
93 *> \param[in] LDA
94 *> \verbatim
95 *> LDA is INTEGER
96 *> The leading dimension of the array A. LDA >= max(1,N).
97 *> \endverbatim
98 *>
99 *> \param[in] VL
100 *> \verbatim
101 *> VL is REAL
102 *> \endverbatim
103 *>
104 *> \param[in] VU
105 *> \verbatim
106 *> VU is REAL
107 *> If RANGE='V', the lower and upper bounds of the interval to
108 *> be searched for eigenvalues. VL < VU.
109 *> Not referenced if RANGE = 'A' or 'I'.
110 *> \endverbatim
111 *>
112 *> \param[in] IL
113 *> \verbatim
114 *> IL is INTEGER
115 *> \endverbatim
116 *>
117 *> \param[in] IU
118 *> \verbatim
119 *> IU is INTEGER
120 *> If RANGE='I', the indices (in ascending order) of the
121 *> smallest and largest eigenvalues to be returned.
122 *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
123 *> Not referenced if RANGE = 'A' or 'V'.
124 *> \endverbatim
125 *>
126 *> \param[in] ABSTOL
127 *> \verbatim
128 *> ABSTOL is REAL
129 *> The absolute error tolerance for the eigenvalues.
130 *> An approximate eigenvalue is accepted as converged
131 *> when it is determined to lie in an interval [a,b]
132 *> of width less than or equal to
133 *>
134 *> ABSTOL + EPS * max( |a|,|b| ) ,
135 *>
136 *> where EPS is the machine precision. If ABSTOL is less than
137 *> or equal to zero, then EPS*|T| will be used in its place,
138 *> where |T| is the 1-norm of the tridiagonal matrix obtained
139 *> by reducing A to tridiagonal form.
140 *>
141 *> Eigenvalues will be computed most accurately when ABSTOL is
142 *> set to twice the underflow threshold 2*SLAMCH('S'), not zero.
143 *> If this routine returns with INFO>0, indicating that some
144 *> eigenvectors did not converge, try setting ABSTOL to
145 *> 2*SLAMCH('S').
146 *>
147 *> See "Computing Small Singular Values of Bidiagonal Matrices
148 *> with Guaranteed High Relative Accuracy," by Demmel and
149 *> Kahan, LAPACK Working Note #3.
150 *> \endverbatim
151 *>
152 *> \param[out] M
153 *> \verbatim
154 *> M is INTEGER
155 *> The total number of eigenvalues found. 0 <= M <= N.
156 *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
157 *> \endverbatim
158 *>
159 *> \param[out] W
160 *> \verbatim
161 *> W is REAL array, dimension (N)
162 *> On normal exit, the first M elements contain the selected
163 *> eigenvalues in ascending order.
164 *> \endverbatim
165 *>
166 *> \param[out] Z
167 *> \verbatim
168 *> Z is COMPLEX array, dimension (LDZ, max(1,M))
169 *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
170 *> contain the orthonormal eigenvectors of the matrix A
171 *> corresponding to the selected eigenvalues, with the i-th
172 *> column of Z holding the eigenvector associated with W(i).
173 *> If an eigenvector fails to converge, then that column of Z
174 *> contains the latest approximation to the eigenvector, and the
175 *> index of the eigenvector is returned in IFAIL.
176 *> If JOBZ = 'N', then Z is not referenced.
177 *> Note: the user must ensure that at least max(1,M) columns are
178 *> supplied in the array Z; if RANGE = 'V', the exact value of M
179 *> is not known in advance and an upper bound must be used.
180 *> \endverbatim
181 *>
182 *> \param[in] LDZ
183 *> \verbatim
184 *> LDZ is INTEGER
185 *> The leading dimension of the array Z. LDZ >= 1, and if
186 *> JOBZ = 'V', LDZ >= max(1,N).
187 *> \endverbatim
188 *>
189 *> \param[out] WORK
190 *> \verbatim
191 *> WORK is COMPLEX array, dimension (MAX(1,LWORK))
192 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
193 *> \endverbatim
194 *>
195 *> \param[in] LWORK
196 *> \verbatim
197 *> LWORK is INTEGER
198 *> The length of the array WORK. LWORK >= 1, when N <= 1;
199 *> otherwise 2*N.
200 *> For optimal efficiency, LWORK >= (NB+1)*N,
201 *> where NB is the max of the blocksize for CHETRD and for
202 *> CUNMTR as returned by ILAENV.
203 *>
204 *> If LWORK = -1, then a workspace query is assumed; the routine
205 *> only calculates the optimal size of the WORK array, returns
206 *> this value as the first entry of the WORK array, and no error
207 *> message related to LWORK is issued by XERBLA.
208 *> \endverbatim
209 *>
210 *> \param[out] RWORK
211 *> \verbatim
212 *> RWORK is REAL array, dimension (7*N)
213 *> \endverbatim
214 *>
215 *> \param[out] IWORK
216 *> \verbatim
217 *> IWORK is INTEGER array, dimension (5*N)
218 *> \endverbatim
219 *>
220 *> \param[out] IFAIL
221 *> \verbatim
222 *> IFAIL is INTEGER array, dimension (N)
223 *> If JOBZ = 'V', then if INFO = 0, the first M elements of
224 *> IFAIL are zero. If INFO > 0, then IFAIL contains the
225 *> indices of the eigenvectors that failed to converge.
226 *> If JOBZ = 'N', then IFAIL is not referenced.
227 *> \endverbatim
228 *>
229 *> \param[out] INFO
230 *> \verbatim
231 *> INFO is INTEGER
232 *> = 0: successful exit
233 *> < 0: if INFO = -i, the i-th argument had an illegal value
234 *> > 0: if INFO = i, then i eigenvectors failed to converge.
235 *> Their indices are stored in array IFAIL.
236 *> \endverbatim
237 *
238 * Authors:
239 * ========
240 *
241 *> \author Univ. of Tennessee
242 *> \author Univ. of California Berkeley
243 *> \author Univ. of Colorado Denver
244 *> \author NAG Ltd.
245 *
246 *> \date November 2011
247 *
248 *> \ingroup complexHEeigen
249 *
250 * =====================================================================
251  SUBROUTINE cheevx( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
252  $ abstol, m, w, z, ldz, work, lwork, rwork,
253  $ iwork, ifail, info )
254 *
255 * -- LAPACK driver routine (version 3.4.0) --
256 * -- LAPACK is a software package provided by Univ. of Tennessee, --
257 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
258 * November 2011
259 *
260 * .. Scalar Arguments ..
261  CHARACTER jobz, range, uplo
262  INTEGER il, info, iu, lda, ldz, lwork, m, n
263  REAL abstol, vl, vu
264 * ..
265 * .. Array Arguments ..
266  INTEGER ifail( * ), iwork( * )
267  REAL rwork( * ), w( * )
268  COMPLEX a( lda, * ), work( * ), z( ldz, * )
269 * ..
270 *
271 * =====================================================================
272 *
273 * .. Parameters ..
274  REAL zero, one
275  parameter( zero = 0.0e+0, one = 1.0e+0 )
276  COMPLEX cone
277  parameter( cone = ( 1.0e+0, 0.0e+0 ) )
278 * ..
279 * .. Local Scalars ..
280  LOGICAL alleig, indeig, lower, lquery, test, valeig,
281  $ wantz
282  CHARACTER order
283  INTEGER i, iinfo, imax, indd, inde, indee, indibl,
284  $ indisp, indiwk, indrwk, indtau, indwrk, iscale,
285  $ itmp1, j, jj, llwork, lwkmin, lwkopt, nb,
286  $ nsplit
287  REAL abstll, anrm, bignum, eps, rmax, rmin, safmin,
288  $ sigma, smlnum, tmp1, vll, vuu
289 * ..
290 * .. External Functions ..
291  LOGICAL lsame
292  INTEGER ilaenv
293  REAL slamch, clanhe
294  EXTERNAL lsame, ilaenv, slamch, clanhe
295 * ..
296 * .. External Subroutines ..
297  EXTERNAL scopy, sscal, sstebz, ssterf, xerbla, csscal,
299  $ cunmtr
300 * ..
301 * .. Intrinsic Functions ..
302  INTRINSIC REAL, max, min, sqrt
303 * ..
304 * .. Executable Statements ..
305 *
306 * Test the input parameters.
307 *
308  lower = lsame( uplo, 'L' )
309  wantz = lsame( jobz, 'V' )
310  alleig = lsame( range, 'A' )
311  valeig = lsame( range, 'V' )
312  indeig = lsame( range, 'I' )
313  lquery = ( lwork.EQ.-1 )
314 *
315  info = 0
316  IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
317  info = -1
318  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
319  info = -2
320  ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
321  info = -3
322  ELSE IF( n.LT.0 ) THEN
323  info = -4
324  ELSE IF( lda.LT.max( 1, n ) ) THEN
325  info = -6
326  ELSE
327  IF( valeig ) THEN
328  IF( n.GT.0 .AND. vu.LE.vl )
329  $ info = -8
330  ELSE IF( indeig ) THEN
331  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
332  info = -9
333  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
334  info = -10
335  END IF
336  END IF
337  END IF
338  IF( info.EQ.0 ) THEN
339  IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
340  info = -15
341  END IF
342  END IF
343 *
344  IF( info.EQ.0 ) THEN
345  IF( n.LE.1 ) THEN
346  lwkmin = 1
347  work( 1 ) = lwkmin
348  ELSE
349  lwkmin = 2*n
350  nb = ilaenv( 1, 'CHETRD', uplo, n, -1, -1, -1 )
351  nb = max( nb, ilaenv( 1, 'CUNMTR', uplo, n, -1, -1, -1 ) )
352  lwkopt = max( 1, ( nb + 1 )*n )
353  work( 1 ) = lwkopt
354  END IF
355 *
356  IF( lwork.LT.lwkmin .AND. .NOT.lquery )
357  $ info = -17
358  END IF
359 *
360  IF( info.NE.0 ) THEN
361  CALL xerbla( 'CHEEVX', -info )
362  return
363  ELSE IF( lquery ) THEN
364  return
365  END IF
366 *
367 * Quick return if possible
368 *
369  m = 0
370  IF( n.EQ.0 ) THEN
371  return
372  END IF
373 *
374  IF( n.EQ.1 ) THEN
375  IF( alleig .OR. indeig ) THEN
376  m = 1
377  w( 1 ) = a( 1, 1 )
378  ELSE IF( valeig ) THEN
379  IF( vl.LT.REAL( A( 1, 1 ) ) .AND. vu.GE.REAL( A( 1, 1 ) ) )
380  $ THEN
381  m = 1
382  w( 1 ) = a( 1, 1 )
383  END IF
384  END IF
385  IF( wantz )
386  $ z( 1, 1 ) = cone
387  return
388  END IF
389 *
390 * Get machine constants.
391 *
392  safmin = slamch( 'Safe minimum' )
393  eps = slamch( 'Precision' )
394  smlnum = safmin / eps
395  bignum = one / smlnum
396  rmin = sqrt( smlnum )
397  rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
398 *
399 * Scale matrix to allowable range, if necessary.
400 *
401  iscale = 0
402  abstll = abstol
403  IF( valeig ) THEN
404  vll = vl
405  vuu = vu
406  END IF
407  anrm = clanhe( 'M', uplo, n, a, lda, rwork )
408  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
409  iscale = 1
410  sigma = rmin / anrm
411  ELSE IF( anrm.GT.rmax ) THEN
412  iscale = 1
413  sigma = rmax / anrm
414  END IF
415  IF( iscale.EQ.1 ) THEN
416  IF( lower ) THEN
417  DO 10 j = 1, n
418  CALL csscal( n-j+1, sigma, a( j, j ), 1 )
419  10 continue
420  ELSE
421  DO 20 j = 1, n
422  CALL csscal( j, sigma, a( 1, j ), 1 )
423  20 continue
424  END IF
425  IF( abstol.GT.0 )
426  $ abstll = abstol*sigma
427  IF( valeig ) THEN
428  vll = vl*sigma
429  vuu = vu*sigma
430  END IF
431  END IF
432 *
433 * Call CHETRD to reduce Hermitian matrix to tridiagonal form.
434 *
435  indd = 1
436  inde = indd + n
437  indrwk = inde + n
438  indtau = 1
439  indwrk = indtau + n
440  llwork = lwork - indwrk + 1
441  CALL chetrd( uplo, n, a, lda, rwork( indd ), rwork( inde ),
442  $ work( indtau ), work( indwrk ), llwork, iinfo )
443 *
444 * If all eigenvalues are desired and ABSTOL is less than or equal to
445 * zero, then call SSTERF or CUNGTR and CSTEQR. If this fails for
446 * some eigenvalue, then try SSTEBZ.
447 *
448  test = .false.
449  IF( indeig ) THEN
450  IF( il.EQ.1 .AND. iu.EQ.n ) THEN
451  test = .true.
452  END IF
453  END IF
454  IF( ( alleig .OR. test ) .AND. ( abstol.LE.zero ) ) THEN
455  CALL scopy( n, rwork( indd ), 1, w, 1 )
456  indee = indrwk + 2*n
457  IF( .NOT.wantz ) THEN
458  CALL scopy( n-1, rwork( inde ), 1, rwork( indee ), 1 )
459  CALL ssterf( n, w, rwork( indee ), info )
460  ELSE
461  CALL clacpy( 'A', n, n, a, lda, z, ldz )
462  CALL cungtr( uplo, n, z, ldz, work( indtau ),
463  $ work( indwrk ), llwork, iinfo )
464  CALL scopy( n-1, rwork( inde ), 1, rwork( indee ), 1 )
465  CALL csteqr( jobz, n, w, rwork( indee ), z, ldz,
466  $ rwork( indrwk ), info )
467  IF( info.EQ.0 ) THEN
468  DO 30 i = 1, n
469  ifail( i ) = 0
470  30 continue
471  END IF
472  END IF
473  IF( info.EQ.0 ) THEN
474  m = n
475  go to 40
476  END IF
477  info = 0
478  END IF
479 *
480 * Otherwise, call SSTEBZ and, if eigenvectors are desired, CSTEIN.
481 *
482  IF( wantz ) THEN
483  order = 'B'
484  ELSE
485  order = 'E'
486  END IF
487  indibl = 1
488  indisp = indibl + n
489  indiwk = indisp + n
490  CALL sstebz( range, order, n, vll, vuu, il, iu, abstll,
491  $ rwork( indd ), rwork( inde ), m, nsplit, w,
492  $ iwork( indibl ), iwork( indisp ), rwork( indrwk ),
493  $ iwork( indiwk ), info )
494 *
495  IF( wantz ) THEN
496  CALL cstein( n, rwork( indd ), rwork( inde ), m, w,
497  $ iwork( indibl ), iwork( indisp ), z, ldz,
498  $ rwork( indrwk ), iwork( indiwk ), ifail, info )
499 *
500 * Apply unitary matrix used in reduction to tridiagonal
501 * form to eigenvectors returned by CSTEIN.
502 *
503  CALL cunmtr( 'L', uplo, 'N', n, m, a, lda, work( indtau ), z,
504  $ ldz, work( indwrk ), llwork, iinfo )
505  END IF
506 *
507 * If matrix was scaled, then rescale eigenvalues appropriately.
508 *
509  40 continue
510  IF( iscale.EQ.1 ) THEN
511  IF( info.EQ.0 ) THEN
512  imax = m
513  ELSE
514  imax = info - 1
515  END IF
516  CALL sscal( imax, one / sigma, w, 1 )
517  END IF
518 *
519 * If eigenvalues are not in order, then sort them, along with
520 * eigenvectors.
521 *
522  IF( wantz ) THEN
523  DO 60 j = 1, m - 1
524  i = 0
525  tmp1 = w( j )
526  DO 50 jj = j + 1, m
527  IF( w( jj ).LT.tmp1 ) THEN
528  i = jj
529  tmp1 = w( jj )
530  END IF
531  50 continue
532 *
533  IF( i.NE.0 ) THEN
534  itmp1 = iwork( indibl+i-1 )
535  w( i ) = w( j )
536  iwork( indibl+i-1 ) = iwork( indibl+j-1 )
537  w( j ) = tmp1
538  iwork( indibl+j-1 ) = itmp1
539  CALL cswap( n, z( 1, i ), 1, z( 1, j ), 1 )
540  IF( info.NE.0 ) THEN
541  itmp1 = ifail( i )
542  ifail( i ) = ifail( j )
543  ifail( j ) = itmp1
544  END IF
545  END IF
546  60 continue
547  END IF
548 *
549 * Set WORK(1) to optimal complex workspace size.
550 *
551  work( 1 ) = lwkopt
552 *
553  return
554 *
555 * End of CHEEVX
556 *
557  END