LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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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*> If RANGE='V', the lower bound of the interval to
103*> be searched for eigenvalues. VL < VU.
104*> Not referenced if RANGE = 'A' or 'I'.
105*> \endverbatim
106*>
107*> \param[in] VU
108*> \verbatim
109*> VU is REAL
110*> If RANGE='V', the upper bound of the interval to
111*> be searched for eigenvalues. VL < VU.
112*> Not referenced if RANGE = 'A' or 'I'.
113*> \endverbatim
114*>
115*> \param[in] IL
116*> \verbatim
117*> IL is INTEGER
118*> If RANGE='I', the index of the
119*> smallest eigenvalue to be returned.
120*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
121*> Not referenced if RANGE = 'A' or 'V'.
122*> \endverbatim
123*>
124*> \param[in] IU
125*> \verbatim
126*> IU is INTEGER
127*> If RANGE='I', the index of the
128*> largest eigenvalue to be returned.
129*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
130*> Not referenced if RANGE = 'A' or 'V'.
131*> \endverbatim
132*>
133*> \param[in] ABSTOL
134*> \verbatim
135*> ABSTOL is REAL
136*> The absolute error tolerance for the eigenvalues.
137*> An approximate eigenvalue is accepted as converged
138*> when it is determined to lie in an interval [a,b]
139*> of width less than or equal to
140*>
141*> ABSTOL + EPS * max( |a|,|b| ) ,
142*>
143*> where EPS is the machine precision. If ABSTOL is less than
144*> or equal to zero, then EPS*|T| will be used in its place,
145*> where |T| is the 1-norm of the tridiagonal matrix obtained
146*> by reducing A to tridiagonal form.
147*>
148*> Eigenvalues will be computed most accurately when ABSTOL is
149*> set to twice the underflow threshold 2*SLAMCH('S'), not zero.
150*> If this routine returns with INFO>0, indicating that some
151*> eigenvectors did not converge, try setting ABSTOL to
152*> 2*SLAMCH('S').
153*>
154*> See "Computing Small Singular Values of Bidiagonal Matrices
155*> with Guaranteed High Relative Accuracy," by Demmel and
156*> Kahan, LAPACK Working Note #3.
157*> \endverbatim
158*>
159*> \param[out] M
160*> \verbatim
161*> M is INTEGER
162*> The total number of eigenvalues found. 0 <= M <= N.
163*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
164*> \endverbatim
165*>
166*> \param[out] W
167*> \verbatim
168*> W is REAL array, dimension (N)
169*> On normal exit, the first M elements contain the selected
170*> eigenvalues in ascending order.
171*> \endverbatim
172*>
173*> \param[out] Z
174*> \verbatim
175*> Z is COMPLEX array, dimension (LDZ, max(1,M))
176*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
177*> contain the orthonormal eigenvectors of the matrix A
178*> corresponding to the selected eigenvalues, with the i-th
179*> column of Z holding the eigenvector associated with W(i).
180*> If an eigenvector fails to converge, then that column of Z
181*> contains the latest approximation to the eigenvector, and the
182*> index of the eigenvector is returned in IFAIL.
183*> If JOBZ = 'N', then Z is not referenced.
184*> Note: the user must ensure that at least max(1,M) columns are
185*> supplied in the array Z; if RANGE = 'V', the exact value of M
186*> is not known in advance and an upper bound must be used.
187*> \endverbatim
188*>
189*> \param[in] LDZ
190*> \verbatim
191*> LDZ is INTEGER
192*> The leading dimension of the array Z. LDZ >= 1, and if
193*> JOBZ = 'V', LDZ >= max(1,N).
194*> \endverbatim
195*>
196*> \param[out] WORK
197*> \verbatim
198*> WORK is COMPLEX array, dimension (MAX(1,LWORK))
199*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
200*> \endverbatim
201*>
202*> \param[in] LWORK
203*> \verbatim
204*> LWORK is INTEGER
205*> The length of the array WORK. LWORK >= 1, when N <= 1;
206*> otherwise 2*N.
207*> For optimal efficiency, LWORK >= (NB+1)*N,
208*> where NB is the max of the blocksize for CHETRD and for
209*> CUNMTR as returned by ILAENV.
210*>
211*> If LWORK = -1, then a workspace query is assumed; the routine
212*> only calculates the optimal size of the WORK array, returns
213*> this value as the first entry of the WORK array, and no error
214*> message related to LWORK is issued by XERBLA.
215*> \endverbatim
216*>
217*> \param[out] RWORK
218*> \verbatim
219*> RWORK is REAL array, dimension (7*N)
220*> \endverbatim
221*>
222*> \param[out] IWORK
223*> \verbatim
224*> IWORK is INTEGER array, dimension (5*N)
225*> \endverbatim
226*>
227*> \param[out] IFAIL
228*> \verbatim
229*> IFAIL is INTEGER array, dimension (N)
230*> If JOBZ = 'V', then if INFO = 0, the first M elements of
231*> IFAIL are zero. If INFO > 0, then IFAIL contains the
232*> indices of the eigenvectors that failed to converge.
233*> If JOBZ = 'N', then IFAIL is not referenced.
234*> \endverbatim
235*>
236*> \param[out] INFO
237*> \verbatim
238*> INFO is INTEGER
239*> = 0: successful exit
240*> < 0: if INFO = -i, the i-th argument had an illegal value
241*> > 0: if INFO = i, then i eigenvectors failed to converge.
242*> Their indices are stored in array IFAIL.
243*> \endverbatim
244*
245* Authors:
246* ========
247*
248*> \author Univ. of Tennessee
249*> \author Univ. of California Berkeley
250*> \author Univ. of Colorado Denver
251*> \author NAG Ltd.
252*
253*> \ingroup complexHEeigen
254*
255* =====================================================================
256 SUBROUTINE cheevx( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
257 $ ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK,
258 $ IWORK, IFAIL, INFO )
259*
260* -- LAPACK driver routine --
261* -- LAPACK is a software package provided by Univ. of Tennessee, --
262* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
263*
264* .. Scalar Arguments ..
265 CHARACTER JOBZ, RANGE, UPLO
266 INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
267 REAL ABSTOL, VL, VU
268* ..
269* .. Array Arguments ..
270 INTEGER IFAIL( * ), IWORK( * )
271 REAL RWORK( * ), W( * )
272 COMPLEX A( LDA, * ), WORK( * ), Z( LDZ, * )
273* ..
274*
275* =====================================================================
276*
277* .. Parameters ..
278 REAL ZERO, ONE
279 PARAMETER ( ZERO = 0.0e+0, one = 1.0e+0 )
280 COMPLEX CONE
281 parameter( cone = ( 1.0e+0, 0.0e+0 ) )
282* ..
283* .. Local Scalars ..
284 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
285 $ WANTZ
286 CHARACTER ORDER
287 INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
288 $ indisp, indiwk, indrwk, indtau, indwrk, iscale,
289 $ itmp1, j, jj, llwork, lwkmin, lwkopt, nb,
290 $ nsplit
291 REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
292 $ SIGMA, SMLNUM, TMP1, VLL, VUU
293* ..
294* .. External Functions ..
295 LOGICAL LSAME
296 INTEGER ILAENV
297 REAL SLAMCH, CLANHE
298 EXTERNAL lsame, ilaenv, slamch, clanhe
299* ..
300* .. External Subroutines ..
301 EXTERNAL scopy, sscal, sstebz, ssterf, xerbla, csscal,
303 $ cunmtr
304* ..
305* .. Intrinsic Functions ..
306 INTRINSIC real, max, min, sqrt
307* ..
308* .. Executable Statements ..
309*
310* Test the input parameters.
311*
312 lower = lsame( uplo, 'L' )
313 wantz = lsame( jobz, 'V' )
314 alleig = lsame( range, 'A' )
315 valeig = lsame( range, 'V' )
316 indeig = lsame( range, 'I' )
317 lquery = ( lwork.EQ.-1 )
318*
319 info = 0
320 IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
321 info = -1
322 ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
323 info = -2
324 ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
325 info = -3
326 ELSE IF( n.LT.0 ) THEN
327 info = -4
328 ELSE IF( lda.LT.max( 1, n ) ) THEN
329 info = -6
330 ELSE
331 IF( valeig ) THEN
332 IF( n.GT.0 .AND. vu.LE.vl )
333 $ info = -8
334 ELSE IF( indeig ) THEN
335 IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
336 info = -9
337 ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
338 info = -10
339 END IF
340 END IF
341 END IF
342 IF( info.EQ.0 ) THEN
343 IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
344 info = -15
345 END IF
346 END IF
347*
348 IF( info.EQ.0 ) THEN
349 IF( n.LE.1 ) THEN
350 lwkmin = 1
351 work( 1 ) = lwkmin
352 ELSE
353 lwkmin = 2*n
354 nb = ilaenv( 1, 'CHETRD', uplo, n, -1, -1, -1 )
355 nb = max( nb, ilaenv( 1, 'CUNMTR', uplo, n, -1, -1, -1 ) )
356 lwkopt = max( 1, ( nb + 1 )*n )
357 work( 1 ) = lwkopt
358 END IF
359*
360 IF( lwork.LT.lwkmin .AND. .NOT.lquery )
361 $ info = -17
362 END IF
363*
364 IF( info.NE.0 ) THEN
365 CALL xerbla( 'CHEEVX', -info )
366 RETURN
367 ELSE IF( lquery ) THEN
368 RETURN
369 END IF
370*
371* Quick return if possible
372*
373 m = 0
374 IF( n.EQ.0 ) THEN
375 RETURN
376 END IF
377*
378 IF( n.EQ.1 ) THEN
379 IF( alleig .OR. indeig ) THEN
380 m = 1
381 w( 1 ) = real( a( 1, 1 ) )
382 ELSE IF( valeig ) THEN
383 IF( vl.LT.real( a( 1, 1 ) ) .AND. vu.GE.real( a( 1, 1 ) ) )
384 $ THEN
385 m = 1
386 w( 1 ) = real( a( 1, 1 ) )
387 END IF
388 END IF
389 IF( wantz )
390 $ z( 1, 1 ) = cone
391 RETURN
392 END IF
393*
394* Get machine constants.
395*
396 safmin = slamch( 'Safe minimum' )
397 eps = slamch( 'Precision' )
398 smlnum = safmin / eps
399 bignum = one / smlnum
400 rmin = sqrt( smlnum )
401 rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
402*
403* Scale matrix to allowable range, if necessary.
404*
405 iscale = 0
406 abstll = abstol
407 IF( valeig ) THEN
408 vll = vl
409 vuu = vu
410 END IF
411 anrm = clanhe( 'M', uplo, n, a, lda, rwork )
412 IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
413 iscale = 1
414 sigma = rmin / anrm
415 ELSE IF( anrm.GT.rmax ) THEN
416 iscale = 1
417 sigma = rmax / anrm
418 END IF
419 IF( iscale.EQ.1 ) THEN
420 IF( lower ) THEN
421 DO 10 j = 1, n
422 CALL csscal( n-j+1, sigma, a( j, j ), 1 )
423 10 CONTINUE
424 ELSE
425 DO 20 j = 1, n
426 CALL csscal( j, sigma, a( 1, j ), 1 )
427 20 CONTINUE
428 END IF
429 IF( abstol.GT.0 )
430 $ abstll = abstol*sigma
431 IF( valeig ) THEN
432 vll = vl*sigma
433 vuu = vu*sigma
434 END IF
435 END IF
436*
437* Call CHETRD to reduce Hermitian matrix to tridiagonal form.
438*
439 indd = 1
440 inde = indd + n
441 indrwk = inde + n
442 indtau = 1
443 indwrk = indtau + n
444 llwork = lwork - indwrk + 1
445 CALL chetrd( uplo, n, a, lda, rwork( indd ), rwork( inde ),
446 $ work( indtau ), work( indwrk ), llwork, iinfo )
447*
448* If all eigenvalues are desired and ABSTOL is less than or equal to
449* zero, then call SSTERF or CUNGTR and CSTEQR. If this fails for
450* some eigenvalue, then try SSTEBZ.
451*
452 test = .false.
453 IF( indeig ) THEN
454 IF( il.EQ.1 .AND. iu.EQ.n ) THEN
455 test = .true.
456 END IF
457 END IF
458 IF( ( alleig .OR. test ) .AND. ( abstol.LE.zero ) ) THEN
459 CALL scopy( n, rwork( indd ), 1, w, 1 )
460 indee = indrwk + 2*n
461 IF( .NOT.wantz ) THEN
462 CALL scopy( n-1, rwork( inde ), 1, rwork( indee ), 1 )
463 CALL ssterf( n, w, rwork( indee ), info )
464 ELSE
465 CALL clacpy( 'A', n, n, a, lda, z, ldz )
466 CALL cungtr( uplo, n, z, ldz, work( indtau ),
467 $ work( indwrk ), llwork, iinfo )
468 CALL scopy( n-1, rwork( inde ), 1, rwork( indee ), 1 )
469 CALL csteqr( jobz, n, w, rwork( indee ), z, ldz,
470 $ rwork( indrwk ), info )
471 IF( info.EQ.0 ) THEN
472 DO 30 i = 1, n
473 ifail( i ) = 0
474 30 CONTINUE
475 END IF
476 END IF
477 IF( info.EQ.0 ) THEN
478 m = n
479 GO TO 40
480 END IF
481 info = 0
482 END IF
483*
484* Otherwise, call SSTEBZ and, if eigenvectors are desired, CSTEIN.
485*
486 IF( wantz ) THEN
487 order = 'B'
488 ELSE
489 order = 'E'
490 END IF
491 indibl = 1
492 indisp = indibl + n
493 indiwk = indisp + n
494 CALL sstebz( range, order, n, vll, vuu, il, iu, abstll,
495 $ rwork( indd ), rwork( inde ), m, nsplit, w,
496 $ iwork( indibl ), iwork( indisp ), rwork( indrwk ),
497 $ iwork( indiwk ), info )
498*
499 IF( wantz ) THEN
500 CALL cstein( n, rwork( indd ), rwork( inde ), m, w,
501 $ iwork( indibl ), iwork( indisp ), z, ldz,
502 $ rwork( indrwk ), iwork( indiwk ), ifail, info )
503*
504* Apply unitary matrix used in reduction to tridiagonal
505* form to eigenvectors returned by CSTEIN.
506*
507 CALL cunmtr( 'L', uplo, 'N', n, m, a, lda, work( indtau ), z,
508 $ ldz, work( indwrk ), llwork, iinfo )
509 END IF
510*
511* If matrix was scaled, then rescale eigenvalues appropriately.
512*
513 40 CONTINUE
514 IF( iscale.EQ.1 ) THEN
515 IF( info.EQ.0 ) THEN
516 imax = m
517 ELSE
518 imax = info - 1
519 END IF
520 CALL sscal( imax, one / sigma, w, 1 )
521 END IF
522*
523* If eigenvalues are not in order, then sort them, along with
524* eigenvectors.
525*
526 IF( wantz ) THEN
527 DO 60 j = 1, m - 1
528 i = 0
529 tmp1 = w( j )
530 DO 50 jj = j + 1, m
531 IF( w( jj ).LT.tmp1 ) THEN
532 i = jj
533 tmp1 = w( jj )
534 END IF
535 50 CONTINUE
536*
537 IF( i.NE.0 ) THEN
538 itmp1 = iwork( indibl+i-1 )
539 w( i ) = w( j )
540 iwork( indibl+i-1 ) = iwork( indibl+j-1 )
541 w( j ) = tmp1
542 iwork( indibl+j-1 ) = itmp1
543 CALL cswap( n, z( 1, i ), 1, z( 1, j ), 1 )
544 IF( info.NE.0 ) THEN
545 itmp1 = ifail( i )
546 ifail( i ) = ifail( j )
547 ifail( j ) = itmp1
548 END IF
549 END IF
550 60 CONTINUE
551 END IF
552*
553* Set WORK(1) to optimal complex workspace size.
554*
555 work( 1 ) = lwkopt
556*
557 RETURN
558*
559* End of CHEEVX
560*
561 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine ssterf(N, D, E, INFO)
SSTERF
Definition: ssterf.f:86
subroutine sstebz(RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK, INFO)
SSTEBZ
Definition: sstebz.f:273
subroutine csscal(N, SA, CX, INCX)
CSSCAL
Definition: csscal.f:78
subroutine cswap(N, CX, INCX, CY, INCY)
CSWAP
Definition: cswap.f:81
subroutine chetrd(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
CHETRD
Definition: chetrd.f:192
subroutine cheevx(JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK, IWORK, IFAIL, INFO)
CHEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices
Definition: cheevx.f:259
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 cunmtr(SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMTR
Definition: cunmtr.f:172
subroutine cstein(N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
CSTEIN
Definition: cstein.f:182
subroutine csteqr(COMPZ, N, D, E, Z, LDZ, WORK, INFO)
CSTEQR
Definition: csteqr.f:132
subroutine cungtr(UPLO, N, A, LDA, TAU, WORK, LWORK, INFO)
CUNGTR
Definition: cungtr.f:123
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79