LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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ssyevx.f
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1*> \brief <b> SSYEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY 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 SSYEVX + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ssyevx.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ssyevx.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ssyevx.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE SSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
22* ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK,
23* 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 A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
33* ..
34*
35*
36*> \par Purpose:
37* =============
38*>
39*> \verbatim
40*>
41*> SSYEVX computes selected eigenvalues and, optionally, eigenvectors
42*> of a real symmetric matrix A. Eigenvalues and eigenvectors can be
43*> selected by specifying either a range of values or a range of indices
44*> for the desired eigenvalues.
45*> \endverbatim
46*
47* Arguments:
48* ==========
49*
50*> \param[in] JOBZ
51*> \verbatim
52*> JOBZ is CHARACTER*1
53*> = 'N': Compute eigenvalues only;
54*> = 'V': Compute eigenvalues and eigenvectors.
55*> \endverbatim
56*>
57*> \param[in] RANGE
58*> \verbatim
59*> RANGE is CHARACTER*1
60*> = 'A': all eigenvalues will be found.
61*> = 'V': all eigenvalues in the half-open interval (VL,VU]
62*> will be found.
63*> = 'I': the IL-th through IU-th eigenvalues will be found.
64*> \endverbatim
65*>
66*> \param[in] UPLO
67*> \verbatim
68*> UPLO is CHARACTER*1
69*> = 'U': Upper triangle of A is stored;
70*> = 'L': Lower triangle of A is stored.
71*> \endverbatim
72*>
73*> \param[in] N
74*> \verbatim
75*> N is INTEGER
76*> The order of the matrix A. N >= 0.
77*> \endverbatim
78*>
79*> \param[in,out] A
80*> \verbatim
81*> A is REAL array, dimension (LDA, N)
82*> On entry, the symmetric matrix A. If UPLO = 'U', the
83*> leading N-by-N upper triangular part of A contains the
84*> upper triangular part of the matrix A. If UPLO = 'L',
85*> the leading N-by-N lower triangular part of A contains
86*> the lower triangular part of the matrix A.
87*> On exit, the lower triangle (if UPLO='L') or the upper
88*> triangle (if UPLO='U') of A, including the diagonal, is
89*> destroyed.
90*> \endverbatim
91*>
92*> \param[in] LDA
93*> \verbatim
94*> LDA is INTEGER
95*> The leading dimension of the array A. LDA >= max(1,N).
96*> \endverbatim
97*>
98*> \param[in] VL
99*> \verbatim
100*> VL is REAL
101*> If RANGE='V', the lower bound of the interval to
102*> be searched for eigenvalues. VL < VU.
103*> Not referenced if RANGE = 'A' or 'I'.
104*> \endverbatim
105*>
106*> \param[in] VU
107*> \verbatim
108*> VU is REAL
109*> If RANGE='V', the upper bound of the interval to
110*> be searched for eigenvalues. VL < VU.
111*> Not referenced if RANGE = 'A' or 'I'.
112*> \endverbatim
113*>
114*> \param[in] IL
115*> \verbatim
116*> IL is INTEGER
117*> If RANGE='I', the index of the
118*> smallest eigenvalue to be returned.
119*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
120*> Not referenced if RANGE = 'A' or 'V'.
121*> \endverbatim
122*>
123*> \param[in] IU
124*> \verbatim
125*> IU is INTEGER
126*> If RANGE='I', the index of the
127*> largest eigenvalue to be returned.
128*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
129*> Not referenced if RANGE = 'A' or 'V'.
130*> \endverbatim
131*>
132*> \param[in] ABSTOL
133*> \verbatim
134*> ABSTOL is REAL
135*> The absolute error tolerance for the eigenvalues.
136*> An approximate eigenvalue is accepted as converged
137*> when it is determined to lie in an interval [a,b]
138*> of width less than or equal to
139*>
140*> ABSTOL + EPS * max( |a|,|b| ) ,
141*>
142*> where EPS is the machine precision. If ABSTOL is less than
143*> or equal to zero, then EPS*|T| will be used in its place,
144*> where |T| is the 1-norm of the tridiagonal matrix obtained
145*> by reducing A to tridiagonal form.
146*>
147*> Eigenvalues will be computed most accurately when ABSTOL is
148*> set to twice the underflow threshold 2*SLAMCH('S'), not zero.
149*> If this routine returns with INFO>0, indicating that some
150*> eigenvectors did not converge, try setting ABSTOL to
151*> 2*SLAMCH('S').
152*>
153*> See "Computing Small Singular Values of Bidiagonal Matrices
154*> with Guaranteed High Relative Accuracy," by Demmel and
155*> Kahan, LAPACK Working Note #3.
156*> \endverbatim
157*>
158*> \param[out] M
159*> \verbatim
160*> M is INTEGER
161*> The total number of eigenvalues found. 0 <= M <= N.
162*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
163*> \endverbatim
164*>
165*> \param[out] W
166*> \verbatim
167*> W is REAL array, dimension (N)
168*> On normal exit, the first M elements contain the selected
169*> eigenvalues in ascending order.
170*> \endverbatim
171*>
172*> \param[out] Z
173*> \verbatim
174*> Z is REAL array, dimension (LDZ, max(1,M))
175*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
176*> contain the orthonormal eigenvectors of the matrix A
177*> corresponding to the selected eigenvalues, with the i-th
178*> column of Z holding the eigenvector associated with W(i).
179*> If an eigenvector fails to converge, then that column of Z
180*> contains the latest approximation to the eigenvector, and the
181*> index of the eigenvector is returned in IFAIL.
182*> If JOBZ = 'N', then Z is not referenced.
183*> Note: the user must ensure that at least max(1,M) columns are
184*> supplied in the array Z; if RANGE = 'V', the exact value of M
185*> is not known in advance and an upper bound must be used.
186*> \endverbatim
187*>
188*> \param[in] LDZ
189*> \verbatim
190*> LDZ is INTEGER
191*> The leading dimension of the array Z. LDZ >= 1, and if
192*> JOBZ = 'V', LDZ >= max(1,N).
193*> \endverbatim
194*>
195*> \param[out] WORK
196*> \verbatim
197*> WORK is REAL array, dimension (MAX(1,LWORK))
198*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
199*> \endverbatim
200*>
201*> \param[in] LWORK
202*> \verbatim
203*> LWORK is INTEGER
204*> The length of the array WORK. LWORK >= 1, when N <= 1;
205*> otherwise 8*N.
206*> For optimal efficiency, LWORK >= (NB+3)*N,
207*> where NB is the max of the blocksize for SSYTRD and SORMTR
208*> returned by ILAENV.
209*>
210*> If LWORK = -1, then a workspace query is assumed; the routine
211*> only calculates the optimal size of the WORK array, returns
212*> this value as the first entry of the WORK array, and no error
213*> message related to LWORK is issued by XERBLA.
214*> \endverbatim
215*>
216*> \param[out] IWORK
217*> \verbatim
218*> IWORK is INTEGER array, dimension (5*N)
219*> \endverbatim
220*>
221*> \param[out] IFAIL
222*> \verbatim
223*> IFAIL is INTEGER array, dimension (N)
224*> If JOBZ = 'V', then if INFO = 0, the first M elements of
225*> IFAIL are zero. If INFO > 0, then IFAIL contains the
226*> indices of the eigenvectors that failed to converge.
227*> If JOBZ = 'N', then IFAIL is not referenced.
228*> \endverbatim
229*>
230*> \param[out] INFO
231*> \verbatim
232*> INFO is INTEGER
233*> = 0: successful exit
234*> < 0: if INFO = -i, the i-th argument had an illegal value
235*> > 0: if INFO = i, then i eigenvectors failed to converge.
236*> Their indices are stored in array IFAIL.
237*> \endverbatim
238*
239* Authors:
240* ========
241*
242*> \author Univ. of Tennessee
243*> \author Univ. of California Berkeley
244*> \author Univ. of Colorado Denver
245*> \author NAG Ltd.
246*
247*> \ingroup realSYeigen
248*
249* =====================================================================
250 SUBROUTINE ssyevx( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
251 $ ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK,
252 $ IFAIL, INFO )
253*
254* -- LAPACK driver routine --
255* -- LAPACK is a software package provided by Univ. of Tennessee, --
256* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
257*
258* .. Scalar Arguments ..
259 CHARACTER JOBZ, RANGE, UPLO
260 INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
261 REAL ABSTOL, VL, VU
262* ..
263* .. Array Arguments ..
264 INTEGER IFAIL( * ), IWORK( * )
265 REAL A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
266* ..
267*
268* =====================================================================
269*
270* .. Parameters ..
271 REAL ZERO, ONE
272 PARAMETER ( ZERO = 0.0e+0, one = 1.0e+0 )
273* ..
274* .. Local Scalars ..
275 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
276 $ WANTZ
277 CHARACTER ORDER
278 INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
279 $ indisp, indiwo, indtau, indwkn, indwrk, iscale,
280 $ itmp1, j, jj, llwork, llwrkn, lwkmin,
281 $ lwkopt, nb, nsplit
282 REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
283 $ SIGMA, SMLNUM, TMP1, VLL, VUU
284* ..
285* .. External Functions ..
286 LOGICAL LSAME
287 INTEGER ILAENV
288 REAL SLAMCH, SLANSY
289 EXTERNAL lsame, ilaenv, slamch, slansy
290* ..
291* .. External Subroutines ..
292 EXTERNAL scopy, slacpy, sorgtr, sormtr, sscal, sstebz,
294* ..
295* .. Intrinsic Functions ..
296 INTRINSIC max, min, sqrt
297* ..
298* .. Executable Statements ..
299*
300* Test the input parameters.
301*
302 lower = lsame( uplo, 'L' )
303 wantz = lsame( jobz, 'V' )
304 alleig = lsame( range, 'A' )
305 valeig = lsame( range, 'V' )
306 indeig = lsame( range, 'I' )
307 lquery = ( lwork.EQ.-1 )
308*
309 info = 0
310 IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
311 info = -1
312 ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
313 info = -2
314 ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
315 info = -3
316 ELSE IF( n.LT.0 ) THEN
317 info = -4
318 ELSE IF( lda.LT.max( 1, n ) ) THEN
319 info = -6
320 ELSE
321 IF( valeig ) THEN
322 IF( n.GT.0 .AND. vu.LE.vl )
323 $ info = -8
324 ELSE IF( indeig ) THEN
325 IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
326 info = -9
327 ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
328 info = -10
329 END IF
330 END IF
331 END IF
332 IF( info.EQ.0 ) THEN
333 IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
334 info = -15
335 END IF
336 END IF
337*
338 IF( info.EQ.0 ) THEN
339 IF( n.LE.1 ) THEN
340 lwkmin = 1
341 work( 1 ) = lwkmin
342 ELSE
343 lwkmin = 8*n
344 nb = ilaenv( 1, 'SSYTRD', uplo, n, -1, -1, -1 )
345 nb = max( nb, ilaenv( 1, 'SORMTR', uplo, n, -1, -1, -1 ) )
346 lwkopt = max( lwkmin, ( nb + 3 )*n )
347 work( 1 ) = lwkopt
348 END IF
349*
350 IF( lwork.LT.lwkmin .AND. .NOT.lquery )
351 $ info = -17
352 END IF
353*
354 IF( info.NE.0 ) THEN
355 CALL xerbla( 'SSYEVX', -info )
356 RETURN
357 ELSE IF( lquery ) THEN
358 RETURN
359 END IF
360*
361* Quick return if possible
362*
363 m = 0
364 IF( n.EQ.0 ) THEN
365 RETURN
366 END IF
367*
368 IF( n.EQ.1 ) THEN
369 IF( alleig .OR. indeig ) THEN
370 m = 1
371 w( 1 ) = a( 1, 1 )
372 ELSE
373 IF( vl.LT.a( 1, 1 ) .AND. vu.GE.a( 1, 1 ) ) THEN
374 m = 1
375 w( 1 ) = a( 1, 1 )
376 END IF
377 END IF
378 IF( wantz )
379 $ z( 1, 1 ) = one
380 RETURN
381 END IF
382*
383* Get machine constants.
384*
385 safmin = slamch( 'Safe minimum' )
386 eps = slamch( 'Precision' )
387 smlnum = safmin / eps
388 bignum = one / smlnum
389 rmin = sqrt( smlnum )
390 rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
391*
392* Scale matrix to allowable range, if necessary.
393*
394 iscale = 0
395 abstll = abstol
396 IF( valeig ) THEN
397 vll = vl
398 vuu = vu
399 END IF
400 anrm = slansy( 'M', uplo, n, a, lda, work )
401 IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
402 iscale = 1
403 sigma = rmin / anrm
404 ELSE IF( anrm.GT.rmax ) THEN
405 iscale = 1
406 sigma = rmax / anrm
407 END IF
408 IF( iscale.EQ.1 ) THEN
409 IF( lower ) THEN
410 DO 10 j = 1, n
411 CALL sscal( n-j+1, sigma, a( j, j ), 1 )
412 10 CONTINUE
413 ELSE
414 DO 20 j = 1, n
415 CALL sscal( j, sigma, a( 1, j ), 1 )
416 20 CONTINUE
417 END IF
418 IF( abstol.GT.0 )
419 $ abstll = abstol*sigma
420 IF( valeig ) THEN
421 vll = vl*sigma
422 vuu = vu*sigma
423 END IF
424 END IF
425*
426* Call SSYTRD to reduce symmetric matrix to tridiagonal form.
427*
428 indtau = 1
429 inde = indtau + n
430 indd = inde + n
431 indwrk = indd + n
432 llwork = lwork - indwrk + 1
433 CALL ssytrd( uplo, n, a, lda, work( indd ), work( inde ),
434 $ work( indtau ), work( indwrk ), llwork, iinfo )
435*
436* If all eigenvalues are desired and ABSTOL is less than or equal to
437* zero, then call SSTERF or SORGTR and SSTEQR. If this fails for
438* some eigenvalue, then try SSTEBZ.
439*
440 test = .false.
441 IF( indeig ) THEN
442 IF( il.EQ.1 .AND. iu.EQ.n ) THEN
443 test = .true.
444 END IF
445 END IF
446 IF( ( alleig .OR. test ) .AND. ( abstol.LE.zero ) ) THEN
447 CALL scopy( n, work( indd ), 1, w, 1 )
448 indee = indwrk + 2*n
449 IF( .NOT.wantz ) THEN
450 CALL scopy( n-1, work( inde ), 1, work( indee ), 1 )
451 CALL ssterf( n, w, work( indee ), info )
452 ELSE
453 CALL slacpy( 'A', n, n, a, lda, z, ldz )
454 CALL sorgtr( uplo, n, z, ldz, work( indtau ),
455 $ work( indwrk ), llwork, iinfo )
456 CALL scopy( n-1, work( inde ), 1, work( indee ), 1 )
457 CALL ssteqr( jobz, n, w, work( indee ), z, ldz,
458 $ work( indwrk ), info )
459 IF( info.EQ.0 ) THEN
460 DO 30 i = 1, n
461 ifail( i ) = 0
462 30 CONTINUE
463 END IF
464 END IF
465 IF( info.EQ.0 ) THEN
466 m = n
467 GO TO 40
468 END IF
469 info = 0
470 END IF
471*
472* Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN.
473*
474 IF( wantz ) THEN
475 order = 'B'
476 ELSE
477 order = 'E'
478 END IF
479 indibl = 1
480 indisp = indibl + n
481 indiwo = indisp + n
482 CALL sstebz( range, order, n, vll, vuu, il, iu, abstll,
483 $ work( indd ), work( inde ), m, nsplit, w,
484 $ iwork( indibl ), iwork( indisp ), work( indwrk ),
485 $ iwork( indiwo ), info )
486*
487 IF( wantz ) THEN
488 CALL sstein( n, work( indd ), work( inde ), m, w,
489 $ iwork( indibl ), iwork( indisp ), z, ldz,
490 $ work( indwrk ), iwork( indiwo ), ifail, info )
491*
492* Apply orthogonal matrix used in reduction to tridiagonal
493* form to eigenvectors returned by SSTEIN.
494*
495 indwkn = inde
496 llwrkn = lwork - indwkn + 1
497 CALL sormtr( 'L', uplo, 'N', n, m, a, lda, work( indtau ), z,
498 $ ldz, work( indwkn ), llwrkn, iinfo )
499 END IF
500*
501* If matrix was scaled, then rescale eigenvalues appropriately.
502*
503 40 CONTINUE
504 IF( iscale.EQ.1 ) THEN
505 IF( info.EQ.0 ) THEN
506 imax = m
507 ELSE
508 imax = info - 1
509 END IF
510 CALL sscal( imax, one / sigma, w, 1 )
511 END IF
512*
513* If eigenvalues are not in order, then sort them, along with
514* eigenvectors.
515*
516 IF( wantz ) THEN
517 DO 60 j = 1, m - 1
518 i = 0
519 tmp1 = w( j )
520 DO 50 jj = j + 1, m
521 IF( w( jj ).LT.tmp1 ) THEN
522 i = jj
523 tmp1 = w( jj )
524 END IF
525 50 CONTINUE
526*
527 IF( i.NE.0 ) THEN
528 itmp1 = iwork( indibl+i-1 )
529 w( i ) = w( j )
530 iwork( indibl+i-1 ) = iwork( indibl+j-1 )
531 w( j ) = tmp1
532 iwork( indibl+j-1 ) = itmp1
533 CALL sswap( n, z( 1, i ), 1, z( 1, j ), 1 )
534 IF( info.NE.0 ) THEN
535 itmp1 = ifail( i )
536 ifail( i ) = ifail( j )
537 ifail( j ) = itmp1
538 END IF
539 END IF
540 60 CONTINUE
541 END IF
542*
543* Set WORK(1) to optimal workspace size.
544*
545 work( 1 ) = lwkopt
546*
547 RETURN
548*
549* End of SSYEVX
550*
551 END
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine ssteqr(COMPZ, N, D, E, Z, LDZ, WORK, INFO)
SSTEQR
Definition: ssteqr.f:131
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 sormtr(SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMTR
Definition: sormtr.f:172
subroutine sstein(N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
SSTEIN
Definition: sstein.f:174
subroutine sorgtr(UPLO, N, A, LDA, TAU, WORK, LWORK, INFO)
SORGTR
Definition: sorgtr.f:123
subroutine ssytrd(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
SSYTRD
Definition: ssytrd.f:192
subroutine ssyevx(JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO)
SSYEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
Definition: ssyevx.f:253
subroutine sswap(N, SX, INCX, SY, INCY)
SSWAP
Definition: sswap.f:82
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79