LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ dsyevx()

 subroutine dsyevx ( character jobz, character range, character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision vl, double precision vu, integer il, integer iu, double precision abstol, integer m, double precision, dimension( * ) w, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info )

DSYEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

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Purpose:
``` DSYEVX computes selected eigenvalues and, optionally, eigenvectors
of a real symmetric matrix A.  Eigenvalues and eigenvectors can be
selected by specifying either a range of values or a range of indices
for the desired eigenvalues.```
Parameters
 [in] JOBZ ``` JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.``` [in] RANGE ``` RANGE is CHARACTER*1 = 'A': all eigenvalues will be found. = 'V': all eigenvalues in the half-open interval (VL,VU] will be found. = 'I': the IL-th through IU-th eigenvalues will be found.``` [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in,out] A ``` A is DOUBLE PRECISION array, dimension (LDA, N) On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A. On exit, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in] VL ``` VL is DOUBLE PRECISION If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.``` [in] VU ``` VU is DOUBLE PRECISION If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.``` [in] IL ``` IL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.``` [in] IU ``` IU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.``` [in] ABSTOL ``` ABSTOL is DOUBLE PRECISION The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*DLAMCH('S'). See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3.``` [out] M ``` M is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.``` [out] W ``` W is DOUBLE PRECISION array, dimension (N) On normal exit, the first M elements contain the selected eigenvalues in ascending order.``` [out] Z ``` Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M)) If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. If JOBZ = 'N', then Z is not referenced. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used.``` [in] LDZ ``` LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).``` [out] WORK ``` WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.``` [in] LWORK ``` LWORK is INTEGER The length of the array WORK. LWORK >= 1, when N <= 1; otherwise 8*N. For optimal efficiency, LWORK >= (NB+3)*N, where NB is the max of the blocksize for DSYTRD and DORMTR returned by ILAENV. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.``` [out] IWORK ` IWORK is INTEGER array, dimension (5*N)` [out] IFAIL ``` IFAIL is INTEGER array, dimension (N) If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in array IFAIL.```

Definition at line 250 of file dsyevx.f.

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 DOUBLE PRECISION ABSTOL, VL, VU
262* ..
263* .. Array Arguments ..
264 INTEGER IFAIL( * ), IWORK( * )
265 DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
266* ..
267*
268* =====================================================================
269*
270* .. Parameters ..
271 DOUBLE PRECISION ZERO, ONE
272 parameter( zero = 0.0d+0, one = 1.0d+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 DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
283 \$ SIGMA, SMLNUM, TMP1, VLL, VUU
284* ..
285* .. External Functions ..
286 LOGICAL LSAME
287 INTEGER ILAENV
288 DOUBLE PRECISION DLAMCH, DLANSY
289 EXTERNAL lsame, ilaenv, dlamch, dlansy
290* ..
291* .. External Subroutines ..
292 EXTERNAL dcopy, dlacpy, dorgtr, dormtr, dscal, dstebz,
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, 'DSYTRD', uplo, n, -1, -1, -1 )
345 nb = max( nb, ilaenv( 1, 'DORMTR', 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( 'DSYEVX', -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 = dlamch( 'Safe minimum' )
386 eps = dlamch( '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 = dlansy( '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 dscal( n-j+1, sigma, a( j, j ), 1 )
412 10 CONTINUE
413 ELSE
414 DO 20 j = 1, n
415 CALL dscal( 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 DSYTRD 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 dsytrd( 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 DSTERF or DORGTR and SSTEQR. If this fails for
438* some eigenvalue, then try DSTEBZ.
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 dcopy( n, work( indd ), 1, w, 1 )
448 indee = indwrk + 2*n
449 IF( .NOT.wantz ) THEN
450 CALL dcopy( n-1, work( inde ), 1, work( indee ), 1 )
451 CALL dsterf( n, w, work( indee ), info )
452 ELSE
453 CALL dlacpy( 'A', n, n, a, lda, z, ldz )
454 CALL dorgtr( uplo, n, z, ldz, work( indtau ),
455 \$ work( indwrk ), llwork, iinfo )
456 CALL dcopy( n-1, work( inde ), 1, work( indee ), 1 )
457 CALL dsteqr( 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 DSTEBZ 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 dstebz( 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 dstein( 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 DSTEIN.
494*
495 indwkn = inde
496 llwrkn = lwork - indwkn + 1
497 CALL dormtr( '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 dscal( 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 dswap( 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 DSYEVX
550*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine dcopy(n, dx, incx, dy, incy)
DCOPY
Definition dcopy.f:82
subroutine dsytrd(uplo, n, a, lda, d, e, tau, work, lwork, info)
DSYTRD
Definition dsytrd.f:192
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:162
subroutine dlacpy(uplo, m, n, a, lda, b, ldb)
DLACPY copies all or part of one two-dimensional array to another.
Definition dlacpy.f:103
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
double precision function dlansy(norm, uplo, n, a, lda, work)
DLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition dlansy.f:122
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine dscal(n, da, dx, incx)
DSCAL
Definition dscal.f:79
subroutine dstebz(range, order, n, vl, vu, il, iu, abstol, d, e, m, nsplit, w, iblock, isplit, work, iwork, info)
DSTEBZ
Definition dstebz.f:273
subroutine dstein(n, d, e, m, w, iblock, isplit, z, ldz, work, iwork, ifail, info)
DSTEIN
Definition dstein.f:174
subroutine dsteqr(compz, n, d, e, z, ldz, work, info)
DSTEQR
Definition dsteqr.f:131
subroutine dsterf(n, d, e, info)
DSTERF
Definition dsterf.f:86
subroutine dswap(n, dx, incx, dy, incy)
DSWAP
Definition dswap.f:82
subroutine dorgtr(uplo, n, a, lda, tau, work, lwork, info)
DORGTR
Definition dorgtr.f:123
subroutine dormtr(side, uplo, trans, m, n, a, lda, tau, c, ldc, work, lwork, info)
DORMTR
Definition dormtr.f:171
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