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

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

CHEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Purpose:
``` CHEEVX computes selected eigenvalues and, optionally, eigenvectors
of a complex Hermitian 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 COMPLEX array, dimension (LDA, N) On entry, the Hermitian 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 REAL 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 REAL 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 REAL 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*SLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('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 REAL array, dimension (N) On normal exit, the first M elements contain the selected eigenvalues in ascending order.``` [out] Z ``` Z is COMPLEX 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 COMPLEX 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 2*N. For optimal efficiency, LWORK >= (NB+1)*N, where NB is the max of the blocksize for CHETRD and for CUNMTR as 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] RWORK ` RWORK is REAL array, dimension (7*N)` [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 256 of file cheevx.f.

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, SROUNDUP_LWORK
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 ) = sroundup_lwork(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 ) = sroundup_lwork(lwkopt)
556*
557 RETURN
558*
559* End of CHEEVX
560*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine scopy(n, sx, incx, sy, incy)
SCOPY
Definition scopy.f:82
subroutine chetrd(uplo, n, a, lda, d, e, tau, work, lwork, info)
CHETRD
Definition chetrd.f:192
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:162
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
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
real function clanhe(norm, uplo, n, a, lda, work)
CLANHE returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition clanhe.f:124
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
real function sroundup_lwork(lwork)
SROUNDUP_LWORK
subroutine csscal(n, sa, cx, incx)
CSSCAL
Definition csscal.f:78
subroutine sscal(n, sa, sx, incx)
SSCAL
Definition sscal.f:79
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 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 ssterf(n, d, e, info)
SSTERF
Definition ssterf.f:86
subroutine cswap(n, cx, incx, cy, incy)
CSWAP
Definition cswap.f:81
subroutine cungtr(uplo, n, a, lda, tau, work, lwork, info)
CUNGTR
Definition cungtr.f:123
subroutine cunmtr(side, uplo, trans, m, n, a, lda, tau, c, ldc, work, lwork, info)
CUNMTR
Definition cunmtr.f:172
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