LAPACK 3.11.0
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

Download CHEEVX + dependencies [TGZ] [ZIP] [TXT]

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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

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
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*
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
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
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
subroutine chetrd(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
CHETRD
Definition: chetrd.f:192
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
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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