LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ cheevr()

 subroutine cheevr ( 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, integer, dimension( * ) ISUPPZ, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO )

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

Purpose:
``` CHEEVR 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.

CHEEVR first reduces the matrix A to tridiagonal form T with a call
to CHETRD.  Then, whenever possible, CHEEVR calls CSTEMR to compute
the eigenspectrum using Relatively Robust Representations.  CSTEMR
computes eigenvalues by the dqds algorithm, while orthogonal
eigenvectors are computed from various "good" L D L^T representations
(also known as Relatively Robust Representations). Gram-Schmidt
orthogonalization is avoided as far as possible. More specifically,
the various steps of the algorithm are as follows.

For each unreduced block (submatrix) of T,
(a) Compute T - sigma I  = L D L^T, so that L and D
define all the wanted eigenvalues to high relative accuracy.
This means that small relative changes in the entries of D and L
cause only small relative changes in the eigenvalues and
eigenvectors. The standard (unfactored) representation of the
tridiagonal matrix T does not have this property in general.
(b) Compute the eigenvalues to suitable accuracy.
If the eigenvectors are desired, the algorithm attains full
accuracy of the computed eigenvalues only right before
the corresponding vectors have to be computed, see steps c) and d).
(c) For each cluster of close eigenvalues, select a new
shift close to the cluster, find a new factorization, and refine
the shifted eigenvalues to suitable accuracy.
(d) For each eigenvalue with a large enough relative separation compute
the corresponding eigenvector by forming a rank revealing twisted
factorization. Go back to (c) for any clusters that remain.

The desired accuracy of the output can be specified by the input
parameter ABSTOL.

For more details, see CSTEMR's documentation and:
- Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
- Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
2004.  Also LAPACK Working Note 154.
- Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
tridiagonal eigenvalue/eigenvector problem",
Computer Science Division Technical Report No. UCB/CSD-97-971,
UC Berkeley, May 1997.

Note 1 : CHEEVR calls CSTEMR when the full spectrum is requested
on machines which conform to the ieee-754 floating point standard.
CHEEVR calls SSTEBZ and CSTEIN on non-ieee machines and
when partial spectrum requests are made.

Normal execution of CSTEMR may create NaNs and infinities and
hence may abort due to a floating point exception in environments
which do not handle NaNs and infinities in the ieee standard default
manner.```
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. For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and CSTEIN are called``` [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. See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3. If high relative accuracy is important, set ABSTOL to SLAMCH( 'Safe minimum' ). Doing so will guarantee that eigenvalues are computed to high relative accuracy when possible in future releases. The current code does not make any guarantees about high relative accuracy, but future releases will. See J. Barlow and J. Demmel, "Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices", LAPACK Working Note #7, for a discussion of which matrices define their eigenvalues to high relative accuracy.``` [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) 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 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] ISUPPZ ``` ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ) The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). This is an output of CSTEMR (tridiagonal matrix). The support of the eigenvectors of A is typically 1:N because of the unitary transformations applied by CUNMTR. Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1``` [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 >= max(1,2*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 sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.``` [out] RWORK ``` RWORK is REAL array, dimension (MAX(1,LRWORK)) On exit, if INFO = 0, RWORK(1) returns the optimal (and minimal) LRWORK.``` [in] LRWORK ``` LRWORK is INTEGER The length of the array RWORK. LRWORK >= max(1,24*N). If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.``` [out] IWORK ``` IWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the optimal (and minimal) LIWORK.``` [in] LIWORK ``` LIWORK is INTEGER The dimension of the array IWORK. LIWORK >= max(1,10*N). If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: Internal error```
Contributors:
Osni Marques, LBNL/NERSC, USA
Ken Stanley, Computer Science Division, University of California at Berkeley, USA
Jason Riedy, Computer Science Division, University of California at Berkeley, USA

Definition at line 354 of file cheevr.f.

357*
358* -- LAPACK driver routine --
359* -- LAPACK is a software package provided by Univ. of Tennessee, --
360* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
361*
362* .. Scalar Arguments ..
363 CHARACTER JOBZ, RANGE, UPLO
364 INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK,
365 \$ M, N
366 REAL ABSTOL, VL, VU
367* ..
368* .. Array Arguments ..
369 INTEGER ISUPPZ( * ), IWORK( * )
370 REAL RWORK( * ), W( * )
371 COMPLEX A( LDA, * ), WORK( * ), Z( LDZ, * )
372* ..
373*
374* =====================================================================
375*
376* .. Parameters ..
377 REAL ZERO, ONE, TWO
378 parameter( zero = 0.0e+0, one = 1.0e+0, two = 2.0e+0 )
379* ..
380* .. Local Scalars ..
381 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
382 \$ WANTZ, TRYRAC
383 CHARACTER ORDER
384 INTEGER I, IEEEOK, IINFO, IMAX, INDIBL, INDIFL, INDISP,
385 \$ INDIWO, INDRD, INDRDD, INDRE, INDREE, INDRWK,
386 \$ INDTAU, INDWK, INDWKN, ISCALE, ITMP1, J, JJ,
387 \$ LIWMIN, LLWORK, LLRWORK, LLWRKN, LRWMIN,
388 \$ LWKOPT, LWMIN, NB, NSPLIT
389 REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
390 \$ SIGMA, SMLNUM, TMP1, VLL, VUU
391* ..
392* .. External Functions ..
393 LOGICAL LSAME
394 INTEGER ILAENV
395 REAL CLANSY, SLAMCH
396 EXTERNAL lsame, ilaenv, clansy, slamch
397* ..
398* .. External Subroutines ..
399 EXTERNAL chetrd, csscal, cstemr, cstein, cswap, cunmtr,
401* ..
402* .. Intrinsic Functions ..
403 INTRINSIC max, min, real, sqrt
404* ..
405* .. Executable Statements ..
406*
407* Test the input parameters.
408*
409 ieeeok = ilaenv( 10, 'CHEEVR', 'N', 1, 2, 3, 4 )
410*
411 lower = lsame( uplo, 'L' )
412 wantz = lsame( jobz, 'V' )
413 alleig = lsame( range, 'A' )
414 valeig = lsame( range, 'V' )
415 indeig = lsame( range, 'I' )
416*
417 lquery = ( ( lwork.EQ.-1 ) .OR. ( lrwork.EQ.-1 ) .OR.
418 \$ ( liwork.EQ.-1 ) )
419*
420 lrwmin = max( 1, 24*n )
421 liwmin = max( 1, 10*n )
422 lwmin = max( 1, 2*n )
423*
424 info = 0
425 IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
426 info = -1
427 ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
428 info = -2
429 ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
430 info = -3
431 ELSE IF( n.LT.0 ) THEN
432 info = -4
433 ELSE IF( lda.LT.max( 1, n ) ) THEN
434 info = -6
435 ELSE
436 IF( valeig ) THEN
437 IF( n.GT.0 .AND. vu.LE.vl )
438 \$ info = -8
439 ELSE IF( indeig ) THEN
440 IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
441 info = -9
442 ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
443 info = -10
444 END IF
445 END IF
446 END IF
447 IF( info.EQ.0 ) THEN
448 IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
449 info = -15
450 END IF
451 END IF
452*
453 IF( info.EQ.0 ) THEN
454 nb = ilaenv( 1, 'CHETRD', uplo, n, -1, -1, -1 )
455 nb = max( nb, ilaenv( 1, 'CUNMTR', uplo, n, -1, -1, -1 ) )
456 lwkopt = max( ( nb+1 )*n, lwmin )
457 work( 1 ) = lwkopt
458 rwork( 1 ) = lrwmin
459 iwork( 1 ) = liwmin
460*
461 IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
462 info = -18
463 ELSE IF( lrwork.LT.lrwmin .AND. .NOT.lquery ) THEN
464 info = -20
465 ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
466 info = -22
467 END IF
468 END IF
469*
470 IF( info.NE.0 ) THEN
471 CALL xerbla( 'CHEEVR', -info )
472 RETURN
473 ELSE IF( lquery ) THEN
474 RETURN
475 END IF
476*
477* Quick return if possible
478*
479 m = 0
480 IF( n.EQ.0 ) THEN
481 work( 1 ) = 1
482 RETURN
483 END IF
484*
485 IF( n.EQ.1 ) THEN
486 work( 1 ) = 2
487 IF( alleig .OR. indeig ) THEN
488 m = 1
489 w( 1 ) = real( a( 1, 1 ) )
490 ELSE
491 IF( vl.LT.real( a( 1, 1 ) ) .AND. vu.GE.real( a( 1, 1 ) ) )
492 \$ THEN
493 m = 1
494 w( 1 ) = real( a( 1, 1 ) )
495 END IF
496 END IF
497 IF( wantz ) THEN
498 z( 1, 1 ) = one
499 isuppz( 1 ) = 1
500 isuppz( 2 ) = 1
501 END IF
502 RETURN
503 END IF
504*
505* Get machine constants.
506*
507 safmin = slamch( 'Safe minimum' )
508 eps = slamch( 'Precision' )
509 smlnum = safmin / eps
510 bignum = one / smlnum
511 rmin = sqrt( smlnum )
512 rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
513*
514* Scale matrix to allowable range, if necessary.
515*
516 iscale = 0
517 abstll = abstol
518 IF (valeig) THEN
519 vll = vl
520 vuu = vu
521 END IF
522 anrm = clansy( 'M', uplo, n, a, lda, rwork )
523 IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
524 iscale = 1
525 sigma = rmin / anrm
526 ELSE IF( anrm.GT.rmax ) THEN
527 iscale = 1
528 sigma = rmax / anrm
529 END IF
530 IF( iscale.EQ.1 ) THEN
531 IF( lower ) THEN
532 DO 10 j = 1, n
533 CALL csscal( n-j+1, sigma, a( j, j ), 1 )
534 10 CONTINUE
535 ELSE
536 DO 20 j = 1, n
537 CALL csscal( j, sigma, a( 1, j ), 1 )
538 20 CONTINUE
539 END IF
540 IF( abstol.GT.0 )
541 \$ abstll = abstol*sigma
542 IF( valeig ) THEN
543 vll = vl*sigma
544 vuu = vu*sigma
545 END IF
546 END IF
547
548* Initialize indices into workspaces. Note: The IWORK indices are
549* used only if SSTERF or CSTEMR fail.
550
551* WORK(INDTAU:INDTAU+N-1) stores the complex scalar factors of the
552* elementary reflectors used in CHETRD.
553 indtau = 1
554* INDWK is the starting offset of the remaining complex workspace,
555* and LLWORK is the remaining complex workspace size.
556 indwk = indtau + n
557 llwork = lwork - indwk + 1
558
559* RWORK(INDRD:INDRD+N-1) stores the real tridiagonal's diagonal
560* entries.
561 indrd = 1
562* RWORK(INDRE:INDRE+N-1) stores the off-diagonal entries of the
563* tridiagonal matrix from CHETRD.
564 indre = indrd + n
565* RWORK(INDRDD:INDRDD+N-1) is a copy of the diagonal entries over
566* -written by CSTEMR (the SSTERF path copies the diagonal to W).
567 indrdd = indre + n
568* RWORK(INDREE:INDREE+N-1) is a copy of the off-diagonal entries over
569* -written while computing the eigenvalues in SSTERF and CSTEMR.
570 indree = indrdd + n
571* INDRWK is the starting offset of the left-over real workspace, and
572* LLRWORK is the remaining workspace size.
573 indrwk = indree + n
574 llrwork = lrwork - indrwk + 1
575
576* IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in SSTEBZ and
577* stores the block indices of each of the M<=N eigenvalues.
578 indibl = 1
579* IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in SSTEBZ and
580* stores the starting and finishing indices of each block.
581 indisp = indibl + n
582* IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
583* that corresponding to eigenvectors that fail to converge in
584* SSTEIN. This information is discarded; if any fail, the driver
585* returns INFO > 0.
586 indifl = indisp + n
587* INDIWO is the offset of the remaining integer workspace.
588 indiwo = indifl + n
589
590*
591* Call CHETRD to reduce Hermitian matrix to tridiagonal form.
592*
593 CALL chetrd( uplo, n, a, lda, rwork( indrd ), rwork( indre ),
594 \$ work( indtau ), work( indwk ), llwork, iinfo )
595*
596* If all eigenvalues are desired
597* then call SSTERF or CSTEMR and CUNMTR.
598*
599 test = .false.
600 IF( indeig ) THEN
601 IF( il.EQ.1 .AND. iu.EQ.n ) THEN
602 test = .true.
603 END IF
604 END IF
605 IF( ( alleig.OR.test ) .AND. ( ieeeok.EQ.1 ) ) THEN
606 IF( .NOT.wantz ) THEN
607 CALL scopy( n, rwork( indrd ), 1, w, 1 )
608 CALL scopy( n-1, rwork( indre ), 1, rwork( indree ), 1 )
609 CALL ssterf( n, w, rwork( indree ), info )
610 ELSE
611 CALL scopy( n-1, rwork( indre ), 1, rwork( indree ), 1 )
612 CALL scopy( n, rwork( indrd ), 1, rwork( indrdd ), 1 )
613*
614 IF (abstol .LE. two*n*eps) THEN
615 tryrac = .true.
616 ELSE
617 tryrac = .false.
618 END IF
619 CALL cstemr( jobz, 'A', n, rwork( indrdd ),
620 \$ rwork( indree ), vl, vu, il, iu, m, w,
621 \$ z, ldz, n, isuppz, tryrac,
622 \$ rwork( indrwk ), llrwork,
623 \$ iwork, liwork, info )
624*
625* Apply unitary matrix used in reduction to tridiagonal
626* form to eigenvectors returned by CSTEMR.
627*
628 IF( wantz .AND. info.EQ.0 ) THEN
629 indwkn = indwk
630 llwrkn = lwork - indwkn + 1
631 CALL cunmtr( 'L', uplo, 'N', n, m, a, lda,
632 \$ work( indtau ), z, ldz, work( indwkn ),
633 \$ llwrkn, iinfo )
634 END IF
635 END IF
636*
637*
638 IF( info.EQ.0 ) THEN
639 m = n
640 GO TO 30
641 END IF
642 info = 0
643 END IF
644*
645* Otherwise, call SSTEBZ and, if eigenvectors are desired, CSTEIN.
646* Also call SSTEBZ and CSTEIN if CSTEMR fails.
647*
648 IF( wantz ) THEN
649 order = 'B'
650 ELSE
651 order = 'E'
652 END IF
653
654 CALL sstebz( range, order, n, vll, vuu, il, iu, abstll,
655 \$ rwork( indrd ), rwork( indre ), m, nsplit, w,
656 \$ iwork( indibl ), iwork( indisp ), rwork( indrwk ),
657 \$ iwork( indiwo ), info )
658*
659 IF( wantz ) THEN
660 CALL cstein( n, rwork( indrd ), rwork( indre ), m, w,
661 \$ iwork( indibl ), iwork( indisp ), z, ldz,
662 \$ rwork( indrwk ), iwork( indiwo ), iwork( indifl ),
663 \$ info )
664*
665* Apply unitary matrix used in reduction to tridiagonal
666* form to eigenvectors returned by CSTEIN.
667*
668 indwkn = indwk
669 llwrkn = lwork - indwkn + 1
670 CALL cunmtr( 'L', uplo, 'N', n, m, a, lda, work( indtau ), z,
671 \$ ldz, work( indwkn ), llwrkn, iinfo )
672 END IF
673*
674* If matrix was scaled, then rescale eigenvalues appropriately.
675*
676 30 CONTINUE
677 IF( iscale.EQ.1 ) THEN
678 IF( info.EQ.0 ) THEN
679 imax = m
680 ELSE
681 imax = info - 1
682 END IF
683 CALL sscal( imax, one / sigma, w, 1 )
684 END IF
685*
686* If eigenvalues are not in order, then sort them, along with
687* eigenvectors.
688*
689 IF( wantz ) THEN
690 DO 50 j = 1, m - 1
691 i = 0
692 tmp1 = w( j )
693 DO 40 jj = j + 1, m
694 IF( w( jj ).LT.tmp1 ) THEN
695 i = jj
696 tmp1 = w( jj )
697 END IF
698 40 CONTINUE
699*
700 IF( i.NE.0 ) THEN
701 itmp1 = iwork( indibl+i-1 )
702 w( i ) = w( j )
703 iwork( indibl+i-1 ) = iwork( indibl+j-1 )
704 w( j ) = tmp1
705 iwork( indibl+j-1 ) = itmp1
706 CALL cswap( n, z( 1, i ), 1, z( 1, j ), 1 )
707 END IF
708 50 CONTINUE
709 END IF
710*
711* Set WORK(1) to optimal workspace size.
712*
713 work( 1 ) = lwkopt
714 rwork( 1 ) = lrwmin
715 iwork( 1 ) = liwmin
716*
717 RETURN
718*
719* End of CHEEVR
720*
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
subroutine chetrd(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
CHETRD
Definition: chetrd.f:192
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 cstemr(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, IWORK, LIWORK, INFO)
CSTEMR
Definition: cstemr.f:338
real function clansy(NORM, UPLO, N, A, LDA, WORK)
CLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: clansy.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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