 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ cheevr_2stage()

 subroutine cheevr_2stage ( 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_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Purpose:
``` CHEEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors
of a complex Hermitian matrix A using the 2stage technique for
the reduction to tridiagonal.  Eigenvalues and eigenvectors can
be selected by specifying either a range of values or a range of
indices for the desired eigenvalues.

CHEEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call
to CHETRD.  Then, whenever possible, CHEEVR_2STAGE calls CSTEMR to compute
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_2STAGE calls CSTEMR when the full spectrum is requested
on machines which conform to the ieee-754 floating point standard.
CHEEVR_2STAGE 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. Not available in this release.``` [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 dimension of the array WORK. If JOBZ = 'N' and N > 1, LWORK must be queried. LWORK = MAX(1, 26*N, dimension) where dimension = max(stage1,stage2) + (KD+1)*N + N = N*KD + N*max(KD+1,FACTOPTNB) + max(2*KD*KD, KD*NTHREADS) + (KD+1)*N + N where KD is the blocking size of the reduction, FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1. If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available 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:
```Inderjit Dhillon, IBM Almaden, USA \n
Osni Marques, LBNL/NERSC, USA \n
Ken Stanley, Computer Science Division, University of
California at Berkeley, USA \n
Jason Riedy, Computer Science Division, University of
California at Berkeley, USA \n
```
Further Details:
```  All details about the 2stage techniques are available in:

Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
Parallel reduction to condensed forms for symmetric eigenvalue problems
using aggregated fine-grained and memory-aware kernels. In Proceedings
of 2011 International Conference for High Performance Computing,
Networking, Storage and Analysis (SC '11), New York, NY, USA,
Article 8 , 11 pages.
http://doi.acm.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013.
An improved parallel singular value algorithm and its implementation
for multicore hardware, In Proceedings of 2013 International Conference
for High Performance Computing, Networking, Storage and Analysis (SC '13).
Article 90, 12 pages.
http://doi.acm.org/10.1145/2503210.2503292

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
A novel hybrid CPU-GPU generalized eigensolver for electronic structure
calculations based on fine-grained memory aware tasks.
International Journal of High Performance Computing Applications.
Volume 28 Issue 2, Pages 196-209, May 2014.
http://hpc.sagepub.com/content/28/2/196 ```

Definition at line 402 of file cheevr_2stage.f.

406*
407 IMPLICIT NONE
408*
409* -- LAPACK driver routine --
410* -- LAPACK is a software package provided by Univ. of Tennessee, --
411* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
412*
413* .. Scalar Arguments ..
414 CHARACTER JOBZ, RANGE, UPLO
415 INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK,
416 \$ M, N
417 REAL ABSTOL, VL, VU
418* ..
419* .. Array Arguments ..
420 INTEGER ISUPPZ( * ), IWORK( * )
421 REAL RWORK( * ), W( * )
422 COMPLEX A( LDA, * ), WORK( * ), Z( LDZ, * )
423* ..
424*
425* =====================================================================
426*
427* .. Parameters ..
428 REAL ZERO, ONE, TWO
429 parameter( zero = 0.0e+0, one = 1.0e+0, two = 2.0e+0 )
430* ..
431* .. Local Scalars ..
432 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
433 \$ WANTZ, TRYRAC
434 CHARACTER ORDER
435 INTEGER I, IEEEOK, IINFO, IMAX, INDIBL, INDIFL, INDISP,
436 \$ INDIWO, INDRD, INDRDD, INDRE, INDREE, INDRWK,
437 \$ INDTAU, INDWK, INDWKN, ISCALE, ITMP1, J, JJ,
438 \$ LIWMIN, LLWORK, LLRWORK, LLWRKN, LRWMIN,
439 \$ LWMIN, NSPLIT, LHTRD, LWTRD, KD, IB, INDHOUS
440 REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
441 \$ SIGMA, SMLNUM, TMP1, VLL, VUU
442* ..
443* .. External Functions ..
444 LOGICAL LSAME
445 INTEGER ILAENV, ILAENV2STAGE
446 REAL SLAMCH, CLANSY
448* ..
449* .. External Subroutines ..
450 EXTERNAL scopy, sscal, sstebz, ssterf, xerbla, csscal,
452* ..
453* .. Intrinsic Functions ..
454 INTRINSIC real, max, min, sqrt
455* ..
456* .. Executable Statements ..
457*
458* Test the input parameters.
459*
460 ieeeok = ilaenv( 10, 'CHEEVR', 'N', 1, 2, 3, 4 )
461*
462 lower = lsame( uplo, 'L' )
463 wantz = lsame( jobz, 'V' )
464 alleig = lsame( range, 'A' )
465 valeig = lsame( range, 'V' )
466 indeig = lsame( range, 'I' )
467*
468 lquery = ( ( lwork.EQ.-1 ) .OR. ( lrwork.EQ.-1 ) .OR.
469 \$ ( liwork.EQ.-1 ) )
470*
471 kd = ilaenv2stage( 1, 'CHETRD_2STAGE', jobz, n, -1, -1, -1 )
472 ib = ilaenv2stage( 2, 'CHETRD_2STAGE', jobz, n, kd, -1, -1 )
473 lhtrd = ilaenv2stage( 3, 'CHETRD_2STAGE', jobz, n, kd, ib, -1 )
474 lwtrd = ilaenv2stage( 4, 'CHETRD_2STAGE', jobz, n, kd, ib, -1 )
475 lwmin = n + lhtrd + lwtrd
476 lrwmin = max( 1, 24*n )
477 liwmin = max( 1, 10*n )
478*
479 info = 0
480 IF( .NOT.( lsame( jobz, 'N' ) ) ) THEN
481 info = -1
482 ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
483 info = -2
484 ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
485 info = -3
486 ELSE IF( n.LT.0 ) THEN
487 info = -4
488 ELSE IF( lda.LT.max( 1, n ) ) THEN
489 info = -6
490 ELSE
491 IF( valeig ) THEN
492 IF( n.GT.0 .AND. vu.LE.vl )
493 \$ info = -8
494 ELSE IF( indeig ) THEN
495 IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
496 info = -9
497 ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
498 info = -10
499 END IF
500 END IF
501 END IF
502 IF( info.EQ.0 ) THEN
503 IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
504 info = -15
505 END IF
506 END IF
507*
508 IF( info.EQ.0 ) THEN
509 work( 1 ) = lwmin
510 rwork( 1 ) = lrwmin
511 iwork( 1 ) = liwmin
512*
513 IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
514 info = -18
515 ELSE IF( lrwork.LT.lrwmin .AND. .NOT.lquery ) THEN
516 info = -20
517 ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
518 info = -22
519 END IF
520 END IF
521*
522 IF( info.NE.0 ) THEN
523 CALL xerbla( 'CHEEVR_2STAGE', -info )
524 RETURN
525 ELSE IF( lquery ) THEN
526 RETURN
527 END IF
528*
529* Quick return if possible
530*
531 m = 0
532 IF( n.EQ.0 ) THEN
533 work( 1 ) = 1
534 RETURN
535 END IF
536*
537 IF( n.EQ.1 ) THEN
538 work( 1 ) = 2
539 IF( alleig .OR. indeig ) THEN
540 m = 1
541 w( 1 ) = real( a( 1, 1 ) )
542 ELSE
543 IF( vl.LT.real( a( 1, 1 ) ) .AND. vu.GE.real( a( 1, 1 ) ) )
544 \$ THEN
545 m = 1
546 w( 1 ) = real( a( 1, 1 ) )
547 END IF
548 END IF
549 IF( wantz ) THEN
550 z( 1, 1 ) = one
551 isuppz( 1 ) = 1
552 isuppz( 2 ) = 1
553 END IF
554 RETURN
555 END IF
556*
557* Get machine constants.
558*
559 safmin = slamch( 'Safe minimum' )
560 eps = slamch( 'Precision' )
561 smlnum = safmin / eps
562 bignum = one / smlnum
563 rmin = sqrt( smlnum )
564 rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
565*
566* Scale matrix to allowable range, if necessary.
567*
568 iscale = 0
569 abstll = abstol
570 IF (valeig) THEN
571 vll = vl
572 vuu = vu
573 END IF
574 anrm = clansy( 'M', uplo, n, a, lda, rwork )
575 IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
576 iscale = 1
577 sigma = rmin / anrm
578 ELSE IF( anrm.GT.rmax ) THEN
579 iscale = 1
580 sigma = rmax / anrm
581 END IF
582 IF( iscale.EQ.1 ) THEN
583 IF( lower ) THEN
584 DO 10 j = 1, n
585 CALL csscal( n-j+1, sigma, a( j, j ), 1 )
586 10 CONTINUE
587 ELSE
588 DO 20 j = 1, n
589 CALL csscal( j, sigma, a( 1, j ), 1 )
590 20 CONTINUE
591 END IF
592 IF( abstol.GT.0 )
593 \$ abstll = abstol*sigma
594 IF( valeig ) THEN
595 vll = vl*sigma
596 vuu = vu*sigma
597 END IF
598 END IF
599
600* Initialize indices into workspaces. Note: The IWORK indices are
601* used only if SSTERF or CSTEMR fail.
602
603* WORK(INDTAU:INDTAU+N-1) stores the complex scalar factors of the
604* elementary reflectors used in CHETRD.
605 indtau = 1
606* INDWK is the starting offset of the remaining complex workspace,
607* and LLWORK is the remaining complex workspace size.
608 indhous = indtau + n
609 indwk = indhous + lhtrd
610 llwork = lwork - indwk + 1
611
612* RWORK(INDRD:INDRD+N-1) stores the real tridiagonal's diagonal
613* entries.
614 indrd = 1
615* RWORK(INDRE:INDRE+N-1) stores the off-diagonal entries of the
616* tridiagonal matrix from CHETRD.
617 indre = indrd + n
618* RWORK(INDRDD:INDRDD+N-1) is a copy of the diagonal entries over
619* -written by CSTEMR (the SSTERF path copies the diagonal to W).
620 indrdd = indre + n
621* RWORK(INDREE:INDREE+N-1) is a copy of the off-diagonal entries over
622* -written while computing the eigenvalues in SSTERF and CSTEMR.
623 indree = indrdd + n
624* INDRWK is the starting offset of the left-over real workspace, and
625* LLRWORK is the remaining workspace size.
626 indrwk = indree + n
627 llrwork = lrwork - indrwk + 1
628
629* IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in SSTEBZ and
630* stores the block indices of each of the M<=N eigenvalues.
631 indibl = 1
632* IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in SSTEBZ and
633* stores the starting and finishing indices of each block.
634 indisp = indibl + n
635* IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
636* that corresponding to eigenvectors that fail to converge in
637* CSTEIN. This information is discarded; if any fail, the driver
638* returns INFO > 0.
639 indifl = indisp + n
640* INDIWO is the offset of the remaining integer workspace.
641 indiwo = indifl + n
642
643*
644* Call CHETRD_2STAGE to reduce Hermitian matrix to tridiagonal form.
645*
646 CALL chetrd_2stage( jobz, uplo, n, a, lda, rwork( indrd ),
647 \$ rwork( indre ), work( indtau ),
648 \$ work( indhous ), lhtrd,
649 \$ work( indwk ), llwork, iinfo )
650*
651* If all eigenvalues are desired
652* then call SSTERF or CSTEMR and CUNMTR.
653*
654 test = .false.
655 IF( indeig ) THEN
656 IF( il.EQ.1 .AND. iu.EQ.n ) THEN
657 test = .true.
658 END IF
659 END IF
660 IF( ( alleig.OR.test ) .AND. ( ieeeok.EQ.1 ) ) THEN
661 IF( .NOT.wantz ) THEN
662 CALL scopy( n, rwork( indrd ), 1, w, 1 )
663 CALL scopy( n-1, rwork( indre ), 1, rwork( indree ), 1 )
664 CALL ssterf( n, w, rwork( indree ), info )
665 ELSE
666 CALL scopy( n-1, rwork( indre ), 1, rwork( indree ), 1 )
667 CALL scopy( n, rwork( indrd ), 1, rwork( indrdd ), 1 )
668*
669 IF (abstol .LE. two*n*eps) THEN
670 tryrac = .true.
671 ELSE
672 tryrac = .false.
673 END IF
674 CALL cstemr( jobz, 'A', n, rwork( indrdd ),
675 \$ rwork( indree ), vl, vu, il, iu, m, w,
676 \$ z, ldz, n, isuppz, tryrac,
677 \$ rwork( indrwk ), llrwork,
678 \$ iwork, liwork, info )
679*
680* Apply unitary matrix used in reduction to tridiagonal
681* form to eigenvectors returned by CSTEMR.
682*
683 IF( wantz .AND. info.EQ.0 ) THEN
684 indwkn = indwk
685 llwrkn = lwork - indwkn + 1
686 CALL cunmtr( 'L', uplo, 'N', n, m, a, lda,
687 \$ work( indtau ), z, ldz, work( indwkn ),
688 \$ llwrkn, iinfo )
689 END IF
690 END IF
691*
692*
693 IF( info.EQ.0 ) THEN
694 m = n
695 GO TO 30
696 END IF
697 info = 0
698 END IF
699*
700* Otherwise, call SSTEBZ and, if eigenvectors are desired, CSTEIN.
701* Also call SSTEBZ and CSTEIN if CSTEMR fails.
702*
703 IF( wantz ) THEN
704 order = 'B'
705 ELSE
706 order = 'E'
707 END IF
708
709 CALL sstebz( range, order, n, vll, vuu, il, iu, abstll,
710 \$ rwork( indrd ), rwork( indre ), m, nsplit, w,
711 \$ iwork( indibl ), iwork( indisp ), rwork( indrwk ),
712 \$ iwork( indiwo ), info )
713*
714 IF( wantz ) THEN
715 CALL cstein( n, rwork( indrd ), rwork( indre ), m, w,
716 \$ iwork( indibl ), iwork( indisp ), z, ldz,
717 \$ rwork( indrwk ), iwork( indiwo ), iwork( indifl ),
718 \$ info )
719*
720* Apply unitary matrix used in reduction to tridiagonal
721* form to eigenvectors returned by CSTEIN.
722*
723 indwkn = indwk
724 llwrkn = lwork - indwkn + 1
725 CALL cunmtr( 'L', uplo, 'N', n, m, a, lda, work( indtau ), z,
726 \$ ldz, work( indwkn ), llwrkn, iinfo )
727 END IF
728*
729* If matrix was scaled, then rescale eigenvalues appropriately.
730*
731 30 CONTINUE
732 IF( iscale.EQ.1 ) THEN
733 IF( info.EQ.0 ) THEN
734 imax = m
735 ELSE
736 imax = info - 1
737 END IF
738 CALL sscal( imax, one / sigma, w, 1 )
739 END IF
740*
741* If eigenvalues are not in order, then sort them, along with
742* eigenvectors.
743*
744 IF( wantz ) THEN
745 DO 50 j = 1, m - 1
746 i = 0
747 tmp1 = w( j )
748 DO 40 jj = j + 1, m
749 IF( w( jj ).LT.tmp1 ) THEN
750 i = jj
751 tmp1 = w( jj )
752 END IF
753 40 CONTINUE
754*
755 IF( i.NE.0 ) THEN
756 itmp1 = iwork( indibl+i-1 )
757 w( i ) = w( j )
758 iwork( indibl+i-1 ) = iwork( indibl+j-1 )
759 w( j ) = tmp1
760 iwork( indibl+j-1 ) = itmp1
761 CALL cswap( n, z( 1, i ), 1, z( 1, j ), 1 )
762 END IF
763 50 CONTINUE
764 END IF
765*
766* Set WORK(1) to optimal workspace size.
767*
768 work( 1 ) = lwmin
769 rwork( 1 ) = lrwmin
770 iwork( 1 ) = liwmin
771*
772 RETURN
773*
774* End of CHEEVR_2STAGE
775*
integer function ilaenv2stage(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV2STAGE
Definition: ilaenv2stage.f:149
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_2stage(VECT, UPLO, N, A, LDA, D, E, TAU, HOUS2, LHOUS2, WORK, LWORK, INFO)
CHETRD_2STAGE
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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