LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ ssyevr_2stage()

subroutine ssyevr_2stage ( character  jobz,
character  range,
character  uplo,
integer  n,
real, dimension( lda, * )  a,
integer  lda,
real  vl,
real  vu,
integer  il,
integer  iu,
real  abstol,
integer  m,
real, dimension( * )  w,
real, dimension( ldz, * )  z,
integer  ldz,
integer, dimension( * )  isuppz,
real, dimension( * )  work,
integer  lwork,
integer, dimension( * )  iwork,
integer  liwork,
integer  info 
)

SSYEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

Download SSYEVR_2STAGE + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 SSYEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors
 of a real symmetric 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.

 SSYEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call
 to SSYTRD.  Then, whenever possible, SSYEVR_2STAGE calls SSTEMR to compute
 the eigenspectrum using Relatively Robust Representations.  SSTEMR
 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 SSTEMR'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 : SSYEVR_2STAGE calls SSTEMR when the full spectrum is requested
 on machines which conform to the ieee-754 floating point standard.
 SSYEVR_2STAGE calls SSTEBZ and SSTEIN on non-ieee machines and
 when partial spectrum requests are made.

 Normal execution of SSTEMR 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
          SSTEIN 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 REAL 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 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 REAL 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.
          Supplying N columns is always safe.
[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 SSTEMR (tridiagonal
          matrix). The support of the eigenvectors of A is typically
          1:N because of the orthogonal transformations applied by SORMTR.
          Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
[out]WORK
          WORK is REAL 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 + 5*N
                                             = N*KD + N*max(KD+1,FACTOPTNB)
                                               + max(2*KD*KD, KD*NTHREADS)
                                               + (KD+1)*N + 5*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 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 (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LWORK.
[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 size of the IWORK array,
          returns this value as the first entry of the IWORK array, and
          no error message related to 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
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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).
  Denver, Colorado, USA, 2013.
  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 378 of file ssyevr_2stage.f.

381*
382 IMPLICIT NONE
383*
384* -- LAPACK driver routine --
385* -- LAPACK is a software package provided by Univ. of Tennessee, --
386* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
387*
388* .. Scalar Arguments ..
389 CHARACTER JOBZ, RANGE, UPLO
390 INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N
391 REAL ABSTOL, VL, VU
392* ..
393* .. Array Arguments ..
394 INTEGER ISUPPZ( * ), IWORK( * )
395 REAL A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
396* ..
397*
398* =====================================================================
399*
400* .. Parameters ..
401 REAL ZERO, ONE, TWO
402 parameter( zero = 0.0e+0, one = 1.0e+0, two = 2.0e+0 )
403* ..
404* .. Local Scalars ..
405 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, VALEIG, WANTZ,
406 $ TRYRAC, TEST
407 CHARACTER ORDER
408 INTEGER I, IEEEOK, IINFO, IMAX, INDD, INDDD, INDE,
409 $ INDEE, INDIBL, INDIFL, INDISP, INDIWO, INDTAU,
410 $ INDWK, INDWKN, ISCALE, J, JJ, LIWMIN,
411 $ LLWORK, LLWRKN, LWMIN, NSPLIT,
412 $ LHTRD, LWTRD, KD, IB, INDHOUS
413 REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
414 $ SIGMA, SMLNUM, TMP1, VLL, VUU
415* ..
416* .. External Functions ..
417 LOGICAL LSAME
418 INTEGER ILAENV, ILAENV2STAGE
419 REAL SLAMCH, SLANSY, SROUNDUP_LWORK
422* ..
423* .. External Subroutines ..
424 EXTERNAL scopy, sormtr, sscal, sstebz, sstemr, sstein,
426* ..
427* .. Intrinsic Functions ..
428 INTRINSIC max, min, sqrt
429* ..
430* .. Executable Statements ..
431*
432* Test the input parameters.
433*
434 ieeeok = ilaenv( 10, 'SSYEVR', 'N', 1, 2, 3, 4 )
435*
436 lower = lsame( uplo, 'L' )
437 wantz = lsame( jobz, 'V' )
438 alleig = lsame( range, 'A' )
439 valeig = lsame( range, 'V' )
440 indeig = lsame( range, 'I' )
441*
442 lquery = ( ( lwork.EQ.-1 ) .OR. ( liwork.EQ.-1 ) )
443*
444 kd = ilaenv2stage( 1, 'SSYTRD_2STAGE', jobz, n, -1, -1, -1 )
445 ib = ilaenv2stage( 2, 'SSYTRD_2STAGE', jobz, n, kd, -1, -1 )
446 lhtrd = ilaenv2stage( 3, 'SSYTRD_2STAGE', jobz, n, kd, ib, -1 )
447 lwtrd = ilaenv2stage( 4, 'SSYTRD_2STAGE', jobz, n, kd, ib, -1 )
448 lwmin = max( 26*n, 5*n + lhtrd + lwtrd )
449 liwmin = max( 1, 10*n )
450*
451 info = 0
452 IF( .NOT.( lsame( jobz, 'N' ) ) ) THEN
453 info = -1
454 ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
455 info = -2
456 ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
457 info = -3
458 ELSE IF( n.LT.0 ) THEN
459 info = -4
460 ELSE IF( lda.LT.max( 1, n ) ) THEN
461 info = -6
462 ELSE
463 IF( valeig ) THEN
464 IF( n.GT.0 .AND. vu.LE.vl )
465 $ info = -8
466 ELSE IF( indeig ) THEN
467 IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
468 info = -9
469 ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
470 info = -10
471 END IF
472 END IF
473 END IF
474 IF( info.EQ.0 ) THEN
475 IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
476 info = -15
477 ELSE IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
478 info = -18
479 ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
480 info = -20
481 END IF
482 END IF
483*
484 IF( info.EQ.0 ) THEN
485* NB = ILAENV( 1, 'SSYTRD', UPLO, N, -1, -1, -1 )
486* NB = MAX( NB, ILAENV( 1, 'SORMTR', UPLO, N, -1, -1, -1 ) )
487* LWKOPT = MAX( ( NB+1 )*N, LWMIN )
488 work( 1 ) = sroundup_lwork(lwmin)
489 iwork( 1 ) = liwmin
490 END IF
491*
492 IF( info.NE.0 ) THEN
493 CALL xerbla( 'SSYEVR_2STAGE', -info )
494 RETURN
495 ELSE IF( lquery ) THEN
496 RETURN
497 END IF
498*
499* Quick return if possible
500*
501 m = 0
502 IF( n.EQ.0 ) THEN
503 work( 1 ) = 1
504 RETURN
505 END IF
506*
507 IF( n.EQ.1 ) THEN
508 work( 1 ) = 26
509 IF( alleig .OR. indeig ) THEN
510 m = 1
511 w( 1 ) = a( 1, 1 )
512 ELSE
513 IF( vl.LT.a( 1, 1 ) .AND. vu.GE.a( 1, 1 ) ) THEN
514 m = 1
515 w( 1 ) = a( 1, 1 )
516 END IF
517 END IF
518 IF( wantz ) THEN
519 z( 1, 1 ) = one
520 isuppz( 1 ) = 1
521 isuppz( 2 ) = 1
522 END IF
523 RETURN
524 END IF
525*
526* Get machine constants.
527*
528 safmin = slamch( 'Safe minimum' )
529 eps = slamch( 'Precision' )
530 smlnum = safmin / eps
531 bignum = one / smlnum
532 rmin = sqrt( smlnum )
533 rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
534*
535* Scale matrix to allowable range, if necessary.
536*
537 iscale = 0
538 abstll = abstol
539 IF (valeig) THEN
540 vll = vl
541 vuu = vu
542 END IF
543 anrm = slansy( 'M', uplo, n, a, lda, work )
544 IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
545 iscale = 1
546 sigma = rmin / anrm
547 ELSE IF( anrm.GT.rmax ) THEN
548 iscale = 1
549 sigma = rmax / anrm
550 END IF
551 IF( iscale.EQ.1 ) THEN
552 IF( lower ) THEN
553 DO 10 j = 1, n
554 CALL sscal( n-j+1, sigma, a( j, j ), 1 )
555 10 CONTINUE
556 ELSE
557 DO 20 j = 1, n
558 CALL sscal( j, sigma, a( 1, j ), 1 )
559 20 CONTINUE
560 END IF
561 IF( abstol.GT.0 )
562 $ abstll = abstol*sigma
563 IF( valeig ) THEN
564 vll = vl*sigma
565 vuu = vu*sigma
566 END IF
567 END IF
568
569* Initialize indices into workspaces. Note: The IWORK indices are
570* used only if SSTERF or SSTEMR fail.
571
572* WORK(INDTAU:INDTAU+N-1) stores the scalar factors of the
573* elementary reflectors used in SSYTRD.
574 indtau = 1
575* WORK(INDD:INDD+N-1) stores the tridiagonal's diagonal entries.
576 indd = indtau + n
577* WORK(INDE:INDE+N-1) stores the off-diagonal entries of the
578* tridiagonal matrix from SSYTRD.
579 inde = indd + n
580* WORK(INDDD:INDDD+N-1) is a copy of the diagonal entries over
581* -written by SSTEMR (the SSTERF path copies the diagonal to W).
582 inddd = inde + n
583* WORK(INDEE:INDEE+N-1) is a copy of the off-diagonal entries over
584* -written while computing the eigenvalues in SSTERF and SSTEMR.
585 indee = inddd + n
586* INDHOUS is the starting offset Householder storage of stage 2
587 indhous = indee + n
588* INDWK is the starting offset of the left-over workspace, and
589* LLWORK is the remaining workspace size.
590 indwk = indhous + lhtrd
591 llwork = lwork - indwk + 1
592
593
594* IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in SSTEBZ and
595* stores the block indices of each of the M<=N eigenvalues.
596 indibl = 1
597* IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in SSTEBZ and
598* stores the starting and finishing indices of each block.
599 indisp = indibl + n
600* IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
601* that corresponding to eigenvectors that fail to converge in
602* SSTEIN. This information is discarded; if any fail, the driver
603* returns INFO > 0.
604 indifl = indisp + n
605* INDIWO is the offset of the remaining integer workspace.
606 indiwo = indifl + n
607
608*
609* Call SSYTRD_2STAGE to reduce symmetric matrix to tridiagonal form.
610*
611*
612 CALL ssytrd_2stage( jobz, uplo, n, a, lda, work( indd ),
613 $ work( inde ), work( indtau ), work( indhous ),
614 $ lhtrd, work( indwk ), llwork, iinfo )
615*
616* If all eigenvalues are desired
617* then call SSTERF or SSTEMR and SORMTR.
618*
619 test = .false.
620 IF( indeig ) THEN
621 IF( il.EQ.1 .AND. iu.EQ.n ) THEN
622 test = .true.
623 END IF
624 END IF
625 IF( ( alleig.OR.test ) .AND. ( ieeeok.EQ.1 ) ) THEN
626 IF( .NOT.wantz ) THEN
627 CALL scopy( n, work( indd ), 1, w, 1 )
628 CALL scopy( n-1, work( inde ), 1, work( indee ), 1 )
629 CALL ssterf( n, w, work( indee ), info )
630 ELSE
631 CALL scopy( n-1, work( inde ), 1, work( indee ), 1 )
632 CALL scopy( n, work( indd ), 1, work( inddd ), 1 )
633*
634 IF (abstol .LE. two*n*eps) THEN
635 tryrac = .true.
636 ELSE
637 tryrac = .false.
638 END IF
639 CALL sstemr( jobz, 'A', n, work( inddd ), work( indee ),
640 $ vl, vu, il, iu, m, w, z, ldz, n, isuppz,
641 $ tryrac, work( indwk ), lwork, iwork, liwork,
642 $ info )
643*
644*
645*
646* Apply orthogonal matrix used in reduction to tridiagonal
647* form to eigenvectors returned by SSTEMR.
648*
649 IF( wantz .AND. info.EQ.0 ) THEN
650 indwkn = inde
651 llwrkn = lwork - indwkn + 1
652 CALL sormtr( 'L', uplo, 'N', n, m, a, lda,
653 $ work( indtau ), z, ldz, work( indwkn ),
654 $ llwrkn, iinfo )
655 END IF
656 END IF
657*
658*
659 IF( info.EQ.0 ) THEN
660* Everything worked. Skip SSTEBZ/SSTEIN. IWORK(:) are
661* undefined.
662 m = n
663 GO TO 30
664 END IF
665 info = 0
666 END IF
667*
668* Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN.
669* Also call SSTEBZ and SSTEIN if SSTEMR fails.
670*
671 IF( wantz ) THEN
672 order = 'B'
673 ELSE
674 order = 'E'
675 END IF
676
677 CALL sstebz( range, order, n, vll, vuu, il, iu, abstll,
678 $ work( indd ), work( inde ), m, nsplit, w,
679 $ iwork( indibl ), iwork( indisp ), work( indwk ),
680 $ iwork( indiwo ), info )
681*
682 IF( wantz ) THEN
683 CALL sstein( n, work( indd ), work( inde ), m, w,
684 $ iwork( indibl ), iwork( indisp ), z, ldz,
685 $ work( indwk ), iwork( indiwo ), iwork( indifl ),
686 $ info )
687*
688* Apply orthogonal matrix used in reduction to tridiagonal
689* form to eigenvectors returned by SSTEIN.
690*
691 indwkn = inde
692 llwrkn = lwork - indwkn + 1
693 CALL sormtr( 'L', uplo, 'N', n, m, a, lda, work( indtau ), z,
694 $ ldz, work( indwkn ), llwrkn, iinfo )
695 END IF
696*
697* If matrix was scaled, then rescale eigenvalues appropriately.
698*
699* Jump here if SSTEMR/SSTEIN succeeded.
700 30 CONTINUE
701 IF( iscale.EQ.1 ) THEN
702 IF( info.EQ.0 ) THEN
703 imax = m
704 ELSE
705 imax = info - 1
706 END IF
707 CALL sscal( imax, one / sigma, w, 1 )
708 END IF
709*
710* If eigenvalues are not in order, then sort them, along with
711* eigenvectors. Note: We do not sort the IFAIL portion of IWORK.
712* It may not be initialized (if SSTEMR/SSTEIN succeeded), and we do
713* not return this detailed information to the user.
714*
715 IF( wantz ) THEN
716 DO 50 j = 1, m - 1
717 i = 0
718 tmp1 = w( j )
719 DO 40 jj = j + 1, m
720 IF( w( jj ).LT.tmp1 ) THEN
721 i = jj
722 tmp1 = w( jj )
723 END IF
724 40 CONTINUE
725*
726 IF( i.NE.0 ) THEN
727 w( i ) = w( j )
728 w( j ) = tmp1
729 CALL sswap( n, z( 1, i ), 1, z( 1, j ), 1 )
730 END IF
731 50 CONTINUE
732 END IF
733*
734* Set WORK(1) to optimal workspace size.
735*
736 work( 1 ) = sroundup_lwork(lwmin)
737 iwork( 1 ) = liwmin
738*
739 RETURN
740*
741* End of SSYEVR_2STAGE
742*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine scopy(n, sx, incx, sy, incy)
SCOPY
Definition scopy.f:82
subroutine ssytrd_2stage(vect, uplo, n, a, lda, d, e, tau, hous2, lhous2, work, lwork, info)
SSYTRD_2STAGE
integer function ilaenv2stage(ispec, name, opts, n1, n2, n3, n4)
ILAENV2STAGE
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:162
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
real function slansy(norm, uplo, n, a, lda, work)
SLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition slansy.f:122
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
real function sroundup_lwork(lwork)
SROUNDUP_LWORK
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 sstein(n, d, e, m, w, iblock, isplit, z, ldz, work, iwork, ifail, info)
SSTEIN
Definition sstein.f:174
subroutine sstemr(jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz, nzc, isuppz, tryrac, work, lwork, iwork, liwork, info)
SSTEMR
Definition sstemr.f:322
subroutine ssterf(n, d, e, info)
SSTERF
Definition ssterf.f:86
subroutine sswap(n, sx, incx, sy, incy)
SSWAP
Definition sswap.f:82
subroutine sormtr(side, uplo, trans, m, n, a, lda, tau, c, ldc, work, lwork, info)
SORMTR
Definition sormtr.f:172
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