 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ dsyevr_2stage()

 subroutine dsyevr_2stage ( character JOBZ, character RANGE, character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO )

DSYEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

Purpose:
``` DSYEVR_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.

DSYEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call
to DSYTRD.  Then, whenever possible, DSYEVR_2STAGE calls DSTEMR to compute
the eigenspectrum using Relatively Robust Representations.  DSTEMR
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 DSTEMR'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 : DSYEVR_2STAGE calls DSTEMR when the full spectrum is requested
on machines which conform to the ieee-754 floating point standard.
DSYEVR_2STAGE calls DSTEBZ and SSTEIN on non-ieee machines and
when partial spectrum requests are made.

Normal execution of DSTEMR 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, DSTEBZ and DSTEIN 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DLAMCH( '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 DOUBLE PRECISION array, dimension (N) The first M elements contain the selected eigenvalues in ascending order.``` [out] Z ``` Z is DOUBLE PRECISION 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 DSTEMR (tridiagonal matrix). The support of the eigenvectors of A is typically 1:N because of the orthogonal transformations applied by DORMTR. Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1``` [out] WORK ``` WORK is DOUBLE PRECISION 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```
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 378 of file dsyevr_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  DOUBLE PRECISION ABSTOL, VL, VU
392 * ..
393 * .. Array Arguments ..
394  INTEGER ISUPPZ( * ), IWORK( * )
395  DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
396 * ..
397 *
398 * =====================================================================
399 *
400 * .. Parameters ..
401  DOUBLE PRECISION ZERO, ONE, TWO
402  parameter( zero = 0.0d+0, one = 1.0d+0, two = 2.0d+0 )
403 * ..
404 * .. Local Scalars ..
405  LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, VALEIG, WANTZ,
406  \$ TRYRAC
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  DOUBLE PRECISION 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  DOUBLE PRECISION DLAMCH, DLANSY
420  EXTERNAL lsame, dlamch, dlansy, ilaenv, ilaenv2stage
421 * ..
422 * .. External Subroutines ..
423  EXTERNAL dcopy, dormtr, dscal, dstebz, dstemr, dstein,
425 * ..
426 * .. Intrinsic Functions ..
427  INTRINSIC max, min, sqrt
428 * ..
429 * .. Executable Statements ..
430 *
431 * Test the input parameters.
432 *
433  ieeeok = ilaenv( 10, 'DSYEVR', 'N', 1, 2, 3, 4 )
434 *
435  lower = lsame( uplo, 'L' )
436  wantz = lsame( jobz, 'V' )
437  alleig = lsame( range, 'A' )
438  valeig = lsame( range, 'V' )
439  indeig = lsame( range, 'I' )
440 *
441  lquery = ( ( lwork.EQ.-1 ) .OR. ( liwork.EQ.-1 ) )
442 *
443  kd = ilaenv2stage( 1, 'DSYTRD_2STAGE', jobz, n, -1, -1, -1 )
444  ib = ilaenv2stage( 2, 'DSYTRD_2STAGE', jobz, n, kd, -1, -1 )
445  lhtrd = ilaenv2stage( 3, 'DSYTRD_2STAGE', jobz, n, kd, ib, -1 )
446  lwtrd = ilaenv2stage( 4, 'DSYTRD_2STAGE', jobz, n, kd, ib, -1 )
447  lwmin = max( 26*n, 5*n + lhtrd + lwtrd )
448  liwmin = max( 1, 10*n )
449 *
450  info = 0
451  IF( .NOT.( lsame( jobz, 'N' ) ) ) THEN
452  info = -1
453  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
454  info = -2
455  ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
456  info = -3
457  ELSE IF( n.LT.0 ) THEN
458  info = -4
459  ELSE IF( lda.LT.max( 1, n ) ) THEN
460  info = -6
461  ELSE
462  IF( valeig ) THEN
463  IF( n.GT.0 .AND. vu.LE.vl )
464  \$ info = -8
465  ELSE IF( indeig ) THEN
466  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
467  info = -9
468  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
469  info = -10
470  END IF
471  END IF
472  END IF
473  IF( info.EQ.0 ) THEN
474  IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
475  info = -15
476  ELSE IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
477  info = -18
478  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
479  info = -20
480  END IF
481  END IF
482 *
483  IF( info.EQ.0 ) THEN
484 * NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
485 * NB = MAX( NB, ILAENV( 1, 'DORMTR', UPLO, N, -1, -1, -1 ) )
486 * LWKOPT = MAX( ( NB+1 )*N, LWMIN )
487  work( 1 ) = lwmin
488  iwork( 1 ) = liwmin
489  END IF
490 *
491  IF( info.NE.0 ) THEN
492  CALL xerbla( 'DSYEVR_2STAGE', -info )
493  RETURN
494  ELSE IF( lquery ) THEN
495  RETURN
496  END IF
497 *
498 * Quick return if possible
499 *
500  m = 0
501  IF( n.EQ.0 ) THEN
502  work( 1 ) = 1
503  RETURN
504  END IF
505 *
506  IF( n.EQ.1 ) THEN
507  work( 1 ) = 7
508  IF( alleig .OR. indeig ) THEN
509  m = 1
510  w( 1 ) = a( 1, 1 )
511  ELSE
512  IF( vl.LT.a( 1, 1 ) .AND. vu.GE.a( 1, 1 ) ) THEN
513  m = 1
514  w( 1 ) = a( 1, 1 )
515  END IF
516  END IF
517  IF( wantz ) THEN
518  z( 1, 1 ) = one
519  isuppz( 1 ) = 1
520  isuppz( 2 ) = 1
521  END IF
522  RETURN
523  END IF
524 *
525 * Get machine constants.
526 *
527  safmin = dlamch( 'Safe minimum' )
528  eps = dlamch( 'Precision' )
529  smlnum = safmin / eps
530  bignum = one / smlnum
531  rmin = sqrt( smlnum )
532  rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
533 *
534 * Scale matrix to allowable range, if necessary.
535 *
536  iscale = 0
537  abstll = abstol
538  IF (valeig) THEN
539  vll = vl
540  vuu = vu
541  END IF
542  anrm = dlansy( 'M', uplo, n, a, lda, work )
543  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
544  iscale = 1
545  sigma = rmin / anrm
546  ELSE IF( anrm.GT.rmax ) THEN
547  iscale = 1
548  sigma = rmax / anrm
549  END IF
550  IF( iscale.EQ.1 ) THEN
551  IF( lower ) THEN
552  DO 10 j = 1, n
553  CALL dscal( n-j+1, sigma, a( j, j ), 1 )
554  10 CONTINUE
555  ELSE
556  DO 20 j = 1, n
557  CALL dscal( j, sigma, a( 1, j ), 1 )
558  20 CONTINUE
559  END IF
560  IF( abstol.GT.0 )
561  \$ abstll = abstol*sigma
562  IF( valeig ) THEN
563  vll = vl*sigma
564  vuu = vu*sigma
565  END IF
566  END IF
567
568 * Initialize indices into workspaces. Note: The IWORK indices are
569 * used only if DSTERF or DSTEMR fail.
570
571 * WORK(INDTAU:INDTAU+N-1) stores the scalar factors of the
572 * elementary reflectors used in DSYTRD.
573  indtau = 1
574 * WORK(INDD:INDD+N-1) stores the tridiagonal's diagonal entries.
575  indd = indtau + n
576 * WORK(INDE:INDE+N-1) stores the off-diagonal entries of the
577 * tridiagonal matrix from DSYTRD.
578  inde = indd + n
579 * WORK(INDDD:INDDD+N-1) is a copy of the diagonal entries over
580 * -written by DSTEMR (the DSTERF path copies the diagonal to W).
581  inddd = inde + n
582 * WORK(INDEE:INDEE+N-1) is a copy of the off-diagonal entries over
583 * -written while computing the eigenvalues in DSTERF and DSTEMR.
584  indee = inddd + n
585 * INDHOUS is the starting offset Householder storage of stage 2
586  indhous = indee + n
587 * INDWK is the starting offset of the left-over workspace, and
588 * LLWORK is the remaining workspace size.
589  indwk = indhous + lhtrd
590  llwork = lwork - indwk + 1
591
592
593 * IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and
594 * stores the block indices of each of the M<=N eigenvalues.
595  indibl = 1
596 * IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and
597 * stores the starting and finishing indices of each block.
598  indisp = indibl + n
599 * IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
600 * that corresponding to eigenvectors that fail to converge in
601 * DSTEIN. This information is discarded; if any fail, the driver
602 * returns INFO > 0.
603  indifl = indisp + n
604 * INDIWO is the offset of the remaining integer workspace.
605  indiwo = indifl + n
606
607 *
608 * Call DSYTRD_2STAGE to reduce symmetric matrix to tridiagonal form.
609 *
610 *
611  CALL dsytrd_2stage( jobz, uplo, n, a, lda, work( indd ),
612  \$ work( inde ), work( indtau ), work( indhous ),
613  \$ lhtrd, work( indwk ), llwork, iinfo )
614 *
615 * If all eigenvalues are desired
616 * then call DSTERF or DSTEMR and DORMTR.
617 *
618  IF( ( alleig .OR. ( indeig .AND. il.EQ.1 .AND. iu.EQ.n ) ) .AND.
619  \$ ieeeok.EQ.1 ) THEN
620  IF( .NOT.wantz ) THEN
621  CALL dcopy( n, work( indd ), 1, w, 1 )
622  CALL dcopy( n-1, work( inde ), 1, work( indee ), 1 )
623  CALL dsterf( n, w, work( indee ), info )
624  ELSE
625  CALL dcopy( n-1, work( inde ), 1, work( indee ), 1 )
626  CALL dcopy( n, work( indd ), 1, work( inddd ), 1 )
627 *
628  IF (abstol .LE. two*n*eps) THEN
629  tryrac = .true.
630  ELSE
631  tryrac = .false.
632  END IF
633  CALL dstemr( jobz, 'A', n, work( inddd ), work( indee ),
634  \$ vl, vu, il, iu, m, w, z, ldz, n, isuppz,
635  \$ tryrac, work( indwk ), lwork, iwork, liwork,
636  \$ info )
637 *
638 *
639 *
640 * Apply orthogonal matrix used in reduction to tridiagonal
641 * form to eigenvectors returned by DSTEMR.
642 *
643  IF( wantz .AND. info.EQ.0 ) THEN
644  indwkn = inde
645  llwrkn = lwork - indwkn + 1
646  CALL dormtr( 'L', uplo, 'N', n, m, a, lda,
647  \$ work( indtau ), z, ldz, work( indwkn ),
648  \$ llwrkn, iinfo )
649  END IF
650  END IF
651 *
652 *
653  IF( info.EQ.0 ) THEN
654 * Everything worked. Skip DSTEBZ/DSTEIN. IWORK(:) are
655 * undefined.
656  m = n
657  GO TO 30
658  END IF
659  info = 0
660  END IF
661 *
662 * Otherwise, call DSTEBZ and, if eigenvectors are desired, DSTEIN.
663 * Also call DSTEBZ and DSTEIN if DSTEMR fails.
664 *
665  IF( wantz ) THEN
666  order = 'B'
667  ELSE
668  order = 'E'
669  END IF
670
671  CALL dstebz( range, order, n, vll, vuu, il, iu, abstll,
672  \$ work( indd ), work( inde ), m, nsplit, w,
673  \$ iwork( indibl ), iwork( indisp ), work( indwk ),
674  \$ iwork( indiwo ), info )
675 *
676  IF( wantz ) THEN
677  CALL dstein( n, work( indd ), work( inde ), m, w,
678  \$ iwork( indibl ), iwork( indisp ), z, ldz,
679  \$ work( indwk ), iwork( indiwo ), iwork( indifl ),
680  \$ info )
681 *
682 * Apply orthogonal matrix used in reduction to tridiagonal
683 * form to eigenvectors returned by DSTEIN.
684 *
685  indwkn = inde
686  llwrkn = lwork - indwkn + 1
687  CALL dormtr( 'L', uplo, 'N', n, m, a, lda, work( indtau ), z,
688  \$ ldz, work( indwkn ), llwrkn, iinfo )
689  END IF
690 *
691 * If matrix was scaled, then rescale eigenvalues appropriately.
692 *
693 * Jump here if DSTEMR/DSTEIN succeeded.
694  30 CONTINUE
695  IF( iscale.EQ.1 ) THEN
696  IF( info.EQ.0 ) THEN
697  imax = m
698  ELSE
699  imax = info - 1
700  END IF
701  CALL dscal( imax, one / sigma, w, 1 )
702  END IF
703 *
704 * If eigenvalues are not in order, then sort them, along with
705 * eigenvectors. Note: We do not sort the IFAIL portion of IWORK.
706 * It may not be initialized (if DSTEMR/DSTEIN succeeded), and we do
707 * not return this detailed information to the user.
708 *
709  IF( wantz ) THEN
710  DO 50 j = 1, m - 1
711  i = 0
712  tmp1 = w( j )
713  DO 40 jj = j + 1, m
714  IF( w( jj ).LT.tmp1 ) THEN
715  i = jj
716  tmp1 = w( jj )
717  END IF
718  40 CONTINUE
719 *
720  IF( i.NE.0 ) THEN
721  w( i ) = w( j )
722  w( j ) = tmp1
723  CALL dswap( n, z( 1, i ), 1, z( 1, j ), 1 )
724  END IF
725  50 CONTINUE
726  END IF
727 *
728 * Set WORK(1) to optimal workspace size.
729 *
730  work( 1 ) = lwmin
731  iwork( 1 ) = liwmin
732 *
733  RETURN
734 *
735 * End of DSYEVR_2STAGE
736 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
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 dstebz(RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK, INFO)
DSTEBZ
Definition: dstebz.f:273
subroutine dsterf(N, D, E, INFO)
DSTERF
Definition: dsterf.f:86
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
subroutine dswap(N, DX, INCX, DY, INCY)
DSWAP
Definition: dswap.f:82
subroutine dstemr(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, IWORK, LIWORK, INFO)
DSTEMR
Definition: dstemr.f:321
subroutine dstein(N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
DSTEIN
Definition: dstein.f:174
subroutine dormtr(SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMTR
Definition: dormtr.f:171
double precision function dlansy(NORM, UPLO, N, A, LDA, WORK)
DLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: dlansy.f:122
subroutine dsytrd_2stage(VECT, UPLO, N, A, LDA, D, E, TAU, HOUS2, LHOUS2, WORK, LWORK, INFO)
DSYTRD_2STAGE
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