zheevr_2stage.f

Go to the documentation of this file.

*> \brief <b> ZHEEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices</b>
*
*  @precisions fortran z -> s d c
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> Download ZHEEVR_2STAGE + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zheevr_2stage.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zheevr_2stage.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zheevr_2stage.f">
*> [TXT]</a>
*
*  Definition:
*  ===========
*
*       SUBROUTINE ZHEEVR_2STAGE( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU,
*                                 IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ,
*                                 WORK, LWORK, RWORK, LRWORK, IWORK,
*                                 LIWORK, INFO )
*
*       IMPLICIT NONE
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, RANGE, UPLO
*       INTEGER            IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK,
*      $                   M, N
*       DOUBLE PRECISION   ABSTOL, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            ISUPPZ( * ), IWORK( * )
*       DOUBLE PRECISION   RWORK( * ), W( * )
*       COMPLEX*16         A( LDA, * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZHEEVR_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.
*>
*> ZHEEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call
*> to ZHETRD.  Then, whenever possible, ZHEEVR_2STAGE calls ZSTEMR to compute
*> eigenspectrum using Relatively Robust Representations.  ZSTEMR
*> 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 ZSTEMR'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 : ZHEEVR_2STAGE calls ZSTEMR when the full spectrum is requested
*> on machines which conform to the ieee-754 floating point standard.
*> ZHEEVR_2STAGE calls DSTEBZ and ZSTEIN on non-ieee machines and
*> when partial spectrum requests are made.
*>
*> Normal execution of ZSTEMR 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.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*>                  Not available in this release.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*>          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
*>          ZSTEIN are called
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is COMPLEX*16 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.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          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'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*>          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'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*>          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'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*>          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'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          The first M elements contain the selected eigenvalues in
*>          ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is COMPLEX*16 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.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] ISUPPZ
*> \verbatim
*>          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 ZSTEMR (tridiagonal
*>          matrix). The support of the eigenvectors of A is typically
*>          1:N because of the unitary transformations applied by ZUNMTR.
*>          Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If N <= 1,               LWORK must be at least 1.
*>          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.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*>          RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
*>          On exit, if INFO = 0, RWORK(1) returns the optimal
*>          (and minimal) LRWORK.
*> \endverbatim
*>
*> \param[in] LRWORK
*> \verbatim
*>          LRWORK is INTEGER
*>          The length of the array RWORK.
*>          If N <= 1, LRWORK >= 1, else LRWORK >= 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.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*>          On exit, if INFO = 0, IWORK(1) returns the optimal
*>          (and minimal) LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK.
*>          If N <= 1, LIWORK >= 1, else LIWORK >= 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.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  Internal error
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup heevr_2stage
*
*> \par 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
*>
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  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
*>
*> \endverbatim
*
*  =====================================================================
      SUBROUTINE zheevr_2stage( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU,
     $                          IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ,
     $                          WORK, LWORK, RWORK, LRWORK, IWORK,
     $                          LIWORK, INFO )
*
      IMPLICIT NONE
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, RANGE, UPLO
      INTEGER            IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK,
     $                   M, N
      DOUBLE PRECISION   ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            ISUPPZ( * ), IWORK( * )
      DOUBLE PRECISION   RWORK( * ), W( * )
      COMPLEX*16         A( LDA, * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO
      PARAMETER          ( ZERO = 0.0d+0, one = 1.0d+0, two = 2.0d+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
     $                   WANTZ, TRYRAC
      CHARACTER          ORDER
      INTEGER            I, IEEEOK, IINFO, IMAX, INDIBL, INDIFL, INDISP,
     $                   INDIWO, INDRD, INDRDD, INDRE, INDREE, INDRWK,
     $                   indtau, indwk, indwkn, iscale, itmp1, j, jj,
     $                   liwmin, llwork, llrwork, llwrkn, lrwmin,
     $                   lwmin, nsplit, lhtrd, lwtrd, kd, ib, indhous
      DOUBLE PRECISION   ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
     $                   SIGMA, SMLNUM, TMP1, VLL, VUU
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV, ILAENV2STAGE
      DOUBLE PRECISION   DLAMCH, ZLANSY
      EXTERNAL           lsame, dlamch, zlansy, ilaenv,
     $                   ilaenv2stage
*     ..
*     .. External Subroutines ..
      EXTERNAL           dcopy, dscal, dstebz, dsterf, xerbla,
     $                   zdscal,
     $                   zhetrd_2stage, zstemr, zstein, zswap, zunmtr
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          dble, max, min, sqrt
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      ieeeok = ilaenv( 10, 'ZHEEVR', 'N', 1, 2, 3, 4 )
*
      lower = lsame( uplo, 'L' )
      wantz = lsame( jobz, 'V' )
      alleig = lsame( range, 'A' )
      valeig = lsame( range, 'V' )
      indeig = lsame( range, 'I' )
*
      lquery = ( ( lwork.EQ.-1 ) .OR. ( lrwork.EQ.-1 ) .OR.
     $         ( liwork.EQ.-1 ) )
*
      kd     = ilaenv2stage( 1, 'ZHETRD_2STAGE', jobz, n, -1, -1,
     $                       -1 )
      ib     = ilaenv2stage( 2, 'ZHETRD_2STAGE', jobz, n, kd, -1,
     $                       -1 )
      lhtrd  = ilaenv2stage( 3, 'ZHETRD_2STAGE', jobz, n, kd, ib,
     $                       -1 )
      lwtrd  = ilaenv2stage( 4, 'ZHETRD_2STAGE', jobz, n, kd, ib,
     $                       -1 )
*
      IF( n.LE.1 ) THEN
         lwmin  = 1
         lrwmin = 1
         liwmin = 1
      ELSE
         lwmin  = n + lhtrd + lwtrd
         lrwmin = 24*n
         liwmin = 10*n
      END IF
*
      info = 0
      IF( .NOT.( lsame( jobz, 'N' ) ) ) THEN
         info = -1
      ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
         info = -2
      ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
         info = -3
      ELSE IF( n.LT.0 ) THEN
         info = -4
      ELSE IF( lda.LT.max( 1, n ) ) THEN
         info = -6
      ELSE
         IF( valeig ) THEN
            IF( n.GT.0 .AND. vu.LE.vl )
     $         info = -8
         ELSE IF( indeig ) THEN
            IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
               info = -9
            ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
               info = -10
            END IF
         END IF
      END IF
      IF( info.EQ.0 ) THEN
         IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
            info = -15
         END IF
      END IF
*
      IF( info.EQ.0 ) THEN
         work( 1 )  = lwmin
         rwork( 1 ) = real( lrwmin )
         iwork( 1 ) = liwmin
*
         IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
            info = -18
         ELSE IF( lrwork.LT.lrwmin .AND. .NOT.lquery ) THEN
            info = -20
         ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
            info = -22
         END IF
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZHEEVR_2STAGE', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      m = 0
      IF( n.EQ.0 ) THEN
         work( 1 ) = 1
         RETURN
      END IF
*
      IF( n.EQ.1 ) THEN
         work( 1 ) = 1
         IF( alleig .OR. indeig ) THEN
            m = 1
            w( 1 ) = dble( a( 1, 1 ) )
         ELSE
            IF( vl.LT.dble( a( 1, 1 ) ) .AND. vu.GE.dble( a( 1, 1 ) ) )
     $           THEN
               m = 1
               w( 1 ) = dble( a( 1, 1 ) )
            END IF
         END IF
         IF( wantz ) THEN
            z( 1, 1 ) = one
            isuppz( 1 ) = 1
            isuppz( 2 ) = 1
         END IF
         RETURN
      END IF
*
*     Get machine constants.
*
      safmin = dlamch( 'Safe minimum' )
      eps    = dlamch( 'Precision' )
      smlnum = safmin / eps
      bignum = one / smlnum
      rmin   = sqrt( smlnum )
      rmax   = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
*
*     Scale matrix to allowable range, if necessary.
*
      iscale = 0
      abstll = abstol
      IF (valeig) THEN
         vll = vl
         vuu = vu
      END IF
      anrm = zlansy( 'M', uplo, n, a, lda, rwork )
      IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
         iscale = 1
         sigma = rmin / anrm
      ELSE IF( anrm.GT.rmax ) THEN
         iscale = 1
         sigma = rmax / anrm
      END IF
      IF( iscale.EQ.1 ) THEN
         IF( lower ) THEN
            DO 10 j = 1, n
               CALL zdscal( n-j+1, sigma, a( j, j ), 1 )
   10       CONTINUE
         ELSE
            DO 20 j = 1, n
               CALL zdscal( j, sigma, a( 1, j ), 1 )
   20       CONTINUE
         END IF
         IF( abstol.GT.0 )
     $      abstll = abstol*sigma
         IF( valeig ) THEN
            vll = vl*sigma
            vuu = vu*sigma
         END IF
      END IF
 
*     Initialize indices into workspaces.  Note: The IWORK indices are
*     used only if DSTERF or ZSTEMR fail.
 
*     WORK(INDTAU:INDTAU+N-1) stores the complex scalar factors of the
*     elementary reflectors used in ZHETRD.
      indtau = 1
*     INDWK is the starting offset of the remaining complex workspace,
*     and LLWORK is the remaining complex workspace size.
      indhous = indtau + n
      indwk   = indhous + lhtrd
      llwork  = lwork - indwk + 1
 
*     RWORK(INDRD:INDRD+N-1) stores the real tridiagonal's diagonal
*     entries.
      indrd = 1
*     RWORK(INDRE:INDRE+N-1) stores the off-diagonal entries of the
*     tridiagonal matrix from ZHETRD.
      indre = indrd + n
*     RWORK(INDRDD:INDRDD+N-1) is a copy of the diagonal entries over
*     -written by ZSTEMR (the DSTERF path copies the diagonal to W).
      indrdd = indre + n
*     RWORK(INDREE:INDREE+N-1) is a copy of the off-diagonal entries over
*     -written while computing the eigenvalues in DSTERF and ZSTEMR.
      indree = indrdd + n
*     INDRWK is the starting offset of the left-over real workspace, and
*     LLRWORK is the remaining workspace size.
      indrwk = indree + n
      llrwork = lrwork - indrwk + 1
 
*     IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and
*     stores the block indices of each of the M<=N eigenvalues.
      indibl = 1
*     IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and
*     stores the starting and finishing indices of each block.
      indisp = indibl + n
*     IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
*     that corresponding to eigenvectors that fail to converge in
*     ZSTEIN.  This information is discarded; if any fail, the driver
*     returns INFO > 0.
      indifl = indisp + n
*     INDIWO is the offset of the remaining integer workspace.
      indiwo = indifl + n
 
*
*     Call ZHETRD_2STAGE to reduce Hermitian matrix to tridiagonal form.
*
      CALL zhetrd_2stage( jobz, uplo, n, a, lda, rwork( indrd ),
     $                    rwork( indre ), work( indtau ),
     $                    work( indhous ), lhtrd,
     $                    work( indwk ), llwork, iinfo )
*
*     If all eigenvalues are desired
*     then call DSTERF or ZSTEMR and ZUNMTR.
*
      test = .false.
      IF( indeig ) THEN
         IF( il.EQ.1 .AND. iu.EQ.n ) THEN
            test = .true.
         END IF
      END IF
      IF( ( alleig.OR.test ) .AND. ( ieeeok.EQ.1 ) ) THEN
         IF( .NOT.wantz ) THEN
            CALL dcopy( n, rwork( indrd ), 1, w, 1 )
            CALL dcopy( n-1, rwork( indre ), 1, rwork( indree ), 1 )
            CALL dsterf( n, w, rwork( indree ), info )
         ELSE
            CALL dcopy( n-1, rwork( indre ), 1, rwork( indree ), 1 )
            CALL dcopy( n, rwork( indrd ), 1, rwork( indrdd ), 1 )
*
            IF (abstol .LE. two*n*eps) THEN
               tryrac = .true.
            ELSE
               tryrac = .false.
            END IF
            CALL zstemr( jobz, 'A', n, rwork( indrdd ),
     $                   rwork( indree ), vl, vu, il, iu, m, w,
     $                   z, ldz, n, isuppz, tryrac,
     $                   rwork( indrwk ), llrwork,
     $                   iwork, liwork, info )
*
*           Apply unitary matrix used in reduction to tridiagonal
*           form to eigenvectors returned by ZSTEMR.
*
            IF( wantz .AND. info.EQ.0 ) THEN
               indwkn = indwk
               llwrkn = lwork - indwkn + 1
               CALL zunmtr( 'L', uplo, 'N', n, m, a, lda,
     $                      work( indtau ), z, ldz, work( indwkn ),
     $                      llwrkn, iinfo )
            END IF
         END IF
*
*
         IF( info.EQ.0 ) THEN
            m = n
            GO TO 30
         END IF
         info = 0
      END IF
*
*     Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
*     Also call DSTEBZ and ZSTEIN if ZSTEMR fails.
*
      IF( wantz ) THEN
         order = 'B'
      ELSE
         order = 'E'
      END IF
 
      CALL dstebz( range, order, n, vll, vuu, il, iu, abstll,
     $             rwork( indrd ), rwork( indre ), m, nsplit, w,
     $             iwork( indibl ), iwork( indisp ), rwork( indrwk ),
     $             iwork( indiwo ), info )
*
      IF( wantz ) THEN
         CALL zstein( n, rwork( indrd ), rwork( indre ), m, w,
     $                iwork( indibl ), iwork( indisp ), z, ldz,
     $                rwork( indrwk ), iwork( indiwo ), iwork( indifl ),
     $                info )
*
*        Apply unitary matrix used in reduction to tridiagonal
*        form to eigenvectors returned by ZSTEIN.
*
         indwkn = indwk
         llwrkn = lwork - indwkn + 1
         CALL zunmtr( 'L', uplo, 'N', n, m, a, lda, work( indtau ),
     $                z,
     $                ldz, work( indwkn ), llwrkn, iinfo )
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
   30 CONTINUE
      IF( iscale.EQ.1 ) THEN
         IF( info.EQ.0 ) THEN
            imax = m
         ELSE
            imax = info - 1
         END IF
         CALL dscal( imax, one / sigma, w, 1 )
      END IF
*
*     If eigenvalues are not in order, then sort them, along with
*     eigenvectors.
*
      IF( wantz ) THEN
         DO 50 j = 1, m - 1
            i = 0
            tmp1 = w( j )
            DO 40 jj = j + 1, m
               IF( w( jj ).LT.tmp1 ) THEN
                  i = jj
                  tmp1 = w( jj )
               END IF
   40       CONTINUE
*
            IF( i.NE.0 ) THEN
               itmp1 = iwork( indibl+i-1 )
               w( i ) = w( j )
               iwork( indibl+i-1 ) = iwork( indibl+j-1 )
               w( j ) = tmp1
               iwork( indibl+j-1 ) = itmp1
               CALL zswap( n, z( 1, i ), 1, z( 1, j ), 1 )
            END IF
   50    CONTINUE
      END IF
*
*     Set WORK(1) to optimal workspace size.
*
      work( 1 )  = lwmin
      rwork( 1 ) = real( lrwmin )
      iwork( 1 ) = liwmin
*
      RETURN
*
*     End of ZHEEVR_2STAGE
*
      END