SRC/ssyevr_2stage.f(3)

NAME
SRC/ssyevr_2stage.f

SYNOPSIS

  Functions/Subroutines
subroutine ssyevr_2stage (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info)
SSYEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

Function/Subroutine Documentation

  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

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
    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.
                      Not available in this release.

    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

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    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.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

    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'.

    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'.

    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'.

    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'.

    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.

    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.

    W

              W is REAL array, dimension (N)
              The first M elements contain the selected eigenvalues in
              ascending order.

    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.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(1,N).

    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

    WORK

              WORK is REAL array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    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.

    IWORK

              IWORK is INTEGER array, dimension (MAX(1,LIWORK))
              On exit, if INFO = 0, IWORK(1) returns the optimal LWORK.

    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.

    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.

Author

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