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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

Download CHEEVR_2STAGE + dependencies [TGZ] [ZIP] [TXT]

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  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:
      !>   Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
      !> - Inderjit Dhillon and Beresford Parlett:  SIAM Journal on Matrix Analysis and Applications, Vol. 25,
      !>   2004.  Also LAPACK Working Note 154.
      !> - Inderjit Dhillon: ,
      !>   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  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,
      !>          , 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 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.
      !>

  [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.
      !>          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.
      !>

  [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.
      !>          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.
      !>

  [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
  Osni Marques, LBNL/NERSC, USA
  Ken Stanley, Computer Science Division, University of California at Berkeley, USA
  Jason Riedy, Computer Science Division, University of California at Berkeley, USA

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
      !>
      !>