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## chbevx_2stage()

subroutine chbevx_2stage | ( | character | jobz,
|  | character | range,
|  | character | uplo,
|  | integer | n,
|  | integer | kd,
|  | complex, dimension( ldab, * ) | ab,
|  | integer | ldab,
|  | complex, dimension( ldq, * ) | q,
|  | integer | ldq,
|  | real | vl,
|  | real | vu,
|  | integer | il,
|  | integer | iu,
|  | real | abstol,
|  | integer | m,
|  | real, dimension( * ) | w,
|  | complex, dimension( ldz, * ) | z,
|  | integer | ldz,
|  | complex, dimension( * ) | work,
|  | integer | lwork,
|  | real, dimension( * ) | rwork,
|  | integer, dimension( * ) | iwork,
|  | integer, dimension( * ) | ifail,
|  | integer | info )

CHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Download CHBEVX_2STAGE + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !> CHBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors
      !> of a complex Hermitian band 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.
      !>

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.
      !>

  [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]

  KD

      !>          KD is INTEGER
      !>          The number of superdiagonals of the matrix A if UPLO = 'U',
      !>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
      !>

  [in,out]

  AB

      !>          AB is COMPLEX array, dimension (LDAB, N)
      !>          On entry, the upper or lower triangle of the Hermitian band
      !>          matrix A, stored in the first KD+1 rows of the array.  The
      !>          j-th column of A is stored in the j-th column of the array AB
      !>          as follows:
      !>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
      !>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
      !>
      !>          On exit, AB is overwritten by values generated during the
      !>          reduction to tridiagonal form.
      !>

  [in]

  LDAB

      !>          LDAB is INTEGER
      !>          The leading dimension of the array AB.  LDAB >= KD + 1.
      !>

  [out]

  Q

      !>          Q is COMPLEX array, dimension (LDQ, N)
      !>          If JOBZ = 'V', the N-by-N unitary matrix used in the
      !>                          reduction to tridiagonal form.
      !>          If JOBZ = 'N', the array Q is not referenced.
      !>

  [in]

  LDQ

      !>          LDQ is INTEGER
      !>          The leading dimension of the array Q.  If JOBZ = 'V', then
      !>          LDQ >= 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 AB to tridiagonal form.
      !>
      !>          Eigenvalues will be computed most accurately when ABSTOL is
      !>          set to twice the underflow threshold 2*SLAMCH('S'), not zero.
      !>          If this routine returns with INFO>0, indicating that some
      !>          eigenvectors did not converge, try setting ABSTOL to
      !>          2*SLAMCH('S').
      !>
      !>          See  by Demmel and
      !>          Kahan, LAPACK Working Note #3.
      !>

  [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 an eigenvector fails to converge, then that column of Z
      !>          contains the latest approximation to the eigenvector, and the
      !>          index of the eigenvector is returned in IFAIL.
      !>          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]

  WORK

      !>          WORK is COMPLEX array, dimension (LWORK)
      !>

  [in]

  LWORK

      !>          LWORK is INTEGER
      !>          The length of the array WORK. LWORK >= 1, when N <= 1;
      !>          otherwise
      !>          If JOBZ = 'N' and N > 1, LWORK must be queried.
      !>                                   LWORK = MAX(1, dimension) where
      !>                                   dimension = (2KD+1)*N + KD*NTHREADS
      !>                                   where KD is the size of the band.
      !>                                   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 (7*N)
      !>

  [out]

  IWORK

      !>          IWORK is INTEGER array, dimension (5*N)
      !>

  [out]

  IFAIL

      !>          IFAIL is INTEGER array, dimension (N)
      !>          If JOBZ = 'V', then if INFO = 0, the first M elements of
      !>          IFAIL are zero.  If INFO > 0, then IFAIL contains the
      !>          indices of the eigenvectors that failed to converge.
      !>          If JOBZ = 'N', then IFAIL is not referenced.
      !>

  [out]

  INFO

      !>          INFO is INTEGER
      !>          = 0:  successful exit
      !>          < 0:  if INFO = -i, the i-th argument had an illegal value
      !>          > 0:  if INFO = i, then i eigenvectors failed to converge.
      !>                Their indices are stored in array IFAIL.
      !>

Author
  Univ. of Tennessee
  Univ. of California Berkeley
  Univ. of Colorado Denver
  NAG Ltd.

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
      !>
      !>