heevd_2stage(3)

NAME
heevd_2stage - {he,sy}evd_2stage: eig, divide and conquer

SYNOPSIS

  Functions
subroutine cheevd_2stage (jobz, uplo, n, a, lda, w, work, lwork, rwork, lrwork, iwork, liwork, info)
CHEEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices
subroutine dsyevd_2stage (jobz, uplo, n, a, lda, w, work, lwork, iwork, liwork, info)
DSYEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
subroutine ssyevd_2stage (jobz, uplo, n, a, lda, w, work, lwork, iwork, liwork, info)
SSYEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
subroutine zheevd_2stage (jobz, uplo, n, a, lda, w, work, lwork, rwork, lrwork, iwork, liwork, info)
ZHEEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Detailed Description

Function Documentation

  subroutine cheevd_2stage (character jobz, character uplo, integer n, complex, dimension( lda, * ) a, integer lda, real, dimension( * ) w, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer lrwork, integer, dimension( * ) iwork, integer liwork, integer info)

CHEEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Purpose:

    !>
    !> CHEEVD_2STAGE computes all eigenvalues and, optionally, eigenvectors of a
    !> complex Hermitian matrix A using the 2stage technique for
    !> the reduction to tridiagonal.  If eigenvectors are desired, it uses a
    !> divide and conquer algorithm.
    !>
    !>

Parameters
    JOBZ

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

    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 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, if JOBZ = 'V', then if INFO = 0, A contains the
    !>          orthonormal eigenvectors of the matrix A.
    !>          If JOBZ = 'N', then 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).
    !>

    W

    !>          W is REAL array, dimension (N)
    !>          If INFO = 0, the eigenvalues in ascending order.
    !>

    WORK

    !>          WORK is COMPLEX 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 N <= 1,               LWORK must be at least 1.
    !>          If JOBZ = 'N' and N > 1, LWORK must be queried.
    !>                                   LWORK = MAX(1, dimension) where
    !>                                   dimension = max(stage1,stage2) + (KD+1)*N + N+1
    !>                                             = N*KD + N*max(KD+1,FACTOPTNB)
    !>                                               + max(2*KD*KD, KD*NTHREADS)
    !>                                               + (KD+1)*N + N+1
    !>                                   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 at least 2*N + N**2
    !>
    !>          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.
    !>

    RWORK

    !>          RWORK is REAL array,
    !>                                         dimension (LRWORK)
    !>          On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
    !>

    LRWORK

    !>          LRWORK is INTEGER
    !>          The dimension of the array RWORK.
    !>          If N <= 1,                LRWORK must be at least 1.
    !>          If JOBZ  = 'N' and N > 1, LRWORK must be at least N.
    !>          If JOBZ  = 'V' and N > 1, LRWORK must be at least
    !>                         1 + 5*N + 2*N**2.
    !>
    !>          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.
    !>

    IWORK

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

    LIWORK

    !>          LIWORK is INTEGER
    !>          The dimension of the array IWORK.
    !>          If N <= 1,                LIWORK must be at least 1.
    !>          If JOBZ  = 'N' and N > 1, LIWORK must be at least 1.
    !>          If JOBZ  = 'V' and N > 1, LIWORK must be at least 3 + 5*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.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  if INFO = i and JOBZ = 'N', then the algorithm failed
    !>                to converge; i off-diagonal elements of an intermediate
    !>                tridiagonal form did not converge to zero;
    !>                if INFO = i and JOBZ = 'V', then the algorithm failed
    !>                to compute an eigenvalue while working on the submatrix
    !>                lying in rows and columns INFO/(N+1) through
    !>                mod(INFO,N+1).
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:
    Modified description of INFO. Sven, 16 Feb 05.

Contributors:
    Jeff Rutter, 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
    !>
    !>

Definition at line 245 of file cheevd_2stage.f.

  subroutine dsyevd_2stage (character jobz, character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) w, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

DSYEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

Purpose:

    !>
    !> DSYEVD_2STAGE computes all eigenvalues and, optionally, eigenvectors of a
    !> real symmetric matrix A using the 2stage technique for
    !> the reduction to tridiagonal. If eigenvectors are desired, it uses a
    !> divide and conquer algorithm.
    !>
    !>

Parameters
    JOBZ

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

    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 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, if JOBZ = 'V', then if INFO = 0, A contains the
    !>          orthonormal eigenvectors of the matrix A.
    !>          If JOBZ = 'N', then 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).
    !>

    W

    !>          W is DOUBLE PRECISION array, dimension (N)
    !>          If INFO = 0, the eigenvalues in ascending order.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array,
    !>                                         dimension (LWORK)
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    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, dimension) where
    !>                                   dimension = max(stage1,stage2) + (KD+1)*N + 2*N+1
    !>                                             = N*KD + N*max(KD+1,FACTOPTNB)
    !>                                               + max(2*KD*KD, KD*NTHREADS)
    !>                                               + (KD+1)*N + 2*N+1
    !>                                   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 at least
    !>                                                1 + 6*N + 2*N**2.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal sizes of the WORK and IWORK
    !>          arrays, returns these values as the first entries of the WORK
    !>          and IWORK arrays, and no error message related to LWORK or
    !>          LIWORK is issued by XERBLA.
    !>

    IWORK

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

    LIWORK

    !>          LIWORK is INTEGER
    !>          The dimension of the array IWORK.
    !>          If N <= 1,                LIWORK must be at least 1.
    !>          If JOBZ  = 'N' and N > 1, LIWORK must be at least 1.
    !>          If JOBZ  = 'V' and N > 1, LIWORK must be at least 3 + 5*N.
    !>
    !>          If LIWORK = -1, then a workspace query is assumed; the
    !>          routine only calculates the optimal sizes of the WORK and
    !>          IWORK arrays, returns these values as the first entries of
    !>          the WORK and IWORK arrays, and no error message related to
    !>          LWORK or 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:  if INFO = i and JOBZ = 'N', then the algorithm failed
    !>                to converge; i off-diagonal elements of an intermediate
    !>                tridiagonal form did not converge to zero;
    !>                if INFO = i and JOBZ = 'V', then the algorithm failed
    !>                to compute an eigenvalue while working on the submatrix
    !>                lying in rows and columns INFO/(N+1) through
    !>                mod(INFO,N+1).
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
     Modified by Francoise Tisseur, University of Tennessee
     Modified description of INFO. Sven, 16 Feb 05.

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 219 of file dsyevd_2stage.f.

  subroutine ssyevd_2stage (character jobz, character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) w, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

SSYEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

Purpose:

    !>
    !> SSYEVD_2STAGE computes all eigenvalues and, optionally, eigenvectors of a
    !> real symmetric matrix A using the 2stage technique for
    !> the reduction to tridiagonal. If eigenvectors are desired, it uses a
    !> divide and conquer algorithm.
    !>
    !>

Parameters
    JOBZ

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

    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, if JOBZ = 'V', then if INFO = 0, A contains the
    !>          orthonormal eigenvectors of the matrix A.
    !>          If JOBZ = 'N', then 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).
    !>

    W

    !>          W is REAL array, dimension (N)
    !>          If INFO = 0, the eigenvalues in ascending order.
    !>

    WORK

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

    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, dimension) where
    !>                                   dimension = max(stage1,stage2) + (KD+1)*N + 2*N+1
    !>                                             = N*KD + N*max(KD+1,FACTOPTNB)
    !>                                               + max(2*KD*KD, KD*NTHREADS)
    !>                                               + (KD+1)*N + 2*N+1
    !>                                   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 at least
    !>                                                1 + 6*N + 2*N**2.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal sizes of the WORK and IWORK
    !>          arrays, returns these values as the first entries of the WORK
    !>          and IWORK arrays, and no error message related to LWORK or
    !>          LIWORK is issued by XERBLA.
    !>

    IWORK

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

    LIWORK

    !>          LIWORK is INTEGER
    !>          The dimension of the array IWORK.
    !>          If N <= 1,                LIWORK must be at least 1.
    !>          If JOBZ  = 'N' and N > 1, LIWORK must be at least 1.
    !>          If JOBZ  = 'V' and N > 1, LIWORK must be at least 3 + 5*N.
    !>
    !>          If LIWORK = -1, then a workspace query is assumed; the
    !>          routine only calculates the optimal sizes of the WORK and
    !>          IWORK arrays, returns these values as the first entries of
    !>          the WORK and IWORK arrays, and no error message related to
    !>          LWORK or 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:  if INFO = i and JOBZ = 'N', then the algorithm failed
    !>                to converge; i off-diagonal elements of an intermediate
    !>                tridiagonal form did not converge to zero;
    !>                if INFO = i and JOBZ = 'V', then the algorithm failed
    !>                to compute an eigenvalue while working on the submatrix
    !>                lying in rows and columns INFO/(N+1) through
    !>                mod(INFO,N+1).
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
     Modified by Francoise Tisseur, University of Tennessee
     Modified description of INFO. Sven, 16 Feb 05.

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 219 of file ssyevd_2stage.f.

  subroutine zheevd_2stage (character jobz, character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) w, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer lrwork, integer, dimension( * ) iwork, integer liwork, integer info)

ZHEEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Purpose:

    !>
    !> ZHEEVD_2STAGE computes all eigenvalues and, optionally, eigenvectors of a
    !> complex Hermitian matrix A using the 2stage technique for
    !> the reduction to tridiagonal.  If eigenvectors are desired, it uses a
    !> divide and conquer algorithm.
    !>
    !>

Parameters
    JOBZ

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

    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 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, if JOBZ = 'V', then if INFO = 0, A contains the
    !>          orthonormal eigenvectors of the matrix A.
    !>          If JOBZ = 'N', then 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).
    !>

    W

    !>          W is DOUBLE PRECISION array, dimension (N)
    !>          If INFO = 0, the eigenvalues in ascending order.
    !>

    WORK

    !>          WORK is COMPLEX*16 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 N <= 1,               LWORK must be at least 1.
    !>          If JOBZ = 'N' and N > 1, LWORK must be queried.
    !>                                   LWORK = MAX(1, dimension) where
    !>                                   dimension = max(stage1,stage2) + (KD+1)*N + N+1
    !>                                             = N*KD + N*max(KD+1,FACTOPTNB)
    !>                                               + max(2*KD*KD, KD*NTHREADS)
    !>                                               + (KD+1)*N + N+1
    !>                                   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 at least 2*N + N**2
    !>
    !>          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.
    !>

    RWORK

    !>          RWORK is DOUBLE PRECISION array,
    !>                                         dimension (LRWORK)
    !>          On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
    !>

    LRWORK

    !>          LRWORK is INTEGER
    !>          The dimension of the array RWORK.
    !>          If N <= 1,                LRWORK must be at least 1.
    !>          If JOBZ  = 'N' and N > 1, LRWORK must be at least N.
    !>          If JOBZ  = 'V' and N > 1, LRWORK must be at least
    !>                         1 + 5*N + 2*N**2.
    !>
    !>          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.
    !>

    IWORK

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

    LIWORK

    !>          LIWORK is INTEGER
    !>          The dimension of the array IWORK.
    !>          If N <= 1,                LIWORK must be at least 1.
    !>          If JOBZ  = 'N' and N > 1, LIWORK must be at least 1.
    !>          If JOBZ  = 'V' and N > 1, LIWORK must be at least 3 + 5*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.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  if INFO = i and JOBZ = 'N', then the algorithm failed
    !>                to converge; i off-diagonal elements of an intermediate
    !>                tridiagonal form did not converge to zero;
    !>                if INFO = i and JOBZ = 'V', then the algorithm failed
    !>                to compute an eigenvalue while working on the submatrix
    !>                lying in rows and columns INFO/(N+1) through
    !>                mod(INFO,N+1).
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:
    Modified description of INFO. Sven, 16 Feb 05.

Contributors:
    Jeff Rutter, 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
    !>
    !>

Definition at line 245 of file zheevd_2stage.f.

Author

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