hetrd_2stage(3)

NAME
hetrd_2stage - {he,sy}trd_2stage: reduction to tridiagonal, 2-stage

SYNOPSIS

  Functions

subroutine chetrd_2stage (vect, uplo, n, a, lda, d, e, tau, hous2, lhous2, work, lwork, info)
CHETRD_2STAGE
subroutine dsytrd_2stage (vect, uplo, n, a, lda, d, e, tau, hous2, lhous2, work, lwork, info)
DSYTRD_2STAGE
subroutine ssytrd_2stage (vect, uplo, n, a, lda, d, e, tau, hous2, lhous2, work, lwork, info)
SSYTRD_2STAGE
subroutine zhetrd_2stage (vect, uplo, n, a, lda, d, e, tau, hous2, lhous2, work, lwork, info)
ZHETRD_2STAGE

Detailed Description

Function Documentation

  subroutine chetrd_2stage (character vect, character uplo, integer n, complex, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( * ) tau, complex, dimension( * ) hous2, integer lhous2, complex, dimension( * ) work, integer lwork, integer info)

CHETRD_2STAGE

Purpose:

    !>
    !> CHETRD_2STAGE reduces a complex Hermitian matrix A to real symmetric
    !> tridiagonal form T by a unitary similarity transformation:
    !> Q1**H Q2**H* A * Q2 * Q1 = T.
    !>

Parameters
    VECT

    !>          VECT is CHARACTER*1
    !>          = 'N':  No need for the Housholder representation,
    !>                  in particular for the second stage (Band to
    !>                  tridiagonal) and thus LHOUS2 is of size max(1, 4*N);
    !>          = 'V':  the Householder representation is needed to
    !>                  either generate Q1 Q2 or to apply Q1 Q2,
    !>                  then LHOUS2 is to be queried and computed.
    !>                  (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, and the strictly lower
    !>          triangular part of A is not referenced.  If UPLO = 'L', the
    !>          leading N-by-N lower triangular part of A contains the lower
    !>          triangular part of the matrix A, and the strictly upper
    !>          triangular part of A is not referenced.
    !>          On exit, if UPLO = 'U', the band superdiagonal
    !>          of A are overwritten by the corresponding elements of the
    !>          internal band-diagonal matrix AB, and the elements above
    !>          the KD superdiagonal, with the array TAU, represent the unitary
    !>          matrix Q1 as a product of elementary reflectors; if UPLO
    !>          = 'L', the diagonal and band subdiagonal of A are over-
    !>          written by the corresponding elements of the internal band-diagonal
    !>          matrix AB, and the elements below the KD subdiagonal, with
    !>          the array TAU, represent the unitary matrix Q1 as a product
    !>          of elementary reflectors. See Further Details.
    !>

    LDA

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

    D

    !>          D is REAL array, dimension (N)
    !>          The diagonal elements of the tridiagonal matrix T.
    !>

    E

    !>          E is REAL array, dimension (N-1)
    !>          The off-diagonal elements of the tridiagonal matrix T.
    !>

    TAU

    !>          TAU is COMPLEX array, dimension (N-KD)
    !>          The scalar factors of the elementary reflectors of
    !>          the first stage (see Further Details).
    !>

    HOUS2

    !>          HOUS2 is COMPLEX array, dimension (MAX(1,LHOUS2))
    !>          Stores the Householder representation of the stage2
    !>          band to tridiagonal.
    !>

    LHOUS2

    !>          LHOUS2 is INTEGER
    !>          The dimension of the array HOUS2.
    !>          LHOUS2 >= 1.
    !>
    !>          If LWORK = -1, or LHOUS2=-1,
    !>          then a query is assumed; the routine
    !>          only calculates the optimal size of the HOUS2 array, returns
    !>          this value as the first entry of the HOUS2 array, and no error
    !>          message related to LHOUS2 is issued by XERBLA.
    !>          If VECT='N', LHOUS2 = max(1, 4*n);
    !>          if VECT='V', option not yet available.
    !>

    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 = 0, LWORK >= 1, else LWORK = MAX(1, dimension).
    !>
    !>          If LWORK = -1, or LHOUS2 = -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.
    !>          LWORK = MAX(1, dimension) where
    !>          dimension   = max(stage1,stage2) + (KD+1)*N
    !>                      = N*KD + N*max(KD+1,FACTOPTNB)
    !>                        + max(2*KD*KD, KD*NTHREADS)
    !>                        + (KD+1)*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.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  Implemented by Azzam Haidar.
    !>
    !>  All details are available on technical report, SC11, SC13 papers.
    !>
    !>  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
    !>
    !>

  subroutine dsytrd_2stage (character vect, character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( * ) tau, double precision, dimension( * ) hous2, integer lhous2, double precision, dimension( * ) work, integer lwork, integer info)

DSYTRD_2STAGE

Purpose:

    !>
    !> DSYTRD_2STAGE reduces a real symmetric matrix A to real symmetric
    !> tridiagonal form T by a orthogonal similarity transformation:
    !> Q1**T Q2**T* A * Q2 * Q1 = T.
    !>

Parameters
    VECT

    !>          VECT is CHARACTER*1
    !>          = 'N':  No need for the Housholder representation,
    !>                  in particular for the second stage (Band to
    !>                  tridiagonal) and thus LHOUS2 is of size max(1, 4*N);
    !>          = 'V':  the Householder representation is needed to
    !>                  either generate Q1 Q2 or to apply Q1 Q2,
    !>                  then LHOUS2 is to be queried and computed.
    !>                  (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, and the strictly lower
    !>          triangular part of A is not referenced.  If UPLO = 'L', the
    !>          leading N-by-N lower triangular part of A contains the lower
    !>          triangular part of the matrix A, and the strictly upper
    !>          triangular part of A is not referenced.
    !>          On exit, if UPLO = 'U', the band superdiagonal
    !>          of A are overwritten by the corresponding elements of the
    !>          internal band-diagonal matrix AB, and the elements above
    !>          the KD superdiagonal, with the array TAU, represent the orthogonal
    !>          matrix Q1 as a product of elementary reflectors; if UPLO
    !>          = 'L', the diagonal and band subdiagonal of A are over-
    !>          written by the corresponding elements of the internal band-diagonal
    !>          matrix AB, and the elements below the KD subdiagonal, with
    !>          the array TAU, represent the orthogonal matrix Q1 as a product
    !>          of elementary reflectors. See Further Details.
    !>

    LDA

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

    D

    !>          D is DOUBLE PRECISION array, dimension (N)
    !>          The diagonal elements of the tridiagonal matrix T.
    !>

    E

    !>          E is DOUBLE PRECISION array, dimension (N-1)
    !>          The off-diagonal elements of the tridiagonal matrix T.
    !>

    TAU

    !>          TAU is DOUBLE PRECISION array, dimension (N-KD)
    !>          The scalar factors of the elementary reflectors of
    !>          the first stage (see Further Details).
    !>

    HOUS2

    !>          HOUS2 is DOUBLE PRECISION array, dimension (MAX(1,LHOUS2))
    !>          Stores the Householder representation of the stage2
    !>          band to tridiagonal.
    !>

    LHOUS2

    !>          LHOUS2 is INTEGER
    !>          The dimension of the array HOUS2.
    !>          LHOUS2 >= 1.
    !>
    !>          If LWORK = -1, or LHOUS2 = -1,
    !>          then a query is assumed; the routine
    !>          only calculates the optimal size of the HOUS2 array, returns
    !>          this value as the first entry of the HOUS2 array, and no error
    !>          message related to LHOUS2 is issued by XERBLA.
    !>          If VECT='N', LHOUS2 = max(1, 4*n);
    !>          if VECT='V', option not yet available.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION 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 = 0, LWORK >= 1, else LWORK = MAX(1, dimension).
    !>
    !>          If LWORK = -1, or LHOUS2 = -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.
    !>          LWORK = MAX(1, dimension) where
    !>          dimension   = max(stage1,stage2) + (KD+1)*N
    !>                      = N*KD + N*max(KD+1,FACTOPTNB)
    !>                        + max(2*KD*KD, KD*NTHREADS)
    !>                        + (KD+1)*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.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  Implemented by Azzam Haidar.
    !>
    !>  All details are available on technical report, SC11, SC13 papers.
    !>
    !>  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
    !>
    !>

  subroutine ssytrd_2stage (character vect, character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, real, dimension( * ) tau, real, dimension( * ) hous2, integer lhous2, real, dimension( * ) work, integer lwork, integer info)

SSYTRD_2STAGE

Purpose:

    !>
    !> SSYTRD_2STAGE reduces a real symmetric matrix A to real symmetric
    !> tridiagonal form T by a orthogonal similarity transformation:
    !> Q1**T Q2**T* A * Q2 * Q1 = T.
    !>

Parameters
    VECT

    !>          VECT is CHARACTER*1
    !>          = 'N':  No need for the Housholder representation,
    !>                  in particular for the second stage (Band to
    !>                  tridiagonal) and thus LHOUS2 is of size max(1, 4*N);
    !>          = 'V':  the Householder representation is needed to
    !>                  either generate Q1 Q2 or to apply Q1 Q2,
    !>                  then LHOUS2 is to be queried and computed.
    !>                  (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, and the strictly lower
    !>          triangular part of A is not referenced.  If UPLO = 'L', the
    !>          leading N-by-N lower triangular part of A contains the lower
    !>          triangular part of the matrix A, and the strictly upper
    !>          triangular part of A is not referenced.
    !>          On exit, if UPLO = 'U', the band superdiagonal
    !>          of A are overwritten by the corresponding elements of the
    !>          internal band-diagonal matrix AB, and the elements above
    !>          the KD superdiagonal, with the array TAU, represent the orthogonal
    !>          matrix Q1 as a product of elementary reflectors; if UPLO
    !>          = 'L', the diagonal and band subdiagonal of A are over-
    !>          written by the corresponding elements of the internal band-diagonal
    !>          matrix AB, and the elements below the KD subdiagonal, with
    !>          the array TAU, represent the orthogonal matrix Q1 as a product
    !>          of elementary reflectors. See Further Details.
    !>

    LDA

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

    D

    !>          D is REAL array, dimension (N)
    !>          The diagonal elements of the tridiagonal matrix T.
    !>

    E

    !>          E is REAL array, dimension (N-1)
    !>          The off-diagonal elements of the tridiagonal matrix T.
    !>

    TAU

    !>          TAU is REAL array, dimension (N-KD)
    !>          The scalar factors of the elementary reflectors of
    !>          the first stage (see Further Details).
    !>

    HOUS2

    !>          HOUS2 is REAL array, dimension (MAX(1,LHOUS2))
    !>          Stores the Householder representation of the stage2
    !>          band to tridiagonal.
    !>

    LHOUS2

    !>          LHOUS2 is INTEGER
    !>          The dimension of the array HOUS2.
    !>          LHOUS2 >= 1.
    !>
    !>          If LWORK = -1, or LHOUS2 = -1,
    !>          then a query is assumed; the routine
    !>          only calculates the optimal size of the HOUS2 array, returns
    !>          this value as the first entry of the HOUS2 array, and no error
    !>          message related to LHOUS2 is issued by XERBLA.
    !>          If VECT='N', LHOUS2 = max(1, 4*n);
    !>          if VECT='V', option not yet available.
    !>

    WORK

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

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.
    !>          If N = 0, LWORK >= 1, else LWORK = MAX(1, dimension).
    !>
    !>          If LWORK = -1, or LHOUS2 = -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.
    !>          LWORK = MAX(1, dimension) where
    !>          dimension   = max(stage1,stage2) + (KD+1)*N
    !>                      = N*KD + N*max(KD+1,FACTOPTNB)
    !>                        + max(2*KD*KD, KD*NTHREADS)
    !>                        + (KD+1)*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.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  Implemented by Azzam Haidar.
    !>
    !>  All details are available on technical report, SC11, SC13 papers.
    !>
    !>  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
    !>
    !>

  subroutine zhetrd_2stage (character vect, character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( * ) tau, complex*16, dimension( * ) hous2, integer lhous2, complex*16, dimension( * ) work, integer lwork, integer info)

ZHETRD_2STAGE

Purpose:

    !>
    !> ZHETRD_2STAGE reduces a complex Hermitian matrix A to real symmetric
    !> tridiagonal form T by a unitary similarity transformation:
    !> Q1**H Q2**H* A * Q2 * Q1 = T.
    !>

Parameters
    VECT

    !>          VECT is CHARACTER*1
    !>          = 'N':  No need for the Housholder representation,
    !>                  in particular for the second stage (Band to
    !>                  tridiagonal) and thus LHOUS2 is of size max(1, 4*N);
    !>          = 'V':  the Householder representation is needed to
    !>                  either generate Q1 Q2 or to apply Q1 Q2,
    !>                  then LHOUS2 is to be queried and computed.
    !>                  (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, and the strictly lower
    !>          triangular part of A is not referenced.  If UPLO = 'L', the
    !>          leading N-by-N lower triangular part of A contains the lower
    !>          triangular part of the matrix A, and the strictly upper
    !>          triangular part of A is not referenced.
    !>          On exit, if UPLO = 'U', the band superdiagonal
    !>          of A are overwritten by the corresponding elements of the
    !>          internal band-diagonal matrix AB, and the elements above
    !>          the KD superdiagonal, with the array TAU, represent the unitary
    !>          matrix Q1 as a product of elementary reflectors; if UPLO
    !>          = 'L', the diagonal and band subdiagonal of A are over-
    !>          written by the corresponding elements of the internal band-diagonal
    !>          matrix AB, and the elements below the KD subdiagonal, with
    !>          the array TAU, represent the unitary matrix Q1 as a product
    !>          of elementary reflectors. See Further Details.
    !>

    LDA

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

    D

    !>          D is DOUBLE PRECISION array, dimension (N)
    !>          The diagonal elements of the tridiagonal matrix T.
    !>

    E

    !>          E is DOUBLE PRECISION array, dimension (N-1)
    !>          The off-diagonal elements of the tridiagonal matrix T.
    !>

    TAU

    !>          TAU is COMPLEX*16 array, dimension (N-KD)
    !>          The scalar factors of the elementary reflectors of
    !>          the first stage (see Further Details).
    !>

    HOUS2

    !>          HOUS2 is COMPLEX*16 array, dimension (MAX(1,LHOUS2))
    !>          Stores the Householder representation of the stage2
    !>          band to tridiagonal.
    !>

    LHOUS2

    !>          LHOUS2 is INTEGER
    !>          The dimension of the array HOUS2.
    !>          LHOUS2 >= 1.
    !>
    !>          If LWORK = -1, or LHOUS2 = -1,
    !>          then a query is assumed; the routine
    !>          only calculates the optimal size of the HOUS2 array, returns
    !>          this value as the first entry of the HOUS2 array, and no error
    !>          message related to LHOUS2 is issued by XERBLA.
    !>          If VECT='N', LHOUS2 = max(1, 4*n);
    !>          if VECT='V', option not yet available.
    !>

    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 = 0, LWORK >= 1, else LWORK = MAX(1, dimension).
    !>
    !>          If LWORK = -1, or LHOUS2 = -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.
    !>          LWORK = MAX(1, dimension) where
    !>          dimension   = max(stage1,stage2) + (KD+1)*N
    !>                      = N*KD + N*max(KD+1,FACTOPTNB)
    !>                        + max(2*KD*KD, KD*NTHREADS)
    !>                        + (KD+1)*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.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  Implemented by Azzam Haidar.
    !>
    !>  All details are available on technical report, SC11, SC13 papers.
    !>
    !>  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
    !>
    !>

Author

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