hegv_2stage(3)

NAME
hegv_2stage - {he,sy}gv_2stage: eig, QR iteration, 2-stage

SYNOPSIS

  Functions

subroutine chegv_2stage (itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, rwork, info)
CHEGV_2STAGE
subroutine dsygv_2stage (itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, info)
DSYGV_2STAGE
subroutine ssygv_2stage (itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, info)
SSYGV_2STAGE
subroutine zhegv_2stage (itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, rwork, info)
ZHEGV_2STAGE

Detailed Description

Function Documentation

  subroutine chegv_2stage (integer itype, character jobz, character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, real, dimension( * ) w, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer info)

CHEGV_2STAGE

Purpose:

    !>
    !> CHEGV_2STAGE computes all the eigenvalues, and optionally, the eigenvectors
    !> of a complex generalized Hermitian-definite eigenproblem, of the form
    !> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
    !> Here A and B are assumed to be Hermitian and B is also
    !> positive definite.
    !> This routine use the 2stage technique for the reduction to tridiagonal
    !> which showed higher performance on recent architecture and for large
    !> sizes N>2000.
    !>

Parameters
    ITYPE

    !>          ITYPE is INTEGER
    !>          Specifies the problem type to be solved:
    !>          = 1:  A*x = (lambda)*B*x
    !>          = 2:  A*B*x = (lambda)*x
    !>          = 3:  B*A*x = (lambda)*x
    !>

    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 triangles of A and B are stored;
    !>          = 'L':  Lower triangles of A and B are stored.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrices A and B.  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
    !>          matrix Z of eigenvectors.  The eigenvectors are normalized
    !>          as follows:
    !>          if ITYPE = 1 or 2, Z**H*B*Z = I;
    !>          if ITYPE = 3, Z**H*inv(B)*Z = I.
    !>          If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
    !>          or the lower triangle (if UPLO='L') of A, including the
    !>          diagonal, is destroyed.
    !>

    LDA

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

    B

    !>          B is COMPLEX array, dimension (LDB, N)
    !>          On entry, the Hermitian positive definite matrix B.
    !>          If UPLO = 'U', the leading N-by-N upper triangular part of B
    !>          contains the upper triangular part of the matrix B.
    !>          If UPLO = 'L', the leading N-by-N lower triangular part of B
    !>          contains the lower triangular part of the matrix B.
    !>
    !>          On exit, if INFO <= N, the part of B containing the matrix is
    !>          overwritten by the triangular factor U or L from the Cholesky
    !>          factorization B = U**H*U or B = L*L**H.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B.  LDB >= 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 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 = 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 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.
    !>

    RWORK

    !>          RWORK is REAL array, dimension (max(1, 3*N-2))
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  CPOTRF or CHEEV returned an error code:
    !>             <= N:  if INFO = i, CHEEV failed to converge;
    !>                    i off-diagonal elements of an intermediate
    !>                    tridiagonal form did not converge to zero;
    !>             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
    !>                    principal minor of order i of B is not positive.
    !>                    The factorization of B could not be completed and
    !>                    no eigenvalues or eigenvectors were computed.
    !>

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
    !>
    !>

  subroutine dsygv_2stage (integer itype, character jobz, character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) w, double precision, dimension( * ) work, integer lwork, integer info)

DSYGV_2STAGE

Purpose:

    !>
    !> DSYGV_2STAGE computes all the eigenvalues, and optionally, the eigenvectors
    !> of a real generalized symmetric-definite eigenproblem, of the form
    !> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
    !> Here A and B are assumed to be symmetric and B is also
    !> positive definite.
    !> This routine use the 2stage technique for the reduction to tridiagonal
    !> which showed higher performance on recent architecture and for large
    !> sizes N>2000.
    !>

Parameters
    ITYPE

    !>          ITYPE is INTEGER
    !>          Specifies the problem type to be solved:
    !>          = 1:  A*x = (lambda)*B*x
    !>          = 2:  A*B*x = (lambda)*x
    !>          = 3:  B*A*x = (lambda)*x
    !>

    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 triangles of A and B are stored;
    !>          = 'L':  Lower triangles of A and B are stored.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrices A and B.  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
    !>          matrix Z of eigenvectors.  The eigenvectors are normalized
    !>          as follows:
    !>          if ITYPE = 1 or 2, Z**T*B*Z = I;
    !>          if ITYPE = 3, Z**T*inv(B)*Z = I.
    !>          If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
    !>          or the lower triangle (if UPLO='L') of A, including the
    !>          diagonal, is destroyed.
    !>

    LDA

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

    B

    !>          B is DOUBLE PRECISION array, dimension (LDB, N)
    !>          On entry, the symmetric positive definite matrix B.
    !>          If UPLO = 'U', the leading N-by-N upper triangular part of B
    !>          contains the upper triangular part of the matrix B.
    !>          If UPLO = 'L', the leading N-by-N lower triangular part of B
    !>          contains the lower triangular part of the matrix B.
    !>
    !>          On exit, if INFO <= N, the part of B containing the matrix is
    !>          overwritten by the triangular factor U or L from the Cholesky
    !>          factorization B = U**T*U or B = L*L**T.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B.  LDB >= 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 (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    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 = max(stage1,stage2) + (KD+1)*N + 2*N
    !>                                             = N*KD + N*max(KD+1,FACTOPTNB)
    !>                                               + max(2*KD*KD, KD*NTHREADS)
    !>                                               + (KD+1)*N + 2*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.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  DPOTRF or DSYEV returned an error code:
    !>             <= N:  if INFO = i, DSYEV failed to converge;
    !>                    i off-diagonal elements of an intermediate
    !>                    tridiagonal form did not converge to zero;
    !>             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
    !>                    principal minor of order i of B is not positive.
    !>                    The factorization of B could not be completed and
    !>                    no eigenvalues or eigenvectors were computed.
    !>

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
    !>
    !>

  subroutine ssygv_2stage (integer itype, character jobz, character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) w, real, dimension( * ) work, integer lwork, integer info)

SSYGV_2STAGE

Purpose:

    !>
    !> SSYGV_2STAGE computes all the eigenvalues, and optionally, the eigenvectors
    !> of a real generalized symmetric-definite eigenproblem, of the form
    !> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
    !> Here A and B are assumed to be symmetric and B is also
    !> positive definite.
    !> This routine use the 2stage technique for the reduction to tridiagonal
    !> which showed higher performance on recent architecture and for large
    !> sizes N>2000.
    !>

Parameters
    ITYPE

    !>          ITYPE is INTEGER
    !>          Specifies the problem type to be solved:
    !>          = 1:  A*x = (lambda)*B*x
    !>          = 2:  A*B*x = (lambda)*x
    !>          = 3:  B*A*x = (lambda)*x
    !>

    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 triangles of A and B are stored;
    !>          = 'L':  Lower triangles of A and B are stored.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrices A and B.  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
    !>          matrix Z of eigenvectors.  The eigenvectors are normalized
    !>          as follows:
    !>          if ITYPE = 1 or 2, Z**T*B*Z = I;
    !>          if ITYPE = 3, Z**T*inv(B)*Z = I.
    !>          If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
    !>          or the lower triangle (if UPLO='L') of A, including the
    !>          diagonal, is destroyed.
    !>

    LDA

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

    B

    !>          B is REAL array, dimension (LDB, N)
    !>          On entry, the symmetric positive definite matrix B.
    !>          If UPLO = 'U', the leading N-by-N upper triangular part of B
    !>          contains the upper triangular part of the matrix B.
    !>          If UPLO = 'L', the leading N-by-N lower triangular part of B
    !>          contains the lower triangular part of the matrix B.
    !>
    !>          On exit, if INFO <= N, the part of B containing the matrix is
    !>          overwritten by the triangular factor U or L from the Cholesky
    !>          factorization B = U**T*U or B = L*L**T.
    !>

    LDB

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

    W

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

    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 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 = max(stage1,stage2) + (KD+1)*N + 2*N
    !>                                             = N*KD + N*max(KD+1,FACTOPTNB)
    !>                                               + max(2*KD*KD, KD*NTHREADS)
    !>                                               + (KD+1)*N + 2*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.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  SPOTRF or SSYEV returned an error code:
    !>             <= N:  if INFO = i, SSYEV failed to converge;
    !>                    i off-diagonal elements of an intermediate
    !>                    tridiagonal form did not converge to zero;
    !>             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
    !>                    principal minor of order i of B is not positive.
    !>                    The factorization of B could not be completed and
    !>                    no eigenvalues or eigenvectors were computed.
    !>

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
    !>
    !>

  subroutine zhegv_2stage (integer itype, character jobz, character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) w, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer info)

ZHEGV_2STAGE

Purpose:

    !>
    !> ZHEGV_2STAGE computes all the eigenvalues, and optionally, the eigenvectors
    !> of a complex generalized Hermitian-definite eigenproblem, of the form
    !> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
    !> Here A and B are assumed to be Hermitian and B is also
    !> positive definite.
    !> This routine use the 2stage technique for the reduction to tridiagonal
    !> which showed higher performance on recent architecture and for large
    !> sizes N>2000.
    !>

Parameters
    ITYPE

    !>          ITYPE is INTEGER
    !>          Specifies the problem type to be solved:
    !>          = 1:  A*x = (lambda)*B*x
    !>          = 2:  A*B*x = (lambda)*x
    !>          = 3:  B*A*x = (lambda)*x
    !>

    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 triangles of A and B are stored;
    !>          = 'L':  Lower triangles of A and B are stored.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrices A and B.  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
    !>          matrix Z of eigenvectors.  The eigenvectors are normalized
    !>          as follows:
    !>          if ITYPE = 1 or 2, Z**H*B*Z = I;
    !>          if ITYPE = 3, Z**H*inv(B)*Z = I.
    !>          If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
    !>          or the lower triangle (if UPLO='L') of A, including the
    !>          diagonal, is destroyed.
    !>

    LDA

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

    B

    !>          B is COMPLEX*16 array, dimension (LDB, N)
    !>          On entry, the Hermitian positive definite matrix B.
    !>          If UPLO = 'U', the leading N-by-N upper triangular part of B
    !>          contains the upper triangular part of the matrix B.
    !>          If UPLO = 'L', the leading N-by-N lower triangular part of B
    !>          contains the lower triangular part of the matrix B.
    !>
    !>          On exit, if INFO <= N, the part of B containing the matrix is
    !>          overwritten by the triangular factor U or L from the Cholesky
    !>          factorization B = U**H*U or B = L*L**H.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B.  LDB >= 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 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 = 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 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.
    !>

    RWORK

    !>          RWORK is DOUBLE PRECISION array, dimension (max(1, 3*N-2))
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  ZPOTRF or ZHEEV returned an error code:
    !>             <= N:  if INFO = i, ZHEEV failed to converge;
    !>                    i off-diagonal elements of an intermediate
    !>                    tridiagonal form did not converge to zero;
    !>             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
    !>                    principal minor of order i of B is not positive.
    !>                    The factorization of B could not be completed and
    !>                    no eigenvalues or eigenvectors were computed.
    !>

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
    !>
    !>

Author

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