hetrd_hb2st(3)

NAME
hetrd_hb2st - {he,sy}trd_hb2st: band to tridiagonal (2nd stage)

SYNOPSIS

  Functions

subroutine chetrd_hb2st (stage1, vect, uplo, n, kd, ab, ldab, d, e, hous, lhous, work, lwork, info)
CHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T
subroutine dsytrd_sb2st (stage1, vect, uplo, n, kd, ab, ldab, d, e, hous, lhous, work, lwork, info)
DSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric tridiagonal form T
subroutine ssytrd_sb2st (stage1, vect, uplo, n, kd, ab, ldab, d, e, hous, lhous, work, lwork, info)
SSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric tridiagonal form T
subroutine zhetrd_hb2st (stage1, vect, uplo, n, kd, ab, ldab, d, e, hous, lhous, work, lwork, info)
ZHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T

Detailed Description

Function Documentation

  subroutine chetrd_hb2st (character stage1, character vect, character uplo, integer n, integer kd, complex, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( * ) hous, integer lhous, complex, dimension( * ) work, integer lwork, integer info)

CHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T

Purpose:

    !>
    !> CHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric
    !> tridiagonal form T by a unitary similarity transformation:
    !> Q**H * A * Q = T.
    !>

Parameters
    STAGE1

    !>          STAGE1 is CHARACTER*1
    !>          = 'N':  : to mention that the stage 1 of the reduction
    !>                  from dense to band using the chetrd_he2hb routine
    !>                  was not called before this routine to reproduce AB.
    !>                  In other term this routine is called as standalone.
    !>          = 'Y':  : to mention that the stage 1 of the
    !>                  reduction from dense to band using the chetrd_he2hb
    !>                  routine has been called to produce AB (e.g., AB is
    !>                  the output of chetrd_he2hb.
    !>

    VECT

    !>          VECT is CHARACTER*1
    !>          = 'N':  No need for the Housholder representation,
    !>                  and thus LHOUS is of size max(1, 4*N);
    !>          = 'V':  the Householder representation is needed to
    !>                  either generate or to apply Q later on,
    !>                  then LHOUS 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.
    !>

    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.
    !>

    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, the diagonal elements of AB are overwritten by the
    !>          diagonal elements of the tridiagonal matrix T; if KD > 0, the
    !>          elements on the first superdiagonal (if UPLO = 'U') or the
    !>          first subdiagonal (if UPLO = 'L') are overwritten by the
    !>          off-diagonal elements of T; the rest of AB is overwritten by
    !>          values generated during the reduction.
    !>

    LDAB

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

    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:
    !>          E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.
    !>

    HOUS

    !>          HOUS is COMPLEX array, dimension (MAX(1,LHOUS))
    !>          Stores the Householder representation.
    !>

    LHOUS

    !>          LHOUS is INTEGER
    !>          The dimension of the array HOUS.
    !>          If N = 0 or KD <= 1, LHOUS >= 1, else LHOUS = MAX(1, dimension).
    !>
    !>          If LWORK = -1, or LHOUS = -1,
    !>          then a query is assumed; the routine
    !>          only calculates the optimal size of the HOUS array, returns
    !>          this value as the first entry of the HOUS array, and no error
    !>          message related to LHOUS is issued by XERBLA.
    !>          LHOUS = MAX(1, dimension) where
    !>          dimension = 4*N if VECT='N'
    !>          not available now if VECT='H'
    !>

    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 or KD <= 1, LWORK >= 1, else LWORK = MAX(1, dimension).
    !>
    !>          If LWORK = -1, or LHOUS = -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   = (2KD+1)*N + KD*NTHREADS
    !>          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_sb2st (character stage1, character vect, character uplo, integer n, integer kd, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( * ) hous, integer lhous, double precision, dimension( * ) work, integer lwork, integer info)

DSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric tridiagonal form T

Purpose:

    !>
    !> DSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric
    !> tridiagonal form T by a orthogonal similarity transformation:
    !> Q**T * A * Q = T.
    !>

Parameters
    STAGE1

    !>          STAGE1 is CHARACTER*1
    !>          = 'N':  : to mention that the stage 1 of the reduction
    !>                  from dense to band using the dsytrd_sy2sb routine
    !>                  was not called before this routine to reproduce AB.
    !>                  In other term this routine is called as standalone.
    !>          = 'Y':  : to mention that the stage 1 of the
    !>                  reduction from dense to band using the dsytrd_sy2sb
    !>                  routine has been called to produce AB (e.g., AB is
    !>                  the output of dsytrd_sy2sb.
    !>

    VECT

    !>          VECT is CHARACTER*1
    !>          = 'N':  No need for the Housholder representation,
    !>                  and thus LHOUS is of size max(1, 4*N);
    !>          = 'V':  the Householder representation is needed to
    !>                  either generate or to apply Q later on,
    !>                  then LHOUS 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.
    !>

    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.
    !>

    AB

    !>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
    !>          On entry, the upper or lower triangle of the symmetric 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, the diagonal elements of AB are overwritten by the
    !>          diagonal elements of the tridiagonal matrix T; if KD > 0, the
    !>          elements on the first superdiagonal (if UPLO = 'U') or the
    !>          first subdiagonal (if UPLO = 'L') are overwritten by the
    !>          off-diagonal elements of T; the rest of AB is overwritten by
    !>          values generated during the reduction.
    !>

    LDAB

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

    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:
    !>          E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.
    !>

    HOUS

    !>          HOUS is DOUBLE PRECISION array, dimension (MAX(1,LHOUS))
    !>          Stores the Householder representation.
    !>

    LHOUS

    !>          LHOUS is INTEGER
    !>          The dimension of the array HOUS.
    !>          If N = 0 or KD <= 1, LHOUS >= 1, else LHOUS = MAX(1, dimension).
    !>
    !>          If LWORK = -1, or LHOUS = -1,
    !>          then a query is assumed; the routine
    !>          only calculates the optimal size of the HOUS array, returns
    !>          this value as the first entry of the HOUS array, and no error
    !>          message related to LHOUS is issued by XERBLA.
    !>          LHOUS = MAX(1, dimension) where
    !>          dimension = 4*N if VECT='N'
    !>          not available now if VECT='H'
    !>

    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 or KD <= 1, LWORK >= 1, else LWORK = MAX(1, dimension).
    !>
    !>          If LWORK = -1, or LHOUS = -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   = (2KD+1)*N + KD*NTHREADS
    !>          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_sb2st (character stage1, character vect, character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) d, real, dimension( * ) e, real, dimension( * ) hous, integer lhous, real, dimension( * ) work, integer lwork, integer info)

SSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric tridiagonal form T

Purpose:

    !>
    !> SSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric
    !> tridiagonal form T by a orthogonal similarity transformation:
    !> Q**T * A * Q = T.
    !>

Parameters
    STAGE1

    !>          STAGE1 is CHARACTER*1
    !>          = 'N':  : to mention that the stage 1 of the reduction
    !>                  from dense to band using the ssytrd_sy2sb routine
    !>                  was not called before this routine to reproduce AB.
    !>                  In other term this routine is called as standalone.
    !>          = 'Y':  : to mention that the stage 1 of the
    !>                  reduction from dense to band using the ssytrd_sy2sb
    !>                  routine has been called to produce AB (e.g., AB is
    !>                  the output of ssytrd_sy2sb.
    !>

    VECT

    !>          VECT is CHARACTER*1
    !>          = 'N':  No need for the Housholder representation,
    !>                  and thus LHOUS is of size max(1, 4*N);
    !>          = 'V':  the Householder representation is needed to
    !>                  either generate or to apply Q later on,
    !>                  then LHOUS 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.
    !>

    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.
    !>

    AB

    !>          AB is REAL array, dimension (LDAB,N)
    !>          On entry, the upper or lower triangle of the symmetric 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, the diagonal elements of AB are overwritten by the
    !>          diagonal elements of the tridiagonal matrix T; if KD > 0, the
    !>          elements on the first superdiagonal (if UPLO = 'U') or the
    !>          first subdiagonal (if UPLO = 'L') are overwritten by the
    !>          off-diagonal elements of T; the rest of AB is overwritten by
    !>          values generated during the reduction.
    !>

    LDAB

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

    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:
    !>          E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.
    !>

    HOUS

    !>          HOUS is REAL array, dimension (MAX(1,LHOUS))
    !>          Stores the Householder representation.
    !>

    LHOUS

    !>          LHOUS is INTEGER
    !>          The dimension of the array HOUS.
    !>          If N = 0 or KD <= 1, LHOUS >= 1, else LHOUS = MAX(1, dimension)
    !>
    !>          If LWORK = -1, or LHOUS = -1,
    !>          then a query is assumed; the routine
    !>          only calculates the optimal size of the HOUS array, returns
    !>          this value as the first entry of the HOUS array, and no error
    !>          message related to LHOUS is issued by XERBLA.
    !>          LHOUS = MAX(1, dimension) where
    !>          dimension = 4*N if VECT='N'
    !>          not available now if VECT='H'
    !>

    WORK

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

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.
    !>          IF N = 0 or KD <= 1, LWORK >= 1, else LWORK = MAX(1, dimension)
    !>
    !>          If LWORK = -1, or LHOUS = -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   = (2KD+1)*N + KD*NTHREADS
    !>          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_hb2st (character stage1, character vect, character uplo, integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( * ) hous, integer lhous, complex*16, dimension( * ) work, integer lwork, integer info)

ZHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T

Purpose:

    !>
    !> ZHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric
    !> tridiagonal form T by a unitary similarity transformation:
    !> Q**H * A * Q = T.
    !>

Parameters
    STAGE1

    !>          STAGE1 is CHARACTER*1
    !>          = 'N':  : to mention that the stage 1 of the reduction
    !>                  from dense to band using the zhetrd_he2hb routine
    !>                  was not called before this routine to reproduce AB.
    !>                  In other term this routine is called as standalone.
    !>          = 'Y':  : to mention that the stage 1 of the
    !>                  reduction from dense to band using the zhetrd_he2hb
    !>                  routine has been called to produce AB (e.g., AB is
    !>                  the output of zhetrd_he2hb.
    !>

    VECT

    !>          VECT is CHARACTER*1
    !>          = 'N':  No need for the Housholder representation,
    !>                  and thus LHOUS is of size max(1, 4*N);
    !>          = 'V':  the Householder representation is needed to
    !>                  either generate or to apply Q later on,
    !>                  then LHOUS 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.
    !>

    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.
    !>

    AB

    !>          AB is COMPLEX*16 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, the diagonal elements of AB are overwritten by the
    !>          diagonal elements of the tridiagonal matrix T; if KD > 0, the
    !>          elements on the first superdiagonal (if UPLO = 'U') or the
    !>          first subdiagonal (if UPLO = 'L') are overwritten by the
    !>          off-diagonal elements of T; the rest of AB is overwritten by
    !>          values generated during the reduction.
    !>

    LDAB

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

    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:
    !>          E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.
    !>

    HOUS

    !>          HOUS is COMPLEX*16 array, dimension (MAX(1,LHOUS))
    !>          Stores the Householder representation.
    !>

    LHOUS

    !>          LHOUS is INTEGER
    !>          The dimension of the array HOUS.
    !>          If N = 0 or KD <= 1, LHOUS >= 1, else LHOUS = MAX(1, dimension).
    !>
    !>          If LWORK = -1, or LHOUS = -1,
    !>          then a query is assumed; the routine
    !>          only calculates the optimal size of the HOUS array, returns
    !>          this value as the first entry of the HOUS array, and no error
    !>          message related to LHOUS is issued by XERBLA.
    !>          LHOUS = MAX(1, dimension) where
    !>          dimension = 4*N if VECT='N'
    !>          not available now if VECT='H'
    !>

    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 or KD <= 1, LWORK >= 1, else LWORK = MAX(1, dimension).
    !>
    !>          If LWORK = -1, or LHOUS = -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   = (2KD+1)*N + KD*NTHREADS
    !>          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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