hetrd_he2hb(3)

NAME
hetrd_he2hb - {he,sy}trd_he2hb: full to band (1st stage)

SYNOPSIS

  Functions
subroutine chetrd_he2hb (uplo, n, kd, a, lda, ab, ldab, tau, work, lwork, info)
CHETRD_HE2HB
subroutine dsytrd_sy2sb (uplo, n, kd, a, lda, ab, ldab, tau, work, lwork, info)
DSYTRD_SY2SB
subroutine ssytrd_sy2sb (uplo, n, kd, a, lda, ab, ldab, tau, work, lwork, info)
SSYTRD_SY2SB
subroutine zhetrd_he2hb (uplo, n, kd, a, lda, ab, ldab, tau, work, lwork, info)
ZHETRD_HE2HB

Detailed Description

Function Documentation

  subroutine chetrd_he2hb (character uplo, integer n, integer kd, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( * ) tau, complex, dimension( * ) work, integer lwork, integer info)

CHETRD_HE2HB

Purpose:

    !>
    !> CHETRD_HE2HB reduces a complex Hermitian matrix A to complex Hermitian
    !> band-diagonal form AB by a unitary similarity transformation:
    !> Q**H * A * Q = AB.
    !>

Parameters
    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 reduced matrix if UPLO = 'U',
    !>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
    !>          The reduced matrix is stored in the array AB.
    !>

    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 diagonal and first superdiagonal
    !>          of A are overwritten by the corresponding elements of the
    !>          tridiagonal matrix T, and the elements above the first
    !>          superdiagonal, with the array TAU, represent the unitary
    !>          matrix Q as a product of elementary reflectors; if UPLO
    !>          = 'L', the diagonal and first subdiagonal of A are over-
    !>          written by the corresponding elements of the tridiagonal
    !>          matrix T, and the elements below the first subdiagonal, with
    !>          the array TAU, represent the unitary matrix Q as a product
    !>          of elementary reflectors. See Further Details.
    !>

    LDA

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

    AB

    !>          AB is COMPLEX array, dimension (LDAB,N)
    !>          On exit, 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).
    !>

    LDAB

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

    TAU

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

    WORK

    !>          WORK is COMPLEX array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, or if LWORK = -1,
    !>          WORK(1) returns the size of LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK which should be calculated
    !>          by a workspace query.
    !>          If N <= KD+1, LWORK >= 1, else LWORK = MAX(1, LWORK_QUERY).
    !>
    !>          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.
    !>          LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD
    !>          where FACTOPTNB is the blocking used by the QR or LQ
    !>          algorithm, usually FACTOPTNB=128 is a good choice otherwise
    !>          putting LWORK=-1 will provide the size of WORK.
    !>

    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
    !>
    !>

!>
!>  If UPLO = 'U', the matrix Q is represented as a product of elementary
!>  reflectors
!>
!>     Q = H(k)**H . . . H(2)**H H(1)**H, where k = n-kd.
!>
!>  Each H(i) has the form
!>
!>     H(i) = I - tau * v * v**H
!>
!>  where tau is a complex scalar, and v is a complex vector with
!>  v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in
!>  A(i,i+kd+1:n), and tau in TAU(i).
!>
!>  If UPLO = 'L', the matrix Q is represented as a product of elementary
!>  reflectors
!>
!>     Q = H(1) H(2) . . . H(k), where k = n-kd.
!>
!>  Each H(i) has the form
!>
!>     H(i) = I - tau * v * v**H
!>
!>  where tau is a complex scalar, and v is a complex vector with
!>  v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in
!>  A(i+kd+2:n,i), and tau in TAU(i).
!>
!>  The contents of A on exit are illustrated by the following examples
!>  with n = 5:
!>
!>  if UPLO = 'U':                       if UPLO = 'L':
!>
!>    (  ab  ab/v1  v1      v1     v1    )              (  ab                            )
!>    (      ab     ab/v2   v2     v2    )              (  ab/v1  ab                     )
!>    (             ab      ab/v3  v3    )              (  v1     ab/v2  ab              )
!>    (                     ab     ab/v4 )              (  v1     v2     ab/v3  ab       )
!>    (                            ab    )              (  v1     v2     v3     ab/v4 ab )
!>
!>  where d and e denote diagonal and off-diagonal elements of T, and vi
!>  denotes an element of the vector defining H(i).
!> .fi

Definition at line 243 of file chetrd_he2hb.f.

  subroutine dsytrd_sy2sb (character uplo, integer n, integer kd, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer lwork, integer info)

DSYTRD_SY2SB

Purpose:

    !>
    !> DSYTRD_SY2SB reduces a real symmetric matrix A to real symmetric
    !> band-diagonal form AB by a orthogonal similarity transformation:
    !> Q**T * A * Q = AB.
    !>

Parameters
    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 reduced matrix if UPLO = 'U',
    !>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
    !>          The reduced matrix is stored in the array AB.
    !>

    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 diagonal and first superdiagonal
    !>          of A are overwritten by the corresponding elements of the
    !>          tridiagonal matrix T, and the elements above the first
    !>          superdiagonal, with the array TAU, represent the orthogonal
    !>          matrix Q as a product of elementary reflectors; if UPLO
    !>          = 'L', the diagonal and first subdiagonal of A are over-
    !>          written by the corresponding elements of the tridiagonal
    !>          matrix T, and the elements below the first subdiagonal, with
    !>          the array TAU, represent the orthogonal matrix Q as a product
    !>          of elementary reflectors. See Further Details.
    !>

    LDA

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

    AB

    !>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
    !>          On exit, 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).
    !>

    LDAB

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

    TAU

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

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, or if LWORK = -1,
    !>          WORK(1) returns the size of LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK which should be calculated
    !>          by a workspace query.
    !>          If N <= KD+1, LWORK >= 1, else LWORK = MAX(1, LWORK_QUERY)
    !>
    !>          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.
    !>          LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD
    !>          where FACTOPTNB is the blocking used by the QR or LQ
    !>          algorithm, usually FACTOPTNB=128 is a good choice otherwise
    !>          putting LWORK=-1 will provide the size of WORK.
    !>

    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
    !>
    !>

!>
!>  If UPLO = 'U', the matrix Q is represented as a product of elementary
!>  reflectors
!>
!>     Q = H(k)**T . . . H(2)**T H(1)**T, where k = n-kd.
!>
!>  Each H(i) has the form
!>
!>     H(i) = I - tau * v * v**T
!>
!>  where tau is a real scalar, and v is a real vector with
!>  v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in
!>  A(i,i+kd+1:n), and tau in TAU(i).
!>
!>  If UPLO = 'L', the matrix Q is represented as a product of elementary
!>  reflectors
!>
!>     Q = H(1) H(2) . . . H(k), where k = n-kd.
!>
!>  Each H(i) has the form
!>
!>     H(i) = I - tau * v * v**T
!>
!>  where tau is a real scalar, and v is a real vector with
!>  v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in
!>  A(i+kd+2:n,i), and tau in TAU(i).
!>
!>  The contents of A on exit are illustrated by the following examples
!>  with n = 5:
!>
!>  if UPLO = 'U':                       if UPLO = 'L':
!>
!>    (  ab  ab/v1  v1      v1     v1    )              (  ab                            )
!>    (      ab     ab/v2   v2     v2    )              (  ab/v1  ab                     )
!>    (             ab      ab/v3  v3    )              (  v1     ab/v2  ab              )
!>    (                     ab     ab/v4 )              (  v1     v2     ab/v3  ab       )
!>    (                            ab    )              (  v1     v2     v3     ab/v4 ab )
!>
!>  where d and e denote diagonal and off-diagonal elements of T, and vi
!>  denotes an element of the vector defining H(i).
!> .fi

Definition at line 243 of file dsytrd_sy2sb.f.

  subroutine ssytrd_sy2sb (character uplo, integer n, integer kd, real, dimension( lda, * ) a, integer lda, real, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) tau, real, dimension( * ) work, integer lwork, integer info)

SSYTRD_SY2SB

Purpose:

    !>
    !> SSYTRD_SY2SB reduces a real symmetric matrix A to real symmetric
    !> band-diagonal form AB by a orthogonal similarity transformation:
    !> Q**T * A * Q = AB.
    !>

Parameters
    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 reduced matrix if UPLO = 'U',
    !>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
    !>          The reduced matrix is stored in the array AB.
    !>

    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 diagonal and first superdiagonal
    !>          of A are overwritten by the corresponding elements of the
    !>          tridiagonal matrix T, and the elements above the first
    !>          superdiagonal, with the array TAU, represent the orthogonal
    !>          matrix Q as a product of elementary reflectors; if UPLO
    !>          = 'L', the diagonal and first subdiagonal of A are over-
    !>          written by the corresponding elements of the tridiagonal
    !>          matrix T, and the elements below the first subdiagonal, with
    !>          the array TAU, represent the orthogonal matrix Q as a product
    !>          of elementary reflectors. See Further Details.
    !>

    LDA

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

    AB

    !>          AB is REAL array, dimension (LDAB,N)
    !>          On exit, 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).
    !>

    LDAB

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

    TAU

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

    WORK

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

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK which should be calculated
    !>          by a workspace query.
    !>          If N <= KD+1, LWORK >= 1, else LWORK = MAX(1, LWORK_QUERY)
    !>
    !>          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.
    !>          LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD
    !>          where FACTOPTNB is the blocking used by the QR or LQ
    !>          algorithm, usually FACTOPTNB=128 is a good choice otherwise
    !>          putting LWORK=-1 will provide the size of WORK.
    !>

    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
    !>
    !>

!>
!>  If UPLO = 'U', the matrix Q is represented as a product of elementary
!>  reflectors
!>
!>     Q = H(k)**T . . . H(2)**T H(1)**T, where k = n-kd.
!>
!>  Each H(i) has the form
!>
!>     H(i) = I - tau * v * v**T
!>
!>  where tau is a real scalar, and v is a real vector with
!>  v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in
!>  A(i,i+kd+1:n), and tau in TAU(i).
!>
!>  If UPLO = 'L', the matrix Q is represented as a product of elementary
!>  reflectors
!>
!>     Q = H(1) H(2) . . . H(k), where k = n-kd.
!>
!>  Each H(i) has the form
!>
!>     H(i) = I - tau * v * v**T
!>
!>  where tau is a real scalar, and v is a real vector with
!>  v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in
!>  A(i+kd+2:n,i), and tau in TAU(i).
!>
!>  The contents of A on exit are illustrated by the following examples
!>  with n = 5:
!>
!>  if UPLO = 'U':                       if UPLO = 'L':
!>
!>    (  ab  ab/v1  v1      v1     v1    )              (  ab                            )
!>    (      ab     ab/v2   v2     v2    )              (  ab/v1  ab                     )
!>    (             ab      ab/v3  v3    )              (  v1     ab/v2  ab              )
!>    (                     ab     ab/v4 )              (  v1     v2     ab/v3  ab       )
!>    (                            ab    )              (  v1     v2     v3     ab/v4 ab )
!>
!>  where d and e denote diagonal and off-diagonal elements of T, and vi
!>  denotes an element of the vector defining H(i).
!> .fi

Definition at line 243 of file ssytrd_sy2sb.f.

  subroutine zhetrd_he2hb (character uplo, integer n, integer kd, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer lwork, integer info)

ZHETRD_HE2HB

Purpose:

    !>
    !> ZHETRD_HE2HB reduces a complex Hermitian matrix A to complex Hermitian
    !> band-diagonal form AB by a unitary similarity transformation:
    !> Q**H * A * Q = AB.
    !>

Parameters
    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 reduced matrix if UPLO = 'U',
    !>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
    !>          The reduced matrix is stored in the array AB.
    !>

    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 diagonal and first superdiagonal
    !>          of A are overwritten by the corresponding elements of the
    !>          tridiagonal matrix T, and the elements above the first
    !>          superdiagonal, with the array TAU, represent the unitary
    !>          matrix Q as a product of elementary reflectors; if UPLO
    !>          = 'L', the diagonal and first subdiagonal of A are over-
    !>          written by the corresponding elements of the tridiagonal
    !>          matrix T, and the elements below the first subdiagonal, with
    !>          the array TAU, represent the unitary matrix Q as a product
    !>          of elementary reflectors. See Further Details.
    !>

    LDA

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

    AB

    !>          AB is COMPLEX*16 array, dimension (LDAB,N)
    !>          On exit, 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).
    !>

    LDAB

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

    TAU

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

    WORK

    !>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, or if LWORK = -1,
    !>          WORK(1) returns the size of LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK which should be calculated
    !>          by a workspace query.
    !>          If N <= KD+1, LWORK >= 1, else LWORK = MAX(1, LWORK_QUERY).
    !>
    !>          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.
    !>          LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD
    !>          where FACTOPTNB is the blocking used by the QR or LQ
    !>          algorithm, usually FACTOPTNB=128 is a good choice otherwise
    !>          putting LWORK=-1 will provide the size of WORK.
    !>

    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
    !>
    !>

!>
!>  If UPLO = 'U', the matrix Q is represented as a product of elementary
!>  reflectors
!>
!>     Q = H(k)**H . . . H(2)**H H(1)**H, where k = n-kd.
!>
!>  Each H(i) has the form
!>
!>     H(i) = I - tau * v * v**H
!>
!>  where tau is a complex scalar, and v is a complex vector with
!>  v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in
!>  A(i,i+kd+1:n), and tau in TAU(i).
!>
!>  If UPLO = 'L', the matrix Q is represented as a product of elementary
!>  reflectors
!>
!>     Q = H(1) H(2) . . . H(k), where k = n-kd.
!>
!>  Each H(i) has the form
!>
!>     H(i) = I - tau * v * v**H
!>
!>  where tau is a complex scalar, and v is a complex vector with
!>  v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in
!>  A(i+kd+2:n,i), and tau in TAU(i).
!>
!>  The contents of A on exit are illustrated by the following examples
!>  with n = 5:
!>
!>  if UPLO = 'U':                       if UPLO = 'L':
!>
!>    (  ab  ab/v1  v1      v1     v1    )              (  ab                            )
!>    (      ab     ab/v2   v2     v2    )              (  ab/v1  ab                     )
!>    (             ab      ab/v3  v3    )              (  v1     ab/v2  ab              )
!>    (                     ab     ab/v4 )              (  v1     v2     ab/v3  ab       )
!>    (                            ab    )              (  v1     v2     v3     ab/v4 ab )
!>
!>  where d and e denote diagonal and off-diagonal elements of T, and vi
!>  denotes an element of the vector defining H(i).
!> .fi

Definition at line 243 of file zhetrd_he2hb.f.

Author

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