zhetrd_he2hb.f(3)

NAME
zhetrd_he2hb.f

SYNOPSIS

  Functions/Subroutines
subroutine zhetrd_he2hb (UPLO, N, KD, A, LDA, AB, LDAB, TAU, WORK, LWORK, INFO)
ZHETRD_HE2HB

Function/Subroutine Documentation

  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)

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 (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. 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 241 of file zhetrd_he2hb.f.

Author

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