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zlahef

 NAME
      ZLAHEF - compute a partial factorization of a complex Hermi-
      tian matrix A using the Bunch-Kaufman diagonal pivoting
      method

 SYNOPSIS
      SUBROUTINE ZLAHEF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW,
                         INFO )

          CHARACTER      UPLO

          INTEGER        INFO, KB, LDA, LDW, N, NB

          INTEGER        IPIV( * )

          COMPLEX*16     A( LDA, * ), W( LDW, * )

 PURPOSE
      ZLAHEF computes a partial factorization of a complex Hermi-
      tian matrix A using the Bunch-Kaufman diagonal pivoting
      method. The partial factorization has the form:

      A  =  ( I  U12 ) ( A11  0  ) (  I    0   )  if UPLO = 'U',
      or:
            ( 0  U22 ) (  0   D  ) ( U12' U22' )

      A  =  ( L11  0 ) (  D   0  ) ( L11' L21' )  if UPLO = 'L'
            ( L21  I ) (  0  A22 ) (  0    I   )

      where the order of D is at most NB. The actual order is
      returned in the argument KB, and is either NB or NB-1, or N
      if N <= NB.  Note that U' denotes the conjugate transpose of
      U.

      ZLAHEF is an auxiliary routine called by ZHETRF. It uses
      blocked code (calling Level 3 BLAS) to update the submatrix
      A11 (if UPLO = 'U') or A22 (if UPLO = 'L').

 ARGUMENTS
      UPLO    (input) CHARACTER*1
              Specifies whether the upper or lower triangular part
              of the Hermitian matrix A is stored:
              = 'U':  Upper triangular
              = 'L':  Lower triangular

      N       (input) INTEGER
              The order of the matrix A.  N >= 0.

      NB      (input) INTEGER
              The maximum number of columns of the matrix A that
              should be factored.  NB should be at least 2 to

              allow for 2-by-2 pivot blocks.

      KB      (output) INTEGER
              The number of columns of A that were actually fac-
              tored.  KB is either NB-1 or NB, or N if N <= NB.

      A       (input/output) 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 con-
              tains 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 tri-
              angular part of A is not referenced.  On exit, A
              contains details of the partial factorization.

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

      IPIV    (output) INTEGER array, dimension (N)
              Details of the interchanges and the block structure
              of D.  If UPLO = 'U', only the last KB elements of
              IPIV are set; if UPLO = 'L', only the first KB ele-
              ments are set.

              If IPIV(k) > 0, then rows and columns k and IPIV(k)
              were interchanged and D(k,k) is a 1-by-1 diagonal
              block.  If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0,
              then rows and columns k-1 and -IPIV(k) were inter-
              changed and D(k-1:k,k-1:k) is a 2-by-2 diagonal
              block.  If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0,
              then rows and columns k+1 and -IPIV(k) were inter-
              changed and D(k:k+1,k:k+1) is a 2-by-2 diagonal
              block.

      W       (workspace) COMPLEX*16 array, dimension (LDW,NB)

      LDW     (input) INTEGER
              The leading dimension of the array W.  LDW >=
              max(1,N).

      INFO    (output) INTEGER
              = 0: successful exit
              > 0: if INFO = k, D(k,k) is exactly zero.  The fac-
              torization has been completed, but the block diago-
              nal matrix D is exactly singular.