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

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

     CHARACTER UPLO

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

     INTEGER IPIV( * )

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



     #include <sunperf.h>

     void clahef(char uplo, int n, int nb, int *kb, complex  *ca,
               int  lda,  int  *ipivot,  complex *w, int ldw, int
               *info);

PURPOSE
     CLAHEF 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.

     CLAHEF is an auxiliary routine called  by  CHETRF.  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 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 tri-
               angular part of the matrix  A,  and  the  strictly
               upper  triangular part of A is not referenced.  On
               exit, A contains details of the partial factoriza-
               tion.

     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  struc-
               ture  of  D.  If UPLO = 'U', only the last KB ele-
               ments of IPIV are set; if UPLO  =  'L',  only  the
               first KB elements 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 interchanged 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 interchanged and D(k:k+1,k:k+1) is a
               2-by-2 diagonal block.

     W         (workspace) COMPLEX 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
               factorization  has  been  completed, but the block
               diagonal matrix D is exactly singular.

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