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NAME

       ZHER2K  -  perform  one  of  the hermitian rank 2k operations   C := alpha*A*conjg( B' ) +
       conjg( alpha )*B*conjg( A' ) + beta*C,

SYNOPSIS

       SUBROUTINE ZHER2K( UPLO, TRANS, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC )

           CHARACTER*1    UPLO, TRANS

           INTEGER        N, K, LDA, LDB, LDC

           DOUBLE         PRECISION BETA

           COMPLEX*16     ALPHA

           COMPLEX*16     A( LDA, * ), B( LDB, * ), C( LDC, * )

PURPOSE

       ZHER2K  performs one of the hermitian rank 2k operations

       or

          C := alpha*conjg( A' )*B + conjg( alpha )*conjg( B' )*A + beta*C,

       where  alpha and beta  are scalars with  beta  real,  C is an  n by n hermitian matrix and
       A and B  are  n by k matrices in the first case and  k by n  matrices in the second case.

PARAMETERS

       UPLO   - CHARACTER*1.
              On   entry,    UPLO  specifies  whether  the  upper  or  lower triangular  part  of
              the  array  C  is to be  referenced  as follows:

              UPLO = 'U' or 'u'   Only the  upper triangular part of  C is to be referenced.

              UPLO = 'L' or 'l'   Only the  lower triangular part of  C is to be referenced.

              Unchanged on exit.

       TRANS  - CHARACTER*1.
              On entry,  TRANS  specifies the operation to be performed as follows:

              TRANS = 'N' or 'n'    C := alpha*A*conjg( B' )          + conjg(  alpha  )*B*conjg(
              A' ) + beta*C.

              TRANS  = 'C' or 'c'    C := alpha*conjg( A' )*B          + conjg( alpha )*conjg( B'
              )*A + beta*C.

              Unchanged on exit.

       N      - INTEGER.
              On entry,  N specifies the order of the  matrix  C.   N  must  be  at  least  zero.
              Unchanged on exit.

       K      - INTEGER.
              On  entry  with   TRANS = 'N' or 'n',  K  specifies  the number of  columns  of the
              matrices  A and B,  and on  entry  with TRANS = 'C'  or  'c',   K   specifies   the
              number  of  rows  of the matrices  A and B.  K must be at least zero.  Unchanged on
              exit.

       ALPHA  - COMPLEX*16      .
              On entry, ALPHA specifies the scalar alpha.  Unchanged on exit.

       A      - COMPLEX*16       array of DIMENSION ( LDA, ka ), where ka is
              k  when  TRANS = 'N' or 'n',  and is  n  otherwise.  Before entry with  TRANS = 'N'
              or  'n',   the   leading   n by k part of the array  A  must contain the matrix  A,
              otherwise the leading  k by n  part of the array  A  must contain   the  matrix  A.
              Unchanged on exit.

       LDA    - INTEGER.
              On  entry,  LDA  specifies  the  first  dimension of A as declared in  the  calling
              (sub)  program.   When  TRANS = 'N' or 'n' then  LDA must be at least  max( 1, n ),
              otherwise  LDA must be at least  max( 1, k ).  Unchanged on exit.

       B      - COMPLEX*16       array of DIMENSION ( LDB, kb ), where kb is
              k  when  TRANS = 'N' or 'n',  and is  n  otherwise.  Before entry with  TRANS = 'N'
              or 'n',  the  leading  n by k part of the array  B  must  contain  the  matrix   B,
              otherwise  the  leading   k by n  part of the array  B  must contain  the matrix B.
              Unchanged on exit.

       LDB    - INTEGER.
              On entry, LDB specifies the first dimension of  B  as  declared  in   the   calling
              (sub)  program.   When  TRANS = 'N' or 'n' then  LDB must be at least  max( 1, n ),
              otherwise  LDB must be at least  max( 1, k ).  Unchanged on exit.

       BETA   - DOUBLE PRECISION.
              On entry, BETA specifies the scalar beta.  Unchanged on exit.

       C      - COMPLEX*16       array of DIMENSION ( LDC, n ).
              Before entry  with  UPLO = 'U' or 'u',  the leading  n by n upper  triangular  part
              of  the  array  C  must contain the upper triangular part  of the  hermitian matrix
              and the strictly lower triangular part of C is not referenced.  On exit, the  upper
              triangular  part of the array  C is overwritten by the upper triangular part of the
              updated matrix.  Before entry  with  UPLO = 'L' or 'l',  the leading  n by n  lower
              triangular  part  of  the  array  C  must contain the lower triangular part  of the
              hermitian matrix  and the strictly upper triangular part of C  is  not  referenced.
              On  exit,  the  lower  triangular  part of the array  C is overwritten by the lower
              triangular part of the updated matrix.   Note  that  the  imaginary  parts  of  the
              diagonal  elements need not be set,  they are assumed to be zero,  and on exit they
              are set to zero.

       LDC    - INTEGER.
              On entry, LDC specifies the first dimension of  C  as  declared  in   the   calling
              (sub)  program.   LDC  must  be  at  least max( 1, n ).  Unchanged on exit.

              Level 3 Blas routine.

              --  Written  on 8-February-1989.  Jack Dongarra, Argonne National Laboratory.  Iain
              Duff, AERE  Harwell.   Jeremy  Du  Croz,  Numerical  Algorithms  Group  Ltd.   Sven
              Hammarling, Numerical Algorithms Group Ltd.