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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.