ZSYRK - perform one of the symmetric rank k operations   C
       := alpha*A*A' + beta*C,


SYNOPSIS

       SUBROUTINE ZSYRK ( UPLO, TRANS, N, K, ALPHA, A, LDA, BETA,
                        C, LDC )

           CHARACTER*1  UPLO, TRANS

           INTEGER      N, K, LDA, LDC

           COMPLEX*16   ALPHA, BETA

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


PURPOSE

       ZSYRK  performs one of the symmetric rank k operations

       or

          C := alpha*A'*A + beta*C,

       where   alpha and beta  are scalars,  C is an  n by n sym­
       metric matrix and  A  is an  n by k  matrix in  the  first
       case and a  k by n  matrix 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*A' + beta*C.

              TRANS = 'T' or 't'   C := alpha*A'*A + beta*C.

              Unchanged on exit.

       N      - INTEGER.
              On entry,  N specifies the order of the  matrix  C.


       K      - INTEGER.
              On  entry  with   TRANS = 'N' or 'n',  K  specifies
              the number of   columns    of   the    matrix    A,
              and   on    entry    with  TRANS  =  'T' or 't',  K
              specifies  the number of rows of the matrix  A.   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  lead­
              ing  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.

       BETA   - COMPLEX*16      .
              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   symmet­
              ric  matrix  and the strictly lower triangular part
              of C is not referenced.  On exit, the upper  trian­
              gular  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  symmet­
              ric matrix  and the strictly upper triangular  part
              of  C is not referenced.  On exit, the lower trian­
              gular part of the array  C is  overwritten  by  the
              lower triangular part of the updated matrix.

       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.


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


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