NAME

       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 symmetric 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.  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
              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   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.

       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   symmetric  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
              symmetric  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.

       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.