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NAME

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

SYNOPSIS

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

           CHARACTER*1  UPLO, TRANS

           INTEGER      N, K, LDA, LDC

           COMPLEX      ALPHA, BETA

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

PURPOSE

       CSYRK  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         .
              On  entry, ALPHA specifies the scalar alpha.  Unchanged on exit.

       A      - COMPLEX          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         .
              On entry, BETA specifies the scalar beta.  Unchanged on exit.

       C      - COMPLEX          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.