NAME

       CSYR2K - perform one of the symmetric rank 2k operations   C := alpha*A*B' + alpha*B*A' + beta*C,

SYNOPSIS

       SUBROUTINE CSYR2K( UPLO, TRANS, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC )

           CHARACTER*1    UPLO, TRANS

           INTEGER        N, K, LDA, LDB, LDC

           COMPLEX        ALPHA, BETA

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

PURPOSE

       CSYR2K  performs one of the symmetric rank 2k operations

       or

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

       where  alpha and beta  are scalars,  C is an  n by n symmetric 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*B' + alpha*B*A' + beta*C.

              TRANS = 'T' or 't'    C := alpha*A'*B + alpha*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 = 'T' or 't',  K  specifies  the number of rows of the  matrices   A
              and B.  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.

       B      - COMPLEX          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   - 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.