NAME

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

SYNOPSIS

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

           CHARACTER*1  UPLO, TRANS

           INTEGER      N, K, LDA, LDC

           DOUBLE       PRECISION ALPHA, BETA

           DOUBLE       PRECISION A( LDA, * ), C( LDC, * )

PURPOSE

       DSYRK  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.

              TRANS = 'C' or 'c'   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' or 'C' or 'c',  K
              specifies  the  number of rows of  the  matrix   A.   K  must  be  at  least  zero.
              Unchanged on exit.

       ALPHA  - DOUBLE PRECISION.
              On entry, ALPHA specifies the scalar alpha.  Unchanged on exit.

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

       C      - DOUBLE PRECISION 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.