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NAME

       DSYR2K  -  perform  one  of  the  symmetric  rank  2k operations   C :=
       alpha*A*B’ + alpha*B*A’ + beta*C,

SYNOPSIS

       SUBROUTINE DSYR2K( UPLO, TRANS, N, K, ALPHA, A, LDA, B, LDB,  BETA,  C,
                          LDC )

           CHARACTER*1    UPLO, TRANS

           INTEGER        N, K, LDA, LDB, LDC

           DOUBLE         PRECISION ALPHA, BETA

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

PURPOSE

       DSYR2K  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.

              TRANS = ’C’ or ’c’   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’ or ’C’ or ’c’,  K  specifies  the  number of rows of
              the matrices  A and B.  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.

       B      - DOUBLE PRECISION 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   - 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.