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   sym­
       metric  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 +


              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   lead­
              ing   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  lead­
              ing  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.


       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   symmet­
              ric  matrix  and the strictly lower triangular part
              of C is not referenced.  On exit, the upper  trian­
              gular  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  symmet­
              ric matrix  and the strictly upper triangular  part
              of  C is not referenced.  On exit, the lower trian­
              gular 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 Har­
              well.  Jeremy Du Croz, Numerical  Algorithms  Group
              Ltd.   Sven  Hammarling, Numerical Algorithms Group
              Ltd.


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