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dsyr2k

 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  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 per-
             formed 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.

 ka is
      A      -
              DOUBLE PRECISION array of DIMENSION ( LDA, ka ), where
             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.

 kb is
      B      -
              DOUBLE PRECISION array of DIMENSION ( LDB, kb ), where
             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 con-
             tain 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 tri-
             angular 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 refer-
             enced.  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.