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NAME

       SGEMM - perform one of the matrix-matrix operations   C := alpha*op( A )*op( B ) + beta*C,

SYNOPSIS

       SUBROUTINE SGEMM ( TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC )

           CHARACTER*1  TRANSA, TRANSB

           INTEGER      M, N, K, LDA, LDB, LDC

           REAL         ALPHA, BETA

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

PURPOSE

       SGEMM  performs one of the matrix-matrix operations

       where  op( X ) is one of

          op( X ) = X   or   op( X ) = X',

       alpha  and  beta  are scalars, and A, B and C are matrices, with op( A ) an m by k matrix,
       op( B )  a  k by n matrix and  C an m by n matrix.

PARAMETERS

       TRANSA - CHARACTER*1.  On entry, TRANSA specifies the form of op( A ) to be  used  in  the
       matrix multiplication as follows:

       TRANSA = 'N' or 'n',  op( A ) = A.

       TRANSA = 'T' or 't',  op( A ) = A'.

       TRANSA = 'C' or 'c',  op( A ) = A'.

       Unchanged on exit.

       TRANSB  -  CHARACTER*1.   On entry, TRANSB specifies the form of op( B ) to be used in the
       matrix multiplication as follows:

       TRANSB = 'N' or 'n',  op( B ) = B.

       TRANSB = 'T' or 't',  op( B ) = B'.

       TRANSB = 'C' or 'c',  op( B ) = B'.

       Unchanged on exit.

       M      - INTEGER.
              On entry,  M  specifies  the number  of rows  of the  matrix op( A )   and  of  the
              matrix  C.  M  must  be at least  zero.  Unchanged on exit.

       N      - INTEGER.
              On entry,  N  specifies the number  of columns of the matrix op( B ) and the number
              of columns of the matrix C. N must be at least zero.  Unchanged on exit.

       K      - INTEGER.
              On entry,  K  specifies  the number of columns of the matrix op( A ) and the number
              of rows of the matrix op( B ). K must be at least  zero.  Unchanged on exit.

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

       A      - REAL             array of DIMENSION ( LDA, ka ), where ka is
              k   when   TRANSA = 'N' or 'n',  and is  m  otherwise.  Before entry with  TRANSA =
              'N' or 'n',  the leading  m by k part of the array  A  must contain the matrix   A,
              otherwise  the  leading   k by m  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   TRANSA  =  'N'  or  'n'  then  LDA  must be at least  max( 1, m ),
              otherwise  LDA must be at least  max( 1, k ).  Unchanged on exit.

       B      - REAL             array of DIMENSION ( LDB, kb ), where kb is
              n  when  TRANSB = 'N' or 'n',  and is  k  otherwise.  Before entry with   TRANSB  =
              'N'  or 'n',  the leading  k by n part of the array  B  must contain the matrix  B,
              otherwise the leading  n by k  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  TRANSB = 'N' or 'n' then  LDB  must  be  at  least   max(  1,  k  ),
              otherwise  LDB must be at least  max( 1, n ).  Unchanged on exit.

       BETA   - REAL            .
              On  entry,  BETA  specifies the scalar  beta.  When  BETA  is supplied as zero then
              C need not be set on input.  Unchanged on exit.

       C      - REAL             array of DIMENSION ( LDC, n ).
              Before entry, the leading  m by n  part of the array  C must contain the matrix  C,
              except when  beta  is zero, in which case C need not be set on entry.  On exit, the
              array  C  is overwritten by the  m by n  matrix ( alpha*op( A )*op( B ) + beta*C ).

       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, m ).  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.