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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.