Optimization_1x4_5

Copy the contents of file MMult_1x4_4.c into a file named MMult_1x4_5.c and change the contents:

from

/* Create macros so that the matrices are stored in column-major order */

#define A(i,j) a[ (j)*lda + (i) ]
#define B(i,j) b[ (j)*ldb + (i) ]
#define C(i,j) c[ (j)*ldc + (i) ]

/* Routine for computing C = A * B + C */

void AddDot1x4( int, double *, int,  double *, int, double *, int )

void MY_MMult( int m, int n, int k, double *a, int lda, 
                                    double *b, int ldb,
                                    double *c, int ldc )
{
  int i, j;

  for ( j=0; j<n; j+=4 ){        /* Loop over the columns of C, unrolled by 4 */
    for ( i=0; i<m; i+=1 ){        /* Loop over the rows of C */
      /* Update C( i,j ), C( i,j+1 ), C( i,j+2 ), and C( i,j+3 ) in
	 one routine (four inner products) */

      AddDot1x4( k, &A( i,0 ), lda, &B( 0,j ), ldb, &C( i,j ), ldc );
    }
  }
}


void AddDot1x4( int k, double *a, int lda,  double *b, int ldb, double *c, int ldc )
{
  /* So, this routine computes four elements of C: 

           C( 0, 0 ), C( 0, 1 ), C( 0, 2 ), C( 0, 3 ).  

     Notice that this routine is called with c = C( i, j ) in the
     previous routine, so these are actually the elements 

           C( i, j ), C( i, j+1 ), C( i, j+2 ), C( i, j+3 ) 
	  
     in the original matrix C.

     In this version, we "inline" AddDot */ 

  int p;

  //  AddDot( k, &A( 0, 0 ), lda, &B( 0, 0 ), &C( 0, 0 ) );
  for ( p=0; p<k; p++ ){
    C( 0, 0 ) += A( 0, p ) * B( p, 0 );     
  }

  //  AddDot( k, &A( 0, 0 ), lda, &B( 0, 1 ), &C( 0, 1 ) );
  for ( p=0; p<k; p++ ){
    C( 0, 1 ) += A( 0, p ) * B( p, 1 );     
  }

  //  AddDot( k, &A( 0, 0 ), lda, &B( 0, 2 ), &C( 0, 2 ) );
  for ( p=0; p<k; p++ ){
    C( 0, 2 ) += A( 0, p ) * B( p, 2 );     
  }

  //  AddDot( k, &A( 0, 0 ), lda, &B( 0, 3 ), &C( 0, 3 ) );
  for ( p=0; p<k; p++ ){
    C( 0, 3 ) += A( 0, p ) * B( p, 3 );     
  }
}

to

/* Create macros so that the matrices are stored in column-major order */

#define A(i,j) a[ (j)*lda + (i) ]
#define B(i,j) b[ (j)*ldb + (i) ]
#define C(i,j) c[ (j)*ldc + (i) ]

/* Routine for computing C = A * B + C */

void AddDot1x4( int, double *, int,  double *, int, double *, int )

void MY_MMult( int m, int n, int k, double *a, int lda, 
                                    double *b, int ldb,
                                    double *c, int ldc )
{
  int i, j;

  for ( j=0; j<n; j+=4 ){        /* Loop over the columns of C, unrolled by 4 */
    for ( i=0; i<m; i+=1 ){        /* Loop over the rows of C */
      /* Update C( i,j ), C( i,j+1 ), C( i,j+2 ), and C( i,j+3 ) in
	 one routine (four inner products) */

      AddDot1x4( k, &A( i,0 ), lda, &B( 0,j ), ldb, &C( i,j ), ldc );
    }
  }
}


void AddDot1x4( int k, double *a, int lda,  double *b, int ldb, double *c, int ldc )
{
  /* So, this routine computes four elements of C: 

           C( 0, 0 ), C( 0, 1 ), C( 0, 2 ), C( 0, 3 ).  

     Notice that this routine is called with c = C( i, j ) in the
     previous routine, so these are actually the elements 

           C( i, j ), C( i, j+1 ), C( i, j+2 ), C( i, j+3 ) 
	  
     in the original matrix C.

     In this version, we accumulate in registers and put A( 0, p ) in a register */

  int p;
  register double 
    /* hold contributions to
       C( 0, 0 ), C( 0, 1 ), C( 0, 2 ), C( 0, 3 ) */
       c_00_reg,   c_01_reg,   c_02_reg,   c_03_reg,  
    /* holds A( 0, p ) */
       a_0p_reg;
    
  c_00_reg = 0.0; 
  c_01_reg = 0.0; 
  c_02_reg = 0.0; 
  c_03_reg = 0.0;
 
  for ( p=0; p<k; p++ ){
    a_0p_reg = A( 0, p );

    c_00_reg += a_0p_reg * B( p, 0 );     
    c_01_reg += a_0p_reg * B( p, 1 );     
    c_02_reg += a_0p_reg * B( p, 2 );     
    c_03_reg += a_0p_reg * B( p, 3 );     
  }

  C( 0, 0 ) += c_00_reg; 
  C( 0, 1 ) += c_01_reg; 
  C( 0, 2 ) += c_02_reg; 
  C( 0, 3 ) += c_03_reg;
}

Change the first lines in the makefile to

OLD  := MMult_1x4_4
NEW  := MMult_1x4_5     

• make run

octave:3> PlotAll        % this will create the plot

This time the performance graph will look something like

Now we start seeing a performance benefit. The reason is that the four loops have been fused and therefore the four inner products are now being performed simultaneously. This has the following benefits:

• The index p needs only be updated once every eight floating point operations.

• Element A( 0, p ) needs only be brought in from memory once instead of four times. (This only becomes a benefit when the matrices no longer fit in the L2 cache.)