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4.2 Blocking

4.2 Blocking. Original Code. Transformed Code. for (jj=0; jj<N; jj = jj+B) for (kk=0; kk<N; kk = kk+B) for (i=0; i<N; i++) for (j=jj; j<min(jj+B,N); j++) {r=0; for(k=kk;j<min(kk+B,N); k++) r = r + y[i][k] * z[k][[j]; x[i][j] = x[i][j] + r; };. for (i=0; i<N; i++)

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4.2 Blocking

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  1. 4.2 Blocking Original Code Transformed Code for (jj=0; jj<N; jj = jj+B) for (kk=0; kk<N; kk = kk+B) for (i=0; i<N; i++) for (j=jj; j<min(jj+B,N); j++) {r=0; for(k=kk;j<min(kk+B,N); k++) r = r + y[i][k] * z[k][[j]; x[i][j] = x[i][j] + r; }; for (i=0; i<N; i++) for (j=0; j<N; j++) {r=0; for (k=0; k<n; k++) r = r + y[i][k] * z[k][[j]; x[i][j] = r; }; A row in a block One block in a column All blocks in a column All columns of blocks • Restructure the loops to improve • Fit in the cache • Improve temporal locality • Solutions now become machine dependent

  2. 4.2 Blocking (cont.) z[k][j] y[i][k] • What is the miss behavior? • Decompose the computation to operate on BxB blocks such that three blocks fit in the cache • Reduce the overall number of worst case misses by a factor of B Compute the partial product for this block Complete computation of all columns (jj) Compute a row in the block (j and k) Complete computation of Block (0,0) (i) Complete computation of Blocks in a column (kk)

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