Lec14-Cache_measurement

Loop k kj j a b c columnwise fixed same

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Unformatted text preview: phen Chong, Harvard University B = 0.25 C = 0.25 Total: 0.5 27 Matrix Multiplication (jki) /* jki */ for (j=0; j<n; j++) { for (k=0; k<n; k++) { r = b[k][j]; for (i=0; i<n; i++) c[i][j] += a[i][k] * r; } } Inner loop: (*,k) (k,j) (*,j) A B C Columnwise Fixed Columnwise • 2 load, 1 store per iteration • Assume cache line size of 32 bytes, so 4 doubles per line • Misses per iteration: A=1 Stephen Chong, Harvard University B=0 C=1 Total: 2 28 Matrix Multiplication (kji) /* kji */ for (k=0; k<n; k++) { for (j=0; j<n; j++) { r = b[k][j]; for (i=0; i<n; i++) c[i][j] += a[i][k] * r; } } Inner loop: (*,k) (k,j) (*,j) A B C Columnwise Fixed • Same as kji, just swapped order of outer loops Columnwise • 2 load, 1 store per iteration • Assume cache line size of 32 bytes, so 4 doubles per line • Misses per iteration: A=1 Stephen Chong, Harvard University B=0 C=1 Total: 2 29 Summary of Matrix Multiplication for (i=0; i<n; i++) { for (j=0; j<n; j++) { sum = 0.0; for (k=0; k<n; k++) sum += a[i][k] * b[k][j]; c[i][j] = sum; } ijk or jik: 2 loads, 0 stores misses/iter = 1.25 } for (k=0; k<n; k++) { for (i=0; i<n; i++) { r = a[i][k]; for (j=0; j<n; j++) c[i][j] += r * b[k][j]; } } for (j=0; j<n; j++) { for (k=0; k<n; k++) { r = b[k][j]; for (i=0; i<n; i++) c[i][j] += a[i][k] * r; } } Stephen Chong, Harvard University kij or ikj: 2 loads, 1 store misses/iter = 0.5 jki or kji: 2 loads, 1 store misses/iter = 2.0 30 Matrix Multiply Performance 100 75 50 1250 1150 1050 950 850 n 750 650 550 450 350 250 0 150 25 50 ijk jik jki kji kij ikj Cycles per loop iteration 125 • Each implementation doing same number of arithmetic operations, but ~20× difference! • Pairs with same number of mem. references and misses per iteration almost identical Stephen Chong, Harvard University 31 Matrix Multiply Performance jki and kji: 2 loads, 1 store misses/iter = 2.0 100 75 ijk and jik: 2 loads, 0 stores misses/iter = 1.25 50 1250 1150 1050 950 850 n 750 650 550 450 350 250 0 150 25 50 ijk jik jki kji kij ikj Cycles per loop iteration 125 kij and ikj: 2 loads, 1 store misses/iter = 0.5 • Miss rate better predictor o...
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