Thinking about Systolic Arrays Peter Cappello This presentation assumes that you have read the Chapter by C. Leiserson and H.-T. Kung

2 Matrix-Vector Product b = Ax, illustrated with a 3X3 matrix, A. _______________________________ b1 = a11*x1 + a12*x2 + a13*x3 b2 = a21*x1 + a22*x2 + a23*x3 b3 = a31*x1 + a32*x2 + a33*x3

3 a31a32a33 a21a22a23 a11a12a13 x1x2x3 x1 x2 x3 y1 y2 y3 x1x2x3

4 a31a32a33 a21a22a23 a11a12a13 x1x2x3 x1 x2 x3 TIME SPACESPACE

5 a31 a32 a33 a2 1 a2 2 a2 3 a1 1 a1 2 a1 3 x1x1 x2x2 x3x3 x1x1 x1x1 x2x2 x2x2 x3x3 x3x3 SPACESPACE TIME

6 a31a32a33 a21a22a23 a11a12a13 x1x2x3 x1 x2 x3 SPACESPACE TIME

7 Matrix Product C = AB, illustrated with a 2X2 matrices. c11 = a11*b11 + a12*b21 c12 = a11*b12 + a12*b22 c21 = a21*b11 + a22*b21 c12 = a21*b12 + a22*b22

8 a21a22 a11a12 b11 b21 T S a21a22 a11a12b12 b21 b12 b22 S

9 a21a22 a11a12 b11 b21 T S a21a22 a11a12b12 b21 b12 b22 S

10 a21a22 a11a12 b11 b21 T S a21a22 a11a12b12 b21 b12 b22 S A 2 X 2 X 2 Mesh ADG (!DAC)