Thinking about Systolic Arrays

1. 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. a31 x1 a32 x2 a33 x3 y3 a21 x1 a22 x2 a23 x3 y2 a11 x1 a12 x2 a13 x3 y1 x1 x2 x3

4. a31 x1 a32 x2 a33 x3 SPACE a21 x1 a22 x2 a23 x3 a11 x1 a12 x2 a13 x3 TIME

5. x1 x2 x3 a31 a33 a32 x1 x2 x3 a22 a21 a23 x1 x2 x3 a11 a12 a13 SPACE TIME

6. x1 x2 x3 a31 a32 a33 SPACE a21 x1 a22 x2 a23 x3 a11 x1 a12 x2 a13 x3 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. S a21 b12 a22 b22 a21 b11 a22 b21 a11 b12 a12 b22 a11 b11 a12 b21 S T

9. S a21 b12 a22 b22 a21 b11 a22 b21 a11 b12 a12 b22 a11 b11 a12 b21 S T

10. A 2 X 2 X 2 Mesh ADG (!DAC) S a21 b12 a22 b22 a21 b11 a22 b21 a11 b12 a12 b22 a11 b11 a12 b21 S T