Systolic Arrays – Matrix-Vector Multiplication

1. Systolic Arrays – Matrix-Vector Multiplication Cathy Yen

2. Introduction • The developments in microelectronics have revolutionized computer design • Component density has been doubling every one to two years • A multiplier can fit on a very large scale integrated (VLSI) circuit chip • It is feasible to build low-cost, special-purpose, peripheral devices to solve sophisticated problems.

3. What is Systolic Arrays? • (By analogy with the regular pumping of blood by the heart) . • Systolic arrays are examples of single-instruction-multiple-data machines . • special purpose machines used mainly in dedicated equipment and not in general purpose computers. . • A systolic array is a form of parallel architecture with a regular, nearest-neighbour interconnection .

4. What do Systolic Arrays do? • data compression; [Royals] • Discrete Cosine Transforms; [Chang, Chang] • Fourier transformations; [Peng, Lippert] • genetic algorithms; [Megson] • image processing and detection systems; [Hwang , Cheng, Ker, Diamantaras, Liu, Gasteratos, Kim]

5. What do Systolic Arrays do? • matrix inversion; [Margaritis] • matrix multiplication; [Wan] • shortest-path problems; [Myoupo] • signal processing; [Li, Robert, Speiser] • tridiagonal banded matrixmultiplication; [Evans, Naritomi] and • sorting.

6. The functioning of a systolic array depends on: • the structure of the individual cells that are replicated across the array; • how the data is moved across the array (a function of how they are connected); and • how the data is feed into the array.

7. a Closer look at Systolic Arrays Processing Elements a y a x y y x x x a y a matrix-vector matrix

8. a Closer look at Systolic Arrays Processor Arrays Linearly connected Orthogonally connected Hexagonally connected

9. Matrix-Vector Multiplication (A*X = Y) a11 a12 a13 x1 y1 a21 a22 a23 x2 = y2 a31 a32 a33 x3 y3

10. Matrix-Vector Multiplication(A*X = Y) Yi(1) = 0, Ts (n)= w*n = (2n-1)*n = 2n2-n Yi(k+1) = yi(k) + aikxk O(wn) Yi = yi(n+1) a11 a12 a13 x1 y1 a21 a22 a23 x2 = y2 a31 a32 a33 x3 y3

11. Matrix-Vector Multiplication(A*X = Y) Bandwidth: w = n+n–1 = 2n-1 O(n) Tp(n) = 2n + w  2n+(2n-1)  4n-1 O(n) a11 a12 a13 x1 y1 a21 a22 a23 x2 = y2 a31 a32 a33 x3 y3

12. Matrix-Vector Multiplication(A*X = Y)

16. Matrix-Vector Multiplication(A*X = Y) Y1 = a11*x1 + a12*x2 + a13*x3 Y2 = a21*x1 + a22*x2 + a23*x3 Y3 = a31*x1 + a32*x2 + a33*x3 a11 a12 a13 x1 y1 a21 a22 a23 x2 = y2 a31 a32 a33 x3 y3

17. Matrix-Vector Multiplication(A*X = Y) • Ts (n)= w*n = (2n-1)*n = 2n2-n O(wn) • Tp(n)= 2n + w  2n+(2n-1)  4n-1 O(n)

18. Conclusion • On the positive side systolic arrays are highly compact, robust, provide high performance, and can be massively and cheaply produced as they are accessible via VLSI technology. • On the negative side the arrays are inflexible to the type of problem they solve and often are inflexible to the size of the problem they address.

19. Conclusion • Systolic arrays may be used to solve many regular problems containing repetitive operations on large arrays of data. They are not yet suited to general purpose computers.

20. References: • H.T. Kung and C.E. Leiserson, “Systolic arrays (for VLSI),” pp. 271-276 • Manohar, S.; Baudet, G.; Computers and Digital Techniques, IEE Proceedings E Volume: 137 Issue: 4 , July 1990 Page(s): 277 –282 • Milovanovic, I.Z.; Milovanovic, E.I.; Tokic, T.I.; Stojanovic, N.M.; Stojcev, M.K.; Modern Problems of Radio Engineering, Telecommunications and Computer Science, 2002. Proceedings of the International Conference , 18-23Feb. 2002 Page(s): 90 -92