E. I. Milovanović, M. P. Bekakos, I. Ž. Milovanović, T. Z. Mirković. 1. Introduction

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1 FACTA UNIVERSITATIS (NIŠ) Ser. Math. Inform. (006) 87 9 SYSTOLIC MATRIX VECTOR ITERATIONS E. I. Milovanović M. P. Bekakos I. Ž. Milovanović T. Z. Mirković Abstract. In this paper we consider computation of iterative process of the form x (t) A x (t ) where A (a ik ) is n n dense matrix on unidirectional linear sstolic arra.. Introduction We consider the sstolic implementation of iterative method defined b (.) x (t) A x (t ) t... m where A (a ik ) is a dense square matrix of order n n x (0) [x (0) x (0)... x (0) n is a given initial vector and m. The main building block for the sstolic implementation of matrix vector iterations is a sstolic arra for matrix vector multiplication. The computation task can be expanded either in space leading to series of cascaded linear sstolic arras performing pipelined iterations or in time leading to an iterative sstolic structure with feedback control [. A combination of the two approaches is possible also. In this paper we consider implementation of matrix vector iterations on linear unidirectional sstolic arra (ULSA) for matrix vector multiplication Received Januar Mathematics Subject Classification. 68M07 68Q35. The authors were supported in part b the Serbian Ministr of Science and Environmental Protection (Project: #44034: Parallel Methods and Algorithms in Discrete Mathematics). T 87

2 88 E. I. Milovanović M. P. Bekakos I. Ž. Milovanović T. Z. Mirković with feedback connection. We give an explicit formulas to design ULSA which is space optimal i.e. has the optimal number od processing elements (PE) for a given matrix dimension and minimal execution time for such number of PEs. The iteration process (.) is implemented on the ULSA for matrix vector multiplication in m iterations such that x (t ) represents input for the computation of x (t) for t... m. Data schedule is defined such that input time of t-th iteration is overlapped with computation of (t )-st iteration for t 3... m reducing the overall computation time. The efficienc of the proposed method is considered also. The problem of computing matrix vector iterations on the BLSA (bidirectional linear SA) was considered in [ and on the ULSA in [. However explicit formulas for the snthesis of these arras are not given. In both cases matrix A was a band matrix with band width less than the dimension of matrix which enables obtaining space optimal arras. If width of matrix band is greater than matrix dimension or full matrix is considered than obtained arras are not space optimal.. Implementation of Matrix Vector Iterations on the ULSA The space optimal ULSA that computes product of matrix A (a ik ) of order n n and vector b [b b... b n T was designed in [3. It consists of Ω n PEs and computes the matrix vector product c A b for time T tot 3n. An example of this arra for the case n 4 is given in Fig.. Fig. : Data flow in the ULSA snthesized in [3 In order to compute matrix vector iterations (.) on the ULSA from [3 efficientl we need to reorder data schedule. Namel elements of vectors

3 Sstolic Matrix Vector Iterations 89 c and b enter the ULSA in mutuall reverse order. Since in the iterative process elements of resulting vector x (t) form t-th iteration will appear as inputs x (t ) in the next iteration it is desirable that both x (t) and x (t ) enter/leave the arra in the same order. If opposite we need to reorder resulting elements x (t) before each new iteration. Second concerns the position of dela elements in the ULSA. Namel there are two alternatives. Either to first dela elements of b and then use it in the computation or use it first then dela. The second solution is better since it will shorten the total execution time for m time units which in the case m is not negligible. In order to obtain equal data schedule for x (t) and x (t ) i.e. c and b we will use matrix Î of order n n defined b Î Matrix Î has some interesting properties which we will use to obtain desired data schedule. First Î Î I. Second for each vector b [b b... b n T and matrix A (a ik ) of order n n the following equalities are valid: b Î b [b n b n... b T and A A Î (a in k+). In other words vector b and matrix A are mirror smmetric to b and A respectivel. According to this properties we have that (.) Now denote with P E c A b (A Î) (Î b). d γ( ) γ {a b c} (x ) coordinates of the PEs dela elements and data elements respectivel. According to the above properties we have [ [ x k + P E d [ k + 3/

4 90 E. I. Milovanović M. P. Bekakos I. Ž. Milovanović T. Z. Mirković [ (.) a(i 0 n k + ) b(0 n k + i) c(i 0) [ i k + i k [ (3 + k i)/ for i... n k... n and a(i 0 t + n) a(i 0 t) a it b(0 t + n) b(0 t) b t. The communication links in the ULSA are implemented along the propagation vectors [ [ e b e a e c / 0 (.3). 0 0 Data schedule in the ULSA snthesized according to (.) and (.3) at the beginning of the computation for the case n 4 is depicted in Fig.. Fig. : Data flow in the ULSA snthesized according to (.) and (.3) for n 4 Table shows the complete step b step diagram for computing matrix vector iterations on the ULSA for the case n 3 and m 3.

5 Sstolic Matrix Vector Iterations 9 Clk PE d PE d PE3 d 0 : x (0) 0 : x (0) x (0) 3 x () : 0 + a 3 x (0) 3 x (0) 0 : x (0) 4 x () : 0 + a x (0) x (0) 3 x () : x () + a x (0) x (0) 5 x () 3 : 0 + a 3 x (0) x (0) x () : x () + a 3 x (0) 3 x (0) x () : x () + a x (0) 6 0 : x () x (0) x () 3 : x () 3 + a 3 x (0) x (0) 3 x () : x () + a x (0) x (0) 7 0 : x () x () 0 : x (0) x (0) x (0) 3 : x () 3 + a 33 x (0) 3 x (0) 8 x () : 0 + a 3 x () 3 x () 0 : x () x (0) 0 : x (0) x (0) 3 9 x () : 0 + a x () x () 3 x () : x () + a x () x () 0 : x (0) x (0) 0 x () 3 : 0 + a 3 x () x () x () : x () + a 3 x () 3 x () : x () + a x () x (0) 0 : x () x () 3 : x () 3 + a 3 x () x () 3 x () : x () + a x () x () 0 : x () 0 : x () x () x () 3 : x () 3 + a 33 x () 3 x () : 0 + a 3 x () 3 x () 0 : x () x () 0 : x () x () 3 4 x (3) : 0 + a x () : x (3) + a 0 : x () x () 5 x (3) 3 : 0 + a 3 x (3) : x (3) + a 3 x () 3 x () : x (3) + a x () x () 6 0 : x (3) x () 3 : x (3) 3 + a 3 x () x () : x (3) + a 7 0 : x (3) 0 : x () x (3) 3 : x (3) 3 + a 33 x () 3 x () 8 x (4) : 0 + a 0 : x (3) x () 0 : x () x () 3 9 x (4) : 0 + a x (3) 3 x (4) : x (4) + a 0 : x () 0 x (4) 3 : 0 + a 3 x (4) : x (4) + a x (4) : x (4) + a x (3) x () 0 : x (3) x (4) 3 : x (4) 3 + a x (3) 3 x (4) : x (4) + a 0 : : x (3) x (4) 3 : x (4) 3 + a 3 Table : Step b step diagram for computing matrix vector iteration x (t) Ax (t ) t... m on the ULSA for the case n 3 and m 4 3. Performance Analsis According to (.) we conclude that ULSA consists of Ω n PEs which is optimal number for a given dimension of matrix A. Computation time for one iteration on the ULSA is t tot t in +t exe +t out 3n where t in n is the initialization time t exe n is active computation time and t out 0 is output time. The important propert of the proposed data schedule is that initialization time of the t-th iteration (i.e. x (t) ) is overlapped with active computation time of the (t )-st iteration (i.e. x (t ) ) for t... m. Having this in mind the time needed to compute x (m) on the ULSA is T tot t in + (m )t exe m(n ) + n (m + )n m. The efficienc of the arra depends on the relation between number of iterations m and matrix dimension n. The values of the efficienc for various m and n are summarized in Table. From Table we conclude that for some fixed m the efficienc decreases as n increases. However reducing of the efficienc is not a substantial and all values are grouped around 0.5 even for relativel small n (n 50) and m 0. Therefore it is convenient to consider a realization of the matrix vector iterations (.) on the fixed

6 9 E. I. Milovanović M. P. Bekakos I. Ž. Milovanović T. Z. Mirković size ULSA. If n is fixed on some constant value the efficienc increases as m increases. The efficienc is greater than 0.5 for all m n. However for n 0 the efficienc is around 0.5 regardless to the number of iterations m. mn n m Table : Efficienc of the ULSA for various m and n R E F E R E N C E S. K. G. Margaritis D. J. Evans: Folding techniques for sstolic iterations. Parallel Algorithms Appl. 7 (995) D. J. Evans K. G. Margaritis: Sstolic designs for eigenvalueeigenvector computations using matrix powers. Parallel Comput. 4 (990) I. Ž. Milovanović E. I. Milovanović M. P. Bekakos: Snthesis of a unidirectional sstolic arra for matrix-vector multiplication. Math. Comput. Modelling (006) Facult of Electronic Engineering Department of Computer Science P.O. Box Niš Serbia ema@elfak.ni.ac.u (E.I. Milovanović) Facult of Electronic Engineering Department of Mathematics P.O. Box Niš Serbia igor@elfak.ni.ac.u (I.Ž. Milovanović) Facult of Science and Mathematics Department of Mathematics 380 Kosovska Mitrovica Serbia