Figure 8 Fic. 9.1. Graph of the performance results of the two algorithms on PARAGON.
Related Figures (9)
LEMMA 3.2. By assuming that the vector x and the matrices A® are correctly defined, the computation Therefore, the recursion step s is obtained from the expressions 4.1-4.4 and is ex- pressed by : 6.1.2. Derivation from algebraic expression. From Proposition 5.3, the ‘omputation of V‘*) from V(*t+) can be naturally expressed as follows: 6.2. Global algorithm. As we have previously said in Section 5, we consider the ordering s = N, N—1,...,1 for our recursions. In addition, because the set of the embedded loops of any step can be considered in any order, we obtain the following global algorithm by chosing an appropriate ordering which allows a maximum use of each entry of the current matrix. a A This algorithm computes the multiplication in an efficient time cost and reason- able space memory. We are now ready to study the parallel issue of our problem. Because, in the context of distributed memory machines, communication cost may often cause severe inefficiency, we shall first consider a parallel approach in which there will be no need of communication between processors. This is also justified by the increasing speed of actual processors. From the fact that T(p1) = T (po) when consider g = 1 in T(p;), the result is obtained by reminding that q > 1. This ends the proof of the theorem. 0 Theoretical speedups for the algorithm involving communications proved, at step s each processor I sends a vector of length Mes" to each of the n,—1—1 processors in I'(/). By considering the ideal case for physical communication network, it appears that the communication cost is given by
