Abstract. The parallelization of many algorithms can be obtained using space-time transformations which are applied on nested do-loops or on recurrence equations. In this paper, we analyze systems of linear recurrence equations, a generalization of uniform recurrence equations. The first part of the paper describes a method for finding automatically whether such a system can be scheduled by an affine timing function, independent of the size parameter of the algorithm. In the second part, we describe a powerful method that makes it possible to transform linear recurrences into uniform recurrence equations. Both parts rely on results on integral convex polyhedra. Our results are illustrated on the Gauss elimination algorithm and on the Gauss-Jordan diagonalization algorithm.
We describe a systematic method for the design of systolic arrays. This method may be used for algorithms that can be expressed as a set of uniform recurrent equations over a convex set D of Cartesian coordinates. Most of the algorithms already considered for systolic implementation may be represented in this way. The methods consists of two steps: finding a timing-function for the computations that is compatible with the dependences introduced by the equations, then mapping the domain D onto another finite set of coordinates, each representing a processor of the systolic array, in such a way that concurrent computations are mapped onto different processors. The scheduling and mapping functions meet conditions that allow the full automation of the method. The method is exemplified on the convolution product and the matrix product.
Our results demonstrate that this new processing approach for RMP seems to be a valid tool for estimating V with sufficient accuracy during lying, sitting and standing and under various exercise conditions.
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