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.

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Abstract. Several recent papers demonstrate the interest of viewing systolic algorithms as while-programs whose statements are synchronous multiple assignments. This approach is based on the classical invariant method and compares favourably with earlier ones, based on recurrence systems and space-time transformations. Our purpose is to use the particularities of the systolic paradigm to reduce the creativity needed to develop a systolic algorithm and its invariant. More precisely, two points are taken into account. First, the architecture is often chosen before the real beginning of the development and, second, the basic operations to be executed by individual cells are also partially known at the beginning. In fact, the development does not start from scratch, but from a "generic systolic array" (gsa), whose parameters have to be instantiated. Most systolic arrays are instances of a simple gsa that is introduced, investigated and illustrated in this paper.

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