Converting affine recurrence equations to quasi-uniform recurrence equations

Abstract

Most work on the problem of synthesizing a systolic array from a system of recurrence equations is restricted to systems of uniform recurrence equations. In this paper, this restriction is relaxed to include systems of affine recurrence equations. A system of uniform recurrence equations typically can be embedded in spacetime so that the distance between a variable and a dependent variable does not depend on the problem size. Systems of affine recurrence equations which are not uniform, do not enjoy this property. A method is presented for converting a system of affine recurrence equations to an equivalent system of recurrence equations that is uniform, except for points near the boundaries of its index sets. A characterization of those systems of affine recurrence equations that can be so converted is given, along with an algorithm that decides if a system is amenable to such a conversion, and a procedure that converts those affine systems which can be converted.