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. Recently, researchers have begun to relax this restriction 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 that 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. Necessary and sufficient conditions are given for an affine system to be amenable to such a conversion, along with an algorithm that checks for these conditions, and a procedure that converts those affine systems which can be converted.The characterization of convertible systems brings together classical ideas in algebraic geometry, number theory, and matrix representations of groups. While the proof of this characterization is complex, the characterization itself is simple, suggesting that the mathematical ideas are well chosen for this difficult problem in array design.