Loop transformation using nonunimodular matrices

Linear transformations are widely used to vectorize and parallelize loops. A subset of these transformations are unimodular transformations. When a unimodular transformation is used, the exact bounds of the transformed loop nest are easily computed and the steps of the loops are equal to 1. Unimodular loop transformations have been widely used since they permit the implementation of many useful loop transformations. Recently, nonunimodular transformations have been proposed to reduce communication requirements or to use the memory hierarchy efficiently. The methods used for unimodular transformations do not work in the case of nonunimodular transformations, since they do not produce the exact bounds of the transformed loop nest. In this paper, we present a method for nested loop transformation which gives the exact bounds for both unimodular and nonunimodular transformations. The basic idea is to use the Hermite Normal Form (HNF) of the transformation matrix