A new approach for finding loop transformation matrices

Abstract

Traditional approach for generating loop transformation matrix, which is based upon the computation of distance vectors or direction vectors, does not work for those nested loops whose distance vectors are uncomputable and direction vectors contain no useful information. In this paper, we present a new technique for generating transformation matrix that is based upon identifying certain types of linear equations or inequalities of distance vectors. Two issues related to this technique are discussed in this paper: 1) Given a nested loop how to identify these linear equations or inequalities; 2) Given such a linear equation or inequality how to generate a legal and unimodular transformation matrix for the purpose of loop parallelization.