A Direct Approach for Finding Loop Transformation Matrices

Loop transformations, such as loop interchange, reversal and skewing, have been unified under linear matrix transformations. A legal transformation matrix is usually generated based upon distance vectors or direction vectors. Unfortunately, for some nested loops, distance vectors may not be computable and direction vectors, on the other hand, may not contain useful information. We propose the use of linear equations or inequalities of distance vectors to approximate data dependence. This approach is advanta…