on Linear Cache Optimization & Vectorization 15 - 411 : Compiler Design André Platzer

We have seen a number of loop transformations, but they all have been different, needing different analysis and implementation. However, a closer look reveals that the previous list of loop transformations (permutation, reversal, skewing) all follow a general pattern of linear loop transformations. Each of those transformations (and combinations and many others) can be represented by unimodular linear transformations. That is, such a transformation on n loops corresponds to an n × n integer matrix U ∈ Zn×n with determinant detU = ±1. Because of the unit determinant detU , they actually form a group, because we can form inverses