A singular loop transformation framework based on non-singular matrices

In this paper, we discuss a loop transformation framework that is based on integer non-singular matrices. The transformations included in this framework are called $\Lambda$-transformations and include permutation, skewing and reversal, as well as a transformation called loop scaling. This framework is more general than the existing ones; however, it is also more difficult to generate code in our framework. This paper shows how integer lattice theory can be used to generate efficient code. An added advantage of our framework over existing ones is that there is a simple completion algorithm which, given a partial transformation matrix, produces a full transformation matrix that satisfies all dependences. This completion procedure has applications in parallelization and in the generation of code for NUMA machines.