An Algorithm to Automate Non-Unimodular Transformations of Loop Nests

This paper provides a solution to the open problem of automatic rewriting loop nests for non-unimodular transformations. We present an algorithm that rewrites a loop nest under any non-singular (unimod-ular or non-unimodular) transformation. The algorithm works nicely with unimodular transformations being treated as a special case. The rst step of the algorithm calculates the loop bounds using the Fourier-Motzkin elimination method. The second step relies on a method based on the theory of Hermite normal form and lattice and is only needed for non-unimodular transformations. It consists of calculating the loop strides and adjusting the loop bounds derived in the rst step so that the \holes" in the image iteration space are skipped. The time complexity of the second step is polynomial, which is the time complexity of calculating the Hermite normal form of an nn matrix, where n is the depth of the loop nest. The adjusted loop bounds are the simplest that can be expected.