A basis approach to loop parallelization and synchronization

Loop transformation is a crucial step in parallelizing compilers. We introduce the concept of positive coordinate basis for deriving loop transformations. The basis serves to find proper loop transformations to change the dependence vectors into the desired forms. We demonstrate how this approach can, systematically extract maximal outer loop parallelism. Based on the concept, we can also construct a minimal set of synchronization vectors, which are deadlock free, to transform the inner serial loops into doacross loops.