Unifying and optimizing parallel linear algebra algorithms

Two issues in linear algebra algorithms for multicomputers are addressed. First, how to unify parallel implementations of the same algorithm in a decomposition-independent way. Second, how to optimize naive parallel programs maintaining the decomposition independence. Several matrix decompositions are viewed as instances of a more general allocation function called subcube matrix decomposition. By this meta-decomposition, a programming environment characterized by general primitives that allow one to design meta-algorithms independently of a particular decomposition. The authors apply such a framework to the parallel solution of dense matrices. This demonstrates that most of the existing algorithms can be derived by suitably setting the primitives used in the meta-algorithm. A further application of this programming style concerns the optimization of parallel algorithms. The idea to overlap communication and computation has been extended from 1-D decompositions to 2-D decompositions. Thus, a first attempt towards a decomposition-independent definition of such optimization strategies is provided