Fill-in reduction in sparse matrix factorizations using hypergraphs

We discuss partitioning methods using hypergraphs to produce fill-reducing orderings of sparse matrices for Cholesky, LU and QR factorizations. For the Cholesky factorization, we investigate a recent result on pattern-wise decomposition of sparse matrices, generalize the result, and develop algorithmic tools to obtain more effective ordering methods. The generalized results help us to develop fill-reducing orderings for LU factorization in a similar way to those for Cholesky factorization, without symmetrizing the given matrix A as |A| + |A | or |A ||A|. For the QR factorization, we adopt a recently proposed technique to use hypergraph models in a fairly standard manner. The method again does not form the possibly much denser matrix |A ||A|. We also discuss alternatives for LU and QR factorization where the symmetrized matrix can be used. We provide comparisons with the most common alternatives in all three cases.