Section: New Results

Fill-in reduction in sparse matrix factorizations using hypergraphs

We discuss the use of hypergraph partitioning based methods in fill-reducing orderings of sparse matrices for Cholesky, LU and QR factorizations [33] . 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 formulate the fill-reducing ordering problem for LU factorization as we do for the Cholesky case, without ever symmetrizing the given matrix $A$ as $\left|A\right|+|A^T|$ or $|A^T\left|\right|A|$. 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 $|A^T\left|\right|A|$. We also discuss alternatives for LU and QR factorization cases where the symmetrized matrix can be used. We provide comparisons with the most common alternatives in all three cases.