Performance of greedy ordering heuristics for sparse Cholesky factorization

Greedy algorithms for ordering sparse matrices for Cholesky factorization can be based on different metrics. Minimum degree, a popular and effective greedy ordering scheme, minimizes the number of nonzero entries in the rank-1 update (degree) at each step of the factorization. Alternatively, minimum deficiency minimizes the number of nonzero entries introduced (deficiency) at each step of the factorization. In this paper we develop two new heuristics: modified minimum deficiency (MMDF) and modified multiple minimum degree (MMMD). The former uses a metric similar to deficiency while the latter uses a degree-like metric. Our experiments reveal that on the average, MMDF orderings result in 21% fewer operations to factor than minimum degree; MMMD orderings result in 15% fewer operations to factor than minimum degree. MMMD requires on the average 7-13% more time than minimum degree, while MMDF requires on the average 33-34% more time than minimum degree.