Performance of Greedy Ordering Heuristics for Sparse Cholesky Factorization

Greedy algorithms for ordering sparse matrices for Cholesky factorization can be based on diierent metrics. Minimum degree, a popular and eeective greedy ordering scheme, minimizes the number of nonzero entries in the rank-1 update (degree) at each step of the factorization. Alternatively, minimum deeciency minimizes the number of nonzero entries introduced (deeciency) at each step of the factorization. In this paper we develop two new heuristics: \modiied minimum deeciency" (MMDF) and \modiied multiple minimum degree" (MMMD). The former uses a metric similar to deeciency while the latter uses a degree-like metric. Our experiments reveal that on the average MMDF has 21% fewer operations to factor than minimum degree; MMMD has 15% fewer operations to factor than minimum degree. MMMD is no more expensive to compute than minimum degree while MMDF requires on the average 30% more time than minimum degree.