Unsymmetric Ordering Using A Constrained Markowitz Scheme

We consider the LU factorization of unsymmetric sparse matrices using a threephase approach (analysis, factorization and triangular solution). Usually the analysis phase first determines a set of potentially good pivot and then orders this set of pivots to decrease the fill-in in the factors. In this paper, we present a preprocessing algorithm that simultaneously achieves the objectives of selecting numerically good pivots and preserving the sparsity. We describe the algorithmic properties and difficulties in implementation. By mixing the two objectives we show that we can reduce the amount of fill in the factors and reduce the number of numerical problems during factorization. On a set of large unsymmetric real problems, we obtain the average gains of 14% in the factorization time, of 12% in the size of the LU factors, and of 21% in the number of operations performed in the factorization phase.