An Assessment of Incomplete-LU Preconditioners for Nonsymmetric Linear Systems1

We report on an extensive experiment to compare an iterative solver preconditioned by several versions of incomplete LU factorization with a sparse direct solver using LU factorization with partial pivoting. Our test suite is 24 nonsymmetric matrices drawn from benchmark sets in the literature. On a few matrices, the best iterative method is more than 5 times as fast and more than 10 times as memory-efficient as the direct method. Nonetheless, in most cases the iterative methods are slower; in many cases they do not save memory; and in general they are less reliable. Our primary conclusion is that a direct method is currently more appropriate than an iterative method for a general-purpose black-box nonsymmetric linear solver. We draw several other conclusions about these nonsymmetric problems: pivoting is even more important for incomplete than for complete factorizations; the best iterative solutions almost always take only 8 to 16 iterations; a drop-tolerance strategy is superior to a column-count strategy; and column MMD ordering is superior to RCM ordering. The reader is advised to keep in mind that our conclusions are drawn from experiments with 24 matrices; other test suites might have given somewhat different results. Nonetheless, we are not aware of any other studies more extensive than ours.