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A Structural Diagnosis of Some IC Orderings

We present a novel analysis of the potential effectiveness of a matrix ordering for the incomplete Cholesky factorization (IC) in terms of just the sparsity structure. By looking at the structure of the approximate inverse implicitly created by IC we can help to explain the success of reverse Cuthill--McKee orderings, the problems IC(0) has under red-black orderings that disappear when extra fill is included, and where fill must be added to make fill-reducing orderings such as minimum degree effective.