# Incomplete Factorizations with S/P and Modified S/P Consistently Ordered M-Factors

## Summary

We report here on the first results obtained in the development of a new approximate factorization technique based on the properties of “S/P consistently ordered” M-matrices. An ordered undirected graph is called S/P consistently ordered if whenever a pair of nodes *i* < *j* has a common neighbour *k* with *k* < *i*, then either it has also a common neighbour ℓ with *j* < ℓ or neither *i* has any neighbour ℓ with ℓ > *i* nor *j* any neighbour ℓ with ℓ > *j*. Incomplete factorizations with (M-matrix) triangular factors whose graphs are S/P consistently ordered have attractive properties which have been analysed. The conclusion of our analysis is that in such cases, the submatrix determined by the nodes without successors should be factored exactly.

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