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Incomplete factorizations can be given in various forms. If (with and nonsingular triangular matrices), solving a system proceeds in the usual way (figure ),
Figure: Preconditioner solve of a system , with
but often incomplete factorizations are given as (with diagonal, and and now strictly triangular matrices, determined through the factorization process). In that case, one could use either of the following equivalent formulations for :
In either case, the diagonal elements are used twice (not three times as the formula for would lead one to expect), and since only divisions with are performed, storing explicitly is the practical thing to do.
Figure: Preconditioner solve of a system , with .
At the cost of some extra storage, one could store or , thereby saving some computation. Solving a system using the first formulation is outlined in figure . The second formulation is slightly harder to implement.