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 :
or
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.