The most common type of incomplete factorization is based on taking a set of matrix positions, and keeping all positions outside this set equal to zero during the factorization. The resulting factorization is incomplete in the sense that fill is suppressed.

The set is usually chosen to encompass all positions for
which . A position that is zero in but not so in an
exact factorization
is called a *fill* position, and if it is
outside , the fill there is said to be ``discarded''.
Often, is chosen to coincide with the set of nonzero positions
in , discarding all fill. This factorization type is called
the factorization: the Incomplete factorization of
level zero.

We can describe an incomplete factorization formally as

Meijerink and Van der Vorst [152] proved that, if is an -matrix, such a factorization exists for any choice of , and gives a symmetric positive definite matrix if is symmetric positive definite. Guidelines for allowing levels of fill were given by Meijerink and Van der Vorst in [153].

- Fill-in strategies
- Simple cases: and -
- Special cases: central differences
- Modified incomplete factorizations
- Vectorization of the preconditioner solve
- Parallelizing the preconditioner solve

Mon Nov 20 08:52:54 EST 1995