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  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 .