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].
and 
-