A broad class of preconditioners is based on incomplete factorizations
of the coefficient matrix. We call a factorization incomplete if
during the factorization process certain fill elements,
nonzero elements in the factorization in positions where the original
matrix had a zero, have been
ignored. Such a preconditioner is then given in factored form 
with 
lower and 
upper triangular. The efficacy of the
preconditioner depends on how well 
approximates 
.
and 
-