Some important applicative problems require the evaluation of functions
Ψ of large and sparse and/or \emph{localized} matrices A. Popular and
interesting techniques for computing Ψ(A) and Ψ(A)v, where
v is a vector, are based on partial fraction expansions. However,
some of these techniques require solving several linear systems whose matrices
differ from A by a complex multiple of the identity matrix I for computing
Ψ(A)v or require inverting sequences of matrices with the same
characteristics for computing Ψ(A). Here we study the use and the
convergence of a recent technique for generating sequences of incomplete
factorizations of matrices in order to face with both these issues. The
solution of the sequences of linear systems and approximate matrix inversions
above can be computed efficiently provided that A−1 shows certain decay
properties. These strategies have good parallel potentialities. Our claims are
confirmed by numerical tests