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Efficient approximation of functions of some large matrices by partial fraction expansions

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 $Ψ(A)\mathbf{v}$, where $\mathbf{v}$ 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 $Ψ(A)\mathbf{v}$ 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.