Abstract
The paper is concerned with efficient numerical methods for solving a linear system ϕ(A)x=b, where ϕ(z) is a ϕ-function and A∈RN×N. In particular in this work we are interested in the computation of ϕ(A)−1b for the case where ϕ(z)=ϕ1(z)=ez−1z and ϕ(z)=ϕ2(z)=ez−1−zz2. Under suitable conditions on the spectrum of A we design fast algorithms for computing both ϕℓ(A)−1 and ϕℓ(A)−1b based on Newton's iteration and Krylov-type methods, respectively. Adaptations of these schemes for structured matrices are considered. In particular the cases of banded and more generally quasiseparable matrices are investigated. Numerical results are presented to show the effectiveness of our proposed algorithms.
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