  The paper is concerned with efficient numerical methods for solving a linear
system $\phi(A) x= b$, where $\phi(z)$ is a $\phi$-function and $A\in \mathbb
R^{N\times N}$. In particular in this work we are interested in the computation
of ${\phi(A)}^{-1} b$ for the case where
$\phi(z)=\phi_1(z)=\displaystyle\frac{e^z-1}{z}, \quad
\phi(z)=\phi_2(z)=\displaystyle\frac{e^z-1-z}{z^2}$. Under suitable conditions
on the spectrum of $A$ we design fast algorithms for computing both
${\phi_\ell(A)}^{-1}$ and ${\phi_\ell(A)}^{-1} b$ 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.
