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.

翻译：本文关注求解线性系统 $\phi(A) x= b$ 的高效数值方法，其中 $\phi(z)$ 为 $\phi$-函数，且 $A\in \mathbb R^{N\times N}$。具体而言，本文针对 $\phi(z)=\phi_1(z)=\displaystyle\frac{e^z-1}{z}$ 及 $\phi(z)=\phi_2(z)=\displaystyle\frac{e^z-1-z}{z^2}$ 的情形，研究 ${\phi(A)}^{-1} b$ 的计算问题。在 $A$ 的谱满足适当条件时，我们分别基于牛顿迭代法与Krylov型方法设计了计算 ${\phi_\ell(A)}^{-1}$ 和 ${\phi_\ell(A)}^{-1} b$ 的快速算法。进一步考虑了这些格式在结构矩阵上的适应性，特别是带状矩阵及更一般的拟可分矩阵情形。数值结果验证了所提算法的有效性。