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:本文涉及解决线性系统$phi(A)x=b$(b$)的有效数字方法, 美元(z)是美元和美元(athbb R ⁇ N) 美元, 特别是这项工作中,我们感兴趣的是计算美元(a) ⁇ 1美元(b) 美元(美元), 美元(z) ⁇ 1(z) displaystone\frac{e ⁇ z-1 ⁇ z}, ⁇ quad\phi(z) ⁇ (z) ⁇ (z) ⁇ dplaystystele\frac{ez- z-z ⁇ 2}美元(美元), 在适当的条件下, 我们设计了美元(A) ⁇ 1美元和美元(A) ⁇ 1美元) 美元(b) 美元(b) 美元(美元), 分别根据牛顿的过滤和克里洛夫- 类型方法计算。 这些结构式矩阵的调整方案得到了考虑。 特别在捆绑和较一般的准可逆矩阵的情况下, 展示了我们提议的算算结果。