Efficient Inversion of Matrix $\phi$-Functions of Low Order

TL;DR

This paper develops efficient numerical algorithms for inverting specific matrix functions called $oldsymbol{ extphi}$-functions, especially for structured matrices, using Newton's iteration and Krylov methods, with demonstrated numerical effectiveness.

Contribution

The paper introduces novel algorithms for fast inversion of $oldsymbol{ extphi}$-functions of matrices, tailored for structured matrices like banded and quasiseparable, using Newton's iteration and Krylov methods.

Findings

Algorithms are effective for matrices with suitable spectral properties.

Structured matrix adaptations improve computational efficiency.

Numerical experiments confirm the algorithms' practicality.

Abstract

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.