Efficient Inversion of Matrix ϕ-Functions of Low Order

2023/02/15 by L. Gemignani, Gemignani, L.
Computer Science · Engineering · Physics and Astronomy · #65F60 #Electromagnetic Scattering and Analysis #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2302.07565

openalex publication_date 2023/02/15 · openalex created_date 2023/02/18 · openalex updated_date 2026/07/28

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∈ \mathbb R^{N× N}. In particular in this work we are interested in the computation of ϕ(A)^-1 b for the case where ϕ(z)=ϕ₁(z)=(e^z-1)/(z), ϕ(z)=ϕ₂(z)=(e^z-1-z)/(z²). Under suitable conditions on the spectrum of A we design fast algorithms for computing both ϕ_ℓ(A)^-1 and ϕ_ℓ(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.

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