Scaling-and-squaring method for computing the inverses of matrix φ-functions

2025/01/17 by Lidia Aceto, Aceto, Lidia, Luca Gemignani +1 · 1 citation
Computer Science · #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2501.10028

openalex publication_date 2025/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

This paper aims to develop efficient numerical methods for computing the inverse of matrix φ-functions, ψ_ℓ(A) := (φ_ℓ(A))^-1, for ℓ =1,2,…, when A is a large and sparse matrix with eigenvalues in the open left half-plane. While φ-functions play a crucial role in the analysis and implementation of exponential integrators, their inverses arise in solving certain direct and inverse differential problems with non-local boundary conditions. We propose an adaptation of the standard scaling-and-squaring technique for computing ψ_ℓ(A), based on the Newton-Schulz iteration for matrix inversion. The convergence of this method is analyzed both theoretically and numerically. In addition, we derive and analyze Padé approximants for approximating ψ₁(A/2^s), where s is a suitably chosen integer, necessary at the root of the squaring process. Numerical experiments demonstrate the effectiveness of the proposed approach.

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• Algorithm with variable coefficients for computing matrix inverses

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