Theoretical error estimates for computing the matrix logarithm by Pad\'e-type approximants

TL;DR

This paper develops theoretical error estimates for computing the matrix logarithm using Padé-type approximants via Gauss--Legendre quadrature, enabling better parameter selection and improved accuracy in numerical computations.

Contribution

It introduces novel a priori error estimates for matrix logarithm approximation, guiding the choice of quadrature points and scaling steps for enhanced precision.

Findings

Error estimates effectively predict approximation accuracy

Guidelines for selecting quadrature points and scaling steps

Numerical experiments confirm the reliability of the estimates

Abstract

In this article, we focus on the error that is committed when computing the matrix logarithm using the Gauss--Legendre quadrature rules. These formulas can be interpreted as Pad\'e approximants of a suitable Gauss hypergeometric function. Empirical observation tells us that the convergence of these quadratures becomes slow when the matrix is not close to the identity matrix, thus suggesting the usage of an inverse scaling and squaring approach for obtaining a matrix with this property. The novelty of this work is the introduction of error estimates that can be used to select a priori both the number of Legendre points needed to obtain a given accuracy and the number of inverse scaling and squaring to be performed. We include some numerical experiments to show the reliability of the estimates introduced.