A fixed point iteration for computing the matrix logarithm

TL;DR

This paper introduces a novel algorithm for computing the matrix logarithm using successive matrix exponentials, offering improved convergence properties over traditional series expansion methods, especially for matrices far from the identity.

Contribution

The paper presents a new fixed point iteration method based on matrix exponentials for calculating the matrix logarithm, addressing convergence issues of existing series-based approaches.

Findings

Convergence demonstrated for a broad class of matrices.

Effective initial matrix choices improve convergence.

Method outperforms traditional series expansion techniques.

Abstract

In various areas of applied numerics, the problem of calculating the logarithm of a matrix A emerges. Since series expansions of the logarithm usually do not converge well for matrices far away from the identity, the standard numerical method calculates successive square roots. In this article, a new algorithm is presented that relies on the computation of successive matrix exponentials. Convergence of the method is demonstrated for a large class of initial matrices and favorable choices of the initial matrix are discussed.