A Krylov subspace method for the approximation of bivariate matrix functions

TL;DR

This paper introduces a new tensorized Krylov subspace method for efficiently approximating bivariate matrix functions, with novel convergence analysis applicable to most instances except Sylvester equations.

Contribution

It presents a novel tensorized Krylov subspace approach for bivariate matrix functions along with new convergence estimates, expanding understanding beyond existing methods.

Findings

The method effectively approximates bivariate matrix functions.

New convergence estimates are derived for most instances.

The approach offers insights into the convergence behavior for various applications.

Abstract

Bivariate matrix functions provide a unified framework for various tasks in numerical linear algebra, including the solution of linear matrix equations and the application of the Fr\'echet derivative. In this work, we propose a novel tensorized Krylov subspace method for approximating such bivariate matrix functions and analyze its convergence. While this method is already known for some instances, our analysis appears to result in new convergence estimates and insights for all but one instance, Sylvester matrix equations.