Low-rank updates of matrix functions II: Rational Krylov methods

TL;DR

This paper introduces advanced rational Krylov methods for efficiently updating large matrix functions after low-rank modifications, with convergence analysis and practical error bounds for functions like the exponential and Markov functions.

Contribution

It extends previous polynomial Krylov methods to rational Krylov methods, providing convergence analysis, error bounds, and novel techniques for matrix sign function updates.

Findings

Derived error bounds guide pole selection in rational Krylov methods.

Developed methods for low-rank updates of matrix exponential and Markov functions.

Connected updates of the matrix sign function with rational Krylov methods for Sylvester equations.

Abstract

This work develops novel rational Krylov methods for updating a large-scale matrix function f(A) when A is subject to low-rank modifications. It extends our previous work in this context on polynomial Krylov methods, for which we present a simplified convergence analysis. For the rational case, our convergence analysis is based on an exactness result that is connected to work by Bernstein and Van Loan on rank-one updates of rational matrix functions. We demonstrate the usefulness of the derived error bounds for guiding the choice of poles in the rational Krylov method for the exponential function and Markov functions. Low-rank updates of the matrix sign function require additional attention; we develop and analyze a combination of our methods with a squaring trick for this purpose. A curious connection between such updates and existing rational Krylov subspace methods for Sylvester matrix equations is pointed out.