Computation of generalized matrix functions with rational Krylov methods

TL;DR

This paper introduces rational Krylov-based algorithms for efficiently computing generalized matrix functions on vectors, leveraging quasiseparable structures and providing error bounds with demonstrated numerical accuracy.

Contribution

It extends existing Golub-Kahan bidiagonalization methods to the rational case, enabling efficient computations for generalized matrix functions with proven error bounds.

Findings

Algorithms effectively compute generalized matrix functions.

Error bounds relate to uniform rational approximation.

Numerical experiments confirm accuracy and efficiency.

Abstract

We present a class of algorithms based on rational Krylov methods to compute the action of a generalized matrix function on a vector. These algorithms incorporate existing methods based on the Golub-Kahan bidiagonalization as a special case. By exploiting the quasiseparable structure of the projected matrices, we show that the basis vectors can be updated using a short recurrence, which can be seen as a generalization to the rational case of the Golub-Kahan bidiagonalization. We also prove error bounds that relate the error of these methods to uniform rational approximation. The effectiveness of the algorithms and the accuracy of the bounds is illustrated with numerical experiments.