A posteriori error bounds for the block-Lanczos method for matrix function approximation

TL;DR

This paper extends a posteriori error bounds for the Lanczos method to the block algorithm, demonstrating robustness and potential for practical stopping criteria through numerical experiments and hyperparameter analysis.

Contribution

It introduces new error bounds for the block-Lanczos method, enhancing understanding and practical applicability in matrix function approximation.

Findings

Error bounds are robust to block size changes

Bounds can serve as practical stopping criteria

Hyperparameter choices influence bound quality

Abstract

We extend the error bounds from [SIMAX, Vol. 43, Iss. 2, pp. 787-811 (2022)] for the Lanczos method for matrix function approximation to the block algorithm. Numerical experiments suggest that our bounds are fairly robust to changing block size and have the potential for use as a practical stopping criteria. Further experiments work towards a better understanding of how certain hyperparameters should be chosen in order to maximize the quality of the error bounds, even in the previously studied block-size one case.