Error Estimates for the Arnoldi Approximation of a Matrix Square Root

TL;DR

This paper develops error estimates for the Arnoldi approximation of a matrix square root, extending existing analysis to non-Hermitian and perturbed matrices, with numerical validation on large-scale structured matrices.

Contribution

It extends error analysis of the Arnoldi method for matrix square roots to non-Hermitian and perturbed matrices, providing practical bounds and validation.

Findings

Error bounds are reliable for structured matrices.

Analysis extends to non-Hermitian matrices.

Numerical results confirm theoretical bounds.

Abstract

The Arnoldi process provides an efficient framework for approximating functions of a matrix applied to a vector, i.e., of the form $f(M)\bm{b}$, by repeated matrix-vector multiplications. In this paper, we derive error estimates for approximating the action of a matrix square root using the Arnoldi process, where the integral representation of the error is reformulated in terms of the error for solving the linear system $M\bm{x}=\bm{b}$. The results extend the error analysis of the Lanczos method for Hermitian matrices in [Chen et al., SIAM J. Matrix Anal. Appl., 2022] to non-Hermitian cases and provide an improved bound for the Hermitian case. Furthermore, in practical settings, the matrix may only be available via approximate or structured representations. Motivated by this, we extend the analysis and establish a generalized error bound for perturbed matrices. The numerical results on matrices with different structures demonstrate that our theoretical analysis yields a reliable upper bound. Finally, simulations on large-scale matrices arising in particulate suspensions, represented in hierarchical matrix form, validate the effectiveness and practicality of the approach.