Error Estimates for the Arnoldi Approximation of a Matrix Square Root

Abstract: The Arnoldi process provides an efficient framework for approximating functions of a matrix applied to a vector, i.e., of the form $f(M)\mathbf{b}$, by repeated matrix-vector multiplications. In this paper, we derive an \textit{a priori} error estimate 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 $M\mathbf{x}=\mathbf{b}$. 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. Furthermore, to make the method applicable to large-scale problems, we assume that the matrices are preprocessed utilizing data-sparse approximations preserving positive definiteness, and then establish a refined error bound in this setting. 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 validate the effectiveness and practicality of the approach.