Error Estimates for the Arnoldi Approximation of a Matrix Square Root

TL;DR

This work develops rigorous error estimates for the Arnoldi approximation of the matrix square root acting on a vector, $M^{1/2}\mathbf{b}$, by deriving an integral (contour) representation of the error and relating it to shifted linear systems. It provides both a posteriori and a priori bounds for general (non-Hermitian) matrices and yields a sharper bound in the Hermitian case, leveraging average eigenvalues. The results are extended to perturbed matrices, enabling error control for data-sparse representations such as hierarchical matrices. Numerical experiments on non-Hermitian and Hermitian matrices with varied spectral structures validate the bounds and demonstrate practical usefulness for large-scale problems, including mobility matrices in particulate suspensions stored in hierarchical form. The approach supports reliable stopping criteria, scalable computation, and applicability to structured, real-world applications where explicit square roots are impractical to form.

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.

Figures (14)

• Figure 1: Keyhole contour with outer circle $\mathcal{C}_R$, inner circle $\mathcal{C}_\epsilon$, and offset $\delta$ above/below the branch cut.
• Figure 1: Eigenvalues of the uniformly-distributed spectrum matrix in the complex plane (left) and their magnitudes (right).
• Figure 2: Error of the Arnoldi process for a non-Hermitian matrix with a uniformly-distributed spectrum.
• Figure 3: Eigenvalues of the clustered spectrum matrix in the complex plane (left) and their magnitudes (right).
• Figure 4: Error of the Arnoldi process for a non-Hermitian matrix with a clustered spectrum.

Theorems & Definitions (19)

• Lemma 3.1
• Proof 1
• Lemma 3.2
• Proof 2
• Theorem 3.3
• Proof 3
• Remark 3.4
• Theorem 3.5
• Proof 4
• Remark 3.6