Error bounds for Lanczos-based matrix function approximation

TL;DR

This paper develops detailed error bounds for the Lanczos-FA method in matrix function approximation, accounting for spectral properties, and demonstrates their effectiveness through numerical experiments.

Contribution

It introduces a new framework based on the Cauchy integral formula to derive spectral-aware error bounds for Lanczos-FA, extending to finite precision and quadratic form approximations.

Findings

Error bounds depend on spectral properties like eigenvalue clustering.

Bounds are applicable in finite precision with existing Lanczos theory.

Numerical experiments confirm the bounds' effectiveness.

Abstract

We analyze the Lanczos method for matrix function approximation (Lanczos-FA), an iterative algorithm for computing $f(\mathbf{A}) \mathbf{b}$ when $\mathbf{A}$ is a Hermitian matrix and $\mathbf{b}$ is a given vector. Assuming that $f : \mathbb{C} \rightarrow \mathbb{C}$ is piecewise analytic, we give a framework, based on the Cauchy integral formula, which can be used to derive a priori and a posteriori error bounds for Lanczos-FA in terms of the error of Lanczos used to solve linear systems. Unlike many error bounds for Lanczos-FA, these bounds account for fine-grained properties of the spectrum of $\mathbf{A}$, such as clustered or isolated eigenvalues. Our results are derived assuming exact arithmetic, but we show that they are easily extended to finite precision computations using existing theory about the Lanczos algorithm in finite precision. We also provide generalized bounds for the Lanczos method used to approximate quadratic forms $\mathbf{b}^\textsf{H} f(\mathbf{A}) \mathbf{b}$, and demonstrate the effectiveness of our bounds with numerical experiments.