Estimating the numerical range with a Krylov subspace

TL;DR

This paper investigates how Krylov subspace methods can be used to estimate the numerical range of a matrix, providing bounds that depend on matrix dimensions and eigenbasis conditioning rather than eigenvalue gaps.

Contribution

It introduces new bounds for numerical range approximation using Krylov subspaces that are independent of eigenvalue gaps and includes tight lower bounds for these estimates.

Findings

Estimates depend only on matrix size, Krylov subspace dimension, and eigenbasis conditioning.

Provided nearly matching lower bounds demonstrating the bounds' tightness.

Results improve understanding of Krylov methods for numerical range estimation.

Abstract

Krylov subspace methods are a powerful tool for efficiently solving high-dimensional linear algebra problems. In this work, we study the approximation quality that a Krylov subspace provides for estimating the numerical range of a matrix. In contrast to prior results, which often depend on the gaps between eigenvalues, our estimates depend only on the dimensions of the matrix and Krylov subspace, and the conditioning of the eigenbasis of the matrix. In addition, we provide nearly matching lower bounds for our estimates, illustrating the tightness of our arguments.