Twice is enough for dangerous eigenvalues

TL;DR

This paper demonstrates that subspace iteration with rational filters can effectively handle dangerous eigenvalues near filter poles, achieving high accuracy in eigenvalue computations.

Contribution

It shows that two iterations suffice for stability in rational-filter eigensolvers with orthogonal eigenvectors, and proposes a restart strategy for Arnoldi methods.

Findings

Two iterations reduce round-off errors to machine precision for certain eigensolvers.

Subspace iteration with rational filters is robust near filter poles.

A restart strategy improves Arnoldi's accuracy for close eigenvalues.

Abstract

We analyze the stability of a class of eigensolvers that target interior eigenvalues with rational filters. We show that subspace iteration with a rational filter is robust even when an eigenvalue is near a filter's pole. These dangerous eigenvalues contribute to large round-off errors in the first iteration, but are self-correcting in later iterations. For matrices with orthogonal eigenvectors (e.g., real-symmetric or complex Hermitian), two iterations is enough to reduce round-off errors to the order of the unit-round off. In contrast, Krylov methods accelerated by rational filters with fixed poles typically fail to converge to unit round-off accuracy when an eigenvalue is close to a pole. In the context of Arnoldi with shift-and-invert enhancement, we demonstrate a simple restart strategy that recovers full precision in the target eigenpairs.