On the Kahan-Parlett-Jiang theorem—A globally optimal backward perturbation error for two-sided invariant subspaces

Abstract

A crucial question in large non-Hermitian eigenproblems is whether or not any residual bounds can be very useful in practical situations. For this problem, the famous Kahan-Parlett-Jiang theorem Kahan et al. (1982) [15] seems to be the best compromise between generality and sharpness. In essence, it presents an a posteriori error bound on approximate solutions to eigenproblems, which provides a powerful tool to evaluate quality of the computed two-sided invariant subspaces of large-scale non-Hermitian matrices. However, the perturbation error determined by this theorem is only locally optimal rather than globally optimal. In this work, we revisit this problem and derive a globally optimal backward perturbation error for given two-sided approximate invariant subspaces. Our theorem enhances the Kahan-Parlett-Jiang theorem, and numerical experiments demonstrate the effectiveness of our theoretical results.