An analysis of the Rayleigh-Ritz and refined Rayleigh-Ritz methods for regular nonlinear eigenvalue problems

TL;DR

This paper develops a comprehensive convergence theory for the Rayleigh-Ritz and refined Rayleigh-Ritz methods applied to regular nonlinear eigenvalue problems, extending linear eigenproblem results.

Contribution

It provides new convergence results, error bounds, and insights into the behavior of Ritz and refined Ritz vectors for nonlinear eigenproblems.

Findings

Unconditional convergence of a Ritz value as deviation approaches zero.

Conditional convergence and potential non-uniqueness of Ritz vectors.

Error bounds for approximate eigenvectors and residual norms.

Abstract

We establish a general convergence theory of the Rayleigh--Ritz method and the refined Rayleigh--Ritz method for computing some simple eigenpair $(\lambda_{*},x_{*})$ of a given analytic regular nonlinear eigenvalue problem (NEP). In terms of the deviation $\varepsilon$ of $x_{*}$ from a given subspace $\mathcal{W}$, we establish a priori convergence results on the Ritz value, the Ritz vector and the refined Ritz vector. The results show that, as $\varepsilon\rightarrow 0$, there exists a Ritz value that unconditionally converges to $\lambda_*$ and the corresponding refined Ritz vector does so too but the Ritz vector converges conditionally and it may fail to converge and even may not be unique. We also present an error bound for the approximate eigenvector in terms of the computable residual norm of a given approximate eigenpair, and give lower and upper bounds for the error of the refined Ritz vector and the Ritz vector as well as for that of the corresponding residual norms. These results nontrivially extend some convergence results on these two methods for the linear eigenvalue problem to the NEP. Examples are constructed to illustrate the main results.