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

Zhongxiao Jia, Qingqing Zheng

TL;DR

This paper develops a general convergence theory for the Rayleigh--Ritz and refined Rayleigh--Ritz methods applied to regular analytic nonlinear eigenvalue problems. By quantifying how the target eigenvector $x_*$ deviates from a subspace $\mathcal{W}$ via $\varepsilon$, the authors prove that a Ritz value $\mu$ converges to the target eigenvalue $\lambda_*$ unconditionally as $\varepsilon\to0$, with a rate governed by the largest Jordan block order $m_{\mu}$. The refined Ritz vector $\widehat{x}$ is shown to converge unconditionally to $x_*$, while the Ritz vector $\widetilde{x}$ may converge conditionally or be nonunique, highlighting the robustness of the refined method. The work also provides residual-based bounds and comparisons between the two approaches, supported by numerical examples that illustrate the potential disparity in accuracy. Overall, the results extend classical linear-EVP convergence insights to NEPs and offer practical guidance for solving large NEPs with projection methods.

Abstract

We establish a general convergence theory of the Rayleigh--Ritz method and the refined Rayleigh--Ritz method for computing some simple eigenpair $(λ_{*},x_{*})$ 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 $λ_*$ 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.

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

TL;DR

This paper develops a general convergence theory for the Rayleigh--Ritz and refined Rayleigh--Ritz methods applied to regular analytic nonlinear eigenvalue problems. By quantifying how the target eigenvector deviates from a subspace via , the authors prove that a Ritz value converges to the target eigenvalue unconditionally as , with a rate governed by the largest Jordan block order . The refined Ritz vector is shown to converge unconditionally to , while the Ritz vector may converge conditionally or be nonunique, highlighting the robustness of the refined method. The work also provides residual-based bounds and comparisons between the two approaches, supported by numerical examples that illustrate the potential disparity in accuracy. Overall, the results extend classical linear-EVP convergence insights to NEPs and offer practical guidance for solving large NEPs with projection methods.

Abstract

We establish a general convergence theory of the Rayleigh--Ritz method and the refined Rayleigh--Ritz method for computing some simple eigenpair of a given analytic regular nonlinear eigenvalue problem (NEP). In terms of the deviation of from a given subspace , we establish a priori convergence results on the Ritz value, the Ritz vector and the refined Ritz vector. The results show that, as , there exists a Ritz value that unconditionally converges to 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.
Paper Structure (6 sections, 13 theorems, 108 equations, 1 figure, 1 table, 2 algorithms)

This paper contains 6 sections, 13 theorems, 108 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Convergence of the Ritz value
  3. Convergence of the Ritz vector
  4. Convergence of the refined Ritz vector
  5. Bounds for the error of the Ritz vector and the refined Ritz vector
  6. Concluding remarks

Key Result

Theorem 2.1

\newlabell10 For a given $m$-dimensional subspace $\mathcal{W}$, let $\varepsilon$ be defined by devia. Then for the projected matrix-valued function $B(\lambda)=W^HT(\lambda) W$, there exists a matrix-valued function $E(\lambda):\lambda\in \Omega\subseteq \mathbb{C}\rightarrow\mathbb{C}^{m\times such that $\lambda_{*}$ is an eigenvalue of the perturbed matrix-valued function $B(\lambda)+E(\lamb

Figures (1)

  • Figure 1: Curves of $\sin\angle(\widetilde{x},x_*)$, $\|\widetilde{r}\|$ and $\sin\angle(\widehat{x},x_*)$, $\|\widehat{r}\|$ for with each of the four $\varepsilon$ and 100 random perturbation matrices, and the upper and lower figures in (a)-(d) are for $\sin\angle(\widetilde{x},x_*)$, $\|\widetilde{r}\|$ and $\sin\angle(\widehat{x},x_*)$, $\|\widehat{r}\|$, respectively, where the '-o' line is the curve of $\sin\angle(\widetilde{x},x_*)$, the dashed dot line is the curve of $\|\widetilde{r}\|$, the dotted line is the curve of $\sin\angle(\widehat{x},x_*)$, and the solid line is the curve of $\|\widehat{r}\|$.

Theorems & Definitions (33)

  • Definition 1.1
  • Theorem 2.1
  • Proof 1
  • Theorem 2.2
  • Proof 2
  • Theorem 2.3
  • Proof 3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • ...and 23 more