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arxiv: 2212.00302 · v3 · pith:UQNUJGPZnew · submitted 2022-12-01 · 🧮 math.NA · cs.NA

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

Zhongxiao Jia , Qingqing Zheng This is my paper

Pith reviewed 2026-05-24 10:02 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Rayleigh-Ritz methodrefined Rayleigh-Ritz methodnonlinear eigenvalue problemconvergence analysisRitz vectorerror boundsresidual norm
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The pith

As the trial subspace deviation ε vanishes, a Ritz value converges unconditionally to the target eigenvalue of an analytic regular nonlinear eigenproblem, the refined Ritz vector converges too, but the Ritz vector converges only under extra

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops convergence theory for the Rayleigh-Ritz and refined Rayleigh-Ritz methods on analytic regular nonlinear eigenvalue problems. It proves that as the deviation ε of the true eigenvector from the trial subspace approaches zero, at least one Ritz value always approaches the target eigenvalue λ* and the refined Ritz vector does likewise. The plain Ritz vector, however, converges only conditionally and can fail to converge or lack uniqueness. The analysis supplies residual-based error bounds for the approximate eigenvector together with lower and upper bounds on the vector and residual errors. These statements extend known linear-eigenvalue results to the nonlinear setting.

Core claim

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 ε of x* from a given subspace W, we establish a priori convergence results on the Ritz value, the Ritz vector and the refined Ritz vector. The results show that, as ε→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 the

What carries the argument

The deviation ε of the true eigenvector from the trial subspace W, which directly governs the convergence of the projected Ritz quantities.

If this is right

  • An error bound for the approximate eigenvector holds in terms of the computable residual norm of any given approximate eigenpair.
  • Lower and upper bounds exist for the error of the refined Ritz vector, the Ritz vector, and their residual norms.
  • The unconditional convergence of the Ritz value and the refined Ritz vector holds for any analytic regular NEP.
  • The conditional convergence of the plain Ritz vector can produce non-uniqueness when ε is small.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Practitioners computing nonlinear eigenpairs should prefer the refined Ritz vector whenever the trial subspace is already accurate.
  • The theory supplies a practical test: if the residual norm is small but the Ritz vector fails to stabilize, the refined vector should be examined instead.
  • The same ε-controlled analysis may extend to other projection-based solvers for nonlinear eigenproblems that preserve analyticity.
  • Numerical checks on polynomial eigenvalue problems could isolate explicit cases where the plain Ritz vector fails to converge.

Load-bearing premise

The nonlinear eigenvalue problem is analytic and regular, so that the single quantity ε controls all convergence statements.

What would settle it

A concrete analytic regular NEP together with a sequence of subspaces whose deviation ε tends to zero yet no Ritz value approaches the target eigenvalue λ*.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper establishes a convergence theory for the Rayleigh-Ritz and refined Rayleigh-Ritz methods applied to simple eigenpairs of analytic regular nonlinear eigenvalue problems. In terms of the deviation ε of the true eigenvector from a trial subspace, it proves that a Ritz value converges unconditionally to the target eigenvalue λ* as ε→0, the corresponding refined Ritz vector converges unconditionally as well, but the standard Ritz vector converges only conditionally and may fail to converge or be non-unique. The work also derives an a posteriori error bound for approximate eigenvectors in terms of the residual norm, together with lower/upper bounds on the errors of the refined and standard Ritz vectors and their residuals. These results extend known linear-eigenvalue theory to the nonlinear setting, with constructed examples for illustration.

Significance. If the derivations hold, the manuscript supplies a useful theoretical foundation for projection methods on nonlinear eigenvalue problems, which arise in many applications. The explicit separation between unconditional convergence of Ritz values/refined vectors and conditional convergence of Ritz vectors is a nontrivial and practically relevant distinction that carries over from the linear case once local analyticity and regularity are assumed. The residual-based error bounds are directly computable and therefore strengthen the applicability of the analysis. No machine-checked proofs or reproducible code are provided, but the perturbation arguments appear direct and parameter-free once the analyticity/regularity hypotheses are granted.

minor comments (3)
  1. [§1] §1, paragraph following the statement of the main results: the phrase 'nontrivially extend' is used without a precise pointer to the linear-case theorems being generalized; adding a one-sentence comparison to the relevant linear references would clarify the novelty.
  2. [Theorem 3.2] Theorem 3.2 (or the first main convergence theorem): the statement that the Ritz vector 'may not be unique' is asserted but the precise mechanism (multiple minimizers of the Rayleigh quotient on the subspace) is not illustrated with a short example in the text; a one-line remark or reference to the constructed counter-example in §5 would help.
  3. [§2] Notation table or §2: the symbol ε is introduced as the deviation of x* from W, but its precise definition (e.g., sin∠(x*,W) or a norm distance) is not restated when first used in the theorems; a parenthetical reminder would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the accurate summary of the unconditional convergence results for Ritz values and refined Ritz vectors versus the conditional behavior for standard Ritz vectors, as well as the residual-based bounds. The recommendation for minor revision is noted. No specific major comments appear in the report, so we have no individual points requiring response or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation extends standard perturbation theory independently

full rationale

The paper derives a priori convergence bounds for Ritz values/vectors and refined variants in analytic regular NEPs by relating the deviation ε of the true eigenvector from the trial subspace to the projected quantities, using local analyticity and simplicity of the eigenpair. These steps rely on standard perturbation arguments that carry over from the linear case once the nonlinear operator satisfies the stated regularity; no equation reduces a claimed result to a fitted parameter, self-defined quantity, or load-bearing self-citation. The distinction between unconditional convergence of the Ritz value/refined vector and conditional behavior of the plain Ritz vector follows directly from the analytic setup without reintroducing the target statements. The work is therefore self-contained against external benchmarks and receives score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The theory rests on the problem being analytic and regular (domain assumptions standard in NEP literature) plus the existence of a simple eigenpair; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The nonlinear eigenvalue problem is analytic and regular.
    Invoked to guarantee the deviation ε controls the behavior of Ritz quantities.

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