arxiv: 2605.12846 · v1 · submitted 2026-05-13 · 🧮 math.NA · cs.NA

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A refined CJ--SS--RR method with a reliable removal approach of spurious Ritz values for the Hermitian eigenvalue problem

Tianhang Liu, Zhongxiao Jia

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Pith reviewed 2026-05-14 19:09 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords Hermitian eigenvalue problemRitz valuesspurious eigenvaluesrefined Rayleigh-Ritzsubspace iterationrestarted algorithmsnumerical linear algebra

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The pith

Refined Rayleigh-Ritz projection enables tune-free removal of spurious Ritz values in Hermitian eigenproblems by exploiting unconditional convergence of refined vectors.

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

When a subspace gets close to the desired eigenvectors of a Hermitian matrix A, standard Ritz vectors can still fail to converge, especially if eigenvalues are close. The paper replaces the usual Rayleigh-Ritz step with its refined version, under which the refined Ritz vectors converge unconditionally once the subspace is accurate enough. This property supplies a reliable, parameter-free test that identifies which Ritz values are genuine and which are spurious duplicates. The authors embed the test in a restarted CJ-SS-RRR algorithm and show through experiments that the new version is both faster and more accurate than the earlier CJ-SS-RR method on large test matrices.

Core claim

Under the hypothesis that the deviations of the desired eigenvectors of A from the underlying subspace tend to zero, the refined SS-RR methods compute eigenpairs by removing spurious Ritz values on the basis of the unconditional convergence of refined Ritz vectors, yielding a restarted CJ-SS-RRR algorithm that outperforms the non-refined restarted CJ-SS-RR algorithm in efficiency and reliability.

What carries the argument

Refined Rayleigh-Ritz projection that produces vectors whose convergence is unconditional when the subspace deviation tends to zero, paired with a removal rule for spurious Ritz values.

If this is right

Spurious Ritz values that arise when multiple Ritz values converge to the same eigenvalue can be discarded without any user-tuned threshold.
The restarted CJ-SS-RRR algorithm converges more reliably than CJ-SS-RR on matrices whose eigenvalues of interest are close.
Residual norms alone are insufficient for identification, but the refined-vector test supplies a rigorous substitute.
The approach extends directly to block versions of the SS-RR family for computing multiple eigenpairs.

Where Pith is reading between the lines

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

The same unconditional-convergence property could be used to design removal steps in other subspace iteration schemes such as block Lanczos or Davidson methods.
The technique may reduce the number of matrix-vector products needed when eigenvalue clusters are present, an effect worth measuring on larger-scale problems.
Because the removal rule is deterministic and theory-supported, it could serve as a stable post-processing step for any Ritz-based solver that generates extra copies of an eigenvalue.

Load-bearing premise

The deviations of the desired eigenvectors of A from the subspace tend to zero as the subspace improves.

What would settle it

If refined Ritz vectors still fail to converge to the true eigenvectors on a sequence of subspaces whose deviation from the desired eigenspace goes to zero, the removal criterion would misclassify genuine and spurious values.

Figures

Figures reproduced from arXiv: 2605.12846 by Tianhang Liu, Zhongxiao Jia.

**Figure 2.** Figure 2: The performance of Algorithm 5.2. (a): the condition number [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 3.** Figure 3: The average numbers of the retained approximate eigenvalues using different [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗

**Figure 4.** Figure 4: The performance of different removal approaches. The horizontal axis is the [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗

read the original abstract

Under the hypothesis that the deviations of the desired eigenvectors of the matrix $A$ from the underlying subspace tend to zero, the Ritz vectors may not converge and have poor or little accuracy. This phenomenon is not unusual and particularly occurs when the associated Ritz values are close, which is independent of the eigenvalue distribution of $A$. For the (block) SS--RR methods, there are possibly {\em more} Ritz values that converge to the same desired eigenvalue(s) counting multiplicity in the region of interest, meaning that some of the Ritz values must be spurious and the corresponding residual norms of the Ritz pairs may not be small. Consequently, the (block) SS--RR methods including the CJ--SS--RR method cannot base on the corresponding residual norms to effectively identify if the Ritz values in the region are genuine or spurious. This paper proposes refined SS--RR, abbreviated as SS--RRR, methods based on the refined Rayleigh--Ritz projection that compute the eigenpairs of large matrices with the eigenvalues located in the given region. We present a new approach to accurately implement the RRR methods more efficiently than ever before for a general subspace.Exploiting the unconditional convergence of the refined Ritz vectors when the subspace is sufficiently accurate, we propose a tune-free removal approach to effectively remove spurious Ritz values with a rigorous theory supported, and develop a restarted CJ--SS--RRR algorithm. Numerical experiments show that the restarted CJ--SS--RRR algorithm is more efficient and effective than the restarted CJ--SS--RR algorithm.

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.

Desk Editor's Note private letter to a colleague

The paper gives a tune-free way to drop spurious Ritz values in CJ-SS-RR via refined projections, but the fix rests on an assumption about subspace accuracy that may not hold tightly in restarted runs.

read the letter

The main takeaway is that this work refines the CJ-SS-RR method into SS-RRR by using refined Rayleigh-Ritz projection and adds a removal rule for spurious Ritz values that needs no tuning parameters. They develop a restarted CJ-SS-RRR algorithm and report that it runs more efficiently than the prior version in their tests. The core idea is to exploit unconditional convergence of the refined Ritz vectors once the subspace is accurate enough, under the hypothesis that desired eigenvectors approach the subspace. This sidesteps the problem that standard residual norms cannot reliably flag spurious values when Ritz values are close, a issue that shows up regardless of the matrix spectrum. The approach is a direct response to a practical weakness in block SS-RR methods, where extra Ritz values can converge to the same eigenvalue and leave some residuals unhelpful for identification. The theory they sketch for the removal step is presented as rigorous, and the experiments back up better performance on the tested cases. The soft spot is the dependence on that single hypothesis holding in the restarted setting. If the subspace is only marginally accurate or if there is clustering or multiplicity, the refined vectors may not deliver the expected separation, and the removal criterion could still produce errors. The abstract does not indicate an algorithmic check or bound that enforces the hypothesis, so the reliability in production codes would need the full proofs and more varied test matrices to confirm. This is aimed at people who implement or use subspace iteration for large Hermitian eigenproblems in scientific computing. It deserves peer review because it targets a known robustness gap with a concrete change and some supporting analysis, even if the assumption requires close examination.

Referee Report

2 major / 1 minor

Summary. The paper introduces refined SS--RRR methods based on the refined Rayleigh-Ritz projection for computing eigenpairs of large Hermitian matrices with eigenvalues in a specified region. It identifies the limitation of standard (block) SS--RR methods, including CJ--SS--RR, where residual norms cannot reliably distinguish genuine from spurious Ritz values when Ritz values are close. Exploiting the unconditional convergence of refined Ritz vectors under the hypothesis that deviations of desired eigenvectors from the underlying subspace tend to zero, the authors propose a tune-free removal approach for spurious Ritz values, supported by rigorous theory, and develop a restarted CJ--SS--RRR algorithm. Numerical experiments claim this restarted variant is more efficient and effective than the restarted CJ--SS--RR algorithm.

Significance. If the hypothesis holds and the removal criterion proves reliable, the work addresses a known practical difficulty in subspace iteration methods for clustered eigenvalues, potentially improving robustness of eigensolvers without parameter tuning. The emphasis on an efficient implementation of refined projections and the provision of theoretical support for the removal step are positive features that could influence subsequent developments in numerical linear algebra for large-scale Hermitian problems.

major comments (2)

[Development of the removal approach and restarted CJ--SS--RRR algorithm] The tune-free removal approach for spurious Ritz values is justified by unconditional convergence of refined Ritz vectors once the subspace is sufficiently accurate, under the single hypothesis that deviations of desired eigenvectors from the subspace tend to zero. However, the manuscript provides no explicit bounds, algorithmic enforcement, or practical test to verify when this hypothesis holds in the restarted setting, especially for close Ritz values where standard Ritz vectors are known to fail. This makes the reliability of the removal criterion dependent on an assumption whose satisfaction is not guaranteed by the algorithm.
[Theoretical justification of refined Ritz vector convergence] The abstract states that the refined approach is supported by rigorous theory, yet the provided description does not include concrete error bounds or a checkable condition linking subspace accuracy to the hypothesis. Without such quantification, it is unclear whether the removal remains effective when the subspace is only marginally accurate or in the presence of multiplicity/clustering.

minor comments (1)

[Abstract and introduction] The abstract refers to 'a new approach to accurately implement the RRR methods more efficiently than ever before for a general subspace' without specifying the prior implementation or quantifying the efficiency gain; a brief comparison in the introduction would clarify the advance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below, providing our strongest honest defense of the manuscript while indicating revisions where the presentation can be strengthened.

read point-by-point responses

Referee: [Development of the removal approach and restarted CJ--SS--RRR algorithm] The tune-free removal approach for spurious Ritz values is justified by unconditional convergence of refined Ritz vectors once the subspace is sufficiently accurate, under the single hypothesis that deviations of desired eigenvectors from the subspace tend to zero. However, the manuscript provides no explicit bounds, algorithmic enforcement, or practical test to verify when this hypothesis holds in the restarted setting, especially for close Ritz values where standard Ritz vectors are known to fail. This makes the reliability of the removal criterion dependent on an assumption whose satisfaction is not guaranteed by the algorithm.

Authors: The hypothesis is the standard one underlying all subspace iteration methods: the contour integration step is constructed precisely to drive the deviation of desired eigenvectors from the trial subspace toward zero. The restarted CJ--SS--RRR algorithm enforces this progressively by augmenting the subspace with new contour-integrated vectors at each restart and then applying the refined projection. The unconditional convergence theorem for refined Ritz vectors (independent of eigenvalue clustering or closeness of Ritz values) is the key property that justifies the tune-free removal; standard RR lacks this property, which is why residuals fail. While the manuscript does not supply matrix-dependent a priori bounds (typical in this literature), we will add an explicit discussion of the restart mechanism's role in guaranteeing monotonic improvement of subspace accuracy and a practical monitoring test based on the change in refined residual norms between successive restarts. This supplies the algorithmic enforcement requested. revision: partial
Referee: [Theoretical justification of refined Ritz vector convergence] The abstract states that the refined approach is supported by rigorous theory, yet the provided description does not include concrete error bounds or a checkable condition linking subspace accuracy to the hypothesis. Without such quantification, it is unclear whether the removal remains effective when the subspace is only marginally accurate or in the presence of multiplicity/clustering.

Authors: The rigorous theory consists of the unconditional convergence result for refined Ritz vectors to the desired eigenvectors as the subspace deviation tends to zero; this result holds without any assumption on the eigenvalue distribution or multiplicity and is the foundation for the removal criterion. We agree that the current write-up would benefit from more explicit quantification. In the revision we will insert concrete error bounds (derived from the refined Rayleigh-Ritz perturbation theory already referenced) that relate the sine of the subspace angle to the error in the refined vector, together with a checkable residual-based condition that can be evaluated after each projection. For marginal accuracy the removal is applied conservatively (only when the refined vector deviates markedly from the standard Ritz vector), and the numerical experiments already demonstrate robustness under clustering; we will add a short remark on this point. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior CJ-SS-RR work; new refined projection and removal logic independent of inputs.

full rationale

The derivation introduces a new refined Rayleigh-Ritz projection (SS-RRR) and a tune-free spurious-value removal criterion based on unconditional convergence of refined Ritz vectors under the explicit hypothesis that eigenvector deviations from the subspace tend to zero. This convergence property and the removal rule are supported by new theory presented in the current manuscript rather than reducing to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The baseline CJ-SS-RR method is cited for context, but the central algorithmic advance and its supporting analysis remain self-contained against external benchmarks and do not collapse to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the stated hypothesis about eigenvector deviations tending to zero and standard properties of Rayleigh-Ritz projections; no free parameters, new entities, or ad-hoc axioms beyond domain-standard assumptions are introduced in the abstract.

axioms (1)

domain assumption Deviations of the desired eigenvectors of A from the underlying subspace tend to zero
Explicitly stated as the hypothesis under which refined Ritz vectors converge unconditionally.

pith-pipeline@v0.9.0 · 5583 in / 1161 out tokens · 57439 ms · 2026-05-14T19:09:11.641218+00:00 · methodology

discussion (0)

Reference graph

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