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arxiv: 2605.20933 · v1 · pith:JQKAUUA6new · submitted 2026-05-20 · 🧮 math.NA · cs.NA

Conditioning and backward errors for nonlinear eigenvalue problems with eigenvector nonlinearities

Vilhelm Peterson Lithell , Victor Janssens , Elias Jarlebring , Karl Meerbergen , Wim Michiels This is my paper

Pith reviewed 2026-05-21 02:19 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords nonlinear eigenvalue problemseigenvector nonlinearitiescondition numbersbackward errorssymmetric perturbationsspectral normFrobenius normnumerical stability
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The pith

Explicit expressions give condition numbers and backward errors for symmetric nonlinear eigenvalue problems with eigenvector nonlinearities.

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

The paper establishes explicit and computable formulas for eigenvalue condition numbers and backward errors in a class of symmetric nonlinear eigenvalue problems where the nonlinearity depends on the eigenvector. These formulas apply when matrix perturbations are measured in the spectral or Frobenius norm and can be evaluated directly from a given eigenpair with little effort. Symmetric perturbations are exploited to simplify the derivations. A sympathetic reader cares because the results supply practical tools to assess the reliability of computed solutions, and the numerical experiments indicate that eigenvector nonlinearities require treatment distinct from standard linear or nonlinear eigenvalue problems.

Core claim

For symmetric nonlinear eigenvalue problems with eigenvector nonlinearities, the eigenvalue condition number and the backward error admit explicit expressions that are computable from the eigenpair when perturbations are measured by the spectral or Frobenius norm, and symmetric perturbations can be used to obtain these expressions.

What carries the argument

The explicit formulas for the condition number and backward error obtained via perturbation analysis that exploits symmetry.

If this is right

  • The conditioning of an eigenpair can be evaluated directly without solving auxiliary optimization problems.
  • Backward errors quantify how close a computed pair is to an exact eigenpair of a nearby perturbed problem.
  • The formulas extend the standard theory by handling the additional nonlinearity coming from the eigenvector.
  • Numerical examples confirm that eigenvector nonlinearities can produce different conditioning behavior than linear or eigenvalue-only nonlinear cases.

Where Pith is reading between the lines

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

  • The expressions could be embedded inside iterative solvers to monitor and control accuracy on the fly.
  • Similar explicit formulas might be derived for non-symmetric problems by replacing the symmetry exploitation step with a different perturbation model.
  • The approach could connect to sensitivity analysis in related areas such as nonlinear structural dynamics or optimization with eigenvector constraints.

Load-bearing premise

The underlying problems are symmetric so that symmetric perturbations can be used in the analysis.

What would settle it

A test case in which the minimal perturbation size found by direct optimization differs from the value predicted by the derived backward-error expression.

Figures

Figures reproduced from arXiv: 2605.20933 by Elias Jarlebring, Karl Meerbergen, Victor Janssens, Vilhelm Peterson Lithell, Wim Michiels.

Figure 1
Figure 1. Figure 1: Eigenvalues of (5.9) as functions of a perturbation parameter δ. The different marks correspond to different branches of the eigenvalues. using Stirling’s formula. Hence, by Theorem 3.2, the eigenvalue λ = n of (5.1) will become exponentially ill-conditioned as n increases, which explains the dramatic behavior observed in [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Condition number of each eigenvalue of ( [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
read the original abstract

We consider eigenvalue condition numbers and backward errors for a class of symmetric nonlinear eigenvalue problems with eigenvector nonlinearities. For both of these quantities, we derive explicit and computable expressions that can be evaluated with little computational effort for a given eigenpair, assuming the matrix perturbations are measured by the spectral or Frobenius norm. We also show how symmetric perturbations can be exploited in the analysis. By means of two numerical experiments we demonstrate that problems incorporating eigenvector nonlinearities potentially need to be treated with additional care, when compared to the linear or eigenvalue-nonlinear theory.

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 manuscript derives explicit and computable expressions for eigenvalue condition numbers and backward errors in symmetric nonlinear eigenvalue problems with eigenvector nonlinearities. These formulas are obtained via first-order perturbation analysis adapted to the eigenvector-dependent setting, with perturbations measured in the spectral or Frobenius norm; symmetry is exploited to restrict the admissible perturbation class. The expressions are claimed to require little computational effort once an eigenpair is known. Two numerical experiments are used to illustrate that such problems may require additional care relative to linear or standard nonlinear eigenvalue problems.

Significance. If the derivations hold, the work supplies practical, low-cost tools for sensitivity analysis in a specialized but relevant class of nonlinear eigenproblems. The explicit formulas extend classical conditioning and backward-error theory while preserving computational tractability, which is valuable for numerical analysts working on applications where eigenvector nonlinearities arise. The numerical demonstrations provide concrete evidence of the practical distinction from simpler problem classes.

minor comments (3)
  1. Abstract: the phrase 'potentially need to be treated with additional care' is imprecise; a short clause summarizing the concrete differences observed in the two experiments (e.g., larger condition numbers or different scaling) would strengthen the claim.
  2. Section 3 (or wherever the main derivations appear): verify that the transition from the general first-order perturbation formula to the explicit expressions for the eigenvector-nonlinear case is fully detailed, including any intermediate steps that rely on the fixed nonlinearity at the given eigenpair.
  3. Numerical experiments section: include a brief comparison of the new explicit formulas against a finite-difference or small-perturbation reference computation to provide independent verification of accuracy.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The report correctly identifies the core contribution: explicit, low-cost formulas for eigenvalue condition numbers and backward errors in the symmetric eigenvector-nonlinear setting, obtained via first-order perturbation analysis with spectral or Frobenius norm perturbations.

Circularity Check

0 steps flagged

No significant circularity detected in derivation

full rationale

The paper derives explicit expressions for eigenvalue condition numbers and backward errors by adapting standard first-order perturbation analysis to the eigenvector-nonlinear setting, with perturbations measured in spectral or Frobenius norm and symmetry exploited. These steps follow directly from the definitions of condition numbers and backward errors applied to the given nonlinear eigenproblem at a fixed eigenpair, without reducing to fitted parameters, self-referential definitions, or load-bearing self-citations. The central claims remain independent of the paper's own inputs and are presented as computable formulas evaluated for given eigenpairs, consistent with external benchmarks in numerical linear algebra.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivations rest on standard properties of matrix norms and the assumption of symmetry to simplify perturbation analysis; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The nonlinear eigenvalue problems under consideration are symmetric.
    Invoked to allow exploitation of symmetric perturbations in the conditioning and backward-error analysis.
  • domain assumption Perturbations are measured in the spectral or Frobenius matrix norm.
    Stated as the setting under which the explicit expressions are derived.

pith-pipeline@v0.9.0 · 5625 in / 1220 out tokens · 41554 ms · 2026-05-21T02:19:39.561926+00:00 · methodology

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