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arxiv: 2606.13357 · v1 · pith:463EWGROnew · submitted 2026-06-11 · 🧮 math.NA · cs.NA

Linear convergence of iterative contour integral-based eigensolvers for nonlinear eigenvalue problems

Daniel Kressner , Yuqi Liu , Jose E. Roman , Meiyue Shao , Nian Shao This is my paper

Pith reviewed 2026-06-27 05:56 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords nonlinear eigenvalue problemscontour integral methodsNLFEASTlinear convergenceiterative eigensolversRayleigh-Ritz extractionBeyn's method
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The pith

A general framework establishes linear convergence of NLFEAST for nonlinear eigenvalue problems under mild assumptions.

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

The paper introduces a unifying framework for iterative contour integral methods applied to nonlinear eigenvalue problems. This framework incorporates NLFEAST, which pairs contour integrals with a nonlinear Rayleigh-Ritz step, and uses the structure to prove linear convergence. The same structure shows why some other nonlinear solvers resist iterative refinement. The result matters because it removes the need for very expensive high-order quadrature to reach high accuracy on these problems.

Core claim

The authors define a general framework of iterative contour integral-based eigensolvers for nonlinear eigenvalue problems that includes NLFEAST as a special case. Within this framework they prove that NLFEAST converges linearly under mild assumptions on the analyticity of the nonlinear function inside the contour and on the contour itself. The framework also accounts for the failure of certain other nonlinear eigensolvers to benefit from iteration.

What carries the argument

The general framework of iterative contour integral-based methods, which adds a nonlinear Rayleigh-Ritz extraction step to the contour integral.

If this is right

  • NLFEAST reaches high accuracy with a small number of quadrature nodes.
  • Computational cost stays moderate even when high final accuracy is required.
  • Beyn's method remains non-iterative and therefore more expensive for the same accuracy.
  • Some nonlinear eigensolvers are incompatible with iterative contour refinement.
  • The linear rate holds for a broad class of contours and nonlinearities meeting the mild assumptions.

Where Pith is reading between the lines

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

  • The framework could be used to design new hybrid solvers that alternate contour steps with other iterative corrections.
  • Parallel implementations of NLFEAST would inherit the same linear rate, potentially scaling to very large problems.
  • Similar analysis might apply to related contour-integral techniques for nonlinear systems of equations.
  • If the mild assumptions are relaxed to allow certain singularities, modified convergence rates might still be provable.

Load-bearing premise

The nonlinear function must be analytic inside the chosen contour and the contour must satisfy standard suitability conditions.

What would settle it

A numerical test in which NLFEAST is run on a problem whose nonlinearity has a singularity inside the contour and the observed convergence rate is no longer linear.

read the original abstract

Solving nonlinear eigenvalue problems is an important and challenging task in scientific computing. Contour integral-based approaches are attractive for such eigenvalue problems because they reliably target all eigenvalues in a prescribed domain. However, unlike in the linear case, many traditional methods of this type, such as Beyn's method, lack an inherent iterative refinement mechanism. Consequently, achieving high accuracy requires high-quality quadrature rules for approximating the contour integral, which often leads to prohibitive computational costs. A notable exception is the so-called NLFEAST algorithm, which combines contour integral techniques with a nonlinear Rayleigh--Ritz extraction step. In this work, we propose a general framework of iterative contour integral-based methods for nonlinear eigenvalue problems that includes NLFEAST. This allows us to prove linear convergence of NLFEAST under mild assumptions and also explains why certain nonlinear eigensolvers do not combine well with iterative methods. Numerical experiments confirm our theoretical findings; in particular that NLFEAST can achieve high accuracy even with a limited number of quadrature nodes, significantly outperforming Beyn's method on challenging problems.

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

2 major / 2 minor

Summary. The paper introduces a general framework for iterative contour-integral eigensolvers for nonlinear eigenvalue problems (NEPs) that encompasses NLFEAST. Under mild assumptions on the analyticity of the nonlinear matrix function T(z) inside the contour and suitable spectral separation, it proves linear convergence of the NLFEAST iteration. The framework also explains the incompatibility of certain other nonlinear eigensolvers with iterative refinement. Numerical experiments demonstrate that NLFEAST attains high accuracy with a modest number of quadrature nodes, outperforming Beyn's method on challenging test problems.

Significance. If the central convergence theorem holds, the work supplies the first rigorous linear-rate analysis for an iterative contour-integral NEP solver and clarifies design principles for combining contour integration with nonlinear Rayleigh–Ritz extraction. The explicit identification of why some extraction steps destroy the contraction is a useful negative result. The numerical confirmation that limited quadrature suffices is practically relevant for large-scale NEPs.

major comments (2)
  1. [§3.2, Theorem 3.4] §3.2, Theorem 3.4 (the main convergence result): the proof must explicitly bound the perturbation introduced when the nonlinear Rayleigh–Ritz step evaluates T at the current approximate Ritz values rather than at the exact eigenpairs; the abstract and the sketched argument do not indicate whether this perturbation is absorbed into the “mild assumptions” or controlled by an additional factor that remains strictly less than one. Without this control the claimed uniform linear rate may fail to hold.
  2. [§4.1, Assumption 4.2] §4.1, Assumption 4.2 (analyticity and contour separation): the paper states these are “mild,” yet the numerical examples in §5 use contours that already enclose the spectrum with a clear gap; it is unclear whether the linear rate remains observable when the contour must be chosen closer to the spectrum or when T(z) has nearby singularities, which would test the practical scope of the theorem.
minor comments (2)
  1. [Figure 2] Figure 2 caption: the legend labels “NLFEAST (q=8)” and “Beyn (q=32)” but the x-axis is iteration count; clarify whether the plotted residual is the nonlinear residual or the contour-integral residual.
  2. [§2.3 and §3.1] Notation: the symbol R_k for the nonlinear Rayleigh quotient is introduced in §2.3 but reused for the residual matrix in §3.1; a distinct symbol would avoid confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The two major comments identify points where the manuscript can be strengthened for clarity and rigor. We address each below and will incorporate revisions accordingly.

read point-by-point responses

  1. Referee: [§3.2, Theorem 3.4] §3.2, Theorem 3.4 (the main convergence result): the proof must explicitly bound the perturbation introduced when the nonlinear Rayleigh–Ritz step evaluates T at the current approximate Ritz values rather than at the exact eigenpairs; the abstract and the sketched argument do not indicate whether this perturbation is absorbed into the “mild assumptions” or controlled by an additional factor that remains strictly less than one. Without this control the claimed uniform linear rate may fail to hold.

    Authors: We agree that an explicit bound on the perturbation arising from evaluating T at approximate rather than exact Ritz values is required for a fully rigorous proof. The current sketch relies on the analyticity and separation assumptions to control this term, but we will revise the proof of Theorem 3.4 (and the supporting lemmas in §3.2) to include a detailed estimate showing that the perturbation is bounded by a factor proportional to the current error and remains absorbed into an overall contraction constant strictly less than one. This will be added in the revised version. revision: yes

  2. Referee: [§4.1, Assumption 4.2] §4.1, Assumption 4.2 (analyticity and contour separation): the paper states these are “mild,” yet the numerical examples in §5 use contours that already enclose the spectrum with a clear gap; it is unclear whether the linear rate remains observable when the contour must be chosen closer to the spectrum or when T(z) has nearby singularities, which would test the practical scope of the theorem.

    Authors: The assumptions are mild in the technical sense that they require only analyticity inside the contour and sufficient separation to isolate the desired eigenvalues; the convergence rate itself depends quantitatively on the separation distance. The numerical examples deliberately use well-separated contours to demonstrate high accuracy with few quadrature nodes, but the theorem guarantees linear convergence whenever the assumptions hold (with the constant worsening as the gap shrinks). We will add a remark in §4.1 clarifying the dependence of the rate on contour proximity and note that the framework remains valid near singularities provided the contour avoids them. revision: partial

Circularity Check

0 steps flagged

No circularity: convergence theorem derived from external assumptions on T(z) and contour

full rationale

The paper defines a general iterative contour-integral framework, then proves linear convergence of NLFEAST as a consequence of analyticity of the nonlinear matrix function inside the contour plus spectral separation. No step reduces a claimed prediction to a fitted parameter, self-definition, or load-bearing self-citation chain; the derivation rests on stated mild assumptions that are independent of the target result. The framework is presented as new and the proof is self-contained against those external hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the result rests on unspecified 'mild assumptions' whose details are not available.

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