pith. sign in

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
0
0 comments X

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.

pith-pipeline@v0.9.1-grok · 5725 in / 1077 out tokens · 23589 ms · 2026-06-27T05:56:28.941929+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

34 extracted references · 32 canonical work pages · 2 internal anchors

  1. [1]

    Araujo C., J.C., Campos, C., Engström, C., Roman, J.E.: Computation of scattering resonances in absorptive and dispersive media with applications to metal-dielectric nano-structures. J. Comput. Phys.407, 109220 (2020) https: //doi.org/10.1016/j.jcp.2019.109220

    work page doi:10.1016/j.jcp.2019.109220 2020

  2. [2]

    JSIAM Lett

    Asakura, J., Sakurai, T., Tadano, H., Ikegami, T., Kimura, K.: A numerical method for nonlinear eigenvalue problems using contour integrals. JSIAM Lett. 1, 52–55 (2009) https://doi.org/10.14495/jsiaml.1.52

    work page doi:10.14495/jsiaml.1.52 2009

  3. [3]

    Baydoun, S.K., Voigt, M., Goderbauer, B., Jelich, C., Marburg, S.: A subspace iteration eigensolver based on Cauchy integrals for vibroacoustic problems in unbounded domains. Int. J. Numer. Methods Eng.122(16), 4250–4269 (2021) https://doi.org/10.1002/nme.6701

    work page doi:10.1002/nme.6701 2021

  4. [4]

    Betcke, M.M., Voss, H.: Stationary Schrödinger equations governing electronic states of quantum dots in the presence of spin-orbit splitting. Appl. Math.52, 267–284 (2007) https://doi.org/10.1007/s10492-007-0014-5

    work page doi:10.1007/s10492-007-0014-5 2007

  5. [5]

    ACM Trans

    Betcke, T., Higham, N.J., Mehrmann, V., Schröder, C., Tisseur, F.: NLEVP: A collection of nonlinear eigenvalue problems. ACM Trans. Math. Software39(2), 1–28 (2013) https://doi.org/10.1145/2427023.2427024

    work page doi:10.1145/2427023.2427024 2013

  6. [6]

    6Convergence history of Algorithm 3 applied to the nine NEPs from Table 1

    Beyn,W.-J.:Anintegralmethodforsolvingnonlineareigenvalueproblems.Linear 23 0 5 10 Iteration 10!12 10!8 10!4 100 Residual spring (8) 0 10 20 Iteration 10!12 10!8 10!4 100 Residual acoustic wave 2d (16) 0 20 Iteration 10!12 10!8 10!4 100 Residual butterfly (16) 0 1 2 Iteration 10!12 10!8 10!4 100 Residual loaded string (16) 0 100 200 Iteration 10!12 10!8 10...

    work page doi:10.1016/j.laa.2011.03 2012

  7. [7]

    SIAM Rev.65(2), 439–470 (2023) https://doi.org/10.1137/20M1389303

    Brennan, M.C., Embree, M., Gugercin, S.: Contour integral methods for nonlinear eigenvalue problems: a systems theoretic approach. SIAM Rev.65(2), 439–470 (2023) https://doi.org/10.1137/20M1389303

    work page doi:10.1137/20m1389303 2023

  8. [8]

    2602.14979

    Brenneck, J., Polizzi, E.: An iterative method for contour-based nonlinear eigen- solvers. arXiv preprint arXiv:2007.03000 (2020) https://doi.org/10.48550/arXiv. 2007.03000

    work page internal anchor Pith review doi:10.48550/arxiv 2007

  9. [9]

    Bruno, O.P., Santana, M., Trefethen, L.N.: Evaluation of resonances: Adaptivity 24 and AAA rational approximation of randomly scalarized boundary integral resol- vents. SIAM J. Sci. Comput.48(3), 1260–1283 (2026) https://doi.org/10.1137/ 24M1690680

    2026

  10. [10]

    Demésy,G.,Nicolet,A.,Gralak,B.,Geuzaine,C.,Campos,C.,Roman,J.E.:Non- linear eigenvalue problems with GetDP and SLEPc: Eigenmode computations of frequency-dispersive photonic open structures. Comput. Phys. Commun.257, 107509 (2020) https://doi.org/10.1016/j.cpc.2020.107509

    work page doi:10.1016/j.cpc.2020.107509 2020

  11. [11]

    Garcia-Vergara, M., Demésy, G., Zolla, F.: Extracting an accurate model for permittivity from experimental data: hunting complex poles from the real line. Opt. Lett.42(6), 1145–1148 (2017) https://doi.org/10.1364/OL.42.001145

    work page doi:10.1364/ol.42.001145 2017

  12. [12]

    Gavin, B., Międlar, A., Polizzi, E.: FEAST eigensolver for nonlinear eigenvalue problems. J. Comput. Sci.27, 107–117 (2018) https://doi.org/10.1016/j.jocs. 2018.05.006

    work page doi:10.1016/j.jocs 2018

  13. [13]

    Ghani,A.,Polifke,W.:Anexceptionalpointswitchesstabilityofathermoacoustic experiment. J. Fluid Mech.920, 3 (2021) https://doi.org/10.1017/jfm.2021.480

    work page doi:10.1017/jfm.2021.480 2021

  14. [14]

    Johns Hopkins University Press, Baltimore, MD, USA (2013)

    Golub, G.H., Van Loan, C.F.: Matrix Computations, 4th edn. Johns Hopkins University Press, Baltimore, MD, USA (2013)

    2013

  15. [15]

    Acta Numer.26, 1–94 (2017) https://doi.org/10.1017/S0962492917000034

    Güttel, S., Tisseur, F.: The nonlinear eigenvalue problem. Acta Numer.26, 1–94 (2017) https://doi.org/10.1017/S0962492917000034

    work page doi:10.1017/s0962492917000034 2017

  16. [16]

    Güttel, S., Kressner, D., Vandereycken, B.: Randomized sketching of nonlinear eigenvalue problems. SIAM J. Sci. Comput.46(5), 3022–3043 (2024) https://doi. org/10.1137/22M153656X

    work page doi:10.1137/22m153656x 2024

  17. [17]

    Güttel, S., Negri Porzio, G.M., Tisseur, F.: Robust rational approximations of nonlinear eigenvalue problems. SIAM J. Sci. Comput.44(4), 2439–2463 (2022) https://doi.org/10.1137/20M1380533

    work page doi:10.1137/20m1380533 2022

  18. [18]

    Güttel, S., Van Beeumen, R., Meerbergen, K., Michiels, W.: NLEIGS: a class of fully rational Krylov methods for nonlinear eigenvalue problems. SIAM J. Sci. Comput.36(6), 2842–2864 (2014) https://doi.org/10.1137/130935045

    work page doi:10.1137/130935045 2014

  19. [19]

    Huang, R., Struthers, A.A., Sun, J., Zhang, R.: Recursive integral method for transmission eigenvalues. J. Comput. Phys.327, 830–840 (2016) https://doi.org/ 10.1016/j.jcp.2016.10.001

    work page doi:10.1016/j.jcp.2016.10.001 2016

  20. [20]

    Russian Mathematical Surveys26(4), 15–44 (1971) https://doi.org/10.1070/RM1971v026n04ABEH003985

    Keldysh, M.V.: On the completeness of the eigenfunctions of some classes of non- selfadjoint linear operators. Russian Mathematical Surveys26(4), 15–44 (1971) https://doi.org/10.1070/RM1971v026n04ABEH003985

    work page doi:10.1070/rm1971v026n04abeh003985 1971

  21. [21]

    Kressner, D.: A block Newton method for nonlinear eigenvalue problems. Numer. 25 Math.114(2), 355–372 (2009) https://doi.org/10.1007/s00211-009-0259-x

    work page doi:10.1007/s00211-009-0259-x 2009

  22. [22]

    Lalanne, P., Yan, W., Gras, A., Sauvan, C., Hugonin, J.-P., Besbes, M., Demésy, G., Truong, M.D., Gralak, B., Zolla, F., Nicolet, A., Binkowski, F., Zschiedrich, L., Burger, S., Zimmerling, J., Remis, R., Urbach, P., Liu, H.T., Weiss, T.: Quasi- normal mode solvers for resonators with dispersive materials. J. Opt. Soc. Am. A36(4), 686–704 (2019) https://d...

    work page doi:10.1364/josaa.36.000686 2019

  23. [23]

    Li,D.,Polizzi,E.:NonlineareigenvaluealgorithmforGWquasiparticleequations. Phys. Rev. B111, 045137 (2025) https://doi.org/10.1103/PhysRevB.111.045137

    work page doi:10.1103/physrevb.111.045137 2025

  24. [24]

    Mehrmann, V., Voss, H.: Nonlinear eigenvalue problems: A challenge for modern eigenvalue methods.GAMM-Mitteilungen27(2), 121–152 (2004)https://doi.org/ 10.1002/gamm.201490007

    work page doi:10.1002/gamm.201490007 2004

  25. [25]

    Neumaier, A.: Residual inverse iteration for the nonlinear eigenvalue problem. SIAM J. Numer. Anal.22(5), 914–923 (1985) https://doi.org/10.1137/0722055

    work page doi:10.1137/0722055 1985

  26. [26]

    Polizzi, E.: Density-matrix-based algorithms for solving eigenvalue problems. Phys. Rev. B79, 115112 (2009) https://doi.org/10.1103/physrevb.79.115112

    work page doi:10.1103/physrevb.79.115112 2009

  27. [27]

    arXiv preprint arXiv:1901.01188 (2019) https:// doi.org/10.48550/arXiv.1901.01188

    Saad, Y., El-Guide, M., Międlar, A.: A rational approximation method for the nonlinear eigenvalue problem. arXiv preprint arXiv:1901.01188 (2019) https:// doi.org/10.48550/arXiv.1901.01188

    work page doi:10.48550/arxiv.1901.01188 1901

  28. [28]

    Sakurai, T., Sugiura, H.: A projection method for generalized eigenvalue problems using numerical integration. J. Comput. Appl. Math.159(1), 119–128 (2003) https://doi.org/10.1016/S0377-0427(03)00565-X

    work page doi:10.1016/s0377-0427(03)00565-x 2003

  29. [29]

    Stabilizing the Rayleigh--Ritz procedure by randomization

    Shao, N.: Stabilizing the Rayleigh–Ritz procedure by randomization. arXiv preprint arXiv:2604.01037 (2026) https://doi.org/10.48550/arXiv.2604.01037

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2604.01037 2026

  30. [30]

    Tang, P.T.P., Polizzi, E.: FEAST as a subspace iteration eigensolver accelerated by approximate spectral projection. SIAM J. Matrix Anal. Appl.35(2), 354–390 (2014) https://doi.org/10.1137/13090866X

    work page doi:10.1137/13090866x 2014

  31. [31]

    Tisseur, F., Higham, N.J.: Structured pseudospectra for polynomial eigenvalue problems, with applications. SIAM J. Matrix Anal. Appl.23(1), 187–208 (2001) https://doi.org/10.1137/S0895479800371451

    work page doi:10.1137/s0895479800371451 2001

  32. [32]

    Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, USA (2013)

    Trefethen, L.N.: Approximation Theory and Approximation Practice. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, USA (2013). https://doi.org/10.1137/1.9781611975949

    work page doi:10.1137/1.9781611975949 2013

  33. [33]

    Van Beeumen, R., Meerbergen, K., Michiels, W.: Compact rational Krylov meth- ods for nonlinear eigenvalue problems. SIAM J. Matrix Anal. Appl.36(2), 820–838 (2015) https://doi.org/10.1137/140976698 26

    work page doi:10.1137/140976698 2015

  34. [34]

    Zhu, P., Knyazev, A.V.: Angles between subspaces and their tangents. J. Numer. Math.21(4), 325–340 (2013) https://doi.org/10.1515/jnum-2013-0013 27

    work page doi:10.1515/jnum-2013-0013 2013