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AUTHOR MANUSCRIPT · SIAM JOURNAL FORMAT · 2026

Preprint · Not peer reviewed or journal accepted

Residual-Budgeted Inexact Contour Moments

for Nonlinear Eigenvalue Problems

Accuracy control for contour-based nonlinear eigenvalue solvers using physical residuals, extraction-aware error budgets, and shared-basis cluster allocation.

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ABSTRACT

From physical residuals to extraction-level accuracy budgets

Contour-integral solvers for analytic nonlinear eigenvalue problems require linear solves at many quadrature nodes, often using shared infinite-GMRES bases. We develop an accuracy-control framework for centered Beyn extraction that converts physical node residuals into computable moment, subspace, and eigenvalue perturbation budgets. The weights reflect the quadrature rule, local resolvent behavior, and extraction objective, enabling node- and cluster-aware work allocation. A nested quadrature comparison estimates the discretization scale, while a singular-subspace guard prevents premature stopping. Reproducible experiments on a nonnormal analytic family and two NLEVP benchmarks empirically support the bounds and show that safeguarded stopping preserves the observed accuracy plateau with less counted Arnoldi-prefix work.
Paired synthetic traces1,10513 levels · 5 seeds · 17 prefixes
Stopping decisions224Hadeler and loaded string
Counted work reduction40%Loaded string · nominal safeguard
Over-relaxation penalty2,294×Loaded-string eigenvalue error

KEY CONTRIBUTIONS

A residual-budgeted theory for inexact contour moments

The analysis keeps solver error, discretization scale, extraction sensitivity, and shared-basis work allocation distinct.
01 · MOMENT CONTROL

Physical residual identity

Derives a residual-to-centered-moment identity and a resolvent-weighted bound for block analytic nonlinear eigenvalue problems.

02 · EXTRACTION

Direction-sensitive budgets

Connects inexact moments to Keldysh subspace perturbations and an exact reduced-operator identity for eigenvalue enclosure.

03 · STOPPING

Two independent guards

Balances solve error against nested quadrature variation while a singular-subspace guard blocks premature rank decisions.

04 · ALLOCATION

Node and cluster awareness

Allocates tolerances across contour nodes and shared infinite-GMRES bases using extraction-aware contribution weights.

ANALYSIS PIPELINE

Five layers from node solves to reported eigenvalues

Each layer has a separate diagnostic, so a small linear residual is not mistaken for a reliable extracted spectrum.
  1. 01

    Physical residuals

    Evaluate the original nonlinear system residual at every contour node, rather than relying on an internal Krylov proxy.

  2. 02

    Moment propagation

    Weight nodewise residuals by quadrature coefficients, centered monomials, and local inverse-operator sensitivity.

  3. 03

    Subspace guard

    Track the extraction-relevant singular subspace and refuse stopping when its separation is too fragile.

  4. 04

    Reduced operator

    Propagate moment error through the centered Beyn reduction to obtain a structured eigenvalue perturbation budget.

  5. 05

    Shared-basis work

    Allocate the remaining budget across nodes and expansion-point clusters that share infinite-GMRES prefixes.

REPRODUCIBLE EXPERIMENTS

Stress tests reveal where residual-only stopping fails

Evidence spans a controlled nonnormal analytic family and the Hadeler and loaded-string NLEVP benchmarks. The figures below are taken from the manuscript experiment pipeline.
Sensitivity of residual, subspace angle, and eigenvalue error to increasing nonnormality
Nonnormality sweep. At prefix m = 24, increasing cond₂(S) from 1 to 10⁶ raises the median solve-induced subspace-angle sine from 5.30×10⁻⁶ to 8.49×10⁻¹. Residual matching reduces the endpoint angle to 3.02×10⁻⁴.
Heatmaps of stopping-rule sensitivity for Hadeler and loaded-string benchmarks
Stopping sensitivity. Across 224 decisions, nominal safeguards save 16.7% of counted prefix work for Hadeler and 40% for loaded string while retaining longest-prefix accuracy.
Node and cluster contribution concentration for the loaded-string benchmark
Contribution heterogeneity. One of eight loaded-string clusters carries 83.8%–90.9% of the evaluated residual bound; the eight largest node terms carry 86.5%–97.8%.
INTERPRETATION BOUNDARY

Reported savings are retrospective counts of retained Arnoldi-prefix iterations, not measured wall-clock speedups. Experimental resolvent bounds use double-precision singular values and are numerical-oracle bounds, not outward-rounded certificates.

CITATION & STATUS

Cite this work as a manuscript/preprint

This page is a stable portfolio record and download location. It is not a journal DOI record.
PLAIN-TEXT CITATION

Y. Liu, Y. Zhan, J. E. Roman, and M. Shao, Residual-budgeted inexact contour moments for nonlinear eigenvalue problems, manuscript/preprint, 2026. Available at: https://yaoyaozhan.site/papers/rbicm-2026.

BIBTEX
@unpublished{liu2026residual,
  author = {Yuqi Liu and Yaoyao Zhan and Jose E. Roman and Meiyue Shao},
  title  = {Residual-Budgeted Inexact Contour Moments for Nonlinear Eigenvalue Problems},
  note   = {Manuscript/preprint},
  year   = {2026},
  url    = {https://yaoyaozhan.site/papers/rbicm-2026}
}
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