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Symbolic and Numerical Tools for $L_{\infty}$-Norm Calculation

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Pith's one-line read Symbolic methods compute L-infinity norms exactly but slowly, and they outperform numerical methods on ill-conditioned or parameterized systems.

arxiv 2505.13980 v1 pith:HI5ZEEJS submitted 2025-05-20 math.OC cs.MScs.SCcs.SYeess.SY

Grace Younes , Alban Quadrat , Fabrice Rouillier This is my paper · ORCID
classification math.OCcs.MScs.SCcs.SYeess.SY
keywords computationinftysymbolicnormnumericalparticularlyanalysisapproaches
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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The reading

The L-infinity norm of a linear system measures its worst-case response to inputs: the largest amplification the system can ever produce. Control engineers use it to judge stability and robustness. Standard numerical methods, like bisection on frequency grids or eigenvalue searches, usually give good answers, but they can fail when the system is nearly singular, has huge or tiny coefficients, or depends on parameters. This paper reviews a different, exact approach. Instead of sampling frequencies, symbolic methods translate the norm into a polynomial equation and then use tools from algebraic geometry: Sturm-Habicht sequences count real roots exactly, rational univariate representations solve polynomial systems, and cylindrical algebraic decomposition handles parameters. The authors demonstrate these tools on small examples and benchmark them against Matlab and Maple. They find that symbolic methods return exact, certified values and remain reliable for extremely small damping values, but are far slower: a 5 by 5 system can take minutes to hours, while numerical methods take seconds. The paper concludes that symbolic methods are best used as a complement to numerical methods, for validation and for parameterized small systems, not as a replacement for large-scale computation.
Extended reading notes

Core claim

The paper's central comparative statement is that symbolic methods are 'robust but computationally expensive' and deliver certified accuracy where numerical methods 'can yield inaccurate results and sometimes fail' (Section 4, Table 1). If true, symbolic computation is a reliable complement to numerical L-infinity norm computation for ill-conditioned or parameterized systems.

Load-bearing premise

The benchmark conclusions rest on the assumption that the reported numerical failures (e.g., Maple NormHinf returning 1.32 in Example 10, Matlab reporting infinity in Example 9) are inherent limitations of numerical methods rather than artifacts of specific implementations or default settings. If those failures are fixable by tuning the solvers, the paper's contrast between symbolic and numerical reliability is overstated.

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Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted in the paper; all quantities are either inputs or computed exactly. The paper relies on standard algebraic facts and on propositions from prior work, cited as axioms. No new physical entities are introduced.

assumptions (4)
  • standard math The maximum modulus principle applies to functions in RH_infinity, so the L-infinity norm over the right half-plane equals the supremum over the imaginary axis.
    Invoked in Section 2.1 to reduce the norm to a frequency-domain supremum; cited to standard references, not proven in the paper.
  • domain assumption Proposition 1 (Kanno and Smith 2006): gamma > ||G||_infinity if and only if gamma > sigma-bar(G(i infinity)) and det(Phi_gamma(i omega)) != 0 for all real omega.
    Basis for the symbolic reformulation in Section 3.1; accepted from the cited literature without proof.
  • domain assumption Proposition 2 (Bouzidi et al. 2021): the L-infinity norm is the maximum of the gamma-projections of the real solutions of the critical system and of the leading coefficient roots.
    Transfers L-infinity norm computation to real root finding for a bivariate polynomial system; cited from prior work.
  • standard math The specialization property of subresultants and Sturm-Habicht sequences holds under ring homomorphisms.
    Used in Section 3.2 to specialize the Sturm-Habicht sequence to parameter values; cited from Gonzalez-Vega et al. 1990/1994.

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Pith. "Pith review of Symbolic and Numerical Tools for $L_{\infty}$-Norm Calculation." pith.science (2026). https://pith.science/paper/HI5ZEEJS

@misc{pith2026250513980,
  author       = {Pith},
  title        = {Pith review of: Symbolic and Numerical Tools for $L_\infty$-Norm Calculation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HI5ZEEJS}},
  note         = {Machine review of arXiv:2505.13980}
}
abstract

The computation of the $L_\infty $-norm is an important issue in $H_{\infty}$ control, particularly for analyzing system stability and robustness. This paper focuses on symbolic computation methods for determining the $L_{\infty} $-norm of finite-dimensional linear systems, highlighting their advantages in achieving exact solutions where numerical methods often encounter limitations. Key techniques such as Sturm-Habicht sequences, Rational Univariate Representations (RUR), and Cylindrical Algebraic Decomposition (CAD) are surveyed, with an emphasis on their theoretical foundations, practical implementations, and specific applicability to $ L_{\infty} $-norm computation. A comparative analysis is conducted between symbolic and conventional numerical approaches, underscoring scenarios in which symbolic computation provides superior accuracy, particularly in parametric cases. Benchmark evaluations reveal the strengths and limitations of both approaches, offering insights into the trade-offs involved. Finally, the discussion addresses the challenges of symbolic computation and explores future opportunities for its integration into control theory, particularly for robust and stable system analysis.

Figures

Figures reproduced from arXiv: 2505.13980 by the authors.

Figure 1
Figure 1. Regions and root functions for ξ > 0. JuliaTime column reports execution time of the sym￾bolic method via PACE.jl on Julia. The matrices and benchmark codes are available upon request. Note that the JuliaTime column reports only execution times; the computed H∞-norm values coincide exactly with those of MapSymNorm column. Size – Degree MatNumNorm MapNumNorm MapSymNorm (Time (s)) JuliaTime (s) 2×2 – 2 3.0116 3.0116 3… view at source ↗

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Works this paper leans on

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