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Approximation of the Pseudospectral Abscissa via Eigenvalue Perturbation Theory

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Pith's one-line read The pseudospectral abscissa is governed by a first-order eigenvalue-perturbation formula for small $\epsilon$, with an $O(\epsilon^3)$ refinement for matrices.

desk verdict A solid, honest paper that makes perturbation-theory-based pseudospectral abscissa approximation practical; the local results are rigorous and the global convergence claim is explicitly heuristic, with the failure rate disclosed. read the letter →

arxiv 2506.05535 v1 pith:LVWV3GRX submitted 2025-06-05 math.NA cs.NA

Waqar Ahmed , Emre Mengi This is my paper
classification math.NAcs.NA MSC 65F1593D0965H1730C15
keywords pseudospectralabscissapseudospectrumeigenvalueperturbationtheorynonlinearproblemsfixed-pointiterationlarge-scalematrixcomputationsquadraticpolynomialsbackwarderror
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's target is the $\epsilon$-pseudospectral abscissa: the largest real part among eigenvalues of all matrices (or matrix-valued functions) within distance $\epsilon$. Its central claim is that for small $\epsilon$ this worst-case point is captured by first-order eigenvalue perturbation theory: each eigenvalue $\mu_0$ of the unperturbed problem moves right at a rate equal to a weighted sum of coefficient magnitudes divided by the left--right eigenvector normalization, so $\alpha_\epsilon(T)$ equals the maximum, over all eigenvalues, of $\operatorname{Re}\mu_0 + \epsilon\cdot(\text{that rate})$, up to $O(\epsilon^2)$. In the matrix case the paper derives a second-order refinement with error $O(\epsilon^3)$, expressed as the spectral abscissa of a specifically constructed rank-one perturbation of $A$. These estimates matter because existing large-scale algorithms for the pseudospectral abscissa are only locally convergent; using the perturbation estimates as starting points, the paper's fixed-point iterations converge to globally rightmost points in the large majority of the numerical experiments.

What carries the argument

The engine of the paper is the derivative of a simple eigenvalue under perturbation, $\mu'(0;\Delta)=-y^*\Delta T(\mu_0)x/(y^*T'(\mu_0)x)$, whose real part is maximized over unit-norm $\Delta$ by the rank-one update $\Delta T_j=w_j t_j(\mu_0)yx^*\big/\sqrt{\sum_\ell w_\ell^2|t_\ell(\mu_0)|^2}$, with the eigenvectors normalized so that $y^*T'(\mu_0)x$ is real and negative. That gives the $O(\epsilon^2)$ estimate. For matrices, a second-order Taylor expansion in the Frobenius norm, with finite-difference corrections to the eigenvectors, produces the rank-one matrix $\Delta^{\mu_0}_*$ whose rightmost eigenvalue carries the $O(\epsilon^3)$ estimate. The fixed-point iterations are driven by the backward-error identity $\varphi(z)=\sigma_{\min}(T(z))\big/\sqrt{\sum_j w_j^2|t_j(z)|^2}$, the minimal perturbation norm needed to make $z$ an eigenvalue; at each step one forms the minimal perturbation that makes the current point $z$ an eigenvalue and then moves to the rightmost eigenvalue of the perturbed problem. The fixed-point analysis characterizes limits as rbvt points, meaning the smallest singular value is simple, $z$ lies on the boundary of $\Lambda_\epsilon$, and the singular-vector projection $u^*v$ is real and positive.

What would settle it

Take a matrix with a defective repeated eigenvalue at the rightmost spectral position, so the left and right eigenvectors are orthogonal and the perturbation-theory denominator vanishes, and compare the right side of (16), or its matrix specialization, with a direct computation of $\alpha_\epsilon(A)$ for several small $\epsilon$; if the estimate is undefined or does not track $\alpha_\epsilon(A)$ to order $\epsilon^2$, the simplicity assumption is load-bearing. A second check is to record, for the failures reported in Section 7.2.3 at $\epsilon=0.5$ with $N=1$, whether the initializer (40) lies outside the connected component of $\Lambda_\epsilon(A)$ containing the global rightmost point.

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Extended reading notes

Core claim

The paper establishes two asymptotic identities. For a regular analytic matrix-valued function $T(\lambda)=\sum_j t_j(\lambda)T_j$ with weighted perturbations, if every eigenvalue is simple and isolated and $\epsilon$ is small enough, then $\alpha_\epsilon(T)=\max_{\mu_0\in\Lambda(T)}\bigl(\operatorname{Re}(\mu_0)+\epsilon\,\sqrt{\sum_j w_j^2|t_j(\mu_0)|^2}\big/|y_{\mu_0}^*T'(\mu_0)x_{\mu_0}|\bigr)+O(\epsilon^2)$, where $x_{\mu_0},y_{\mu_0}$ are unit right and left eigenvectors. For a matrix $A$, using Frobenius-norm perturbations and a finite-difference correction to the eigenvectors, the paper proves $\alpha_\epsilon(A)=\max_{\mu_0\in\Lambda(A)}\alpha(A+\epsilon\,\Delta^{\mu_0}_*)+O(\epsilon^3)$, with $\Delta^{\mu_0}_*$ an explicit rank-one matrix built from the left and right eigenvectors of $\mu_0$ and of $A+h y_{\mu_0}x_{\mu_0}^*$. The paper further shows that the fixed-point maps built from these perturbation directions have a useful property: any nondegenerate limit must be an rbvt point, meaning a right boundary point with a vertical tangent, and the unique global rightmost point is always a fixed point. Numerically, initializing these locally convergent iterations with the perturbation-theory estimates returns the same values as globally convergent criss-cross algorithms on the tested quadratic matrix polynomials and matrices, in a fraction of the time.

Load-bearing premise

The load-bearing premise is that each eigenvalue is distinct and stays distinct under perturbation, with $\epsilon$ small enough that every eigenvalue follows one smooth path; the reported global convergence of the fixed-point iterations also leans on the unproved heuristic that the perturbation-theory starting point lies in the same pseudospectrum region as the global rightmost point.

Editorial extensions

If this is right

  • For small $\epsilon$, $\alpha_\epsilon(T)$ can be estimated from one full eigenvalue and eigenvector computation plus a closed-form scalar evaluation per eigenvalue, with no criss-cross or pseudospectral boundary iteration.
  • In the matrix case the $O(\epsilon^3)$ formula needs, for each eigenvalue, only one additional eigenvector solve at $A+h y_{\mu_0}x_{\mu_0}^*$ before computing the rightmost eigenvalue of a rank-one-updated matrix; this is the basis of the reported large-scale speedups.
  • Theorem 5.1 gives a provable reason the initialization works: when the first-order model is exact, the maximizing eigenvalue and a global rightmost point lie in the same connected component of $\Lambda_\epsilon(T)$.
  • If the fixed-point iteration converges to a nondegenerate point, that point is always an rbvt point; a second-derivative check at such a point decides whether it is locally rightmost.
  • On the tested quadratic matrix polynomials and on dense and sparse matrices, the initialized iterations match the values returned by globally convergent criss-cross algorithms while using fewer iterations and less runtime.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The first-order metric $\operatorname{Re}\mu_0+\epsilon/|y_{\mu_0}^*x_{\mu_0}|$ could itself be used as a screening function: for large sparse $A$, a Krylov method that targets the few largest values of this metric would bypass the full-spectrum bottleneck the paper identifies in its concluding remarks.
  • The rbvt characterization suggests replacing the fixed-point iteration by a Newton-type boundary tracker on the two equations $\sigma_{\min}(T(z))\sqrt{\sum_j w_j^2|t_j(z)|^2}=\epsilon$ and $S(z)=0$, which could converge quadratically instead of linearly.
  • The matrix second-order formula is derived with Frobenius-norm perturbations; the same two-term Taylor argument can likely be specialized to structured perturbations such as real, sparse, or banded ones, yielding structured pseudospectral abscissa estimates with the same $O(\epsilon^3)$ order.
  • Table 10's restart experiments quantify how often the perturbation-theory initializer misses the global component; an adaptive restart rule based on gaps in the first-order metric, rather than a fixed $N$, would make the heuristic self-certifying.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper develops eigenvalue-perturbation-based approximations and fixed-point iterations for the pseudospectral abscissa. For a general nonlinear eigenvalue problem, a first-order formula with O(ε^2) error is derived (Theorem 2.3, Corollary 2.1), and for the standard eigenvalue problem a second-order formula with claimed O(ε^3) error is given (Theorem 4.2). Two fixed-point iterations (Algorithms 1 and 3) and their matrix specialization (Algorithm 4) are proposed, with fixed-point analyses showing that convergent nondegenerate limits must be rbvt points (Theorems 3.3, 3.4, 4.3). Section 5 discusses initialization using the perturbation formulas, Theorem 5.1 proving that the exact maximizer μ* of R(ε; μ0) lies in a connected component containing a global rightmost point. Numerical experiments on damping problems, stabilization examples, and EigTool matrices compare favorably with criss-cross and Guglielmi–Overton methods, including a 1000-matrix dataset reporting success rates in Table 10. MATLAB implementations are publicly available.

Significance. If the central results hold, the paper provides a cheap and accurate way to estimate global pseudospectral abscissae in the small-ε regime, and a new initialization strategy that appears to make locally convergent fixed-point iterations find the global rightmost point in a vast majority of tested cases. The second-order O(ε^3) formula for matrices is a new contribution. The paper is careful to state its assumptions (simplicity, small ε, nondegeneracy) and does not overclaim a global-convergence theorem. It also ships reproducible MATLAB code and validates against independent implementations (criss-cross, GO), which strengthens the empirical claims. The main weaknesses are the sketch-level derivation of the second-order formula and the unproved component-selection heuristic underlying the practical global claim; both are acknowledged or implicitly present, but need more explicit treatment.

major comments (2)
  1. [§4.2, Eqs. (33)–(36), Theorem 4.2] The derivation of the O(ε^3) approximation rests on several unquantified approximation steps: the maximizer is claimed to be of the form Δ* = yx* + O(ε), the eigenvector derivatives are replaced by finite differences with step h = O(ε), and the resulting matrix eΔ* is asserted to achieve the maximum up to O(ε^2) in the scaled objective. The text jumps from these approximations to the final O(ε^3) error without proving that the O(ε) errors in eΔ* and in the finite-difference derivative estimates do not propagate into O(ε^2) errors in Re{μ(ε; eΔ*)}. A detailed error analysis, for instance using first-order optimality and Lipschitz continuity of the eigenvector derivatives under the simplicity assumption, is needed to justify the O(ε^3) claim. This is load-bearing for the second-order formula (37) and for the hybrid initialization strategy built on it.
  2. [§5, Theorem 5.1 vs. Eqs. (39)–(40)] Theorem 5.1 guarantees that an exact maximizer μ* of R(ε; μ0) lies in a connected component of Λε(T) containing a global rightmost point, but the implemented initializer eμ* is the maximizer of the first-order surrogate in (39) or (40), not of R(ε; μ0). No result bounds the distance between eμ* and μ* or ensures that eμ* lands in the same component; the paper's own Table 10 shows failure rates up to 7.5% at ε = 0.5 with N = 1, which is consistent with this gap. Since the paper's practical claim is explicitly only that the method converges in a 'vast majority' of cases, this is not a fatal flaw, but the limitation should be stated openly in Section 5 (rather than left as 'seems plausible') and the abstract/conclusion should be worded to keep the empirical and proven statements distinct.
minor comments (6)
  1. [Abstract] The phrase 'built on the the perturbation theory ideas' contains a duplicated 'the'.
  2. [§2.3, proof of Theorem 2.3] In the sentence 'there must be (ΔT1, ..., ΔTκ) ∈ Sε such that z* ∈ Λε(T + ΔT)', the set should be Λ(T + ΔT), the spectrum of the perturbed function, not the ε-pseudospectrum.
  3. [§5, Theorem 5.1] The notation 'arg max_{μ0∈Λε(T)} R(ε; μ0)' is inconsistent with the domain of R(ε; ·), which is Λ(T); it should read 'arg max_{μ0∈Λ(T)} R(ε; μ0)'.
  4. [§7.2.2] The word 'Morover' is a typo for 'Moreover'.
  5. [References] Reference [22] spells the publisher as 'Princetion' (should be 'Princeton'), and reference [5] gives 'John Hopkins' (should be 'Johns Hopkins').
  6. [Proof of Theorem 3.3] In the interior case, 'Take any vector bv ∈ C' should read 'bv ∈ C^n'; the same notational issue appears with the expansion coefficients c, c2, ..., cn.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the perturbation-theory estimates are derived from first-order eigenvalue expansions and are validated against independent algorithms.

full rationale

The paper's central derivation is self-contained. The first-order estimate in Corollary 2.1 and Theorem 2.3 is obtained by differentiating the perturbed eigenvector equation (equation (10)), maximizing the resulting derivative L'_Delta(0) over the compact perturbation set S (Theorem 2.2), and then proving alpha_epsilon(T) = max_{mu0} R(epsilon; mu0) by elementary set-inclusion arguments that do not presuppose the estimate. The matrix second-order formula in Theorem 4.2 is derived by an explicit Taylor expansion of the analytic eigenvalue function, with the O(epsilon^3) error tracked through equations (33)-(35); no parameter is fitted to alpha_epsilon(A). The numerical validation compares against the independent criss-cross algorithms of Burke-Lewis-Overton and Mehrmann-Mengi and the Guglielmi-Overton fixed-point iteration, and the self-citations ([1], [2], [11]) are software releases, benchmark problems, or a standard singular-value derivative lemma that does not contain the target result. The practical initialization from (39)/(40) is explicitly presented as an empirical heuristic, and the paper reports its failure rates (e.g., Table 10, 7.5% failures at epsilon=0.5 with N=1), so the gap between local convergence and observed global behavior is a stated limitation rather than a circular reduction of the main claims.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The central claim depends on standard complex analysis (Rouché's theorem, implicit function theorem), explicit simplicity and small-ε assumptions, and two algorithmic controls (restart count N and step h). No weights or constants are fitted to α_ε, and no invented entities are introduced.

free parameters (2)
  • restart count N = N=1 default; N=7 used in Table 10
    User-set parameter controlling how many eigenvalue-based initializations are tried. Accuracy improves with N (Table 10), but N is an algorithm control, not a constant fitted to data.
  • finite-difference step h = half machine precision in Remark 4.1
    Used to approximate eigenvector derivatives in the O(ε³) formula (36). The theorem only requires h=O(ε), and the implementation chooses a very small h; again an algorithmic choice, not a fitted scientific constant.
assumptions (7)
  • standard math Rouché's theorem
    Used in Theorem 2.1 to guarantee that a unique simple eigenvalue persists under small perturbations.
  • standard math Analytic implicit function theorem and Rellich's theorem
    Used throughout to obtain analytic eigenvalue and eigenvector functions μ(η), x(η), y(η), e.g., Section 2.1 and Section 4.2.
  • domain assumption All eigenvalues of T (or A) are simple and isolated
    Assumed in Theorem 2.3, Theorem 4.1, and Theorem 4.2; no treatment is given for repeated or defective eigenvalues.
  • domain assumption ε is small enough that [0,ε]⊂U for every eigenvalue, and Assumption 2.1 attainment
    Restricts the O(ε²)/O(ε³) expansions to the asymptotic regime; for larger ε the formulas are used heuristically to initialize iterations.
  • standard math Frobenius-norm and 2-norm pseudospectra of a matrix coincide
    Section 4.2 replaces 2-norm perturbations by Frobenius-norm perturbations; this is true because the minimal perturbation is rank-one with both norms equal to σ_min.
  • standard math Eckart-Young theorem
    Used in Theorem 3.1 and in the derivation of Algorithm 4 to identify the minimal perturbation that makes a given z an eigenvalue.
  • domain assumption Nondegeneracy of fixed points
    Theorems 3.3, 3.4, and 4.3 require the smallest singular value to be simple and S(z)≠0; otherwise the fixed-point update can be undefined.

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Pith. "Pith review of Approximation of the Pseudospectral Abscissa via Eigenvalue Perturbation Theory." pith.science (2026). https://pith.science/paper/LVWV3GRX

@misc{pith2026250605535,
  author       = {Pith},
  title        = {Pith review of: Approximation of the Pseudospectral Abscissa via Eigenvalue Perturbation Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LVWV3GRX}},
  note         = {Machine review of arXiv:2506.05535}
}
abstract

Reliable and efficient computation of the pseudospectral abscissa in the large-scale setting is still not settled. Unlike the small-scale setting where there are globally convergent criss-cross algorithms, all algorithms in the large-scale setting proposed to date are at best locally convergent. We first describe how eigenvalue perturbation theory can be put in use to estimate the globally rightmost point in the $\epsilon$-pseudospectrum if $\epsilon$ is small. Our treatment addresses both general nonlinear eigenvalue problems, and the standard eigenvalue problem as a special case. For small $\epsilon$, the estimates by eigenvalue perturbation theory are quite accurate. In the standard eigenvalue case, we even derive a formula with an ${\mathcal O}(\epsilon^3)$ error. For larger $\epsilon$, the estimates can be used to initialize the locally convergent algorithms. We also propose fixed-point iterations built on the the perturbation theory ideas for large $\epsilon$ that are suitable for the large-scale setting. The proposed fixed-point iterations initialized by using eigenvalue perturbation theory converge to the globally rightmost point in the pseudospectrum in a vast majority of the cases that we experiment with.

Figures

Figures reproduced from arXiv: 2506.05535 by the authors.

Figure 1
Figure 1. Approximation of αϵ(P(·; ν)) with Rϵ(P(·; ν)) for the damping optimization problem in Example 2.1. For the plot on the left, ϵ = 0.1. Cint = 2ξM1/2 √ M−1/2KM−1/2M1/2 , ξ = 0.005 from [11, Example 5.2], and with the weights wm = wc = wk = 1. The example concerns a mass-spring-damper system consisting of twenty masses tied together through springs and with a damper on the second mass. The parameter ν is a nonnegative … view at source ↗
Figure 2
Figure 2. The boundary of Λϵ(P) for the ma￾trix polynomial P in Example 3.1. Black and red dots represent eigenvalues and computed right￾most points in Λϵ(P). ϵ Alg. 1 Alg. 3 0.1 7 10 0.2 9 13 k Alg. 1 Alg. 3 0 −0.03863 + 7.72651i −0.03863 + 7.72651i 1 0.30088 + 7.70127i 0.30310 + 7.73108i 2 0.30492 + 7.75204i 0.30308 + 7.75194i 3 0.30493 + 7.75204i 0.30492 + 7.75193i 4 0.30493 + 7.75204i 0.30492 + 7.75204i k Alg. 1 Alg. 3 0 … view at source ↗
Figure 3
Figure 3. The errors of the first-order approximation [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The approximation of αϵ(A(ν)) with R (2) ϵ (A(ν)) for ν ∈ [−1, 1], where A(ν) = A+νBCT is the NN18 example from the COMPleib collection. In the left-plot, ϵ = 0.2. approximation ideas of the previous two subsections can be employed to initialize the fixed-point iterati…
Figure 5
Figure 5. Figure 5: The boundary of Λϵ(A) for the matrix A from the NN18 example. Black and red dots correspond to eigenvalues and computed right￾most points in Λϵ(A). k Alg. 4 0 −1.00000 + 400.00000i 1 38.82335 + 400.00000i 2 39.00000 + 400.00000i k Alg. 4 0 −1.00000 + 400.00000i 1 78.64…
Figure 6
Figure 6. Figure 6: This figure concerns the one-parameter damping problem in §7.1.1. (Left) The computed [PITH_FULL_IMAGE:figures/full_fig_p031_6.png]
Figure 7
Figure 7. Figure 7: The figure concerns the two-parameter damping problem in §7.1.2. (Left) Plot of [PITH_FULL_IMAGE:figures/full_fig_p032_7.png]
Figure 8
Figure 8. Figure 8: The plot of computed αϵ(ν) by Algorithm 4 (left plot) and its error (right plot) as a function of ν = (ν1, ν2) ∈ [−1, 1] × [−1, 1] for the HF1 example in §7.2.1. error time iterations Example αϵ(A) Alg. 4 GO Alg. 4 GO CC Alg. 4 GO grcar 3.1252 -2.0e-07 -5.0e-07 0.38 2.…

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