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Vector Fitting

T0 review · 1 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Vector Fitting solves rational approximation from frequency samples by iterating linear least squares and relocating poles in closed form.

desk verdict An honest, well-crafted tutorial on Vector Fitting that breaks no new ground but delivers a usable reference with code and worked examples. read the letter →

arxiv 1908.08977 v1 pith:TPSE56GP submitted 2019-08-23 physics.comp-ph

Piero Triverio This is my paper · ORCID
classification physics.comp-ph
keywords VectorFittingrationalapproximationreduced-ordermodelfrequency-domainsamplingSanathanan-Koerneriterationpolerelocationlinearleastsquaresdata-drivenmodeling
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

This chapter presents Vector Fitting as a practical method for building reduced-order models of linear time-invariant systems from sampled frequency responses, without access to the system's equations. The technical problem is rational approximation of the measured transfer function, a nonlinear least-squares problem that is hard to solve directly; Vector Fitting replaces it with a sequence of linear least-squares problems whose poles are relocated in closed form at each iteration. The chapter's claim is that, when implemented with the pole initialization, stability enforcement, and order-adaptation routines it describes, the method is robust enough for real engineering use, typically converging in four or five iterations. This matters because a reduced model obtained this way can be converted to state-space, pole-residue, or equivalent-circuit form and used in simulation, so engineers and biomedical researchers can model devices known only through measurements.

What carries the argument

The load-bearing mechanism is the partial-fraction basis $1/(s-p_n^{(0)})$ with a pole set that is relocated at every iteration, combined with the eigenvalue update $\{p_n^{(i)}\} = \mathrm{eig}(A^{(i-1)} - b_w (c_w^{(i)})^T)$. Whereas the earlier Sanathanan-Koerner iteration used monomials $s^n$ and an explicit, often ill-conditioned frequency-dependent weight, Vector Fitting keeps condition numbers under control by using partial fractions and applies the weight implicitly through pole relocation. The matrix $A^{(i-1)}$ is the diagonal matrix of previous poles, $b_w$ is a vector of ones, and $c_w^{(i)}$ holds the weighting coefficients found by the linear least-squares solve; the eigenvalues of this rank-one perturbed diagonal matrix are exactly the zeros of the new denominator, hence the new poles. This machinery turns a nonlinear rational fit into repeated linear solves, with a final residue-only least-squares fit over the converged poles providing the model.

What would settle it

The claim would be refuted by a physically measured frequency response on which Algorithm 3.1, with standard pole initialization and a generous iteration limit, fails to meet the user's error threshold, for example by stalling at an error floor or by cycling between pole estimates while the linearized cost (17) keeps decreasing. A reader can test this by running the chapter's open-source implementation on a corpus of measured impedance or scattering-parameter datasets and counting convergence failures.

Watch

Extended reading notes

Core claim

The central claim is that the Vector Fitting iteration solves the rational approximation problem (4)—find a rational $\tilde H(s)$ matching samples $H_k = H(\mathrm{j}\omega_k)$—by minimizing a linearized least-squares error (17) instead of the original nonlinear error (6). At each iteration, the numerator and a weighting function are built from partial fractions over the previous poles, and the weighting coefficients $c_w^{(i)}$ feed the eigenvalue update $\{p_n^{(i)}\} = \mathrm{eig}(A^{(i-1)} - b_w (c_w^{(i)})^T)$, equation (18), which relocates the poles. On convergence the weight $w^{(i)}(s)$ tends to 1, so the linearized objective becomes the true nonlinear objective. The chapter maintains that this procedure, implemented as the given pseudocode, produces a stable, causal rational model for single-input and multi-input systems, with a final residue-only fit (29) over the converged poles used both as convergence test and as the actual model.

Load-bearing premise

The load-bearing premise is that the VF iteration, though not covered by a convergence theorem, will in practice reach an acceptable fit on real data within the user's iteration limit; the chapter concedes in Section 3.2 that contrived examples show convergence is not guaranteed.

Editorial extensions

If this is right

  • An engineer can implement Algorithm 3.1 and obtain a reduced-order model from frequency samples alone, then convert the pole-residue form (58) into state-space, impulse-response, or equivalent-circuit representations for simulation.
  • The same iteration covers multiple inputs and outputs by fitting all transfer-function entries with a shared pole set, and the fast variant in Section 3.6 reduces the cost enough to handle systems with hundreds of ports.
  • Stability and causality can be enforced during the iterations, and passivity can be imposed afterwards, so the resulting model is usable in time-domain transient simulations of circuits and interconnects.
  • With noisy measurement data, convergence slows and the achievable error is bounded by the noise floor; the adding-and-skimming and relaxed-normalization variants are the chapter's standard remedies.

Reading between the lines

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

  • Editorial extension: the same pole-relocation idea could be adapted to other rational fitting problems, such as spectral densities, filter design, or parameter-dependent models, where the denominator nonlinearity is the main obstacle.
  • Editorial extension: the chapter's practical-convergence claim rests on anecdotal experience; a quantitative benchmark across many measured datasets, recording the distribution of iterations to convergence and failure rates, would make it a testable statistical statement.
  • Editorial extension: because convergence is not guaranteed, one could combine VF with a multistart strategy over different initial pole distributions to increase robustness; the chapter does not explore such a safeguard.
  • Editorial extension: the explicit admission that no strong convergence theory exists suggests a useful theoretical target is a local convergence result under a noise or data-quality model, with the eigenvalue update (18) providing a concrete map to analyze.
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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

1 major / 6 minor

Summary. This manuscript is a draft book chapter on the Vector Fitting (VF) algorithm for constructing reduced-order models of linear time-invariant systems from sampled frequency responses. After motivating the data-driven reduction problem, the chapter reviews the Levy and Sanathanan–Koerner approaches, then derives VF as an iteratively reweighted linear least-squares method with pole relocation via an eigenvalue problem. It presents the SISO and MIMO formulations, a real-valued fast implementation with pseudocode, model realization via Gilbert's algorithm, stability/causality/passivity considerations, and a survey of time-domain, parametric, and distributed-system extensions. The chapter includes three worked examples: a synthetic rational function, aortic impedance data, and a multiport PCB measurement.

Significance. The chapter is a competent and useful survey/tutorial. Its algebraic derivations in Secs. 3.1–3.5 are consistent with the established VF literature, the pseudocode is detailed and accompanied by an open-source implementation (Sec. 3.8), and the examples illustrate typical behavior, including a synthetic case reaching machine-precision error. The main limitation is the empirical nature of the robustness/convergence claim, which the chapter itself openly acknowledges in Sec. 3.2. For a handbook chapter this limitation is not disqualifying, but the abstract and conclusion should carry the same caveat. Overall, the chapter will be a valuable reference if the framing is adjusted.

major comments (1)
  1. [Abstract, Sec. 3.2, Sec. 5] The chapter's central practical claim—that VF converges quickly and reliably—is empirically grounded rather than proven, as the chapter itself states in Sec. 3.2: no theoretical convergence results exist and contrived examples show nonconvergence. However, the abstract and Sec. 5 state the robustness and 'handful of iterations' claim without this caveat, and Algorithm 3.1 simply returns 'Failure' after imax iterations with no guidance on how to set imax, how to restart, or how to recognize datasets for which VF is likely to fail. Since this robustness claim is load-bearing for the chapter's value as a practitioner's guide, please qualify the abstract and conclusion and add a short practical paragraph (in Sec. 3.2 or 3.11) on handling nonconvergence. This is a framing issue rather than an error in the algorithm exposition.
minor comments (6)
  1. [Sec. 3.2, convergence criterion 1] 'The calculation of the norm of w′' should read 'the norm of w(i)'; the prime appears to be a typographical artifact.
  2. [Sec. 3.4] 'Conditioning number' should be 'condition number'.
  3. [Sec. 3.11.2, Eq. (75)] The constraint row omits the real-part operator from (74). As written, the row enforces (β/¯k) Σ_k w(i)(jω_k) = β on complex values rather than on their real parts; the correct constraint is on Re{w(i)(jω_k)}. Please fix or clarify the complex arithmetic.
  4. [Sec. 3.10] The statement that condition (70) 'becomes a condition for causality' is imprecise. Stability (Re p ≤ 0) and causality are distinct: a right-half-plane pole can still yield a causal (but unstable) impulse response under the standard right-sided ROC. Please rephrase to state that (70) enforces stability, and that a causal realization follows from the usual right-sided ROC.
  5. [Sec. 3.8, after Eq. (52)] The sentence 'The obtained system, which has real coefficients and unknowns will ensure...' should be split or rephrased for grammatical clarity.
  6. [Algorithm 3.1] Consider suggesting a default value or heuristic for imax, since the failure exit at line 17 is otherwise left unspecified and the convergence caveat in Sec. 3.2 makes this exit a realistic outcome.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the chapter is a self-contained tutorial whose derivations are algebraic transformations and whose robustness caveat is openly acknowledged.

full rationale

The paper's derivation chain is not circular. The core claim—that VF solves the rational approximation problem (4) by iterating the linearized least-squares problem (17) and pole relocation (18)—is a direct algebraic derivation: equations (15) and (16) express the Sanathanan-Koerner weighting and model numerator in partial fractions, and substituting them into (12) yields (17); equation (18) follows from the factored form in (15). These are identities defining the algorithm, not predictions tuned to data. The examples validate against measured or independently generated data (a known rational function in Sec. 3.3, aortic impedance measurements in Sec. 3.4, PCB scattering measurements in Sec. 3.7) and are compared to external samples, not to the fitted model by construction. The practical robustness claim is not disguised: Sec. 3.2 states explicitly that "no one has been able to support this experimental evidence with strong theoretical results on VF convergence" and cites contrived counterexamples [53,72], and Algorithms 3.1 and 3.2 return Failure if imax is exceeded. This is an honest limitation rather than a circular justification. Self-references, mainly to the companion book [35] and to the author's extension papers, are used for implementation details and extensions, not as the load-bearing evidence for the core derivation.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

This expository chapter introduces no fitted scientific claim, so the ledger records the assumptions behind the algorithm's exposition: LTI and rational transfer function assumptions, heuristic initial pole selection, and the unproven stability-enforcement claim. No invented entities are introduced.

free parameters (3)
  • Initial pole damping factor alpha = 0.01 (Eqs. 19-20)
    Chosen by hand as a default; affects conditioning of the least-squares systems and thus the practical reliability of VF, but is not part of the mathematical statement.
  • Model order n-bar = user-specified or adapted by adding-and-skimming
    The chapter notes the order is generally unknown; examples use n=2..8 or adapt to 60..226, illustrating that output depends on this choice.
  • Convergence thresholds epsilon_w, epsilon_H, imax
    User-defined stopping criteria that determine whether VF reports success or failure, but do not change the fitted model if convergence is achieved.
assumptions (4)
  • domain assumption The system under modeling is linear, time-invariant, and its transfer function is rational or well approximated by a rational function.
    Sec. 2.1 assumes LTI and Eq. (5) fixes rational form; the claim that rational functions are universal approximators is stated without proof.
  • domain assumption Initial poles can be assigned with heuristic (19)-(20) and the final result is not critically sensitive to them.
    Sec. 3.2 asserts the distribution 'works well for most cases' without proof, and convergence is known to fail on contrived examples.
  • domain assumption Stable and causal models can be enforced by flipping unstable poles without accuracy penalty on error-free samples.
    Sec. 3.10 states this claim but provides no derivation or supporting analysis.
  • standard math QR decomposition and eigenvalue computations provide reliable numerical solutions to the least-squares and pole-update problems.
    Used throughout Secs. 3.2, 3.6, and 3.8 to solve systems (21), (38), and (55).

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Cite this review

Pith. "Pith review of Vector Fitting." pith.science (2026). https://pith.science/paper/TPSE56GP

@misc{pith2026190808977,
  author       = {Pith},
  title        = {Pith review of: Vector Fitting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TPSE56GP}},
  note         = {Machine review of arXiv:1908.08977}
}
read the original abstract

We introduce the Vector Fitting algorithm for the creation of reduced-order models from the sampled response of a linear time-invariant system. This data-driven approach to reduction is particularly useful when the system under modeling is known only through experimental measurements. The theory behind Vector Fitting is presented for single- and multiple-input systems, together with numerical details, pseudocodes, and an open-source implementation. We discuss how the reduced model can be made stable and converted to a variety of forms for use in virtually any modeling context. Finally, we survey recent extensions of the Vector Fitting algorithm geared towards time-domain, parametric and distributed systems modeling.

Figures

Figures reproduced from arXiv: 1908.08977 by the authors.

Figure 1
Figure 1. Left panel: initial poles p 0 n used by VF in the first iteration. Right panel: poles of the final model He(s) compared to the exact poles of the original transfer function. 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 ω [rad/s] Magnitude Samples Model 0 1 2 3 4 5 6 7 8 9 10 −100 −50 0 ω [rad/s] Phase [deg] Samples Model [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Example of Sec. 3.3: magnitude (top) and phase (bottom) of samples [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Example of Sec. 3.3: worst-case fitting error [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Impedance seen into the ascending aorta of the pediatric patient [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: Interconnect network on a printed circuit board considered in Sec. 3.7. [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: example of Sec. 3.7: comparison between samples [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: example of Sec. 3.7: comparison between samples 20 [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: example of Sec. 3.7: VF error as a function of iteration, compared to [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 8
Figure 8. Figure 8: If error e is still too high, the order is increased until either VF converges or it becomes evident that no further error reduction can be achieved, as in the last four iterations in [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.