{"id":"38d400c6-93bd-46d2-9bca-e6dda5725a40","arxiv_id":"1908.08977","paper_version":1,"verdict":"UNVERDICTED","confidence":"HIGH","novelty_score":2.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"This draft book chapter explains the established Vector Fitting algorithm, including pseudocode and extensions, and presents no new research result.","lead":"Vector Fitting is a widely used method for building compact models of electrical and mechanical systems from measured frequency data. This draft book chapter explains the algorithm, gives pseudocode, and surveys its extensions, but introduces no new science.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The practical robustness claim rests on an unproven convergence property: Sec. 3.2 admits no theoretical guarantees and known counterexamples exist, so VF can fail by exiting after imax; this is the load-bearing soft spot.","rationale":"I read the chapter as an expository handbook entry, not a new research result. The derivation of the VF iteration is internally consistent: equations (15) and (16) correctly re-express the Sanathanan-Koerner error with implicit weighting, and (18) follows from the pole-zero form of w(i). The real-valued formulation in Sec. 3.8 and the Fast VF condensed system (42) are standard and reproducible, and the three worked examples give concrete evidence of typical behavior. The only point on which the central claim could fail is convergence: the algorithm's fixed-point map has no proven domain of attraction, and the chapter explicitly cites counterexamples. Because the chapter discloses this limitation and frames the robustness claim as empirical, I do not think the reader's UNVERDICTED verdict should change; the concern is real but already acknowledged. The proposed test would quantify how much of the practical robustness claim survives on realistic data.","tokens_in":26678,"tokens_out":10453,"duration_ms":107561,"concrete_test":"Run the linked open-source implementation [75] on the published nonconvergence examples of Lefteriu & Antoulas (IEEE TMTT 61(4):1435–1443, 2013) and Shi (IEEE TCAS-II 63(8):718–722, 2016), and on a Monte-Carlo suite of 100 noisy synthetic datasets with the noise levels of Sec. 3.4 (relative std 0.66–14.5%) for orders 2–8, recording success rate within imax=50. If VF fails on realistic noisy or ill-conditioned data at a nontrivial rate, the blanket 'remarkably robust' claim is overstated; if failures occur only on contrived examples, the chapter's practical claim stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"VF's central claim—that an engineer can rely on it to converge in a handful of iterations and produce an accurate reduced model—depends on the fixed-point iteration (21)/(18) converging to a useful minimum of the nonlinear least-squares problem (6). The chapter itself states in Sec. 3.2 that no strong theoretical convergence results exist, that contrived examples (Lefteriu & Antoulas [53]; Shi [72]) show convergence is not guaranteed, and that the robustness claim rests on anecdotal experience. That is the weakest load-bearing point: the algorithm can return Failure after imax iterations (Algorithm 3.1, line 17), and the chapter offers no criterion for recognizing data on which VF will be unreliable beyond trial and error. This is not an internal inconsistency; the math in (15)-(18) is coherent and the examples are illustrative. It is an explicit, honest limitation that nevertheless conditions every downstream claim of practical robustness.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":26860,"tokens_out":11890,"duration_ms":112961,"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":[{"comment":"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.","section":"Abstract, Sec. 3.2, Sec. 5"}],"minor_comments":[{"comment":"'The calculation of the norm of w′' should read 'the norm of w(i)'; the prime appears to be a typographical artifact.","section":"Sec. 3.2, convergence criterion 1"},{"comment":"'Conditioning number' should be 'condition number'.","section":"Sec. 3.4"},{"comment":"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.","section":"Sec. 3.11.2, Eq. (75)"},{"comment":"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.","section":"Sec. 3.10"},{"comment":"The sentence 'The obtained system, which has real coefficients and unknowns will ensure...' should be split or rephrased for grammatical clarity.","section":"Sec. 3.8, after Eq. (52)"},{"comment":"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.","section":"Algorithm 3.1"}],"recommendation":"minor_revision","confidential_remarks":"This is an arXiv preprint of a book chapter for a handbook; the journal should confirm that a survey of this nature is within its scope. The chapter leans heavily on the author's own book [35] for extensions and implementation details, but the core VF algorithm is independently established in the literature, so the self-citation pattern is not a circularity concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThis is a draft book chapter, not a research paper. You should read it as a tutorial/reference on Vector Fitting, and it's a good one. It makes no claim to novelty and explicitly credits the original work of Gustavsen and Semlyen and the book by Grivet-Talocia and Gustavsen. What it does well: it gives a coherent derivation from the Sanathanan-Koerner iteration to the pole-relocation step, includes pseudocode that matches an open-source implementation, and illustrates the method with three examples, including a measured PCB multiport and an aortic impedance dataset. The synthetic example reaches machine precision. The chapter is honest about the central limitation: convergence of VF is not proven, and Section 3.2 states this clearly, mentioning known counterexamples and the possibility of exiting after imax without a model.\n\nThe soft spots are modest and mostly inherited from the topic. The 'typically converges in 4-5 iterations' claim is heuristic, not theorem-backed; the author says so. The chapter relies heavily on [35] for details, so it is not fully self-contained despite its tutorial tone. The real-valued fast VF section is dense and will be tough going for someone seeing VF for the first time, but the pseudocode helps. Some references are self-citations, but they point to the standard literature, so I don't see that as a problem.\n\nOverall, this is a solid reference artifact. A serious referee should engage with it as a book chapter—there are no load-bearing math errors, and the examples are reproducible in principle. I would not cite it in my own work, since I'd cite the original sources, but I'd point students to it as an entry point.\n\nRecommendation: accept for peer review as a book chapter.","headline":"An honest, well-crafted tutorial on Vector Fitting that breaks no new ground but delivers a usable reference with code and worked examples.","tokens_in":27365,"tokens_out":2680,"would_cite":false,"duration_ms":25819,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Vector Fitting solves rational approximation from frequency samples by iterating linear least squares and relocating poles in closed form.","keywords":["Vector Fitting","rational approximation","reduced-order model","frequency-domain sampling","Sanathanan-Koerner iteration","pole relocation","linear least squares","data-driven modeling"],"falsifier":"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.","tokens_in":26461,"feed_emoji":"⚡","tokens_out":13289,"duration_ms":119336,"temperature":0.7,"pith_summary":"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.","feed_headline":"Sampled frequency data become reduced-order models in five iterations","feed_subtitle":"Vector Fitting builds models of chips, circuits, and arteries from measurements alone, no first-principles equations.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces the partial-fraction basis, the implicit weighting through pole relocation, and the eigenvalue update that form the core of VF.","marker":"[42]"},{"why":"Provides the comprehensive theory, numerical details, pole initialization rules, and realization variants the chapter relies on.","marker":"[35]"},{"why":"Defines the iterative linearized weighted least-squares scheme that VF improves on.","marker":"[69]"},{"why":"Gives the original linearized rational fitting formulation that VF and the later iterative scheme build upon.","marker":"[54]"},{"why":"Relaxes the weighting-function normalization to improve VF convergence on noisy data.","marker":"[41]"},{"why":"Supplies the adding-and-skimming order adaptation and spurious-pole removal used in the noisy-data example.","marker":"[33]"},{"why":"Provides the aortic input impedance measurements used to demonstrate VF on real, uncertain data.","marker":"[71]"},{"why":"Exhibits a counterexample showing VF convergence is not guaranteed, which the chapter cites to delimit its practical robustness claim.","marker":"[53]"}],"fun_headline_variants":["Vector Fitting: data-only reduced-order models from samples","Five iterations from measurements to stable rational models","No equations needed: Vector Fitting learns from sampled responses","From frequency samples to reduced-order models via Vector Fitting"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Vector Fitting: data-only reduced-order models from samples","Five iterations from measurements to stable rational models","No equations needed: Vector Fitting learns from sampled responses","From frequency samples to reduced-order models via Vector Fitting"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001016,"raw_usage":{"total_tokens":4242,"prompt_tokens":848,"completion_tokens":3394,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":464,"completion_tokens_details":{"reasoning_tokens":3330}},"tokens_in":464,"tokens_out":3394,"duration_ms":26389,"temperature":1.0,"reasoning_tokens":3330,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:24:30.367260+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Gustavsen and A","cited_arxiv_id":null,"evidence_quote":"Introduces the partial-fraction basis, the implicit weighting through pole relocation, and the eigenvalue update that form the core of VF."},{"cited_title":"Grivet-Talocia and B","cited_arxiv_id":null,"evidence_quote":"Provides the comprehensive theory, numerical details, pole initialization rules, and realization variants the chapter relies on."},{"cited_title":"Sanathanan and J","cited_arxiv_id":null,"evidence_quote":"Defines the iterative linearized weighted least-squares scheme that VF improves on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the original linearized rational fitting formulation that VF and the later iterative scheme build upon."},{"cited_title":"Gustavsen","cited_arxiv_id":null,"evidence_quote":"Relaxes the weighting-function normalization to improve VF convergence on noisy data."},{"cited_title":"Grivet-Talocia and M","cited_arxiv_id":null,"evidence_quote":"Supplies the adding-and-skimming order adaptation and spurious-pole removal used in the noisy-data example."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the aortic input impedance measurements used to demonstrate VF on real, uncertain data."},{"cited_title":"Lefteriu and A","cited_arxiv_id":null,"evidence_quote":"Exhibits a counterexample showing VF convergence is not guaranteed, which the chapter cites to delimit its practical robustness claim."}],"review_version":1}