Structured vector fitting framework for mechanical systems

The paper correctly formulates two structured VF variants (SOVF1 with partially structured numerator, SOVF2 with fully structured numerator/denominator), derives the weighted LS systems A_so1 and A_so2, and prescribes the standard VF update via eigenvalues of Â−ĜĈ for SOVF1 and of a quadratic pencil for SOVF2 (see the paper’s equations and algorithms around (12)–(19) and Algorithm 2–3 ). However, proofs are only sketched or omitted (e.g., the determinant-lemma steps and fixed-point implications). The candidate solution supplies the missing derivations: (i) the determinant-lemma proof that the denominator zeros match the cited eigenvalue problems in both SOVF1 and SOVF2; and (ii) a fixed-point argument for SOVF2 showing that if the expansion points coincide with the denominator zeros, then the denominator weights must vanish, reducing to a diagonal second-order ROM as in SOVF1. These additions are correct and consistent with the paper’s statements and intent. The only minor quibble is the claim that row-weighting “does not change the set of minimizers,” which is not generally true for LS; it is, however, standard within the SK/VF linearization framework used by the paper (cf. the weighting defined in (5) and used throughout ).