Error Estimation and Stopping Criteria for Krylov-Based Model Order Reduction in Acoustics

The paper proves two claims: (i) ||G(s)−Gr(s)|| = ||Gr+1(s)−Gr(s)|| + O(|s−s0|^{r+1}) via moment matching plus the reverse triangle inequality, and (ii) under G(s0)≠0 and Gq(s0)≠0 (q=r,r+1), the relative-error estimators Êr(s) and Ẽr(s) approximate Er(s) to order r+1 near s0. These are stated as Theorems 4.1–4.2 and justified using the Taylor/moment expansions of G, Gr, Gr+1 about s0 and basic norm inequalities . The setup and moment-matching properties for second-order Krylov reductions are given earlier (definitions of transfer functions and the moment interpretation as negative Taylor coefficients) . The candidate solution follows the same structure: it expands G, Gr, Gr+1 around s0, notes equality of the first r (or r+1) moments, factors the differences as powers of Δ=s−s0, applies the reverse triangle inequality for part (a), and then uses continuity and lower bounds on ||G||, ||Gr||, ||Gr+1|| to prove the O(|Δ|^{r+1}) accuracy of Êr and Ẽr in part (b). The reasoning matches the paper’s, with slightly more explicit bounding of analytic remainders, so both are correct and essentially the same proof.