The Shamanskii Frontier: Sharp Order Barriers and Dimensional Saturation

Shamanskii’s frozen-Jacobian method is a natural benchmark for assessing whether an iterative method for nonlinear systems gains enough local convergence order to justify its computational cost. We call the corresponding order–cost benchmark the Shamanskii frontier and ask which algorithmic constructions can improve on it. To address this question, we study an explicitly typed operational class of deterministic, causal methods without intercycle memory in which new directions are obtained from a single frozen inverse. Arbitrary adaptive scalar rules, functional and differential evaluations along accessible directions, and exact reuse of inverse responses are permitted. For a budget of s < n new inverse queries, we prove the sharp universal local-order bound p ≤ s + 1 and the dimensional law Smin(p, n) = min{p − 1, n} for every integer target p ≥ 2. The lower bound is established by a fixed quadratic vector field, accessible-subspace filtrations, and a bounded separating functional; the matching construction uses exact response reconstruction. We also determine the role of compatible initial information, distinguish rank from the cost of preparing the data, and examine external vector resources and transpose solves. In the latter setting, a sharp result is established for up to three total direct or transpose solves. For the conventional dense cost normalization, we characterize the integer optimum in the Shamanskii family and show that, with fixed normalized elementary costs, the optimal number of corrections satisfies m∗(n) ∼ n/(3 log n). We then identify classes of extrapolation, derivative-assisted, defect-polynomial, and learned low-rank corrections that cannot exceed the Shamanskii order bound with the same inverse-query budget; a cost comparison gives a further criterion for excluding improvements in the modeled efficiency index. Neither criterion asserts the universal optimality of Shamanskii under unrestricted matrix processing.