SYSTEMATIC REVIEW OF NEWTON-SCHULZ ITERATIONS WITH UNIFIED FACTORIZATIONS : INTEGRATION IN THE RICHARDSON METHOD AND APPLICATION TO ROBUST FAILURE DETECTION IN ELECTRICAL NETWORKS

The paper defines the two-step combination (Γ_k, L_k, G_k) and states the error identities F_k = Γ_k^n F_{k-1}^n (with a short derivation) and the closed form F_k = F_0^{k n^{k+1} + n^k} for n > 1 (its derivation deferred to “explicit evaluation”), see equations (16)–(22) and (21) in the paper . The candidate solution proves the same results in full detail via (i) an induction showing L_k A = I − Γ_k^n, (ii) multiplying the G_k update by A and using the geometric-sum identity to obtain F_k = Γ_k^n F_{k-1}^n, and (iii) solving the resulting exponent recurrence to get F_k = F_0^{k n^{k+1} + n^k}. Both arguments rely on the same algebraic identities and the structure Γ_k = Γ_{k-1}^n; the model simply fills in details the paper sketches. Convergence conditions (e.g., ρ(F_0) < 1) are noted in the paper for Newton–Schulz methods (equation (8)) but are not needed for the finite algebra used here .