Performance Enhancement of the Recursive Least Squares Algorithms with Rank Two Updates

The paper states the RLSR2 updates (Ak = λAk−1 + Qk D Qk^T and the Γk, θk recursions) and then asserts the error model Ek = (I − Γk−1 Qk S−1 Qk^T) Ek−1 and θ̃k = (I − Γk−1 Qk S−1 Qk^T) θ̃k−1, followed by the Lyapunov one-step identity Vk − Vk−1 = −λ θ̃T k−1 Qk S−1 Qk^T θ̃k−1 − (1−λ)Vk−1, with the bound Vk ≤ λ^k V0 under a mild nonnegativity condition. These are exactly the statements the model proves, using the same update equations and the same assumption Ek = 0 for the Lyapunov step; the model simply fills in the missing algebra via identities implied by S = λD + Qk^T Γk−1 Qk. See the paper’s equations (3)–(5)/(10)–(11) and the stated error model (12)–(13) and one-step identity (14) for confirmation .