Acceleration of convergence of general linear sequences by the Shanks transformation

摘要

The Shanks transformation is a powerful nonlinear extrapolation method that is used to accelerate the convergence of slowly converging, and even diverging, sequences {A_n}. It generates a two-dimensional array of approximations A_n~((j)) to the limit or anti-limit of {A_n} defined as solutions of the linear systems Where β? are additional unknowns. In this work, we study the convergence and stability properties of A_n~((j)), as j → ∞ with n fixed, derived from general linear sequences {A_n}, where as n → ∞, where ζ_k ≠ 1 are distinct and {pipe}ζ_1{pipe}=... ={pipe}ζ_m{pipe}= θ, and γ_k ≠ 0, 1, 2,.... Here A is the limit or the anti-limit of {A_n}. Such sequences arise, for example, as partial sums of Fourier series of functions that have finite jump discontinuities and/or algebraic branch singularities. We show that definitive results are obtained with those values of n for which the integer programming problems have unique (integer) solutions for s_1,..., s_m. A special case of our convergence result concerns the situation in which and n = mν with ν = 1, 2,..., for which the integer programming problems above have unique solutions, and it reads A_n~((j)) - A = O(θ~j j~(α-2v)) as j → ∞. When compared with A_j - A = O(θ~j j~α) as j → ∞, this result shows that the Shanks transformation is a true convergence acceleration method for the sequences considered. In addition, we show that it is stable for the case being studied, and we also quantify its stability properties. The results of this work are the first ones pertaining to the Shanks transformation on general linear sequences with m > 1.