Asymptotic analysis of a generalized Richardson extrapolation process on linear sequences

Abstract

In this work, we give a detailed convergence and stability analysis for the author's generalized Richardson extrapolation process GREP (m) as this is being applied to linearly convergent or divergent infinite sequences {A n }, where A n ~ A + ∑ m k=1 ζ n k ∑ ∞ i=0 β ki n γk-i as n → ∞, ζ k ≠ 1 being distinct. The quantity we would like to compute is A, whether it is the limit or antilimit of {A n }. Such sequences arise, for example, as partial sums of power series and of Fourier series of functions that have algebraic and/or logarithmic branch singularities. Specifically, we define the GREP (m) approximation A n (m,j) to A, with n = (n 1, ..., n m ) and α > 0, via the linear systems A l = A n (m,j) + ∑ m k=1 ζ l k ∑ n i=0 -1 β ki (α + l) γk-1 , j ≤ l ≤ j + ∑ m k=1 n k, where β ki are additional unknowns. We study the convergence and stability properties of A n (m,j) as j → ∞. We show, in particular, that A n (m,j) -A = ∑ m k=1 O(ζ j k j γk-2nk ) as j → ∞. When compared with A j - A = ∑ m k=1 O(ζ j k j γk ) as j → ∞, this result shows that GREP (m) is a true convergence acceleration method for the sequences considered. In addition, we show that GREP (m) 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 GREP (m) with m > 1.