Acceleration of convergence of general linear sequences by the Shanks transformation

Abstract

The Shanks transformation is a powerful nonlinear extrapolation method that is used to accelerate the convergence of slowly converging, and even diverging, sequences {An}. It generates a two-dimensional array of approximations A j) n to the limit or anti-limit of {An} defined as solutions of the linear systems Al = A j) n + n ∑ k=1 β̄k( Al+k−1), j ≤ l ≤ j + n, where β̄k are additional unknowns. In this work, we study the convergence and stability properties of A j) n , as j → ∞ with n fixed, derived from general linear sequences {An}, where An ∼ A +mk=1 ζ n k ∑∞ i=0 βki nγk−i as n → ∞, where ζk = 1 are distinct and |ζ1| = · · · = |ζm | = θ, and γk = 0, 1, 2, . . .. Here A is the limit or the anti-limit of {An}. 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 max s1,...,sm m ∑ k=1 [ ( γk)sk − sk(sk − 1) ] , subject to s1 ≥ 0, . . . , sm ≥ 0 and m ∑ k=1 sk = n, A. Sidi (B) Computer Science Department, Technion-Israel Institute of Technology, Haifa 32000, Israel e-mail: asidi@cs.technion.ac.il URL: http://www.cs.technion.ac.il/∼asidi

DOI: 10.1007/s00211-011-0398-8

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Cite this paper

@article{Sidi2011AccelerationOC, title={Acceleration of convergence of general linear sequences by the Shanks transformation}, author={Avram Sidi}, journal={Numerische Mathematik}, year={2011}, volume={119}, pages={725-764} }