A simultaneous process for convergence acceleration and error control

摘要

Let (xn) be a real sequence, converging to the limit x∗ and such that Δxn=xn+1−xn =λnnvΣi⩾0aini (a0≠0, λ≠0) for n large enough.Here we propose a repeated process, using two transformations (namely Aitken's Δ2 process and Brezinski's θ2 algorithm) which provide a sequence of intervals asymptotically containing x*.In particular, we introduce, for λ=1, a version of the modified iterated Δ2 process, based on the convergence orders, so generalizing a method proposed by Sablonnière. We establish that the order vp, used in the pth iteration, is vp−1−kp−1+1, kp−1⩾3 being an integer. We give an estimation method for vp confirming the choice of kp−1.