Convergence acceleration of some logarithmic sequences

摘要

Let (Sn) be some real sequence defined as Sn+1=⨍(Sn for∈N,where ⨍(x)=x+∑i⩾1αp+i(x−S)p+i, with p∈N, p≠0 and αp+1<0, S0given. For S0 well chosen, it converges to S and limn→+∞(Sn+1−S)/(Sn−S) = 1 (logarithmic convergence). By asymptotic analysis, we show that different algorithms, modified iterated versions of Aitken's Δ2 process, iterated θ2-algorithm, modified ϵ-algorithms and θ-algorithm, accelerate the convergence of this type of sequences and we estimate the errors on the transformed sequences. For sequences of type (1) with p = 1, 2, we give more precise results and an efficient algorithm combining the modified Δ2 and θ2-algorithm. Finally, we apply these algorithms to some series and integrals.