Comparison of four nonlinear transforms on some classes of logarithmic fixed point sequences

Let LOG be the set of logarithmic sequences, i.e., of convergent sequences (Sn) satisfying lim(en+1en) = 1, where en = Sn − S and S = limSn. This paper is devoted to the comparison of four nonlinear sequence transforms on some subsets LF(r) of LOG. For any positive integer r, LF(r) denotes the set of fixed point sequences (Sn) whose associated error sequences en = Sn − S (S = limSn) have an asymptotic expansion of the following type: en+1 = en + ∑i⩾1αien1+ir, where α1 < 0 and the quantity c0(r) = 12(r + 1) − α2α1−2 is different from zero. For (Sn ∈ LF(r), there holds en = a1n−1r + O(n−(1+1r)logn), with a1 = (−rα1)−1r. The four are, respectively, modifications of the ϵ-algorithm and of the iterated Δ2 transform, the iterated Lubkinʹs transform and the θ-algorithm of Brezinski. All of them accelerate the convergence of sequences in LF(r). We give accurate results concerning the asymptotic behaviour of transformed sequences. We also study two algorithms combining the second and third transforms. Numerical examples are given to illustrate theoretical results.