Convergence acceleration of some logarithmic sequences
Let \((S_ n)\) be some real sequence defined as \((1)\quad S_{n+1}=f(S_ n)\text{ for } n\in N,\text{ where } f(x)=x+\sum_{i\geq 1}\alpha_{p+i}(x-S)^{p+i}\) with \(p\in N\), \(p\neq 0\) and \(\alpha_{p+1}<0\), \(S_ 0\) given. For \(S_ 0\) well chosen, it converges to S and \(\lim_{n\to \infty}(S_{n+1}-S)/(S_ n-S)=1\) (logarithmic convergence). The author proposes a specific study of some algorithms (modified iterated versions of Aitken's \(\Delta^ 2\) process, iterated \(\theta_ 2\)-algorithm, modified \(\epsilon\)-algorithms and \(\theta\)-algorithm) which accelerate the convergence of this type of sequences and also estimate the errors of the transformed sequences. For sequences of type (1), with \(p=1,2\), more precise results and an efficient algorithm combining the modified \(\Delta^ 2\) and \(\theta_ 2\)-algorithm are given. Finally, some numerical results are considered in order to compare the different algorithms.
scientific article; zbMATH DE number 4195007