A hierarchically consistent, iterative sequence transformation
Sequences \(s_n\) which converge very slowly to some limits \(s\) are abundant in mathematics and its applications in science and in engineering. Thus, there is a need for methods to accelerate their convergence. In other cases, divergent expansions occur which have to be summed to some finite number \(s\) if possible. Hence, there are the problems of convergence acceleration, of extrapolation, and of summation. It turns out that no single method is able to solve efficiently all such problems which arise in practice. There are numerous methods known. Many of them have been derived on the basis of so-called model sequences. In a previous paper the author introduced a sequence transformation. This transformation proved to be able to accelerate quite efficiently a number of test sequences. The computational effort to calculate it is low. In this paper this sequence transformation is derived by hierarchically consistent iteration of some basic transformation. Hierarchical consistency is proposed as an approach to control the well-known problem that the basic transformation can be generalized in many ways. Properties of the transformation are studied. It is shown that it can be implemented quite easily. The kernel of the transformation is derived. The expression for the kernel is relatively compact and does not depend on any lower-order transforms. It is shown that several important other sequence transformations can be computed in an economical way using this transformation.
scientific article; zbMATH DE number 698815
scientific article