Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series

\v{C}\'{\i}\v{z}ek, Zamastil, and Sk\'{a}la [J. Math. Phys. \textbf{44}, 962 - 968 (2003)] introduced in connection with the summation of the divergent perturbation expansion of the hydrogen atom in an external magnetic field a new sequence transformation which uses as input data not only the elements of a sequence $\{s_n \}_{n=0}^{\infty}$ of partial sums, but also explicit estimates $\{\omega_n \}_{n=0}^{\infty}$ for the truncation errors. The explicit incorporation of the ...

The practical usefulness of Levin-type nonlinear sequence transformations as numerical tools for the summation of divergent series or for the convergence acceleration of slowly converging series, is nowadays beyond dispute. Weniger's transformation, in particular, is able to accomplish spectacular results when used to overcome resummation problems, often outperforming better known resummation techniques, the most known being Pad\'e approximants. However, our understanding of ...

The aim of this review, based on a series of four lectures held at the 22nd "Saalburg" Summer School (2016), is to cover selected topics in the theory of perturbation series and their summation. The first part is devoted to strategies for accelerating the rate of convergence of convergent series, namely Richardson extrapolation, and the Shanks transformations, and also covers a few techniques for accelerating the convergence of Fourier series. The second part focuses on diver...

The paper considers a universal approach that allows one to quite simply obtain nonlinear asymptotic estimates of various summation functions. It is shown the application of this approach to the asymptotic estimation of divergent Dirichlet series. Several assertions have been proven and numerous examples have been considered.

This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein's "An efficient algorithm for computing the Riemann zeta function", to more general series. The algorithm provides a rapid means of evaluating Li_s(z) for general values of complex s and the regi...

In a recent paper (Appl. Math. Comput. 215, 1622--1645, 2009), the authors proposed a method of summation of some slowly convergent series. The purpose of this note is to give more theoretical analysis for this transformation, including the convergence acceleration theorem in the case of summation of generalized hypergeometric series. Some new theoretical results and illustrative numerical examples are given.

Since more than three centuries Kepler's equation continues to represents an important benchmark for testing new computational techniques. In the present paper, the classical Kapteyn series solution of Kepler's equation originally conceived by Lagrange and Bessel will be revisited from a different perspective, offered by the relatively new and still largely unexplored framework of the so-called nonlinear sequence transformations. The main scope of the paper is to provide nu...

The paper investigates the properties of a nonlinear recursive sequence which includes several ones studied formerly in the literature.

Sequence transformations are valuable numerical tools that have been used with considerable success for the acceleration of convergence and the summation of diverging series. However, our understanding of their theoretical properties is far from satisfactory. The Euler series $\mathcal{E}(z) \sim \sum_{n=0}^{\infty} (-1)^n n! z^n$ is a very important model for the ubiquitous factorially divergent perturbation expansions in physics. In this article, we analyze the summation of...

A new general and unified method of summation, which is both regular and consistent, is invented. It is based on the idea concerning a way of integers reordering. The resulting theory includes a number of explicit and closed form summation formulas, and assigns limits to certain unbounded or oscillating functions. Some problems and future lines of research are briefly discussed.