Nonlinear Sequence Transformations: Computational Tools for the Acceleration of Convergence and the Summation of Divergent Series

This paper presents a family of rapidly convergent summation formulas for various finite sums of analytic functions. These summation formulas are obtained by applying a series acceleration transformation involving Stirling numbers of the first kind to the asymptotic, but divergent, expressions for the corresponding sums coming from the Euler-Maclaurin summation formula. While it is well-known that the expressions obtained from the Euler-Maclaurin summation formula diverge, ou...

A novel type of approximants is introduced, being based on the ideas of self-similar approximation theory. The method is illustrated by the examples possessing the structure typical of many problems in applied mathematics. Good numerical convergence is demonstrated for the cases that can be compared with exact solutions, when these are available. The method is shown to be not less and as a rule essentially more accurate than that of Pade approximants. Comparison with other ap...

The present paper presents some reflections of the author on divergent series and their role and place in mathematics over the centuries. The point of view presented here is limited to differential equations and dynamical systems.

A variant of self-similar approximation theory is suggested, permitting an easy and accurate summation of divergent series consisting of only a few terms. The method is based on a power-law algebraic transformation, whose powers play the role of control functions governing the fastest convergence of the renormalized series. A striking relation between the theory of critical phenomena and optimal control theory is discovered: The critical indices are found to be directly relat...

The theory of summability of divergent series is a major branch of mathematical analysis that has found important applications in engineering and science. It addresses methods of assigning natural values to divergent sums, whose prototypical examples include the Abel summation method, the Cesaro means, and Borel summability method. In this paper, we introduce a new summability method for divergent series, and derive an asymptotic expression to its error term. We show that it ...

In this work we present new methods for transforming and solving finite series by using the Laplace transform. In addition we introduce both an alternative method based on the Fourier transform and a simplified approach. The latter allows a quick solution in some cases.

The method of self-similar factor approximants is completed by defining the approximants of odd orders, constructed from the power series with the largest term of an odd power. It is shown that the method provides good approximations for transcendental functions. In some cases, just a few terms in a power series make it possible to reconstruct a transcendental function exactly. Numerical convergence of the factor approximants is checked for several examples. A special attenti...

A method is suggested for treating the well-known deficiency in the use of Pade approximants that are well suited for approximating rational functions, but confront problems in approximating irrational functions. We develop the approach of self-similarly corrected Pade approximants, making it possible to essentially increase the class of functions treated by these approximants. The method works well even in those cases, where the standard Pade approximants are not applicable,...

Prompted by an observation about the integral of exponential functions of the form $f(x)=\lambda\mathrm{e}^{\alpha x}$, we investigate the possibility to exactly integrate families of functions generated from a given function by scaling or by affine transformations of the argument using nonlinear generalizations of quadrature formulae. The main result of this paper is that such formulae can be explicitly constructed for a wide class of functions, and have the same accuracy as...

In this article we construct a family of expressions $\varepsilon(n)$. For each element E(n) from $\varepsilon(n)$, the convergence of the series $\sum_{n \ge n_E}{E(n)}$ can be determined in accordance to the theorems of this article. Some applications are also presented.