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

Convergence problems occur abundantly in all branches of mathematics or in the mathematical treatment of the sciences. Sequence transformations are principal tools to overcome convergence problems of the kind. They accomplish this by converting a slowly converging or diverging input sequence $\{s_n \}_{n=0}^{\infty}$ into another sequence $\{s^{\prime}_n \}_{n=0}^{\infty}$ with hopefully better numerical properties. Pad\'{e} approximants, which convert the partial sums of a power series to a doubly indexed sequence of rational functions, are the best known sequence transformations, but the emphasis of the review will be on alternative sequence transformations which for some problems provide better results than Pad\'{e} approximants.Comment: 29 pages, LaTeX, 0 figure

THE d2-TRANSFORMATION FOR INFINITE DOUBLE SERIES AND THE D2-TRANSFORMATION FOR INFINITE DOUBLE INTEGRALS

Abstract. New transformations for accelerating the convergence of infinite double series and infinite double integrals are presented. These transformations are generalizations of the univariate d- andD-transformations. The D2transformation for infinite double integrals is efficient if the integrand satisfies a p.d.e. of a certain type. Similarly, the d2-transformation for double series works well for series whose terms satisfy a difference equation of a certain type. In both cases, the application of the transformation does not require an explicit knowledge of the differential or the difference equation. Asymptotic expansions for the remainders in the infinite double integrals and series are derived, and nonlinear transformations based upon these expansions are presented. Finally, numerical examples which demonstrate the efficiency of these transformations are given. 1

The d2-Transformation For Infinite Double Series And The D2-Transformation For Infinite Double Integrals

New transformations for accelerating the convergence of infinite double series and infinite double integrals are presented. These transformations are generalizations of the univariate d- and D-transformations. The D2-transformation for infinite double integrals is efficient if the integrand satisfies a p.d.e. of a certain type. Similarly, the d2-transformation for double series works well for series whose terms satisfy a difference equation of a certain type. In both cases, the application of the transformation does not require an explicit knowledge of the differential or the difference equation. Asymptotic expansions for the remainders in the infinite double integrals and series are derived, and nonlinear transformations based upon these expansions are presented. Finally, numerical examples which demonstrate the efficiency of these transformations are given

The d2-Transformation for Infinite Double Series and the D2-Transformation for Infinite Double Integrals

New transformations for accelerating the convergence of infinite double series and infinite double integrals are presented. These transformations are generalizations of the univariate d- and D- transformations. The D 2 -transformation for infinite double integrals is efficient if the integrand satisfies a p.d.e of a certain type. Similarly, the d 2 -transformation for double series works well for series whose terms satisfy a difference equation of a certain type. In both cases, the application of the transformation does not require an explicit knowledge of the differential or the difference equation. Asymptotic expansions for the remainders in the infinite double integrals and series are derived, and nonlinear transformations based upon these expansions are presented. Finally, numerical examples which demonstrate the efficiency of these transformations are given. 1. Introduction We discuss the problem of accelerating the convergence of infinite double integrals and infinite double se..