Survey of numerical stability issues in convergence acceleration

Abstract

An important issue that arises in application of convergence acceleration (extrapolation) methods is that of stability in the presence of floating-point arithmetic. This issue turns out to be critical because numerical instability is inherent, even built in, when convergence acceleration methods are applied to certain types of sequences that occur commonly in practice. If methods are applied without taking this issue into account, the attainable accuracy is limited, and eventually destroyed completely, as more terms are added in the process. Therefore, it is important to understand the origin of the problem and to propose practical ways to solve it effectively. In this work, we present a general discussion of the issue of stability within the context of a generalization of the Richardson extrapolation process, and review some of the recent developments that have taken place in the theoretical study of many of the known acceleration methods. We discuss approaches that have been proposed to cope with built-in instabilities when applying various methods, and illustrate the effectiveness of these strategies with some numerical examples.