The genesis and early developments of Aitken's process, Shanks' transformation, the \(\varepsilon\)-algorithm, and related fixed point methods
Operators applied to sequences for the purpose of convergence acceleration are a fundamental method of numerical analysis. Many convergence phenomena and limits could not be observed or computed efficiently (or, in fact, not at all) without them. Several authors have made celebrated contributions to this work, and, among many others, Aitken is an important name. Steffenson is another famous researcher who should be mentioned in this context. Prony should also be mentioned due to his early contribution to the subject by the algorithm that is named after him. The sequences are by no means restricted to scalars, by the way, they may be sequences of vectors, tensors and matrices as well. Convergence acceleration algorithms are closely related to Padé approximations too and these are all reviewed in great historical detail in this long article. This connection and its description in the paper is true for the \(\varepsilon\)-algorithm and certain related fixed-point theorems as well. Not only the mathematical developments are shown meticulously but also all the personal history of the mathematicians involved. The mentioned convergence acceleration methods are in effect extrapolation schemes and this is detailed in the paper as well. A particular fascinating aspect of this large body of work is its connection with continued fractions which is an interesting link with pure mathematics. The article gives a long and detailed historical survey of the methods, including the connections between pure and applied mathematics, their development in the context of their mathematics and their applications. Moreover the lifes of the mathematicians who are the most well-known contributors are included into the picture. Very many of the methods are named after them. A chapter of future developemts and the relation to other well-known schemes of numerical analysis completes this very helpful contribution to the history and numerics of mathematical algorithms.
scientific article; zbMATH DE number 7005707
scientific article