The genesis and early developments of Aitken's process, Shanks' transformation, the \(\varepsilon\)-algorithm, and related fixed point methods
type: Item
instance of: scholarly article
title: The genesis and early developments of Aitken's process, Shanks' transformation, the \(\varepsilon\)-algorithm, and related fixed point methods
zbMATH Open document ID: 1477.65011
author: Michela Redivo-Zaglia
author: Claude Brezinski
published in: Numerical Algorithms
publication date: 2019-01-23T00:00:00Z
review text: 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.
reviewed by: Martin D. Buhmann
Mathematics Subject Classification ID: 65B05
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 65Q10
Mathematics Subject Classification ID: 65-03
zbMATH DE Number: 7005707
zbMATH Keywords: extrapolation
zbMATH Keywords: Aitken's process
zbMATH Keywords: Shanks' transformations
zbMATH Keywords: epsilon-algorithms
zbMATH Keywords: MMPE
zbMATH Keywords: MPE
zbMATH Keywords: RRE
zbMATH Keywords: fixed points
zbMATH Keywords: Pulay mixing
zbMATH Keywords: Anderson acceleration
describes a project that uses: SQUAREM
describes a project that uses: ALGOL 60
describes a project that uses: Anderson
describes a project that uses: EPSfun
describes a project that uses: GRPIA
describes a project that uses: Matlab
describes a project that uses: Maple
MaRDI profile type: Publication
full work available at URL: s11075-018-0567-2
OpenAlex ID: W2888051195
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The genesis and early developments of Aitken's process, Shanks' transformation, the \(\varepsilon\)-algorithm, and related fixed point methods (cites work)
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The genesis and early developments of Aitken's process, Shanks' transformation, the \(\varepsilon\)-algorithm, and related fixed point methods (DOI)
DOI: 10.1007/S11075-018-0567-2