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The genesis and early developments of Aitken’s process, Shanks’ transformation, the ε–algorithm, and related fixed point methods

Authors
  • Brezinski, Claude1
  • Redivo–Zaglia, Michela2
  • 1 Université de Lille, Laboratoire Paul Painlevé, UMR CNRS 8524, UFR de Mathématiques, Villeneuve d’Ascq cedex, 59655, France , Villeneuve d’Ascq cedex (France)
  • 2 Università degli Studi di Padova, Dipartimento di Matematica “Tullio Levi-Civita”, Via Trieste 63, Padova, 35121, Italy , Padova (Italy)
Type
Published Article
Journal
Numerical Algorithms
Publisher
Springer US
Publication Date
Aug 23, 2018
Volume
80
Issue
1
Pages
11–133
Identifiers
DOI: 10.1007/s11075-018-0567-2
Source
Springer Nature
Keywords
License
Yellow

Abstract

In this paper, we trace back the genesis of Aitken’s Δ2 process and Shanks’ sequence transformation. These methods, which are extrapolation methods, are used for accelerating the convergence of sequences of scalars, vectors, matrices, and tensors. They had, and still have, many important applications in numerical analysis and in applied mathematics. They are related to continued fractions and Padé approximants. We go back to the roots of these methods and analyze the original contributions. New and detailed explanations on the building and properties of Shanks’ transformation and its kernel are provided. We then review their historical algebraic and algorithmic developments. We also analyze how they were involved in the solution of systems of linear and nonlinear equations, in particular in the methods of Steffensen, Pulay, and Anderson. Testimonies by various actors of the domain are given. The paper can also serve as an introduction to this domain of numerical analysis.

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