Numerical Algorithms

, Volume 80, Issue 1, pp 11–133 | Cite as

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

  • Claude BrezinskiEmail author
  • Michela Redivo–Zaglia
Original Paper


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.


Extrapolation Aitken’s process Shanks’ transformations Epsilon–algorithms MMPE MPE RRE Fixed points Pulay mixing Anderson acceleration 

Mathematics Subject Classification (2010)

65B05 65F10 65H10 65Q10 01–08 65–03 


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We are indebted to many colleagues for their help during the preparation of this paper. We would like to thank Stan Cabay, William Ford, Juan Antonio Pérez, Avram Sidi, David Smith, Yossi Stein, and Héctor René Vega–Carrillo for sharing with us their memories on their work, and allowing us to quote what they wrote us. We are also grateful to Matania Ben–Artzi, Michel Bercovier, Achiya Dax, Arieh Iserles, George Labahn and Stig Skelboe for their help in our search for information. Luc Vinet, Director of the Centre de Recherches Mathématiques of the Université de Montréal asked Vincent Masciotra, head of the Administration of this Center, to retrieve and sent us the administrative documents concerning Peter Wynn. We are grateful to them, and to Michel Delfour for his help. Thank you to Reuvena Shalhevet–Kaniel for sending us a photo of her husband. Peter Pulay was kind enough to allow us to reproduce what he wrote us on his work. We would like to thank Donald G.M. Anderson for sending us a long commentary about the discovery of his acceleration method, and, moreover, to carefully reading our paper and suggesting many important improvements. Susan Virginia Welby was quite helpful during our exchanges with Donald G.M. Anderson. We thank her. We appreciated the help of Gérard Meurant who carefully checked the paper, and whose remarks greatly helped us to improve it. Hervé Le Ferrand also suggested many useful modifications. Our paper substantially benefited from the relevant remarks of Ernst Joachim Weniger. We are grateful to him. Jean–Claude Miellou reminded us some interesting references. With Michel Crouzeix we looked again at our common paper of 1970, and we produced a better proof of one of our results. We thank him. Finally, we are indebted to Stefano Cipolla who was able to find on the internet all the references we needed. Without his help, it would have been much more difficult to complete this paper.

Funding information

The work of C.B. was supported by the Labex CEMPI (ANR-11-LABX-0007-01). M.R.–Z. is a member of the INdAM Research group GNCS, which partially supported the research.


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Authors and Affiliations

  1. 1.Laboratoire Paul Painlevé, UMR CNRS 8524, UFR de MathématiquesUniversité de LilleVilleneuve d’Ascq cedexFrance
  2. 2.Dipartimento di Matematica “Tullio Levi-Civita”Università degli Studi di PadovaPadovaItaly

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