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and to new results. He also discussed applications to linear systems. In fact, as showed by Sidi [133],
and Jbilou and Sadok [75], vector sequence transformations are closely related to projection methods
for the solution of systems of equations. In particular, the RPA, a vector sequence transformation
deÃ¿ned by Brezinski [20] was extensively studied by Messaoudi who showed its connections to
direct and iterative methods for solving systems of linear equations [98,99].
Vector sequence transformations lead to new methods for the solution of systems of nonlinear
equations. They also have other applications. First of all, it is quite important to accelerate the con-
vergence of iterative methods for the solution of systems of linear equations, see [32,33,36]. Special
vector extrapolation techniques were designed for the regularization of ill-posed linear systems in
[43] and the idea of extrapolation was used in [35] to obtain estimates of the norm of the error
when solving a system of linear equations by an arbitrary method, direct or iterative.
General theoretical results similar to those obtained in the scalar case are still lacking in the vector
case although some partial results have been obtained. Relevant results on quasilinear transformations
are in the papers by Sadok [123] and Benazzouz [8]. The present author proposed a mechanism for
vector sequence transformations in [45,34].

4. Conclusions and perspectives

In this paper, I have tried to give a survey of the development of convergence acceleration
methods for scalar and vector sequences in the 20th century. These methods are based on the idea
of extrapolation. Since a universal algorithm for accelerating the convergence of all sequences cannot
exist (and this is even true for some restricted classes of sequences), it was necessary to deÃ¿ne and
study a large variety of algorithms, each of them being appropriate for some special subsets of
sequences.
It is, of course, always possible to construct other convergence acceleration methods for scalar
sequences. However, to be of interest, such new processes must provide a major improvement
over existing ones. For scalar sequence transformations, the emphasis must be placed on the theory
rather than on special devices (unless a quite powerful one is found) and on the application of new
16 C. Brezinski / Journal of Computational and Applied Mathematics 122 (2000) 1â€“21

methods to particular algorithms in numerical analysis and to various domains of applied sciences. In
particular, the connection between convergence acceleration algorithms and continuous and discrete
integrable systems brings a di erent and fresh look to both domains and could be of beneÃ¿t to them.
An important problem in numerical analysis is the solution of large, sparse systems of linear
equations. Most of the methods used nowadays are projection methods. Often the iterates obtained
in such problems must be subject to acceleration techniques. However, many of the known vector
convergence acceleration algorithms require the storage of too many vectors to be useful. New
and cheaper acceleration algorithms are required. This di cult project, in my opinion, o ers many
opportunities for future research.
In this paper, I only brie y mentioned the con uent algorithms whose aim is the computation of
the limit of a function when the variable tends to inÃ¿nity (the continuous analog of the problem
of convergence acceleration for a sequence). This subject and its applications will provide fertile
ground for new discoveries.

Acknowledgements

I would like to thank Jet Wimp for his careful reading of the paper. He corrected my English
in many places, he asked me to provide more explanations when needed, and suggested many
improvements in the presentation. I am also indebted to Naoki Osada for his informations about
Takakazu Seki.

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