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Extrapolation and rational approximation. The works of the main contributors. (English) Zbl 1477.65006

Cham: Springer (ISBN 978-3-030-58417-7/hbk; 978-3-030-58420-7/pbk; 978-3-030-58418-4/ebook). viii, 406 p. (2020).
Linear and nonlinear extrapolation methods are widely used for accelerating the convergence of sequences and series in numerical analysis. This impressive monograph describes this important topic in its historical context. Further this book can serve as an introduction to extrapolation methods, Padé approximation, formal orthogonal polynomials, and continued fractions.
After an overview of the content, Chapter 2 presents an introduction to the main topics. In Chapter 3, the authors describe the research until 1950, where the Richardson extrapolation method, the Romberg integration, the Aitken process for accelerating convergence, and the Shanks transformation were developed. Chapters 4 and 5 are devoted to the life of the British mathematician Peter Wynn (1931–2017) and to the analysis of his important works on nonlinear extrapolation methods, rational approximation, continued fractions, and formal orthogonal polynomials. In Chapter 6, the further developments affected by the results of P. Wynn are discussed. In Chapter 7, several contemporary researchers whose works are related to extrapolation methods present personal testimonies which reflect their own views on the domain. Brief biographies of mathematicians with important results in extrapolation methods is given in Chapter 8. This book ends with a conclusion and a bibliography of 816 items. Here an important branch of numerical analysis is impressively described in its historical context, opening a way to new developments.

MSC:

65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65-03 History of numerical analysis
65B05 Extrapolation to the limit, deferred corrections
40A15 Convergence and divergence of continued fractions
41A20 Approximation by rational functions
65F10 Iterative numerical methods for linear systems
65H10 Numerical computation of solutions to systems of equations

Biographic References:

Wynn, Peter
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