As a Conclusion This book proposed a tour d’horizon, a fresco, on linear and nonlinear extrapolation methods, Padé approximation, formal orthogonal polynomials, continued fractions, and related topics. It also has the unique feature of mixing mathematical concepts and results, their history, a detailed analysis of important publications, personal testimonies by researchers involved in research on these domains, and information, anecdotes, and exclusive remembrances on some actors. After a mathematical introduction to these themes, we gave an account of their status of knowledge around 1950, that is, at the time most scholars, Aitken, Richard- son, Shanks, Wynn, and others, began their research. In the course of the analysis of these works, and in particular of Peter Wynn’s papers and reports (all written alone), some more mathematical concepts were introduced and explained. Then the recent developments of the domains were presented. In published papers and books, research is most of the time introduced in an impersonal way, although developed by people who have existed or are still alive. The human side of research is missing. This is why we asked researchers who personally played (and are still playing) a role in these fields, and whose works are related to those described herein, to send us their testimonies. They show how research in a domain starts and evolves. For the same reason, biographies of other scholars encountered have been included. The human side of research plays a fundamental part in its development. This is why congresses form an important feature of our profession, since they allow researchers to meet other specialists and to build relationships that often last a lifetime. In the domains concerned here or related to them, many congresses, or special sessions in more general ones, or even summer schools, were organized. We cannot list them here, because they were too numerous. The proceedings of many of them were published and can easily be found. We are aware of not having cited all the interesting extensions of the works mentioned above, an almost impossible task. We also apologize to those who are not quoted here. It does not mean that their achievements were less interesting than those mentioned, but a choice had to be made. We are responsible for this choice, for the unintentional omissions, and for the misprints and even the mistakes contained in this book. © Springer Nature Switzerland AG 2020 339 C. Brezinski, M. Redivo-Zaglia, Extrapolation and Rational Approximation, https://doi.org/10.1007/978-3-030-58418-4 340 As a Conclusion If we look back to the General Introduction of this book, we put forward the names of five pioneering mathematicians, who, in chronological order of their works, are Henri Eugène Padé, Lewis Fry Richardson, Alexander Craig Aitken, Daniel Shanks, and Peter Wynn. Let us now try to draw a conclusion about the accomplishments of each of them, although it is quite a presumptuous task. Continued fractions and Padé approximants were known and used long before Henri Padé defended his thesis in 1892. Almost any mathematical book of the nineteenth century, in English, French, German, Italian, etc., contains a chapter on continued fractions. However, it was his thesis, although written in French (but at that time French was one of the most widely used languages for science), that brought light to the topic, since it was the first systematic study of these approximants. Padé arranged them into a double entry table, and he gave a closed formula for the approximants of the exponential function. He also related them to continued fractions. Let us mention that Charles Hermite, Padé’s thesis advisor, was a little bit skeptical about the results obtained, since in his report at this occasion, he wrote (see [483]):1 While thinking that the theory of continued fractions could be generalized under other points of view, opening a more interesting and more fruitful path, and that the importance of the results which he has achieved does not correspond as much as one would like to his long and conscientious work, we judge that the thesis of Mr. Padé is worthy of the approval of the Faculty. Padé published several papers after his thesis, but overwhelmed by his administrative duties, he quite rapidly abandoned research. His papers are gathered in [483]to- gether with their contemporary analyses. However, his contribution certainly opened the way to further work, and it was rapidly known outside France, as illustrated, for example, by van Vleck’s talk at the 1903 colloquium of the American Mathematical Society [665] (see Sect. 3.4). Thus, the work of Padé was quite influential in the de- velopment, in approximation theory, of continued fractions and of the approximants named after him. It was for improving the accuracy of a finite difference approximation to a partial differential equation that Lewis Fry Richardson introduced extrapolation in 1910 [520]. He noticed that the error had an expansion in even powers of the step size h, and that by combining two approximations with h and 2h, the term in h2, and some years later (in 1927 [523]), also the term in h4 can be eliminated. Although the idea could be traced back to the improvement of old approximations of π by inscribed and circumscribed polygons, it was the first time it was used for solving a numerical analysis problem. Then extrapolation rapidly found its place in the toolbox of numerical analysts, and it became a favorite method for treating the case of a polynomial expansion of the error. It seems that Richardson, interested in many other topics, was not fully aware of the potentialities of the method he found (nor was he of his discovery of fractals!). 1 Tout en pensant que la théorie des fractions continues pourrait se généraliser sous d’autres points de vue ouvrant une voie plus intéressante et plus féconde, et que l’importance des résultats auxquels il est parvenu ne répond pas autant qu’on le désirerait à son long et consciencieux travail, nous jugeons que la thèse de Mr. Padé est digne de l’approbation de la Faculté. As a Conclusion 341 Although it can also be traced back to at least the seventeenth century (see [476]), the Δ2 process introduced in 1926 by Alexander Craig Aitken was the first nonlinear convergence acceleration method used in numerical analysis [5]. It rapidly became known and used, but was not immediately studied from the theoretical point of view, in particular for its interpretation as an extrapolation method. However, despite the fact that Aitken did not fully exploit its possibilities, the process finally gained recognition, and it had an important impact on the opening of the new field of convergence acceleration and extrapolation methods in numerical analysis. The purpose of Daniel Shanks when he built his transformation in 1949 [567] was to accelerate the convergence of slowly convergent sequences and series, and to assign a value, the antilimit as he called it, to divergent ones. His sequence trans- formation was based on the analogy between transients (a sum of exponentials) and mathematical sequences, and it was expressed as a ratio of two Hankel determinants. Shanks knew that it was a generalization of the Aitken process, but he was not aware that his transformation had a larger kernel containing other sequences than pure sums of exponentials. He studied the connections of his transformation with continued fractions and Padé approximants, and gave several numerical examples. Shanks produced a more detailed study of his transformation in 1955 [568]. As he explained to C.B., he was implementing it by computing separately the Hankel determinants of their numerators and their denominators by their recurrence relation given in Sect. 3.2.4. Although it is a quite difficult exercise, let us now try to express our feelings about the work of Peter Wynn. He obviously had an extended mathematical culture running from computer languages and the skills required for the implementation of algorithms, to classical numerical analysis, complex analysis, and abstract algebra. He also had a vast knowledge of the mathematical literature in English, French, German, Italian, and Russian. At that time, it was difficult to be aware even of the existence of several references he cited, not to mention that they had to be found in libraries at a period before the internet existed. Even if his papers are not quoted as much as they should be, they are quite influential. Recall that the Shanks’s paper appeared in an applied mathematics and mathematical physics journal. Wynn’s paper on the scalar ε-algorithm was published in 1956 in the only journal devoted, at that time, to numerical methods [713]. Without Wynn’s paper, the Shanks transformation could have remained unknown, and if not, at least useless in practice. Once known, the name of Wynn sparked interest, and researchers began to look at his other papers, even if they were not directly working on the topic treated. Thus, due to his work, some themes that were, if not completely forgotten, at least neglected, were revived. Padé approximants are a particular case, since they were used by theoretical physicists for some time, but not so much studied by numerical and complex analysts. The same is true for continued fractions, which were mostly used by number theorists. The influence of Wynn on the further development of these domains cannot be denied. In particular, the field of convergence acceleration and extrapolation methods could have remained in an embryonic state without him. Moreover, many of his discoveries have not yet been fully exploited, and we hope that this book will inspire some readers to pursue them and to study the manuscripts found after his death and analyzed in 342 As a Conclusion [122].