Extrapolation algorithms and Pade approximations: A historical survey

In numerical analysis there are many methods producing sequences. Such is the case of iterative methods, of methods involving series expansions, of discretization methods, that is methods depending on a parameter such that the approximate solution tends to the exact one when the parameter tends to zero, of perturbation methods, etc. Sometimes, the convergence of these sequences is so slow that their effective use is quite limited. The aim of extrapolation methods is to construct a new sequence converging to the same limit faster than the initial one. Among these methods, the most well known are certainly Richardson's extrapolation algorithm and Aitken's Delta(2) process. In many branches of applied sciences, the solution of a given problem is often obtained as a power series expansion. The question is then trying to approximate the function from its series expansion. A possible answer is to construct a rational function whose series expansion matches the original one as far as possible. Such rational functions are called Pade' approximants. These two subjects, which have some connections, go quite deep and far into the history of mathematics. They are related to continued fractions (a field which goes back to the Greek antiquity), orthogonal polynomials, the moment problem, etc., they played an important role in the development of mathematics (such as the transcendence of e and pi) and they have many applications. This paper will give a short historical overview of these two subjects. Of course, we do not pretend to be exhaustive nor even to quote every important contribution. We refer the interested reader to the literature and, in particular, to the recent books [5,22,29,24,38,46,48,68,78,131], For an extensive bibliography, see [23].