Interpolation, approximation and rational extrapolation

An important problem in numerical analysis and applied mathematics concerns the approximation of functions known only by certain information. Interpolation and approximation are two techniques that enable us to represent an unknown function, for which we either know the values at a certain number of points, or some other information such as the beginning of its Taylor series development, by a simple function, but in an approximate way. The simplest function to use for this is, of course, a polynomial. But a polynomial won't always be able to adequately represent, for example, points coming from an exponential over a large interval or a function admitting poles. For reasons such as these, we turn to rational fractions.

Let's consider a second frequently encountered problem. Many methods used in numerical analysis and, more generally, in applied mathematics are iterative. They produce a sequence which, in the best cases, rapidly converges to the solution of the problem under consideration. Other methods provide an approximation of the solution which depends on a parameter and, as this parameter tends towards a limit (usually zero or infinity), this approximation tends towards the exact solution of the problem. By considering a sequence of these parameters converging towards their limit, we obtain a sequence of approximations to the solution that converges towards the desired answer. However, in both cases, convergence can be slow, making the method difficult to use in practice. On the other hand, the sequence (or approximation) may come from a black box, making it impossible to modify the manufacturing process. The idea is then to transform this slow sequence into a new sequence converging, under certain conditions, more rapidly towards the same limit. Such methods are based on the idea of linear or, better still, rational extrapolation.

The aim of this article is to provide an introduction to interpolation and approximation by rational functions, as well as to rational extrapolation. Examples of the application of these techniques will be given.

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