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Tricks of the Trade
Edited by Paul Abbott

Generalized Padé Approximation

It is often the case that, while an exact solution is difficult or impossible, series or asymptotic solutions to a particular problem can be obtained without too much difficulty. As is well known, Padé approximations (see mathworld.wolfram.com/PadeApproximant.html) are usually superior to Taylor expansions when functions contain poles, because the use of rational functions allows them to be well represented.

Define and as polynomials of degrees and , with .

The coefficients in the Padé approximant to are determined by requiring that the Taylor expansion of is of order .

Suppose that you have obtained the following Taylor series up to for a function of interest.

Here is the Padé approximant for this series.

Verify that the Taylor expansion of is of order .

A plot shows that the Taylor series diverges from the Padé approximant near .

Suppose now that you have also computed the following asymptotic series for the same function.

A plot shows that the Padé approximant about and the asymptotic expansion intersect for .

How can you incorporate information from expansions of a function about two or more points? One approach is to generalize the idea of computing Padé approximants [1]. Suppose has the asymptotic expansions

in the neighborhoods of distinct points and , respectively. To compute a two-point Padé approximant to , we require that the rational function agrees with up to order about and to order about , where .

To use all the information at hand, note that for , , and for , .

The diagonal Padé approximant then has .

It is straightforward to determine the coefficients and in terms of and . Here we obtain these coefficients using series arithmetic via Solve.

Here is the two-point Padé approximant to .

The two-point Padé approximant agrees with the Taylor series about , the one-point Padé approximant about , and the asymptotic expansion about .



     
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