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Volume 9, Issue 4


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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 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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