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## Re: [Axiom-developer] evaluator for functional equations / CHALLENGE!

 From: Martin Rubey Subject: Re: [Axiom-developer] evaluator for functional equations / CHALLENGE! Date: 12 Sep 2006 09:14:21 +0200 User-agent: Gnus/5.09 (Gnus v5.9.0) Emacs/21.3

```A followup on my own post, containing some material for fast algorithms.

> I need an operation evalADE that takes a functional equation of the form
>
> f(x) = g(f(x), D(f(x),x), D(f(x),x,2),...),
>
> where g is any "nice" expression, some initial values, and an integer n.
>
> The result of the operation should be the n-th coefficient of the taylor
> expansion of f, if it exists.
>
> Even more important, suppose that the functional equation is of the form
>
> p(f(x), D(f(x),x), D(f(x),x,2), ...)
>
> where p is a polynomial. These f are called differentially algebraic.
>
> The algorithm does not need to be especially fast, but it would be nice to be
> a
> able to compute the first fifty to hundred coefficients in a reasonable time.
>
> Note that Axiom provides an operation seriesSolve, which provides a partial
> solution. However, it is very buggy and gives up even for certain algebraic
> equations.

In the case of expansion around an ordinary point of f(x) satisfying a *linear*
differential equation, i.e.,

a0(x) f(x) + a1(x) D(f(x),x) + a2(x) D(f(x),x,2) + ... + a_k(x) D(f(x),x,k) = 0

with a_k(x0)<>0,

a fast algorithm has been proposed by

Alin Bostan, Frédéric Chyzak, François Ollivier, Bruno Salvy, Éric Schost,
Alexandre Sedoglavic

available at http://arxiv.org/ps/cs/0604101

There is a paper by Nedialkov and Pryce

http://www.cas.mcmaster.ca/~nedialk/PAPERS/DAEs/taylcoeff_I/

that proposes an algorithm for the general problem. Maybe that's the one we
want...

Martin

```