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

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

Martin Rubey <[email protected]> writes:


> 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