[Axiom-math] evaluator for functional equations / CHALLENGE!

[Martin Rubey]

Dear all!

next week I'm going to present my guessing package at MathInfo 06. It works
quite well meanwhile :-)

Hower, there is one thing I don't really want to program - in fact, I won't -
although it would be really really useful. Maybe somebody else can do it. I
offer a price, ok?

The challenge is as follows:

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.

Price is negotiable.

Martin

[Martin Rubey]

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

Post by Martin Rubey
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

[David Joyner]

Hello:

Does anyone on this list know about the PAFF package for AXIOM
written by Gaétan Haché? It does Brill-Noether computations, Riemann-Roch
computations, etc.

- David Joyner