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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.
Martin Rubey <address@hidden> 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