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 _______________________________________________ Axiom-developer mailing list Axiom-developer@nongnu.org http://lists.nongnu.org/mailman/listinfo/axiom-developer
