Over the years I've looked at Corliss/Chang and Barton/Willers and
Henrici papers on using Taylor series
for solving ODEs, specifically initial value problems.
(e.g. Corliss & Chang,
Solving Ordinary Differential
Using Taylor Series,
ACM TOMS Vol 8, No. 2, June 1982, Pages 114-144)
I think that starting with the technology in Maxima allows for vast
simplification in the setup.
Basically, you don't need a separate compiler in language 1 to produce
code in language 2. (FORTRAN? C?)
You can write a program to generate the needed Taylor series coefficients
using Maxima, starting with the initial condition and the DE. You can
then evaluate
the series as needed, and also run whatever code you had in mind to estimate
the radius of convergence. In Maxima. If you want more speed, there are
some
tricks that can be used that do not require generating FORTRAN etc.
I suggest that writing a Maxima program to generate Taylor series
solutions to ODEs would
be a nice project, and probably can be demonstrated using a half-page of
code.
Doing a fancy job would naturally require more code.
The basic problem with these methods is that people who solve ODEs for a
living
have their own methods that are not directly comparable to Taylor series
methods,
and generally show no interest in changing.
(I have not found the code to be readable, but I doubt that any sizable
percentage of the time
need be spent computing factorials.)
RJF