Why I am trying what I am trying in my ode program

First there is probably a way to symbolically derive equations for the terms of Taylor series terms from the differential equations in terms of the initial conditions. But then you will still have to numerically evaluate these equations. The equations are likely to be very complex if you derive very many terms. Also I don't think the total amount of numerical calculation could be reduced by postponing that evaluation. I don't think the Taylor function would do this, unless you already had a solution to the equation say by integrating, which is not always possible.

The method, I am using, uses a theoretically exact derivation of the Taylor terms. And you can compute as many terms as you need (given enough space & time.) I usually use 30, but don't really know yet how to optimize that. Also need to optimize increment size. Both of those involve many factors, including how long it will take. I am just beginning to investigate those things. I need to make as sure as I can that the program is correct. 

The other numerical methods I know about use 'divided differences'. They are approximations to the first few (or perhaps several) terms of the Taylor series. And I suspect may (note 'differences') be subject to subtraction error. (When you subtract numbers close to each other significant digits cancel resulting in fewer significant digits.) 

So I have more and more accurate terms. So I am interested in trying to make it work. Dr Chang worked on it, but didn't use a good language for the code generation. He used FORTRAN. Snobol was around then. I used SPITBOL in classes at UNL where he was then, but he only used FORTRAN. I think that was a mistake.  

There is a small bibliography on my sourceforge web page. Some of those books contain more extensive bibliographies. So there is some interest in the subject. Unless I do something, more than I can expect, I won't even be noticed. Most of the material in the books I have don't give me enough detail to help much. And I enjoy working out the details for myself anyway.

Dennis J. Darland
dennis.darland at yahoo.com
http://dennisdarland.com/
http://dennisdarland.com/dennisdaze/
http://dennisdarland.com/philosophy/
http://sode.sourceforge.net/
"According to the World Health Organization, the warming of the planet caused an additional 140,000 deaths in 2004, as compared with the number of deaths there would have been had average global temperatures remained as they were during the period 1961 to 1990. This means that climate change is already causing, every week, as many deaths as occurred in the terrorist attacks on September 11, 2001"
-- Peter Singer _Practical Ethics, Third Edition_, p. 216.