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

My only comment is that I am not, as I have said at least twice before, trying to translate a FORTRAN program. (or translate any program) I looked at code generated by Chang's program (not the program itself mind you) once in the last year. I was having trouble with his equation for arccos (found in the appendix to his draft of his book), All I learned was that arccos was not implemented in his program. (I did locate a typo in my code a little later.) I know I could get this ACM article by signing up for a service with ACM (about $100/year I think). I seriously doubt I could find the other articles locally. Probably closest is Iowa City - about 60 miles, but I have not driven that far in about 15 years, and have no intent to now.

I didn't find a book by Henrici on Taylor Series on Amazon.

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.


--- On Fri, 8/10/12, Richard Fateman <fateman at eecs.berkeley.edu> wrote:

> From: Richard Fateman <fateman at eecs.berkeley.edu>
> Subject: Re: [Maxima] Why I am trying what I am trying in my ode program
> To: "Dennis Darland" <dennis.darland at yahoo.com>
> Cc: "math maxima" <maxima at math.utexas.edu>
> Date: Friday, August 10, 2012, 9:27 AM
> On 8/10/2012 12:40 AM, Dennis Darland
> wrote:
> > 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.
> Yes, Barton Willis showed how.? David Barton (&
> others) showed how to do this a few decades ago.
> > But then you will still have to numerically evaluate
> these equations.
> Not if you are happy with the semi-symbolic representation
> of a Taylor series.? For example,? Here is
> an approximate solution to a differential equation:?
> 1+x+x^2/2.
> Do you need to numerically evaluate?? Maybe.
> >???The equations are likely to be very
> complex if you derive very many terms.
> Not really, the solutions have a very specific format that
> is generally not complex.
> > 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.
> If you have an equation y'=f(y,x)? then you have an
> integration formula.
> > 
> > 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.)
> This is, in Maxima, a program that fits on half a page.
> >???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.
> Have you read the papers
> 1 BARTON, D. On Taylor series and stiff equations. ACM
> Trans. Math. Softw. 6, 3(Sept. 1980),
> 280-294.
> 2. BARTON, D., WILLERS, I.M, AND ZAHAR, R V.M. The automatic
> solution of ordinary differential
> equations by the method of Taylor series. Comput. J 14
> (1971), 243-248
> 3. BARTON, D., WILLERS, I.M., AND ZAHAR, R.V.M. Taylor
> series methods for ordinary differential
> equations--An e~caluation. In Mathematical Software, John
> Rice (Ed.), Academm Press, New
> York, 1971, pp. 369-390.
> 
> attempting to translate a fortran program that you don't
> understand into languages that
> you don't understand is not such a good idea.
> > 
> > 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.)
> It should be possible for you to learn about numerical
> methods (for ODEs, etc) by reading about them,
> or maybe taking courses.
> It is not necessary for you to reveal your unfamiliarity
> with these methods.
> > 
> > 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.
> Macsyma, Reduce, and other computer algebra languages, as
> well as Lisp, were around for
> a decade or two by the time Chang tried to do this work.
> >???
> > 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.
> I suggest, once again, that you take the Barton, Willers,
> Zahar paper.? Read it carefully and see how to
> implement
> the ideas in Maxima (or Maple or Mathematica or ...) in a
> page or so of code.? You don't need to declare or
> compile anything at all.
> 
> Then read the stuff in Henrici's book about Taylor series
> and singularities and radius of convergence and errors.
> 
> You may realize that there will be more than one Taylor
> series of interest, expanded at a single point, and that
> there are mathematical (algebraic) questions about what you
> are trying to do.
> 
> Finally, if you have everything "working"? you can
> start worrying about numerical stability, or computing
> Taylor series
> of many many terms.
> 
> 
> 
>