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

"Try first thyself, and after call in God; For to the worker God himself
lends aid." -- Euripedes (via Wikipedia)

If you're unwilling to take the trouble to read the published literature --
with a nice full bibliographic citation provided by a senior researcher in
the field -- I don't see why we should trouble to help you.  If you don't
know how to order copies of articles for free or near-free, talk to a
librarian (public or university).  Some of them even have telephones -- you
apparently have Internet service, so I'm guessing that you have telephone
service that will reach Iowa City -- if you're unfamiliar with that, ask
your local telephone operator about "Long Distance" -- it's a marvel!

In fact, if you ask nicely, some people on this list might even email you
PDFs of the articles.

             -s

On Fri, Aug 10, 2012 at 1:09 PM, Dennis Darland <dennis.darland at yahoo.com>wrote:

> 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.
> >
> >
> >
> >
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