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Subject: Re: rounding issue
Date: Sat, 21 Mar 2009 18:28:01 +0000 (UTC)
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"John D'Errico" <woodchips@rochester.rr.com> wrote in message <gq2efk$src$1@fred.mathworks.com>...
> "Bruno Luong" <b.luong@fogale.findmycountry> wrote in message <gq2ak0$gtp$1@fred.mathworks.com>...
> > Was the above sum approximates an integral? How alpha is chosen? I guess it is related to a timestep? Have you analyzed what kind of error induced for the alpha you are using?
> > 
> > Sometime it's better to go back to the blackboard and take a new look on the original problem instead of persisting on the bad discretization.
> > 
> > Bruno
> 
> I'd very much agree with Bruno. One solution, IMHO
> the wrong one, is to immediately look for higher
> precision. The problem is, this just pushes the problem
> out a little ways. And then you need to go to higher
> precision yet. And high precision arithmetic is slow.
> 
> It is better to look for a different approach. Many
> times it is something as simple as scaling the problem
> so that the parameter lies in the domain where the
> series is nicely convergent.
> 
> John

  Bruno and John, what you say may be true.  However, it remains a fact that the function defined by Gavrilo's infinite series is a very respectable entire analytic function, f(z), (analytic over the entire complex plane) using z = t^alpha/(R*C).  For alpha = 1, it is the function

 f(z) = 1-exp(-z)

and for alpha = 2, it is

 f(z) = 1-cos(z^(1/2))

both of them entire analytic functions.  The only troublesome thing about them is that their Taylor series is expanded about the wrong point.  About a nearer point there should be an accurate way to compute them.  The problem seems to be one of applying the wrong numerical technique for its evaluation, not something intrinsic in the function itself.

Roger Stafford