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Re: FindRoots?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg112227] Re: FindRoots?
  • From: "Ingolf Dahl" <ingolf.dahl at telia.com>
  • Date: Sun, 5 Sep 2010 05:29:05 -0400 (EDT)
  • References: <201009040758.DAA25856@smc.vnet.net>
  • Reply-to: <ingolf.dahl at telia.com>

Andrzej, 
I must ask some question related to your answer. Better to ask and appear
unwise than not to ask and remain unwise.
The questions are interlaced below.

> -----Original Message-----
> From: Andrzej Kozlowski [mailto:akozlowski at gmail.com]
> Sent: den 4 september 2010 09:58
> To: mathgroup at smc.vnet.net
> Subject: [mg112182] Re: FindRoots?
> 
> (snipped)
> 
> Reduce used exact methods so you have to rationalize the output or use
equivalent
> approaches and it works fine in such cases. You are right that you can't
use it with
> Interpolating functions since of course they are not analytic.

I want to put a question mark on "of course". Say, an interpolation
polynomial of fifth degree - is that not "analytic"? Or a RBF (radial basis
function) interpolation, with a Gaussian radial basis function. And most
other interpolation methods (also those returned by NDSolve) are
differentiable any number of times everywhere except in isolated points or
along some lines. Then it should be a bookkeeping problem to analyze each
interval or region, one at a time? It should not be too difficult to
exchange the interpolation method, if that is preferred.

> 
Gianluca Gorni wrote:
> > I have made some interactive panels where I can change the Locators
> > with the mouse and I get in real time the roots of the interpolating
> > function as big Points in the plot: I can do this with RootSearch,
> > but not with Reduce.
> > Unfortunately, I can't give these panels to other users, because
> > I can't assume that they have RootSearch installed.
> >
> > I endorse the wish that the functionality of RootSearch were available
> > in the kernel.

In some cases it should as an alternative be possible to interpolate the
inverse function to find roots. That requires in general scattered-point
interpolation, "available in a package designed specifically for dealing
with interpolating functions ", maybe found at
http://www.familydahl.se/mathematica . (It is possible that we have already
have solved the hard part of the problem when we have found that there is an
inverse function for some region.)

Andrzej Kozlowski  wrote:
> I concede that you make a reasonable case for this sort of capability.
However,
> computations of this kind with Interpolating functions are generally quite
unreliable, since
> the conditions required by FindRoot to work are often not satisfied. I
don't think this sort
> of hit or miss approach is appropriate for a mathematical solver. Perhaps
this sort of
> capabilities should be available in a package designed specifically for
dealing with
> interpolating functions (which I don't consider as "mathematical
objects").

What are interpolating functions then? Please elaborate in detail, because I
am curious!

I think that different point sets in abstract vector spaces are important in
many places for our description of the world and of the universe of
thoughts. Interpolation functions make it possible to apply various
mathematical methods to these point sets. And I think that abstract point
set tools might become important in the further development of machine
intelligence.

Best regards

Ingolf Dahl



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