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

**References**:**Re: FindRoots?***From:*Andrzej Kozlowski <akozlowski@gmail.com>