eigen value vs Newton's Method vs.... also Matlab Symbolic Toolkit

eigen value vs Newton's Method vs.... also Matlab Symbolic Toolkit

The Numerical Algorithms Group (NAG) does this kind of thing for
a living; they have a Laguerre method as their primary algorithm
it seems.

http://www.nag.com/numeric/FL/manual/pdf/C02/c02_intro.pdf

I would not be surprised if Matlab previously had a Jenkins-Traub
algorithm and discarded it.

The Matlab symbolic toolkit, since you mention it, is something
I have been playing with. And I think it is more appropriate
to discuss that here, since it is peripheral to most Matlab
activities.

In order to convert a Matlab value to a symbolic one, you can
add sym(0) to it.
Thus
simplify(3+sym(0))  is ok, and returns 3.  While simplify(3) is
illegal: an attempt to call a symbolic routine "simplify" on a
traditional Matlab value, the 1X1 array of double-precision numbers,
namely  3.    This is a misfeature, certainly. Why not add sym(0)
and see what happens instead of the mysterious error message.

But here is an anomaly...

simplify(pi^2+sym(1))   returns a number 3059521645650671/281474976710656
but
simplify(pi+sym(1))  returns  pi+1

Cheers..
RJF

> If you and/or the original poster would like to suggest that MATLAB use
> these algorithms for polynomial root finding instead of or in addition to
> the routine currently used in roots.m, please suggest it to

> future version of MATLAB.  Also (to the original poster) while sometimes
> questions about MATLAB's Symbolic Math Toolbox are posted to
> sci.math.symbolic, you're more likely to get a discussion about MATLAB going
> on the MATLAB newsgroup comp.soft-sys.matlab.

> --
> Steve Lord

I'll give more details and some limitations later
(I had a detailed mail when netscape crashed;
now I have to reconstruct everything I wrote...)

Since you are from Mathworks, why don't you suggest it to them?

I have bad experience with suggesting changes to the Matlab team.
None of several suggestions had any effect. For example,
many years ago I mentioned that fzero fails to solve a simple
quadratic equation. But it still does, as anyone can see who tries
x=fzero('x^2-0.009',1)
(The reply I got from Mathworks was that it works with a different
starting value!)
There are a number of very simple remedies, one of them being described
in my recent numerical analysis book
http://www.mat.univie.ac.at/~neum/index.html#numbook
[~ is a tilde]

Arnold Neumaier