Understood that this is not a solved problem. However, it would be useful to give users who aren't familiar with Jenkins-Traub some guidance.

I was also surprised that sols:allroots(eq:x^243+23*x-24) gives a warning:
allroots: unexpected error; treat results with caution.
allroots: only 65 out of 243 roots found.
and the first element of the result list is a reduced polynomial.

This behavior isn't warned against in the doc at all.

I was somewhat less surprised that some values for subst(sol,eq) were pretty big -- that's the nature of the beast....

             -s

On Mon, Jul 7, 2014 at 6:04 PM, Richard Fateman <fateman@berkeley.edu> wrote:
On 7/7/2014 12:54 PM, Raymond Toy wrote:
>>>>>> "Stavros" == Stavros Macrakis <(Σταῦρος Μακράκης)" <macrakis@alum.mit.edu>> writes:
>      Stavros> ? allroots and ? bfallroots (unlike ? realroots) say
>      Stavros> nothing about the precision of the result, or how to
>      Stavros> control it (except the unhelpful "allroots may give
>      Stavros> inaccurate results in case of multiple roots"). It would
>      Stavros> be helpful if someone knowledgeable about the algorithms
>      Stavros> could add some information about precision to the docs.
>
> allroots (and bfallroots) use Jenkins-Traub to compute the
> roots. AFAIK, there are no guarantees on accuracy. I do know accuracy
> degrades for multiple roots (or near multiples). And since deflation
> is used, the accuracy of the later roots is worse than the earlier
> roots.
This is a current research topic.  (Not Jenkins-Traub, but rootfinding
methods).
Given that you cannot necessarily evaluate a polynomial accurately, it
can be
difficult  to find its roots accurately.  There are methods that find
all roots at the
same time, perhaps more accurately and more slowly.

>
> Having said that, I think the algorithm does produce a single root, r,
> of the polynomial, P, such that |P(r)| < eps where eps is
> approximately roundoff in computing the polynomial.
>
> Ray
>
>
>
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