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:This is a current research topic. (Not Jenkins-Traub, but rootfinding

>>>>>> "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.

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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