How to find all roots of this high order polynomial equation?

[iwannafly]

How to find all roots of this high order polynomial equation?

Hi all,

I need to find all real $u$'s, such that

$$\sum_{i=1}^{n}{\left(\frac{w_{i}}{u-e_{i}}\right)^{2}}=1$$

where $$\sum_{i=1}^{n}{w_{i}^2}=1$$ and w_{i}'s and e_{i}'s are given.

My questions are: are there systematic way of finding all possible
solutions $u?$

$u$ is unconstrained... all the rest are given...

For n large (thousands), what's the most numerically reliable way to
systematically find all the real roots?

I am thinking of reversing the eigenvalue decomposition thru
characteristic polynomial...

But is that method stable/reliable?

[Edit] One step further - are there any shortcuts?

Now I need to find a number $u$, such that

$$\sum_{i=1}^{n}{\left(\frac{w_{i}}{u-e_{i}}\right)^{2}}=1$$

And I am looking for real numbers $u$...

And after finding all these roots $u$'s,

I would like to compare all of the following:

$$\sum_{i=1}^{n}{\left(\frac{w_{i}}{u-e_{i}}\right)^{2}/e_{i}}$$

and find one of the roots u* which maximizes the above expression?

Any possible shortcuts?

Thanks!

[Axel Vogt]

I do not read Tex and would prefer Maple's syntax.

In that generality there is no symbolic solution.

It is a numerical question (as your degree is high), so you may wish
to look for a root finder. Maple can do it, but may need long time.
The stability will depend on your data, reliability will depend on
used working precision.