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networm
Posts:
327
Registered:
10/6/05
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How to find all roots of this high order polynomial equation?
Posted:
Apr 18, 2012 11:47 PM
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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!
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