I am given that:
$\sum_{i=1}^{n}{w_{i}}=1\\$
and that a set of numbers $e_i$, where $i$ can range from $1$ to $n$.
Now I need to find a number $u$, such that
$\sum_{i=1}^{n}{\left(\frac{w_{i}}{u-e_{i}}\right)^{2}}=1$
My questions are: are there systematic way of finding all possible solutions $u?$
And is the number of solution related to $n?$
I am thinking of maybe for $n=2$, the number of solutions $u$ is $1?$
$u$ is unconstrained... all the rest are given...
and yes, $w_i >$ or = $0$ for all $i$...
And for general $n$, the number of solutions $u$ is $n-1$?
Thanks a lot!
[Edit]
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:
$\sum_{i=1}^{n}{\left(\frac{w_{i}}{u-e_{i}}\right)^{2}/e_{i}^{2}}$
and find one of the roots u* which maximizes the above expression?
Any possible shortcuts?
Thanks