1
$\begingroup$

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

  • 1
    It looks like you forgot something. What should $\sum_{i = 1}^n \frac{w_i}{(u-e_i)^2}$ be?2012-04-18
  • 0
    If you are given the result of the sum, $\sum_{i = 1}^n \frac{w_i}{(u-e_i)^2}=k$, for $n=1$ you have a quadratic, so will generally have two solutions. Depending upon the parameters, they may be complex.2012-04-18
  • 0
    I've added some new info... any more thoughts? thx2012-04-18
  • 0
    No constraints on the $e_i$?2012-04-18

3 Answers 3