We have a problem to find $x$ such that $ \sum_{i}\left(\frac{a_i}{x}\right)^\lambda= \sum_{j}\left(\frac{b_j}{x}\right)^{-\lambda} $
$a_i < x < b_j$
$a_i \approx x \approx b_j$
We get analytic solution: $ x = \left[\frac{\sum_{i}a_i^\lambda}{\sum_{j}b_j^{-\lambda}}\right]^{\frac{1}{2\lambda}} $
But since $a_i,b_j,x,\lambda \approx 10^3$ we cannot evaluate this formula because of numerical overflow ($a_i ^ \lambda \approx 1000^{1000}$)
Please advise. Thanks.