# Numerical computation stability issue

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 ($a_i ^ \lambda \approx 1000^{1000}$)

The expression for $x$ is invariant under change of scale: $a'_i= s a_i$, $b'_j= s b_j$. So, use $s=10^{-3}$ and get those numbers close to 1. This probably helps.
This is similar to what lhf said, but it might help. Set $A=\max_i a_i$ and $B=\max_i b_i$, then $$x=\sqrt{\frac{A}{B}}\left[\frac{\sum_i\left(\frac{a_i}{A}\right)^\lambda}{\sum_i\left(\frac{b_i}{B}\right)^\lambda}\right]^\frac{1}{2\lambda}$$ I think that this should get you decent stability and keep the computation from overflowing.