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So I'm trying to simplify this problem $c = \operatorname{mean}\left(\frac{a_i}{a_i+b_ix}\right)$ to solve for $x$. Right now the way I'm doing it is basically trying a bunch of $x$ values and seeing which one is closest to $c$, but I was wondering if there's a way to solve for $x$ directly. I tried this: $$ c = \frac 1 n \sum_i^n \frac{a_i}{a_i+b_ix} \\ cn = \sum_i^n \frac{a_i}{a_i+b_ix} \\ cn = \sum_i^n a_i \prod_{j \neq i} (a_j + b_jx) $$

But for large $i$ this would be a large, intractable polynomial, right? Are there any other ideas that might help? Thanks.

Mohsen Shahriari
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sw_buddy
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    Any additional constraints? E.g., are perhaps all $a_i,b_i$ positive? – Hagen von Eitzen Aug 27 '21 at 20:15
  • maybe you could use Newton instead of just trying a bunch? – trula Aug 27 '21 at 20:31
  • Let $,c_i=-a_i/b_i,$ and assume WLOG that the sequence is ascending. The equation will have one root in each interval $,(c_i,c_{i+1}),$ and one more root elsewhere, see this for example. You can first find which interval $,c,$ falls into, then find the closest roots to it numerically. – dxiv Aug 27 '21 at 20:59

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