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I would like to estimate the largest solution (the solution with largest modulus) of the equation $$1 = \sum_{i=1}^n \frac{a_i}{b_i - \lambda},$$ where $a_i$ and $b_i$ are known, and $\lambda$ is the unknown. All variables are complex.

In the case where $a_i$ and $b_i$ are real and all of the same sign, an acceptable estimate would be $\max_i \lvert b_i - a_i \rvert$. The generalization to complex numbers is not so clear, or not even the generalization to the case where the $a_i$ and $b_i$ are real numbers with varying sign.

It would of course be possible to convert the problem to finding the largest root of an $n$-th order polynomial in $\mathbb{C}$, but I don't think this would make the problem any easier.

The equation arises from trying to find the largest eigenvalue of a matrix $A - B$, where $A = \operatorname{diag}(a)$, $a = (a_1, \dotsc, a_n)$, and $$ B = \begin{bmatrix} b_1 & b_2 & \cdots & b_n \\ b_1 & b_2 & \cdots & b_n \\ &&\vdots \\ b_1 & b_2 & \cdots & b_n \\ \end{bmatrix}. $$

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  • $\begingroup$ Just an idea (maybe useless): It could be that the polynomial (after conversion) is the characteristic polynomial of a normal nxn matrix. If so, the problem reduces to estimating the spectral norm of it. You can also have a look here: en.m.wikipedia.org/wiki/… $\endgroup$ – Friedrich Philipp Mar 8 '16 at 15:12
  • $\begingroup$ Thank you, you are actually quite close to the problem where the equation came from. I have added some more detail in the question. $\endgroup$ – Håkon Marthinsen Mar 8 '16 at 15:26
  • $\begingroup$ math.stackexchange.com/questions/506761/… may be very relevant. $\endgroup$ – Håkon Marthinsen Mar 9 '16 at 9:19

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