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In short, I had this problem that required me to solve a system of simultaneous linear equations which turned out to be of the form $$w_j = \frac{1}{n_j}\sum_{k=1}^N \frac{n_{jk}c_{jk}}{1-c_{jk}}w_{k},$$

where $w_j$ is the only unknown for $j = 1,\dots, N$. We can formulate it as an eigenvalue problem by defining $A_{jk} = \frac{n_{jk}c_{jk}}{1-c_{jk}}$ which means $A \boldsymbol w = \boldsymbol w$. Nice and clean.

But then I altered my mathematical model. So this is a new problem which is independent of the first. I only included the first to illustrate the analogy to an easy linear eigenvector problem. This is the system of nonlinear equations that the new model needs to satisfy:

$$\begin{align} w_j = \frac{1}{n_j}\sum_k \frac{n_{jk}w_k c_{jk}}{2w_k c_{jk}-w_k-c_{jk}}. \end{align} $$

So, $w_k$ shows up in both numerator and denominator, which means I can't view this sum as a scalar product. But it sort of has a similar flavor to the easy problem above. Is this just a case of "standard" satisfyability of nonlinear systems problem, or does this possess more structure? Is it an instance of a well-studied problem? Any references appreciated.

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  • $\begingroup$ I assume $w_{jk}$ are independent of $w_{j}$ and are fixed? It's just at first glance your notation could cause that confusion. How did you end up with the second expression, can you show your workings please? If the $n_{jk}$ and $w_{jk}$ are known and independent of the $w_{k}$ you want the $w_{k}$ which correspond to the "eigenvalue = 1" eigenvector of your matrix A. $\endgroup$ – AlphaNumeric Sep 14 '16 at 9:21
  • $\begingroup$ Everything is fixed except for $w_j$. I don't need help with finding the eigenvector of the first system. It's the nonlinear system that causes concerns. $\endgroup$ – Benjamin Lindqvist Sep 14 '16 at 9:23
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references: a book on scientific computing, many techniques, clear examples... scientific computing an introductory survey by micheal heath

you must linearize your termz. produce final eqn that looks like Ax=b solve using likely approximate method of your choice. also if all of right hand side is known... then like whatever? your eqn here is just a constraint. you must read more to understand and write your question better.

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