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I have the following problem. I am trying to solve the following non-linear system of equations in MATLAB

$\mathbf{b}=\mathrm{diag}\left(\mathbf{x}\right)\left(\mathbf{I}+\mathbf{A'A}\right)^{-1}\left(\mathbf{A'1}-\mathbf{x}\right)$

where $\mathbf{b}$ (known) and $\mathbf{x}$ (to be found) are $N\times1$ column vectors, $\mathbf{A'A}$ (known) is a $N\times N$ cosine similarity matrix (symmetric positive definite). $\mathbf{I}$ is the identity matrix and $\mathbf{1}$ is a $N\times1$ vector of ones.

The obvious reference would seem this one: Solving Non Linear System of Equations with MATLAB

the problem is I'm not sure how to take the Jacobian. It seems to me that $\mathrm{diag}(\mathbf{x})$ and $(\mathbf{I+A'A})^{-1}$ should commute, so perhaps using that could simplify computation of the Jacobian.

But perhaps I'm wrong and that's not the way to go. Perhaps another possibility would be to write

$\mathbf{b}=\mathrm{diag}\left(\mathbf{x_{t}}\right)\left(\mathbf{I}+\mathbf{A'A}\right)^{-1}\left(\mathbf{A'1}-\mathbf{x_{t+1}}\right)$

make a guess for $\mathbf{x}_1$ and iterate till convergence. Any ideas? Thanks in advance!

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