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Given symmetric positive-definite matrix $\mathbf A$ and (real-valued) vectors $\mathbf q$ and $\mathbf z$, does the following iteration converge to a (unique) fixed point for $n\to \infty$?

$$ \mathbf q_{n+1} \gets \mathbf z + \tfrac 1 2 \operatorname{diag}\left[(\mathbf A + \operatorname{diag}[e^{\mathbf q_n}])^{-1}\right] $$

This iteration comes from inside some log-Gaussian Cox process code. It seems to converge for all the values I've tried, but I can't seem to prove it.

Edit:

If it helps, $\mathbf A$ can be circulant (it is in my application anyway).

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  • $\begingroup$ How is defined diag of a matrix ? $\endgroup$
    – GreginGre
    Commented Nov 20, 2021 at 7:51
  • $\begingroup$ ah! extracts main diagonal as a vector; is there a more standard notation? $\endgroup$
    – MRule
    Commented Nov 20, 2021 at 11:19

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