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Given the following recursion relation \begin{equation} E^{(n)}=(E^{(n-1)}-\alpha_1)\,e^{-\alpha_2\,(\alpha_3E^{(n-1)}+b)} \end{equation}

where $\alpha_i$'s and $b$ are some constants.

I am trying to find $E^{(n)}$ as a function of $E^{(1)}$, but I am not sure if there is a clear way to do so, what do you guys think?

Hawi
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  • Maybe try with some trivial values of the constants and see if you can solve that. E.g., an analytical form for the recurrence $E_n=E_{n-1}\exp(-E_{n-1})$. Though this alone seems quite hard, which suggests that you should probably content yourself with giving sufficient conditions for non-blowup and numerical computation. – parsiad May 29 '16 at 15:36
  • sure, one can consider those constant such that reduce the main relation to the one you wrote but as you pointed out still not easy! – Hawi May 31 '16 at 02:47
  • That's exactly my point. Maybe it is not worth your while looking for an analytical solution. Perhaps its best to characterize non-blowup and compute the recurrence numerically. – parsiad May 31 '16 at 14:15

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