I am dealing with the following recurrent form: \begin{equation} \forall t \geq 1 \,;\, \boldsymbol{x}_t = \boldsymbol{x}_{t-1} - \epsilon \frac{\boldsymbol{a} \circ \boldsymbol{x}_{t-1}}{\| \boldsymbol{a} \circ \boldsymbol{x}_{t-1}\|} \end{equation} Here $\epsilon>0$ and both $\boldsymbol{x}_t$ (for all $t$) and $\boldsymbol{a}$ are vectors, respectively in $\mathbb{R}^n$ and $\mathbb{R}^n_+$ (latter meaning strictly positive components), and $\circ$ shows the element-wise product (hence the result will be again a vector). Given a base solution $\boldsymbol{x}_0$, is it possible to solve this recurrent equation and find a closed form for $\boldsymbol{x}_t$?
Thanks,
Golabi