I'm tring to solve a system of recurrence equations to then formalize a formula depending on the number of recursions to calculate this number of recursions to reach the stationarity of the system knowing an initial point.
$\left[ {\begin{array}{*{20}{c}}{{{\rm{A}}_{{\rm{t + 1}}}}}\\{{{\rm{B}}_{{\rm{t + 1}}}}}\end{array}} \right]{\rm{ = }}\left[ {\begin{array}{*{20}{c}}{\rm{c}}&{\rm{d}}\\{\rm{e}}&{\rm{f}}\end{array}} \right]{\rm{.}}\left[ {\begin{array}{*{20}{c}}{{{\rm{A}}_{\rm{t}}}}\\{{{\rm{B}}_{\rm{t}}}}\end{array}} \right]{\rm{ + }}\left[ {\begin{array}{*{20}{c}}{\rm{g}}\\{\rm{h}}\end{array}} \right]$
This system allows to calculate a third equation: ${C_t} = \frac{{{A_t} - {B_t}}}{{1 - {B_t}}}$ resulting in ${C_{t + 1}} = \frac{{\left( {c - e} \right).{A_t} + \left( {d - f} \right).{B_t} + g - h}}{{1 - e.{A_t} - f.{B_t} - h}}$
I search to obtain the equation that can provide numerically the number of recursions needed from a starting point $\left[ {\begin{array}{*{20}{c}}{{{\rm{A}}_{\rm{0}}}}\\{{{\rm{B}}_{\rm{0}}}}\end{array}} \right]$ to reach the stationarity of ${C_t}$ (...so ${C_t}={C_{t+1}}$). How can I do that? I search for courses and online training/teaching that can learn me the way but I failed: no example or demonstration to obtain the number of recursions to reach a stationarity point. Thanks for your incoming pedagogic helps.
