# Solving systems of reccurence equations to get number of recursions to reach a stationarity point?

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.

• Such a system will give you a solution that might converge to a fixed point, but unless you start the system at it, it won't ever reach it. So asking for the number of steps to "reach the stationary point" makes no sense. Commented Aug 26, 2015 at 12:59