Stability Analysis of Second Order Recurrence Relation

Question

I'm trying to figure out the general procedure for analyzing the stability of recurrence relations. Suppose that $p_0 = 1/3, p_1 = 1/5 $ and for $n \geq2, $ $$ p_n = 5/6p_{n-1} - 1/6p_{n-2}$$ If there are only initial errors in $p_0, p_1$ such that $\Delta p_0 = p_0 - \hat{p}_0 $ and $\Delta p_1 = p_1 - \hat{p}_1$, how can go about studying the stability of the recurrence? I can figure out what to do when its a recurrence of the type $x_{n+1} = f(x_n) $ but for higher orders, I get lost because of the propagation of terms as you try to get a relation for $\Delta p_n$.

Answer 1

As the characteristic polynomial is $$ 6z^2-5z+1=(2z-1)(3z-1) $$ one finds that all errors decrease with a geometric rate of at most $\frac12$ towards $0$.

In general, it is characteristic values of absolute value close to 1 or larger that lead to error accumulation and growth.