 Solution

Zero-stability means that perturbations of the starting values of the zero-step recurrence remain bounded by a fixed multiple of their initial size, uniformly as the number of steps tends to infinity on a fixed time interval. In particular solutions of ${\sum }_{j=0}^k{\alpha }_j{e}_{n+j}=0$ obey ${\max }_{n\le N}|e_n|\le C{\max }_{j<k}|e_j|$ with $C$ independent of $N$.

The equivalent root condition for a multistep method isA root outside the unit circle gives exponential growth of a recurrence relation mode; a repeated root on it gives polynomial growth. Repeated roots strictly inside it are harmless because their polynomial factor is dominated by decay.