Zero-stability

Zero-stability requires the roots of the first characteristic polynomial to lie in the closed unit disc, with unit-modulus roots simple.

The root condition is the polynomial criterion equivalent to zero-stability for a linear multistep method.

The polynomial $\rho (w)=(w-1)(w^2+w+a)$ satisfies the root condition for a multistep method exactly for $0\le a\le 1$. For $a<0$ a root is below $-1$; for $a>1$ the quadratic root product forces a root outside the unit disc. The endpoints have simple unit-modulus roots, and the repeated root at $a=1/4$ is strictly inside. A consistent method with this polynomial is convergent throughout the closed interval.