= Root stability of a cubic multistep polynomial
{title2=$\rho(w)=(w-1)(w^2+w+a)$}
The polynomial $\rho(w)=(w-1)(w^2+w+a)$ satisfies the <root condition for a multistep method> exactly for $0\leq a\leq1$. 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.
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