Convergence of linear multistep methods with multiple roots (Q1069276)

It is generally believed that the first associated polynomial \(\rho\) (\(\xi)\) of a convergent linear multistep method cannot have multiple roots on the unit circle. This is true generally, but the authors indicate that there does exist a class of linear multistep methods which are convergent but have multiple roots on the unit circle. However, their global convergence rate is \(h^{p-q+1}\) if there are q roots of modulus one, which is less than their local consistence error \(h^ p\). Furthermore, they are absolutely unstable, although relatively stable.