On the definition of the stability region of multistep methods

The usual definition of the stability region of implicit multistep methods often implies that there are some isolated points of stability within the region of instability of the numerical method. These isolated stable points may appear when the leading coefficient of the characteristic polynomial of the method vanishes---they cannot be detected by the well-known root locus method, and their existence renders many results about stability regions problematic. It is suggested that the definition of the stability region should exclude such isolated points.