On the definition of the stability region of multistep methods
For numerical analysts using multistep methods, this clarifies a subtle but important flaw in the definition of stability regions, though the fix is incremental.
The paper identifies that the standard definition of the stability region for implicit multistep methods can include isolated stable points within the instability region, which are undetectable by the root locus method and invalidate many stability results. The authors propose excluding such isolated points from the definition.
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.