Trapezoidal-BDF two-step family ($\rho (\zeta )={\zeta }^2-\alpha \zeta -(1-\alpha ),\sigma (\zeta )=(1-\alpha /4){\zeta }^2+1-3\alpha /4$)

This real-parameter linear multistep method has formal order two except at the degenerate value $\alpha =2$, where its local symbol vanishes to fourth degree. The zero-stability roots are $1$ and $\alpha -1$, so it is convergent exactly for $0\le \alpha <2$. Every convergent member is A-stable: the boundary-locus test for multistep A-stability gives nonnegative real part $\alpha (2-\alpha )(1-\cos \theta {)}^2/(2|\sigma (e^{i\theta }){|}^2)$, with no unit-circle denominator zero for $0<\alpha <2$. At $\alpha =0$ the two interlaced sequences obey a trapezoidal step of length $2h$. At $\alpha =4/3$ it is the BDF2 method. The endpoint two fails the root condition for a multistep method even though canceling a common polynomial factor resembles the trapezoidal rule.