A-stable third-order two-step multiderivative method ($7y_{n+2}-6h{y}_{n+2}^{\prime }+2h^2{y}_{n+2}^{\prime \prime }=8y_{n+1}-y_n$)

The endpoint expansion of this multiderivative multistep method has unscaled defect $h^4{y}^{(4)}/3+O(h^5)$ and zero-stability roots $1,1/7$, giving third-order convergence with suitably accurate starting values. On the Dahlquist test equation its amplification roots obey $(7-6z+2z^2){\zeta }^2-8\zeta +1=0$. The complex quadratic Schur criterion puts both roots in the open unit disk for $\mathrm{Re}z<0$, and only the simple root one reaches the boundary at $z=0$. Thus it is A-stable; its higher derivative places it outside the hypotheses of the Second Dahlquist barrier.