= Two-step family with a third-order member
{title2=$\rho(w)=(w-1)(w-a),\quad\sigma(w)=\tfrac12(1+a)w^2+\tfrac12(1-3a)w$}
The <exponential-symbol order criterion for a multistep method> gives leading defect $-(1+5a)z^3/12$, so the family has order two except at $a=-1/5$, where it has order three. The <root condition for a multistep method> holds exactly for $-1\leq a<1$, and the convergent <A-stable> members are exactly $0\leq a<1$. At $a=0$ it reduces to the <trapezoidal rule> after removal of a zero root. At $a=1$ a repeated root one destroys <convergence of a numerical method> of the original recurrence despite a common <polynomial> factor and a formally vanishing second-order residual.
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