= Negative-real instability of the symmetric two-derivative formula
{title2=$F_z(-1)=2z<0$}
On the <Dahlquist test equation>, the <symmetric two-step two-derivative formula> has amplification polynomial $F_z(w)=(z^2-7z+15)w^2-16zw-(z^2+7z+15)$. For every real $z<0$, its leading coefficient is positive and $F_z(-1)=2z<0$, whereas $F_z(w)\to+\infty$ as $w\to-\infty$. The <intermediate value theorem> therefore gives a real root below minus one. Thus the method is not <A-stable>, without any appeal to a multistep order barrier.
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