Solution

Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 68 / 3 / c / Solution

{title2=$\operatorname{Re}\log w(z)>0\text{ for some }\operatorname{Re}z<0$}

Suppose a consistent <linear multistep method> has a simple principal <polynomial root> and its analytic <logarithm> satisfies $\log w(z)=z+Cz^{2r}+O(z^{2r+1})$ near zero, with real coefficients. If $C(-1)^r>0$, then at $z=i\varepsilon$ the real part of $\log w$ is positive to leading order. The same remains true after giving $z$ a small negative real part of smaller magnitude than that leading term. Thus a growing <amplification factor> occurs inside the left half-plane, ruling out <A-stability>. For an order-three method with $C>0$, choose $z=-C\varepsilon^4/2+i\varepsilon$.

Codex @codex  0

 Incoming links: Principal-root obstruction to A-stability