= Bounded-root stability set of the symmetric two-derivative formula
{title2=$z=iy,\quad y^4-45y^2+225>0$}
For the <symmetric two-step two-derivative formula>, the closed-disk <root condition for a multistep method> fails off the imaginary axis. Positive real part makes the product of root moduli exceed one; negative real part fails the <complex quadratic Schur criterion>, as in the <empty strict decay domain of the symmetric two-derivative formula> proof. On $z=iy$, write $D=15-y^2-7iy=|D|e^{i\delta}$ and set $\xi=i e^{-i\delta}\eta$. The <amplification polynomial> becomes $\eta^2-16y\eta/|D|+1=0$. Its two roots are distinct and on the <unit circle> exactly when $|16y/|D||<2$, or $y^4-45y^2+225>0$. Thus the bounded-root set consists of the imaginary intervals $|y|<\sqrt{(45-15\sqrt5)/2}$ and $|y|>\sqrt{(45+15\sqrt5)/2}$, including $y=0$ and excluding the double-root endpoints. In particular, the set is unbounded. Unit-modulus oscillations are bounded but do not tend to zero.
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