= Symmetric two-step two-derivative formula
{title2=$\alpha=7/15,\quad\beta=1/15,\quad\gamma=16/15$}
For the <multiderivative multistep method> $y_{n+1}-\alpha h f_{n+1}+\beta h^2g_{n+1}=\gamma h f_n+y_{n-1}+\alpha h f_{n-1}+\beta h^2g_{n-1}$, with $g=f'f$, centered expansion gives defect coefficients $2-2\alpha-\gamma$, $1/3-\alpha+2\beta$, and $1/60-\alpha/12+\beta/3$ at powers $h,h^3,h^5$. Their simultaneous vanishing gives the displayed parameters. The next coefficient is $1/4725$ at $h^7$, so the formal order is six. Reflection symmetry removes every even defect power. The method satisfies fifth-order conditions but cannot have exact order five.
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