Fourth-order conditions for a Runge-Kutta method

With $c=Ae$ and $C=\mathrm{diag}(c)$, a Runge-Kutta method has order at least four exactly when the eight Butcher order conditionshold; powers of $c$ are componentwise. These conditions match the numerical and exact coefficients indexed by rooted trees through order four. Matching the scalar Dahlquist test equation alone is insufficient to check the nonlinear Butcher order conditions.