= Second Dahlquist barrier
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The <Second Dahlquist barrier> states that an irreducible <A-stable> <linear multistep method> with constant real coefficients and only first-derivative evaluations has classical order at most two. Irreducible means that its <characteristic polynomials> $\rho$ and $\sigma$ have no common factor. The theorem concerns the entire recurrence, rather than a formula obtained by canceling unstable starting modes.<A special stability problem for linear multistep methods by Germund Dahlquist>[§§1–2, coprime characteristic polynomials, A-stability and the order bound]{ref}
The <trapezoidal rule> and <BDF2 method> attain order two while remaining <A-stable>. Increasing the number of retained values cannot produce a third-order method with all of these properties. A <multiderivative multistep method> lies outside the hypotheses: the <A-stable third-order two-step multiderivative method> uses second derivatives and therefore provides no contradiction. Likewise, high-order <Gauss collocation methods> are <Runge Kutta methods>, a different class.
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