Let $P,Q \in \mathbb{C}[z]$ be polynomials such that $Q$ is not the zero polynomial and $P$ and $Q$ are relatively prime, meaning they have no common nonconstant divisor in $\mathbb{C}[z]$. Define
With A-stability understood to mean that $R$ has no pole in $\overline{\mathbb{C}}_{-}$ and satisfies $|R(z)| \le 1$ for every $z \in \overline{\mathbb{C}}_{-}$, the method is A-stable if and only if
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\begin{align*}
Q(z) \neq 0 \qquad \text{for every } z \in \overline{\mathbb{C}}_{-}
\end{align*}
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and
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\begin{align*}
|P(z)| \le |Q(z)| \qquad \text{for every } z \in \overline{\mathbb{C}}_{-}.
\end{align*}