= Maximal-order Runge-Kutta methods are A-stable
{title2=$p=2s\Longrightarrow A\text{-stability}$}
An $s$-stage <Runge-Kutta method> has <stability function> $R(z)=1+zb^T(I-zA)^{-1}\mathbf1$, a <rational function> with numerator and denominator degrees at most $s$. Order $2s$ forces $R(z)-e^z=O(z^{2s+1})$, so uniqueness of the diagonal <Padé approximant> identifies $R$ with $[s/s]_{e^z}$. That approximant has no poles in the closed left half-plane and modulus at most one there. Hence every such maximal-order method is <A-stable>. Since the <Padé approximant> numerator and denominator are coprime and both have degree $s$, the original stage <determinant> cannot have hidden canceled poles in that half-plane.
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