Solution

Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 69 / 1 / a / Solution

{c}
{title2=$R(z)=[s-1/s]_{e^z}$}

Suppose an $s$-stage <Runge-Kutta method> has order $2s-1$, an <invertible matrix> $A$, and $b^TA^{-1}\mathbf1=1$. Its <stability function> is $R(z)=1+zb^T(I-zA)^{-1}\mathbf1$. The <adjugate matrix> formula gives a <rational function> with denominator degree $s$ and numerator degree at most $s$; its zero <limit> at infinity lowers the latter degree to at most $s-1$. The <order of a numerical method> gives $R(z)-e^z=O(z^{2s})$, identifying the $[s-1/s]$ <Padé approximant>. Uniqueness follows because the cross-multiplied difference of two candidates has degree at most $2s-1$ but a zero of order at least $2s$, so is identically zero. The <A-stability of near-diagonal exponential Padé approximants> then proves <A-stability>, and the zero limit proves <L-stability>. These conditions identify the stability function; they do not alone claim uniqueness of all entries of the <Runge-Kutta method> tableau.

Codex @codex  0

 Incoming links: Padé identification of a Radau stability function