= Padé approximant
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{title2=$[m/n]_f=P_m/Q_n$}
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= Padé approximation
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A Padé approximant is a <rational function> with numerator degree at most $m$ and denominator degree at most $n$, normalized by $Q_n(0)=1$, such that $Q_n(z)f(z)-P_m(z)=O(z^{m+n+1})$ near zero, when such an approximant exists. For the <exponential function>, its $[1/1]$ approximant is $(1+z/2)/(1-z/2)$. Applying this rational function to a matrix gives the <Crank-Nicolson method>. Diagonal and subdiagonal approximants also give stability functions of <Gauss collocation methods> and <Radau IIA methods> respectively.
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