Padé approximant ($[m/n{]}_f=P_m/Q_n$)

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.