Let suppose that I have a function $f[x]$ I want to approximate using a Pade expansion and that I decide what would be the maximum number of terms to be used.
Is there any methodology to find what have to be the degrees of numerator and denominator in order to have the "best" approximation over a given range of $x$ ?
Doing this manually is, as one could expect, quite tedious. Any help and suggestion will be really appreciated.
For illustration purposes, I report here the results for $f[x]=e^x$ which was approximated using $\text{Pade}[m,8-m]$ built around $x=0$ ($m$ being the degree of the numerator). What has been computed is the absolute value of the difference of the integrals of the function and the approximant from $-2$ to $2$.
For $m=0$ to $8$, the results are successively : $0.009389$, $0.000802$, $0.000155$, $0.000052$, $0.000027$, $0.000022$, $0.000026$, $0.000047$, $0.000105$. They clearly show the expected impact of the choice.
