Rational Approximation
Source inspiration: (Mathew 2000-2019).
Animations
The animation below compares the diagonal Padé approximant \(R_{[n/n]}(x)\) with the Taylor polynomial \(T_{2n}(x)\) of equal total degree for \(f(x) = e^x\) on \([-6,6]\), as the order \(n\) increases from 1 to 4.
Julia source scripts that generated these animations are linked under each case.
Case 1 — Padé \([n/n]\) vs Taylor \(T_{2n}\) for \(e^x\), \(n=1,\ldots,4\)
Behavior: Both approximations use \(2n+1\) parameters. The Taylor polynomial is accurate near \(x=0\) but diverges away from the origin. The Padé rational approximant distributes its accuracy more evenly, remaining close to \(e^x\) over a much wider interval. For \(n=4\) the Padé \([4/4]\) tracks \(e^x\) well past \(x=\pm5\), where \(T_8\) has already diverged.
Derivation Notes (Planned)
Short derivations will be added to explain the core equations and assumptions.