Pade Approximations convergence acceleration

Question

Why Pade Approximatoins accelerate the convergence of series?

Generally speaking, what is there an advantage in the sence of convergence acceleration using rational interpolation? Thanks much in advance!!!

Answer 1

To expand in vadim123's answer. With truncations of a series (the trivial way to obtain approximations to its sum) you are using a number $n$ of terms and you get a remainder which is just 'killing' those $n$ terms from the series. With Pade approximants, since a rational function has more parameters, you can 'kill' more terms of the series (this is just what is done to define Pade's approximants) and in that way, hopefully, closer to the sum of the series (or what is the same get a smaller remainder).

Accelerating also depends on how you compare. A truncation of a series up to degree $n$ is a polynomial of degree $n$. Pade's approximants are quotients of two polynomials. Should we compare this truncation of degree $n$ to the Pade approximant that uses a quotient of two polynomials of degree $n$, which is going to capture the information of $2n$ terms of the series and it is likely to give a better approximation? Or should we compare to the Pade approximant that is a quotient of two $n/2$ polynomials, which captures the information of only $n$ terms of the series, the same as the truncation of degree $n$?

Answer 2

The set of rational functions contains the set of polynomials, so a rational approximation will always be at least as good as a polynomial approximation, and will typically be better.