Pade Approximant

The Pade Approximant is a rational polynomial approximation to a function. In some sense this approximation is optimal for many problems because it can model different asymptotic behavior more easily. In the application of a Taylor Series there is often a high-degree polynomial this polynomial will inevitably go to positive or negative infinity. So if you have function, with a logistic growth rate, you are out of luck. The Pade Approximant can mitigate this error and actually recover the function. This method can recover information from the coefficients of a Taylor Expansion which the expansion itself cannot even use. Pade Approximants show us the power of using a different representation. If at first your approximation fails... try another representation!

An Example

How do we compute this thing actually?

There are four main approaches to calculating the Pade Approximant that I could find. The Euclidean Algorithm, The Generalized Euclidean Aglorithm, Gaussian Elimination, and Robust Pade.

The Euclidean Algorithm is the same as the one used for the Greatest Common Denominator except it is applied to polynomials. The polynomial

What if I want to fit over an interval?

Pade Approximant is a terminology usually left for the approximant based on the Taylor Series. Well, sometimes you may want a rational approximation over the interval. This requires a bit more work. I will refer you to the section on rational function approximation.