Padé approximant

Definition

About a general point and for a given order

Suppose $f$ is a function, $x_{0}$ is a point in the domain of $f$ , and $m, n$ are (possibly equal, possibly distinct) nonnegative integers. Suppose further that $f$ is at least $m + n$ times differentiable at $x_{0}$ .

The Padé approximant to $f$ of order $[m / n]$ at $x_{0}$ is a rational function of the form:

$R (x) = \frac{a_{0} + a_{1} (x - x_{0}) + a_{2} (x - x_{0})^{2} + \dots + a_{m} (x - x_{0})^{m}}{1 + b_{1} (x - x_{0}) + \dots + b_{n} (x - x_{0})^{n}}$

where $a_{i}, b_{j}$ are all real numbers, and where $f^{(j)} (x_{0}) = R^{(j)} (x_{0})$ for $j \in {0, 1, 2, \dots, m + n}$ .

About the point 0 and for a given order

This definition adapts the previous one for the case $x_{0} = 0$ .

Suppose $f$ is a function and $m, n$ are (possibly equal, possibly distinct) nonnegative integers. Suppose further that $f$ is at least $m + n$ times differentiable at 0.

The Padé approximant to $f$ of order $[m / n]$ at 0 is a rational function of the form:

$R (x) = \frac{a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{m} x^{m}}{1 + b_{1} x + \dots + b_{n} x^{n}}$

where $a_{0}, a_{1}, \dots, a_{m}, b_{1}, b_{2}, \dots, b_{n}$ are all real numbers, and where $f^{(j)} (0) = R^{(j)} (0)$ for $j \in {0, 1, 2, \dots, m + n}$ .