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 $j\in \{0,1,2,\dots ,m+n\}$.

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