Padé approximant: Difference between revisions

Latest revision as of 00:57, 2 May 2014

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

This page is under construction. In the interim, please check the corresponding Wikipedia page.

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 ===About a general point and for a given order===
-Suppose <math>f</math> is a function, <math>x_0</math> is a point in the domain of <math>f</math>, and <math>m,n</math> are (possibly equal, possibly distinct) nonnegative integers. The '''Padé approximant''' to <math>f</math> of order <math>[m/n]</math> at <math>x_0</math> is a rational function of the form:
+Suppose <math>f</math> is a function, <math>x_0</math> is a point in the domain of <math>f</math>, and <math>m,n</math> are (possibly equal, possibly distinct) nonnegative integers. Suppose further that <math>f</math> is at least <math>m + n</math> times differentiable at <math>x_0</math>.
+The '''Padé approximant''' to <math>f</math> of order <math>[m/n]</math> at <math>x_0</math> is a rational function of the form:
 <math>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}</math>
 where <math>a_i,b_j</math> are all real numbers, and where <math>f^{(j)}(x_0) = R^{(j)}(x_0)</math> for <math>j \in \{ 0, 1,2,\dots,m+n\}</math>.
+===About the point 0 and for a given order===
+This definition adapts the previous one for the case <math>x_0 = 0</math>.
+Suppose <math>f</math> is a function and <math>m,n</math> are (possibly equal, possibly distinct) nonnegative integers. Suppose further that <math>f</math> is at least <math>m + n</math> times differentiable at 0.
+The '''Padé approximant''' to <math>f</math> of order <math>[m/n]</math> at 0 is a rational function of the form:
+<math>R(x) = \frac{a_0 + a_1x + a_2x^2 + \dots + a_mx^m}{1 + b_1x + \dots + b_nx^n}</math>
+where <math>a_0,a_1,\dots,a_m,b_1,b_2,\dots,b_n</math> are all real numbers, and where <math>f^{(j)}(0) = R^{(j)}(0)</math> for <math>j \in \{ 0, 1,2,\dots,m+n\}</math>.
+{{under construction check wikipedia}}