<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">  <!--Converted with LaTeX2HTML 99.2beta6 (1.42) original version by:  Nikos Drakos, CBLU, University of Leeds * revised and updated by:  Marcus Hennecke, Ross Moore, Herb Swan * with significant contributions from:   Jens Lippmann, Marek Rouchal, Martin Wilck and others --> <HTML> <HEAD> <TITLE>Interpolation polynmiale de Lagrange</TITLE> <META NAME="description" CONTENT="Interpolation polynmiale de Lagrange"> <META NAME="keywords" CONTENT="analnum"> <META NAME="resource-type" CONTENT="document"> <META NAME="distribution" CONTENT="global">  <META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1"> <META NAME="Generator" CONTENT="LaTeX2HTML v99.2beta6"> <META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">  <LINK REL="STYLESHEET" HREF="analnum.css">  <LINK REL="next" HREF="node29.html"> <LINK REL="previous" HREF="node27.html"> <LINK REL="up" HREF="node27.html"> <LINK REL="next" HREF="node29.html"> </HEAD>  <BODY BGCOLOR=#ffffff> <!--Navigation Panel--> <A NAME="tex2html346"   HREF="node29.html"> <IMG WIDTH="30" HEIGHT="30" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.gif"></A>  <A NAME="tex2html344"   HREF="node27.html"> <IMG WIDTH="30" HEIGHT="30" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.gif"></A>  <A NAME="tex2html338"   HREF="node27.html"> <IMG WIDTH="30" HEIGHT="30" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="prev.gif"></A>    <BR> <B> suivant:</B> <A NAME="tex2html347"   HREF="node29.html">Interpolation d'Hermite</A> <B> monter:</B> <A NAME="tex2html345"   HREF="node27.html">Interpolation</A> <B> pr&eacute;c&eacute;dent:</B> <A NAME="tex2html339"   HREF="node27.html">Interpolation</A> <BR> <BR> <!--End of Navigation Panel-->  <H2><A NAME="SECTION00051000000000000000"> Interpolation polynmiale de Lagrange</A> </H2>  <P> Soient des rels distincts : <!-- MATH  $x_0<x_1<\dots<x_n$  --> <IMG  WIDTH="139" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"  SRC="img297.gif"  ALT="$x_0&lt;x_1&lt;\dots&lt;x_n$"> et <IMG  WIDTH="52" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"  SRC="img298.gif"  ALT="$\alpha_i\in R$">, <IMG  WIDTH="72" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"  SRC="img299.gif"  ALT="$0\le i\le n$"> .  <P> Il existe un polyn&#244;me <IMG  WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"  SRC="img300.gif"  ALT="$P$"> unique de degr <IMG  WIDTH="31" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"  SRC="img301.gif"  ALT="$\le n$"> tel que <!-- MATH  $P(x_i)=\alpha_i$  --> <IMG  WIDTH="81" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"  SRC="img302.gif"  ALT="$P(x_i)=\alpha_i$"> <IMG  WIDTH="72" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"  SRC="img303.gif"  ALT="$0\le i\le n$">.  <P> <!-- MATH  $P=\sum_{i=0}^{i=n}\alpha_iL_i$  --> <IMG  WIDTH="109" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"  SRC="img304.gif"  ALT="$P=\sum_{i=0}^{i=n}\alpha_iL_i$"> o&#249; les <IMG  WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"  SRC="img305.gif"  ALT="$L_i$"> sont les polyn&#244;mes   satisfaisant <!-- MATH  $L_i(x_j)=\delta_{ij}$  --> <IMG  WIDTH="88" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"  SRC="img306.gif"  ALT="$L_i(x_j)=\delta_{ij}$"> : <BR><P></P> <DIV ALIGN="CENTER"> <!-- MATH  \begin{displaymath} L_i(x)=\prod_{j=0 j\ne i}^{j=n}\left(\frac{x-x_j}{x_i-x_j}\right) \end{displaymath}  -->  <IMG  WIDTH="174" HEIGHT="54" BORDER="0"  SRC="img307.gif"  ALT="\begin{displaymath}L_i(x)=\prod_{j=0 j\ne i}^{j=n}\left(\frac{x-x_j}{x_i-x_j}\right)\end{displaymath}"> </DIV> <BR CLEAR="ALL"> <P></P>  <P> Pour <!-- MATH  $f\in {\cal C}^{n+1}([a,b])$  --> <IMG  WIDTH="112" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"  SRC="img308.gif"  ALT="$f\in {\cal C}^{n+1}([a,b])$"> on note <IMG  WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"  SRC="img309.gif"  ALT="$L_f$"> le polyn&#244;me de degr <IMG  WIDTH="31" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"  SRC="img310.gif"  ALT="$\le n$"> tel que  <BR><P></P> <DIV ALIGN="CENTER"> <!-- MATH  \begin{displaymath} L_f(x_i)=f(x_i)\qquad 0\le i\le n \end{displaymath}  -->  <IMG  WIDTH="206" HEIGHT="29" BORDER="0"  SRC="img311.gif"  ALT="\begin{displaymath}L_f(x_i)=f(x_i)\qquad 0\le i\le n\end{displaymath}"> </DIV> <BR CLEAR="ALL"> <P></P>  <P> <I>Mthode de Newton</I>  <P> On forme les diffrences divises de <IMG  WIDTH="14" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"  SRC="img312.gif"  ALT="$f$"> : <BR><P></P> <DIV ALIGN="CENTER"> <!-- MATH  \begin{displaymath} \delta[y]=f(y) \end{displaymath}  -->  <IMG  WIDTH="78" HEIGHT="29" BORDER="0"  SRC="img313.gif"  ALT="\begin{displaymath}\delta[y]=f(y)\end{displaymath}"> </DIV> <BR CLEAR="ALL"> <P></P> <BR><P></P> <DIV ALIGN="CENTER"> <!-- MATH  \begin{displaymath} \delta[y_1,y_2]= \frac{f(y_1)-f(y_2)}{y_1-y_2}\ ,\dots \end{displaymath}  -->  <IMG  WIDTH="205" HEIGHT="44" BORDER="0"  SRC="img314.gif"  ALT="\begin{displaymath}\delta[y_1,y_2]= \frac{f(y_1)-f(y_2)}{y_1-y_2}\ ,\dots\end{displaymath}"> </DIV> <BR CLEAR="ALL"> <P></P> <BR><P></P> <DIV ALIGN="CENTER"> <!-- MATH  \begin{displaymath} \delta[y_1,\dots,y_k]= \frac{\delta[y_1,\dots,y_{k-1}]-\delta[y_2,\dots,y_k]}{y_1-y_k} \end{displaymath}  -->  <IMG  WIDTH="311" HEIGHT="44" BORDER="0"  SRC="img315.gif"  ALT="\begin{displaymath}\delta[y_1,\dots,y_k]= \frac{\delta[y_1,\dots,y_{k-1}]-\delta[y_2,\dots,y_k]}{y_1-y_k}\end{displaymath}"> </DIV> <BR CLEAR="ALL"> <P></P> On a alors <BR><P></P> <DIV ALIGN="CENTER"> <!-- MATH  \begin{displaymath} \delta[y_1,\dots,y_k]=\sum_{i=1}^{i=k} \frac{f(y_i)}{\prod_{j\ne i}(y_i-y_j)} \end{displaymath}  -->  <IMG  WIDTH="223" HEIGHT="51" BORDER="0"  SRC="img316.gif"  ALT="\begin{displaymath}\delta[y_1,\dots,y_k]=\sum_{i=1}^{i=k} \frac{f(y_i)}{\prod_{j\ne i}(y_i-y_j)}\end{displaymath}"> </DIV> <BR CLEAR="ALL"> <P></P>  <BR><P></P> <DIV ALIGN="CENTER"> <!-- MATH  \begin{displaymath} L_f(x)=\sum_{k=0}^{k=n-1}\delta[x_0,\dots,x_k]\prod_{j=0}^{j=k-1}(x-x_j) \end{displaymath}  -->  <IMG  WIDTH="286" HEIGHT="54" BORDER="0"  SRC="img317.gif"  ALT="\begin{displaymath}L_f(x)=\sum_{k=0}^{k=n-1}\delta[x_0,\dots,x_k]\prod_{j=0}^{j=k-1}(x-x_j)\end{displaymath}"> </DIV> <BR CLEAR="ALL"> <P></P>  <P> <I>Approximation de <IMG  WIDTH="14" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"  SRC="img318.gif"  ALT="$f$"></I>  <P> <!-- MATH  $\forall x\in [a,b]\quad \exists c\in ]a,b[$  --> <IMG  WIDTH="156" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"  SRC="img319.gif"  ALT="$\forall x\in [a,b]\quad \exists c\in ]a,b[$">   tel que <BR><P></P> <DIV ALIGN="CENTER"> <!-- MATH  \begin{displaymath} f(x)-L_f(x)=\frac{1}{(n+1)!}   \prod_{i=0}^{i=n}(x-x_i)f^{(n+1)}(c) \end{displaymath}  -->  <IMG  WIDTH="315" HEIGHT="51" BORDER="0"  SRC="img320.gif"  ALT="\begin{displaymath}f(x)-L_f(x)=\frac{1}{(n+1)!} \prod_{i=0}^{i=n}(x-x_i)f^{(n+1)}(c)\end{displaymath}"> </DIV> <BR CLEAR="ALL"> <P></P>  <P> Soit <IMG  WIDTH="43" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"  SRC="img321.gif"  ALT="$a&gt;0$"> et <!-- MATH  $f:[-a,a]\mapsto R$  --> <IMG  WIDTH="113" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"  SRC="img322.gif"  ALT="$f:[-a,a]\mapsto R$"> dfinie par  <!-- MATH  $\displaystyle f(x)=\frac{1}{1+x^2}$  --> <IMG  WIDTH="106" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"  SRC="img323.gif"  ALT="$\displaystyle f(x)=\frac{1}{1+x^2}$">  <P> Soit <IMG  WIDTH="44" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"  SRC="img324.gif"  ALT="$h&gt;0$"> et <IMG  WIDTH="76" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"  SRC="img325.gif"  ALT="$0&lt;\alpha &lt;a$"> &nbsp;. On dfinit   <BR><P></P> <DIV ALIGN="CENTER"> <!-- MATH  \begin{displaymath} \{x_i ;i=0,\dots ,n\}=\{x\in [0,a];x=\alpha +(k+\frac{1}{2})h, k\in Z\} \end{displaymath}  -->  <IMG  WIDTH="404" HEIGHT="39" BORDER="0"  SRC="img326.gif"  ALT="\begin{displaymath}\{x_i ;i=0,\dots ,n\}=\{x\in [0,a];x=\alpha +(k+\frac{1}{2})h, k\in Z\}\end{displaymath}"> </DIV> <BR CLEAR="ALL"> <P></P> Le polyn&#244;me   <IMG  WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"  SRC="img327.gif"  ALT="$P$"> satisfaisant <IMG  WIDTH="102" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"  SRC="img328.gif"  ALT="$P(x_i)=f(x_i)$"> et <!-- MATH  $P(-x_i)=f(-x_i)$  --> <IMG  WIDTH="129" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"  SRC="img329.gif"  ALT="$P(-x_i)=f(-x_i)$">, <IMG  WIDTH="72" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"  SRC="img330.gif"  ALT="$0\le i\le n$">   est pair et de degr <IMG  WIDTH="40" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"  SRC="img331.gif"  ALT="$\le 2n$"> &nbsp;.  <P> Quand <!-- MATH  $h\rightarrow 0$  --> <IMG  WIDTH="48" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"  SRC="img332.gif"  ALT="$h\rightarrow 0$"> on a <!-- MATH  $\vert f(\alpha )-P(\alpha )\vert\rightarrow 0$  --> <IMG  WIDTH="137" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"  SRC="img333.gif"  ALT="$\vert f(\alpha )-P(\alpha )\vert\rightarrow 0$"> si <IMG  WIDTH="74" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"  SRC="img334.gif"  ALT="$0&lt;\alpha&lt;b$"> et <!-- MATH  $\vert f(\alpha )-P(\alpha )\vert\rightarrow\infty$  --> <IMG  WIDTH="146" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"  SRC="img335.gif"  ALT="$\vert f(\alpha )-P(\alpha )\vert\rightarrow\infty$"> si <IMG  WIDTH="74" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"  SRC="img336.gif"  ALT="$b&lt;\alpha&lt;a$">  <P> <HR> <!--Navigation Panel--> <A NAME="tex2html346"   HREF="node29.html"> <IMG WIDTH="30" HEIGHT="30" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.gif"></A>  <A NAME="tex2html344"   HREF="node27.html"> <IMG WIDTH="30" HEIGHT="30" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.gif"></A>  <A NAME="tex2html338"   HREF="node27.html"> <IMG WIDTH="30" HEIGHT="30" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="prev.gif"></A>    <BR> <B> suivant:</B> <A NAME="tex2html347"   HREF="node29.html">Interpolation d'Hermite</A> <B> monter:</B> <A NAME="tex2html345"   HREF="node27.html">Interpolation</A> <B> pr&eacute;c&eacute;dent:</B> <A NAME="tex2html339"   HREF="node27.html">Interpolation</A> <!--End of Navigation Panel-->  </BODY> </HTML> 
