<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">  <!--Converted with LaTeX2HTML 99.1 release (March 30, 1999) 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>Sur la convergence de l'interpolation de Lagrange</TITLE> <META NAME="description" CONTENT="Sur la convergence de l'interpolation de Lagrange"> <META NAME="keywords" CONTENT="support"> <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 v99.1 release"> <META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">  <LINK REL="STYLESHEET" HREF="support.css">  <LINK REL="previous" HREF="node3.html"> <LINK REL="up" HREF="node2.html"> <LINK REL="next" HREF="node5.html"> </HEAD>  <BODY >  <A NAME="tex2html75"  HREF="node5.html"> <IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"  SRC="/home/picasso/tex/latex2html/icons.gif/next_motif.gif"></A>  <A NAME="tex2html73"  HREF="node2.html"> <IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"  SRC="/home/picasso/tex/latex2html/icons.gif/up_motif.gif"></A>  <A NAME="tex2html69"  HREF="node3.html"> <IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"  SRC="/home/picasso/tex/latex2html/icons.gif/previous_motif.gif"></A>   <A NAME="tex2html1"  HREF="support.html"><IMG  ALIGN="BOTTOM" BORDER="0" SRC="contents_motif.gif"  ALT="58"></A> <BR> <B> Next:</B> <A NAME="tex2html76"  HREF="node5.html">Un exemple pour lequel</A> <B>Up:</B> <A NAME="tex2html74"  HREF="node2.html">Chap. 1 : Probl&#232;mes</A> <B> Previous:</B> <A NAME="tex2html70"  HREF="node3.html">Position du probl&#232;me</A> <BR> <P>  <!--End of Navigation Panel-->  <H1><A NAME="SECTION00220000000000000000"> Sur la convergence de l'interpolation de Lagrange</A> </H1>  <P> Soit <IMG  WIDTH="35" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"  SRC="img19.gif"  ALT="$ [a,b]$"> un intervalle,  soit  <!-- MATH  $f:[a,b]\to\mathbb{R}$  --> <IMG  WIDTH="94" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"  SRC="img20.gif"  ALT="$ f:[a,b]\to\mathbb{R}$"> une fonction continue donn&#233;e,  soit <IMG  WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"  SRC="img21.gif"  ALT="$ n$"> un entier positif donn&#233; et soit   <!-- MATH  $t_0, t_1, t_2,\ldots,t_n$  --> <IMG  WIDTH="109" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"  SRC="img22.gif"  ALT="$ t_0, t_1, t_2,\ldots,t_n$"> des points de <IMG  WIDTH="35" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"  SRC="img19.gif"  ALT="$ [a,b]$"> distincts donn&#233;s. Nous notons <IMG  WIDTH="21" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"  SRC="img23.gif"  ALT="$ p_n$"> le polyn&#244;me de degr&#233; <IMG  WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"  SRC="img21.gif"  ALT="$ n$"> interpolant <IMG  WIDTH="14" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"  SRC="img24.gif"  ALT="$ f$">  aux points  <!-- MATH  $t_0,t_1,t_2,\ldots,t_n$  --> <IMG  WIDTH="109" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"  SRC="img22.gif"  ALT="$ t_0, t_1, t_2,\ldots,t_n$"> (voir section 1.4 du livre). Consid&#233;rons le cas o&#249; les points d'interpolation sont &#233;quidistants, c'est-&#224;-dire <IMG  WIDTH="79" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"  SRC="img25.gif"  ALT="$ t_i=a+ih$">,  <!-- MATH  $i=0,\ldots,n$  --> <IMG  WIDTH="86" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"  SRC="img26.gif"  ALT="$ i=0,\ldots,n$">, avec <IMG  WIDTH="99" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"  SRC="img27.gif"  ALT="$ h=(b-a)/n$">.  <P> Le th&#233;or&#232;me 1.1 du livre nous assure que, si <IMG  WIDTH="14" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"  SRC="img24.gif"  ALT="$ f$"> est continue et que toutes ses d&#233;riv&#233;es sont continues jusqu'&#224; l'ordre <IMG  WIDTH="41" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"  SRC="img28.gif"  ALT="$ n+1$">, alors  <P></P> <DIV ALIGN="CENTER"><A NAME="conv_interp"></A> <!-- MATH  \begin{equation} \max\limits_{t\in[a,b]} 	\vert f(t)-p_n(t)\vert \le \dfrac{1}{2(n+1)} 	\left(\dfrac{b-a}{n}\right)^{(n+1)} \max\limits_{t\in[a,b]}         \vert f^{n+1}(t)\vert. \end{equation}  --> <TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER"> <TR VALIGN="MIDDLE"> <TD NOWRAP ALIGN="CENTER"><IMG  WIDTH="424" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"  SRC="img29.gif"  ALT="$\displaystyle \max\limits_{t\in[a,b]} \vert f(t)-p_n(t)\vert \le \dfrac{1}{2(n+... ...ft(\dfrac{b-a}{n}\right)^{(n+1)} \max\limits_{t\in[a,b]} \vert f^{n+1}(t)\vert.$"></TD> <TD NOWRAP WIDTH="10" ALIGN="RIGHT"> (1.1)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> Nous pourrions nous attendre a priori &#224; ce que <P></P> <DIV ALIGN="CENTER"> <!-- MATH  \begin{equation*} \max\limits_{t\in[a,b]}\vert f(t)-p_n(t)\vert \to 0         \qquad\text{lorsque}\qquad n\to\infty. \end{equation*}  --> <TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER"> <TR VALIGN="MIDDLE"> <TD NOWRAP ALIGN="CENTER"><IMG  WIDTH="167" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"  SRC="img30.gif"  ALT="$\displaystyle \max\limits_{t\in[a,b]}\vert f(t)-p_n(t)\vert \to 0$">&nbsp; &nbsp;lorsque<IMG  WIDTH="91" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"  SRC="img31.gif"  ALT="$\displaystyle \qquad n\to\infty.$"></TD> <TD NOWRAP WIDTH="10" ALIGN="RIGHT"> &nbsp;&nbsp;&nbsp;</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> Ceci n'est malheureusement pas souvent le cas car les d&#233;riv&#233;es <IMG  WIDTH="38" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"  SRC="img32.gif"  ALT="$ f^{n+1}$"> de <IMG  WIDTH="14" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"  SRC="img24.gif"  ALT="$ f$"> peuvent grandir tr&#232;s vite lorsque <IMG  WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"  SRC="img21.gif"  ALT="$ n$"> cro&#238;t. Dans la suite, nous pr&#233;sentons deux exemples, un pour lequel l'interpolation de Lagrange converge, l'autre pour lequel l'interpolation de Lagrange diverge.  <P> <BR> <HR> <!--Table of Child-Links--> <A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>  <UL> <LI><A NAME="tex2html77"  HREF="node5.html">Un exemple pour lequel l'interpolation de Lagrange converge</A> <LI><A NAME="tex2html78"  HREF="node6.html">Un exemple pour lequel l'interpolation de Lagrange diverge</A> <LI><A NAME="tex2html79"  HREF="node7.html">Une simulation interactive libre</A> <LI><A NAME="tex2html80"  HREF="node8.html">Conclusion</A> </UL> <!--End of Table of Child-Links--> <HR> <A NAME="tex2html75"  HREF="node5.html"> <IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"  SRC="/home/picasso/tex/latex2html/icons.gif/next_motif.gif"></A>  <A NAME="tex2html73"  HREF="node2.html"> <IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"  SRC="/home/picasso/tex/latex2html/icons.gif/up_motif.gif"></A>  <A NAME="tex2html69"  HREF="node3.html"> <IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"  SRC="/home/picasso/tex/latex2html/icons.gif/previous_motif.gif"></A>   <A NAME="tex2html1"  HREF="support.html"><IMG  ALIGN="BOTTOM" BORDER="0" SRC="contents_motif.gif"  ALT="58"></A> <BR> <B> Next:</B> <A NAME="tex2html76"  HREF="node5.html">Un exemple pour lequel</A> <B>Up:</B> <A NAME="tex2html74"  HREF="node2.html">Chap. 1 : Probl&#232;mes</A> <B> Previous:</B> <A NAME="tex2html70"  HREF="node3.html">Position du probl&#232;me</A>  <!--End of Navigation Panel--> <ADDRESS> <I>EPFL-IACS-ASN</I> </ADDRESS> </BODY> </HTML> 
