<div class="wims_chemin">\reload{<img src="gifs/doc/etoile.gif" alt="rechargez" width="20" height="20" border=0>}\link{main}{Rsolution numrique de l'quation \( f ( x ) = 0 \)} <img src="gifs/arrows/right3.32.gif" alt=" ---> " width="25" height="15" border=0 valign="bottom"> \link{mainS3}{III  Mthode de point fixe} <img src="gifs/arrows/right3.32.gif" alt=" ---> " width="25" height="15" border=0 valign="bottom"> \link{mainS3S3}{III-3  Point rpulsif} <img src="gifs/arrows/right3.32.gif" alt=" ---> " width="25" height="15" border=0 valign="bottom"> III-3-1  Thorme de non-convergence</div><table width=100%><tr><td valign=top><div class="left_toc"><p>
\link{mainS1}{I  Introduction}

\link{mainS2}{II  Mthode de dichotomie}

<div class="left_selection">\link{mainS3}{III  Mthode de point fixe}</div>

\link{mainS4}{IV  Mthode de Newton}

\link{mainS5}{V  Mthode de Lagrange}

\link{mainS6}{VI  Bibliographie}

\link{mainS7}{VII  Exercices}


\link{index}{Index}</div></td><td valign=top align=left width=100%><div class="wimsdoc">
<h2 class="thm">Thorme</h2><div class="thm">
Soit \( \displaystyle g:I = \left[ a, \; b \right] \longrightarrow \left[ a, \; b \right] \) de
classe \( \mathcal{ C}^1  \). On suppose que \( g \) admet un unique point fixe \( \alpha
\in \left[ a, \; b \right] \) vrifiant  \( |g'(\alpha)|>1  \). Alors il existe un voisinage \( V_{\alpha} \) de \( \alpha \) dans \( I \) tel
que la suite \( (x_n) \) dfinie par:

<div class="math">\(
\displaystyle \left\{
\begin{matrix}  
x_0 \in V_{\alpha} \setminus \left\{ \alpha \right\} & \\ 
x_{n+1}=g(x_n) ;                                  &  \;  \forall \; n \geq 0
\end{matrix}  
\right.
\)</div> 

\noindent ne converge pas vers \( \alpha  \).

</div>




\fold{mainS3S3S1F_preu1}{<span class="preu">Preuve</span>

}



<h2 class="defn">Dfinition</h2><div class="defn">
Le rel \( \alpha \) vrifiant les hypothses du thorme prcdent est
appel <em><font color="green"> point fixe rpulsif</font></em>   <a name="point fixe! rpulsif"> de \( g  \). 
</div>

</div></td><td valign=top align=right> <div class="right_toc">
<div class="right_selection">\link{mainS3S3S1}{III-3-1  Thorme de non-convergence}</div>

\link{mainS3S3S2}{III-3-2  Illustration graphique}

\link{mainS3S3S3}{III-3-3  Remarque sur la convergence}
</div><center>\reload{<img src="gifs/doc/etoile.gif" alt="rechargez" width="20" height="20" border=0>}</center></td></tr></table>