ajout de partiels

diff --git a/partiels/12mesi/main.aux b/partiels/12mesi/main.aux
new file mode 100644 (file)
index 0000000..9246ac7
--- /dev/null
+++ b/partiels/12mesi/main.aux
@@ -0,0 +1,11 @@
+\relax 
+\catcode`:\active
+\catcode`;\active
+\catcode`!\active
+\catcode`?\active
+\select@language{french}
+\@writefile{toc}{\select@language{french}}
+\@writefile{lof}{\select@language{french}}
+\@writefile{lot}{\select@language{french}}
+\newlabel{itm:1}{{1}{1}}[None]
+\newlabel{itm:2}{{2}{1}}[None]

diff --git a/partiels/12mesi/main.log b/partiels/12mesi/main.log
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index 0000000..2fc1523
--- /dev/null
+++ b/partiels/12mesi/main.log
@@ -0,0 +1,1166 @@
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diff --git a/partiels/12mesi/main.pdf b/partiels/12mesi/main.pdf
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+\documentclass[10pt,a4paper,french]{article}
+\usepackage[francais]{babel}
+\usepackage[utf8]{inputenc}
+\usepackage{a4}
+\usepackage{amsmath}
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+\usepackage{amssymb}
+\usepackage{framed}
+\usepackage{dsfont}
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+\usepackage{calc}
+\usepackage{tabls}
+\usepackage{slashbox}
+\usepackage{times}
+\usepackage{multicol}
+\usepackage{tabularx}
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+
+\usepackage[a4paper]{geometry}
+
+
+\date{}
+\geometry{hmargin=1cm, tmargin=1cm,bmargin=1.5cm}
+\begin{document}
+\title{UE MESI, Master IMR 2ème année.\\
+  Novembre 2012 (durée 1h).  J.-F.  Couchot,}
+
+\maketitle
+\vspace{-5em}
+\begin{tabular}{ll}
+Nom:& ........................................\\
+Prénom:& ........................................\\
+\end{tabular}
+
+
+On s'intéresse à résoudre une équation de la forme $f(x)=0$ par la 
+méthode de Müller. Dans cette méthode, on considère que l'on 
+a le triplet de points $(x_{n-2},x_{n-1},x_{n})$. Pour calculer 
+$x_{n+1}$, on fait comme suit:
+\begin{enumerate}
+\item \label{itm:1} on approche $f(x)$ par un polynôme $P(x)$ aux points
+  $(x_{n-2},x_{n-1},x_{n})$,
+\item\label{itm:2} on résout l'équation $P(x)=0$. La racine la plus proche de $x_n$ est $x_{n+1}$;
+\item on recommence avec le triplet $(x_{n-1},x_{n},x_{n+1})$\ldots
+\end{enumerate}
+
+
+
+
+
+
+Vos réponses seront données directement ci-dessous.
+
+\begin{enumerate}
+
+\item En utilisant une base de Lagrange, montrer que le polynôme $P(x)$ 
+  obtenu à l'étape 1. de la première itération est défini par 
+$$
+P(x) =  \dfrac{(x-x_{n-1})(x-x_{n-2})f(x_n)}{(x_n-x_{n-1})(x_n-x_{n-2})}  +
+\dfrac{(x-x_{n})(x-x_{n-2})f(x_{n-1})}{(x_{n-1}-x_{n})(x_{n-1}-x_{n-2})} +
+\dfrac{(x-x_{n})(x-x_{n-1})f(x_{n-2})}{(x_{n-2}-x_{n})(x_{n-2}-x_{n-1})} 
+$$   
+
+\vspace{4cm}
+
+\item Montrer que le polynôme de la question précédente est de degré 2. 
+  Est-ce cohérent avec le fait qu'on veuille approximer $f$ en trois points? 
+\vspace{4cm}
+
+\item Montrer que le polynôme de la première question peut s'écrire 
+  sous la forme $P(x) = a_n x^2 + b_n x + c $ où 
+\begin{eqnarray*}
+a_n & = & \dfrac{f(x_n)}{(x_n-x_{n-1})(x_n-x_{n-2})} +
+\dfrac{f(x_{n-1})}{(x_{n-1}-x_{n})(x_{n-1}-x_{n-2})} +
+\dfrac{f(x_{n-2})}{(x_{n-2}-x_{n})(x_{n-2}-x_{n-1})} \\
+b_n & = &-\dfrac{f(x_n)(x_{n-1}+x_{n-2})}{(x_n-x_{n-1})(x_n-x_{n-2})} -
+\dfrac{f(x_{n-1})(x_{n}+x_{n-2})}{(x_{n-1}-x_{n})(x_{n-1}-x_{n-2})} -
+\dfrac{f(x_{n-2})(x_{n}+x_{n-1})}{(x_{n-2}-x_{n})(x_{n-2}-x_{n-1})} \\
+c_n & = & \dfrac{f(x_n)x_{n-1}x_{n-2}}{(x_n-x_{n-1})(x_n-x_{n-2})} +
+\dfrac{f(x_{n-1})x_{n}x_{n-2}}{(x_{n-1}-x_{n})(x_{n-1}-x_{n-2})} +
+\dfrac{f(x_{n-2})x_{n}x_{n-1}}{(x_{n-2}-x_{n})(x_{n-2}-x_{n-1})} 
+\end{eqnarray*}
+\vspace{8cm}
+
+
+\item Exprimer les deux racines $x'_{n}$ et $x''_{n}$ du polynôme précédent
+en fonctions de $a_n$, $b_n$ et  $c_n$ lorsqu'on itère dans les réels.
+\vspace{3cm}
+
+\item Comment est alors défini $x_{n+1}$?
+\vspace{3cm}
+
+\item On pourrait montrer que l'ordre de la convergence est 1,84. Comparer cette vitesse de convergence avec celle de Newton et celle de Lagrange.
+\vspace{3cm}
+
+
+\item Donner le code Python de la fonction 
+  $\verb+[n,X] = iteration_muller(x+_{\verb+0+},\verb+x+_{\verb+1+},\verb+x+_{\verb+2+}\verb+,m,epsilon,f)+$ où
+\begin{itemize}
+\item $\verb+x+_{\verb+0+}$, $\verb+x+_{\verb+1+}$ et $\verb+x+_{\verb+2+}$ sont les trois premières valeurs des itérés, \verb+m+
+  est le nombre maximal 
+  d'itérations, \texttt{epsilon} est la précision souhaitée 
+  et \verb+f+ la fonction à itérer;
+\item \verb+n+ est le nombre d'itérations réalisées pour que 
+\verb+f(+$\verb+x+_{\verb+n+}$\verb+)+=0 ou que 
+$|\verb+x+_{\verb+n+}- \verb+x+_{\verb+n-1+}| \leq \verb+epsilon+$, \verb+n+ étant inférieur à \verb+m+ et \verb+X+ est 
+  le vecteur contenant les 
+  valeurs $\verb+x+_{\verb+0+},\ldots,\verb+x+_{\verb+n+}$.
+\end{itemize}
+\end{enumerate}
+
+
+\end{document}
\ No newline at end of file

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diff --git a/partiels/13mesi/main.pdf b/partiels/13mesi/main.pdf
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diff --git a/partiels/13mesi/main.tex b/partiels/13mesi/main.tex
new file mode 100644 (file)
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--- /dev/null
+++ b/partiels/13mesi/main.tex
@@ -0,0 +1,121 @@
+\documentclass[10pt,a4paper,french]{article}
+\usepackage[francais]{babel}
+\usepackage[utf8]{inputenc}
+\usepackage{a4}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{framed}
+\usepackage{dsfont}
+\usepackage[amsmath,thmmarks,thref,framed]{ntheorem}
+\usepackage[dvips]{graphicx}
+\usepackage{epsfig}
+\usepackage{calc}
+\usepackage{tabls}
+\usepackage{times}
+\usepackage{multicol}
+\usepackage{tabularx}
+\usepackage{textcomp}
+
+\usepackage{pst-all}
+
+\usepackage[a4paper]{geometry}
+
+
+\date{}
+\geometry{hmargin=1cm, tmargin=1cm,bmargin=1.5cm}
+\begin{document}
+\title{UE MESI, Master IMR 2ème année.\\
+  Novembre 2013 (durée 45 mn).  J.-F.  Couchot,}
+
+\maketitle
+\vspace{-5em}
+% \begin{tabular}{ll}
+% Nom:& ........................................\\
+% Prénom:& ........................................\\
+% \end{tabular}
+
+
+\begin{minipage}{0.6\textwidth}
+
+On cherche à résoudre une équation de la forme $f(x)=0$ par la 
+méthode de la fausse position.
+Cette méthode commence par deux points $a_0$ et $b_0$
+tels que $f(a_0)$ et $f(b_0)$ sont de signes opposés.
+Comme la fonction $f$ est continue, elle possède au moins une racine
+dans l’intervalle $[a_0, b_0]$.
+
+Dans cette méthode, on considère que l'on 
+a l'intervalle $[a_n,b_n]$. Pour calculer 
+$[a_{n+1},b_{n+1}]$, on fait comme suit:
+
+\begin{enumerate}
+\item \label{itm:1} on construit la droite $(D_n)$ qui passe par les points
+$(a_n,f(a_n))$ et $(b_n,f(b_n))$;
+\item\label{itm:2} on construit $X(x_n,0)$ le point d'intersection entre 
+  la droite $(D_n)$ et l'axe de $x$;
+\item si $f(a_n)f(x_n)\leq 0$, alors $a_{n+1} =  a_{n}$ et $b_{n+1}=x_n$;
+\item si $f(a_n)f(x_n)> 0$, alors $a_{n+1} =  x_{n}$ et $b_{n+1}=b_n$.
+\end{enumerate}
+
+
+\end{minipage}
+\begin{minipage}{0.35\textwidth}
+%\begin{figure}
+%\includegraphics[width=0.99\textwidth]{Regula_falsi_method.png}
+\includegraphics[width=0.99\textwidth]{False_position_method}
+%\caption{Deux itérations de la méthode de la fausse postion.}\label{fig:iter}
+%\end{figure}
+\end{minipage}
+
+
+
+
+
+
+
+
+Vos réponses seront données directement ci-dessous.
+
+\begin{enumerate}
+
+\item (2pts) Montrer que l'équation de la droite $(D_n)$ est  
+$y - f(b_n) = \frac{f(b_n)-f(a_n)}{b_n-a_n} (x-b_n)$. 
+
+\vspace{4cm}
+
+\item (2pts) Montrer que le nombre $x_n$ est donné par l'équation
+$x_n = a_n - \frac{a_n-b_n}{f(a_n)-f(b_n)} f(a_n)$.
+ 
+\vspace{4cm}
+
+
+\item (2pts) En moyenne, l'ordre de cette méthode est 1,618. 
+  Comparer cet ordre avec celui des autres méthodes du cours. 
+\vspace{4cm}
+
+\item (5pts)  Quelle partie de cette méthode est commune avec la 
+  méthode par dichotomie? Est-elle toujours plus efficace? 
+  Comparer les approches par exemple 
+  sur l'intervalle $[-1,1]$ avec fonction $f$ définie sur $\mathds{R}$ par
+  $f(x)= 2x^3-4x^2+3x$. 
+\vspace{6cm}
+
+\item  (4pts) Quelle partie de cette méthode est commune avec la méthode de Lagrange? 
+Est-ce la même méthode? Si ce n'est pas le cas, Expliquer ce qui diffère.
+\vspace{4cm}
+
+
+\item (5pts) Donner le code d'un programme qui implanterait cette méthode.
+
+\vspace{4cm}
+
+
+
+\end{enumerate}
+
+
+
+
+
+\end{document}
\ No newline at end of file

diff --git a/partiels/13mesi/main.tex.bak b/partiels/13mesi/main.tex.bak
new file mode 100644 (file)
index 0000000..dc9bddf
--- /dev/null
+++ b/partiels/13mesi/main.tex.bak
@@ -0,0 +1,121 @@
+\documentclass[10pt,a4paper,french]{article}
+\usepackage[francais]{babel}
+\usepackage[utf8]{inputenc}
+\usepackage{a4}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{framed}
+\usepackage{dsfont}
+\usepackage[amsmath,thmmarks,thref,framed]{ntheorem}
+\usepackage[dvips]{graphicx}
+\usepackage{epsfig}
+\usepackage{calc}
+\usepackage{tabls}
+\usepackage{times}
+\usepackage{multicol}
+\usepackage{tabularx}
+\usepackage{textcomp}
+
+\usepackage{pst-all}
+
+\usepackage[a4paper]{geometry}
+
+
+\date{}
+\geometry{hmargin=1cm, tmargin=1cm,bmargin=1.5cm}
+\begin{document}
+\title{UE MESI, Master IMR 2ème année.\\
+  Novembre 2013 (durée 45 mn).  J.-F.  Couchot,}
+
+\maketitle
+\vspace{-5em}
+% \begin{tabular}{ll}
+% Nom:& ........................................\\
+% Prénom:& ........................................\\
+% \end{tabular}
+
+
+\begin{minipage}{0.6\textwidth}
+
+On cherche à résoudre une équation de la forme $f(x)=0$ par la 
+méthode de la fausse position.
+Cette méthode commence par deux points $a_0$ et $b_0$
+tels que $f(a_0)$ et $f(b_0)$ sont de signes opposés.
+Comme la fonction $f$ est continue, elle possède au moins une racine
+dans l’intervalle $[a_0, b_0]$.
+
+Dans cette méthode, on considère que l'on 
+a l'intervalle $[a_n,b_n]$. Pour calculer 
+$[a_{n+1},b_{n+1}]$, on fait comme suit:
+
+\begin{enumerate}
+\item \label{itm:1} on construit la droite $(D_n)$ qui passe par les points
+$(a_n,f(a_n))$ et $(b_n,f(b_n))$;
+\item\label{itm:2} on construit $X(x_n,0)$ le point d'intersection entre 
+  la droite $(D_n)$ et l'axe de $x$;
+\item si $f(a_n)f(x_n)\leq 0$, alors $a_{n+1} =  a_{n}$ et $b_{n+1}=x_n$;
+\item si $f(a_n)f(x_n)> 0$, alors $a_{n+1} =  x_{n}$ et $b_{n+1}=b_n$.
+\end{enumerate}
+
+
+\end{minipage}
+\begin{minipage}{0.35\textwidth}
+%\begin{figure}
+%\includegraphics[width=0.99\textwidth]{Regula_falsi_method.png}
+\includegraphics[width=0.99\textwidth]{False_position_method}
+%\caption{Deux itérations de la méthode de la fausse postion.}\label{fig:iter}
+%\end{figure}
+\end{minipage}
+
+
+
+
+
+
+
+
+Vos réponses seront données directement ci-dessous.
+
+\begin{enumerate}
+
+\item (2pts) Montrer que l'équation de la droite $(D_n)$ est  
+$y - f(b_n) = \frac{f(b_n)-f(a_n)}{b_n-a_n} (x-b_n)$. 
+
+\vspace{4cm}
+
+\item (2pts) Montrer que le nombre $x_n$ est donné par l'équation
+$x_n = a_n - \frac{a_n-b_n}{f(a_n)-f(b_n)} f(a_n)$.
+ 
+\vspace{4cm}
+
+
+\item (2pts) En moyenne, l'ordre de cette méthode est 1,618. 
+  Comparer cet ordre avec celui des autres methodes du cours. 
+\vspace{4cm}
+
+\item (5pts)  Quelle partie de cette méthode est commune avec la 
+  méthode par dichotomie? Est-elle toujours plus efficace? 
+  Comparer les approches par exemple 
+  sur l'intervalle $[-1,1]$ avec fonction $f$ définie sur $\mathds{R}$ par
+  $f(x)= 2x^3-4x^2+3x$. 
+\vspace{6cm}
+
+\item  (4pts) Quelle partie de cette méthode est commune avec la méthode de Lagrange? 
+Est-ce la même méthode? Si ce n'est pas le cas, Expliquer ce qui difère.
+\vspace{4cm}
+
+
+\item (5pts) Donner le code d'un programme qui implanterait cette méthode.
+
+\vspace{4cm}
+
+
+
+\end{enumerate}
+
+
+
+
+
+\end{document}
\ No newline at end of file

diff --git a/partiels/13mesi/main.tex~ b/partiels/13mesi/main.tex~
new file mode 100644 (file)
index 0000000..57c391d
--- /dev/null
+++ b/partiels/13mesi/main.tex~
@@ -0,0 +1,117 @@
+\documentclass[10pt,a4paper,french]{article}
+\usepackage[francais]{babel}
+\usepackage[utf8]{inputenc}
+\usepackage{a4}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{framed}
+\usepackage{dsfont}
+\usepackage[amsmath,thmmarks,thref,framed]{ntheorem}
+\usepackage[dvips]{graphics}
+\usepackage{epsfig}
+\usepackage{calc}
+\usepackage{tabls}
+\usepackage{slashbox}
+\usepackage{times}
+\usepackage{multicol}
+\usepackage{tabularx}
+\usepackage{textcomp}
+
+\usepackage{pst-all}
+
+\usepackage[a4paper]{geometry}
+
+
+\date{}
+\geometry{hmargin=1cm, tmargin=1cm,bmargin=1.5cm}
+\begin{document}
+\title{UE MESI, Master IMR 2ème année.\\
+  Novembre 2012 (durée 1h).  J.-F.  Couchot,}
+
+\maketitle
+\vspace{-5em}
+\begin{tabular}{ll}
+Nom:& ........................................\\
+Prénom:& ........................................\\
+\end{tabular}
+
+
+On s'intéresse à résoudre une équation de la forme $f(x)=0$ par la 
+méthode de Müller. Dans cette méthode, on considère que l'on 
+a le triplet de points $(x_{n-2},x_{n-1},x_{n})$. Pour calculer 
+$x_{n+1}$, on fait comme suit:
+\begin{enumerate}
+\item \label{itm:1} on approche $f(x)$ par un polynôme $P(x)$ aux points
+  $(x_{n-2},x_{n-1},x_{n})$,
+\item\label{itm:2} on résout l'équation $P(x)=0$. La racine la plus proche de $x_n$ est $x_{n+1}$;
+\item on recommence avec le triplet $(x_{n-1},x_{n},x_{n+1})$\ldots
+\end{enumerate}
+
+
+
+
+
+
+Vos réponses seront données directement ci-dessous.
+
+\begin{enumerate}
+
+\item En utilisant une base de Lagrange, montrer que le polynôme $P(x)$ 
+  obtenu à l'étape 1. de la première itération est défini par 
+$$
+P(x) =  \dfrac{(x-x_{n-1})(x-x_{n-2})f(x_n)}{(x_n-x_{n-1})(x_n-x_{n-2})}  +
+\dfrac{(x-x_{n})(x-x_{n-2})f(x_{n-1})}{(x_{n-1}-x_{n})(x_{n-1}-x_{n-2})} +
+\dfrac{(x-x_{n})(x-x_{n-1})f(x_{n-2})}{(x_{n-2}-x_{n})(x_{n-2}-x_{n-1})} 
+$$   
+
+\vspace{4cm}
+
+\item Montrer que le polynôme de la question précédente est de degré 2. 
+  Est-ce cohérent avec le fait qu'on veuille approximer $f$ en trois points? 
+\vspace{4cm}
+
+\item Montrer que le polynôme de la première question peut s'écrire 
+  sous la forme $P(x) = a_n x^2 + b_n x + c $ où 
+\begin{eqnarray*}
+a_n & = & \dfrac{f(x_n)}{(x_n-x_{n-1})(x_n-x_{n-2})} +
+\dfrac{f(x_{n-1})}{(x_{n-1}-x_{n})(x_{n-1}-x_{n-2})} +
+\dfrac{f(x_{n-2})}{(x_{n-2}-x_{n})(x_{n-2}-x_{n-1})} \\
+b_n & = &-\dfrac{f(x_n)(x_{n-1}+x_{n-2})}{(x_n-x_{n-1})(x_n-x_{n-2})} -
+\dfrac{f(x_{n-1})(x_{n}+x_{n-2})}{(x_{n-1}-x_{n})(x_{n-1}-x_{n-2})} -
+\dfrac{f(x_{n-2})(x_{n}+x_{n-1})}{(x_{n-2}-x_{n})(x_{n-2}-x_{n-1})} \\
+c_n & = & \dfrac{f(x_n)x_{n-1}x_{n-2}}{(x_n-x_{n-1})(x_n-x_{n-2})} +
+\dfrac{f(x_{n-1})x_{n}x_{n-2}}{(x_{n-1}-x_{n})(x_{n-1}-x_{n-2})} +
+\dfrac{f(x_{n-2})x_{n}x_{n-1}}{(x_{n-2}-x_{n})(x_{n-2}-x_{n-1})} 
+\end{eqnarray*}
+\vspace{8cm}
+
+
+\item Exprimer les deux racines $x'_{n}$ et $x''_{n}$ du polynôme précédent
+en fonctions de $a_n$, $b_n$ et  $c_n$ lorsqu'on itère dans les réels.
+\vspace{3cm}
+
+\item Comment est alors défini $x_{n+1}$?
+\vspace{3cm}
+
+\item On pourrait montrer que l'ordre de la convergence est 1,84. Comparer cette vitesse de convergence avec celle de Newton et celle de Lagrange.
+\vspace{3cm}
+
+
+\item Donner le code Python de la fonction 
+  $\verb+[n,X] = iteration_muller(x+_{\verb+0+},\verb+x+_{\verb+1+},\verb+x+_{\verb+2+}\verb+,m,epsilon,f)+$ où
+\begin{itemize}
+\item $\verb+x+_{\verb+0+}$, $\verb+x+_{\verb+1+}$ et $\verb+x+_{\verb+2+}$ sont les trois premières valeurs des itérés, \verb+m+
+  est le nombre maximal 
+  d'itérations, \texttt{epsilon} est la précision souhaitée 
+  et \verb+f+ la fonction à itérer;
+\item \verb+n+ est le nombre d'itérations réalisées pour que 
+\verb+f(+$\verb+x+_{\verb+n+}$\verb+)+=0 ou que 
+$|\verb+x+_{\verb+n+}- \verb+x+_{\verb+n-1+}| \leq \verb+epsilon+$, \verb+n+ étant inférieur à \verb+m+ et \verb+X+ est 
+  le vecteur contenant les 
+  valeurs $\verb+x+_{\verb+0+},\ldots,\verb+x+_{\verb+n+}$.
+\end{itemize}
+\end{enumerate}
+
+
+\end{document}
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