From d1778cd1db87ac86b892c6d65935e6ee1abc95a3 Mon Sep 17 00:00:00 2001 From: couchot Date: Mon, 4 Nov 2013 10:11:25 +0100 Subject: [PATCH] ajout de partiels --- partiels/12mesi/main.aux | 11 + partiels/12mesi/main.log | 1166 ++++++++++++++++ partiels/12mesi/main.pdf | Bin 0 -> 77459 bytes partiels/12mesi/main.tex | 117 ++ partiels/12mesi/main.thm | 0 ...False_position_method-eps-converted-to.pdf | 0 partiels/13mesi/False_position_method.pdf | Bin 0 -> 36227 bytes partiels/13mesi/False_position_method.svg | 293 ++++ partiels/13mesi/Regula_falsi_method.png | Bin 0 -> 46484 bytes partiels/13mesi/main.aux | 11 + partiels/13mesi/main.log | 1188 +++++++++++++++++ partiels/13mesi/main.pdf | Bin 0 -> 113686 bytes partiels/13mesi/main.tex | 121 ++ partiels/13mesi/main.tex.bak | 121 ++ partiels/13mesi/main.tex~ | 117 ++ partiels/13mesi/main.thm | 0 16 files changed, 3145 insertions(+) create mode 100644 partiels/12mesi/main.aux create mode 100644 partiels/12mesi/main.log create mode 100644 partiels/12mesi/main.pdf create mode 100644 partiels/12mesi/main.tex create mode 100644 partiels/12mesi/main.thm create mode 100644 partiels/13mesi/False_position_method-eps-converted-to.pdf create mode 100644 partiels/13mesi/False_position_method.pdf create mode 100644 partiels/13mesi/False_position_method.svg create mode 100644 partiels/13mesi/Regula_falsi_method.png create mode 100644 partiels/13mesi/main.aux create mode 100644 partiels/13mesi/main.log create mode 100644 partiels/13mesi/main.pdf create mode 100644 partiels/13mesi/main.tex create mode 100644 partiels/13mesi/main.tex.bak create mode 100644 partiels/13mesi/main.tex~ create mode 100644 partiels/13mesi/main.thm diff --git a/partiels/12mesi/main.aux b/partiels/12mesi/main.aux new file mode 100644 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} 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V!|BJl?CcEeEYPH+qVi(U{{t(Pq22%h literal 0 HcmV?d00001 diff --git a/partiels/12mesi/main.tex b/partiels/12mesi/main.tex new file mode 100644 index 0000000..57c391d --- /dev/null +++ b/partiels/12mesi/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} \ No newline at end of file diff --git a/partiels/12mesi/main.thm b/partiels/12mesi/main.thm new file mode 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zLDQna&EbWye1pGA!-<5hLyadmvgW`q-r9GrV$Y!seAvX+QL#8$%CW|v;p9DljrpV2iI@)(TnpIdkS7F9tQE3DBhr>6rz`L`ikiHph!P9wf1PN z_*JM!_}ir|jVU3F9pvRKKAAiU?Qc4Z{x5|Y&;KY#v{ZoRNGvk;W_1> z8>grU4>v#rAj%~sCdMuZ5R^76+btnnw}{XG=06aB+T(Xghvwa_~;BSS(kFfdSty>LWi zS)cNhV#MWKBotl54f`c@QTpRMJ5OvGr9JXvV!wYn2#rxTX>X2 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 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 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} \ No newline at end of file diff --git a/partiels/13mesi/main.thm b/partiels/13mesi/main.thm new file mode 100644 index 0000000..e69de29 -- 2.39.5