From 2964b9a0a78739f287e9fe5a354499f36cd851fa Mon Sep 17 00:00:00 2001 From: mschiess <mathieu.schiess@posteo.ch> Date: Wed, 9 Apr 2025 10:24:28 +0200 Subject: [PATCH] =?UTF-8?q?ajout=20d'exercices=20sur=20le=20NB=20sur=20la?= =?UTF-8?q?=20m=C3=A9thode=20du=20point=20fixe?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- format_ipynb.sh | 1 + zeros/figures/fig_canette.png | Bin 0 -> 178342 bytes zeros/figures/fig_nageur.pdf | Bin 0 -> 21581 bytes zeros/point_fixe.ipynb | 295 +++++++++++++++++++++++++++++++++- 4 files changed, 295 insertions(+), 1 deletion(-) create mode 100755 format_ipynb.sh create mode 100644 zeros/figures/fig_canette.png create mode 100644 zeros/figures/fig_nageur.pdf diff --git a/format_ipynb.sh b/format_ipynb.sh new file mode 100755 index 0000000..f86a433 --- /dev/null +++ b/format_ipynb.sh @@ -0,0 +1 @@ +for file in *.ipynb; do jq --sort-keys --indent 1 . 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zq@-j!!YX3*gEOmAxr{CJ!=nuqj1-Jw@#JztBa}|H>M3i!rUa3ei;^y`jtA0LOEbzJ zbW8Gi@j@e_Des`@2E}cu9v|l4dG_SoS@r}OTNV#jr7#yJgMSROL>|0;dt%<V*&P07 zj)eR;!#;hEwf8n9fnvd~r4b)4?S2;(z5V&y>~NJNG56Wo_oA=c`k9&QNbfFtbl?^H jA4V1(-ZhwkS6q@<R8motn#N^hXk=o<rK;-c@5TiHxUJs( literal 0 HcmV?d00001 diff --git a/zeros/point_fixe.ipynb b/zeros/point_fixe.ipynb index 1b59e0e..08054e7 100644 --- a/zeros/point_fixe.ipynb +++ b/zeros/point_fixe.ipynb @@ -1 +1,294 @@ -{"cells":[{"metadata":{},"cell_type":"markdown","source":"# Méthode du Point fixe\n\n\nSoit $x \\mapsto g(x)$ une fonction continue. Tout nombre réel $a$ tel que $g(a)=a$ est appelé **point fixe** de $g$.\n\nPour une valeur de démarrage $x_0$ donnée, la méthode qui consiste à construire la suite de nombres\n\n$$x_1 = g (x_0 ),\\ x_2 = g(x_1),\\ x_3 = g(x_2),\\ x_4 = g(x_3 ), \\ldots$$\n\nest appelée **méthode itérative du point fixe**. La fonction $g$ est appelée fonction d'itération.\n\nDans la suite nous allons montrer que si la suite $\\left(x_{n}\\right)_{n\\in \\mathbf{N}}$ converge vers un nombre $a$, cela a pour conséquence que $g(a) = a$ (ceci est garanti uniquement si $g$ est continue), autrement dit que $a$ est une solution de l'équation $x = g(x)$.\n\n\n## Motivation\n\nLa résolution d'une équation du type $f(x)=0$ peut se ramener à la recherche d'un point fixe. En effet, nous pouvons définir une fonction $g$ avec $g(x)=f(x)+x$. Si on trouve un point fixe $x$ de $g$ alors par définition on a\n\n$$g(x)=x \\Leftrightarrow f(x)+x = x \\Leftrightarrow f(x) = 0 .$$\n\nEn fonction de l'expression de $f$, il peut être judicieux de proposer d'autre construction pour $g$ afin des s'assurer que la méthode du point fixe converge.\n\n\n\n## Interprétation graphique\n\nGraphiquement les points fixes de $g$ sont situés sur l'intersection de la courbe $y=g(x)$ avec la droite $y=x$.\n\n### Exemple\nLa fonction $g:x\\mapsto x^2+2x-6$ possède deux point fixe en $x=-3$ et $x=2$.\n\nOn peut vérifier cette affirmation en résolvant \n$$g(x)=x \\Leftrightarrow x^2+2x-6=x \\Leftrightarrow x^2+x-6=0 \\Leftrightarrow (x-2)(x+3)=0 \\Leftrightarrow \\ x=2 \\text{ ou }x=-3$$\n\nCe sont bien les abscisses des point d'intersection de la courbe $y=g(x)$ avec la droite $y=x$, comme on peut le voir en exécutant le progragge suivant:"},{"metadata":{"trusted":true},"cell_type":"code","source":"import numpy as np\nimport matplotlib.pyplot as plt\n\na, b = -6, 6 # l'intervalle sur lequel on trace le graphique\n\nX = np.linspace(a, b, 100)\nY = [x**2+2*x-6 for x in X]\n\nfig, ax = plt.subplots()\n\nplt.plot(X, Y)\nplt.plot(X, X)\nplt.xlim(a, b)\nplt.ylim(-8, 5)\n\nax.axhline(linewidth=2, color=\"k\")\nax.axvline(linewidth=2, color=\"k\")\n\nplt.xticks(np.arange(a, b, step=1))\nplt.yticks(np.arange(-8, 5, step=1))\n\nplt.grid(True)\n\nplt.show()","execution_count":30,"outputs":[{"output_type":"display_data","data":{"application/javascript":"element.append(window._basthonBypassBus.pop(0));"},"metadata":{}}]},{"metadata":{"trusted":true,"scrolled":true},"cell_type":"code","source":"from matplotlib import pyplot as plt\n\ndef f(x):\n return 8*x/(1 + 2*x)\n\ndef cobweb(x0, n, ax):\n xs = [x0]\n ys = [0]\n for i in range(1,n):\n if i % 2 == 0:\n xs.append(ys[-1])\n ys.append(ys[-1])\n else:\n xs.append(xs[-1])\n ys.append(f(xs[-1]))\n ax.plot(xs, ys, 'k--', lw=2.0)\n\nx = np.linspace(0, 4, 100)\n\nfig, ax = plt.subplots()\n\nax.plot(x, x, 'k', lw=2.0)\nax.plot(x, f(x), 'r', lw=2.0)\ncobweb(0.5, 50, ax)\n\nax.set_xlabel('$x$')\nax.set_ylabel('$f(x)$')\n\nplt.grid()\nplt.show()","execution_count":2,"outputs":[{"output_type":"display_data","data":{"application/javascript":"element.append(window._basthonBypassBus.pop(0));"},"metadata":{}}]},{"metadata":{},"cell_type":"markdown","source":"## Etude de la convergence\n\nA chaque itération on construit la prochaine valeur de la suite avec \n\n$$x_{n}=g(x_{n-1})$$\n\nPrenons deux point successifs $x_{n-1}$ et $x_{n}$. Si la fonction est continument dérivable, on a que d'après le théorème des acroissements finis qu'il existe un point $\\xi$ dans l'intervalle délimité par $x_{n}$ et $x_{n-1}$ telle que\n\n$$\\frac{g\\left( x_{n}\\right)-g\\left( x_{n-1}\\right)}{x_{n}-x_{n-1}}=g'\\left( \\xi \\right)$$\n\nmais comme $g\\left( x_{n}\\right)=x_{n+1}$ et $g\\left( x_{n-1}\\right)=x_{n}$, on a \n\n$$\\frac{x_{n+1}-x_{n}}{x_{n}-x_{n-1}}=g'\\left( \\xi \\right)\n\\Leftrightarrow \nx_{n+1}-x_{n} = g'\\left( \\xi \\right)\\cdot \\left( x_{n}-x_{n-1} \\right)\n$$\n\nSi on prend les valeurs absolu on aura l'égalité \n\n$$\\lvert x_{n+1}-x_{n} \\rvert = \\lvert g'\\left( \\xi \\right) \\rvert \\cdot \\lvert x_{n}-x_{n-1} \\rvert $$\n\nDonc si $\\lvert g'\\left( \\xi \\right) > 1 $ la distance entre $x_{n+1}$ et $x_{n}$ va augmenter (la suite va avoir tendance à diverger), tandis que si $\\lvert g'\\left( \\xi \\right) < 1 $ la distance entre $x_{n+1}$ et $x_{n}$ va dimunuer (la suite va avoir tendance à converger).\n\nPar conséquent, l'étude de $g'$ va nous permettre de déterminer si la suite $\\left(x_{n}\\right)_{n\\in \\mathbf{N}}$ converge ou pas.\n\n!!! note Définition\nSoit $a$ un point fixe d'une fonction $g$. On dit que le point fixe $a$ est **attractif** si $|g′(a)|<1$, tandis qu'il est dit **répulsif** si $|g′(a)|>1$.\n!!!\n"},{"metadata":{},"cell_type":"markdown","source":"### Point fixe attractif en $x=a$ avec $0<g'(a)<1$\n\nPrenons un point fixe attractif en $x=a$ avec $0<g'(a)<1$. Supposons que l'on parte d'un point $x_0 $ qui se situe dans un voisinage $I$ de $a$ où $\\forall x \\in I $ on a $0<g'(x)\\leq K$ avec $0<K<1$.\n\nAprès chaque itération $x_n=f \\left( x_{n-1}\\right)$, d'après le théorème des acroissements finis, il existe un $\\xi$ dans l'intervalle délimité par $a$ et $x_n$ telle que\n\n$$ g(a) - g(x_n) = g'(\\xi) \\left(a - x_n \\right)$$\n\nmais comme $ 0< g'(\\xi)\\leq K <1$ et que $g(a) = a$ et $g(x_n)=x_{n+1}$ on a \n\n$$ a - x_{n+1} < K \\left( a - x_n \\right)$$\n\nPar récurrence, on peut montrer que \n\n$$ a - x_{n+1} < K^n \\left( a - x_0 \\right)$$\n\nDans ce cas la suite $\\left(x_{n}\\right)_{n\\in \\mathbf{N}}$ va converger vers $a$. De plus si, $x_0<a$ alors la suite sera strictement croissante tandis que si $x_0>a$ la suite sera strictement décroissante\n\n#### Exemple avec $x_0 < a$ "},{"metadata":{"trusted":true},"cell_type":"code","source":"from matplotlib import pyplot as plt\n\n\ndef plot_point_fixe(f, x0, a, b, n =10):\n '''\n Permet d'afficher un graphique qui montre l'évolution des points\n par la méthode du point fixe ou x_{n+1} := f(x_{n}) sur \n l'intervalle [a,b]\n Args:\n - f (function) : la fonction pour laquelle on cherche un point fixe\n - x0 (float) : le point de départ\n - a (float) : le debut de l'intervalle sur lequel on affiche la fonction\n - a (float) : la fin de l'intervalle sur lequel on affiche la fonction\n - n (int) : le nombre d'itérations de la méthode du point fixe (par\n défaut n vaut 100)\n '''\n X = np.linspace(a, b, 100) # 100 points entre a et b\n Y = [f(x) for x in X]\n \n fig, ax = plt.subplots()\n \n # calcul les coordonnées des points pour la méthode du point fixe\n xs = [x0]\n ys = [0]\n for i in range(1,2*n+1):\n if i % 2 == 0:\n xs.append(ys[-1])\n ys.append(ys[-1])\n else:\n xs.append(xs[-1])\n ys.append(f(xs[-1]))\n ax.plot(xs, ys, 'k--', lw=2.0)\n\n \n ax.plot(X, X, '0.4', lw=2.0, ) # affiche la droite y=x\n ax.plot(X, Y, 'r', lw=2.0) # affiche y=f(x\n \n ax.set_xlabel('$x$')\n ax.set_ylabel('$f(x)$')\n \n ax.axhline(linewidth=2, color=\"k\")\n ax.axvline(linewidth=2, color=\"k\")\n\n plt.xlim(a, b)\n \n plt.grid()\n plt.show()\n\n\ndef g(x):\n return 8*x/(1 + 2*x)\n \nplot_point_fixe(g, 0.5, 0, 4)","execution_count":36,"outputs":[{"output_type":"display_data","data":{"application/javascript":"element.append(window._basthonBypassBus.pop(0));"},"metadata":{}}]},{"metadata":{},"cell_type":"markdown","source":"#### Exemple avec $x_0 > a$"},{"metadata":{"trusted":true,"scrolled":true},"cell_type":"code","source":"from math import log\n\ndef g(x):\n return log(x)+1\n\nplot_point_fixe(g, 4, 0.1, 4.2)","execution_count":4,"outputs":[{"output_type":"display_data","data":{"application/javascript":"element.append(window._basthonBypassBus.pop(0));"},"metadata":{}}]},{"metadata":{},"cell_type":"markdown","source":"### Point fixe attractif en $x=a$ avec $-1<g'(a)<0$\n\nPrenons un point fixe attractif en $x=a$ avec $-1<g'(a)<0$. Supposons que l'on parte d'un point $x_0 < a $ qui se situe dans un voisinage $I$ de $a$ où $\\forall x \\in I $ on a $K\\leq g'(x) < 0$ avec $-1<K<0$.\n\nComme auparavant on aura\n$$ a - x_{n+1} = g'(\\xi) \\left(a - x_n \\right)$$\n\nmais ici $ -1<K\\leq g'(\\xi)<0$. Donc à chaque itération, on aura que le signe de $a - x_{n+1}$ va changer. Cepandant la convergence sera assuré car en prenant les valeurs absolu on aura\n\n$$ \\lvert a - x_{n+1} \\rvert < \\lvert K \\rvert \\cdot \\lvert a - x_n \\rvert$$\n\nDe même que\n\n$$ \\lvert a - x_{n+1} \\rvert < \\lvert K \\rvert ^n \\cdot \\lvert a - x_0 \\rvert$$\n\nPar conséquent, on aura une suite convergente dont les termes vont osciller autour de $a$"},{"metadata":{"trusted":true,"scrolled":true},"cell_type":"code","source":"def g(x):\n return -8*x/(1 + 2*x+1)\n \nplot_point_fixe(g, -3, -8, -2)","execution_count":5,"outputs":[{"output_type":"display_data","data":{"application/javascript":"element.append(window._basthonBypassBus.pop(0));"},"metadata":{}}]},{"metadata":{},"cell_type":"markdown","source":"### Point fixe répulsif en $x=a$ avec $g'(a)>1$\n\nPrenons un point fixe répulsif en $x=a$ ce qui veut dire que $\\lvert g'(a)\\rvert >1$. Supposons que l'on parte d'un point $x_0 $ qui se situe dans un voisinage $I$ de $a$ où $\\forall x \\in I $ on a $ g'(x) \\geq K$ avec $K>1$.\n\nComme auparavant on aura\n$$ a - x_{1} = g'(\\xi) \\left(a - x_0 \\right)$$\n\nmais ici $g'(\\xi) \\geq K > 1$. Donc à la première itération, on aura que le point $x_1$ s'éloigne de $a$ car\n\n$$ \\lvert a - x_{1} \\rvert > \\lvert K \\rvert \\cdot \\lvert a - x_0 \\rvert$$\n\nDonc à priori, mis-à-part le cas où la suite devient stationnaire, c'est-à-dire si \"par chance\" on obtient un point $x_n$ où $f(x_n)$ la suite ne convergera pas vers $a$.\n\n### Exemple de divergence"},{"metadata":{"trusted":true},"cell_type":"code","source":"def g(x):\n return x**2+2*x-6\n \nplot_point_fixe(g, 2.1, -4, 8,3)","execution_count":37,"outputs":[{"output_type":"display_data","data":{"application/javascript":"element.append(window._basthonBypassBus.pop(0));"},"metadata":{}}]},{"metadata":{},"cell_type":"markdown","source":"### Exemple d'une suite stationnaire\n\nOn reprend l'exemple de la fonction $g:x\\mapsto x^2+2x-6$ qui a deux points fixes $a=2$ et $a=-3$. Si on part de $x_0=-4$ alors $x_1 = \\underbrace{g(-4)}_{=2}$ est un point fixe."},{"metadata":{"trusted":true},"cell_type":"code","source":"def g(x):\n return x**2+2*x-6\n \nplot_point_fixe(g, -4, -5, 3,3)","execution_count":40,"outputs":[{"output_type":"display_data","data":{"application/javascript":"element.append(window._basthonBypassBus.pop(0));"},"metadata":{}}]},{"metadata":{"trusted":true},"cell_type":"code","source":"","execution_count":null,"outputs":[]}],"metadata":{"kernelspec":{"name":"python3","display_name":"Python 3","language":"python"}},"nbformat":4,"nbformat_minor":2} \ No newline at end of file +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": "# Méthode du Point fixe\n\n\nSoit $x \\mapsto g(x)$ une fonction continue. Tout nombre réel $a$ tel que $g(a)=a$ est appelé **point fixe** de $g$.\n\nPour une valeur de démarrage $x_0$ donnée, la méthode qui consiste à construire la suite de nombres\n\n$$x_1 = g (x_0 ),\\ x_2 = g(x_1),\\ x_3 = g(x_2),\\ x_4 = g(x_3 ), \\ldots$$\n\nest appelée **méthode itérative du point fixe**. La fonction $g$ est appelée fonction d'itération.\n\nDans la suite nous allons montrer que si la suite $\\left(x_{n}\\right)_{n\\in \\mathbf{N}}$ converge vers un nombre $a$, cela a pour conséquence que $g(a) = a$ (ceci est garanti uniquement si $g$ est continue), autrement dit que $a$ est une solution de l'équation $x = g(x)$.\n\n\n## Motivation\n\nLa résolution d'une équation du type ($f(x) = 0$ peut se ramener à la recherche d'un point fixe. En effet, on peut définir une fonction $ g $ telle que $g(x) = f(x) + x $. Si on trouve un point fixe $ x$ de $g $ alors par définition on a\n\n$$ g(x) = x \\Leftrightarrow f(x) + x = x \\Leftrightarrow f(x) = 0. $$\n\nIl est crucial de choisir une fonction $g$ non seulement pour reformuler le problème en termes de recherche de point fixe, mais aussi pour s'assurer que la méthode itérative converge ( voir plus bas.\n\n\n## Interprétation graphique\n\nGraphiquement les points fixes de $g$ sont situés sur l'intersection de la courbe $y=g(x)$ avec la droite $y=x$.\n\n### Exemple\nLa fonction $g:x\\mapsto x^2+2x-6$ possède deux point fixe en $x=-3$ et $x=2$.\n\nOn peut vérifier cette affirmation en résolvant \n$$g(x)=x \\Leftrightarrow x^2+2x-6=x \\Leftrightarrow x^2+x-6=0 \\Leftrightarrow (x-2)(x+3)=0 \\Leftrightarrow \\ x=2 \\text{ ou }x=-3$$\n\nCe sont bien les abscisses des point d'intersection de la courbe $y=g(x)$ avec la droite $y=x$, comme on peut le voir en exécutant le progragge suivant:" + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": { + "trusted": true + }, + "outputs": [ + { + "data": { + "application/javascript": "element.append(window._basthonBypassBus.pop(0));" + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": "import numpy as np\nimport matplotlib.pyplot as plt\n\na, b = -6, 6 # l'intervalle sur lequel on trace le graphique\n\ndef f(x):\n return x**2+2*x-6\n\nX = np.linspace(a, b, 100)\nY = [f(x) for x in X]\n\nfig, ax = plt.subplots()\n\nplt.plot(X, Y)\nplt.plot(X, X)\nplt.xlim(a, b)\nplt.ylim(-8, 5)\n\nax.axhline(linewidth=2, color=\"k\")\nax.axvline(linewidth=2, color=\"k\")\n\nplt.xticks(np.arange(a, b, step=1))\nplt.yticks(np.arange(-8, 5, step=1))\n\nplt.grid(True)\n\nplt.title(\"$h:x\\mapsto x^2+2x-6$\")\n\nplt.show()" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": "## Etude de la convergence\n\nA chaque itération on construit la prochaine valeur de la suite avec \n\n$$x_{n}=g(x_{n-1})$$\n\nPrenons deux point successifs $x_{n-1}$ et $x_{n}$. Si la fonction est continument dérivable, on a que d'après le théorème des acroissements finis qu'il existe un point $\\xi$ dans l'intervalle délimité par $x_{n}$ et $x_{n-1}$ telle que\n\n$$\\frac{g\\left( x_{n}\\right)-g\\left( x_{n-1}\\right)}{x_{n}-x_{n-1}}=g'\\left( \\xi \\right)$$\n\nmais comme $g\\left( x_{n}\\right)=x_{n+1}$ et $g\\left( x_{n-1}\\right)=x_{n}$, on a \n\n$$\\frac{x_{n+1}-x_{n}}{x_{n}-x_{n-1}}=g'\\left( \\xi \\right)\n\\Leftrightarrow \nx_{n+1}-x_{n} = g'\\left( \\xi \\right)\\cdot \\left( x_{n}-x_{n-1} \\right)\n$$\n\nSi on prend les valeurs absolu on aura l'égalité \n\n$$\\lvert x_{n+1}-x_{n} \\rvert = \\lvert g'\\left( \\xi \\right) \\rvert \\cdot \\lvert x_{n}-x_{n-1} \\rvert $$\n\nDonc si $\\lvert g'\\left( \\xi \\right) > 1 $ la distance entre $x_{n+1}$ et $x_{n}$ va augmenter (la suite va avoir tendance à diverger), tandis que si $\\lvert g'\\left( \\xi \\right) < 1 $ la distance entre $x_{n+1}$ et $x_{n}$ va dimunuer (la suite va avoir tendance à converger).\n\nPar conséquent, l'étude de $g'$ va nous permettre de déterminer si la suite $\\left(x_{n}\\right)_{n\\in \\mathbf{N}}$ converge ou pas.\n\n!!! note Définition\nSoit $a$ un point fixe d'une fonction $g$. On dit que le point fixe $a$ est **attractif** si $|g′(a)|<1$, tandis qu'il est dit **répulsif** si $|g′(a)|>1$.\n!!!\n" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": "### Point fixe attractif en $x=a$ avec $0<g'(a)<1$\n\nPrenons un point fixe attractif en $x=a$ avec $0<g'(a)<1$. Supposons que l'on parte d'un point $x_0 $ qui se situe dans un voisinage $I$ de $a$ où $\\forall x \\in I $ on a $0<g'(x)\\leq K$ avec $0<K<1$.\n\nAprès chaque itération $x_n=f \\left( x_{n-1}\\right)$, d'après le théorème des acroissements finis, il existe un $\\xi$ dans l'intervalle délimité par $a$ et $x_n$ telle que\n\n$$ g(a) - g(x_n) = g'(\\xi) \\left(a - x_n \\right)$$\n\nmais comme $ 0< g'(\\xi)\\leq K <1$ et que $g(a) = a$ et $g(x_n)=x_{n+1}$ on a \n\n$$ a - x_{n+1} < K \\left( a - x_n \\right)$$\n\nPar récurrence, on peut montrer que \n\n$$ a - x_{n+1} < K^n \\left( a - x_0 \\right)$$\n\nDans ce cas la suite $\\left(x_{n}\\right)_{n\\in \\mathbf{N}}$ va converger vers $a$. De plus si, $x_0<a$ alors la suite sera strictement croissante tandis que si $x_0>a$ la suite sera strictement décroissante\n\n#### Exemple avec $x_0 < a$ " + }, + { + "cell_type": "code", + "execution_count": 36, + "metadata": { + "trusted": false + }, + "outputs": [ + { + "data": { + "application/javascript": "element.append(window._basthonBypassBus.pop(0));" + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": "from matplotlib import pyplot as plt\n\n\ndef plot_point_fixe(f, x0, a, b, n =10):\n '''\n Permet d'afficher un graphique qui montre l'évolution des points\n par la méthode du point fixe ou x_{n+1} := f(x_{n}) sur \n l'intervalle [a,b]\n Args:\n - f (function) : la fonction pour laquelle on cherche un point fixe\n - x0 (float) : le point de départ\n - a (float) : le debut de l'intervalle sur lequel on affiche la fonction\n - a (float) : la fin de l'intervalle sur lequel on affiche la fonction\n - n (int) : le nombre d'itérations de la méthode du point fixe (par\n défaut n vaut 100)\n '''\n X = np.linspace(a, b, 100) # 100 points entre a et b\n Y = [f(x) for x in X]\n \n fig, ax = plt.subplots()\n \n # calcul les coordonnées des points pour la méthode du point fixe\n xs = [x0]\n ys = [0]\n for i in range(1,2*n+1):\n if i % 2 == 0:\n xs.append(ys[-1])\n ys.append(ys[-1])\n else:\n xs.append(xs[-1])\n ys.append(f(xs[-1]))\n ax.plot(xs, ys, 'k--', lw=2.0)\n\n \n ax.plot(X, X, '0.4', lw=2.0, ) # affiche la droite y=x\n ax.plot(X, Y, 'r', lw=2.0) # affiche y=f(x\n \n ax.set_xlabel('$x$')\n ax.set_ylabel('$f(x)$')\n \n ax.axhline(linewidth=2, color=\"k\")\n ax.axvline(linewidth=2, color=\"k\")\n\n plt.xlim(a, b)\n \n plt.grid()\n plt.show()\n\n\ndef g(x):\n return 8*x/(1 + 2*x)\n \nplot_point_fixe(g, 0.5, 0, 4)" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": "#### Exemple avec $x_0 > a$" + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "scrolled": true, + "trusted": false + }, + "outputs": [ + { + "data": { + "application/javascript": "element.append(window._basthonBypassBus.pop(0));" + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": "from math import log\n\ndef g(x):\n return log(x)+1\n\nplot_point_fixe(g, 4, 0.1, 4.2)" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": "### Point fixe attractif en $x=a$ avec $-1<g'(a)<0$\n\nPrenons un point fixe attractif en $x=a$ avec $-1<g'(a)<0$. Supposons que l'on parte d'un point $x_0 < a $ qui se situe dans un voisinage $I$ de $a$ où $\\forall x \\in I $ on a $K\\leq g'(x) < 0$ avec $-1<K<0$.\n\nComme auparavant on aura\n$$ a - x_{n+1} = g'(\\xi) \\left(a - x_n \\right)$$\n\nmais ici $ -1<K\\leq g'(\\xi)<0$. Donc à chaque itération, on aura que le signe de $a - x_{n+1}$ va changer. Cepandant la convergence sera assuré car en prenant les valeurs absolu on aura\n\n$$ \\lvert a - x_{n+1} \\rvert < \\lvert K \\rvert \\cdot \\lvert a - x_n \\rvert$$\n\nDe même que\n\n$$ \\lvert a - x_{n+1} \\rvert < \\lvert K \\rvert ^n \\cdot \\lvert a - x_0 \\rvert$$\n\nPar conséquent, on aura une suite convergente dont les termes vont osciller autour de $a$" + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "scrolled": true, + "trusted": false + }, + "outputs": [ + { + "data": { + "application/javascript": "element.append(window._basthonBypassBus.pop(0));" + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": "def g(x):\n return -8*x/(1 + 2*x+1)\n \nplot_point_fixe(g, -3, -8, -2)" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": "### Point fixe répulsif en $x=a$ avec $g'(a)>1$\n\nPrenons un point fixe répulsif en $x=a$ ce qui veut dire que $\\lvert g'(a)\\rvert >1$. Supposons que l'on parte d'un point $x_0 $ qui se situe dans un voisinage $I$ de $a$ où $\\forall x \\in I $ on a $ g'(x) \\geq K$ avec $K>1$.\n\nComme auparavant on aura\n$$ a - x_{1} = g'(\\xi) \\left(a - x_0 \\right)$$\n\nmais ici $g'(\\xi) \\geq K > 1$. Donc à la première itération, on aura que le point $x_1$ s'éloigne de $a$ car\n\n$$ \\lvert a - x_{1} \\rvert > \\lvert K \\rvert \\cdot \\lvert a - x_0 \\rvert$$\n\nDonc à priori, mis-à-part le cas où la suite devient stationnaire, c'est-à-dire si \"par chance\" on obtient un point $x_n$ où $f(x_n)$ la suite ne convergera pas vers $a$.\n\n### Exemple de divergence" + }, + { + "cell_type": "code", + "execution_count": 37, + "metadata": { + "trusted": false + }, + "outputs": [ + { + "data": { + "application/javascript": "element.append(window._basthonBypassBus.pop(0));" + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": "def g(x):\n return x**2+2*x-6\n \nplot_point_fixe(g, 2.1, -4, 8,3)" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": "### Exemple d'une suite stationnaire\n\nOn reprend l'exemple de la fonction $g:x\\mapsto x^2+2x-6$ qui a deux points fixes $a=2$ et $a=-3$. Si on part de $x_0=-4$ alors $x_1 = \\underbrace{g(-4)}_{=2}$ est un point fixe." + }, + { + "cell_type": "code", + "execution_count": 40, + "metadata": { + "trusted": false + }, + "outputs": [ + { + "data": { + "application/javascript": "element.append(window._basthonBypassBus.pop(0));" + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": "def g(x):\n return x**2+2*x-6\n \nplot_point_fixe(g, -4, -5, 3,3)" + }, + { + "cell_type": "markdown", + "metadata": { + "trusted": true + }, + "source": "## Influence de le fonction auxilaire $h$ dans la convergence\n\nOn sait maintenant que pour que la métode itérative du point fixe converge, il faut que le point fixe $g(a)=a$ soit attractif, et que le point de départ $x_0$ soit situé dans un intervalle $I$ contenant $a$ tel que $\\lvert f'(x) \\rvert <1$ pout tout $x\\in I$.\n\nSi l'on souhaite résoudre un problème de recherche de zéro d'une fonction $f$ en utilisant la méthode du point fixe sur une foction $g$ définie à partir de $f$, alors on cherchera à construire $g$ de telle manière que la dérivée autour du point fixe soit, en valeur absolu, plus petite que 1. On peux par exemple tracer le graphique de $g$, $g'$ et de la droite $y=x$ pour estimer cette condition.\n\n### Exemple\n\nOn souaiterésoudre l'équation $e^x -2x-3=0$, ce qui est équivalent à trouver les zéros de $f:x\\mapsto e^x -2x-3$. Si on définit la fonction auxiliaire $g$ avec $g(x)=f(x)+x=e^x -x-3$, alors comme observé sur la représentation graphique ci-dessous, on aurra un point fixe répulsif en $x\\approx 1.9$ et un point fixe attractif en $x\\approx -1.5$." + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": { + "trusted": true + }, + "outputs": [ + { + "data": { + "application/javascript": "element.append(window._basthonBypassBus.pop(0));" + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": "import numpy as np\nimport matplotlib.pyplot as plt\nfrom math import exp\n\ndef h(x):\n return exp(x)-x-3\n\ndef dh(x):\n return exp(x)-1\n\na, b = -6, 6 # l'intervalle sur lequel on trace le graphique\n\nX = np.linspace(a, b, 100)\nY = [h(x) for x in X]\nDY = [dh(x) for x in X]\n\nfig, ax = plt.subplots()\n\nplt.plot(X, Y, label=\"y=h(x)\")\nplt.plot(X, X)\nplt.plot(X, DY, label=\"y=h'(x)\")\nplt.xlim(a, b)\nplt.ylim(-8, 5)\n\nax.axhline(linewidth=2, color=\"k\")\nax.axvline(linewidth=2, color=\"k\")\n\nplt.xticks(np.arange(a, b, step=1))\nplt.yticks(np.arange(-8, 5, step=1))\n\nplt.grid(True)\n\nplt.legend(loc=\"lower right\")\n\nplt.title(\"$h:x\\mapsto e^x-x-3$\")\n\nplt.show()" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": "Prenons un choix différent pour $h$. Par exemple, on a \n\n$$e^x -2x-3=0 \\Leftrightarrow e^x =2x+3 \\Leftrightarrow x = \\ln \\left( 2x+3 \\right)$$\n\nDonc les solutions de l'équation sont les points fixes de la fonction $\\hat{h}:x\\mapsto \\ln \\left( 2x+3 \\right)$. La fonction $\\hat{h}$ possède les mêmes point fixe que $h$ mais comme on peut le voir sur le graphique ci-dessous, la situation est inversée. Le point fixe en $x\\approx 1.9$ est maintenant attractif tandi que le point fixe en $x\\approx -1.5$ est devenu répulsif.\n\n\n*Remarque : pour tracer le graphique on a besoin de voir que $D_\\hat{h} = ]-3/2, + \\infty[$.*" + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": { + "scrolled": true, + "trusted": true + }, + "outputs": [ + { + "data": { + "application/javascript": "element.append(window._basthonBypassBus.pop(0));" + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": "import numpy as np\nimport matplotlib.pyplot as plt\nfrom math import log\n\ndef h(x):\n return log(2*x+3)\n\ndef dh(x):\n return 1/(2*x+3)\n\na, b = -1.49, 6 # l'intervalle sur lequel on trace le graphique\n\nX = np.linspace(a, b, 100)\nY = [h(x) for x in X]\nDY = [dh(x) for x in X]\n\nfig, ax = plt.subplots()\n\nplt.plot(X, Y, label=\"y=h(x)\")\nplt.plot(X, X)\nplt.plot(X, DY, label=\"y=h'(x)\")\nplt.xlim(a, b)\nplt.ylim(-8, 5)\n\nax.axhline(linewidth=2, color=\"k\")\nax.axvline(linewidth=2, color=\"k\")\n\nplt.xticks(np.arange(a, b, step=1))\nplt.yticks(np.arange(-8, 5, step=1))\n\nplt.grid(True)\n\nplt.legend(loc=\"lower right\")\n\nplt.title(\"$h:x\\mapsto \\log(2x+3)$\")\n\nplt.show()" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": "Pour résumer un utilisera $h$ et $\\hat{h}$ pour trouver les solutions de $e^x -2x-3=0$." + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": { + "trusted": true + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": "il y a deux solution à l'équation ex eˣ -2x-3=0 \nune première solution en x = -1.373799270699159\nune seconde solution en x = 1.923729843806448\n" + } + ], + "source": "def point_fixe(h, x0, tol=0.001, iter_max=100):\n '''\n Cherche un point fixe x de la fonction h (i.e. un solution de h(x)=x)e\n n utilisant la méthode itérative du point fixe\n Args:\n - f (function) : la fonction pour laquelle on cherche le point\n fixe.\n - x0 (float) : la valeur de démarrage de la méthode.\n - tol (float) : l'erreur minimal à partir de laquelle on\n arrête la recherche (|x_n-x_{n+1}|<tol).\n Par défaut tol = 0.001\n - iter_max(int): Le nombre d'itérations maximal dans le cas où \n la méthode diverge ou converge trop lentement\n \n return:\n - x1(float) : une approximation du point fixe qui se trouve à\n distance tol de la solution ou le iter_max termes\n si la méthode diverge.\n \n '''\n x1 = x0\n for _ in range(iter_max):\n x0 = x1\n x1 = h(x0)\n \n # print(x1)\n \n # on stop la recherche si on est suffisament proche de la solution\n if abs(x0-x1) < tol:\n return x1\n\n return x1\n\nprint(\"il y a deux solution à l'équation ex eˣ -2x-3=0 \")\n\n\n# première solution avec h\ndef h(x):\n return exp(x)-x-3\n\nprint(\"une première solution en x = \",point_fixe(h, -1))\n\n# Deusième solution avec ĥ\ndef h(x):\n return log(2*x+3)\n\nprint(\"une seconde solution en x = \",point_fixe(h, 1))" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": "## Exercices\n\n### Exercice 87\nOn désire fabriquer une canette en aluminium qui ait la forme d’un cylindre circulaire droit de hauteur h et de rayon r fermé aux deux extrémités, d'un volume de 33 cl (voir les figures).\n\n\n\nEn utilisant le calcul différentiel, répondre aux questions suivantes :\n\n1. Quelles sont les cotes h et r en cm qui minimisent le coût de la canette si l'aluminium employé pour les disques coûte 0,002 Fr/cm2 et celui utilisé pour le cylindre circulaire droit, 0,001 Fr/cm² ?\n\n Quelle est son coût minimum ?" + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "trusted": true + }, + "outputs": [], + "source": "# Afficher la fonction d'intérêt pour estimer la localisation \n# du point fixe \n" + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "trusted": true + }, + "outputs": [], + "source": "# Trouver le point fixe\n" + }, + { + "cell_type": "raw", + "metadata": {}, + "source": "Réponse: Le coût minumum d'une canette est de ... Fr et il faut que h=.... [cm] et r=....[cm] " + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": "2. Par souci d'écologie, le fabriquant aimerait construire cette canette avec un minimum d'aluminium. Est-ce compatible avec un coût minimum ? " + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "trusted": false + }, + "outputs": [], + "source": "" + }, + { + "cell_type": "raw", + "metadata": {}, + "source": "Réponse : \n\nExplication :" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": "### Exercice 90\n\nUn nageur se trouve à une distance de $0,57$ kilomètres du rivage et aimerait rejoindre sa cabane.\n \nIl est positionné tel que le point $A$ est sa projection orthogonale sur le rivage et la distance séparant le point $A$ de la cabane est de $2,8\\; km$.\n\n\nLe nageur peut nager a une vitesse de $3\\;km/h$ et marcher sur le rivage à une vitesse de $5km/h$.\n\n\n\nOn souhaite déterminer à quelle distance de la cabane doit se trouver le point $P$, correspondant à l'endroit où le nageur doit sortir de l'eau, pour rejoindre la cabane en un minimum de temps." + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "trusted": true + }, + "outputs": [], + "source": "# Afficher la fonction d'intérêt pour estimer la localisation \n# du point fixe \n\n" + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "trusted": true + }, + "outputs": [], + "source": "# Trouver le point fixe\n\n" + }, + { + "cell_type": "raw", + "metadata": {}, + "source": "Réponse: La cabane de trouve à .... [km] du point $P$." + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} -- GitLab