{ "cells": [ { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "IPython.notebook.set_autosave_interval(300000)" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Autosaving every 300 seconds\n" ] } ], "source": [ "%reset -f\n", "%matplotlib inline\n", "%autosave 300\n", "#import sys #only needed to determine Python version number\n", "#import matplotlib #only needed to determine Matplotlib version number\n", "\n", "#print('Python version ' + sys.version)\n", "#print('Matplotlib version ' + matplotlib.__version__)\n", "from matplotlib.pylab import *" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Nous allons comparer une fonction $f$ de $\\mathbb R$ dans $\\mathbb R$ avec son développement limité (un polynôme!) en un point à différents ordre. Nous devons d'abord définir la fonction $f$ puis calculer ses dérivées à la main por enfin construire le polynôme d'approximation. Nous allons ensuite comparer les valeurs de $f$ et de son polynôme d'approximation sur un voisinnage du point de développement." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Construction d'une fonction et de son graphe\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Calcul des dérivées et initiation à la syntaxe LaTeX\n", "Soit la fonction $f$ telle que $f(x)=\\sin(x)(x-2)^2$. Proposer les expressions de la dérivée première, seconde et troisième de $f$ dans la cellule Markdown suivante." ] }, { "cell_type": "markdown", "metadata": {}, "source": [] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left(x - 2\\right)^{2} \\cos{\\left(x \\right)} + \\left(2 x - 4\\right) \\sin{\\left(x \\right)}$" ], "text/plain": [ "(x - 2)**2*cos(x) + (2*x - 4)*sin(x)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle - \\left(x - 2\\right)^{2} \\sin{\\left(x \\right)} + 2 \\left(2 x - 4\\right) \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)}$" ], "text/plain": [ "-(x - 2)**2*sin(x) + 2*(2*x - 4)*cos(x) + 2*sin(x)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left(4 - 2 x\\right) \\sin{\\left(x \\right)} - \\left(x - 2\\right)^{2} \\cos{\\left(x \\right)} - \\left(4 x - 8\\right) \\sin{\\left(x \\right)} + 6 \\cos{\\left(x \\right)}$" ], "text/plain": [ "(4 - 2*x)*sin(x) - (x - 2)**2*cos(x) - (4*x - 8)*sin(x) + 6*cos(x)" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "(x - 2)**2*cos(x) + (2*x - 4)*sin(x)\n", "-(x - 2)**2*sin(x) + 2*(2*x - 4)*cos(x) + 2*sin(x)\n", "(4 - 2*x)*sin(x) - (x - 2)**2*cos(x) - (4*x - 8)*sin(x) + 6*cos(x)\n" ] } ], "source": [ "import sympy as sy\n", "x=sy.symbols('x')\n", "f=sy.sin(x)*(x-2)**2\n", "fp=sy.diff(f,x)\n", "fpp=sy.diff(fp,x)\n", "fppp=sy.diff(fpp,x)\n", "display(fp)\n", "display(fpp)\n", "display(fppp)\n", "print(fp)\n", "print(fpp)\n", "print(fppp)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Graphe de la fonction\n", "Nous allons maintenant tracer le graphe de la fonction $f$ sur l'intervalle $[-1,5]$. \n", "### fonction Python\n", "Définissez la fonction $f$ en Python à l'aide de la syntaxe `def` ou `lambda`. " ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "-3.027209981231713\n", "0.18421579309275324\n" ] } ], "source": [ "def f(x):\n", " return sin(x)*(x-2)**2\n", "print(f(4))\n", "print(f(pi/2))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Liste de compréhension\n", "Proposez une liste de compréhension pour les valeurs de $x$ dans l'intervalle $[-1,5]$ avec un pas de $0.01$. Puis proposez une liste de compréhension pour les valeurs de $f(x)$ pour ces valeurs de $x$. Faites alors le graphes de $f$ sur l'intervalle $[-1,5]$.\n", "### Tableau numpy\n", "On va refaire le même exercice à l'aide de la librairie numpy déjà importée par matplotlib dans la première cellule de ce Notebook. Utiliser alors la commande `linspace` pour créer un tableau de valeurs de $x$ et le tableau des valeurs de $f(x)$ correspondantes. On choisira un nombre de points du graphe du même ordre qu'à l'exercice précédent. Refaites alors le graphes de $f$ sur l'intervalle $[-1,5]$." ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "dx=0.01\n", "La=[-1+dx*i for i in range(601)]\n", "Lfo=[f(x) for x in La]\n", "plot(La,Lfo)" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "xabsc=linspace(-1,5,601)\n", "yf=f(xabsc)\n", "plot(xabsc,yf)" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [], "source": [ "def fp(x):\n", " return (x - 2)**2*cos(x) + (2*x - 4)*sin(x)\n", "def fsec(x):\n", " return -(x - 2)**2*sin(x) + 2*(2*x - 4)*cos(x) + 2*sin(x)\n", "def ftierc(x):\n", " return (4 - 2*x)*sin(x) - (x - 2)**2*cos(x) - (4*x - 8)*sin(x) + 6*cos(x)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Développement limité en $0$\n", "Nous allons maintenant développer la fonction $f$ en $0$ à l'ordre $1$, $2$ puis $3$.\n", "### Codez les dérivées de $f$ dans des fonctions Python.\n", "### Développement limité à l'ordre $1$\n", "Proposez le développement limité de $f$ en $0$ à l'ordre $1$ et renseignez le dans une fonction python `dev_lim_1(x)`.\n", "### Développement limité à l'ordre $2$\n", "Proposez le développement limité de $f$ en $0$ à l'ordre $2$ et renseignez le dans une fonction python `dev_lim_2(x)`.\n", "### Développement limité à l'ordre $3$\n", "Proposez le développement limité de $f$ en $0$ à l'ordre $3$ et renseignez le dans une fonction python `dev_lim_3(x)`.\n", "### Graphes des développements limités\n", "Faites les graphes des développements limités de $f$ en $0$ à l'ordre $1$, $2$ puis $3$ sur l'intervalle $[-1,1]$.\n" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [], "source": [ "def dev_lim_1(x):\n", " return f(0)+fp(0)*x\n", "def dev_lim_2(x):\n", " return dev_lim_1(x)+0.5*fsec(0)*(x)**2\n", "def dev_lim_3(x):\n", " return dev_lim_2(x)+1/6*ftierc(0)*x**3" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "xabsc=linspace(-1,1,601)\n", "ordo=f(xabsc)\n", "plot(xabsc,ordo)\n", "plot(xabsc,dev_lim_1(xabsc))\n", "plot(xabsc,dev_lim_2(xabsc))\n", "plot(xabsc,dev_lim_3(xabsc))\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Developpements limités en différents points\n", "Proposez une fonction python `dev_lim_ordre_3(x,a)` qui renvoie le développement limité de $f$ en $a$ à l'ordre $3$. Faites alors les graphes des développements limités de $f$ en chaque point entier de l'intervalle $]-1;5[$ sur un intervalle de longueur $\\frac 3 4$ centré en chacun des points du développement. Faites ces graphes de couleur bleue et en pointillés et faites le graphe de $f$ en rouge sur l'intervalle $[-1,5]$." ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "def dev_lim_ordre_1(x,a):\n", " return f(a)+fp(a)*(x-a)\n", "def dev_lim_ordre_2(x,a):\n", " return dev_lim_ordre_1(x,a)+0.5*fsec(a)*(x-a)**2\n", "def dev_lim_ordre_3(x,a):\n", " return dev_lim_ordre_2(x,a)+1/6*ftierc(a)*(x-a)**3\n", "\n", "\n", "absc=linspace(-1,5,1000)\n", "#plot(absc,f(absc),'r')\n", "for i in range(-1,6):\n", " a=i\n", " absc=linspace(a-3/8,a+3/8,500)\n", " plot(absc,dev_lim_ordre_3(absc,a),'-.')\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Calcul d'erreur\n", "Nous allons maintenant calculer l'erreur d'approximation de $f$ par son développement limité en $0$ à l'ordre $3$ sur l'intervalle $[-1,1]$. Pour cela, proposez une fonction python `erreur_dev_lim_3()` qui renvoie la valeur absolue de l'écart maximale entre $f$ et son développement limité en $0$ à l'ordre $3$ pour $1000$ valeurs de $x$ équireparties dans l'intervalle $[-1,1]$. \n", "\n", "### Réduction de l'intervalle\n", "Faites le même travail sur l'intervalle $[-\\varepsilon,\\varepsilon]$ pour le développement limité en $0$ à l'ordre $3$. Proposez une fonction python `erreur_dev_lim_3_eps(eps)` qui renvoie la valeur absolue de l'écart maximale entre $f$ et son développement limité en $0$ à l'ordre $3$ pour $1000$ valeurs de $x$ équireparties dans l'intervalle $[-\\varepsilon,\\varepsilon]$.\n", "### Calcul d'erreur en fonction de $\\varepsilon$\n", "En commençant par $\\varepsilon=1$, puis en le divisant par $3$ à chaque fois, faites le graphique de l'erreur `erreur_dev_lim_3_eps(eps)` en fonction de $\\varepsilon$ sur l'intervalle $[10^{-10},1]$, en échelle $log$.\n", "### Quelle est la pente de la courbe? Commentez." ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.7600944700622652\n", "0.7600944700622652\n", "0.008730107566627732\n", "0.00010380425856798547\n", "1.263656420863457e-06\n", "1.552517001762732e-08\n", "1.913553378252164e-10\n", "2.3611173780024863e-12\n", "2.914422175814835e-14\n", "3.5973828083069037e-16\n", "4.4994390158148434e-18\n" ] }, { "data": { "text/plain": [ "[]" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "def erreur_dev_lim_3():\n", " absc=linspace(-1,1,1000)\n", " return max(abs(f(absc)-dev_lim_ordre_3(absc,0)))\n", "print(erreur_dev_lim_3())\n", "\n", "\n", "def erreur_dev_lim_3(eps):\n", " absc=linspace(-eps,eps,1000)\n", " return max(abs(f(absc)-dev_lim_ordre_3(absc,0)))\n", "\n", "eps=3\n", "epstab=[]\n", "errtab=[]\n", "for k in range(10):\n", " eps/=3\n", " print(erreur_dev_lim_3(eps))\n", " epstab.append(eps)\n", " errtab.append(erreur_dev_lim_3(eps))\n", "plot(log(epstab),log(errtab),'o')\n", "plot(log(epstab),4*log(epstab))\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Interpolation de Lagrange\n", "Nous allons proposer maintenant de nouvelles \n", "approximations polynômiales qui ne s'appuient pas sur le développement limité de la fonction. Nous allons introduire l'interpolation de Lagrange qui consiste à construire l'unique polynôme $p$ de $\\mathbb R^{k}[x]$ qui satisfait $(p(x_\\ell)=f(x_\\ell))_{0\\le \\ell\\le k}$ pour $x_0