{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "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": [
    "from IPython.display import display, Latex\n",
    "from IPython.core.display import HTML\n",
    "%reset -f\n",
    "%matplotlib inline\n",
    "%autosave 300\n",
    "from matplotlib.pylab import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# M25 - Cours 1\n",
    "## Introduction Jupyter Notebook\n",
    "Le document présenté ici est au format \".ipynb\" pour profiter d'un document qui contient aussi bien des écritures scientifiques que des codes python3 exécutables. Le but de ces documents est de les rendre lisibles et dynamiques. Les mathématiques seront rédigées dans des cellules \"Markdown\" et les codes python3 dans des cellules \"Code\". \n",
    "\n",
    "### Installation Jupyter\n",
    "Pour installer Jupyter, le plus simple est d'installer Anaconda (quel que soit le système) puis de lancer l'outil Jupyter Notebook à travers la suite de logiciels proposée.\n",
    "Le lien pour Anconda est le suivant\n",
    "https://www.anaconda.com/products/individual\n",
    "\n",
    "**En salle machine de l'université, il convient d'ouvrir une fenêtre terminal puis de lancer le logiciel en tapant \"jupyter notebook &\" suivi d'\"entrée\".**\n",
    "\n",
    "### Premiers pas\n",
    "L'exécution des cellules (\"Markdown\" et \"Code\") s'obtient en cliquant sur le bouton \"Exécuter\". Pour éditer le code source Markdown, s'il est déjà exécuté (le rendu est lisible par tous, y compris les formules mathématiques), il suffit de doubleclicker sur le texte scientifique (la cellule Markdown). Les formules mathématiques sont alors lisibles dans un language (Markdown) proche de celui de Latex. Une initiation plus pousée peut se trouver sur le web. Quelques points sont abordés ici. On remarque déjà l'utilisation de \\# pour structurer section sous-section...\n",
    "\n",
    "#### Faire un tableau\n",
    "| Titre 1       |     Titre 2     |        Titre 3 |\n",
    "| :------------ | :-------------: | -------------: |\n",
    "| Colonne       |     Colonne     |        Colonne |\n",
    "| Alignée à     |   Alignée au    |      Alignée à |\n",
    "| Gauche        |     Centre      |         Droite |\n",
    "\n",
    "#### Ecrire des Mathématiques\n",
    "Les formules mathématiques en ligne sont encadrées par des $\\$$, les formules centrées sont encadrées par des doubles $\\$$.\n",
    "Les exemples qui suivent permettent de comprendre la gestion des indices, exposants, fraction, intégrale, systèmes... Pensez à basculer du source à l'exécution de la cellule pour comprendre.\n",
    "\n",
    "Voici la fonction $x \\to \\sin x$ définie de $\\mathbb R$ dans $[-1,1]$.\n",
    "\n",
    "Voici une formule centrée avec des indices, des exposants et une fraction. Les acollades ouvrante et fermante servent à délimiter un bloc qui pourra être placé en indice, en exposant, au numérateur ou dénominateur d'une fraction... Si le bloc ne contient qu'un seul caractère, on peut omettre les acollades.\n",
    "$$\n",
    " x_{n+1} = -x_n+2 x_n^2+\\frac {12}{2+x_n }.\n",
    "$$\n",
    "Voici une intégrale,\n",
    "$$\n",
    "\\int_{0}^{\\pi}\\cos (x) \\, dx=[\\sin (x)]_0^{\\pi}=0.\n",
    "$$\n",
    "Voici comment présenter un système... une façon parmi d'autres:\n",
    "$$\n",
    "\\begin{cases}\n",
    "x_0 = 1\\\\\n",
    "    \\text{pour } n \\geq 0, \\quad x_{n+1} = -x_n+\\frac 3 {|x_n|+2},\\\\\n",
    "    \\text{on peut rajouter autant de ligne que l'on souhaite.}\n",
    "\\end{cases}\n",
    "$$\n",
    "### Cellules de code\n",
    "On remarque qu'en haut à droite, le language python 3 est selectionné, les cellules de code seront écrites en ce language. Les cellules de code peuvent être nombreuses dans un tel document et l'ordre d'exécution est choisie par la personne qui manipule de document. Les grandeurs calculées restent en mémoire vive. Les cellules peuvent donc s'enchainer les unes aux autres en s'appuyant sur les résultats antérieurs. Il est recommandé de respecter l'ordre chronologique d'exécution en suivant l'avancement du document. Ce document est donc dynamique et peut planter si le code exécuté est faux ou ne pas rendre la main dans une cellule si une boucle infinie est présente. Les autres cellules ne pourront plus s'exécuter, il faudra alors interompre le noyau avec le bouton carré au centre du bandeau (à droite \"d'Exécuter\"). Il faudra alors relancer les cellules car toute l'information en mémoire vive a été perdue. Plusieurs options sont possible sur les boutons \"Cellule\" et \"Noyau\"."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0, 1, 4, 9, 16, 25, 36, 49, 64, 81]\n"
     ]
    }
   ],
   "source": [
    "#exemple de calcul \n",
    "L=[i**2 for i in range(10)]\n",
    "print(L)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On profite de cette cellule Markdown pour rappeler qu'on vient de faire une liste de compréhension dans la cellule de code précédente. La prochaine cellule de code va permettre de calculer la somme des carrés des 10 premiers entiers en se servant de la cellule précédente déjà exécutée. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "La somme des carrés des 10 premiers entiers  positifs est 285 .\n"
     ]
    }
   ],
   "source": [
    "print(\"La somme des carrés des 10 premiers entiers  positifs est\",sum(L),\".\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rappel de la formule de Taylor\n",
    "On suppose une fonction $f$ de $\\mathbb R$ dans $\\mathbb R$ de classe $\\mathbf C^{k+1}$,\n",
    "Pour comparer cette fonction $f$ à un polynôme $p$ au voisinage de $x_0\\in \\mathbb R$, on dispose de la formule de Taylor, proposée ici avec reste de Lagrange. \n",
    "$$\n",
    "\\begin{cases}\n",
    "\\exists \\xi \\in [\\min(x,x_0),\\max(x,x_0)] \\text { tel que,}\\\\\n",
    "f(x)=p(x)+\\frac {f^{(k+1)}(\\xi)}{(k+1)!}(x-x_0)^{k+1}\\\\\n",
    "p(x)=f(x_0)+f'(x_0)(x-x_0)+\\frac {f''(x_0)}{2}(x-x_0)^2+\\cdots+\\frac {f^{(k)}(x_0)}{k!}(x-x_0)^k.\n",
    "\\end {cases}\n",
    "$$\n",
    "Si on applique cette formule pour la fonction $\\sin$ au voisinage de $0$ pour $k=4$, on a\n",
    "$$\n",
    "p(x)=x-\\frac {x^3}{6}.\n",
    "$$\n",
    "On va d'abord comparer le graphe des deux fonctions au voisinage de $0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def p(x):\n",
    "    return x-x**3/6\n",
    "absc=linspace(-pi,pi,200)\n",
    "ord_f=sin(absc)\n",
    "ord_p=p(absc)\n",
    "plot(absc,ord_f,absc,ord_p);\n",
    "grid();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On constate que les deux graphes sont presque superposés. On va calculer l'erreur pour la norme du $\\sup$ entre $f$ et $p$ sur des voisinages $]-\\varepsilon, \\varepsilon[$ de plus en plus petit et observer l'erreur en $\\mathcal O(\\varepsilon^5)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.0 0.008137651474563135\n",
      "0.5 0.0002588719375363202\n",
      "0.25 8.125921189594543e-06\n",
      "0.125 2.542185610215908e-07\n",
      "0.0625 7.946546864978643e-09\n",
      "0.03125 2.483469124792137e-10\n",
      "0.015625 7.760975889725685e-12\n",
      "0.0078125 2.425316891763174e-13\n",
      "0.00390625 7.57900686654267e-15\n",
      "0.001953125 2.3678975447083417e-16\n"
     ]
    }
   ],
   "source": [
    "def erreur(eps):\n",
    "    pts=linspace(-eps,eps,1000)\n",
    "    return max(abs(sin(pts)-p(pts)))\n",
    "\n",
    "epstab=[]\n",
    "erreurtab=[]\n",
    "eps=2\n",
    "for loop in range (10):\n",
    "    eps*=0.5\n",
    "    print(eps,erreur(eps))\n",
    "    epstab.append(eps)\n",
    "    erreurtab.append(erreur(eps))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Il est intéressant de regarder l'erreur en fonction de $\\varepsilon$ en échelle logarithmique car le log de l'erreur (erreur censée être en $O(\\varepsilon^5)$) doit s'écrire $C+5\\log (\\varepsilon)$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fc90977ba20>]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fc90977b978>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "eps_arr=array(epstab)\n",
    "err_arr=array(erreurtab)\n",
    "plot(log(eps_arr),log(err_arr),log(eps_arr),log(eps_arr**5))\n",
    "plot(log(eps_arr),log(err_arr),'o')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Le parallélisme de ces deux droites de pente $5$ nous confirme le comportement de l'erreur au voisnage de $0$ entre la fonction $\\sin$ et son polynôme obtenu par le développement limité à l'ordre $4$.\n",
    "\n",
    "## Polynôme de Lagrange\n",
    "On s'intéresse à de nouvelles approximation de fonction à l'aide de polynôme, en s'appuyant sur plusieurs points du graphe de la fonction, mais cette fois sans tenir compte des dérivées.\n",
    "\n",
    "**Exemple:** On veut interpoler les points d'un graphe suivants: $(1,2)$, $(2,0)$, $(3,0)$, $(4,0)$. On cherche un degré minimal pour selectionner un polynôme candidat $p$.\n",
    "On remarque que ce polynôme $p$ est factorisable à l'aide des points d'annulation ($p(2)=p(3)=p(4)=0$). On a $p(x)=(x-2)(x-3)(x-4)q(x)$. On va choisir $q$ constant pour limiter le degré et assurer $p(1)=2$. On a $p(1)=-6 q(1)$. On trouve $q=\\frac {-1} 3$.\n",
    "\n",
    "Soit $n+1$ points de $\\mathbb R^2$ de coordonnées $(x_i,f_i=f(x_i))$, $i=0\\cdots n$, avec $x_i\\neq x_j$ si $i\\neq j$.\n",
    "On cherche à interpoler $f$ par un polynôme $p$:\n",
    "\n",
    "**Théorème :**\n",
    "Il existe un unique polynôme $p$ de degré au plus $n$ tel que \n",
    "$$ \\forall 0\\le i\\le n,\\quad p(x_i)=f_i.$$\n",
    "Le polynôme $p$ s'appelle le polynôme de Lagrange (associé aux points $(x_i,f_i=f(x_i))$, $i=0\\cdots n$). On dira que $p$ interpole $f$ en $x_i$, $i=0\\cdots n$.\n",
    "\n",
    "Preuve: on note $l_i$ le polynôme défini par \n",
    "$$\n",
    "l_i(x)=\\frac {\\displaystyle \\prod_{\\begin {array} c j=0\\cdots n\\\\j\\neq i\\end {array}}(x-x_j)}{\\displaystyle \\prod_{\\begin {array} c j=0\\cdots n\\\\j\\neq i\\end {array}}(x_i-x_j)}.\n",
    "$$\n",
    "On remarque que $l_i$ est de degré $n$ et que \n",
    "$$\n",
    "l_i(x_j)=\\delta_{ij}.\n",
    "$$\n",
    "On rappelle que l'espace vectoriel $\\mathbb P_n$ des polynômes de degré au plus $n$ est de dimension $n+1$. La famille $(l_i)_{i=0\\cdots n}$ est une famille libre de $P_n$: si,\n",
    "$$\n",
    "\\forall x\\in \\mathbb R,\\quad \\sum_{i=0}^n \\alpha_i l_i(x)=0,\n",
    "$$\n",
    "alors, en choisissant $x=x_j$ pour tout $j=0\\cdots n$, on a $\\alpha_i l_i(x_j)=\\alpha_i\\delta_{ij}$, donc $\\sum_{i=0}^n \\alpha_i l_i(x)=\\alpha_j=0$.\n",
    "\n",
    "La famille $(l_i)_{i=0\\cdots n}$ constituée de $n+1$ éléments est donc libre et constitue une base de $\\mathbb P_n$. Ainsi, pour tout $q\\in \\mathbb P_n$, $q$ se décompose de façon unique  sur la base $(l_i)_{i=0\\cdots n}$. Finalement, le poynôme $ p=\\sum_{i=0}^n f_i l_i$ est l'unique polynôme de $\\mathbb P_n$ tel que $p(x_i)=f_i$ pour $i=0\\cdots n$.\n",
    "\n",
    "### Estimation d'erreur d'interpolation\n",
    "Soit une fonction $f$ donnée, et soit $p$ le polynôme de Lagrange interpolant  $f$ en $x_i$, $i=0\\cdots n$, $x_0<x_1<\\cdots<x_n$. On cherche à quantifier l'écart entre $f$ et $p$ en fonction de $x$ afin de mesurer l'erreur d'interpolation.\n",
    "\n",
    "**Théorème:**\n",
    "On suppose que $x_i\\in [a,b]$ pour $i=0\\cdots n$.\n",
    "\n",
    "Si $f$ est $C^{n+1}([a,b])$, alors il existe $\\xi\\in [\\min(x,x_0),max(x,x_n)]$ tel que \n",
    "$$\n",
    "f(x)-p(x)= \\prod_{i=0}^n \\frac{(x-x_i)}{(n+1)!}f^{(n+1)}(\\xi).\n",
    "$$\n",
    "\n",
    "\n",
    "Remarque: si les points $x_i$ sont voisins,  le produit $\\prod_{i=0}^n (x-x_i)$ est petit pour $x\\in [\\min(x_i),\\max(x_i)]$. On pourrait alors être tenté d'augmenter le nombre de points d'interpolation pour réduire l'erreur, mais l'erreur croît avec l'étalement des points d'interpolation et le facteur $\\frac {f^{(n+1)}(\\xi)}{(n+1)!}$ peut croître (comme décroître) selon $f$.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# M25-Cours 2\n",
    "## Notion de Différences finies \n",
    "### Approximation de la dérivée\n",
    "On peut calculer, à partir de la définition du nombre dérivée $$f^\\prime(x_0)=\\lim_{h\\to 0} \\frac {f(x_0+h)-f(x_0)}{h},$$ une approximation de $f^\\prime(x_0)$ en évaluant le taux d'accroissement pour $h$ \"petit\". \n",
    "\n",
    "Exemple $h=10^{-3}$, $f'(x_0)\\sim \\frac {f(x_0+h)-f(x_0)}{h}$.\n",
    "A l'aide du développement limité de $f$ en $x_0$, on peut estimer l'erreur commise:\n",
    "$$\n",
    "f(x_0+h)   =  f(x_0) +  f'(x_0)h+\\frac {h^2} 2 f''(x_0)+O(h^3).\n",
    "$$\n",
    "On remarque que \n",
    "$$\n",
    "f(x_0+h)   -  f(x_0) =f'(x_0)h+ O(h^2),\n",
    "$$\n",
    "Soit, après division par $h$,\n",
    "$$\n",
    "f'(x_0)=\\frac {f(x_0+h)-f(x_0)}{h}+O(h).\n",
    "$$\n",
    "On peut chercher une formule plus précise, en se servant d'autres points pour écrire le développement en $x_0$:\n",
    "$$\n",
    "f(x_0-h)   =  f(x_0) -  f'(x_0)h+\\frac {h^2} 2 f''(x_0)+O(h^3).\n",
    "$$\n",
    "Par différence des développements en $x_0-h$ et $x_0+h$, on obtient:\n",
    "$$\n",
    "f(x_0+h) -f(x_0-h)   =  2  f'(x_0)h+0 f''(x_0)+O(h^3).\n",
    "$$\n",
    "On a ainsi trouvé une formule plus précise pour approcher la dérivée en un point à l'aide de l'évaluation de la fonction enseulement 2 points:\n",
    "$$\n",
    "f'(x_0)=\\frac {f(x_0+h)-f(x_0-h)}{2h}+O(h^2).\n",
    "$$\n",
    "### Approximation de la dérivée seconde\n",
    "De la même manière que ce qui a été fait pour approcher la dérivée, on peut établir que\n",
    "$$\n",
    "f^{\\prime \\prime}(x_0) =\\lim_{h\\to 0} \\frac {f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2}\n",
    "$$\n",
    "et plus précisément:\n",
    "$$\n",
    "f^{\\prime \\prime}(x_0) = \\frac {f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2}+O(h^2).\n",
    "$$\n",
    "(preuve en TP).\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "erreur pour h= 0.1 : 0.0009000536983791996\n",
      "erreur pour h= 0.01 : 9.004993400729688e-06\n",
      "erreur pour h= 0.001 : 9.00503946965614e-08\n",
      "erreur pour h= 0.0001 : 9.004295087322589e-10\n"
     ]
    }
   ],
   "source": [
    "#approximation de la dérivée de sin en 1 (exacte=cos(1))\n",
    "def fpri(x,h):\n",
    "    return 0.5*(sin(x+h)-sin(x-h))/h\n",
    "h=1.e-1\n",
    "print('erreur pour h=',h,':',abs(cos(1)-fpri(1,h)))\n",
    "h=1.e-2\n",
    "print('erreur pour h=',h,':',abs(cos(1)-fpri(1,h)))\n",
    "h=1.e-3\n",
    "print('erreur pour h=',h,':',abs(cos(1)-fpri(1,h)))\n",
    "h=1.e-4\n",
    "print('erreur pour h=',h,':',abs(cos(1)-fpri(1,h)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On constate bien que l'erreur est 100 fois plus petite quand $h$ est 10 fois plus petit d'où le comportement de l'erreur en $O(h^2)$.\n",
    "## Formule de quadrature\n",
    "On vient de voir que l'on sait approcher une dérivée, on va maintenant essayer d'approcher une intégrale.\n",
    "L'objectif ici est de proposer un calcul approché d'intégrale sans avoir à intégrer la fonction considérée, chose que l'on ne sait pas toujours faire. Exemple, savez-vous calculer l'intégrale suivante?\n",
    "$$\n",
    "\\int_{0}^{1} e^{-x^2}\\, dx \n",
    "$$\n",
    "En effet, on ne connaît pas la primitive de la fonction $e^{-x^2}$.  \n",
    "\n",
    "L'idée du calcul approché repose sur ce qu'on nome une **formule de quadrature** et qui prend la forme\n",
    "$$\n",
    "\\sum_{k=0}^n A_k f(x_k),\n",
    "$$\n",
    "où les points $(x_k)_{0\\le k \\le n}$ sont répartis sur l'intervalle d'intégration $[a,b]$ et les poids $(A_k)_{0\\le k \\le n}$ sont choisis de façon pertinente pour approcher (lorsque $n$ est grand) l'intégrale\n",
    "$$\n",
    "\\int_a^b f(x)\\, dx.\n",
    "$$\n",
    "Les coefficients $(A_k)_{0\\le k \\le n}$ et les points $(x_k)_{0\\le k \\le n}$ changent bien-sûr lorsque $n$ change.\n",
    "### Un premier exemple de formule de quadrature\n",
    "La première idée qui vient à l'esprit pour approcher l'intégrale est d'approcher $f$ par une fonction constante par morceau et de calculer la somme des surfaces des rectangles (batonnets) proposés sur le graphe suivant. La surface verte est proche de la surface sous la courbe rouge."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fa7533238d0>]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa7520d8ac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "NB=11\n",
    "x = linspace(0,1,NB)\n",
    "pas=1/(NB-1)\n",
    "xfin = linspace(0,1,200)\n",
    "hei=sin(xfin) \n",
    "height = sin(x-pas/2)\n",
    "width= 1/(NB-1)\n",
    "plt.bar(x,height, width, color='g')\n",
    "plot(xfin,hei,'r')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fa75520d898>]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa755204a58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "NB=41\n",
    "x = linspace(0,1,NB)\n",
    "pas=1/(NB-1)\n",
    "xfin = linspace(0,1,200)\n",
    "hei=sin(xfin) \n",
    "height = sin(x-pas/2)\n",
    "width= 1/(NB-1)\n",
    "plt.bar(x,height, width, color='g')\n",
    "plot(xfin,hei,'r')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On peut choisir de prendre pour hauteur du batonnet l'évaluation de la fonction à droite du batonnet et non à gauche comme sur le précédent graphe."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fa755139240>]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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zA2Me3wbcltjSRFLLyuH6GcHWd8t2hVN7wxtNo65KZNvoWr8IUG8NPD02uNN07CHQrzOs2CXqqkS2ncJdZMIEPngUapfBVWfBY/lo6zvJeAp3yQnxNqquvgH+OAlufBuW1YPTLoE5e0dQnEgSKNwlJzVbDiPGwFFLYfDRcFN7+HHHqKsSSRyFu+QWh96z4JFCKKsG51wI4w6NuiiRxFO4S86o/WOwUXWP2TClSbCn6eLdoq5KJDkU7pIT8hfDqOeh8Sq4/VS490Qo1w7CksUU7pLd3OFvf2P6P2FpLfhVH5ihZe0kByjcJWtUnhGz51p46iXo9Am8fAj06QLf7hxRcSIppnCXrHT8l8ESAvW+h2vPhEfaoLnrklMU7pJVrBxumQ73TIaFu8PxfeG9faOuSiT1FO6SNequgWfGwhmfwagW0L+TlueV3KVwl+wwZQrvPwZ7/AD9z4bHj0LDMJLTFO6SMeItIbBDOdzxJgx8A1btCWf0gg/3iaA4kTSjcJeMVf87+NeLcMoXMLwlXH0WfL9T1FWJpIdQt3GYWQczm2dm883s1i2062pmbmb5iStRZFPt58Osx6DNYuh9DvQ+T8EuEqvKcDezPGAwcCbQHOhhZs3jtKsF/Bp4J9FFivyk2kb442sw4Vn4uiYcfTkMPzLqqkTST5ieextgvrsvcPd1wEigS5x29wD3AT8msD6RnzVYBVOegtumwdDWcEw/+Khe1FWJpKcw4d4AWBRzXFLx3M/MrBXQyN3HJ7A2kf+ZOJGZQ+DIr6DH+XBFZ/ihetRFiaSvMBdU400o859fNNsBeJAQm2KbWX+gP0Djxo3DVSg5pfKMmB3Kg5kwv3sDvqoH3brBvLoRFSeSQcL03EuA2KWWGgJLYo5rAYcBr5vZF8CxQEG8i6ruPtTd8909v25d/YTKltVdAxOegTvfgOFHBMMwCnaRcML03IuAZmbWFFgMdAd6/vSiu68C6vx0bGavAze5e3FiS5VccuLCYInePX6EPp3hydZRVySSWarsubv7BmAAMAH4CBjt7nPMbJCZdU52gZJbrBxunhZcOF1THY7tp2AX2RahbmJy90KgsNJzAzfTtu32lyU5acUKXhoJnT+B0c2hX2dYrbVhRLaJ9qKR9FBUBK1b02F+sETvhd0U7CLbQ8sPSMr9YkaMw9VF8MAE+KomXNAH/tswutpEsoXCXSJTswweL4Duc2B8M7jkXFi5S9RViWQHhbtE4rCv4YXRcOAKuPU0uO8EcA0SiiSMwl1SrvdM+MfL8G0NOLU3vNkk6opEso/CXVLnhx/gmmt4ahxMago9z4dlNaMuSiQ7Kdwl4eJtqrH/ChgzClp+DX84Ce5qC+UahhFJGoW7JN1Z8+DZF6Hc4KyL4JVmUVckkv0U7pI0O5TD3VPgjqnwbn3oegF8sUfUVYnkBoW7JMVe38NzY6D9AniiFQzoCGU7Rl2VSO5QuEvCHV0STHOs9z307QzDtDaMSMop3CVx3GHIEKYNg8W14YS+8N6+URclkpsU7rJNKs+I2XkdPPoy9H4fXjsQep2nu01FoqRwl+12wDcwZjQc/jXc2RbuOUl3m4pETeEu26XTxzB8LGzcATpeBBM0zVEkLSjcZZvsUA6DpsDtU6G4YprjQk1zFEkboX55NrMOZjbPzOab2a1xXr/SzD40s1lmNs3Mmie+VEkbpaVMeCYI9qGt4cQ+CnaRdFNluJtZHjAYOBNoDvSIE97Pufvh7n4kcB/wQMIrlfTwzjvQujUnfgmXdYErOmv+ukg6CjMs0waY7+4LAMxsJNAFmPtTA3f/Lqb9roAnskhJvU3Wh3G4qgj+9kowzfH8vjBT0xxF0laYcG8ALIo5LgGOqdzIzK4BbgCqA6cmpDpJCzuvg8fGwyUfwMvN4GJtqiGS9sKMuW+6xF+cnrm7D3b3A4BbgDvivpFZfzMrNrPi0tLSratUItFkJUwfBr0+gN+dAp16KNhFMkGYcC8BGsUcNwSWbKH9SOCceC+4+1B3z3f3/Lp164avUiLRbj68OwSafBus5vj7kzV/XSRThPlRLQKamVlTM6sOdAcKYhuYWezs5rOATxNXoqScO7dOhVeehZLakN9fy/SKZJoqx9zdfYOZDQAmAHnAMHefY2aDgGJ3LwAGmNnpwHpgJdA7mUVLEq1eDZddxp8mwYjDoF9nWFs96qJEZGuFuonJ3QuBwkrPDYx5fF2C65IUqDwjptlyGDsKDlkOv2kPDxxH/CsuIpL2dIeqAMEyAs+MhXV50O5imLJ/1BWJyPbQ5bEcZ+Vw1xQoGAmf7glH9Vewi2QD9dxz2G4/BHubnv0pPHkkXH0W/Ki7TUWygsI9V82eTdHjwTTHqzvCo0ej8XWRLKJwz3KbLCMAdJsNw8ZBzZ2g7aXwVuPU1yUiyaVwzyF5G+EPk+GW6fBWw2CZ3qW1o65KRJJB4Z4j9voeRoyBdgvg0Xy4rgOs16cvkrX0450DWi2BF0fBPmugT2d4snXUFYlIsincs9zFs2DIeFi+C/yqDxQ3iLoiEUkFhXu2Wr8ebryR4S/BlCZwYVcorRl1USKSKgr3DBdvNszeq2H083DSl/DAsfCbdrAxL4LiRCQyCvcsc3RJsD7MHj9Az/NgRMuoKxKRKCjcs0jvmcH4+pJacFw/+GCfqCsSkago3LNAtY3w14lw3TswqSlc0A1WaLckkZymcM9we30fjK+f+oXG10Xkf0KtCmlmHcxsnpnNN7Nb47x+g5nNNbMPzGySme2X+FJlE7NmUTwUjl8El5wDN3ZQsItIoMqeu5nlAYOBdgT7qRaZWYG7z41pNhPId/e1ZnYVcB9wYTIKzkXxZsRc+GGwPkzeLnBiH3hX89dFJEaYnnsbYL67L3D3dQQbYHeJbeDuU9x9bcXhDIJNtCUJdiiHe1+FkWPgvfrB/qYKdhGpLMyYewNgUcxxCXDMFtr3Bf6zPUVJfLv/AM+NgTPnw2NHwa/P1PowIhJfmGiIt8q3x21o1gvIB07ezOv9gf4AjRtrndmtcegyGDcS9vsWrjgbhuZHXZGIpLMwwzIlQKOY44bAksqNzOx04Hags7uXxXsjdx/q7vnunl+3bt1tqTcndf4Y3nkCapXBKZcq2EWkamHCvQhoZmZNzaw60B0oiG1gZq2AIQTBvizxZeao8nK4+27GjYSP6gTj69pYQ0TCqHJYxt03mNkAYAKQBwxz9zlmNggodvcC4C9ATeB5MwP40t07J7HurBJvNkzNMhg+Fs79GJ4+IhiKKdP+piISUqjLce5eCBRWem5gzOPTE1xXTjvgm2B8/eDlwaYafz8G7W8qIltFcy3SzBmfBjsmbTQ442KYvH/UFYlIJgp1h6qkgMPN0+Dl5+DL3eDo/gp2Edl26rmngZ3XwRMF0HM2jGoBfbrA2upRVyUimUzhHrWFC5k+DI74Cm49Df58IhpfF5HtpnBPkXgzYk7+HJ5/HppuhLN7wn8OiqAwEclKGnOPgsOAd+C14VC6C7S5XMEuIomlnnuKVd8Aj46HPrNg3MFw8bmwukbUVYlItlG4p9Deq+HFUXB8CQw6Ce5qC67fnUQkCRTuKdJ6Cbw0Evb8Abp2gzEtoq5IRLKZwj0VRoxg2jBYtiuc0Aferx91QSKS7RTuCRJvNoyVw+8nw2+nQVFj6HoBlNaMoDgRyTkK9ySp9SP860Xo9AkMOQqu1cYaIpJCipskOOAbKBgBB30DV3eER49GNyaJSEop3BPstM9g9PPgBu0vhilaH0ZEIqCJeInicO0MeOVZWFwbjr5cwS4i0VHPPRHKyni8APrNhJcOhovPgzU7RV2UiOSyUOFuZh2Ahwh2YnrC3e+t9PpJwN+AlkB3d38h0YWmg3gzYuqtgTGjoN8iuOckuLOtbkwSkehVGe5mlgcMBtoRbJZdZGYF7j43ptmXwKXATckoMl21qrgxqc5auKArPH9Y1BWJiATC9NzbAPPdfQGAmY0EugA/h7u7f1HxWnkSakxL3WbDUy/B8l3ghL4wSzcmiUgaCTOA0ABYFHNcUvHcVjOz/mZWbGbFpaWl2/IWkbNyuGcSjH4B3qsf7JikYBeRdBMm3OPN0PZt+WLuPtTd8909v27dutvyFpGqWRYs/HXHVHiiFZzWG5bpjlMRSUNhhmVKgEYxxw2BJckpJ40tWMDbT8Ahy4O7TR9pg25MEpG0FSbci4BmZtYUWAx0B3omtaoIxZsR0/ZzeGE07OvQoRdMOiCCwkREtkKVwzLuvgEYAEwAPgJGu/scMxtkZp0BzOxoMysBugFDzGxOMotOGYer/wuvDoevagY7JinYRSQThJrn7u6FQGGl5wbGPC4iGK7JGjtugEcKof97UHAQ9DpPOyaJSObQHapx1F0DY0bDr76EP/wKfneKbkwSkcyicK/kiKUwbiTU+x56nA8jD4+6IhGRradwj/XCC0wfBitrwIl94L19oy5IRGTbaLABoLwc7rwTunXj/b0hv7+CXUQyW8713CtPddy1DIaPhfM+hiePhCvPhnU597ciItkmp2OsyUoYNwJalML/nQEPHYtuTBKRrJCz4X5yxY1JeQ5nXgSvHhh1RSIiiZOTY+5XFsGrz0DprsGNSQp2Eck2udVzX7+ef4yHq4rh5WbQ83z4TjcmiUgWyp2ee2kptGvHVcVw7wnQuYeCXUSyV1b23CvPiDn8q+DGpPqroe958FzLiAoTEUmRrAz3WOfODaY6rqoBv+oDxdu0zYiISGbJ2mEZK4eBr8OLo2F2veDGJAW7iOSKrOy577IOnh4LXT+Cp4+AK86Gsh2jrkpEJHWyL9wXLmT6P+HwZXBDe3jwOHRjkojknFDDMmbWwczmmdl8M7s1zus7mdmoitffMbMmiS40lKlTIT+fJt/CWT3hweNRsItITqqy525mecBgoB3BfqpFZlbg7nNjmvUFVrr7gWbWHfgzcGEyCob4W+FdXgyDC2HBHtD5cvikTrK+uohI+gvTc28DzHf3Be6+DhgJdKnUpgvwdMXjF4DTzCwlfeZqG+Hhl2HoeHhtfzimn4JdRCTMmHsDYFHMcQlwzObauPsGM1sF7AUsT0SRm+Xw7+egw2fwl+Ph1tOhPGvn/4iIhBcm3OP1wH0b2mBm/YH+FYdrzGxeiK+/RWcG/6nDWyznre19t4xQh2T/o5lecul8c+lcIbfO9xfnandt18DGfmEahQn3EqBRzHFDYMlm2pSYWTVgN2BF5Tdy96HA0DCFbQ0zK3b3/ES/bzrKpXOF3DrfXDpXyK3zjeJcwwxiFAHNzKypmVUHugMFldoUAL0rHncFJrv7Jj13ERFJjSp77hVj6AOACUAeMMzd55jZIKDY3QuAfwLPmNl8gh5792QWLSIiWxbqJiZ3LwQKKz03MObxj0C3xJa2VRI+1JPGculcIbfON5fOFXLrfFN+rqbRExGR7KOJgyIiWSijwj1jlkFIgBDneoOZzTWzD8xskpmFmh6Vrqo635h2Xc3MzSxjZ1mEOVczu6Di851jZs+lusZECfF93NjMppjZzIrv5Y5R1JkIZjbMzJaZ2ezNvG5m9veKv4sPzKx1Ugty94z4Q3Ax9zNgf6A68D7QvFKbq4HHKh53B0ZFXXcSz/UUYJeKx1dl6rmGPd+KdrWAN4EZQH7UdSfxs20GzAT2qDiuF3XdSTzXocBVFY+bA19EXfd2nO9JQGtg9mZe7wj8h+C+oGOBd5JZTyb13NN6GYQEq/Jc3X2Ku6+tOJxBcP9Bpgrz2QLcA9wH/JjK4hIszLleDgx295UA7r4sxTUmSphzdaB2xePd2PQemozh7m8S5/6eGF2A4R6YAexuZvWTVU8mhXu8ZRAqb7/xi2UQgJ+WQcg0Yc41Vl+CHkGmqvJ8zawV0Mjdx6eysCQI89keBBxkZtPNbIaZdUhZdYkV5lzvAnqZWQnBjLxrU1NaJLb253q7ZNJ67glbBiEDhD4PM+sF5AMnJ7Wi5Nri+ZrZDsCDwKWpKiiJwny21QiGZtoS/EY21cwOc/dvk1xbooU51x7AU+5+v5kdR3C/zGHuXp788lIupfmUST33rVkGgS0tg5ABwpwrZnY6cDvQ2d3LUlRbMlR1vrWAw4DXzewLgvHKggy9qBr2+3icu6/bQKzwAAABKUlEQVR398+BeQRhn2nCnGtfYDSAu78N1CBYhyUbhfq5TpRMCvdcWgahynOtGKYYQhDsmTom+5Mtnq+7r3L3Ou7exN2bEFxj6OzuxdGUu13CfB+/RHDBHDOrQzBMsyClVSZGmHP9EjgNwMwOJQj30pRWmToFwCUVs2aOBVa5+9KkfbWorzBv5dXojsAnBFfgb694bhDBDzoE3xjPA/OB/wL7R11zEs/1NeBrYFbFn4Koa07m+VZq+zoZOlsm5GdrwAPAXOBDoHvUNSfxXJsD0wlm0swC2kdd83ac6whgKbCeoJfeF7gSuDLmcx1c8XfxYbK/h3WHqohIFsqkYRkREQlJ4S4ikoUU7iIiWUjhLiKShRTuIiJZSOEuIpKFFO4iIllI4S4ikoX+H0p/Dg0JmfxNAAAAAElFTkSuQmCC",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa7551c7dd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "height = sin(x+pas/2)\n",
    "width= 1/(NB-1)\n",
    "plt.bar(x,height, width, color='g')\n",
    "plot(xfin,hei,'r')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Le résultat sera d'autant plus probant que la largeur des batonnets sera petite, c'est à dire pour $n$ grand. La surface du rectangle correspond à l'intégration de la fonction constante (hauteur du batonnet) sur l'intervalle qui correspond à la largeur du batonnet. Pour construire ces batonnets, on introduit une subdivision (ou discrétisation) du segment $[a,b]$ comme par exemple\n",
    "$$\n",
    "x_0=a,~x_1=a+h,\\cdots,x_i=a+ih,\\cdots, x_n=a+nh=b,\n",
    "$$\n",
    "avec $h=\\frac {b-a}{n}$ la largeur du batonnet. On obtient alors la formule de quadrature suivante (on choisit ici $f(x_k)$ comme hauteur du batonnet sur $[x_k,x_{k+1}]$)\n",
    "$$\n",
    "F=\\sum_{k=0}^{n-1} h f(x_k)\\sim \\int_a^b f(x)\\, dx.\n",
    "$$\n",
    "On appellera cette formule la formule des rectangles à gauche car la hauteur du rectangle est évaluée par $f$ à gauche de l'intervalle considéré. \n",
    "### Une construction plus élaborée\n",
    "Plutôt que d'approcher la fonction $f$ par une fonction constante par morceaux, nous allons approcher $f$ par des polynômes par morceaux car nous savons intégrer les polynômes:\n",
    "$$\n",
    "F=\\sum_{k=0}^{n-1} \\int_{x_k}^{x_{k+1}} p_{[x_k,x_{k+1}]}(x) \\,dx\\sim \\int_a^b f(x)\\, dx.\n",
    "$$\n",
    "On pourra choisir pour le polynôme $p_{[x_k,x_{k+1}]}$ un polynôme de Lagrange d'interpolation de $f$ sur $[x_k,x_{k+1}]$. Une interpolation à 1 point induit une méthode des rectangles (polynôme constant sur $[x_k,x_{k+1}]$).\n",
    "\n",
    "Une interpolation à 2 points, par exemple les points $x_k$ et $x_{k+1}$, permet d'avoir une interpolation affine et l'integration d'une telle fonction revient à calculer l'aire d'un trapèze."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot([0,0.5],[sin(0),sin(0.5)])\n",
    "plot([0.5,1],[sin(0.5),sin(1)])\n",
    "x=linspace(0,1,100)\n",
    "plot(x,sin(x))\n",
    "plot([1,1],[0,sin(1)],'-.')\n",
    "plot([0.5,0.5],[0,sin(0.5)],'-.')\n",
    "grid();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On observe que cette expression représente une formule de calcul d'aire d'un trapèze, de largeur $h$ et de hauteurs $f(x_k)$ et $f(x_{k+1})$. La hauteur moyenne est donc $\\frac {f(x_k)+f(x_{k+1})}2$. L'aire du trapèze qui approche l'aire sous la graphe est donc $h\\frac {f(x_k)+f(x_{k+1})}2$.\n",
    "\n",
    "Dans le cas d'une interpolation de Lagrange à plus de points, la construction est plus technique mais elle aboutit car on sait toujours intégrer un polynôme: \n",
    "$$\\int_{x_k}^{x_{k+1}} f(x) \\,dx\\sim \\int_{x_k}^{x_{k+1}} p_{[x_k,x_{k+1}]}(x) \\,dx.$$ \n",
    "Si on note $(y_m)_{0\\le m< M}$ les $M$ points de $[x_k,x_{k+1}]$ qui définissent l'interpolation de Lagrange, on a:\n",
    "$$\n",
    "p_{[x_k,x_{k+1}]}(x)=\\sum_{l=0}^{M-1}f(y_m)l_m(x),\n",
    "$$\n",
    "$$\\int_{x_k}^{x_{k+1}} f(x) \\,dx\\sim \\sum_{l=0}^{M-1}f(y_m)\\int_{x_k}^{x_{k+1}}l_m(x)\\,dx.\n",
    "$$\n",
    "Ensuite, il reste à produire la sommation de ces intégrales de polynôme. Ce procédé itératif se programme aisément."
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