{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "7820d9ef",
   "metadata": {},
   "outputs": [],
   "source": [
    "# importing libraries used in this book\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "from NonLinearEquationsLib import plotPhi, FixedPoint, plotPhiIterations"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "db6d1492",
   "metadata": {},
   "source": [
    "On considère les méthodes de point fixe $x^{(n+1)} = g_i(x^{(n)})\n",
    "\\quad (i=1,2,3)$ avec:\n",
    "\n",
    "$$g_1(x^{(n)}) = \\dfrac{1}{2} e^{x^{(n)}/2} , \\qquad g_2(x^{(n)}) = -\\dfrac{1}{2}\n",
    "  e^{x^{(n)}/2}, \\qquad g_3(x^{(n)}) = 2 \\ln(2x^{(n)}),$$\n",
    "dont les fonctions d'itération $g_i(x)$ sont visualisées sur la figure plus bas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5a067c29",
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.rcParams['figure.figsize'] = [20, 5]\n",
    "\n",
    "plt.subplot(1,3,1)\n",
    "[a,b] = [-2,5]\n",
    "phi1 = lambda x : np.exp(x/2)/2\n",
    "plotPhi(a,b,phi1,'$g_1$')\n",
    "\n",
    "plt.subplot(1,3,2)\n",
    "[a,b] = [-2,5]\n",
    "phi2 = lambda x : - np.exp(x/2)/2\n",
    "plotPhi(a,b,phi2,'$g_2$')\n",
    "\n",
    "plt.subplot(1,3,3)\n",
    "[a,b] = [1e-1,5]\n",
    "phi3 = lambda x : 2*np.log(2*x)\n",
    "plotPhi(a,b,phi3,'$g_3$')\n",
    "\n",
    "plt.show()\n",
    "\n",
    "# Graph of the fonction $f(x)=e^x-4x^2$\n",
    "N = 100\n",
    "z = np.linspace(a,b,N)\n",
    "f = lambda x : np.exp(x) - 4*x*x\n",
    "\n",
    "plt.subplot(1,3,1)\n",
    "plt.plot(z,f(z),'k-')\n",
    "\n",
    "plt.xlabel('x'); plt.ylabel('f(x)');\n",
    "# Plot the x,y-axis \n",
    "plt.plot([a,b], [0,0], 'k-',linewidth=0.1)\n",
    "plt.plot([0,0], [np.min(f(z)),np.max(f(z))], 'k-',linewidth=0.1)\n",
    "plt.legend(['f(x)'])\n",
    "plt.title('Graph of $f(x)=e^x-4x^2$')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "72db30e2",
   "metadata": {},
   "source": [
    "**Partie 1**\n",
    "*Pour chaque point fixe $\\bar{x}$ de la fonction d'itération $g_i$ ($i=1,2,3$), on suppose choisir une valeur \n",
    "initiale $x^{(0)}$ proche de $\\bar{x}$. Etudiez si la méthode converge vers $\\bar{x}$.*\n",
    "\n",
    "On va utiliser la fonction `FixedPoint` qui se trouve dans `NonLinearEquationsLib.py`\n",
    "```python\n",
    "def  FixedPoint( phi, x0, a, b tol, nmax ) :\n",
    "    '''\n",
    "    FixedPoint Find the fixed point of a function by iterative iterations\n",
    "    FixedPoint( PHI,X0,a,b, TOL,NMAX) tries to find the fixedpoint X of the a\n",
    "    continuous function PHI nearest to X0 using \n",
    "    the fixed point iterations method. \n",
    "    [a,b] : if the iterations exit the interval, the method stops\n",
    "    If the search fails an error message is displayed.\n",
    "     \n",
    "    Outputs : [x, r, n, inc, x_sequence]\n",
    "    x : the approximated fixed point of the function\n",
    "    r : the absolute value of the residual in X : |phi(x) - x|\n",
    "    n : the number of iterations N required for computing X and\n",
    "    x_sequence : the sequence computed by Newton\n",
    "    \n",
    "    ...\n",
    "    return xk1, rk1, n, np.array(x)    \n",
    "    '''\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "dac0c670",
   "metadata": {},
   "outputs": [],
   "source": [
    "tol  = 1e-2\n",
    "nmax = 10\n",
    "\n",
    "# Choose fonction phi\n",
    "[a,b] = [-2,5]\n",
    "phi = phi1\n",
    "label = '$phi_1$'\n",
    "\n",
    "# Initial Point\n",
    "x0   = 3\n",
    "zero, residual, niter, x = FixedPoint(phi, x0, a,b, tol, nmax)\n",
    "\n",
    "plt.subplot(131)\n",
    "plotPhi (a,b,phi,label)\n",
    "plt.plot(x,phi(x), 'rx')\n",
    "\n",
    "# plot the graphical interpretation of the Fixed Point method\n",
    "plotPhiIterations(x)\n",
    "\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c1e6f036",
   "metadata": {},
   "source": [
    "Avec $\\phi_1$, la méthode de point fixe converge. Que se passe-t-il avec $\\phi_2$ et $\\phi_3$ ?\n",
    "Esssayez avec Python :\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "dd28a994",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Réutilisez le code plus haut et observez se qu'il se passe pour \n",
    "# phi = phi1 et phi2\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "fad5fc0c",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9711e211",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "7fccf249",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.11.5"
  },
  "unianalytics_cell_mapping": [
   [
    "7820d9ef",
    "7820d9ef"
   ],
   [
    "db6d1492",
    "db6d1492"
   ],
   [
    "5a067c29",
    "5a067c29"
   ],
   [
    "72db30e2",
    "72db30e2"
   ],
   [
    "dac0c670",
    "dac0c670"
   ],
   [
    "c1e6f036",
    "c1e6f036"
   ],
   [
    "dd28a994",
    "dd28a994"
   ],
   [
    "fad5fc0c",
    "fad5fc0c"
   ],
   [
    "9711e211",
    "9711e211"
   ],
   [
    "7fccf249",
    "7fccf249"
   ]
  ],
  "unianalytics_notebook_id": "99def5d6-d70b-4453-b608-8b79bcec7155"
 },
 "nbformat": 4,
 "nbformat_minor": 5
}