Solving-non-linear-equations.ipynb

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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "\n",
    "from bokeh.io import show, output_notebook\n",
    "from bokeh.plotting import figure\n",
    "from bokeh.layouts import row\n",
    "output_notebook(hide_banner=True)\n",
    "\n",
    "from scipy.optimize import newton"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Méthode de Newton"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Implémentation de la méthode"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 185,
   "metadata": {},
   "outputs": [],
   "source": [
    "def my_newton(f, df, x0, xstar, tol=1.e-12, nitmax=50):\n",
    "   \n",
    "    # initialisation\n",
    "    x = np.zeros(nitmax+1)\n",
    "    x[0] = x0\n",
    "    \n",
    "    # iteration de Newton        \n",
    "    for i in range(1, nitmax+1):\n",
    "        x[i] = x[i-1] - f(x[i-1])/df(x[i-1])\n",
    "        #if ( abs(x[i] - xstar) < tol) : break\n",
    "        if ( abs(f(x[i])) < tol ): break\n",
    "        if (abs(x[i]-x[i-1]) < tol) : break\n",
    "\n",
    "    return x[0:i+1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Ordre de convergence"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Image élément par élément d'un vecteur par une fonction de variable réelle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 186,
   "metadata": {},
   "outputs": [],
   "source": [
    "def elt_par_elt(f,x) :\n",
    "    y = np.zeros(x.size)\n",
    "    for i in range(x.size) :\n",
    "        y[i] = f(x[i])\n",
    "    return y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Test de la fonction : $f(x) = e^x - 2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 187,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return np.exp(x) - 2\n",
    "\n",
    "def df(x):\n",
    "    return np.exp(x) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 190,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Convergence de l'algorithme de Newton en 5 itérations\n",
      "--> solution de f(x) = 0 obtenue pour  x = 0.6931471805600254\n"
     ]
    },
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Application\",\"version\":\"1.3.4\"}};\n",
       "  var render_items = [{\"docid\":\"0d5ff790-d9a2-4a9a-9810-a63582c5ac2a\",\"roots\":{\"151018\":\"224859e7-248d-40aa-b3ef-a1b0a92da531\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "151018"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xstar = np.log(2)\n",
    "xsol = my_newton(f, df, x0=0., xstar=xstar, tol=1.e-12, nitmax=50)\n",
    "\n",
    "print(f\"Convergence de l'algorithme de Newton en {xsol.size-1} itérations\")\n",
    "print(f\"--> solution de f(x) = 0 obtenue pour  x = {xsol[-1]}\")\n",
    "\n",
    "err = np.abs(xsol - xstar)\n",
    "im = elt_par_elt(f,xsol)\n",
    "\n",
    "fig1 = figure(width=490, height=300, y_axis_type=\"log\")\n",
    "fig1.x(range(xsol.size), err, size=5)\n",
    "\n",
    "fig2 = figure(width=490, height=300, x_axis_type=\"log\", y_axis_type=\"log\")\n",
    "fig2.x(err[:-1], err[1:], size=10) #err(n+1) en fctn de err(n)\n",
    "fig2.x(im[:-1], im[1:], size=10, color=\"orange\", legend=\"erreur résiduelle\") #f(x_n+1) en fctn de f(x_n)\n",
    "fig2.line(err[:-1], err[:-1],  color=\"yellow\", legend=\"ordre 1\")\n",
    "fig2.line(err[:-1], err[:-1]*err[:-1],  color=\"red\", legend=\"ordre 2\")\n",
    "fig2.line(err[:-1], err[:-1]*err[:-1]*err[:-1],  color=\"green\", legend=\"ordre 3\")\n",
    "\n",
    "show(row(fig1, fig2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Test de la fonction : $g(x) = \\displaystyle \\frac{x}{\\sqrt{1+x^2}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 191,
   "metadata": {},
   "outputs": [],
   "source": [
    "def g(x):\n",
    "    return x/(np.sqrt(1+x*x))\n",
    "\n",
    "def dg(x):\n",
    "    return 1/(np.power(1+x*x, 1/3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 193,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Convergence de l'algorithme de Newton en 3 itérations\n",
      "--> solution de g(x) = 0 obtenue pour  x = 1.73429995950318e-19\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"da311eab-32ec-44c5-917e-e325cc6e7515\" data-root-id=\"155858\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
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    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
       "  var docs_json = 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Application\",\"version\":\"1.3.4\"}};\n",
       "  var render_items = [{\"docid\":\"45960696-5d9e-48b0-8814-fdc8e0c8ee0b\",\"roots\":{\"155858\":\"da311eab-32ec-44c5-917e-e325cc6e7515\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "155858"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xstar = 0.\n",
    "xsol = my_newton(g, dg, x0=0.5, xstar=xstar, tol=1.e-12, nitmax=50)\n",
    "print(f\"Convergence de l'algorithme de Newton en {xsol.size-1} itérations\")\n",
    "print(f\"--> solution de g(x) = 0 obtenue pour  x = {xsol[-1]}\")\n",
    "\n",
    "err = np.abs(xsol - xstar)\n",
    "im = elt_par_elt(g,xsol)\n",
    "\n",
    "fig1 = figure(width=490, height=300, y_axis_type=\"log\")\n",
    "fig1.x(range(xsol.size), err, size=5)\n",
    "\n",
    "fig2 = figure(width=490, height=300, x_axis_type=\"log\", y_axis_type=\"log\")\n",
    "fig2.x(err[:-1], err[1:], size=10) #err(n+1) en fctn de err(n)\n",
    "fig2.x(im[:-1], im[1:], size=5, color=\"orange\", legend=\"erreur résiduelle\") #g(x_n+1) en fctn de g(x_n)\n",
    "\n",
    "fig2.line(err[:-1], err[:-1],  color=\"yellow\", legend=\"ordre 1\")\n",
    "fig2.line(err[:-1], err[:-1]*err[:-1],  color=\"red\", legend=\"ordre 2\")\n",
    "fig2.line(err[:-1], err[:-1]*err[:-1]*err[:-1],  color=\"green\", legend=\"ordre 3\")\n",
    "\n",
    "show(row(fig1, fig2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Test de la fonction : $h(x) = x^3$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 194,
   "metadata": {},
   "outputs": [],
   "source": [
    "def h(x):\n",
    "    return x*x*x\n",
    "\n",
    "def dh(x):\n",
    "    return 3*x*x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 195,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Convergence de l'algorithme de Newton en 22 itérations\n",
      "--> solution de h(x) = 0 obtenue pour  x = 6.682859107149278e-05\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"7e2d415c-708b-4979-8b75-403868cc0431\" data-root-id=\"158317\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
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       "  var render_items = [{\"docid\":\"77076ded-c624-4c3f-b39c-7963d545c0d6\",\"roots\":{\"158317\":\"7e2d415c-708b-4979-8b75-403868cc0431\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
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   "source": [
    "xstar = 0.\n",
    "xsol = my_newton(h, dh, x0=0.5, xstar=xstar, tol=1.e-12, nitmax=100)\n",
    "print(f\"Convergence de l'algorithme de Newton en {xsol.size-1} itérations\")\n",
    "print(f\"--> solution de h(x) = 0 obtenue pour  x = {xsol[-1]}\")\n",
    "\n",
    "err = np.abs(xsol - xstar)\n",
    "im = elt_par_elt(h,xsol)\n",
    "\n",
    "fig1 = figure(width=490, height=300, y_axis_type=\"log\")\n",
    "fig1.x(range(xsol.size), err, size=5)\n",
    "\n",
    "fig2 = figure(width=490, height=300, x_axis_type=\"log\", y_axis_type=\"log\")\n",
    "fig2.x(err[:-1], err[1:], size=10) #err(n+1) en fctn de err(n)\n",
    "fig2.x(im[:-1], im[1:], size=10, color=\"orange\",legend=\"erreur résiduelle\") #h(x_n+1) en fctn de h(x_n)\n",
    "\n",
    "fig2.line(err[:-1], err[:-1],  color=\"yellow\", legend=\"ordre 1\")\n",
    "fig2.line(err[:-1], err[:-1]*err[:-1],  color=\"red\", legend=\"ordre 2\")\n",
    "fig2.line(err[:-1], err[:-1]*err[:-1]*err[:-1],  color=\"green\", legend=\"ordre 3\")\n",
    "\n",
    "\n",
    "\n",
    "show(row(fig1, fig2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commenter ces résulats et observer également l'erreur résiduelle.** "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "J'ai défini au début de cette partie une fonction elt_par_elt qui prend en argument une fonction $f$ à variable réelle et un vecteur $X$, et renvoie le vecteur des images des éléments de $X$ par $f$. (pour observer l'erreur résiduelle)\n",
    "\n",
    "Ensuite j'ai représenté sur la figure en échelle log-log chacune des droites correspondant aux ordres 1, 2 et 3, ainsi que l'érreur résiduelle (croix oranges).\n",
    "\n",
    "Cette échelle nous permet bien de vérifier l'ordre de la méthode utilisée car on a\n",
    "$\\epsilon_{n+1} \\leq C\\epsilon_n^{\\mu}$, et par conséquent $log(\\epsilon_{n+1}) \\leq log(C)+\\mu .log(\\epsilon_n)$.\n",
    "L'ordre devient alors la pente de la courbe.\n",
    "\n",
    "Les courbes obtenues permettent d'affirmer les résultats théoriques : les ordres de $f$, $g$ et $h$ sont respéctivement 2, 3 et 1. Et on remarque de plus que pour $f$ et $g$, ayant des dérivées non nulles en leur zéros, l'erreur résiduelle a le même ordre de convergence que $\\epsilon_n$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Comparer ces résultats avec la fonction newton de Scipy (https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.newton.html)**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 196,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "pour la fonction f :  (0.6931471805599453,       converged: True\n",
      "           flag: 'converged'\n",
      " function_calls: 12\n",
      "     iterations: 6\n",
      "           root: 0.6931471805599453) \n",
      "\n",
      "pour la fonction f :  (0.0,       converged: True\n",
      "           flag: 'converged'\n",
      " function_calls: 8\n",
      "     iterations: 4\n",
      "           root: 0.0) \n",
      "\n",
      "pour la fonction f :  (2.0097272630703606e-08,       converged: True\n",
      "           flag: 'converged'\n",
      " function_calls: 84\n",
      "     iterations: 42\n",
      "           root: 2.0097272630703606e-08) \n",
      "\n"
     ]
    }
   ],
   "source": [
    "xsol = newton(f, x0=0, fprime=df, args=(), tol=1.e-12, maxiter=50, fprime2=None, x1=None, rtol=0.0, full_output=True, disp=True)\n",
    "print( \"pour la fonction f : \",xsol,\"\\n\")\n",
    "\n",
    "xsol = newton(g, x0=0.5, fprime=dg, args=(), tol=1.e-12, maxiter=50, fprime2=None, x1=None, rtol=0.0, full_output=True, disp=True)\n",
    "print( \"pour la fonction f : \",xsol,\"\\n\")\n",
    "\n",
    "xsol = newton(h, x0=0.5, fprime=dh, args=(), tol=1.48e-08, maxiter=50, fprime2=None, x1=None, rtol=0.0, full_output=True, disp=True)\n",
    "print( \"pour la fonction f : \",xsol,\"\\n\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Les nombres d'itérations de my_newton() et de numpy.scipy.newton() sont du même ordre de grandeur, ce qui montre que les ordres de convergence obtenus par les deux fonctions pour chacune des fonctions $f$, $g$ et $h$ sont les mêmes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Cas dégénérés"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Implémenter l'algorithme de Newton modifié sous la forme $x_{n+1} = x_n-m\\frac{f(x_n)}{f'(x_n)} $**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 197,
   "metadata": {},
   "outputs": [],
   "source": [
    "def my_newton_modif(f, df, m, x0, xstar, tol=1.e-12, nitmax=50):\n",
    "   \n",
    "    # initialisation\n",
    "    x = np.zeros(nitmax+1)\n",
    "    x[0] = x0\n",
    "    \n",
    "    # iteration du Newton modifié       \n",
    "    for i in range(1, nitmax+1):\n",
    "        x[i] = x[i-1] - m*f(x[i-1])/df(x[i-1])\n",
    "        if ( abs(x[i] - xstar) < tol) : break\n",
    "        if ( abs(f(x[i])) < tol ): break\n",
    "        if (abs(x[i]-x[i-1]) < tol) : break\n",
    "\n",
    "    return x[0:i+1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Vérifier numériquement le taux de convergence de la méthode de Newton modifié pour la racine de la fonction : $k(x) = x^2(x^2+2)$. Commenter**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 198,
   "metadata": {},
   "outputs": [],
   "source": [
    "def k(x):\n",
    "    return (x*x)*(x*x+2)\n",
    "\n",
    "def dk(x):\n",
    "    return 4*x*x*x + 4*x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 200,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Convergence de l'algorithme de Newton en 3 itérations\n",
      "--> solution de k(x) = 0 obtenue pour  x = 1.2115933527723916e-13\n"
     ]
    },
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r\\u00e9siduelle\"},\"renderers\":[{\"id\":\"163233\",\"type\":\"GlyphRenderer\"}]},\"id\":\"163245\",\"type\":\"LegendItem\"},{\"attributes\":{},\"id\":\"163166\",\"type\":\"BasicTicker\"}],\"root_ids\":[\"163303\"]},\"title\":\"Bokeh Application\",\"version\":\"1.3.4\"}};\n",
       "  var render_items = [{\"docid\":\"09e97b8f-abd5-4da6-8ccf-c45c0308b864\",\"roots\":{\"163303\":\"0029efe3-3426-4b8b-808e-09c2ee9b9324\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "163303"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xstar = 0.\n",
    "xsol = my_newton_modif(k, dk, 2, x0=0.5, xstar=xstar, tol=1.e-15, nitmax=50)\n",
    "print(f\"Convergence de l'algorithme de Newton en {xsol.size-1} itérations\")\n",
    "print(f\"--> solution de k(x) = 0 obtenue pour  x = {xsol[-1]}\")\n",
    "\n",
    "err = np.abs(xsol - xstar)\n",
    "im = elt_par_elt(k,xsol)\n",
    "\n",
    "fig1 = figure(width=490, height=300, y_axis_type=\"log\")\n",
    "fig1.x(range(xsol.size), err, size=5)\n",
    "\n",
    "fig2 = figure(width=490, height=300, x_axis_type=\"log\", y_axis_type=\"log\")\n",
    "fig2.x(err[:-1], err[1:], size=10) #err(n+1) en fctn de err(n)\n",
    "fig2.x(im[:-1], im[1:], size=5, color=\"orange\",legend=\"erreur résiduelle\") #g(x_n+1) en fctn de g(x_n)\n",
    "\n",
    "fig2.line(err[:-1], err[:-1],  color=\"yellow\", legend=\"ordre 1\")\n",
    "fig2.line(err[:-1], err[:-1]*err[:-1],  color=\"red\", legend=\"ordre 2\")\n",
    "fig2.line(err[:-1], err[:-1]*err[:-1]*err[:-1],  color=\"green\", legend=\"ordre 3\")\n",
    "\n",
    "show(row(fig1, fig2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La convergence obtenue est d'ordre 3 ! et donc mieux même qu'une convergence quadratique, ce qui affirme le résultat théorique."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Méthode de la sécante"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Implémentation de la méthode"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 209,
   "metadata": {},
   "outputs": [],
   "source": [
    "def taux_accroissement(f,x,y) :\n",
    "    return (f(x)-f(y))/(x-y)\n",
    "\n",
    "def my_secante(f, x0, x1, xstar, tol=1.e-12, nitmax=50):\n",
    "   \n",
    "    # initialisation\n",
    "    x = np.zeros(nitmax+2)\n",
    "    x[0] = x0\n",
    "    x[1] = x1\n",
    "    \n",
    "    # itération de la methode de la secante\n",
    "    for i in range(2,nitmax+2) :\n",
    "        x[i] = x[i-1] - f(x[i-1])/taux_accroissement(f,x[i-1],x[i-2])\n",
    "        if abs(x[i]-xstar)<tol : break\n",
    "    \n",
    "    return x[0:i+1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Vérifier si la méthode de la sécante a un ordre de convergence d'au moins $\\rho  = \\displaystyle \\frac{\\sqrt{5}+1}{2}$ pour les fonctions $f(x)$, $g(x)$ et $h(x)$. Commenter.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Test de la fonction $f$ :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 211,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Convergence de la méthode de la sécante en 5 itérations\n",
      "--> solution de f(x) = 0 obtenue pour  x = 0.6931471805599452\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"7e5b79f3-3144-4ed8-9356-889a7861b540\" data-root-id=\"184905\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
       "  var docs_json = {\"6fe6ac2d-3f18-4fd6-8b31-3abc8f25c7f0\":{\"roots\":{\"references\":[{\"attributes\":{\"below\":[{\"id\":\"184914\",\"type\":\"LogAxis\"}],\"center\":[{\"id\":\"184918\",\"type\":\"Grid\"},{\"id\":\"184923\",\"type\":\"Grid\"},{\"id\":\"184956\",\"type\":\"Legend\"}],\"left\":[{\"id\":\"184919\",\"type\":\"LogAxis\"}],\"plot_height\":300,\"plot_width\":490,\"renderers\":[{\"id\":\"184940\",\"type\":\"GlyphRenderer\"},{\"id\":\"184945\",\"type\":\"GlyphRenderer\"}],\"title\":{\"id\":\"184947\",\"type\":\"Title\"},\"toolbar\":{\"id\":\"184930\",\"type\":\"Toolbar\"},\"x_range\":{\"id\":\"184906\",\"type\":\"DataRange1d\"},\"x_scale\":{\"id\":\"184910\",\"type\":\"LogScale\"},\"y_range\":{\"id\":\"184908\",\"type\":\"DataRange1d\"},\"y_scale\":{\"id\":\"184912\",\"type\":\"LogScale\"}},\"id\":\"184905\",\"subtype\":\"Figure\",\"type\":\"Plot\"},{\"attributes\":{\"callback\":null,\"data\":{\"x\":{\"__ndarray__\":\"4DU4Xn7Ytz+AOxa2sxF8PwBA7NRVNzU/AADgQPiWsj4AAAAAyKboPQ==\",\"dtype\":\"float64\",\"shape\":[5]},\"y\":{\"__ndarray__\":\"gDsWtrMRfD8AQOzUVTc1PwAA4ED4lrI+AAAAAMim6D0AAAAAAACgPA==\",\"dtype\":\"float64\",\"shape\":[5]}},\"selected\":{\"id\":\"184953\",\"type\":\"Selection\"},\"selection_policy\":{\"id\":\"184954\",\"type\":\"UnionRenderers\"}},\"id\":\"184937\",\"type\":\"ColumnDataSource\"},{\"attributes\":{},\"id\":\"187288\",\"type\":\"UnionRenderers\"},{\"attributes\":{\"data_source\":{\"id\":\"184937\",\"type\":\"ColumnDataSource\"},\"glyph\":{\"id\":\"184938\",\"type\":\"X\"},\"hover_glyph\":null,\"muted_glyph\":null,\"nonselection_glyph\":{\"id\":\"184939\",\"type\":\"X\"},\"selection_glyph\":null,\"view\":{\"id\":\"184941\",\"type\":\"CDSView\"}},\"id\":\"184940\",\"type\":\"GlyphRenderer\"},{\"attributes\":{\"formatter\":{\"id\":\"184949\",\"type\":\"LogTickFormatter\"},\"ticker\":{\"id\":\"184915\",\"type\":\"LogTicker\"}},\"id\":\"184914\",\"type\":\"LogAxis\"},{\"attributes\":{\"text\":\"\"},\"id\":\"184947\",\"type\":\"Title\"},{\"attributes\":{\"active_drag\":\"auto\",\"active_inspect\":\"auto\",\"active_multi\":null,\"active_scroll\":\"auto\",\"active_tap\":\"auto\",\"tools\":[{\"id\":\"184924\",\"type\":\"PanTool\"},{\"id\":\"184925\",\"type\":\"WheelZoomTool\"},{\"id\":\"184926\",\"type\":\"BoxZoomTool\"},{\"id\":\"184927\",\"type\":\"SaveTool\"},{\"id\":\"184928\",\"type\":\"ResetTool\"},{\"id\":\"184929\",\"type\":\"HelpTool\"}]},\"id\":\"184930\",\"type\":\"Toolbar\"},{\"attributes\":{\"fill_color\":{\"value\":\"#1f77b4\"},\"line_color\":{\"value\":\"#1f77b4\"},\"size\":{\"units\":\"screen\",\"value\":5},\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"184938\",\"type\":\"X\"},{\"attributes\":{\"dimension\":1,\"ticker\":{\"id\":\"184920\",\"type\":\"LogTicker\"}},\"id\":\"184923\",\"type\":\"Grid\"},{\"attributes\":{},\"id\":\"184910\",\"type\":\"LogScale\"},{\"attributes\":{\"formatter\":{\"id\":\"184951\",\"type\":\"LogTickFormatter\"},\"ticker\":{\"id\":\"184920\",\"type\":\"LogTicker\"}},\"id\":\"184919\",\"type\":\"LogAxis\"},{\"attributes\":{},\"id\":\"184953\",\"type\":\"Selection\"},{\"attributes\":{\"callback\":null},\"id\":\"184906\",\"type\":\"DataRange1d\"},{\"attributes\":{\"num_minor_ticks\":10},\"id\":\"184920\",\"type\":\"LogTicker\"},{\"attributes\":{\"source\":{\"id\":\"184937\",\"type\":\"ColumnDataSource\"}},\"id\":\"184941\",\"type\":\"CDSView\"},{\"attributes\":{},\"id\":\"187287\",\"type\":\"Selection\"},{\"attributes\":{\"line_color\":\"green\",\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"184943\",\"type\":\"Line\"},{\"attributes\":{\"label\":{\"value\":\"ordre rho\"},\"renderers\":[{\"id\":\"184945\",\"type\":\"GlyphRenderer\"}]},\"id\":\"184957\",\"type\":\"LegendItem\"},{\"attributes\":{\"callback\":null,\"data\":{\"x\":{\"__ndarray__\":\"4DU4Xn7Ytz+AOxa2sxF8PwBA7NRVNzU/AADgQPiWsj4AAAAAyKboPQ==\",\"dtype\":\"float64\",\"shape\":[5]},\"y\":{\"__ndarray__\":\"phtoWXf/lT9oyazxdaU0P4GwZMv+7MI+S9aZ57DE7z0tkgLHbnyoPA==\",\"dtype\":\"float64\",\"shape\":[5]}},\"selected\":{\"id\":\"187287\",\"type\":\"Selection\"},\"selection_policy\":{\"id\":\"187288\",\"type\":\"UnionRenderers\"}},\"id\":\"184942\",\"type\":\"ColumnDataSource\"},{\"attributes\":{},\"id\":\"184929\",\"type\":\"HelpTool\"},{\"attributes\":{},\"id\":\"184924\",\"type\":\"PanTool\"},{\"attributes\":{\"line_alpha\":0.1,\"line_color\":\"#1f77b4\",\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"184944\",\"type\":\"Line\"},{\"attributes\":{\"ticker\":null},\"id\":\"184951\",\"type\":\"LogTickFormatter\"},{\"attributes\":{},\"id\":\"184928\",\"type\":\"ResetTool\"},{\"attributes\":{},\"id\":\"184954\",\"type\":\"UnionRenderers\"},{\"attributes\":{\"num_minor_ticks\":10},\"id\":\"184915\",\"type\":\"LogTicker\"},{\"attributes\":{\"data_source\":{\"id\":\"184942\",\"type\":\"ColumnDataSource\"},\"glyph\":{\"id\":\"184943\",\"type\":\"Line\"},\"hover_glyph\":null,\"muted_glyph\":null,\"nonselection_glyph\":{\"id\":\"184944\",\"type\":\"Line\"},\"selection_glyph\":null,\"view\":{\"id\":\"184946\",\"type\":\"CDSView\"}},\"id\":\"184945\",\"type\":\"GlyphRenderer\"},{\"attributes\":{},\"id\":\"184912\",\"type\":\"LogScale\"},{\"attributes\":{\"ticker\":null},\"id\":\"184949\",\"type\":\"LogTickFormatter\"},{\"attributes\":{\"ticker\":{\"id\":\"184915\",\"type\":\"LogTicker\"}},\"id\":\"184918\",\"type\":\"Grid\"},{\"attributes\":{\"items\":[{\"id\":\"184957\",\"type\":\"LegendItem\"}]},\"id\":\"184956\",\"type\":\"Legend\"},{\"attributes\":{\"overlay\":{\"id\":\"184955\",\"type\":\"BoxAnnotation\"}},\"id\":\"184926\",\"type\":\"BoxZoomTool\"},{\"attributes\":{\"fill_alpha\":{\"value\":0.1},\"fill_color\":{\"value\":\"#1f77b4\"},\"line_alpha\":{\"value\":0.1},\"line_color\":{\"value\":\"#1f77b4\"},\"size\":{\"units\":\"screen\",\"value\":5},\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"184939\",\"type\":\"X\"},{\"attributes\":{},\"id\":\"184925\",\"type\":\"WheelZoomTool\"},{\"attributes\":{\"bottom_units\":\"screen\",\"fill_alpha\":{\"value\":0.5},\"fill_color\":{\"value\":\"lightgrey\"},\"left_units\":\"screen\",\"level\":\"overlay\",\"line_alpha\":{\"value\":1.0},\"line_color\":{\"value\":\"black\"},\"line_dash\":[4,4],\"line_width\":{\"value\":2},\"render_mode\":\"css\",\"right_units\":\"screen\",\"top_units\":\"screen\"},\"id\":\"184955\",\"type\":\"BoxAnnotation\"},{\"attributes\":{\"callback\":null},\"id\":\"184908\",\"type\":\"DataRange1d\"},{\"attributes\":{},\"id\":\"184927\",\"type\":\"SaveTool\"},{\"attributes\":{\"source\":{\"id\":\"184942\",\"type\":\"ColumnDataSource\"}},\"id\":\"184946\",\"type\":\"CDSView\"}],\"root_ids\":[\"184905\"]},\"title\":\"Bokeh Application\",\"version\":\"1.3.4\"}};\n",
       "  var render_items = [{\"docid\":\"6fe6ac2d-3f18-4fd6-8b31-3abc8f25c7f0\",\"roots\":{\"184905\":\"7e5b79f3-3144-4ed8-9356-889a7861b540\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "184905"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "rho = (1+5**0.5)/2\n",
    "\n",
    "xstar = np.log(2)\n",
    "xsol = my_secante(f, x0=0.6,x1=0.7, xstar=xstar, tol=1.e-12, nitmax=50)\n",
    "print(f\"Convergence de la méthode de la sécante en {xsol.size-1} itérations\")\n",
    "print(f\"--> solution de f(x) = 0 obtenue pour  x = {xsol[-1]}\")\n",
    "\n",
    "err = np.abs(xsol - xstar)\n",
    "\n",
    "fig2 = figure(width=490, height=300, x_axis_type=\"log\", y_axis_type=\"log\")\n",
    "fig2.x(err[:-1], err[1:], size=5)\n",
    "fig2.line(err[:-1], err[:-1]**rho,  color=\"green\", legend=\"ordre rho\")\n",
    "\n",
    "show(fig2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Test de la fonction $g$ :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 212,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Convergence de la méthode de la sécante en 6 itérations\n",
      "--> solution de g(x) = 0 obtenue pour  x = -2.638092230933279e-23\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"be387bc9-8230-4d0e-b63f-55ef8ba5987d\" data-root-id=\"187356\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
       "  var docs_json = {\"5322e145-ef12-462c-a918-a1cf35050f14\":{\"roots\":{\"references\":[{\"attributes\":{\"below\":[{\"id\":\"187365\",\"type\":\"LogAxis\"}],\"center\":[{\"id\":\"187369\",\"type\":\"Grid\"},{\"id\":\"187374\",\"type\":\"Grid\"},{\"id\":\"187407\",\"type\":\"Legend\"}],\"left\":[{\"id\":\"187370\",\"type\":\"LogAxis\"}],\"plot_height\":300,\"plot_width\":490,\"renderers\":[{\"id\":\"187391\",\"type\":\"GlyphRenderer\"},{\"id\":\"187396\",\"type\":\"GlyphRenderer\"}],\"title\":{\"id\":\"187398\",\"type\":\"Title\"},\"toolbar\":{\"id\":\"187381\",\"type\":\"Toolbar\"},\"x_range\":{\"id\":\"187357\",\"type\":\"DataRange1d\"},\"x_scale\":{\"id\":\"187361\",\"type\":\"LogScale\"},\"y_range\":{\"id\":\"187359\",\"type\":\"DataRange1d\"},\"y_scale\":{\"id\":\"187363\",\"type\":\"LogScale\"}},\"id\":\"187356\",\"subtype\":\"Figure\",\"type\":\"Plot\"},{\"attributes\":{\"callback\":null,\"data\":{\"x\":{\"__ndarray__\":\"MzMzMzMz0z+amZmZmZnZP1gUi6PJdKU/oOK6YIF5Zz8AEE+Ve6fDPgAArL/fIqU9\",\"dtype\":\"float64\",\"shape\":[6]},\"y\":{\"__ndarray__\":\"mpmZmZmZ2T9YFIujyXSlP6DiumCBeWc/ABBPlXunwz4AAKy/3yKlPQAAAACA5D87\",\"dtype\":\"float64\",\"shape\":[6]}},\"selected\":{\"id\":\"187404\",\"type\":\"Selection\"},\"selection_policy\":{\"id\":\"187405\",\"type\":\"UnionRenderers\"}},\"id\":\"187388\",\"type\":\"ColumnDataSource\"},{\"attributes\":{\"items\":[{\"id\":\"187408\",\"type\":\"LegendItem\"}]},\"id\":\"187407\",\"type\":\"Legend\"},{\"attributes\":{\"line_alpha\":0.1,\"line_color\":\"#1f77b4\",\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"187395\",\"type\":\"Line\"},{\"attributes\":{},\"id\":\"187404\",\"type\":\"Selection\"},{\"attributes\":{},\"id\":\"187375\",\"type\":\"PanTool\"},{\"attributes\":{},\"id\":\"189749\",\"type\":\"Selection\"},{\"attributes\":{},\"id\":\"187376\",\"type\":\"WheelZoomTool\"},{\"attributes\":{\"data_source\":{\"id\":\"187393\",\"type\":\"ColumnDataSource\"},\"glyph\":{\"id\":\"187394\",\"type\":\"Line\"},\"hover_glyph\":null,\"muted_glyph\":null,\"nonselection_glyph\":{\"id\":\"187395\",\"type\":\"Line\"},\"selection_glyph\":null,\"view\":{\"id\":\"187397\",\"type\":\"CDSView\"}},\"id\":\"187396\",\"type\":\"GlyphRenderer\"},{\"attributes\":{\"ticker\":null},\"id\":\"187402\",\"type\":\"LogTickFormatter\"},{\"attributes\":{},\"id\":\"187363\",\"type\":\"LogScale\"},{\"attributes\":{\"overlay\":{\"id\":\"187406\",\"type\":\"BoxAnnotation\"}},\"id\":\"187377\",\"type\":\"BoxZoomTool\"},{\"attributes\":{\"dimension\":1,\"ticker\":{\"id\":\"187371\",\"type\":\"LogTicker\"}},\"id\":\"187374\",\"type\":\"Grid\"},{\"attributes\":{\"text\":\"\"},\"id\":\"187398\",\"type\":\"Title\"},{\"attributes\":{},\"id\":\"187378\",\"type\":\"SaveTool\"},{\"attributes\":{},\"id\":\"187379\",\"type\":\"ResetTool\"},{\"attributes\":{\"fill_color\":{\"value\":\"#1f77b4\"},\"line_color\":{\"value\":\"#1f77b4\"},\"size\":{\"units\":\"screen\",\"value\":5},\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"187389\",\"type\":\"X\"},{\"attributes\":{\"callback\":null,\"data\":{\"x\":{\"__ndarray__\":\"MzMzMzMz0z+amZmZmZnZP1gUi6PJdKU/oOK6YIF5Zz8AEE+Ve6fDPgAArL/fIqU9\",\"dtype\":\"float64\",\"shape\":[6]},\"y\":{\"__ndarray__\":\"zufnWQw/wj+vIVHT8A/NPz3a+Zv0KXg/kkS2ArclFD90JAVGXK0KPu4hxwwmhjs8\",\"dtype\":\"float64\",\"shape\":[6]}},\"selected\":{\"id\":\"189749\",\"type\":\"Selection\"},\"selection_policy\":{\"id\":\"189750\",\"type\":\"UnionRenderers\"}},\"id\":\"187393\",\"type\":\"ColumnDataSource\"},{\"attributes\":{\"ticker\":null},\"id\":\"187400\",\"type\":\"LogTickFormatter\"},{\"attributes\":{\"callback\":null},\"id\":\"187359\",\"type\":\"DataRange1d\"},{\"attributes\":{\"source\":{\"id\":\"187393\",\"type\":\"ColumnDataSource\"}},\"id\":\"187397\",\"type\":\"CDSView\"},{\"attributes\":{},\"id\":\"187380\",\"type\":\"HelpTool\"},{\"attributes\":{},\"id\":\"187405\",\"type\":\"UnionRenderers\"},{\"attributes\":{\"ticker\":{\"id\":\"187366\",\"type\":\"LogTicker\"}},\"id\":\"187369\",\"type\":\"Grid\"},{\"attributes\":{\"label\":{\"value\":\"ordre rho\"},\"renderers\":[{\"id\":\"187396\",\"type\":\"GlyphRenderer\"}]},\"id\":\"187408\",\"type\":\"LegendItem\"},{\"attributes\":{\"data_source\":{\"id\":\"187388\",\"type\":\"ColumnDataSource\"},\"glyph\":{\"id\":\"187389\",\"type\":\"X\"},\"hover_glyph\":null,\"muted_glyph\":null,\"nonselection_glyph\":{\"id\":\"187390\",\"type\":\"X\"},\"selection_glyph\":null,\"view\":{\"id\":\"187392\",\"type\":\"CDSView\"}},\"id\":\"187391\",\"type\":\"GlyphRenderer\"},{\"attributes\":{\"formatter\":{\"id\":\"187400\",\"type\":\"LogTickFormatter\"},\"ticker\":{\"id\":\"187366\",\"type\":\"LogTicker\"}},\"id\":\"187365\",\"type\":\"LogAxis\"},{\"attributes\":{\"active_drag\":\"auto\",\"active_inspect\":\"auto\",\"active_multi\":null,\"active_scroll\":\"auto\",\"active_tap\":\"auto\",\"tools\":[{\"id\":\"187375\",\"type\":\"PanTool\"},{\"id\":\"187376\",\"type\":\"WheelZoomTool\"},{\"id\":\"187377\",\"type\":\"BoxZoomTool\"},{\"id\":\"187378\",\"type\":\"SaveTool\"},{\"id\":\"187379\",\"type\":\"ResetTool\"},{\"id\":\"187380\",\"type\":\"HelpTool\"}]},\"id\":\"187381\",\"type\":\"Toolbar\"},{\"attributes\":{\"num_minor_ticks\":10},\"id\":\"187366\",\"type\":\"LogTicker\"},{\"attributes\":{},\"id\":\"187361\",\"type\":\"LogScale\"},{\"attributes\":{\"callback\":null},\"id\":\"187357\",\"type\":\"DataRange1d\"},{\"attributes\":{\"formatter\":{\"id\":\"187402\",\"type\":\"LogTickFormatter\"},\"ticker\":{\"id\":\"187371\",\"type\":\"LogTicker\"}},\"id\":\"187370\",\"type\":\"LogAxis\"},{\"attributes\":{\"fill_alpha\":{\"value\":0.1},\"fill_color\":{\"value\":\"#1f77b4\"},\"line_alpha\":{\"value\":0.1},\"line_color\":{\"value\":\"#1f77b4\"},\"size\":{\"units\":\"screen\",\"value\":5},\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"187390\",\"type\":\"X\"},{\"attributes\":{\"num_minor_ticks\":10},\"id\":\"187371\",\"type\":\"LogTicker\"},{\"attributes\":{\"line_color\":\"green\",\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"187394\",\"type\":\"Line\"},{\"attributes\":{},\"id\":\"189750\",\"type\":\"UnionRenderers\"},{\"attributes\":{\"bottom_units\":\"screen\",\"fill_alpha\":{\"value\":0.5},\"fill_color\":{\"value\":\"lightgrey\"},\"left_units\":\"screen\",\"level\":\"overlay\",\"line_alpha\":{\"value\":1.0},\"line_color\":{\"value\":\"black\"},\"line_dash\":[4,4],\"line_width\":{\"value\":2},\"render_mode\":\"css\",\"right_units\":\"screen\",\"top_units\":\"screen\"},\"id\":\"187406\",\"type\":\"BoxAnnotation\"},{\"attributes\":{\"source\":{\"id\":\"187388\",\"type\":\"ColumnDataSource\"}},\"id\":\"187392\",\"type\":\"CDSView\"}],\"root_ids\":[\"187356\"]},\"title\":\"Bokeh Application\",\"version\":\"1.3.4\"}};\n",
       "  var render_items = [{\"docid\":\"5322e145-ef12-462c-a918-a1cf35050f14\",\"roots\":{\"187356\":\"be387bc9-8230-4d0e-b63f-55ef8ba5987d\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "187356"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xstar = 0\n",
    "xsol = my_secante(g, x0=0.3,x1=0.4, xstar=xstar, tol=1.e-12, nitmax=50)\n",
    "print(f\"Convergence de la méthode de la sécante en {xsol.size-1} itérations\")\n",
    "print(f\"--> solution de g(x) = 0 obtenue pour  x = {xsol[-1]}\")\n",
    "\n",
    "err = np.abs(xsol - xstar)\n",
    "\n",
    "\n",
    "fig2 = figure(width=490, height=300, x_axis_type=\"log\", y_axis_type=\"log\")\n",
    "fig2.x(err[:-1], err[1:], size=5)\n",
    "fig2.line(err[:-1], err[:-1]**rho,  color=\"green\", legend=\"ordre rho\")\n",
    "\n",
    "show(fig2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Test pour la fonction $h$ :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 213,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Convergence de la méthode de la sécante en 51 itérations\n",
      "--> solution de h(x) = 0 obtenue pour  x = 2.517346967800027e-07\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"2ce57183-7e57-4658-a75a-d53271c4d4ef\" data-root-id=\"189818\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
       "  var docs_json = 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Application\",\"version\":\"1.3.4\"}};\n",
       "  var render_items = [{\"docid\":\"b15b3f6c-efa5-4191-8ac7-4d350b97c3bc\",\"roots\":{\"189818\":\"2ce57183-7e57-4658-a75a-d53271c4d4ef\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "189818"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xstar = 0\n",
    "xsol = my_secante(h, x0=0.3,x1=0.4, xstar=xstar, tol=1.e-12, nitmax=50)\n",
    "print(f\"Convergence de la méthode de la sécante en {xsol.size-1} itérations\")\n",
    "print(f\"--> solution de h(x) = 0 obtenue pour  x = {xsol[-1]}\")\n",
    "\n",
    "err = np.abs(xsol - xstar)\n",
    "\n",
    "fig1 = figure(width=490, height=300, x_axis_type=\"log\", y_axis_type=\"log\")\n",
    "fig1.x(err[:-1], err[1:], size=5)\n",
    "fig1.line(err[:-1], err[:-1],  color=\"yellow\", legend=\"ordre 1\")\n",
    "fig1.line(err[:-1], err[:-1]**rho,  color=\"green\", legend=\"ordre rho\")\n",
    "\n",
    "show(fig1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nous obtenons effectivement un ordre d'au moins $\\rho$ pour $g$ et $f$ car $f(0) = 0$, $f'(0) \\ne 0$ et $g(0) = 0$, $g'(0) \\ne 0$.\n",
    "\n",
    "Par contre, nous obtenons un ordre linéaire pour $h$, ce qui ne contredit pas la théorie car $h(0)=0$ et $h'(0)=0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bissection-Trissection-hybride"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Implémentation de la méthode"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 214,
   "metadata": {},
   "outputs": [],
   "source": [
    "def my_bissection(f, a, b, xstar=None, tol=1.e-12, nitmax=50):\n",
    "   \n",
    "    # initialisation\n",
    "    x = np.zeros(nitmax+1)\n",
    "    x[0] = (a+b)/2\n",
    "    \n",
    "    # iteration de la méthode de bissection\n",
    "    for i in range(1,nitmax+1) :\n",
    "        if f(x[i-1])*f(a)<=0 :\n",
    "            b = x[i-1]\n",
    "        else :\n",
    "            a = x[i-1]\n",
    "        x[i] = (a+b)/2\n",
    "        if b-a<tol : break\n",
    "        if abs(x[i]-xstar)<tol : break\n",
    "        if abs(f(x[i]))<tol : break\n",
    "\n",
    "    return x[0:i+1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Tester la méthode avec la fonction $m(x) = ln(x)+2$. Commenter.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 215,
   "metadata": {},
   "outputs": [],
   "source": [
    "def m(x) :\n",
    "    return np.log(x)+2\n",
    "\n",
    "def dm(x) :\n",
    "    return 1/x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 218,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Convergence de la méthode de la bissection en 38 itérations\n",
      "--> solution de m(x) = 0 obtenue pour  x = 0.13533528323672728\n",
      "\n",
      "Convergence de l'algorithme de Newton en 6 itérations\n",
      "--> solution de m(x) = 0 obtenue pour  x = 0.1353352832336045\n",
      "\n",
      "l'erreur entre les solutions trouvées par deux méthodes est :  3.122779812514409e-12\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"9c1056b0-33d2-4114-b46f-b1d45c0c6ffe\" data-root-id=\"197435\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
       "  var docs_json = 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1\"},\"renderers\":[{\"id\":\"197499\",\"type\":\"GlyphRenderer\"}]},\"id\":\"197512\",\"type\":\"LegendItem\"},{\"attributes\":{},\"id\":\"197510\",\"type\":\"UnionRenderers\"},{\"attributes\":{\"data_source\":{\"id\":\"197481\",\"type\":\"ColumnDataSource\"},\"glyph\":{\"id\":\"197482\",\"type\":\"X\"},\"hover_glyph\":null,\"muted_glyph\":null,\"nonselection_glyph\":{\"id\":\"197483\",\"type\":\"X\"},\"selection_glyph\":null,\"view\":{\"id\":\"197485\",\"type\":\"CDSView\"}},\"id\":\"197484\",\"type\":\"GlyphRenderer\"}],\"root_ids\":[\"197435\"]},\"title\":\"Bokeh Application\",\"version\":\"1.3.4\"}};\n",
       "  var render_items = [{\"docid\":\"cc1b9bef-f9ee-4fb6-85ff-db953a2d8fef\",\"roots\":{\"197435\":\"9c1056b0-33d2-4114-b46f-b1d45c0c6ffe\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
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      "application/vnd.bokehjs_exec.v0+json": {
       "id": "197435"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xstar = np.exp(-2)\n",
    "\n",
    "## Methode de la bissection\n",
    "xsol = my_bissection(m, 0.01, 1, xstar, tol=1.e-12, nitmax=50)\n",
    "print(f\"Convergence de la méthode de la bissection en {xsol.size-1} itérations\")\n",
    "print(f\"--> solution de m(x) = 0 obtenue pour  x = {xsol[-1]}\\n\")\n",
    "err = np.abs(xsol - xstar)\n",
    "\n",
    "## Methode de Newton\n",
    "xsol_newton = my_newton(m, dm, x0=0.01, xstar=xstar, tol=1.e-10, nitmax=100)\n",
    "print(f\"Convergence de l'algorithme de Newton en {xsol_newton.size-1} itérations\")\n",
    "print(f\"--> solution de m(x) = 0 obtenue pour  x = {xsol_newton[-1]}\\n\")\n",
    "err_newton = np.abs(xsol_newton - xstar)\n",
    "\n",
    "err_sol = abs(xsol[-1]-xsol_newton[-1])\n",
    "print (\"l'erreur entre les solutions trouvées par deux méthodes est : \",err_sol)\n",
    "\n",
    "fig1 = figure(width=490, height=300, x_axis_type=\"log\", y_axis_type=\"log\")\n",
    "fig1.x(err[:-1], err[1:], size=5, legend=\"Bissection\")\n",
    "fig1.x(err_newton[:-1], err_newton[1:],color=\"orange\", size=5, legend=\"Newton\")\n",
    "fig1.line(err[:-1], err[:-1],  color=\"yellow\", legend=\"ordre 1\")\n",
    "fig1.line(err[:-1], err[:-1]*err[:-1],  color=\"red\", legend=\"ordre 2\")\n",
    "\n",
    "show(fig1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "L'erreur entre les solutions obtenues par chacune des deux méthode est de l'ordre de $10^{-12}$, on aboutit donc au même bon résultat. Toutefois, avec les conditions initiales que j'ai choisies, la méthode de Newton converge en 6 itérations alors que la méthode dichotomique converge en 38 itérations.\n",
    "\n",
    "En effet, La méthode de Newton est d'ordre 2, et la méthode de bissection est d'ordre 1."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# algorithme hybride Bissection-Newton"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Idée de l'algorithme hybride :** Cet algorithme consiste à appliquer la méthode de Newton tout en garantissant la convergence, en imposant qu'on s'approche de la racine $x_*$ après chaque itération grâce à une méthode dichotomique.\n",
    "\n",
    "En terme de qualité, cet algorithme convergence au moins au même ordre de convergence que l'algorithme de Newton, et donc pour une fonction de classe $C^2$ ayant un zéro non dégénéré, la convergence est au moins quadratique."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 220,
   "metadata": {},
   "outputs": [],
   "source": [
    "def my_hybride(f, df, a, b, x0, tol=1.e-12, nitmax=50) :\n",
    "    \n",
    "    # initialisation\n",
    "    x = np.zeros(nitmax+1)\n",
    "    x[0] = x0\n",
    "    \n",
    "    #itération\n",
    "    for i in range(1,nitmax+1) :\n",
    "        y = x[i-1]-f(x[i-1])/df(x[i-1])\n",
    "        # construction de x[i]\n",
    "        if a<=y and y<=b : x[i]=y\n",
    "        else : x[i]=(a+b)/2\n",
    "        # modification de a et b\n",
    "        if f(a)*f(x[i])<=0 : b=x[i]\n",
    "        else : a=x[i]\n",
    "        \n",
    "        if ( abs(f(x[i])) < tol ): break\n",
    "        if (abs(x[i]-x[i-1]) < tol) : break\n",
    "\n",
    "    return x[0:i+1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Considérer la fonction $f(x) = x^5-x+1$ sur l'intervalle $[-2.5,2.5]$.\n",
    "* Tester la méthode de Newton pour cette fonction avec $x_0=-1$ et $x_0=1$. En cas de non convergence, expliquer le comportement observé.\n",
    "* Tester votre algorithme hybride Bissection-Newton sur la fonction $f$. Représenter graphiquement la courbe de convergence et commenter le comportement de votre algorithme."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 221,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(x) :\n",
    "    return x**5-x+1\n",
    "\n",
    "def df(x) :\n",
    "    return 5*x**4-1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 222,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pour x0 = 1 : convergence de l'algorithme de Newton en 100 itérations\n",
      "--> solution de f(x) = 0 obtenue pour  x* = 0.7503218281592572\n",
      "--> Erreur résiduelle : f(x*) =  0.4874924386834848\n",
      "Pour x0 = -1 : convergence de l'algorithme de Newton en 5 itérations\n",
      "--> solution de f(x) = 0 obtenue pour  x* = -1.1673039782614396\n",
      "--> Erreur résiduelle : f(x*) =  -1.7341683644644945e-13\n"
     ]
    }
   ],
   "source": [
    "# Pour x0 = 1 :\n",
    "xsol = my_newton(f, df, x0=1, xstar=None, tol=1.e-12, nitmax=100)\n",
    "print(f\"Pour x0 = 1 : convergence de l'algorithme de Newton en {xsol.size-1} itérations\")\n",
    "print(f\"--> solution de f(x) = 0 obtenue pour  x* = {xsol[-1]}\")\n",
    "print(\"--> Erreur résiduelle : f(x*) = \",f(xsol[-1]))\n",
    "\n",
    "# Pour x0 = -1 :\n",
    "xsol2 = my_newton(f, df, x0=-1, xstar=None, tol=1.e-12, nitmax=50)\n",
    "print(f\"Pour x0 = -1 : convergence de l'algorithme de Newton en {xsol2.size-1} itérations\")\n",
    "print(f\"--> solution de f(x) = 0 obtenue pour  x* = {xsol2[-1]}\")\n",
    "print(\"--> Erreur résiduelle : f(x*) = \",f(xsol2[-1]))"
   ]
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   "source": [
    "* Pour $x_0 = -1$, on a convergence de l'algorithme, et le résultat obtenu est satisfaisant car l'erreur résiduelle est suffisamment petite.\n",
    "* Pour $x_0 = 1$, l'algorithme ne converge pas.\n",
    "\n",
    "Ce comportement est dû à la distance entre le point initial et le zéro de la fonction : -1 est assez proche de $x_* \\simeq -1.167$, mais 1 ne l'est pas."
   ]
  },
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   "source": [
    "**Test de l'algorithme hybride Bissection-Newton pour la fonction $f$**"
   ]
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   "metadata": {},
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     "text": [
      "Pour x0 = 1 : convergence de l'algorithme de Newton en 8 itérations\n",
      "--> solution de f(x) = 0 obtenue pour  x* = -1.1673039782614187\n",
      "--> Erreur résiduelle : f(x*) =  -6.661338147750939e-16\n"
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   "source": [
    "xsol = my_hybride(f, df, a=-2.5, b=2.5, x0=1, tol=1.e-12, nitmax=100)\n",
    "\n",
    "print(f\"Pour x0 = 1 : convergence de l'algorithme de Newton en {xsol.size-1} itérations\")\n",
    "print(f\"--> solution de f(x) = 0 obtenue pour  x* = {xsol[-1]}\")\n",
    "print(\"--> Erreur résiduelle : f(x*) = \",f(xsol[-1]))\n",
    "\n",
    "xstar = xsol[-1]\n",
    "err = np.abs(xsol - xstar)\n",
    "\n",
    "\n",
    "fig2 = figure(width=490, height=300, x_axis_type=\"log\", y_axis_type=\"log\")\n",
    "fig2.x(err[:-1], err[1:], size=10) #err(n+1) en fctn de err(n)\n",
    "fig2.line(err[:-1], err[:-1],  color=\"yellow\", legend=\"ordre 1\")\n",
    "fig2.line(err[:-1], err[:-1]*err[:-1],  color=\"red\", legend=\"ordre 2\")\n",
    "fig2.line(err[:-1], err[:-1]*err[:-1]*err[:-1],  color=\"green\", legend=\"ordre 3\")\n",
    "\n",
    "show(fig2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On remarque que l'algorithme hybride bissection-Newton résout bien le problème de convergence rencontré avec l'algorithme de Newton : l'algorithme converge quelque soit la condition initiale $x_0$.\n",
    "\n",
    "En plus, sa convergence est d'ordre **quadratique**."
   ]
  }
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