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    "## MT09 - TP2 Préparation - Automne 2024\n",
    "### Elimination de Gauss et factorisation LU\n",
    "\n",
    "1. Ecrire une fonction python \n",
    "```\n",
    "x = solsup(U,b)\n",
    "```\n",
    "qui, étant donné $U$ une matrice triangulaire supérieure de taille $n\\times n$ et $b$ vecteur colonne de taille $n$, calcule $x$ solution de $Ux=b$."
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    "import numpy as np\n",
    "\n",
    "def solsup(U, b):\n",
    "    \"\"\" Résolution du système triangulaire supérieur Ux=b \"\"\"\n",
    "    # A COMPLETER\n",
    "    n = np.size(b);\n",
    "    x = np.zeros((n,1));\n",
    "    # ...\n",
    "    return x"
   ]
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    "Application : utiliser la fonction `solsup()` pour déterminer la solution $x$ de \n",
    "\n",
    "$$\n",
    "\\begin{pmatrix} 1 & 2 & 3 \\\\ 0 & 4 & 8 \\\\ 0 & 0 & 5 \\end{pmatrix} x = \\begin{pmatrix} 6 \\\\ 16 \\\\ 15 \\end{pmatrix}.\n",
    "$$\n",
    "\n",
    "Vérifiez que $Ux-b=0$. Comparer aussi à la solution donnée par ```solve(U,b)``` du module\n",
    "```numpy.linalg```."
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    "U = np.array([[1, 2, 3], [0, 4, 8], [0, 0, 5]], dtype=np.float64);\n",
    "b = np.array([[6, 16, 15]], dtype=np.float64).T;\n",
    "x = solsup(U, b);\n",
    "print('x = ', x)\n",
    "# Verif\n",
    "print(U@x, b)\n",
    "print(\"Ux-b = \", U@x - b)\n",
    "import numpy.linalg as la\n",
    "xverif = la.solve(U,b)\n",
    "print(\"xverif = \", xverif)"
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    "2. Ecrire une fonction python\n",
    "```\n",
    "U, e = trisup(A,b)\n",
    "```\n",
    "qui, étant donné $A$ matrice carrée de taille $n\\times n$ et $b$ vecteur colonne de taille $n$, déterminer par la méthode d'élimination de Gauss, quand c'est possible, la matrice $U$ triangulaire supérieure et le vecteur colonne $e$ tel que $Ax=b$\n",
    "$\\Leftrightarrow$ $Ux = e$.\n",
    "\n",
    "On rappelle l'algorithme d'élimination de Gauss:"
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   "source": [
    "#### Algorithme de Gauss"
   ]
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   "source": [
    "import sys\n",
    "def trisup(A, b):\n",
    "# Méthode d'elimination de Gauss\n",
    "    Am = np.copy(A);\n",
    "    bm = np.copy(b) \n",
    "    n = np.size(b);\n",
    "    for k in np.arange(0,n):\n",
    "        if # A COMPLETER:\n",
    "            print(\"ERROR : Pivot nul, k=\", k, '\\n');\n",
    "            sys.exit(1)\n",
    "        for i in np.arange(k+1, n):\n",
    "            # A COMPLETER\n",
    "    if # A COMPLETER\n",
    "        print(\"ERROR : Pivot A_nn nul\\n\");\n",
    "        sys.exit(1);\n",
    "    return Am, bm"
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    "Application : utiliser la fonction `trisup()` pour déterminer un système d'équations triangulaire $Ux=e$ équivalent au système $Ax=b$ suivant :\n",
    "\n",
    "$$\n",
    "\\begin{pmatrix} 3 & 1 & 2 \\\\ 3 & 2 & 6 \\\\ 6 & 1 & -1 \\end{pmatrix} x = \\begin{pmatrix} 2 \\\\ 1 \\\\ 4 \\end{pmatrix}.\n",
    "$$\n",
    "\n",
    "Affichez $U$ et $e$. Utilisez ```trisup()``` pour résoudre $Ux=e$.\n",
    "Affichez la solution $x$ trouvée.\n",
    "Vérifiez que $Ux-e = 0$ et $Ax-b=0$."
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