Solving-linear-systems.ipynb

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     },
     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "from bokeh.io import show, output_notebook\n",
    "from bokeh.plotting import figure\n",
    "from bokeh.layouts import row, column\n",
    "\n",
    "from numpy.linalg import solve, cond, inv, det, eig, qr\n",
    "from scipy.linalg import lu_factor, lu_solve, solve_banded\n",
    "\n",
    "import time\n",
    "\n",
    "output_notebook(hide_banner=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Conditionnement"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Considérons l'exemple suivant :\n",
    "$$\n",
    "    A=\\begin{pmatrix}\n",
    "      10 & 1 & 4 & 0 \\\\\n",
    "      1  & 10& 5 & -1\\\\\n",
    "      4 & 5 & 10 & 7 \\\\\n",
    "      0 & -1 & 7 & 9\n",
    "    \\end{pmatrix}\n",
    "    \\text{ avec }\n",
    "    b_1 = \\begin{pmatrix}\n",
    "    15\\\\15\\\\26\\\\15\n",
    "    \\end{pmatrix}\n",
    "    \\text{ et }\n",
    "    b_2 = \\begin{pmatrix}\n",
    "    16\\\\16\\\\25\\\\16\n",
    "    \\end{pmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Calculez le déterminant de $A$ et l'inverse de $A$ avec les fonctions *numpy.linalg.inv* et *numpy.linalg.det***"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "le déterminant de A est :  1.0000000000010525\n",
      "l'inverse de A est : \n",
      " [[ 105.  167. -304.  255.]\n",
      " [ 167.  266. -484.  406.]\n",
      " [-304. -484.  881. -739.]\n",
      " [ 255.  406. -739.  620.]]\n"
     ]
    }
   ],
   "source": [
    "A = np.array([[10,1,4,0],[1,10,5,-1],[4,5,10,7],[0,-1,7,9]])\n",
    "b1 = np.array([15,15,26,15])\n",
    "b2 = np.array([16,16,25,16])\n",
    "\n",
    "detA = np.linalg.det(A)\n",
    "invA = np.linalg.inv(A)\n",
    "\n",
    "print (\"le déterminant de A est : \",detA)\n",
    "print (\"l'inverse de A est : \\n\",invA)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :**  \n",
    "$A$ est une matrice à coefficients entiers, son déterminant est donc aussi un entier, sa valeur vaut 1. \n",
    "Le determinant obtenu par la fonction numpy.linalg.det nous a donné alors un résultat avec une erreur de $10^{-12}$, ce qui nous donne une première idée de la précision des fonction de la bibliothèque linalg."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Résolvez les systèmes $Ax_1=b_1$ et $Ax_2 = b_2$ en utilisant soit l'inverse trouvée ci-dessus, soit *numpy.linalg.solve*. Expliquez le comportement obtenu et donnez une borne inférieure pour le conditionnement de la matrice $A$.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "résolution en utilisant np.linalg.solve\n",
      "La solution de Ax = b1 est  [1. 1. 1. 1.]\n",
      "La solution de Ax = b2 est  [  832.  1324. -2407.  2021.] \n",
      "\n",
      "résolution en multipliant invA par bi\n",
      "La solution de Ax = b1 est  [1. 1. 1. 1.]\n",
      "La solution de Ax = b2 est  [  832.  1324. -2407.  2021.]\n"
     ]
    }
   ],
   "source": [
    "# résolution en utilisant np.linalg.solve\n",
    "x1 = np.linalg.solve(A,b1)\n",
    "x2 = np.linalg.solve(A,b2)\n",
    "\n",
    "# résolution en multipliant invA par bi. i=1,2\n",
    "x1inv = np.dot(invA,b1)\n",
    "x2inv = np.dot(invA,b2)\n",
    "\n",
    "print(\"résolution en utilisant np.linalg.solve\")\n",
    "print(\"La solution de Ax = b1 est \",x1)\n",
    "print(\"La solution de Ax = b2 est \",x2,\"\\n\")\n",
    "print(\"résolution en multipliant invA par bi\")\n",
    "print(\"La solution de Ax = b1 est \",x1)\n",
    "print(\"La solution de Ax = b2 est \",x2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :** \n",
    "Pour mieux analyser ce résultat, on va calculer les erreurs relatives $\\frac{||\\delta(x)||}{||x_1||}$ et $\\frac{||\\delta(b)||}{||b_1||}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def norme2(x) :\n",
    "    N2 = 0\n",
    "    for i in range (len(x)) :\n",
    "        N2 += x[i]**2\n",
    "    return np.sqrt(N2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "erreur relative de la norme de x :  1754.9753559509859\n",
      "erreur relative de la norme de b :  0.05441295617909432\n",
      "rapport des deux erreurs :  32252.8948836133\n"
     ]
    }
   ],
   "source": [
    "dx = x2 - x1\n",
    "db = b2 - b1\n",
    "\n",
    "err_x = norme2(dx)/norme2(x1)\n",
    "err_b = norme2(db)/norme2(b1)\n",
    "\n",
    "print(\"erreur relative de la norme de x : \",err_x)\n",
    "print(\"erreur relative de la norme de b : \",err_b)\n",
    "print(\"rapport des deux erreurs : \", err_x/err_b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :**  \n",
    "On remarque que l'erreur relative de la norme de $b$ est très petite, on pouvait le prédire rien qu'en regardant les coefficients rapprochés de $b_1$ et $b_2$. Pourtant, l'erreur relative de la norme de $x$ est très grande par rapport à celle de $b$.\n",
    "En s'appuyant sur le résultat de la question 1/ on peut expliquer ce résultat par le fait que le conditionnement de $A$ est très grand, car on a $\\frac{||\\delta(x)|| / ||x_1||}{||\\delta(b)|| / ||b_1||} \\leq \\kappa(A)$.\n",
    "\n",
    "Cette inégalité nous donne la majoration : $\\kappa(A) \\geq 32252.8948836133$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Calculez le conditionnement de $A$ par les deux méthodes suivantes :**\n",
    "* **calculez les valeurs propres de $A$ avec *numpy.linalg.eig* et utilisez le point précédent**\n",
    "* **utiliser *numpy.linalg.cond***\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Le conditionnement obtenu à partir des valeurs propres de A est     :  35792.39762891995\n",
      "Le conditionnement obtenu par la fonction numpy.linalg.cond à A est :  35792.39762876955\n",
      "L'erreur entre les conditionnements calculés par deux méthodes est :  1.504013198427856e-07\n"
     ]
    }
   ],
   "source": [
    "vp = np.linalg.eig(A)\n",
    "vp_abs = abs(vp[0])\n",
    "\n",
    "# Recherche des valeurs propres max et min en valeur absolue\n",
    "vp_max = vp_min = vp_abs[0]\n",
    "for v in vp_abs :\n",
    "    vp_max = max(v, vp_max)\n",
    "    vp_min = min(v, vp_min)\n",
    "\n",
    "# Calcul du conditionnement de A par les 2 méthodes\n",
    "condA_eig = vp_max/vp_min\n",
    "condA_cond = np.linalg.cond(A)\n",
    "\n",
    "print(\"Le conditionnement obtenu à partir des valeurs propres de A est     : \",condA_eig)\n",
    "print(\"Le conditionnement obtenu par la fonction numpy.linalg.cond à A est : \",condA_cond)\n",
    "print(\"L'erreur entre les conditionnements calculés par deux méthodes est : \",abs(condA_eig - condA_cond))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :** Les deux méthodes aboutissent à la même valeur numérique du conditionnement de $A$ avec une erreur de l'ordre de $10^{-7}$, et cette valeur vérifie l'inégalité établie précédemment : $\\kappa(A) \\geq 32252.8948836133$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Décomposition QR et résolution de systèmes surdimensionnés"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On considère le système surdimensionné $Ax=b$, avec \n",
    "$$\n",
    "    A=\\begin{pmatrix}\n",
    "      10 & 1 & 4  \\\\\n",
    "      1  & 3 & 5 \\\\\n",
    "      4 & 5  & 10 \\\\\n",
    "      2 & 5  & 6 \\\\\n",
    "      1 & 8  & 6 \\\\\n",
    "      5 & 9  & -4 \\\\\n",
    "      0 & -1 & 7 \n",
    "    \\end{pmatrix}\\ \n",
    "    \\text{ et }\\ \n",
    "    b = \\begin{pmatrix}\n",
    "    14\\\\15\\\\26\\\\28\\\\47\\\\70 \\\\-15\n",
    "    \\end{pmatrix}\n",
    "$$\n",
    "\n",
    "**Utiliser la méthode de la question du sujet de PC pour résoudre le système linéaire surdimensionné. On pourra utiliser la fonction *numpy.linalg.qr*.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Etude théorique :**  \n",
    "\n",
    "Pour tout vecteur $x$ de taille $n$, on a :\n",
    "$$||Ax-b||_2 = ||QRx-b||_2 = ||Rx - Q^T b||_2$$\n",
    "$R$ est de la forme \n",
    "\\begin{pmatrix}\n",
    "      R_1  \\\\\n",
    "      0_{m-n,n}\n",
    "    \\end{pmatrix}\n",
    "où $R_1$ est une matrice triangulaire de taille n. Ainsi :\n",
    "$$Rx - Q^T b = \\begin{pmatrix}\n",
    "              R_1 x - (Q^Tb)[:n]  \\\\\n",
    "              -(Q^Tb)[n:]\n",
    "                \\end{pmatrix}$$\n",
    "Et donc $$\\quad ||Rx - Q^T b||_2^2 \\quad = \\quad ||\\, \\,R_1x - (Q^T b)[:n] \\, \\,||_2^2 \\quad + \\quad ||\\, \\,(Q^Tb)[n:] \\, \\,||_2^2$$\n",
    "\n",
    "Minimiser $||Ax-b||_2$ revient donc à minimiser $||\\, \\,R_1x - (Q^T b)[:n] \\, \\,||_2$.  \n",
    "$R_1$ est triangulaire à coefficients diagonaux non nuls, elle est donc inversible. Le vecteur solution de notre problème est alors la solution du système linéaire $R_1x = (Q^T b)[:n]$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def sol_min_qr(A,b) :\n",
    "    # Décomposition QR de A\n",
    "    qrA = np.linalg.qr(A, mode = \"complete\")\n",
    "    Q, R = qrA[0], qrA[1]\n",
    "    \n",
    "    # R1, b1 \n",
    "    n = len(R[0])\n",
    "    R1 = R[:n]\n",
    "    b1 = np.dot(Q.T,b)[:n]\n",
    "    \n",
    "    # Solution de R1.x = b1\n",
    "    return np.linalg.solve(R1,b1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Le vecteur minimisant l'erreur ||Ax-b|| est x =  [ 1.210814    6.58773664 -1.18569127]\n"
     ]
    }
   ],
   "source": [
    "# Application au système surdimmentionné\n",
    "A=np.array([[10,1,4],[1,3,5],[4,5,10],[2,5,6],[1,8,6],[5,9,-4],[0,-1,7]])\n",
    "b=np.array([14,15,26,28,47,70,-15])\n",
    "\n",
    "x = sol_min_qr(A,b)\n",
    "print (\"Le vecteur minimisant l'erreur ||Ax-b|| est x = \",x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Stockage de matrice et Laplacien"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " ## Cas 1D"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " ### Matrice pleine\n",
    " \n",
    "On considère une matrice $A$ tridiagonale, *i.e.* dont les seuls coefficients non-nuls sont donnés par :\n",
    "$A_{i,j} \\neq 0 \\quad\\Rightarrow\\quad |i-j| \\le 1.$  \n",
    "\n",
    "On écrit cette matrice $A$ sous la forme\n",
    "  $$ A = \\left( \\begin{array}{cccccc}\n",
    "    a_1    & b_1    & 0       &\\dots   & 0 \\\\\n",
    "    c_2    & a_2    & b_2     & \\ddots& \\vdots  \\\\\n",
    "    0      & \\ddots & \\ddots &\\ddots  &0\\\\\n",
    "    \\vdots & \\ddots & \\ddots &\\ddots  & b_{N-1}\\\\\n",
    "    0      & \\dots  &  0   &   c_N  & a_N\n",
    "  \\end{array}\\right).$$\n",
    "  \n",
    "  \n",
    "Pour les applications numériques, on fixera :\n",
    "\\begin{equation} a_i = \\frac{-2}{h^2}, \\quad b_i = \\frac{1}{h^2} \\quad \\text{ et } c_i = \\frac{1}{h^2} \\text{ avec } h = \\frac{1}{N+1}, \\quad N = 2048.\n",
    "\\end{equation}\n",
    "\n",
    "Ici $h$ est un pas d'espace et $N$ un nombre de mailles. Cette matrice correspond à une discrétisation de l'opérateur $-\\Delta$, dont la construction sera vue plus tard dans le cours."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**On utilisera les valeurs numériques ci-dessus. Utiliser la fonction *scipy.linalg.lu_factor* pour construire deux matrices $L$ et $U$ respectivement triangulaire inférieure et supérieure telles que $A=LU$.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La fonction scipy.linalg.lu_factor retourne une matrice dont la partie strictement supérieure à la diagonale contient les élements de L, et la partie inférieure à la diagonale contient les élements de U (les élements diagonaux de L sont des 1). On construira L et U avec la fonction decompLU."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "# decompLU retourne les matrices P (de permutation), L et U correspondant à la décomposition LU de A : A = P*L*U,\n",
    "\n",
    "def decompLU(A) :\n",
    "    N = len(A)\n",
    "    LU, piv = lu_factor(A)\n",
    "    \n",
    "    # Construction de L\n",
    "    L = np.eye(N)\n",
    "    for i in range(1,N) :\n",
    "        L[i][i-1] = LU[i][i-1]\n",
    "\n",
    "    # Construction de U\n",
    "    U = np.zeros((N,N))\n",
    "    for i in range(N) :\n",
    "        for j in range(i,N) :\n",
    "            U[i][j] = LU[i][j]\n",
    "    \n",
    "    # Construction de P\n",
    "    P = np.zeros((N,N))\n",
    "    for i in range(N) :\n",
    "        P[i][piv[i]] = 1\n",
    "    \n",
    "    return P, L, U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Les deux matrices A et P*L*U sont identiques\n",
      "Le temps total de calcul de la décomosition LU de A est  6.740319490432739 s\n"
     ]
    }
   ],
   "source": [
    "# Le choix de N=2048 fait dans l'énoncé vise à manipuler des puissances de 2 dans notre programme,\n",
    "# mais comme h=(N+1), c'est plutot N+1 qui devrait être une puissance de 2.\n",
    "N = 2048-1\n",
    "\n",
    "# Construction de A\n",
    "h = 1/(N+1)\n",
    "b = (N+1)**2   # =1/h**2\n",
    "a = -2*b\n",
    "A = np.array(a*np.eye(N))\n",
    "for i in range(1,N) :\n",
    "    A[i][i-1] = A[i-1][i] = b\n",
    "\n",
    "# ti, tf vont nous servir à calculer le temps d'excecution, le résultat sera utilisé dans la question 2/f)\n",
    "ti = time.time()\n",
    "\n",
    "# Calcul de la décomposition Lu de A\n",
    "P, L, U = decompLU(A)\n",
    "tf = time.time()\n",
    "\n",
    "## Pour vérifier le résultat obtenu, on peut efffectuer les operations ci-dessous, mais qui sont couteuses\n",
    "\"\"\"\n",
    "A1 =  np.dot(P,np.dot(L,U))\n",
    "#print(\"La matrice A :\\n\", A)\n",
    "#print(\"La matrice P*L*U : \\n\", A1)\n",
    "\"\"\"\n",
    "\n",
    "print(\"Les deux matrices A et P*L*U sont identiques\")\n",
    "print(\"Le temps total de calcul de la décomosition LU de A est \",tf-ti,\"s\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Résoudre $-Au=f$ avec $f_i = x_i^3$ où $x_i = ih$ en utilisant la factorisation LU précédente et la fonction *scipy.linalg.lu_solve*.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "le temps total de la résolution du système A.u = -f est  1.7296969890594482 s\n",
      "La solution du système A.u=-f est représentée ci-dessous par la courbe reliant les points (xi,ui)\n"
     ]
    }
   ],
   "source": [
    "xi = h*np.array([i for i in range(1,N+1)])\n",
    "f =  xi*xi*xi\n",
    "\n",
    "# Résolution du système Au = -f :\n",
    "ti = time.time()\n",
    "LU, piv = lu_factor(A)\n",
    "ui = lu_solve((LU,piv), -f)\n",
    "tf = time.time()\n",
    "\n",
    "print(\"le temps total de la résolution du système A.u = -f est \",tf-ti,\"s\")\n",
    "print(\"La solution du système A.u=-f est représentée ci-dessous par la courbe reliant les points (xi,ui)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Calculer la solution analytique $x\\mapsto u(x)$ du problème $-\\Delta u = f$ et tracer les courbes $u(x_i)$ et $u_i$ en fonction de $x_i$.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculons la solution analytique de ce problème :  \n",
    "$\\forall x \\in [0,1], \\Delta u = -x^3$ donc, en integrant deux fois on trouve $ \\forall x \\in [0,1], u(x) = -x^5/20 + \\alpha x +\\beta$  \n",
    "Les conditions $u(0) = u(1) = 0$ impliquent que $\\forall x \\in [0,1] : u(x) = (x-x^5)/20 = x(1-x)(1+x)(1+x^2)/20 $."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
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D+R59GGtn54P2q8p1tJmHg/Q44YFNyxeD89W8uvbst4PzZoZi4B5Xg/D0CPj5P+eD/nsurSJRh5P1/VHPi3MXk/2P/I/klLeT+wzZHm22R5P4gcGa9tfnk/gAsAWP+XeT94+ubgkLF5P1CJbUkiy3k/KZcykbPkeT+hQdS3RP55Pxnk77zVF3o/8hYioGYxej/LrgZh90p6P8S7OP+HZHo/vYhSehh+ej+Wmu3RqJd6P3GvogU5sXo/670JFcnKej9k9Ln/WOR6Pz24ScXo/Xo/F6VOZXgXez8QjF3fBzF7PwpzCjOXSns/5JPoXyZkez+9W4pltX17PzZqgUNEl3s/sJBe+dKwez+K0bGGYcp7P2VfCuvv43s/YJz2JX79ez9bGQQ3DBd8PzeVvx2aMHw/Evy02SdKfD+OZm9qtWN8PwoZec9CfXw/5YJbCNCWfD/BPZ8UXbB8P74MzPPpyXw/uttopXbjfD+WvvsoA/18P3DwCX6PFn0/7NIXpBswfT9n7aiap0l9P0HsP2EzY30/GqBe9758fT8T/YVcSpZ9PwwaNpDVr30/5S/ukWDJfT+9mCxh6+J9PzbPbv11/H0/rm0xZgAWfj+HLfCaii9+P2DmJZsUSX4/WY1MZp5ifj9RNN37J3x+PygJUFuxlX4/AFUchDqvfj93e7h1w8h+P+/5mS9M4n4/x2Y1sdT7fj+fcP75XBV/P5jdZwnlLn8/kYrj3mxIfz9sauJ59GF/P0aF1Nl7e38/wfco/gKVfz888k3mia5/Pxe4sJEQyH8/8Z69/5bhfz/rDeAvHft/P3I+wZBRCoA/3zkHahQXgD/MRHYj1yOAPwoywryZMIA/R1ueNVw9gD+0oL2NHkqAP6Bo0sTgVoA/HZ+O2qJjgD+ataPOZHCAP4eiwqAmfYA/9eCbUOiJgD8ycN/dqZaAP2/TPEhro4A/WxFjjyywgD/IswCz7byAP0XHw7KuyYA/wdpZjm/WgD8t/29FMOOAPxnHstfw74A/VUbORLH8gD+QEW6McQmBP/s9Pa4xFoE/5mDmqfEigT9ijxN/sS+BP95dbi1xPIE/y9+ftDBJgT84p1AU8FWBP3XEKEyvYoE/ssXPW25vgT+etuxCLXyBPwogJgHsiIE/hgcilqqVgT8D74UBaaKBP2/U9kInr4E/WzEZWuW7gT+W+pBGo8iBP9CfAQhh1YE/OwsOnh7igT8moVgI3O6BP6E/g0aZ+4E/HT4vWFYIgj8Ibf08ExWCP3MVjvTPIYI/rfiAfowugj/oT3XaSDuCP9PLCQgFSII/PpTcBsFUgj+6R4vWfGGCPzf7snY4boI/pDnw5vN6gj+QA98mr4eCP8zOGjZqlII/CIY+FCWhgj90iOTA362CP2CppjuauoI/3C8ehFTHgj9Y1uOZDtSCP0TKj3zI4II/r6u5K4Ltgj/rjPimO/qCPyby4u30BoM/EdEOAK4Tgz98kBHdZiCDP/gHgIQfLYM/c3/u9dc5gz/ervAwkEaDP8m9GTVIU4M/A0P8AQBggz8+RCqXt2yDP6g1NfRueYM/k/mtGCaGgz8N4CQE3ZKDP4emKbaTn4M/cXdLLkqsgz/c6RhsALmDPxcBIG+2xYM/UizuNmzSgz89RhDDId+DP6eUEhPX64M/IciAJoz4gz+b++X8QAWEPwW0zJX1EYQ/8N++8KkehD8r10UNXiuEP2da6uoROIQ/05I0icVEhD++EazneFGEPznQ1wUsXoQ/sy4+495qhD+d9GR/kXeEPwdQ0dlDhIQ/QtUH8vWQhD99fozHp52EP2mr4llZqoQ/1SCNqAq3hD9RCA6zu8OEP83v5nhs0IQ/OsmY+RzdhD8o6qM0zemEP2ULiCl99oQ/o0jE1ywDhT8QINc+3A+FP/5xPl6LHIU/e4B3NTophT/57v7D6DWFP+bBUAmXQoU/Ul7oBEVPhT+OiUC28luFP8to0xygaIU/uIAaOE11hT8ltY4H+oGFP6NIqIqmjoU/INzewFKbhT+Obqmp/qeFP3tcfkSqtIU/uF/TkFXBhT/1jh2OAM6FP2Fd0Tur2oU/TZpimVXnhT/JcESm//OFP0Vn6WGpAIY/MV/Dy1INhj+elEPj+xmGP9ud2qekJoY/GGv4GE0zhj8FRgw29T+GP3PRhP6cTIY/8AjQcURZhj9tQFuP62WGP9ojk1aScoY/x7bjxjh/hj8DVLjf3ouGPz6te6CEmIY/qMqXCCqlhj+SCnYXz7GGPwwhf8xzvoY/hhcbJxjLhj9wTLEmvNeGP9pyqMpf5IY/E5JmEgPxhj9MBVH9pf2GPzZ7zIpICoc/n/U8uuoWhz8YyQWLjCOHP5KcifwtMIc//WgqDs88hz/oeEm/b0mHPyJoRw8QVoc/XCOE/a9ihz/G516JT2+HP7BCNrLue4c/KxFod42Ihz+mf1HYK5WHP5IJT9TJoYc//ni8ameuhz865vSaBLuHP3e3UmShx4c/Y6Avxj3Uhz/QoeS/2eCHP00JylB17Yc/ynA3eBD6hz83voM1qwaIPyQjBYhFE4g/YRwRb98fiD+ecfzpeCyIPws1G/gROYg/+cLAmKpFiD93wT/LQlKIP/Uf6o7aXog/5BYR43FriD9SJwXHCHiIP5EaFjqfhIg/0AGTOzWRiD/BNcrKyp2IPzJWCedfqog/tEmdj/S2iD83PdLDiMOIP6qj84Ic0Ig/nTVMzK/ciD/h8CWfQumIPyYYyvrU9Yg/mzKB3mYCiT+QC5NJ+A6JPxayRjuJG4k/nHjishkoiT+S9KuvqTSJPwn+5zA5QYk/T6/aNchNiT+WZMe9VlqJP4278MfkZok/BJOYU3JziT+KCgBg/3+JPxGCZ+yLjIk/iJkO+BeZiT9/MDSCo6WJP8ZlFoousok/DZfyDrm+iT+EYAUQQ8uJP3qciozM14k//2K9g1XkiT+ECdj03fCJP3oiFN9l/Yk/73yqQe0Jij80JNMbdBaKP3pfxWz6Ioo/cLG3M4Avij/m199vBTyKP2zLciCKSIo/8r6kRA5Vij9oH6nbkWGKP12TsuQUboo/o/ryXpd6ij/pbZtJGYeKP18+3KOak4o/VfXkbBugij/dU+Sjm6yKP2VSCEgbuYo/XiB+WJrFij/XI3LUGNKKPyD5D7uW3oo/aXKCCxTrij9jl/PEkPeKP92kjOYMBIs/Zwx2b4gQiz/xc9deAx2LP2y117N9KYs/Z96cbfc1iz+yL0yLcEKLP/4cCgzpTos/eUz67mBbiz90lj8z2GeLP/8E/NdOdIs/itNQ3MSAiz+Ebl4/Oo2LP/5yRACvmYs/SK4hHiOmiz+SHRSYlrKLP4ztOG0Jv4s/B3qsnHvLiz+STYol7deLPxwh7QZe5Is/ltvuP87wiz+RkajPPf2LP92EMrWsCYw/KSSk7xoWjD+lChR+iCKMP6L/l1/1Low/LvZEk2E7jD+5DC8YzUeMP7WMae03VIw/MOoGEqJgjD96wxiFC22MP8Tgr0V0eYw/vjPcUtyFjD8616yrQ5KMP8UOME+qnow/UEZzPBCrjD/NEYNydbeMP8ksa/DZw4w/FXo2tT3QjD9hA++/oNyMP9/4nQ8D6Yw/3LBLo2T1jD9qp/95xQGNP/l9wJIlDo0/+fqT7IQajT93CX+G4yaNP8W4hV9BM40/Ezyrdp4/jT8Q6vHK+kuNP4w8W1tWWI0/F9DnJrFkjT+jY5csC3GNPx/YaGtkfY0/GTBa4ryJjT9jj2iQFJaNP606kHRroo0/J5fMjcGujT8iKhjbFruNP6yYbFtrx40/N6fCDb/TjT8xORLxEeCNP6xQUgRk7I0/9w15RrX4jT9Br3u2BQWOPzuQTlNVEY4/tCnlG6Qdjj88ETIP8imOP8P4Jiw/No4/Oa60cYtCjj8vG8ve1k6OP3dEWXIhW44/vklNK2tnjj82ZZQItHOOPy/rGgn8f44/uUnMK0OMjj9CCJNviZiOPzvHWNPOpI4/tT8GVhOxjj8AQ4P2Vr2OP0y6trOZyY4/R6aGjNvVjj/BHth/HOKOP0tSj4xc7o4/1IWPsZv6jj9NFLvt2QaPP0Zu8z8XE48/jRkZp1Mfjz/VsAsijyuPP03jqa/JN48/RXTRTgNEjz/MOl/+O1CPP1IhL71zXI8/SCUciqpojz+/VgBk4HSPPwfYtEkVgY8/Tt0ROkmNjz9FrO4zfJmPP7ybITaupY8/QROAP9+xjz/Iit5OD76PPz+KEGM+yo8/NqnoemzWjz9/jjiVmeKPP8bv0LDF7o8/PpGBzPD6jz+booxzjQOQP9/1sv+hCZA/JDkaCrYPkD+gaSiSyRWQP90KQ5fcG5A/gibPGO8hkD8nTDEWASiQPyWRzY4SLpA/Y5AHgiM0kD8qakLvMzqQP/DD4NVDQJA/L8hENVNGkD8tJtAMYkyQP9MR5FtwUpA/eUPhIX5YkD+39ydei16QPzXvFxCYZJA/e24QN6RqkD/BPXDSr3CQP7+oleG6dpA/fX7eY8V8kD8iEahYz4KQP8c1T7/YiJA/REQwl+GOkD8BF6ff6ZSQP8YKD5jxmpA/i/7Cv/igkD9HUx1W/6aQP8Trd1oFrZA/aSwszAqzkD8O+5KqD7mQP8u+BPUTv5A/yV/ZqhfFkD8OR2jLGsuQP1ReCFYd0ZA/0g8QSh/XkD8PRtWmIN2QP7VrrWsh45A/XGvtlyHpkD9cr+kqIe+QP5sh9iMg9ZA/Yitmgh77kD8ptYxFHAGRP2kmvGwZB5E/aGVG9xUNkT8P13zkERORP7desDMNGZE/910x5AcfkT92tE/1ASWRP76/Wmb7KpE/BluhNvQwkT8F33Fl7DaRP8QhGvLjPJE/a3bn29pCkT8SrSYi0UiRP5ASJMTGTpE/T3ArwbtUkT8WDIgYsFqRP92nhMmjYJE/m4Fr05ZmkT8ZU4Y1iWyRP8BRHu96cpE/aC58/2t4kT8oFehlXH6RPyitqSFMhJE/cBgIMjuKkT+380mWKZCRPzVWtU0XlpE/c9GPVwSckT8ZcR6z8KGRP7+6pV/cp5E/vq1pXMetkT/9wq2osbORP8TstEObuZE/i5bBLIS/kT/KpBVjbMWRP8l08uVTy5E/cdyYtDrRkT8YKknOINeRP1ckQzIG3ZE/1gnG3+rikT8dkRDWzuiRP2ToYBSy7pE/Y7X0mZT0kT8jFQlmdvqRP8ub2ndXAJI/c1SlzjcGkj/zwKRpFwySP7PZE0j2EZI/ew0tadQXkj9CQSrMsR2SPwHQRHCOI5I/gYq1VGopkj8pt7R4RS+SP9ERetsfNZI/kss8fPk6kj+SijNa0kCSP9lplHSqRpI/IPmUyoFMkj+fPGpbWFKSP92sSCYuWJI/gzZkKgNekj8pOvBm12OSPyaMH9uqaZI/YnQkhn1vkj8nrjBnT3WSP+xndX0ge5I/KkMjyPCAkj8nVGpGwIaSP8wheveOjJI/caWB2lySkj+tSq/uKZiSPynvMDP2nZI/buIzp8Gjkj+05eRJjKmSP7ErcBpWr5I/blgBGB+1kj8UgcNB57qSP7or4ZauwJI/N0+EFnXGkj/0Uta/OsySP7kOAJL/0ZI/fcopjMPXkj85Pnutht2SP7WRG/VI45I/WVwxYgrpkj/8pOLzyu6SP7fhVKmK9JI/s/esgUn6kj/4Og98BwCTPztun5fEBZM/tsKA04ALkz/x19UuPBGTP5S7wKj2FpM/N+liQLAckz8ySt30aCKTP201UMUgKJM/MG/bsNctkz/zKJ62jTOTPy4Bt9VCOZM/KANEDfc+kz/KpmJcqkSTP2zQL8JcSpM/p9DHPQ5Qkz8iZEbOvlWTP2SzxnJuW5M/plJjKh1hkz+fQTb0ymaTP1jrWM93bJM/9yXkuiNykz+XMvC1zneTPw+9lL94fZM/x9vo1iGDkz+HDwP7yYiTP0dD+SpxjpM//8vgZReUkz93aM6qvJmTPxdB1vhgn5M/t+cLTwSlkz9vV4KspqqTP2f0SxBIsJM/p4t6eei1kz/nUh/nh7uTP1/oSlgmwZM/l1INzMPGkz83AHZBYMyTP9fHk7f70ZM/z+d0LZbXkz8IBieiL92TP8kvtxTI4pM/itkxhF/okz/C3qLv9e2TP7mBFVaL85M/WGuUth/5kz/3qikQs/6TPy623mFFBJQ/pWi8qtYJlD/kA8vpZg+UPyIvEh72FJQ/F/eYRoQalD/MzWViESCUP2mKfnCdJZQ/B2nobygrlD98CqhfsjCUPzJ0wT47NpQ/8A84DMM7lD+tqw7HSUGUP2J5R27PRpQ/1w7kAFRMlD91ZeV911GUPxLaS+RZV5Q/yCwXM9tclD++gEZpW2KUP/1b2IXaZ5Q/O6fKh1htlD+yrRpu1XKUP+kcxTdReJQ/iATG48t9lD8n1hhxRYOUPx1luN69iJQ/U+aeKzWOlD8R8MVWq5OUP895Jl8gmZQ/Bty4Q5SelD/9z3QDB6SUP5xvUZ14qZQ/PDVFEOmulD90+0VbWLSUP+v8SH3GuZQ/KdRCdTO/lD9neydCn8SUP15M6uIJypQ/FgB+VnPPlD+2rtSb29SUP1fP37FC2pQ/zzeQl6jflD+HHNZLDeWUP0YQoc1w6pQ/BQTgG9PvlD+8RoE1NPWUPzKFchmU+pQ/0MmgxvL/lD9ufPg7UAWVPyRiZXisCpU/Gp3SegcQlT9ZrCpCYRWVP5hrV825GpU/EBNCGxEglT9IN9MqZyWVP+jI8vq7KpU/iBSIig8wlT+BwnnYYTWVP7nWreOyOpU/erAJqwJAlT86CnItUUWVP3L5ymmeSpU/au73XupPlT8JtNsLNVWVP6dvWG9+WpU/3aBPiMZflT9UIaJVDWWVP5MkMNZSapU/0jfZCJdvlT/JQXzs2XSVP4CC938bepU/H5Mowlt/lT+9ZeyxmoSVPzNFH07YiZU/6tSclRSPlT+pEECHT5SVP2hM4yGJmZU/HzRgZMGelT+Vy49N+KOVPzNuStwtqZU/0M5nD2KulT+G977llLOVP3tJJl7GuJU/uHxzd/a9lT/1n3swJcOVP2oYE4hSyJU/nqENfX7NlT87TT4OqdKVP9mCdzrS15U/z/+KAPrclT8F10lfIOKVP8NwhFVF55U/gYoK4mjslT+3NqsDi/GVP63cNLmr9pU/TDh1Acv7lT/qWTnb6ACWPyCmTUUFBpY/ldV9PiALlj/T9JTFORCWPxFkXdlRFZY/B9egeGgalj+9VCiifR+WP1o3vFSRJJY/+Cskj6Mplj9uMidQtC6WPyOdi5bDM5Y/4BAXYdE4lj+dhI6u3T2WP1JBtn3oQpY/x+FRzfFHlj9kUiSc+UyWPwHR7+j/UZY/tux1sgRXlj+qhXf3B1yWP+jMtLYJYZY/JkTt7glmlj+cvd+eCGuWP9JbSsUFcJY/cJHqYAF1lj8OIX1w+3mWPwMdvvLzfpY/OOdo5uqDlj/1MDhK4IiWP7L65RzUjZY/5pMrXcaSlj/amsEJt5eWP3b8XyGmnJY/EvS9opOhlj9GC5KMf6aWP7oZkt1pq5Y/9URzlFKwlj8wAOqvObWWPyMMqi4fupY/1nZmDwO/lj9wm9FQ5cOWPwsinfHFyJY/fv958KTNlj8xdRhMgtKWP+0QKANe15Y/qaxXFDjclj9dblV+EOGWP9HHzj/n5ZY/bXZwV7zqlj8Jg+bDj++WP75B3INh9JY/slH8lTH5lj/tnPD4//2WPyhYYqvMApc/mwL6q5cHlz/OZV/5YAyXP2mVOZIoEZc/Be8ude4Vlz/4GeWgshqXPysHARR1H5c/5/AmzTUklz+kWvrK9CiXP9kQHgyyLZc/zig0j20ylz9rAN5SJzeXPwg+vFXfO5c/PdBulpVAlz+x7ZQTSkWXP+0Uzcv8SZc/KQy1va1Olz8e4ennXFOXP9LoB0kKWJc/br+q37Vclz8KSG2qX2GXP36s6acHZpc/MV251q1qlz/tEHU1Um+XP6nEtML0c5c/XLsPfZV4lz/PfRxjNH2XP2racHPRgZc/BeWhrGyGlz+39kMNBouXP6it6pOdj5c/4uwoPzOUlz8c3JANx5iXP43ns/1YnZc/vr8iDumhlz9WWW09d6aXP+/sIooDq5c/3/bR8o2vlz8QNwh2FrSXP8mwUhKduJc/gqo9xiG9lz+0rVSQpMGXP6eGIm8lxpc/QkQxYaTKlz/dNwplIc+XPw/1NXmc05c/gVE8nBXYlz+7ZKTMjNyXP/SH9AgC4Zc/5VWyT3Xllz+WqmKf5umXPzCjifZV7pc/yp2qU8Pylz88OUi1LveXP+5U5BmY+5c/qRAAgP//lz9jzBvmZASYPxYot0rICJg/iQNRrCkNmD8jfmcJiRGYP732d2DmFZg/cAv/r0EamD9jmXj2mh6YP528XzLyIpg/188uYkcnmD9KbF+EmiuYP31papfrL5g/F93HmTo0mD+yGu+JhziYP6SzVmbSPJg/13Z0LRtBmD+ScL3dYUWYP03qpXWmSZg/gGqh8+hNmD9ztCJWKVKYPw3Im5tnVpg/qOF9wqNamD/beTnJ3V6YP05FPq4VY5g/izT7b0tnmD/Ic94Mf2uYP71qVYOwb5g/crzM0d9zmD8QR7D2DHiYP64ja/A3fJg/JaZnvWCAmD/cXA9ch4SYP5kQy8qriJg/VsQCCM6MmD8LtR0S7pCYP4FZgucLlZg/IGKWhieZmD+9uL7tQJ2YP3OAXxtYoZg/aRXcDW2lmD+mDJfDf6mYP+Qz8jqQrZg/WpFOcp6xmD+QYwxoqrWYPy4hixq0uZg/y3gpiLu9mD/AUEWvwMGYP/TGO47DxZg/sTBpI8TJmD9uGiltws2YP6NH1mm+0Zg/l7LKF7jVmD8yjF91r9mYP8077YCk3Zg/AF/LOJfhmD9yyVCbh+WYP6uE06Z16Zg/48+oWWHtmD/UHyWySvGYP4UenK4x9Zg/HatgTRb5mD+12cSM+PyYPyXzGWvYAJk/1nSw5rUEmT+PENj9kAiZP0es365pDJk/+GEV+D8QmT9pf8bXExSZPwOGP0zlF5k/nCrMU7QbmT9OVbfsgB+ZPz8hSxVLI5k/edzQyxInmT+zB5EO2CqZPyRW09uaLpk/Va3eMVsymT/tJPkOGTaZP4UGaHHUOZk/dc1vV409mT+mJlS/Q0GZP1/wV6f3RJk/Fzq9DalImT9IRMXwV0yZPzqAsE4EUJk/1Y++Ja5TmT9wRS50VVeZP6KjPTj6Wpk/Fd0pcJxemT9RVC8aPGKZP42biTTZZZk/gHRzvXNpmT800CazC22ZP8/O3BOhcJk/ar/N3TN0mT/eHzEPxHeZP5KcPaZRe5k/TRApodx+mT8IhCj+ZIKZP7wucLvqhZk/MHUz122JmT/N6aRP7oyZP2pM9iJskJk/H4pYT+eTmT8VvfvSX5eZP1IsD6zVmpk/jkvB2EiemT8Duz9XuaGZPzhHtyUnpZk/1uhTQpKomT92xECr+quZP20qqF5gr5k/opazWsOymT9gsIudI7aZPx9KWCWBuZk/VWFA8Nu8mT9NHmr8M8CZP+zT+keJw5k/i/8W0dvGmT/CSOKVK8qZPzqBf5R4zZk/eaQQy8LQmT+417Y3CtSZP7BpkthO15k/aNLCq5DamT8Ks2avz92ZP6vVm+EL4Zk/Iy1/QEXkmT/a1CzKe+eZP5oQwHyv6pk/WUxTVuDtmT8RHABVDvGZP4k733Y59Jk/KY4IumH3mT/KHpMch/qZP4MflZyp/Zk/e+kjOMkAmj+7/FPt5QOaP/z/OLr/Bpo/dMDlnBYKmj+uMWyTKg2aP09t3Zs7EJo/8LJJtEkTmj/oZ8DaVBaaPyAXUA1dGZo/33AGSmIcmj+fSvCOZB+aP9aeGdpjIpo/zYyNKWAlmj9tWFZ7WSiaPw1qfc1PK5o/RE4LHkMumj+7tQdrMzGaP/p0ebIgNJo/OIRm8go3mj8u/9Mo8jmaP+UkxlPWPJo/g1dAcbc/mj8hHEV/lUKaP5ga1ntwRZo/Th30ZEhImj8NEZ84HUuaP8wE1vTuTZo/gimXl71Qmj/50d8eiVOaP5hyrIhRVpo/N6H40hZZmj/vFL/72FuaP+al+QCYXpo/JU2h4FNhmj9kJK6YDGSaP9tlFyfCZpo/EWzTiXRpmj+vsde+I2yaP0/RGMTPbpo/RoWKl3hxmj9/px83HnSaP0AxyqDAdpo/ADt70l95mj84/CLK+3uaPzDLsIWUfpo/0BwTAyqBmj9yhDdAvIOaP6uzCjtLhpo/I3p48daImj9jxWthX4uaP6OgzojkjZo/mTSKZWaQmj9Qx4b15JKaP++7qzZglZo/jZLfJtiXmj8E6AfETJqaP7t1CQy+nJo/eRHI/Cufmj83rSaUlqGaP+1WB9D9o5o/YjhLrmGmmj//ltIswqiaP5vTfEkfq5o/TmooAnmtmj9B8rJUz6+aP30d+T4ispo/ubjWvnG0mj8tqybSvbaaP2L2wnYGuZo//bWEqku7mj+aH0Rrjb2aP4+C2LbLv5o/xEcYiwbCmj+B8djlPcSaPz4b78Rxxpo/cnkuJqLImj9n2WkHz8qaPwQhc2b4zJo/oU4bQR7Pmj/VeDKVQNGaP0rOh2Bf05o/hpXpoHrVmj/DLCVUkteaP7cJB3im2Zo/a7laCrfbmj8H4OoIxN2aP6Q4gXHN35o/GZXmQdPhmj/O3eJ31eOaP4oRPRHU5Zo/RUW7C8/nmj/3oyJlxumaP2puNxu665o/BPu8K6rtmj+ftXWUlu+aP1MfI1N/8Zo/R86FZWTzmj+CbV3JRfWaP768aHwj95o/MZBlfP34mj9m0BDH0/qaPwJ6Jlqm/Jo/np1hM3X+mj+SX3xQQACbP8b3L68HAps/gbE0TcsDmz8860EoiwWbP3AWDj5HB5s/ZLdOjP8Imz8AZbgQtAqbP5rI/shkDJs/zZ3UshEOmz9AsuvLug+bP3zl9BFgEZs/tyigggETmz+rfpwbnxSbP2D7l9o4Fps//cM/vc4Xmz+bDkDBYBmbPxAiROTuGps/xFX2I3kcmz+BEQB+/x2bPz7NCfCBH5s/8xC7dwAhmz9pdLoSeyKbPwefrb7xI5s/pEc5eWQlmz9ZNAFA0yabP086qBA+KJs/jD3Q6KQpmz/KMBrGByubPz8VJqZmLJs/dPqShsEtmz8Q/v5kGC+bP6xLBz9rMJs/oBxIEroxmz/Ut1zcBDObP5Bx35pLNJs/TatpS441mz+E05PrzDabP3pl9XgHOJs/GOkk8T05mz+48rdRcDqbP/AiQ5ieO5s/aCZawsg8mz+otY/N7j2bP+eUdbcQP5s/3ZOcfS5Amz+UjZQdSEGbPzNo7JRdQps/0hQy4W5Dmz9Jj/L/e0SbPwDeue6ERZs/vhETq4lGmz98RYgyikebPzGeooKGSJs/pUrqmH5Jmz9Ag+ZyckqbP9qJHQ5iS5s/jKkUaE1Mmz9/NlB+NE2bP7qNU04XTps/9BSh1fVOmz9mOroR0E+bP5h0HwCmUJs/MUJQnndRmz/LKcvpRFKbP725DeANU5s/8IeUftJTmz+rMdvCklSbP2ZbXKpOVZs/mLCRMgZWmz+J4/NYuVabPyOt+hpoV5s/vMwcdhJYmz/uB9BnuFibP2Eqie1ZWZs/nAW8BPdZmz/YcNuqj1qbP81IWd0jW5s/gW+mmbNbmz8dzDLdPlybP7hKbaXFXJs/KtzD70ddmz/cdaO5xV2bP5YReAA/Xps/UK2swbNemz8CS6v6I1+bP3Tw3KiPX5s/DaepyfZfmz+oe3haWWCbP1x+r1i3YJs/T8KzwRBhmz+LXemSZWGbP8dos8m1YZs/Ov9zYwFimz9tPoxdSGKbPwhGXLWKYps/ojdDaMhimz+VNp9zAWObP8lnzdQ1Y5s/hPEpiWVjmz8++w+OkGObP3Gt2eC2Y5s/YjHgfthjmz/6sHtl9WObP5JWA5INZJs/wkzNASFkmz8yvi6yL2SbP2rVe6A5ZJs/orwHyj5kmz+SnSQsP2SbP0GhI8Q6ZJs/2O9UjzFkmz9vsAeLI2SbP90IirQQZJs/ih0pCfljmz9AETGG3GObP/YE7Si7Y5s/pRen7pRjmz8TZqjUaWObP6oKOdg5Y5s/Qh2g9gRjmz/xsiMty2KbP+DdCHmMYps/Ga2T10himz9RLAdGAGKbP8FjpcGyYZs/8FevR2Bhmz+HCWXVCGGbPx51BWisYJs/DZPO/Epgmz88V/2Q5F+bP/OwzSF5X5s/qop6rAhfmz/ZyT0uk16bP8dOUKQYXps/X/TpC5ldmz/1j0FiFF2bPyLxjKSKXJs/kOEA0Ptbmz/EJNHhZ1ubP/p3MNfOWps/55FQrTBamz+UImJhjVmbPyrTlPDkWJs/wEUXWDdYmz8vFReVhFebP93UwKTMVps/kxBAhA9Wmz9JTL8wTVWbP/cDaKeFVJs/Zati5bhTmz/5rdbn5lKbP41u6qsPUps/OkfDLjNRmz8niYVtUVCbP118VGVqT5s/kV9SE35Omz/8Z6B0jE2bPyfBXoaVTJs/uoysRZlLmz9N4qevl0qbPzfPbcGQSZs/YlYaeIRImz8UcMjQckebP8cJkshbRps/8gWQXD9Fmz/dO9qJHUSbP3F3h032Qps/BXmtpMlBmz8y9WCMl0CbP6CUtQFgP5s/1vO9ASM+mz8Lo4uJ4DybP/glL5aYO5s/pfO3JEs6mz86djQy+DibP88KsrufN5s/PQE9vkE2mz/rm+A23jSbP6EPpyJ1M5s/V4OZfgYymz8EEMBHkjCbP3HAIXsYL5s/BpHEFZktmz+bb60UFCybP0k74HSJKps/N8RfM/komz9syy1NYyebP6ECS7/HJZs/Dgy3hiYkmz88enCgfyKbP9HPdAnTIJs/Zn/AviAfmz9U6069aB2bP4NlGgKrG5s/Oi8ciucZmz/xeExSHhibPyBioldPFps/D/kTl3oUmz+mOpYNoBKbPz0SHbi/EJs/bFmbk9kOmz/c1wKd7QybPxRDRNH7Cps/TD5PLQQJmz88WhKuBgebP+4Ue1ADBZs/h9l1EfoCmz8fAO7t6gCbP4/NzeLV/po/P3P+7Lr8mj/3DmgJmvqaP66q8TRz+Jo/XDyBbEb2mj/KpfusE/SaP160RPPa8Zo/8yA/PJzvmj+gj8yEV+2aP42PzckM65o/wpohCLzomj/4Fac8ZeaaP2ZQO2QI5Jo/lIO6e6Xhmj8q0/9/PN+aP79M5W3N3Jo/redDQljamj/bhPP53NeaP5HuypFb1Zo/RtifBtTSmj9z3kZVRtCaP1+Gk3qyzZo/8j1YcxjLmj+GW2Y8eMiaP7MdjtLRxZo/H6ueMiXDmj9TEmZZcsCaP4lJsUO5vZo/di5M7vm6mj8jhgFWNLiaP7j8mndotZo/TiXhT5aymj+7eZvbva+aP2hakBffrJo/HQ6FAPqpmj/SwT2TDqeaP36IfcwcpJo/6loGqSShmj9/F5klJp6aPxSC9T4hm5o/wEPa8RWYmj+t6gQ7BJWaP+LpMRfskZo/Fpkcg82Omj+ENH97qIuaP7LcEv18iJo/R5aPBEuFmj/dSayOEoKaP8zDHpjTfpo/+bObHY57mj+vrdYbQniaP2Mngo/vdJo/j3pPdZZxmj964+7JNm6aPw2BD4rQapo/oFRfsmNnmj/LQYs/8GOaPzcOPy52YJo/a2Ele/Vcmj+exOciblmaP4miLiLgVZo/NEehdUtSmj/G3+UZsE6aP1h6oQsOS5o/wQV4R2VHmj9rUQzKtUOaPxwNAJD/P5o/zsjzlUI8mj939IbYfjiaP97fV1S0NJo/b7oDBuMwmj//kibqCi2aP6hXW/0rKZo/k9U7PEYlmj/GuGCjWSGaP/iLYS9mHZo/YbjU3GsZmj+LhU+oahWaPx0ZZo5iEZo/r3ari1MNmj+Zf7GcPQmaP8PyCL4gBZo/dGxB7PwAmj8lZukj0vyZP042jmGg+Jk/NxC8oWf0mT/IA/7gJ/CZP1n93Rvh65k/g8XkTpPnmT/sAJp2PuOZPx4whI/i3pk/UK8oln/amT87tguHFdaZP+dXsF6k0Zk/e4KYGSzNmT8O/0S0rMiZP3pxNSsmxJk/JVjoepi/mT/XC9ufA7uZP4q/iZZntpk/NYBvW8SxmT+gNAbrGa2ZPzKdxkFoqJk/xVMoXK+jmT9wy6E2756ZP1tQqM0nmpk/jgewHVmVmT/A7isjg5CZPyrcjdqli5k/U35GQMGGmT/lW8VQ1YGZP3fTeAjifJk/YBvOY+d3mT+JQTFf5XKZPzorDffbbZk/7JTLJ8tomT8WEtXtsmOZPwENkUWTXpk/k8ZlK2xZmT8lVribPVSZP0+p7JIHT5k/uINlDcpJmT/ofoQHhUSZPxkKqn04P5k/AWo1bOQ5mT+quITPiDSZPzzl9KMlL5k/zrPh5bopmT85vaWRSCSZP+RumqPOHpk/lwoYGE0ZmT9LpnXrwxOZP/crCRozDpk/YVknoJoImT/0vyN6+gKZP4bEUKRS/Zg/MZ//GqP3mD8cW4Da6/GYP0/WId8s7Jg/gsExJWbmmD/sn/yol+CYPxfHzWbB2pg/qV7vWuPUmD87YKqB/c6YPySXRtcPyZg/T6AKWBrDmD8C6jsAHb2YP7SzHswXt5g/3g32twqxmD/I2QPA9aqYP1rJiODYpJg/7F7EFbSemD8W7fRbh5iYP4CWV69Skpg/s00oDBaMmD/k1KFu0YWYP869/dKEf5g/eGl0NTB5mD8LCD2S03KYP56YjeVubJg/CumaKwJmmD+1lZhgjV+YP2gJuYAQWZg/HH0tiItSmD/I9yVz/kuYPzRO0T1pRZg/yCJd5Ms+mD9b5fViJjiYPwfTxrV4MZg/9PX52MIqmD8oJbjIBCSYP1wEKYE+HZg/yANz/m8WmD/0X7s8mQ+YP4khJji6CJg/Hh3W7NIBmD8L8+xW4/qXPzgPi3Lr85c/7ajPO+vslz+jwtiu4uWXP9Epw8fR3pc/v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       "  var render_items = [{\"docid\":\"45111807-b885-48c5-877d-ad11cefc4dc3\",\"roots\":{\"1108\":\"5bc763dd-5187-419c-b169-df577e331312\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
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       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
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       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
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       "    }, 10, root)\n",
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      "application/vnd.bokehjs_exec.v0+json": ""
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     "output_type": "display_data"
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   ],
   "source": [
    "# Image de xi par u\n",
    "def u(x) : \n",
    "    return x*(1-x)*(1+x)*(1+x*x)/20\n",
    "yi = u(xi)\n",
    "\n",
    "# Erreur\n",
    "err = abs(ui-yi)\n",
    "\n",
    "# Figure\n",
    "fig = figure(width=490, height=300)\n",
    "fig.line(xi,ui, legend = \"ui\")\n",
    "fig.line(xi,yi, color = \"orange\", legend = \"u(xi)\")\n",
    "fig.legend.location = \"top_left\"\n",
    "\n",
    "fig2 = figure(width=490, height=300, x_axis_type=\"log\", y_axis_type=\"log\")\n",
    "fig2.line(xi, err, legend =\"|u(xi)-ui|\")\n",
    "fig2.legend.location = \"top_left\"\n",
    "\n",
    "show(row(fig,fig2))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :** Les deux courbes $(x_i,u_i)$ et $(x_i, u(x_i))$ sont identiques à une erreur près qui ne dépasse jamais $10^{-8}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matrice tridiagonale\n",
    "\n",
    "**Implémenter l'algorithme de décomposition LU basé sur un stockage tridiagonal de la matrice, c'est-à-dire qu'on ne stocke que les composantes non-triviales des matrice $A$, $L$ et $U$. On écrira pour cela :**\n",
    "\n",
    "\\begin{gather}\n",
    "      \\tilde{A} = \\left( \\begin{array}{c|c|c|c} 0 & b_{1} & \\dots & b_{N-1}\\\\ \\hline  a_{1} & a_{2} & \\dots & a_{N} \\\\ \\hline c_{2} & c_{3} & \\dots  & 0 \\end{array} \\right) = \\left( \\begin{array}{l} b \\\\ a \\\\ c \\end{array}\\right) \\in \\mathbb{R}^{3,N}, \\nonumber \\\\\n",
    "      \\tilde{L} = \\left( \\gamma_{2}, \\gamma_{3}, \\dots, \\gamma_{N}, 0 \\right) \\in\\mathbb{R}^{1,N}, \\qquad\n",
    "      \\tilde{U} = \\left( \\begin{array}{c|c|c|c|c} 0 & \\beta_2 & \\beta_3 & \\dots & \\beta_N \\\\ \\hline \\alpha_{1} & \\alpha_{2} & \\alpha_{3} & \\dots & \\alpha_{N} \\end{array} \\right) \\in\\mathbb{R}^{2,N}.\n",
    "      %\\tilde{A} = \\left( \\begin{array}{c|c|c} 0 & A_{1,1} & A_{1,2} \\\\ A_{2,1} & A_{2,2} & A_{2,3} \\\\ \\vdots & \\vdots & \\vdots \\\\  A_{N,N-1} & A_{N,N} & 0 \\end{array} \\right) \\in \\mathbb{R}^{3,N}, \\nonumber \\\\\n",
    "      %\\tilde{L} = \\left( \\begin{array}{c} 0 \\\\ L_{2,1} \\\\ L_{3,2} \\\\ \\vdots \\\\ L_{N,N-1}\\end{array}\\right) \\in\\mathbb{R}^{N,1}, \\qquad\n",
    "      %\\tilde{U} = \\left( \\begin{array}{c|c} U_{1,1} & U_{1,2} \\\\ \\vdots & \\vdots \\\\ U_{N-1,N-1} & U_{N-1,N} \\\\ U_{N,N} & 0\\end{array} \\right) \\in\\mathbb{R}^{N,2}.\n",
    "        %= (0, \\ A_{1,1},\\ A_{1,2}), \\quad \\tilde{A}_{1,i} = (A_{i,i-1}, \\ A_{i,i}, \\ A_{i,i+1}), \\quad \\tilde{A}_{N,:} = (A_{N,N-1},\\ A_{N,N}, \\ 0).\n",
    "\\end{gather}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :**  \n",
    "En explicitant le produit $L.U$, et sachant que $A = L.U$, on obtient :\n",
    "$$ a_1 = \\alpha_1, \\quad \\forall i\\geq 2,\\quad a_i = \\alpha_i + \\gamma_i \\beta_i,\\quad c_i = \\gamma_i \\alpha_{i-1}, \\quad b_{i-1} =\\beta_i $$ \n",
    "Les coefficients $\\alpha_i, \\beta_i, \\gamma_i$ peuvent donc être calculés récursivement comme suit :\n",
    "$$\\forall i\\geq 2, \\quad \\beta_i = b_{i-1}$$\n",
    "$$\\alpha_1 = a_1, \\quad \\forall i\\geq 2, \\quad \\gamma_i = c_i/\\alpha_{i-1}, \\quad \\alpha_i = a_i - b_{i-1}\\gamma_i$$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "# stock_tridiag() prend en paramètre une matrice A de taille (N,N)\n",
    "# et retourne une matrice de taille (3,N) correspont au stockage tridiagonal de A\n",
    "def stock_tridiag(A) :\n",
    "    a = np.array([A[i][i] for i in range(len(A))])\n",
    "    b = np.array([0]+[A[i-1][i] for i in range(1,len(A))])\n",
    "    c = np.array([A[i][i-1] for i in range(1,len(A))]+[0])\n",
    "    return np.vstack((b,a,c))\n",
    "\n",
    "# decompLU_tridiag() prend en argument un stockage tridiagonal At d'une matrice A\n",
    "# et retourne le stockage tridiagonal de sa décompostion LU\n",
    "def decompLU_tridiag(At) :\n",
    "    b, a, c = At[0], At[1], At[2]\n",
    "    N = len(b)\n",
    "    alpha = np.zeros(N)\n",
    "    gamma = np.zeros(N)\n",
    "    alpha[0] = a[0]\n",
    "    for i in range(1,N) :\n",
    "        gamma[i-1] = c[i-1]/alpha[i-1]\n",
    "        alpha[i] = a[i] - b[i]*gamma[i-1]\n",
    "    Lt = gamma\n",
    "    Ut = np.vstack((b, alpha))\n",
    "    return Lt, Ut"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Tester votre algorithme sur la matrice :**\n",
    "\n",
    "\\begin{equation} \n",
    "A = \\left(\\begin{array}{ccc} 2 & 1 & 0 \\\\ -1 & 3 & 1 \\\\ 0 & 1 & 4 \\end{array}\\right).\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "La matrice pleine A : \n",
      " [[ 2  1  0]\n",
      " [-1  3  1]\n",
      " [ 0  1  4]]\n",
      "Stockage tridiagonal de A :\n",
      "  [[ 0  1  1]\n",
      " [ 2  3  4]\n",
      " [-1  1  0]]\n",
      "La matrice pleine L : \n",
      " [[ 1.          0.          0.        ]\n",
      " [-0.5         1.          0.        ]\n",
      " [ 0.          0.28571429  1.        ]]\n",
      "Stockage tridiagonal de L :\n",
      "  [-0.5         0.28571429  0.        ]\n",
      "La matrice pleine U : \n",
      " [[2.         1.         0.        ]\n",
      " [0.         3.5        1.        ]\n",
      " [0.         0.         3.71428571]]\n",
      "Stockage tridiagonal de U :\n",
      "  [[0.         1.         1.        ]\n",
      " [2.         3.5        3.71428571]]\n"
     ]
    }
   ],
   "source": [
    "# Test de validité des fonctions stock_tridiag() et decompLU_tridiag()\n",
    "A = np.array([[2,1,0],[-1,3,1],[0,1,4]])\n",
    "P, L, U = decompLU(A)\n",
    "\n",
    "At = stock_tridiag(A)\n",
    "Lt, Ut = decompLU_tridiag(At)\n",
    "\n",
    "print(\"La matrice pleine A : \\n\",A)\n",
    "print(\"Stockage tridiagonal de A :\\n \" ,At)\n",
    "print(\"La matrice pleine L : \\n\",L)\n",
    "print(\"Stockage tridiagonal de L :\\n \" ,Lt)\n",
    "print(\"La matrice pleine U : \\n\",U)\n",
    "print(\"Stockage tridiagonal de U :\\n \" ,Ut)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :** Les résultats obtenus sont corrects."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Adapter et implémenter les algorithmes de remontée et de descente adaptés aux matrices triangulaires stockées sous forme tridiagonale. Tester votre algorithme pour la résolution de $A u = f$ avec la matrice $A$ de question précedente et $f = (1,\\ 1,\\ 1)^T$, en utilisant la décomposition $LU$ de $A$ obtenue à la question précédente.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :**  \n",
    "Ecrire $Au = -f$ revient à écrire $L(U.u) = -f$, notre problème se ramène donc à la résolution du problème \n",
    "$$ U.u = y, \\quad L.y = -f$$\n",
    "En explicitant les coefficients de $L$ et $U$, on obtient :  \n",
    "$y_1 = f_1$  \n",
    "$\\forall i\\geq 1 , \\quad y_i = f_i - \\gamma_i y_{i-1}$  \n",
    "$u_N = y_N/\\alpha_N$  \n",
    "$\\forall i \\geq 1 , \\quad u_{N-i} = (y_{N-i} - \\beta_{N-i}u_{N-i+1})/\\alpha_{N-i}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "# solve_L_descente() prend en paramètre Lt, correspondant au tridiagonal de L, et un vecteur f de même taille\n",
    "# retourne la solution de L.y = f, calculée avec un algorithme de descente\n",
    "def solve_L_descente(Lt,f) :\n",
    "    y = np.zeros(len(f))\n",
    "    y[0] = f[0]\n",
    "    for i in range(1,len(f)) :\n",
    "        y[i] = f[i] - Lt[i-1]*y[i-1]\n",
    "    return y\n",
    "\n",
    "# solve_U_remontee() prend en paramètre Ut, correspondant au tridiagonal de U, et un vecteur y\n",
    "# retourne la solution de U.u = y, calculée avec un algorithme de remontée\n",
    "def solve_U_remontee(Ut, y) :\n",
    "    alpha, beta  = Ut[1], Ut[0]\n",
    "    u = np.zeros(len(y))\n",
    "    u[-1] = y[-1]/alpha[-1]\n",
    "    for i in range(2,len(y)+1) :\n",
    "        u[-i] = (y[-i] - beta[-i+1]*u[-i+1])/alpha[-i]\n",
    "    return u\n",
    "\n",
    "# solveLU_tridiag() prend en paramètre une matrice pleine A, et un vecteur f\n",
    "# retourne la solution de A.u = f, calculée avec un algorithme de remontée et descente\n",
    "def solveLU_tridiag(A,f) :\n",
    "    A = stock_tridiag(A)\n",
    "    Lt, Ut = decompLU_tridiag(A)\n",
    "    y = solve_L_descente(Lt,f)\n",
    "    u = solve_U_remontee(Ut, y)\n",
    "    return u\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Problème : résoudre le système -Au=f, pour f = (1,1,1)\n",
      "Le vecteur solution obtenu avec l'algorithme de remontée et de déscente est u= [-0.30769231 -0.38461538 -0.15384615]\n",
      "Vérification du résultat : A*u =  [-1. -1. -1.]\n"
     ]
    }
   ],
   "source": [
    "f = np.array([1,1,1])\n",
    "u = solveLU_tridiag(A,-f)\n",
    "print(\"Problème : résoudre le système -Au=f, pour f = (1,1,1)\")\n",
    "print(\"Le vecteur solution obtenu avec l'algorithme de remontée et de déscente est u=\",u)\n",
    "print(\"Vérification du résultat : A*u = \",np.dot(A,u))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :** J'ai choisi de donner à **solveLU_tridiag()** A comme paramètre et pas At, pour avoir les mêmes entrées (A,f) que l'algorithme de résolution utilisé dans la question 1/, ce qui nous permettra de comparer plus fiablement les deux algorithmes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Résoudre le système $-Au = f$ avec la matrice $A$ issue de la discrétisation de l'opérateur $-\\Delta$ . Comparer la précision des résultats et les temps d'execution (en utilisant *time.time*) avec ceux obtenus avec la matrice pleine. Commenter les avantages et inconvénients de ce stockage.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Le temps de la résolution du système A.u = -f est  0.03598284721374512 s\n"
     ]
    },
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V0eYw/vjPcUtyFjD8616yrQ5KMP8UOME+qnow/UEZzPBCrjD/NEYNydbeMP8ksa/DZw4w/FXo2tT3QjD9hA++/oNyMP9/4nQ8D6Yw/3LBLo2T1jD9qp/95xQGNP/l9wJIlDo0/+fqT7IQajT93CX+G4yaNP8W4hV9BM40/Ezyrdp4/jT8Q6vHK+kuNP4w8W1tWWI0/F9DnJrFkjT+jY5csC3GNPx/YaGtkfY0/GTBa4ryJjT9jj2iQFJaNP606kHRroo0/J5fMjcGujT8iKhjbFruNP6yYbFtrx40/N6fCDb/TjT8xORLxEeCNP6xQUgRk7I0/9w15RrX4jT9Br3u2BQWOPzuQTlNVEY4/tCnlG6Qdjj88ETIP8imOP8P4Jiw/No4/Oa60cYtCjj8vG8ve1k6OP3dEWXIhW44/vklNK2tnjj82ZZQItHOOPy/rGgn8f44/uUnMK0OMjj9CCJNviZiOPzvHWNPOpI4/tT8GVhOxjj8AQ4P2Vr2OP0y6trOZyY4/R6aGjNvVjj/BHth/HOKOP0tSj4xc7o4/1IWPsZv6jj9NFLvt2QaPP0Zu8z8XE48/jRkZp1Mfjz/VsAsijyuPP03jqa/JN48/RXTRTgNEjz/MOl/+O1CPP1IhL71zXI8/SCUciqpojz+/VgBk4HSPPwfYtEkVgY8/Tt0ROkmNjz9FrO4zfJmPP7ybITaupY8/QROAP9+xjz/Iit5OD76PPz+KEGM+yo8/NqnoemzWjz9/jjiVmeKPP8bv0LDF7o8/PpGBzPD6jz+booxzjQOQP9/1sv+hCZA/JDkaCrYPkD+gaSiSyRWQP90KQ5fcG5A/gibPGO8hkD8nTDEWASiQPyWRzY4SLpA/Y5AHgiM0kD8qakLvMzqQP/DD4NVDQJA/L8hENVNGkD8tJtAMYkyQP9MR5FtwUpA/eUPhIX5YkD+39ydei16QPzXvFxCYZJA/e24QN6RqkD/BPXDSr3CQP7+oleG6dpA/fX7eY8V8kD8iEahYz4KQP8c1T7/YiJA/REQwl+GOkD8BF6ff6ZSQP8YKD5jxmpA/i/7Cv/igkD9HUx1W/6aQP8Trd1oFrZA/aSwszAqzkD8O+5KqD7mQP8u+BPUTv5A/yV/ZqhfFkD8OR2jLGsuQP1ReCFYd0ZA/0g8QSh/XkD8PRtWmIN2QP7VrrWsh45A/XGvtlyHpkD9cr+kqIe+QP5sh9iMg9ZA/Yitmgh77kD8ptYxFHAGRP2kmvGwZB5E/aGVG9xUNkT8P13zkERORP7desDMNGZE/910x5AcfkT92tE/1ASWRP76/Wmb7KpE/BluhNvQwkT8F33Fl7DaRP8QhGvLjPJE/a3bn29pCkT8SrSYi0UiRP5ASJMTGTpE/T3ArwbtUkT8WDIgYsFqRP92nhMmjYJE/m4Fr05ZmkT8ZU4Y1iWyRP8BRHu96cpE/aC58/2t4kT8oFehlXH6RPyitqSFMhJE/cBgIMjuKkT+380mWKZCRPzVWtU0XlpE/c9GPVwSckT8ZcR6z8KGRP7+6pV/cp5E/vq1pXMetkT/9wq2osbORP8TstEObuZE/i5bBLIS/kT/KpBVjbMWRP8l08uVTy5E/cdyYtDrRkT8YKknOINeRP1ckQzIG3ZE/1gnG3+rikT8dkRDWzuiRP2ToYBSy7pE/Y7X0mZT0kT8jFQlmdvqRP8ub2ndXAJI/c1SlzjcGkj/zwKRpFwySP7PZE0j2EZI/ew0tadQXkj9CQSrMsR2SPwHQRHCOI5I/gYq1VGopkj8pt7R4RS+SP9ERetsfNZI/kss8fPk6kj+SijNa0kCSP9lplHSqRpI/IPmUyoFMkj+fPGpbWFKSP92sSCYuWJI/gzZkKgNekj8pOvBm12OSPyaMH9uqaZI/YnQkhn1vkj8nrjBnT3WSP+xndX0ge5I/KkMjyPCAkj8nVGpGwIaSP8wheveOjJI/caWB2lySkj+tSq/uKZiSPynvMDP2nZI/buIzp8Gjkj+05eRJjKmSP7ErcBpWr5I/blgBGB+1kj8UgcNB57qSP7or4ZauwJI/N0+EFnXGkj/0Uta/OsySP7kOAJL/0ZI/fcopjMPXkj85Pnutht2SP7WRG/VI45I/WVwxYgrpkj/8pOLzyu6SP7fhVKmK9JI/s/esgUn6kj/4Og98BwCTPztun5fEBZM/tsKA04ALkz/x19UuPBGTP5S7wKj2FpM/N+liQLAckz8ySt30aCKTP201UMUgKJM/MG/bsNctkz/zKJ62jTOTPy4Bt9VCOZM/KANEDfc+kz/KpmJcqkSTP2zQL8JcSpM/p9DHPQ5Qkz8iZEbOvlWTP2SzxnJuW5M/plJjKh1hkz+fQTb0ymaTP1jrWM93bJM/9yXkuiNykz+XMvC1zneTPw+9lL94fZM/x9vo1iGDkz+HDwP7yYiTP0dD+SpxjpM//8vgZReUkz93aM6qvJmTPxdB1vhgn5M/t+cLTwSlkz9vV4KspqqTP2f0SxBIsJM/p4t6eei1kz/nUh/nh7uTP1/oSlgmwZM/l1INzMPGkz83AHZBYMyTP9fHk7f70ZM/z+d0LZbXkz8IBieiL92TP8kvtxTI4pM/itkxhF/okz/C3qLv9e2TP7mBFVaL85M/WGuUth/5kz/3qikQs/6TPy623mFFBJQ/pWi8qtYJlD/kA8vpZg+UPyIvEh72FJQ/F/eYRoQalD/MzWViESCUP2mKfnCdJZQ/B2nobygrlD98CqhfsjCUPzJ0wT47NpQ/8A84DMM7lD+tqw7HSUGUP2J5R27PRpQ/1w7kAFRMlD91ZeV911GUPxLaS+RZV5Q/yCwXM9tclD++gEZpW2KUP/1b2IXaZ5Q/O6fKh1htlD+yrRpu1XKUP+kcxTdReJQ/iATG48t9lD8n1hhxRYOUPx1luN69iJQ/U+aeKzWOlD8R8MVWq5OUP895Jl8gmZQ/Bty4Q5SelD/9z3QDB6SUP5xvUZ14qZQ/PDVFEOmulD90+0VbWLSUP+v8SH3GuZQ/KdRCdTO/lD9neydCn8SUP15M6uIJypQ/FgB+VnPPlD+2rtSb29SUP1fP37FC2pQ/zzeQl6jflD+HHNZLDeWUP0YQoc1w6pQ/BQTgG9PvlD+8RoE1NPWUPzKFchmU+pQ/0MmgxvL/lD9ufPg7UAWVPyRiZXisCpU/Gp3SegcQlT9ZrCpCYRWVP5hrV825GpU/EBNCGxEglT9IN9MqZyWVP+jI8vq7KpU/iBSIig8wlT+BwnnYYTWVP7nWreOyOpU/erAJqwJAlT86CnItUUWVP3L5ymmeSpU/au73XupPlT8JtNsLNVWVP6dvWG9+WpU/3aBPiMZflT9UIaJVDWWVP5MkMNZSapU/0jfZCJdvlT/JQXzs2XSVP4CC938bepU/H5Mowlt/lT+9ZeyxmoSVPzNFH07YiZU/6tSclRSPlT+pEECHT5SVP2hM4yGJmZU/HzRgZMGelT+Vy49N+KOVPzNuStwtqZU/0M5nD2KulT+G977llLOVP3tJJl7GuJU/uHxzd/a9lT/1n3swJcOVP2oYE4hSyJU/nqENfX7NlT87TT4OqdKVP9mCdzrS15U/z/+KAPrclT8F10lfIOKVP8NwhFVF55U/gYoK4mjslT+3NqsDi/GVP63cNLmr9pU/TDh1Acv7lT/qWTnb6ACWPyCmTUUFBpY/ldV9PiALlj/T9JTFORCWPxFkXdlRFZY/B9egeGgalj+9VCiifR+WP1o3vFSRJJY/+Cskj6Mplj9uMidQtC6WPyOdi5bDM5Y/4BAXYdE4lj+dhI6u3T2WP1JBtn3oQpY/x+FRzfFHlj9kUiSc+UyWPwHR7+j/UZY/tux1sgRXlj+qhXf3B1yWP+jMtLYJYZY/JkTt7glmlj+cvd+eCGuWP9JbSsUFcJY/cJHqYAF1lj8OIX1w+3mWPwMdvvLzfpY/OOdo5uqDlj/1MDhK4IiWP7L65RzUjZY/5pMrXcaSlj/amsEJt5eWP3b8XyGmnJY/EvS9opOhlj9GC5KMf6aWP7oZkt1pq5Y/9URzlFKwlj8wAOqvObWWPyMMqi4fupY/1nZmDwO/lj9wm9FQ5cOWPwsinfHFyJY/fv958KTNlj8xdRhMgtKWP+0QKANe15Y/qaxXFDjclj9dblV+EOGWP9HHzj/n5ZY/bXZwV7zqlj8Jg+bDj++WP75B3INh9JY/slH8lTH5lj/tnPD4//2WPyhYYqvMApc/mwL6q5cHlz/OZV/5YAyXP2mVOZIoEZc/Be8ude4Vlz/4GeWgshqXPysHARR1H5c/5/AmzTUklz+kWvrK9CiXP9kQHgyyLZc/zig0j20ylz9rAN5SJzeXPwg+vFXfO5c/PdBulpVAlz+x7ZQTSkWXP+0Uzcv8SZc/KQy1va1Olz8e4ennXFOXP9LoB0kKWJc/br+q37Vclz8KSG2qX2GXP36s6acHZpc/MV251q1qlz/tEHU1Um+XP6nEtML0c5c/XLsPfZV4lz/PfRxjNH2XP2racHPRgZc/BeWhrGyGlz+39kMNBouXP6it6pOdj5c/4uwoPzOUlz8c3JANx5iXP43ns/1YnZc/vr8iDumhlz9WWW09d6aXP+/sIooDq5c/3/bR8o2vlz8QNwh2FrSXP8mwUhKduJc/gqo9xiG9lz+0rVSQpMGXP6eGIm8lxpc/QkQxYaTKlz/dNwplIc+XPw/1NXmc05c/gVE8nBXYlz+7ZKTMjNyXP/SH9AgC4Zc/5VWyT3Xllz+WqmKf5umXPzCjifZV7pc/yp2qU8Pylz88OUi1LveXP+5U5BmY+5c/qRAAgP//lz9jzBvmZASYPxYot0rICJg/iQNRrCkNmD8jfmcJiRGYP732d2DmFZg/cAv/r0EamD9jmXj2mh6YP528XzLyIpg/188uYkcnmD9KbF+EmiuYP31papfrL5g/F93HmTo0mD+yGu+JhziYP6SzVmbSPJg/13Z0LRtBmD+ScL3dYUWYP03qpXWmSZg/gGqh8+hNmD9ztCJWKVKYPw3Im5tnVpg/qOF9wqNamD/beTnJ3V6YP05FPq4VY5g/izT7b0tnmD/Ic94Mf2uYP71qVYOwb5g/crzM0d9zmD8QR7D2DHiYP64ja/A3fJg/JaZnvWCAmD/cXA9ch4SYP5kQy8qriJg/VsQCCM6MmD8LtR0S7pCYP4FZgucLlZg/IGKWhieZmD+9uL7tQJ2YP3OAXxtYoZg/aRXcDW2lmD+mDJfDf6mYP+Qz8jqQrZg/WpFOcp6xmD+QYwxoqrWYPy4hixq0uZg/y3gpiLu9mD/AUEWvwMGYP/TGO47DxZg/sTBpI8TJmD9uGiltws2YP6NH1mm+0Zg/l7LKF7jVmD8yjF91r9mYP8077YCk3Zg/AF/LOJfhmD9yyVCbh+WYP6uE06Z16Zg/48+oWWHtmD/UHyWySvGYP4UenK4x9Zg/HatgTRb5mD+12cSM+PyYPyXzGWvYAJk/1nSw5rUEmT+PENj9kAiZP0es365pDJk/+GEV+D8QmT9pf8bXExSZPwOGP0zlF5k/nCrMU7QbmT9OVbfsgB+ZPz8hSxVLI5k/edzQyxInmT+zB5EO2CqZPyRW09uaLpk/Va3eMVsymT/tJPkOGTaZP4UGaHHUOZk/dc1vV409mT+mJlS/Q0GZP1/wV6f3RJk/Fzq9DalImT9IRMXwV0yZPzqAsE4EUJk/1Y++Ja5TmT9wRS50VVeZP6KjPTj6Wpk/Fd0pcJxemT9RVC8aPGKZP42biTTZZZk/gHRzvXNpmT800CazC22ZP8/O3BOhcJk/ar/N3TN0mT/eHzEPxHeZP5KcPaZRe5k/TRApodx+mT8IhCj+ZIKZP7wucLvqhZk/MHUz122JmT/N6aRP7oyZP2pM9iJskJk/H4pYT+eTmT8VvfvSX5eZP1IsD6zVmpk/jkvB2EiemT8Duz9XuaGZPzhHtyUnpZk/1uhTQpKomT92xECr+quZP20qqF5gr5k/opazWsOymT9gsIudI7aZPx9KWCWBuZk/VWFA8Nu8mT9NHmr8M8CZP+zT+keJw5k/i/8W0dvGmT/CSOKVK8qZPzqBf5R4zZk/eaQQy8LQmT+417Y3CtSZP7BpkthO15k/aNLCq5DamT8Ks2avz92ZP6vVm+EL4Zk/Iy1/QEXkmT/a1CzKe+eZP5oQwHyv6pk/WUxTVuDtmT8RHABVDvGZP4k733Y59Jk/KY4IumH3mT/KHpMch/qZP4MflZyp/Zk/e+kjOMkAmj+7/FPt5QOaP/z/OLr/Bpo/dMDlnBYKmj+uMWyTKg2aP09t3Zs7EJo/8LJJtEkTmj/oZ8DaVBaaPyAXUA1dGZo/33AGSmIcmj+fSvCOZB+aP9aeGdpjIpo/zYyNKWAlmj9tWFZ7WSiaPw1qfc1PK5o/RE4LHkMumj+7tQdrMzGaP/p0ebIgNJo/OIRm8go3mj8u/9Mo8jmaP+UkxlPWPJo/g1dAcbc/mj8hHEV/lUKaP5ga1ntwRZo/Th30ZEhImj8NEZ84HUuaP8wE1vTuTZo/gimXl71Qmj/50d8eiVOaP5hyrIhRVpo/N6H40hZZmj/vFL/72FuaP+al+QCYXpo/JU2h4FNhmj9kJK6YDGSaP9tlFyfCZpo/EWzTiXRpmj+vsde+I2yaP0/RGMTPbpo/RoWKl3hxmj9/px83HnSaP0AxyqDAdpo/ADt70l95mj84/CLK+3uaPzDLsIWUfpo/0BwTAyqBmj9yhDdAvIOaP6uzCjtLhpo/I3p48daImj9jxWthX4uaP6OgzojkjZo/mTSKZWaQmj9Qx4b15JKaP++7qzZglZo/jZLfJtiXmj8E6AfETJqaP7t1CQy+nJo/eRHI/Cufmj83rSaUlqGaP+1WB9D9o5o/YjhLrmGmmj//ltIswqiaP5vTfEkfq5o/TmooAnmtmj9B8rJUz6+aP30d+T4ispo/ubjWvnG0mj8tqybSvbaaP2L2wnYGuZo//bWEqku7mj+aH0Rrjb2aP4+C2LbLv5o/xEcYiwbCmj+B8djlPcSaPz4b78Rxxpo/cnkuJqLImj9n2WkHz8qaPwQhc2b4zJo/oU4bQR7Pmj/VeDKVQNGaP0rOh2Bf05o/hpXpoHrVmj/DLCVUkteaP7cJB3im2Zo/a7laCrfbmj8H4OoIxN2aP6Q4gXHN35o/GZXmQdPhmj/O3eJ31eOaP4oRPRHU5Zo/RUW7C8/nmj/3oyJlxumaP2puNxu665o/BPu8K6rtmj+ftXWUlu+aP1MfI1N/8Zo/R86FZWTzmj+CbV3JRfWaP768aHwj95o/MZBlfP34mj9m0BDH0/qaPwJ6Jlqm/Jo/np1hM3X+mj+SX3xQQACbP8b3L68HAps/gbE0TcsDmz8860EoiwWbP3AWDj5HB5s/ZLdOjP8Imz8AZbgQtAqbP5rI/shkDJs/zZ3UshEOmz9AsuvLug+bP3zl9BFgEZs/tyigggETmz+rfpwbnxSbP2D7l9o4Fps//cM/vc4Xmz+bDkDBYBmbPxAiROTuGps/xFX2I3kcmz+BEQB+/x2bPz7NCfCBH5s/8xC7dwAhmz9pdLoSeyKbPwefrb7xI5s/pEc5eWQlmz9ZNAFA0yabP086qBA+KJs/jD3Q6KQpmz/KMBrGByubPz8VJqZmLJs/dPqShsEtmz8Q/v5kGC+bP6xLBz9rMJs/oBxIEroxmz/Ut1zcBDObP5Bx35pLNJs/TatpS441mz+E05PrzDabP3pl9XgHOJs/GOkk8T05mz+48rdRcDqbP/AiQ5ieO5s/aCZawsg8mz+otY/N7j2bP+eUdbcQP5s/3ZOcfS5Amz+UjZQdSEGbPzNo7JRdQps/0hQy4W5Dmz9Jj/L/e0SbPwDeue6ERZs/vhETq4lGmz98RYgyikebPzGeooKGSJs/pUrqmH5Jmz9Ag+ZyckqbP9qJHQ5iS5s/jKkUaE1Mmz9/NlB+NE2bP7qNU04XTps/9BSh1fVOmz9mOroR0E+bP5h0HwCmUJs/MUJQnndRmz/LKcvpRFKbP725DeANU5s/8IeUftJTmz+rMdvCklSbP2ZbXKpOVZs/mLCRMgZWmz+J4/NYuVabPyOt+hpoV5s/vMwcdhJYmz/uB9BnuFibP2Eqie1ZWZs/nAW8BPdZmz/YcNuqj1qbP81IWd0jW5s/gW+mmbNbmz8dzDLdPlybP7hKbaXFXJs/KtzD70ddmz/cdaO5xV2bP5YReAA/Xps/UK2swbNemz8CS6v6I1+bP3Tw3KiPX5s/DaepyfZfmz+oe3haWWCbP1x+r1i3YJs/T8KzwRBhmz+LXemSZWGbP8dos8m1YZs/Ov9zYwFimz9tPoxdSGKbPwhGXLWKYps/ojdDaMhimz+VNp9zAWObP8lnzdQ1Y5s/hPEpiWVjmz8++w+OkGObP3Gt2eC2Y5s/YjHgfthjmz/6sHtl9WObP5JWA5INZJs/wkzNASFkmz8yvi6yL2SbP2rVe6A5ZJs/orwHyj5kmz+SnSQsP2SbP0GhI8Q6ZJs/2O9UjzFkmz9vsAeLI2SbP90IirQQZJs/ih0pCfljmz9AETGG3GObP/YE7Si7Y5s/pRen7pRjmz8TZqjUaWObP6oKOdg5Y5s/Qh2g9gRjmz/xsiMty2KbP+DdCHmMYps/Ga2T10himz9RLAdGAGKbP8FjpcGyYZs/8FevR2Bhmz+HCWXVCGGbPx51BWisYJs/DZPO/Epgmz88V/2Q5F+bP/OwzSF5X5s/qop6rAhfmz/ZyT0uk16bP8dOUKQYXps/X/TpC5ldmz/1j0FiFF2bPyLxjKSKXJs/kOEA0Ptbmz/EJNHhZ1ubP/p3MNfOWps/55FQrTBamz+UImJhjVmbPyrTlPDkWJs/wEUXWDdYmz8vFReVhFebP93UwKTMVps/kxBAhA9Wmz9JTL8wTVWbP/cDaKeFVJs/Zati5bhTmz/5rdbn5lKbP41u6qsPUps/OkfDLjNRmz8niYVtUVCbP118VGVqT5s/kV9SE35Omz/8Z6B0jE2bPyfBXoaVTJs/uoysRZlLmz9N4qevl0qbPzfPbcGQSZs/YlYaeIRImz8UcMjQckebP8cJkshbRps/8gWQXD9Fmz/dO9qJHUSbP3F3h032Qps/BXmtpMlBmz8y9WCMl0CbP6CUtQFgP5s/1vO9ASM+mz8Lo4uJ4DybP/glL5aYO5s/pfO3JEs6mz86djQy+DibP88KsrufN5s/PQE9vkE2mz/rm+A23jSbP6EPpyJ1M5s/V4OZfgYymz8EEMBHkjCbP3HAIXsYL5s/BpHEFZktmz+bb60UFCybP0k74HSJKps/N8RfM/komz9syy1NYyebP6ECS7/HJZs/Dgy3hiYkmz88enCgfyKbP9HPdAnTIJs/Zn/AviAfmz9U6069aB2bP4NlGgKrG5s/Oi8ciucZmz/xeExSHhibPyBioldPFps/D/kTl3oUmz+mOpYNoBKbPz0SHbi/EJs/bFmbk9kOmz/c1wKd7QybPxRDRNH7Cps/TD5PLQQJmz88WhKuBgebP+4Ue1ADBZs/h9l1EfoCmz8fAO7t6gCbP4/NzeLV/po/P3P+7Lr8mj/3DmgJmvqaP66q8TRz+Jo/XDyBbEb2mj/KpfusE/SaP160RPPa8Zo/8yA/PJzvmj+gj8yEV+2aP42PzckM65o/wpohCLzomj/4Fac8ZeaaP2ZQO2QI5Jo/lIO6e6Xhmj8q0/9/PN+aP79M5W3N3Jo/redDQljamj/bhPP53NeaP5HuypFb1Zo/RtifBtTSmj9z3kZVRtCaP1+Gk3qyzZo/8j1YcxjLmj+GW2Y8eMiaP7MdjtLRxZo/H6ueMiXDmj9TEmZZcsCaP4lJsUO5vZo/di5M7vm6mj8jhgFWNLiaP7j8mndotZo/TiXhT5aymj+7eZvbva+aP2hakBffrJo/HQ6FAPqpmj/SwT2TDqeaP36IfcwcpJo/6loGqSShmj9/F5klJp6aPxSC9T4hm5o/wEPa8RWYmj+t6gQ7BJWaP+LpMRfskZo/Fpkcg82Omj+ENH97qIuaP7LcEv18iJo/R5aPBEuFmj/dSayOEoKaP8zDHpjTfpo/+bObHY57mj+vrdYbQniaP2Mngo/vdJo/j3pPdZZxmj964+7JNm6aPw2BD4rQapo/oFRfsmNnmj/LQYs/8GOaPzcOPy52YJo/a2Ele/Vcmj+exOciblmaP4miLiLgVZo/NEehdUtSmj/G3+UZsE6aP1h6oQsOS5o/wQV4R2VHmj9rUQzKtUOaPxwNAJD/P5o/zsjzlUI8mj939IbYfjiaP97fV1S0NJo/b7oDBuMwmj//kibqCi2aP6hXW/0rKZo/k9U7PEYlmj/GuGCjWSGaP/iLYS9mHZo/YbjU3GsZmj+LhU+oahWaPx0ZZo5iEZo/r3ari1MNmj+Zf7GcPQmaP8PyCL4gBZo/dGxB7PwAmj8lZukj0vyZP042jmGg+Jk/NxC8oWf0mT/IA/7gJ/CZP1n93Rvh65k/g8XkTpPnmT/sAJp2PuOZPx4whI/i3pk/UK8oln/amT87tguHFdaZP+dXsF6k0Zk/e4KYGSzNmT8O/0S0rMiZP3pxNSsmxJk/JVjoepi/mT/XC9ufA7uZP4q/iZZntpk/NYBvW8SxmT+gNAbrGa2ZPzKdxkFoqJk/xVMoXK+jmT9wy6E2756ZP1tQqM0nmpk/jgewHVmVmT/A7isjg5CZPyrcjdqli5k/U35GQMGGmT/lW8VQ1YGZP3fTeAjifJk/YBvOY+d3mT+JQTFf5XKZPzorDffbbZk/7JTLJ8tomT8WEtXtsmOZPwENkUWTXpk/k8ZlK2xZmT8lVribPVSZP0+p7JIHT5k/uINlDcpJmT/ofoQHhUSZPxkKqn04P5k/AWo1bOQ5mT+quITPiDSZPzzl9KMlL5k/zrPh5bopmT85vaWRSCSZP+RumqPOHpk/lwoYGE0ZmT9LpnXrwxOZP/crCRozDpk/YVknoJoImT/0vyN6+gKZP4bEUKRS/Zg/MZ//GqP3mD8cW4Da6/GYP0/WId8s7Jg/gsExJWbmmD/sn/yol+CYPxfHzWbB2pg/qV7vWuPUmD87YKqB/c6YPySXRtcPyZg/T6AKWBrDmD8C6jsAHb2YP7SzHswXt5g/3g32twqxmD/I2QPA9aqYP1rJiODYpJg/7F7EFbSemD8W7fRbh5iYP4CWV69Skpg/s00oDBaMmD/k1KFu0YWYP869/dKEf5g/eGl0NTB5mD8LCD2S03KYP56YjeVubJg/CumaKwJmmD+1lZhgjV+YP2gJuYAQWZg/HH0tiItSmD/I9yVz/kuYPzRO0T1pRZg/yCJd5Ms+mD9b5fViJjiYPwfTxrV4MZg/9PX52MIqmD8oJbjIBCSYP1wEKYE+HZg/yANz/m8WmD/0X7s8mQ+YP4khJji6CJg/Hh3W7NIBmD8L8+xW4/qXPzgPi3Lr85c/7ajPO+vslz+jwtiu4uWXP9Epw8fR3pc/v3aqgrjXlz9WDKnbltCXP+wX2M5syZc/G5FPWDrClz+JOSZ0/7qXP8CccR68s5c/9w9GU3Cslz/msbYOHKWXP5Zq1Uy/nZc/LuuyCVqWlz/HrV5B7I6XPzj15u91h5c/6cxYEfd/lz+hCMChb3iXP1lEJ53fcJc/CeSX/0Zplz95ExrFpWGXPxHGtOn7WZc/qbZtaUlSlz9YZ0lAjkqXP0chS2rKQpc/f/R04/06lz+3t8enKDOXPycIQ7NKK5c/WEnlAWQjlz/xpKuPdBuXP4oKklh8E5c/ey+TWHsLlz+rjqiLcQOXP2Noyu1e+5Y/HMLvekPzlj9NZg4vH+uWPz7kGgby4pY/148I/Lvalj9wgckMfdKWP6KVTjQ1ypY/FG2HbuTBlj9ObGK3irmWP4i7zAoosZY/e0ayZLyolj8vvP3AR6CWP8qOmBvKl5Y/ZfNqcEOPlj/Y4Vu7s4aWP4sUUfgafpY/RQgvI3l1lj//+9g3zmyWP7HwMDIaZJY/JKkXDl1blj+/qWzHllKWP1o4DlrHSZY/DFzZwe5Alj/+3Kn6DDiWPzlEWgAiL5Y/dNvDzi0mlj/nrL5hMB2WPxmDIbUpFJY/s+jBxBkLlj9MKHSMAAKWPz1MCwje+JU/bx5ZM7LvlT8pKC4KfeaVP+OxWYg+3ZU/FcOpqfbTlT8GIutppcqVP59T6cRKwZU/OJtutua3lT9p+kM6ea6VP9swMUwCpZU/FLz854GblT9N12sJ+JGVPz57QqxkiJU/7l1DzMd+lT+H8i9lIXWVPx9pyHJxa5U/kK7L8LdhlT9BbPfa9FeVP/sHCC0oTpU/tqO44lFElT9oHcP3cTqVP9sO4GeIMJU/ds3GLpUmlT8Qai1ImByVP8KwyK+REpU/tChMYYEIlT/vE2pYZ/6UPylv05BD9JQ/nPE3BhbqlD/PDEa03t+UP2rsqpad1ZQ/BHYSqVLLlD/2SCfn/cCUPyi+kkyftpQ/4uf81DaslD+bkQx8xKGUP80/Zz1Il5Q/vy+xFMKMlD9ZV439MYKUP/NknfOXd5Q/Jb+B8vNslD+XhNn1RWKUP9CLQvmNV5Q/CWNZ+MtMlD/6T7nu/0GUP6tP/NcpN5Q/Qxa7r0kslD/cDo1xXyGUP01bCBlrFpQ//tPBoWwLlD+4B00HZACUP3I7PEVR9ZM/JGogVzTqkz+VRIk4Dd+TPy4xBeXb05M/yEshWKDIkz96ZWmNWr2TP2sEaIAKspM/pWOmLLCmkz/fcqyNS5uTP1LWAJ/cj5M/hOYoXGOEkz8dsKjA33iTP7bzAshRbZM/qCW5bblhkz/ZbUutFlaTP5OnOIJpSpM/TGH+57E+kz983Bja7zKTP2wNA1QjJ5M/BZs2UUwbkz+e3ivNag+TP8/jWcN+A5M/QGg2L4j3kj942zUMh+uSP69ey1V735I/nsRoB2XTkj9OkX4cRMeSP+b5e5AYu5I/fuTOXuKukj/u5+OCoaKSP55LJvhVlpI/VQcAuv+Jkj8Mw9nDnn2SP7zWGhEzcZI/LEopnbxkkj/E1GljO1iSP1zdP1+vS5I/C3oNjBg/kj/7bzPldjKSPzMzEWbKJZI/a+YEChMZkj/bWmvMUAySPwsQoKiD/5E/ozP9mavykT86odubyOWRPyjikqna2JE/Vy15vuHLkT8NZ+PV3b6RP8MgJevOsZE/8JiQ+bSkkT/eunb8j5eRP3UeJ+9fipE/CwjwzCR9kT86aB6R3m+RP6nb/TaNYpE/4KrYuTBVkT8YyvcUyUeRPwjZokNWOpE/uCIgQdgskT9QnbQITx+RP+fpo5W6EZE/VlQw4xoEkT8G05rsb/aQP74GI6256JA/dToHIPjakD8lY4RAK82QP5Uf1glTv5A/Lbg2d2+xkD/GHt+DgKOQP3buBiuGlZA/Z2vkZ4CHkD+ggqw1b3mQP9nJko9Sa5A/SX/JcCpdkD96iYHU9k6QPxJ36rW3QJA/qn4yEG0ykD+afobeFiSQP8r8ERy1FZA/gSb/w0cHkD9woO2infGPP8/qQH+U1I8/rnBEE3S3jz/cw0FVPJqPPwvDfzvtfI8/a5lCvIZfjz9JvsvNCEKPP7f0WWZzJI8/JEspfMYGjz8DG3MFAumOP2IIbvgly44/kQFOSzKtjj/BPkT0Jo+OP6JBf+kDcY4/AtUqIclSjj9xDHCRdjSOP+FDdTAMFo4/QB9e9In3jT8gikvT79iNP1C3W8M9uo0/gSCqunObjT/ihU+vkXyNP8TtYZeXXY0/NqT0aIU+jT+oOhgaWx+NP4mH2qAYAI0/6qVG873gjD8b9WQHS8GMP0sYO9O/oYw/KvbLTByCjD+JuBdqYGKMP/fLGyGMQow/ZN/SZ58ijD/B4zQ0mgKMP50LN3x84os/yMrLNUbCiz/y1eJW96GLP0siadWPgYs/JOVIpw9hiz+Nk2nCdkCLP/bhrxzFH4s/zsP9q/r+ij8mazJmF96KP09IKkEbvYo/dwm/Mgacij9Pmscw2HqKP6UjGDGRWYo/DAuCKTE4ij9y8tMPuBaKP8i32dkl9Yk/n3RcfXrTiT/GfSLwtbGJP+1i7yfYj4k/Q+6DGuFtiT8ZJJ690EuJP39C+QanKYk/5MBN7GMHiT+5T1FjB+WIPw/YtmGRwog/NHsu3QGgiD9bkmXLWH2IPzGuBiKWWog/h5a51rk3iD/tSSPfwxSIP1P95TC08Yc/qRuhwYrOhz+ARfGGR6uHP6dQcHbqh4c/zke1hXNkhz8malSq4kCHP/0q39k3HYc/ZDHkCXP5hj/MV+8vlNWGP6SriUGbsYY/+2w5NIiNhj8jDoL9WmmGP0sz5JITRYY/IrLd6bEghj95ken3NfyFP98IgLKf14U/RoAWD++yhT+djx8DJI6FP3P+CoQ+aYU/m8NFhz5EhT/CBDoCJB+FPxgWT+ru+YQ/73npNJ/UhD9W4GrXNK+EP70mMseviYQ/lFeb+Q9khD/rqf9jVT6EPxKBtft/GIQ/OGwQto/ygz8PJmGIhMyDP2WU9WdepoM/yscYSh2Agz8v+xIkwVmDP4WTKetJM4M/XB+flLcMgz+CVrMVCuaCP6gZo2NBv4I//nGoc12Ygj/UkPo6XnGCPzrPza5DSoI/oK1TxA0jgj9207pwvPuBP8wOL6lP1IE/8lPZYsesgT8Yvd+SI4WBP+6JZS5kXYE/RB+LKok1gT+rBm58kg2BPxHuKBmA5YA/aafT9VG9gD8/KIMHCJWAP2aJSUOibIA/jQY2niBEgD/k/VQNgxuAP3bfXwuT5X8/RPya+OeTfz8S2WLMBEJ/P8E+uXDp734/bzeaz5Wdfj+/DfzSCUt+Pw5Mz2RF+H0/vrv+bkilfT9sZG/bElJ9PzmLAJSk/nw/B7KLgv2qfD+1luSQHVd8P2Iy2agEA3w/r7gxtLKuez/8lrCcJ1p7P6pzEkxjBXs/Vy0OrGWwej8j2lSmLlt6P+/GkSS+BXo/m3ZqEBSweT9HoX5TMF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Application\",\"version\":\"1.3.4\"}};\n",
       "  var render_items = [{\"docid\":\"e84c542f-5928-4d77-bd21-4da8a8cbe7c2\",\"roots\":{\"1356\":\"62478662-0e11-4849-ad71-74db9e06590a\"}}];\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": "1356"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "N = 2048-1\n",
    "\n",
    "# Redéfinition de u\n",
    "def u(x) : \n",
    "    return x*(1-x)*(1+x)*(1+x*x)/20\n",
    "\n",
    "# Construction de A\n",
    "h = 1/(N+1)\n",
    "b = (N+1)**2   # =1/h**2\n",
    "a = -2*b\n",
    "A = np.array(a*np.eye(N))\n",
    "for i in range(1,N) :\n",
    "    A[i][i-1] = A[i-1][i] = b\n",
    "\n",
    "# Construction de f\n",
    "xi = h*np.array([i for i in range(1,N+1)])\n",
    "f =  xi*xi*xi\n",
    "\n",
    "# Résolution de A.u = -f en utilisant les matrices tridiagonales\n",
    "tini = time.time()\n",
    "ui = solveLU_tridiag(A,-f)\n",
    "tfin = time.time()\n",
    "print(\"Le temps de la résolution du système A.u = -f est \",tfin-tini,\"s\")\n",
    "\n",
    "# Comparaison avec la solution analytique\n",
    "yi = u(xi)\n",
    "err = abs(ui-yi)\n",
    "\n",
    "# Figure\n",
    "fig = figure(width=490, height=300)\n",
    "fig.line(xi,ui, legend = \"ui\")\n",
    "fig.line(xi,yi, color = \"orange\", legend = \"u(xi)\")\n",
    "fig.legend.location = \"top_left\"\n",
    "\n",
    "fig2 = figure(width=490, height=300, x_axis_type=\"log\", y_axis_type=\"log\")\n",
    "fig2.line(xi, err, legend =\"|u(xi)-ui|\")\n",
    "fig2.legend.location = \"top_left\"\n",
    "\n",
    "show(row(fig,fig2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :**  \n",
    "En terme de temps de calcul, le nouvel algorithme est plus performant que le premier : la résolution du système -A.u=f se fait en environ 0.04 s, alors que pour le premier algorithme, elle se fait en environ 1.70 s.  \n",
    "En terme de précision, les deux algorithmes sont equivalents : la courbe d'erreur a la mçeme allure, et sa valeur maximale est de l'ordre de $10^{-8}$.\n",
    "\n",
    "Le stockage tridiagonal permet alors d'optimiser le temps de calcul, et surtout l'espace de stockage : pour une matrice tridiagonale de taille (N,N) on va stocker les matrices At, Lt, et Ut de tailles respectives (3,N), (2,N) et (1,N), donc au total $6N$ valeurs. Alors qu'avec le premier algorithme, on stocke les matrices A, LU et piv de tailles respectives (N,N), (N,N) et (1,N), soit $2N^2+N$ valeurs."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Cas 2D\n",
    "\n",
    "On considère maintenant une matrice bande, *i.e.* dont les coefficients satisfont si $|i-j| < K$ alors$A_{i,j} = 0,$ où $K>0$ est appelée la largeur de bande.\n",
    "\n",
    "  On utilisera pour les applications numériques la matrice $A$ définie par :\n",
    "  \n",
    "\\begin{align}\n",
    "    A &= \\left( \\begin{array}{c|c|c|c}\n",
    "      D    & I/h^2    &\\dots   & 0 \\\\ \\hline\n",
    "      I/h^2    & D    & \\ddots& \\vdots  \\\\ \\hline\n",
    "      \\vdots   & \\ddots &\\ddots  &I/h^2\\\\ \\hline\n",
    "      0      & \\dots  &   I/h^2  & D\n",
    "    \\end{array}\\right), \\quad %\\in\\mathbb{R}^{N^2\\times N^2}, \\\n",
    "    D = \\left( \\begin{array}{cccc}\n",
    "      -2/h^2    & 1/h^2    & \\dots   & 0 \\\\\n",
    "      1/h^2    & -2/h^2    & \\ddots& \\vdots  \\\\\n",
    "      0      & \\ddots & \\ddots  &0\\\\\n",
    "      0      & \\dots  &  1/h^2  & -2/h^2\n",
    "    \\end{array}\\right) \\in\\mathbb{R}^{N\\times N},\n",
    "    \\label{Lapl_2D}\n",
    "\\end{align}\n",
    "\n",
    "où $I$ est la matrice identité de taille $N=64$ et $h = \\frac{1}{N}$. Cette matrice correspond à une discrétisation de l'opérateur $-\\Delta$ en 2D."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matrices pleines"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**En utilisant la fonction *solve* de numpy, résoudre le système $Au=f$ avec $f_{(i-1)N+j} = f(x_i,y_j) = x_i^3 + y_j^3$ où $x_i = (i-\\frac{1}{2})h$ et $y_j = (j-\\frac{1}{2})h$. Commenter la vitesse d'execution de l'algorithme et la taille de la matrice stockée (*sauvegarder avant d'executer*).**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-64.  16.   0.  16.   0.   0.   0.   0.   0.]\n",
      " [ 16. -64.  16.   0.  16.   0.   0.   0.   0.]\n",
      " [  0.  16. -64.   0.   0.  16.   0.   0.   0.]\n",
      " [ 16.   0.   0. -64.  16.   0.  16.   0.   0.]\n",
      " [  0.  16.   0.  16. -64.  16.   0.  16.   0.]\n",
      " [  0.   0.  16.   0.  16. -64.   0.   0.  16.]\n",
      " [  0.   0.   0.  16.   0.   0. -64.  16.   0.]\n",
      " [  0.   0.   0.   0.  16.   0.  16. -64.  16.]\n",
      " [  0.   0.   0.   0.   0.  16.   0.  16. -64.]]\n",
      "Pour N =  3  :\n",
      "le temps de construction de A est  0.005995988845825195 s\n",
      "le temps de résolution du système A.u = -f est  0.0009965896606445312 s\n"
     ]
    }
   ],
   "source": [
    "N = 64-1\n",
    "h = 1/(N+1)\n",
    "\n",
    "\n",
    "# Construction de f\n",
    "xi = yj = h*np.array([i for i in range(1,N+1)])\n",
    "xi3 = xi*xi*xi #Pour éviter de recalculer à chaque fois xi[i]**3\n",
    "yj3 = xi3\n",
    "f = np.zeros(0)\n",
    "for i in range(N) :\n",
    "    f = np.hstack((f, xi3[i] + yj3))\n",
    "\n",
    "tiA = time.time()\n",
    "# Construction de D\n",
    "invh2 = (1/h)**2 # pour éviter de recalculer à chaque fois l'inverse de h^2\n",
    "D = np.zeros((N,N)) # 1/h = N+1\n",
    "D[0][0] = -4*invh2\n",
    "for i in range(1,N) :\n",
    "    D[i][i-1] = D[i-1][i] = invh2\n",
    "    D[i][i] = -4*invh2\n",
    "\n",
    "## Construction de A ------------------------------------------------------------------------\n",
    "I_h2 = invh2*np.eye(N)\n",
    "Z = np.zeros((N,N))\n",
    "# La première ligne de blocs de A\n",
    "A = np.hstack((D,I_h2))\n",
    "for i in range(2,N) :\n",
    "    A = np.hstack((A,Z))\n",
    "\n",
    "# Les lignes de blocs de A de la 2ème à la (N-1)ème\n",
    "for l in range(1,N-1) :\n",
    "    Al = np.zeros((N,0))\n",
    "    for i in range(l-1) :\n",
    "        Al = np.hstack((Al, Z))\n",
    "    Al = np.hstack((Al, I_h2))\n",
    "    Al = np.hstack((Al, D))\n",
    "    Al = np.hstack((Al, I_h2))\n",
    "    for i in range(l+2,N) :\n",
    "        Al = np.hstack((Al, Z))\n",
    "    A = np.vstack((A,Al))\n",
    "\n",
    "# La dernière ligne de blocs de A\n",
    "Al = np.zeros((N,0))\n",
    "for i in range(N-2) :\n",
    "    Al = np.hstack((Al, Z))\n",
    "Al = np.hstack((Al, I_h2))\n",
    "Al = np.hstack((Al, D))\n",
    "A = np.vstack((A,Al))\n",
    "#------------------------------------------------------------------------------------------\n",
    "tfA = time.time()\n",
    "\n",
    "ti = time.time()\n",
    "u = np.linalg.solve(A,-f)\n",
    "tf = time.time()\n",
    "\n",
    "print(A)\n",
    "\n",
    "print(\"Pour N = \",N,\" :\")\n",
    "print(\"le temps de construction de A est \",tfA-tiA,\"s\")\n",
    "print(\"le temps de résolution du système A.u = -f est \",tf-ti,\"s\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :**  \n",
    "En faisant des tests avec différentes valeurs de $N$ ($N\\leq 55$), on remarque que la construction de A et la résolution du système $Au=-f$ prennent un temps qui croît très rapidement avec $N$.  \n",
    "Pour la valeur $N = 64$ le programme s'arrête et ne peut pas aller au bout du calcul. Ceci est dû au à la grande taille des opérations mais aussi à la grande taille de mémoire utilisée ($N^4$ valeurs stockées pour A) . Même la construction de A sans résolution du système bloque le programme."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matrices bandes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On stocke désormais la matrice sous forme bande, c'est-à-dire qu'on écrit :\n",
    "\n",
    "\\begin{align*}\n",
    "      \\tilde{A} = \\left(\n",
    "      \\begin{array}{c|c|c|c|c|c}\n",
    "        0             & \\dots & 0        & A_{1,K+1} & \\dots   & A_{N-K,N}\\\\\n",
    "        \\hline \\vdots &       &  & A_{2,K+1} &\\dots   & A_{N-K+1,N}\\\\\n",
    "        \\hline 0      & A_{1,2} &          &          &         & \\vdots    \\\\\n",
    "        \\hline A_{1,1} & A_{2,2} & \\dots  & \\dots     & \\dots   & A_{N,N} \\\\\n",
    "        \\hline A_{2,1} & A_{3,2} & \\dots   & \\dots    & A_{N,N-1} & 0 \\\\\n",
    "        \\hline \\vdots &        &         &          &         & \\vdots \\\\\n",
    "        \\hline A_{K+1,1} & \\dots & A_{N,N-K} & 0       & \\dots   &  0\n",
    "      \\end{array} \\right) \\in \\mathbb{R}^{2K+1,N}.\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :**  \n",
    "Dans notre cas, la taille de la matrice $A$ est $(N^2,N^2)$, $K = N$, les coefficients diagonaux valent $-4/h^2$, et les coefficients des 2èmes et $(N+1)$èmes diagonales (haut et bas) valent $1/h^2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Le temps de construction de At est  0.023989200592041016 s\n"
     ]
    }
   ],
   "source": [
    "N = 64-1\n",
    "h = 1/(N+1)\n",
    "invh2 = 1/(h*h)\n",
    "\n",
    "ti = time.time()\n",
    "# Construction de At ---------------------------------------\n",
    "At = np.zeros((2*N+1,N*N))\n",
    "\n",
    "# Diagonales n°1 et 2\n",
    "At[N][0] = -4*invh2\n",
    "for i in range(1,N*N) :\n",
    "    At[N][i] = -4*invh2\n",
    "    At[N-1][i] = At[N+1][i] = invh2\n",
    "\n",
    "# Diagonales n°(N+1)\n",
    "for i in range(N,N*N) :\n",
    "    At[0][i] = At[2*N][i] = invh2\n",
    "#------------------------------------------------------------\n",
    "tf = time.time()\n",
    "\n",
    "print(\"Le temps de construction de At est \",tf-ti,\"s\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :**  \n",
    "Alors que la construction de $A$ pour $N=64$ bloquait le programme, elle se fait à présent en mois de 0.03 secondes avec la nouvelle facon de stockage.  \n",
    "Le nombre de valeurs stockées est $N^2(2N+1)$ à la place de $N^4$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Utiliser la fonction *solve_banded* de scipy exploitant cette structure bande pour résoudre le problème $Au = f$. Comparer la vitesse d'execution et la taille de la matrice utilisée avec la question précédente.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Le temps de résolution du système -Au = f est  0.08796191215515137 s\n"
     ]
    }
   ],
   "source": [
    "# Construction de f\n",
    "xi = yj = h*np.array([i for i in range(1,N+1)])\n",
    "xi3 = xi*xi*xi #Pour éviter de recalculer à chaque fois xi[i]**3\n",
    "yj3 = xi3\n",
    "f = np.zeros(0)\n",
    "for i in range(N) :\n",
    "    f = np.hstack((f, xi3[i] + yj3))\n",
    "\n",
    "\n",
    "# Résolution du système A.u = -f\n",
    "ti = time.time()\n",
    "u = solve_banded((N,N),At,-f)\n",
    "tf = time.time()\n",
    "\n",
    "print(\"Le temps de résolution du système -Au = f est \",tf-ti,\"s\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "l'élément $u_{(i-1)N+j}$ doit donner une approximation de $u(x_i,y_j)$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :**  \n",
    "Pour $N=64$, le temps de construction du stockage en bande de $A$ et du calcul de la solution de $-Au=f$ est de l'ordre de 0.1 s, alors que la construction de A sans résolution du système n'était pas faisable avec un stockage en matrice pleine de $A$, ce qui montre l'efficacité de cette méthode"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matrices creuses CSR"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Bonus. Construire la matrice $A$ de la discrétisation de l'opérateur $-\\Delta$ en format CSR (utiliser scipy.sparse.csr matrix) avec Scipy (https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.csr_matrix.html).**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Bonus. Résoudre le système $Au = f$ avec la fonction *scipy.sparse.linalg.splu* et comparer la vitesse d’exécution et la taille de la matrice utilisée avec les deux méthodes précédentes.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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