Ill conditioning, machine precision.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "7bd01c39",
   "metadata": {},
   "source": [
    "# Direct Methods for the solution of Linear Systems."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2d414245",
   "metadata": {},
   "source": [
    "1. Given a matrix $A \\in {R}^{n×n}$ and the vector $x_{true} = {(1, 1, . . . , 1)}^{T} \\in R^n$, write a script that:\n",
    "\n",
    "    • Computes the right-hand side of the linear system $b = Ax_{true}$.\n",
    "\n",
    "    • Computes the condition number in 2-norm of the matrix A. It is ill-conditioned? What if we use\n",
    "    the $\\infty$-norm instead of the 2-norm?\n",
    "\n",
    "    • Solves the linear system $Ax = b$ with the function np.linalg.solve().\n",
    "\n",
    "    • Computes the relative error between the solution computed before and the true solution $x_{true}$.\n",
    "    Remember that the relative error between $x_{true}$ and x in R\n",
    "    n can be computed as\n",
    "    $E(x_{true}, x) = \\frac{||x − x_{true}||_2}{||x_{true}||_2}$\n",
    "\n",
    "    • Plot a graph (using matplotlib.pyplot) with the relative errors as a function of n and (in a new\n",
    "    window) the condition number in 2-norm $K_2(A)$ and in $\\infty$-norm $K_{\\infty}(A)$, as a function of n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "e2470f6d",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy.linalg\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "plt.rcParams['figure.figsize'] = [6, 7]\n",
    "\n",
    "n = 100\n",
    "np.random.seed(12)\n",
    "v_1 = np.linspace(10, 100, 10, dtype=int)\n",
    "v_2 = np.linspace(5, 30, 6, dtype=int)\n",
    "v_3 = np.linspace(4, 12, 8, dtype=int)\n",
    "\n",
    "\n",
    "def solve_linear(matrix_type, v):\n",
    "    errors = []\n",
    "    cond_infnorm = []\n",
    "    cond_2norm = []\n",
    "    for n_1 in v:\n",
    "        if matrix_type == 'random':\n",
    "            A = np.random.rand(n_1, n_1)\n",
    "            s = \"random matrixes\"\n",
    "        elif matrix_type == 'vandermonde':\n",
    "            x = np.linspace(1, n_1, n_1)\n",
    "            A = np.vander(x)\n",
    "            s = \"vandermonde matrixes\"\n",
    "        elif matrix_type == 'hilbert':\n",
    "            A = scipy.linalg.hilbert(n_1)\n",
    "            s = \"hilbert matrixes\"\n",
    "        else:\n",
    "            ValueError(\"Type 0 for random matrix, 1 for vander matrix, 2 for hilbert matrix\")\n",
    "\n",
    "        x_true = np.ones((n_1, 1))\n",
    "        b_1 = A @ x_true\n",
    "        # print(np.linalg.norm(A,2))\n",
    "        # print(np.linalg.norm(A,np.inf))\n",
    "        x = np.linalg.solve(A, b_1)\n",
    "        error = (np.linalg.norm(x - x_true, 2)) / (np.linalg.norm(x_true, 2))\n",
    "        cond2norm = np.linalg.cond(A, 2)\n",
    "        condinfnorm = np.linalg.cond(A, np.inf)\n",
    "        errors.append(error)\n",
    "        cond_infnorm.append(cond2norm)\n",
    "        cond_2norm.append(condinfnorm)\n",
    "\n",
    "    fig, (ax1, ax2) = plt.subplots(1,2, figsize=(10,4))\n",
    "    fig.tight_layout(pad=3.0) \n",
    "    \n",
    "    ax1.title.set_text(s + \" solution relative error\")\n",
    "    ax1.plot(v, errors, label=\"error\")\n",
    "    ax1.legend(loc='upper left')\n",
    "    ax1.set_xlabel('matrix dimension')\n",
    "    ax1.set_yscale('log')\n",
    "    \n",
    "    ax2.title.set_text(s + \" conditioning numbers\")\n",
    "    ax2.plot(v, cond_infnorm, label=\"infinity norm\")\n",
    "    ax2.plot(v, cond_2norm, label='2-norm')\n",
    "    ax2.legend(loc='upper left')\n",
    "    ax2.set_xlabel('matrix dimension')\n",
    "    ax2.set_yscale('log')\n",
    "    \n",
    "    plt.show()\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d8681af3",
   "metadata": {},
   "source": [
    "2. Test the program above with the following choices of $A \\in {R}^{n×n}$:\n",
    "\n",
    "    • A random matrix (created with the function **np.random.rand()**) with size varying with $n = \\{ 10,20,30,...,100\\}$\n",
    "\n",
    "    • The Vandermonde matrix (**np.vander**) of dimension $n = \\{ 5,10,15,20,25,30\\}$ with respect to the vector $x = \\{ 1,2,3,...,n\\}$\n",
    "\n",
    "    • The Hilbert matrix (**scipy.linalg.hilbert**) of dimension $n = \\{ 4,5,6,...,12\\}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "8c9958ea",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 720x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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d6+r0O0DlJjm+zWlMTAy+vr5uX2mKCL6+vnr2QqlcRERmAH8DNUUkTEQGG2MSgCeB5cBeYLYxZndWxZBb6lhXp98BKjfJ8ckpkGsqzdzyOpVypOPno3l1fjAx8YnODuWWGWMeNMaUNcZ4GWP8jDE/2Kb/ZoypYYypaox5N6vj0LrHNehxUC7pUhgsfhriHffDyS2SU6WUSikhMYnv/jpEp8/+YnHQCfac0pGUcqpWrW58Y6u1a9dSt25dAgICOHHiBL169brhOnfddRcXL17k4sWLfPPNN44IVancZfdCGN8Kds2D08EO26wmp9kkMTEx3eepMcaQlJSUVSEp5bZ2HL9It6/WM3bpPtpWL8nK59vRuGIxZ4elMmjDhg03XGb69Om88MILBAUFUb58eebOnXvDdX777TeKFi3qlOT0Zr4DlHJZsZGw6AmYMwB8q8HwNVChqcM2r8mpg/z00080a9aMgIAAhg8fTmJiIoUKFeKNN96gefPm/P333/95/umnn1KvXj3q1avHuHHjAAgNDaV27do8/vjjNG7cmOPHj6dfsFIqWWRsAmMW7+b+b9YTERXLtw83YUL/QMr65Hd2aCoTChUqBMDq1atp3749vXr1olatWvTr1w9jDBMnTmT27Nm89dZb9OvXj9DQUOrVs+72OmXKFHr06EGXLl2oXr06L730UvJ2/f39OXfuHK+88gqHDh0iICCAF198kUceeYRFixYlL9evXz8WL158XUxpxQKwatUqGjVqRP369Rk0aBCxsbHJ5b311lu0bt2aOXPm4O/vz2uvvUbLli0JDAxk27ZtdO7cmapVq/Ltt99m6T5VKsNObIPv2sL26dDmBRi0HHyrOrSIHN9b396bv+xmz0nHXrqrU64Io++tm+4ye/fuZdasWaxfvx4vLy8ef/xxpk+fTlRUFPXq1eOtt94CuO751q1bmTx5Mps2bcIYQ/PmzWnXrh3FihVj//79TJ48WS8zKXULVu45w/8W7eL05RgeaVGJFzrXpIi3DrnjSM6qY+1t376d3bt3U65cOW677TbWr1/PkCFDWLduHffccw+9evUiNDT0unWCgoLYvn07+fLlo2bNmjz11FNUqPDvaFzvv/8+u3btIigoCIC//vqLzz77jO7du3Pp0iU2bNjAjz/+eFOxBAYGMnDgQFatWkWNGjXo378/48ePZ+TIkYA1HNS6desAeOWVV6hQoQJ///03zz77LAMHDmT9+vXExMRQt25dRoz4z7C1SjlPUhJs+Bz+eAcKlYaBv4J/6ywpSs+cOsCqVavYunUrTZs2JSAggFWrVnH48GE8PT3p2bNn8nL2z9etW8f9999PwYIFKVSoED169GDt2rUAVKpUiRYtWjjltSiV05y5HMNjP21lyNQtFPH2Yt5jrXirez1NTN1Us2bN8PPzw8PDg4CAgP8koqm544478PHxwdvbmzp16nD0aJq39AagXbt2hISEEB4ezowZM+jZsyd58vz3XE5qsezfv5/KlStTo0YNAAYMGMCaNWuS1+nTp8912+jWrRsA9evXp3nz5hQuXJiSJUvi7e3NxYsXb/jalMoWl07A1G6wcgzUugceW39dYnrtqoGjuNWZ01v59e1IxhgGDBjA2LFjr5v+8ccf4+npmfzc29s7+Xl6B7JgwYJZE6hSbiQpyfDzP8f4YOk+4hKTeLFzTYa1rYKXp/7mzirOqmPt5cuXL/mxp6cnCQkJWbLOI488wvTp05k5cyaTJk266e3e6Es6Zf1+bRseHh7Xbc/Dw+Om4lQqy+1ZDIufgsR46P41BPQDu5EjNh2O4L2l+5g0IBDfQvnS2dDN01rcAe644w7mzp1LeHg4AOfPn7/hL/O2bduycOFCoqOjiYqKYsGCBbRp0yY7wlUqxztw5gq9v/ub1xfuokEFH5aPbMsTHappYqoypHDhwly5cuW6aQMHDkzuC1C37s0n5bVq1SI0NJSQkBAApk2bRrt27RwWq1LZJi7KGiJq9iNQvDKMWAuNHk5OTOMSkvhg2T76fr+RS9FxRETFOaxotzpz6ix16tThnXfe4c477yQpKQkvLy++/vrrdNdp3LgxAwcOpFmzZgAMGTKERo0a3dQlKqVyq5j4RL7+M4Rv/zpEoXx5+KR3Q3o0Lq/jP6pM8fX15bbbbqNevXp07dqVjz76iNKlS1O7dm3uu+++W9qWt7c3kydPpnfv3iQkJNC0aVNtO6pynpNBMG8IRIRA62eh/WuQJ2/y7JDwSJ6dFUTwiUv0CazAG/fWoWA+x6WU4uh2AlkpMDDQbNmy5bppe/fupXbt2k6KKPvltter1DUbDp1j1IJdHDkXRY/G5Xn97joUL5j3xitmkIhsNcYEZlkBLkjr2H9FR0dTv359tm3bho+Pj7PDSZZbj4fKJklJ8PeXsOptKFgSenwHldsmzzbGMH3TMd5Zsof8Xp6M7dGALvXKZKio9OpYPXOqlHJpF6LiePe3vczdGkYl3wJMH9Kc26qVcHZYyo2tXLmSQYMG8dxzz7lUYqpUlrp8ChYMhyN/Qe174d4voEDx5NnnImN5Zd5OVu4Np031EnzcuyGli3hnSSianCqlXJIxhoVBJ3j7171cvhrPEx2q8tTt1fH28rzxykplQseOHTl27Jizw1Aq++xbAouehIQYKylt3P+6Tk9/7gvnxbk7uByTwBv31GFgK388PLKuOZUmp0opl3M0IorXF+5i7cFzNKpYlLE96lOrTBFnh6WUUu4lLhqWvwZbJ0PZhtDzByhRPXn21bhE3vttL9M2HqVWmcL8NKR5ttTFbpGcGmNyRYeInNQ+WKmMiE9MYuLaI4xbeQAvTw/e7l6Xh5pXwjMLf6ErpVSudGonzBsM5w5Aq6fh9v9d1+lp14lLjJwVREh4JENaV+aFzjWz7cpVjk9Ovb29iYiIwNfX160TVGMMEREReHtnTfsOpZwt6PhFXpm3k32nr9ClbhnGdKtLGR99vyullEMlJcHGb2DVm1DAF/ovgirtk2cnJhm+X3uYT1bsp3jBvPw0uDmtq2dvO/8cn5z6+fkRFhbG2bNnnR1KlvP29sbPz8/ZYSjlUFdi4vlkxQF+/DuU0oW9+e6RJnSum7Hen0oppdJx5TQsfAwO/WHd6anbl9d1ejp58SrPzQ5i4+HzdK1Xhvfur0+xLBwVJS05Pjn18vKicuXKzg5DKZUBK3af5o1FuzlzJYYBLf15/s4aFNbbjqoUjh8/Tv/+/Tl9+jQeHh4MGzaMZ555xtlhKZWz7F8Ki56w2pne8xk0efS6Tk+Ld5zk9QXBJCYZPuzVgN5N/Jx2RdqpyamI3AfcDZQCvjbGrHBmPEqp7HH6UgyjF+9i+e4z1CpTmPEPN6ZRxWLODku5qDx58vDJJ5/QuHFjrly5QpMmTejUqRN16tRxWBkJCQnkyZPjz9co9V/xV2HF67B5IpSpb3V6KlkzefblmHhGL9rNgu0naFSxKOP6BFDJ17m3Uc/wJ1FEJgH3AOHGmHp207sAnwOewERjzPtpbcMYsxBYKCLFgI8BTU6VcmNJSYbpm47ywbL9xCcm8XKXWgxpU1lvO6rSVbZsWcqWLQtYtxqtXbs2J06cuC45DQ0NpWvXrrRu3ZoNGzZQvnx5Fi1aRP78+QkKCmLEiBFER0dTtWpVJk2aRLFixWjfvj2tWrVi/fr1dOvWjV9++YVGjRqxdetWzp49y9SpUxk7dizBwcH06dOHd955x1m7QKmMOb3L6vR0dh+0fBLueAPy5EuevTn0PCNnBnH6cgwjO1bnyQ7VyOMC9XFmfiZOAb4Cpl6bICKewNdAJyAM2Cwii7ES1bEp1h9kjAm3PX7dtp5Syk3tO32ZV+cHs/3YRdpUL8E799Vz+q9zdYuWvgKngx27zTL1oWua5zD+IzQ0lO3bt9O8efP/zDt48CAzZszg+++/54EHHmDevHk8/PDD9O/fny+//JJ27drxxhtv8OabbzJu3DgALl68yF9//QXAL7/8Qt68eVmzZg2ff/453bt3Z+vWrRQvXpyqVavy7LPP4uvr65CXrVSWSkqCTd/CytGQvxg8PB+q3ZE8Oz4xiXErDzB+9SH8ihVg9vCWNKnkOlevMpycGmPWiIh/isnNgBBjzGEAEZkJdDfGjMU6y3odsRozvA8sNcZsy2gsSinXFROfyJd/HOS7vw5TJL8Xn/VpyH0B5d16dA2VNSIjI+nZsyfjxo2jSJH/jrVYuXJlAgICAGjSpAmhoaFcunSJixcv0q5dOwAGDBhA7969k9fp06fPddvo1q0bAPXr16du3brJZ2yrVKnC8ePHNTlVri8y3Or0FLISanSF7l9BwX972x8+G8mzs4LYEXaJ3k38GN2tLoXyuVaTFkdHUx44bvc8DPjvz9t/PQV0BHxEpJox5tuUC4jIMGAYQMWKFR0YqlIqq60POceoBcGERkTTq4kfo+6q7ZSen8pBbuEMp6PFx8fTs2dP+vXrR48ePTh+/Dj33nsvACNGjKBLly7ky/fv5UpPT0+uXr16w+0WLHj92ftr2/Dw8Lhuex4eHiQkJDjipSiVdQ4sh4WPQ1wk3P0JBA5O7vRkjGHGP8d5+9c95M3jwfh+jelav6yTA06do5PT1E6FpDlyvDHmC+CL9DZojJkATAAIDAzUUeiVygHOR8XxzpI9zN92An/fAvw8pDmtqmXvOHm5jYhUAUYBPsaYXrZp7YG3gd3ATGPMamfFlxnGGAYPHkzt2rV57rnnAKhQoQJBQUHJy4SGhqa6ro+PD8WKFWPt2rW0adOGadOmJZ9FVcptxF+F39+AfyZA6XrQcyKUqp08OyIyllfmB/P7njO0rlaCj3s3dOlxpB2dnIYBFeye+wEnHVyGUspFGWOYv+0E7yzZw5WYBJ7sUI0nb6+WbXcVcTe30vHU1pxqsIjMtduEASIBb6z6OUdav34906ZNo379+smX7d977z3uuuuum1r/xx9/TO4QVaVKFSZPnpyF0SqVzc7ssTo9he+BFo/DHaPB69/Ec/X+cF6cu5NL0fG8fndtBt1WGQ8Xv+ueZOaWmLY2p79eqzRFJA9wALgDOAFsBh4yxuzOfKjWmdMtW7Y4YlNKKQcLPRfFqIXBrA+JoHHFoozt0YCaZQo7O6wME5GtxphAJ8fQFiu5nGpXz3pi1bPJHU+BB40xe2zz59qdOfUwxiSJSGngU2NMv/TKS62O3bt3L7Vr105jDZXd9HioZMZYZ0pX/A+8feC+8VC9Y/LsmPhE3l+6jykbQqlZujDj+gZQu+x/22o7S3p1bGaGkpoBtAdKiEgYMNoY84OIPAksx/pFP8lRialSyjXFJyYxYc1hvlh1kLyeHrxzXz0ealbR5X+Z5wS30vEU2JPK+km2hxeAfCnn29bXdv1K5TSRZ2HR43BwBVS/E7p/A4VKJs/effISI2cGcTA8kkdv8+flLrVy1BWszPTWfzCN6b8Bv2U4IqVUjrHt2AVemx/MvtNX6FqvDGO61aV0Eddtx+QmUu14KiK+wLtAIxF51RgzVkR6AJ2BolhD//2HtutXKoc5uBIWjoCYy9D1I2g2NLnTU1KSYeK6w3y8/AA+BbyYOqgZbWuUvMEGXY9rjR2glMoRrsTE89Hy/UzbeJQyRbz5vn8gneqUdnZYuUWqHU+NMRHAiBQT5wPzsyUqpVTWio+BlWNg03goVQf6L4bS/96I4tSlqzw/ewcbDkXQuW5pxvZoQPEcOjqKJqdKqVuybNdpRi/eRfiVWAa28uf5O2u63Bh5bi7bO54aY3RcWheQmT4iKocL3wvzhsCZXdBsOHR6E7zyJ89esvMUry0IJj4xiQ961ueBwAo5+jOr3yhKqZty6tJVRi/azYo9Z6hdtggTHgmkYYWizg4rN9oMVBeRylgdT/sCD2VVYd7e3kRERODr65ujv+xyOmMMEREReHtrs5lcxRjYPBFWvA75CsNDc6DGncmzr8TEM2bxHuZtC6NhhaKM6xNA5RI5/857mpwqpdKVmGT4aeNRPlq+n4SkJF7tWotBrSvj5QL3X3Z3rtDx1M/Pj7CwMM6ePZtVRaib5O3tjZ+fn7PDUNkl6hwsehIOLIVqHa3e+IVKJc/eevQ8I2cFceLCVZ6+ozpP3V7NbeplTU6VUmnae+oyr84PJuj4RdrWKMm799WjQvECzg4r13CFjqdeXl5Urlw5O4pSSl0Tssq6BenVC9DlfetSvoeVeMYnJvHlqoN89WcI5YvlZ86IljSpVNzJATuWJqdKqf+IiU/k81UH+X7NYXzye/F53wC6NSynl3WVUiorJcTCyjdh49dQshY8PB/KJN9/g9BzUTwzK4gdxy/Ss7EfY7rVobC3lxMDzhqanCqlrrP24FlGLdjFsfPRPBDox2t31aZogZzZ41MppXKMs/th7mA4EwxNh8Kdbyd3ejLGMHvLcd78ZQ9enh58/VBj7m5Q1skBZx1NTpVSgHXv5XeW7GXB9hNUKVGQGUNb0LKqr7PDUkop92YMbJkEy0dB3gLw4Eyo2TV59vmoOF6dv5Plu8/QqqovnzzQkLI++dPZYM6nyalSuZwxhnnbTvDOkj1ExSbw9O3VeLxDtRx1NxGllMqRoiJg8VOwfwlUvd3q9FS4TPLsNQfO8sKcHVyMjmfUXbUZ3Lpyrrj7nianSuViR85FMWpBMBsORRBYqRhje9SneunCzg5LKaXc36E/YcEIuHoeOr8HzR9L7vQUE5/IB8v2MXl9KNVLFWLKo82oU66IkwPOPpqcKpULxSUkMWHNIb74I4R8eTx49/56PNi0Yq74Ra6UUk6VEAd/vAUbvoQSNaDfHCjbIHn23lOXGTkziP1nrjCwlT+vdK2V665kaXKqVC6z9eh5Xp0fzIEzkdxdvyyj761DqSI6sLdSSmW5swdg3mA4vRMCB8Gd71rtTIGkJMOk9Uf4cNl+iuT3YsqjTWlfs9QNNuieNDlVKpe4HBPPh8v2MX3TMcoW8eaHAYHcUbu0s8NSSin3Zwxs+xGWvmL1wO/7M9S6O3n26UsxvDBnB+tCztGxdmk+6Fkf30L5nBiwc2lyqpSbM8awbNdpRi/ezbnIWAbdVpnnOtWgYD79+CulVJaLPm91etr3K1RuB/d/B0X+HQZqafApXl0QTGx8EmN71Kdv0wq5fkxp/XZSyo1diYnnpbk7WbrrNHXKFmHigEAa+BV1dlhKKZU7HP7L6vQUdRY6vQ0tn0zu9BQZm8Cbi3czZ2sYDfx8GNcngColCzk5YNegyalSbupoRBRDp27h0NkoXu5Si6FtKpPHTe67rJRSLi0hDv58F9Z/Dr7V4MEZUC4gefa2Yxd4dlYQx89H82SHajzTsTpeWj8n0+RUKTe0PuQcT/y8DYBpg5rRqloJJ0eklFK5xLkQq9PTqSBoPAC6jIW8BQFISEziyz9C+OrPEMoU8WbW8JY09S/u3HhdkCanSrkRYwxT/z7KW7/uoWrJgkzs35SKvgWcHZZSSrk/Y2D7NFj6MuTJB31+gtr3Js8+GhHFyFlBbD92kfsblefN7nUp4u3lxIBdlyanSrmJuIQkRi/exYx/jtOxdmnG9Q2gkHZ6UkqprHf1AvzyDOxZBP5toMcEKFIOsE4azNkaxpuLd+PpIXzxYCO6NSzn5IBdm35zKeUGzkXG8thPW9kceoEnO1TjuU41dEB9pZTKDqHrYP4wiDwDHcdAq6fBwxo0/0JUHK8tCGbprtO0qFKcTx8IoFzR/M6NNwfQ5FSpHG73yUsMm7qViKhYvnywEffqL3KllMp6CXGw+j1YNw6KV4HBv0P5xsmz1x08x/NzgjgfFccrXWsxtE0VPPWkwU1xenIqIgWBNcBoY8yvzo5HqZzkt+BTPD97B0ULeDFneCvq+/k4OySllHJ/5w7aOj3tgMb9ofNYyGcNAxUTn8hHy/fzw7ojVCtViB8GNKVeea2bb0WGk1MRmQTcA4QbY+rZTe8CfA54AhONMe/fYFMvA7MzGodSuVFSkmHcqoN8seogTSoVY/zDjSlVWG9BqpRSWcoY2DIJlo+y7vSUotPT/tNXeGbmdvadvkL/lpV4tWtt8uf1dGLAOVNmzpxOAb4Cpl6bICKewNdAJyAM2Cwii7ES1bEp1h8ENAD2APqtqtRNiopN4LnZQSzffYbeTfx45/565MujlZ9SSmWpqHOw6Ek4sBSqdID7xiff6SkpyTBlQyjvL9tHEe88TB7YlA61Sjk54Jwrw8mpMWaNiPinmNwMCDHGHAYQkZlAd2PMWKyzrNcRkQ5AQaAOcFVEfjPGJGU0JqXc3fHz0QyduoUDZ67wxj11ePQ2/1x/mzullMpyB1fCwscg5qJ1Cb/5iOQ7PYVfjuH5OTtYe/Acd9QqxQe9GlCiUD7nxpvDObrNaXnguN3zMKB5WgsbY0YBiMhA4FxqiamIDAOGAVSsWNGRsSqVo2w8HMHj07eRkJjElEeb0bZGSWeHpFyIiFQBRgE+xphetmm1gWeAEsAqY8x4J4aoVM4TfxVWjoFN30KpOvDIAiiT3JKR5btP88q8nVyNT+Td++vxULOKesLAARydnKZ2RMyNVjLGTEln3gRgAkBgYOANt6WUO/pp41HGLN5NJd8CTBzQlMolCjo7JJUNbqVtv+2K1WARmXttOWPMXmCEiHgA32dv9ErlcKd3wbwhcHYvNH/MGibKy2qFGB2XwNu/7mXGP8eoV74I4/o0olqpQs6N1404OjkNAyrYPfcDTjq4DKVyjfjEJN78ZTc/bTxGh5ol+fzBRnpHkdxlCjfZtt8Ysye1DYhIN+AV23aUUjeSlASbxltnTPMXg4fnQbWOybODwy7xzMztHImIYkS7qjzXqQZ583g4L1435OjkdDNQXUQqAyeAvsBDDi5DqVzhfFQcj/20lU1HzjOiXVVe7FxTx8jLZW6lbT9W59LUtrEYWCwiS4CfszBcpXK+y6dg4Qg4vBpq3g3dvoCCJQBITDJMWHOYT1bsp2ThfEwf0pxWVUs4N143lZmhpGYA7YESIhKGNU7pDyLyJLAc63LTJGPMbodEqlQusvfUZYZO3UL4lVjG9QngvkblnR2Sch2ptu0XEV/gXaCRiLxqjBkrIu2BHkA+4LfUNqbt+pWy2fsLLH4KEmLhnnHQZCDY2o+evHiV52YHsfHwee6uX5b37q+PTwG9ipVVMtNb/8E0pv9GGpWgUurGlu06zXOzgyjsnYc5w1vSsEJRZ4ekXEuqbfuNMRHAiBQTVwOr09uYtutXuV5sJCx7BbZPg7IB0HMilKiePHvJzlO8On8niUmGj3o1oFcTP+30lMWcfocopZTFGMOXf4Tw6e8HaFihKBMeaULpIjoEsPoPbduvlKOEbYX5Q+D8EWj9HLR/FfLkBSAyNoHRi3Yzb1sYARWKMq5PAP7aGTVbaHKqlAuIjkvgxTk7WRJ8ih6NyvNej/p4e+nA+ipV2rZfqcxKSoR1n8KfY6FwWRj4K/i3Tp697dgFRs4MIuxCNE/fXo2n7qiOl6d2esoumpwq5WRhF6IZNnUr+05fZtRdtRnSprJeMlKAtu1XKktcOAoLhsOxv6FeT7j7U8hfFICExCS+/vMQX/xxkLI+3swa3pKm/sWdG28upMmpUk60OfQ8I6ZtJS4xiR8GNqVDTb3dnfqXtu1XysF2zoYlz1uP758ADR5I7vR0/Hw0I2cFsfXoBe5vVJ43u9fVofucRJNTpZxk5j/H+N+iXVQoVoDvBwRStaQO4KyUUlni6kUrKd01Fyq0gB4ToFil5NkLtofxv4W7EeDzvgF0D9ARUpxJk1Olsll8YhLvLtnLlA2htK1Rki/7NtIhSZRSKquErrcu418+CR1eh9bPgqeV/ly6Gs//Fu5i8Y6TNPUvxqcPBFCheAEnB6w0OVUqG12IiuOJn7ex4VAEQ9tU5uUutcijjeyVUsrxEuJg9VhY9xkUrwyDV4BfYPLsf46c59lZQZy+HMMLd9bgsfbV9EYnLkKTU6WyyYEzVxg6dQunLsbwce+G9Gri5+yQlFLKPZ07CPOGwKkgaPQIdHkf8llNp+ITk/h85UG+WR1CheIFmDuiJY0qFnNuvOo6mpwqlQ1W7jnDMzO3UyBfHmYOb0FjrQiVUsrxjIFtP8KyVyFPPnhgGtTpljz7yLkoRs7czo6wSzwQ6Mfoe+tSMJ+mQq5Gj4hSWcgYwzerD/Hxiv3UK+fDhP5NKOuT39lhKaWU+4mKsG4/un8JVGkP942HIuUAqy6esyWMMb/sxsvTg2/6Neau+mWdG69KkyanSmWR+MQkXpsfzJytYXRrWI4PezXQgfWVUiorhKyEhY/D1QvQ+T1o/hh4WO35L0TF8dqCYJbuOk3LKr582qehniRwcZqcKpUFrsTE8/j0baw9eI5n7qjOyI7VdWB9pZRytPgYWDkGNo2HkrXh4XlQpn7y7PUh53hudhDno+J4tWsthrapgod2enJ5mpwq5WCnL8Xw6JTNHDxzhQ97NeCBwAo3XkkppdStObPb6vQUvgeaDYdOb4KXdUY0NiGRT1ccYMLaw1QuUZAfBjSlXnkfJwesbpYmp0o50L7Tl3l08mYuX41n0sCmtK1R0tkhKaWUezEGNn0Hv78B3j7Qby5U75Q8OyT8Cs/MDGL3ycs83KIio+6qQ/682qQqJ9HkVCkH2RByjuHTtlIgnyezR7Skbjn9la6UUg515QwsetxqY1qjC3T7CgpZJwGMMUzfdIx3luyhQN48TOwfSMc6pZ0csMoITU6VcoD528J4ed5OqpQoxORHm1KuqDa2V0oph9q/FBY9AXFRcPcnEDgYbG35z0XG8sq8nazcG07bGiX5uHcDShX2dnLAKqM0OVUqE4wxfP1nCB+vOECrqr6Mf7gJPvn1VqRKKeUwcdGwYhRsmWR1dur5A5SsmTx79f5wXpizk8sx8Yy+tw4DWvprp6ccTpNTpTIoPjGJ/y3cxczNx+nRqDzv92xA3jx6K1KllHKYUzusTk/nDkCrp+D2/1mD6wMx8Ym8v3QfUzaEUrN0YX4a0oxaZYo4OWDlCJqcKpUBkbEJPDF9G38dOMtTt1fjuU41dKgopZRylKQk+PtLWPU2FCwB/RdZA+vb7Dt9mWdmBLH/zBUevc2fl7vU0nGk3Ygmp0rdovDL1lBR+05fYWyP+jzYrKKzQ1JKKfdx6QQsHAFH1kDte+HeL6BAcQCSkgyTN4TywdJ9+BTw4sdBzWino6K4HU1OlboFB89cYeDkzVyIjmPigEA61Czl7JCUUsp97FkEi5+GxHirJ36jh5M7PZ25HMMLc3aw9uA5OtYuzQc96+NbKJ+TA1ZZwanJqYh4AG8DRYAtxpgfnRmPUun5+1AEw6ZtwdvLk9nDW+qAzkop5SixkbDsZdj+E5RrDD0ngm/V5NnLd5/mlXk7uRqfyLv31+OhZhW1KZUby3ByKiKTgHuAcGNMPbvpXYDPAU9gojHm/XQ20x0oD5wHwjIai1JZbVHQCV6cs5OKvgWY8mhT/IoVcHZISinlHsK2WJ2eLoRCmxeg/SvgaY16Eh2XwNu/7mXGP8eoV74I4/o0olqpQs6NV2W5zJw5nQJ8BUy9NkFEPIGvgU5YyeZmEVmMlaiOTbH+IKAm8Lcx5jsRmQusykQ8SjmcMYbxfx3iw2X7aV65OBMeCcSngA4VpZRSmZaUCGs/hdVjoUg5GLgE/G9Lnr0z7CIjZwZxJCKKEe2q8lynGjoiSi6R4eTUGLNGRPxTTG4GhBhjDgOIyEyguzFmLNZZ1uuISBgQZ3uamFo5IjIMGAZQsaJ2PFHZJyExidGLdzN90zG6NSzHR70bkC+P9gZVrklEqgCjAB9jTK+0pinlEi4eg/nD4NjfUK+XNah+/qIAJCYZvv3rEJ/9foCShfPx85AWtKzq69x4VbZy9E+Q8sBxu+dhtmlpmQ90FpEvgTWpLWCMmWCMCTTGBJYsqT3yVPaIik1g2LStTN90jMfaV2VcnwBNTFW2E5FJIhIuIrtSTO8iIvtFJEREXgEwxhw2xgy2Xy61aUo53c45MP42OL0LenwPvX5ITkxPXLzKg99v5KPl++lctwzLnmmriWku5OgOUam1TjZpLWyMiQa04lQuJfxKDIOnbGH3yUu8c189Hm5RydkhqdxrCjfZfMoYs8cpESp1s2IuwZIXIHg2VGgOPSZAMf/k2b/sOMlrC4JJSjJ83LshPRuX105PuZSjk9MwoILdcz/gpIPLUCrLhIRHMnDyP0RExvF9/0DuqF3a2SGpXOxWmk8Bmpwq13X0b+sy/uUT0GEUtH4OPK0U5EpMPKMX7Wb+9hM0qliUcX0CqORb0MkBK2dy9GX9zUB1EaksInmBvsBiB5ehVJb458h5eo7fQEx8IrOGt9DEVLmqVJtPiYiviHwLNBKRVwFSm5aSiAwTkS0isuXs2bNZHrzKZRLj4Y93YMpd4OEBg5ZDu5eSE9OtR89z1xdrWRh0gpEdqzNneEtNTFWmhpKaAbQHStg6No02xvwgIk8Cy7F66E8yxux2SKRKZaFfdpzk+dk78Cuenx8fbUaF4jpUlHJZqTafMsZEACNSTPzPtFRWnABMAAgMDEyzGZZStyziEMwfCie2QkA/6PoB5CsMWB1Ov/gjhK/+OEj5YvmZM6IlTSoVd3LAylVkprf+g2lM/w34LcMRKZWNjDF8v/Yw7/22j6b+xfi+fyBFC+R1dlhKpUebTynXZgwETYffbGdIe02Gej2SZx+NiGLkrCC2H7tIj8blebNbXQp76xB96l96+1KVayUmGd78ZTdT/z7K3Q3K8knvhnh7aY985fKSm08BJ7CaTz3k3JCUsok+D7+OtG5D6t8G7v8WfPwA62TA3K1hjFm8G08P4csHG3Fvw3LOjVe5JE1OVa50NS6Rp2ZsZ+XeMwxvW4WXu9TCw0N7hSrXos2nVI5yZA3MHw5R4dDxTWj1FHhYP/gvRscxasEulgSfonnl4nzWJ4ByRfM7OWDlqjQ5VbnOuchYBv+4heCwi7zVvS79W/o7OySlUqXNp1SOkBAHf74D678A36rw4Eoo1yh59oZD53hu1g7ORcbycpdaDGtbBU89GaDSocmpylUOn41k4OTNhF+J4duHm3Bn3TLODkkppXKucwdh3mA4tQOaPAqd34W8Vm/7uIQkPvl9PxPWHKayb0EWPH4b9f18nBywygk0OVW5xpbQ8wyZugVPEWYMbUGjisWcHZJSSuVMxsD2abD0ZcjjDX1/hlp3J88OCY9k5Kzt7DpxmYeaV+T1u2tTIK+mHOrm6DtF5QpLg0/xzKwgyhfNz5RHm+o4ekoplVFXL8AvI2HPQqjcDu7/DoqUBaxOT9M3HeOdJXsokDcPEx7RK1Tq1mlyqtzexLWHefe3vTSuaA0VVbygDhWllFIZcvRva+zSK6eg4xho9Yw1uD5We/5X5u1k5d5w2lQvwSe9G1KqiLdz41U5kianym1tP3aBT38/wNqD5+harwyf9QnQoaKUUiojEhNgzUew5kMoWgkGrQC/JsmzV+8P54U5O7kcE88b99RhYCt/HQFFZZgmp8rt7Ay7yGe/H+DP/WcpXjAvr99dm0G3VdaKUimlMuLiMZg3FI5vhAZ94e6Pk+/0FBOfyPtL9zFlQyg1SxfmpyHNqFWmiJMDVjmdJqfKbew6cYlxKw+ycu8Zihbw4uUutejfshIF8+nbXCmlMmT3Alj8DJgk6PE9NHggedbeU5d5ZuZ2DpyJ5NHb/Hm5Sy29OqUcQr+1VY637/Rlxv1+kGW7T1PEOw8v3FmDAa389XZ4SimVUXFRVk/87dOgfBPoORGKVwEgKckwaf0RPly2H58CXvw4qBntapR0csDKnWhyqnKsA2eu8PnKgywJPkXhfHkY2bE6g1pXpogmpUoplXGndsDcwRARAq2fgw6vgadVr565HMPzs3ewLuQcneqU5v0e9fEtlM/JASt3o8mpynFCwiP5YtVBftl5kgJenjx1ezWGtK6CTwFNSpVSKsOSkmDTeFg5Bgr4Qv9FUKVd8uxlu07xyvxgYuOTGNujPn2bVkBE2/Irx9PkVOUYR85F8eWqgywMOoG3lyePtavK0DZVKKZDQymlVOZEhsPCxyBkJdS8C7p9BQV9AYiKTeCtX/Ywa8tx6pf34fO+AVQpWcjJASt3psmpcnnHIqL58o+DzN9+Ai9PYWibKgxrW0UvJSmllCMcXAkLR0DsFbjrY2g6BGxnRIOOX2TkzO0cPR/N4+2rMrJjDfLm8XBywMrdaXKqXFbYhWi++iOEuVvD8PQQBrbyZ3i7KpQqrIM6K6VUpiXEwso3YePXUKoO9F8MpesAkJhk+ObPEMatOkiZIt7MHNqC5lV8nRywyi00OVUu5+TFq3z9ZwiztxxHEB5uUYnH2leltN5pRCmlHOPcQZg7CE7vhKZD4c63wSs/AMfPR/PsrCC2HL1A94ByvNW9Hj75tU2/yj6anCqXceZyDN/8GcKMf45jMPRtWpHHO1SlrE9+Z4emlFLuwRhreKilL0Meb+g7A2rdZZtlWBh0gv8t3I0A4/oEcF+j8s6NV+VKmpwqpwu/EsP41YeYvukYSUmG3oEVeKJDVfyKFXB2aEop5T6uXoRfnoE9C6FyW7j/OyhSDoBLV+N5feEuftlxkqb+xfj0gQAqFNc6WDmHJqfKac5FxvLdX4eYtvEo8YmGno3L89Tt1bVCVEopRzv6N8wfCldOwR2j4bZnwMO6m9PGwxE8NyuI8CuxvNi5JiPaVcVTb/esnEiTU5XtzkfFMWHNYX7cEEpsQiL3NSrP07dXx79EQWeHppRS7iUxAdZ+DH99AEUrwqAV4NcEgLiEJD5beYBv/zqEv29B5j3WioYVijo3XqVwcnIqIhWBr4BzwAFjzPvOjEdlrYvRcUxce4TJ648QHZ9It4blePqO6lTV8fKUUsrxLh63zpYe+xsa9LGGifIuAsChs5GMnBlE8IlL9G1agf/dU4eC+fR8lXINGX4nisgk4B4g3BhTz256F+BzwBOYeIOEswawxBjznYhMzWgsyrVduhrPD+uOMHndEa7EJnBPg7I8c0d1qpcu7OzQlFLKPe1eCL88DUmJcP8EaNgHsDo9/fzPMd7+dQ/5vTz59uEmdKlXxrmxKpVCZn4mTcE665mcVIqIJ/A10AkIAzaLyGKsRHVsivUHAduBUSLSB5iWiViUCzLGMPXvo3yyYj+XYxLoWq8Mz3SsTq0yRZwdmlJKuae4KKsn/vZpUL4J9JwIxasAEBEZy8vzglm59wxtqpfg494NdYg+5ZIynJwaY9aIiH+Kyc2AEGPMYQARmQl0N8aMxTrLeh0ReQEYbdvWXGByKssMA4YBVKxYMaPhqmwWHZfAy/OC+WXHSdrWKMnLXWpSt5yPs8NSKlcRkTrAGCACWGWMmevciFSWOrUD5g6GiBBo/Sx0GAWe1vikq/eH88KcnVy+Gs//7qnDo6388dBOT8pFObqBSXnguN3zMKB5OssvA8aIyENAaGoLGGMmABMAAgMDjWPCVFnpyLkoRkzbysHwK7zUpSaPtauKiFaCSjnCLTap6gp8aYxZa7uKpcmpO0pKgk3jYeUYyF8c+i+CKu0AiIlP5P2l+5iyIZSapQszbXAzapfVq1fKtTk6OU0tA0kzoTTG7AJ6OTgG5UQr95zh2VlB5PEUpg5qTuvqJZwdklLuZgo336RqGjBaRLoBeu9JdxQZDgsfg5CVUKMrdP8aClqHes/Jy4yctZ0DZyJ59DZ/Xu5SC28vTycHrNSNOTo5DQMq2D33A046uAzlghKTDONWHuDLP0KoX96H8Q831kH0lcoCGWhS9YQteZ2f2va06VQOdnCllZjGXLJ64jcdAiIkJRkmrT/Ch8v241PAix8HNaNdjZLOjlapm+bo5HQzUF1EKgMngL7AQw4uQ7mYi9FxPDMziL8OnKVPYAXe7F5Xf50rlb1SbVJlS2JfAwoCH6W2ojadyoESYmHlm7DxayhVB/ovhNJ1ATh9KYYX5uxgXcg5OtUpzfs96uNbKJ9z41XqFmVmKKkZQHughIiEYXVs+kFEngSWY7V7mmSM2e2QSJVL2nXiEo9N38qZS7GM7VGfB5vpmRelnCDVJlXGmFBsZ0WVmzi73+r0dCYYmg6FO98Gr/wALNt1ilfmBxMbn8TYHvXp27SCtvdXOVJmeus/mMb034DfMhyRyjHmbQ3jtQXBFC+Yl9kjWhKgdxZRylm0SZW7Mwa2ToFlr1rJ6IMzoWZXAKJiE3jzl93M3hJGAz8fxvUJoIre3ETlYHo7CHXL4hKSePvXPUzbeJSWVXz58qFGlNDLRko5kzapcmfR52HxU7DvV6jSHu77FoqUBSDo+EVGztzO0fPRPNGhKiM71sDL08O58SqVSZqcqlty+lIMj0/fyrZjFxnetgovdq5JHq0Ilco22qQqlzmyBuYPh6iz0OltaPkkeHiQkJjEN6sP8fmqg5Qp4s3MoS1oXkUHZFDuQZNTddM2HY7giZ+3cTUukW/6Neau+mWdHZJSuY42qcolEuPhz/dg3WfWHZ6G/A7lGgFwLCKaZ2cHsfXoBboHlOOt7vXwye/l5ICVchxNTtUNGWOYtD6U937bSyXfAswc1oJqpQo7OyyllHJP5w/DvCFwYis0egS6vA/5CmGMYd62E4xZvBsBPu8bQPeA8s6OVimH0+RUpcv+NqSd65bm494NKeytv9CVUsrhjIGds2DJ8+DhCb2nQN37AWvIvlELdrEk+BTNKhfn0wca6ljSym1pcqrSpLchVUqpbBJzyUpKg+dAxVbQYwIUtQZg2BByjudm7+BcZCwvdanJ8LZV8fTQuli5L01OVar0NqRKKZVNjv8D8wbDpRPQYRS0sc6cxiYk8vHy/Xy/9ghVShZkQf/bqO/n4+xolcpympyq6+htSJVSKpskJcLaT2H1WPApD48uhYrNAThw5grPzAxi76nL9GtekdfvrkP+vHrnPZU7aHKqkl2MjuPpmUGs0duQKqVU1rp4HOYPg2MboF5PuOcz8PbBGMOPG0IZu3QfhfLlYWL/QDrWKe3saJXKVpqcKkBvQ6qUUtlm90L45WnrzOl930LDviBC+JUYXpyzk78OnKVDzZJ82KshJQvrDU5U7qPJqdLbkCqlVHaIi4KlL8P2aVCuMfScCL5VAVix+zSvzA8mKjaBt7vX5eEWlbQDqsq1NDnNxfQ2pEoplU1OBlmdniIOQevnoMNr4OlFdFwCb/+6hxn/HKduuSJ83jdAx5FWuZ4mp7mU3oZUKaWyQVIS/P0VrHoLCpaEAYuhclsAdhy/yMhZQYRGRDGiXVWe61SDvHm0HlZKk9NcaOPhCJ7U25AqpVTWunIaFoyAw39CrXug25dQoDiJSYbxq0MYt/IgpQrn4+chLWhZ1dfZ0SrlMjQ5zUWMMfyw7ghjl+6jUvECzBjaguql9fKRUko53P5lsOhxiIu2euI3eRREOH4+mmdnBbHl6AXubViOd7rXw6eA3nVPKXuanOYSehtSpZTKBvFX4fc34J8JULoe9PwBStXCGMOCbWG8sWg3AnzWpyH3BZTXTk9KpUKT01xAb0OqlFLZ4Mweq9NT+B5o8TjcMRq8vLkUHc+ohcH8uvMUTf2L8ekDAVQorjc3USotmpy6uWW7TvPinB16G1KllMoqxsDmibB8FHgXgX5zoXonAP4+FMHzs4MIvxLLi51rMqJdVTw99OSAUunR5NRN7TpxifeX7mNdyDm9DalSSmWVqAhY9AQcWArVOsJ946FQKeISkvjk9/1MWHOYyr4FmfdYKxrqGNJK3RRNTt3MsYhoPl6xn8U7TlKsgBf/u6cOD7eoSL48ehtSpZRyqIMrrU5PVy9A57HQfAR4eBASfoWnZwSx59RlHmxWkf/dU5sCefXrVqmbpZ8WNxERGcuXf4QwfdNRPD2EJztUY1i7KhTRTk9KKeVYcdFWp6fN30PJWtZl/LINMMYw7e9Q3l2yl4L58vB9/0A61Snt7GiVynGyLTkVkSrAKMDHGNPLNq0g8A0QB6w2xkzPrnjcRVRsAj+sO8KENYe5Gp/IA4EVGNmxOqWLeDs7NKWUcj8ng2D+UDh3AJo/Bh1Hg1d+zl6J5aW5O/hz/1na1SjJR70bUKqw1sNKZcRNJaciMgm4Bwg3xtSzm94F+BzwBCYaY95PaxvGmMPAYBGZaze5BzDXGPOLiMwCNDm9SfGJSczafJxxKw9yLjKWznVL82LnWlQrVcjZoSmllPtJSoR1n8Hqsdadnh5ZAFVvB2DlnjO8PG8nkbEJvNmtLv1bVtIRUZTKhJs9czoF+AqYem2CiHgCXwOdgDBgs4gsxkpUx6ZYf5AxJjyV7foBwbbHiTcfdu5ljGHprtN8tHw/R85F0dS/GN890oQmlYo5OzSllIsRkTZAP6y6vo4xppWTQ8qZLoTC/OFwfCPUvR/u/hQKFCc6LoF3l+xl+qZj1C5bhBl9A6ihNzZRKtNuKjk1xqwREf8Uk5sBIbYzoojITKC7MWYs1lnWmxGGlaAGAaneUFhEhgHDACpWrHiTm3VPGw9HMHbpPnYcv0j1UoWY2D+QO2qX0l/oSuUit3IlyxizFlgrIvcBm50Rb45mDARNh6Uvg3jA/ROgwQMgws6wi4ycGcSRiCiGta3C83fW0I6nSjlIZtqclgeO2z0PA5qntbCI+ALvAo1E5FVbEjsf+EpE7gZ+SW09Y8wEYAJAYGCgyUS8Oda+05f5YOk+/tx/lrI+3nzYqwE9G/vpWHlK5U5TuMkrWcaYPbZFHgKGZHOcOVtUBPzyNOz7FSrdBvd/C0UrkphkGP/nQcatPEjJwvmYPqQ5rarq+NFKOVJmktPUMqM0k0djTAQwIsW0KODRTMTg1k5cvMqnKw4wf3sYhfPl4ZWutRjYyh9vL/11rlRudStXsoA9IlIRuGSMuZy9keZg14aIij4PHd+EVk+BhyfHz0fz7Kwgthy9wD0NyvLuffXxKaAjoijlaJlJTsOACnbP/YCTmQtHAVyMjuOb1YeYsiEUgGFtqvBY+6oULZDXuYEppVxVeleyBgOT01pRm07ZuW6IqNrXDRE1b2sYYxbvRoBxfQLoHlBOm1QplUUyk5xuBqqLSGXgBNAX69KRyqCY+EQmrw/lm9UhRMYm0LOxH892qkH5ovmdHZpSyrWleSXLGDM6vRW16ZTNye0wf5g1RFSLx+GO0eDlzYWoOEYtDOa34NM0q1ycTx9oqHfbUyqL3exQUjOA9kAJEQkDRhtjfhCRJ4HlWA3wJxljdmdZpG4sITGJ+dtO8OnvBzh9OYbba5XipS41qVWmiLNDU0rlDHolK6OSEmHdp7D6fdsQUQuhagcA1h48ywtzdnA+Ko6Xu9RiWNsq2tZfqWxws731H0xj+m/Abw6NKBcxxrBybzgfLtvHwfBIAioUZVzfAFpU8XV2aEqpnEWvZGVEGkNExcQn8uGy/Uxaf4SqJQvyw4Cm1Cvv4+xolco19PalTrL16HneX7qPzaEXqFKiIOP7NaZLvTLahkkplS69kuUA6QwRtffUZUbODGL/mSsMaFmJV7rWJn9e7YSqVHbS5DSbhYRH8uGyfazYc4aShfPx7v31eCCwAl6eqQ7zqpRS19ErWZl03RBRreH+8VC0IklJhknrDvPhsv0Uye/F5Eeb0qFmKWdHq1SupMlpNjlzOYZxKw8wa/NxCuTNw/OdajC4TWUK5NVDoJRS2eLg77DoCWuIqE5vQcsnwcOTU5eu8vzsHWw4FEGnOqV5v0d9fAvlc3a0SuVamhllscsx8Xz31yF+WHeExCRD/5b+PHV7Na34lFIqu6QxRBTArztP8tr8YOITDe/3qE+fphW0eZVSTqbJaRbaduwCg6ds5kJ0PN0DyvF8p5pU9NUhSJRSKtukMUTU5Zh4xizazfztJ2hYoSjj+gRQuURBZ0erlEKT0yyz68QlBkz6h+IF8zJtcHPt6amUUtnpuiGiSl03RNQ/R87z7KwgTl26yjN3VOfJ26tpu3+lXIgmp1lg/+krPPzDJop4e/Hz0BY6iL5SSmW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      "text/plain": [
       "<Figure size 720x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "solve_linear('random', v_1)\n",
    "solve_linear('vandermonde', v_2)\n",
    "solve_linear('hilbert', v_3)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ddcf26a3",
   "metadata": {},
   "source": [
    "The condition number of a problem is the ratio of the relative change in the solution to the relative change in the input. A problem is well-conditioned if small relative changes in input result in small relative changes in the solution. Otherwise, the problem is ill-conditioned.\n",
    "\n",
    "\n",
    "A matrix can be defined **ill conditioned** depending on the value of the conditioning number $K(A) = \\| A \\| \\| A^{-1} \\|$. Higher the conditioning number, higher the resulting perturbation on the solution. \n",
    "\n",
    "\n",
    "$$\\frac{\\|\\Delta x \\|}{\\|x\\|} \\leq K(A) \\left( \\frac{\\|\\Delta b \\|}{\\|b\\|}+ \\frac{\\|\\Delta A \\|}{\\|A\\|}\\right) $$\n",
    "\n",
    "This is a property of the matrix itself, therefore it doesn't depend on the algorithm used to solve the system.\n",
    "\n",
    "\n",
    "As it is possible to see from the graphs, **Hilbert matrixes** are slightly worse than **Vandermonde matrixes**, but both of those matrix conditioning numbers become really high over a certain dimension. Otherwise, **Random Matrixes conditioning number can vary depending on the seed**. This proofs that Vandermonde and Hilbert matrix are ill conditioned. \n",
    "\n",
    "The $2$-norm is in most of the cases above the $\\infty$-norm, but they approximately have the same behavior, in fact we know that in finite spaces holds the equivalence between the two norms.\n",
    "\n",
    "Recall that two norms $p$ and $q$ are said \"equivalent\" if: \n",
    "$$c_{pq} \\|x\\|_q \\leq \\|x\\|_p\\leq C_{pq} \\| x\\|_q \\quad \\forall x \\in V$$\n",
    "\n",
    "where $c_{pq}$ and $C_{pq}$ are two positive constants and $V$ is a vector space\n",
    "\n",
    "We can even show that they induce the same topology:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "e5d91eb1",
   "metadata": {},
   "outputs": [
    {
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       "</g>\n",
       "\t</a>\n",
       "\t<a xlink:title=\"Plot #10\">\n",
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       "\t</a>\n",
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       "\t</a>\n",
       "\t<a xlink:title=\"Plot #12\">\n",
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       "</g>\n",
       "\t</a>\n",
       "\t<a xlink:title=\"Plot #13\">\n",
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       "</g>\n",
       "\t</a>\n",
       "\t<a xlink:title=\"Plot #14\">\n",
       "<g style=\"fill:none; color:red; stroke:currentColor; stroke-width:2.00; stroke-linecap:butt; stroke-linejoin:miter; stroke-dasharray:8,2,2,2\">\n",
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       "</g>\n",
       "\t</a>\n",
       "<g style=\"fill:none; color:black; stroke:currentColor; stroke-width:0.50; stroke-linecap:butt; stroke-linejoin:miter\">\n",
       "\t<path d=\"M107.0,16.7 L107.0,444.0 L534.3,444.0 L534.3,16.7 L107.0,16.7 Z \"/>\n",
       "</g>\n",
       "</svg>"
      ],
      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from IPython.display import SVG, display\n",
    "display(SVG(url='https://upload.wikimedia.org/wikipedia/commons/d/d4/Vector-p-Norms_qtl1.svg'))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a46ee856",
   "metadata": {},
   "source": [
    "###### Other factors\n",
    "*note on the implementation of **np.linalg.solve(A, b)**: this method does not compute the inverse of A. Instead it calls one of the gesv LAPACK routines, which first factorizes A using LU decomposition, then solves for x using forward and backward substitution. This, as we know, is an unstable algorithm* that can introduce errors in the solution"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a803093d",
   "metadata": {},
   "source": [
    "# Floating point arithmetic"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ebe39371",
   "metadata": {},
   "source": [
    "In Python just double precision floating point numbers are implemented"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "f59a0076",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<img src=\"https://media.geeksforgeeks.org/wp-content/uploads/Double-Precision-IEEE-754-Floating-Point-Standard-1024x266.jpg\"/>"
      ],
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from IPython.display import Image, display\n",
    "display(Image(url='https://media.geeksforgeeks.org/wp-content/uploads/Double-Precision-IEEE-754-Floating-Point-Standard-1024x266.jpg'))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2dc5e95f",
   "metadata": {},
   "source": [
    "## Part 1 - Machine precision\n",
    "The Machine epsilon $\\epsilon$ is the distance between 1 and the next floating point number. Compute $\\epsilon$, which\n",
    "is defined as the smallest floating point number such that it holds:\n",
    "fl(1 + $\\epsilon$) > 1\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "61124169",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epsilon value: 2.220446049250313e-16\n",
      "iterations performed:  53\n"
     ]
    }
   ],
   "source": [
    "diff = 1\n",
    "i=0\n",
    "while 1 + diff > 1:\n",
    "    i = i +  1\n",
    "    diff = diff / 2\n",
    "diff = diff * 2\n",
    "print(\"epsilon value: \" + str(diff))\n",
    "print(\"iterations performed: \",i)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f4896b06",
   "metadata": {},
   "source": [
    "That, as we can see, matches with the one shown by the specific python bundled function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "c16f6456",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.220446049250313e-16\n"
     ]
    }
   ],
   "source": [
    "import sys\n",
    "print(np.finfo(float).eps)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fb9d153f",
   "metadata": {},
   "source": [
    "We have divided by 2 because in computers numbers are usually in binary representation. The number of performed iterations attests that $\\epsilon$ has became non-representable in the addition after 53 iterations: that's because in the IEEE standard (which is applied in python), mantissa has 52 values, so that if we fix an exponent, the epsilon value shifts one position for each iteration, unless it becomes too small to be inside the mantissa. \n",
    "\n",
    "$$\\epsilon = 2.220446049250313 \\cdot {10}^{-16}$$\n",
    "$$iterations = 53$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5b926e7e",
   "metadata": {},
   "source": [
    "## Part 2 - Sequence limit\n",
    "Let’s consider the sequence $a_n = {(1 + \\frac{1}{n})}^n$\n",
    "n. It is well known that:\n",
    "$\\lim_{n \\to +\\infty} a(n) = e$\n",
    "where e is the Euler costant. Choose different values for $n$, compute $a_n$ and compare it to the real\n",
    "value of the Euler costant. What happens if you choose a large value of $n$? Guess the reason"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "d75e962e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "e = []\n",
    "x = []\n",
    "i = 1\n",
    "a_n = (1 + (1 / i)) ** i\n",
    "\n",
    "# iterations to do over the reaching of a_n = 1, just to show the points in the plot\n",
    "over_iter = 10\n",
    "\n",
    "while (a_n != 1) or (over_iter>0):\n",
    "    x.append(i)\n",
    "    a_n = (1 + (1/i)) ** i\n",
    "    e.append(a_n)\n",
    "    i = i * 2\n",
    "    \n",
    "    if (a_n == 1):\n",
    "        if (over_iter == 10):\n",
    "            x_pos = x[-1]\n",
    "            y_pos = e[-1]\n",
    "        over_iter = over_iter - 1\n",
    "    \n",
    "    \n",
    "\n",
    "plt.plot(x, e, '.-', label=\"e (calculated in this implementation)\")\n",
    "plt.plot(x, np.ones(len(e)) * np.e,label=\"e (from numpy library)\")\n",
    "plt.title(\"Calculating 'e' as a succession\")\n",
    "plt.xlabel(\"n\")\n",
    "plt.ylabel(\"Value\")\n",
    "plt.xscale(\"log\")\n",
    "#hold on;\n",
    "plt.plot(x_pos,y_pos,'ro', label = 'a_no')\n",
    "plt.legend()\n",
    "plt.show()\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "02ca6be2",
   "metadata": {},
   "source": [
    "$$a_n = {(1 + \\frac{1}{n})}^n$$\n",
    "\n",
    "$$\\lim_{n \\to +\\infty} a(n) = e$$\n",
    "\n",
    "In practice, when $\\frac{1}{n}$ becomes smaller than the machine precision it is no longer representable and that same fraction becomes undistinguishable from 0, therefore for each $n > n_o$ (with $n_o$ following this relation: $\\frac{1}{n_o} < \\epsilon$) we have $$a_n = (1)^n = 1$$  \n",
    "What happens is similar to the previous example, therefore this is caused by finite arithmetics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "374e4b0d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "err = abs((np.ones(len(e))* np.e) - e)\n",
    "\n",
    "plt.plot(x, err, '.-', label=\"e (error in absolute value)\")\n",
    "plt.title(\"e (error in absolute value)\")\n",
    "plt.xlabel(\"Length of list (number)\")\n",
    "plt.ylabel(\"Value\")\n",
    "plt.xscale(\"log\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "364a1980",
   "metadata": {},
   "source": [
    "## Part 3 - Rank & eigenvalues\n",
    "Let's consider the matrices:$$ A = \\left(\\begin{array}{cc} 4 & 2\\\\ 1 & 3 \\end{array}\\right), B = \\left(\\begin{array}{cc} 4 & 2\\\\ 2 & 1 \\end{array}\\right)$$ $$ $$ Compute the rank of **A** and **B** and their eigenvalues. Are **A** and **B** full-rank matrices? Can you infer some relationship between the values of the eigenvalues and the full-rank condition? Please, corroborate your deduction with other examples.$$ $$ *Tips:* Please, have a look at **np.linalg**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "08b3c91d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Matrix A   Rank: 2         Maximum rank possible: 2  Eigenvalues: [5. 2.]\n",
      "Matrix B   Rank: 1         Maximum rank possible: 2  Eigenvalues: [5. 0.]\n",
      "\n",
      "\n",
      "Matrix C   Rank: 2         Maximum rank possible: 3  Eigenvalues: [0. 4. 1.]\n",
      "Matrix D   Rank: 3         Maximum rank possible: 3  Eigenvalues: [ 9.  2. -2.]\n"
     ]
    }
   ],
   "source": [
    "\n",
    "A = np.array([[4, 2], [1, 3]])\n",
    "B = np.array([[4, 2], [2, 1]])\n",
    "\n",
    "print(\"Matrix A   Rank: \" + str(np.linalg.matrix_rank(A))+ \"         Maximum rank possible: \" + str(min(A.shape)) + \"  Eigenvalues: \" + str(np.linalg.eig(A)[0]))\n",
    "print(\"Matrix B   Rank: \" + str(np.linalg.matrix_rank(B))+ \"         Maximum rank possible: \" + str(min(B.shape)) + \"  Eigenvalues: \" + str(np.linalg.eig(B)[0]))\n",
    "print(\"\\n\")\n",
    "\n",
    "C = np.array([[0, 0, 1], [0, 1, 0], [0, 4, 4]])\n",
    "D = np.array([[0, 4, 0], [1, 0, 0], [3, 7, 9]])\n",
    "print(\"Matrix C   Rank: \" + str(np.linalg.matrix_rank(C))+ \"         Maximum rank possible: \" + str(min(C.shape)) + \"  Eigenvalues: \" + str(np.linalg.eig(C)[0]))\n",
    "print(\"Matrix D   Rank: \" + str(np.linalg.matrix_rank(D))+ \"         Maximum rank possible: \" + str(min(D.shape)) + \"  Eigenvalues: \" + str(np.linalg.eig(D)[0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "246707aa",
   "metadata": {},
   "source": [
    "Here, the maximum rank is 2, but just $rank(A) = rank_{max}(A)$, while $rank(B) < rank_{max}(B)$\n",
    " \n",
    "We know that given a matrix $M$, \n",
    "$$ rank(M)=rank_{max}(M) \\Leftrightarrow det(M)\\neq 0$$ \n",
    "\n",
    "and $$det(M) = \\prod_{i \\in \\sigma_M}{\\lambda_i}$$ so \n",
    "$$rank(M) = rank_{max}(A)\\Rightarrow det(M) \\neq 0 \\Rightarrow \\lambda_i \\neq 0 \\quad \\forall i$$\n",
    "and\n",
    "$$rank(M) \\neq rank_{max}(A)\\Rightarrow det(M) = 0 \\Rightarrow \\exists i \\quad \\lambda_i = 0  \\quad $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3807ea07",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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