ex2.3.2.ipynb

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    "<h1>Table of Contents<span class=\"tocSkip\"></span></h1>\n",
    "<div class=\"toc\" style=\"margin-top: 1em;\"><ul class=\"toc-item\"><li><ul class=\"toc-item\"><li><span><a href=\"#Set-$x_0-=--1.0$\" data-toc-modified-id=\"Set-$x_0-=--1.0$-0.1\"><span class=\"toc-item-num\">0.1&nbsp;&nbsp;</span>Set $x_0 = -1.0$</a></span></li><li><span><a href=\"#$x_0-=-0$-can-not-be-used\" data-toc-modified-id=\"$x_0-=-0$-can-not-be-used-0.2\"><span class=\"toc-item-num\">0.2&nbsp;&nbsp;</span>$x_0 = 0$ can not be used</a></span></li></ul></li></ul></div>"
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   "source": [
    "Let $f(x) = −x^3 − \\cos x$ and $p_0 = −1$. Use Newton’s method to find $p_2$. Could $p_0 = 0$ be used?"
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    "import numpy as np\n",
    "from numpy import linalg\n",
    "from abc import abstractmethod\n",
    "import pandas as pd\n",
    "import math\n",
    "\n",
    "pd.options.display.float_format = '{:,.8f}'.format\n",
    "np.set_printoptions(suppress=True, precision=8)\n",
    "\n",
    "TOR = pow(10.0, -9)\n",
    "MAX_ITR = 150"
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   "source": [
    "class NewtonMethod(object):\n",
    "\n",
    "    def __init__(self):\n",
    "        return\n",
    "\n",
    "    @abstractmethod\n",
    "    def f(self, x):\n",
    "        return NotImplementedError('Implement f()!')\n",
    "\n",
    "    @abstractmethod\n",
    "    def jacobian(self, x):\n",
    "        return NotImplementedError('Implement jacobian()!')\n",
    "\n",
    "    @abstractmethod\n",
    "    def run(self, x):\n",
    "        return NotImplementedError('Implement run()!')"
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   "source": [
    "class Newton1D(NewtonMethod):\n",
    "\n",
    "    def __init__(self):\n",
    "        super(NewtonMethod, self).__init__()\n",
    "\n",
    "    def f(self, x):\n",
    "        return -pow(x, 3) - math.cos(x)\n",
    "\n",
    "    def jacobian(self, x):\n",
    "        return -3 * pow(x, 2) + math.sin(x)\n",
    "\n",
    "    def run(self, x0):\n",
    "        df = pd.DataFrame(columns=['(NT) f(x)'])\n",
    "        row = len(df)\n",
    "        x = x0\n",
    "        df.loc[row] = [x]\n",
    "        for k in range(MAX_ITR):\n",
    "            try:\n",
    "                y = x - self.f(x) / self.jacobian(x)\n",
    "            except ValueError:\n",
    "                break\n",
    "            residual = math.fabs(x - y)\n",
    "            x = y\n",
    "\n",
    "            row = len(df)\n",
    "            df.loc[row] = [y]\n",
    "            if residual < TOR or x > 1e9:\n",
    "                break\n",
    "        return df"
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   "source": [
    "$21^{\\frac{1}{3}} = 2.75892417...$"
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   "source": [
    "## Set $x_0 = -1.0$"
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>(NT) f(x)</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-1.00000000</td>\n",
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       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-0.88033290</td>\n",
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       "      <th>2</th>\n",
       "      <td>-0.86568416</td>\n",
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       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-0.86547408</td>\n",
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       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-0.86547403</td>\n",
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       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>-0.86547403</td>\n",
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      "text/plain": [
       "    (NT) f(x)\n",
       "0 -1.00000000\n",
       "1 -0.88033290\n",
       "2 -0.86568416\n",
       "3 -0.86547408\n",
       "4 -0.86547403\n",
       "5 -0.86547403"
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    "Newton1D().run(-1.0).astype(np.float64)"
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    "## $x_0 = 0$ can not be used"
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     "ename": "ZeroDivisionError",
     "evalue": "float division by zero",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mZeroDivisionError\u001b[0m                         Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-5-138d74d6419d>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mNewton1D\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrun\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mastype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfloat64\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m<ipython-input-3-0e9bdfb02559>\u001b[0m in \u001b[0;36mrun\u001b[0;34m(self, x0)\u001b[0m\n\u001b[1;32m     17\u001b[0m         \u001b[0;32mfor\u001b[0m \u001b[0mk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mMAX_ITR\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     18\u001b[0m             \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 19\u001b[0;31m                 \u001b[0my\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mx\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m/\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mjacobian\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     20\u001b[0m             \u001b[0;32mexcept\u001b[0m \u001b[0mValueError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     21\u001b[0m                 \u001b[0;32mbreak\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mZeroDivisionError\u001b[0m: float division by zero"
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    "Newton1D().run(0).astype(np.float64)"
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