ex10.2.5.ipynb

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   "source": [
    "<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=\"#(a)\" data-toc-modified-id=\"(a)-0.1\"><span class=\"toc-item-num\">0.1&nbsp;&nbsp;</span>(a)</a></span></li><li><span><a href=\"#(b)\" data-toc-modified-id=\"(b)-0.2\"><span class=\"toc-item-num\">0.2&nbsp;&nbsp;</span>(b)</a></span></li><li><span><a href=\"#(c)\" data-toc-modified-id=\"(c)-0.3\"><span class=\"toc-item-num\">0.3&nbsp;&nbsp;</span>(c)</a></span></li><li><span><a href=\"#(d)\" data-toc-modified-id=\"(d)-0.4\"><span class=\"toc-item-num\">0.4&nbsp;&nbsp;</span>(d)</a></span></li></ul></li></ul></div>"
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   "source": [
    "Use the answers obtained in Exercise 3 as initial approximations to Newton’s method. Iterate until $\\|x^{(k)} - x^{(k+1)} \\|_{\\infty} < 10^{-6}$."
   ]
  },
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   "source": [
    "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, -6)\n",
    "MAX_ITR = 150"
   ]
  },
  {
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   "execution_count": 2,
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     "end_time": "2018-02-05T04:49:01.191770Z",
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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()!')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## (a) \n",
    "$$\\begin{align*}\n",
    "4x_1^2 - 20 x_1 + \\frac{1}{4} x_2^2 + 8 &= 0 \\\\\n",
    "\\frac{1}{2}x_1x_2^2+2x_1-5x_2+8 &= 0\n",
    "\\end{align*}$$"
   ]
  },
  {
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   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
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   "source": [
    "class Newton(NewtonMethod):\n",
    "\n",
    "    def __init__(self):\n",
    "        super(NewtonMethod, self).__init__()\n",
    "\n",
    "    def f(self, x):\n",
    "        sol = np.zeros(len(x))\n",
    "        sol[0] = 4 * pow(x[0], 2) - 20 * x[0] + pow(x[1], 2) / 4 + 8\n",
    "        sol[1] = x[0] * pow(x[1], 2) / 2 + 2 * x[0] - 5 * x[1] + 8\n",
    "        return sol\n",
    "\n",
    "    def jacobian(self, x):\n",
    "        jac = np.zeros(shape=(2, 2))\n",
    "        jac[0][0] = 8 * x[0] - 20\n",
    "        jac[0][1] = x[1] / 2\n",
    "        jac[1][0] = pow(x[1], 2) / 2\n",
    "        jac[1][1] = x[0] * x[1] - 5\n",
    "        return jac\n",
    "\n",
    "    def run(self, x):\n",
    "        \"\"\"\n",
    "        given x_0 in R^3 as a starting point.\n",
    "\n",
    "        :param x: x_0 as described\n",
    "        :return: the minimizer x* of f\n",
    "        \"\"\"\n",
    "        df = pd.DataFrame(columns=['x' + str(i + 1) for i in range(len(x))] + ['residual', 'actual-residual'])\n",
    "\n",
    "        row = len(df)\n",
    "        df.loc[row] = [xe for xe in x] + [np.nan, np.nan]\n",
    "\n",
    "        for k in range(MAX_ITR):\n",
    "            jac = self.jacobian(x)\n",
    "            f = -self.f(x)\n",
    "            y = linalg.solve(jac, f)\n",
    "            nx = x + y\n",
    "            residual = linalg.norm(x - nx, np.inf)\n",
    "            x = nx\n",
    "\n",
    "            row = len(df)\n",
    "            df.loc[row] = [nxe for nxe in nx] + [residual, np.nan]\n",
    "            if residual < TOR:\n",
    "                break\n",
    "\n",
    "        for i in range(len(df)):\n",
    "            xk = np.array([df.loc[i][j] for j in range(len(x))])\n",
    "            df.loc[i][3] = linalg.norm(x - xk, np.inf)\n",
    "\n",
    "        print(self.f(x))\n",
    "        return df"
   ]
  },
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     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.          0.00000003]\n"
     ]
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       "      <th></th>\n",
       "      <th>x1</th>\n",
       "      <th>x2</th>\n",
       "      <th>residual</th>\n",
       "      <th>actual-residual</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.00000000</td>\n",
       "      <td>0.00000000</td>\n",
       "      <td>nan</td>\n",
       "      <td>1.99999999</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.40000000</td>\n",
       "      <td>1.60000000</td>\n",
       "      <td>1.60000000</td>\n",
       "      <td>0.39999999</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.49180328</td>\n",
       "      <td>1.92786885</td>\n",
       "      <td>0.32786885</td>\n",
       "      <td>0.07213114</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.49961276</td>\n",
       "      <td>1.99517417</td>\n",
       "      <td>0.06730532</td>\n",
       "      <td>0.00482582</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.49998699</td>\n",
       "      <td>1.99979768</td>\n",
       "      <td>0.00462351</td>\n",
       "      <td>0.00020232</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>0.49999958</td>\n",
       "      <td>1.99999328</td>\n",
       "      <td>0.00019560</td>\n",
       "      <td>0.00000671</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>0.49999999</td>\n",
       "      <td>1.99999978</td>\n",
       "      <td>0.00000650</td>\n",
       "      <td>0.00000021</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>0.50000000</td>\n",
       "      <td>1.99999999</td>\n",
       "      <td>0.00000021</td>\n",
       "      <td>0.00000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
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       "          x1         x2   residual  actual-residual\n",
       "0 0.00000000 0.00000000        nan       1.99999999\n",
       "1 0.40000000 1.60000000 1.60000000       0.39999999\n",
       "2 0.49180328 1.92786885 0.32786885       0.07213114\n",
       "3 0.49961276 1.99517417 0.06730532       0.00482582\n",
       "4 0.49998699 1.99979768 0.00462351       0.00020232\n",
       "5 0.49999958 1.99999328 0.00019560       0.00000671\n",
       "6 0.49999999 1.99999978 0.00000650       0.00000021\n",
       "7 0.50000000 1.99999999 0.00000021       0.00000000"
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   ],
   "source": [
    "x0 = np.array([0, 0])\n",
    "Newton().run(x0).astype(np.float64)"
   ]
  },
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   "execution_count": 5,
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      "[ 0.         -0.00000011]\n"
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       "      <th></th>\n",
       "      <th>x1</th>\n",
       "      <th>x2</th>\n",
       "      <th>residual</th>\n",
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       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>5.00000000</td>\n",
       "      <td>5.00000000</td>\n",
       "      <td>nan</td>\n",
       "      <td>3.90328030</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>4.60338983</td>\n",
       "      <td>2.47288136</td>\n",
       "      <td>2.52711864</td>\n",
       "      <td>3.56805156</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>4.69198568</td>\n",
       "      <td>-0.53299962</td>\n",
       "      <td>3.00588098</td>\n",
       "      <td>6.57393254</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4.60332934</td>\n",
       "      <td>2.22707567</td>\n",
       "      <td>2.76007529</td>\n",
       "      <td>3.81385725</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>4.71202036</td>\n",
       "      <td>-1.15390322</td>\n",
       "      <td>3.38097889</td>\n",
       "      <td>7.19483614</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>4.62993924</td>\n",
       "      <td>1.36361900</td>\n",
       "      <td>2.51752222</td>\n",
       "      <td>4.67731392</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>4.99493808</td>\n",
       "      <td>-10.12174698</td>\n",
       "      <td>11.48536598</td>\n",
       "      <td>16.16267990</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>4.73632277</td>\n",
       "      <td>-4.52005172</td>\n",
       "      <td>5.60169526</td>\n",
       "      <td>10.56098464</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>4.70445978</td>\n",
       "      <td>-1.18282107</td>\n",
       "      <td>3.33723065</td>\n",
       "      <td>7.22375399</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>4.63066852</td>\n",
       "      <td>1.33147149</td>\n",
       "      <td>2.51429256</td>\n",
       "      <td>4.70946143</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>5.04173521</td>\n",
       "      <td>-11.60004439</td>\n",
       "      <td>12.93151588</td>\n",
       "      <td>17.64097732</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>4.72172141</td>\n",
       "      <td>-5.39751533</td>\n",
       "      <td>6.20252907</td>\n",
       "      <td>11.43844825</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>4.72139126</td>\n",
       "      <td>-1.68409963</td>\n",
       "      <td>3.71341569</td>\n",
       "      <td>7.72503255</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>4.64613169</td>\n",
       "      <td>0.82159148</td>\n",
       "      <td>2.50569111</td>\n",
       "      <td>5.21934144</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>4.25761533</td>\n",
       "      <td>13.18340434</td>\n",
       "      <td>12.36181286</td>\n",
       "      <td>7.14247142</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>5.14062941</td>\n",
       "      <td>5.41251671</td>\n",
       "      <td>7.77088762</td>\n",
       "      <td>4.04390971</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>4.60791029</td>\n",
       "      <td>2.84002810</td>\n",
       "      <td>2.57248862</td>\n",
       "      <td>3.51119059</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>4.67000785</td>\n",
       "      <td>0.13811694</td>\n",
       "      <td>2.70191116</td>\n",
       "      <td>5.90281598</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>4.54874052</td>\n",
       "      <td>3.97114682</td>\n",
       "      <td>3.83302988</td>\n",
       "      <td>3.45202081</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>4.63433916</td>\n",
       "      <td>1.38508888</td>\n",
       "      <td>2.58605794</td>\n",
       "      <td>4.65584404</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>4.96652929</td>\n",
       "      <td>-9.26156889</td>\n",
       "      <td>10.64665777</td>\n",
       "      <td>15.30250181</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>4.73895859</td>\n",
       "      <td>-4.01650656</td>\n",
       "      <td>5.24506233</td>\n",
       "      <td>10.05743948</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>4.69531716</td>\n",
       "      <td>-0.87789217</td>\n",
       "      <td>3.13861439</td>\n",
       "      <td>6.91882509</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>4.61920964</td>\n",
       "      <td>1.70489026</td>\n",
       "      <td>2.58278243</td>\n",
       "      <td>4.33604266</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>4.79369275</td>\n",
       "      <td>-3.74882256</td>\n",
       "      <td>5.45371282</td>\n",
       "      <td>9.78975548</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>4.68994628</td>\n",
       "      <td>-0.73250318</td>\n",
       "      <td>3.01631939</td>\n",
       "      <td>6.77343610</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>4.61287594</td>\n",
       "      <td>1.90874104</td>\n",
       "      <td>2.64124421</td>\n",
       "      <td>4.13219188</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>4.75071340</td>\n",
       "      <td>-2.38484268</td>\n",
       "      <td>4.29358372</td>\n",
       "      <td>8.42577560</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>4.66365342</td>\n",
       "      <td>0.22927973</td>\n",
       "      <td>2.61412241</td>\n",
       "      <td>5.81165319</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>4.53573772</td>\n",
       "      <td>4.37613657</td>\n",
       "      <td>4.14685685</td>\n",
       "      <td>3.43901802</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>68</th>\n",
       "      <td>5.06813110</td>\n",
       "      <td>-12.43745664</td>\n",
       "      <td>13.75365827</td>\n",
       "      <td>18.47838956</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>69</th>\n",
       "      <td>4.70588917</td>\n",
       "      <td>-5.90694250</td>\n",
       "      <td>6.53051414</td>\n",
       "      <td>11.94787542</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>70</th>\n",
       "      <td>4.73288617</td>\n",
       "      <td>-1.95796224</td>\n",
       "      <td>3.94898026</td>\n",
       "      <td>7.99889516</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>71</th>\n",
       "      <td>4.65343773</td>\n",
       "      <td>0.57766586</td>\n",
       "      <td>2.53562811</td>\n",
       "      <td>5.46326706</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>72</th>\n",
       "      <td>4.44872350</td>\n",
       "      <td>7.13548476</td>\n",
       "      <td>6.55781890</td>\n",
       "      <td>3.35200380</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>73</th>\n",
       "      <td>4.58685606</td>\n",
       "      <td>3.47144885</td>\n",
       "      <td>3.66403590</td>\n",
       "      <td>3.49013636</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>74</th>\n",
       "      <td>4.64595354</td>\n",
       "      <td>0.92540206</td>\n",
       "      <td>2.54604679</td>\n",
       "      <td>5.11553086</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75</th>\n",
       "      <td>3.99769473</td>\n",
       "      <td>21.44514831</td>\n",
       "      <td>20.51974625</td>\n",
       "      <td>15.40421539</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>76</th>\n",
       "      <td>3.83436434</td>\n",
       "      <td>11.65374930</td>\n",
       "      <td>9.79139901</td>\n",
       "      <td>5.61281638</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>77</th>\n",
       "      <td>15.04921088</td>\n",
       "      <td>-13.02364976</td>\n",
       "      <td>24.67739906</td>\n",
       "      <td>19.06458268</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>78</th>\n",
       "      <td>8.79559068</td>\n",
       "      <td>-8.79892792</td>\n",
       "      <td>6.25362020</td>\n",
       "      <td>14.83986084</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>79</th>\n",
       "      <td>5.91767382</td>\n",
       "      <td>-5.17402301</td>\n",
       "      <td>3.62490490</td>\n",
       "      <td>11.21495593</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>80</th>\n",
       "      <td>4.88078952</td>\n",
       "      <td>-2.05662388</td>\n",
       "      <td>3.11739914</td>\n",
       "      <td>8.09755680</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>81</th>\n",
       "      <td>4.66353434</td>\n",
       "      <td>0.46415738</td>\n",
       "      <td>2.52078126</td>\n",
       "      <td>5.57677554</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>82</th>\n",
       "      <td>4.48759377</td>\n",
       "      <td>5.92714880</td>\n",
       "      <td>5.46299142</td>\n",
       "      <td>3.39087406</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>83</th>\n",
       "      <td>4.59832985</td>\n",
       "      <td>2.77363392</td>\n",
       "      <td>3.15351489</td>\n",
       "      <td>3.50161015</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>84</th>\n",
       "      <td>4.67433031</td>\n",
       "      <td>0.02561003</td>\n",
       "      <td>2.74802389</td>\n",
       "      <td>6.01532289</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>85</th>\n",
       "      <td>4.56187034</td>\n",
       "      <td>3.55452030</td>\n",
       "      <td>3.52891028</td>\n",
       "      <td>3.46515064</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>86</th>\n",
       "      <td>4.64578310</td>\n",
       "      <td>0.99550904</td>\n",
       "      <td>2.55901127</td>\n",
       "      <td>5.04542388</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>87</th>\n",
       "      <td>3.46412355</td>\n",
       "      <td>38.40226283</td>\n",
       "      <td>37.40675380</td>\n",
       "      <td>32.36132991</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>88</th>\n",
       "      <td>3.45321294</td>\n",
       "      <td>19.89723709</td>\n",
       "      <td>18.50502574</td>\n",
       "      <td>13.85630417</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>89</th>\n",
       "      <td>3.11284644</td>\n",
       "      <td>11.55296937</td>\n",
       "      <td>8.34426772</td>\n",
       "      <td>5.51203645</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>90</th>\n",
       "      <td>1.42181503</td>\n",
       "      <td>9.89463247</td>\n",
       "      <td>1.69103141</td>\n",
       "      <td>3.85369955</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>91</th>\n",
       "      <td>1.28679045</td>\n",
       "      <td>7.20822221</td>\n",
       "      <td>2.68641025</td>\n",
       "      <td>1.16728929</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>92</th>\n",
       "      <td>1.13381436</td>\n",
       "      <td>6.27543662</td>\n",
       "      <td>0.93278559</td>\n",
       "      <td>0.23450370</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>93</th>\n",
       "      <td>1.09663016</td>\n",
       "      <td>6.04675670</td>\n",
       "      <td>0.22867992</td>\n",
       "      <td>0.00582378</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>94</th>\n",
       "      <td>1.09672751</td>\n",
       "      <td>6.04096473</td>\n",
       "      <td>0.00579197</td>\n",
       "      <td>0.00003181</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>95</th>\n",
       "      <td>1.09671907</td>\n",
       "      <td>6.04093055</td>\n",
       "      <td>0.00003418</td>\n",
       "      <td>0.00000237</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>96</th>\n",
       "      <td>1.09671976</td>\n",
       "      <td>6.04093313</td>\n",
       "      <td>0.00000258</td>\n",
       "      <td>0.00000021</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>97</th>\n",
       "      <td>1.09671970</td>\n",
       "      <td>6.04093292</td>\n",
       "      <td>0.00000021</td>\n",
       "      <td>0.00000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>98 rows × 4 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "            x1           x2    residual  actual-residual\n",
       "0   5.00000000   5.00000000         nan       3.90328030\n",
       "1   4.60338983   2.47288136  2.52711864       3.56805156\n",
       "2   4.69198568  -0.53299962  3.00588098       6.57393254\n",
       "3   4.60332934   2.22707567  2.76007529       3.81385725\n",
       "4   4.71202036  -1.15390322  3.38097889       7.19483614\n",
       "5   4.62993924   1.36361900  2.51752222       4.67731392\n",
       "6   4.99493808 -10.12174698 11.48536598      16.16267990\n",
       "7   4.73632277  -4.52005172  5.60169526      10.56098464\n",
       "8   4.70445978  -1.18282107  3.33723065       7.22375399\n",
       "9   4.63066852   1.33147149  2.51429256       4.70946143\n",
       "10  5.04173521 -11.60004439 12.93151588      17.64097732\n",
       "11  4.72172141  -5.39751533  6.20252907      11.43844825\n",
       "12  4.72139126  -1.68409963  3.71341569       7.72503255\n",
       "13  4.64613169   0.82159148  2.50569111       5.21934144\n",
       "14  4.25761533  13.18340434 12.36181286       7.14247142\n",
       "15  5.14062941   5.41251671  7.77088762       4.04390971\n",
       "16  4.60791029   2.84002810  2.57248862       3.51119059\n",
       "17  4.67000785   0.13811694  2.70191116       5.90281598\n",
       "18  4.54874052   3.97114682  3.83302988       3.45202081\n",
       "19  4.63433916   1.38508888  2.58605794       4.65584404\n",
       "20  4.96652929  -9.26156889 10.64665777      15.30250181\n",
       "21  4.73895859  -4.01650656  5.24506233      10.05743948\n",
       "22  4.69531716  -0.87789217  3.13861439       6.91882509\n",
       "23  4.61920964   1.70489026  2.58278243       4.33604266\n",
       "24  4.79369275  -3.74882256  5.45371282       9.78975548\n",
       "25  4.68994628  -0.73250318  3.01631939       6.77343610\n",
       "26  4.61287594   1.90874104  2.64124421       4.13219188\n",
       "27  4.75071340  -2.38484268  4.29358372       8.42577560\n",
       "28  4.66365342   0.22927973  2.61412241       5.81165319\n",
       "29  4.53573772   4.37613657  4.14685685       3.43901802\n",
       "..         ...          ...         ...              ...\n",
       "68  5.06813110 -12.43745664 13.75365827      18.47838956\n",
       "69  4.70588917  -5.90694250  6.53051414      11.94787542\n",
       "70  4.73288617  -1.95796224  3.94898026       7.99889516\n",
       "71  4.65343773   0.57766586  2.53562811       5.46326706\n",
       "72  4.44872350   7.13548476  6.55781890       3.35200380\n",
       "73  4.58685606   3.47144885  3.66403590       3.49013636\n",
       "74  4.64595354   0.92540206  2.54604679       5.11553086\n",
       "75  3.99769473  21.44514831 20.51974625      15.40421539\n",
       "76  3.83436434  11.65374930  9.79139901       5.61281638\n",
       "77 15.04921088 -13.02364976 24.67739906      19.06458268\n",
       "78  8.79559068  -8.79892792  6.25362020      14.83986084\n",
       "79  5.91767382  -5.17402301  3.62490490      11.21495593\n",
       "80  4.88078952  -2.05662388  3.11739914       8.09755680\n",
       "81  4.66353434   0.46415738  2.52078126       5.57677554\n",
       "82  4.48759377   5.92714880  5.46299142       3.39087406\n",
       "83  4.59832985   2.77363392  3.15351489       3.50161015\n",
       "84  4.67433031   0.02561003  2.74802389       6.01532289\n",
       "85  4.56187034   3.55452030  3.52891028       3.46515064\n",
       "86  4.64578310   0.99550904  2.55901127       5.04542388\n",
       "87  3.46412355  38.40226283 37.40675380      32.36132991\n",
       "88  3.45321294  19.89723709 18.50502574      13.85630417\n",
       "89  3.11284644  11.55296937  8.34426772       5.51203645\n",
       "90  1.42181503   9.89463247  1.69103141       3.85369955\n",
       "91  1.28679045   7.20822221  2.68641025       1.16728929\n",
       "92  1.13381436   6.27543662  0.93278559       0.23450370\n",
       "93  1.09663016   6.04675670  0.22867992       0.00582378\n",
       "94  1.09672751   6.04096473  0.00579197       0.00003181\n",
       "95  1.09671907   6.04093055  0.00003418       0.00000237\n",
       "96  1.09671976   6.04093313  0.00000258       0.00000021\n",
       "97  1.09671970   6.04093292  0.00000021       0.00000000\n",
       "\n",
       "[98 rows x 4 columns]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([5, 5])\n",
    "Newton().run(x0).astype(np.float64)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## (b) \n",
    "$$\\begin{align*}\n",
    "\\sin(4\\pi x_1 x_2)-2x_2-x_1 &= 0 \\\\\n",
    "\\left( \\frac{4\\pi-1}{4\\pi} \\right)(\\exp^{2x_1}-\\exp)+4\\exp x_2^2 -2\\exp x_1 &= 0\n",
    "\\end{align*}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T04:49:01.616176Z",
     "start_time": "2018-02-05T04:49:01.525432Z"
    },
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class Newton(NewtonMethod):\n",
    "\n",
    "    def __init__(self):\n",
    "        super(NewtonMethod, self).__init__()\n",
    "\n",
    "    def f(self, x):\n",
    "        sol = np.zeros(len(x))\n",
    "        sol[0] = math.sin(4 * math.pi * x[0] * x[1]) - 2 * x[1] - x[0]\n",
    "        sol[1] = ((4 * math.pi - 1) / (4 * math.pi)) * (math.exp(2 * x[0]) - math.e) + 4 * math.e * pow(x[1], 2) - 2 * math.e * x[0]\n",
    "        return sol\n",
    "\n",
    "    def jacobian(self, x):\n",
    "        jac = np.zeros(shape=(2, 2))\n",
    "        jac[0][0] = 4 * math.pi * x[1] * math.cos(4 * math.pi * x[0] * x[1]) - 1\n",
    "        jac[0][1] = 4 * math.pi * x[0] * math.cos(4 * math.pi * x[0] * x[1]) - 2\n",
    "        jac[1][0] = 2 * (4 * math.pi - 1) / (4 * math.pi) * math.exp(2 * x[0]) - 2 * math.e\n",
    "        jac[1][1] = 8 * math.e * x[1]\n",
    "        return jac\n",
    "\n",
    "    def run(self, x):\n",
    "        df = pd.DataFrame(columns=['x' + str(i + 1) for i in range(len(x))] + ['residual', 'actual-residual'])\n",
    "\n",
    "        row = len(df)\n",
    "        df.loc[row] = [xe for xe in x] + [np.nan, np.nan]\n",
    "\n",
    "        for k in range(MAX_ITR):\n",
    "            jac = self.jacobian(x)\n",
    "            f = -self.f(x)\n",
    "            y = linalg.solve(jac, f)\n",
    "            nx = x + y\n",
    "            residual = linalg.norm(x - nx, np.inf)\n",
    "            x = nx\n",
    "\n",
    "            row = len(df)\n",
    "            df.loc[row] = [nxe for nxe in nx] + [residual, np.nan]\n",
    "            if residual < TOR:\n",
    "                break\n",
    "\n",
    "        for i in range(len(df)):\n",
    "            xk = np.array([df.loc[i][j] for j in range(len(x))])\n",
    "            df.loc[i][3] = linalg.norm(x - xk, np.inf)\n",
    "        print(self.f(x))\n",
    "        return df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T04:49:01.671984Z",
     "start_time": "2018-02-05T04:49:01.617825Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.  0.]\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<div>\n",
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>x1</th>\n",
       "      <th>x2</th>\n",
       "      <th>residual</th>\n",
       "      <th>actual-residual</th>\n",
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       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.00000000</td>\n",
       "      <td>0.00000000</td>\n",
       "      <td>nan</td>\n",
       "      <td>0.37369822</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-0.43984123</td>\n",
       "      <td>0.21992062</td>\n",
       "      <td>0.43984123</td>\n",
       "      <td>0.16365413</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-0.51316159</td>\n",
       "      <td>-0.01837622</td>\n",
       "      <td>0.23829684</td>\n",
       "      <td>0.13946338</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-0.38821176</td>\n",
       "      <td>0.04285363</td>\n",
       "      <td>0.12494983</td>\n",
       "      <td>0.01451354</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-0.37423804</td>\n",
       "      <td>0.05590018</td>\n",
       "      <td>0.01397372</td>\n",
       "      <td>0.00053982</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>-0.37369869</td>\n",
       "      <td>0.05626611</td>\n",
       "      <td>0.00053935</td>\n",
       "      <td>0.00000048</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>-0.37369822</td>\n",
       "      <td>0.05626649</td>\n",
       "      <td>0.00000048</td>\n",
       "      <td>0.00000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "           x1          x2   residual  actual-residual\n",
       "0  0.00000000  0.00000000        nan       0.37369822\n",
       "1 -0.43984123  0.21992062 0.43984123       0.16365413\n",
       "2 -0.51316159 -0.01837622 0.23829684       0.13946338\n",
       "3 -0.38821176  0.04285363 0.12494983       0.01451354\n",
       "4 -0.37423804  0.05590018 0.01397372       0.00053982\n",
       "5 -0.37369869  0.05626611 0.00053935       0.00000048\n",
       "6 -0.37369822  0.05626649 0.00000048       0.00000000"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([0, 0])\n",
    "Newton().run(x0).astype(np.float64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T04:49:01.726823Z",
     "start_time": "2018-02-05T04:49:01.673740Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.  0.]\n"
     ]
    },
    {
     "data": {
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       "      <th></th>\n",
       "      <th>x1</th>\n",
       "      <th>x2</th>\n",
       "      <th>residual</th>\n",
       "      <th>actual-residual</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-10.00000000</td>\n",
       "      <td>-10.00000000</td>\n",
       "      <td>nan</td>\n",
       "      <td>10.40809566</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-15.17335965</td>\n",
       "      <td>-4.63217129</td>\n",
       "      <td>5.36782871</td>\n",
       "      <td>15.58145532</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-28.23026900</td>\n",
       "      <td>-0.81732534</td>\n",
       "      <td>13.05690934</td>\n",
       "      <td>28.63836466</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>3.91251166</td>\n",
       "      <td>-1.74617256</td>\n",
       "      <td>32.14278066</td>\n",
       "      <td>3.50441600</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>3.40844539</td>\n",
       "      <td>-1.92025834</td>\n",
       "      <td>0.50406628</td>\n",
       "      <td>3.00034972</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>2.88739806</td>\n",
       "      <td>-2.24320033</td>\n",
       "      <td>0.52104733</td>\n",
       "      <td>2.47930240</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>2.27217530</td>\n",
       "      <td>-2.82656081</td>\n",
       "      <td>0.61522276</td>\n",
       "      <td>2.33393142</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>-0.05629811</td>\n",
       "      <td>-6.60187856</td>\n",
       "      <td>3.77531774</td>\n",
       "      <td>6.10924916</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>-2.34889137</td>\n",
       "      <td>-3.24995620</td>\n",
       "      <td>3.35192236</td>\n",
       "      <td>2.75732681</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>-5.88932554</td>\n",
       "      <td>-1.20807018</td>\n",
       "      <td>3.54043417</td>\n",
       "      <td>6.29742121</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>-24.57730048</td>\n",
       "      <td>4.38678463</td>\n",
       "      <td>18.68797494</td>\n",
       "      <td>24.98539615</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>-51.82949682</td>\n",
       "      <td>-0.73410982</td>\n",
       "      <td>27.25219633</td>\n",
       "      <td>52.23759248</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>2.88963192</td>\n",
       "      <td>-1.50783899</td>\n",
       "      <td>54.71912874</td>\n",
       "      <td>2.48153626</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>2.37084280</td>\n",
       "      <td>-1.56456980</td>\n",
       "      <td>0.51878912</td>\n",
       "      <td>1.96274714</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>1.74482391</td>\n",
       "      <td>-1.91618209</td>\n",
       "      <td>0.62601889</td>\n",
       "      <td>1.42355269</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>-0.36721223</td>\n",
       "      <td>-3.30424971</td>\n",
       "      <td>2.11203614</td>\n",
       "      <td>2.81162032</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>-0.67067210</td>\n",
       "      <td>-1.63378560</td>\n",
       "      <td>1.67046412</td>\n",
       "      <td>1.14115620</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>-0.59693595</td>\n",
       "      <td>-0.78819793</td>\n",
       "      <td>0.84558766</td>\n",
       "      <td>1.00503161</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>-0.89264046</td>\n",
       "      <td>-0.25029229</td>\n",
       "      <td>0.53790564</td>\n",
       "      <td>1.30073612</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>0.20953910</td>\n",
       "      <td>-0.70321542</td>\n",
       "      <td>1.10217955</td>\n",
       "      <td>0.21058603</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>0.37806004</td>\n",
       "      <td>-0.52725485</td>\n",
       "      <td>0.17596058</td>\n",
       "      <td>0.03462545</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>0.40440273</td>\n",
       "      <td>-0.49359298</td>\n",
       "      <td>0.03366187</td>\n",
       "      <td>0.00369293</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>0.40810006</td>\n",
       "      <td>-0.49263613</td>\n",
       "      <td>0.00369732</td>\n",
       "      <td>0.00000674</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>0.40809566</td>\n",
       "      <td>-0.49262939</td>\n",
       "      <td>0.00000674</td>\n",
       "      <td>0.00000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>0.40809566</td>\n",
       "      <td>-0.49262939</td>\n",
       "      <td>0.00000000</td>\n",
       "      <td>0.00000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "             x1           x2    residual  actual-residual\n",
       "0  -10.00000000 -10.00000000         nan      10.40809566\n",
       "1  -15.17335965  -4.63217129  5.36782871      15.58145532\n",
       "2  -28.23026900  -0.81732534 13.05690934      28.63836466\n",
       "3    3.91251166  -1.74617256 32.14278066       3.50441600\n",
       "4    3.40844539  -1.92025834  0.50406628       3.00034972\n",
       "5    2.88739806  -2.24320033  0.52104733       2.47930240\n",
       "6    2.27217530  -2.82656081  0.61522276       2.33393142\n",
       "7   -0.05629811  -6.60187856  3.77531774       6.10924916\n",
       "8   -2.34889137  -3.24995620  3.35192236       2.75732681\n",
       "9   -5.88932554  -1.20807018  3.54043417       6.29742121\n",
       "10 -24.57730048   4.38678463 18.68797494      24.98539615\n",
       "11 -51.82949682  -0.73410982 27.25219633      52.23759248\n",
       "12   2.88963192  -1.50783899 54.71912874       2.48153626\n",
       "13   2.37084280  -1.56456980  0.51878912       1.96274714\n",
       "14   1.74482391  -1.91618209  0.62601889       1.42355269\n",
       "15  -0.36721223  -3.30424971  2.11203614       2.81162032\n",
       "16  -0.67067210  -1.63378560  1.67046412       1.14115620\n",
       "17  -0.59693595  -0.78819793  0.84558766       1.00503161\n",
       "18  -0.89264046  -0.25029229  0.53790564       1.30073612\n",
       "19   0.20953910  -0.70321542  1.10217955       0.21058603\n",
       "20   0.37806004  -0.52725485  0.17596058       0.03462545\n",
       "21   0.40440273  -0.49359298  0.03366187       0.00369293\n",
       "22   0.40810006  -0.49263613  0.00369732       0.00000674\n",
       "23   0.40809566  -0.49262939  0.00000674       0.00000000\n",
       "24   0.40809566  -0.49262939  0.00000000       0.00000000"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([-10, -10])\n",
    "Newton().run(x0).astype(np.float64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T04:49:01.912769Z",
     "start_time": "2018-02-05T04:49:01.727967Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0. -0.]\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>x1</th>\n",
       "      <th>x2</th>\n",
       "      <th>residual</th>\n",
       "      <th>actual-residual</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1.00000000</td>\n",
       "      <td>-10.00000000</td>\n",
       "      <td>nan</td>\n",
       "      <td>9.72003816</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1.56845109</td>\n",
       "      <td>-4.98388580</td>\n",
       "      <td>5.01611420</td>\n",
       "      <td>4.70392397</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>2.41143348</td>\n",
       "      <td>-2.11060844</td>\n",
       "      <td>2.87327736</td>\n",
       "      <td>1.83064661</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1.53244984</td>\n",
       "      <td>-3.18102125</td>\n",
       "      <td>1.07041281</td>\n",
       "      <td>2.90105942</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>2.74408498</td>\n",
       "      <td>-0.86612274</td>\n",
       "      <td>2.31489851</td>\n",
       "      <td>1.71101351</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>2.27815747</td>\n",
       "      <td>-0.41820592</td>\n",
       "      <td>0.46592751</td>\n",
       "      <td>1.24508600</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>1.83492424</td>\n",
       "      <td>-0.48685897</td>\n",
       "      <td>0.44323322</td>\n",
       "      <td>0.80185277</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>1.38464703</td>\n",
       "      <td>-0.85145937</td>\n",
       "      <td>0.45027722</td>\n",
       "      <td>0.57149753</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>0.54605429</td>\n",
       "      <td>-1.25801143</td>\n",
       "      <td>0.83859274</td>\n",
       "      <td>0.97804959</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>0.77911351</td>\n",
       "      <td>-0.72826963</td>\n",
       "      <td>0.52974180</td>\n",
       "      <td>0.44830779</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>0.92462365</td>\n",
       "      <td>-0.48309119</td>\n",
       "      <td>0.24517844</td>\n",
       "      <td>0.20312935</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>1.74146955</td>\n",
       "      <td>0.08549685</td>\n",
       "      <td>0.81684590</td>\n",
       "      <td>0.70839807</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>1.41188844</td>\n",
       "      <td>0.02377367</td>\n",
       "      <td>0.32958110</td>\n",
       "      <td>0.37881697</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>1.20216742</td>\n",
       "      <td>0.08703281</td>\n",
       "      <td>0.20972103</td>\n",
       "      <td>0.36699465</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>1.09647272</td>\n",
       "      <td>0.26914065</td>\n",
       "      <td>0.18210785</td>\n",
       "      <td>0.54910249</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>1.13474208</td>\n",
       "      <td>0.09891392</td>\n",
       "      <td>0.17022674</td>\n",
       "      <td>0.37887575</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>0.97204270</td>\n",
       "      <td>0.87675467</td>\n",
       "      <td>0.77784075</td>\n",
       "      <td>1.15671650</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>0.12413828</td>\n",
       "      <td>0.83970404</td>\n",
       "      <td>0.84790441</td>\n",
       "      <td>1.11966588</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>0.35913379</td>\n",
       "      <td>0.56881412</td>\n",
       "      <td>0.27088992</td>\n",
       "      <td>0.84877596</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>0.29379580</td>\n",
       "      <td>0.48313718</td>\n",
       "      <td>0.08567694</td>\n",
       "      <td>0.76309902</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>0.20330209</td>\n",
       "      <td>0.45577939</td>\n",
       "      <td>0.09049371</td>\n",
       "      <td>0.82976938</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>0.39680337</td>\n",
       "      <td>0.50454721</td>\n",
       "      <td>0.19350128</td>\n",
       "      <td>0.78450905</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>0.28929530</td>\n",
       "      <td>0.47802954</td>\n",
       "      <td>0.10750807</td>\n",
       "      <td>0.75799138</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>0.19223038</td>\n",
       "      <td>0.45296888</td>\n",
       "      <td>0.09706493</td>\n",
       "      <td>0.84084110</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>0.34353228</td>\n",
       "      <td>0.49130323</td>\n",
       "      <td>0.15130190</td>\n",
       "      <td>0.77126506</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>0.25439471</td>\n",
       "      <td>0.46855546</td>\n",
       "      <td>0.08913757</td>\n",
       "      <td>0.77867676</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>0.06269203</td>\n",
       "      <td>0.42063370</td>\n",
       "      <td>0.19170268</td>\n",
       "      <td>0.97037945</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>0.22138102</td>\n",
       "      <td>0.46514864</td>\n",
       "      <td>0.15868899</td>\n",
       "      <td>0.81169045</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>0.85897349</td>\n",
       "      <td>0.61924484</td>\n",
       "      <td>0.63759247</td>\n",
       "      <td>0.89920668</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>1.75042118</td>\n",
       "      <td>0.14202936</td>\n",
       "      <td>0.89144769</td>\n",
       "      <td>0.71734970</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>76</th>\n",
       "      <td>5.65818367</td>\n",
       "      <td>-6.01193260</td>\n",
       "      <td>0.50087109</td>\n",
       "      <td>5.73197076</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>77</th>\n",
       "      <td>5.15522611</td>\n",
       "      <td>-6.66075709</td>\n",
       "      <td>0.64882449</td>\n",
       "      <td>6.38079526</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>78</th>\n",
       "      <td>4.64619549</td>\n",
       "      <td>-6.97029317</td>\n",
       "      <td>0.50903062</td>\n",
       "      <td>6.69033133</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>79</th>\n",
       "      <td>4.09775741</td>\n",
       "      <td>-10.03423954</td>\n",
       "      <td>3.06394637</td>\n",
       "      <td>9.75427771</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>80</th>\n",
       "      <td>3.09345724</td>\n",
       "      <td>-20.52636540</td>\n",
       "      <td>10.49212586</td>\n",
       "      <td>20.24640356</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>81</th>\n",
       "      <td>5.52998480</td>\n",
       "      <td>-4.44637991</td>\n",
       "      <td>16.07998549</td>\n",
       "      <td>4.49691333</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>82</th>\n",
       "      <td>5.02796124</td>\n",
       "      <td>-4.98097729</td>\n",
       "      <td>0.53459738</td>\n",
       "      <td>4.70101545</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>83</th>\n",
       "      <td>4.52074302</td>\n",
       "      <td>-5.59794538</td>\n",
       "      <td>0.61696809</td>\n",
       "      <td>5.31798355</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>84</th>\n",
       "      <td>3.99690412</td>\n",
       "      <td>-6.04281820</td>\n",
       "      <td>0.52383891</td>\n",
       "      <td>5.76285637</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>85</th>\n",
       "      <td>3.41889312</td>\n",
       "      <td>-6.41943865</td>\n",
       "      <td>0.57801100</td>\n",
       "      <td>6.13947681</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>86</th>\n",
       "      <td>2.48820148</td>\n",
       "      <td>-8.62014074</td>\n",
       "      <td>2.20070209</td>\n",
       "      <td>8.34017891</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>87</th>\n",
       "      <td>4.80888130</td>\n",
       "      <td>-0.44778215</td>\n",
       "      <td>8.17235860</td>\n",
       "      <td>3.77580983</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>88</th>\n",
       "      <td>4.30965844</td>\n",
       "      <td>-0.67891234</td>\n",
       "      <td>0.49922286</td>\n",
       "      <td>3.27658697</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>89</th>\n",
       "      <td>3.81140263</td>\n",
       "      <td>-0.70811759</td>\n",
       "      <td>0.49825581</td>\n",
       "      <td>2.77833116</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>90</th>\n",
       "      <td>3.31479543</td>\n",
       "      <td>-0.85767999</td>\n",
       "      <td>0.49660720</td>\n",
       "      <td>2.28172396</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>91</th>\n",
       "      <td>2.82019184</td>\n",
       "      <td>-0.98174430</td>\n",
       "      <td>0.49460359</td>\n",
       "      <td>1.78712037</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>92</th>\n",
       "      <td>2.32226795</td>\n",
       "      <td>-1.14902243</td>\n",
       "      <td>0.49792390</td>\n",
       "      <td>1.28919648</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>93</th>\n",
       "      <td>1.77719025</td>\n",
       "      <td>-1.40674842</td>\n",
       "      <td>0.54507769</td>\n",
       "      <td>1.12678659</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>94</th>\n",
       "      <td>0.37932920</td>\n",
       "      <td>-2.74175283</td>\n",
       "      <td>1.39786106</td>\n",
       "      <td>2.46179099</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>95</th>\n",
       "      <td>0.61974116</td>\n",
       "      <td>-1.42053358</td>\n",
       "      <td>1.32121925</td>\n",
       "      <td>1.14057174</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>96</th>\n",
       "      <td>1.64354354</td>\n",
       "      <td>-0.76688234</td>\n",
       "      <td>1.02380238</td>\n",
       "      <td>0.61047207</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>97</th>\n",
       "      <td>1.12129690</td>\n",
       "      <td>-0.96473059</td>\n",
       "      <td>0.52224665</td>\n",
       "      <td>0.68476875</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>98</th>\n",
       "      <td>1.71142225</td>\n",
       "      <td>-0.14427843</td>\n",
       "      <td>0.82045215</td>\n",
       "      <td>0.67835078</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>99</th>\n",
       "      <td>1.38065733</td>\n",
       "      <td>-0.21792701</td>\n",
       "      <td>0.33076492</td>\n",
       "      <td>0.34758586</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>100</th>\n",
       "      <td>1.15890071</td>\n",
       "      <td>-0.25646690</td>\n",
       "      <td>0.22175661</td>\n",
       "      <td>0.12582924</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>101</th>\n",
       "      <td>1.05625065</td>\n",
       "      <td>-0.27482913</td>\n",
       "      <td>0.10265006</td>\n",
       "      <td>0.02317918</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>102</th>\n",
       "      <td>1.03407542</td>\n",
       "      <td>-0.27971352</td>\n",
       "      <td>0.02217523</td>\n",
       "      <td>0.00100395</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>103</th>\n",
       "      <td>1.03307350</td>\n",
       "      <td>-0.27996131</td>\n",
       "      <td>0.00100192</td>\n",
       "      <td>0.00000203</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>104</th>\n",
       "      <td>1.03307147</td>\n",
       "      <td>-0.27996184</td>\n",
       "      <td>0.00000203</td>\n",
       "      <td>0.00000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>105</th>\n",
       "      <td>1.03307147</td>\n",
       "      <td>-0.27996184</td>\n",
       "      <td>0.00000000</td>\n",
       "      <td>0.00000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>106 rows × 4 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "            x1           x2    residual  actual-residual\n",
       "0   1.00000000 -10.00000000         nan       9.72003816\n",
       "1   1.56845109  -4.98388580  5.01611420       4.70392397\n",
       "2   2.41143348  -2.11060844  2.87327736       1.83064661\n",
       "3   1.53244984  -3.18102125  1.07041281       2.90105942\n",
       "4   2.74408498  -0.86612274  2.31489851       1.71101351\n",
       "5   2.27815747  -0.41820592  0.46592751       1.24508600\n",
       "6   1.83492424  -0.48685897  0.44323322       0.80185277\n",
       "7   1.38464703  -0.85145937  0.45027722       0.57149753\n",
       "8   0.54605429  -1.25801143  0.83859274       0.97804959\n",
       "9   0.77911351  -0.72826963  0.52974180       0.44830779\n",
       "10  0.92462365  -0.48309119  0.24517844       0.20312935\n",
       "11  1.74146955   0.08549685  0.81684590       0.70839807\n",
       "12  1.41188844   0.02377367  0.32958110       0.37881697\n",
       "13  1.20216742   0.08703281  0.20972103       0.36699465\n",
       "14  1.09647272   0.26914065  0.18210785       0.54910249\n",
       "15  1.13474208   0.09891392  0.17022674       0.37887575\n",
       "16  0.97204270   0.87675467  0.77784075       1.15671650\n",
       "17  0.12413828   0.83970404  0.84790441       1.11966588\n",
       "18  0.35913379   0.56881412  0.27088992       0.84877596\n",
       "19  0.29379580   0.48313718  0.08567694       0.76309902\n",
       "20  0.20330209   0.45577939  0.09049371       0.82976938\n",
       "21  0.39680337   0.50454721  0.19350128       0.78450905\n",
       "22  0.28929530   0.47802954  0.10750807       0.75799138\n",
       "23  0.19223038   0.45296888  0.09706493       0.84084110\n",
       "24  0.34353228   0.49130323  0.15130190       0.77126506\n",
       "25  0.25439471   0.46855546  0.08913757       0.77867676\n",
       "26  0.06269203   0.42063370  0.19170268       0.97037945\n",
       "27  0.22138102   0.46514864  0.15868899       0.81169045\n",
       "28  0.85897349   0.61924484  0.63759247       0.89920668\n",
       "29  1.75042118   0.14202936  0.89144769       0.71734970\n",
       "..         ...          ...         ...              ...\n",
       "76  5.65818367  -6.01193260  0.50087109       5.73197076\n",
       "77  5.15522611  -6.66075709  0.64882449       6.38079526\n",
       "78  4.64619549  -6.97029317  0.50903062       6.69033133\n",
       "79  4.09775741 -10.03423954  3.06394637       9.75427771\n",
       "80  3.09345724 -20.52636540 10.49212586      20.24640356\n",
       "81  5.52998480  -4.44637991 16.07998549       4.49691333\n",
       "82  5.02796124  -4.98097729  0.53459738       4.70101545\n",
       "83  4.52074302  -5.59794538  0.61696809       5.31798355\n",
       "84  3.99690412  -6.04281820  0.52383891       5.76285637\n",
       "85  3.41889312  -6.41943865  0.57801100       6.13947681\n",
       "86  2.48820148  -8.62014074  2.20070209       8.34017891\n",
       "87  4.80888130  -0.44778215  8.17235860       3.77580983\n",
       "88  4.30965844  -0.67891234  0.49922286       3.27658697\n",
       "89  3.81140263  -0.70811759  0.49825581       2.77833116\n",
       "90  3.31479543  -0.85767999  0.49660720       2.28172396\n",
       "91  2.82019184  -0.98174430  0.49460359       1.78712037\n",
       "92  2.32226795  -1.14902243  0.49792390       1.28919648\n",
       "93  1.77719025  -1.40674842  0.54507769       1.12678659\n",
       "94  0.37932920  -2.74175283  1.39786106       2.46179099\n",
       "95  0.61974116  -1.42053358  1.32121925       1.14057174\n",
       "96  1.64354354  -0.76688234  1.02380238       0.61047207\n",
       "97  1.12129690  -0.96473059  0.52224665       0.68476875\n",
       "98  1.71142225  -0.14427843  0.82045215       0.67835078\n",
       "99  1.38065733  -0.21792701  0.33076492       0.34758586\n",
       "100 1.15890071  -0.25646690  0.22175661       0.12582924\n",
       "101 1.05625065  -0.27482913  0.10265006       0.02317918\n",
       "102 1.03407542  -0.27971352  0.02217523       0.00100395\n",
       "103 1.03307350  -0.27996131  0.00100192       0.00000203\n",
       "104 1.03307147  -0.27996184  0.00000203       0.00000000\n",
       "105 1.03307147  -0.27996184  0.00000000       0.00000000\n",
       "\n",
       "[106 rows x 4 columns]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([1, -10])\n",
    "Newton().run(x0).astype(np.float64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T04:49:01.935108Z",
     "start_time": "2018-02-05T04:49:01.914000Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.  0.]\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
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       "    .dataframe thead th {\n",
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       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>x1</th>\n",
       "      <th>x2</th>\n",
       "      <th>residual</th>\n",
       "      <th>actual-residual</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-1.00000000</td>\n",
       "      <td>1.00000000</td>\n",
       "      <td>nan</td>\n",
       "      <td>0.94373351</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-2.02974173</td>\n",
       "      <td>0.11368626</td>\n",
       "      <td>1.02974173</td>\n",
       "      <td>1.65604351</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-0.37403377</td>\n",
       "      <td>0.21864891</td>\n",
       "      <td>1.65570796</td>\n",
       "      <td>0.16238242</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-0.49557423</td>\n",
       "      <td>-0.00045536</td>\n",
       "      <td>0.21910426</td>\n",
       "      <td>0.12187602</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-0.38335717</td>\n",
       "      <td>0.04651631</td>\n",
       "      <td>0.11221707</td>\n",
       "      <td>0.00975017</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>-0.37398164</td>\n",
       "      <td>0.05608409</td>\n",
       "      <td>0.00956778</td>\n",
       "      <td>0.00028342</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>-0.37369834</td>\n",
       "      <td>0.05626639</td>\n",
       "      <td>0.00028330</td>\n",
       "      <td>0.00000012</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>-0.37369822</td>\n",
       "      <td>0.05626649</td>\n",
       "      <td>0.00000012</td>\n",
       "      <td>0.00000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "           x1          x2   residual  actual-residual\n",
       "0 -1.00000000  1.00000000        nan       0.94373351\n",
       "1 -2.02974173  0.11368626 1.02974173       1.65604351\n",
       "2 -0.37403377  0.21864891 1.65570796       0.16238242\n",
       "3 -0.49557423 -0.00045536 0.21910426       0.12187602\n",
       "4 -0.38335717  0.04651631 0.11221707       0.00975017\n",
       "5 -0.37398164  0.05608409 0.00956778       0.00028342\n",
       "6 -0.37369834  0.05626639 0.00028330       0.00000012\n",
       "7 -0.37369822  0.05626649 0.00000012       0.00000000"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([-1, 1])\n",
    "Newton().run(x0).astype(np.float64)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## (c) \n",
    "$$\\begin{align*}\n",
    "x_1(1-x_1)+4x_2 &= 12 \\\\\n",
    "(x_1 - x_2)^2 + (2x_2-3)^2 &= 25\n",
    "\\end{align*}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T04:49:02.008262Z",
     "start_time": "2018-02-05T04:49:01.936625Z"
    },
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class Newton(NewtonMethod):\n",
    "\n",
    "    def __init__(self):\n",
    "        super(NewtonMethod, self).__init__()\n",
    "\n",
    "    def f(self, x):\n",
    "        sol = np.zeros(len(x))\n",
    "        sol[0] = x[0] * (1 - x[0]) + 4 * x[1] - 12\n",
    "        sol[1] = pow(x[0] - x[1], 2) + pow(2 * x[1] - 3, 2) - 25\n",
    "        return sol\n",
    "\n",
    "    def jacobian(self, x):\n",
    "        jac = np.zeros(shape=(2, 2))\n",
    "        jac[0][0] = 1 - 2 * x[0]\n",
    "        jac[0][1] = 4\n",
    "        jac[1][0] = 2 * (x[0] - x[1])\n",
    "        jac[1][1] = -2 * (x[0] * x[1]) + 4 * (2 * x[1] - 3)\n",
    "        return jac\n",
    "\n",
    "    def run(self, x):\n",
    "        df = pd.DataFrame(columns=['x' + str(i + 1) for i in range(len(x))] + ['residual', 'actual-residual'])\n",
    "\n",
    "        row = len(df)\n",
    "        df.loc[row] = [xe for xe in x] + [np.nan, np.nan]\n",
    "\n",
    "        for k in range(MAX_ITR):\n",
    "            jac = self.jacobian(x)\n",
    "            f = -self.f(x)\n",
    "            y = linalg.solve(jac, f)\n",
    "            nx = x + y\n",
    "            residual = linalg.norm(x - nx, np.inf)\n",
    "            x = nx\n",
    "\n",
    "            row = len(df)\n",
    "            df.loc[row] = [nxe for nxe in nx] + [residual, np.nan]\n",
    "            if residual < TOR:\n",
    "                break\n",
    "\n",
    "        for i in range(len(df)):\n",
    "            xk = np.array([df.loc[i][j] for j in range(len(x))])\n",
    "            df.loc[i][3] = linalg.norm(x - xk, np.inf)\n",
    "        print(self.f(x))\n",
    "        return df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T04:49:02.100398Z",
     "start_time": "2018-02-05T04:49:02.009554Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.          0.00000038]\n"
     ]
    },
    {
     "data": {
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       "      <th></th>\n",
       "      <th>x1</th>\n",
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       "      <th>residual</th>\n",
       "      <th>actual-residual</th>\n",
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       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.00000000</td>\n",
       "      <td>0.00000000</td>\n",
       "      <td>nan</td>\n",
       "      <td>3.18782896</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>17.33333333</td>\n",
       "      <td>-1.33333333</td>\n",
       "      <td>17.33333333</td>\n",
       "      <td>17.83399104</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>8.31420636</td>\n",
       "      <td>-2.13320757</td>\n",
       "      <td>9.01912697</td>\n",
       "      <td>8.81486407</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>2.49537953</td>\n",
       "      <td>-4.53180157</td>\n",
       "      <td>5.81882683</td>\n",
       "      <td>7.71963053</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-1.59183172</td>\n",
       "      <td>-0.14488396</td>\n",
       "      <td>4.38691761</td>\n",
       "      <td>3.33271292</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>4.48355091</td>\n",
       "      <td>-2.32289906</td>\n",
       "      <td>6.07538263</td>\n",
       "      <td>5.51072802</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>2.72651237</td>\n",
       "      <td>3.40504322</td>\n",
       "      <td>5.72794228</td>\n",
       "      <td>3.22717008</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>0.23884959</td>\n",
       "      <td>1.40743335</td>\n",
       "      <td>2.48766278</td>\n",
       "      <td>1.78039561</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>-11.73969307</td>\n",
       "      <td>4.51865057</td>\n",
       "      <td>11.97854265</td>\n",
       "      <td>11.23903536</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>-5.77552583</td>\n",
       "      <td>3.89023341</td>\n",
       "      <td>5.96416724</td>\n",
       "      <td>5.27486812</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>-2.77577493</td>\n",
       "      <td>3.37054899</td>\n",
       "      <td>2.99975090</td>\n",
       "      <td>2.27511722</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>-1.25566718</td>\n",
       "      <td>3.13040992</td>\n",
       "      <td>1.52010775</td>\n",
       "      <td>0.75500947</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>-0.63085111</td>\n",
       "      <td>3.15960728</td>\n",
       "      <td>0.62481607</td>\n",
       "      <td>0.13019340</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>-0.51150876</td>\n",
       "      <td>3.18972684</td>\n",
       "      <td>0.11934235</td>\n",
       "      <td>0.01085105</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>-0.50020550</td>\n",
       "      <td>3.18757082</td>\n",
       "      <td>0.01130326</td>\n",
       "      <td>0.00045221</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>-0.50072653</td>\n",
       "      <td>3.18786333</td>\n",
       "      <td>0.00052103</td>\n",
       "      <td>0.00006882</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>-0.50064855</td>\n",
       "      <td>3.18782438</td>\n",
       "      <td>0.00007799</td>\n",
       "      <td>0.00000916</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>-0.50065891</td>\n",
       "      <td>3.18782956</td>\n",
       "      <td>0.00001036</td>\n",
       "      <td>0.00000120</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>-0.50065753</td>\n",
       "      <td>3.18782887</td>\n",
       "      <td>0.00000138</td>\n",
       "      <td>0.00000018</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>-0.50065771</td>\n",
       "      <td>3.18782896</td>\n",
       "      <td>0.00000018</td>\n",
       "      <td>0.00000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "             x1          x2    residual  actual-residual\n",
       "0    0.00000000  0.00000000         nan       3.18782896\n",
       "1   17.33333333 -1.33333333 17.33333333      17.83399104\n",
       "2    8.31420636 -2.13320757  9.01912697       8.81486407\n",
       "3    2.49537953 -4.53180157  5.81882683       7.71963053\n",
       "4   -1.59183172 -0.14488396  4.38691761       3.33271292\n",
       "5    4.48355091 -2.32289906  6.07538263       5.51072802\n",
       "6    2.72651237  3.40504322  5.72794228       3.22717008\n",
       "7    0.23884959  1.40743335  2.48766278       1.78039561\n",
       "8  -11.73969307  4.51865057 11.97854265      11.23903536\n",
       "9   -5.77552583  3.89023341  5.96416724       5.27486812\n",
       "10  -2.77577493  3.37054899  2.99975090       2.27511722\n",
       "11  -1.25566718  3.13040992  1.52010775       0.75500947\n",
       "12  -0.63085111  3.15960728  0.62481607       0.13019340\n",
       "13  -0.51150876  3.18972684  0.11934235       0.01085105\n",
       "14  -0.50020550  3.18757082  0.01130326       0.00045221\n",
       "15  -0.50072653  3.18786333  0.00052103       0.00006882\n",
       "16  -0.50064855  3.18782438  0.00007799       0.00000916\n",
       "17  -0.50065891  3.18782956  0.00001036       0.00000120\n",
       "18  -0.50065753  3.18782887  0.00000138       0.00000018\n",
       "19  -0.50065771  3.18782896  0.00000018       0.00000000"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([0, 0])\n",
    "Newton().run(x0).astype(np.float64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T04:49:02.369425Z",
     "start_time": "2018-02-05T04:49:02.102181Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ -1.58601288e+00   2.96889928e+08]\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>x1</th>\n",
       "      <th>x2</th>\n",
       "      <th>residual</th>\n",
       "      <th>actual-residual</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1.00000000</td>\n",
       "      <td>5.00000000</td>\n",
       "      <td>nan</td>\n",
       "      <td>7,736.88321259</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2.14285714</td>\n",
       "      <td>3.28571429</td>\n",
       "      <td>1.71428571</td>\n",
       "      <td>7,738.59749831</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-2.99016813</td>\n",
       "      <td>-0.60416871</td>\n",
       "      <td>5.13302527</td>\n",
       "      <td>7,742.48738131</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1.41885065</td>\n",
       "      <td>-1.71129000</td>\n",
       "      <td>4.40901877</td>\n",
       "      <td>7,743.59450260</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-21.29127300</td>\n",
       "      <td>-7.28503427</td>\n",
       "      <td>22.71012365</td>\n",
       "      <td>7,749.16824686</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>-9.49344969</td>\n",
       "      <td>-6.89239956</td>\n",
       "      <td>11.79782331</td>\n",
       "      <td>7,748.77561215</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>-2.76037894</td>\n",
       "      <td>-5.73854274</td>\n",
       "      <td>6.73307075</td>\n",
       "      <td>7,747.62175533</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>2.64679065</td>\n",
       "      <td>-3.21969320</td>\n",
       "      <td>5.40716958</td>\n",
       "      <td>7,745.10290579</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>-2.38955071</td>\n",
       "      <td>-1.31630774</td>\n",
       "      <td>5.03634136</td>\n",
       "      <td>7,743.19952033</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>2.03855877</td>\n",
       "      <td>-1.37274763</td>\n",
       "      <td>4.42810949</td>\n",
       "      <td>7,743.25596023</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>-7.97071573</td>\n",
       "      <td>-4.17063778</td>\n",
       "      <td>10.00927451</td>\n",
       "      <td>7,746.05385038</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>-2.21394037</td>\n",
       "      <td>-3.50624759</td>\n",
       "      <td>5.75677537</td>\n",
       "      <td>7,745.38946018</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>2.70316531</td>\n",
       "      <td>-1.89349771</td>\n",
       "      <td>4.91710568</td>\n",
       "      <td>7,743.77671031</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>-3.65847655</td>\n",
       "      <td>-2.85688999</td>\n",
       "      <td>6.36164186</td>\n",
       "      <td>7,744.74010259</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>0.82470434</td>\n",
       "      <td>-2.06086949</td>\n",
       "      <td>4.48318089</td>\n",
       "      <td>7,743.94408209</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>55.01081507</td>\n",
       "      <td>11.76109099</td>\n",
       "      <td>54.18611073</td>\n",
       "      <td>7,730.12212161</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>28.07723622</td>\n",
       "      <td>11.70907195</td>\n",
       "      <td>26.93357885</td>\n",
       "      <td>7,730.17414064</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>14.95375640</td>\n",
       "      <td>12.10883783</td>\n",
       "      <td>13.12347982</td>\n",
       "      <td>7,729.77437476</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>9.19581572</td>\n",
       "      <td>13.55333254</td>\n",
       "      <td>5.75794068</td>\n",
       "      <td>7,728.32988005</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>8.16841727</td>\n",
       "      <td>17.37476895</td>\n",
       "      <td>3.82143641</td>\n",
       "      <td>7,724.50844364</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>9.82440601</td>\n",
       "      <td>23.98806219</td>\n",
       "      <td>6.61329324</td>\n",
       "      <td>7,717.89515040</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>11.26535639</td>\n",
       "      <td>31.39164006</td>\n",
       "      <td>7.40357787</td>\n",
       "      <td>7,710.49157253</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>12.71494045</td>\n",
       "      <td>39.71336907</td>\n",
       "      <td>8.32172901</td>\n",
       "      <td>7,702.16984353</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>14.13289889</td>\n",
       "      <td>48.89883150</td>\n",
       "      <td>9.18546243</td>\n",
       "      <td>7,692.98438109</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>15.53251771</td>\n",
       "      <td>58.94191394</td>\n",
       "      <td>10.04308245</td>\n",
       "      <td>7,682.94129865</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>16.91640617</td>\n",
       "      <td>69.83331104</td>\n",
       "      <td>10.89139710</td>\n",
       "      <td>7,672.04990155</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>18.28755528</td>\n",
       "      <td>81.56676820</td>\n",
       "      <td>11.73345716</td>\n",
       "      <td>7,660.31644439</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>19.64810753</td>\n",
       "      <td>94.13722989</td>\n",
       "      <td>12.57046168</td>\n",
       "      <td>7,647.74598270</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>20.99973428</td>\n",
       "      <td>107.54055270</td>\n",
       "      <td>13.40332281</td>\n",
       "      <td>7,634.34265990</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>22.34374840</td>\n",
       "      <td>121.77324259</td>\n",
       "      <td>14.23268990</td>\n",
       "      <td>7,620.10997000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>121</th>\n",
       "      <td>139.89245872</td>\n",
       "      <td>4,860.10376897</td>\n",
       "      <td>87.55384802</td>\n",
       "      <td>2,881.77944362</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>122</th>\n",
       "      <td>141.15427908</td>\n",
       "      <td>4,948.44600804</td>\n",
       "      <td>88.34223907</td>\n",
       "      <td>2,793.43720455</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>123</th>\n",
       "      <td>142.41598998</td>\n",
       "      <td>5,037.57657419</td>\n",
       "      <td>89.13056615</td>\n",
       "      <td>2,704.30663840</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>124</th>\n",
       "      <td>143.67759343</td>\n",
       "      <td>5,127.49540457</td>\n",
       "      <td>89.91883038</td>\n",
       "      <td>2,614.38780802</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>125</th>\n",
       "      <td>144.93909141</td>\n",
       "      <td>5,218.20243746</td>\n",
       "      <td>90.70703289</td>\n",
       "      <td>2,523.68077513</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>126</th>\n",
       "      <td>146.20048581</td>\n",
       "      <td>5,309.69761222</td>\n",
       "      <td>91.49517475</td>\n",
       "      <td>2,432.18560037</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>127</th>\n",
       "      <td>147.46177849</td>\n",
       "      <td>5,401.98086924</td>\n",
       "      <td>92.28325703</td>\n",
       "      <td>2,339.90234335</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>128</th>\n",
       "      <td>148.72297125</td>\n",
       "      <td>5,495.05214997</td>\n",
       "      <td>93.07128073</td>\n",
       "      <td>2,246.83106262</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>129</th>\n",
       "      <td>149.98406586</td>\n",
       "      <td>5,588.91139684</td>\n",
       "      <td>93.85924686</td>\n",
       "      <td>2,152.97181576</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>130</th>\n",
       "      <td>151.24506403</td>\n",
       "      <td>5,683.55855324</td>\n",
       "      <td>94.64715641</td>\n",
       "      <td>2,058.32465935</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>131</th>\n",
       "      <td>152.50596742</td>\n",
       "      <td>5,778.99356355</td>\n",
       "      <td>95.43501031</td>\n",
       "      <td>1,962.88964904</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>132</th>\n",
       "      <td>153.76677766</td>\n",
       "      <td>5,875.21637304</td>\n",
       "      <td>96.22280949</td>\n",
       "      <td>1,866.66683956</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>133</th>\n",
       "      <td>155.02749633</td>\n",
       "      <td>5,972.22692789</td>\n",
       "      <td>97.01055485</td>\n",
       "      <td>1,769.65628470</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>134</th>\n",
       "      <td>156.28812498</td>\n",
       "      <td>6,070.02517517</td>\n",
       "      <td>97.79824728</td>\n",
       "      <td>1,671.85803742</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>135</th>\n",
       "      <td>157.54866511</td>\n",
       "      <td>6,168.61106281</td>\n",
       "      <td>98.58588764</td>\n",
       "      <td>1,573.27214978</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>136</th>\n",
       "      <td>158.80911818</td>\n",
       "      <td>6,267.98453957</td>\n",
       "      <td>99.37347676</td>\n",
       "      <td>1,473.89867302</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>137</th>\n",
       "      <td>160.06948564</td>\n",
       "      <td>6,368.14555503</td>\n",
       "      <td>100.16101546</td>\n",
       "      <td>1,373.73765756</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>138</th>\n",
       "      <td>161.32976886</td>\n",
       "      <td>6,469.09405957</td>\n",
       "      <td>100.94850454</td>\n",
       "      <td>1,272.78915303</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>139</th>\n",
       "      <td>162.58996922</td>\n",
       "      <td>6,570.83000434</td>\n",
       "      <td>101.73594478</td>\n",
       "      <td>1,171.05320825</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>140</th>\n",
       "      <td>163.85008805</td>\n",
       "      <td>6,673.35334128</td>\n",
       "      <td>102.52333693</td>\n",
       "      <td>1,068.52987132</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>141</th>\n",
       "      <td>165.11012663</td>\n",
       "      <td>6,776.66402303</td>\n",
       "      <td>103.31068176</td>\n",
       "      <td>965.21918956</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>142</th>\n",
       "      <td>166.37008624</td>\n",
       "      <td>6,880.76200301</td>\n",
       "      <td>104.09797997</td>\n",
       "      <td>861.12120958</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>143</th>\n",
       "      <td>167.62996812</td>\n",
       "      <td>6,985.64723530</td>\n",
       "      <td>104.88523229</td>\n",
       "      <td>756.23597729</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>144</th>\n",
       "      <td>168.88977347</td>\n",
       "      <td>7,091.31967470</td>\n",
       "      <td>105.67243940</td>\n",
       "      <td>650.56353789</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>145</th>\n",
       "      <td>170.14950347</td>\n",
       "      <td>7,197.77927669</td>\n",
       "      <td>106.45960199</td>\n",
       "      <td>544.10393590</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>146</th>\n",
       "      <td>171.40915927</td>\n",
       "      <td>7,305.02599740</td>\n",
       "      <td>107.24672071</td>\n",
       "      <td>436.85721519</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>147</th>\n",
       "      <td>172.66874200</td>\n",
       "      <td>7,413.05979362</td>\n",
       "      <td>108.03379622</td>\n",
       "      <td>328.82341897</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>148</th>\n",
       "      <td>173.92825277</td>\n",
       "      <td>7,521.88062277</td>\n",
       "      <td>108.82082914</td>\n",
       "      <td>220.00258983</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>149</th>\n",
       "      <td>175.18769264</td>\n",
       "      <td>7,631.48844287</td>\n",
       "      <td>109.60782011</td>\n",
       "      <td>110.39476972</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>150</th>\n",
       "      <td>176.44706267</td>\n",
       "      <td>7,741.88321259</td>\n",
       "      <td>110.39476972</td>\n",
       "      <td>0.00000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>151 rows × 4 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "              x1             x2     residual  actual-residual\n",
       "0     1.00000000     5.00000000          nan   7,736.88321259\n",
       "1     2.14285714     3.28571429   1.71428571   7,738.59749831\n",
       "2    -2.99016813    -0.60416871   5.13302527   7,742.48738131\n",
       "3     1.41885065    -1.71129000   4.40901877   7,743.59450260\n",
       "4   -21.29127300    -7.28503427  22.71012365   7,749.16824686\n",
       "5    -9.49344969    -6.89239956  11.79782331   7,748.77561215\n",
       "6    -2.76037894    -5.73854274   6.73307075   7,747.62175533\n",
       "7     2.64679065    -3.21969320   5.40716958   7,745.10290579\n",
       "8    -2.38955071    -1.31630774   5.03634136   7,743.19952033\n",
       "9     2.03855877    -1.37274763   4.42810949   7,743.25596023\n",
       "10   -7.97071573    -4.17063778  10.00927451   7,746.05385038\n",
       "11   -2.21394037    -3.50624759   5.75677537   7,745.38946018\n",
       "12    2.70316531    -1.89349771   4.91710568   7,743.77671031\n",
       "13   -3.65847655    -2.85688999   6.36164186   7,744.74010259\n",
       "14    0.82470434    -2.06086949   4.48318089   7,743.94408209\n",
       "15   55.01081507    11.76109099  54.18611073   7,730.12212161\n",
       "16   28.07723622    11.70907195  26.93357885   7,730.17414064\n",
       "17   14.95375640    12.10883783  13.12347982   7,729.77437476\n",
       "18    9.19581572    13.55333254   5.75794068   7,728.32988005\n",
       "19    8.16841727    17.37476895   3.82143641   7,724.50844364\n",
       "20    9.82440601    23.98806219   6.61329324   7,717.89515040\n",
       "21   11.26535639    31.39164006   7.40357787   7,710.49157253\n",
       "22   12.71494045    39.71336907   8.32172901   7,702.16984353\n",
       "23   14.13289889    48.89883150   9.18546243   7,692.98438109\n",
       "24   15.53251771    58.94191394  10.04308245   7,682.94129865\n",
       "25   16.91640617    69.83331104  10.89139710   7,672.04990155\n",
       "26   18.28755528    81.56676820  11.73345716   7,660.31644439\n",
       "27   19.64810753    94.13722989  12.57046168   7,647.74598270\n",
       "28   20.99973428   107.54055270  13.40332281   7,634.34265990\n",
       "29   22.34374840   121.77324259  14.23268990   7,620.10997000\n",
       "..           ...            ...          ...              ...\n",
       "121 139.89245872 4,860.10376897  87.55384802   2,881.77944362\n",
       "122 141.15427908 4,948.44600804  88.34223907   2,793.43720455\n",
       "123 142.41598998 5,037.57657419  89.13056615   2,704.30663840\n",
       "124 143.67759343 5,127.49540457  89.91883038   2,614.38780802\n",
       "125 144.93909141 5,218.20243746  90.70703289   2,523.68077513\n",
       "126 146.20048581 5,309.69761222  91.49517475   2,432.18560037\n",
       "127 147.46177849 5,401.98086924  92.28325703   2,339.90234335\n",
       "128 148.72297125 5,495.05214997  93.07128073   2,246.83106262\n",
       "129 149.98406586 5,588.91139684  93.85924686   2,152.97181576\n",
       "130 151.24506403 5,683.55855324  94.64715641   2,058.32465935\n",
       "131 152.50596742 5,778.99356355  95.43501031   1,962.88964904\n",
       "132 153.76677766 5,875.21637304  96.22280949   1,866.66683956\n",
       "133 155.02749633 5,972.22692789  97.01055485   1,769.65628470\n",
       "134 156.28812498 6,070.02517517  97.79824728   1,671.85803742\n",
       "135 157.54866511 6,168.61106281  98.58588764   1,573.27214978\n",
       "136 158.80911818 6,267.98453957  99.37347676   1,473.89867302\n",
       "137 160.06948564 6,368.14555503 100.16101546   1,373.73765756\n",
       "138 161.32976886 6,469.09405957 100.94850454   1,272.78915303\n",
       "139 162.58996922 6,570.83000434 101.73594478   1,171.05320825\n",
       "140 163.85008805 6,673.35334128 102.52333693   1,068.52987132\n",
       "141 165.11012663 6,776.66402303 103.31068176     965.21918956\n",
       "142 166.37008624 6,880.76200301 104.09797997     861.12120958\n",
       "143 167.62996812 6,985.64723530 104.88523229     756.23597729\n",
       "144 168.88977347 7,091.31967470 105.67243940     650.56353789\n",
       "145 170.14950347 7,197.77927669 106.45960199     544.10393590\n",
       "146 171.40915927 7,305.02599740 107.24672071     436.85721519\n",
       "147 172.66874200 7,413.05979362 108.03379622     328.82341897\n",
       "148 173.92825277 7,521.88062277 108.82082914     220.00258983\n",
       "149 175.18769264 7,631.48844287 109.60782011     110.39476972\n",
       "150 176.44706267 7,741.88321259 110.39476972       0.00000000\n",
       "\n",
       "[151 rows x 4 columns]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([1, 5])\n",
    "Newton().run(x0).astype(np.float64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T04:49:02.441482Z",
     "start_time": "2018-02-05T04:49:02.371397Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.         -0.00000105]\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>x1</th>\n",
       "      <th>x2</th>\n",
       "      <th>residual</th>\n",
       "      <th>actual-residual</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-1.00000000</td>\n",
       "      <td>500.00000000</td>\n",
       "      <td>nan</td>\n",
       "      <td>496.81217108</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-260.65781151</td>\n",
       "      <td>198.24335863</td>\n",
       "      <td>301.75664137</td>\n",
       "      <td>260.15715388</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-131.55657623</td>\n",
       "      <td>195.89009339</td>\n",
       "      <td>129.10123528</td>\n",
       "      <td>192.70226447</td>\n",
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       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-68.38866968</td>\n",
       "      <td>191.80359803</td>\n",
       "      <td>63.16790655</td>\n",
       "      <td>188.61576910</td>\n",
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       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-39.22113366</td>\n",
       "      <td>184.69332540</td>\n",
       "      <td>29.16753602</td>\n",
       "      <td>181.50549647</td>\n",
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       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>-27.94625393</td>\n",
       "      <td>173.45411248</td>\n",
       "      <td>11.27487973</td>\n",
       "      <td>170.26628355</td>\n",
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       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>-24.71148944</td>\n",
       "      <td>159.22637457</td>\n",
       "      <td>14.22773790</td>\n",
       "      <td>156.03854565</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>-23.35009994</td>\n",
       "      <td>144.68097140</td>\n",
       "      <td>14.54540317</td>\n",
       "      <td>141.49314248</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>-22.14596000</td>\n",
       "      <td>130.78488779</td>\n",
       "      <td>13.89608362</td>\n",
       "      <td>127.59705886</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>-20.95154502</td>\n",
       "      <td>117.62303912</td>\n",
       "      <td>13.16184867</td>\n",
       "      <td>114.43521020</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>-19.75981566</td>\n",
       "      <td>105.19747798</td>\n",
       "      <td>12.42556114</td>\n",
       "      <td>102.00964906</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>-18.57061047</td>\n",
       "      <td>93.50599373</td>\n",
       "      <td>11.69148425</td>\n",
       "      <td>90.31816480</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>-17.38411325</td>\n",
       "      <td>82.54593276</td>\n",
       "      <td>10.96006097</td>\n",
       "      <td>79.35810383</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>-16.20054522</td>\n",
       "      <td>72.31434436</td>\n",
       "      <td>10.23158840</td>\n",
       "      <td>69.12651543</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>-15.02015201</td>\n",
       "      <td>62.80794759</td>\n",
       "      <td>9.50639677</td>\n",
       "      <td>59.62011866</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>-13.84320532</td>\n",
       "      <td>54.02308380</td>\n",
       "      <td>8.78486378</td>\n",
       "      <td>50.83525488</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>-12.67000577</td>\n",
       "      <td>45.95566369</td>\n",
       "      <td>8.06742011</td>\n",
       "      <td>42.76783477</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>-11.50088693</td>\n",
       "      <td>38.60111209</td>\n",
       "      <td>7.35455161</td>\n",
       "      <td>35.41328316</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>-10.33622241</td>\n",
       "      <td>31.95431818</td>\n",
       "      <td>6.64679391</td>\n",
       "      <td>28.76648926</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>-9.17644182</td>\n",
       "      <td>26.00960879</td>\n",
       "      <td>5.94470939</td>\n",
       "      <td>22.82177987</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>-8.02207178</td>\n",
       "      <td>20.76078429</td>\n",
       "      <td>5.24882450</td>\n",
       "      <td>17.57295537</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>-6.87384813</td>\n",
       "      <td>16.20130470</td>\n",
       "      <td>4.55947959</td>\n",
       "      <td>13.01347577</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>-5.73303533</td>\n",
       "      <td>12.32481891</td>\n",
       "      <td>3.87648579</td>\n",
       "      <td>9.13698998</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>-4.60237275</td>\n",
       "      <td>9.12645245</td>\n",
       "      <td>3.19836646</td>\n",
       "      <td>5.93862353</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>-3.48901041</td>\n",
       "      <td>6.60565710</td>\n",
       "      <td>2.52079535</td>\n",
       "      <td>3.41782818</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>-2.41398436</td>\n",
       "      <td>4.77140595</td>\n",
       "      <td>1.83425115</td>\n",
       "      <td>1.91332673</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>-1.44199083</td>\n",
       "      <td>3.64413925</td>\n",
       "      <td>1.12726671</td>\n",
       "      <td>0.94133320</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>-0.74553005</td>\n",
       "      <td>3.20407187</td>\n",
       "      <td>0.69646078</td>\n",
       "      <td>0.24487242</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>-0.51586649</td>\n",
       "      <td>3.18230984</td>\n",
       "      <td>0.22966357</td>\n",
       "      <td>0.01520886</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>-0.50215984</td>\n",
       "      <td>3.18853412</td>\n",
       "      <td>0.01370665</td>\n",
       "      <td>0.00150221</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>30</th>\n",
       "      <td>-0.50047167</td>\n",
       "      <td>3.18773518</td>\n",
       "      <td>0.00168817</td>\n",
       "      <td>0.00018596</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>31</th>\n",
       "      <td>-0.50068266</td>\n",
       "      <td>3.18784144</td>\n",
       "      <td>0.00021099</td>\n",
       "      <td>0.00002503</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>32</th>\n",
       "      <td>-0.50065437</td>\n",
       "      <td>3.18782729</td>\n",
       "      <td>0.00002830</td>\n",
       "      <td>0.00000326</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>33</th>\n",
       "      <td>-0.50065813</td>\n",
       "      <td>3.18782917</td>\n",
       "      <td>0.00000376</td>\n",
       "      <td>0.00000050</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>34</th>\n",
       "      <td>-0.50065763</td>\n",
       "      <td>3.18782892</td>\n",
       "      <td>0.00000050</td>\n",
       "      <td>0.00000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "              x1           x2     residual  actual-residual\n",
       "0    -1.00000000 500.00000000          nan     496.81217108\n",
       "1  -260.65781151 198.24335863 301.75664137     260.15715388\n",
       "2  -131.55657623 195.89009339 129.10123528     192.70226447\n",
       "3   -68.38866968 191.80359803  63.16790655     188.61576910\n",
       "4   -39.22113366 184.69332540  29.16753602     181.50549647\n",
       "5   -27.94625393 173.45411248  11.27487973     170.26628355\n",
       "6   -24.71148944 159.22637457  14.22773790     156.03854565\n",
       "7   -23.35009994 144.68097140  14.54540317     141.49314248\n",
       "8   -22.14596000 130.78488779  13.89608362     127.59705886\n",
       "9   -20.95154502 117.62303912  13.16184867     114.43521020\n",
       "10  -19.75981566 105.19747798  12.42556114     102.00964906\n",
       "11  -18.57061047  93.50599373  11.69148425      90.31816480\n",
       "12  -17.38411325  82.54593276  10.96006097      79.35810383\n",
       "13  -16.20054522  72.31434436  10.23158840      69.12651543\n",
       "14  -15.02015201  62.80794759   9.50639677      59.62011866\n",
       "15  -13.84320532  54.02308380   8.78486378      50.83525488\n",
       "16  -12.67000577  45.95566369   8.06742011      42.76783477\n",
       "17  -11.50088693  38.60111209   7.35455161      35.41328316\n",
       "18  -10.33622241  31.95431818   6.64679391      28.76648926\n",
       "19   -9.17644182  26.00960879   5.94470939      22.82177987\n",
       "20   -8.02207178  20.76078429   5.24882450      17.57295537\n",
       "21   -6.87384813  16.20130470   4.55947959      13.01347577\n",
       "22   -5.73303533  12.32481891   3.87648579       9.13698998\n",
       "23   -4.60237275   9.12645245   3.19836646       5.93862353\n",
       "24   -3.48901041   6.60565710   2.52079535       3.41782818\n",
       "25   -2.41398436   4.77140595   1.83425115       1.91332673\n",
       "26   -1.44199083   3.64413925   1.12726671       0.94133320\n",
       "27   -0.74553005   3.20407187   0.69646078       0.24487242\n",
       "28   -0.51586649   3.18230984   0.22966357       0.01520886\n",
       "29   -0.50215984   3.18853412   0.01370665       0.00150221\n",
       "30   -0.50047167   3.18773518   0.00168817       0.00018596\n",
       "31   -0.50068266   3.18784144   0.00021099       0.00002503\n",
       "32   -0.50065437   3.18782729   0.00002830       0.00000326\n",
       "33   -0.50065813   3.18782917   0.00000376       0.00000050\n",
       "34   -0.50065763   3.18782892   0.00000050       0.00000000"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([-1, 500])\n",
    "Newton().run(x0).astype(np.float64)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## (d) \n",
    "$$\\begin{align*}\n",
    "5x_1^2-x_2^2 &= 0 \\\\\n",
    "x_2 - 0.25(\\sin x_1 + \\cos x_2) &= 0\n",
    "\\end{align*}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T04:59:13.845295Z",
     "start_time": "2018-02-05T04:59:13.774773Z"
    },
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class Newton(NewtonMethod):\n",
    "\n",
    "    def __init__(self):\n",
    "        super(NewtonMethod, self).__init__()\n",
    "\n",
    "    def f(self, x):\n",
    "        sol = np.zeros(len(x))\n",
    "        sol[0] = 5 * pow(x[0], 2) - pow(x[1], 2)\n",
    "        sol[1] = x[1] - 0.25 * (math.sin(x[0]) + math.cos(x[1]))\n",
    "        return sol\n",
    "\n",
    "    def jacobian(self, x):\n",
    "        jac = np.zeros(shape=(2, 2))\n",
    "        jac[0][0] = 10 * x[0]\n",
    "        jac[0][1] = -2 * x[1]\n",
    "        jac[1][0] = -0.25 * math.cos(x[0])\n",
    "        jac[1][1] = 1 + 0.25 * math.sin(x[1])\n",
    "        return jac\n",
    "\n",
    "    def run(self, x):\n",
    "        df = pd.DataFrame(columns=['x' + str(i + 1) for i in range(len(x))] + ['residual', 'actual-residual'])\n",
    "\n",
    "        row = len(df)\n",
    "        df.loc[row] = [xe for xe in x] + [np.nan, np.nan]\n",
    "\n",
    "        for k in range(MAX_ITR):\n",
    "            jac = self.jacobian(x)\n",
    "            f = -self.f(x)\n",
    "            y = linalg.solve(jac, f)\n",
    "            nx = x + y\n",
    "            residual = linalg.norm(x - nx, np.inf)\n",
    "            x = nx\n",
    "\n",
    "            row = len(df)\n",
    "            df.loc[row] = [nxe for nxe in nx] + [residual, np.nan]\n",
    "            if residual < TOR:\n",
    "                break\n",
    "\n",
    "        for i in range(len(df)):\n",
    "            xk = np.array([df.loc[i][j] for j in range(len(x))])\n",
    "            df.loc[i][3] = linalg.norm(x - xk, np.inf)\n",
    "        print(self.f(x))\n",
    "        return df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T05:02:25.077710Z",
     "start_time": "2018-02-05T05:02:25.060315Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "ename": "LinAlgError",
     "evalue": "Singular matrix",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mLinAlgError\u001b[0m                               Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-18-b4f907dd2a10>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[0mx0\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marray\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0mNewton\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[0mx0\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-15-aa4106d0aeaa>\u001b[0m in \u001b[0;36mrun\u001b[0;34m(self, x)\u001b[0m\n\u001b[1;32m     27\u001b[0m             \u001b[0mjac\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[1;32m     28\u001b[0m             \u001b[0mf\u001b[0m \u001b[0;34m=\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[0m\n\u001b[0;32m---> 29\u001b[0;31m             \u001b[0my\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mlinalg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msolve\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mjac\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     30\u001b[0m             \u001b[0mnx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mx\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0my\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     31\u001b[0m             \u001b[0mresidual\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mlinalg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mnorm\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mnx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0minf\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/bin/anaconda3/lib/python3.6/site-packages/numpy/linalg/linalg.py\u001b[0m in \u001b[0;36msolve\u001b[0;34m(a, b)\u001b[0m\n\u001b[1;32m    373\u001b[0m     \u001b[0msignature\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m'DD->D'\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0misComplexType\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mt\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32melse\u001b[0m \u001b[0;34m'dd->d'\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    374\u001b[0m     \u001b[0mextobj\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mget_linalg_error_extobj\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0m_raise_linalgerror_singular\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 375\u001b[0;31m     \u001b[0mr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mgufunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msignature\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0msignature\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mextobj\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mextobj\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    376\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    377\u001b[0m     \u001b[0;32mreturn\u001b[0m \u001b[0mwrap\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mastype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresult_t\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcopy\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/bin/anaconda3/lib/python3.6/site-packages/numpy/linalg/linalg.py\u001b[0m in \u001b[0;36m_raise_linalgerror_singular\u001b[0;34m(err, flag)\u001b[0m\n\u001b[1;32m     88\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     89\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0m_raise_linalgerror_singular\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0merr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mflag\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 90\u001b[0;31m     \u001b[0;32mraise\u001b[0m \u001b[0mLinAlgError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"Singular matrix\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     91\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     92\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0m_raise_linalgerror_nonposdef\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0merr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mflag\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mLinAlgError\u001b[0m: Singular matrix"
     ]
    }
   ],
   "source": [
    "x0 = np.array([0, 0])\n",
    "Newton().run(x0).astype(np.float64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T05:02:34.619672Z",
     "start_time": "2018-02-05T05:02:34.593489Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.  0.]\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>x1</th>\n",
       "      <th>x2</th>\n",
       "      <th>residual</th>\n",
       "      <th>actual-residual</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1.00000000</td>\n",
       "      <td>1.00000000</td>\n",
       "      <td>nan</td>\n",
       "      <td>0.87875809</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.48024079</td>\n",
       "      <td>0.40120393</td>\n",
       "      <td>0.59879607</td>\n",
       "      <td>0.35899887</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.25766868</td>\n",
       "      <td>0.30562837</td>\n",
       "      <td>0.22257211</td>\n",
       "      <td>0.13642676</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.15905219</td>\n",
       "      <td>0.28019426</td>\n",
       "      <td>0.09861649</td>\n",
       "      <td>0.03781028</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.12609042</td>\n",
       "      <td>0.27225827</td>\n",
       "      <td>0.03296176</td>\n",
       "      <td>0.00484851</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>0.12134459</td>\n",
       "      <td>0.27112951</td>\n",
       "      <td>0.00474583</td>\n",
       "      <td>0.00010268</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>0.12124196</td>\n",
       "      <td>0.27110517</td>\n",
       "      <td>0.00010263</td>\n",
       "      <td>0.00000005</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>0.12124191</td>\n",
       "      <td>0.27110516</td>\n",
       "      <td>0.00000005</td>\n",
       "      <td>0.00000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          x1         x2   residual  actual-residual\n",
       "0 1.00000000 1.00000000        nan       0.87875809\n",
       "1 0.48024079 0.40120393 0.59879607       0.35899887\n",
       "2 0.25766868 0.30562837 0.22257211       0.13642676\n",
       "3 0.15905219 0.28019426 0.09861649       0.03781028\n",
       "4 0.12609042 0.27225827 0.03296176       0.00484851\n",
       "5 0.12134459 0.27112951 0.00474583       0.00010268\n",
       "6 0.12124196 0.27110517 0.00010263       0.00000005\n",
       "7 0.12124191 0.27110516 0.00000005       0.00000000"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([1, 1])\n",
    "Newton().run(x0).astype(np.float64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T05:02:43.471567Z",
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