ex10.4.1.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>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use the method of Steepest Descent with $TOL = 0.05$ to approximate the solutions of the following nonlinear systems."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The nonlinear system continuous with ex10.3.3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
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    "ExecuteTime": {
     "end_time": "2018-02-05T06:01:07.138620Z",
     "start_time": "2018-02-05T06:01:06.891299Z"
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   "outputs": [],
   "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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T06:01:07.152257Z",
     "start_time": "2018-02-05T06:01:07.140016Z"
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   "outputs": [],
   "source": [
    "class SteepestDescentMethod(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 g(self, x):\n",
    "        return NotImplementedError('Implement g()!')\n",
    "\n",
    "    @abstractmethod\n",
    "    def grad_g(self, x):\n",
    "        return NotImplementedError('Implement grad_g()!')\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",
    "$$\\begin{align*}\n",
    "4x_1^2 - 20 x_1 + \\frac{1}{4} x_2^2 + 8 &= 0 \\\\\n",
    "\\frac{1}{2}x_1x_2^2+2x_1-5x_2+8 &= 0\n",
    "\\end{align*}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T06:01:07.354431Z",
     "start_time": "2018-02-05T06:01:07.153626Z"
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   "outputs": [],
   "source": [
    "class SteepestDescent(SteepestDescentMethod):\n",
    "\n",
    "    def __init__(self):\n",
    "        super(SteepestDescentMethod, 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 g(self, x):\n",
    "        sol = self.f(x)\n",
    "        return sum([e * e for e in sol])\n",
    "\n",
    "    def grad_g(self, x):\n",
    "        return 2 * self.jacobian(x).transpose().dot(self.f(x))\n",
    "\n",
    "    def run(self, x):\n",
    "        df = pd.DataFrame(columns=['x' + str(i + 1) for i in range(len(x))] + ['g', 'residual', 'actual-residual'])\n",
    "\n",
    "        row = len(df)\n",
    "        df.loc[row] = [xe for xe in x] + [self.g(x), np.nan, np.nan]\n",
    "\n",
    "        while True:\n",
    "            prev_x = x\n",
    "            g1 = self.g(x)\n",
    "            z = self.grad_g(x)\n",
    "            z0 = linalg.norm(z, 2)\n",
    "            if z0 == 0.0:\n",
    "                print('Zero gradient')\n",
    "                return x\n",
    "\n",
    "            z /= z0\n",
    "            alpha3 = 1\n",
    "            g3 = self.g(x - alpha3 * z)\n",
    "            while g3 >= g1:\n",
    "                alpha3 /= 2.0\n",
    "                g3 = self.g(x - alpha3 * z)\n",
    "                if alpha3 < TOR / 2.0:\n",
    "                    print('No likely improvement')\n",
    "                    return x\n",
    "\n",
    "            alpha2 = alpha3 / 2.0\n",
    "            g2 = self.g(x - alpha2 * z)\n",
    "\n",
    "            h1 = (g2 - g1) / alpha2\n",
    "            h2 = (g3 - g2) / (alpha3 - alpha2)\n",
    "            h3 = (h2 - h1) / alpha3\n",
    "\n",
    "            alpha0 = (alpha2 - h1 / h3) / 2.0\n",
    "            g0 = self.g(x - alpha0 * z)\n",
    "\n",
    "            alpha = alpha0\n",
    "            g = g0\n",
    "            if g3 < g:\n",
    "                alpha = alpha3\n",
    "                g = g3\n",
    "\n",
    "            x = x - alpha * z\n",
    "            residual = linalg.norm(x - prev_x, np.inf)\n",
    "            row = len(df)\n",
    "            df.loc[row] = [nxe for nxe in x] + [g, residual, np.nan]\n",
    "            if math.fabs(g - g1) < 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][4] = linalg.norm(xk - x, np.inf)\n",
    "\n",
    "        print(self.f(x))\n",
    "        return df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T06:01:07.438271Z",
     "start_time": "2018-02-05T06:01:07.356017Z"
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.00020007  0.0009075 ]\n"
     ]
    },
    {
     "data": {
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       "    }\n",
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       "        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>g</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>128.00000000</td>\n",
       "      <td>nan</td>\n",
       "      <td>1.99977134</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.40926180</td>\n",
       "      <td>0.10231545</td>\n",
       "      <td>69.27847372</td>\n",
       "      <td>0.40926180</td>\n",
       "      <td>1.89745589</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.66283555</td>\n",
       "      <td>1.38995524</td>\n",
       "      <td>18.19552276</td>\n",
       "      <td>1.28763979</td>\n",
       "      <td>0.60981610</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.43320141</td>\n",
       "      <td>1.45993792</td>\n",
       "      <td>4.49808135</td>\n",
       "      <td>0.22963413</td>\n",
       "      <td>0.53983342</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.48412240</td>\n",
       "      <td>1.51291181</td>\n",
       "      <td>3.86258028</td>\n",
       "      <td>0.05297389</td>\n",
       "      <td>0.48685953</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>0.42827597</td>\n",
       "      <td>1.60719656</td>\n",
       "      <td>2.54955032</td>\n",
       "      <td>0.09428475</td>\n",
       "      <td>0.39257478</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>0.47108338</td>\n",
       "      <td>1.62644824</td>\n",
       "      <td>2.06973978</td>\n",
       "      <td>0.04280741</td>\n",
       "      <td>0.37332310</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>0.48109243</td>\n",
       "      <td>1.97435531</td>\n",
       "      <td>0.07833432</td>\n",
       "      <td>0.34790707</td>\n",
       "      <td>0.02541603</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>0.49695350</td>\n",
       "      <td>1.97377989</td>\n",
       "      <td>0.00917044</td>\n",
       "      <td>0.01586108</td>\n",
       "      <td>0.02599146</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>0.50008950</td>\n",
       "      <td>1.97979996</td>\n",
       "      <td>0.00706617</td>\n",
       "      <td>0.00602008</td>\n",
       "      <td>0.01997138</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>0.49729223</td>\n",
       "      <td>1.98172777</td>\n",
       "      <td>0.00453200</td>\n",
       "      <td>0.00279727</td>\n",
       "      <td>0.01804357</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>0.49874287</td>\n",
       "      <td>1.98289485</td>\n",
       "      <td>0.00404281</td>\n",
       "      <td>0.00145064</td>\n",
       "      <td>0.01687650</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>0.49719839</td>\n",
       "      <td>1.98804950</td>\n",
       "      <td>0.00243194</td>\n",
       "      <td>0.00515466</td>\n",
       "      <td>0.01172184</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>0.49879265</td>\n",
       "      <td>1.98845033</td>\n",
       "      <td>0.00177743</td>\n",
       "      <td>0.00159425</td>\n",
       "      <td>0.01132101</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>0.50081297</td>\n",
       "      <td>1.99587343</td>\n",
       "      <td>0.00068363</td>\n",
       "      <td>0.00742310</td>\n",
       "      <td>0.00389791</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>0.49950850</td>\n",
       "      <td>1.99627369</td>\n",
       "      <td>0.00018477</td>\n",
       "      <td>0.00130447</td>\n",
       "      <td>0.00349765</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>0.49982952</td>\n",
       "      <td>1.99665175</td>\n",
       "      <td>0.00016205</td>\n",
       "      <td>0.00037806</td>\n",
       "      <td>0.00311959</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>0.49949120</td>\n",
       "      <td>1.99714635</td>\n",
       "      <td>0.00011605</td>\n",
       "      <td>0.00049460</td>\n",
       "      <td>0.00262499</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>0.49974839</td>\n",
       "      <td>1.99727223</td>\n",
       "      <td>0.00009986</td>\n",
       "      <td>0.00025719</td>\n",
       "      <td>0.00249912</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>0.49981488</td>\n",
       "      <td>1.99971788</td>\n",
       "      <td>0.00000733</td>\n",
       "      <td>0.00244565</td>\n",
       "      <td>0.00018334</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>0.49996597</td>\n",
       "      <td>1.99971383</td>\n",
       "      <td>0.00000108</td>\n",
       "      <td>0.00015109</td>\n",
       "      <td>0.00005751</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>0.49999821</td>\n",
       "      <td>1.99977134</td>\n",
       "      <td>0.00000086</td>\n",
       "      <td>0.00005751</td>\n",
       "      <td>0.00000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "           x1         x2            g   residual  actual-residual\n",
       "0  0.00000000 0.00000000 128.00000000        nan       1.99977134\n",
       "1  0.40926180 0.10231545  69.27847372 0.40926180       1.89745589\n",
       "2  0.66283555 1.38995524  18.19552276 1.28763979       0.60981610\n",
       "3  0.43320141 1.45993792   4.49808135 0.22963413       0.53983342\n",
       "4  0.48412240 1.51291181   3.86258028 0.05297389       0.48685953\n",
       "5  0.42827597 1.60719656   2.54955032 0.09428475       0.39257478\n",
       "6  0.47108338 1.62644824   2.06973978 0.04280741       0.37332310\n",
       "7  0.48109243 1.97435531   0.07833432 0.34790707       0.02541603\n",
       "8  0.49695350 1.97377989   0.00917044 0.01586108       0.02599146\n",
       "9  0.50008950 1.97979996   0.00706617 0.00602008       0.01997138\n",
       "10 0.49729223 1.98172777   0.00453200 0.00279727       0.01804357\n",
       "11 0.49874287 1.98289485   0.00404281 0.00145064       0.01687650\n",
       "12 0.49719839 1.98804950   0.00243194 0.00515466       0.01172184\n",
       "13 0.49879265 1.98845033   0.00177743 0.00159425       0.01132101\n",
       "14 0.50081297 1.99587343   0.00068363 0.00742310       0.00389791\n",
       "15 0.49950850 1.99627369   0.00018477 0.00130447       0.00349765\n",
       "16 0.49982952 1.99665175   0.00016205 0.00037806       0.00311959\n",
       "17 0.49949120 1.99714635   0.00011605 0.00049460       0.00262499\n",
       "18 0.49974839 1.99727223   0.00009986 0.00025719       0.00249912\n",
       "19 0.49981488 1.99971788   0.00000733 0.00244565       0.00018334\n",
       "20 0.49996597 1.99971383   0.00000108 0.00015109       0.00005751\n",
       "21 0.49999821 1.99977134   0.00000086 0.00005751       0.00000000"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([0, 0])\n",
    "SteepestDescent().run(x0).astype(np.float64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T06:01:07.457345Z",
     "start_time": "2018-02-05T06:01:07.440046Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "No likely improvement\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([ 4.50137356,  1.15513658])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([5, 5])\n",
    "SteepestDescent().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": "2018-02-05T06:01:07.703022Z",
     "start_time": "2018-02-05T06:01:07.458822Z"
    },
    "collapsed": true
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   "outputs": [],
   "source": [
    "class SteepestDescent(SteepestDescentMethod):\n",
    "\n",
    "    def __init__(self):\n",
    "        super(SteepestDescentMethod, 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 g(self, x):\n",
    "        sol = self.f(x)\n",
    "        return sum([e * e for e in sol])\n",
    "\n",
    "    def grad_g(self, x):\n",
    "        return 2 * self.jacobian(x).transpose().dot(self.f(x))\n",
    "\n",
    "    def run(self, x):\n",
    "        df = pd.DataFrame(columns=['x' + str(i + 1) for i in range(len(x))] + ['g', 'residual', 'actual-residual'])\n",
    "\n",
    "        row = len(df)\n",
    "        df.loc[row] = [xe for xe in x] + [self.g(x), np.nan, np.nan]\n",
    "\n",
    "        while True:\n",
    "            prev_x = x\n",
    "            g1 = self.g(x)\n",
    "            z = self.grad_g(x)\n",
    "            z0 = linalg.norm(z, 2)\n",
    "            if z0 == 0.0:\n",
    "                print('Zero gradient')\n",
    "                return x\n",
    "\n",
    "            z /= z0\n",
    "            alpha3 = 1\n",
    "            g3 = self.g(x - alpha3 * z)\n",
    "            while g3 >= g1:\n",
    "                alpha3 /= 2.0\n",
    "                g3 = self.g(x - alpha3 * z)\n",
    "                if alpha3 < TOR / 2.0:\n",
    "                    print('No likely improvement')\n",
    "                    return x\n",
    "\n",
    "            alpha2 = alpha3 / 2.0\n",
    "            g2 = self.g(x - alpha2 * z)\n",
    "\n",
    "            h1 = (g2 - g1) / alpha2\n",
    "            h2 = (g3 - g2) / (alpha3 - alpha2)\n",
    "            h3 = (h2 - h1) / alpha3\n",
    "\n",
    "            alpha0 = (alpha2 - h1 / h3) / 2.0\n",
    "            g0 = self.g(x - alpha0 * z)\n",
    "\n",
    "            alpha = alpha0\n",
    "            g = g0\n",
    "            if g3 < g:\n",
    "                alpha = alpha3\n",
    "                g = g3\n",
    "\n",
    "            x = x - alpha * z\n",
    "            residual = linalg.norm(x - prev_x, np.inf)\n",
    "            row = len(df)\n",
    "            df.loc[row] = [nxe for nxe in x] + [g, residual, np.nan]\n",
    "            if math.fabs(g - g1) < 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][4] = linalg.norm(xk - x, np.inf)\n",
    "\n",
    "        print(self.f(x))\n",
    "        return df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T06:01:07.757005Z",
     "start_time": "2018-02-05T06:01:07.705080Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.00000966  0.00000187]\n"
     ]
    },
    {
     "data": {
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       "      <td>0.00000000</td>\n",
       "      <td>0.00000000</td>\n",
       "      <td>2.50128555</td>\n",
       "      <td>nan</td>\n",
       "      <td>0.37369823</td>\n",
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       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-0.35792588</td>\n",
       "      <td>0.00000000</td>\n",
       "      <td>0.13938957</td>\n",
       "      <td>0.35792588</td>\n",
       "      <td>0.05626797</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-0.36100921</td>\n",
       "      <td>0.05788368</td>\n",
       "      <td>0.00331642</td>\n",
       "      <td>0.05788368</td>\n",
       "      <td>0.01268902</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-0.37370318</td>\n",
       "      <td>0.05683275</td>\n",
       "      <td>0.00001418</td>\n",
       "      <td>0.01269398</td>\n",
       "      <td>0.00056478</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-0.37365538</td>\n",
       "      <td>0.05627157</td>\n",
       "      <td>0.00000004</td>\n",
       "      <td>0.00056118</td>\n",
       "      <td>0.00004285</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>-0.37369823</td>\n",
       "      <td>0.05626797</td>\n",
       "      <td>0.00000000</td>\n",
       "      <td>0.00004285</td>\n",
       "      <td>0.00000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "           x1         x2          g   residual  actual-residual\n",
       "0  0.00000000 0.00000000 2.50128555        nan       0.37369823\n",
       "1 -0.35792588 0.00000000 0.13938957 0.35792588       0.05626797\n",
       "2 -0.36100921 0.05788368 0.00331642 0.05788368       0.01268902\n",
       "3 -0.37370318 0.05683275 0.00001418 0.01269398       0.00056478\n",
       "4 -0.37365538 0.05627157 0.00000004 0.00056118       0.00004285\n",
       "5 -0.37369823 0.05626797 0.00000000 0.00004285       0.00000000"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([0, 0])\n",
    "SteepestDescent().run(x0).astype(np.float64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T06:01:07.843782Z",
     "start_time": "2018-02-05T06:01:07.758446Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.00000295 -0.00001662]\n"
     ]
    },
    {
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       "      <th>0</th>\n",
       "      <td>-10.00000000</td>\n",
       "      <td>-10.00000000</td>\n",
       "      <td>1,298,622.87059286</td>\n",
       "      <td>nan</td>\n",
       "      <td>10.05626676</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-9.85030322</td>\n",
       "      <td>-6.23166204</td>\n",
       "      <td>224,460.45002276</td>\n",
       "      <td>3.76833796</td>\n",
       "      <td>9.47660872</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-9.73850971</td>\n",
       "      <td>-3.63703626</td>\n",
       "      <td>38,002.93010292</td>\n",
       "      <td>2.59462578</td>\n",
       "      <td>9.36481520</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-9.57475611</td>\n",
       "      <td>-1.79627799</td>\n",
       "      <td>7,352.61856965</td>\n",
       "      <td>1.84075827</td>\n",
       "      <td>9.20106160</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-9.42728129</td>\n",
       "      <td>-0.48084698</td>\n",
       "      <td>2,744.48879915</td>\n",
       "      <td>1.31543101</td>\n",
       "      <td>9.05358678</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>-9.31427307</td>\n",
       "      <td>0.07822576</td>\n",
       "      <td>2,402.52613683</td>\n",
       "      <td>0.55907274</td>\n",
       "      <td>8.94057856</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>-9.28263021</td>\n",
       "      <td>-0.04270285</td>\n",
       "      <td>2,373.03991323</td>\n",
       "      <td>0.12092861</td>\n",
       "      <td>8.90893570</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>-8.96090564</td>\n",
       "      <td>0.34004158</td>\n",
       "      <td>2,313.21884492</td>\n",
       "      <td>0.38274444</td>\n",
       "      <td>8.58721114</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>-8.95674868</td>\n",
       "      <td>0.34677641</td>\n",
       "      <td>2,309.37009128</td>\n",
       "      <td>0.00673483</td>\n",
       "      <td>8.58305417</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>-7.40302032</td>\n",
       "      <td>-0.55298912</td>\n",
       "      <td>1,775.72712083</td>\n",
       "      <td>1.55372835</td>\n",
       "      <td>7.02932582</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>-7.20657046</td>\n",
       "      <td>0.09341813</td>\n",
       "      <td>1,390.58547268</td>\n",
       "      <td>0.64640725</td>\n",
       "      <td>6.83287596</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>-7.08460806</td>\n",
       "      <td>-0.12816988</td>\n",
       "      <td>1,351.19429618</td>\n",
       "      <td>0.22158801</td>\n",
       "      <td>6.71091356</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>-6.91123672</td>\n",
       "      <td>0.15981110</td>\n",
       "      <td>1,281.19588883</td>\n",
       "      <td>0.28798098</td>\n",
       "      <td>6.53754222</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>-1.86143734</td>\n",
       "      <td>0.49402202</td>\n",
       "      <td>108.92164043</td>\n",
       "      <td>5.04979938</td>\n",
       "      <td>1.48774284</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>-1.48788784</td>\n",
       "      <td>-0.12742161</td>\n",
       "      <td>39.67203025</td>\n",
       "      <td>0.62144363</td>\n",
       "      <td>1.11419334</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>-0.55681873</td>\n",
       "      <td>-0.49226446</td>\n",
       "      <td>13.53254966</td>\n",
       "      <td>0.93106911</td>\n",
       "      <td>0.54853122</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>-0.35917137</td>\n",
       "      <td>0.08853882</td>\n",
       "      <td>0.04307231</td>\n",
       "      <td>0.58080329</td>\n",
       "      <td>0.03227206</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>-0.36078680</td>\n",
       "      <td>0.05728080</td>\n",
       "      <td>0.00342227</td>\n",
       "      <td>0.03125803</td>\n",
       "      <td>0.01290771</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>-0.37344633</td>\n",
       "      <td>0.05748443</td>\n",
       "      <td>0.00006459</td>\n",
       "      <td>0.01265953</td>\n",
       "      <td>0.00121767</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>-0.37346477</td>\n",
       "      <td>0.05628353</td>\n",
       "      <td>0.00000113</td>\n",
       "      <td>0.00120090</td>\n",
       "      <td>0.00022973</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>-0.37369422</td>\n",
       "      <td>0.05628677</td>\n",
       "      <td>0.00000002</td>\n",
       "      <td>0.00022945</td>\n",
       "      <td>0.00002001</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>-0.37369450</td>\n",
       "      <td>0.05626676</td>\n",
       "      <td>0.00000000</td>\n",
       "      <td>0.00002001</td>\n",
       "      <td>0.00000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "             x1           x2                  g   residual  actual-residual\n",
       "0  -10.00000000 -10.00000000 1,298,622.87059286        nan      10.05626676\n",
       "1   -9.85030322  -6.23166204   224,460.45002276 3.76833796       9.47660872\n",
       "2   -9.73850971  -3.63703626    38,002.93010292 2.59462578       9.36481520\n",
       "3   -9.57475611  -1.79627799     7,352.61856965 1.84075827       9.20106160\n",
       "4   -9.42728129  -0.48084698     2,744.48879915 1.31543101       9.05358678\n",
       "5   -9.31427307   0.07822576     2,402.52613683 0.55907274       8.94057856\n",
       "6   -9.28263021  -0.04270285     2,373.03991323 0.12092861       8.90893570\n",
       "7   -8.96090564   0.34004158     2,313.21884492 0.38274444       8.58721114\n",
       "8   -8.95674868   0.34677641     2,309.37009128 0.00673483       8.58305417\n",
       "9   -7.40302032  -0.55298912     1,775.72712083 1.55372835       7.02932582\n",
       "10  -7.20657046   0.09341813     1,390.58547268 0.64640725       6.83287596\n",
       "11  -7.08460806  -0.12816988     1,351.19429618 0.22158801       6.71091356\n",
       "12  -6.91123672   0.15981110     1,281.19588883 0.28798098       6.53754222\n",
       "13  -1.86143734   0.49402202       108.92164043 5.04979938       1.48774284\n",
       "14  -1.48788784  -0.12742161        39.67203025 0.62144363       1.11419334\n",
       "15  -0.55681873  -0.49226446        13.53254966 0.93106911       0.54853122\n",
       "16  -0.35917137   0.08853882         0.04307231 0.58080329       0.03227206\n",
       "17  -0.36078680   0.05728080         0.00342227 0.03125803       0.01290771\n",
       "18  -0.37344633   0.05748443         0.00006459 0.01265953       0.00121767\n",
       "19  -0.37346477   0.05628353         0.00000113 0.00120090       0.00022973\n",
       "20  -0.37369422   0.05628677         0.00000002 0.00022945       0.00002001\n",
       "21  -0.37369450   0.05626676         0.00000000 0.00002001       0.00000000"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([-10, -10])\n",
    "SteepestDescent().run(x0).astype(np.float64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T06:01:07.908386Z",
     "start_time": "2018-02-05T06:01:07.845417Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "[-0.00094747  0.0003531 ]\n"
     ]
    },
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       "      <td>1.00000000</td>\n",
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       "      <td>1,180,137.68151253</td>\n",
       "      <td>nan</td>\n",
       "      <td>9.50736624</td>\n",
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       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.89932050</td>\n",
       "      <td>-6.32337970</td>\n",
       "      <td>187,552.64298358</td>\n",
       "      <td>3.67662030</td>\n",
       "      <td>5.83074593</td>\n",
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       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.77395070</td>\n",
       "      <td>-3.88312257</td>\n",
       "      <td>26,152.76944100</td>\n",
       "      <td>2.44025712</td>\n",
       "      <td>3.39048881</td>\n",
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       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.75284256</td>\n",
       "      <td>-2.25227309</td>\n",
       "      <td>2,788.10119944</td>\n",
       "      <td>1.63084948</td>\n",
       "      <td>1.75963933</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.66257732</td>\n",
       "      <td>-1.16190450</td>\n",
       "      <td>148.55882322</td>\n",
       "      <td>1.09036859</td>\n",
       "      <td>0.66927074</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>0.56349580</td>\n",
       "      <td>-0.41724529</td>\n",
       "      <td>0.69904636</td>\n",
       "      <td>0.74465921</td>\n",
       "      <td>0.15564059</td>\n",
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       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>0.56255023</td>\n",
       "      <td>-0.46020087</td>\n",
       "      <td>0.39844694</td>\n",
       "      <td>0.04295558</td>\n",
       "      <td>0.15469502</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>0.38500066</td>\n",
       "      <td>-0.45931872</td>\n",
       "      <td>0.16652615</td>\n",
       "      <td>0.17754957</td>\n",
       "      <td>0.03331504</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>0.38647846</td>\n",
       "      <td>-0.49257640</td>\n",
       "      <td>0.00763522</td>\n",
       "      <td>0.03325768</td>\n",
       "      <td>0.02137675</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>0.39463347</td>\n",
       "      <td>-0.49622444</td>\n",
       "      <td>0.00419297</td>\n",
       "      <td>0.00815501</td>\n",
       "      <td>0.01322174</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>0.39628233</td>\n",
       "      <td>-0.49271157</td>\n",
       "      <td>0.00237254</td>\n",
       "      <td>0.00351287</td>\n",
       "      <td>0.01157289</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>0.40052114</td>\n",
       "      <td>-0.49468541</td>\n",
       "      <td>0.00133635</td>\n",
       "      <td>0.00423881</td>\n",
       "      <td>0.00733408</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>0.40146215</td>\n",
       "      <td>-0.49270545</td>\n",
       "      <td>0.00076249</td>\n",
       "      <td>0.00197996</td>\n",
       "      <td>0.00639306</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>0.40376760</td>\n",
       "      <td>-0.49380838</td>\n",
       "      <td>0.00043609</td>\n",
       "      <td>0.00230545</td>\n",
       "      <td>0.00408761</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>0.40430988</td>\n",
       "      <td>-0.49268312</td>\n",
       "      <td>0.00025074</td>\n",
       "      <td>0.00112526</td>\n",
       "      <td>0.00354533</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>0.40560811</td>\n",
       "      <td>-0.49331015</td>\n",
       "      <td>0.00014428</td>\n",
       "      <td>0.00129823</td>\n",
       "      <td>0.00224710</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>0.40592062</td>\n",
       "      <td>-0.49266404</td>\n",
       "      <td>0.00008318</td>\n",
       "      <td>0.00064611</td>\n",
       "      <td>0.00193459</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>0.40666278</td>\n",
       "      <td>-0.49302316</td>\n",
       "      <td>0.00004796</td>\n",
       "      <td>0.00074216</td>\n",
       "      <td>0.00119243</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>0.40684283</td>\n",
       "      <td>-0.49265075</td>\n",
       "      <td>0.00002767</td>\n",
       "      <td>0.00037241</td>\n",
       "      <td>0.00101238</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>0.40726973</td>\n",
       "      <td>-0.49285710</td>\n",
       "      <td>0.00001596</td>\n",
       "      <td>0.00042690</td>\n",
       "      <td>0.00058548</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>0.40737346</td>\n",
       "      <td>-0.49264222</td>\n",
       "      <td>0.00000921</td>\n",
       "      <td>0.00021488</td>\n",
       "      <td>0.00048176</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>0.40761926</td>\n",
       "      <td>-0.49276082</td>\n",
       "      <td>0.00000531</td>\n",
       "      <td>0.00024580</td>\n",
       "      <td>0.00023595</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>0.40767914</td>\n",
       "      <td>-0.49263689</td>\n",
       "      <td>0.00000307</td>\n",
       "      <td>0.00012393</td>\n",
       "      <td>0.00017607</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>0.40782063</td>\n",
       "      <td>-0.49270528</td>\n",
       "      <td>0.00000177</td>\n",
       "      <td>0.00014149</td>\n",
       "      <td>0.00007152</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>0.40785521</td>\n",
       "      <td>-0.49263376</td>\n",
       "      <td>0.00000102</td>\n",
       "      <td>0.00007152</td>\n",
       "      <td>0.00000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "           x1           x2                  g   residual  actual-residual\n",
       "0  1.00000000 -10.00000000 1,180,137.68151253        nan       9.50736624\n",
       "1  0.89932050  -6.32337970   187,552.64298358 3.67662030       5.83074593\n",
       "2  0.77395070  -3.88312257    26,152.76944100 2.44025712       3.39048881\n",
       "3  0.75284256  -2.25227309     2,788.10119944 1.63084948       1.75963933\n",
       "4  0.66257732  -1.16190450       148.55882322 1.09036859       0.66927074\n",
       "5  0.56349580  -0.41724529         0.69904636 0.74465921       0.15564059\n",
       "6  0.56255023  -0.46020087         0.39844694 0.04295558       0.15469502\n",
       "7  0.38500066  -0.45931872         0.16652615 0.17754957       0.03331504\n",
       "8  0.38647846  -0.49257640         0.00763522 0.03325768       0.02137675\n",
       "9  0.39463347  -0.49622444         0.00419297 0.00815501       0.01322174\n",
       "10 0.39628233  -0.49271157         0.00237254 0.00351287       0.01157289\n",
       "11 0.40052114  -0.49468541         0.00133635 0.00423881       0.00733408\n",
       "12 0.40146215  -0.49270545         0.00076249 0.00197996       0.00639306\n",
       "13 0.40376760  -0.49380838         0.00043609 0.00230545       0.00408761\n",
       "14 0.40430988  -0.49268312         0.00025074 0.00112526       0.00354533\n",
       "15 0.40560811  -0.49331015         0.00014428 0.00129823       0.00224710\n",
       "16 0.40592062  -0.49266404         0.00008318 0.00064611       0.00193459\n",
       "17 0.40666278  -0.49302316         0.00004796 0.00074216       0.00119243\n",
       "18 0.40684283  -0.49265075         0.00002767 0.00037241       0.00101238\n",
       "19 0.40726973  -0.49285710         0.00001596 0.00042690       0.00058548\n",
       "20 0.40737346  -0.49264222         0.00000921 0.00021488       0.00048176\n",
       "21 0.40761926  -0.49276082         0.00000531 0.00024580       0.00023595\n",
       "22 0.40767914  -0.49263689         0.00000307 0.00012393       0.00017607\n",
       "23 0.40782063  -0.49270528         0.00000177 0.00014149       0.00007152\n",
       "24 0.40785521  -0.49263376         0.00000102 0.00007152       0.00000000"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([1, -10])\n",
    "SteepestDescent().run(x0).astype(np.float64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T06:01:07.950099Z",
     "start_time": "2018-02-05T06:01:07.909953Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.000012   -0.00009894]\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>g</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>195.10867067</td>\n",
       "      <td>nan</td>\n",
       "      <td>0.94373274</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-0.74472188</td>\n",
       "      <td>0.03313233</td>\n",
       "      <td>3.25901614</td>\n",
       "      <td>0.96686767</td>\n",
       "      <td>0.37104555</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-0.12489440</td>\n",
       "      <td>0.22405344</td>\n",
       "      <td>0.75954690</td>\n",
       "      <td>0.61982748</td>\n",
       "      <td>0.24878193</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-0.15513335</td>\n",
       "      <td>0.23488793</td>\n",
       "      <td>0.71990015</td>\n",
       "      <td>0.03023895</td>\n",
       "      <td>0.21854298</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-0.24127452</td>\n",
       "      <td>0.00019728</td>\n",
       "      <td>0.44483196</td>\n",
       "      <td>0.23469065</td>\n",
       "      <td>0.13240182</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>-0.35561465</td>\n",
       "      <td>0.05707686</td>\n",
       "      <td>0.00672077</td>\n",
       "      <td>0.11434013</td>\n",
       "      <td>0.01806169</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>-0.37282006</td>\n",
       "      <td>0.05857054</td>\n",
       "      <td>0.00023516</td>\n",
       "      <td>0.01720541</td>\n",
       "      <td>0.00230327</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>-0.37304367</td>\n",
       "      <td>0.05628978</td>\n",
       "      <td>0.00000889</td>\n",
       "      <td>0.00228076</td>\n",
       "      <td>0.00063267</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>-0.37366873</td>\n",
       "      <td>0.05634828</td>\n",
       "      <td>0.00000030</td>\n",
       "      <td>0.00062507</td>\n",
       "      <td>0.00008102</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>-0.37367633</td>\n",
       "      <td>0.05626726</td>\n",
       "      <td>0.00000001</td>\n",
       "      <td>0.00008102</td>\n",
       "      <td>0.00000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "           x1         x2            g   residual  actual-residual\n",
       "0 -1.00000000 1.00000000 195.10867067        nan       0.94373274\n",
       "1 -0.74472188 0.03313233   3.25901614 0.96686767       0.37104555\n",
       "2 -0.12489440 0.22405344   0.75954690 0.61982748       0.24878193\n",
       "3 -0.15513335 0.23488793   0.71990015 0.03023895       0.21854298\n",
       "4 -0.24127452 0.00019728   0.44483196 0.23469065       0.13240182\n",
       "5 -0.35561465 0.05707686   0.00672077 0.11434013       0.01806169\n",
       "6 -0.37282006 0.05857054   0.00023516 0.01720541       0.00230327\n",
       "7 -0.37304367 0.05628978   0.00000889 0.00228076       0.00063267\n",
       "8 -0.37366873 0.05634828   0.00000030 0.00062507       0.00008102\n",
       "9 -0.37367633 0.05626726   0.00000001 0.00008102       0.00000000"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([-1, 1])\n",
    "SteepestDescent().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": "2018-02-05T06:01:08.190876Z",
     "start_time": "2018-02-05T06:01:07.951771Z"
    },
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class SteepestDescent(SteepestDescentMethod):\n",
    "\n",
    "    def __init__(self):\n",
    "        super(SteepestDescentMethod, 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 g(self, x):\n",
    "        sol = self.f(x)\n",
    "        return sum([e * e for e in sol])\n",
    "\n",
    "    def grad_g(self, x):\n",
    "        return 2 * self.jacobian(x).transpose().dot(self.f(x))\n",
    "\n",
    "    def run(self, x):\n",
    "        df = pd.DataFrame(columns=['x' + str(i + 1) for i in range(len(x))] + ['g', 'residual', 'actual-residual'])\n",
    "\n",
    "        row = len(df)\n",
    "        df.loc[row] = [xe for xe in x] + [self.g(x), np.nan, np.nan]\n",
    "\n",
    "        while True:\n",
    "            prev_x = x\n",
    "            g1 = self.g(x)\n",
    "            z = self.grad_g(x)\n",
    "            z0 = linalg.norm(z, 2)\n",
    "            if z0 == 0.0:\n",
    "                print('Zero gradient')\n",
    "                return x\n",
    "\n",
    "            z /= z0\n",
    "            alpha3 = 1\n",
    "            g3 = self.g(x - alpha3 * z)\n",
    "            while g3 >= g1:\n",
    "                alpha3 /= 2.0\n",
    "                g3 = self.g(x - alpha3 * z)\n",
    "                if alpha3 < TOR / 2.0:\n",
    "                    print('No likely improvement')\n",
    "                    return x\n",
    "\n",
    "            alpha2 = alpha3 / 2.0\n",
    "            g2 = self.g(x - alpha2 * z)\n",
    "\n",
    "            h1 = (g2 - g1) / alpha2\n",
    "            h2 = (g3 - g2) / (alpha3 - alpha2)\n",
    "            h3 = (h2 - h1) / alpha3\n",
    "\n",
    "            alpha0 = (alpha2 - h1 / h3) / 2.0\n",
    "            g0 = self.g(x - alpha0 * z)\n",
    "\n",
    "            alpha = alpha0\n",
    "            g = g0\n",
    "            if g3 < g:\n",
    "                alpha = alpha3\n",
    "                g = g3\n",
    "\n",
    "            x = x - alpha * z\n",
    "            residual = linalg.norm(x - prev_x, np.inf)\n",
    "            row = len(df)\n",
    "            df.loc[row] = [nxe for nxe in x] + [g, residual, np.nan]\n",
    "            if math.fabs(g - g1) < 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][4] = linalg.norm(xk - x, np.inf)\n",
    "\n",
    "        print(self.f(x))\n",
    "        return df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T06:01:08.224207Z",
     "start_time": "2018-02-05T06:01:08.192218Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "No likely improvement\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([ 0.80017438, -0.71487857])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([0, 0])\n",
    "SteepestDescent().run(x0).astype(np.float64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T06:01:08.236858Z",
     "start_time": "2018-02-05T06:01:08.225836Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "No likely improvement\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([ 1.59191552,  3.7219658 ])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([1, 5])\n",
    "SteepestDescent().run(x0).astype(np.float64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T06:01:08.267228Z",
     "start_time": "2018-02-05T06:01:08.238329Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "No likely improvement\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([  31.12282349,  340.09020578])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([-1, 500])\n",
    "SteepestDescent().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": "2018-02-05T06:01:08.466763Z",
     "start_time": "2018-02-05T06:01:08.268628Z"
    },
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class SteepestDescent(SteepestDescentMethod):\n",
    "\n",
    "    def __init__(self):\n",
    "        super(SteepestDescentMethod, 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 g(self, x):\n",
    "        sol = self.f(x)\n",
    "        return sum([e * e for e in sol])\n",
    "\n",
    "    def grad_g(self, x):\n",
    "        return 2 * self.jacobian(x).transpose().dot(self.f(x))\n",
    "\n",
    "    def run(self, x):\n",
    "        df = pd.DataFrame(columns=['x' + str(i + 1) for i in range(len(x))] + ['g', 'residual', 'actual-residual'])\n",
    "\n",
    "        row = len(df)\n",
    "        df.loc[row] = [xe for xe in x] + [self.g(x), np.nan, np.nan]\n",
    "\n",
    "        while True:\n",
    "            prev_x = x\n",
    "            g1 = self.g(x)\n",
    "            z = self.grad_g(x)\n",
    "            z0 = linalg.norm(z, 2)\n",
    "            if z0 == 0.0:\n",
    "                print('Zero gradient')\n",
    "                return x\n",
    "\n",
    "            z /= z0\n",
    "            alpha3 = 1\n",
    "            g3 = self.g(x - alpha3 * z)\n",
    "            while g3 >= g1:\n",
    "                alpha3 /= 2.0\n",
    "                g3 = self.g(x - alpha3 * z)\n",
    "                if alpha3 < TOR / 2.0:\n",
    "                    print('No likely improvement')\n",
    "                    return x\n",
    "\n",
    "            alpha2 = alpha3 / 2.0\n",
    "            g2 = self.g(x - alpha2 * z)\n",
    "\n",
    "            h1 = (g2 - g1) / alpha2\n",
    "            h2 = (g3 - g2) / (alpha3 - alpha2)\n",
    "            h3 = (h2 - h1) / alpha3\n",
    "\n",
    "            alpha0 = (alpha2 - h1 / h3) / 2.0\n",
    "            g0 = self.g(x - alpha0 * z)\n",
    "\n",
    "            alpha = alpha0\n",
    "            g = g0\n",
    "            if g3 < g:\n",
    "                alpha = alpha3\n",
    "                g = g3\n",
    "\n",
    "            x = x - alpha * z\n",
    "            residual = linalg.norm(x - prev_x, np.inf)\n",
    "            row = len(df)\n",
    "            df.loc[row] = [nxe for nxe in x] + [g, residual, np.nan]\n",
    "            if math.fabs(g - g1) < 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][4] = linalg.norm(xk - x, np.inf)\n",
    "\n",
    "        print(self.f(x))\n",
    "        return df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T06:01:08.496036Z",
     "start_time": "2018-02-05T06:01:08.468601Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.00007182  0.00002307]\n"
     ]
    },
    {
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       "      <td>-0.08643053</td>\n",
       "      <td>0.20997065</td>\n",
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       "      <td>0.03035299</td>\n",
       "      <td>0.01167556</td>\n",
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       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-0.09563509</td>\n",
       "      <td>0.22095960</td>\n",
       "      <td>0.00001039</td>\n",
       "      <td>0.01098894</td>\n",
       "      <td>0.00247101</td>\n",
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       "      <th>4</th>\n",
       "      <td>-0.09776686</td>\n",
       "      <td>0.21914366</td>\n",
       "      <td>0.00000028</td>\n",
       "      <td>0.00213177</td>\n",
       "      <td>0.00039189</td>\n",
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       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>-0.09810610</td>\n",
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       "      <td>0.00039189</td>\n",
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       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "           x1         x2          g   residual  actual-residual\n",
       "0  0.00000000 0.00000000 0.06250000        nan       0.21953554\n",
       "1 -0.05607755 0.22431019 0.00122590 0.22431019       0.04202855\n",
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       "5 -0.09810610 0.21953554 0.00000001 0.00039189       0.00000000"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([0, 0])\n",
    "SteepestDescent().run(x0).astype(np.float64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T06:01:08.528191Z",
     "start_time": "2018-02-05T06:01:08.497454Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.00000376 -0.00001162]\n"
     ]
    },
    {
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       "      <td>nan</td>\n",
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       "      <td>0.67405563</td>\n",
       "      <td>0.90223862</td>\n",
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       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.47028574</td>\n",
       "      <td>0.96715963</td>\n",
       "      <td>0.53594490</td>\n",
       "      <td>0.15456997</td>\n",
       "      <td>0.74766865</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-0.30202986</td>\n",
       "      <td>0.33192060</td>\n",
       "      <td>0.14855017</td>\n",
       "      <td>0.77231559</td>\n",
       "      <td>0.20387434</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.19752416</td>\n",
       "      <td>0.35303426</td>\n",
       "      <td>0.00977782</td>\n",
       "      <td>0.49955401</td>\n",
       "      <td>0.29567968</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>0.15314729</td>\n",
       "      <td>0.34371460</td>\n",
       "      <td>0.00492878</td>\n",
       "      <td>0.04437687</td>\n",
       "      <td>0.25130281</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>0.16160171</td>\n",
       "      <td>0.30898780</td>\n",
       "      <td>0.00216867</td>\n",
       "      <td>0.03472680</td>\n",
       "      <td>0.25975723</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>-0.08211628</td>\n",
       "      <td>0.25329631</td>\n",
       "      <td>0.00193674</td>\n",
       "      <td>0.24371799</td>\n",
       "      <td>0.03380532</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>-0.09440997</td>\n",
       "      <td>0.21789093</td>\n",
       "      <td>0.00001539</td>\n",
       "      <td>0.03540538</td>\n",
       "      <td>0.00374555</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>-0.09810260</td>\n",
       "      <td>0.21952376</td>\n",
       "      <td>0.00000001</td>\n",
       "      <td>0.00369262</td>\n",
       "      <td>0.00005292</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>-0.09815552</td>\n",
       "      <td>0.21949098</td>\n",
       "      <td>0.00000000</td>\n",
       "      <td>0.00005292</td>\n",
       "      <td>0.00000000</td>\n",
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       "  </tbody>\n",
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      ],
      "text/plain": [
       "            x1         x2           g   residual  actual-residual\n",
       "0   1.00000000 1.00000000 16.42844444        nan       1.09815552\n",
       "1   0.32594437 1.12172961  1.39940755 0.67405563       0.90223862\n",
       "2   0.47028574 0.96715963  0.53594490 0.15456997       0.74766865\n",
       "3  -0.30202986 0.33192060  0.14855017 0.77231559       0.20387434\n",
       "4   0.19752416 0.35303426  0.00977782 0.49955401       0.29567968\n",
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       "8  -0.09440997 0.21789093  0.00001539 0.03540538       0.00374555\n",
       "9  -0.09810260 0.21952376  0.00000001 0.00369262       0.00005292\n",
       "10 -0.09815552 0.21949098  0.00000000 0.00005292       0.00000000"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([1, 1])\n",
    "SteepestDescent().run(x0).astype(np.float64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T06:01:08.556994Z",
     "start_time": "2018-02-05T06:01:08.529685Z"
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    "scrolled": true
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      "[-0.00005302  0.00006253]\n"
     ]
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       "      <td>0.71280410</td>\n",
       "      <td>0.55268215</td>\n",
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       "      <th>4</th>\n",
       "      <td>-0.17706816</td>\n",
       "      <td>-0.00505512</td>\n",
       "      <td>0.06909515</td>\n",
       "      <td>0.41661199</td>\n",
       "      <td>0.29829051</td>\n",
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       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>0.09920689</td>\n",
       "      <td>0.25108387</td>\n",
       "      <td>0.00044220</td>\n",
       "      <td>0.27627505</td>\n",
       "      <td>0.02201547</td>\n",
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       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>0.12065048</td>\n",
       "      <td>0.27272920</td>\n",
       "      <td>0.00000609</td>\n",
       "      <td>0.02164533</td>\n",
       "      <td>0.00156999</td>\n",
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       "      <td>0.00120908</td>\n",
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       "      <th>8</th>\n",
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       "      <td>0.27115921</td>\n",
       "      <td>0.00000001</td>\n",
       "      <td>0.00043423</td>\n",
       "      <td>0.00000000</td>\n",
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      ],
      "text/plain": [
       "           x1          x2           g   residual  actual-residual\n",
       "0 -1.00000000 -1.00000000 16.85508457        nan       1.27115921\n",
       "1 -0.32655598 -1.12277883  1.85368655 0.67344402       1.39393804\n",
       "2 -0.47326027 -0.98288612  1.03890416 0.14670429       1.25404533\n",
       "3  0.23954383 -0.28152293  0.38067552 0.71280410       0.55268215\n",
       "4 -0.17706816 -0.00505512  0.06909515 0.41661199       0.29829051\n",
       "5  0.09920689  0.25108387  0.00044220 0.27627505       0.02201547\n",
       "6  0.12065048  0.27272920  0.00000609 0.02164533       0.00156999\n",
       "7  0.12165658  0.27152013  0.00000019 0.00120908       0.00043423\n",
       "8  0.12122236  0.27115921  0.00000001 0.00043423       0.00000000"
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     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([-1, -1])\n",
    "SteepestDescent().run(x0).astype(np.float64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-02-05T06:01:44.146039Z",
     "start_time": "2018-02-05T06:01:08.558866Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.00003071 -0.00001606]\n"
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    },
    {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>x1</th>\n",
       "      <th>x2</th>\n",
       "      <th>g</th>\n",
       "      <th>residual</th>\n",
       "      <th>actual-residual</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>100.00000000</td>\n",
       "      <td>-100.00000000</td>\n",
       "      <td>1,600,010,017.80558038</td>\n",
       "      <td>nan</td>\n",
       "      <td>100.21947968</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>71.57253153</td>\n",
       "      <td>-105.68541049</td>\n",
       "      <td>208,632,572.14402342</td>\n",
       "      <td>28.42746847</td>\n",
       "      <td>105.90489017</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>56.31336362</td>\n",
       "      <td>-110.19162532</td>\n",
       "      <td>13,804,240.15237501</td>\n",
       "      <td>15.25916791</td>\n",
       "      <td>110.41110500</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>50.94467543</td>\n",
       "      <td>-112.29234463</td>\n",
       "      <td>147,541.08674494</td>\n",
       "      <td>5.36868819</td>\n",
       "      <td>112.51182431</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>50.33699733</td>\n",
       "      <td>-112.55977060</td>\n",
       "      <td>12,722.52334879</td>\n",
       "      <td>0.60767810</td>\n",
       "      <td>112.77925028</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>50.33806702</td>\n",
       "      <td>-112.55877959</td>\n",
       "      <td>12,721.94341635</td>\n",
       "      <td>0.00106969</td>\n",
       "      <td>112.77825928</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>50.33514049</td>\n",
       "      <td>-112.55562070</td>\n",
       "      <td>12,721.36308737</td>\n",
       "      <td>0.00315889</td>\n",
       "      <td>112.77510038</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>50.33621103</td>\n",
       "      <td>-112.55462893</td>\n",
       "      <td>12,720.78227655</td>\n",
       "      <td>0.00107054</td>\n",
       "      <td>112.77410861</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>50.33328211</td>\n",
       "      <td>-112.55146737</td>\n",
       "      <td>12,720.20106991</td>\n",
       "      <td>0.00316156</td>\n",
       "      <td>112.77094705</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>50.33435351</td>\n",
       "      <td>-112.55047483</td>\n",
       "      <td>12,719.61938192</td>\n",
       "      <td>0.00107140</td>\n",
       "      <td>112.76995451</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>50.33142220</td>\n",
       "      <td>-112.54731060</td>\n",
       "      <td>12,719.03729884</td>\n",
       "      <td>0.00316423</td>\n",
       "      <td>112.76679028</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>50.33249445</td>\n",
       "      <td>-112.54631729</td>\n",
       "      <td>12,718.45473493</td>\n",
       "      <td>0.00107226</td>\n",
       "      <td>112.76579698</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>50.32956076</td>\n",
       "      <td>-112.54315041</td>\n",
       "      <td>12,717.87177668</td>\n",
       "      <td>0.00316689</td>\n",
       "      <td>112.76263009</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>50.33063387</td>\n",
       "      <td>-112.54215634</td>\n",
       "      <td>12,717.28833812</td>\n",
       "      <td>0.00107311</td>\n",
       "      <td>112.76163602</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>50.32769779</td>\n",
       "      <td>-112.53898680</td>\n",
       "      <td>12,716.70450599</td>\n",
       "      <td>0.00316954</td>\n",
       "      <td>112.75846648</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>50.32877175</td>\n",
       "      <td>-112.53799197</td>\n",
       "      <td>12,716.12019408</td>\n",
       "      <td>0.00107396</td>\n",
       "      <td>112.75747165</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>50.32583331</td>\n",
       "      <td>-112.53481978</td>\n",
       "      <td>12,715.53548937</td>\n",
       "      <td>0.00317219</td>\n",
       "      <td>112.75429946</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>50.32690811</td>\n",
       "      <td>-112.53382419</td>\n",
       "      <td>12,714.95030544</td>\n",
       "      <td>0.00107481</td>\n",
       "      <td>112.75330387</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>50.32396730</td>\n",
       "      <td>-112.53064936</td>\n",
       "      <td>12,714.36472949</td>\n",
       "      <td>0.00317483</td>\n",
       "      <td>112.75012904</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>50.32504295</td>\n",
       "      <td>-112.52965301</td>\n",
       "      <td>12,713.77867489</td>\n",
       "      <td>0.00107566</td>\n",
       "      <td>112.74913269</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>50.32209977</td>\n",
       "      <td>-112.52647554</td>\n",
       "      <td>12,713.19222907</td>\n",
       "      <td>0.00317747</td>\n",
       "      <td>112.74595522</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>50.32317628</td>\n",
       "      <td>-112.52547844</td>\n",
       "      <td>12,712.60530519</td>\n",
       "      <td>0.00107650</td>\n",
       "      <td>112.74495812</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>50.32023074</td>\n",
       "      <td>-112.52229833</td>\n",
       "      <td>12,712.01799088</td>\n",
       "      <td>0.00318010</td>\n",
       "      <td>112.74177802</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>50.32130808</td>\n",
       "      <td>-112.52130048</td>\n",
       "      <td>12,711.43019912</td>\n",
       "      <td>0.00107735</td>\n",
       "      <td>112.74078016</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>50.31836019</td>\n",
       "      <td>-112.51811775</td>\n",
       "      <td>12,710.84201776</td>\n",
       "      <td>0.00318273</td>\n",
       "      <td>112.73759743</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>50.31943838</td>\n",
       "      <td>-112.51711913</td>\n",
       "      <td>12,710.25335955</td>\n",
       "      <td>0.00107819</td>\n",
       "      <td>112.73659882</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>50.31648813</td>\n",
       "      <td>-112.51393378</td>\n",
       "      <td>12,709.66431258</td>\n",
       "      <td>0.00318535</td>\n",
       "      <td>112.73341347</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>50.31756716</td>\n",
       "      <td>-112.51293442</td>\n",
       "      <td>12,709.07478938</td>\n",
       "      <td>0.00107903</td>\n",
       "      <td>112.73241410</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>50.31461458</td>\n",
       "      <td>-112.50974645</td>\n",
       "      <td>12,708.48487828</td>\n",
       "      <td>0.00318797</td>\n",
       "      <td>112.72922614</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>50.31569444</td>\n",
       "      <td>-112.50874634</td>\n",
       "      <td>12,707.89449158</td>\n",
       "      <td>0.00107987</td>\n",
       "      <td>112.72822602</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19393</th>\n",
       "      <td>1.24381644</td>\n",
       "      <td>-2.75159190</td>\n",
       "      <td>7.62864601</td>\n",
       "      <td>0.02857456</td>\n",
       "      <td>2.97107158</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19394</th>\n",
       "      <td>1.19567273</td>\n",
       "      <td>-2.71772754</td>\n",
       "      <td>7.46840975</td>\n",
       "      <td>0.04814372</td>\n",
       "      <td>2.93720723</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19395</th>\n",
       "      <td>1.21578685</td>\n",
       "      <td>-2.68877165</td>\n",
       "      <td>7.30722951</td>\n",
       "      <td>0.02895590</td>\n",
       "      <td>2.90825133</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19396</th>\n",
       "      <td>1.16723242</td>\n",
       "      <td>-2.65477448</td>\n",
       "      <td>7.15102363</td>\n",
       "      <td>0.04855443</td>\n",
       "      <td>2.87425416</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19397</th>\n",
       "      <td>1.18753077</td>\n",
       "      <td>-2.62539071</td>\n",
       "      <td>6.99380362</td>\n",
       "      <td>0.02938377</td>\n",
       "      <td>2.84487039</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19398</th>\n",
       "      <td>1.13854904</td>\n",
       "      <td>-2.59128942</td>\n",
       "      <td>6.84152504</td>\n",
       "      <td>0.04898173</td>\n",
       "      <td>2.81076910</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19399</th>\n",
       "      <td>1.15904121</td>\n",
       "      <td>-2.56142401</td>\n",
       "      <td>6.68814671</td>\n",
       "      <td>0.02986541</td>\n",
       "      <td>2.78090369</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19400</th>\n",
       "      <td>1.10734003</td>\n",
       "      <td>-2.52567924</td>\n",
       "      <td>6.53980348</td>\n",
       "      <td>0.05170119</td>\n",
       "      <td>2.74515892</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19401</th>\n",
       "      <td>1.12912278</td>\n",
       "      <td>-2.49532972</td>\n",
       "      <td>6.38112454</td>\n",
       "      <td>0.03034952</td>\n",
       "      <td>2.71480940</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19402</th>\n",
       "      <td>1.07386215</td>\n",
       "      <td>-2.45533409</td>\n",
       "      <td>6.22783422</td>\n",
       "      <td>0.05526063</td>\n",
       "      <td>2.67481377</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19403</th>\n",
       "      <td>1.09705882</td>\n",
       "      <td>-2.42430453</td>\n",
       "      <td>6.06332600</td>\n",
       "      <td>0.03102956</td>\n",
       "      <td>2.64378422</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19404</th>\n",
       "      <td>1.03786258</td>\n",
       "      <td>-2.37965893</td>\n",
       "      <td>5.90466294</td>\n",
       "      <td>0.05919625</td>\n",
       "      <td>2.59913861</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19405</th>\n",
       "      <td>1.06259171</td>\n",
       "      <td>-2.34773022</td>\n",
       "      <td>5.73408368</td>\n",
       "      <td>0.03192871</td>\n",
       "      <td>2.56720990</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19406</th>\n",
       "      <td>0.99914399</td>\n",
       "      <td>-2.29815663</td>\n",
       "      <td>5.56997061</td>\n",
       "      <td>0.06344772</td>\n",
       "      <td>2.51763631</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19407</th>\n",
       "      <td>1.02551124</td>\n",
       "      <td>-2.26508304</td>\n",
       "      <td>5.39347532</td>\n",
       "      <td>0.03307358</td>\n",
       "      <td>2.48456273</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19408</th>\n",
       "      <td>0.95760293</td>\n",
       "      <td>-2.21051138</td>\n",
       "      <td>5.22431001</td>\n",
       "      <td>0.06790831</td>\n",
       "      <td>2.42999106</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19409</th>\n",
       "      <td>0.98569255</td>\n",
       "      <td>-2.17601487</td>\n",
       "      <td>5.04253638</td>\n",
       "      <td>0.03449651</td>\n",
       "      <td>2.39549455</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19410</th>\n",
       "      <td>0.91326450</td>\n",
       "      <td>-2.11667035</td>\n",
       "      <td>4.86927410</td>\n",
       "      <td>0.07242805</td>\n",
       "      <td>2.33615003</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19411</th>\n",
       "      <td>0.94313273</td>\n",
       "      <td>-2.08043044</td>\n",
       "      <td>4.68336147</td>\n",
       "      <td>0.03623991</td>\n",
       "      <td>2.29991012</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19412</th>\n",
       "      <td>0.86629740</td>\n",
       "      <td>-2.01689096</td>\n",
       "      <td>4.50749328</td>\n",
       "      <td>0.07683533</td>\n",
       "      <td>2.23637064</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19413</th>\n",
       "      <td>0.89797144</td>\n",
       "      <td>-1.97852707</td>\n",
       "      <td>4.31899167</td>\n",
       "      <td>0.03836390</td>\n",
       "      <td>2.19800675</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19414</th>\n",
       "      <td>0.81699241</td>\n",
       "      <td>-1.91171583</td>\n",
       "      <td>4.14237140</td>\n",
       "      <td>0.08097903</td>\n",
       "      <td>2.13119551</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19415</th>\n",
       "      <td>0.85047863</td>\n",
       "      <td>-1.87075694</td>\n",
       "      <td>3.95302397</td>\n",
       "      <td>0.04095889</td>\n",
       "      <td>2.09023662</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19416</th>\n",
       "      <td>0.76569685</td>\n",
       "      <td>-1.80185285</td>\n",
       "      <td>3.77755232</td>\n",
       "      <td>0.08478178</td>\n",
       "      <td>2.02133253</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19417</th>\n",
       "      <td>0.80100005</td>\n",
       "      <td>-1.75768720</td>\n",
       "      <td>3.58898421</td>\n",
       "      <td>0.04416564</td>\n",
       "      <td>1.97716688</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19418</th>\n",
       "      <td>0.01618746</td>\n",
       "      <td>-1.13795403</td>\n",
       "      <td>3.22814608</td>\n",
       "      <td>0.78481259</td>\n",
       "      <td>1.35743371</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19419</th>\n",
       "      <td>-0.00997232</td>\n",
       "      <td>-0.13829626</td>\n",
       "      <td>0.14735508</td>\n",
       "      <td>0.99965777</td>\n",
       "      <td>0.35777594</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19420</th>\n",
       "      <td>-0.10203206</td>\n",
       "      <td>0.21536377</td>\n",
       "      <td>0.00004370</td>\n",
       "      <td>0.35366003</td>\n",
       "      <td>0.00411591</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19421</th>\n",
       "      <td>-0.09846021</td>\n",
       "      <td>0.21971613</td>\n",
       "      <td>0.00000013</td>\n",
       "      <td>0.00435236</td>\n",
       "      <td>0.00027463</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19422</th>\n",
       "      <td>-0.09818558</td>\n",
       "      <td>0.21947968</td>\n",
       "      <td>0.00000000</td>\n",
       "      <td>0.00027463</td>\n",
       "      <td>0.00000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>19423 rows × 5 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                x1            x2                      g    residual  \\\n",
       "0     100.00000000 -100.00000000 1,600,010,017.80558038         nan   \n",
       "1      71.57253153 -105.68541049   208,632,572.14402342 28.42746847   \n",
       "2      56.31336362 -110.19162532    13,804,240.15237501 15.25916791   \n",
       "3      50.94467543 -112.29234463       147,541.08674494  5.36868819   \n",
       "4      50.33699733 -112.55977060        12,722.52334879  0.60767810   \n",
       "5      50.33806702 -112.55877959        12,721.94341635  0.00106969   \n",
       "6      50.33514049 -112.55562070        12,721.36308737  0.00315889   \n",
       "7      50.33621103 -112.55462893        12,720.78227655  0.00107054   \n",
       "8      50.33328211 -112.55146737        12,720.20106991  0.00316156   \n",
       "9      50.33435351 -112.55047483        12,719.61938192  0.00107140   \n",
       "10     50.33142220 -112.54731060        12,719.03729884  0.00316423   \n",
       "11     50.33249445 -112.54631729        12,718.45473493  0.00107226   \n",
       "12     50.32956076 -112.54315041        12,717.87177668  0.00316689   \n",
       "13     50.33063387 -112.54215634        12,717.28833812  0.00107311   \n",
       "14     50.32769779 -112.53898680        12,716.70450599  0.00316954   \n",
       "15     50.32877175 -112.53799197        12,716.12019408  0.00107396   \n",
       "16     50.32583331 -112.53481978        12,715.53548937  0.00317219   \n",
       "17     50.32690811 -112.53382419        12,714.95030544  0.00107481   \n",
       "18     50.32396730 -112.53064936        12,714.36472949  0.00317483   \n",
       "19     50.32504295 -112.52965301        12,713.77867489  0.00107566   \n",
       "20     50.32209977 -112.52647554        12,713.19222907  0.00317747   \n",
       "21     50.32317628 -112.52547844        12,712.60530519  0.00107650   \n",
       "22     50.32023074 -112.52229833        12,712.01799088  0.00318010   \n",
       "23     50.32130808 -112.52130048        12,711.43019912  0.00107735   \n",
       "24     50.31836019 -112.51811775        12,710.84201776  0.00318273   \n",
       "25     50.31943838 -112.51711913        12,710.25335955  0.00107819   \n",
       "26     50.31648813 -112.51393378        12,709.66431258  0.00318535   \n",
       "27     50.31756716 -112.51293442        12,709.07478938  0.00107903   \n",
       "28     50.31461458 -112.50974645        12,708.48487828  0.00318797   \n",
       "29     50.31569444 -112.50874634        12,707.89449158  0.00107987   \n",
       "...            ...           ...                    ...         ...   \n",
       "19393   1.24381644   -2.75159190             7.62864601  0.02857456   \n",
       "19394   1.19567273   -2.71772754             7.46840975  0.04814372   \n",
       "19395   1.21578685   -2.68877165             7.30722951  0.02895590   \n",
       "19396   1.16723242   -2.65477448             7.15102363  0.04855443   \n",
       "19397   1.18753077   -2.62539071             6.99380362  0.02938377   \n",
       "19398   1.13854904   -2.59128942             6.84152504  0.04898173   \n",
       "19399   1.15904121   -2.56142401             6.68814671  0.02986541   \n",
       "19400   1.10734003   -2.52567924             6.53980348  0.05170119   \n",
       "19401   1.12912278   -2.49532972             6.38112454  0.03034952   \n",
       "19402   1.07386215   -2.45533409             6.22783422  0.05526063   \n",
       "19403   1.09705882   -2.42430453             6.06332600  0.03102956   \n",
       "19404   1.03786258   -2.37965893             5.90466294  0.05919625   \n",
       "19405   1.06259171   -2.34773022             5.73408368  0.03192871   \n",
       "19406   0.99914399   -2.29815663             5.56997061  0.06344772   \n",
       "19407   1.02551124   -2.26508304             5.39347532  0.03307358   \n",
       "19408   0.95760293   -2.21051138             5.22431001  0.06790831   \n",
       "19409   0.98569255   -2.17601487             5.04253638  0.03449651   \n",
       "19410   0.91326450   -2.11667035             4.86927410  0.07242805   \n",
       "19411   0.94313273   -2.08043044             4.68336147  0.03623991   \n",
       "19412   0.86629740   -2.01689096             4.50749328  0.07683533   \n",
       "19413   0.89797144   -1.97852707             4.31899167  0.03836390   \n",
       "19414   0.81699241   -1.91171583             4.14237140  0.08097903   \n",
       "19415   0.85047863   -1.87075694             3.95302397  0.04095889   \n",
       "19416   0.76569685   -1.80185285             3.77755232  0.08478178   \n",
       "19417   0.80100005   -1.75768720             3.58898421  0.04416564   \n",
       "19418   0.01618746   -1.13795403             3.22814608  0.78481259   \n",
       "19419  -0.00997232   -0.13829626             0.14735508  0.99965777   \n",
       "19420  -0.10203206    0.21536377             0.00004370  0.35366003   \n",
       "19421  -0.09846021    0.21971613             0.00000013  0.00435236   \n",
       "19422  -0.09818558    0.21947968             0.00000000  0.00027463   \n",
       "\n",
       "       actual-residual  \n",
       "0         100.21947968  \n",
       "1         105.90489017  \n",
       "2         110.41110500  \n",
       "3         112.51182431  \n",
       "4         112.77925028  \n",
       "5         112.77825928  \n",
       "6         112.77510038  \n",
       "7         112.77410861  \n",
       "8         112.77094705  \n",
       "9         112.76995451  \n",
       "10        112.76679028  \n",
       "11        112.76579698  \n",
       "12        112.76263009  \n",
       "13        112.76163602  \n",
       "14        112.75846648  \n",
       "15        112.75747165  \n",
       "16        112.75429946  \n",
       "17        112.75330387  \n",
       "18        112.75012904  \n",
       "19        112.74913269  \n",
       "20        112.74595522  \n",
       "21        112.74495812  \n",
       "22        112.74177802  \n",
       "23        112.74078016  \n",
       "24        112.73759743  \n",
       "25        112.73659882  \n",
       "26        112.73341347  \n",
       "27        112.73241410  \n",
       "28        112.72922614  \n",
       "29        112.72822602  \n",
       "...                ...  \n",
       "19393       2.97107158  \n",
       "19394       2.93720723  \n",
       "19395       2.90825133  \n",
       "19396       2.87425416  \n",
       "19397       2.84487039  \n",
       "19398       2.81076910  \n",
       "19399       2.78090369  \n",
       "19400       2.74515892  \n",
       "19401       2.71480940  \n",
       "19402       2.67481377  \n",
       "19403       2.64378422  \n",
       "19404       2.59913861  \n",
       "19405       2.56720990  \n",
       "19406       2.51763631  \n",
       "19407       2.48456273  \n",
       "19408       2.42999106  \n",
       "19409       2.39549455  \n",
       "19410       2.33615003  \n",
       "19411       2.29991012  \n",
       "19412       2.23637064  \n",
       "19413       2.19800675  \n",
       "19414       2.13119551  \n",
       "19415       2.09023662  \n",
       "19416       2.02133253  \n",
       "19417       1.97716688  \n",
       "19418       1.35743371  \n",
       "19419       0.35777594  \n",
       "19420       0.00411591  \n",
       "19421       0.00027463  \n",
       "19422       0.00000000  \n",
       "\n",
       "[19423 rows x 5 columns]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.array([100, -100])\n",
    "SteepestDescent().run(x0).astype(np.float64)"
   ]
  }
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