Section 2.6 - Zeros of polynomials_Fixed.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "# 2.6 Zeros of polynomials and Horner's method \n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A polynomial of degree $n$ has the form\n",
    "$$P(x)=P_n(x)=a_nx^n+a_{n-1}x^{n-1}+\\cdots +a_1x+a_0$$\n",
    "where the $a_i$’s, called the coefficients of $P_n$ and $a_n\\neq 0$. \n",
    "\n",
    "The problem is as follows: Find $x$ such that\n",
    "\n",
    "$$P(x) = 0.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> ### Fundamental Theorem of Algebra\n",
    "> If $P(x)$ is a polynomial of degree $n \\ge 1$ with real or complex coefficients, \n",
    ">then $P(x) = 0$ has at least one ( possibly complex) root."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example: \n",
    "Determine all the zeros of the polynomial $P(x) = x^3 - 5x^2 + 17x - 13$.\n",
    "\n",
    "Note that $P(1)=0.$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "P (generic function with 1 method)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P(x) = x^3 - 5x^2 + 17x - 13"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Zs69Qb6/YJ0macW1PNbd67C2/UfDYCE9GRoby8/OVlpamgIAABQQEaOnSpXr55ZcVEBCguLg4VVVVqaioqM798vPz3aM68fHxx43kFBUVqbq6+riRHwAAUKuy2qkHP9ogw5BG922jCzq3NDuS6TxWeC666CJt3LhR69atc1/69u2rm266yf19YGCglixZ4r5PTk6ONm3apIEDB0qSBgwYoE2bNiknJ8e9z+LFi2W1WpWWluap6AAANGovfrVTew6VKTbMqj80oeUjfonHxrfCwsLUo0ePOttCQ0MVHR3t3n777bfr/vvvV3R0tKKiovTAAw+oZ8+e7gnO6enpSklJ0bhx4zRz5kwVFhbqgQce0IQJExQeHu6p6AAANFrrs4o159vdkqQnr+mpiOBAkxN5B1M/0HvhhRcUEBCg0aNHq6KiQhdddJH+/ve/y9+/9pA5f39//ec//9Fdd92lQYMGKTg4WGPGjNGzzz5rZmwAALySo6b2oyyXIV3ZK0EXpzD94yiLYRiG2SE8zW63KyIiQjabjZEhAIDPevqLbZr9v92KDg3S4vsuUHTzxn0AT32+f7OWFgAAPiBjf6FeX/rTR1mNvezUNwoPAACNXJmjRlP/sV4uQ7q2T2td0iPe7Eheh8IDAEAjN2PRVu0/XK6EiGZ67IruZsfxShQeAAAasaU7CvTeykxJ0sxRvTgq6yQoPAAANFK28mo9+NF6SdL4ge01qFOMyYm8F4UHAIBG6k8LNynP7lCHmFA9dEnTXivr11B4AABohP69IVufrsuWv59Fz1/fW8FB/mZH8moUHgAAGpkDReWa9slGSdJdQzuqd2ILkxN5PwoPAACNiNNl6L7561RSWaPeiS00+aJksyM1ChQeAAAakb9+s0ur9xWpuTVAL9+QqkB/3spPBa8SAACNRMb+Qr309U5J0p+v7q620SEmJ2o8KDwAADQC9spq3TtvnZwuQ1f3TtA1qW3MjtSoUHgAAPByhmHo0QWbdKCoQolRwXri6h5mR2p0KDwAAHi5BWsPauH62kPQX7w+VeHNOJvy6aLwAADgxXYXlOqP/9okSZpyUbLS2kWanKhxovAAAOClKqudmvT+jyqrcqp/UpTuGtbJ7EiNFoUHAAAv9fjCzdqWW6KY5kF65cZU+ftZzI7UaFF4AADwQgvWHtC81VmyWKSXbkhVbHgzsyM1ahQeAAC8zK78Ej3ySe28nckXJrMKej2g8AAA4EXKq2p01/s/qqLaqUGdolk6op5QeAAA8CJ/+nSzduSVqmWYVS9ez7yd+kLhAQDAS/xjdZY+yjggP4v08g2pahlmNTuSz6DwAADgBdZnFevRT2vn7Uy9uLMGdIw2OZFvofAAAGCyQ6UO/e69DFXVuDS8W5zuGsr5duobhQcAABPVOF26+4MflW2rVIeYUD1/fS/5MW+n3lF4AAAw0VOLtmnlnkKFBvnr9XFprJPlIRQeAABMsnB9tt5YtleS9OyoXkqOCzM5ke+i8AAAYIKtOXY99NEGSdLvhnbUpT1bmZzIt1F4AABoYEVlVbrz3QxVVDs1ODlGD6R3MTuSz6PwAADQgKpqXJr4XoYyC8uVGBWsl2/g5IINgcIDAEADMQxDjy3cpB/2Fqq5NUBv3tJPkaFBZsdqEig8AAA0kLe+36cPV2XJzyK9cmOqOjNJucFQeAAAaADfbM/XX/6zRZL0yGXdNKxrrMmJmhYKDwAAHrYrv0STP1grlyFd3zdRt5+fZHakJofCAwCABxWVVen2t9eoxFGjc5Oi9Oere8hiYZJyQ6PwAADgIZXVTt3xzhrtP1x7RNZrY9MUFMBbrxl41QEA8ACXy9DUf6xTxv4ihTcL0N9u6acojsgyDYUHAAAPmP75Vn2+MVdB/n6ac3Nflo0wGYUHAIB69tb3e91rZM0cdY7O6xBtciJQeAAAqEdfbMrVE/+uPfz8oUu66qrerU1OBInCAwBAvfkxs0j3zlsrw5Bu6t9WE4d0MDsSjqDwAABQD3bll+i2v6+Wo8alC7vG6v+u7M7h516EwgMAwFk6UFSusW+sUnF5tXq1idArN6YqwJ+3WG/CnwYAAGfhUKlDN7+5Srn2SnWKba6/33quQq0BZsfCz1B4AAA4QyWV1Rr/1irtOVSm1i2C9e7t57L6uZei8AAAcAYqq52a8M4abTpoV3RokN69/Vy1igg2OxZOgsIDAMBpqnG6NPnDtVq5p1DNrQF6+7Zz1aFlc7Nj4RdQeAAAOA1Ol6Gp/1ivxVvyFBTgp7k391WP1hFmx8KvoPAAAHCKXC5DD328QQvXZyvAz6JXx/TRgI6cRbkxoPAAAHAKXC5Df/jXRn2UcUD+fha9cmOqhqfEmR0Lp4jCAwDArzAMQ49/tlkfrsqSn0V6fnQvXdqzldmxcBooPAAA/ALDMPTkf7bqnRX7ZbFIz1zXi/WxGiEKDwAAJ2EYhp7+Yrt75fMZ1/TUdWltTE6FM8GpIAEAOAHDMPTnf2/V376vLTtPXNVdN5zb1uRUOFMUHgAAfsblMvSnhZv03spMSdKfr+qucQPamxsKZ4XCAwDAMZwuQ498slHz12TJYpGeuranru/HyE5j59E5PDNmzFC/fv0UFham2NhYXX311dq+fXudfRwOh+655x7FxMQoNDRUV155pQ4cOFBnn8zMTF1xxRUKDQ1VTEyMJk+erKqqKk9GBwA0QTVOl37/z/Wav6b2aKznRvWi7PgIjxaepUuXatKkSVq5cqWWLFmimpoapaenq6yszL3PlClTtGDBAs2bN0/Lli1TaWmpLr/8cjmdTkmS0+nUyJEjVVZWpmXLlmnevHn6+OOPdf/993syOgCgiamqcene+ev0ydqD8vez6OUbU3VtHyYo+wqLYRhGQz1ZQUGBYmNjtXTpUl1wwQWy2Wxq2bKl3n33XV1//fWSpOzsbCUmJurzzz/XiBEjtGjRIl1++eXKyspSQkKCJGnevHkaP3688vPzFR4eftzzOBwOORwO93W73a7ExETZbLYT7g8AaNrKq2r0u/d+1NIdBQr0t2jWmD4a0T3e7FhNnt1uV0RERL28fzfoYek2m02SFBUVJUnKyMhQdXW10tPT3fskJCSoR48eWr58uSRpxYoV6tGjh7vsSNKIESPkcDiUkZFxwueZMWOGIiIi3JfExERP/UgAgEauuLxKY9/4QUt3FKhZoJ/m3NyXsuODGqzwGIahqVOn6vzzz1ePHj0kSbm5uQoKClJkZGSdfePi4pSbm+veJy6u7qm7IyMjFRQU5N7n56ZNmyabzea+ZGVleeAnAgA0drm2So1+fYV+zCxWeLMAvX9Hfw3rEmt2LHhAgx2ldffdd2vDhg1atmzZr+5rGIYsFov7+rHfn2yfY1mtVlmt1jMPCwDweXsKSjXuzVU6WFyhuHCr3rmtv7rEh5kdCx7SICM899xzjxYuXKhvvvlGbdr8NAEsPj5eVVVVKioqqrN/fn6+e1QnPj7+uJGcoqIiVVdXHzfyAwDAqdh4wKZRr63QweIKtY8O0UcTB1J2fJxHC49hGLr77rv1ySef6L///a+SkpLq3J6WlqbAwEAtWbLEvS0nJ0ebNm3SwIEDJUkDBgzQpk2blJOT495n8eLFslqtSktL82R8AIAP+nprnq6fs0KHy6rUPSFcH/1uoBKjQsyOBQ/z6EdakyZN0gcffKBPP/1UYWFh7pGaiIgIBQcHKyIiQrfffrvuv/9+RUdHKyoqSg888IB69uyp4cOHS5LS09OVkpKicePGaebMmSosLNQDDzygCRMmcMQVAOC0vLtinx5buFkuQzq/U4xmj+2jsGaBZsdCA/DoYeknm2Pz1ltvafz48ZKkyspK/f73v9cHH3ygiooKXXTRRXr11VfrHFmVmZmpu+66S//9738VHBysMWPG6Nlnnz3leTr1eVgbAKDxcbkMzVi0VXO/q10Xa3TfNnrymp4K9GcNbW9Wn+/fDXoeHrNQeACg6aqsduq++eu0aFPtpwwPpHfWpGGdTvqfcniP+nz/Zi0tAIDPyi+p1J3vZmhtZrGC/P00c9Q5uqp3a7NjwQQUHgCAT9pwoFh3vpuhHFulIoIDNWdcmvp3iDY7FkxC4QEA+JxP1x3Ugx9tkKPGpY4tQzX35r7q0LK52bFgIgoPAMBnOF2GZn65Xa8t3S1JurBrrF68obfCORKryaPwAAB8gr2yWvd+uFbfbC+QJP1uaEc9kN5F/n5MTgaFBwDgA7Zk23XX+xnad7hc1gA/PXMdk5NRF4UHANCo/XNNlh791yY5alxq3SJYr41NU882EWbHgpeh8AAAGqXKaqceX7hZ81ZnSZKGdmmpF0b3VmRokMnJ4I0oPACARmf/4TLd9f6P2pxtl8UiTR1eezJBP+br4CQoPACARuWz9dl6ZMFGlVTWKCo0SC/fkKrzk2PMjgUvR+EBADQKZY4aPbZwsz7KOCBJSmsXqVljUtUqItjkZGgMKDwAAK+38YBNk+et1d5DZbJYpLuHddK9FyUrgMU/cYooPAAAr+VyGXpj2R7N/HK7qp2GWkU00wvX99Z5LBGB00ThAQB4pazCcv3+o/VauadQknRJ93g99ZueahHCUVg4fRQeAIBXMQxDH6zK1PT/bFVZlVPBgf764+UpuvHcRFksHIWFM0PhAQB4jeziCj308QZ9t/OQJKlf+0jNvK6X2seEmpwMjR2FBwBgOsMw9FHGAT3x2RaVOGpkDfDT70d00a2DklgLC/WCwgMAMFXm4XI9+ukmfbujdtHP3okt9NzoXurYsrnJyeBLKDwAAFNUO12a+90evfTVTjlqXAoK8NOU4cn67eAOHG6OekfhAQA0uIz9RXrkk43anlciSRrYMVp/ubqHOjCqAw+h8AAAGoytvFozF2/T+z9kyjCkyJBAPToyRdf2ac0RWPAoCg8AwOOcLkPzVmfqucU7VFhWJUm6Lq2NHrmsm6JY3RwNgMIDAPCoVXsL9fjCzdqSY5ckdYptrieu6q6BHVnwEw2HwgMA8Ijs4grNWLRNn63PliSFNQvQfcM7a9yAdgpkUjIaGIUHAFCvSiqrNefbPZr73R5VVrtksUg39GurB9I7K7q51ex4aKIoPACAeuGocer9lZl65b87VVReLan2TMmPXdFdPVpHmJwOTR2FBwBwVlwuQwvXZ+vZxdt1oKhCktShZageHNFVI7rHcfQVvAKFBwBwRgzD0Fdb8/XCkh3uCcmxYVbdd3FnjUprw8kD4VUoPACA02IYhpZsydNLX+/U5uzaohNmDdDEoR1126AkBQf5m5wQOB6FBwBwSgzD0OIteXr5mKITGuSvWwa214TBHRTJ+XTgxSg8AIBfVON06fNNuZr9v93amvNT0Rk/qL3uOJ+ig8aBwgMAOKHyqhrNX52lN5ftdU9GDg3y162DknT7+UkUHTQqFB4AQB2HSh16e/k+vbtyv4qPHF4eFRqk8QPba9x57Sg6aJQoPAAASdKGA8V6e/l+fbYhW1U1LklSu+gQTRjcQdeltVGzQCYjo/Gi8ABAE1ZZ7dTnG3P09or9Wp9V7N7eq02EJg7pqPTu8fL34zw6aPwoPADQBGUVluvDVZmatzrLvXp5kL+fRp7TSuMGtFNqYgtOGAifQuEBgCaistqpLzfn6h9rsvT9rsPu7a0immnsee10fb9ExbDWFXwUhQcAfJhhGNpwwKZ/rMnSwvXZKqmscd92fqcYjT2vnYZ3i+WsyPB5FB4A8EGZh8v12YZsfbruoHbklbq3t24RrFF92+g3fdooMSrExIRAw6LwAICPyLNX6t8bcrRwfXadCcjWAD9d2iNeo/sm6rwO0fJjEjKaIAoPADRiObYKLdmSp8835uiHvYUyjNrtfhZpYMcYXdkrQSN6xCsiONDcoIDJKDwA0IgYhqEdeaVasiVXi7fkacMBW53b09pF6speCbq0Z7xiw5qZlBLwPhQeAPByjhqnMvYV6Zvt+Vq8JU/7D5e7b7NYpLS2kUrvHqfLerZSm0jm5QAnQuEBAC9jGIb2HS7XtzsK9O2OAq3Yc1jlVU737UEBfhrcKUbp3eN0Ydc4tQzjUHLg11B4AMALHCp16Ic9hVqx55C+3XFImYXldW6PaW7VBZ1jdHG3OF3QuaUB+qUsAAAgAElEQVRCrfz6Bk4H/2IAwAT59kqt3FuoH/Yc1g97C7Urv7TO7YH+FvVtF6ULOrfUkM4t1a1VGGc+Bs4ChQcAPMzpMrQjr0Trsoq1LrNYq/cVas+hsuP26xofpv5JURqc3FIDOkYzigPUI/41AUA9y7dXam1WsdZmFmtdVpE2HrCp7Jg5OFLtZOOUVuHqnxSt/h2idG77KEWGBpmUGPB9FB4AOEMul6H9heXakm3X1hy7tuTYtSXbrlx75XH7NrcG6Jw2EUpt20J92kaqb/sozo0DNCAKDwCcgqKyKu0qKNXOvFJ3udmWYz9u5EaqPelf57gw9U5sodS2LdQ7MVKdYpvLnzMcA6ah8ADAEYZhKNtWqV35pdqdX6pdBaXu7w+XVZ3wPtYAP3WND1NKQri6tQpXSqtwdW0VrubMvwG8Cv8iATQp1U6XsosrtP9wufYXlivzcJn2Hy5XZmHtpfwEIzZHtW4RrE6xzdW1VZhSjpSbpJhQVhoHGgEKDwCfUlntVK6tUtm2CuUUVyrHVqGDxZXKKizX/sIyZRdXyukyTnr/AD+L2kWHqFNsc/clOTZMHVqGKiSIX5lAY8W/XgCNRkWVU/kllcovcSjHVqmc4grl2CqVfeRrjq1Ch0pP/NHTsawBfmobFaJ20SFqGxVa+zU6RG2jai+BjNgAPofCAzRFTqf03XdSTo7UqpU0eLDk729KFJfLUFF5lfJLHCoocRzztdJ9/dCRr6WOmlN6zGaBfkqICFZCi2C1imimVi2ClRgZrHbRoWobFaLYMKv8mEAMNCmNpvC8+uqrmjlzpnJyctS9e3e9+OKLGjx4sNmxgMbnk0+ke++VDhz4aVubNtJLL0nXXlsvT2EYhmwV1TpU6lBBSZUOlTp0uNShQ6VVR7Y5VFDqUL7doUOlDtX8wkdMP9cs0E8tw6yKD292pNAEK6FFM/fXhIhgtQgJ5KzEAOpoFIVn/vz5mjJlil599VUNGjRIr7/+ui699FJt2bJFbdu2NTse0Hh88ol03XWS8bOCcfBg7faPPjpp6XG6DBWW1RaW2gJzpLyUOnSopO72w2UOVTtPvcRIUlRokGLDrGp5zCU2rNmRr1b31+bWAMoMgNNmMYyf/+bzPv3791efPn00e/Zs97Zu3brp6quv1owZM371/na7XREREcrOzlZ4eLgnowLey+lUcEqKLAcP6kR1wbBYVBEbr7+/vUR5FTU6VFqlwrJqHS6r0uGyKhWVV+s0BmIkSeHNAhQVEqjo5kGKDg1SdGhg7dfmQWp55BJz5DbmzQD4ObvdroSEBNlstrN+//b6EZ6qqiplZGTo4YcfrrM9PT1dy5cvP+F9HA6HHA6H+7rdbpckJSQkeC4o4G0sfvJvHqWA8JbyD4vWEJdLnx88ePLdDUMheTn69u2FWtn2nBPuYxguuSpK5CwrlrOsWK7yIvf3znKbXGVFcpYfvV4sOU9tzg0AeJrXF55Dhw7J6XQqLi6uzva4uDjl5uae8D4zZszQ//3f/zVEPMA0lgCr/MNbKiAiVgHhLWuLTfiR7yNi5R8WI4vfTxORO21ZKu048X8SjhW+7XvZcnfVlpejZaasWM7yIrnK7ZLh8uSPBQAe4fWF56iff2ZvGMZJP8efNm2apk6d6r5ut9uVmJjIR1podGwV1cosqlBm4TGXI9cLy6t/9f4BfhbFh1sVF25Ve0uS9NmvP+dLT07SCxdcUA/pAeDsHP1Iqz54feGJiYmRv7//caM5+fn5x436HGW1WmW1Wo/bHhoaqtDQUI/kBM5UjdOl/YXl2plXop15pdpdUKq9h8u1/3CZin+l1DS3Bqh1i2C1jgxW6xa1h2HXft9MrVuEqGWY9af1m5wDpPem105QPtHUPYtFatNGwenpph2iDgDHcjpPfubz0+X1hScoKEhpaWlasmSJrrnmGvf2JUuW6KqrrjIxGXB6XC5Dew+XaUduiXbml2pHXol25ZdqT0GZqpwn/5goNsyq9tG1J8drHxPq/j4xKuT0Vtv296899Py662rLzbGl5+ho6YsvUnYA+CSvLzySNHXqVI0bN059+/bVgAEDNGfOHGVmZmrixIlmRwNOyFHj1M68Um3Otmlztl2bs+3ammM/6TpNwYH+R5YwaK6Osc3VISZU7Y4Um9D6XITy2mtrDz0/0Xl4Xnyx3s7DAwDeplEUnuuvv16HDx/WE088oZycHPXo0UOff/652rVrZ3Y0QE6XoZ35JVqbWay1mUXacMCmXfmlJzyZXrNAP3WOC1NybJg6xzVXclztOk2tWwQ33Jl/r71WuuoqrznTMgA0hEZxHp6zdfQ8PPVxHD9wuNRRW26yirQ2s1jrs4pVdoKRmxYhgeqeEK7uCRFHvoYrKab5T3NqAAC/qD7fvxvFCA9gpvySSv2wp1Ar9xzWyj2Htbug7Lh9QoP81bttC6UmRuqcNhHq3jpCCRHNOCMwAHgJCg/wMwUlDi3ffUg/7K0tOXtOUHCSY5srtW0LpbaNVJ+2keoUy8gNAHgzCg+avGqnSxn7i/TtjgIt3VGgzdn2OrdbLFJKq3Cd1yFa/ZOidG5SlFqEBJmUFgBwJig8aJKyiyv0zfZ8Ld1eoOW7D6vUUXcJhO4JtQXnvA7ROrd9lCJCTuPwbwCA16HwoEkwDEM780u1eHOuvtycp40HbXVujwoN0gXJMRrSpaUGJ7dUTPPjT1wJAGi8KDzwWS6XobVZxVq8OVeLt+Rp76Gf5uJYLFKftpEa1qWlhnSOVfeE8IY7LBwA0OAoPPAphmFo00G7Fq4/qM/W5yjXXum+LcjfT+cnxyg9JU7DU+IYxQGAJoTCA5+wK79UC9dn67P12XVGcsKsARrWNVYjusdrSJeWal6fZy0GADQa/PZHo2Urr9a/1h3UPzOytOngT0dWWQP8NDwlTlf2StDQLi1lDeAMwgDQ1FF40Ki4XIaW7z6s+Wuy9OXmXFXV1C666e9n0QXJMbqyd4IuTolnJAcAUAfvCmgUcmwV+sfqA/pnRpYOFFW4t3eND9P1/RJ1Ve/Wigrl3DgAgBOj8MBrGYahlXsK9c6KfVq8JU/OI4txhjUL0NW9W2t030T1aB3O8g0AgF9F4YHXKa+q0YK1B/XO8v3anlfi3t4/KUpj+rfViO7xahbIvBwAwKmj8MBrHCyu0N+W7dU/1mSppLL2zMfBgf66tk9r3TygvbrEh5mcEADQWFF4YLqtOXbN+XaPFq7Pdn9s1T46ROMGtNd1aW0UEcyyDgCAs0PhgSmOzs95/dvd+t/2Avf2gR2jNWFwBw3p3JIzHwMA6g2FBw3KMAz9b0eBXvpqp9ZlFUuS/CzSpT1b6c4LOuicNi1MTggA8EUUHjSIo0Xnxa92av2RomMN8NOovm10x/kd1D4m1OSEAABfRuGBRxmGoaVHis7REZ1mgX66eUB7TRjcQS3DWM8KAOB5FB54zKq9hXpq0Vb9mEnRAQCYi8KDerczr0RPf7FNX23Nl1RbdMad106/vaAjRQcAYAoKD+pNjq1CLyzZoY8yDshl1K5vdUO/RN07PFmxYc3MjgcAaMIoPDhrZY4avfq/XXrju71yHFnM85Lu8fr9JV3UsWVzk9MBAEDhwVkwDEOfrsvWjEVblWd3SJL6tY/Uw5d2U1q7SJPTAQDwEwoPzsimgzY9vnCz1uwvkiS1jQrRH0Z2U3pKHIt5AgC8DoUHp6WwrEozv9yueaszZRi1a13dfWEn3X5+Egt6AgC8FoUHp8QwDP0z44Cmf75VxeXVkqQreyVo2mVd1Soi2OR0AAD8MgoPftWeglI9smCjVu4plCR1jQ/TE1f10LlJUSYnAwDg1FB4cFJVNS69vnS3Xvlml6pqXGoW6KepF3fWbYOSFODvZ3Y8AABOGYUHJ5Sxv1APf7xRO/NLJUkXdG6pJ6/uocSoEJOTAQBw+ig8qKOy2qnnFm/XG8v2yjCk6NAg/emKFF3ZK4GjrwAAjRaFB27rs4p1/z/Xa9eRUZ3f9GmjP17eTS1CgkxOBgDA2aHwQFU1Ls3670799X+75XQZimlu1VPX9tTwlDizowEAUC8oPE3ctly7ps5fry05dknSFb0S9MSV3RUZyqgOAMB3UHiaKMMw9M6K/Xry862qqnEpMiRQf766hy4/J8HsaAAA1DsKTxNUVFal33+0QV9tzZMkXdg1Vk/9picrmgMAfBaFp4lZsfuw7pu/Trn2SgX5++mRy7rqloHtOQILAODTKDxNRI3TpZe+3qlZ3+ySYUgdWobqlRtT1T0hwuxoAAB4HIWnCcizV+ruD37U6n21K5uP7ttGj1/ZXSFB/PEDAJoG3vF83Mo9h3X3B2t1qNShMGuAnry2p67sxcRkAEDTQuHxUYZhaO53e/T0F9vldBnqGh+m2WPTlBQTanY0AAAaHIXHB5VUVuvBjzZo0aZcSdI1qa01/ZqeCg7yNzkZAADmoPD4mF35pfrtu2u0p6BMgf4W/enyFI09rx1HYQEAmjQKjw/5Znu+Jn+wViWOGrWKaKa/3tRHfdpGmh0LAADTUXh8gGEYenPZXk3/fKtchtSvfaRmj01TTHOr2dEAAPAKFJ5GzlHj1B8WbNJHGQckSdf3TdSfr+6hoAA/k5MBAOA9KDyNWEGJQxPfy1DG/iL5WaRHR6bo1kGcNRkAgJ+j8DRSu/JLdMvfVutgcYXCmgVo1pg+GtK5pdmxAADwShSeRuiHPYc14Z01slfWqH10iN4c308dWzY3OxYAAF6LwtPILFyfrQf+sV5VTpf6tG2hN27pp6jQILNjAQDg1Sg8jYRhGHr92z16atE2SdKI7nF66YZUNQvkZIIAAPwaCk8j4HQZenzhZr27cr8k6dZB7fXoyBT5+zE5GQCAU0Hh8XKOGqfum79On2/MleXIkVi3n59kdiwAABoVCo8XK3PUaOJ7Gfpu5yEF+fvpxRt667KercyOBQBAo0Ph8VLF5VUa/9ZqrcsqVkiQv+aM66vzk2PMjgUAQKNE4fFCefZKjXvzB+3IK1VEcKD+fms/pbImFgAAZ8xj6w/s27dPt99+u5KSkhQcHKyOHTvqscceU1VVVZ39Nm7cqCFDhig4OFitW7fWE088IcMw6uzz8ccfKyUlRVarVSkpKVqwYIGnYptu/+Ey/Wb2cu3IK1VcuFX/nDiAsgMAwFny2AjPtm3b5HK59Prrr6tTp07atGmTJkyYoLKyMj377LOSJLvdrosvvljDhg3T6tWrtWPHDo0fP16hoaG6//77JUkrVqzQ9ddfrz//+c+65pprtGDBAo0ePVrLli1T//79PRXfFLsLSnXjnJXKL3GoXXSI3ru9vxKjQsyOBQBAo2cxfj6c4kEzZ87U7NmztWfPHknS7NmzNW3aNOXl5clqrV3Z+6mnntIrr7yiAwcOyGKx6Prrr5fdbteiRYvcj3PJJZcoMjJSH3744Sk9r91uV0REhGw2m8LDw+v/B6sHO/NKdOPcH3So1KEucWF6945zFRvWzOxYAACYpj7fvxt0SW2bzaaoqCj39RUrVmjIkCHusiNJI0aMUHZ2tvbt2+feJz09vc7jjBgxQsuXLz/p8zgcDtnt9joXb7Yt164b5qzUoVKHusaH6YMJ/Sk7AADUowYrPLt379Yrr7yiiRMnurfl5uYqLi6uzn5Hr+fm5v7iPkdvP5EZM2YoIiLCfUlMTKyvH6Pebc626cY5K3W4rErdE8L14YTzFN3c+ut3BAAAp+y0C8/jjz8ui8Xyi5c1a9bUuU92drYuueQSjRo1SnfccUed2yyWumcLPvoJ27HbT7TPz7cda9q0abLZbO5LVlbW6f6YDWLjAZvGzP1BReXV6tUmQh/ccZ4iWRcLAIB6d9qTlu+++27dcMMNv7hP+/bt3d9nZ2dr2LBhGjBggObMmVNnv/j4+ONGavLz8yX9NNJzsn1+PupzLKvVWudjMm+06aBNN72xUvbKGqW2baG3bztX4c0CzY4FAIBPOu3CExMTo5iYUzsB3sGDBzVs2DClpaXprbfekp9f3QGlAQMG6JFHHlFVVZWCgmpHNhYvXqyEhAR3aRowYICWLFmi++67z32/xYsXa+DAgacb3Wtsy7Vr7Js/yF5Zo77tIvXWrf0URtkBAMBjPDaHJzs7W0OHDlViYqKeffZZFRQUKDc3t85ozZgxY2S1WjV+/Hht2rRJCxYs0PTp0zV16lT3R1b33nuvFi9erKefflrbtm3T008/ra+++kpTpkzxVHSP2pVfqrFv/KDi8mr1SmxB2QEAoAF47Dw8ixcv1q5du7Rr1y61adOmzm1H5+lERERoyZIlmjRpkvr27avIyEhNnTpVU6dOde87cOBAzZs3T48++qj++Mc/qmPHjpo/f36jPAfP/sNluumNlTpUWqWUVuF659ZzKTsAADSABj0Pj1m84Tw8B4srNPq1FTpYXKHk2Oaa91uOxgIA4Jc02vPwNFV59kqNmbtSB4srlBQTqvcn9KfsAADQgCg8HlZcXqVxb/6g/YfL1SYyWO/fwUkFAQBoaBQeD6qocuq2v6/WjrxSxYZZ9eGE85TQItjsWAAANDkUHg+pdrp01/sZ+jGzWOHNAvTO7eeyECgAACah8HiAy2XowY826JvtBbIG+Olv4/upa7x3LloKAEBTQOGpZ4Zh6MnPt2rB2oPy97No9tg+6ts+6tfvCAAAPIbCU89eW7pHby7bK0l65jfn6MKuJ18CAwAANAwKTz1asPaAnv5imyTp0ZHd9Ju0Nr9yDwAA0BAoPPVkxe7DevCjDZKk317QQXcM7mByIgAAcBSFpx7syi/Rne+uUbXT0GU94/XwJV3NjgQAAI5B4TlLBSUOjX9rteyVNerTtoWeH91bfn4Ws2MBAIBjUHjOQkWVU3e8vVoHiirULjpEc2/uq2aB/mbHAgAAP0PhOUNOl6F7563V+gM2RYYE6u+3nsv6WAAAeCkKzxl6atFWLd6Sp6AAP829ua+SYkLNjgQAAE6CwnMG/rkmS3O/qz3XzrOjenFiQQAAvByF5zSt2VeoPyzYJEmafFGyruyVYHIiAADwayg8p+FgcYUmvpehKqdLl/aI15SLks2OBAAATgGF5xSVV9XojrfX6FBplVJaheu50b04/BwAgEaCwnMKXC5D9/9jvbbm2BXTPEhzb+mrkKAAs2MBAIBTROE5BS99vVOLNuUq0N+i18amqXWLYLMjAQCA00Dh+RWLN+fqpa93SpKmX9OTI7IAAGiEKDy/YE9Bqe7/x3pJ0viB7TWqb6LJiQAAwJmg8JxEmaNGE9/LUImjRue2j9IfRnYzOxIAADhDFJ4TMAxDD328QTvyShUbZtWsm1IV6M9LBQBAY8W7+Am8uWyv/r0hRwF+Fr16Ux/FhjUzOxIAADgLFJ6fWbH7sGYs2iZJenRkNyYpAwDgAyg8x8i1VeqeD3+U02Xo6t4JumVge7MjAQCAekDhOaLG6dI9H/6oQ6VV6hofphnXniOLhTMpAwDgCyg8Rzy/ZIdW7ytSc2uAXhubpuAgf7MjAQCAekLhkbR0R4Fe/d9uSdJTv+mp9jGhJicCAAD1qckXnlxbpe6bv06SNPa8trr8nASTEwEAgPrWpAtPjdOlyfPWqrCsdgX0R0emmB0JAAB4QJMuPC99vVOr9hYqNMhff72pj5oFMm8HAABf1GQLz3c7CzTrm12SpBm/OUdJzNsBAMBnNcnCU1Di0H3z18kwpBvPbasrezFvBwAAX9bkCo9hGHrwo/U6VFqlLnFheuwK5u0AAODrmlzheWfFfn2zvUBBAX56+cZU5u0AANAENKnCszOvRE9+vlWS9MilXdUlPszkRAAAoCE0qcLz0McbVFXj0rAuLVknCwCAJqRJFZ4deaWKaR6kZ67rxTpZAAA0IU2q8EjSzOt6qWWY1ewYAACgATWpwjPm3EQN6xprdgwAANDAmlThmZrexewIAADABE2q8HAIOgAATVOTKjwAAKBpovAAAACfR+EBAAA+j8IDAAB8HoUHAAD4PAoPAADweRQeAADg8yg8AADA51F4AACAz6PwAAAAn0fhAQAAPo/CAwAAfB6FBwAA+DwKDwAA8HkBZgdoCIZhSJLsdrvJSQAAwKk6+r599H38bDSJwlNSUiJJSkxMNDkJAAA4XYcPH1ZERMRZPYbFqI/a5OVcLpeys7MVFhYmi8ViSga73a7ExERlZWUpPDzclAzegNehFq9DLV6HWrwOtXgdavE6/MRms6lt27YqKipSixYtzuqxmsQIj5+fn9q0aWN2DElSeHh4k/8LLPE6HMXrUIvXoRavQy1eh1q8Dj/x8zv7KcdMWgYAAD6PwgMAAHye/+OPP/642SGaCn9/fw0dOlQBAU3ik8ST4nWoxetQi9ehFq9DLV6HWrwOP6mv16JJTFoGAABNGx9pAQAAn0fhAQAAPo/CAwAAfB6FBwAA+DwKDwAA8HkUHhM5HA717t1bFotF69atMztOg9q3b59uv/12JSUlKTg4WB07dtRjjz2mqqoqs6N53KuvvqqkpCQ1a9ZMaWlp+u6778yO1OBmzJihfv36KSwsTLGxsbr66qu1fft2s2OZbsaMGbJYLJoyZYrZURrcwYMHNXbsWEVHRyskJES9e/dWRkaG2bEaVE1NjR599FH378UOHTroiSeekMvlMjuaR3377be64oorlJCQIIvFon/96191bjcMQ48//rgSEhIUHBysoUOHavPmzaf9PBQeEz344INKSEgwO4Yptm3bJpfLpddff12bN2/WCy+8oNdee02PPPKI2dE8av78+ZoyZYr+8Ic/aO3atRo8eLAuvfRSZWZmmh2tQS1dulSTJk3SypUrtWTJEtXU1Cg9PV1lZWVmRzPN6tWrNWfOHJ1zzjlmR2lwRUVFGjRokAIDA7Vo0SJt2bJFzz333FmvndTYPP3003rttdc0a9Ysbd26Vc8884xmzpypV155xexoHlVWVqZevXpp1qxZJ7z9mWee0fPPP69Zs2Zp9erVio+P18UXX+xeGPyUGTDF559/bnTt2tXYvHmzIclYu3at2ZFM98wzzxhJSUlmx/Coc88915g4cWKdbV27djUefvhhkxJ5h/z8fEOSsXTpUrOjmKKkpMRITk42lixZYgwZMsS49957zY7UoB566CHj/PPPNzuG6UaOHGncdtttdbZde+21xtixY01K1PAkGQsWLHBfd7lcRnx8vPHUU0+5t1VWVhoRERHGa6+9dlqPzQiPCfLy8jRhwgS9++67CgkJMTuO17DZbIqKijI7hsdUVVUpIyND6enpdbanp6dr+fLlJqXyDjabTZJ8+s//l0yaNEkjR47U8OHDzY5iioULF6pv374aNWqUYmNjlZqaqrlz55odq8Gdf/75+vrrr7Vjxw5J0vr167Vs2TJddtllJiczz969e5Wbm1vn96bVatWQIUNO+/cm56xuYIZhaPz48Zo4caL69u2rffv2mR3JK+zevVuvvPKKnnvuObOjeMyhQ4fkdDoVFxdXZ3tcXJxyc3NNSmU+wzA0depUnX/++erRo4fZcRrcvHnzlJGRoTVr1pgdxTR79uzR7NmzNXXqVD3yyCNatWqVJk+eLKvVqptvvtnseA3moYceks1mU9euXeXv7y+n06knn3xSN954o9nRTHP0d+OJfm/u37//tB6LEZ568vjjj8tisfziZc2aNXrllVdkt9s1bdo0syN7xKm+DsfKzs7WJZdcolGjRumOO+4wKXnDsVgsda4bhnHctqbk7rvv1oYNG/Thhx+aHaXBZWVl6d5779X777+vZs2amR3HNC6XS3369NH06dOVmpqqO++8UxMmTNDs2bPNjtag5s+fr/fee08ffPCBfvzxR7399tt69tln9fbbb5sdzXT18XuTEZ56cvfdd+uGG274xX3at2+vv/zlL1q5cqWsVmud2/r27aubbrqp0f/FPtXX4ajs7GwNGzZMAwYM0Jw5czyczlwxMTHy9/c/bjQnPz//uP+9NBX33HOPFi5cqG+//VZt2rQxO06Dy8jIUH5+vtLS0tzbnE6nvv32W82aNUsOh0P+/v4mJmwYrVq1UkpKSp1t3bp108cff2xSInP8/ve/18MPP+z+HdqzZ0/t379fM2bM0C233GJyOnPEx8dLqh3padWqlXv7mfzepPDUk5iYGMXExPzqfi+//LL+8pe/uK9nZ2drxIgRmj9/vvr37+/JiA3iVF8HqfYw1GHDhiktLU1vvfWW/Px8e8AxKChIaWlpWrJkia655hr39iVLluiqq64yMVnDMwxD99xzjxYsWKD//e9/SkpKMjuSKS666CJt3LixzrZbb71VXbt21UMPPdQkyo4kDRo06LjTEuzYsUPt2rUzKZE5ysvLj/s96O/v7/OHpf+SpKQkxcfHa8mSJUpNTZVUOx9y6dKlevrpp0/rsSg8Daxt27Z1rjdv3lyS1LFjxyb1P9zs7GwNHTpUbdu21bPPPquCggL3bUcbvS+aOnWqxo0bp759+7pHtTIzMzVx4kSzozWoSZMm6YMPPtCnn36qsLAw96hXRESEgoODTU7XcMLCwo6btxQaGqro6OgmNZ/pvvvu08CBAzV9+nSNHj1aq1at0pw5c3x+1PfnrrjiCj355JNq27atunfvrrVr1+r555/XbbfdZnY0jyotLdWuXbvc1/fu3at169YpKipKbdu21ZQpUzR9+nQlJycrOTlZ06dPV0hIiMaMGXN6T1Qfh5HhzO3du7dJHpb+1ltvGZJOePF1f/3rX4127doZQUFBRp8+fZrkodj/374d2zYIhGEYTqpjBCoXrig8A6swAgNQsAeNh2AAhvAmdFRfirRJEUUJ1vl5FrhfOun0Fvd/d/f3+/3s0U73imvpSbKua263W0op6bouy7KcPdK/2/c94zjmcrmkaZpcr9dM05TjOM4e7U9t2/blezAMQ5LP1fR5ntO2bUop6fs+j0fUQ2IAAABJSURBVMfjx+e8J8mv0gwA4MnV/WkCAOBN8AAAL0DwAADVEzwAQPUEDwBQPcEDAFRP8AAA1RM8AED1BA8AUD3BAwBUT/AAANX7AEBJhHn/ZsWpAAAAAElFTkSuQmCC",
      "text/plain": [
       "Figure(PyObject <Figure size 640x480 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "using PyPlot\n",
    "x = range(-5, stop=10, length=1000)\n",
    "plot(x, P.(x))\n",
    "hlines(0, -5, 10)\n",
    "plot(1, 0, \"ro\")\n",
    "xlim(-5, 10);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Horner's Method\n",
    "To use Newton’s method, we need to evaluate $P(x)$ and $P^\\prime(x)$ at specified values. \n",
    "\n",
    "Computational efficiency requires that the evaluation of polynomials be done in the **nested** manner discussed before. Horner’s (Ruffini) method incorporates this nesting technique, and, as a consequence, requires only $n$ multiplications and $n$ additions to evaluate an arbitrary $n$th-degree polynomial."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">## Theorem (Horner’s Method)\n",
    ">Let $$P(x)=a_nx^n+a_{n-1}x^{n-1}+\\cdots +a_1x+a_0.$$\n",
    "Define $b_n = a_n$ and\n",
    "$$ b_k =a_k +b_{k+1}x_0,\\qquad k=n−1,n−2,\\ldots,1,0.$$\n",
    "Then $b_0 = P(x_0)$. Moreover, if\n",
    "$$Q(x)=b_nx^{n−1} +b_{n−1}x^{n−2} +\\cdots+b_2x+b_1,$$\n",
    "then $$P(x)=(x-x_0)Q(x)+b_0.$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Polynomials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "search: \u001b[0m\u001b[1mP\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ml\u001b[22m\u001b[0m\u001b[1my\u001b[22m\u001b[0m\u001b[1mn\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1mm\u001b[22m\u001b[0m\u001b[1mi\u001b[22m\u001b[0m\u001b[1ma\u001b[22m\u001b[0m\u001b[1ml\u001b[22m\u001b[0m\u001b[1ms\u001b[22m\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "No documentation found.\n",
       "\n",
       "No docstring found for module \\texttt{Polynomials}.\n",
       "\n"
      ],
      "text/markdown": [
       "No documentation found.\n",
       "\n",
       "No docstring found for module `Polynomials`.\n"
      ],
      "text/plain": [
       "  No documentation found.\n",
       "\n",
       "  No docstring found for module \u001b[36mPolynomials\u001b[39m."
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "? Polynomials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "&#45;13 &#43; 17&#8729;x &#45; 5&#8729;x<sup>2</sup> &#43; x<sup>3</sup>"
      ],
      "text/latex": [
       "$-13 + 17\\cdot x - 5\\cdot x^{2} + x^{3}$"
      ],
      "text/plain": [
       "Poly(-13 + 17*x - 5*x^2 + x^3)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Poly([-13,17,-5,1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Array{Complex{Float64},1}:\n",
       " 0.15384615384615374 - 0.23076923076923064im\n",
       " 0.15384615384615374 + 0.23076923076923064im\n",
       "  1.0000000000000007 + 0.0im                "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "roots(Poly([1, -5, 17, -13]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.683"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = Poly([1, -5, 17, -13])\n",
    "polyval(p, -0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "horner (generic function with 1 method)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function horner(coefs, x)\n",
    "    s = copy(coefs[end])\n",
    "    for k in length(coefs)-1:-1:1\n",
    "        s = copy(coefs[k]) + x * s  # b_k = a_k + x_0*b_{k+1}\n",
    "    end\n",
    "    return s\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "1 &#45; 5&#8729;x &#43; 17&#8729;x<sup>2</sup> &#45; 13&#8729;x<sup>3</sup>"
      ],
      "text/latex": [
       "$1 - 5\\cdot x + 17\\cdot x^{2} - 13\\cdot x^{3}$"
      ],
      "text/plain": [
       "Poly(1 - 5*x + 17*x^2 - 13*x^3)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Poly([1, -5, 17, -13])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.683"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "horner([1, -5, 17, -13], -0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.002947 seconds (5 allocations: 176 bytes)\n",
      "  0.002949 seconds (5 allocations: 176 bytes)\n",
      "  0.002935 seconds (5 allocations: 176 bytes)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "0.7842767176684162"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# We are beating JULIA!!!\n",
    "coefs=rand(1000001)\n",
    "p = Poly(coefs)\n",
    "x0=-0.2345\n",
    "@time polyval(p, x0)\n",
    "@time p(x0)\n",
    "@time horner(coefs, x0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example: \n",
    "Use Horner's method to evaluate\n",
    "$$P(x) = 2x^4 − 3x^2 + 3x − 4 \\qquad \\mbox{at}\\quad x_0 = −2.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An additional advantage of using the Horner  procedure is that, since\n",
    "$$P(x) = (x − x_0)Q(x) + b_0,$$ \n",
    "\n",
    "$$P^\\prime(x)=Q(x)+(x−x )Q^\\prime(x)\\qquad \\mbox{and} \\qquad\n",
    "P^\\prime(x_0)=Q(x_0).$$\n",
    "\n",
    "When the Newton's method is being used to find an approximate zero of a polynomial, $P(x)$ and $P^\\prime(x)$ can be evaluated."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "hornerVect (generic function with 1 method)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function hornerVect(coefs, x)\n",
    "    coefsnew = copy(coefs)\n",
    "    for k in length(coefs)-1:-1:1\n",
    "        coefsnew[k] = copy(coefs[k]) + x * coefsnew[k+1]  # b_k = a_k + x_0*b_{k+1}\n",
    "    end\n",
    "    return coefsnew\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Array{Int64,1}:\n",
       " 10\n",
       " -7\n",
       "  5\n",
       " -4\n",
       "  2"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = Poly([-4,3, -3,0, 2])\n",
    "b=hornerVect([-4,3, -3,0, 2],-2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p(-2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "&#45;7 &#43; 5&#8729;x &#45; 4&#8729;x<sup>2</sup> &#43; 2&#8729;x<sup>3</sup>"
      ],
      "text/latex": [
       "$-7 + 5\\cdot x - 4\\cdot x^{2} + 2\\cdot x^{3}$"
      ],
      "text/plain": [
       "Poly(-7 + 5*x - 4*x^2 + 2*x^3)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q = Poly(b[2:length(b)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-49"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = horner(b[2:length(b)],-2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-49"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q(-2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "plotnewtonH (generic function with 1 method)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using PyPlot\n",
    "using Printf\n",
    "\n",
    "function plotnewtonH(p,q, x0, a, b; tol=1e-6, maxiter=10)    \n",
    "    xspan = range(a, stop=b, length=1000)\n",
    "    plot(xspan, p.(xspan), \"b\")\n",
    "    hlines(0, a, b)\n",
    "    xlim(a, b)    \n",
    "    k = 0\n",
    "    x = x0\n",
    "    px = p(x)\n",
    "    #q=polyder(p)\n",
    "    #qx = q(x)\n",
    "    xold, xnew = x, Inf\n",
    "    @printf \"%4s %22s %22s\\n\" \"k\" \"x\" \"p(x)\"\n",
    "    while abs(xnew - xold) > tol*(1 + abs(xold)) && k < maxiter\n",
    "        k += 1\n",
    "        xnew = x - px/q(x)\n",
    "        vlines(x, 0, px, linestyles=\":\")\n",
    "        plot([xnew, x], [0, px], \"r\")\n",
    "        plot(x, px, \"bo\")\n",
    "        xold = x\n",
    "        x = xnew\n",
    "        px = p(x)\n",
    "        #qx = q(x)\n",
    "        @printf \"%4d %22.15e %22.15e\\n\" k x px\n",
    "    end  \n",
    "    return x  \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "&#45;4 &#43; 3&#8729;x &#45; 3&#8729;x<sup>2</sup> &#43; 2&#8729;x<sup>4</sup>"
      ],
      "text/latex": [
       "$-4 + 3\\cdot x - 3\\cdot x^{2} + 2\\cdot x^{4}$"
      ],
      "text/plain": [
       "Poly(-4 + 3*x - 3*x^2 + 2*x^4)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p=Poly([-4,3, -3,0, 2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "3 &#45; 6&#8729;x &#43; 8&#8729;x<sup>3</sup>"
      ],
      "text/latex": [
       "$3 - 6\\cdot x + 8\\cdot x^{3}$"
      ],
      "text/plain": [
       "Poly(3 - 6*x + 8*x^3)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q=polyder(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   k                      x                   p(x)\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Figure(PyObject <Figure size 640x480 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   1 -1.795918367346939e+00  1.741690649859379e+00\n",
      "   2 -1.742432916750542e+00  9.995592022432565e-02\n",
      "   3 -1.738970235336167e+00  4.002870896355759e-04\n",
      "   4 -1.738956256679049e+00  6.504574656673867e-09\n",
      "   5 -1.738956256451892e+00  8.881784197001252e-16\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "-1.738956256451892"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = plotnewtonH(p,q, -2.0, -2.5, -1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Array{Complex{Float64},1}:\n",
       " -1.7389562564518912 + 0.0im               \n",
       " 0.24203718580880035 - 0.9262454872675303im\n",
       " 0.24203718580880035 + 0.9262454872675303im\n",
       "  1.2548818848342933 + 0.0im               "
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "roots(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "newtonH (generic function with 1 method)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Polynomials\n",
    "using Printf\n",
    "function newtonH(pcoefs, x0; tol=1e-6, maxiter=10)   \n",
    "    k=0\n",
    "    @printf \"%4s %22s %22s  %22s\\n\" \"k\" \"x\" \"p(x)\" \"q(x)\" \n",
    "    x = x0\n",
    "    p=Poly(pcoefs)\n",
    "    q=polyder(p)\n",
    "    px = p(x)\n",
    "    qx = q(x)   \n",
    "    xold, xnew = x, Inf\n",
    "    @printf \"%4d %22.15e %22.15e\\n\" k x px \n",
    "    while abs(xnew - xold) > tol*(1 + abs(xold)) && k < maxiter\n",
    "        k += 1\n",
    "        xnew = x - px/qx\n",
    "        xold = x\n",
    "        x = xnew\n",
    "        px = p(x)\n",
    "        qx = q(x) \n",
    "        @printf \"%4d %22.15e %22.15e\\n\" k x px \n",
    "    end  \n",
    "    return x  \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   k                      x                   p(x)                    q(x)\n",
      "   0  2.000000000000000e+01  3.900000000000000e+02\n",
      "   1  1.025000000000000e+01  9.506250000000000e+01\n",
      "   2  5.612804878048781e+00  2.150357859904819e+01\n",
      "   3  3.697222645433951e+00  3.669455289909621e+00\n",
      "   4  3.200977809537825e+00  2.462589371535735e-01\n",
      "   5  3.162511604551991e+00  1.479648926011379e-03\n",
      "   6  3.162277668821308e+00  5.472592690125566e-08\n",
      "   7  3.162277660168380e+00  1.776356839400250e-15\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "3.1622776601683795"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a=[-10,0, 1]\n",
    "newtonH(a, 20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## An application: Minimizing a function in one variable\n",
    "\n",
    "We can use the root-finding methods described above to find the **minimum** or **maximum** value of a function $\\phi \\in C^2[a,b]$.\n",
    "\n",
    "Recall that $x^* \\in (a,b)$ is a **critical point** of $\\phi$ if \n",
    "\n",
    "$$\\phi'(x^*) = 0.$$\n",
    "\n",
    "We can find $x^*$ by applying Newton's method to this nonlinear equation to obtain:\n",
    "\n",
    "$$x_{k+1} = x_k - \\frac{\\phi'(x_k)}{\\phi''(x_k)}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Another interpretation for Newton's method\n",
    "\n",
    "We can also obtain this by considering the **second-order (quadratic) approximation** of $\\phi$ around the point $x_k$:\n",
    "\n",
    "$$\\phi(x) \\approx \\phi(x_k) + \\phi'(x_k)(x-x_k) + \\frac{\\phi''(x_k)}{2}(x-x_k)^2, \\quad \\text{for all $x \\approx x_k$}.$$\n",
    "\n",
    "If $x_k$ is close to $x^*$, we expect the minimum/maximum of $\\phi$ to be near the minimum/maximum of the **quadratic approximation** of $\\phi$:\n",
    "\n",
    "$$q(x) = \\phi(x_k) + \\phi'(x_k)(x-x_k) + \\frac{\\phi''(x_k)}{2}(x-x_k)^2.$$\n",
    "\n",
    "Thus we should choose $x_{k+1}$ to be the critical point of $q$, so we want to find $x_{k+1}$ such that $q'(x_{k+1}) = 0$. Note that\n",
    "\n",
    "$$q'(x) = \\phi'(x_k) + \\phi''(x_k)(x-x_k).$$\n",
    "\n",
    "Thus $q'(x_{k+1}) = 0$ gives us\n",
    "\n",
    "$$x_{k+1} = x_k - \\frac{\\phi'(x_k)}{\\phi''(x_k)}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example\n",
    "\n",
    "Let's find the minimum value of \n",
    "\n",
    "$$\\phi(x) = 10\\cosh(x/4) - x$$\n",
    "\n",
    "by applying Newton's method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "plotnewton (generic function with 1 method)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using PyPlot\n",
    "\n",
    "function plotnewton(f, fp, x0, a, b; tol=1e-6, maxiter=10)\n",
    "    \n",
    "    xspan = range(a, stop=b, length=1000)\n",
    "    plot(xspan, f.(xspan), \"b\")\n",
    "    hlines(0, a, b)\n",
    "    xlim(a, b)\n",
    "    \n",
    "    k = 0\n",
    "    x = x0\n",
    "    fx = f(x)\n",
    "    xold, xnew = x, Inf\n",
    "    @printf \"%4s %22s %22s\\n\" \"k\" \"x\" \"f(x)\"\n",
    "    while abs(xnew - xold) > tol*(1 + abs(xold)) && k < maxiter\n",
    "        k += 1\n",
    "        xnew = x - fx/fp(x)\n",
    "        \n",
    "        # Plotting stuff\n",
    "        vlines(x, 0, fx, linestyles=\":\")\n",
    "        plot([xnew, x], [0, fx], \"r\")\n",
    "        plot(x, fx, \"bo\")\n",
    "        \n",
    "        xold = x\n",
    "        x = xnew\n",
    "        fx = f(x)\n",
    "        @printf \"%4d %22.15e %22.15e\\n\" k x fx\n",
    "    end\n",
    "    \n",
    "    return x\n",
    "    \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ϕ (generic function with 1 method)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ϕ(x) = 10cosh(x/4) - x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ϕp (generic function with 1 method)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ϕp(x) = 2.5sinh(x/4) - 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ϕpp (generic function with 1 method)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ϕpp(x) = 0.625cosh(x/4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Figure(PyObject <Figure size 640x480 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   k                      x                   f(x)\n",
      "   1  2.454132846259612e+00  6.318888130288225e-01\n",
      "   2  1.607515395183571e+00  3.196055878741744e-02\n",
      "   3  1.560247213739166e+00  7.130979857405961e-05\n",
      "   4  1.560141279603856e+00  3.507054646689767e-10\n",
      "   5  1.560141279082861e+00 -1.110223024625157e-16\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "1.560141279082861"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0, a, b = 5., 0., 10.\n",
    "\n",
    "x = plotnewton(ϕp, ϕpp, x0, a, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.673145600891813"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ϕpp(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "PyPlot.Figure(PyObject <matplotlib.figure.Figure object at 0x114ed1310>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xspan = range(-3, stop=7, length=1000)\n",
    "\n",
    "plot(xspan, ϕ.(xspan))\n",
    "plot(x, ϕ(x), \"o\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus \n",
    "\n",
    "$$x^* = 1.560141279082861\\ldots.$$ \n",
    "\n",
    "Note that $\\phi'(x^*) = 0$, as required. Since $\\phi''(x^*) > 0$, we know that \n",
    "\n",
    "$$\\phi(x^*) = 9.210188335186146\\ldots$$ \n",
    "\n",
    "is a **local minimum value** of $\\phi$.\n",
    "\n",
    "In fact, since $\\phi''(x) > 0$ for all $x$, we know that $\\phi(x^*)$ is the **global minimum value** of $\\phi$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Julia 1.3.1",
   "language": "julia",
   "name": "julia-1.3"
  },
  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "1.3.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}