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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Sensitivity of polynomial roots"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Consider the polynomial\n",
      "$$\n",
      "x^3 - 21 x^2 + 120 x - 100 = 0\n",
      "$$\n",
      "The roots are 1, 10, 10."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy import N, roots\n",
      "from sympy import Integer as Int"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "r= roots([1,-21,120,-100])\n",
      "for x in r:\n",
      "    print x"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "1\n",
        "10\n"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Perturb the coefficient of $x^3$ \n",
      "$$\n",
      "\\frac{99}{100}x^3 - 21 x^2 + 120 x - 100 = 0\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "r= roots([Int(99)/100,-21,120,-100])\n",
      "for x in r:\n",
      "    print N(x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "1.0001235059284 + 0.e-20*I\n",
        "9.0412990764462 + 0.e-20*I\n",
        "11.1706986297466 + 0.e-20*I\n"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The double roots are very sensitive. Now perturb the coefficient in the other direction\n",
      "$$\n",
      "\\frac{101}{100}x^3 - 21 x^2 + 120 x - 100 = 0\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "r= roots([Int(101)/100,-21,120,-100])\n",
      "for x in r:\n",
      "    print N(x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "9.89610130781283 + 1.04369535224477*I\n",
        "9.89610130781283 - 1.04369535224477*I\n",
        "0.999876592295142\n"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The double roots become complex, again showing they are very sensitive to the coefficient of $x^3$."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}