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   "cell_type": "markdown",
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   "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",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import N, roots\n",
    "from sympy import Integer as Int"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Roots =  {1: 1, 10: 2}\n",
      "1\n",
      "10\n"
     ]
    }
   ],
   "source": [
    "r = roots([1,-21,120,-100])\n",
    "print('Roots = ',r)\n",
    "for x in r:\n",
    "    print(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The output shows that the root 1 has multiplicity one and root 10 has multiplicity of two.\n",
    "\n",
    "**Note**: The type of return value `r` is `dict`."
   ]
  },
  {
   "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",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.0001235059284 - 0.e-22*I\n",
      "11.1706986297466 - 0.e-22*I\n",
      "9.0412990764462 + 0.e-22*I\n"
     ]
    }
   ],
   "source": [
    "r = roots([Int(99)/100,-21,120,-100])\n",
    "for x in r:\n",
    "    print(N(x))"
   ]
  },
  {
   "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",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.999876592295142\n",
      "9.89610130781283 - 1.04369535224477*I\n",
      "9.89610130781283 + 1.04369535224477*I\n"
     ]
    }
   ],
   "source": [
    "r = roots([Int(101)/100,-21,120,-100])\n",
    "for x in r:\n",
    "    print(N(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The double roots become complex, again showing they are very sensitive to the coefficient of $x^3$."
   ]
  }
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