{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "d73c7ab5",
   "metadata": {},
   "source": [
    "Quiz 2 MCS 471, on Friday 2 September 2022, due 10:50am."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "81cf59ec",
   "metadata": {},
   "source": [
    "Consider the polynomials\n",
    "\n",
    "$$\n",
    "    p(x) = (x-1)(x-2)(x-3)\n",
    "$$\n",
    "\n",
    "and\n",
    "\n",
    "$$\n",
    "    q(x) = (x-1.99999)(x-2)(x-2.00001).\n",
    "$$\n",
    "\n",
    "Compute the condition number for 2 as root of $p$ and 2 as root of $q$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "552c338c",
   "metadata": {},
   "source": [
    "Write one sentence to compare the condition numbers."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b3cbfa83",
   "metadata": {},
   "source": [
    "# Answer"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3c990cfd",
   "metadata": {},
   "source": [
    "We evaluate the formula\n",
    "\n",
    "$$\n",
    "    \\kappa = \\frac{|f|}{|2 f'(2)|}.\n",
    "$$\n",
    "\n",
    "for $f = p$ and $f = q$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "3de2c263",
   "metadata": {},
   "outputs": [],
   "source": [
    "using Polynomials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "ed116108",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "&#45;6 &#43; 11&#8729;x &#45; 6&#8729;x<sup>2</sup> &#43; x<sup>3</sup>"
      ],
      "text/latex": [
       "$-6 + 11\\cdot x - 6\\cdot x^{2} + x^{3}$"
      ],
      "text/plain": [
       "Polynomial(-6 + 11*x - 6*x^2 + x^3)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = fromroots([1, 2, 3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "c7205191",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "11 &#45; 12&#8729;x &#43; 3&#8729;x<sup>2</sup>"
      ],
      "text/latex": [
       "$11 - 12\\cdot x + 3\\cdot x^{2}$"
      ],
      "text/plain": [
       "Polynomial(11 - 12*x + 3*x^2)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dp = derivative(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "4db54563",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6.0"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mp = maximum([abs(p(x)) for x=0.0:0.01:4.0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "39cef47e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.0"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "kp = abs(mp/(2*dp(2)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "d2f698d9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "&#45;7.9999999998 &#43; 11.999999999900002&#8729;x &#45; 6.0&#8729;x<sup>2</sup> &#43; 1.0&#8729;x<sup>3</sup>"
      ],
      "text/latex": [
       "$-7.9999999998 + 11.999999999900002\\cdot x - 6.0\\cdot x^{2} + 1.0\\cdot x^{3}$"
      ],
      "text/plain": [
       "Polynomial(-7.9999999998 + 11.999999999900002*x - 6.0*x^2 + 1.0*x^3)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q = fromroots([1.99999, 2, 2.00001])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "bb3bcb3d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "11.999999999900002 &#45; 12.0&#8729;x &#43; 3.0&#8729;x<sup>2</sup>"
      ],
      "text/latex": [
       "$11.999999999900002 - 12.0\\cdot x + 3.0\\cdot x^{2}$"
      ],
      "text/plain": [
       "Polynomial(11.999999999900002 - 12.0*x + 3.0*x^2)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dq = derivative(q)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "894d2b36",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.999999999800007"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mq = maximum([abs(q(x)) for x = 0.0:0.01:4.0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "2f95e711",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.0000707244625626e10"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "kq = abs(mq/(2*dq(2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2b9d6786",
   "metadata": {},
   "source": [
    "Observe that the condition number of 2 as root of $q$ is $4 \\times 10^{10}$, whereas the condition number of 2 as root of $p$ is 3."
   ]
  }
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