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   "source": [
    "Quiz 4, on Friday 16 September 2022, due 10:50am."
   ]
  },
  {
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   "metadata": {},
   "source": [
    "Consider the linear system $A {\\bf x} = {\\bf b}$, $A \\in {\\mathbb R}^{n \\times n}$\n",
    "and ${\\bf b} \\in {\\mathbb R}^n$.\n",
    "\n",
    "Suppose the condition number of $A$ is $10^{37}$.\n",
    "\n",
    "Let $\\overline{\\bf x}$ be a numerically computed solution,\n",
    "computed with a precision equal to 56 decimal places.\n",
    "\n",
    "1. How many decimal places in $\\overline{\\bf x}$ are correct?\n",
    "\n",
    "2. Justify your answer."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f0050137",
   "metadata": {},
   "source": [
    "# Answer to question 1"
   ]
  },
  {
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   "source": [
    "The number of correct decimal places equals $56-37 = 19$."
   ]
  },
  {
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   "metadata": {},
   "source": [
    "# Answer to question 2"
   ]
  },
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   "id": "43af62f6",
   "metadata": {},
   "source": [
    "As justification, consider the following inequality:\n",
    "\n",
    "$$\n",
    "    \\frac{\\|{\\bf x} - \\overline{\\bf x}\\|}{\\|{\\bf x}\\|}\n",
    "    \\leq\n",
    "    \\mbox{cond}(A)\n",
    "    \\frac{\\|A - \\overline{A}\\|}{\\|A\\|}\n",
    "$$\n",
    "\n",
    "where $\\overline{A}$ is the representation of $A$ with 56 decimal places accuracy, or with error $10^{-56}$ which is magnified by the condition number $\\mbox{cond}(A) = 10^{37}$, so we obtain as the bound $10^{-56} \\times 10^{37} = 10^{-19}$ on the relative error on the computed solution $\\overline{\\bf x}$."
   ]
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