Exercises/03 Exercises.ipynb

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    "## Exercise 03.1\n",
    "\n",
    "Compare the computed values of \n",
    "\n",
    "$$ \n",
    "d_0 = a \\cdot b + a  \\cdot  c\n",
    "$$\n",
    "\n",
    "and \n",
    "\n",
    "$$ \n",
    "d_1 = a \\cdot (b + c)\n",
    "$$\n",
    "\n",
    "when $a = 100$, $b = 0.1$ and $c = 0.2$. Store $d_{0}$ in the variable `d0` and $d_{1}$ in the variable `d1`.\n",
    "\n",
    "\n",
    "Try checking for equality, e.g. `print(d0 == d1)`. "
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    "# YOUR CODE HERE\n",
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    "assert d0 == 30.0\n",
    "assert d1 != 30.0\n",
    "assert d0 != d1"
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    "## Exercise 03.2\n",
    "\n",
    "For the polynomial \n",
    "\\begin{align}\n",
    "f(x, y) &= (x + y)^{6} \n",
    "\\\\\n",
    "&=  x^6 + 6x^{5}y + 15x^{4}y^{2} + 20x^{3}y^{3} + 15x^{2}y^{4} + 6xy^{5} + y^{6}\n",
    "\\end{align}\n",
    "compute $f$ using: (i) the compact form $(x + y)^{6}$; and (ii) the expanded form for:\n",
    "\n",
    "(a) $x = 10$ and $y = 10.1$\n",
    "\n",
    "(b) $x = 10$ and $y = -10.1$\n",
    "\n",
    "and compare the number of significant digits for which the answers are the same.\n",
    "Store the answer for the compact version using the variable `f0`, and using the variable `f1` for the expanded version.\n",
    "\n",
    "For case (b), compare the computed and analytical solutions and consider the relative error.\n",
    "Which approach would you recommend for computing this expression?"
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   "source": [
    "#### (a) $x = 10$ and $y = 10.1$"
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    "x = 10.0\n",
    "y = 10.1\n",
    "\n",
    "# YOUR CODE HERE\n",
    "raise NotImplementedError()"
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    "import math\n",
    "assert math.isclose(f0, 65944160.60120103, rel_tol=1e-10)\n",
    "assert math.isclose(f1, 65944160.601201, rel_tol=1e-10)"
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   "source": [
    "#### (b) $x = 10$ and $y = -10.1$"
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    "x = 10.0\n",
    "y = -10.1\n",
    "\n",
    "# YOUR CODE HERE\n",
    "raise NotImplementedError()"
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    "import math\n",
    "assert math.isclose(f0, 1.0e-6, rel_tol=1e-10)\n",
    "assert math.isclose(f1, 1.0e-6, rel_tol=1e-2)"
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    "## Exercise 03.3\n",
    "\n",
    "Consider the expression\n",
    "\n",
    "$$\n",
    "f = \\frac{1}{\\sqrt{x^2 - 1} - x}\n",
    "$$\n",
    "\n",
    "When $x$ is very large, the denominator approaches zero, which can cause problems.\n",
    "\n",
    "Try rephrasing the problem and eliminating the fraction by multiplying the numerator and denominator by $\\sqrt{x^2 - 1} + x$ and evaluate the two versions of the expression when:\n",
    "\n",
    "(a) $x = 1 \\times 10^{7}$\n",
    "\n",
    "(b) $x = 1 \\times 10^{9}$ (You may get a Python error for this case. Why?)"
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    "#### (a) $x = 1 \\times 10^{7}$"
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    "#### (b) $x = 1 \\times 10^{9}$"
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