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    "# CDS: Numerical Methods Assignments\n",
    "\n",
    "- See lecture notes and documentation on Brightspace for Python and Jupyter basics. If you are stuck, try to google or get in touch via Discord.\n",
    "\n",
    "- Solutions must be submitted via the Jupyter Hub.\n",
    "\n",
    "- Make sure you fill in any place that says `YOUR CODE HERE` or \"YOUR ANSWER HERE\".\n",
    "\n",
    "## Submission\n",
    "\n",
    "1. Name all team members in the the cell below\n",
    "2. make sure everything runs as expected\n",
    "3. **restart the kernel** (in the menubar, select Kernel$\\rightarrow$Restart)\n",
    "4. **run all cells** (in the menubar, select Cell$\\rightarrow$Run All)\n",
    "5. Check all outputs (Out[\\*]) for errors and **resolve them if necessary**\n",
    "6. submit your solutions  **in time (before the deadline)**"
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    "## Composite Numerical Integration: Trapezoid and Simpson Rules\n",
    "\n",
    "In the following we will implement the composite trapezoid and Simpson rules to calculate definite integrals. These rules are defined by\n",
    "\n",
    "\\begin{align}\n",
    "\t\\int_a^b \\, f(x)\\, dx &\\approx \\frac{h}{2} \\left[ f(a) + 2 \\sum_{j=1}^{n-1} f(x_j) + f(b) \\right] \n",
    "                          &\\text{trapezoid} \\\\\n",
    "                          &\\approx \\frac{h}{3} \\left[ f(a) + 2 \\sum_{j=1}^{n/2-1} f(x_{2j}) + 4 \\sum_{j=1}^{n/2} f(x_{2j-1}) + f(b) \\right]\t \n",
    "                          &\\text{Simpson}\n",
    "\\end{align}\n",
    "    \n",
    "with $a = x_0 < x_1 < \\dots < x_{n-1} < x_n = b$ and $x_k = a + kh$. Here $k = 0, \\dots, n$ and $h = (b-a) / n$ is the step size."
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    "# Import packages here ...\n",
    "\n",
    "# YOUR CODE HERE\n",
    "raise NotImplementedError()"
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    "### Task 1\n",
    "\n",
    "Implement both integration schemes as Python functions $\\text{trapz(yk, dx)}$ and $\\text{simps(yk, dx)}$. The argument $\\text{yk}$ is an array of length $n+1$ representing $y_k = f(x_k)$ and $\\text{dx}$ is the step size $h$. Compare your results with Scipy's functions $\\text{scipy.integrate.trapz(yk, xk)}$ and $\\text{scipy.integrate.simps(yk, xk)}$ for a $f(x_k)$ of your choice.\n",
    "\n",
    "Try both even and odd $n$. What do you see? Why?\n",
    "\n",
    "Hint: go to the Scipy documentation. How are even and odd $n$ handled there?"
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    "def trapz(yk, dx):\n",
    "    # YOUR CODE HERE\n",
    "    raise NotImplementedError()\n",
    "    \n",
    "def simps(yk, dx):\n",
    "    # YOUR CODE HERE\n",
    "    raise NotImplementedError()"
   ]
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    "# Compare your results here ...\n",
    "\n",
    "# YOUR CODE HERE\n",
    "raise NotImplementedError()"
   ]
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   "source": [
    "### Task 2\n",
    "\n",
    "Implement at least one test function for each of your integration functions."
   ]
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    "def test_trapz():\n",
    "    # YOUR CODE HERE\n",
    "    raise NotImplementedError()\n",
    "    \n",
    "def test_simps():\n",
    "    # YOUR CODE HERE\n",
    "    raise NotImplementedError()\n",
    "    \n",
    "test_trapz()\n",
    "test_simps()"
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   "source": [
    "### Task 3\n",
    "\n",
    "Study the accuracy of these integration routines by calculating the following integrals for a variety of step sizes $h$:\n",
    "\n",
    "- $\\int_0^1 \\, x\\, dx$\n",
    "- $\\int_0^1 \\, x^2\\, dx$\n",
    "- $\\int_0^1 \\, x^\\frac{1}{2}\\, dx$\n",
    "\n",
    "The integration error is defined as the difference (not the absolute difference) between your numerical results and the exact results. Plot the integration error as a function of $h$ for both integration routines and all listed functions. Comment on the comparison between both integration routines. Does the sign of the error match your expectations? Why?"
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   "source": [
    "# YOUR CODE HERE\n",
    "raise NotImplementedError()"
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