314 lines
8.3 KiB
Plaintext
314 lines
8.3 KiB
Plaintext
{
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"cells": [
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"cell_type": "markdown",
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"source": [
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"# CDS: Numerical Methods Assignments\n",
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"\n",
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"- 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",
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"\n",
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"- Solutions must be submitted via the Jupyter Hub.\n",
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"\n",
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"- Make sure you fill in any place that says `YOUR CODE HERE` or \"YOUR ANSWER HERE\".\n",
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"\n",
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"## Submission\n",
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"\n",
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"1. Name all team members in the the cell below\n",
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"2. make sure everything runs as expected\n",
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"3. **restart the kernel** (in the menubar, select Kernel$\\rightarrow$Restart)\n",
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"4. **run all cells** (in the menubar, select Cell$\\rightarrow$Run All)\n",
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"5. Check all outputs (Out[\\*]) for errors and **resolve them if necessary**\n",
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"6. submit your solutions **in time (before the deadline)**"
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]
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},
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{
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"cell_type": "raw",
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"metadata": {},
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"source": [
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"team_members = \"\""
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"deletable": false,
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"editable": false,
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"nbgrader": {
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"cell_type": "markdown",
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"checksum": "177d489a4104e3c95a4de1a4c7768c01",
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"grade": false,
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"grade_id": "cell-1e89a94d71771bb6",
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"locked": true,
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}
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},
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"source": [
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"## Composite Numerical Integration: Trapezoid and Simpson Rules\n",
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"\n",
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"In the following we will implement the composite trapezoid and Simpson rules to calculate definite integrals. These rules are defined by\n",
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"\n",
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"\\begin{align}\n",
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"\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",
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" &\\text{trapezoid} \\\\\n",
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" &\\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",
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" &\\text{Simpson}\n",
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"\\end{align}\n",
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" \n",
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"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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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"deletable": false,
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"nbgrader": {
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"cell_type": "code",
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"checksum": "a60a63e0450a3157dd421b394288f18a",
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"grade": true,
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"grade_id": "cell-44d29c12deac7ed7",
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"locked": false,
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"points": 0,
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"schema_version": 3,
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"solution": true,
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"task": false
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}
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},
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"outputs": [],
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"source": [
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"# Import packages here ...\n",
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"\n",
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"# YOUR CODE HERE\n",
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"raise NotImplementedError()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"deletable": false,
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"editable": false,
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"nbgrader": {
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"cell_type": "markdown",
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"checksum": "e11bb7c15d840e7a9397f209769ebb66",
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"grade": false,
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"grade_id": "cell-ce9a56067e726f36",
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"locked": true,
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"schema_version": 3,
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"solution": false,
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"task": false
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}
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},
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"source": [
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"### Task 1\n",
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"\n",
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"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",
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"\n",
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"Try both even and odd $n$. What do you see? Why?\n",
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"\n",
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"Hint: go to the Scipy documentation. How are even and odd $n$ handled there?"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"deletable": false,
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"nbgrader": {
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"cell_type": "code",
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"checksum": "2bc6bd3985c2b7ab4ab051ebe94496f9",
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"grade": true,
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"grade_id": "cell-59f0de06f77dce3e",
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"locked": false,
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"points": 6,
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"schema_version": 3,
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"solution": true,
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"task": false
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}
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},
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"outputs": [],
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"source": [
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"def trapz(yk, dx):\n",
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" # YOUR CODE HERE\n",
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" raise NotImplementedError()\n",
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" \n",
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"def simps(yk, dx):\n",
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" # YOUR CODE HERE\n",
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" raise NotImplementedError()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"deletable": false,
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"nbgrader": {
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"cell_type": "code",
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"checksum": "9599217f233689affb19148157e62b41",
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"grade": true,
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"grade_id": "cell-ff04b1d785ea895f",
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"locked": false,
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"points": 1,
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"schema_version": 3,
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"solution": true,
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"task": false
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}
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},
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"outputs": [],
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"source": [
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"# Compare your results here ...\n",
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"\n",
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"# YOUR CODE HERE\n",
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"raise NotImplementedError()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"deletable": false,
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"editable": false,
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"nbgrader": {
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"cell_type": "markdown",
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"checksum": "f3a2f1f2b9ba3ffeb8646c346797d95a",
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"grade": false,
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"grade_id": "cell-1a7e33464e3be83b",
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"locked": true,
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"schema_version": 3,
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"solution": false,
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"task": false
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}
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},
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"source": [
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"### Task 2\n",
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"\n",
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"Implement at least one test function for each of your integration functions."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"deletable": false,
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"nbgrader": {
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"cell_type": "code",
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"checksum": "c3b7ee0d3ced97054590e89bba97e031",
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"grade": true,
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"grade_id": "cell-d8f2e0aa55860e08",
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"locked": false,
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"points": 6,
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"schema_version": 3,
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"solution": true,
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"task": false
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}
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},
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"outputs": [],
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"source": [
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"def test_trapz():\n",
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" # YOUR CODE HERE\n",
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" raise NotImplementedError()\n",
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" \n",
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"def test_simps():\n",
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" # YOUR CODE HERE\n",
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" raise NotImplementedError()\n",
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" \n",
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"test_trapz()\n",
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"test_simps()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"deletable": false,
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"editable": false,
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"nbgrader": {
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"cell_type": "markdown",
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"checksum": "ead1d68798b82e5c9c5dba354a255abb",
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"grade": false,
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"grade_id": "cell-71d20f6b94c6ed05",
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"locked": true,
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"schema_version": 3,
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"solution": false,
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"task": false
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}
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},
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"source": [
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"### Task 3\n",
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"\n",
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"Study the accuracy of these integration routines by calculating the following integrals for a variety of step sizes $h$:\n",
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"\n",
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"- $\\int_0^1 \\, x\\, dx$\n",
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"- $\\int_0^1 \\, x^2\\, dx$\n",
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"- $\\int_0^1 \\, x^\\frac{1}{2}\\, dx$\n",
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"\n",
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"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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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"deletable": false,
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"nbgrader": {
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"cell_type": "code",
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"checksum": "90eaf3d9beb2347589518aba4e8ad3c4",
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"grade": true,
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"grade_id": "cell-b0bb51b7eae7769b",
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"locked": false,
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"points": 4,
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"schema_version": 3,
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"solution": true,
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"task": false
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}
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},
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"outputs": [],
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"source": [
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"# YOUR CODE HERE\n",
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"raise NotImplementedError()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"outputs": [],
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.8.10"
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}
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