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| 1 | { |
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1 | { | |
| 2 | "cells": [ |
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2 | "cells": [ | |
| 3 | { |
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3 | { | |
| 4 | "cell_type": "markdown", |
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4 | "cell_type": "markdown", | |
| 5 | "metadata": {}, |
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5 | "metadata": {}, | |
| 6 | "source": [ |
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6 | "source": [ | |
| 7 | "# Basic Numerical Integration: the Trapezoid Rule" |
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7 | "# Basic Numerical Integration: the Trapezoid Rule" | |
| 8 | ] |
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8 | ] | |
| 9 | }, |
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9 | }, | |
| 10 | { |
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10 | { | |
| 11 | "cell_type": "markdown", |
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11 | "cell_type": "markdown", | |
| 12 | "metadata": {}, |
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12 | "metadata": {}, | |
| 13 | "source": [ |
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13 | "source": [ | |
| 14 | "A simple illustration of the trapezoid rule for definite integration:\n", |
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14 | "A simple illustration of the trapezoid rule for definite integration:\n", | |
| 15 | "\n", |
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15 | "\n", | |
| 16 | "$$\n", |
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16 | "$$\n", | |
| 17 | "\\int_{a}^{b} f(x)\\, dx \\approx \\frac{1}{2} \\sum_{k=1}^{N} \\left( x_{k} - x_{k-1} \\right) \\left( f(x_{k}) + f(x_{k-1}) \\right).\n", |
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17 | "\\int_{a}^{b} f(x)\\, dx \\approx \\frac{1}{2} \\sum_{k=1}^{N} \\left( x_{k} - x_{k-1} \\right) \\left( f(x_{k}) + f(x_{k-1}) \\right).\n", | |
| 18 | "$$\n", |
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18 | "$$\n", | |
| 19 | "<br>\n", |
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19 | "<br>\n", | |
| 20 | "First, we define a simple function and sample it between 0 and 10 at 200 points" |
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20 | "First, we define a simple function and sample it between 0 and 10 at 200 points" | |
| 21 | ] |
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21 | ] | |
| 22 | }, |
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22 | }, | |
| 23 | { |
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23 | { | |
| 24 | "cell_type": "code", |
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24 | "cell_type": "code", | |
| 25 | "execution_count": 1, |
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25 | "execution_count": 1, | |
| 26 | "metadata": { |
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26 | "metadata": { | |
| 27 | "collapsed": false |
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27 | "collapsed": false | |
| 28 | }, |
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28 | }, | |
| 29 | "outputs": [], |
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29 | "outputs": [], | |
| 30 | "source": [ |
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30 | "source": [ | |
| 31 | "%matplotlib inline\n", |
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31 | "%matplotlib inline\n", | |
| 32 | "import numpy as np\n", |
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32 | "import numpy as np\n", | |
| 33 | "import matplotlib.pyplot as plt" |
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33 | "import matplotlib.pyplot as plt" | |
| 34 | ] |
|
34 | ] | |
| 35 | }, |
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35 | }, | |
| 36 | { |
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36 | { | |
| 37 | "cell_type": "code", |
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37 | "cell_type": "code", | |
| 38 | "execution_count": 2, |
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38 | "execution_count": 2, | |
| 39 | "metadata": { |
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39 | "metadata": { | |
| 40 |
"collapsed": |
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40 | "collapsed": false | |
| 41 | }, |
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41 | }, | |
| 42 | "outputs": [], |
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42 | "outputs": [], | |
| 43 | "source": [ |
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43 | "source": [ | |
| 44 | "def f(x):\n", |
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44 | "def f(x):\n", | |
| 45 | " return (x-3)*(x-5)*(x-7)+85\n", |
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45 | " return (x-3)*(x-5)*(x-7)+85\n", | |
| 46 | "\n", |
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46 | "\n", | |
| 47 | "x = np.linspace(0, 10, 200)\n", |
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47 | "x = np.linspace(0, 10, 200)\n", | |
| 48 | "y = f(x)" |
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48 | "y = f(x)" | |
| 49 | ] |
|
49 | ] | |
| 50 | }, |
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50 | }, | |
| 51 | { |
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51 | { | |
| 52 | "cell_type": "markdown", |
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52 | "cell_type": "markdown", | |
| 53 | "metadata": {}, |
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53 | "metadata": {}, | |
| 54 | "source": [ |
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54 | "source": [ | |
| 55 | "Choose a region to integrate over and take only a few points in that region" |
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55 | "Choose a region to integrate over and take only a few points in that region" | |
| 56 | ] |
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56 | ] | |
| 57 | }, |
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57 | }, | |
| 58 | { |
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58 | { | |
| 59 | "cell_type": "code", |
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59 | "cell_type": "code", | |
| 60 | "execution_count": 3, |
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60 | "execution_count": 3, | |
| 61 | "metadata": { |
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61 | "metadata": { | |
| 62 | "collapsed": true |
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62 | "collapsed": true | |
| 63 | }, |
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63 | }, | |
| 64 | "outputs": [], |
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64 | "outputs": [], | |
| 65 | "source": [ |
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65 | "source": [ | |
| 66 | "a, b = 1, 9\n", |
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66 | "a, b = 1, 8 # the left and right boundaries\n", | |
| 67 | "xint = x[np.logical_and(x>=a, x<=b)][::30]\n", |
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67 | "N = 5 # the number of points\n", | |
| 68 | "yint = y[np.logical_and(x>=a, x<=b)][::30]" |
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68 | "xint = np.linspace(a, b, N)\n", | |
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69 | "yint = f(xint)" | |||
| 69 | ] |
|
70 | ] | |
| 70 | }, |
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71 | }, | |
| 71 | { |
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72 | { | |
| 72 | "cell_type": "markdown", |
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73 | "cell_type": "markdown", | |
| 73 | "metadata": {}, |
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74 | "metadata": {}, | |
| 74 | "source": [ |
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75 | "source": [ | |
| 75 | "Plot both the function and the area below it in the trapezoid approximation" |
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76 | "Plot both the function and the area below it in the trapezoid approximation" | |
| 76 | ] |
|
77 | ] | |
| 77 | }, |
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78 | }, | |
| 78 | { |
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79 | { | |
| 79 | "cell_type": "code", |
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80 | "cell_type": "code", | |
| 80 | "execution_count": 4, |
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81 | "execution_count": 4, | |
| 81 | "metadata": { |
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82 | "metadata": { | |
| 82 | "collapsed": false |
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83 | "collapsed": false | |
| 83 | }, |
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84 | }, | |
| 84 | "outputs": [ |
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85 | "outputs": [ | |
| 85 | { |
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86 | { | |
| 86 | "data": { |
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87 | "data": { | |
| 87 | "image/png": [ |
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88 | "image/png": [ | |
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316 | "zBizqd9uopRSKqR+7XIxxrSJyN8AL2EtpPUbTeZKKTU4+rWFrpRSyjmDOlN0ACYd9TWe+0SkUkQ+\n", | |
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320 | "xY5nF5DkdBynxfQA8PlO//3inY7ptPhcwCEgx+E4soGdQLRdfhy41eGYSoGPAY+dR1cD+aHqDmYL\n", | |
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321 | "vb8nHfWZMWYNcMzJGEIxxhw2xqyz39cDm7D2UnWUMabBfhuF9Uu53cFwABCRUVhDXO8Fuh0BMMjC\n", | |
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323 | "77dJRyOJiORhfYt4x9lIQERcIrIOqAReNsa853RMwE+AfyQMfrl0YoBXROQvIvLXZ6098MYAR0Rk\n", | |
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324 | "pYh8ICL3iIjP6aBOcz3wiNNBGGMOAP+FtffCQaDWGPOKs1GxAZgrIkn2f7dLgJDj9gczoevT13Mk\n", | |
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325 | "IjHAKuAOu6XuKGNMuzFmMtY/ppkiMt7JeETkUqDKGPMhYdQiBmYbY6YAS4CviMhch+OJAKYCvzDG\n", | |
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328 | "f6dvVKuwEny4WAK8b/99OW0RsMsYU22MaQN+j/XvzFHGmPuMMdONMfOBWqznamcYzIT+F6BQRPLs\n", | |
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331 | "+BOye6q/13LpkgnDSUci8igwH0gWkX3At40xK52MyTYba2OPj0TkQ/vYncaYFx2MKRN4wF4i2QU8\n", | |||
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332 | "box53sF4QgmHbr104A9WLiACeNgY87KzIQHwt8DDdmNqB7DC4XiAT3/pLeLUNpOOMsa8KyKrsNaP\n", | |||
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333 | "arNff+1sVACsEpFkoBW43RhzIlQlnViklFLDxFDYgk4ppVQPaEJXSqlhQhO6UkoNE5rQlVJqmNCE\n", | |||
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334 | "rpRSw4QmdKWUGiY0oSul1DChCV0ppYaJ/w8u1j+cuZEJ1QAAAABJRU5ErkJggg==\n" | |||
| 329 | ], |
|
335 | ], | |
| 330 | "text/plain": [ |
|
336 | "text/plain": [ | |
| 331 |
"<matplotlib.figure.Figure at 0x |
|
337 | "<matplotlib.figure.Figure at 0x7f47175e3898>" | |
| 332 | ] |
|
338 | ] | |
| 333 | }, |
|
339 | }, | |
| 334 | "metadata": {}, |
|
340 | "metadata": {}, | |
| 335 | "output_type": "display_data" |
|
341 | "output_type": "display_data" | |
| 336 | } |
|
342 | } | |
| 337 | ], |
|
343 | ], | |
| 338 | "source": [ |
|
344 | "source": [ | |
| 339 | "plt.plot(x, y, lw=2)\n", |
|
345 | "plt.plot(x, y, lw=2)\n", | |
| 340 |
"plt.axis([0, |
|
346 | "plt.axis([0, 9, 0, 140])\n", | |
| 341 | "plt.fill_between(xint, 0, yint, facecolor='gray', alpha=0.4)\n", |
|
347 | "plt.fill_between(xint, 0, yint, facecolor='gray', alpha=0.4)\n", | |
| 342 | "plt.text(0.5 * (a + b), 30,r\"$\\int_a^b f(x)dx$\", horizontalalignment='center', fontsize=20);" |
|
348 | "plt.text(0.5 * (a + b), 30,r\"$\\int_a^b f(x)dx$\", horizontalalignment='center', fontsize=20);" | |
| 343 | ] |
|
349 | ] | |
| 344 | }, |
|
350 | }, | |
| 345 | { |
|
351 | { | |
| 346 | "cell_type": "markdown", |
|
352 | "cell_type": "markdown", | |
| 347 | "metadata": {}, |
|
353 | "metadata": {}, | |
| 348 | "source": [ |
|
354 | "source": [ | |
| 349 | "Compute the integral both at high accuracy and with the trapezoid approximation" |
|
355 | "Compute the integral both at high accuracy and with the trapezoid approximation" | |
| 350 | ] |
|
356 | ] | |
| 351 | }, |
|
357 | }, | |
| 352 | { |
|
358 | { | |
| 353 | "cell_type": "code", |
|
359 | "cell_type": "code", | |
| 354 | "execution_count": 5, |
|
360 | "execution_count": 5, | |
| 355 | "metadata": { |
|
361 | "metadata": { | |
| 356 | "collapsed": false |
|
362 | "collapsed": false | |
| 357 | }, |
|
363 | }, | |
| 358 | "outputs": [ |
|
364 | "outputs": [ | |
| 359 | { |
|
365 | { | |
| 360 | "name": "stdout", |
|
366 | "name": "stdout", | |
| 361 | "output_type": "stream", |
|
367 | "output_type": "stream", | |
| 362 | "text": [ |
|
368 | "text": [ | |
| 363 |
"The integral is: |
|
369 | "The integral is: 565.2499999999999 +/- 6.275535646693696e-12\n", | |
| 364 |
"The trapezoid approximation with |
|
370 | "The trapezoid approximation with 5 points is: 559.890625\n" | |
| 365 | ] |
|
371 | ] | |
| 366 | } |
|
372 | } | |
| 367 | ], |
|
373 | ], | |
| 368 | "source": [ |
|
374 | "source": [ | |
| 369 | "from __future__ import print_function\n", |
|
375 | "from __future__ import print_function\n", | |
| 370 |
"from scipy.integrate import quad |
|
376 | "from scipy.integrate import quad\n", | |
| 371 |
"integral, error = quad(f, |
|
377 | "integral, error = quad(f, a, b)\n", | |
|
|
378 | "integral_trapezoid = sum( (xint[1:] - xint[:-1]) * (yint[1:] + yint[:-1]) ) / 2\n", | |||
| 372 | "print(\"The integral is:\", integral, \"+/-\", error)\n", |
|
379 | "print(\"The integral is:\", integral, \"+/-\", error)\n", | |
| 373 |
"print(\"The trapezoid approximation with\", len(xint), \"points is:\", |
|
380 | "print(\"The trapezoid approximation with\", len(xint), \"points is:\", integral_trapezoid)" | |
| 374 | ] |
|
381 | ] | |
| 375 | } |
|
382 | } | |
| 376 | ], |
|
383 | ], | |
| 377 |
"metadata": { |
|
384 | "metadata": { | |
|
|
385 | "kernelspec": { | |||
|
|
386 | "display_name": "Python 3", | |||
|
|
387 | "name": "python3" | |||
|
|
388 | }, | |||
|
|
389 | "language_info": { | |||
|
|
390 | "codemirror_mode": { | |||
|
|
391 | "name": "ipython", | |||
|
|
392 | "version": 3 | |||
|
|
393 | }, | |||
|
|
394 | "file_extension": ".py", | |||
|
|
395 | "mimetype": "text/x-python", | |||
|
|
396 | "name": "python", | |||
|
|
397 | "nbconvert_exporter": "python", | |||
|
|
398 | "pygments_lexer": "ipython3", | |||
|
|
399 | "version": "3.4.2" | |||
|
|
400 | } | |||
|
|
401 | }, | |||
| 378 | "nbformat": 4, |
|
402 | "nbformat": 4, | |
| 379 | "nbformat_minor": 0 |
|
403 | "nbformat_minor": 0 | |
| 380 | } No newline at end of file |
|
404 | } | |
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