making prompt_tolkit history search configurable
making prompt_tolkit history search configurable

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Trapezoid Rule.ipynb
971 lines | 73.7 KiB | text/plain | TextLexer
Brian E. Granger
Moving examples.
r16076 {
Min RK
upate exmaple notebooks to nbformat v4
r18669 "cells": [
Brian E. Granger
Moving examples.
r16076 {
Min RK
upate exmaple notebooks to nbformat v4
r18669 "cell_type": "markdown",
"metadata": {},
"source": [
"# Basic Numerical Integration: the Trapezoid Rule"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"A simple illustration of the trapezoid rule for definite integration:\n",
"\n",
"$$\n",
"\\int_{a}^{b} f(x)\\, dx \\approx \\frac{1}{2} \\sum_{k=1}^{N} \\left( x_{k} - x_{k-1} \\right) \\left( f(x_{k}) + f(x_{k-1}) \\right).\n",
"$$\n",
"<br>\n",
"First, we define a simple function and sample it between 0 and 10 at 200 points"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"%matplotlib inline\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
Andreas Amann
use better integration boundaries...
r19878 "collapsed": false
Min RK
upate exmaple notebooks to nbformat v4
r18669 },
"outputs": [],
"source": [
"def f(x):\n",
" return (x-3)*(x-5)*(x-7)+85\n",
"\n",
"x = np.linspace(0, 10, 200)\n",
"y = f(x)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Choose a region to integrate over and take only a few points in that region"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
Andreas Amann
use better integration boundaries...
r19878 "a, b = 1, 8 # the left and right boundaries\n",
"N = 5 # the number of points\n",
"xint = np.linspace(a, b, N)\n",
"yint = f(xint)"
Min RK
upate exmaple notebooks to nbformat v4
r18669 ]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Plot both the function and the area below it in the trapezoid approximation"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
Brian E. Granger
Moving examples.
r16076 {
Min RK
upate exmaple notebooks to nbformat v4
r18669 "data": {
"image/png": [
Min RK
pass on IPython Kernel example notebooks...
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r18669 ],
"text/plain": [
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r20547 "<matplotlib.figure.Figure at 0x10664cfd0>"
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r18669 ]
},
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r20547 "metadata": {
"image/png": {
"height": 381,
"width": 595
}
},
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r18669 "output_type": "display_data"
}
],
"source": [
"plt.plot(x, y, lw=2)\n",
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r19878 "plt.axis([0, 9, 0, 140])\n",
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r18669 "plt.fill_between(xint, 0, yint, facecolor='gray', alpha=0.4)\n",
"plt.text(0.5 * (a + b), 30,r\"$\\int_a^b f(x)dx$\", horizontalalignment='center', fontsize=20);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Compute the integral both at high accuracy and with the trapezoid approximation"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
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r16076 {
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r18669 "name": "stdout",
"output_type": "stream",
"text": [
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r19878 "The integral is: 565.2499999999999 +/- 6.275535646693696e-12\n",
"The trapezoid approximation with 5 points is: 559.890625\n"
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r16076 ]
}
],
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r18669 "source": [
"from __future__ import print_function\n",
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r19878 "from scipy.integrate import quad\n",
"integral, error = quad(f, a, b)\n",
"integral_trapezoid = sum( (xint[1:] - xint[:-1]) * (yint[1:] + yint[:-1]) ) / 2\n",
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r18669 "print(\"The integral is:\", integral, \"+/-\", error)\n",
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r19878 "print(\"The trapezoid approximation with\", len(xint), \"points is:\", integral_trapezoid)"
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r18669 ]
Brian E. Granger
Moving examples.
r16076 }
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r18669 ],
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r19878 "metadata": {
"kernelspec": {
"display_name": "Python 3",
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r20278 "language": "python",
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r19878 "name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.4.2"
}
},
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r18669 "nbformat": 4,
"nbformat_minor": 0
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r20278 }