{ "cells": [ { "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", "
\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": { "collapsed": false }, "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": [ "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)" ] }, { "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": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAABKcAAAL7CAYAAAAoHObLAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAAWJQAAFiUBSVIk8AAAIABJREFUeJzs3Xl8Y3d97//3kSxvs9mzL8lMMjOZSSaTPZokJBOI5tfS\n", "ABfK0hPWtKUXSOlKi4ByS2+BB8tFQG9a2gu0hVLupeRQthZStggCpSwKEBISkkkyOvKMxx7b0pHs\n", "sbXr/P6QBpQztmexrWPZr+fjcR6yP9+vdD55KCTkPd/v9xiu6woAAAAAAADwQ8DvBgAAAAAAALB8\n", "EU4BAAAAAADAN4RTAAAAAAAA8A3hFAAAAAAAAHxDOAUAAAAAAADfEE4BAAAAAADAN4RTAAAAAAAA\n", "8A3hFAAAAAAAAHxDOAUAAAAAAADfEE4BAAAAAADAN4RTAAAAAAAA8A3hFAAAAAAAAHxDOAUAAAAA\n", "AADfEE4BAAAAAADANx1z/QDTNLdK+rKkiyzL6j+L+V+S9BxJb7As6+5pxl8i6c2S9kqalHS/pLda\n", "lnVkrr0CAAAAAABgcZnTyinTNPdL+r6kKyW5ZzH/v0m6vvHrafNN03y9pHskPSDp5ZKiknZL+q5p\n", "mjvm0isAAAAAAAAWn/NeOWWa5iFJn5X0mKQvSHrVGeZ3S/orSW+R9LFpxrdJer+kd1uW9bam+udU\n", "D6s+KOnF59svAAAAAAAAFp+5rJy6U9KPJB2SlDmL+W+WlLYs659m+byypPc0Fy3LmpL0XknPN01z\n", 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"output_type": "stream", "text": [ "The integral is: 565.2499999999999 +/- 6.275535646693696e-12\n", "The trapezoid approximation with 5 points is: 559.890625\n" ] } ], "source": [ "from __future__ import print_function\n", "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", "print(\"The integral is:\", integral, \"+/-\", error)\n", "print(\"The trapezoid approximation with\", len(xint), \"points is:\", integral_trapezoid)" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "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" } }, "nbformat": 4, "nbformat_minor": 0 }