{ "metadata": { "name": "", "signature": "sha256:6b68098d91d017a5150871b0459fce035287ed523f3112af21ecc50abed9d15e" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Error in finite difference approximation of derivative" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Compute derivative of\n", "$$\n", "f(x) = \\sin(x)\n", "$$\n", "at $x=2\\pi$ using finite difference\n", "$$\n", "\\frac{f(x+h) - f(x)}{h}\n", "$$\n", "for $h=10^{-1},10^{-2},\\ldots,10^{-14}$." ] }, { "cell_type": "code", "collapsed": false, "input": [ "from numpy import sin,arange,zeros,pi,abs\n", "from matplotlib.pyplot import loglog,xlabel,ylabel" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 24 }, { "cell_type": "code", "collapsed": false, "input": [ "def f(x):\n", " return sin(x)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 34 }, { "cell_type": "code", "collapsed": false, "input": [ "h = 10.0**arange(-1,-15,-1)\n", "df= zeros(len(h))\n", "x = 2.0*pi\n", "f0= f(x)\n", "for i in range(len(h)):\n", " f1 = f(x+h[i])\n", " df[i] = (f1 - f0)/h[i]\n", "loglog(h,abs(df-1.0),'o-')\n", "xlabel('h')\n", "ylabel('Error in derivative')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 35, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "svg": [ "\n", "\n", "\n", "\n" ], "text": [ "" ] } ], "prompt_number": 35 } ], "metadata": {} } ] }