{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%plot gnuplot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$f'(x_0) \\approx \\frac{f(x_0 + h) -f(x_0)}{h} + O(h)$\n",
    "\n",
    "error = $|f'(x_0) - \\frac{f(x_0 + h) -f(x_0)}{h}| \\approx O(h)$ \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "c1 =\n",
      "\n",
      "   1.002263431950136  -0.740850628132612\n",
      "\n"
     ]
    }
   ],
   "source": [
    "clear all\n",
    "format long\n",
    "f = @(x) sin(x);\n",
    "x0=1.2;\n",
    "f0=sin(x0);\n",
    "fp=cos(x0);\n",
    "i=-6:0.5:0;\n",
    "h=10.^i;\n",
    "err=abs(fp-(sin(x0+h)-f0)./h);\n",
    "subplot(1,2,1)\n",
    "loglog(h,err,'-*');\n",
    "xlabel('h')\n",
    "ylabel('error = |f(x_0) (exact) - f(x_0) (numeric)|')\n",
    "c1 = polyfit(log(h),log(abs(err)),1)\n",
    "hold on\n",
    "subplot(1,2,2)\n",
    "\n",
    "plot(log(h), c1(1)*log(h)+c1(2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p> c1(1) = 1.002263431950136 means the order of error is $\\approx 1$.</p>\n",
    "<br>\n",
    "<img src=\"images/derivativeorder.png\" alt=\"Derivative\" style=\"width:500px;height:300px;\">"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "f = @(x) sin(x);\n",
    "x0 = 0:0.001:20;\n",
    "f0 = f(x0);\n",
    "J = length(f0);\n",
    "dx=x0(2)-x0(1);\n",
    "slope = [(f0(2:J)-f0(1:J-1))/dx, (f0(J)-f0(J-1))/dx]; % Forward\n",
    "x1 = 0:0.5:20;\n",
    "figure;\n",
    "plot(x0,slope,'r',x1,cos(x1),'ob')\n",
    "legend('exact','numerical')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "<img src=\"images/derivativevalue.png\" alt=\"Derivative\" style=\"width:500px;height:300px;\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Try yourself\n",
    "Find second derivative, $f''(x_0) \\approx \\frac{f(x_0 + h) -2 f(x_0) +f(x_0 - h)}{h^2} + O(h^2)$. <br>\n",
    "Verify the order of error. "
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Octave",
   "language": "octave",
   "name": "octave"
  },
  "language_info": {
   "file_extension": ".m",
   "help_links": [
    {
     "text": "MetaKernel Magics",
     "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
    }
   ],
   "mimetype": "text/x-octave",
   "name": "octave",
   "version": "0.16.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}