{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Truncation Error vs Rounding Error\n",
    "\n",
    "Copyright (C) 2010-2020 Luke Olson<br>\n",
    "Copyright (C) 2020 Andreas Kloeckner\n",
    "\n",
    "<details>\n",
    "<summary>MIT License</summary>\n",
    "Permission is hereby granted, free of charge, to any person obtaining a copy\n",
    "of this software and associated documentation files (the \"Software\"), to deal\n",
    "in the Software without restriction, including without limitation the rights\n",
    "to use, copy, modify, merge, publish, distribute, sublicense, and/or sell\n",
    "copies of the Software, and to permit persons to whom the Software is\n",
    "furnished to do so, subject to the following conditions:\n",
    "\n",
    "The above copyright notice and this permission notice shall be included in\n",
    "all copies or substantial portions of the Software.\n",
    "\n",
    "THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n",
    "IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n",
    "FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE\n",
    "AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n",
    "LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,\n",
    "OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN\n",
    "THE SOFTWARE.\n",
    "</details>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this notebook, we'll investigate two common sources of error: Truncation error and rounding error."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as pt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task:** Approximate a function (here: a parabola, by a line)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-28.0, 2.2497704315886136)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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30312d/ac3MOcf8yhdrnaf7j/yy/hqafg00+haVOXihQpoNRPyUWz1vLxho+pOaYmta6u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\nCnkRkRCmkBcRCWEKeRGREKaQFxEJYf8PBAJm/JrSotYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f35e995a7f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "center = -1\n",
    "width = 6\n",
    "\n",
    "def f(x):\n",
    "    return - x**2 + 3*x\n",
    "\n",
    "def df(x):\n",
    "    return -2*x + 3\n",
    "\n",
    "grid = np.linspace(center-width/2, center+width/2, 100)\n",
    "\n",
    "fx = f(grid)\n",
    "pt.plot(grid, fx)\n",
    "pt.plot(grid, f(center) + df(center) * (grid-center))\n",
    "\n",
    "pt.xlim([grid[0], grid[-1]])\n",
    "pt.ylim([np.min(fx), np.max(fx)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* What's the error we see?\n",
    "* What if we make `width` smaller?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.8.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}