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| { | |
| "cells": [ | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "# Solution to Form #1" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 21, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "ans = 14\r\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "[1,2,3]*[1;2;3]" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 8, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "[1,2,3]*[1;2;3]=?" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": null, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "# The first source of error is roundoff error\n", | |
| "## Just storing a number in a computer requires rounding" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 11, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "realmax = 1.79769313486231570815e+308\n", | |
| "realmin = 2.22507385850720138309e-308\n", | |
| "maximum relative error = 2.22044604925031308085e-16\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "fprintf('realmax = %1.20e\\n',realmax)\n", | |
| "fprintf('realmin = %1.20e\\n',realmin)\n", | |
| "fprintf('maximum relative error = %1.20e\\n',eps)\n" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 13, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "ans = 1\r\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "s=1;\n", | |
| "for i=1:1000\n", | |
| " s=s+eps/10;\n", | |
| "end\n", | |
| "s==1" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "# Freefall Model (revisited)\n", | |
| "## Octave solution (will run same on Matlab)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Set default values in Octave for linewidth and text size" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 14, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "%plot --format svg" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 15, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "set (0, \"defaultaxesfontname\", \"Helvetica\")\n", | |
| "set (0, \"defaultaxesfontsize\", 18)\n", | |
| "set (0, \"defaulttextfontname\", \"Helvetica\")\n", | |
| "set (0, \"defaulttextfontsize\", 18) \n", | |
| "set (0, \"defaultlinelinewidth\", 4)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Define time from 0 to 12 seconds with `N` timesteps \n", | |
| "function defined as `freefall`" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 16, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "function [v_analytical,v_terminal,t]=freefall(N)\n", | |
| " t=linspace(0,12,N)';\n", | |
| " c=0.25; m=60; g=9.81; v_terminal=sqrt(m*g/c);\n", | |
| "\n", | |
| " v_analytical = v_terminal*tanh(g*t/v_terminal);\n", | |
| " v_numerical=zeros(length(t),1);\n", | |
| " delta_time =diff(t);\n", | |
| " for i=1:length(t)-1\n", | |
| " v_numerical(i+1)=v_numerical(i)+(g-c/m*v_numerical(i)^2)*delta_time(i);\n", | |
| " end\n", | |
| " % Print values near 0,2,4,6,8,10,12 seconds\n", | |
| " indices = round(linspace(1,length(t),7));\n", | |
| " fprintf('time (s)|vel analytical (m/s)|vel numerical (m/s)\\n')\n", | |
| " fprintf('-----------------------------------------------\\n')\n", | |
| " M=[t(indices),v_analytical(indices),v_numerical(indices)];\n", | |
| " fprintf('%7.1f | %18.2f | %15.2f\\n',M(:,1:3)');\n", | |
| " plot(t,v_analytical,'-',t,v_numerical,'o-')\n", | |
| "end\n", | |
| " " | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 19, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "time (s)|vel analytical (m/s)|vel numerical (m/s)\n", | |
| "-----------------------------------------------\n", | |
| " 0.0 | 0.00 | 0.00\n", | |
| " 2.0 | 18.76 | 18.82\n", | |
| " 4.0 | 32.64 | 32.80\n", | |
| " 6.1 | 40.79 | 40.97\n", | |
| " 8.0 | 44.80 | 44.94\n", | |
| " 10.0 | 46.84 | 46.93\n", | |
| " 12.0 | 47.77 | 47.82\n" | |
| ] | |
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| "[v_analytical,v_terminal,t]=freefall(120);" | |
| ] | |
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| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "# Types of error\n", | |
| "## Freefall is example of \"truncation error\"\n", | |
| "### Truncation error results from approximating exact mathematical procedure" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "We approximated the derivative as $\\delta v/\\delta t\\approx\\Delta v/\\Delta t$\n", | |
| "\n", | |
| "Can reduce error by decreasing step size -> $\\Delta t$=`delta_time`" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## Another example of truncation error is a Taylor series (or Maclaurin if centered at a=0)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Taylor series:\n", | |
| "$f(x)=f(a)+f'(a)(x-a)+\\frac{f''(a)}{2!}(x-a)^{2}+\\frac{f'''(a)}{3!}(x-a)^{3}+...$\n", | |
| "\n", | |
| "We can approximate the next value in a function by adding Taylor series terms:\n", | |
| "\n", | |
| "|Approximation | formula |\n", | |
| "|---|-------------------------|\n", | |
| "|$0^{th}$-order | $f(x_{i+1})=f(x_{i})+R_{1}$ |\n", | |
| "|$1^{st}$-order | $f(x_{i+1})=f(x_{i})+f'(x_{i})h+R_{2}$ |\n", | |
| "|$2^{nd}$-order | $f(x_{i+1})=f(x_{i})+f'(x_{i})h+\\frac{f''(x_{i})}{2!}h^{2}+R_{3}$|\n", | |
| "|$n^{th}$-order | $f(x_{i+1})=f(x_{i})+f'(x_{i})h+\\frac{f''(x_{i})}{2!}h^{2}+...\\frac{f^{(n)}}{n!}h^{n}+R_{n}$|\n", | |
| "\n", | |
| "Where $R_{n}=\\frac{f^{(n+1)}(\\xi)}{(n+1)!}h^{n+1}$ is the error associated with truncating the approximation at order $n$.\n", | |
| "\n", | |
| "The $n^{th}$-order approximation estimates that the unknown function, $f(x)$, is equal to an $n^{th}$-order polynomial. \n", | |
| "\n", | |
| "In the Freefall example, we estimated the function with a $1^{st}$-order approximation, so \n", | |
| "\n", | |
| "$v(t_{i+1})=v(t_{i})+v'(t_{i})(t_{i+1}-t_{i})+R_{1}$\n", | |
| "\n", | |
| "$v'(t_{i})=\\frac{v(t_{i+1})-v(t_{i})}{t_{i+1}-t_{i}}-\\frac{R_{1}}{t_{i+1}-t_{i}}$\n", | |
| "\n", | |
| "$\\frac{R_{1}}{t_{i+1}-t_{i}}=\\frac{v''(\\xi)}{2!}(t_{i+1}-t_{i})$\n", | |
| "\n", | |
| "or the truncation error for a first-order Taylor series approximation is\n", | |
| "\n", | |
| "$\\frac{R_{1}}{t_{i+1}-t_{i}}=O(\\Delta t)$\n" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "1. digital representation of a number is rarely exact\n", | |
| "\n", | |
| "2. arithmetic (+,-,/,\\*) causes roundoff error" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 20, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "3.14159265358979311600\n", | |
| "3.14159274101257324219\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "fprintf('%1.20f\\n',double(pi))\n", | |
| "fprintf('%1.20f\\n',single(pi))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": null, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [] | |
| } | |
| ], | |
| "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", | |
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| "version": "0.19.14" | |
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