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ME3255F2017/02_Roundoff-Truncation errors/.ipynb_checkpoints/lecture_02-checkpoint.ipynb

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"%plot --format svg"
]
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"cell_type": "markdown",
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"source": [
"# Errors in Numerical Modeling\n",
"\n",
"## 1 - Roundoff \n",
"## 2 - Truncation"
]
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"|Question|\tAnswered Right|\tAnswered Wrong\t|Total % Right\n",
"|-----------------------------|---|---|---|\n",
"|What is the 32-bit representation of e \"exp(1)\"?\t|37|\t26|\t58.7%|\n",
"|The command \">10+eps-10\" results in:\t|24|\t39\t|38.1%|\n",
"|Using a first-order Taylor series expansion and a time step, h, of 0.1 s, the order of expected truncation error is:\t|37\t|26|\t58.7%|"
]
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{
"cell_type": "markdown",
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"source": [
"# 1- Roundoff"
]
},
{
"cell_type": "markdown",
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"slide_type": "subslide"
}
},
"source": [
"## Just storing a number in a computer requires rounding\n",
"\n",
"1. digital representation of a number is rarely exact\n",
"\n",
"2. arithmetic (+,-,/,\\*) causes roundoff error"
]
},
{
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"text": [
"3.14159265358979311600\n",
"3.14159274101257324219\n"
]
}
],
"source": [
"fprintf('%1.20f\\n',double(pi)) % 64-bit\n",
"fprintf('%1.20f\\n',single(pi)) % 32-bit"
]
},
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"execution_count": 36,
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"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": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Machine epsilon\n",
"\n",
"Smallest number that can be added to 1 and change the value in a computer"
]
},
{
"cell_type": "code",
"execution_count": 41,
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{
"name": "stdout",
"output_type": "stream",
"text": [
"ans = 0\n",
"ans = 1.1102e-12\n"
]
}
],
"source": [
"s=1;\n",
"for i=1:10000\n",
" s=s+eps/2;\n",
"end\n",
"s-1\n",
"10000*eps/2"
]
},
{
"cell_type": "markdown",
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"slideshow": {
"slide_type": "slide"
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"source": [
"# 2- Truncation error\n",
"## Freefall is example of \"truncation error\"\n",
"### Truncation error results from approximating exact mathematical procedure\n",
"\n",
"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": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Truncation error as a Taylor series "
]
},
{
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"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 $R_{n}=O(h^{n+1})$ is the error associated with truncating the approximation at order $n$."
]
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"![3](https://media.giphy.com/media/xA7G2n20MzTOw/giphy.gif)\n",
"\n",
"$n^{th}$-order approximation equivalent to \n",
"an $n^{th}$-order polynomial. "
]
},
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"source": [
"# Thanks"
]
},
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"slideshow": {
"slide_type": "slide"
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},
"source": [
"# Freefall Model (revisited)\n",
"## Octave solution (will run same on Matlab)\n",
"\n",
"## Create function called `freefall.m`\n",
"\n",
"Define time from 0 to 12 seconds with `N` timesteps \n",
"function defined as `freefall`\n",
"\n",
"m=60 kg, c=0.25 kg/m"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Set default values in Octave for linewidth and text size"
]
},
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"set (0, \"defaultaxesfontsize\", 18)\n",
"set (0, \"defaulttextfontsize\", 18) \n",
"set (0, \"defaultlinelinewidth\", 4)"
]
},
{
"cell_type": "markdown",
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},
"source": [
"## Freefall example\n",
"\n",
"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",
"$R_{1}=O(\\Delta t^{2})$\n"
]
},
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"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",
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"time (s)|vel analytical (m/s)|vel numerical (m/s)\n",
"-----------------------------------------------\n",
" 0.0 | 0.00 | 0.00\n",
" 2.2 | 20.12 | 20.88\n",
" 4.4 | 34.33 | 36.32\n",
" 6.5 | 42.10 | 44.08\n",
" 7.6 | 44.29 | 45.95\n",
" 9.8 | 46.72 | 47.69\n",
" 12.0 | 47.77 | 48.26\n"
]
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