312 lines
95 KiB
Plaintext
312 lines
95 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {
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"collapsed": false
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},
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"source": [
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"## Load Packages"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {
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"collapsed": false,
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"execution": {
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"iopub.execute_input": "2020-09-25T11:59:02.029Z",
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"iopub.status.busy": "2020-09-25T11:59:01.320Z",
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"iopub.status.idle": "2020-09-25T11:59:08.538Z"
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}
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"printyellow (generic function with 1 method)"
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]
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},
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"execution_count": 1,
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"metadata": {
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},
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"output_type": "execute_result"
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}
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],
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"source": [
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"using Printf, Distributions\n",
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"include(\"jlFiles/printmat.jl\")"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {
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"collapsed": false,
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"execution": {
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"iopub.execute_input": "2020-09-25T11:59:08.640Z",
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"iopub.status.busy": "2020-09-25T11:59:08.559Z",
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"iopub.status.idle": "2020-09-25T11:59:25.236Z"
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}
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},
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"outputs": [
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],
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"source": [
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"using Plots\n",
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"gr(size=(480,480))\n",
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"default(fmt = :svg)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"collapsed": true
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},
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"source": [
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"# Find $\\pi$\n",
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"\n",
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"Find $\\pi$ by checking how often a pair of random variables are inside a circle. \n",
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"\n",
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"Clearly, this is not the smartest way to find $pi$, but the steps require some useful skills. Who knows, maybe it's smarter to remember how many letters there are in each of the words in \"How I wish I could calculate pi\" (3.141592).\n",
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"\n",
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"The next few cells will give you a useful starting point."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {
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"collapsed": false,
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"execution": {
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"iopub.execute_input": "2020-09-25T11:59:40.900Z",
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"iopub.status.busy": "2020-09-25T11:59:40.890Z",
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"iopub.status.idle": "2020-09-25T11:59:42.059Z"
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}
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},
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"outputs": [
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{
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"data": {
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|
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},
|
|
"execution_count": 3,
|
|
"metadata": {
|
|
},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"θ = range(0,2*pi,length=201) #angles (in radians) from 0 to 2*pi\n",
|
|
"\n",
|
|
"xCirc = cos.(θ)\n",
|
|
"yCirc = sin.(θ)\n",
|
|
"\n",
|
|
"p1 = plot( xCirc,yCirc,\n",
|
|
" legend=nothing,\n",
|
|
" title = \"A circle with radius of 1 and center at (0,0)\" )\n",
|
|
"display(p1)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 4,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
" x y x.^2 + y.^2 v\n",
|
|
" 0.492 0.961 1.166 0 \n",
|
|
" -0.118 -0.275 0.089 1 \n",
|
|
" -0.227 0.742 0.602 1 \n",
|
|
" -0.558 -0.404 0.475 1 \n",
|
|
" 0.183 0.431 0.219 1 \n",
|
|
" -0.769 -0.847 1.309 0 \n",
|
|
" 0.526 0.731 0.811 1 \n",
|
|
" 0.777 0.530 0.885 1 \n",
|
|
" 0.427 -0.490 0.423 1 \n",
|
|
" -0.940 0.348 1.005 0 \n",
|
|
" 0.999 0.006 0.999 1 \n",
|
|
" -0.833 0.594 1.046 0 \n",
|
|
" -0.885 0.907 1.606 0 \n",
|
|
" 0.665 0.225 0.492 1 \n",
|
|
" -0.326 -0.178 0.138 1 \n",
|
|
" 0.875 -0.778 1.372 0 \n",
|
|
" 0.688 -0.440 0.667 1 \n",
|
|
" 0.630 0.141 0.416 1 \n",
|
|
" 0.092 0.742 0.559 1 \n",
|
|
" -0.447 -0.630 0.596 1 \n",
|
|
" -0.596 0.589 0.703 1 \n",
|
|
" 0.532 0.403 0.446 1 \n",
|
|
" -0.805 -0.740 1.195 0 \n",
|
|
" -0.017 -0.562 0.316 1 \n",
|
|
" -0.899 -0.723 1.332 0 \n",
|
|
"\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"N = 25\n",
|
|
"\n",
|
|
"dist = Uniform(-1,1)\n",
|
|
"x = rand(dist,N) #draw uniformly [-1,1] distributed numbers\n",
|
|
"y = rand(dist,N)\n",
|
|
"\n",
|
|
"v = (x.^2 + y.^2) .<= 1 #true if inside\n",
|
|
"\n",
|
|
"printmat(x,y,x.^2 + y.^2,v,colNames=[\"x\",\"y\",\"x.^2 + y.^2\",\"v\"],width=14)"
|
|
]
|
|
},
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{
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"cell_type": "code",
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"execution_count": 5,
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|
"metadata": {
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"collapsed": false,
|
|
"execution": {
|
|
"iopub.execute_input": "2020-09-25T11:59:43.020Z",
|
|
"iopub.status.busy": "2020-09-25T11:59:43.009Z",
|
|
"iopub.status.idle": "2020-09-25T11:59:43.481Z"
|
|
}
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
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|
|
},
|
|
"execution_count": 5,
|
|
"metadata": {
|
|
},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"p1 = plot( xCirc,yCirc,\n",
|
|
" legend = nothing,\n",
|
|
" title = \"What fraction is inside the circle?\",\n",
|
|
" xlabel = \"x\",\n",
|
|
" ylabel = \"y\" )\n",
|
|
"scatter!(x[v],y[v],color=:blue)\n",
|
|
"scatter!(x[.!v],y[.!v],color=:red)\n",
|
|
"display(p1)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"source": [
|
|
"## Hints\n",
|
|
"\n",
|
|
"Draw many more random numbers (1_000_000 or more) and calculate $pi$ as the fraction of points inside the circle times the area of the box.\n",
|
|
"\n",
|
|
"The area of the box is $2x2=4$, and a pair is inside or on the circle if $x^2 + y^2 \\le 1$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 7,
|
|
"metadata": {
|
|
"collapsed": false,
|
|
"execution": {
|
|
"iopub.execute_input": "2020-09-25T11:59:43.499Z",
|
|
"iopub.status.busy": "2020-09-25T11:59:43.491Z"
|
|
}
|
|
},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
" 0.785 3.141 π\n",
|
|
"\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"N = 25_000_000\n",
|
|
"\n",
|
|
"x = rand(dist,N)\n",
|
|
"y = rand(dist,N)\n",
|
|
"\n",
|
|
"FreqInCircle = mean((x.^2 + y.^2) .<= 1)\n",
|
|
"\n",
|
|
"piEstimate1 = FreqInCircle*4\n",
|
|
"\n",
|
|
"printmat(FreqInCircle,piEstimate1,pi)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 8,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"false"
|
|
]
|
|
},
|
|
"execution_count": 8,
|
|
"metadata": {
|
|
},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"1 > 2"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 0,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [
|
|
],
|
|
"source": [
|
|
]
|
|
}
|
|
],
|
|
"metadata": {
|
|
"@webio": {
|
|
"lastCommId": null,
|
|
"lastKernelId": null
|
|
},
|
|
"anaconda-cloud": {
|
|
},
|
|
"kernelspec": {
|
|
"display_name": "Julia 1.6",
|
|
"env": {
|
|
"JULIA_DEPOT_PATH": "/home/user/.julia/:/ext/julia/depot/",
|
|
"JULIA_PROJECT": "/home/user/.julia/environment/v1.6"
|
|
},
|
|
"language": "julia",
|
|
"metadata": {
|
|
"cocalc": {
|
|
"description": "The Julia Programming Language",
|
|
"priority": 10,
|
|
"url": "https://julialang.org/"
|
|
}
|
|
},
|
|
"name": "julia-1.6",
|
|
"resource_dir": "/ext/jupyter/kernels/julia-1.6"
|
|
},
|
|
"language_info": {
|
|
"file_extension": ".jl",
|
|
"mimetype": "application/julia",
|
|
"name": "julia",
|
|
"version": "1.6.2"
|
|
},
|
|
"nteract": {
|
|
"version": "0.25.0"
|
|
}
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 4
|
|
} |