{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Eliminate from (general) A to (upper triangular) U"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Demonstrate Gaussian Elimination (page 51 of GS)\n",
"\n",
"(Reminder: Julia programming syntax will not be tested in this class, but that doesn't mean you can't learn from\n",
"watching the code being executed)\n",
"\n",
"(If you are familiar with Jupyter notebooks and wish to execute yourself, you can download this notebook \n",
"using the download button in the upper right, but WARNING: right click to \"download as\", or OPTION-click (MAC) or ALT-Click (Linux and Windows I think) on the download icon, or you won't get an ipynb file.) You can then drag into Jupyter."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"using Plots"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"lookat (generic function with 1 method)"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# nice viz for matrices\n",
"function lookat(A; redrow=0, rounding=2, showtext=true)\n",
" n = size(A,1)\n",
" plot(legend=false, axis=false)\n",
" rowcolor = redrow > 0 ? :red : :black \n",
" for i=1:n, j=1:n \n",
" scatter!( [j],[i], ann= showtext ? (j,i,round(A[i,j],digits=rounding), :white ) : (j,i,\"\") ,\n",
" color=abs(A[i,j]) > .0001 ? (i==redrow ? rowcolor : :black) : :white, \n",
" marker=:square, markersize=30, aspectratio=1, yflip=true, yaxis=[.5,n+.5],xaxis=[.5,n+.5])\n",
" end\n",
" plot!()\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4×4 Array{Float64,2}:\n",
" 4.0 9.0 4.0 9.0\n",
" 4.0 7.0 6.0 6.0\n",
" 2.0 9.0 8.0 6.0\n",
" 7.0 1.0 6.0 8.0"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = rand(1.0:9,4,4)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4×4 Array{Float64,2}:\n",
" 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L = fill(0.0,4,4)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"lookat(A)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## What multiple of row 1 must we subtract from row 2 to zero out the (2,1) entry?"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1.0"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L[2,1] = A[2,1] / A[1,1]"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# A[2,:] = A[2,:] - L[2,1] * A[1,:]\n",
"A[2,:] -= L[2,1] * A[1,:] # subtract that multiple of the first row from the second\n",
"lookat(A, redrow=2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## What multiple of row 1 must we subtract from row 3 to zero out the (3,1) entry?"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.5"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L[3,1] = A[3,1] / A[1,1]"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A[3,:] -= L[3,1] * A[1,:]\n",
"lookat(A, redrow=3)"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L[4,1] = A[4,1] / A[1,1]\n",
"A[4,:] -= L[4,1] * A[1,:]\n",
"lookat(A, redrow=4)"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-2.25"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L[3,2] = A[3,2] / A[2,2]"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A[3,:] -= L[3,2] * A[2,:]; lookat(A, redrow=3)"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L[4,2] = A[4,2] / A[2,2]; A[4,:] -= L[4,2] * A[2,:]; lookat(A, redrow=4)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
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},
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"n = 5\n",
"A = randn(n,n)\n",
"L = fill(0.0,n,n)\n",
"display(lookat(A))\n",
"for j=1:n, i=(j+1):n\n",
" L[i,j] = A[i,j]/A[j,j]\n",
" A[i,:] -= L[i,j] * A[j,:]\n",
" display(lookat(A,redrow=i))\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"n = 5\n",
"A = randn(n,n)\n",
"L = fill(0.0,n,n)\n",
"Akeep = [copy(A)]\n",
"row = [0]\n",
"display(lookat(A))\n",
"for j=1:n, i=(j+1):n\n",
" L[i,j] = A[i,j]/A[j,j]\n",
" A[i,:] -= L[i,j] * A[j,:]\n",
" push!(Akeep,copy(A))\n",
" push!(row,i)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
" Unable to load WebIO. Please make sure WebIO works for your Jupyter client.\n",
" For troubleshooting, please see \n",
" the WebIO/IJulia documentation.\n",
" \n",
"\n"
],
"text/plain": [
"HTML{String}(\"\\n\\n Unable to load WebIO. Please make sure WebIO works for your Jupyter client.\\n For troubleshooting, please see \\n the WebIO/IJulia documentation.\\n \\n\\n\")"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"using Interact"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
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"application/vnd.webio.node+json": {
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"Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Scope(Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Node{WebIO.DOM}(WebIO.DOM(:html, :label), Any[nothing], Dict{Symbol,Any}(:className => \"interact \",:style => Dict{Any,Any}(:padding => \"5px 10px 0px 10px\")))], Dict{Symbol,Any}(:className => \"interact-flex-row-left\")), Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Node{WebIO.DOM}(WebIO.DOM(:html, :input), Any[], Dict{Symbol,Any}(:max => 11,:min => 1,:attributes => Dict{Any,Any}(:type => \"range\",Symbol(\"data-bind\") => \"numericValue: index, valueUpdate: 'input', event: {change: function (){this.changes(this.changes()+1)}}\",\"orient\" => \"horizontal\"),:step => 1,:className => \"slider slider is-fullwidth\",:style => Dict{Any,Any}()))], Dict{Symbol,Any}(:className => \"interact-flex-row-center\")), Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Node{WebIO.DOM}(WebIO.DOM(:html, :p), Any[], Dict{Symbol,Any}(:attributes => Dict(\"data-bind\" => \"text: formatted_val\")))], Dict{Symbol,Any}(:className => \"interact-flex-row-right\"))], Dict{Symbol,Any}(:className => \"interact-flex-row interact-widget\")), Dict{String,Tuple{Observables.AbstractObservable,Union{Nothing, Bool}}}(\"changes\" => (Observable{Int64} with 1 listeners. Value:\n",
"0, nothing),\"index\" => (Observable{Int64} with 2 listeners. Value:\n",
"1, nothing)), Set(String[]), nothing, Asset[Asset(\"js\", \"knockout\", \"/Users/alanedelman/.julia/packages/Knockout/1sDlc/src/../assets/knockout.js\"), Asset(\"js\", \"knockout_punches\", \"/Users/alanedelman/.julia/packages/Knockout/1sDlc/src/../assets/knockout_punches.js\"), Asset(\"js\", nothing, \"/Users/alanedelman/.julia/packages/InteractBase/9mFwe/src/../assets/all.js\"), Asset(\"css\", nothing, \"/Users/alanedelman/.julia/packages/InteractBase/9mFwe/src/../assets/style.css\"), Asset(\"css\", nothing, \"/Users/alanedelman/.julia/packages/Interact/SbgIk/src/../assets/bulma_confined.min.css\")], Dict{Any,Any}(\"changes\" => Any[WebIO.JSString(\"(function (val){return (val!=this.model[\\\"changes\\\"]()) ? (this.valueFromJulia[\\\"changes\\\"]=true, this.model[\\\"changes\\\"](val)) : undefined})\")],\"index\" => Any[WebIO.JSString(\"(function (val){return (val!=this.model[\\\"index\\\"]()) ? (this.valueFromJulia[\\\"index\\\"]=true, this.model[\\\"index\\\"](val)) : undefined})\")]), WebIO.ConnectionPool(Channel{Any}(sz_max:32,sz_curr:0), Set(AbstractConnection[]), Base.GenericCondition{Base.AlwaysLockedST}(Base.InvasiveLinkedList{Task}(Task (runnable) @0x000000010ae28250, Task (runnable) @0x000000010ae28250), Base.AlwaysLockedST(1))), WebIO.JSString[WebIO.JSString(\"function () {\\n var handler = (function (ko, koPunches) {\\n ko.punches.enableAll();\\n ko.bindingHandlers.numericValue = {\\n init: function(element, valueAccessor, allBindings, data, context) {\\n var stringified = ko.observable(ko.unwrap(valueAccessor()));\\n stringified.subscribe(function(value) {\\n var val = parseFloat(value);\\n if (!isNaN(val)) {\\n valueAccessor()(val);\\n }\\n });\\n valueAccessor().subscribe(function(value) {\\n var str = JSON.stringify(value);\\n if ((str == \\\"0\\\") && ([\\\"-0\\\", \\\"-0.\\\"].indexOf(stringified()) >= 0))\\n return;\\n if ([\\\"null\\\", \\\"\\\"].indexOf(str) >= 0)\\n return;\\n stringified(str);\\n });\\n ko.applyBindingsToNode(\\n element,\\n {\\n value: stringified,\\n valueUpdate: allBindings.get('valueUpdate'),\\n },\\n context,\\n );\\n }\\n };\\n var json_data = {\\\"formatted_vals\\\":[\\\"1\\\",\\\"2\\\",\\\"3\\\",\\\"4\\\",\\\"5\\\",\\\"6\\\",\\\"7\\\",\\\"8\\\",\\\"9\\\",\\\"10\\\",\\\"11\\\"],\\\"changes\\\":WebIO.getval({\\\"name\\\":\\\"changes\\\",\\\"scope\\\":\\\"17084213940907812968\\\",\\\"id\\\":\\\"ob_10\\\",\\\"type\\\":\\\"observable\\\"}),\\\"index\\\":WebIO.getval({\\\"name\\\":\\\"index\\\",\\\"scope\\\":\\\"17084213940907812968\\\",\\\"id\\\":\\\"ob_09\\\",\\\"type\\\":\\\"observable\\\"})};\\n var self = this;\\n function AppViewModel() {\\n for (var key in json_data) {\\n var el = json_data[key];\\n this[key] = Array.isArray(el) ? ko.observableArray(el) : ko.observable(el);\\n }\\n \\n [this[\\\"formatted_val\\\"]=ko.computed( function(){\\n return this.formatted_vals()[parseInt(this.index())-(1)];\\n }\\n,this)]\\n [this[\\\"changes\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"changes\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"changes\\\",\\\"scope\\\":\\\"17084213940907812968\\\",\\\"id\\\":\\\"ob_10\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"changes\\\"]=false}),self),this[\\\"index\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"index\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"index\\\",\\\"scope\\\":\\\"17084213940907812968\\\",\\\"id\\\":\\\"ob_09\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"index\\\"]=false}),self)]\\n \\n }\\n self.model = new AppViewModel();\\n self.valueFromJulia = {};\\n for (var key in json_data) {\\n self.valueFromJulia[key] = false;\\n }\\n ko.applyBindings(self.model, self.dom);\\n}\\n);\\n (WebIO.importBlock({\\\"data\\\":[{\\\"name\\\":\\\"knockout\\\",\\\"type\\\":\\\"js\\\",\\\"url\\\":\\\"/assetserver/2618192f019dfcc40a9013ef31cb09a511d8ea26-knockout.js\\\"},{\\\"name\\\":\\\"knockout_punches\\\",\\\"type\\\":\\\"js\\\",\\\"url\\\":\\\"/assetserver/df29ad18e8884e98d1d36dd0c339f642f729bf96-knockout_punches.js\\\"}],\\\"type\\\":\\\"async_block\\\"})).then((imports) => handler.apply(this, imports));\\n}\\n\")])], Dict{Symbol,Any}(:className => \"field interact-widget\")), Observable{Any} with 0 listeners. Value:\n",
"Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Plot{Plots.GRBackend() n=25}], Dict{Symbol,Any}(:className => \"interact-flex-row interact-widget\"))], Dict{Symbol,Any}())"
]
},
"execution_count": 24,
"metadata": {
"application/vnd.webio.node+json": {
"kernelId": "491971e1-5e20-4d24-b021-08a27ca47da3"
}
},
"output_type": "execute_result"
}
],
"source": [
"@manipulate for i=slider(1:length(Akeep),value=1)\n",
" lookat(Akeep[i],redrow=row[i],showtext=true)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [],
"source": [
"using LinearAlgebra"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"5×5 Array{Float64,2}:\n",
" 1.0 0.0 0.0 0.0 0.0\n",
" 11.2268 1.0 0.0 0.0 0.0\n",
" 1.65127 0.100262 1.0 0.0 0.0\n",
" 22.1736 2.04623 -7.9508 1.0 0.0\n",
" -7.45666 -0.817478 -7.32798 1.03327 1.0"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"U = Akeep[end]\n",
"A = Akeep[1]\n",
"L += I"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"5×5 Array{Float64,2}:\n",
" 0.0737129 -0.934928 0.686715 0.756193 0.665156\n",
" 0.827557 -0.250188 0.22969 -0.451796 -0.067432\n",
" 0.12172 -0.516531 0.479388 -1.64916 0.735843\n",
" 1.63448 0.234953 -0.837002 1.12689 -2.08616 \n",
" -0.549652 -1.40446 0.295059 2.6386 -1.12414 "
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L*U"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"5×5 Array{Float64,2}:\n",
" 0.0737129 -0.934928 0.686715 0.756193 0.665156\n",
" 0.827557 -0.250188 0.22969 -0.451796 -0.067432\n",
" 0.12172 -0.516531 0.479388 -1.64916 0.735843\n",
" 1.63448 0.234953 -0.837002 1.12689 -2.08616 \n",
" -0.549652 -1.40446 0.295059 2.6386 -1.12414 "
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"true"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L*U ≈ A"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"@webio": {
"lastCommId": "eae5c4a469c94994884bdc3fd65a736a",
"lastKernelId": "6c0e5fe2-1c02-40cc-b0c7-24e073bb3e7d"
},
"kernelspec": {
"display_name": "Julia 1.3.1",
"language": "julia",
"name": "julia-1.3"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.3.1"
}
},
"nbformat": 4,
"nbformat_minor": 2
}