{
"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 watching the code being executed)"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"using Plots"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"lookat (generic function with 1 method)"
]
},
"execution_count": 4,
"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": 9,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4×4 Array{Float64,2}:\n",
" 2.0 12.0 4.0 7.0\n",
" 29.0 12.0 69.0 24.0\n",
" 7.0 7.0 7.0 7.0\n",
" 3.0 7.0 5.0 2.0"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = [2 12 4 7; 29 12 69 24 ; 7 7 7 7; 3 7 5 2]\n",
"A = A*1.0"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"lookat(A)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## What multiple of the first row must i subtract from the second row to zero out the (2,1) entry?"
]
},
{
"cell_type": "code",
"execution_count": 6,
"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": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L = fill(0.0, 4,4)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"14.5"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L[2,1] = A[2,1]/A[1,1]"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A[2,:] -= L[2,1]*A[1,:]; # subtract the multiplier times the first row from the second row\n",
"lookat(A, redrow=2)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3.5"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L[3,1] = A[3,1]/A[1,1] # the diagonal entry that we divide by is called the \"pivot\""
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A[3,:] -= L[3,1]*A[1,:]; # subtract the multiplier times the first row from the second row\n",
"lookat(A, redrow=3)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L[4,1] = A[4,1]/A[1,1] \n",
"A[4,:] -= L[4,1]*A[1,:]; # subtract the multiplier times the first row from the second row\n",
"lookat(A, redrow=4)"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L[3,2] = A[3,2]/A[2,2] \n",
"A[3,:] -= L[3,2]*A[2,:]; # subtract the multiplier times the first row from the second row\n",
"lookat(A, redrow=3)"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L[4,2] = A[4,2]/A[2,2] \n",
"A[4,:] -= L[4,2]*A[2,:]; # subtract the multiplier times the first row from the second row\n",
"lookat(A, redrow=4)"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L[4,3] = A[4,3]/A[3,3] \n",
"A[4,:] -= L[4,3]*A[3,:]; # subtract the multiplier times the first row from the second row\n",
"lookat(A, redrow=4)"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [],
"source": [
"n = 6\n",
"A = rand(n,n)\n",
"keepA = [copy(A)]\n",
"row = [0]\n",
"L = fill(0.0,n,n)\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!(keepA,copy(A))\n",
" push!(row,i)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [],
"source": [
"using Interact"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [
{
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"mount_callbacks": [
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"systemjs_options": null
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"Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Plot{Plots.GRBackend() n=36}], Dict{Symbol,Any}(:className => \"interact-flex-row interact-widget\"))], Dict{Symbol,Any}())"
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},
"execution_count": 29,
"metadata": {
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],
"source": [
"@manipulate for i =1: length(keepA)\n",
" lookat( keepA[i], redrow=row[i])\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {},
"outputs": [],
"source": [
"using LinearAlgebra"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6×6 Array{Float64,2}:\n",
" 0.601848 0.719816 0.0549105 0.664714 0.184826 0.856158\n",
" 0.373551 0.503678 0.964187 0.613921 0.840482 0.349087\n",
" 0.139154 0.146278 0.0863354 0.315124 0.982249 0.226666\n",
" 0.176656 0.560782 0.473594 0.625323 0.164821 0.457171\n",
" 0.919038 0.751709 0.0796167 0.569861 0.793574 0.418587\n",
" 0.288685 0.0699438 0.0534446 0.0420711 0.788685 0.921932"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = keepA[1]"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6×6 Array{Float64,2}:\n",
" 0.601848 0.719816 0.0549105 0.664714 0.184826 0.856158 \n",
" -5.55112e-17 0.0569073 0.930105 0.201351 0.725765 -0.182307 \n",
" -1.96573e-17 0.0 0.403003 0.232736 1.19652 -0.0358447\n",
" 8.46105e-17 0.0 0.0 2.22829 11.2549 0.858134 \n",
" 3.25219e-17 0.0 0.0 0.0 0.685981 -0.537103 \n",
" 3.55852e-18 0.0 0.0 0.0 -5.55112e-17 1.1249 "
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"U = keepA[end]"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6×6 Array{Float64,2}:\n",
" 0.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.620673 0.0 0.0 0.0 0.0 0.0\n",
" 0.231211 -0.354114 0.0 0.0 0.0 0.0\n",
" 0.293522 6.14156 -13.0392 0.0 0.0 0.0\n",
" 1.52703 -6.10589 14.0815 -1.11881 0.0 0.0\n",
" 0.479664 -4.83816 11.2334 -0.860311 0.660471 0.0"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6×6 Array{Float64,2}:\n",
" 1.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.620673 1.0 0.0 0.0 0.0 0.0\n",
" 0.231211 -0.354114 1.0 0.0 0.0 0.0\n",
" 0.293522 6.14156 -13.0392 1.0 0.0 0.0\n",
" 1.52703 -6.10589 14.0815 -1.11881 1.0 0.0\n",
" 0.479664 -4.83816 11.2334 -0.860311 0.660471 1.0"
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L += I"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## First view of the LU decomposition"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6×6 Array{Float64,2}:\n",
" 0.601848 0.719816 0.0549105 0.664714 0.184826 0.856158\n",
" 0.373551 0.503678 0.964187 0.613921 0.840482 0.349087\n",
" 0.139154 0.146278 0.0863354 0.315124 0.982249 0.226666\n",
" 0.176656 0.560782 0.473594 0.625323 0.164821 0.457171\n",
" 0.919038 0.751709 0.0796167 0.569861 0.793574 0.418587\n",
" 0.288685 0.0699438 0.0534446 0.0420711 0.788685 0.921932"
]
},
"execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6×6 Array{Float64,2}:\n",
" 0.601848 0.719816 0.0549105 0.664714 0.184826 0.856158\n",
" 0.373551 0.503678 0.964187 0.613921 0.840482 0.349087\n",
" 0.139154 0.146278 0.0863354 0.315124 0.982249 0.226666\n",
" 0.176656 0.560782 0.473594 0.625323 0.164821 0.457171\n",
" 0.919038 0.751709 0.0796167 0.569861 0.793574 0.418587\n",
" 0.288685 0.0699438 0.0534446 0.0420711 0.788685 0.921932"
]
},
"execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L * U"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"true"
]
},
"execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A ≈ L * U"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4×4 Array{Int64,2}:\n",
" 2 12 4 7\n",
" 29 12 69 24\n",
" 7 7 7 7\n",
" 3 7 5 2"
]
},
"execution_count": 40,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = [2 12 4 7; 29 12 69 24 ; 7 7 7 7; 3 7 5 2]"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4×4 Adjoint{Int64,Array{Int64,2}}:\n",
" 2 29 7 3\n",
" 12 12 7 7\n",
" 4 69 7 5\n",
" 7 24 7 2"
]
},
"execution_count": 41,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A'\n"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2×3 Array{Int64,2}:\n",
" 1 2 3\n",
" 4 5 6"
]
},
"execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A= [ 1 2 3; 4 5 6]"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3×2 Adjoint{Int64,Array{Int64,2}}:\n",
" 1 4\n",
" 2 5\n",
" 3 6"
]
},
"execution_count": 43,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A'\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Adjoint{Int64,Array{Int64,2}}\n",
" parent: Array{Int64}((2, 3)) [1 2 3; 4 5 6]\n"
]
}
],
"source": [
"dump(A')"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2×3×4 Array{Int64,3}:\n",
"[:, :, 1] =\n",
" 2 8 3\n",
" 7 5 3\n",
"\n",
"[:, :, 2] =\n",
" 7 5 8\n",
" 3 4 5\n",
"\n",
"[:, :, 3] =\n",
" 4 2 8\n",
" 8 4 2\n",
"\n",
"[:, :, 4] =\n",
" 8 3 8\n",
" 7 4 9"
]
},
"execution_count": 45,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = rand( 1:9, 2,3,4)"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4×3×2 Array{Int64,3}:\n",
"[:, :, 1] =\n",
" 2 8 3\n",
" 7 5 8\n",
" 4 2 8\n",
" 8 3 8\n",
"\n",
"[:, :, 2] =\n",
" 7 5 3\n",
" 3 4 5\n",
" 8 4 2\n",
" 7 4 9"
]
},
"execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"permutedims(A, [3 2 1])"
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {},
"outputs": [
{
"ename": "DimensionMismatch",
"evalue": "DimensionMismatch(\"matrix is not square: dimensions are (4, 6)\")",
"output_type": "error",
"traceback": [
"DimensionMismatch(\"matrix is not square: dimensions are (4, 6)\")",
"",
"Stacktrace:",
" [1] inv(::Array{Float64,2}) at /Users/julia/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/LinearAlgebra.jl:221",
" [2] top-level scope at In[47]:1"
]
}
],
"source": [
"inv( rand(4,6))"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3×3 Array{Float64,2}:\n",
" 0.961711 0.705796 0.334855 \n",
" 0.825322 0.76569 0.0211825\n",
" 0.22131 0.162853 0.357397 "
]
},
"execution_count": 48,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = rand(3,3)"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3×3 Array{Float64,2}:\n",
" 0.466144 0.282758 0.176453 \n",
" 0.804355 0.748308 0.9467 \n",
" 0.764508 0.457893 0.0473501"
]
},
"execution_count": 49,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"B = rand(3,3)"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3×3 Array{Float64,2}:\n",
" 110.378 -81.0345 -112.35 \n",
" -191.261 138.822 201.288 \n",
" 50.3087 -34.295 -57.3957"
]
},
"execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"inv( A*B )"
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3×3 Array{Float64,2}:\n",
" 110.378 -81.0345 -112.35 \n",
" -191.261 138.822 201.288 \n",
" 50.3087 -34.295 -57.3957"
]
},
"execution_count": 51,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"inv(B) * inv(A)"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1:10"
]
},
"execution_count": 52,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = 1:10\n"
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"10×10 Array{Int64,2}:\n",
" 1 2 3 4 5 6 7 8 9 10\n",
" 2 4 6 8 10 12 14 16 18 20\n",
" 3 6 9 12 15 18 21 24 27 30\n",
" 4 8 12 16 20 24 28 32 36 40\n",
" 5 10 15 20 25 30 35 40 45 50\n",
" 6 12 18 24 30 36 42 48 54 60\n",
" 7 14 21 28 35 42 49 56 63 70\n",
" 8 16 24 32 40 48 56 64 72 80\n",
" 9 18 27 36 45 54 63 72 81 90\n",
" 10 20 30 40 50 60 70 80 90 100"
]
},
"execution_count": 53,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x * x'"
]
},
{
"cell_type": "code",
"execution_count": 63,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4-element Array{Int64,1}:\n",
" 1\n",
" 2\n",
" 4\n",
" 5"
]
},
"execution_count": 63,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x =[ 1 ,2 ,4, 5]"
]
},
{
"cell_type": "code",
"execution_count": 64,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3-element Array{Int64,1}:\n",
" 2\n",
" 4\n",
" 6"
]
},
"execution_count": 64,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"y = [ 2, 4, 6]"
]
},
{
"cell_type": "code",
"execution_count": 66,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4×3 Array{Int64,2}:\n",
" 2 4 6\n",
" 4 8 12\n",
" 8 16 24\n",
" 10 20 30"
]
},
"execution_count": 66,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x * y'"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"@webio": {
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"kernelspec": {
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