elvis/cmdla
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity
Files
3ad6f7f86fc0bd1952264f2dbe40d5b9388d2504
cmdla/Lessons/10-27/lesson.ipynb

331 lines
8.1 KiB
Plaintext
Raw Normal View History

first commit with current lessons
2023-10-29 02:06:02 +01:00
{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"id": "40e2ecf6-a1ee-4d82-924a-e2f763915652",
"metadata": {},
"outputs": [],
"source": [
"using LinearAlgebra, Plots"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "89746093-dc10-4bb2-9646-c84d5db0d8f8",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"householder_vector (generic function with 2 methods)"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function householder_vector(x::Vector{<:AbstractFloat})::Tuple{Vector, AbstractFloat}\n",
" # returns the normalized vector u such that H*x is a multiple of e_1\n",
"\n",
" s = norm(x)\n",
" if x[1] ≥ 0\n",
" s = -s\n",
" end\n",
" u = copy(x)\n",
" u[1] -= s\n",
" u ./= norm(u)\n",
" return u, s\n",
"end\n",
"\n",
"function householder_vector(x::Matrix{<:AbstractFloat})::Tuple{Matrix, AbstractFloat}\n",
" # returns the normalized vector u such that H*x is a multiple of e_1\n",
"\n",
" s = norm(x)\n",
" if x[1] ≥ 0\n",
" s = -s\n",
" end\n",
" u = copy(x)\n",
" u[1] -= s\n",
" u ./= norm(u)\n",
" return u, s\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "262a769a-aa42-4929-bbf4-7f5a97783810",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"([0.7960091839647405, 0.3357514552548967, 0.503627182882345], -3.7416573867739413)"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"3-element Vector{Float64}:\n",
" 1.0\n",
" 2.0\n",
" 3.0"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"x = [1., 2, 3]\n",
"householder_vector(x) |> display\n",
"x |> display\n",
"\n",
"# better with copy and division in place\n",
"# @benchmark householder_vector(randn(100_000))"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "457b3bcf-a077-42cb-9a6c-f6d1d6e00504",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"5-element Vector{Float64}:\n",
fix
2023-11-20 15:30:01 +01:00
" -3.256101502501094\n",
" 3.4967785872994334e-17\n",
" 4.405003154616756e-17\n",
" -3.1688882202679996e-17\n",
" 2.3942585187350707e-16"
first commit with current lessons
2023-10-29 02:06:02 +01:00
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = randn(5, 4)\n",
"\n",
"# first step of QR factorization\n",
"R1 = A\n",
"(u1, s1) = householder_vector(R1[1:end,1])\n",
"\n",
"H1 = I - 2 * u1 * u1'\n",
"\n",
"Q1 = H1\n",
"\n",
"Q1 * R1[1:end, 1] |> display # what we expect -> a multiple of e_1"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "fd98a89e-01b9-4403-8bca-b9e1443b2eea",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"5×4 Matrix{Float64}:\n",
fix
2023-11-20 15:30:01 +01:00
" -3.2561 0.130609 -0.788793 -0.0946472\n",
" -1.81593e-16 -1.31508 0.468296 0.501379\n",
" 3.27736e-17 -8.21875e-19 0.548462 -1.5704\n",
" 7.34914e-17 6.58877e-17 0.563737 -0.53087\n",
" 1.48462e-16 -1.40266e-16 -0.454505 0.883403"
first commit with current lessons
2023-10-29 02:06:02 +01:00
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# second step\n",
"R2 = Q1 * R1\n",
"\n",
"(u2, s2) = householder_vector(R2[2:end, 2])\n",
"H2 = I - 2 * u2 * u2'\n",
"\n",
"# there is no blkdiag method in julia\n",
"# (maybe look into https://github.com/JuliaArrays/BlockDiagonals.jl)\n",
"# there are 2 methods (blocks is an array of blocks):\n",
"### METHOD 1:\n",
"# cat(blocks..., dims=(1,2))\n",
"### METHOD 2:\n",
"# using SparseArrays\n",
"# blockdiag(SparseMatrixCSC.(blocks)...)\n",
"## method 2 is slightly faster with subsequent matrix multiplication\n",
"# performance is ignored in this step\n",
"Q2 = cat(1, H2, dims=(1, 2))\n",
"\n",
"Q2 * R2"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "a563f47e-7eda-46c7-9804-080335bcb8a3",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"5×4 Matrix{Float64}:\n",
fix
2023-11-20 15:30:01 +01:00
" -3.2561 0.130609 -0.788793 -0.0946472\n",
" -1.81593e-16 -1.31508 0.468296 0.501379\n",
" 8.88587e-18 -1.10573e-16 -0.908398 1.71961\n",
" 6.42479e-17 2.34191e-17 -1.02761e-16 0.742212\n",
" 1.55915e-16 -1.06026e-16 7.81748e-17 -0.143001"
first commit with current lessons
2023-10-29 02:06:02 +01:00
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# third step\n",
"\n",
"R3 = Q2 * R2\n",
"\n",
"(u3, s3) = householder_vector(R3[3:end, 3])\n",
"H3 = I - 2 * u3 * u3'\n",
"\n",
"Q3 = cat(Diagonal(ones(2)), H3, dims=(1,2))\n",
"Q3 * R3"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "85f5eb54-f3fe-40c8-86d0-90e80895b753",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"5×4 Matrix{Float64}:\n",
fix
2023-11-20 15:30:01 +01:00
" -3.2561 0.130609 -0.788793 -0.0946472\n",
" -1.81593e-16 -1.31508 0.468296 0.501379\n",
" 8.88587e-18 -1.10573e-16 -0.908398 1.71961\n",
" -3.35902e-17 -4.30553e-17 1.15696e-16 -0.755862\n",
" 1.65254e-16 -9.96807e-17 5.73216e-17 7.46032e-18"
first commit with current lessons
2023-10-29 02:06:02 +01:00
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# fourth step\n",
"\n",
"R4 = Q3 * R3\n",
"\n",
"(u4, s4) = householder_vector(R4[4:end, 4])\n",
"H4 = I - 2 * u4 * u4'\n",
"\n",
"Q4 = cat(Diagonal(ones(3)), H4, dims=(1,2))\n",
"Q4 * R4\n",
"\n",
"# done because we arrived at the second dimension of A"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "eff90e91-7856-4fd7-b2a5-79c0f681147a",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"qrfactorization (generic function with 1 method)"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function qrfactorization(A::Matrix{<:AbstractFloat})::Tuple{Matrix{<:AbstractFloat}, Matrix{<:AbstractFloat}}\n",
" (m, n) = size(A)\n",
" R = copy(A)\n",
" Q = Diagonal(ones(eltype(A), m))\n",
"\n",
" for k ∈ 1:n\n",
" (u, s) = householder_vector(R[k:end, k])\n",
" # construct R\n",
" R[k, k] = s\n",
" R[k+1:end, k] .= 0\n",
" R[k:end, k+1:end] -= 2 * u * (u' * R[k:end, k+1:end])\n",
" # contruct the new H\n",
" H = I - 2 * u * u'\n",
" # contruct the Q\n",
" Q = Q * cat(Diagonal(ones(eltype(A), k-1)), H, dims=(1,2)) # very inefficient (maybe simply send back the list of u_i)\n",
" end\n",
" return (Q, R)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "4dbe13ff-44f5-4bd2-8452-a9e1477c80ff",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"true"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"true"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = randn(Float32, 1000, 20)\n",
"(Q, R) = qrfactorization(A)\n",
"(norm(A - Q*R) ≤ size(A)[1] * 2^-23 * norm(A)) |> display\n",
"(norm(I - Q*Q') ≤ size(A)[1] * 2^-23) |> display"
]
}
],
"metadata": {
"kernelspec": {
fix
2023-11-20 15:30:01 +01:00
"display_name": "Julia 1.9.4",
first commit with current lessons
2023-10-29 02:06:02 +01:00
"language": "julia",
"name": "julia-1.9"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
fix
2023-11-20 15:30:01 +01:00
"version": "1.9.4"
first commit with current lessons
2023-10-29 02:06:02 +01:00
}
},
"nbformat": 4,
"nbformat_minor": 5
}
Reference in New Issue Copy Permalink