first commit with current lessons
This commit is contained in:
elvis
330
10-27/lesson.ipynb
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330
10-27/lesson.ipynb
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "40e2ecf6-a1ee-4d82-924a-e2f763915652",
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"metadata": {},
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"outputs": [],
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"source": [
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"using LinearAlgebra, Plots"
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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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"id": "89746093-dc10-4bb2-9646-c84d5db0d8f8",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"householder_vector (generic function with 2 methods)"
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]
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},
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"execution_count": 2,
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"metadata": {},
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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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"function householder_vector(x::Vector{<:AbstractFloat})::Tuple{Vector, AbstractFloat}\n",
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" # returns the normalized vector u such that H*x is a multiple of e_1\n",
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"\n",
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" s = norm(x)\n",
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" if x[1] ≥ 0\n",
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" s = -s\n",
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" end\n",
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" u = copy(x)\n",
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" u[1] -= s\n",
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" u ./= norm(u)\n",
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" return u, s\n",
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"end\n",
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"\n",
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"function householder_vector(x::Matrix{<:AbstractFloat})::Tuple{Matrix, AbstractFloat}\n",
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" # returns the normalized vector u such that H*x is a multiple of e_1\n",
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"\n",
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" s = norm(x)\n",
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" if x[1] ≥ 0\n",
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" s = -s\n",
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" end\n",
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" u = copy(x)\n",
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" u[1] -= s\n",
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" u ./= norm(u)\n",
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" return u, s\n",
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"end"
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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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"id": "262a769a-aa42-4929-bbf4-7f5a97783810",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"([0.7960091839647405, 0.3357514552548967, 0.503627182882345], -3.7416573867739413)"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/plain": [
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"3-element Vector{Float64}:\n",
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" 1.0\n",
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" 2.0\n",
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" 3.0"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"x = [1., 2, 3]\n",
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"householder_vector(x) |> display\n",
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"x |> display\n",
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"\n",
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"# better with copy and division in place\n",
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"# @benchmark householder_vector(randn(100_000))"
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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": 4,
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"id": "457b3bcf-a077-42cb-9a6c-f6d1d6e00504",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"5-element Vector{Float64}:\n",
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" 3.983324158315414\n",
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" -5.457417586244235e-16\n",
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" -4.598436127585974e-17\n",
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" -1.4573782140540267e-16\n",
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" 3.2867357162787514e-16"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"A = randn(5, 4)\n",
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"\n",
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"# first step of QR factorization\n",
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"R1 = A\n",
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"(u1, s1) = householder_vector(R1[1:end,1])\n",
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"\n",
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"H1 = I - 2 * u1 * u1'\n",
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"\n",
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"Q1 = H1\n",
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"\n",
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"Q1 * R1[1:end, 1] |> display # what we expect -> a multiple of e_1"
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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": 5,
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"id": "fd98a89e-01b9-4403-8bca-b9e1443b2eea",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"5×4 Matrix{Float64}:\n",
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" 3.98332 -1.23134 1.8182 0.590776\n",
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" 3.80466e-16 1.52712 -1.85383 -0.709258\n",
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" -3.09958e-16 -1.11852e-17 -0.563749 -1.85235\n",
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" 4.05338e-16 -4.61156e-17 -0.174643 1.27511\n",
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" 1.55202e-16 1.4856e-17 1.73802 0.266999"
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]
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},
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"execution_count": 5,
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"metadata": {},
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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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"# second step\n",
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"R2 = Q1 * R1\n",
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"\n",
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"(u2, s2) = householder_vector(R2[2:end, 2])\n",
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"H2 = I - 2 * u2 * u2'\n",
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"\n",
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"# there is no blkdiag method in julia\n",
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"# (maybe look into https://github.com/JuliaArrays/BlockDiagonals.jl)\n",
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"# there are 2 methods (blocks is an array of blocks):\n",
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"### METHOD 1:\n",
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"# cat(blocks..., dims=(1,2))\n",
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"### METHOD 2:\n",
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"# using SparseArrays\n",
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"# blockdiag(SparseMatrixCSC.(blocks)...)\n",
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"## method 2 is slightly faster with subsequent matrix multiplication\n",
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"# performance is ignored in this step\n",
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"Q2 = cat(1, H2, dims=(1, 2))\n",
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"\n",
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"Q2 * R2"
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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": 6,
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"id": "a563f47e-7eda-46c7-9804-080335bcb8a3",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"5×4 Matrix{Float64}:\n",
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" 3.98332 -1.23134 1.8182 0.590776\n",
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" 3.80466e-16 1.52712 -1.85383 -0.709258\n",
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" 2.03593e-16 2.18903e-17 1.83549 0.700423\n",
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" 4.4272e-16 -4.37079e-17 6.42625e-17 1.46093\n",
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" -2.16817e-16 -9.104e-18 -4.23002e-16 -1.58224"
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]
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},
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"execution_count": 6,
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"metadata": {},
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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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"# third step\n",
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"\n",
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"R3 = Q2 * R2\n",
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"\n",
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"(u3, s3) = householder_vector(R3[3:end, 3])\n",
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"H3 = I - 2 * u3 * u3'\n",
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"\n",
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"Q3 = cat(Diagonal(ones(2)), H3, dims=(1,2))\n",
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"Q3 * R3"
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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": 7,
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"id": "85f5eb54-f3fe-40c8-86d0-90e80895b753",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"5×4 Matrix{Float64}:\n",
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" 3.98332 -1.23134 1.8182 0.590776\n",
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" 3.80466e-16 1.52712 -1.85383 -0.709258\n",
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" 2.03593e-16 2.18903e-17 1.83549 0.700423\n",
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" -4.5963e-16 2.29618e-17 -3.54379e-16 -2.15356\n",
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" 1.78186e-16 -3.82887e-17 -2.39742e-16 1.50034e-16"
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]
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},
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"execution_count": 7,
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"metadata": {},
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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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"# fourth step\n",
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"\n",
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"R4 = Q3 * R3\n",
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"\n",
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"(u4, s4) = householder_vector(R4[4:end, 4])\n",
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"H4 = I - 2 * u4 * u4'\n",
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"\n",
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"Q4 = cat(Diagonal(ones(3)), H4, dims=(1,2))\n",
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"Q4 * R4\n",
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"\n",
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"# done because we arrived at the second dimension of A"
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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": 8,
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"id": "eff90e91-7856-4fd7-b2a5-79c0f681147a",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"qrfactorization (generic function with 1 method)"
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]
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},
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"execution_count": 8,
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"metadata": {},
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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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"function qrfactorization(A::Matrix{<:AbstractFloat})::Tuple{Matrix{<:AbstractFloat}, Matrix{<:AbstractFloat}}\n",
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" (m, n) = size(A)\n",
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" R = copy(A)\n",
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" Q = Diagonal(ones(eltype(A), m))\n",
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"\n",
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" for k ∈ 1:n\n",
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" (u, s) = householder_vector(R[k:end, k])\n",
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" # construct R\n",
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" R[k, k] = s\n",
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" R[k+1:end, k] .= 0\n",
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" R[k:end, k+1:end] -= 2 * u * (u' * R[k:end, k+1:end])\n",
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" # contruct the new H\n",
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" H = I - 2 * u * u'\n",
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" # contruct the Q\n",
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" 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",
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" end\n",
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" return (Q, R)\n",
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"end"
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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": 9,
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"id": "4dbe13ff-44f5-4bd2-8452-a9e1477c80ff",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"true"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/plain": [
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"true"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"A = randn(Float32, 1000, 20)\n",
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"(Q, R) = qrfactorization(A)\n",
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"(norm(A - Q*R) ≤ size(A)[1] * 2^-23 * norm(A)) |> display\n",
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"(norm(I - Q*Q') ≤ size(A)[1] * 2^-23) |> display"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Julia 1.9.3",
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"language": "julia",
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"name": "julia-1.9"
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},
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"language_info": {
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"file_extension": ".jl",
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"mimetype": "application/julia",
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"name": "julia",
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"version": "1.9.3"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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