lec07.ipynb

{
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   "cell_type": "code",
   "execution_count": 1,
   "id": "22c3320f",
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    {
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     "text": [
      "\u001b[32m\u001b[1m    Updating\u001b[22m\u001b[39m registry at `~/.julia/registries/General.toml`\n",
      "\u001b[32m\u001b[1m   Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/srv/rep/22-course-cse6643/mybinder/Project.toml`\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/srv/rep/22-course-cse6643/mybinder/Manifest.toml`\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "LogLevel(1)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import Pkg; Pkg.add([\"Plots\", \"Interact\"]);\n",
    "import Logging; Logging.disable_logging(Logging.Info)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "8f0eae62",
   "metadata": {},
   "outputs": [],
   "source": [
    "const STATIC_LECTURE = false;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "ed67d067",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "draw (generic function with 1 method)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function draw(plt)\n",
    "    if STATIC_LECTURE\n",
    "        filename = Base.Filesystem.tempname() * \".png\"\n",
    "        savefig(filename)\n",
    "        return display(\"image/png\", read(filename))\n",
    "    else\n",
    "        return plt\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "3ac47923",
   "metadata": {},
   "outputs": [
    {
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      "text/html": [
       "<div style=\"padding: 1em; background-color: #f8d6da; border: 1px solid #f5c6cb; font-weight: bold;\">\n",
       "<p>The WebIO Jupyter extension was not detected. See the\n",
       "<a href=\"https://juliagizmos.github.io/WebIO.jl/latest/providers/ijulia/\" target=\"_blank\">\n",
       "    WebIO Jupyter integration documentation\n",
       "</a>\n",
       "for more information.\n",
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      "text/plain": [
       "WebIO._IJuliaInit()"
      ]
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    {
     "data": {
      "text/plain": [
       "imshow (generic function with 2 methods)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using LinearAlgebra;\n",
    "using Plots; default(l=2, size=(900,550), label=false);\n",
    "using Interact;\n",
    "\n",
    "imshow(x, _clim=:nothing) = heatmap(x\n",
    "    , aspect_ratio=:equal\n",
    "    , framestyle=:none\n",
    "    , yflip=true\n",
    "    , c=:diverging_gwv_55_95_c39_n256\n",
    "    , clim=if _clim != :nothing\n",
    "          _clim\n",
    "      elseif maximum(abs.(x)) > 1.e-12\n",
    "          (-maximum(abs.(x)),maximum(abs.(x)))\n",
    "      else \n",
    "          (-1,1)\n",
    "      end\n",
    "    , size=(500,500)\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "384d2ca1",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**(Follow along: <https://tinyurl.com/gtcse6643lec7>)**\n",
    "\n",
    "### Warmup (T&B Exercise 7.3)\n",
    "\n",
    "Let $A$ be an $m \\times m$ matrix, and let $a_j$ be its $j$th column.  Give an algebraic proof of _Hadamard's inequality_:\n",
    "\n",
    "$$|\\det A| \\leq \\prod_{j=1}^m \\| a_j \\|_2.$$\n",
    "\n",
    "Also give a geometric interpretation of this result, making use of the fact that the determinant equals the volume of a parallelipiped.\n",
    "\n",
    "Hints:\n",
    "\n",
    "- $|\\det A| = |\\det (QR)| = |\\det(Q)| |\\det(R)|$\n",
    "- How does $|r_{jj}|$ compare to $\\|a_j\\|_2$?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "44cd5be9",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Householder QR\n",
    "\n",
    "**1. Review & what Householder QR does**\n",
    "\n",
    "**2. How Householder QR works**\n",
    "\n",
    "**3. Householder QR performance**\n",
    "\n",
    "**4. Backup: RRQR, Parallelism**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "de60574f",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Review"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3dd88613",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Classical Gram-Schmidt vs. Modified Gram-Schmidt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "eb1d293b",
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [],
   "source": [
    "function cgs(A)\n",
    "    _, n = size(A); Q = copy(A);\n",
    "    R = zeros(eltype(A), n, n)\n",
    "    for j in 1:n\n",
    "        R[1:j-1,j] = Q[:,1:j-1]' * Q[:,j]\n",
    "        Q[:,j] -= Q[:,1:j-1] * R[1:j-1,j]\n",
    "        R[j,j] = norm(Q[:,j])\n",
    "        Q[:,j] /= R[j,j]\n",
    "    end\n",
    "    return Q, R\n",
    "end;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "a618b080",
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [],
   "source": [
    "function mgs(A)\n",
    "    _, n = size(A); Q = copy(A);\n",
    "    R = zeros(eltype(A), n, n)\n",
    "    for i in 1:n\n",
    "        R[i,i] = norm(Q[:,i])\n",
    "        Q[:,i] /= R[i,i]\n",
    "        R[i,i+1:n] = Q[:,i]' * Q[:,i+1:n]\n",
    "        Q[:,i+1:n] -= Q[:,[i]] * R[[i],i+1:n]\n",
    "    end\n",
    "    return Q, R\n",
    "end;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "78c5a0cc",
   "metadata": {
    "cell_style": "split",
    "hideCode": true
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   "outputs": [
    {
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\" />"
      ],
      "text/plain": [
       "Plots.AnimatedGif(\"/home/tisaac/srv/rep/22-course-cse6643/mybinder/tmp.gif\")"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 7; n = 5;\n",
    "Q = ones(m,n);\n",
    "err = ones(n,n);\n",
    "anim = Animation()\n",
    "function q_frame!(Q, E, anim)\n",
    "    qplot = imshow(Q)\n",
    "    plot!(qplot, colorbar=:legend, legend=false, title=\"Q\")\n",
    "    eplot = imshow(E)\n",
    "    plot!(eplot, colorbar=:legend, legend=false, title=\"||I-Q'Q||\")\n",
    "    p = plot(qplot, eplot, layout=(1,2))\n",
    "    frame(anim, p)\n",
    "end\n",
    "q_frame!(Q, err, anim)\n",
    "for j in 1:n\n",
    "    Q[:,j] .= -1.0+0.01\n",
    "    err[1:j,1:j] .= 0\n",
    "    q_frame!(Q, err, anim)\n",
    "end\n",
    "q_frame!(Q, err, anim)\n",
    "gif(anim, fps=0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "f724f666",
   "metadata": {
    "cell_style": "split",
    "hideCode": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<img 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\" />"
      ],
      "text/plain": [
       "Plots.AnimatedGif(\"/home/tisaac/srv/rep/22-course-cse6643/mybinder/tmp.gif\")"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q = ones(m,n);\n",
    "err = ones(n,n);\n",
    "anim = Animation()\n",
    "q_frame!(Q, err, anim)\n",
    "for j in 1:n\n",
    "    Q[:,j] .= -1+0.02\n",
    "    Q[:,j+1:n] .= 1.0\n",
    "    Q[(mod(j,2)+1):2:end,j+1:n] .-= 0.1 \n",
    "    err[j,:] .= 0\n",
    "    err[:,j] .= 0\n",
    "    q_frame!(Q, err, anim)\n",
    "end\n",
    "q_frame!(Q, err, anim)\n",
    "gif(anim, fps=0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "02fbfebc",
   "metadata": {
    "hideCode": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Both algorithms construct $\\hat{Q}$ by a sequence of triangular factorizations $Z_i$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "726eef61",
   "metadata": {
    "cell_style": "split",
    "hideCode": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"Classical Gram-Schmidt\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
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X1h/bUhvb+heWCiHexhwfWhff5iFgXhwgwhKheiCh9hwif/oX4tYgynYXXc4EXl4fI2ocY9oYGt4c5kIcptIQ2VYS59ocrUVdZ0Ic6XIcqfodakYcas4c60od6+Ib7EYdDRneLX4bbeIdLk4eZG4g5M4XZUoEZdIe71Ydb/IfLuIZMkodsuIRqPISs+IdmMHTs0IZNVod9doRtMoStvYd93ISNmIY+GoeONIRt+ISefIddG1jo7UjoT3juX4ZMNYh8OHXPBISPIYefQYjEF4j/hoffpYj1UmkAMpfgUJkEeIkAm5av8Yh3KogXT4kLhYW/uIR/2ITtuVkWy0kefUkQa5ZQ5pkb0WkQA4kXM4Em9nktz4Wx7ZRSD5TSLJkE1Yki7/+WwoiYAqmTk4mZMyVJMS2ZNaWJFAKY4wOZJh9pNH+UJCmZJEGYJM2ZQt9JQ8GZV7N5VUaYc7+YBYmZVbuJW9t5BDGZUz6U1W6ZVfuT5aKZZpeYFryZZhKZbQR5ZQuZZnaU1v6YFx+YJGSZe3Z5dXiZdtuZV7WYJ9mZfVdJgsmJiFSZWMOYOOOZeAqZBJaZNT+JhNGZk5OJl/WZl1eZllSZSK2Uyc+YOeyZKgGZiieZdfWZrMdJpFmJoi0ZKraZkYqZRnpplHKZtLSJshYZu3CZGCqZaESZnDOWxdCZdxCZvE5JtRCJwgIZzJeZLFyZzH+ZnVeWvLyZfNyZtACZ1e/yidMnOMocdB2phB2Wkz5ul46GmO6vma8YmV72mPONOelTefZqmfpMmfPVmfB3mf5Bkz+Ol6/qmSAEqSAvqdC7qeNVOgwXegE5mgS9mg8mmh9CmhgKihhsihjOihkoih+ymi/Umi/wmiwmiiCIqiAamiE8qiDemiGyqjHUqjH2qjIXozEEp9VtOjPvqjQBqkQjqkRFqkRnqkSJqkSrqkTNqkTvqkUBqlUjqlVFqlVnqlWJqlWrqlXNqlXvqlYBqmYjqmZFqmZnqmaJqmarqmbNqmbvqmcBqncjqndFqndnqneJqnerqnfNqnfvqngBqogjqohFqohnqoiJqoirqojE/aqI76qJAaqZI6qZRaqZZ6qZiaqZq6qZzaqZ76qaAaqqI6qqRaqqZ6qqiaqqq6qqzaqq76qrAaq7I6q7Raq7Z6q7iaq7q6q7zaq75KMQEBACH5BAXIAAEALBoAaAAlAScBAAj/APOMGEiwoMGDCBMeDMCwocOHECNKnEixosWLGDNq3Mixo8ePHQUqHEmSJMiTKFOqXMmypUuNIkvKnDnipc2bOHPq3GkxJs2fCXkKHUq0qNGIPoEqJXi0qdOnUD0mXao0qtWrWKFOpfozq9evYG1u5TozrNmzaGGSpZq2rdu3Y9eafEu3rte4chXa3cvXKd68CPsKHrzzL2CDhBMrZmn4MNPFkCNzbOy4puTLmCdSdpy5s+cAmw9/Hi05NGDSqBWbzpu6teDVcl3Ltgt77ezbbmuTxc37rG6uvYN//c1WuHGrxJceX/40eVXm0Is6Bxq9utDpXa1rx4md5vbvL7uX/wVPXqV4meXTnzxfUr37kJXRv5+Pkf1c+vg1x2+fvz9E+yP5J2BDAOo1oIAFBnWgfwkGtmB/DS70YH4RIjYhfhUWdCGG+923IWFlVCHiiCSWaOKJKJqY4WMfgpjiizDGuOJALSYWYow45ijijJbVKNiNOgaZIo8+DgakkEiSSGSRfR2ZZJJLMrmXk08KGaWUdVFZpY5XYvmWllvi2KWXbYEZJoxjkomWmWcO2WGAan7ZppVvGhhnmXMGmeadYLGZZ4l78umVn3+OGKigWBFaaBWHImqVooU26ihUkP4p6aROVZrnpZgepemcnHZa1KdthirqUKSeaeqpPKUa5qqs6v/k6pawxorTrFXWaqtNuD6p664u9QplnQoC+5SwSP5q7ErI0kmshMs21ayez0IbrVHTclmthddiu6iM22rYrbffvqjsuB9lm+O56Hakrpjhsthuq+WaGy+N8wr1Lrj35qtvvW726+9O+6J5b48D3wowiuwmbFHB9grs8E0QBxzvxAovrOLBGC/WcMfIcQwyYR+P3JzIJvdVcspHrcyydCi/XJfLMl8Xc8253YxzWjTvnFPPPt8EdNDh6Uw0WEMfzZjRSmeVdNMpPQ31ekxPHZXUVktVddZ+bc11y15/DbPEYl8ddtk2k41212qvDXbbbo99cdxsz03323bfLXe4euP/zXffe28LeODVDk4U1oYTeHbi5i3OOEqIMx554pMbXvnglwOeed+b69353Z/THXrco7td+tqno5162auL3frXr3Mde9azW1371LdDnXvTuyvd+9G/Ex180MP7XPzOx+OcfM3Ly9z8y8+zHH3K05tc/cjXg5x9x9tj3P3E3zscfsLjD1y+v+fnm/6867fbPrrvjxt/t/NfW3+09y+bv7H7A9v/rv+zVQBjNUBWFfBUBxRVAju1QEw1cFIPdFQEETVBQVWQTxe8UwbjtEE1dZBMH/RSCLE0QimVkEknLFIKfbTCGrWwRS/8UAw3NMML1XBCN3xQDhe0wwP1cEA/RJDj9R4HkiAyaIhE1Brckri0JTKxcU58IuSQKMWMGBFCVKziRa5IoSxqsSJc5FAUvwifMZJxI2GkTxrns8b3tNE9b1RPHNMzx/LUkTx3BE8ev7PH7fRRO3+0TiCrM8joFBI6h2ROIpezyOM00jiPFE4kgzPJ3lSSN5fETSZvs8nZdFI2n3RNKFszytSUEjWnJE0qR7PKz7TSM6/sTCwzM0vM1PIytyyNF8+IlF3y8iG5jEwwITNMj/nyl4ozIzJ7csxlFlM1zUTmMxMzTZJF85fVHEw2X3NNXm5TZd084zf5Ms69lJM24STjOWeWzi+uky7vhEs7tRgQACH5BAXIAAEALEUAaAAkAScBAAj/AEcIHEiwoMGDCA0GWMiwocOHECNKnEixosWLGDNq3Mixo0eOCUOKHPmxpMmTKFOqXMky48iXMAe2nEmzps2bOCvG3Ckyp8+fQIMKhciz6MGhSJMqXdrRqFOZTKNKnar06VOqWLNqnWnV6davYMO67FpUrNmzaMmWRcu2LVa1PN3KnZsU7k66ePPitBtTr9+/KvnCBEy4MEjBJA0rXiwRcWLGkCE77hm5suLJIS1rJow54ebPejsjBE16ruijpVOnPV1QtWuxrFu/nq01NkHauKnahpq7d9XdI3wLRwo8+PDjP4sjX74XOPPnNJVDnx7YOfXrJqVj3354N/fvGrWD/x8/UTz58w7No1+vfv359u7Hw4//fT797fbvX8+vfzr//iZ5gcSABBZo4IEIJnigCAw26OCDEEYoIYT/AfiRgApmqKGGE3bo4YcVWtgRhhuWaCISH6aoYoMhirgRiSfGmOCKNHrYoosZwSjjjgTW6GOEN+J4kY487vjjkSxaJ6RKRBYZI5JIBrkkRU06aSKUR0o5pURVWrkhlj9quSVEXXqZIZg+ijmmQ2WaOSOaNKq5JkNturkgnCvKOWcAddpZIJ55KrnniH6eCKiKes7ZZ6EoHgqioIO+yGiJjj7qXaSETsphpR0muuaihXLaKaSYYgSqn6JO6OmYp9qZqoSrbv/ZqpuvAklqqRbNamatFN6KK5Wabsprkpf+amqwZw5LrG3G5oisgsouG1uzxz6LYLQMxjqlrl5iK4K2S3JrpbfgCimuk+T6Su1D5xaZbrHrTtQuj+8yGy+w1t4Zbbk4zmsktvy66K+M9U57r7z56qtswCIO/CTA6h7MZ8IGFsyaxFxS/CfE8GLckMOGcmyvx2xq3KPIBpP8sckDWnyayiWz7LJoMK8sM8oX14xUFTz37PPPQActNNAM60zR0EgnrXTRRkuk9NNQ98x00xBFbXXSU1Pt0NVcB5211gx1LbbUEYPN0thjf2022mKrDTbbXbutNdxcy0013Vfb3TTeVuv/bTTfUfutM+BQC14z4U8bDjPiS5dttkqMY+344yhFjrTiKls+NOYkay405x577vXklJskOtGkl/7R6T+DjjHrPrsuMexkd6z62bRXIfvBueue+u0c9b77vcL/DrxGxdt+fOW5Dx9v8iMvD3nzxktvEfQpW8887c6vi33O2m8Pe/fUfv9y+OKzTn6z5tOM/kntd/Y+/NQrPz9Y69//UP76N8R//wv5HwAF2D8C6s+A90Pg/BT4Pgaiz4Hhg6D2JGg9CkrPgsvD4PE0CDwO3s6DqgNh6URIORI+zoRmQyHYVKg1FlLNhU2DodFkqDMa1syGMMOhynRIMh56zIcYA6LE/4R4MCLey4jxQuK6lEgtJjbLicaC4q+kiCsqlsqKmMJipLQ4KC7uyYtzAuOaxDgmMm7JjFNC45LUKCQ24siNLoKjiORoIToCyI79waN+9HgfPtLHj/EBpHsEyZ7qAXAohERPIt9jyEMGZZHkgaR8GunI5FCykjmRJHg0WZ9LYvImnOROKPHjyU/WZJTYQeV+SmlKrrCylSxRJXVk6Z9XwrI69rslcWypy5PQEjq/fE4wmTPM5RQTOcc8TjKHs0zhNNM3z+xNNHMzTdxUkzbXnE02X7NN13RTNd9MTThLM07SlBM05/xMOjezTs200zLvrEw8IzNPyfCylyWpJ2P0uY4Yfl7mnvj0iD8NM9DCFJQzAA1od6KnUEQmtKFjySVEmyPRidrkoIDB6F806heOhuahFtUJSENanpGSNCIezUtK8bJSurTUNCY96f5iKlP/0bSmAbwpTl8qF566xadtASpbhLqaiuI0JUQ9S1LNslTY6LSmTQ1LVPH3VJlO9StX3UpWa1PVk241K1/FSkAAACH5BAXIAAEALG8AaAAkAScBAAj/AEcIHEiwoMGDCA0GWMiwocOHECNKnEixosWLGDNq3Mixo0eOCUOKHPmxpMmTKFOqXMky48iXMAe2nEmzps2bOCvG3Ckyp8+fQIMKhciz6MGhSJMqXdrRqFOZTKNKnar06VOqWLNqnWnV6davYMO67FpUrNmzaMmWRcu2LVa1PN3KnZsU7k66ePPitBtTr9+/KvnCBEy4MEjBJA0rXiwRcWLGkCE77hm5suLJIS1rJow54ebPejsjBE16ruijpVOnPV1QtWuxrFu/nq01NkHauKnahpq7d9XdI3wLRwo8+PDjP4sjX74XOPPnNJVDnx7YOfXrJqVj3354N/fvGrWD/x8/UTz58w7No1+vfv359u7Hw4//fT797fbvX8+vfzr//ietIeCABBZo4IEIGijCggw26OCDEEb44H8AlpTghRhmKOGGHHZIYYUeZSjiiAN2aOKJDH4IIkcktoghijByqOKKGrlo44Ex5gjhjDRidOOPJeoo5II89mgRkEAOOWSRRlKE5I9KCslkkxI9eWOUOk5JJURW2ohljlpu6VCXLn4ZY5hiMkRmi2bCiGaaAaxJYpsovpmmnCPSeaKdYuIpop4m8rmlnxoCKqN1cH5E6IuGbigolYte2KijiCbaUaQJTirho01iiqCmEXJqpKc4gjphpZZuRKqCpjooao+rFv/YqquopppRrATO2uCrNOIapK4i8LqirwICS2Sttl5E7BrGBotsshUt26ywIEprLLUVWgsstgBqqyu3/Xk7K7j6idsqufeZayq69KkLKrvxuaspvO7JOym969nbKL7o6Wsov+f5CyjA5AmsJ8HjGUwnwuAp3CbD3yEh8cQUV2zxxRhbPO2z0E6U8ccgh7yxdx1nFPLJKE88sm0lm5zyyx+vHFvLGMFs88Uys0bzRTf3rPK1HO/ckM8+53ya0BUR3bPRoiFNkdI3M92Z0x5DDbPUmFEtkdVXA02y1g9x/TLWk4ENkdgpk+2Y2WGjfbLaiLHtkNtve82y3AzRLbLdM+P/vZDeIMMtmN9/A56x4HwRHoDhh/OtM+GMY4y4XYpHjrPjR0NuecWTw1X55hR3rtbnoEssOlmkl356V6mDvrpVrW/++lWal47E7F7VrjrmTevuOu9T+y478FkLbznuRsV+PPFlGx858mv5bbvpzK/tPOPQx3W94dnftT3g3ff1vd7hDzY+3eW/pPzz1cd9vtvpP6a4T1XUb//9+Oev//75Q6w1/wAMoAD9RzUBGvCA9iOg0xDIwAAqEGkNjKD+Hig0CVowgUGbX0sueEEK7oyDFvQgzUAoQRG2jIQRNGHJUNhAFXaMhQx0IbRgiEAZJouGB7ShrXBoQB2miocDzKAG/1cCRAcKcYgpKSIAfWgpJfKPiYly4v6gCCcpTvCISDyJFfuHxSyWZIv4o2KawHg/MYqJjBj8mhdpgsb6mXFLbazCG6kUxzk2qY5dXGNH8KhGPbKEj3fz4wbbaEcjAbJvgvwjIfOYSIwc8nGNVMkjMxfJJC6yj5X84iUDmUmTTLJ3nfTkJhEZyo98MnilNOUoIZlKsxQyka8UZCz9OEs91nKNt/RiLrO4SyT2coi/1GAw5zdMxRWTcMf0WzLxtky5NZNtzzRbNME2Ta1Vk2rXdFo2kbZNoXVzZ9+kWThbNs6SlbNj54RWOpO1Tlu1M1XvtFQ8EzVPONUzTfcUUz63tP9PKvWzSf80UkB7NFAaFXRFBwVRQiu0UAA1tD8P1U9E7zNR+lQ0Phd1T0bZw8hW/gaTHt3KRtEz0vd0NKRDKSl5VCqfk6I0KCwFT0zr49KXJqemNs3JTLmzU/zgNKc36Sl2hLqfnwK1JkSlTlL9Y9SjcqWpTmXJUqEz1edUlTlXXU5WkbPV43R1OF8VTlh9M9belDU3Z8VNWmmz1tm09TVvdU1cVTPX1NS1NHclTV5Bs9fP9HUzf9VMYC0z2MoUNjKHlQxUo1odkDK2Lot97EkSyxjKLsayl4msZEuCWcN0tjCf5YxmN+uR0ALGtH9BrV9UG5rRkrY7nHxtSl0r27EKOLa2OqUtbi0SEAAh+QQFyAABACyZAGgAJAEnAQAI/wBHCBxIsKDBgwgNBljIsKHDhxAjSpxIsaLFixgzatzIsaNHjglDihz5saTJkyhTqlzJMuPIlzAHtpxJs6bNmzgrxtwpMqfPn0CDCoXIs+jBoUiTKl3a0ahTmUyjSp2q9OlTqlizap1p1enWr2DDuuxaVKzZs2jJlkXLti1WtTzdyp2bFO5Ounjz4rQbU6/fvyr5wgRMuDBIwSQNK14sEXFixpAhO+4ZubLiySEtayaMOeHmz3o7IwRNeq7oo6VTpz1dULVrsaxbv56tNTZB2rip2oaau3fV3SN8C0cKPPjw4z+LI1++Fzjz5zSVQ58e2Dn16yalY99+eDf37xq1g/8fP1E8+fMOzaNfr379+fbux8OP/30+/e3271/Pr386//4nrSHggAQWaOCBCBoowoIMNujggxBG+OB/AJaU4IUYZijhhhx2SGGFHmUo4ogDdmjiiQx+CCJHJLaIIYowcqjiihq5aOOBMeYI4Yw0YnTjjyXqKOSCPPZoEZBADjlkkUZShOSPSgrJZJMSPXljlDpOSSVEVtqIZY5abulQly5+GWOYYjJEZotmwohmmgGsSWKbKL6Zppwj0nminWLiKaKeJvK5pZ8aAiqjdXB+ROiLhm4oKJWLXtioo4gm2lGkCU4q4aNNYoqgphFyaqSnOII6YaWWbkSqgqY6KGqPqxb/2KqrqKaaUawEztrgqzTiGqSuIvC6oq8CAktkrbZeROwaxgaLbLIVLdussCBKayy1FVoLLLYAaqsrt/15Oyu4+onbKrn3mWsquvSpCyq78bmrKbzuyTspvevZ2yi+6OlrKL/n+QsowOQJrCfB4xlMJ8LgKdwmw985bCbE3En8JcXbWYwlxthpHCXH13msJMjUibzks9BOZLKUKKcc0cpZtuzyQzCDKfPMDdV85s04L6Szmzz3/HOdQeM89J5Fz3x0oEm7vLSHTaf89KHe9RwtsdNGDe3UlFZttcpYX6t1slxvOratZYd6dqpp77i2pW2f6vXXL4e97duJxk3r3HTT/2z3t3jDqfeugd/597iF93n4uYkPuvi6jUP6+LuRdzr5vJWPevm9mcO6+b6d9/r5v6EPO/rApVd7+sGpZ7v6wq13+/rDsYc7+8S1l3v7xbmnu/vGvdOHxPDEF2/88cgnf3zWfPfdkPLQRy8987Y5H5H02GdPPPWxWQ+R9uBHzz1r3j8U/vnIj39a+Q6h7/72Yjfv/fvvqy8a+8/Tj779neHPkP77i1/1/BcAAJ6Pf5ghYAENCD4ETkaBDGygALtHwAhqz4GOgaAFsYdBxGhwg+KbIPkqCMIQ3k1+1iuhCQGHQuepEHodFMwHX7g8Ea6PhDSs4QkH6L8cps+G98OhD//ht0MK9nCIxYshX2aIRCXahYlDdCJcoOhDKaqFijm0IlmwSEMtdoWLL/SiVcCoQjFeRYhNBGL/0BhFNSaQjVV04wPhmEU5ZpCOXbSjB/EYRj3KkI9l9OMSAVlCM3qFkCA0pFHIWEhBPhGRG1TkWo6IxOFJMi4KBEoVNsnJTnryk6AM5SdJ1jNRmvKUqCQlzlDJylZyUpUzc6UsTwlLl83ylqCsZcpwyctXBi+TveylLqEVTF4OM1nFxOUxbZXMWy4zVc2c5TMtFU1ZTjNR1XTlNeGUzVZuM03dZOU3xRTOVP5SgeWk5TkJmE5TjnNL7RTlO6kUz1DOs0n1zOU6/Zf/z1HuE3/99OQ9jRTQTg60RwX1ZQszOZOEbvKgNHJoFSC6IolSFEQW/Sf7MrpQhrKEozz06E1AakSR2oSkIzTpSR160Qqh9IYqrclLgxhTmsx0jTVtKEs1Wr6bvjGnLfHpHIH60Z12lKgdEeodkUqXlqrUqSaFqkil6lGqMtSqmcSqArVKQK76z6v4Ayv7xFo+snrPrNZDq/PU2je20s2tX4Or1eTaM7rizK4zw6vL9JoyvkLLr8kCrK0EmyrCWsqwiUIsnBSbJsaKybFbgiyVJNskyhrJsj3CLI00uyLOgsizFQItgETbH9Lqx7T3QS19VBsf1rrHtezhKVNLAlv0Z9T2PbKdrUduSx7eyie3uu1OSIO7mqMSNyu+BU9y6wPc42JkudyBLn6a61yLSBc7190PdatLkexSx7v+2S53GyPe8RKlvOZND3rTyxDwQse9z4Evc+S7HPoix77Hwe9w9Csc/vomIAAAIfkEBcgAAQAsGwBoAMwBJwEACP8ARQgcSLCgwYMIEypcyLChw4cQI0qcSDGAxYsYM2rcyLGjx48gQ4ocSbKkyZMoU56kyLKly5cwY8psqbKmzZs4c+rcyZPkzJ9AgwodKrOn0aNIkypdCpKo06dQowJlSrWq1atYN0rdyrWr14JZw4odSxbl17No084sy7atW7Jq48qdy/Ct3bt4jdLdy3du3r+AA/vsS7gwV8GIEys2zLjxUMWQI991TLnyS8mYM4u1zLnzQ82gQy/1TLq0QdGoU+s0zbq06tewV7aebTm27dseaet2jLu37wC7gxf+TTy28ON7iytPjbx53OXQQTuffja6dcnUsx++zh2x9u9Qu4v/Bwy+/OPx6O2aX/8zvfu27OPDfE9/rPz7LOvrx4q/f8T9AFLl34ANBWhgUgQmmNCBDPak4INgNSghThBWKMKEGNZkIYQZdijbhgl6KOJIICo44okflRgiiixqpCKBLcZ40YsDyigjjf7ZGCOO/enYIo/4+cgikPcJiSKR8hl5IpLxKZnUGlBGKeWUVFZpJZVMZvmZk0dd6eWXYGop5kJcdgnmmWhCOeaaB5VpVJpwfsnmnAO52VOceFZJJ5128pTnn1HuOWefOwEKqKBsEqqToX8iuqaiOTGap6NjQoqTpHhSKqalN2Eap6ZacmqTp3CCmqWoNZGapqlMoqqSqmiy/4qkqynBeqasRNKKkq1h4sqjrifxKqevOAJrkrBeElussSQhe6WyNDLbrLN6QquitCNRW621IGIrkrZYctuttyCBO6W445LrkblSoruhuh+xG6i7FcK7rrxr0FuvvRzhm6++D/LbL74AByywRv4WbOLBGSWsMIwMY+TwwzlGfNHEFAdpsUUYZ5zkxgF07DF7IIdM8MhFgiwyyuWVvDLL37l8MszrySwvzSSrPDPO4NnMLs/m+Wwu0C3rfDPRMRv9M9LaCQ0u000rPTTU1DmtLdVVS/001s5ZTS3XXWt9NdjIee0s2WWL/TXawpmNLNttq3023Lu5LSzddcv9Nt602f/NK99963034K35bSvhhQv+N+KmGQ4r440rfjjkpDmuKuWVS/445p1ZTirnnWt+Oei1if456ZV57inqqZu+Ouu8uY4p7LFv/DLtdKk+O+6G6S4p773L/jvwhPnOKPHFC3888nwZbyjzzSv/PPS5S38o9X5Z3yj2cjl/Pfdqeb89+GmJPyn55Wt/PvrVqZ8p++3bvjP828l/NP1dmf8+/vVbfDv/57Hf0gAoFf19ioAFdN8BERgeBZaKgQ0U4NQg6BQDPpCCRLHgqjCYQQdukINC0WCsQBhCD46QhFMx4a1QmEIJbo2Fa1Fhr2BYFBkOi4YxEeEKcTgfGyaLhz104dj/gOgSHc6QiPnx4bOQSBMlWomJTRTi2qA4ESPekIr/ceK2sAgRK/6Qi13UYrjAuCUpzo2MDvHiEtFYIDGei41tNOPe4EgmN7aLjnWU4+DwuCA7zouPCFHjEwEZSD+qiZBtMuS/EBkhPS6OkQQR5BYheSFFUjKSIEOCJjfJyU568pOg9OQl65TJUJrylKccpUBKhspWunKTqqzkxl5Jy1SqkpW1zKUob1lKXfoSCbHE5S91GcxeDrOWxZzlMXOZTIstk5m8VOYzX9nMiE2TltVk2DWpGU1nbrOV2TzYN8HZTWuO05ajFOY5QRlOga3TlO3k1ztDGU97zZOd5dTmPXeZ/05j7lOT9YTXPzsZUHUNlJMFJddBYZlPcS4UmA1150MT6q2HQrSf0jwoRbFl0Y1Kq6MRledEQ2rPkWLUmwv1KLNAelJzppSkAjXpJdU5UJUai6Uz9ec/bQosnFKSpjuFacl6UoWiGvWoSE2qUpeaVP4N1SpMjapUp+rUp1JlqljNqlGratWlaPWrUuVqV5MC1rIqVaxjPYpZ17pV/KVVKWxlK1rfypO4rnWudNWJXc2K17ziZK9l7atfbQJYsAp2sCop7FcPi1iUKFarjG2sSR6b1chKliSUxaplLyuSzFLVrZy9iWfDCtrQ1mS0Ud2saT2CWqaqdrUcae1SXwtbjf/I9qylre1kb4tU2ur2IrztbW5/O5LgHtW3xDVuW+lH3N0qF7m/VW5Roatb6VaBurW1LnZhq93hNhck3WXud4sr3e2uNrzwGy95n+td9ca2vO11r23hK175spa+6bXvfdlbX/1uBL3s829HAIw+Ab+Xv/k1cEYITD4FwyW+DhYQhCM8mglTGEEWvvBRzKvhDyW4wxLuL4grLOIRY7jEJt5whlNMoRWz2CYcfnGKXCzjlMS4xhy5MY5dROMdl0THPp5Rj4MsEiAT2chBRrKPlbxjJuPYyTWGsoyl/GIqs9jKKcayibU8Yi6D2MsdBrOGxXxhMlPYzBFGs4PVrGA2G9gHzQKGs38DAgAh+QQFyAABACzyAfIBAQABAAAIBAADBAQAOw==\" />"
      ],
      "text/plain": [
       "Plots.AnimatedGif(\"/home/tisaac/srv/rep/22-course-cse6643/mybinder/tmp.gif\")"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 7; n = 5;\n",
    "Q = ones(m,n);\n",
    "U = -0.5*Matrix(1.0*I,n,n);\n",
    "anim = Animation();\n",
    "function u_frame!(Q, U, anim)\n",
    "    qplot = imshow(Q)\n",
    "    plot!(qplot, colorbar=:legend, legend=false, title=\"Q\")\n",
    "    eplot = imshow(U, (-1,1))\n",
    "    plot!(eplot, colorbar=:legend, legend=false, title=\"Zᵢ\")\n",
    "    p = plot(qplot, eplot, layout=(1,2))\n",
    "    frame(anim, p)\n",
    "end\n",
    "u_frame!(Q, U, anim)\n",
    "for j in 1:n\n",
    "    Q[:,j] .= -1.0+0.01\n",
    "    U[:,:] = -0.5*Matrix(1.0*I,n,n)\n",
    "    U[j,j] = 1.0\n",
    "    U[1:j-1,j] .= -1.0\n",
    "    u_frame!(Q, U, anim)\n",
    "end\n",
    "u_frame!(Q, U, anim)\n",
    "display(\"Classical Gram-Schmidt\")\n",
    "gif(anim, fps=0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "75acdb08",
   "metadata": {
    "cell_style": "split",
    "hideCode": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"Modified Gram-Schmidt\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<img 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\" />"
      ],
      "text/plain": [
       "Plots.AnimatedGif(\"/home/tisaac/srv/rep/22-course-cse6643/mybinder/tmp.gif\")"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 7; n = 5;\n",
    "Q = ones(m,n);\n",
    "U = -0.5*Matrix(1.0*I,n,n);\n",
    "anim = Animation();\n",
    "u_frame!(Q, U, anim)\n",
    "for j in 1:n\n",
    "    Q[:,j:end] .= 1.0\n",
    "    Q[:,j] .= -1.0+0.01\n",
    "    Q[(mod(j,2)+1):2:end,j+1:end] .-= 0.1\n",
    "    U[:,:] = -0.5*Matrix(1.0*I,n,n)\n",
    "    U[j,j] = 1.0\n",
    "    U[j,j+1:n] .= -1.0\n",
    "    u_frame!(Q, U, anim)\n",
    "end\n",
    "u_frame!(Q, U, anim)\n",
    "display(\"Modified Gram-Schmidt\")\n",
    "gif(anim, fps=0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7377c41a",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### CGS is unstable; MGS is better but not as good as `qr`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "8fda5818",
   "metadata": {
    "hideCode": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n",
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"1300\" height=\"800\" viewBox=\"0 0 5200 3200\">\n",
       "<defs>\n",
       "  <clipPath id=\"clip660\">\n",
       "    <rect x=\"0\" y=\"0\" width=\"5200\" height=\"3200\"/>\n",
       "  </clipPath>\n",
       "</defs>\n",
       "<path clip-path=\"url(#clip660)\" d=\"\n",
       "M0 3200 L5200 3200 L5200 0 L0 0  Z\n",
       "  \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n",
       "<defs>\n",
       "  <clipPath id=\"clip661\">\n",
       "    <rect x=\"1040\" y=\"0\" width=\"3641\" height=\"3200\"/>\n",
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      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function kahan(m, θ)\n",
    "    s = sin(θ)\n",
    "    c = cos(θ)\n",
    "    S = Diagonal(s .^ collect(0:m-1))\n",
    "    K = UnitUpperTriangular(-c * ones(m,m))\n",
    "    return S * K\n",
    "end\n",
    "orthonormality_error(Q) = norm(I - Q[:,:]' * Q[:,:], 2)\n",
    "reconstruction_error(A, Q, R) = norm(A - Q * R, 2)\n",
    "ortho_errs_cgs = Float64[]\n",
    "recon_errs_cgs = Float64[]\n",
    "ortho_errs_mgs = Float64[]\n",
    "recon_errs_mgs = Float64[]\n",
    "rang = π ./ (2:.1:20)\n",
    "for θ in rang\n",
    "    M = kahan(10, θ)\n",
    "    ϕ = eps(Float64)\n",
    "    M_p = copy(M[:,:])\n",
    "    rot = [cos(ϕ) sin(ϕ); -sin(ϕ) cos(ϕ)]\n",
    "    M_p[1:2,:] = rot * M[1:2,:]\n",
    "    Q_cgs, R_cgs = cgs(M_p)\n",
    "    Q_mgs, R_mgs = mgs(M_p)\n",
    "    push!(ortho_errs_cgs, orthonormality_error(Q_cgs))\n",
    "    push!(recon_errs_cgs, reconstruction_error(M_p, Q_cgs, R_cgs))\n",
    "    push!(ortho_errs_mgs, orthonormality_error(Q_mgs))\n",
    "    push!(recon_errs_mgs, reconstruction_error(M_p, Q_mgs, R_mgs))\n",
    "end\n",
    "ymin=Inf\n",
    "ymax=0.\n",
    "for series in [ortho_errs_cgs, ortho_errs_mgs, recon_errs_cgs, recon_errs_mgs]\n",
    "    series[:] = max.(series[:], 0.001*eps())\n",
    "    ymin = min(ymin, minimum(series))\n",
    "    ymax = max(ymax, maximum(series))\n",
    "end\n",
    "ymin *= 0.5\n",
    "ymax *= 2.0\n",
    "p1 = plot( rang .^-1, [ortho_errs_cgs recon_errs_cgs]\n",
    "    , yscale=:log10\n",
    "    , label=[\"orthonormality error\" \"reconstruction error\"]\n",
    "    , xlabel=\"Kahan matrix θ⁻¹\"\n",
    "    , title=\"Classical Gram-Schmidt\"\n",
    "    , size=(1300,800)\n",
    "    , ylim=(ymin,ymax)\n",
    ")\n",
    "p2 = plot( rang .^-1, [ortho_errs_mgs recon_errs_mgs]\n",
    "    , yscale=:log10\n",
    "    , label=[\"orthonormality error\" \"reconstruction error\"]\n",
    "    , xlabel=\"Kahan matrix θ⁻¹\"\n",
    "    , title=\"Modified Gram-Schmidt\"\n",
    "    , size=(1300,800)\n",
    "    , ylim=(ymin,ymax)\n",
    ")\n",
    "draw(plot(p1,p2,layout=(1,2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2d902aa5",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Rounding errors affect the orthogonality\n",
    "\n",
    "If we start with $\\hat{R}^{(0)} = I$ and $\\hat{Q}^{(0)} = A$, step $i$ is like\n",
    "\n",
    "$$\n",
    "A = \\hat{Q}^{(i-1)} \\hat{R}^{(i-1)} = \\hat{Q}^{(i-1)} Z_{i} Z_{i}^{-1} \\hat{R}^{(i-1)} = \\underbrace{(\\hat{Q}^{(i-1)} Z_{i})}_{\\hat{Q}^{(i)}} \\underbrace{(Z_i^{-1} \\hat{R}^{(i-1)})}_{\\hat{R}^{(i)}}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bbd6b7e0",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "- We saw last time that rounding errors have the biggest effect when\n",
    "\n",
    "$$|r_{ii}| \\ll \\max_j |r_{ji}|$$\n",
    "\n",
    "- When this happens\n",
    "\n",
    "$$\\underbrace{\\|Z_i\\|_2 \\|Z_i^{-1}\\|_2}_{\\text{(condition number of } Z_i\\text{)}} \\gg 1$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a6144796",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "- **Matrices with large condition numbers blow-up rounding errors**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5e2f018c",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "**Exercise:**  What type of matrix $Y$ has\n",
    "\n",
    "$$\\|Y\\|_2 \\|Y^{-1}\\|_2 = 1?$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "81b406db",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Householder QR: produce a _full QR factorization_ by  _unitary transformations_ to $R$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5c5a2461",
   "metadata": {
    "cell_style": "center"
   },
   "source": [
    "- Start with $Q^{(0)} = I$ and $R^{(0)} = A$\n",
    "\n",
    "- At step $i$: $\\quad \\quad A = Q^{(i-1)} R^{(i-1)} = Q^{(i-1)} U_i^{-1} U_i R^{(i-1)} = \\underbrace{(Q^{(i-1)} U_{i-1}^*)}_{Q^{(i)}} \\underbrace{(U_{i-1} R^{(i-1)})}_{R^{(i)}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "d44deead",
   "metadata": {
    "cell_style": "split",
    "hideCode": true
   },
   "outputs": [
    {
     "data": {
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\" />"
      ],
      "text/plain": [
       "Plots.AnimatedGif(\"/home/tisaac/srv/rep/22-course-cse6643/mybinder/tmp.gif\")"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 7; n = 5;\n",
    "R = ones(m,n);\n",
    "E = zeros(n,n);\n",
    "anim = Animation();\n",
    "function h_frame!(R, E, anim)\n",
    "    rplot = imshow(R)\n",
    "    plot!(rplot, colorbar=:legend, legend=false, title=\"R\")\n",
    "    eplot = imshow(E, (-1,1))\n",
    "    plot!(eplot, colorbar=:legend, legend=false, title=\"||I-Q'Q||\")\n",
    "    p = plot(rplot, eplot, layout=(1,2), plot_title=\"Householder QR\")\n",
    "    frame(anim, p)\n",
    "end\n",
    "h_frame!(R, E, anim)\n",
    "for j in 1:n\n",
    "    R[:] = ones(m,n)\n",
    "    R[j:2:end,:] .-= 0.1\n",
    "    R[1:j,:] .= -1.0 + 0.01\n",
    "    for i in 1:j\n",
    "        R[i+1:end,i] .= 0.\n",
    "    end\n",
    "    h_frame!(R, E, anim)\n",
    "end\n",
    "h_frame!(R, E, anim)\n",
    "gif(anim, fps=0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "949ea4e4",
   "metadata": {
    "cell_style": "split"
   },
   "source": [
    "- $I - Q^{(i)*} Q^{(i)} = 0\\ $ **at each step** because $Q^{(i)} = Q^{(i-1)} U_i$, and\n",
    "\n",
    "$$\\text{unitary} \\cdot \\text{unitary} = \\text{unitary}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e0230174",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "> Gram-Schmidt: triangular orthogonalization\n",
    ">\n",
    "> Householder: orthogonal triangularization"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "295c0c6c",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Householder QR: how it's done"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a187c64a",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Consider the first step\n",
    "\n",
    "$$R^{(1)} \\gets U_1 R^{(0)}$$\n",
    "\n",
    "- Action on the left $\\Rightarrow$ acts on the rows $\\Rightarrow$ same action applied to each column"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "27ba731f",
   "metadata": {
    "cell_style": "split",
    "hideCode": true
   },
   "outputs": [
    {
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      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 7; n = 5;\n",
    "R0 = ones(m, n);\n",
    "R1 = ones(m, n);\n",
    "R1[2:2:end,:] .-= 0.1;\n",
    "R1[1,:] .= -1;\n",
    "R1[2:end,1] .= 0;\n",
    "plot(imshow(R0), imshow(R1), colorbar=false, layout=(1,2), plot_title =\"R⁽⁰⁾            ⇒            R⁽¹⁾\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "049c92cc",
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     "slide_type": "-"
    }
   },
   "source": [
    "- We want an action such that\n",
    "\n",
    "$$U_1 \\underbrace{r_1^{(0)}}_{a_1} = \\alpha e_1$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2094101d",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "If\n",
    "\n",
    "$$\n",
    "U_1 r_1^{(0)} = \\alpha e_1,\n",
    "$$\n",
    "\n",
    "what can $\\alpha$ be?\n",
    "\n",
    "(Hint: unitary transformations preserve the 2-norm)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b2a9b970",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$ \\|\\alpha e_1\\|_2 = \\| U_1 r_1^{(0)} \\|_2 = \\| r_1^{(0)} \\|_2$$\n",
    "\n",
    "$$ \\|\\alpha e_1 \\|_2 = |\\alpha| \\|e_1\\|_2 = |\\alpha|$$\n",
    "\n",
    "$$ \\Rightarrow |\\alpha| = \\| r_1^{(0)} \\|_2 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "358dbc9e",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Geometrically\n",
    "\n",
    "In $\\mathbb{R}$, there are only two possibilities:"
   ]
  },
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   "cell_type": "markdown",
   "id": "109b4efa",
   "metadata": {
    "cell_style": "split"
   },
   "source": [
    "![householder1.svg](attachment:householder1.svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "090ddf03",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### The simplest unitary matrix that maps $a_1$ into $\\mathrm{span}\\{e_1\\}$ is a _Householder reflector_\n",
    "\n",
    "First defined algebraically:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "efd6fe4c",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
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   },
   "source": [
    "$$\n",
    "\\begin{aligned}\n",
    "\\phi &= \\mathrm{sign}(a_{11}) & [(\\phi e_1)^* a_1 = |a_{11}|]\n",
    "\\\\\n",
    "\\tilde{u}_1 &= \\frac{\\overbrace{-\\phi \\|a_1\\| e_1}^{\\text{target}} - a_1}{2} & \\text{[half the difference]} \\\\\n",
    "\\tilde{v}_1 &= \\frac{-\\phi \\|a_1\\| e_1 + a_1}{2} & \\text{[midpoint]} \\\\\n",
    "u_1 &= \\frac{\\tilde{u}_1}{\\|\\tilde{u}_1\\|} \\\\\n",
    "U_1 &= I - 2 u_1 u_1^*\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "12aa70d7",
   "metadata": {
    "cell_style": "split",
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     "slide_type": "fragment"
    }
   },
   "source": [
    "- $\\tilde{v}_1 + \\tilde{u}_1 = -\\phi \\|a_1\\| e_1$\n",
    "\n",
    "- $\\tilde{v}_1 - \\tilde{u}_1 = a_1$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2f6bc38d",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- $\\tilde{u}_1$ and $\\tilde{v}_1$ are orthogonal: $\\tilde{u}_1^* \\tilde{v}_1$ =\n",
    "\n",
    "$$\\begin{gather}\n",
    "\\frac{1}{4}(\\|a_1\\|^2 \\underbrace{(\\phi e_1)^*(\\phi e_1)}_{1} \\\\ - \\|a_1\\| \\underbrace{(\\phi e_1)^* a_1}_{|a_{11}|}  + \\|a_1\\| \\underbrace{a_1^*(\\phi e_1)}_{|a_{11}|}\\\\  - a_1^* a_1) \\\\ = 0\\end{gather}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "82674a0f",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Claim: $U_1$ is unitary and $U_1 a_1 = - \\phi \\|a_1\\|e_1$\n",
    "\n",
    "_Proof._"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aa7b7ea5",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "$$\n",
    "\\begin{aligned}\n",
    "\\phi &= \\mathrm{sign}(a_{11}) & [(\\phi e_1)^* a_1 = |a_{11}|]\n",
    "\\\\\n",
    "\\tilde{u}_1 &= \\frac{-\\phi \\|a_1\\| e_1 - a_1}{2} \\\\\n",
    "\\tilde{v}_1 &= \\frac{-\\phi \\|a_1\\| e_1 + a_1}{2} \\\\\n",
    "u_1 &= \\frac{\\tilde{u}_1}{\\|\\tilde{u}_1\\|} \\\\\n",
    "U_1 &= I - 2 u_1 u_1^*\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3efe0252",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- $U_1$ is unitary because it is equal to $(I - 2P_1)$ where $P_1$ is the orthogonal projector onto $\\mathrm{span}\\{u_1\\}$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6fd25d44",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- $U_1 \\tilde{u}_1 = (I - 2P_1) \\tilde{u}_1 = \\tilde{u}_1 - 2 \\underbrace{P_1 \\tilde{u}_1}_{\\tilde{u}_1} = - \\tilde{u}_1$\n",
    "- $U_1 \\tilde{v}_1 = (I - 2P_1) \\tilde{v}_1\n",
    "= \\tilde{v}_1 - 2 \\underbrace{P_1 \\tilde{v}_1}_{0} = \\tilde{v}_1$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "db48f3fd",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\\Rightarrow U_1 a_1 = U_1 (\\tilde v_1 - \\tilde u_1) = \\tilde v_1 + \\tilde u_1 = -\\phi \\|a_1\\|e_1$$"
   ]
  },
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   "cell_type": "markdown",
   "id": "b9ae7fe7",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Geometrically\n",
    "\n",
    "![householder4.svg](attachment:householder4.svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a31c9eca",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Exercise (T&B Exercise 10.1)\n",
    "\n",
    "Determine the (a) eigenvalues, (b) determinant, and (c) singular values of a Householder reflector.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ba69529f",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### In action"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "bb015476",
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "householder_vector (generic function with 1 method)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function householder_vector(x)\n",
    "    \"\"\"\n",
    "    Compute the vector of the Householder\n",
    "    reflector that sends x to a multiple of\n",
    "    e₁\n",
    "    \"\"\"\n",
    "    u = copy(x)\n",
    "    xnorm = norm(x)\n",
    "    phi = sign(u[1])\n",
    "    u[1] += phi * xnorm # u = x + sign(x₁)||x|| e₁\n",
    "    unorm = norm(u)\n",
    "    u /= unorm\n",
    "    return u\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "7d634232",
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "householder_reflect (generic function with 1 method)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function householder_reflect(u,v)\n",
    "    \"\"\"\n",
    "    Apply the Householder reflector defined by\n",
    "    u to v\n",
    "    \"\"\"\n",
    "    return v - 2*u*(u'*v)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "fde5806e",
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "householder_reflect_left (generic function with 1 method)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function householder_reflect_left(vstar,u)\n",
    "    \"\"\"\n",
    "    Apply the Householder reflector defined by\n",
    "    u to row vector vstar\n",
    "    \"\"\"\n",
    "    return vstar - 2*(vstar*u)*u'\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1346f733",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Why $x \\mapsto - \\mathrm{sign}(x_1)\\|x\\|_2 e_1$ and not $x \\mapsto \\mathrm{sign}(x_1)\\|x\\|_2 e_1$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "aa4401e7",
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [],
   "source": [
    "function householder_vector(x)\n",
    "    u = copy(x)\n",
    "    xnorm = norm(x)\n",
    "    phi = sign(u[1])\n",
    "    u[1] += phi * xnorm #  |u[1]| ≥  |x[1]|\n",
    "    unorm = norm(u)     # || u || ≥ || x ||\n",
    "    u /= unorm # errors in denominator here\n",
    "    return u\n",
    "end;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "5cdc602b",
   "metadata": {
    "cell_style": "split",
    "hideCode": false,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "function householder_vector_unstable(x)\n",
    "    u = copy(x)\n",
    "    xnorm = norm(x)\n",
    "    phi = sign(u[1])\n",
    "    u[1] -= phi * xnorm #  |u[1]| ≤  |x[1]|\n",
    "    unorm = norm(u)     # || u || ≤ || x ||\n",
    "    u /= unorm # errors in denominator here\n",
    "    return u\n",
    "end;"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "50a4b203",
   "metadata": {},
   "source": [
    "Suppose we have made a previous error and in place of $x_1$ we have $x_1 + \\delta$:\n",
    "\n",
    "$$u_1 \\gets (x_1+\\delta) + \\mathrm{sign}(x_1) \\|x\\|; \\quad \\check u_1 \\gets (x_1 + \\delta) - \\mathrm{sign}(x_1) \\|x\\|.$$\n",
    "\n",
    "In either case we have a $|\\delta|$ error in $u_1$ or $\\check u_1$, but it is a larger _relative error_ for $\\check u_1$ than $u_1$:\n",
    "\n",
    "$$ \\frac{|\\delta|}{\\underbrace{|u_1|}_{|x_1| + \\|x\\|}} \\leq \\frac{|\\delta|}{\\underbrace{|\\check u_1|}_{|x_1| - \\|x\\|}} $$\n",
    "\n",
    "This matters when we normalize by $\\|u\\|$ on line 7."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "caea152d",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### From step 1 to step $j$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "f1248041",
   "metadata": {
    "cell_style": "split",
    "hideCode": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
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      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 7; n = 5; j = 2;\n",
    "R = ones(m,n)\n",
    "R[j:2:end,:] .-= 0.1\n",
    "R[1:j,:] .= -1.0 + 0.01\n",
    "for i in 1:j\n",
    "    R[i+1:end,i] .= 0.\n",
    "end\n",
    "Rj = copy(R);\n",
    "j += 1\n",
    "R = ones(m,n)\n",
    "R[j:2:end,:] .-= 0.1\n",
    "R[1:j,:] .= -1.0 + 0.01\n",
    "for i in 1:j\n",
    "    R[i+1:end,i] .= 0.\n",
    "end\n",
    "Rjp1 = copy(R)\n",
    "plot(imshow(Rj), imshow(Rjp1), colorbar=false, layout=(1,2), plot_title =\"R⁽ʲ⁻¹⁾            ⇒            R⁽ʲ⁾\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "489d0ea7",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "- Let $x^{(j)}$ be the _tail_ of column $j$,\n",
    "\n",
    "$$x^{(j)} = r_{j:m,j}^{(j-1)} \\in \\mathbb{C}^{m-(j-1)}$$\n",
    "\n",
    "- Compute the Householder vector $u^{(j)}$ for $x^{(j)}$ that makes the unitary matrix\n",
    "$\\tilde{U}_j$\n",
    "\n",
    "- Make $U_J$ by putting $\\tilde{U}_j$ on lower diagonal block"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8a65cab7",
   "metadata": {
    "cell_style": "center",
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "& & & & & & \\\\\n",
    "& & & & & & \\\\\n",
    "& & & & & & \\\\\n",
    "& & & U_j & & & \\\\\n",
    "& & & & & & \\\\\n",
    "& & & & & & \\\\\n",
    "& & & & & & \\\\\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\left[\n",
    "\\begin{array}{cccc|ccc}\n",
    "1 & & & & & & \\\\\n",
    "& 1 & & & & & \\\\\n",
    "& & \\ddots & & & & \\\\\n",
    "& & & 1 & & & \\\\ \\hline\n",
    "& & & & & & \\\\\n",
    "& & & & & \\tilde{U}_j & \\\\\n",
    "& & & & & & \\\\\n",
    "\\end{array}\n",
    "\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5fc402eb",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### All together"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "17e19a01",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "householder_literal (generic function with 1 method)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function householder_literal(A)\n",
    "    m, n = size(A)\n",
    "    Q = Matrix{eltype(A)}(I, m, m)\n",
    "    R = copy(A)\n",
    "    for i = 1:n\n",
    "        x = R[i:m,i]\n",
    "        u = householder_vector(x)\n",
    "        R[i:m,i:n] = householder_reflect(u, R[i:m ,i:n]) # apply reflector to trailing rows of R to make r_i upper\n",
    "        R[i+1:m,i] .= 0\n",
    "        Q[:,i:m] = householder_reflect_left(Q[:,i:m], u) # apply reflector to trailing columns of Q to multiply in U_i\n",
    "    end\n",
    "    return Q, R\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f24af7d2",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Recall our tests from last time"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "5fb90892",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "qr_test (generic function with 4 methods)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reconstruction_error(A, Q, R) = norm(A - Q * R, 2)\n",
    "orthonormality_error(Q) = norm(I - Q[:,:]' * Q[:,:], 2)\n",
    "upper_triangularity_error(R) = norm(R - UpperTriangular(R), 2)\n",
    "function qr_test(A, Q, R, recon_tol=1.e-12, ortho_tol=1.e-12, triu_tol=0.0)\n",
    "    m, n = size(A)\n",
    "    p = min(m,n)\n",
    "    recon_err = reconstruction_error(A, Q, R)\n",
    "    ortho_err = orthonormality_error(Q)\n",
    "    triu_err = upper_triangularity_error(R[1:p,1:p])\n",
    "    println(\"Reconstruction error || A - Q R ||_F = $recon_err\")\n",
    "    println(\"Orthonormality error || I - Q'Q ||_F = $ortho_err\")\n",
    "    println(\"Upper-triangularity error || R - triu(R) ||_F = $triu_err\")\n",
    "    return (recon_err <= recon_tol) && (ortho_err <= ortho_tol) && (triu_err <= triu_tol) \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "739c4a22",
   "metadata": {
    "cell_style": "center",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Reconstruction error || A - Q R ||_F = 7.511540944239318e-15\n",
      "Orthonormality error || I - Q'Q ||_F = 2.1031818359611525e-15\n",
      "Upper-triangularity error || R - triu(R) ||_F = 0.0\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 7; n = 5;\n",
    "A = randn(m, n)\n",
    "Q, R = householder_literal(A);\n",
    "qr_test(A, Q, R)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e9acbdc2",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### `householder_literal` is on par with `qr`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "7c052c93",
   "metadata": {
    "hideCode": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
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      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ortho_errs_hh = Float64[]\n",
    "recon_errs_hh = Float64[]\n",
    "ortho_errs_qr = Float64[]\n",
    "recon_errs_qr = Float64[]\n",
    "rang = π ./ (2:.1:20)\n",
    "for θ in rang\n",
    "    M = kahan(10, θ)\n",
    "    ϕ = eps(Float64)\n",
    "    M_p = copy(M[:,:])\n",
    "    rot = [cos(ϕ) sin(ϕ); -sin(ϕ) cos(ϕ)]\n",
    "    M_p[1:2,:] = rot * M[1:2,:]\n",
    "    Q_hh, R_hh = householder_literal(M_p)\n",
    "    Q_qr, R_qr = qr(M_p)\n",
    "    push!(ortho_errs_hh, orthonormality_error(Q_hh))\n",
    "    push!(recon_errs_hh, reconstruction_error(M_p, Q_hh, R_hh))\n",
    "    push!(ortho_errs_qr, orthonormality_error(Q_qr))\n",
    "    push!(recon_errs_qr, reconstruction_error(M_p, Q_qr, R_qr))\n",
    "end\n",
    "ymin=Inf\n",
    "ymax=0.\n",
    "for series in [ortho_errs_hh, ortho_errs_hh, recon_errs_qr, recon_errs_qr]\n",
    "    series[:] = max.(series[:], 0.001*eps())\n",
    "    ymin = min(ymin, minimum(series))\n",
    "    ymax = max(ymax, maximum(series))\n",
    "end\n",
    "ylo = eps()*0.01\n",
    "ymin = max(ymin, ylo)\n",
    "ymin *= 0.5\n",
    "ymax *= 2.0\n",
    "p1 = plot( rang .^-1, [max.(ylo,ortho_errs_hh) max.(ylo,recon_errs_hh)]\n",
    "    , yscale=:log10\n",
    "    , label=[\"orthonormality error\" \"reconstruction error\"]\n",
    "    , xlabel=\"Kahan matrix θ⁻¹\"\n",
    "    , title=\"householder_literal\"\n",
    "    , size=(1300,800)\n",
    "    , ylim=(ymin,2.0)\n",
    ")\n",
    "p2 = plot( rang .^-1, [max.(ylo,ortho_errs_qr) max.(ylo,recon_errs_qr)]\n",
    "    , yscale=:log10\n",
    "    , label=[\"orthonormality error\" \"reconstruction error\"]\n",
    "    , xlabel=\"Kahan matrix θ⁻¹\"\n",
    "    , title=\"julia qr\"\n",
    "    , size=(1300,800)\n",
    "    , ylim=(ymin,2.0)\n",
    ")\n",
    "draw(plot(p1,p2,layout=(1,2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6d91f54d",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Householder QR performance"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "101270e6",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Work of applying one reflector\n",
    "\n",
    "- The work of applying\n",
    "\n",
    "  $$v \\gets v - 2 u (u^*v) = v - u (2*(u^*v))$$\n",
    "\n",
    "  is just one more flop than the work of applying a projection\n",
    "\n",
    "  $$v \\gets v - u (u^* v).$$\n",
    "  \n",
    "  This is $4 m$ flops if $u \\in \\mathbb{C}^m$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6588089c",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- If we apply the reflector to a matrix $A \\in \\mathbb{C}^{m \\times k}$, the work is $4mk$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "7fd6e8db",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "householder_literal (generic function with 1 method)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function householder_literal(A)\n",
    "    m, n = size(A)\n",
    "    Q = Matrix{eltype(A)}(I, m, m)\n",
    "    R = copy(A)\n",
    "    for i = 1:n\n",
    "        x = R[i:m,i]                                     # x has length (m - (i-1))\n",
    "        u = householder_vector(x)                        # O(m - (i-1)) (lower order term)\n",
    "        R[i:m,i:n] = householder_reflect(u, R[i:m,i:n])  # length: (m - (i-1)); # vectors: (n - (i-1)) ⇒ 4(m-(i-1))(n-(i-1))\n",
    "        R[i+1:m,i] .= 0                                  # ignore\n",
    "        Q[:,i:m] = householder_reflect_left(Q[:,i:m], u) # length: (m - (i-1)); # vectors: m ⇒ 4(m-(i-1))m\n",
    "    end\n",
    "    return Q, R\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "623dd3bf",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "- Work applying reflectors to $R$: $$\\sum_{i=1}^n 4(m-(i-1))(n-(i-1)) = \\text{(change of variable)} \\sum_{k=1}^n 4(k + (m-n))k$$\n",
    "\n",
    "- Work building $Q$: $$\\sum_{i=1}^n 4 m(m -(i-1)) = \\sum_{k=1}^n 4m(k + (m-n))$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "12a0a7e3",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Trick for getting the leading coefficients work summations: calculus\n",
    "\n",
    "- If you remember a rule for the indefinite integral like $\\int f(x) dx = F(x) + C$, then\n",
    "\n",
    "  $$\\sum_{k=0,1}^n f(k) = F(n) - F(0,1) + \\text{lower order terms}$$\n",
    "  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f1958142",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "- So for the work of apply reflectors to $R$:\n",
    "\n",
    "  $$\\begin{gather}\n",
    "  \\sum_{k=1}^n 4(k + (m-n))k \\approx \\int_0^n 4k^2 + 4(m-n)k\\ dk = \\frac{4}{3}n^3 + \\frac{4}{2}(m-n)n^2 + \\text{l.o.t.} \\\\ = 2mn^2 - \\frac{2}{3}n^3 + \\text{l.o.t.}\\end{gather}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8495dd5a",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Work building $Q$:\n",
    "\n",
    "$$\\begin{gather}\\sum_{k=1}^n 4m( k + (m-n) ) \\approx \\int_0^n 4mk + 4m(m-n)\\ dk = \\frac{4}{2}m n^2 + 4m(m-n)n + \\text{l.o.t.}\\\\ = 4m^2n - 2 mn^2 + \\text{l.o.t.} \\end{gather}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4ae527f7",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Building $Q$ ($O(m^2n)$) is more expensive than builiding $R$ ($O(mn^2)$), so avoid it\n",
    "\n",
    "- Collect the reflectors $\\{u_1, \\dots, u_n\\}$ (total cost of constructing all these: $O(mn)$)\n",
    "\n",
    "- The vectors are often stored in the _lower trapezoidal part_ of a 2D array $U$\n",
    "\n",
    "- $U_i$s were accumulated into $Q$ on the right,\n",
    "\n",
    "  $$Q = U_1 \\cdot U_2 \\cdot \\cdots \\cdot U_n,$$\n",
    "  \n",
    "  so apply the reflectors in reverse order to apply $Q$,\n",
    "  \n",
    "  $$Qv = U_1 (U_2 ( \\cdots ( U_n v ) \\cdots )).$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "0b42c948",
   "metadata": {
    "cell_style": "center"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "apply_Q (generic function with 1 method)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function apply_Q(U, v)\n",
    "    m, n = size(U)\n",
    "    y = copy(v)\n",
    "    for i in n:-1:1\n",
    "        y[i:end] -= U[i:end,i] * (2 * (U[i:end,i]' * y[i:end]))\n",
    "    end\n",
    "    return y\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9a3d5aaf",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "- To apply $Q^*$,\n",
    "\n",
    "  $$Q^* = U_n^* \\cdot U_{n-1}^* \\cdot \\cdots \\cdot U_1^*.$$\n",
    "  \n",
    "- But reflectors are hermitian, so to apply $Q^*$ apply reflectors in original order,\n",
    "\n",
    "  $$Q^* v = U_n(U_{n-1}(\\cdots(U_1 v) \\cdots )).$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "8c80abba",
   "metadata": {
    "cell_style": "center"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "apply_Qstar (generic function with 1 method)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function apply_Qstar(U, v)\n",
    "    m, n = size(U)\n",
    "    y = copy(v)\n",
    "    for i in 1:n\n",
    "        y[i:end] -= U[i:end,i] * (2 * (U[i:end,i]' * y[i:end]))\n",
    "    end\n",
    "    return y\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a9ad58d3",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Non-literal Householder QR: return reflectors $U$ with $R$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "5804b2b7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "householder_qr (generic function with 1 method)"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function householder_qr(A)\n",
    "    m, n = size(A)\n",
    "    U = zeros(eltype(A), m, n)\n",
    "    R = copy(A)\n",
    "    for i = 1:n\n",
    "        U[i:m,i] = householder_vector(R[i:m,i])\n",
    "        R[i:m,i:n] = householder_reflect(U[i:m,i], R[i:m,i:n])\n",
    "        R[i+1:m,i] .= 0\n",
    "    end\n",
    "    return U, R\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "b3f04648",
   "metadata": {
    "cell_style": "center",
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 100; n = 50;\n",
    "A = randn(m, n);\n",
    "Q, R_lit = householder_literal(A)\n",
    "U, R_hh = householder_qr(A)\n",
    "norm(R_lit - R_hh, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "ac766f02",
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6.229237174668418e-15"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = randn(m)\n",
    "Qv_lit = Q * v\n",
    "Qv_hh = apply_Q(U,v)\n",
    "norm(Qv_lit - Qv_hh, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "4ce2d28d",
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.873855566419749e-15"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Qstarv_lit = Q' * v\n",
    "Qstarv_hh = apply_Qstar(U,v)\n",
    "norm(Qstarv_lit - Qstarv_hh, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9895b27d",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### `householder_qr` is very close to `qr`\n",
    "\n",
    "- The `LinearAlgebra.QRCompactWY` type of `qr().Q` is essentially the matrix $U$ containing the reflectors\n",
    "\n",
    "- The [LAPACK `*geqrf` fortran routines](http://netlib.org/lapack/lapack-3.1.1/html/dgeqrf.f.html) that higher-level languages call also are essentially Householder QR, but they pack (most of) $U$ and $R$ into the same output array:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "49872eba",
   "metadata": {
    "hideCode": true
   },
   "outputs": [
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      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 7; n = 5;\n",
    "A = ones(m,n)\n",
    "for i = 1:n\n",
    "    A[i+1:end,i] .= -1.0\n",
    "end\n",
    "p = imshow(A)\n",
    "plot!(p, colorbar=false, title = \"LAPACK *GEQRF R (upper) and reflectors (lower)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7c8b6cb5",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Notes on HW2, problem 5\n",
    "\n",
    "The point is to see that for some triangular matrices the singular values are very different from the diagonal values, using the example of a particular _Toeplitz_ (constant diagonals) matrix.\n",
    "\n",
    "- Part (a): straightforward given previous work on triangular matrices in lectures and homework 1.\n",
    "- Part (b): guess that $A^{-1}$ is _also_ Toeplitz; determine what each diagonal must be; check that it is the inverse.\n",
    "- Part (c): How does $\\|A^{-1}\\|_2$ relate to the smallest singular value of $A$? Consider $\\|A^{-1} e_k\\|_2$ for different choices of $k$."
   ]
  }
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