lec08static.ipynb

{
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   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
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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\", \"Polyhedra\", \"GLPK\"]);\n",
    "import Logging; Logging.disable_logging(Logging.Info)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "const STATIC_LECTURE = true;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "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,
   "metadata": {},
   "outputs": [
    {
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       "<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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       "WebIO._IJuliaInit()"
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     "data": {
      "text/plain": [
       "imshow (generic function with 1 method)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using LinearAlgebra;\n",
    "using Plots; plotlyjs(); default(l=5, size=(900,550), label=false);\n",
    "using Interact;\n",
    "using Polyhedra;\n",
    "using Printf;\n",
    "\n",
    "import GLPK\n",
    "lib = DefaultLibrary{Float64}(GLPK.Optimizer)\n",
    "\n",
    "prettyprint(a) = map(x -> @sprintf(\"%+.2f\", x), v)\n",
    "\n",
    "function imshow(x; kwargs...)\n",
    "    return heatmap(x\n",
    "        , aspect_ratio=:equal\n",
    "        , framestyle=:none\n",
    "        , yflip=true\n",
    "        , c=:diverging_gwv_55_95_c39_n256\n",
    "        ; kwargs...\n",
    "    )\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**(Follow along: <https://tinyurl.com/gtcse6643lec8>)**\n",
    "\n",
    "### Warmup\n",
    "\n",
    "- Look over HW2 and prepare questions for office hours / piazza"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Parallelism, Least Squares Problems\n",
    "\n",
    "**1. Parallelism and QR**\n",
    "\n",
    "**2. Least Squares Problems**\n",
    "\n",
    "**3. Orthogonal Projection and the Normal Equations**\n",
    "\n",
    "**4. The Pseudoinverse**\n",
    "\n",
    "**5. Solving the Least Squares Problem**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Parallelism and QR"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### A summary comparison of algorithms\n",
    "\n",
    "| algorithm | acts on | stability | behavior | cost |\n",
    "| :-: | :-: | :-: | :-: | :-: |\n",
    "| classical Gram-Schmidt | $$Q$$ | ❌ | left-looking | $$\\sim 2mn^2$$ |\n",
    "| modified Gram-Schmidt | $$Q$$ | ⚠️ | right-looking | $$\\sim 2mn^2$$ |\n",
    "| Householder QR | $$R$$ | ✅ | down-looking | $$\\sim 2mn^2 - \\textstyle{\\frac{2}{3}}n^3$$ |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### The missing column(s)?\n",
    "\n",
    "| algorithm | acts on | stability | behavior | cost | **_parallelism_** |\n",
    "| :-: | :-: | :-: | :-: | :-: | :-: |\n",
    "| classical Gram-Schmidt | $$Q$$ | ❌ | left-looking | $$\\sim 2mn^2$$ | **?** |\n",
    "| modified Gram-Schmidt | $$Q$$ | ⚠️ | right-looking | $$\\sim 2mn^2$$ | **?** |\n",
    "| Householder QR | $$R$$ | ✅ | down-looking | $$\\sim 2mn^2 - \\textstyle{\\frac{2}{3}}n^3$$ | **?** |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### The importance of parallelism\n",
    "\n",
    "- Most of the performance of modern computers is tied up in their ability to do multiple things **in parallel**\n",
    "- The work (flops) is sometimes overshadowed by the cost of **moving data (communicating)** between the parallel parts of the machine"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Concurrency\n",
    "\n",
    "- Roughly speaking, the **concurrency** of an algorithm is how much it exploits the paralleism of a computer:\n",
    "\n",
    "  - High concurrency $\\Rightarrow$ adding parallelism will make the implementation faster\n",
    "  \n",
    "  - Low concurrency $\\Rightarrow$ adding parallelism does not speed up the calculation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### A simple model of parallelism for today: factoring lots of distributed data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "$$\n",
    "\\begin{aligned}\n",
    "&\\begin{bmatrix} \\hline\n",
    "\\\\\n",
    "A_1 \\\\\n",
    "\\\\ \\hline\n",
    "\\end{bmatrix}\n",
    "&&\\begin{bmatrix} \\hline\n",
    "\\\\\n",
    "Q_1 \\\\\n",
    "\\\\ \\hline\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\\hline \\strut R \\\\ \\hline\n",
    "\\end{bmatrix}\n",
    "\\\\\n",
    "&\\begin{bmatrix} \\hline\n",
    "\\\\\n",
    "A_2 \\\\\n",
    "\\\\ \\hline\n",
    "\\end{bmatrix}\n",
    "&&\\begin{bmatrix} \\hline\n",
    "\\\\\n",
    "Q_2 \\\\\n",
    "\\\\ \\hline\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\\hline \\strut R \\\\ \\hline\n",
    "\\end{bmatrix}\\\\\n",
    "&\\quad \\vdots \\quad \\quad \\quad=& &\\quad \\vdots\\\\\n",
    "&\\begin{bmatrix} \\hline\n",
    "\\\\\n",
    "A_p \\\\\n",
    "\\\\ \\hline\n",
    "\\end{bmatrix}\n",
    "&&\\begin{bmatrix} \\hline\n",
    "\\\\\n",
    "Q_p \\\\\n",
    "\\\\ \\hline\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\\hline \\strut R \\\\ \\hline\n",
    "\\end{bmatrix}\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "split"
   },
   "source": [
    "- $m \\gg n$, so each _processor_ has a portion of each column of $A$ and $Q$\n",
    "\n",
    "- Copies of $R$ are small and can be duplicated for each processor\n",
    "\n",
    "- We want to count **rounds of communication**: one round of communication for each matrix matrix product $A B$ where _both_ $A$ and $B$ have an $m$ dimension:\n",
    "\n",
    "  - $Q \\cdot R$, $Q \\in \\mathbb{C}^{m\\times n}$, $R \\in \\mathbb{C}^{n\\times n}$: 0 rounds\n",
    "  \n",
    "  - $Q^* Q$, $Q \\in \\mathbb{C}^{m \\times n}$: 1 round\n",
    "  \n",
    "  - $Q y$, $Q \\in \\mathbb{C}^{m \\times n}$, $y \\in \\mathbb{C}^n$: 0 rounds\n",
    "  - $v^*w$, $v, w \\in \\mathbb{C}^m$: 1 round"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Rounds of communication in CGS / MGS?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [],
   "source": [
    "function cgs(A)\n",
    "    m, n = size(A); Q = copy(A); R = zeros(n, n);\n",
    "    for j = 1:n\n",
    "        R[1:j-1,j] = Q[:,1:j-1]' * Q[:,j] # 1\n",
    "        Q[:,j] -= Q[:,1:j-1] * R[1:j-1,j] # 0\n",
    "        R[j,j] = norm(Q[:,j])             # 1\n",
    "        Q[:,j] /= R[j,j]                  # 0\n",
    "    end\n",
    "    return Q, R\n",
    "end;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [],
   "source": [
    "function mgs(A)\n",
    "    m, n = size(A); Q = copy(A); R = zeros(n, n);\n",
    "    for i = 1:n\n",
    "        R[i,i] = norm(Q[:,i])             # 1\n",
    "        Q[:,i] /= R[i,i]                  # 0\n",
    "        R[i,i+1:n] = Q[:,i]' * Q[:,i+1:n] # 1\n",
    "        Q[:,j] -= Q[:,[i]] * R[[i],i+1:n] # 0\n",
    "    end\n",
    "    return Q, R\n",
    "end;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Each of classical and modified Gram-Schmidt has two round of communication per iteration\n",
    "\n",
    "$$\\Rightarrow 2n \\text{ rounds of communication}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Rounds of communication in Householder?\n",
    "\n",
    "(products subvectors of $m$-vectors will incur a round of communication, even if the length is $m - (i-1) < m$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [],
   "source": [
    "function householder_vector(x)\n",
    "    u = copy(x)\n",
    "    xnorm_tail = norm(x[2:end])           # 1 (broadcast x[1] in the same communication)\n",
    "    xnorm = (xnorm_tail^2 + conj(u[1])*u[1])^(1/2) # 0\n",
    "    phi = sign(u[1])                      # 0\n",
    "    u[1] += phi * xnorm                   # 0\n",
    "    unorm = (xnorm_tail^2 + u[1]^2)^(1/2) # 0\n",
    "    u /= unorm                            # 0\n",
    "    return u\n",
    "end;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "householder_reflect (generic function with 1 method)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function householder_reflect(u,v)\n",
    "    return v - 2*u*(u'*v) # 1\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "householder_reflect_left (generic function with 1 method)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function householder_reflect_left(vstar,u)\n",
    "    return vstar - 2*(vstar*u)*u' # 1\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "householder_qr (generic function with 1 method)"
      ]
     },
     "execution_count": 10,
     "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])                # 1\n",
    "        R[i:m,i:n] = householder_reflect(U[i:m,i], R[i:m,i:n]) # 1\n",
    "        R[i+1:m,i] .= 0\n",
    "    end\n",
    "    return U, R\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\Rightarrow 2n \\text{ rounds of communication}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "| algorithm | acts on | stability | behavior | cost | comm. rounds |\n",
    "| :-: | :-: | :-: | :-: | :-: | :-: |\n",
    "| classical Gram-Schmidt | $$Q$$ | ❌ | left-looking | $$\\sim 2mn^2$$ | $$2n$$ |\n",
    "| modified Gram-Schmidt | $$Q$$ | ⚠️ | right-looking | $$\\sim 2mn^2$$ | $$2n$$ |\n",
    "| Householder QR | $$R$$ | ✅ | down-looking | $$\\sim 2mn^2 - \\textstyle{\\frac{2}{3}}n^3$$ | $$2n$$ |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Online variant\n",
    "\n",
    "How much communication is required if a new vector $v$ must be orthogonalized after the others have already been orthogonalized?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "$$\n",
    "\\begin{aligned}\n",
    "&\\begin{bmatrix} \\hline\n",
    "& & \\\\\n",
    "& Q_1 & \\\\\n",
    "\\\\ \\hline\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\\hline \\strut R \\\\ \\hline\n",
    "\\end{bmatrix},\n",
    "\\begin{bmatrix} \\hline\n",
    "\\\\\n",
    "v_{1}\\\\\n",
    "\\\\ \\hline\n",
    "\\end{bmatrix}\n",
    "\\\\\n",
    "&\\begin{bmatrix} \\hline\n",
    "& & \\\\\n",
    "& Q_2 & \\\\\n",
    "& & \\\\ \\hline\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\\hline \\strut R \\\\ \\hline\n",
    "\\end{bmatrix},\\begin{bmatrix} \\hline\n",
    "\\\\\n",
    "v_{2}\\\\\n",
    "\\\\ \\hline\n",
    "\\end{bmatrix}\\\\\n",
    "&\\quad \\vdots\\\\\n",
    "&\\begin{bmatrix} \\hline\n",
    "& & \\\\\n",
    "& Q_p &  \\\\\n",
    "& & \\\\ \\hline\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\\hline \\strut R \\\\ \\hline\n",
    "\\end{bmatrix},\\begin{bmatrix} \\hline\n",
    "\\\\\n",
    "v_{p}\\\\\n",
    "\\\\ \\hline\n",
    "\\end{bmatrix}\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "$$\n",
    "\\begin{aligned}\n",
    "&\\left[\\begin{array}{ccc|c} \\hline\n",
    "& & & \\\\\n",
    "& Q_1 & & q_1 \\\\\n",
    "& & & \\\\ \\hline\n",
    "\\end{array}\n",
    "\\right]\n",
    "\\begin{bmatrix}\\hline \\strut \\tilde{R} \\\\ \\hline\n",
    "\\end{bmatrix}\n",
    "\\\\\n",
    "&\\left[\\begin{array}{ccc|c} \\hline\n",
    "& & & \\\\\n",
    "& Q_p & & q_p \\\\\n",
    "& & & \\\\ \\hline\n",
    "\\end{array}\n",
    "\\right]\n",
    "\\begin{bmatrix}\\hline \\strut \\tilde{R} \\\\ \\hline\n",
    "\\end{bmatrix}\\\\\n",
    "&\\quad \\vdots\\\\\n",
    "&\\left[\\begin{array}{ccc|c} \\hline\n",
    "& & & \\\\\n",
    "& Q_p & & q_p \\\\\n",
    "& & & \\\\ \\hline\n",
    "\\end{array}\n",
    "\\right]\n",
    "\\begin{bmatrix}\\hline \\strut \\tilde{R} \\\\ \\hline\n",
    "\\end{bmatrix}\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Classical Gram-Schmidt vs. Modified Gram-Schmidt, online communication"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "cgs_new_vec! (generic function with 1 method)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function cgs_new_vec!(Q,R,v)\n",
    "    # store the orthogonalization of v as the last\n",
    "    # columns of Q and R\n",
    "    R[1:end-1,end] = Q[:,1:end-1]' * v           # 1\n",
    "    Q[:,end] = v - Q[:,1:end-1] * R[1:end-1,end] # 0\n",
    "    R[end,end] = norm(Q[:,end])                  # 1\n",
    "    Q[:,end] /= R[end,end]                       # 0\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "mgs_new_vec! (generic function with 1 method)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function mgs_new_vec!(Q,R,v)\n",
    "    # store the orthogonalization of v as the last\n",
    "    # columns of Q and R\n",
    "    Q[:,end] = v\n",
    "    for i = 1:size(Q)[2]-1\n",
    "        R[i,end] = Q[:,i]' * Q[:,end] # 1\n",
    "        Q[:,end] -= Q[:,i] * R[i,end] # 0\n",
    "    end\n",
    "    R[end,end] = norm(Q[:,end]) # 1\n",
    "    Q[:,end] /= R[end,end]      # 0\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Classical Gram-Schmidt: 2 rounds of communication for column $k$ $\\Rightarrow$ $2n$ total"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Modified Gram-Schmidt: $k$ rounds of communication for column $k$ $\\Rightarrow$ $(1/2)n^2$ total"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Householder QR: $k$ rounds of communication for column $k$  $\\Rightarrow$ $(1/2)n^2$ total"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "| algorithm | acts on | stability | behavior | cost | static comm. rounds | online comm. rounds |\n",
    "| :-: | :-: | :-: | :-: | :-: | :-: | :-: |\n",
    "| classical Gram-Schmidt | $$Q$$ | ❌ | left-looking | $$\\sim 2mn^2$$ | $$2n$$ | $$2n$$ |\n",
    "| modified Gram-Schmidt | $$Q$$ | ⚠️ | right-looking | $$\\sim 2mn^2$$ | $$2n$$ | $$\\sim\\textstyle\\frac{1}{2}n^2$$ |\n",
    "| Householder QR | $$R$$ | ✅ | down-looking | $$\\sim 2mn^2 - \\textstyle{\\frac{2}{3}}n^3$$ | $$2n$$ | $$\\sim\\textstyle\\frac{1}{2}n^2$$ |\n",
    "\n",
    "- Classical Gram-Schmidt is widely used in online parallel orthogonalization.  It's stability deficiencies can be overcome using the same idea as HW2, Problem 8."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Tall Skinny QR <https://arxiv.org/abs/0806.2159>\n",
    "\n",
    "Consider just 2 processors:\n",
    "\n",
    "1. Each processor does a QR factorization on its portion of the columns of $A$:\n",
    "\n",
    "   $$\\begin{aligned}\n",
    "   A_1 &= \\tilde{Q}_1 \\hat{R}_1, \\\\\n",
    "   A_2 &= \\tilde{Q}_2 \\hat{R}_2. \\\\\n",
    "   \\end{aligned}$$\n",
    "   \n",
    "   These separate equations can be restated as one matrix equation,\n",
    "   \n",
    "   $$\n",
    "   A = \\begin{bmatrix} A_1 \\\\ \\hline A_2 \\end{bmatrix} = \n",
    "   \\underbrace{\\left[\\begin{array}{c|c} \\tilde{Q}_1 & \\\\ \\hline & \\tilde{Q}_2 \\end{array}\\right]}_{\\text{orthonormal columns}} \\begin{bmatrix} \\hat{R}_1 \\\\ \\hline \\hat{R}_2 \\end{bmatrix}.\n",
    "   $$\n",
    "   "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "2. The processors exchange $\\hat{R}_1$ and $\\hat{R}_2$ with each other (1 round of communiciation) and each does a duplicate QR factorization of the vertical stack:\n",
    "\n",
    "   $$\\begin{bmatrix} \\hat{R}_1 \\\\ \\hline \\hat{R}_2 \\end{bmatrix} = \\tilde{U} \\hat{R} = \n",
    "   \\begin{bmatrix} \\tilde{U}_1  \\\\ \\hline \\tilde{U}_2 \\end{bmatrix} \\hat{R}$$\n",
    "   \n",
    "3. Each process can form its portion of the columns of $\\hat{Q}$:\n",
    "\n",
    "   $$A = \\begin{bmatrix} A_1 \\\\ \\hline A_2 \\end{bmatrix} = \\left[\\begin{array}{c|c} \\tilde{Q}_1 & \\\\ \\hline & \\tilde{Q}_2 \\end{array}\\right] \\begin{bmatrix} \\hat{R}_1 \\\\ \\hline \\hat{R}_2 \\end{bmatrix} =\n",
    "   \\left[\\begin{array}{c|c} \\tilde{Q}_1 & \\\\ \\hline & \\tilde{Q}_2 \\end{array}\\right]\\begin{bmatrix} \\tilde{U}_1  \\\\ \\hline \\tilde{U}_2 \\end{bmatrix} \\hat{R} =\n",
    "   \\begin{bmatrix} \\tilde{Q}_1\\tilde{U}_1  \\\\ \\hline \\tilde{Q}_2 \\tilde{U}_2 \\end{bmatrix} \\hat{R}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Tall Skinny QR is recursive\n",
    "\n",
    "We can generalize this idea to halve the number of processes in each iteration for a total of\n",
    "\n",
    "$$\\log p\\text{ round of communication}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "| algorithm | acts on | stability | behavior | cost | static comm. rounds | online comm. rounds |\n",
    "| :-: | :-: | :-: | :-: | :-: | :-: | :-: |\n",
    "| classical Gram-Schmidt | $$Q$$ | ❌ | left-looking | $$\\sim 2mn^2$$ | $$2n$$ | $$2n$$ |\n",
    "| modified Gram-Schmidt | $$Q$$ | ⚠️ | right-looking | $$\\sim 2mn^2$$ | $$2n$$ | $$\\sim\\textstyle\\frac{1}{2}n^2$$ |\n",
    "| Householder QR | $$R$$ | ✅ | down-looking | $$\\sim 2mn^2 - \\textstyle{\\frac{2}{3}}n^3$$ | $$2n$$ | $$\\sim\\textstyle\\frac{1}{2}n^2$$ |\n",
    "| Tall Skinny QR (TSQR) | $$R$$ | ✅ | recursive | $$\\sim 2mn^2 + \\textstyle{\\frac{2}{3}}n^3 p \\log p$$ | $$\\log p$$ | $$-$$ |\n",
    "| Cholesky QR | $$A^*A$$ | ❌ | Cholesky | $$\\sim 2mn^2 + \\textstyle{\\frac{1}{3}}n^3p$$ | $$1$$ | $$n$$ |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- There are tradeoffs in stability, work, and parallelism: different situations call for different QRs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Least Squares Problems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### The problem\n",
    "\n",
    "The linear problem\n",
    "\n",
    "$$A x= b, \\quad x\\in \\mathbb{C}^n, b \\in \\mathbb{C}^m, m > n,$$\n",
    "\n",
    "is _overdetermined:_ in the general case ($b \\not \\in \\mathrm{range}(A)$), there is no solution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### The residual\n",
    "\n",
    "Restated: for a particular $x$, the _residual_ vector, \n",
    "\n",
    "$$r = b - A x \\in \\mathbb{C}^m,$$\n",
    "\n",
    "is not zero if $b \\not \\in \\mathrm{range}(A)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Residual visualized"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "hideCode": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
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"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plotlyjs();\n",
    "function plot_residual(A,x,b)\n",
    "    corners = Float64[[-1,-1,-1] [ 1,-1,-1] [-1, 1,-1] [ 1, 1,-1] [-1,-1, 1] [ 1,-1, 1] [-1, 1, 1] [ 1, 1, 1]];\n",
    "    cube = polyhedron(vrep(corners'), lib)\n",
    "    triangle = [[2.0, 0.0] [-1.0, sqrt(3.)] [-1.0, -sqrt(3)]]*4\n",
    "    Ax = A * x\n",
    "    Q, R = qr(A, ColumnNorm())\n",
    "    scale = maximum(abs.([b Ax]))\n",
    "    if scale == 0\n",
    "        scale = 1\n",
    "    end\n",
    "    scale *= 1.1\n",
    "    p = plot(b[[1]], b[[2]], b[[3]]\n",
    "        , m=:o\n",
    "        , lim=(-scale, scale)\n",
    "        , size=(1000,1000)\n",
    "        , colorbar=:legend\n",
    "        , label=\"b\"\n",
    "        , hover=\"b:<br>$(@sprintf(\"%+.2f\", b[1]))<br>$(@sprintf(\"%+.2f\", b[2]))<br>$(@sprintf(\"%+.2f\", b[3]))\"\n",
    "    )\n",
    "    plot!(p,Ax[[1]], Ax[[2]], Ax[[3]]\n",
    "        , m=:square\n",
    "        , label=\"Ax\"\n",
    "        , hover=\"x:<br>$(@sprintf(\"%+.2f\", x[1]))<br>$(@sprintf(\"%+.2f\", x[2]))<br>Ax:<br>$(@sprintf(\"%+.2f\", Ax[1]))<br>$(@sprintf(\"%+.2f\", Ax[2]))<br>$(@sprintf(\"%+.2f\", Ax[3]))\"\n",
    "    )\n",
    "    plot!(p,[Ax[1],b[1]],[Ax[2],b[2]],[Ax[3],b[3]]\n",
    "        , label=\"r = b - Ax\"\n",
    "        , hover=\"r:<br>$(@sprintf(\"%+.2f\", b[1] - Ax[1]))<br>$(@sprintf(\"%+.2f\", b[2] - Ax[2]))<br>$(@sprintf(\"%+.2f\", b[3] - Ax[3]))\"\n",
    "        )\n",
    "    if (abs(R[1,1]) != 0.0)\n",
    "        if abs(R[2,2]) > eps() * 100. * abs(R[1,1])\n",
    "            # plot plane \n",
    "            Qtrivert = Q[:,1:2]*triangle\n",
    "            Qtri = polyhedron(vrep((Q[:,1:2]*triangle)'), lib)\n",
    "            isect = hcat(points(vrep(intersect(cube, Qtri)))...)*scale\n",
    "            plot!(p,isect[1,:],isect[2,:],isect[3,:],st=:mesh3d\n",
    "                , l = 0\n",
    "                , fillalpha=0.5\n",
    "                , label=\"range(A)\"\n",
    "                )\n",
    "            q1 = A[:,1]\n",
    "            q1max = maximum(abs.(q1))\n",
    "            q1 *= scale / q1max\n",
    "            plot!(p,[-q1[1], q1[1]],[-q1[2],q1[2]],[-q1[3],q1[3]]\n",
    "                , label = \"a₁\"\n",
    "                , hover=\"a₁:<br>$(@sprintf(\"%+.2f\", A[1,1]))<br>$(@sprintf(\"%+.2f\", A[2,1]))<br>$(@sprintf(\"%+.2f\", A[3,1]))\"\n",
    "            )\n",
    "            q2 = A[:,2]\n",
    "            q2max = maximum(abs.(q2))\n",
    "            q2 *= scale / q2max\n",
    "            plot!(p,[-q2[1], q2[1]],[-q2[2],q2[2]],[-q2[3],q2[3]]\n",
    "                , label = \"a₂\"\n",
    "                , hover=\"a₂:<br>$(@sprintf(\"%+.2f\", A[1,2]))<br>$(@sprintf(\"%+.2f\", A[2,2]))<br>$(@sprintf(\"%+.2f\", A[3,2]))\"\n",
    "            )\n",
    "        else\n",
    "            # plot line\n",
    "            q = Q[:,1]\n",
    "            qmax = maximum(abs.(q))\n",
    "            q *= scale / qmax\n",
    "            plot!(p,[-q[1], q[1]],[-q[2],q[2]],[-q[3],q[3]]\n",
    "            )\n",
    "        end\n",
    "    end\n",
    "    return p\n",
    "end;\n",
    "draw(plot_residual(randn(3,2), randn(2), randn(3)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Residual minimization\n",
    "\n",
    "If $Ax \\neq b$, we settle for the $x$ that makes the residual the smallest."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- \"Smallest\" implies a choice of norm.  As elsewhere, the Euclidean / 2-norm is our default choice.  The least squares problem:\n",
    "\n",
    "$$\\mathrm{minimize}_{x\\in \\mathbb{C}^n}\\ \\| Ax - b \\|_2$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- We more often see the equivalent minimization problem (hence the name _least squares_):\n",
    "\n",
    "$$\\mathrm{minimize}_{x \\in \\mathbb{C}^n}\\ {\\textstyle\\frac 1 2} \\| A x - b \\|_2^2.$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "hideCode": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
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"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "function plot_smallest_residual(A,b)\n",
    "    corners = Float64[[-1,-1,-1] [ 1,-1,-1] [-1, 1,-1] [ 1, 1,-1] [-1,-1, 1] [ 1,-1, 1] [-1, 1, 1] [ 1, 1, 1]];\n",
    "    cube = polyhedron(vrep(corners'), lib)\n",
    "    triangle = [[2.0, 0.0] [-1.0, sqrt(3.)] [-1.0, -sqrt(3)]]*4\n",
    "    x = A \\ b\n",
    "    Ax = A * x\n",
    "    scale = maximum(abs.([b Ax]))\n",
    "    if scale == 0\n",
    "        scale = 1\n",
    "    end\n",
    "    scale *= 1.1\n",
    "    p = plot(b[[1]], b[[2]], b[[3]]\n",
    "        , m=:o\n",
    "        , lim=(-scale, scale)\n",
    "        , size=(1000,1000)\n",
    "        , colorbar=:legend\n",
    "        , label=\"b\"\n",
    "        , hover=\"b:<br>$(@sprintf(\"%+.2f\", b[1]))<br>$(@sprintf(\"%+.2f\", b[2]))<br>$(@sprintf(\"%+.2f\", b[3]))\"\n",
    "    )\n",
    "    plot!(p,Ax[[1]], Ax[[2]], Ax[[3]]\n",
    "        , m=:square\n",
    "        , label=\"Ax\"\n",
    "        , hover=\"x:<br>$(@sprintf(\"%+.2f\", x[1]))<br>$(@sprintf(\"%+.2f\", x[2]))<br>Ax:<br>$(@sprintf(\"%+.2f\", Ax[1]))<br>$(@sprintf(\"%+.2f\", Ax[2]))<br>$(@sprintf(\"%+.2f\", Ax[3]))\"\n",
    "    )\n",
    "    plot!(p,[Ax[1],b[1]],[Ax[2],b[2]],[Ax[3],b[3]]\n",
    "        , label=\"r = b - Ax\"\n",
    "        , hover=\"r:<br>$(@sprintf(\"%+.2f\", b[1] - Ax[1]))<br>$(@sprintf(\"%+.2f\", b[2] - Ax[2]))<br>$(@sprintf(\"%+.2f\", b[3] - Ax[3]))\"\n",
    "        )\n",
    "    Q, R = qr(A, ColumnNorm())\n",
    "    if (abs(R[1,1]) != 0.0)\n",
    "        if abs(R[2,2]) > eps() * 100. * abs(R[1,1])\n",
    "            # plot plane \n",
    "            Qtrivert = Q[:,1:2]*triangle\n",
    "            Qtri = polyhedron(vrep((Q[:,1:2]*triangle)'), lib)\n",
    "            isect = hcat(points(vrep(intersect(cube, Qtri)))...)*scale\n",
    "            plot!(p,isect[1,:],isect[2,:],isect[3,:],st=:mesh3d\n",
    "                , l = 0\n",
    "                , fillalpha=0.5\n",
    "                , label=\"range(A)\"\n",
    "                )\n",
    "            q1 = A[:,1]\n",
    "            q1max = maximum(abs.(q1))\n",
    "            q1 *= scale / q1max\n",
    "            plot!(p,[-q1[1], q1[1]],[-q1[2],q1[2]],[-q1[3],q1[3]]\n",
    "                , label = \"a₁\"\n",
    "                , hover=\"a₁:<br>$(@sprintf(\"%+.2f\", A[1,1]))<br>$(@sprintf(\"%+.2f\", A[2,1]))<br>$(@sprintf(\"%+.2f\", A[3,1]))\"\n",
    "            )\n",
    "            q2 = A[:,2]\n",
    "            q2max = maximum(abs.(q2))\n",
    "            q2 *= scale / q2max\n",
    "            plot!(p,[-q2[1], q2[1]],[-q2[2],q2[2]],[-q2[3],q2[3]]\n",
    "                , label = \"a₂\"\n",
    "                , hover=\"a₂:<br>$(@sprintf(\"%+.2f\", A[1,2]))<br>$(@sprintf(\"%+.2f\", A[2,2]))<br>$(@sprintf(\"%+.2f\", A[3,2]))\"\n",
    "            )\n",
    "        else\n",
    "            # plot line\n",
    "            q = Q[:,1]\n",
    "            qmax = maximum(abs.(q))\n",
    "            q *= scale / qmax\n",
    "            plot!(p,[-q[1], q[1]],[-q[2],q[2]],[-q[3],q[3]]\n",
    "            )\n",
    "        end\n",
    "    end\n",
    "    #plot!(p, corners[1,:], corners[2,:], corners[3,:], l=:none)\n",
    "    return p\n",
    "end;\n",
    "draw(plot_smallest_residual(randn(3,2), randn(3)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Example: polynomial interpolation (degree $n-1$ through $n$ points) can be a bad fit "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "cell_style": "split",
    "hideCode": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"||V||₂ ||V⁻¹||₂ = 1.6e+03\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "gr();\n",
    "agnesi(x) = 1 ./ (1 .+ 25 * x .^ 2)\n",
    "for n = 9\n",
    "    p = n - 1\n",
    "    x = LinRange(-1.0,1.0,p+1)\n",
    "    y = agnesi(x)\n",
    "    pows = 0:p\n",
    "    V = hcat([ x .^ q for q in pows]...)\n",
    "    c = V \\ y\n",
    "    xfull = LinRange(-1.0,1.0,200)\n",
    "    yfull = map(x -> c' * (x .^ pows), xfull)\n",
    "    display(\"||V||₂ ||V⁻¹||₂ = $(@sprintf(\"%.1e\",cond(V)))\")\n",
    "    p = plot(xfull, agnesi(xfull))\n",
    "    plot!(p, xfull, yfull, ylim=(-2,2))\n",
    "    plot!(p, x, y, l = 0, m = :o)\n",
    "    draw(p)\n",
    "    #vbox(txt, p)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "split"
   },
   "source": [
    "Recall the Vandermonde matrix equation that solves for coefficients from interpolation values\n",
    "\n",
    "$$\n",
    "\\underbrace{\n",
    "\\begin{bmatrix}\n",
    "1 & x_1 & x_1^2 & & x_1^{n-1} \\\\\n",
    "1 & x_2 & x_2^2 & \\dots &x_2^{n-1} \\\\\n",
    "1 & x_3 & x_3^2 & & x_3^{n-1} \\\\\n",
    "  & \\vdots & & & \\vdots \\\\\n",
    "1 & x_n & x_n^2 & & x_n^{n-1}\n",
    "\\end{bmatrix}\n",
    "}_{V}\n",
    "\\begin{bmatrix}\n",
    "c_0 \\\\ c_1 \\\\ c_2 \\\\ \\vdots \\\\ c_{n-1}\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    "y_0 \\\\ y_1 \\\\ y_2 \\\\ \\vdots \\\\ y_{n-1}\n",
    "\\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Polynomial least-squares (degree $n-1$ through $m>n$ points) is more stable"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "cell_style": "split",
    "hideCode": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"||V||₂ ||V⁺||₂ = 5.3e+02\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "gr();\n",
    "agnesi(x) = 1 ./ (1 .+ 25 * x .^ 2)\n",
    "for n = 9, m = 21\n",
    "    p = n - 1\n",
    "    x = LinRange(-1.0,1.0,m)\n",
    "    y = agnesi(x)\n",
    "    pows = 0:p\n",
    "    V = hcat([ x .^ q for q in pows]...)\n",
    "    c = V \\ y\n",
    "    xfull = LinRange(-1.0,1.0,200)\n",
    "    yfull = map(x -> c' * (x .^ pows), xfull)\n",
    "    display(\"||V||₂ ||V⁺||₂ = $(@sprintf(\"%.1e\",cond(V)))\")\n",
    "    p = plot(xfull, agnesi(xfull))\n",
    "    plot!(p, xfull, yfull, ylim=(-2,2))\n",
    "    plot!(p, x, y, l = 0, m = :o)\n",
    "    draw(p)\n",
    "    #vbox(txt, p)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$$\\mathrm{minimize}_{p \\in \\mathcal{P}^{n-1}}\\ {\\textstyle\\frac 1 2} \\sum_{i=1}^m |p(x_i) - y_i|^2$$\n",
    "\n",
    "In terms of solving for the coefficients:\n",
    "\n",
    "$$\\mathrm{minimize}_{c \\in \\mathbb{C}^n}\\ \n",
    "{\\textstyle\\frac 1 2} \\| V c - y \\|_2^2$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Least-squares over evenly spaced appoints approximately minimizes the $L^2$ inner product\n",
    "\n",
    "If $\\{x_i\\}$ are evenly spaced throughout $[a, b]$, then (roughly speaking)\n",
    "\n",
    "$$\n",
    "\\lim_{m \\to \\infty} \\frac{1}{m} \\sum_{i=1}^m |f(x_i) - g(x_i)|^2 = \\| f - g \\|_{L^2[a,b]}^2.\n",
    "$$\n",
    "\n",
    "Make the equivalent of $V$ for the given basis functions and solve for the coefficients as a least-squares problem."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Orthogonal Projection and Normal Equations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Go back and look at the first 3D residual plot\n",
    "\n",
    "Your visual intuition is that $Ax$ can't be the closest to $b$ if $r$ is oblique to the plane of $\\mathrm{range}(A)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Translated into math, the best choice of $x$ has\n",
    "\n",
    "$$r \\perp \\mathrm{range}(A) \\quad \\Leftrightarrow \\quad A^*r = 0.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "**T&B Theorem 11.1.** Let $A \\in \\mathbb{C}^{m \\times n}$ ($m \\geq n$) and $b \\in \\mathbb{C}^m$ be given.\n",
    "\n",
    "$$x = \\arg\\min_{\\hat x} {\\textstyle\\frac 1 2}\\|b - A\\hat x\\|_2^2 \\quad \\stackrel{i}{\\Leftrightarrow} \\quad A^*\\underbrace{r}_{b - A x} = 0 \\quad \\stackrel{ii}{\\Leftrightarrow} \\quad \\underbrace{A^*A x = A^*b}_{\\text{normal equations}} \\quad \\stackrel{iii}{\\Leftrightarrow} \\quad \\underbrace{Pb = Ax}_{P = \\text{orth. proj. onto }\\mathrm{range}(A)}$$\n",
    "\n",
    "_Proof._ (just _i_, others follow from definitions)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "_i._ ($\\Rightarrow$) We can write $r = r_{\\parallel} + r_{\\perp},$ an orthogonal decompostion with $A^* r_{\\perp} = 0$ and $r_{\\parallel} = Ay \\in \\mathrm{range}(A)$:\n",
    "\n",
    "$$\n",
    "\\|b - A(x + y)\\|_2^2 = \\|r - r_\\parallel\\|_2^2 = \\|r\\|_2^2 - \\|r_\\parallel\\|_2^2.$$\n",
    "\n",
    "By the assumption that $\\|r\\|_2$ is minimal $\\|r_\\parallel\\|_2 = 0 \\Rightarrow r_\\parallel = 0.$\n",
    "\n",
    "($\\Leftarrow$) For every $z$ in $\\mathbb{C}^n$,\n",
    "\n",
    "$$\\|b - Az \\|_2^2 = \\|r - A(z-x)\\|_2^2 = \\|r\\|_2^2 + \\|A(z-x)\\|_2^2 \\geq \\|r\\|_2^2.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### This equivalence is important\n",
    "\n",
    "$$\\underbrace{A^*A x = A^*b}_{\\text{normal equations}} \\quad \\Leftrightarrow \\quad \\underbrace{Pb = Ax}_{P = \\text{orth. proj. onto }\\mathrm{range}(A)}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## The Pseudoinverse"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### If we assume the normal equations are solvable\n",
    "\n",
    "The matrix that maps $b \\to x$ is the _pseudoinverse_ $A^+\\in \\mathbb{C}^{n \\times m}$,\n",
    "\n",
    "$$\n",
    "x = \\underbrace{(A^* A)^{-1} A^*}_{\\text{pseudoinverse }A^+} b.\n",
    "$$\n",
    "\n",
    "$$A^+ A = (A^* A)^{-1} (A^* A) = I$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Recall a fact about projectors\n",
    "\n",
    "Given $A$ full rank,\n",
    "\n",
    "$$P = A (A^* A)^{-1} A^*$$\n",
    "\n",
    "is the orthogonal projector onto $\\mathrm{range}(A)$ (we did not have to construct an orthonormal basis of $\\mathrm{range}(A)$ to build the projector)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "But this is\n",
    "\n",
    "$$P = A A^+.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### The pseudoinverse and the SVD\n",
    "\n",
    "Let $\\hat{U} \\hat{\\Sigma} V^*$ be the reduced SVD of $A$ (full rank, $\\hat{\\Sigma}$ invertible):\n",
    "\n",
    "$$\n",
    "\\begin{aligned}A^+ &= (A^*A)^{-1} A^*\n",
    "\\\\&= (V \\hat{\\Sigma}^* \\hat{U}^* \\hat{U}\\hat{\\Sigma} V^*)^{-1} V \\hat\\Sigma \\hat U^*\n",
    "\\\\&= (V \\hat{\\Sigma}^2 V^*)^{-1} V \\hat\\Sigma \\hat{U}^*\n",
    "\\\\&= V \\hat{\\Sigma}^{-2} V^* V \\hat\\Sigma \\hat{U}^*\n",
    "\\\\&= V \\hat{\\Sigma}^{-1} \\hat{U}^*.\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Exercise\n",
    "\n",
    "**T&B Exercise 11.1.** Suppose the $m \\times n$ matrix $A$ has the form\n",
    "\n",
    "$$A = \\begin{bmatrix} A_1 \\\\ A_2 \\end{bmatrix},$$\n",
    "\n",
    "where $A_1$ is a nonsingular matrix of dimension $n \\times n$ and $A_2$ is an arbitrary matrix of dimension $(m-n) \\times n$.  Prove that\n",
    "\n",
    "$$(1 / \\sigma_{\\min}(A))\\leq (1 / \\sigma_{\\min} (A_1)).$$\n",
    "\n",
    "_Hint._\n",
    "\n",
    "1. From the previous slide, every singular value of $A^+$ is the reciprocal of a singular value of $A$.\n",
    "2. $\\sigma_{\\min}(A)^2 = \\min_{\\|x\\|_2 = 1} \\|Ax\\|_2^2 = \\|A_1 x\\|_2^2 + \\|A_2 x\\|_2^2$.\n",
    "3. $\\sigma_{\\min}(A_1)^2 = \\min_{\\|x\\|_2 = 1} \\|A_1 x\\|_2^2.$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Solving the Least Squares Problem"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Method 0: backslash\n",
    "\n",
    "- Algorithm?\n",
    "- Work?\n",
    "- Stability?\n",
    "- Parallelism?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
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    {
     "data": {
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       "1.8452387626045089e-13"
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     "execution_count": 17,
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   "source": [
    "m = 1000; n = 50;\n",
    "\n",
    "A = randn(m, n); b = randn(m);\n",
    "\n",
    "x_backslash = A \\ b\n",
    "\n",
    "orthogonality_error(A,x,b) = norm(A'*(b - A*x),2)\n",
    "orthogonality_error(A, x_backslash, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Method 1: Cholesky factorization of $A^*A$ in normal equations\n",
    "\n",
    "If $A$ is full rank, $A^*A$ has a Cholesky factorization, $A^*A = LL^*$, $L$ lower triangular and invertible.\n",
    "\n",
    "1. Compute $A^*A$ (symmetry saves work, $\\sim mn^2$)\n",
    "2. Compute Cholesky factorization $LL^* = A^*A$, ($\\sim \\frac{1}{3}n^3$)\n",
    "3. Compute $z = A^* b$ ($\\sim 2mn$)\n",
    "4. Solve $Lw = z$ ($\\sim n^2$)\n",
    "5. Solve $L^*x = w$ ($\\sim n^2$)\n",
    "\n",
    "Total work $\\sim mn^2 + \\frac{1}{3}n^3$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.5962401177184377e-13"
      ]
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     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "K = A'*A\n",
    "L = cholesky(K).L\n",
    "z = A'*b\n",
    "w = L \\ z\n",
    "x_cholesky = L' \\ w\n",
    "\n",
    "orthogonality_error(A, x_cholesky, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Method 2: QR Factorization of $A$\n",
    "\n",
    "1. Compute reduced QR factorization $A = \\hat{Q}\\hat{R}$ (most work efficient we've seen is Householder, $\\sim 2mn^2 - \\frac{2}{3}n^3$)\n",
    "2. Compute $z = \\hat Q^*b$ ($\\sim 4mn - 2n^2$ with Householder reflectors)\n",
    "3. Solve $\\hat{R}x = z$ ($\\sim n^2$)\n",
    "\n",
    "Total work $\\sim 2mn^2 - \\frac{2}{3}n^3$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.8077103907520088e-13"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q, R = qr(A)\n",
    "z = (Q'*b)[1:n]\n",
    "x_qr = R \\ z\n",
    "# note: norm((Q'*b)[n+1:m]) = ||r||_2\n",
    "\n",
    "orthogonality_error(A, x_qr, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Method 3: SVD of $A$\n",
    "\n",
    "1. Compute the reduced SVD $A = \\hat{U} \\hat \\Sigma V^*$ (approx $\\sim 2mn^2 + 11n^3$)\n",
    "2. Compute $z = \\hat U^* b$ ($\\sim 2mn$)\n",
    "3. Solve $\\hat \\Sigma w = z$ ($\\sim n$)\n",
    "4. Compute $x = V w$ ($\\sim 2n^2$)\n",
    "\n",
    "Total work $\\sim 2mn^2 + 11n^3$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.062259078795485e-13"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U, σ , V = svd(A) # reduced by default\n",
    "z = U'*b\n",
    "w = z ./ σ\n",
    "x_svd = V * w\n",
    "\n",
    "orthogonality_error(A, x_svd, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Comparison\n",
    "\n",
    "| algorithm | work | stability | parallelism |\n",
    "| :-: | :-: | :-: | :-: |\n",
    "| Cholesky | $$\\sim mn^2 + {\\textstyle\\frac{1}{3}}n^3$$ | ❌? | best |\n",
    "| QR | $$\\sim 2mn^2 - {\\textstyle\\frac{2}{3}{n^3}}$$ | ✅? | good--better |\n",
    "| SVD | $$\\sim 2mn^2 + 11n^3$$ | ✅✅? | bad--good |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The next section of our class is making the stability of these algorithms _quantifiable_."
   ]
  }
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