lec11.ipynb

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      "\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"
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   "source": [
    "import Pkg; Pkg.add([\"Plots\", \"SpecialFunctions\", \"Polynomials\"]);\n",
    "import Logging; Logging.disable_logging(Logging.Info)"
   ]
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    "const STATIC_LECTURE = false;"
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   "source": [
    "using LinearAlgebra;\n",
    "using Plots; plotlyjs(); default(l=5, size=(900,550), label=false);\n",
    "using Printf;\n",
    "using Polynomials;\n",
    "\n",
    "BLAS.set_num_threads(1)\n",
    "\n",
    "#default(fmt=:svg)\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\n",
    "p = plot!([0,1],[0,1]);"
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   "source": [
    "**(follow along: <https://tinyurl.com/gtcse6643lec11>)**\n",
    "\n",
    "### Warmup\n",
    "\n",
    "You have to sum a list of nonnegative floating point numbers as accurately as possible.  What's your best option?\n",
    "\n",
    "1. Large to small\n",
    "2. Small to large\n",
    "3. It doesn't matter\n",
    "\n",
    "(See also: [Kahan summation](https://en.wikipedia.org/wiki/Kahan_summation_algorithm))"
   ]
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    "total = zero(Float16)\n",
    "\n",
    "for i = 1:65519\n",
    "    total += one(Float16) / Float16(i)\n",
    "end\n",
    "total"
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    "total"
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    "# Backward Error Analysis\n",
    "\n",
    "**1. Review & backward stability of floating point**\n",
    "\n",
    "**2. Examples of stable and unstable algorithms**\n",
    "\n",
    "**3. Backward Error Analysis**\n",
    "\n",
    "**4. Housesholder QR Stability**"
   ]
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   "source": [
    "## Review"
   ]
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    "### Relative conditioning ...\n",
    "\n",
    "Given problem $f: X \\to Y$, the _relative condition number_ $\\kappa(x)$ is\n",
    "\n",
    "$$ \\kappa(x) =  \\lim_{\\rho\\to 0} \\sup_{\\|\\delta x\\| \\leq \\rho} \\frac{\\|f(x + \\delta x) - f(x)\\| / \\|f(x)\\|}{\\|\\delta x\\| / \\|x\\|}$$"
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    "### ... of $A: x \\mapsto Ax$, $A^{-1}: b \\to A^{-1} b$, or $b: A \\mapsto A^{-1} b$ ($A\\in\\mathbb{C}^{m \\times m}$ invertible)\n",
    "\n",
    "For each of these problems, the condition number is the matrix condition number of $A$:\n",
    "\n",
    "$$\\kappa(A) = \\|A\\|\\|A^{-1}\\| \\geq 1.$$"
   ]
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    "### If $\\|\\cdot\\| = \\|\\cdot\\|_2$,\n",
    "\n",
    "$$\\kappa(A) = \\|A\\|_2 \\|A^{-1}\\|_2 = \\frac{\\sigma_1}{\\sigma_m}.$$"
   ]
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    "n = 100; θ = LinRange(0,2π,n+1); x = cos.(θ); y = sin.(θ); xy = [x y]';\n",
    "\n",
    "A1 = 2 * I; A2 = 0.5 * I;\n",
    "\n",
    "xy1 = A1 * xy; xy2 = A2 * xy;\n",
    "p = plot(x, y, l=(2,:black))\n",
    "plot!(p, [xy1[1,:] xy2[1,:]], [xy1[2,:] xy2[2,:]]\n",
    "    , st = :shape\n",
    "    , aspect_ratio = :equal\n",
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    "    , fillalpha = 0.5\n",
    "    , title=\"κ(A) = 1\"\n",
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    "\n",
    "σ = [1, 1/10] \n",
    "U3 = qr(randn(2,2)).Q; V3 = qr(randn(2,2)).Q\n",
    "U4 = qr(randn(2,2)).Q; V4 = qr(randn(2,2)).Q\n",
    "\n",
    "A3 = 2 * U3 * diagm(σ) * V3';\n",
    "A4 = 0.5 * U4 * diagm(σ) * V4';\n",
    "\n",
    "xy3 = A3 * xy; xy4 = A4 * xy;\n",
    "q = plot(x, y, l=(2,:black))\n",
    "plot!(q, [xy3[1,:] xy4[1,:]], [xy3[2,:] xy4[2,:]]\n",
    "    , st = :shape\n",
    "    , aspect_ratio = :equal\n",
    "    , framestyle = :zerolines\n",
    "    , fillalpha = 0.5\n",
    "    , title=\"κ(A) = 10\"\n",
    ")\n",
    "plot(p, q, layout=(1,2), size=(1300,800))"
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   "cell_type": "markdown",
   "id": "080c8383",
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    "### Floating point arithmetic\n",
    "\n",
    "We treat a floating point system $\\mathbf{F}\\subset\\mathbb{R}$ as though there is one number $\\epsilon_{\\text{machine}}$ that:"
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   "source": [
    "1. bounds the _relative error_ of projection $\\mathrm{fl}: \\mathbb{R} \\to \\mathbf{F}$:\n",
    "\n",
    "$$\\frac{|\\mathrm{fl}(x) - x|}{|x|} \\leq \\epsilon_{\\text{machine}}.$$"
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   "cell_type": "code",
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   "source": [
    "function floating_point_bytes(exponent_digits)\n",
    "    @assert exponent_digits >= 2 && exponent_digits <= 7\n",
    "    mantissa_digits = 7 - exponent_digits\n",
    "    e_max = 2^(exponent_digits - 1) - 1\n",
    "    e_min = 1 - e_max\n",
    "    bias = e_max\n",
    "    V = Vector{Float64}(undef, 0)\n",
    "    for sign in [-1, 1.]\n",
    "        # subnormal numbers\n",
    "        for mantissa in 0:2^mantissa_digits - 1\n",
    "            frac = mantissa / 2^mantissa_digits\n",
    "            push!(V, sign * 2. ^e_min * frac)\n",
    "        end\n",
    "        # normal numbers\n",
    "        for ex in e_min:e_max\n",
    "            for mantissa in 0:2^mantissa_digits - 1\n",
    "                frac = (2^mantissa_digits + mantissa) / 2^mantissa_digits\n",
    "                push!(V, sign * 2. ^ex * frac)\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "    return sort(V)\n",
    "end\n",
    "p = plot(title = \"Absolute error spacing\", ylim=(-1,1))\n",
    "V = filter(x -> x > 0., floating_point_bytes(4))\n",
    "plot!(p, V\n",
    "    , zeros(length(V))\n",
    "    , line = nothing\n",
    "    , marker = :cross\n",
    ")\n",
    "q = plot(xlabel = \"ℝ\", title = \"Relative error spacing\", xscale=:log10, ylim=(-1,1))\n",
    "V = filter(x -> x > 0., floating_point_bytes(4))\n",
    "plot!(q, V\n",
    "    , zeros(length(V))\n",
    "    , line = nothing\n",
    "    , marker = :cross\n",
    ")\n",
    "plot(p, q, layout=(2,1)\n",
    "    , plot_title = \"Positive 8-bit floating point values for exponent\"\n",
    "    , size=(1300,450)\n",
    "    , bottom_margin=3Plots.mm\n",
    ")"
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   "source": [
    "2. (Fundamental axiom) For every basic operation $\\ast\\in\\{+, -, \\times, \\div\\}: \\mathbb{R}\\otimes \\mathbb{R} \\to \\mathbb{R}$ with floating point approximation $\\circledast\\in\\{\\bigcirc\\!\\!\\!\\!\\!+\\,, \\bigcirc\\!\\!\\!\\!\\!-\\,, \\bigcirc\\!\\!\\!\\!\\!\\times\\,, \\bigcirc\\!\\!\\!\\!\\!\\div\\,\\}: \\mathbf{F} \\otimes \\mathbf{F} \\to \\mathbf{F}$,\n",
    "\n",
    "$$x \\circledast y = (x \\ast y)(1 + \\epsilon), \\quad |\\epsilon| \\leq \\epsilon_{\\text{machine}}.$$"
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   "source": [
    "$$\\Rightarrow \\frac{|x \\circledast y - x \\ast y|}{|x \\ast y|} \\leq \\epsilon_{\\text{machine}}.$$"
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     },
     "execution_count": 8,
     "metadata": {},
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    }
   ],
   "source": [
    "n_samples = 100;\n",
    "x = randn(Float32, n_samples)\n",
    "y = randn(Float32, n_samples)\n",
    "z_true = Float64.(x) - Float64.(y)\n",
    "z_fl32 = x - y\n",
    "r = abs.(Float64.(z_fl32) - z_true) ./ abs.(z_true)\n",
    "plot(z_true, r, st=:scatter, m=:utriangle)\n",
    "plot!([minimum(z_true),maximum(z_true)], eps(Float32)*[1,1], l=(2,:black,:dash), text=[\"\", \"ϵ\"], ylim=1.1*eps(Float32), size=(650,500), xlabel=\"x-y\", title = \"x-y floating point relative error, 32-bit\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "367ecbad",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Accuracy, Stability, and Backward Stability\n",
    "\n",
    "of _algorithm_ $\\tilde{f}_{\\epsilon}: X \\to Y$ that approximates problem $f: X \\to Y$ with floating point precision $\\epsilon$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8c57fa43",
   "metadata": {
    "cell_style": "split"
   },
   "source": [
    "**Accurate:**\n",
    "\n",
    "$$\\frac{\\|\\tilde{f}_{\\epsilon}(x) - f(x)\\|}{\\|f(x)\\|} = O(\\epsilon) \\text{ uniformly in }x.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "749af96a",
   "metadata": {
    "cell_style": "split"
   },
   "source": [
    "**Stable:**  For every $x$ there is $\\tilde{x}_\\epsilon$, \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\frac{\\|\\tilde{f}_\\epsilon(x) - f(\\tilde{x}_\\epsilon)\\|}{\\|f(\\tilde{x}_\\epsilon)\\|}\n",
    "&= O(\\epsilon)\\text{ uniformly in }x, \\\\\n",
    "\\frac{\\|\\tilde{x}_\\epsilon - x\\|}{\\|x\\|} \n",
    "&= O(\\epsilon)\\text{ uniformly in }x.\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d017b425",
   "metadata": {
    "cell_style": "split"
   },
   "source": [
    "**Backward stable:** For every x there is $\\tilde{x}_\\epsilon$,\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\tilde{f}_\\epsilon &= f(\\tilde{x}_\\epsilon) \\\\\n",
    "\\frac{\\|\\tilde{x}_\\epsilon - x\\|}{\\|x\\|} \n",
    "&= O(\\epsilon)\\text{ uniformly in }x.\n",
    "\\end{aligned}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f14de2d5",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "(The choice of norm doesn't matter.)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f9eb6db4",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Backward stability of floating point arithmetic (example: subtraction)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "76f6418f",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\\tilde{f}(x, y) = \\mathrm{fl}(x) \\bigcirc\\!\\!\\!\\!\\!\\!-\\ \\ \\mathrm{fl}(y)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7f9babe5",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$=x(1 + \\delta_1) \\bigcirc\\!\\!\\!\\!\\!\\!-\\ \\ y(1 + \\delta_2), \\quad |\\delta_1|,|\\delta_2| \\leq \\epsilon_{\\text{machine}},$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "46ba3e0a",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$= (x(1 + \\delta_1) - y(1 + \\delta_2))(1 + \\delta_3), \\quad |\\delta_3| \\leq \\epsilon_{\\text{machine}},$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "19a2bdb9",
   "metadata": {},
   "source": [
    "$$(1 + \\delta_1)(1 + \\delta_3) = 1 + \\underbrace{\\delta_1 + \\delta_3 + \\delta_1 \\delta_3}_{\\delta_4}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cc1b02fd",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$= \\underbrace{x(1 + \\delta_4)}_{\\tilde{x}} - \\underbrace{y(1 + \\delta_5)}_{\\tilde{y}}, \\quad |\\delta_4|,|\\delta_5| \\leq 2 \\epsilon_{\\text{machine}} + O(\\epsilon_{\\text{machine}}^2),$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "26bf5ac4",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$= f(\\tilde{x},\\tilde{y}).$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c5d0f6d4",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Verifying backward stability: backward error $O(\\epsilon_{\\text{machine}})?$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9016cade",
   "metadata": {},
   "source": [
    "$$\\tilde{f}(x,y) = f(\\tilde{x},\\tilde{y}), \\quad \\tilde{x} = x(1 + \\delta_4), \\tilde{y} = y(1 + \\delta_5), \\quad |\\delta_4|,|\\delta_5| \\leq 2 \\epsilon_{\\text{machine}} + O(\\epsilon_{\\text{machine}}^2)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "616be6c8",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\\frac{\\max\\{|x - \\tilde{x}|,|y - \\tilde{y}|\\}}{\\max\\{|x|,|y|\\}} \\leq \\frac{\\max\\{|x|,|y|\\} \\cdot\n",
    "\\max\\{|\\delta_4|,|\\delta_5|\\}}{\\max\\{|x|,|y|\\}} \\leq 2 \\epsilon_{\\text{machine}} + O(\\epsilon_{\\text{machine}}^2)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ac75a9b4",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "There must be $\\rho > 0$ such that $\\max\\{|\\delta_4|,|\\delta_5|\\} \\leq 3 \\epsilon_{\\text{machine}}$ whenever $\\epsilon_{\\text{machine}} < \\rho$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a036921b",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\\Rightarrow \\frac{\\max\\{|x - \\tilde{x}|,|y - \\tilde{y}|\\}}{\\max\\{|x|,|y|\\}} = O(\\epsilon_{\\text{machine}}).$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "59acce58",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### On HW3 3\n",
    "\n",
    "**3(a) restated:**  There are $g(\\epsilon) = O(\\epsilon)$ and $h(\\epsilon)$ such that $f(\\epsilon) = (1 + g(\\epsilon))(1 + h(\\epsilon))$.  Prove that $f(\\epsilon) - 1 = O(\\epsilon)$.\n",
    "\n",
    "From the definitions, there exist, $\\rho_g$, $\\rho_h$, $C_g$, and $C_h$ such that\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "|g(\\epsilon)| &\\leq C_g |\\epsilon|, & |\\epsilon| \\leq \\rho_g, \\\\\n",
    "|h(\\epsilon)| &\\leq C_h |\\epsilon|, & |\\epsilon| \\leq \\rho_h.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "Prove that similar constants exist for $(f(\\epsilon) - 1)$.\n",
    "\n",
    "**3(b) restated:** There is $g(\\epsilon) = O(\\epsilon)$ such that $f(\\epsilon) = 1 / (1 + g(\\epsilon))$.  Prove that $f(\\epsilon) - 1 = O(\\epsilon).$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "45f04ab8",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### On HW3 4\n",
    "\n",
    "**4(a):** Use Pascal's triangle AKA binomial expansion:\n",
    "\n",
    "$$(1 + x)^n = \\sum_{k=0}^n \\binom{n}{k} x^k.$$\n",
    "\n",
    "**4(b):** Many ways, my strategy:\n",
    "\n",
    "1. Show that $\\lim_{\\epsilon \\to 0} e_n(\\epsilon) / \\gamma_n(\\epsilon) = 1.$\n",
    "\n",
    "2. Show that $\\partial_\\epsilon \\{e_n(\\epsilon) / \\gamma_n(\\epsilon)\\} < 0$ for all $0 < \\epsilon < 1/n.$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2ba2e494",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Stability examples and counterexamples"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3aea8914",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Inner product"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "50e223f5",
   "metadata": {
    "cell_style": "center",
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Input space $X$?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "69b61c63",
   "metadata": {
    "cell_style": "center",
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\\mathbb{C}^m \\otimes \\mathbb{C}^m$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "68f98873",
   "metadata": {
    "cell_style": "center",
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Output space $Y$?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "16315c6b",
   "metadata": {
    "cell_style": "center",
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\\mathbb{C}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a23718aa",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- $f: X \\to Y$?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4c566777",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$f(x,y) = x^*y$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e2dabe3e",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Inner product algorithm\n",
    "\n",
    "- $\\tilde{f}: X \\to Y$?  Here for the first time we have to reckon with the fact that **floating point addition is not associative**:\n",
    "\n",
    "$$(x \\oplus y) \\oplus z \\neq x \\oplus (y \\oplus z)$$"
   ]
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   "cell_type": "code",
   "execution_count": 9,
   "id": "e5bd8d21",
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    "cell_style": "center",
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       "    \"height\": 500,\n",
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       ");\n",
       "\n",
       "    });\n",
       "    </script>\n"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n_samples = 100;\n",
    "x = randn(n_samples)\n",
    "y = randn(n_samples)\n",
    "z = randn(n_samples)\n",
    "w_left = (x + y) + z\n",
    "w_right = x + (y + z)\n",
    "r = abs.(w_left - w_right) ./ max.(abs.(w_left),abs.(w_right))\n",
    "plot(w_left, r, st=:scatter, m=:utriangle)\n",
    "plot!([minimum(w_left),maximum(w_right)], eps(Float64)*[1,1], l=(2,:black,:dash), text=[\"\", \"ϵ\"], ylim=1.1*eps(Float32), size=(650,500), xlabel=\"x+y+z\", title = \"((x ⊕ y) ⊕ z) - (x ⊕ (y ⊕ z)), relative error\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "678f1513",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Inner product algorithm\n",
    "\n",
    "$$\\tilde{f}(x,y) = \\tilde{f}(x_{1:m-1},y_{1:m-1}) \\oplus (\\mathrm{fl}(\\overline{x_m}) \\otimes \\mathrm{fl}(y_m)).$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d017c73f",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "**Base case ($m=1$):** just floating point multiplication $\\mathrm{fl}(\\overline{x_1}) \\otimes \\mathrm{fl}(y_1)$, which we said is backward stable:\n",
    "\n",
    "$$\n",
    "(\\mathrm{fl}(\\overline{x_1}) \\otimes \\mathrm{fl}(y_1)) = \\overline{x_1}(y_1 + \\delta y_1), \\quad\n",
    "|\\delta y_1| / |y_1| = O(\\epsilon_{\\text{machine}}).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bdb3236c",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "**Inductive case:** Assume $\\tilde{f}(x_{1:m-1},y_{1:m-1}) = x_{1:m-1}^*(y_{1:m-1} + \\delta y_{1:m-1}),$\n",
    "$\\|\\delta y_{1:m-1}\\| / \\|y_{1:m-1}\\| = O(\\epsilon_{\\text{machine}}).$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b9c3fd98",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "**Determine backward error:**\n",
    "\n",
    "$$\\tilde{f}(x_{1:m-1},y_{1:m-1}) \\oplus (\\mathrm{fl}(\\overline{x_m}) \\otimes \\mathrm{fl}(y_m))\n",
    "= x_{1:m-1}^*(y_{1:m-1} + \\delta y_{1:m-1}) \\oplus \\overline{x_m}(y_m + \\delta y_m)$$\n",
    "\n",
    "$$\n",
    "= \\big(x_{1:m-1}^*(y_{1:m-1} + \\delta y_{1:m-1}) + \\overline{x_m}(y_m + \\delta y_m)\\big)(1 + \\hat{\\delta}), \\quad\n",
    "|\\hat{\\delta}| \\leq \\epsilon_{\\text{machine}}.\n",
    "$$\n",
    "\n",
    "$$\n",
    "=x_{1:m-1}^*(y_{1:m-1} + \\underbrace{\\hat{\\delta}\\cdot y_{1:m-1} + \\delta y_{1:m-1}(1 + \\hat{\\delta})}_{\\delta y_{1:m-1}^{\\text{new}}})\n",
    "+ \\overline{x_m}(y_m + \\underbrace{\\hat{\\delta}\\cdot y_m + \\delta y_m (1 + \\hat{\\delta})}_{\\delta y_m^{\\text{new}}})\n",
    "$$\n",
    "\n",
    "$$\n",
    "=\n",
    "x^*(y + \\delta y^{\\text{new}}).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "31074b8e",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "**Prove that $\\|\\delta y^{\\text{new}}\\| / \\|y\\| = O(\\epsilon_{\\text{machine}})$:** Exercise."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c2f146a1",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "**Backward stable**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fecc3a93",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Example: Outer Product\n",
    "\n",
    "$$f(x,y) = xy^*$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b0bd7075",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Input space?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "97951937",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$ \\mathbb{C}^m \\otimes \\mathbb{C}^m $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd071a3c",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Output space?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c19c5d9b",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\\mathbb{C}^{m \\times m}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "055a29c1",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Outer product algorithm\n",
    "\n",
    "$$(\\tilde{f}(x,y))_{ij} = \\mathrm{fl}(x_i) \\otimes \\mathrm{fl}(\\overline{y_j})$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5864d379",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Each entry is individually backward stable, because it is just floating point multiplication:\n",
    "$$\\mathrm{fl}(x_i) \\otimes \\mathrm{fl}(\\overline{y_j}) = x_i \\overline{y_j} + \\delta_{ij}, \\quad |\\delta_{ij}| / |x_i \\overline{y_j}| = O(\\epsilon_{\\text{machine}}).$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bf98f417",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- But for backward stability we would require $$\n",
    "\\tilde{f}(x,y) = (x + \\delta x) (y + \\delta y)^*$$\n",
    "for some $\\delta x, \\delta y$, which would mean $\\tilde{f}(x,y)$ is still a rank-1 matrix."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dad02c2a",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- The matrix of errors $\\Delta$ looks more like a matrix of i.i.d. errors, which is almost surely full rank (HW3 Problem 1)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "27764660",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- **Outer product is not backward stable** (but it is stable)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5d7b4f19",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "> As a rule, for problems where the dimension of the solution space $Y$ is greater than that of the problem space $X$, backward stability is rare.\n",
    "\n",
    "(whiteboard)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "582bf8c7",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Example: $f(x) = x + 1$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b4d2c94b",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Input space $X$?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e78d3447",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\\mathbb{C}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "51a45028",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Output space $y$?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "559dab7b",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\\mathbb{C}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "460c3b1f",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- $\\tilde{f}?$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "405b1006",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\\begin{aligned}\n",
    "\\tilde{f}(x) &= \\mathrm{fl}(x) \\oplus 1 = (x(1 + \\delta_1) + 1)(1 + \\delta_2) \\\\\n",
    "&= x + 1 + \\underbrace{\\delta_1 x + \\delta_2 + \\delta_1 \\delta_2 x}_{\\tilde{x} - x}\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b899dcdd",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Checking the backward error magnitude\n",
    "\n",
    "$$\n",
    "\\frac{|\\tilde{x} - x|}{|x|} = \\frac{|\\delta_1 x + \\delta_2 + \\delta_1 \\delta_2 x|}{|x|} \\neq O(\\epsilon_{\\text{machine}}).$$\n",
    "\n",
    "We are looking for $C$ (independent of $x$) and some $\\rho$ (ditto), such that whenever $\\epsilon < \\rho$\n",
    "\n",
    "$$ \\frac{|\\delta_1 x + \\delta_2 + \\delta_1 \\delta_2 x|}{|x|} \\leq C |\\epsilon|. $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6a4eaa2f",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- $|\\delta_2|$ cannot be bounded relative to $|x|$: **not backward stable**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "48a8206e",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- But just plain **stable**? Yes: choose\n",
    "$$\n",
    "\\tilde{x} = \\begin{cases}\n",
    "x + \\delta_1 x + \\delta_1\\delta_2 x, & |x| < 1/2, \\\\\n",
    "x + \\delta_1 x + \\delta_1\\delta_2 x + \\delta_2, & |x| > 1/2.\n",
    "\\end{cases}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9063e9b9",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- In general if $f(0) \\neq 0$, backward stability is rare."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "98c4981b",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Exercise: why does this function exist?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "daa3f312",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{verbatim}\n",
       "log1p(x)\n",
       "\\end{verbatim}\n",
       "Accurate natural logarithm of \\texttt{1+x}. Throws \\href{@ref}{\\texttt{DomainError}} for \\href{@ref}{\\texttt{Real}} arguments less than -1.\n",
       "\n",
       "\\section{Examples}\n",
       "\\begin{verbatim}\n",
       "julia> log1p(-0.5)\n",
       "-0.6931471805599453\n",
       "\n",
       "julia> log1p(0)\n",
       "0.0\n",
       "\n",
       "julia> log1p(-2)\n",
       "ERROR: DomainError with -2.0:\n",
       "log1p will only return a complex result if called with a complex argument. Try log1p(Complex(x)).\n",
       "Stacktrace:\n",
       " [1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31\n",
       "[...]\n",
       "\\end{verbatim}\n"
      ],
      "text/markdown": [
       "```\n",
       "log1p(x)\n",
       "```\n",
       "\n",
       "Accurate natural logarithm of `1+x`. Throws [`DomainError`](@ref) for [`Real`](@ref) arguments less than -1.\n",
       "\n",
       "# Examples\n",
       "\n",
       "```jldoctest; filter = r\"Stacktrace:(\\n \\[[0-9]+\\].*)*\"\n",
       "julia> log1p(-0.5)\n",
       "-0.6931471805599453\n",
       "\n",
       "julia> log1p(0)\n",
       "0.0\n",
       "\n",
       "julia> log1p(-2)\n",
       "ERROR: DomainError with -2.0:\n",
       "log1p will only return a complex result if called with a complex argument. Try log1p(Complex(x)).\n",
       "Stacktrace:\n",
       " [1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31\n",
       "[...]\n",
       "```\n"
      ],
      "text/plain": [
       "\u001b[36m  log1p(x)\u001b[39m\n",
       "\n",
       "  Accurate natural logarithm of \u001b[36m1+x\u001b[39m. Throws \u001b[36mDomainError\u001b[39m for \u001b[36mReal\u001b[39m arguments\n",
       "  less than -1.\n",
       "\n",
       "\u001b[1m  Examples\u001b[22m\n",
       "\u001b[1m  ≡≡≡≡≡≡≡≡≡≡\u001b[22m\n",
       "\n",
       "\u001b[36m  julia> log1p(-0.5)\u001b[39m\n",
       "\u001b[36m  -0.6931471805599453\u001b[39m\n",
       "\u001b[36m  \u001b[39m\n",
       "\u001b[36m  julia> log1p(0)\u001b[39m\n",
       "\u001b[36m  0.0\u001b[39m\n",
       "\u001b[36m  \u001b[39m\n",
       "\u001b[36m  julia> log1p(-2)\u001b[39m\n",
       "\u001b[36m  ERROR: DomainError with -2.0:\u001b[39m\n",
       "\u001b[36m  log1p will only return a complex result if called with a complex argument. Try log1p(Complex(x)).\u001b[39m\n",
       "\u001b[36m  Stacktrace:\u001b[39m\n",
       "\u001b[36m   [1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31\u001b[39m\n",
       "\u001b[36m  [...]\u001b[39m"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using SpecialFunctions\n",
    "\n",
    "@doc log1p"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9200cd53",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Unstable example: 2x2 matrix eigenvalues by root finding\n",
    "\n",
    "$O(\\epsilon_{\\text{machine}})$ errors in coefficients lead to $O(\\sqrt{\\epsilon_{\\text{machine}}})$ errors in roots (think quadratic formula)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "a170dee2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "mat_2x2_roots_by_char_poly (generic function with 1 method)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function mat_2x2_roots_by_char_poly(A)\n",
    "    # det(λ I - A)\n",
    "    return roots(Polynomial(Float64[det(A), -tr(A), 1]))\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "e06e2aac",
   "metadata": {
    "hideCode": true
   },
   "outputs": [
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   "source": [
    "n_samples = 100;\n",
    "rel_errs = Float64[]\n",
    "for i in 1:n_samples\n",
    "    C = [[1 + 1.e-14, 0.0] [randn(), 1.]]\n",
    "    X = randn(2,2)\n",
    "    A = X \\ (C * X)\n",
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    "    eigs = eigvals(A)\n",
    "    diff = norm(eigs_roots - eigs)\n",
    "    diff = min(diff, norm(eigs_roots - eigs[[2,1]]))\n",
    "    rel_err = diff / norm(eigs)\n",
    "    push!(rel_errs, rel_err)\n",
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       "mat_3x3_roots_by_char_poly (generic function with 1 method)"
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     "execution_count": 13,
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   "source": [
    "function mat_3x3_roots_by_char_poly(A)\n",
    "    # det(λ I - A)\n",
    "    return roots(Polynomial(Float64[-det(A), (tr(A)^2-tr(A^2))/2, -tr(A), 1]))\n",
    "end"
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     "execution_count": 14,
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   ],
   "source": [
    "n_samples = 100;\n",
    "rel_errs = Float64[]\n",
    "for i in 1:n_samples\n",
    "    C = [[1 + 1.e-14, 0.0, 0.0] [randn(), 1., 0.] [randn(), randn(), 1.]]\n",
    "    X = randn(3,3)\n",
    "    A = X \\ (C * X)\n",
    "    eigs_roots = mat_3x3_roots_by_char_poly(A)\n",
    "    eigs = eigvals(A)\n",
    "    diff = norm(eigs_roots - eigs)\n",
    "    for order in [[1,2,3],[1,3,2],[2,1,3],[2,3,1],[3,1,2],[3,2,1]]\n",
    "        diff = min(diff, norm(eigs_roots - eigs[order]))\n",
    "    end\n",
    "    rel_err = diff / norm(eigs)\n",
    "    push!(rel_errs, rel_err)\n",
    "end\n",
    "p = plot(1:n_samples, rel_errs\n",
    "    , st = :scatter\n",
    "    , ylim=(1.e-16,1.e-0)\n",
    "    , yscale=:log10\n",
    "    , title=\"relative error in eigenvalues via characteristics polynomial roots\")\n",
    "plot!(p,[1,n_samples],[eps(),eps()],l=(2,:black,:dash))\n",
    "annotate!(p, n_samples /2, log10(eps())+0.5, (\"ϵ\"))\n",
    "plot!(p,[1,n_samples],[eps()^0.5,eps()^0.5],l=(2,:black,:dash))\n",
    "annotate!(p, n_samples /2, log10(eps()^0.5)+0.5, (\"√ϵ\"))\n",
    "plot!(p,[1,n_samples],[eps()^(1/3),eps()^(1/3)],l=(2,:black,:dash))\n",
    "annotate!(p, n_samples /2, log10(eps()^(1/3))+0.5, (\"∛ϵ\"))"
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  },
  {
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   "id": "c86b81bc",
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   "source": [
    "## Accuracy of a Backward Stable Algorithm\n",
    "\n",
    "If $\\tilde{f}$ is backward stable, then **accuracy is proportional to conditioning**\n",
    "\n",
    "**T&B Theorem 15.1.** Suppose a backward stable algorithm is applied to solve a problem $f: X \\to Y$ with condition number $\\kappa$ on a computer satisfying the floating point axioms.  Then the relative errors satisfy\n",
    "\n",
    "$$\n",
    "\\frac{\\|\\tilde{f}(x) - f(x)\\|}{\\|f(x)\\|} = O(\\kappa(x) \\epsilon_{\\text{machine}}).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2b843694",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "_Proof._"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e8e77799",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\\begin{aligned}\n",
    "\\frac{\\|\\tilde{f}(x) - f(x)\\|}{\\|f(x)\\|}\n",
    "&=\n",
    "\\frac{\\|\\overbrace{\\tilde{f}(x) - f(\\tilde x)}^0 +  f(\\tilde x) - f(x)\\|}{\\|f(x)\\|} \\\\\n",
    "&=\n",
    "\\frac{\\|f(\\tilde x) - f(x)\\|}{\\|f(x)\\|} \\\\\n",
    "&\\leq (\\kappa(x) + o(1)) \\frac{\\|\\tilde x - x\\|}{\\|x\\|} \\\\\n",
    "&= O(\\kappa(x) \\epsilon_{\\text{machine}}).\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aec22ba0",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### This is Backward Error Analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "70daf26a",
   "metadata": {},
   "source": [
    "$$\\frac{\\|\\tilde{f}(x) - f(x)\\|}{\\|f(x)\\|}\n",
    "\\leq\n",
    "\\frac{\\|\\tilde{f}(x) - f(\\tilde x)\\|}{\\|f(x)\\|} +  \\frac{\\|f(\\tilde x) - f(x)\\|}{\\|f(x)\\|}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5c6410d3",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- The first term is controlled by the **stability of the algorithm** (in fact zero if it is backward stable)\n",
    "\n",
    "- The second term is controlled by the **conditioning of the problem**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8deca112",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Stability of Householder Triangularization"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e52ad72e",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Householder QR is backward stable: Experiment"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "7c8457e7",
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [],
   "source": [
    "m = 50\n",
    "\n",
    "R = triu(randn(m,m));\n",
    "\n",
    "Q = qr(randn(m,m)).Q;\n",
    "\n",
    "A = Q * R;\n",
    "\n",
    "Q2, R2 = qr(A);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "3ab467cf",
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.685758271146226"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "0.3376193521462045"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(norm(Q2 - Q))\n",
    "display(norm(R2-R) / norm(R))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8b8474b8",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Repeat, but with the convention that the diagonal of `R` and `R2` are non-negative"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "90cd3822",
   "metadata": {
    "cell_style": "center"
   },
   "outputs": [],
   "source": [
    "m = 50\n",
    "\n",
    "R = triu(randn(m,m));\n",
    "\n",
    "Q = qr(randn(m,m)).Q;\n",
    "\n",
    "Q = Q * diagm(sign.(diag(R))); R = diagm(sign.(diag(R))) * R;\n",
    "\n",
    "A = Q * R;\n",
    "\n",
    "Q2, R2 = qr(A);\n",
    "\n",
    "Q2 = Q2 * diagm(sign.(diag(R2))); R2 = diagm(sign.(diag(R2))) * R2;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "82aed908",
   "metadata": {
    "cell_style": "center"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.887542971285266"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "0.14121340893232875"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(norm(Q2 - Q))\n",
    "display(norm(R2-R) / norm(R))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "18e945ba",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "What have we learned about accuracy / stability / backward stability of Householder QR?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a1b06e3b",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Observe the reconstruction error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "dbcca759",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6.930347925734405e-16"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(norm(A - Q2 * R2) / norm(A))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "caa49c24",
   "metadata": {},
   "source": [
    "Now what have we learned about accuracy / stability / backward stability?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "26e2244b",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Householder in the stability analysis terms\n",
    "\n",
    "$$f: A \\mapsto (Q, R)$$\n",
    "\n",
    "$$\\tilde{f}: A \\mapsto (\\tilde{Q}, \\tilde{R})$$\n",
    "\n",
    "$$\\max\\{\\|\\tilde{Q} - Q\\|/\\|Q\\|, \\|\\tilde{R} - R\\| / \\|R\\|\\} = \\frac{\\|\\tilde{f}(x) - f(x)\\|}{\\|f(x)\\|}$$\n",
    "\n",
    "$$\\frac{\\|A - \\tilde{Q}\\tilde{R}\\|}{\\|A\\|} = \\frac{\\|A - \\tilde{A}\\|}{\\|A\\|} = \\frac{\\|x - \\tilde x\\|}{\\|x\\|}.$$\n",
    "\n",
    "We have observed large forward errors (the problem is ill-conditioned), but small backward errors (the algorithm is stable? needs proof)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0aa412ba",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### T&B Theorem 16.1\n",
    "\n",
    "Let the QR factorization $A = QR$ of a matrix $A \\in \\mathbb{C}^{m \\times n}$ be computed by Householder triangularization on a computer satisfying the floating point axioms, and let the computed $\\tilde{Q}$ be\n",
    "\n",
    "$$\n",
    "\\tilde{Q} = \\tilde{Q}_1 \\tilde{Q}_2 \\dots \\tilde{Q}_n,\n",
    "$$\n",
    "\n",
    "where $\\tilde{Q}_k$ is the Householder reflector of the computed reflector $\\tilde{v}_k$.  Then\n",
    "\n",
    "$$\\tilde{Q}\\tilde{R} = A + \\delta A, \\quad \\frac{\\|\\delta A\\|}{\\|A\\|} = O(\\epsilon_{\\text{machine}})$$\n",
    "\n",
    "for some $\\delta A \\in \\mathbb{C}^{m \\times n}$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b8243cbf",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Exercise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "a5ce4900",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "svd_accuracy_test (generic function with 2 methods)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function svd_accuracy_test(m, pow=1)\n",
    "    U = qr(randn(m,m)).Q # random unitary U\n",
    "    V = qr(randn(m,m)).Q # random unitary V\n",
    "    σ = rand(m) # random singular values\n",
    "    A = U * diagm(σ) * V' # construct A\n",
    "    U2, σ2, V2 = svd(A) # refactorize\n",
    "    # compute errors\n",
    "    return [norm(U-U2) norm(V-V2) norm(σ-σ2)/norm(σ) norm(A - U2 * diagm(σ2) * V2')/norm(A) cond(A)]\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "c5046b2d",
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      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 50\n",
    "n_tests = 100\n",
    "data = vcat([svd_accuracy_test(m, 6) for i in 1:n_tests]...);\n",
    "data = log10.(data)\n",
    "uplot = plot(data[:,5], data[:,1], st=:scatter, xlabel=\"log₁₀(cond(A))\", title=\"log₁₀||U - U₂||\")\n",
    "vplot = plot(data[:,5], data[:,2], st=:scatter, xlabel=\"log₁₀(cond(A))\", title=\"log₁₀||V - V₂||\")\n",
    "splot = plot(data[:,5], data[:,3], st=:scatter, xlabel=\"log₁₀(cond(A))\", title=\"log₁₀||Σ - Σ₂||/||Σ||\")\n",
    "aplot = plot(data[:,5], data[:,4], st=:scatter, xlabel=\"log₁₀(cond(A))\", title=\"log₁₀||A - U₂Σ₂V₂'||/||A||\")\n",
    "plot(uplot, vplot, splot, aplot, layout=(2,2), plot_title=\"m=$m\")"
   ]
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