psets/pset6sol.ipynb

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   "source": [
    "# 18.S190/6.S090 Problem Set 6 Solutions"
   ]
  },
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   "source": [
    "## Problem 1 (2+5+5+5 points)\n",
    "\n",
    "Simpson’s method for integration of ODEs $\\frac{dv}{dt} = f(v, t)$ is given by:\n",
    "\n",
    "$$\n",
    "\\frac{v^{n+1} - v^{n-1}}{2\\Delta t} = \\frac{1}{6} \\left[ f(v^{n+1}, t^{n+1}) + 4f(v^{n}, t^{n}) + f(v^{n-1}, t^{n-1}) \\right].\n",
    "$$\n",
    "\n",
    "1. Is the method explicit or implicit?\n",
    "2. Using a Taylor series analysis, calculate the local truncation error for the method. Specifically, what is the order of accuracy?\n",
    "3. Is the method convergent? Why or why not?\n",
    "4. Plot the linear-stability region ($f(v,t) = \\lambda v$) for this integration method as a function of $\\lambda \\Delta t$ in the complex plane. For purely real, negative $\\lambda < 0$, what is the largest timestep $\\Delta t$ that can be taken while remaining stable for this method?"
   ]
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   "source": [
    "### Solutions:\n",
    "\n",
    "**(1.1)** This is an **implicit** method because $f(v^{n+1})$ appears on the right-hand side.  (So you will need to solve for $v^{n+1}$, possibly a complicated system of equations depending on $f$.)\n",
    "\n",
    "**(1.2)** Suppose we plug in an exact solution $v^n = v(n\\Delta t)$, where $v(t)$ satisfies $dv/dt = f(v,t)$.  We now plug in the Taylor expansion around $n\\Delta t$:\n",
    "$$\n",
    "v^{n\\pm 1} = v^n \\pm v'^n \\Delta t + v'' ^n \\Delta t^2/2 \\pm v''' ^n \\Delta t^3/6 + v'''' ^n \\Delta t^4/24 + O(\\Delta t^5)\n",
    "$$\n",
    "which gives:\n",
    "$$\n",
    "\\frac{v^{n+1} - v^{n-1}}{2\\Delta t} = v'^n + v''' ^n \\Delta t^2/6 + O(\\Delta t^4)\n",
    "$$\n",
    "on the left-hand side, where all of the other terms cancel.  For the right-hand side, let $g(t) = f(v(t), t)$ and let $g^n = g(n\\Delta t)$.  Then, Taylor-expanding $g^{n\\pm 1}$ exactly as for $v^{n\\pm 1}$, the right-hand side becomes:\n",
    "$$\n",
    "\\frac{1}{6} \\left[ g^{n+1}) + 4g^{n} + g^{n-1} \\right] = g^n + g'' ^n \\Delta t^2/6 + O(\\Delta t^4)\n",
    "$$\n",
    "where all of the other terms cancel.   The error is the difference between the right- and left-hand sides.  Since $v'^n = g^n$ by assumption ($v$ satisfies the ODE), we are left with:\n",
    "$$\n",
    "\\text{error} = \\left( v''' ^n - g'' ^n \\right)\\Delta t^2/6  O(\\Delta t^4)\n",
    "$$\n",
    "But $v'(t) = f(v(t), t) = g(t)$, so $v''' = g''$ and hence the $v''' ^n - g'' ^n$ cancels!  We are left with a $\\boxed{O(\\Delta t^4)} local truncation error!\n",
    "\n",
    "***Not required:*** It's always a good idea to do a quick numerical check to make sure we didn't make an algebra error.  For example, we could check this numerically for a simple test function, e.g. let $v(t) = e^{-t^2}$ and let $f(v,t) = dv/dt = -2t v$.  We'll then try plotting the numerical difference between the left- and right-hand sides around $t =n\\Delta t = 1$ as a function of $\\Delta t$.   As expected, it shows $O(\\Delta t^4)$ error."
   ]
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   "execution_count": 1,
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   "outputs": [
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      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots, LaTeXStrings\n",
    "Δt = exp10.(range(-5,0,length=20)) # log-spaced points from 10⁻⁵ to 10⁰\n",
    "v(t) = exp(-t^2)\n",
    "function simpson_rel_err(v, f, t, Δt)\n",
    "    tⁿ⁺¹, tⁿ, tⁿ⁻¹ = t + Δt, t, t - Δt\n",
    "    vⁿ⁺¹, vⁿ, vⁿ⁻¹ = v(tⁿ⁺¹), v(tⁿ), v(tⁿ⁻¹)\n",
    "    lhs = (vⁿ⁺¹ - vⁿ⁻¹) / 2Δt\n",
    "    rhs = (f(vⁿ⁺¹,tⁿ⁺¹) + 4f(vⁿ, tⁿ) + f(vⁿ⁻¹,tⁿ⁻¹)) / 6\n",
    "    return abs(rhs - lhs) / abs(f(vⁿ, tⁿ))\n",
    "end\n",
    "rel_err = simpson_rel_err.(t -> exp(-t^2), (v,t) -> -2t*v, 1, Δt)\n",
    "scatter(Δt, rel_err, xaxis=:log10, yaxis=:log10, label=\"simpson\")\n",
    "plot!(Δt, Δt.^4, ylim=(1e-16, 1), label=L\"\\Delta t^4\", legend=:topleft,\n",
    "      title=L\"problem 1.2: example $v = e^{-t^2}$ Simpson error\", xlabel=L\"\\Delta t\", ylabel=\"relative error\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d6c1c3aa-4a12-4b28-b2ef-4790e2616fe4",
   "metadata": {},
   "source": [
    "**(3)** We already showed in 1.2 that the method is *consistent* (the *local* truncation error $\\to 0$ as $\\Delta t \\to 0$).  By the Dahlquist equivalence theorem, convergence follows as long as the method is **zero stable**.  This corresponds to setting $f = 0$, in which case the recurrence becomes $v^{n+1} = v^{n-1}$, which can be written in matrix form as:\n",
    "$$\n",
    "\\begin{pmatrix} v^{n+1} \\\\ v^n \\end{pmatrix} = \\underbrace{\\begin{pmatrix} 0 & 1 \\\\ 1 & 0\\end{pmatrix}}_A \\begin{pmatrix} v^{n} \\\\ v^{n-1} \\end{pmatrix}\n",
    "$$\n",
    "which is indeed stable because the matrix $A$ has eigenvalues $\\pm 1$ (absolute value $\\le 1$).  (In fact, this analysis is identical to the analysis of the midpoint method, from class!)\n",
    "\n",
    "**(4)** For linear stability, we let $f(v, t) = \\lambda v$, which gives the recurrence:\n",
    "$$\n",
    "v^{n+1} = v^{n-1} + \\frac{\\lambda \\Delta t}{3} \\left[ v^{n+1} + 4v^n + v^{n-1} \\right]\n",
    "$$\n",
    "which can be solved for $v^{n+1}$ to obtain:\n",
    "$$\n",
    "v^{n+1} = \\left(1 - \\frac{\\lambda \\Delta t}{3}\\right)^{-1} \\left[ 4\\frac{\\lambda \\Delta t}{3} v^n + \\left(1 + \\frac{\\lambda \\Delta t}{3}\\right)  v^{n-1}\\right] = \\mu_-^{-1} \\left[ 4\\frac{\\lambda \\Delta t}{3} v^n + \\mu_+ v^{n-1} \\right]\n",
    "$$\n",
    "letting $\\mu_\\pm = 1 \\pm \\frac{\\lambda \\Delta t}{3}$.  In matrix form:\n",
    "$$\n",
    "\\begin{pmatrix} v^{n+1} \\\\ v^n \\end{pmatrix} = \\underbrace{\\begin{pmatrix} 4\\frac{\\lambda \\Delta t}{3 \\mu_-} & \\frac{\\mu_+}{\\mu_-} \\\\ 1 & 0\\end{pmatrix}}_{A(\\lambda \\Delta t)} \\begin{pmatrix} v^{n} \\\\ v^{n-1} \\end{pmatrix}\n",
    "$$\n",
    "Note that $A(0)$ gives the matrix from the zero-stability analysis, as expected.   Linear stability is obtained when $|\\text{eigval}| \\le 1$ for all eigenvalues of $A$.  This could be done by hand, since $A$ is a $2\\times 2$ matrix whose eigenvalues are obtained from a quadratic formula, but is rather tedious.  Doing it computationally is straightforward, however:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "9cda7c1b-fd9e-42f5-8af1-67b90ca2db21",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using LinearAlgebra\n",
    "\n",
    "function Aeig(λΔt)\n",
    "    μ₊, μ₋ = 1 + λΔt/3, 1 - λΔt/3\n",
    "    A = [ 4λΔt/3μ₋  μ₊/μ₋\n",
    "              1       0    ]\n",
    "    return eigvals(A)\n",
    "end\n",
    "Aeigmax(λΔt) = maximum(abs, Aeig(λΔt))\n",
    "\n",
    "ReλΔt = range(-2.5, 2.5, length=201)\n",
    "ImλΔt = range(-3, 3, length=201)\n",
    "Astable = Aeigmax.(complex.(ReλΔt, ImλΔt')) .≤ 1 + 1e-8\n",
    "heatmap(ReλΔt, ImλΔt, Astable', xlabel=\"Re λΔt\", ylabel=\"Im λΔt\", aspect_ratio=:equal, colorbar=false,\n",
    "        title=\"problem 1.4: linear stability region\")\n",
    "annotate!(0.7, 0, (\"stable\", :yellow, 12))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "6d4278e8-26d7-4679-8335-9bd44bf1ecd0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(ReλΔt, Aeigmax.(complex.(ReλΔt, 0)), label=\"real λΔt\", ylim=(0,2),\n",
    "     title=\"problem 1.4: linear stability on real/imag axes\", xlabel=\"real or imaginary λΔt\",\n",
    "     ylabel=\"maximum |eigenvalue| of A\")\n",
    "plot!(ImλΔt, Aeigmax.(complex.(0, ImλΔt)), label=\"imaginary λΔt\", color=:red)\n",
    "annotate!(0.8, 0.9, (\"stable |eig|=1\", :red, 12))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fef6ab7c-9d0d-4264-8de9-076d43c75768",
   "metadata": {},
   "source": [
    "From the above plot, for a **real λ**, it is *never* stable for *any* $\\Delta t > 0$.\n",
    "\n",
    "However, for an **imaginary λ**, is is stable if $|\\operatorname{Im} \\lambda \\Delta t|$ is sufficiently small.  From the plot,\n",
    "the threshold value for $|\\operatorname{Im} \\lambda \\Delta t|$ is somewhere between 1 and 2.   It turns out that the threshold\n",
    "happens when $|\\operatorname{Im} \\lambda \\Delta t|^2 = 3$, as can be seen by plotting the stability vs $|\\operatorname{Im} \\lambda \\Delta t|^2$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "4a5519c4-b502-4d1a-8ae6-08b3cf21b24a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ImλΔt_fine = range(0,3, length=1000)\n",
    "plot(ImλΔt_fine .^ 2, Aeigmax.(complex.(0, ImλΔt_fine)), label=\"imaginary λΔt\", color=:red,\n",
    "      title=\"problem 1.4: linear stability on (imaginary axis)²\", xlabel=\"(Im λΔt)²\",\n",
    "    xticks=1:9, ylim=(0,2.5),\n",
    "     ylabel=\"maximum |eigenvalue| of A\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c6f04f3a-009e-42be-8157-8efbfaa636b0",
   "metadata": {},
   "source": [
    "So, for **purely imaginary λ**, it is stable for $\\boxed{\\Delta t \\le \\sqrt{3} / |\\operatorname{Im} \\lambda|}$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ab979e06-22c6-4832-8ba9-f70bf5b5142b",
   "metadata": {},
   "source": [
    "## Problem 2 (5+5+5 points)\n",
    "\n",
    "\n",
    "Consider the integral\n",
    "$$\n",
    "I = \\int_{-1}^{1} \\sin\\left( x - 0.5 \\right) \\, dx,\n",
    "$$\n",
    "which can be recast as the initial value problem (IVP)\n",
    "$$\n",
    "\\frac{dI}{dx} = \\sin\\left( x - 0.5 \\right), \\quad I(-1) = 0.\n",
    "$$\n",
    "\n",
    "1. **Forward Euler:**  Discretize the ODE using the Forward Euler method. Compute the integral with $n = 10, 100, 1000$ steps, and analyze the convergence rate by comparing with the exact value.\n",
    "    \n",
    "2. **RK4 Method:** Implement the 4th-order Runge-Kutta (RK4) method for the same ODE and compute the integral for $n = 10, 100, 1000$ steps. Compare its accuracy and computational cost with Forward Euler, and determine the observed order of convergence.\n",
    "\n",
    "3.  Explain why ODE solvers like Forward Euler and RK4 are generally not used for numerical integration, addressing issues of stability, accuracy, and computational efficiency."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b78ed67d-a817-44e3-809d-771d07f8b792",
   "metadata": {},
   "source": [
    "### Solution\n",
    "\n",
    "**(1) and (2)**\n",
    "\n",
    "We implement Euler's method and RK4 below, and plot the error $I(T) - I$, where $I = \\cos(-1.5) - \\cos(0.5)$ is the exact integral, versus the number of steps $n$.  You can search online for descriptions of the standard RK4 steps, e.g. [this description from MIT 10.001](https://web.mit.edu/10.001/Web/Course_Notes/Differential_Equations_Notes/node5.html).  Note that the algorithm simplifies quite a bit because we have $dI/dx = f(x)$, where the right-hand-side $f(x)$ does not depend on $I$.\n",
    "\n",
    "In fact, Euler's method in this case is equivalent to a [rectangle rule (left Riemann sum)](https://en.wikipedia.org/wiki/Riemann_sum), and RK4 in this case is equivalent to [Simpson's quadrature rule](https://en.wikipedia.org/wiki/Simpson%27s_rule)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "97fb36bb-b86d-4ece-97e1-7cc9f355b8ba",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "rk4_integral (generic function with 1 method)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function euler_integral(f, a, b, n)\n",
    "    Δx = (b - a) / n\n",
    "    I = 0.0 # initial condition\n",
    "    for i = 0:n-1\n",
    "        I += Δx * f(a + i*Δx) # Euler step = rectangle rule\n",
    "    end\n",
    "    return I\n",
    "end\n",
    "\n",
    "function rk4_integral(f, a, b, n)\n",
    "    Δx = (b - a) / n\n",
    "    I = 0.0 # initial condition\n",
    "    for i = 0:n-1\n",
    "        x = a + i*Δx\n",
    "        k₁ = Δx * f(x)\n",
    "        k₂ = k₃ = Δx * f(x + Δx/2) # k₂ = k₃ since no I dependence\n",
    "        k₄ = Δx * f(x + Δx)\n",
    "        I += (k₁ + 2k₂ + 2k₃ + k₄)/6 # RK4 step = Simpson rule\n",
    "    end\n",
    "    return I\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "b8b33b9a-d7ba-45a8-b81a-31c7bef13c03",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.8304873781285464e-6, 2.0640070812059235e-14)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I = cos(-1.5) - cos(0.5) # exact integral\n",
    "n = 10 .^ (1:6)\n",
    "euler_err = abs.(euler_integral.(x -> sin(x - 0.5), -1, 1, n) .- I) ./ abs(I)\n",
    "rk4_err = abs.(rk4_integral.(x -> sin(x - 0.5), -1, 1, n) .- I) ./ abs(I)\n",
    "\n",
    "euler_err[end], rk4_err[end]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "1a74ae40-0ec3-4451-95f3-87a185721d79",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots, LaTeXStrings\n",
    "\n",
    "scatter(n, euler_err, color=:red, label=\"2.1: Euler\")\n",
    "plot!(n, 1 ./ n, color=:red, label=L\"2.1: $1/n$\")\n",
    "scatter!(n, rk4_err, color=:blue, label=\"2.2: RK4\")\n",
    "plot!(n, 1 ./ float(n).^4, color=:blue, linestyle=:dash, label=L\"2.2: $1/n^4$\")\n",
    "\n",
    "plot!(title=\"problem 2: Euler and RK4 convergence\", xaxis=:log10, yaxis=:log10, \n",
    "      xlabel=L\"n\", ylabel=\"relative error\", legend=:bottomleft, ylim=(1e-16, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "139f95fa-544b-47f5-855e-e4a4826dfe03",
   "metadata": {},
   "source": [
    "What we see is that Euler has $O(1/n)$ (first-order) convergence and RK4 has $O(1/n^4)$ (fourth-order) convergence, as expected.\n",
    "\n",
    "**(3)** For smooth integrands like this one, we have quadrature algorithms that are **exponentially better** than this, such as Clenshaw–Curtis or Gaussian quadrature, with $O(e^{-\\alpha n})$ convergence.  (Actually, for integrating the sine function, they are even faster than exponential because this function is [\"entire\"](https://en.wikipedia.org/wiki/Entire_function)!)   And we also have quadrature methods that are adaptive, like adaptive ODE methods.  So you are nearly always going to be *far* better off with a specialized quadrature algorithm than treating integration as an initial-value problem.\n",
    "\n",
    "Why is this?  The basic reason is that **ODEs are generally much harder than quadrature**, because you normally have $dy/dx = f(y,x)$ where the right-hand side depends on the *unknown solution* $y(x)$.   So an ODE solver has no choice but to build up its solution \"in sequence\" starting at the initial $x$ and then working \"forwards\" in steps $\\Delta x$.    In contrast, if the right-hand side depends *only on x*, as in integration, the quadrature algorithm has much more freedom — it can evaluate the right-hand side at *any set of x values* it wants to, *in any order*.  So they have the freedom to e.g. evaluate the right-hand side at Chebyshev points and then interpolate $f$ to arbitrary order.  ODE methods can't do this."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0cbc1b8d-56bc-4c20-bce2-260f716b2e29",
   "metadata": {},
   "source": [
    "## Problem 3 (5+5+5 points)\n",
    "\n",
    "Consider the $5 \\times 5$ matrix:\n",
    "$$\n",
    "A = \\begin{pmatrix}\n",
    "1 & -1 &  &  &  \\\\\n",
    "2 & -1 & 1 &  &  \\\\\n",
    " & 3 & 4 & -2 &  \\\\\n",
    " &  & -2 & 5 & -2 \\\\\n",
    " &  &  & -1 & 3\n",
    "\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "**(a)**\n",
    "\n",
    "Compute (by hand) its LU factorization $A = LU$ via Gaussian elimination (i.e. give both $L$ and $U$).\n",
    "\n",
    "Notice any special pattern to the nonzero entries?\n",
    "\n",
    "**(b)** Explain why you will *always* get this pattern of nonzero entries for LU factorization of an $m \\times m$ tridiagonal matrix, assuming for simplicity that no row swaps are required.   Show that LU factorization of an $m \\times m$ tridiagonal matrix therefore requires $O(m)$ work.\n",
    "\n",
    "**(c)** Given such an LU factorization of a tridiagonal matrix $A$, explain why solving $Ax =b$ then requires $O(m)$ work (instead of $O(m^2)$ for ordinary LU factors) whereas computing $A^{-1}$ requires $O(m^2)$ work (instead of $O(m^3)$)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ea1aa091-a3c4-4297-a4dd-dd57522fbd22",
   "metadata": {},
   "source": [
    "### Solutions\n",
    "\n",
    "**(a)** We perform the Gaussian elimination as usual, but you should quickly notice that it is a *lot* easier than usual because so many of the entries are already zero.  This is quantified in part (c).\n",
    "\n",
    "We start by multiplying row 1 by 2 and subtracting it from row 2, denoted \"$r_2 - 2r_1$\", and proceed from there (putting boxes around the pivots and marking the multipliers in red):\n",
    "$$\n",
    "A\n",
    "\\stackrel{r_2 - \\color{red}{2}r_1}{\\longrightarrow}\n",
    "\\begin{pmatrix}\n",
    "\\boxed{1} & -1 &  &  &  \\\\\n",
    " & \\boxed{1} & 1 &  &  \\\\\n",
    " & 3 & 4 & -2 &  \\\\\n",
    " &  & -2 & 5 & -2 \\\\\n",
    " &  &  & -1 & 3\n",
    "\\end{pmatrix}\n",
    "\\stackrel{r_3 - \\color{red}{3}r_2}{\\longrightarrow}\n",
    "\\begin{pmatrix}\n",
    "\\boxed{1} & -1 &  &  &  \\\\\n",
    " & \\boxed{1} & 1 &  &  \\\\\n",
    " &  & \\boxed{1} & -2 &  \\\\\n",
    " &  & -2 & 5 & -2 \\\\\n",
    " &  &  & -1 & 3.\n",
    " \\end{pmatrix}\n",
    "\\stackrel{r_4 - \\color{red}{(-2)}r_3}{\\longrightarrow}\n",
    " \\begin{pmatrix}\n",
    "\\boxed{1} & -1 &  &  &  \\\\\n",
    " & \\boxed{1} & 1 &  &  \\\\\n",
    " &  & \\boxed{1} & -2 &  \\\\\n",
    " &  &  & \\boxed{1} & -2 \\\\\n",
    " &  &  & -1 & 3\n",
    "\\end{pmatrix}\n",
    "\\stackrel{r_5 - \\color{red}{(-1)}r_4}{\\longrightarrow}\n",
    "\\boxed{\n",
    "\\underbrace{\\begin{pmatrix}\n",
    "\\boxed{1} & -1 &  &  &  \\\\\n",
    " & \\boxed{1} & 1 &  &  \\\\\n",
    " &  & \\boxed{1} & -2 &  \\\\\n",
    " &  &  & \\boxed{1} & -2 \\\\\n",
    " &  &  &  & \\boxed{1}\n",
    "\\end{pmatrix}}_U} \\, .\n",
    "$$\n",
    "(The fact that the pivots all $=1$ is *very unusual* and doesn't happen for most matrices!)\n",
    "\n",
    "We also need $L$, but as emphasized in class, this requires no computation.\n",
    "In particular, $L$ has all 1s on its diagonal while the terms below the diagonal are the multipliers coming from the Gaussian elimination process (flipping subtraction to addition). This gives\n",
    "$$\n",
    "\\boxed{\n",
    "L=\n",
    "\\begin{pmatrix}\n",
    "1 &  &  &  &  \\\\\n",
    "\\color{red}{2} & 1 &  &  &  \\\\\n",
    " &  \\color{red}{3}& 1 &  &  \\\\\n",
    " &  &  \\color{red}{-2}& 1 & \\\\\n",
    " &  &  &  \\color{red}{-1}& 1\n",
    " \\end{pmatrix} } \\, .\n",
    "$$\n",
    "\n",
    "The pattern of nonzeros entries in this matrix $A$ is called [tridiagonal](https://en.wikipedia.org/wiki/Tridiagonal_matrix), and such matrices are important in many practical settings.   Moreover, we see that the $L$ and $U$ resulting from Gaussian elimination are [bidiagonal matrices](https://en.wikipedia.org/wiki/Bidiagonal_matrix), which have even fewere nonzero entries.    (These are all examples of structured [sparse matrices](https://en.wikipedia.org/wiki/Sparse_matrix), which are mostly zero in a regular pattern, and you can often do linear algebra *much* more quickly with such matrices than you could with an arbitrary matrix.)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4af5f8a3-f0de-4458-8e6f-75099fd49c32",
   "metadata": {},
   "source": [
    "**(b)** Assuming no row swaps, we will always get this pattern because you only need to **eliminate one element below each diagonal**.   Furthermore, when you subtract the pivot row from the row beneath it, the tridiagonal pattern is preserved: the pivot element cancels the element beneath it, while the nonzero to the right of the pivot combines with the diagonal of the next row.  So **new nonzero entries are never introduced** in the upper triangle of $A$.\n",
    "\n",
    "Hence, $U$ will always be bidiagonal with one entry above the diagonal: elimination eliminates the entries below the diagonal, and doesn't introduce new entries above the diagonal.   And $L$'s bidiagonal pattern reflects the fact that each pivot  was only subtracted from the row immediately beneath it.\n",
    "\n",
    "**(c)** The linear $O(m)$ complexity for solving $Ax = b$ can be seen in two steps:\n",
    "\n",
    "1. Gaussian elimination is $O(m)$, since there is only one row subtraction per column, and this row subtraction only involves two nonzero elements.\n",
    "2. Back and forward substitution are $O(m)$.  For example, consider forward substitution to solve $Lc = b$: working from top to bottom, since each row of $L$ has only two nonzero entries, each forward-substitution step is an equation with only two terms, which takes $O(1)$ work per row.  Similarly for back-substitution on $Ux = c$, except that we work from bottom to top.\n",
    "\n",
    "The inverse matrix $A^{-1}$ is generically \"dense\" (all nonzero) for a tridiagonal matrix.  This is easy to see: even if you start with a right-hand-side consisting of a column of $I$ (with only one nonzero entry), forward substition will make all of the *subsequent* entries nonzero, and then backsubstitution will then make all of the *previous* entries nonzero.   \n",
    "\n",
    "To *compute* the inverse matrix, we simply need to solve $AX = I$ as usual: perform Gaussian elimination once to get $A = LU$, then perform forward/back-solves on each column of $I$ to get the corresponding column of $X = A^{-1}$.   Unlike a general matrix, however, for a tridiagonal matrix we just showed that $LU$ has $O(m)$ cost, and the forward/back-solves have $O(m)$ cost *per column*, for $O(m^2)$ cost overall."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a640287e-c46c-4ce9-8322-73ebf0d0cae9",
   "metadata": {},
   "source": [
    "## Problem 4 (5+5 points)\n",
    "\n",
    "Consider the matrix\n",
    "$$\n",
    "A = \\begin{pmatrix} 10^{-100} & 1 \\\\ 1 & 1 \\end{pmatrix} \\, .\n",
    "$$\n",
    "Suppose we want to solve $Ax = b$ for the right-hand side $b = \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix}$.  The *exact* solution is $x = \\frac{1}{1 - 10^{-100}} \\begin{pmatrix} -1 \\\\ 1 \\end{pmatrix}$.\n",
    "\n",
    "**(a)** Solve this $Ax=b$ for $x$ using Gaussian elimination by hand, but pretend you are doing standard floating-point arithmetic: round the result of every arithmetic operation to 16 significant digits.  Perform *no row swaps*: they are not strictly required since $10^{-100} \\ne 0$.  Show that the resulting approximate solution $\\tilde{x}$ is *completely wrong* compared to the exact solution.\n",
    "\n",
    "**(b)** On a computer, Gaussian elimination uses \"partial pivoting\": it swaps rows to make the pivot element as big in magnitude as possible (looking at all the entries under each pivot before doing elimination under each column).  Carry out this algorithm by hand on the same $Ax=b$ above, again rounding the result of each arithmetic operation.  (Note that any row swaps must be performed on both the matrix and the right-hand-side.)  Show that you get a *much more accurate answer*.\n",
    "\n",
    "This simple example helps explain why partial pivoting is crucial to make Gaussian elimination (i.e. LU factorization) robust in floating-point arithmetic.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "91e4821d-c074-4205-9ada-580e5135356e",
   "metadata": {},
   "source": [
    "### Solution\n",
    "\n",
    "**(a)** If we carry out Gaussian elimination without row swaps, we only need to do one step, using $10^{-100}$ as the first pivot:\n",
    "$$\n",
    "\\begin{pmatrix} \\boxed{10^{-100}} & 1 \\\\ 1 & 1 \\end{pmatrix}\n",
    "\\stackrel{r_2 - \\color{red}{10^{100}}r_1}{\\longrightarrow}\n",
    "\\begin{pmatrix} \\boxed{10^{-100}} & 1 \\\\ 0 & 1 - 10^{100}  \\end{pmatrix}\n",
    "\\approx\n",
    "\\begin{pmatrix} \\boxed{10^{-100}} & 1 \\\\ 0 & - 10^{100} \\end{pmatrix} = \\tilde{U}\n",
    "$$\n",
    "where we have rounded $1 - 10^{100} \\approx - 10^{100}$, since the $1$ only affects the 100th significant digit — this will be lost in 16-digit floating-point arithmetic!\n",
    "\n",
    "We also do the same row operation to our right-hand side\n",
    "$$\n",
    "\\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix}\n",
    "\\stackrel{r_2 - \\color{red}{10^{100}}r_1}{\\longrightarrow}\n",
    "\\begin{pmatrix} 1 \\\\ - 10^{100} \\end{pmatrix} = c\n",
    "$$\n",
    "\n",
    "Now, we solve $\\tilde{U} \\tilde{x} = c$ by back-substitution to get our approximate solution $\\tilde{x}$.  The last row gives $- 10^{100} \\tilde{x}_2 = - 10^{100} \\implies \\tilde{x}_2 = 1$.  The first row gives $10^{-100} \\tilde{x}_1 + \\tilde{x}_2 = 1 \\implies  \\tilde{x}_1 = 0$, i.e. we get the \"solution\"\n",
    "$$\n",
    " \\tilde{x} = \\boxed{\\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} } \\, ,\n",
    "$$\n",
    "which is **completely wrong** — it is not even close to our exact solution!   A tiny roundoff error in the 100th significant digit has led to catastrophe: this is a symptom of \"numerical instability\".\n",
    "\n",
    "We can also reproduce this in Julia, using the `NoPivot()` option to `lu` to prevent pivoting:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "7464a00f-bb3f-4759-8ee9-8bb3d86f6026",
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    {
     "data": {
      "text/plain": [
       "2-element Vector{Float64}:\n",
       " 0.0\n",
       " 1.0"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = [1e-100 1\n",
    "     1      1]\n",
    "b = [1,0]\n",
    "\n",
    "x̃ = lu(A, NoPivot()) \\ b"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a0ea86ab-035f-4c93-93f5-d25d5c74148a",
   "metadata": {},
   "source": [
    "which indeed is the same *completely wrong* answer.\n",
    "\n",
    "**(b)** Now, let's try it with \"partial pivoting\" — at each step, we swap rows to make the pivot as big as possible in magnitude.  Here, clearly $1 > 10^{-100}$ so we will start with a row swap to make $1$ the pivot, and then proceed with Gaussian elimination as usual:\n",
    "$$\n",
    "\\begin{pmatrix} 10^{-100} & 1 \\\\ 1 & 1 \\end{pmatrix}\n",
    "\\stackrel{r_2 \\leftrightarrow r_1}{\\longrightarrow}\n",
    "\\begin{pmatrix} \\boxed{1} & 1 \\\\ 10^{-100} & 1 \\end{pmatrix}\n",
    "\\stackrel{r_2 - \\color{red}{10^{-100}}r_1}{\\longrightarrow}\n",
    "\\begin{pmatrix} \\boxed{1} & 1 \\\\ 0 & 1 - 10^{-100} \\end{pmatrix}\n",
    "\\approx\n",
    "\\begin{pmatrix} \\boxed{1} & 1 \\\\ 0 & 1 \\end{pmatrix} = \\hat{U}\n",
    "$$\n",
    "where we have rounded $1 - 10^{-100} \\approx 1$, since the $10^{-100}$ only affects the 100th significant digit — again, this will be lost in 16-digit floating-point arithmetic.\n",
    "\n",
    "We also do the same row operation to our right-hand side\n",
    "$$\n",
    "\\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix}\n",
    "\\stackrel{r_2 \\leftrightarrow r_1}{\\longrightarrow}\n",
    "\\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix}\n",
    "\\stackrel{r_2 - \\color{red}{10^{-100}}r_1}{\\longrightarrow}\n",
    "\\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} = \\hat{c}\n",
    "$$\n",
    "\n",
    "Finally, we solve $\\hat{U} \\hat{x} = \\hat{c}$ to get our *new* approximate solution $\\hat{x}$.  The last row gives $\\hat{x}_2 = 1$, and the first row gives $\\hat{x}_2 + \\hat{x}_2 = 0 \\implies \\hat{x}_1 = -1$, for an approximate solution of:\n",
    "$$\n",
    "\\hat{x} = \\boxed{ \\begin{pmatrix} -1 \\\\ 1 \\end{pmatrix} } \\, ,\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "02282066-83da-4434-94ea-a2f7d0c97c2c",
   "metadata": {},
   "source": [
    "which is **nearly correct**: it only differs from the exact solution in the 100th significant digits!  A tiny roundoff error during the computations led to only a tiny roundoff error in the solution, which we should have hoped for since the condition number of this $A$ matrix is small.  This is a **numerically stable** algorithm."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "5fca3373-8f18-4f8d-9996-a264e1035c42",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.6180339887498953"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cond(A) # condition number of A is small — we should hope for an accurate answer"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e1147380-f688-409b-8b39-551006832875",
   "metadata": {},
   "source": [
    "Indeed, partial pivoting is precisely the algorithm that Julia (and other good-quality linear-algebra libraries) employs:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "47ca8028-17d1-4a70-b59d-dd66d4473441",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Vector{Float64}:\n",
       " -1.0\n",
       "  1.0"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LU = lu(A) # uses partial pivoting by default\n",
    "x̂ = LU \\ b # accurate answer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "f8629433-6079-4501-ac3b-e199db7abf1a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Vector{Int64}:\n",
       " 2\n",
       " 1"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LU.p # the row permutation used for this LU factorization"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a196addd-b54b-4f51-828e-f82e86b642a3",
   "metadata": {},
   "source": [
    "We can see that Julia indeed swapped the rows, and gave the same (accurate) approximate answer as our hand computation."
   ]
  }
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