psets/pset1sol.ipynb

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   "source": [
    "# 18.S097/16.S092 Problem Set 1 Solutions"
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    "## Problem 1 (5+5+5+5 points)\n",
    "\n",
    "Suppose that you are solving $Ax = b$ where $A$ is an $m \\times m$ (real) matrix of the form\n",
    "$$\n",
    "A = S + uv^T\n",
    "$$\n",
    "where $S$ is a *sparse* matrix (with $K$ nonzero entries) and $u,v$ are two $m$-component vectors.  That is, $A$ is the sum of a sparse matrix and a **rank-1** matrix.   We want to solve this efficiently for large $m$, even though $A$ itself is not sparse!\n",
    "\n",
    "**(a)** For example, give possible vectors $u,v$ if \n",
    "$$\n",
    "uv^T = \\alpha \\begin{pmatrix} 1 & 1 & 1 & 1 \\\\ 1 & 1 & 1 & 1 \\\\ 1 & 1 & 1 & 1 \\\\ 1 & 1 & 1 & 1 \\end{pmatrix}\n",
    "$$\n",
    "or\n",
    "$$\n",
    "uv^T = \\alpha \\begin{pmatrix} 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 1 & 1 & 1 & 1 \\end{pmatrix}\n",
    "$$\n",
    "for some scalar $\\alpha$.\n",
    "\n",
    "**(b)** Explain why you can efficiently multiply $A$ by any vector $x$, in cost *much* better than $O(m^2)$.  Give the complexity of your algorithm $O(\\cdots)$ in terms of $K$ and $m$.\n",
    "\n",
    "**(c)** Show by explicit multiplication $AA^{-1} = I$ that $A^{-1} = S^{-1} - \\frac{S^{-1} u v^T S^{-1}}{1 + v^T S^{-1} u}$.\n",
    "\n",
    "**(d)** Assume that $S$ has an efficient (low-fill-in) sparse LU factorization $PSP^T = LU$, with $P$ being a permutation of the rows and columns ($P^{-1} = P^T$, like any permutation).   Explain how you can exploit this to efficiently solve $Ax = b$ (using the formula from part **c**).\n",
    "\n",
    "*Note:* Don't try to compute $S^{-1}$ explicitly — it is probably not sparse!  Note that solving a triangular system like $Ly = c$ or $Uy = c$ is very efficient given sparse $L$ and $U$ vectors.  Also, permuting a vector, like $Py$ or $P^Ty$, can be done in $O(m)$ operations."
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    "### Solutions:\n",
    "\n",
    "**(a)** These can be factored as:\n",
    "\n",
    "$$\n",
    "\\alpha \\begin{pmatrix} 1 & 1 & 1 & 1 \\\\ 1 & 1 & 1 & 1 \\\\ 1 & 1 & 1 & 1 \\\\ 1 & 1 & 1 & 1 \\end{pmatrix} =\n",
    "\\boxed{\\underbrace{\\alpha \\begin{pmatrix} 1 \\\\ 1 \\\\ 1 \\\\ 1 \\end{pmatrix}}_u \\underbrace{\\begin{pmatrix} 1 & 1 & 1 & 1 \\end{pmatrix}}_v}\n",
    "$$\n",
    "and\n",
    "$$\n",
    "\\alpha \\begin{pmatrix} 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 1 & 1 & 1 & 1 \\end{pmatrix} =\n",
    "\\boxed{\\underbrace{\\alpha \\begin{pmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 1 \\end{pmatrix}}_u \\underbrace{\\begin{pmatrix} 1 & 1 & 1 & 1 \\end{pmatrix}}_v} \\, ,\n",
    "$$\n",
    "respectively.  Of course, the choice of $u,v$ here is not unique — we can rescale them in various ways, as long as the scale factors multiply out to $\\alpha$.\n"
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    "**(b)** If we just put parentheses in the right places to indicate the order of operations, we get:\n",
    "$$\n",
    "\\boxed{Ax = Sx + u(v^T x)} \\, .\n",
    "$$\n",
    "Computing $Sx$ is $O(K)$ because you only have to multiply/add nonzero entries, computing $v^T x$ is $O(m)$ ($m$ multiplications and $m-1$ additions) then multiplying the resulting scalar by $u$ is also $m$ multiplications.  So the overall cost is $\\boxed{O(K + m)}$."
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    "**(c)** This is the famous [Sherman–Morrison formula](https://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula) for the inverse of a rank-1 update.  By explicit multiplication, we get:\n",
    "\\begin{aligned}\n",
    "AA^{-1} &= \\left( S + uv^T \\right) \\left(S^{-1} - \\frac{S^{-1} u v^T S^{-1}}{1 + v^T S^{-1} u}\\right) \\\\\n",
    "&= \\cancelto{I}{S S^{-1}} + uv^T S^{-1} - \\frac{\\cancelto{I}{S S^{-1}} u v^T S^{-1}}{1 + v^T S^{-1} u} \n",
    "- \\frac{u(v^T S^{-1} u) v^T S^{-1}}{1 + v^T S^{-1} u} \\\\\n",
    "&= I + uv^T S^{-1} - \\frac{u v^T S^{-1}}{1 + v^T S^{-1} u} \n",
    "- \\frac{u v^T S^{-1} (v^T S^{-1} u)}{1 + v^T S^{-1} u} \\\\ \n",
    "&= I + uv^T S^{-1}  \n",
    "- \\frac{u v^T S^{-1} \\cancel{(1 + v^T S^{-1} u)}}{\\cancel{1 + v^T S^{-1} u}}  = \\boxed{I} \\, .\n",
    "\\end{aligned}\n",
    "A key step here is realizing that $v^T S^{-1} u$ is a scalar, so it can be commuted freely with other multiplications (unlike matrices)."
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    "**(d)** We want to compute\n",
    "$$\n",
    "x = A^{-1}b = \\boxed{\\overbrace{(S^{-1} b)}^c - \\frac{\\overbrace{(S^{-1} u)}^d \\left[v^T \\overbrace{(S^{-1} b)}^c\\right]}{1 + v^T \\underbrace{(S^{-1} u)}_d}} \\, ,\n",
    "$$\n",
    "where I've used parentheses to indicate an efficient order of operations.  Since $PSP^T = LU$, we have $S = P^T L U P \\implies S^{-1} = P^T U^{-1} L^{-1} P$.  This allows us to do:\n",
    "\n",
    "1. First, compute $c = S^{-1} b = P^T (U^{-1} (L^{-1} (P b)))$, performing the operations from right-to-left as indicated by parentheses: a permutation $Pb$ followed by two sparse triangular solves (forward- then back-substitution), followed by a permuation $P^T$.  The permuations are linear time, and the triangular solves are efficient as long as there wasn't too much fill-in, as noted.\n",
    "2. Then compute $d = S^{-1} u$ in the same way: permutation, two sparse triangular solves, permutation.\n",
    "3. Finally, compute $x = c + \\frac{d (v^T c)}{1 + v^T d}$, which is $O(m)$ (just some dot products, vector-scalar multiplications, and vector addition)."
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    "## Problem 2 (10+10 points)\n",
    "\n",
    "[Niu et al. (2014)](https://aapm.onlinelibrary.wiley.com/doi/10.1118/1.4866386) describe an optimization problem (equation 12) to solve for an unknown vector $x \\in \\mathbb{R}^{2N}$ whose entries describe the pixels of a medical image to be reconstructed from measurements.\n",
    "\n",
    "You *don't* need to understand any of the physics, statistics, or interpretation of this paper.  You just need to be able to decipher the *math* of sections 2.A–2.C and **translate it into linear algebra**.  This is an important skill!\n",
    "\n",
    "**(a)** Show that equation 12 of this paper is equivalent to a quadratic optimization problem\n",
    "$$\n",
    "\\min_{x} \\left[ x^T M x - b^T x + \\alpha \\right]\n",
    "$$\n",
    "for some symmetric matrix $M = M^T$ and vector $b$ and a scalar $\\alpha$  (all independent of $x$), each of which can be computed efficiently.  (In fact, it is a *convex* quadratic problem because all of the terms are strictly nonnegative, but you need not show that.)\n",
    "\n",
    "Hence, by taking the gradient $\\nabla_x (x^T M x - b^T x + \\alpha)$ and setting it to zero, we obtain a simple linear system $Mx = b/2$ that we could solve for $x$.\n",
    "\n",
    "**(b)** Explain why your matrix $M$ is very *sparse* matrix (only a small number of nonzero entries per row).\n",
    "\n",
    "Hence you could solve for $x$ efficiently assuming we have a good way to solve sparse linear systems (e.g. a sparse-direct solver, though in the paper they eventually use a conjugate-gradient algorithm instead)."
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    "### Solution:\n",
    "\n",
    "**(a)** Equation 12 in the paper is:\n",
    "$$\n",
    "\\min_x (A x - \\mu)^T V^{-1} (A x - \\mu) + \\lambda R(x) \\,\n",
    "$$\n",
    "where $A, V$ are given matrices ($V = V^T$ is diagonal), $\\mu$ is a given vector, $\\lambda$ is a scalar, and $R(x)$ is given by equation (13) in the paper:\n",
    "$$\n",
    "R(x) = \\frac{1}{2} \\sum_i \\sum_{k \\in N_i} e_{i,k} (x_i - x_k)^2 \\,\n",
    "$$\n",
    "where $N_i$ is the set of 4 neighbors of pixel $i$, and $e_{i,k}$ are given numbers.  It should be pretty obvious from looking at these equations that they are quadratic (and in fact convex quadratic).   To put them in the requested $x^T M x - b^T x + \\alpha$ form, we just need to do a little re-arranging.  The first terms are:\n",
    "$$\n",
    "x^T (A^T V^{-1} A) x - (2 A^T V^{-1} \\mu)^T x + \\mu^T V^{-1} \\mu\n",
    "$$\n",
    "which is now in the desired form.  To put $R(x)$ in the requested form, it is convenient to define a matrix $D$ such that $Dx$ gives the vector of all differences $x_i - x_k$ between $x_i$ components and the nearest neighbors $N_i$.   Note that $D$ will not be square, it will be a \"tall\" matrix $(\\sum_i |N_i|) \\times 2N = 8N \\times 2N$ since the paper says that each pixel $i$ has 4 neighbors (or maybe slightly fewer at image edges? not our problem).  In terms of $Dx$, the regularization $R(x)$ is simply:\n",
    "$$\n",
    "R(x) = \\frac{1}{2} (Dx)^T E (Dx)\n",
    "$$\n",
    "where $E$ is a diagonal matrix of the $e_{i,k}$ \"edge\" weights.\n",
    "\n",
    "Hence, we have the requested form $x^T M x - b^T x + \\alpha$ where\n",
    "1. $M = \\boxed{A^T V^{-1} A + \\frac{\\lambda}{2} D^T E D}$\n",
    "2. $b = \\boxed{2 A^T V^{-1} \\mu}$\n",
    "3. $\\alpha = \\boxed{\\mu^T V^{-1} \\mu}$"
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    "**(b)** If you peruse the paper, it is easy to see that all of the matrices appearing in the above expressions are sparse:\n",
    "\n",
    "1. From equation (3) in the paper, $A = \\begin{pmatrix} \\mu_{\\text{1H}} I & \\mu_{\\text{2H}} I \\\\ \\mu_{\\text{1L}} I & \\mu_{\\text{2L}} I \\end{pmatrix}$ consists of four scaled copies of the identity matrix, with only 2 nonzero entries per row.\n",
    "2. From equation (11) in the paper $V$ (and hence $V^{-1}$) is a diagonal matrix.\n",
    "3. Our difference matrix $D$ has only two nonzero entries per row ($\\pm 1$), since each row take the difference of two pixels.\n",
    "4. Our matrix $E$ is diagonal.\n",
    "\n",
    "Hence the matrix $M$, which consists of sums of products of sparse matrices, is also sparse.  (And $b$ is easy to compute from sparse matrix–vector multiplication.)"
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    "## Problem 3 (5 + 5 + 10 points)\n",
    "\n",
    "Suppose that $A = A^T $ is the symmetric $m \\times m$ \"banded\" matrix:\n",
    "$$\n",
    "A = \\begin{pmatrix} a_1 & b_1 & c_1 & & & & & & &\\\\\n",
    "b_1 & a_2 & b_2 & c_2 & & & & &\\\\\n",
    "c_1 & b_2 & a_3 & b_3 & c_3 & & & & & \\\\\n",
    "& c_2 & b_3 & a_4 & b_4 & c_4 & & & & \\\\\n",
    "& & c_3 & b_4 & a_5 & b_5 & c_5 & & & \\\\\n",
    "& & & \\ddots & \\ddots & \\ddots & \\ddots & \\ddots & & \\\\\n",
    "& & &  & c_{m-4} & b_{m-3} & a_{m-2} & b_{m-2} & c_{m-2} \\\\\n",
    "& & & &  & c_{m-3} & b_{m-2} & a_{m-1} & b_{m-1} \\\\\n",
    "& & & & &  & c_{m-2} & b_{m-1} & a_m\n",
    "\\end{pmatrix}\n",
    "$$\n",
    "which is defined in terms of three vectors $a \\in \\mathbb{R}^m$, $c \\in \\mathbb{R}^{m-1}$, and $c \\in \\mathbb{R}^{m-2}$.\n",
    "\n",
    "**(a)** Sketch the sparsity pattern of this matrix as a graph, as defined in class ($m$ nodes representing rows and columns, with edges representing nonzero entries).   You can assume all of the elements of $a,b,c$ are nonzero.\n",
    "\n",
    "**(b)** Write code (in Julia using the `SparseArrays` package, or in Python with `scipy.sparse`) that creates an (unstructured) sparse matrix $A$ given vectors $a,b,c$.   (The function `spdiagm` in Julia or `diags_array` in Python might be helpful.)  Plot the sparsity pattern (e.g. with `pyplot.spy`) for $m = 20$ to make sure it looks right.\n",
    "\n",
    "**(c)** Benchmark the performance (for a set of logarithmically spaced $m$ values up to $m = 10^7$) of solving $A x = y$ (e.g with `x = A \\ y` in Julia or `spsolve` in Python) for random $a,b,c,y$ and plot it as a function of $m$ on a log–log scale.  What power-law do you observe?  Does it make sense given the material from class?"
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"
    }
   },
   "cell_type": "markdown",
   "id": "21377dd2-5572-43dc-8264-ea54869cc36c",
   "metadata": {},
   "source": [
    "### Solutions\n",
    "\n",
    "**(a)** The graph has $m$ nodes, most naturally arranged into a line corresponding to rows/columns $1,2,\\ldots,m$, and each node is connected to its neighbors (by $b_k$) and to its second neighbors (by $c_k$), except for the beginning and ending nodes which have fewer neighbors.   One could also show edges connecting each node to itself, corresponding to the diagonal elements $a_k$, but as mentioned in class we typically omit those from such diagrams.\n",
    "\n",
    "![image.png](attachment:395e3112-bfd7-496f-a0d3-7dc7443c7dc8.png)\n",
    "\n",
    "Since it is a symmetric matrix, each edge has two arrows, or equivalently we could omit the arrows and show it as an undirected graph."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4e9b8273-66b8-4002-a9ef-aca172359009",
   "metadata": {},
   "source": [
    "**(b)** Note that a banded matrix could be viewed as a structured sparse matrix, because the pattern of nonzero entries is very regular, but we will store it an an *unstructured sparse-matrix format*, i.e. a format that could handle *any* nonzero pattern and doesn't \"know\" the banded structure.  The resulting `A \\ b` sparse-direct solver algorithm will therefore be generic to any sparsity pattern (though it will try to apply graph-analysis algorithms to take advantage of whatever structure it can find).   (In contrast, the Julia [BandedMatrices.jl package](https://github.com/JuliaLinearAlgebra/BandedMatrices.jl) defines structured-sparse formats and solver algorithms that are specialized to banded matrices.)\n",
    "\n",
    "We'll implement this in Julia, writing a function `A3(a,b,c)` to generate such a matrix given $a,b,c$ via the `spdiagm` function as suggested.  We'll also define `A3(m)` to generate an $m \\times m$ such matrix with Gaussian random entries.\n",
    "\n",
    "(If you are using the `scipy.sparse.diags_array` function in Python, you must pass a `format=...` argument to specify an unstructured sparse format like `'csc'` or `'coo'`.  Otherwise it may default to a `'dia'` format that is specialized to banded matrices.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "a2c28227-208e-4ad5-9514-6b64c9aac06d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "A3 (generic function with 2 methods)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using SparseArrays\n",
    "\n",
    "function A3(a::AbstractVector{<:Real}, b::AbstractVector{<:Real}, c::AbstractVector{<:Real})\n",
    "    m = length(a)\n",
    "    m == length(b)+1 == length(c)+2 || throw(DimensionMismatch())\n",
    "    return spdiagm(m, m, 0=>a, 1=>b, -1=>b, 2=>c, -2=>c)\n",
    "end\n",
    "\n",
    "A3(m::Integer) = A3(randn(m), randn(m-1), randn(m-2))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "46773140-b676-4207-aebd-4328365a223a",
   "metadata": {},
   "source": [
    "Before we plot the sparsity pattern, let's just look at it for a $6 \\times 6$ case where `a = 1:6`, `b=7:11`, and `c = -4:-1`, so that we can see the values directly:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "2b64e51c-6cf0-4078-9d92-2c1bd83ce4fc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6×6 SparseMatrixCSC{Int64, Int64} with 24 stored entries:\n",
       "  1   7  -4   ⋅   ⋅   ⋅\n",
       "  7   2   8  -3   ⋅   ⋅\n",
       " -4   8   3   9  -2   ⋅\n",
       "  ⋅  -3   9   4  10  -1\n",
       "  ⋅   ⋅  -2  10   5  11\n",
       "  ⋅   ⋅   ⋅  -1  11   6"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A3(1:6, 7:11, -4:-1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "90c04b4b-365d-4823-ae31-59e3f7cc1d00",
   "metadata": {},
   "source": [
    "Looks good!  Now let's do the requested $20 \\times 20$ plot, using the `spy` function in Julia's Plots.jl library:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "132a7f99-31d4-4cdd-90a8-91ac6ffb5c2f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots, LaTeXStrings\n",
    "spy(A3(ones(20), ones(19), ones(18)), legend=nothing,\n",
    "    title=\"Problem 3b: banded sparsity pattern\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ea32991f-4fd4-4e96-b6c8-40bb5976ac71",
   "metadata": {},
   "source": [
    "Looks good!\n",
    "\n",
    "**(c)** Now, let's do the suggested benchmark, for 20 log-spaced $m$ values from $m=100$ to $10^7$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "619bf918-05a4-4708-9386-7e51b4b72cb4",
   "metadata": {},
   "outputs": [],
   "source": [
    "using Chairmarks\n",
    "\n",
    "# log-spaced values from 10² to 10⁷, rounded to nearest integer\n",
    "ms = round.(Int, logrange(10^2, 10^7, 20))\n",
    "ts = map(ms) do m\n",
    "    A, b = A3(m), b = randn(m) # random banded matrix and rhs\n",
    "    bench = @b (A3(m),randn(m)) args -> (\\)(args...)\n",
    "    bench.time\n",
    "end;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "b1113b5f-8609-457e-be51-cb1507550afe",
   "metadata": {},
   "outputs": [
    {
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      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scatter(ms, ts, xaxis=:log10, yaxis=:log10, label=\"time for A \\\\ b\")\n",
    "plot!(ms, ms * 1e-9, label=L\"$O(m)$ reference\", legend=:bottomright,\n",
    "      title=\"Problem 3c: Time for banded solve\",\n",
    "      xlabel=L\"matrix size $m$\", ylabel=\"time (seconds)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0b3fd36d-a64d-4993-a4cb-be0bb9c674df",
   "metadata": {},
   "source": [
    "From the plot, the scaling is roughly **linear** $\\boxed{O(m)}$ time.\n",
    "\n",
    "This indeed makes sense from the material in class, since a banded matrix is a \"1d\" graph: it can be partitioned into two disconnected pieces by removing only $O(1)$ nodes (in fact, only 2), independent of $m$, so that we expect nested-dissection and related sparse-direct algorithms to have linear scaling."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3ff0c62d",
   "metadata": {},
   "source": [
    "## Problem 4 (10 + 10 points)\n",
    "\n",
    "In class (lecture 2) we defined the compressed-sparse-column (CSC) format, in which a sparse matrix is defined by three arrays (denoted in Matlab notation as `Pr, Ir, Jc` in the slides) of nonzero values by column (`Pr`), row indices (`Ir`), and the starting (1-based) index of each column (`Jc`).\n",
    "\n",
    "As an example, `Pr = [31,41,59,26,53]`, `Ir = [1,3,2,3,1]`, and `Jc = [1,3,5,6]` describe the $3 \\times 3$ \"sparse\" matrix:\n",
    "$$\n",
    "A = \\begin{pmatrix} 31 & \\cdot & 53 \\\\ \\cdot & 59 & \\cdot \\\\ 41 & 26 & \\cdot \\end{pmatrix}\n",
    "$$\n",
    "(where $\\cdot$ indicates zero values that are not stored).\n",
    "\n",
    "**(a)** Write a function `matvec(Pr, Ir, Jc, x)`, in the language of your choice, that returns the matrix-vector product `A*x` given the `Pr,Ir,Jc` description of a sparse matrix $A$.  The cost of your function should be proportional to the number of nonzero entries, and you should allocate no arrays other than the output vector.   (You may *not* use a pre-existing sparse-matrix library — write `matvec` from scratch.)   Test it by multipling the $3 \\times 3$ matrix $A$ above by the 3 columns of $I$ and show that you obtain the columns of the same matrix as shown.\n",
    "\n",
    "**(b)** Write a function `matvecT(Pr, Ir, Jc, x)` similar to `matvec`, above, but which returns $A^T x$.  Again, the cost should be proportional to the number of nonzero entries, and you should allocate no arrays other than the output vector.  (This is equivalent to multiplying by $A^T$ stored in compressed sparse *row* format.)  Again, test it with the same $3 \\times 3$ matrix $A$ by the 3 columns of $I$ and show that you obtain the columns of $A^T$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "61e1cfe0",
   "metadata": {},
   "source": [
    "### Solutions:\n",
    "\n",
    "Note that technically this problem is somewhat underspecified: the arrays `Pr, Ir, Jc` do not provide enough information to know the number of rows in your matrix.  (The maximum row index `maximum(Ir)` is a *lower bound* on the number of rows.  `length(Jc)+1` is the number of columns.)   So, we should either add the number of rows `m` as an additional argument, if you want for the sake of this problem you can assume the matrix is square.\n",
    "\n",
    "**(a)** The algorithm here is straightforward: we want to loop over each column, and add its contributions to the result vector $y$ (which is pre-allocated to an array of zeros).  In Julia:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "ee417051-a840-4ce7-b758-63ce7b28228b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matvec (generic function with 1 method)"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function matvec(m::Integer,\n",
    "                Pr::AbstractVector{<:Number},\n",
    "                Ir::AbstractVector{<:Integer},\n",
    "                Jc::AbstractVector{<:Integer},\n",
    "                x::AbstractVector{<:Number})\n",
    "    # some sanity checks\n",
    "    length(Pr) == length(Ir) || throw(DimensionMismatch())\n",
    "    length(x)+1 == length(Jc) || throw(DimensionMismatch())\n",
    "    n = length(x)\n",
    "    \n",
    "    y = fill(zero(eltype(x)) * zero(eltype(Pr)), m) # array of zeros of appropriate type\n",
    "    for j = 1:n # loop over columns\n",
    "        for k = Jc[j]:Jc[j+1]-1  # indices of nonzero entries\n",
    "            i = Ir[k] # row index for k-th nonzero, in column j\n",
    "            y[i] += Pr[k] * x[j]\n",
    "        end\n",
    "    end\n",
    "    return y\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9482de14-98aa-4dc7-bbe0-35bad46ad6d6",
   "metadata": {},
   "source": [
    "Now, let's try it with the example matrix, multiplying by the columns of $I$ as suggested, and stacking them back into a matrix with Julia's `stack` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "c01c06bf-93bb-4305-8b47-1f279f328ae1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Matrix{Int64}:\n",
       " 31   0  53\n",
       "  0  59   0\n",
       " 41  26   0"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using LinearAlgebra\n",
    "\n",
    "# given 3×3 example matrix\n",
    "Pr = [31,41,59,26,53]\n",
    "Ir = [1,3,2,3,1]\n",
    "Jc = [1,3,5,6]\n",
    "\n",
    "stack(map(x -> matvec(3, Pr, Ir, Jc, x), eachcol(I(3))))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "935ea28c-62da-4fd8-a0bc-c5e6096ebe36",
   "metadata": {},
   "source": [
    "Yup, it works!  We've recovered the desired matrix!"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d13568de-45f1-460b-82f8-f3afe1cf9718",
   "metadata": {},
   "source": [
    "**(b)** For `matvecT`, the process is basically the same, except now looping over the *columns* of $A$ is equivalent to looping over the *rows* of $A^T$, so our `i` and `j` indices are swapped:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "819e5a92-4099-4932-ad3f-6e99783c1735",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matvecT (generic function with 1 method)"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function matvecT(m::Integer,\n",
    "                Pr::AbstractVector{<:Number},\n",
    "                Ir::AbstractVector{<:Integer},\n",
    "                Jc::AbstractVector{<:Integer},\n",
    "                x::AbstractVector{<:Number})\n",
    "    # some sanity checks\n",
    "    length(Pr) == length(Ir) || throw(DimensionMismatch())\n",
    "    m == length(x) || throw(DimensionMismatch())\n",
    "    n = length(Jc)-1\n",
    "    \n",
    "    y = fill(zero(eltype(x)) * zero(eltype(Pr)), n) # array of zeros of appropriate type\n",
    "    for j = 1:n # loop over columns\n",
    "        for k = Jc[j]:Jc[j+1]-1  # indices of nonzero entries\n",
    "            i = Ir[k] # row index for k-th nonzero, in column j\n",
    "            y[j] += Pr[k] * x[i]\n",
    "        end\n",
    "    end\n",
    "    return y\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e00028c6-0f1f-4bb0-8782-aec9ff542b96",
   "metadata": {},
   "source": [
    "Again, let's try it out with the example matrix and the columns of $I$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "897ea619-bef2-4547-bb80-86a3d95a6652",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Matrix{Int64}:\n",
       " 31   0  41\n",
       "  0  59  26\n",
       " 53   0   0"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stack(map(x -> matvecT(3, Pr, Ir, Jc, x), eachcol(I(3))))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3f2ab414-99e3-4501-a7c7-c75b6a0ba4fe",
   "metadata": {},
   "source": [
    "Hooray!  As desired, this is indeed $A^T$."
   ]
  }
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