hw2_2023.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "bca33f70-0324-4594-8998-08352cc65c9d",
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "cell-06d3e5bf55c941ae",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "# Homework set 2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bfc44ed9-3b17-4f38-94fe-0f4d709ed1d6",
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "cell-b13bc5ed16bce8e0",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "# Exercise 0\n",
    "\n",
    "\n",
    "- Amir Sahrani (12661651)\n",
    "- Jonas Schäfer (14039982)\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "71d93788-1857-48f1-bfff-8cf004cf1a5d",
   "metadata": {},
   "source": [
    "## Importing packages"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "0c56160e-301a-4888-b439-a45bc02b8171",
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "import numpy as np\n",
    "from numpy.linalg import norm, inv\n",
    "from scipy.sparse import csr_array\n",
    "from scipy.sparse.linalg import spsolve\n",
    "from scipy.linalg import solve\n",
    "import timeit\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import warnings\n",
    "warnings.filterwarnings('ignore')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "967cec69-1602-4c3c-9bff-871eabca2c8b",
   "metadata": {
    "tags": []
   },
   "source": [
    "# Sparse matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "651b9158-f983-4dd6-a478-2dd0456961ca",
   "metadata": {},
   "source": [
    "A matrix is called sparse if only a small fraction of the entries is nonzero. For such matrices, special data formats exist. `scipy.sparse` is the scipy package that implements such data formats and provides functionality such as the LU decomposition (in the subpackage `scipy.sparse.linalg`).\n",
    "\n",
    "As an example, we create the matrix \n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "1 & 0 & 2 & 0 \\\\ \n",
    "0 & 3 & 0 & 0 \\\\\n",
    "0 & 0 & 4 & 5 \\\\\n",
    "0 & 0 & 0 & 6 \n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "in the so called compressed sparse row (CSR) format. As you can see, the arrays `row`, `col`, `data` contain the row and column coordinate and the value of each nonzero element respectively."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "d9090c65-724c-4bfc-a19c-85cd7314ca06",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1. 0. 2. 0.]\n",
      " [0. 3. 0. 0.]\n",
      " [0. 0. 4. 5.]\n",
      " [0. 0. 0. 6.]]\n"
     ]
    }
   ],
   "source": [
    "# a sparse matrix with 6 nonzero entries\n",
    "row = np.array([0, 0, 1, 2, 2, 3])\n",
    "col = np.array([0, 2, 1, 2, 3, 3])\n",
    "data = np.array([1.0, 2, 3, 4, 5, 6])\n",
    "sparseA = csr_array((data, (row, col)), shape=(4, 4))\n",
    "\n",
    "# convert to a dense matrix. This allows us to print to screen in regular formatting\n",
    "denseA = sparseA.toarray()\n",
    "print(denseA)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7bcb2618-8ec2-4cd2-86b3-786946719c20",
   "metadata": {},
   "source": [
    "For sparse matrices, a sparse data format is much more efficient in terms of storage than the standard array format. Because of this efficient storage, very large matrices of size $n \\times n$ with $n = 10^7$ or more can be stored in RAM for performing computations on regular computers. Often the number of nonzero elements per row is quite small, such as 10's or 100's nonzero elements per row. In a regular, dense format, such matrices would require a supercomputer or could not be stored.\n",
    "\n",
    "In the second exercise you have to use the package `scipy.sparse`, please look up the functions you need (or ask during class)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "485a0788-7ac3-41e5-afbb-2957220c6a2a",
   "metadata": {},
   "source": [
    "# Heath computer exercise 2.1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d40e9755-32ad-42e6-bd4a-9066e36cf4f9",
   "metadata": {},
   "source": [
    "## (a)\n",
    "Show that the matrix\n",
    "$$ A = \\begin{bmatrix} \n",
    "0.1 & 0.2 & 0.3 \\\\\n",
    "0.4 & 0.5 & 0.6 \\\\\n",
    "0.7 & 0.8 & 0.9\n",
    "\\end{bmatrix}.$$\n",
    "is singular. Describe the set of solutions to the system $A x = b$ if\n",
    "$$ b = \\begin{bmatrix} 0.1 \\\\ 0.3 \\\\ 0.5 \\end{bmatrix}. $$\n",
    "(N.B. this is a pen-and-paper question.)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "044019bb-6f29-446c-8cf4-a65c7033cf6c",
   "metadata": {},
   "source": [
    "$$\\begin{aligned}\n",
    "    \\det A &=0.1\\cdot\\det\n",
    "    \\begin{bmatrix}\n",
    "        0.5&0.6\\\\\n",
    "        0.8&0.9\n",
    "    \\end{bmatrix} -0.2\\cdot\\det\n",
    "    \\begin{bmatrix}\n",
    "        0.4&0.6\\\\\n",
    "        0.7&0.9\n",
    "    \\end{bmatrix}+0.3\n",
    "    \\begin{bmatrix}\n",
    "    0.4&0.5\\\\\n",
    "    0.7&0.8\n",
    "    \\end{bmatrix}\\\\\n",
    "    &=0.1\\cdot(0.45-0.48)-0.2\\cdot(0.36-0.42)+0.3\\cdot(0.32-0.35)\\\\\n",
    "    &=0.1\\cdot(-0.03)-0.2\\cdot(-0.06)+0.3\\cdot(-0.03)\\\\\n",
    "    &=-0.003+0.012-0.009\\\\\n",
    "    &=0\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "14074eea-48d7-49e6-81b1-5fadeaeec111",
   "metadata": {},
   "source": [
    "## (b)\n",
    "If we were to use Gaussian elimination with partial pivoting to solve this system using exact arithmetic, at what point would the process fail?\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "05aa07a1-da53-4f43-bad8-8be6c55e0c39",
   "metadata": {},
   "source": [
    "$$\\begin{bmatrix} \n",
    "    0.1&0.2&0.3&0.1\\\\\n",
    "    0.4&0.5&0.4&0.5\\\\\n",
    "    0.7&0.8&0.9&0.5\n",
    "\\end{bmatrix} =\n",
    "\\begin{bmatrix} \n",
    "    0.1&0.2&0.3&0.1\\\\\n",
    "    0.0&-0.3&-0.6&-0.1\\\\\n",
    "    0.0&-0.6&-1.2&-0.2\n",
    "\\end{bmatrix} =\n",
    "\\begin{bmatrix}\n",
    "    1&2&3&1\\\\\n",
    "    0&3&6&1\\\\\n",
    "    0&0&0&0\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "$$\n",
    "\\Rightarrow\n",
    "\\left\\{\n",
    "    \\begin{array}{l}\n",
    "        x_1 + 2x_2 + 3x_3 = 1 \\\\\n",
    "        3x_2 + 6x_3 = 1\n",
    "    \\end{array} \n",
    "\\right.\n",
    "\\Leftrightarrow\n",
    "\\left\\{\n",
    "    \\begin{array}{l}\n",
    "        x_1 = 1 - 2x_2 - 3x_3 \\\\\n",
    "        x_2 = \\frac{1}{3} - 2x_3\n",
    "    \\end{array} \n",
    "\\right.\n",
    "$$\n",
    "The result would fail when trying to get a pivot in the last row, because this row and the row above it will be identical, you would be left with no pivots in to bottom row and have infinetly many solutions"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "85c1ae38-f504-47eb-a1d6-eec193e32878",
   "metadata": {},
   "source": [
    "## (c)\n",
    "Because some of the entries of $A$ are not exactly representable in a binary floating point system, the matrix is no longer exactly singular when entered into a computer; thus, solving the system by Gaussian elimination will not necessarily fail. Solve this system on a computer using a library routine for Gaussian elimination. Compare the computed solution with your description of the solution set in part (a). What is the estimated value for $\\text{cond}(A)$? How many digits of accuracy in the solution would this lead you to expect?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "f3747e27",
   "metadata": {},
   "outputs": [],
   "source": [
    "def solve_cantilevered_beam_matrix(A, b, n, method):\n",
    "    match(method):\n",
    "        case 'default':\n",
    "            sol = solve(A, b)\n",
    "            return sol, A\n",
    "        case 'sparse':\n",
    "            sparse_A = csr_array(A)\n",
    "            sol = spsolve(sparse_A, b)\n",
    "            return sol, sparse_A\n",
    "        case _:\n",
    "            raise ValueError('method must be either default or sparse')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "1d6808a3-aecd-4ebe-aa79-5ddcbff09c17",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "solution: [ 0.06327986  0.87344029 -0.27005348]\n",
      "Condition number: 6.830e+16\n",
      "Number of significant digits lost in base 2: 55.923\n",
      "Number of significant digits lost in base 10: 16.834\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = np.matrix([[0.1, 0.2, 0.3],\n",
    "               [0.4, 0.5, 0.6],\n",
    "               [0.7, 0.8, 0.9]])\n",
    "b = np.array([0.1, 0.3, 0.5])\n",
    "cond = norm(A)*norm(inv(A))\n",
    "sol = solve(A, b)\n",
    "print(f\"solution: {sol}\")\n",
    "print(f'Condition number: {cond:.3e}')\n",
    "\n",
    "print(f'Number of significant digits lost in base 2: {np.log2(cond):.3f}')\n",
    "print(f'Number of significant digits lost in base 10: {np.log10(cond):.3f}')\n",
    "\n",
    "\n",
    "diffs = []\n",
    "for i in range(-10,10):\n",
    "    b_hat = b + i*10**-8\n",
    "    sol_hat = solve_cantilevered_beam_matrix(A, b_hat, 0, 'sparse')[0]\n",
    "    diffs.append(sum(sol_hat-sol))\n",
    "\n",
    "    \n",
    "plt.plot(np.arange(-10,10)*10**-8, diffs, marker='o', linestyle='-', color='#93bfb6', label='Differences')\n",
    "plt.title('The effect of a change in $\\mathbf{b}$ on $\\mathbf{x}$' )\n",
    "plt.xlabel('change in $\\mathbf{b}$')\n",
    "plt.ylabel('Change in $\\mathbf{x}$')\n",
    "plt.grid(True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "32537b21-ef0e-4d21-83ec-630d7c6ef4ac",
   "metadata": {},
   "source": [
    "We find that\n",
    "$$x = \\begin{bmatrix}0.06327986\\\\0.87344029\\\\-0.27005348\\end{bmatrix}$$\n",
    "with a condition number of $6.8 \\times 10^{16}$ using the product of the norm of matrix $A$ with the norm of its inverse. This results in the 16 digits of lost accuaracy in base 10, or roughly 60 in base 2.This means that in any floating point computation we can expect about 55 significant digits of lost precision to be the upperbound of our accuracy loss. \n",
    "\n",
    "Looking at the plot, we can see that our results around b are highly unstable, and the change of x is on the scale of our loss of accuracy."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "46c99b9d-1ab5-49ae-9fa9-805bb984e4a5",
   "metadata": {},
   "source": [
    "# Heath computer exercise 2.17"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aad5ecf6-4902-4e39-af7c-20fae6a41eb5",
   "metadata": {},
   "source": [
    "Consider a horizontal cantilevered beam that is clamped at one end but free along the remainder of its length. A discrete model of the forces on the beam yields a system of linear equations $A x = b$, where the $n \\times n$ matrix $A$ has the banded form\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    " 9 & -4     &  1 &  0 & \\ldots & \\ldots & 0 \\\\\n",
    "-4 &  6     & -4 &  1 & \\ddots && \\vdots \\\\\n",
    " 1 & -4     &  6 & -4 &  1 & \\ddots & \\vdots \\\\\n",
    " 0 & \\ddots & \\ddots & \\ddots & \\ddots & \\ddots & 0 \\\\\n",
    " \\vdots & \\ddots & 1 & -4 &  6 & -4 &  1 \\\\ \n",
    " \\vdots && \\ddots    &  1 & -4 &  5 & -2 \\\\\n",
    " 0 & \\ldots & \\ldots & 0 & 1 & -2 & 1 \n",
    "\\end{bmatrix}, $$\n",
    "the $n$-vector $b$ is the known load on the bar (including its own weight), and the $n$-vector $x$ represents the resulting deflection of the bar that is to be determined. We will take the bar to be uniformly loaded, with $b_i = 1/n^4$ for each component of the load vector.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0c3221df-1706-4536-9d13-900af379a450",
   "metadata": {},
   "source": [
    "## (a)\n",
    "Make a python function that creates the matrix $A$ given the size $n$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "c9753d31-5888-4de1-8ebe-39b55f413932",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 9. -4.  1.  0.  0.  0.]\n",
      " [-4.  6. -4.  1.  0.  0.]\n",
      " [ 1. -4.  6. -4.  1.  0.]\n",
      " [ 0.  1. -4.  6. -4.  1.]\n",
      " [ 0.  0.  1. -4.  5. -2.]\n",
      " [ 0.  0.  0.  1. -2.  1.]]\n"
     ]
    }
   ],
   "source": [
    "def cantilevered_beam_matrix(n):\n",
    "    main_diag = 6 * np.ones(n)\n",
    "    main_diag[[0, -2, -1]] = [9, 5, 1]\n",
    "    middle_diag = -4 * np.ones(n-1)\n",
    "    middle_diag[-1] = -2\n",
    "    upper_diag = np.ones(n-2)\n",
    "    A = np.diag(main_diag) + np.diag(middle_diag, k=1) + np.diag(upper_diag,\n",
    "                                                                 k=2) + np.diag(middle_diag, k=-1) + np.diag(upper_diag, k=-2)\n",
    "    return A\n",
    "\n",
    "\n",
    "def get_b(n):\n",
    "    return np.ones(n)/n**4\n",
    "\n",
    "print(cantilevered_beam_matrix(6))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3a7511aa-c209-4803-96a4-6743b5f52626",
   "metadata": {},
   "source": [
    "## (b)\n",
    "\n",
    "Solve this linear system using both a standard library routine for dense linear systems and a library routine designed for sparse linear systems. Take $n=100$ and $n=1000$. How do the two routines compare in the time required to compute the solution? And in the memory occupied by the LU decomposition? (Hint: as part of this assignment, look for the number of nonzero elements in the matrices $L$ and $U$ of the sparse LU decomposition.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "7e2da1b5-74b5-4490-b590-9604040c8aba",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Difference of solutions between default and sparse matrix decomposition\n",
      "for n=100: 5.632e-11\n",
      "for n=1000: 4.315e-06\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>method</th>\n",
       "      <th>n</th>\n",
       "      <th>nonzero entries</th>\n",
       "      <th>average runtime (in ms)</th>\n",
       "      <th>memory (in bytes)</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>default</td>\n",
       "      <td>100</td>\n",
       "      <td>494</td>\n",
       "      <td>0.140</td>\n",
       "      <td>80000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>sparse</td>\n",
       "      <td>100</td>\n",
       "      <td>494</td>\n",
       "      <td>0.365</td>\n",
       "      <td>3952</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>default</td>\n",
       "      <td>1000</td>\n",
       "      <td>4994</td>\n",
       "      <td>24.306</td>\n",
       "      <td>8000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>sparse</td>\n",
       "      <td>1000</td>\n",
       "      <td>4994</td>\n",
       "      <td>8.010</td>\n",
       "      <td>39952</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    method     n  nonzero entries average runtime (in ms)  memory (in bytes)\n",
       "0  default   100              494                   0.140              80000\n",
       "1   sparse   100              494                   0.365               3952\n",
       "2  default  1000             4994                  24.306            8000000\n",
       "3   sparse  1000             4994                   8.010              39952"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "\n",
    "def cantilevered_comparison(N_sim=100):\n",
    "    global A\n",
    "    global b\n",
    "    A = cantilevered_beam_matrix(100)\n",
    "    b = get_b(100)\n",
    "    time_default_100 = timeit.timeit(\"solve_cantilevered_beam_matrix(A, b, 100, 'default')\", number=N_sim, globals=globals()) / N_sim\n",
    "    sol_100_default, A_100_default = solve_cantilevered_beam_matrix(A, b, 100, 'default')\n",
    "    mem_100_default, nonzero_100_default = A_100_default.data.nbytes, np.count_nonzero(A_100_default.data)\n",
    "    time_sparse_100 = timeit.timeit(\"solve_cantilevered_beam_matrix(A, b, 100, 'sparse')\", number=N_sim, globals=globals()) / N_sim\n",
    "    sol_100_sparse, A_100_sparse = solve_cantilevered_beam_matrix(A, b, 100, 'sparse')\n",
    "    A = cantilevered_beam_matrix(1000)\n",
    "    b = get_b(1000)\n",
    "    mem_100_sparse, nonzero_100_sparse = A_100_sparse.data.nbytes, np.count_nonzero(A_100_sparse.data)\n",
    "    time_default_1000 = timeit.timeit(\"solve_cantilevered_beam_matrix(A, b, 1000, 'default')\", number=N_sim, globals=globals()) / N_sim\n",
    "    sol_1000_default, A_1000_default = solve_cantilevered_beam_matrix(A, b, 1000, 'default')\n",
    "    mem_1000_default, nonzero_1000_default = A_1000_default.data.nbytes, np.count_nonzero(A_1000_default.data)\n",
    "    time_sparse_1000 = timeit.timeit(\"solve_cantilevered_beam_matrix(A, b, 1000, 'sparse')\", number=N_sim, globals=globals()) / N_sim\n",
    "    sol_1000_sparse, A_1000_sparse = solve_cantilevered_beam_matrix(A, b, 1000, 'sparse')\n",
    "    mem_1000_sparse, nonzero_1000_sparse = A_1000_sparse.data.nbytes, np.count_nonzero(A_1000_sparse.data)\n",
    "    data_table = pd.DataFrame({'method': ['default', 'sparse', 'default', 'sparse'],\n",
    "                            'n': [100, 100, 1000, 1000],\n",
    "                            'nonzero entries': [nonzero_100_default, nonzero_100_sparse, nonzero_1000_default, nonzero_1000_sparse],\n",
    "                            'average runtime (in ms)': [f'{time*1000:.3f}' for time in [time_default_100, time_sparse_100, time_default_1000, time_sparse_1000]],\n",
    "                            'memory (in bytes)': [mem_100_default, mem_100_sparse, mem_1000_default, mem_1000_sparse],})\n",
    "    print('Difference of solutions between default and sparse matrix decomposition')\n",
    "    print(f'for n=100: {norm(sol_100_default-sol_100_sparse):.3e}\\nfor n=1000: {norm(sol_1000_default-sol_1000_sparse):.3e}')\n",
    "    return data_table\n",
    "\n",
    "data_table = cantilevered_comparison(100)\n",
    "data_table"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9decc62c-feb5-4afb-bfba-0b8a19af5be3",
   "metadata": {},
   "source": [
    "As we can observe from the table above, we find that the LU decomposition using the sparse matrix both cuts our average runtime roughly in half compared to our default method, as well as massively decreases the memory required due to the low number of non-zero entries."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8ebb2590-2f08-4362-aef6-f334eedbf306",
   "metadata": {},
   "source": [
    "## (c)\n",
    "For $n=100$, what is the condition number? What accuracy do you expect based on the condition number?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "d71e4897-2b95-44de-88a0-029c89a24609",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Condition number: 2.142e+13\n",
      "Number of significant digits lost in base 2: 44.284\n",
      "Number of significant digits lost in base 10: 13.331\n"
     ]
    }
   ],
   "source": [
    "def condition_number(A):\n",
    "    return norm(A) * norm(inv(A))\n",
    "\n",
    "A = cantilevered_beam_matrix(1000)\n",
    "cond = condition_number(A)\n",
    "print(f'Condition number: {cond:.3e}')\n",
    "\n",
    "print(f'Number of significant digits lost in base 2: {np.log2(cond):.3f}')\n",
    "print(f'Number of significant digits lost in base 10: {np.log10(cond):.3f}')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6f316041-7f14-48dd-a458-3a01fe87daa3",
   "metadata": {},
   "source": [
    "A large condition number shows us that A is ill-conditioned and decreases the numerical stability of the decompostion and thus reliability of the computed results to our system of equations. We can clearly observe a relatively high condition number, altough not as high as our singular matrix. We expect to lose at most 9 significant digits in base 10, and 30 in base 2. This is an upperbound for our loss in accuracy, and does not guarantee inaccurate results. But in the worst case we could still lose a lot of accuracy."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c2a94aca-a8cc-407c-9212-3b09e9bd850e",
   "metadata": {},
   "source": [
    "## (d)\n",
    "How well do the answers of (b) agree with each other (make an appropriate quantitative comparison)?\n",
    "\n",
    "Should we be worried about the fact that the two answers are different?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "9d8d9237",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 1200x500 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def get_diffs(n, method):\n",
    "    b = get_b(n)\n",
    "    A = cantilevered_beam_matrix(n)\n",
    "    sol = solve_cantilevered_beam_matrix(A, b, 0, method)[0]\n",
    "    diffs = []\n",
    "    for i in range(-10,10):\n",
    "        b_hat = b + i*10**-8\n",
    "        sol_hat = solve_cantilevered_beam_matrix(A, b_hat, 0, method)[0]\n",
    "        diffs.append(sum(sol_hat-sol)/n)\n",
    "    return diffs\n",
    "\n",
    "diffs_sparse_100 = get_diffs(100, 'sparse')\n",
    "diffs_sparse_1000 = get_diffs(1000, 'sparse') \n",
    "diffs_dense_100 = get_diffs(100, 'default')\n",
    "diffs_dense_1000 = get_diffs(1000, 'default')\n",
    "\n",
    "fig, axs = plt.subplots(1, 2, figsize=(12, 5))\n",
    "\n",
    "axs[0].plot(np.arange(-10, 10) * 10**-8, diffs_sparse_100, marker='o', linestyle='-', color='#93bfb6', label='Sparse')\n",
    "axs[0].plot(np.arange(-10, 10) * 10**-8, diffs_dense_100, marker='o', linestyle='-', color='#4f7942', label='Dense')\n",
    "axs[0].set_title('Effect of a change in $\\\\mathbf{b}$ on $\\\\mathbf{x}$ (n=100)')\n",
    "axs[0].set_xlabel('Change in $\\\\mathbf{b}$')\n",
    "axs[0].set_ylabel('Change in $\\\\mathbf{x}$')\n",
    "axs[0].legend()\n",
    "axs[0].grid(True)\n",
    "\n",
    "axs[1].plot(np.arange(-10, 10) * 10**-8, diffs_sparse_1000, marker='o', linestyle='-', color='#c78387', label='Sparse')\n",
    "axs[1].plot(np.arange(-10, 10) * 10**-8, diffs_dense_1000, marker='o', linestyle='-', color='#d6616b', label='Dense')\n",
    "axs[1].set_title('Effect of a change in $\\\\mathbf{b}$ on $\\\\mathbf{x}$ (n=1000)')\n",
    "axs[1].set_xlabel('Change in $\\\\mathbf{b}$')\n",
    "axs[1].set_ylabel('Change in $\\\\mathbf{x}$')\n",
    "axs[1].legend()\n",
    "axs[1].grid(True)\n",
    "\n",
    "# plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "02317207-3944-4be0-a8e6-29d81d007b00",
   "metadata": {},
   "source": [
    "As we see in our statistical analysis of (b), for $n=100$ we observed a difference of `5.632e-11` and for $n=1000$ a difference of `4.315e-06`. Although these errors might be large in certain contexts, for the stated problem the change of x is very insentive to a change in b, a change in $\\mathbf{b}$ of the order $10^{-7}$ results in a change in $\\mathbf{x}$ in the order of $10^{-9}$. This order of magnitude is smaller than the difference we found between the sparse and dense matrices, but it seem that this is not preventing us from getting stable and accurate results around $\\mathbf{b}$. These numerical differences are very small, but the differences in memory usage and computation time for large matrices is substantial. This all leads us to conclude that whereever possible, sparse matrices should be used."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.11.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}