notebooks/labwork-06.ipynb

{
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "ChEn-3170: Computational Methods in Chemical Engineering Spring 2020 UMass Lowell; Prof. V. F. de Almeida **24Feb20**\n",
    "\n",
    "# Laboratory Work 06 (25Feb20)\n",
    "$\n",
    "  \\newcommand{\\Amtrx}{\\boldsymbol{\\mathsf{A}}}\n",
    "  \\newcommand{\\Bmtrx}{\\boldsymbol{\\mathsf{B}}}\n",
    "  \\newcommand{\\Cmtrx}{\\boldsymbol{\\mathsf{C}}}\n",
    "  \\newcommand{\\Mmtrx}{\\boldsymbol{\\mathsf{M}}}\n",
    "  \\newcommand{\\Imtrx}{\\boldsymbol{\\mathsf{I}}}\n",
    "  \\newcommand{\\Pmtrx}{\\boldsymbol{\\mathsf{P}}}\n",
    "  \\newcommand{\\Qmtrx}{\\boldsymbol{\\mathsf{Q}}}\n",
    "  \\newcommand{\\Lmtrx}{\\boldsymbol{\\mathsf{L}}}\n",
    "  \\newcommand{\\Umtrx}{\\boldsymbol{\\mathsf{U}}}\n",
    "  \\newcommand{\\xvec}{\\boldsymbol{\\mathsf{x}}}\n",
    "  \\newcommand{\\yvec}{\\boldsymbol{\\mathsf{y}}}\n",
    "  \\newcommand{\\zvec}{\\boldsymbol{\\mathsf{z}}}\n",
    "  \\newcommand{\\avec}{\\boldsymbol{\\mathsf{a}}}\n",
    "  \\newcommand{\\bvec}{\\boldsymbol{\\mathsf{b}}}\n",
    "  \\newcommand{\\cvec}{\\boldsymbol{\\mathsf{c}}}\n",
    "  \\newcommand{\\rvec}{\\boldsymbol{\\mathsf{r}}}\n",
    "  \\newcommand{\\norm}[1]{\\bigl\\lVert{#1}\\bigr\\rVert}\n",
    "  \\DeclareMathOperator{\\rank}{rank}\n",
    "  \\DeclareMathOperator{\\abs}{abs}\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Name: `your name`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rubric for each assignment: \n",
    "\n",
    "|         Context           |  Points |\n",
    "| -----------------------     | ------- |\n",
    "| Precision of the answer     |   80%   |\n",
    "| Answer Markdown readability |   10%   |\n",
    "| Code readability            |   10%   |\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <span style=\"color:red\">Guidance:</span>\n",
    " +  <span style=\"color:red\"> \n",
    "    Save your work frequently to a file locally to your computer.\n",
    "    </span>\n",
    " +  <span style=\"color:red\">\n",
    "    Before submitting, `Kernel` -> `Restart & Run All`, to verify your notebook runs correctly.\n",
    "    </span>\n",
    " +  <span style=\"color:red\">\n",
    "    Save your file again.\n",
    "    </span>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "### Table of Contents\n",
    "* [Assignment 1 (30pts)](#a1) $\\Amtrx=\\Lmtrx\\Umtrx$ factorization code.\n",
    " - [1.1)](#a11) Code the $\\Lmtrx\\Umtrx$ in a `Python` function.\n",
    " - [1.2)](#a12) Import image as matrix.\n",
    " - [1.3)](#a13) Verify the $\\Lmtrx\\Umtrx$ factorization of $\\Amtrx$.\n",
    " - [1.4)](#a14) Further verify the $\\Lmtrx\\Umtrx$ factorization of $\\Amtrx$ against `NumPy`.\n",
    "* [Assignment 2 (35pts)](#a2) $\\Pmtrx\\Amtrx=\\Lmtrx\\Umtrx$ factorization code.\n",
    " - [2.1)](#a21) Code the $\\Lmtrx\\Umtrx$ factorization in a `Python` function.\n",
    " - [2.2)](#a22) Verify the $\\Lmtrx\\Umtrx$ factorization of $\\Pmtrx\\Amtrx$.\n",
    " - [2.3)](#a23) Further verify the $\\Lmtrx\\Umtrx$ factorization of $\\Pmtrx\\Amtrx$ against `NumPy`.\n",
    "* [Assignment 3 (35pts)](#a3) $\\Pmtrx\\Amtrx\\Qmtrx=\\Lmtrx\\Umtrx$ factorization code.\n",
    " - [3.1)](#a31) Code the $\\Lmtrx\\Umtrx$ factorization in a `Python` function.\n",
    " - [3.2)](#a32) Verify the $\\Lmtrx\\Umtrx$ factorization of $\\Pmtrx\\Amtrx\\Qmtrx$.\n",
    " - [3.3)](#a33) Further verify the $\\Lmtrx\\Umtrx$ factorization of $\\Pmtrx\\Amtrx\\Qmtrx$ against `NumPy`.\n",
    " ---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <span style=\"color:blue\">Assignment 1 (30 pts): For each item below respond in a separate notebook cell.</span><a id=\"a1\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<span style=\"color:blue\">\n",
    "1.1) Program (in a Python function) a $\\Lmtrx\\,\\Umtrx$ factorization algorithm <b>(without using pivoting)</b> for a matrix $\\overset{(m \\times m)}{\\Amtrx}$. The factorization is obtained by elimination steps $k = 1,\\ldots,m-1$ so that \n",
    "    \n",
    "\\begin{equation*}\n",
    "    A^{(k+1)}_{i,j} = A^{(k)}_{i,j} - A^{(k)}_{k,j}\\, m_{i,k} \\quad\\quad \\forall \\quad\\quad i=k+1,\\ldots,m \\quad\\quad \\text{and} \\quad\\quad j=k+1,\\ldots,m\n",
    "\\end{equation*}\n",
    "\n",
    "where the multipliers $m_{i,k}$ are given by $m_{i,k} = \\frac{A^{(k)}_{i,k}}{A^{(k)}_{k,k}}$. When $k = m-1$, $A^{(m)}_{i,j}$, is upper triangular, that is, $U_{i,j} = A^{(m)}_{i,j}$ . The lower triangular matrix is obtained using the multipliers $m_{i,k}$, that is $L_{i,j} = m_{i,j} \\ \\forall \\ i>j$,  $L_{i,i}=1$, and $L_{i,j}=0 \\ \\forall \\ i<j$.\n",
    "</span><a id=\"a11\"></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "code_folding": [],
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "'''1.1) LU factorization function'''\n",
    "\n",
    "def lu_factorization( mtrx, pivoting_option=None ):\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<span style=\"color:blue\">\n",
    "1.2) Import the following image URL: \n",
    "        \n",
    " + https://raw.githubusercontent.com/dpploy/chen-3170/master/notebooks/images/cermet.png\n",
    "\n",
    "as a matrix $\\Amtrx$ (need internet connection).\n",
    "</span><a id=\"a12\"></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 1440x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n"
     ]
    }
   ],
   "source": [
    "'''1.2) Import matrix'''\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " <span style=\"color:blue\">\n",
    " 1.3) Compute the $\\Lmtrx\\Umtrx$ factorization of $\\overset{(113 \\times 113)}{\\Amtrx}$ and verify that your factorization is correct.\n",
    " </span><a id=\"a13\"></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "max(abs((A - LU)) =  8.52651e-14\n"
     ]
    }
   ],
   "source": [
    "'''1.3) Perform LU factorization and verify'''\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " <span style=\"color:blue\">\n",
    "1.4) Print the diagonal entry of the $\\Umtrx$ factor with smallest absolute value, compute the $\\rank(\\Amtrx)$ and the $\\det(\\Amtrx)$ using the $\\Lmtrx\\,\\Umtrx$ factors.\n",
    "</span><a id=\"a14\"></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "min(abs(diag(U)) =  1.25385e-03\n",
      "my rank(A) = 113\n",
      "linalg rank(A) = 113\n",
      "my det(A) = 2.72809e-142\n",
      "linalg det(A) = 2.72809e-142\n"
     ]
    }
   ],
   "source": [
    "'''1.4) U diagonal entry w/ smallest magnitude, rank(A), and det(A)'''\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <span style=\"color:blue\">Assignment 2 (35 pts): For each item below respond in a separate notebook cell.</span><a id=\"a2\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<span style=\"color:blue\">\n",
    "2.1) Program (in a Python function; hint: extend the function from Assignment 1) a $\\Lmtrx\\,\\Umtrx$ factorization algorithm (<b>using partial pivoting</b>) for a matrix $\\overset{(m \\times m)}{\\Amtrx}$ and compute the $\\Pmtrx\\,\\Lmtrx\\,\\Umtrx$ factors. The factorization is obtained by elimination steps $k = 1,\\ldots,m-1$ so that $A^{(k+1)}_{i,j} = A^{(k)}_{i,j} - A^{(k)}_{k,j}\\, m_{i,k} \\ \\forall\\ i=k+1,\\ldots,m \\ \\text{and}\\ j=k+1,\\ldots,m$ where the multipliers $m_{i,k}$ are given by $m_{i,k} = \\frac{A^{(k)}_{i,k}}{A^{(k)}_{k,k}}$. When $k = m-1$, $A^{(m)}_{i,j}$, is upper triangular, that is, $U_{i,j} = A^{(m)}_{i,j}$ . The lower triangular matrix is obtained using the multipliers $m_{i,k}$, that is $L_{i,j} = m_{i,j} \\ \\forall \\ i>j$,  $L_{i,i}=1$, and $L_{i,j}=0 \\ \\forall \\ i<j$. However, every $k$-step selects a pivot $A^{(k)}_{k,k}$ of maximum absolute value via row\n",
    "exchanges recorded in the permutation matrix $\\Pmtrx$.\n",
    "</span><a id=\"a21\"></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'hint: extend 1.1) and avoid generating additional code here'"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "'''2.1) LU factorization function'''\n",
    "'''hint: extend 1.1) and avoid generating additional code here'''\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " <span style=\"color:blue\">\n",
    " 2.2) Compute the  $\\Lmtrx\\Umtrx$  factorization of $\\overset{(113 \\times 113)}{\\Amtrx}$ using partial pivoting and verify that your factorization is correct. Compare your result to the result obtained in 1.3) and make comments.\n",
    " </span><a id=\"a22\"></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "max(abs((PA - LU)) =  8.88178e-16\n"
     ]
    }
   ],
   "source": [
    "'''2.2) Perform LU factorization and verify'''\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**2.2 Comments on comparison of results:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " <span style=\"color:blue\">\n",
    "2.3) Print the diagonal entry of the $\\Umtrx$ factor with smallest absolute value, compute the $\\rank(\\Amtrx)$ and the $\\det(\\Amtrx)$ using the $\\Pmtrx\\,\\Lmtrx\\,\\Umtrx$ factors. Compare with the results obtained without pivoting, 1.4), and make comments on the value of the diagonal entry of $\\Umtrx$ with smallest magnitude.\n",
    "</span><a id=\"a23\"></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "min(abs(diag(U)) =  2.29442e-02\n",
      "my rank(A) = 113\n",
      "linalg rank(A) = 113\n",
      "my det(A) = 2.72809e-142\n",
      "linalg det(A) = 2.72809e-142\n"
     ]
    }
   ],
   "source": [
    "'''2.3) U diagonal entry w/ smallest magnitude, rank(A), and det(A)'''\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**2.3 Comment on the diagonal of the U entry with smallest absolute value:**\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <span style=\"color:blue\">Assignment 3 (35 pts): For each item below respond in a separate notebook cell.</span><a id=\"a3\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<span style=\"color:blue\">\n",
    "3.1) Program (in a Python function; hint: extend the function from Assignment 1) a $\\Lmtrx\\,\\Umtrx$ factorization algorithm (<b>using complete pivoting</b>) for a square matrix $\\overset{(m \\times m)}{\\Amtrx}$ and compute the $\\Pmtrx\\,\\Qmtrx\\,\\Lmtrx\\,\\Umtrx$ factors. The factorization is obtained by elimination steps $k = 1,\\ldots,m-1$ so that $A^{(k+1)}_{i,j} = A^{(k)}_{i,j} - A^{(k)}_{k,j}\\, m_{i,k} \\ \\forall\\ i=k+1,\\ldots,m \\ \\text{and}\\ j=k+1,\\ldots,m$ where the multipliers $m_{i,k}$ are given by $m_{i,k} = \\frac{A^{(k)}_{i,k}}{A^{(k)}_{k,k}}$. When $k = m-1$, $A^{(m)}_{i,j}$, is upper triangular, that is, $U_{i,j} = A^{(m)}_{i,j}$ . The lower triangular matrix is obtained using the multipliers $m_{i,k}$, that is $L_{i,j} = m_{i,j} \\ \\forall \\ i>j$,  $L_{i,i}=1$, and $L_{i,j}=0 \\ \\forall \\ i<j$. However, every $k$-step selects a pivot $A^{(k)}_{k,k}$ of maximum absolute value via row\n",
    "exchanges recorded in the permutation matrix $\\Pmtrx$ and column exchanges recorded in the permutation matrix $\\Qmtrx$.\n",
    "</span><a id=\"a31\"></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'hint: extend 1.1) and avoid generating additional code here'"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "'''3.1) LU factorization function'''\n",
    "'''hint: extend 1.1) and avoid generating additional code here'''\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " <span style=\"color:blue\">\n",
    " 3.2) Compute the  $\\Lmtrx\\Umtrx$ factorization of $\\overset{(113 \\times 113)}{\\Amtrx}$ using complete pivoting and verify that your factorization is correct. Compare your result to the result obtained in 2.2) and make comments.\n",
    " </span><a id=\"a32\"></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
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   "outputs": [
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     "name": "stdout",
     "output_type": "stream",
     "text": [
      "max(abs((PAQ - LU)) =  9.99201e-16\n"
     ]
    }
   ],
   "source": [
    "'''3.2 Perform LU factorization and verify'''\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**3.2 Comments on comparison of results:**\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " <span style=\"color:blue\">\n",
    "3.3) Print the diagonal entry of the $\\Umtrx$ factor with smallest absolute value, compute the $\\rank(\\Amtrx)$ and the $\\det(\\Amtrx)$ using the $\\Pmtrx\\,\\Qmtrx,\\Lmtrx\\,\\Umtrx$ factors. Compare with the results obtained with partial pivoting, 2.3), and make comments on the value of the diagonal entry of $\\Umtrx$ with smallest magnitude.\n",
    "</span><a id=\"a33\"></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "min(abs(diag(U)) =  7.26158e-03\n",
      "my rank(A) = 113\n",
      "linalg rank(A) = 113\n",
      "my det(A) = 2.72809e-142\n",
      "linalg det(A) = 2.72809e-142\n"
     ]
    }
   ],
   "source": [
    "'''3.3 U diagonal entry w/ smallest magnitude, rank(A), and det(A)'''\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**3.3 Comment on the diagonal of the U entry with smallest absolute value:**\n"
   ]
  }
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