01_linear_systems.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy.linalg as la\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Linear Systems of Equations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solve linear systems of equations $A \\mathbf{x} = \\mathbf{b}$:\n",
    "\n",
    "* Create NumPy arrays to represent $A$ and $\\mathbf{b}$\n",
    "* Compute the solution $\\boldsymbol{x}$ using the SciPy function `scipy.linalg.solve`\n",
    "\n",
    "[Learn about NumPy arrays](https://www.math.ubc.ca/~pwalls/math-python/scipy/numpy/) and the [SciPy Linear Algebra package](https://www.math.ubc.ca/~pwalls/math-python/linear-algebra/linear-algebra-scipy/)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 1\n",
    "\n",
    "Compute the solution of the system $A \\mathbf{x} = \\mathbf{b}$ where\n",
    "$$\n",
    "A = \\begin{bmatrix} 2 & 1 & 1 \\\\ 2 & 0 & 2 \\\\ 4 & 3 & 4 \\end{bmatrix}\n",
    "\\hspace{10mm}\n",
    "\\mathbf{b} = \\left[ \\begin{array}{r} -1 \\\\ 1 \\\\ 1 \\end{array} \\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = np.array([[2,1,1],[2,0,2],[4,3,4]])\n",
    "b = np.array([[-1],[1],[1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[2 1 1]\n",
      " [2 0 2]\n",
      " [4 3 4]]\n"
     ]
    }
   ],
   "source": [
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-1]\n",
      " [ 1]\n",
      " [ 1]]\n"
     ]
    }
   ],
   "source": [
    "print(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "numpy.ndarray"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = la.solve(A,b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-1.16666667]\n",
      " [-0.33333333]\n",
      " [ 1.66666667]]\n"
     ]
    }
   ],
   "source": [
    "print(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Due to rounding errors in the computation, our solution $\\hat{\\mathbf{x}}$ is an approximation of the exact solution $$\n",
    "\\mathbf{x} = \\left[ \\begin{array}{r} -7/6 \\\\-1/3 \\\\ 5/3 \\end{array} \\right]\n",
    "$$\n",
    "Compute the norm of the residual $\\| \\mathbf{b} - A \\mathbf{x} \\|$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "r = la.norm(b - A @ x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0\n"
     ]
    }
   ],
   "source": [
    "print(r)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 2\n",
    "\n",
    "Compute the solution of the system $A \\mathbf{x} = \\mathbf{b}$ for\n",
    "$$\n",
    "A = \n",
    "\\left[\n",
    "\\begin{array}{cccccccc}\n",
    "2R & -R & 0 & 0 & \\cdots & 0 & 0 & 0 \\\\\n",
    "-R & 2R & -R & 0 & & 0 & 0  & 0 \\\\\n",
    "0 & -R & 2R & -R & \\cdots & 0 & 0 & 0 \\\\\n",
    "\\vdots &  & \\vdots &  & \\ddots & & \\vdots & \\\\\n",
    "0 & 0 & 0 & 0 & \\cdots & -R & 2R & -R \\\\\n",
    "0 & 0 & 0 & 0 & \\cdots & 0 & -R & 2R \\\\\n",
    "\\end{array}\n",
    "\\right]\n",
    "\\hspace{10mm}\n",
    "\\mathbf{b} = \\left[ \\begin{array}{r} V \\\\ \\vdots \\\\ V \\end{array} \\right]\n",
    "$$\n",
    "where $A$ is a square matrix of size $N$, and $R$ and $V$ are some positive constants."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 10\n",
    "R = 1\n",
    "V = 1\n",
    "A1 = 2*R*np.eye(N)\n",
    "A2 = np.diag(-R*np.ones(N-1),1)\n",
    "A = A1 + A2 + A2.T\n",
    "b = V*np.ones([N,1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 2. -1.  0.  0.  0.  0.  0.  0.  0.  0.]\n",
      " [-1.  2. -1.  0.  0.  0.  0.  0.  0.  0.]\n",
      " [ 0. -1.  2. -1.  0.  0.  0.  0.  0.  0.]\n",
      " [ 0.  0. -1.  2. -1.  0.  0.  0.  0.  0.]\n",
      " [ 0.  0.  0. -1.  2. -1.  0.  0.  0.  0.]\n",
      " [ 0.  0.  0.  0. -1.  2. -1.  0.  0.  0.]\n",
      " [ 0.  0.  0.  0.  0. -1.  2. -1.  0.  0.]\n",
      " [ 0.  0.  0.  0.  0.  0. -1.  2. -1.  0.]\n",
      " [ 0.  0.  0.  0.  0.  0.  0. -1.  2. -1.]\n",
      " [ 0.  0.  0.  0.  0.  0.  0.  0. -1.  2.]]\n"
     ]
    }
   ],
   "source": [
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1.]\n",
      " [1.]\n",
      " [1.]\n",
      " [1.]\n",
      " [1.]\n",
      " [1.]\n",
      " [1.]\n",
      " [1.]\n",
      " [1.]\n",
      " [1.]]\n"
     ]
    }
   ],
   "source": [
    "print(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = la.solve(A,b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x,'b.')\n",
    "plt.xlabel('Loop current at index i')\n",
    "plt.ylabel('Loop currents (Amp)')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 5.]\n",
      " [ 9.]\n",
      " [12.]\n",
      " [14.]\n",
      " [15.]\n",
      " [15.]\n",
      " [14.]\n",
      " [12.]\n",
      " [ 9.]\n",
      " [ 5.]]\n"
     ]
    }
   ],
   "source": [
    "print(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}