{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 42  60]\n",
      " [-20 -28]] \n",
      "\n",
      " [[ 4 -8]\n",
      " [ 6 18]] \n",
      "\n",
      " [[ 4 -4]\n",
      " [-4 10]]\n"
     ]
    }
   ],
   "source": [
    "A = np.array([[42, 60], [-20, -28]]);\n",
    "B = np.array([[4, -8], [6, 18]])\n",
    "C = np.array([[4, -4], [-4, 10]])\n",
    "print(A, \"\\n\\n\", B, \"\\n\\n\", C)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# This powermethod function outputs a pair l, w where l is the list of approximations converging to the eigenvalue\n",
    "# and w is the eigenvector\n",
    "\n",
    "def powermethod(A, v, n): \n",
    "    l = np.zeros(n)\n",
    "    w=(1/np.linalg.norm(v))*v\n",
    "    for k in range(n):\n",
    "        v = A @ w\n",
    "        newl = np.dot(v,w)\n",
    "        l[k] = newl\n",
    "        w=(1/np.linalg.norm(v))*v\n",
    "    return l,w"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "v0 = np.array([1,1])\n",
    "alist, avect = powermethod(A,v0,20)\n",
    "blist, bvect = powermethod(B,v0,20)\n",
    "clist, cvect = powermethod(C,v0,20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "╒═════════════════════════╤═════════════════════════╤═════════════════════════╕\n",
      "│                       A │                       B │                       C │\n",
      "╞═════════════════════════╪═════════════════════════╪═════════════════════════╡\n",
      "│ 26.99999999999999644729 │  9.99999999999999822364 │  2.99999999999999955591 │\n",
      "├─────────────────────────┼─────────────────────────┼─────────────────────────┤\n",
      "│ 13.89801699716713834221 │ 17.94594594594594383352 │ 10.00000000000000000000 │\n",
      "├─────────────────────────┼─────────────────────────┼─────────────────────────┤\n",
      "│ 12.27385210813345217673 │ 15.92126945376868718540 │ 11.93103448275861921957 │\n",
      "├─────────────────────────┼─────────────────────────┼─────────────────────────┤\n",
      "│ 12.04463972920328629357 │ 14.57122787699872112910 │ 11.99807135969142102283 │\n",
      "├─────────────────────────┼─────────────────────────┼─────────────────────────┤\n",
      "│ 12.00741281251859504664 │ 13.79568504688623420407 │ 11.99994641661084671114 │\n",
      "├─────────────────────────┼─────────────────────────┼─────────────────────────┤\n",
      "│ 12.00123471795347640523 │ 13.31327114795233690359 │ 11.99999851156477070901 │\n",
      "├─────────────────────────┼─────────────────────────┼─────────────────────────┤\n",
      "│ 12.00020576548456929800 │ 12.99157810669037260709 │ 11.99999995865457158573 │\n",
      "├─────────────────────────┼─────────────────────────┼─────────────────────────┤\n",
      "│ 12.00003429366859286631 │ 12.76578040016991977268 │ 11.99999999885151602541 │\n",
      "├─────────────────────────┼─────────────────────────┼─────────────────────────┤\n",
      "│ 12.00000571559537121402 │ 12.60121892656163389290 │ 11.99999999996809663116 │\n",
      "├─────────────────────────┼─────────────────────────┼─────────────────────────┤\n",
      "│ 12.00000095259877852527 │ 12.47787220080673442624 │ 11.99999999999911359794 │\n",
      "├─────────────────────────┼─────────────────────────┼─────────────────────────┤\n",
      "│ 12.00000015876644532398 │ 12.38342297374618539152 │ 11.99999999999997690736 │\n",
      "├─────────────────────────┼─────────────────────────┼─────────────────────────┤\n",
      "│ 12.00000002646106800341 │ 12.30989778774626763891 │ 12.00000000000000000000 │\n",
      "├─────────────────────────┼─────────────────────────┼─────────────────────────┤\n",
      "│ 12.00000000441020375774 │ 12.25191626143521439474 │ 12.00000000000000177636 │\n",
      "├─────────────────────────┼─────────────────────────┼─────────────────────────┤\n",
      "│ 12.00000000073502981479 │ 12.20572152181351377465 │ 11.99999999999999822364 │\n",
      "├─────────────────────────┼─────────────────────────┼─────────────────────────┤\n",
      "│ 12.00000000012252243664 │ 12.16861480266319439636 │ 12.00000000000000000000 │\n",
      "├─────────────────────────┼─────────────────────────┼─────────────────────────┤\n",
      "│ 12.00000000002041034008 │ 12.13861092354648008040 │ 11.99999999999999822364 │\n",
      "├─────────────────────────┼─────────────────────────┼─────────────────────────┤\n",
      "│ 12.00000000000340349970 │ 12.11422027447587090876 │ 12.00000000000000177636 │\n",
      "├─────────────────────────┼─────────────────────────┼─────────────────────────┤\n",
      "│ 12.00000000000057731597 │ 12.09430624571495904718 │ 11.99999999999999644729 │\n",
      "├─────────────────────────┼─────────────────────────┼─────────────────────────┤\n",
      "│ 12.00000000000010125234 │ 12.07798926573218167846 │ 12.00000000000000000000 │\n",
      "├─────────────────────────┼─────────────────────────┼─────────────────────────┤\n",
      "│ 12.00000000000002486900 │ 12.06458053992080792227 │ 12.00000000000000000000 │\n",
      "╘═════════════════════════╧═════════════════════════╧═════════════════════════╛\n"
     ]
    }
   ],
   "source": [
    "from tabulate import tabulate\n",
    "data = zip(alist, blist, clist)\n",
    "print(tabulate(data, headers=[\"A\", \"B\", \"C\"], tablefmt='fancy_grid',floatfmt=\".20f\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Interpretation\n",
    "It's straightforward to check that the eigenvalues for $A$ are $12$ and $2$, the eigenvalues for $B$ are $12$ and $10$, and the eigenvalues for $C$ are $12$ and $2$.  The power method to find the principle eigenvalue for $A$ and $B$ will converge to $12$ with a rate of $\\frac{\\lambda_2}{\\lambda_1} = \\frac{2}{12}$ and $\\frac{\\lambda_2}{\\lambda_1} = \\frac{10}{12}$ respectively.  Because $C$ is symmetric, convergence to $12$ will be at the rate $\\left(\\frac{\\lambda_2}{\\lambda_1}\\right)^2 = \\left(\\frac{2}{12}\\right)^2$ (much faster!)"
   ]
  }
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