{
 "metadata": {
  "name": ""
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import numpy.linalg as la"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "eps = 1e-2  # set to 1e-5, 1e-10\n",
      "\n",
      "A = np.array([\n",
      "        [1, 1],\n",
      "        [eps, 0],\n",
      "        [0, eps],\n",
      "        ])\n",
      "\n",
      "np.set_printoptions(precision=20)\n",
      "print A\n",
      "print np.dot(A.T, A)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[ 1.                      1.                    ]\n",
        " [ 0.01000000000000000021  0.                    ]\n",
        " [ 0.                      0.01000000000000000021]]\n",
        "[[ 1.00009999999999998899  1.                    ]\n",
        " [ 1.                      1.00009999999999998899]]\n"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Problem 1 is ...?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n = 5\n",
      "\n",
      "A = np.random.randn(5, 5) * 10**-np.linspace(0, -5, n)\n",
      "la.cond(A)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 3,
       "text": [
        "60358.505352514032"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "la.cond(np.dot(A.T, A))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 4,
       "text": [
        "3643149168.4817424"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Problem 2 is ...?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "General bound: $\\operatorname{cond}(AB)\\le \\dots$?"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}