{
 "metadata": {
  "name": ""
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from __future__ import division\n",
      "\n",
      "import numpy as np\n",
      "import numpy.linalg as la"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = np.array([\n",
      "        [3, 4, 5],\n",
      "        [2, 3, 1],\n",
      "        [9, 2, 7]\n",
      "        ], dtype=np.float64)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = np.array([\n",
      "        [1e-20, 1],\n",
      "        [1, 1],\n",
      "        ])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n = A.shape[0]\n",
      "\n",
      "elim_matrices = []\n",
      "\n",
      "W = A\n",
      "\n",
      "for i in xrange(n-1):\n",
      "    M = np.eye(n)\n",
      "    M[i+1:, i] = -W[i+1:, i]/W[i,i]\n",
      "    \n",
      "    elim_matrices.append(M)\n",
      "    \n",
      "    W = np.dot(M, W)\n",
      "    print \"--------------------------------------\"\n",
      "    print \"column %d\" % i\n",
      "    print \"--------------------------------------\"\n",
      "    print \"elimination mat:\"\n",
      "    print M\n",
      "    print \"left over:\"\n",
      "    print W"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "--------------------------------------\n",
        "column 0\n",
        "--------------------------------------\n",
        "elimination mat:\n",
        "[[  1.00000000e+00   0.00000000e+00]\n",
        " [ -1.00000000e+20   1.00000000e+00]]\n",
        "left over:\n",
        "[[  1.00000000e-20   1.00000000e+00]\n",
        " [  0.00000000e+00  -1.00000000e+20]]\n"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "L = reduce(np.dot, [la.inv(M) for M in elim_matrices])\n",
      "U = W\n",
      "\n",
      "print \"L\"\n",
      "print L\n",
      "print \"U\"\n",
      "print U"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "L\n",
        "[[  1.00000000e+00  -0.00000000e+00]\n",
        " [  1.00000000e+20   1.00000000e+00]]\n",
        "U\n",
        "[[  1.00000000e-20   1.00000000e+00]\n",
        " [  0.00000000e+00  -1.00000000e+20]]\n"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A_rebuilt = np.dot(L, U)\n",
      "A_rebuilt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "array([[  1.00000000e-20,   1.00000000e+00],\n",
        "       [  1.00000000e+00,   0.00000000e+00]])"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "la.norm(A_rebuilt - A)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "1.0"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Why is this a terrible algorithm? What is its cost?"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}