{
 "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 scipy.linalg as la"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stderr",
       "text": [
        "/usr/lib/pymodules/python2.7/numpy/oldnumeric/__init__.py:11: ModuleDeprecationWarning: The oldnumeric module will be dropped in Numpy 1.9\n",
        "  warnings.warn(_msg, ModuleDeprecationWarning)\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n = 5\n",
      "A = np.random.randn(n, n)\n",
      "u = np.random.randn(n)\n",
      "v = np.random.randn(n)\n",
      "\n",
      "b = np.random.randn(n)\n",
      "\n",
      "Ahat = A + np.outer(u, v)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "LU, piv = la.lu_factor(A)\n",
      "print LU\n",
      "print piv"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[-1.34619591 -2.40396885  1.34168773 -1.26156511 -0.57082875]\n",
        " [-0.65336062 -4.03983814  1.8543316   0.62740517  0.04102615]\n",
        " [-0.12411925  0.35016924 -3.01157725  1.71780721 -0.10249761]\n",
        " [-0.23344541 -0.05001819  0.08638383 -1.58556641  0.49576436]\n",
        " [-0.01497017  0.02626202 -0.16798668 -0.09674568  2.0533571 ]]\n",
        "[1 4 2 4 4]\n"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def solveA(b):\n",
      "    return la.lu_solve((LU, piv), b)\n",
      "\n",
      "la.norm(np.dot(A, solveA(b)) - b)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 4,
       "text": [
        "4.002966042486721e-16"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$(A+uv^T)^{-1} = A^{-1} - {A^{-1}uv^T A^{-1} \\over 1 + v^T A^{-1}u}$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "xhat = la.solve(Ahat, b)\n",
      "\n",
      "xhat2 = ...\n",
      "\n",
      "print la.norm(xhat - xhat2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "1.57009245868e-16\n"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<!--\n",
      "xhat2 = solveA(b) - solveA(u)*np.dot(v, solveA(b))/(1+np.dot(v, solveA(u)))\n",
      "-->\n",
      "(Edit this cell to see solution.)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "What's the cost of the Sherman-Morrison procedure?"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}