{
 "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": [
      "A = np.random.randn(5, 3)\n",
      "b = np.random.randn(5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Solve $Ax\\cong b$ using the normal equations:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x1 = ...\n",
      "x1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 4,
       "text": [
        "array([ 0.23982628, -0.26470109, -0.18776136])"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<!--\n",
      "x1 = la.solve(np.dot(A.T, A), np.dot(A.T, b))\n",
      "-->\n",
      "(Edit this cell for solution.)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Solve $Ax\\cong b$ using the pseudoinverse:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "U, sigma, VT = la.svd(A)\n",
      "print U\n",
      "print sigma\n",
      "print VT"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[-0.40467972 -0.33714129 -0.63422814  0.20634795  0.52701543]\n",
        " [ 0.42804578  0.47808154  0.02477487 -0.195988    0.74107338]\n",
        " [-0.06483149 -0.50603238  0.15333028 -0.83497764  0.13795024]\n",
        " [ 0.80377855 -0.49127211 -0.27409423  0.1695712  -0.09332588]\n",
        " [-0.0524219  -0.40044568  0.70604748  0.4393941   0.38121514]]\n",
        "[ 2.93921603  2.18290554  1.39834826]\n",
        "[[-0.75141123  0.65519253 -0.07812746]\n",
        " [ 0.62293553  0.74344157  0.24340493]\n",
        " [-0.21756029 -0.13422882  0.96677306]]\n"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Sigma_inv = np.zeros_like(A.T)\n",
      "Sigma_inv[:3,:3] = np.diag(1/sigma)\n",
      "Sigma_inv"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "array([[ 0.34022678,  0.        ,  0.        ,  0.        ,  0.        ],\n",
        "       [ 0.        ,  0.45810503,  0.        ,  0.        ,  0.        ],\n",
        "       [ 0.        ,  0.        ,  0.71512943,  0.        ,  0.        ]])"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "pinv = ...\n",
      "x2 = np.dot(pinv, b)\n",
      "x2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "array([ 0.23982628, -0.26470109, -0.18776136])"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<!--\n",
      "pinv = np.dot(np.dot(VT.T, Sigma_inv), U.T)\n",
      "-->\n",
      "(Edit this cell for solution.)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "la.norm(x1-x2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 8,
       "text": [
        "1.7554167342883506e-16"
       ]
      }
     ],
     "prompt_number": 8
    }
   ],
   "metadata": {}
  }
 ]
}