{
 "metadata": {
  "language": "Julia",
  "name": "Lecture04.ipynb"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": "Gauss Jordan, LU"
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "A Bit More about Blocks -- Consider 1x1 by 1x2"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "2*[3 4]",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 195,
       "text": "1x2 Array{Int64,2}:\n 6  8"
      }
     ],
     "prompt_number": 195
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "[2*3 2*4]",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 196,
       "text": "1x2 Array{Int64,2}:\n 6  8"
      }
     ],
     "prompt_number": 196
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "Blocks work the same way"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A=rand(1:3,2,4); B=rand(1:3,4,3); C=rand(1:3,4,2);\nprintln( A*[B C])\nprintln( [A*B A*C])",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "15\t16\t13\t15\t16\n15\t16\t13\t17\t14\n\n"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "15\t16\t13\t15\t16\n15\t16\t13\t17\t14\n\n"
      }
     ],
     "prompt_number": 197
    },
    {
     "cell_type": "raw",
     "metadata": {},
     "source": "Question: What dimensions of A and B and C allow this to work\n    \nThere are a number of ways to check that with the right dimensions  A[B C]=[AB AC]\n1.  The highest level way is the block view\n    Block view = if blocks conform then block matmul always works like elementwise matmul\n2. Also worthwhile to consider the column view\n    The columns of A[B C] are all the matrix-vector products of A*the columns of [B C]\n    Why is this the same as [AB AC], the columns of AB concatanated with the columns of AC?\n3. You might try other views of matmul too\n   You might think some are easier on the brain than others, and also some views while\n   less easy on the brain, are more powerful.  You get to decide what you like best.\n                                                                            "
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "Gauss Jordan in high level block form  M[A I]=[I M]  so M is a left inverse (algorithm hidden!)"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "rand3(n)=rand(1:3,n,n);  # define rand3 (A little shorthand)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 198,
       "text": "# methods for generic function rand3\nrand3(n) at In[198]:1"
      }
     ],
     "prompt_number": 198
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "I=eye(3); A=rand3(3); M=inv(A);     # Might be singular, careful (single exception or large numbers indicate singularity)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 200,
       "text": "3x3 Array{Float64,2}:\n  0.4  -0.2   0.0\n -0.6  -0.2   1.0\n  0.4   0.8  -1.0"
      }
     ],
     "prompt_number": 200
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "I=eye(3);\nshowcompact(\n            M*[A I]    );  println('\\n')\nshowcompact(\n            [ I M] \n            )",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "3x6 Array{Float64,2}:"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n 1.0  0.0  0.0   0.4  -0.2   0.0\n 0.0  1.0  0.0  -0.6  -0.2   1.0\n 0.0  0.0  1.0   0.4   0.8  -1.0\n\n3x6 Array{Float64,2}:\n 1.0  0.0  0.0   0.4  -0.2   0.0\n 0.0  1.0  0.0  -0.6  -0.2   1.0\n 0.0  0.0  1.0   0.4   0.8  -1.0"
      }
     ],
     "prompt_number": 209
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "Gauss Jordan details -- do this with pencil and paper once or twice"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Z=[ A I]\n",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 210,
       "text": "3x6 Array{Float64,2}:\n 3.0  1.0  1.0  1.0  0.0  0.0\n 1.0  2.0  2.0  0.0  1.0  0.0\n 2.0  2.0  1.0  0.0  0.0  1.0"
      }
     ],
     "prompt_number": 210
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Z[2,:] -= (Z[2,1]/Z[1,1])*Z[1,:];Z    # Method 1. (Easier for Programming) Elimination via row operations",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 211,
       "text": "3x6 Array{Float64,2}:\n 3.0  1.0      1.0       1.0       0.0  0.0\n 0.0  1.66667  1.66667  -0.333333  1.0  0.0\n 2.0  2.0      1.0       0.0       0.0  1.0"
      }
     ],
     "prompt_number": 211
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Z=[ A I];",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 211
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "E=copy(I); E[2,1]=-Z[2,1]/Z[1,1];E  # Method 2: Create an elimination matrix with the negative multiplier in the (2,1) position, say",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 212,
       "text": "3x3 Array{Float64,2}:\n  1.0       0.0  0.0\n -0.333333  1.0  0.0\n  0.0       0.0  1.0"
      }
     ],
     "prompt_number": 212
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Z=E*Z",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 213,
       "text": "3x6 Array{Float64,2}:\n 3.0  1.0      1.0       1.0       0.0  0.0\n 0.0  1.66667  1.66667  -0.333333  1.0  0.0\n 2.0  2.0      1.0       0.0       0.0  1.0"
      }
     ],
     "prompt_number": 213
    },
    {
     "cell_type": "heading",
     "level": 5,
     "metadata": {},
     "source": "Continue with method 1"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Z[3,:] -= (Z[3,1]/Z[1,1])*Z[1,:];Z",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 214,
       "text": "3x6 Array{Float64,2}:\n 3.0  1.0      1.0        1.0       0.0  0.0\n 0.0  1.66667  1.66667   -0.333333  1.0  0.0\n 0.0  1.33333  0.333333  -0.666667  0.0  1.0"
      }
     ],
     "prompt_number": 214
    },
    {
     "cell_type": "heading",
     "level": 5,
     "metadata": {},
     "source": "Column 2"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Z[1,:] -= (Z[1,2]/Z[2,2])*Z[2,:];Z",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 215,
       "text": "3x6 Array{Float64,2}:\n 3.0  0.0      0.0        1.2       -0.6  0.0\n 0.0  1.66667  1.66667   -0.333333   1.0  0.0\n 0.0  1.33333  0.333333  -0.666667   0.0  1.0"
      }
     ],
     "prompt_number": 215
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Z[3,:] -= (Z[3,2]/Z[2,2])*Z[2,:];Z",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 216,
       "text": "3x6 Array{Float64,2}:\n 3.0  0.0       0.0       1.2       -0.6  0.0\n 0.0  1.66667   1.66667  -0.333333   1.0  0.0\n 0.0  0.0      -1.0      -0.4       -0.8  1.0"
      }
     ],
     "prompt_number": 216
    },
    {
     "cell_type": "heading",
     "level": 5,
     "metadata": {},
     "source": "Column 3"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Z[1,:] -= (Z[1,3]/Z[3,3])*Z[3,:];Z",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 217,
       "text": "3x6 Array{Float64,2}:\n 3.0  0.0       0.0       1.2       -0.6  0.0\n 0.0  1.66667   1.66667  -0.333333   1.0  0.0\n 0.0  0.0      -1.0      -0.4       -0.8  1.0"
      }
     ],
     "prompt_number": 217
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Z[2,:] -= (Z[2,3]/Z[3,3])*Z[3,:];Z",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 218,
       "text": "3x6 Array{Float64,2}:\n 3.0  0.0       0.0   1.2  -0.6       0.0    \n 0.0  1.66667   0.0  -1.0  -0.333333  1.66667\n 0.0  0.0      -1.0  -0.4  -0.8       1.0    "
      }
     ],
     "prompt_number": 218
    },
    {
     "cell_type": "heading",
     "level": 5,
     "metadata": {},
     "source": "Divide out the pivots"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Z[1,:]/=Z[1,1];Z",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 219,
       "text": "3x6 Array{Float64,2}:\n 1.0  0.0       0.0   0.4  -0.2       0.0    \n 0.0  1.66667   0.0  -1.0  -0.333333  1.66667\n 0.0  0.0      -1.0  -0.4  -0.8       1.0    "
      }
     ],
     "prompt_number": 219
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Z[2,:]/=Z[2,2];Z",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 220,
       "text": "3x6 Array{Float64,2}:\n 1.0  0.0   0.0   0.4  -0.2  0.0\n 0.0  1.0   0.0  -0.6  -0.2  1.0\n 0.0  0.0  -1.0  -0.4  -0.8  1.0"
      }
     ],
     "prompt_number": 220
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Z[3,:]/=Z[3,3];Z",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 221,
       "text": "3x6 Array{Float64,2}:\n  1.0   0.0  0.0   0.4  -0.2   0.0\n  0.0   1.0  0.0  -0.6  -0.2   1.0\n -0.0  -0.0  1.0   0.4   0.8  -1.0"
      }
     ],
     "prompt_number": 221
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inv(A)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 222,
       "text": "3x3 Array{Float64,2}:\n  0.4  -0.2   0.0\n -0.6  -0.2   1.0\n  0.4   0.8  -1.0"
      }
     ],
     "prompt_number": 222
    },
    {
     "cell_type": "heading",
     "level": 5,
     "metadata": {},
     "source": "We dit it!"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 223
    },
    {
     "cell_type": "raw",
     "metadata": {},
     "source": "Point of view: we solved Ax =[e1 e2 e3] simultaneously giving a right inverse\n          but  the block view shows us it is also a left inverse\n         hence a left invese is a right inverse"
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": "Elimination = Factorization = LU"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A=[ 6.0 5 10; 8 13 8; 3 14 7]; Aoriginal=copy(A);",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 276
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A[2,:]-= (4/3)*A[1,:]\nA",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 262,
       "text": "3x3 Array{Float64,2}:\n 6.0   5.0      10.0    \n 0.0   6.33333  -5.33333\n 3.0  14.0       7.0    "
      }
     ],
     "prompt_number": 262
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A[3,:] -= ( 0.5 )*A[1,:]\nA",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 263,
       "text": "3x3 Array{Float64,2}:\n 6.0   5.0      10.0    \n 0.0   6.33333  -5.33333\n 0.0  11.5       2.0    "
      }
     ],
     "prompt_number": 263
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A[3,:]-=  (11.5/6.333333   )*A[2,:];A",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 264,
       "text": "3x3 Array{Float64,2}:\n 6.0   5.0         10.0    \n 0.0   6.33333     -5.33333\n 0.0  -6.05263e-7  11.6842 "
      }
     ],
     "prompt_number": 264
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "U=ans",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 265,
       "text": "3x3 Array{Float64,2}:\n 6.0   5.0         10.0    \n 0.0   6.33333     -5.33333\n 0.0  -6.05263e-7  11.6842 "
      }
     ],
     "prompt_number": 265
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "(11.5/6.333333 )",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 266,
       "text": "1.8157895692520827"
      }
     ],
     "prompt_number": 266
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "69/38\n\n\n\n\n",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 267,
       "text": "1.8157894736842106"
      }
     ],
     "prompt_number": 267
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "U",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 268,
       "text": "3x3 Array{Float64,2}:\n 6.0   5.0         10.0    \n 0.0   6.33333     -5.33333\n 0.0  -6.05263e-7  11.6842 "
      }
     ],
     "prompt_number": 268
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "L=eye(3)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 269,
       "text": "3x3 Array{Float64,2}:\n 1.0  0.0  0.0\n 0.0  1.0  0.0\n 0.0  0.0  1.0"
      }
     ],
     "prompt_number": 269
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "L[2,1]=4/3; L[3,1]=0.5; L[3,2]=69/38;L",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 270,
       "text": "3x3 Array{Float64,2}:\n 1.0      0.0      0.0\n 1.33333  1.0      0.0\n 0.5      1.81579  1.0"
      }
     ],
     "prompt_number": 270
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "L*U",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 271,
       "text": "3x3 Array{Float64,2}:\n 6.0   5.0  10.0\n 8.0  13.0   8.0\n 3.0  14.0   7.0"
      }
     ],
     "prompt_number": 271
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Aoriginal",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 277,
       "text": "3x3 Array{Float64,2}:\n 6.0   5.0  10.0\n 8.0  13.0   8.0\n 3.0  14.0   7.0"
      }
     ],
     "prompt_number": 277
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "E",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 258,
       "text": "3x3 Array{Float64,2}:\n  1.0       0.0  0.0\n -0.333333  1.0  0.0\n  0.0       0.0  1.0"
      }
     ],
     "prompt_number": 258
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inv(E)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 237,
       "text": "3x3 Array{Float64,2}:\n  1.0        0.0  0.0\n  0.333333   1.0  0.0\n -0.0       -0.0  1.0"
      }
     ],
     "prompt_number": 237
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "M=rand(3,3)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 238,
       "text": "3x3 Array{Float64,2}:\n 0.0385084  0.992836   0.113299\n 0.399386   0.757624   0.419412\n 0.428451   0.0878158  0.299829"
      }
     ],
     "prompt_number": 238
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "E*M",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 239,
       "text": "3x3 Array{Float64,2}:\n 0.0385084  0.992836   0.113299\n 0.38655    0.426679   0.381646\n 0.428451   0.0878158  0.299829"
      }
     ],
     "prompt_number": 239
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inv(E)* (E*M)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 240,
       "text": "3x3 Array{Float64,2}:\n 0.0385084  0.992836   0.113299\n 0.399386   0.757624   0.419412\n 0.428451   0.0878158  0.299829"
      }
     ],
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    {
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     "input": "",
     "language": "python",
     "metadata": {},
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   "metadata": {}
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 ]
}