London jlUG

{
 "metadata": {
  "language": "Julia",
  "name": "London jlUG Mar 2014"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": "# <center><img src=\"http://www.sengupta.net/images/julia_logo.png\" width=\"250\" alt=\"Julia\"/></center>\n## <center>Julia : The Power of Multiple Dispatch</center>"
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": "<center><img src=\"http://www.sengupta.net/images/languages.png\" alt=\"technical computing systems\" width=\"60%\"/></center>"
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": "### What is a numerical language?\n\n- Obviously, it's one that specializes in numerical work.\n\nBut specialisation means different things\n\n- **Matlab**\n  - originally, everything was a matrix of complex doubles\n  - everything else has been extended from that\n- **R**\n  - data frames as basic type\n  - support for \"NA\" values everywhere\n- **Mathematica**\n  - symbolic evaluation semantics\n\nEvery system picks its own angle/niche to specialize."
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": "### Julia does NOT have a numerical speciality\n\n- instead, it is designed so you can *define numeric types and their behaviors yourself*"
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": "## Design Overview\n\n- high-level & **dynamic**\n- expressive **type-system**\n  - parametric, dependent, invariant\n  - concrete types are final\n    - but extensive hierarchy of abstract supertypes\n    - lots of generic programming with abstract types\n  - **unobtrusive** \u2013\u00a0don't need to mention types\n- ubiquitous **multiple dispatch**\n  - everything is a generic function\n    - even really basic, performance-cricical functions\n  - diagonal dispatch\n- metaprogramming\n  - **homoiconic:** code represented as `Expr` & `Symbol` objects, etc.\n  - can be constructed, manipulated, eval'd\n  - **macros:** `@time sleep(1)`\n- concurrency & parallelism\n  - lightweight **coroutine**-based I/O\n  - distributed global address space\n    - easy run code on a cluster of instances\n    - first-class **remote references**\n"
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": "#Syntax Basics"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "function fib(n) \n    if n < 2 \n        return n\n    else\n        return fib(n-1) + fib(n-2)\n    end\nend",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 1,
       "text": "fib (generic function with 1 method)"
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "fib(n) = n < 2 ? n : fib(n - 1) + fib(n - 2)",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 2,
       "text": "fib (generic function with 1 method)"
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "fib(30)",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 53,
       "text": "832040"
      }
     ],
     "prompt_number": 53
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "@time fib(26)",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "elapsed time: 0."
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "006557479 seconds (64 bytes allocated)\n"
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 51,
       "text": "121393"
      }
     ],
     "prompt_number": 51
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A = randn(10,10)",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": "10x10 Array{Float64,2}:\n -1.30976    -1.24665   -1.98481   \u2026   1.8024     0.972884   2.22496  \n -1.73245    -1.06488    0.131453     -0.26486    1.31325    0.509545 \n  0.68138     0.840067  -0.317617      0.206785   0.108507  -0.409318 \n  0.219705   -0.924119   1.7471        0.856954  -0.59808    1.33977  \n -0.813242   -1.07687   -0.750896      0.423604  -0.522527  -0.225528 \n  0.793055   -2.54525    0.557789  \u2026   1.22375   -0.146579  -0.0398623\n -0.309985    0.152875  -0.895848      1.41556   -0.894667  -0.783496 \n -0.4108     -0.223782  -0.600858     -0.503635   0.757326   0.356692 \n -0.749092    0.578509  -0.153763     -0.2358     1.58472    0.873386 \n -0.0380719   0.434442   0.996071     -1.0903     0.862904  -0.414622 "
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "b = randn(10,10)",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": "10x10 Array{Float64,2}:\n  0.497573   -1.16141    -0.623559   \u2026  -0.405122   -0.408625   0.778652 \n -0.0733248   0.500723    0.0851337      0.12818     0.423034  -0.179    \n -0.304402    1.17565    -1.15891       -0.301865   -0.677907   0.0910984\n  0.551543   -0.821189    1.12574       -0.292132   -1.35895   -0.866446 \n  2.02635    -0.584961    0.610451       0.218641    0.671659   1.04507  \n  0.335287   -0.820849   -0.446622   \u2026  -0.170684    1.40318   -1.51586  \n -1.19941     1.61175     0.293387       0.0950038   1.51991    0.382434 \n  2.0922      0.845295    0.158825      -1.6935     -0.844484   0.460787 \n -1.17735     0.509996    0.600545       0.733072    0.142897  -0.514896 \n -1.4766      0.0797037   0.200661      -1.44351    -1.02833   -0.345725 "
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "x = A \\ b",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": "10x10 Array{Float64,2}:\n  -0.387223   -0.969381   -0.364462  \u2026    0.727768    0.535889  -0.654995\n  20.1103     13.7354    -13.0452       -13.3273    -21.0828     7.63941 \n   3.79951     3.27369    -2.4064        -2.42035    -4.36997    0.899417\n -12.6121     -7.63838     7.54477        8.02675    12.319     -4.51312 \n   2.41019     0.864194    0.171613      -2.09304    -2.15821    0.925504\n  -0.0433425  -0.713409    0.907706  \u2026    1.51948     1.0485    -0.231239\n  -5.03231    -2.7277      3.74034        3.36796     5.6027    -2.1193  \n  13.5295     10.4981     -9.59809       -9.56239   -14.2715     4.83205 \n  12.2784      9.08601    -8.53129       -8.78272   -12.7905     4.04945 \n   4.63908     1.46401    -2.22968       -2.46468    -4.65656    1.51594 "
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A*x - b",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 8,
       "text": "10x10 Array{Float64,2}:\n -4.77396e-15   4.21885e-15   1.22125e-15  \u2026   1.81521e-14  -3.9968e-15 \n -1.27676e-15  -1.11022e-16  -2.35922e-16     -1.33227e-15   1.94289e-16\n  5.4956e-15    4.44089e-16   2.88658e-15      3.33067e-15   1.19349e-15\n  5.66214e-15   1.22125e-15   5.55112e-15     -2.22045e-15  -1.33227e-15\n -1.06581e-14   1.11022e-15  -4.44089e-15     -3.66374e-15  -4.44089e-16\n -1.05471e-14   1.11022e-15   4.44089e-16  \u2026  -4.21885e-15   3.33067e-15\n -1.9984e-15   -3.10862e-15  -1.33227e-15      2.66454e-15  -2.22045e-15\n -4.88498e-15   2.22045e-16  -2.88658e-15     -7.10543e-15   3.88578e-16\n  4.21885e-15   4.21885e-15   2.88658e-15      9.40914e-15  -1.22125e-15\n  5.9952e-15    2.02616e-15  -8.32667e-17     -8.65974e-15   8.32667e-16"
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "eps()",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": "2.220446049250313e-16"
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "round(A*x-b, 12)",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": "10x10 Array{Float64,2}:\n -0.0   0.0   0.0  -0.0   0.0  -0.0  -0.0  -0.0   0.0  -0.0\n -0.0  -0.0  -0.0   0.0  -0.0  -0.0  -0.0   0.0  -0.0   0.0\n  0.0   0.0   0.0  -0.0  -0.0   0.0   0.0   0.0   0.0   0.0\n  0.0   0.0   0.0  -0.0   0.0   0.0  -0.0   0.0  -0.0  -0.0\n -0.0   0.0  -0.0   0.0  -0.0   0.0  -0.0  -0.0  -0.0  -0.0\n -0.0   0.0   0.0  -0.0   0.0   0.0   0.0   0.0  -0.0   0.0\n -0.0  -0.0  -0.0   0.0  -0.0  -0.0  -0.0  -0.0   0.0  -0.0\n -0.0   0.0  -0.0   0.0  -0.0  -0.0  -0.0  -0.0  -0.0   0.0\n  0.0   0.0   0.0   0.0   0.0   0.0   0.0   0.0   0.0  -0.0\n  0.0   0.0  -0.0  -0.0   0.0   0.0  -0.0   0.0  -0.0   0.0"
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "fft(A)",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 11,
       "text": "10x10 Array{Complex{Float64},2}:\n       3.05301+0.0im   -7.12504+9.72478im  \u2026   -7.12504-9.72478im\n   9.83729+3.78063im    3.85146-7.76011im       1.60801+6.18073im\n  -12.6461+1.77685im     2.93565+13.769im       -8.0573-7.12585im\n -0.509551+7.44926im     7.97449+8.8103im      -13.9814-1.56542im\n   6.99391-6.73352im   0.207496+8.01103im       1.95074-4.46963im\n       8.51634+0.0im  -8.42778+0.469558im  \u2026  -8.42778-0.469558im\n   6.99391+6.73352im    1.95074+4.46963im      0.207496-8.01103im\n -0.509551-7.44926im   -13.9814+1.56542im        7.97449-8.8103im\n  -12.6461-1.77685im    -8.0573+7.12585im        2.93565-13.769im\n   9.83729-3.78063im    1.60801-6.18073im       3.85146+7.76011im"
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": "#Multiple Dispatch\n\n##Basic Dispatch"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "f(a, b) = \"fallback\"\nf(a::Number, b::Number) = \"a and b are both numbers\"\nf(a::Number, b) = \"a is a number\"\nf(a, b::Number) = \"b is a number\"\nf(a::Integer, b::Integer) = \"a and b are both integers\"",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": "f (generic function with 5 methods)"
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "f(1.5,2)",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 13,
       "text": "\"a and b are both numbers\""
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "f(1, \"bar\")",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 14,
       "text": "\"a is a number\""
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "f(1,2)",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 15,
       "text": "\"a and b are both integers\""
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "f(\"foo\", [1,2])",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 16,
       "text": "\"fallback\""
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "methods(f)",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "html": "5 methods for generic function <b>f</b>:<ul><li> f(a::<b>Integer</b>,b::<b>Integer</b>) at In[12]:5<li> f(a::<b>Number</b>,b::<b>Number</b>) at In[12]:2<li> f(a::<b>Number</b>,b) at In[12]:3<li> f(a,b::<b>Number</b>) at In[12]:4<li> f(a,b) at In[12]:1</ul>",
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 17,
       "text": "# 5 methods for generic function \"f\":\nf(a::Integer,b::Integer) at In[12]:5\nf(a::Number,b::Number) at In[12]:2\nf(a::Number,b) at In[12]:3\nf(a,b::Number) at In[12]:4\nf(a,b) at In[12]:1"
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": "##Diagonal dispatch"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "f{T<:Number}(a::T, b::T) = \"a and b are both $(T)s\"",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 18,
       "text": "f (generic function with 6 methods)"
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "f(big(1.5),big(2.5))",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 19,
       "text": "\"a and b are both BigFloats\""
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "f(big(1),big(2)) #<== integer rule is more specific",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 20,
       "text": "\"a and b are both integers\""
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "f(\"foo\",\"bar\") #<== still doesn't apply to non-numbers",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 21,
       "text": "\"fallback\""
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": "## Varargs Methods"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "f(args::Number...) = \"$(length(args))-ary heterogeneous call\"\nf{T<:Number}(args::T...) = \"$(length(args))-ary homogeneous call\"",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 22,
       "text": "f (generic function with 8 methods)"
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "f(1)",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 23,
       "text": "\"1-ary homogeneous call\""
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "f(1,2,3)",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 24,
       "text": "\"3-ary homogeneous call\""
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "f(1,1.5,2)",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 25,
       "text": "\"3-ary heterogeneous call\""
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "f() #==> heterogeneous because we can't bind T",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 26,
       "text": "\"0-ary heterogeneous call\""
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "f(1,2) #<== previous 2-arg method is more specific",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 27,
       "text": "\"a and b are both integers\""
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "f(\"foo\") #<== doesn't apply to non-numbers",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "ename": "LoadError",
       "evalue": "no method f(ASCIIString)\nwhile loading In[28], in expression starting on line 1",
       "output_type": "pyerr",
       "traceback": [
        "no method f(ASCIIString)\nwhile loading In[28], in expression starting on line 1"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": "### The Expression Problem\n\nMads Torgersen's paper [The Expression Problem Revisited][2]:\n\n> To which degree can your application be structured in such a way that both the data model and the set of virtual operations over it can be extended without the need to modify existing code, without the need for code repetition and without runtime type errors.\n\n[2]: http://www.daimi.au.dk/~madst/ecoop04/main.pdf\n\nBasically you want to be able to add:\n\n1. **new types** to which you can apply existing operations \u2192 easy in o.o., hard in functional\n2. **new operations** which you can apply to existing types \u2192 easy in functional, hard in o.o."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "module ModInts\nexport ModInt\nimport Base: convert, promote_rule, show, showcompact\n\nimmutable ModInt{n} <: Integer\n    k::Int\n    ModInt(k) = new(k % n)\nend\n\n-{n2}(a::ModInt{n2}) = ModInt{n2}(-a.k)\n+{n}(a::ModInt{n}, b::ModInt{n}) = ModInt{n}(a.k+b.k)\n-{n}(a::ModInt{n}, b::ModInt{n}) = ModInt{n}(a.k-b.k)\n*{n}(a::ModInt{n}, b::ModInt{n}) = ModInt{n}(a.k*b.k)\n\nconvert{n}(::Type{ModInt{n}}, i::Int) = ModInt{n}(i)\npromote_rule{n}(::Type{ModInt{n}}, ::Type{Int}) = ModInt{n}\n\nshow{n}(io::IO, k::ModInt{n}) = print(io, \"$(k.k) mod $n\")\nshowcompact(io::IO, k::ModInt) = print(io, k.k)\n\nend # module",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "outputs": [],
     "prompt_number": 29
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "using ModInts",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [],
     "prompt_number": 30
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "ModInt",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 31,
       "text": "ModInt{n} (constructor with 0 methods)"
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "ModInt{11}",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 32,
       "text": "ModInt{11} (constructor with 1 method)"
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "a=ModInt{11}(12345532425)",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 33,
       "text": "6 mod 11"
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "b=ModInt{11}(23445134156)",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 34,
       "text": "4 mod 11"
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "a+b",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 35,
       "text": "10 mod 11"
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "a+b+b",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 36,
       "text": "3 mod 11"
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "a+1",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 37,
       "text": "7 mod 11"
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "2a",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 38,
       "text": "1 mod 11"
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "2a+1",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 39,
       "text": "2 mod 11"
      }
     ],
     "prompt_number": 39
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A = map(ModInt{11}, rand(1:1000, 5,5))",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 40,
       "text": "5x5 Array{ModInt{11},2}:\n 10  7   3  8  10\n  7  0   7  0   1\n  9  5  10  8   8\n  2  4  10  7   5\n  8  1   1  9   0"
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A^2",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 41,
       "text": "5x5 Array{ModInt{11},2}:\n 8   6   1  8  6\n 9   8   4  0  5\n 9  10   8  5  6\n 5   9   0  3  7\n 4   9  10  3  2"
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A^10000000",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 52,
       "text": "5x5 Array{ModInt{11},2}:\n 4  1  9  8  3\n 8  9  5  8  8\n 9  2  7  6  1\n 3  1  2  3  5\n 1  1  3  2  9"
      }
     ],
     "prompt_number": 52
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "B = map(ModInt{11}, rand(1:1000, 5,5))",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 43,
       "text": "5x5 Array{ModInt{11},2}:\n 2   8  8  5  7\n 1  10  5  8  4\n 0   5  3  6  6\n 1   3  5  3  8\n 1   3  4  7  8"
      }
     ],
     "prompt_number": 43
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A+B",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 44,
       "text": "5x5 Array{ModInt{11},2}:\n 1   4  0   2  6\n 8  10  1   8  5\n 9  10  2   3  3\n 3   7  4  10  2\n 9   4  5   5  8"
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "foo{n}(a::ModInt{n}, b::ModInt{n}) = a^2 + 2b - 1\nfoo{n}(a::ModInt{n}, b::Int) = foo(a, ModInt{n}(b))\nfoo{n}(a::Int, b::ModInt{n}) = foo(ModInt{n}(a), b)\nfoo(a::Int, b::Int) = foo(ModInt{11}(a), ModInt{11}(b)).k",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 45,
       "text": "foo (generic function with 4 methods)"
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "foo(3,4)",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 46,
       "text": "5"
      }
     ],
     "prompt_number": 46
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "code_llvm(foo,(Int,Int))",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\ndefine i64 @julia_foo15929(i64, i64) {\ntop:\n  %2 = srem i64 %0, 11, !dbg !2947\n  %3 = insertvalue %ModInt undef, i64 %2, 0, !dbg !2947, !julia_type !2948\n  %4 = call %ModInt @julia_power_by_squaring15922(%ModInt %3, i64 2), !dbg !2947, !julia_type !2948\n  %5 = extractvalue %ModInt %4, 0, !dbg !2947\n  %6 = srem i64 %1, 11, !dbg !2947\n  %7 = shl i64 %6, 1, !dbg !2947\n  %8 = srem i64 %7, 11, !dbg !2947\n  %9 = add i64 %8, %5, !dbg !2947\n  %10 = srem i64 %9, 11, !dbg !2947\n  %11 = add i64 %10, -1, !dbg !2947\n  %12 = srem i64 %11, 11, !dbg !2947\n  ret i64 %12, !dbg !2947\n}\n"
      }
     ],
     "prompt_number": 47
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "using Gadfly\n@plot(cos(x)/x, 5, 25)",
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "outputs": [
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bbox.x) / bbox.width;\n        var root = getplotroot(d3.select(this));\n        var new_scale;\n        if (xclick < xpos) {\n            new_scale = Math.max(MIN_SCALE, ctx.scale / 1.5);\n            zoom_step(root, ctx, new_scale);\n        }\n        else {\n            new_scale = Math.min(MAX_SCALE, ctx.scale * 1.5);\n            zoom_step(root, ctx, new_scale);\n        }\n        d3.event.stopPropagation();\n    });\n};\n\n\n// Mouseover effects for zoom slider\nvar zoomslider_button_mouseover = function(destcolor) {\n    return (function() {\n        d3.select(this)\n          .selectAll(\".button_logo\")\n          .transition()\n          .duration(150)\n          .attr(\"fill\", destcolor);\n    });\n};\n\n\nvar zoomslider_thumb_mouseover = function(destcolor) {\n    return (function() {\n        d3.select(this)\n          .transition()\n          .duration(150)\n          .attr(\"fill\", destcolor);\n    });\n};\n\n//@ sourceURL=gadfly.js\n</script>",
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     "source": "##Acknowledgements\n\n**Stefan Karpinski**  for inspiration for most of this presentation\n\n**Stefan, with Jeff Bezanson, Viral B Shah and Alan Edelman** for creating Julia \n\n**Daniel C Jones** for creating Gadfly"
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