Workshop1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Stochastic Differential Equations, Quantum Phase Space, and Julia\n",
    "\n",
    "$$\n",
    "\\def\\julia{\\texttt{julia}}\n",
    "\\def\\dashint{{\\int\\!\\!\\!\\!\\!\\!-\\,}}\n",
    "\\def\\infdashint{\\dashint_{\\!\\!\\!-\\infty}^{\\,\\infty}}\n",
    "\\def\\D{\\,{\\rm d}}\n",
    "\\def\\E{{\\rm e}}\n",
    "\\def\\dx{\\D x}\n",
    "\\def\\dt{\\D t}\n",
    "\\def\\dz{\\D z}\n",
    "\\def\\C{{\\mathbb C}}\n",
    "\\def\\R{{\\mathbb R}}\n",
    "\\def\\CC{{\\cal C}}\n",
    "\\def\\HH{{\\cal H}}\n",
    "\\def\\I{{\\rm i}}\n",
    "\\def\\qqqquad{\\qquad\\qquad}\n",
    "\\def\\qqfor{\\qquad\\hbox{for}\\qquad}\n",
    "\\def\\qqwhere{\\qquad\\hbox{where}\\qquad}\n",
    "\\def\\Res_#1{\\underset{#1}{\\rm Res}}\\,\n",
    "\\def\\sech{{\\rm sech}\\,}\n",
    "\\def\\vc#1{{\\mathbf #1}}\n",
    "$$\n",
    "\n",
    "Dr. Ashton Bradley\n",
    "<br>\n",
    "ashton.bradley@otago.ac.nz\n",
    "<br>\n",
    "http://amoqt.otago.ac.nz"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Intro to julia"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This laboratory will serve as an introduction to julia, and provide some essential background for stochastic processes.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Workshop 1 (1.5 hours): $\\julia$ in a nutshell\n",
    "Getting started in $\\julia$: the basics\n",
    "\n",
    "- Familiarity with $\\julia$ computing environments: REPL, Jupyter, Juno\n",
    "- Variables, functions \n",
    "- Using packages and plotting\n",
    "- How does $\\julia$ solve the two-language problem?\n",
    "\n",
    "References\n",
    "\n",
    "- [julialang.org](https://julialang.org)\n",
    "- [juliaobserver](https://juliaobserver.com)\n",
    "- *Julia: A Fresh Approach to Numerical Computing*, Jeff Bezanson, Alan Edelman, Stefan Karpinski and Viral B. Shah [SIAM Review, __59__, 65–98 (2017)](http://julialang.org/publications/julia-fresh-approach-BEKS.pdf)\n",
    "- [Fun and sometimes useful julia packages](http://amoqt.otago.ac.nz/resources/juliajam)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Motivation: the two-language problem\n",
    "Why another language? \n",
    "\n",
    "- Compiled languages\n",
    "    - Fast execution\n",
    "    - Slow to read/write/debug\n",
    "    - Tend to get isolated from latest conceptual developments \n",
    "- Interpreted languages \n",
    "    - Slow exection\n",
    "    - Easy to read/write/debug\n",
    "    - Not well suited for high performance computing\n",
    "- This is a ___major___ issue in technical computing. \n",
    "- This conflict motivated the development of $\\julia$ at MIT (1.0 released 2018). \n",
    "- [next big thing?](https://www.techrepublic.com/article/is-julia-the-next-big-programming-language-mit-thinks-so-as-version-1-0-lands/)\n",
    "- [Julia joined the \"petaflop club\" in 2017](https://www.hpcwire.com/off-the-wire/julia-joins-petaflop-club/), the only high level scripting language ever to do so."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Julia microbenchmarks\n",
    "[https://julialang.org/benchmarks/](https://julialang.org/benchmarks/)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"media/juliabench.png\" width=\"600\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"media/packages.png\" width=\"600\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Julia observer\n",
    "\n",
    "An up to date package listing is provided at [juliaobserver.com](https://juliaobserver.com/packages)\n",
    "\n",
    "## Language \n",
    "Julia is powered by several unique language innovations that developers love, and I think you will too once you get to know a bit more about them:\n",
    "\n",
    "- Multiple dispatch: methods are not owned by classes.\n",
    "- Duck typing: you can define your own data types by their properties; they translate across all of julia \n",
    "- Different julia modules can call each other without any loss of performance\n",
    "\n",
    "Coming from a different language, you may find a list of [noteworthy differences to other languages](https://docs.julialang.org/en/v0.7.0/manual/noteworthy-differences/)  useful."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# $\\julia$ environments"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Read, Evaluate, Print, Loop (REPL)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"media/repl.png\" width=\"500\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Package manager"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To access Pkg mode in the REPL, type\n",
    "```julia\n",
    "julia> ]\n",
    "```\n",
    "which will give you the prompt\n",
    "\n",
    "```julia\n",
    "(@v1.4) pkg>\n",
    "```\n",
    "To add a package\n",
    "\n",
    "```julia\n",
    "(@v1.4) pkg> add IJulia \n",
    "```\n",
    "Then type backspace to go back to REPL mode. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that REPL commands are just $\\julia$ commands, so julia REPL and jupyter are identical, except that the Pkg mode can't be accessed in jupyter."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[32m\u001b[1mStatus\u001b[22m\u001b[39m `~/.julia/environments/v1.4/Project.toml`\n",
      " \u001b[90m [537997a7]\u001b[39m\u001b[37m AbstractPlotting v0.10.1\u001b[39m\n",
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      " \u001b[90m [f013a474]\u001b[39m\u001b[37m FourierGPE v0.1.0 #master (https://github.com/AshtonSBradley/FourierGPE.jl.git)\u001b[39m\n",
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      " \u001b[90m [0e21426a]\u001b[39m\u001b[37m PhaseSpaceTools v0.1.0 #master (https://github.com/AshtonSBradley/PhaseSpaceTools.jl.git)\u001b[39m\n",
      " \u001b[90m [5ad8b20f]\u001b[39m\u001b[37m PhysicalConstants v0.2.0\u001b[39m\n",
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      " \u001b[90m [37e2e46d]\u001b[39m\u001b[37m LinearAlgebra \u001b[39m\n",
      " \u001b[90m [de0858da]\u001b[39m\u001b[37m Printf \u001b[39m\n",
      " \u001b[90m [2f01184e]\u001b[39m\u001b[37m SparseArrays \u001b[39m\n",
      " \u001b[90m [10745b16]\u001b[39m\u001b[37m Statistics \u001b[39m\n"
     ]
    }
   ],
   "source": [
    "#how to execute Pkg commands in jupyter\n",
    "using Pkg\n",
    "pkg\"status\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
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      "\u001b[90m [no changes]\u001b[39m\n",
      "\u001b[32m\u001b[1m  Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m   Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.4/Project.toml`\n",
      "\u001b[90m [no changes]\u001b[39m\n",
      "\u001b[32m\u001b[1m   Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.4/Manifest.toml`\n",
      "\u001b[90m [no changes]\u001b[39m\n",
      "\u001b[32m\u001b[1m  Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m   Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.4/Project.toml`\n",
      "\u001b[90m [no changes]\u001b[39m\n",
      "\u001b[32m\u001b[1m   Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.4/Manifest.toml`\n",
      "\u001b[90m [no changes]\u001b[39m\n"
     ]
    }
   ],
   "source": [
    "# let's add all packages for these workshops\n",
    "pkgs = [:Plots,:DifferentialEquations,:GSL, :SpecialFunctions,\n",
    "    :Statistics, :Plots, :LaTeXStrings, :Revise,:SymPy,:BenchmarkTools]\n",
    "\n",
    "for pkg in pkgs\n",
    "    Pkg.add(String(pkg))\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To load a package\n",
    "```julia\n",
    "julia> using IJulia\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`IJulia` provides the jupyter environment for julia. \n",
    "    \n",
    "To launch jupyter, enter this command in the REPL:\n",
    "```julia\n",
    "julia> notebook()\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You should now have access to the lectures as jupyter notebooks."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Jupyter\n",
    "Web browser front end calls a $\\julia$ kernel, evaluates, returns result; quite low overhead."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Plots.GRBackend()"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "gr(titlefontsize=12,grid=false,legend=false,size=(600,200)) # set defaults for GR backend"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = LinRange(0,10,500);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
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       "  1974,110.532 1978.03,113.74 1982.06,117.048 1986.09,120.453 1990.12,123.955 1994.15,127.551 1998.18,131.241 2002.21,135.023 2006.25,138.896 2010.28,142.857 \n",
       "  2014.31,146.906 2018.34,151.041 2022.37,155.259 2026.4,159.56 2030.43,163.942 2034.46,168.402 2038.49,172.939 2042.52,177.552 2046.55,182.238 2050.58,186.996 \n",
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     "execution_count": 8,
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     "output_type": "execute_result"
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   ],
   "source": [
    "# first plot will take a few seconds\n",
    "plot(x,sin.(x)) \n",
    "xlabel!(\"x\");ylabel!(\"sin(x)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Dot-calls: (large) extension of Matlab's local array operation syntax.\n",
    "- `xlabel!()`: `!` at the end of the function name is a julia convention that declares: \"modifies the input\", also referred to as a ___mutating function___; here the input is the current plot which doesn't neet to be declared. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
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       "  1087.27,236.483 1091.3,242.304 1095.33,248.077 1099.36,253.794 1103.39,259.444 1107.42,265.02 1111.45,270.511 1115.48,275.909 1119.51,281.206 1123.54,286.393 \n",
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       "  1651.55,140.568 1655.58,145.791 1659.61,151.121 1663.64,156.55 1667.67,162.069 1671.71,167.67 1675.74,173.342 1679.77,179.078 1683.8,184.868 1687.83,190.702 \n",
       "  1691.86,196.572 1695.89,202.468 1699.92,208.379 1703.95,214.298 1707.98,220.214 1712.01,226.118 1716.04,232 1720.07,237.852 1724.1,243.662 1728.13,249.423 \n",
       "  1732.16,255.125 1736.19,260.759 1740.23,266.315 1744.26,271.786 1748.29,277.161 1752.32,282.433 1756.35,287.592 1760.38,292.631 1764.41,297.542 1768.44,302.316 \n",
       "  1772.47,306.947 1776.5,311.426 1780.53,315.746 1784.56,319.901 1788.59,323.883 1792.62,327.687 1796.65,331.306 1800.68,334.734 1804.72,337.967 1808.75,340.998 \n",
       "  1812.78,343.823 1816.81,346.437 1820.84,348.836 1824.87,351.017 1828.9,352.975 1832.93,354.708 1836.96,356.213 1840.99,357.487 1845.02,358.529 1849.05,359.336 \n",
       "  1853.08,359.908 1857.11,360.243 1861.14,360.341 1865.17,360.202 1869.2,359.826 1873.24,359.214 1877.27,358.366 1881.3,357.284 1885.33,355.97 1889.36,354.425 \n",
       "  1893.39,352.653 1897.42,350.656 1901.45,348.438 1905.48,346.001 1909.51,343.35 1913.54,340.489 1917.57,337.423 1921.6,334.156 1925.63,330.695 1929.66,327.043 \n",
       "  1933.69,323.208 1937.72,319.196 1941.76,315.012 1945.79,310.664 1949.82,306.158 1953.85,301.502 1957.88,296.704 1961.91,291.77 1965.94,286.71 1969.97,281.53 \n",
       "  1974,276.24 1978.03,270.848 1982.06,265.362 1986.09,259.792 1990.12,254.146 1994.15,248.433 1998.18,242.663 2002.21,236.844 2006.25,230.987 2010.28,225.1 \n",
       "  2014.31,219.194 2018.34,213.276 2022.37,207.358 2026.4,201.448 2030.43,195.557 2034.46,189.692 2038.49,183.865 2042.52,178.084 2046.55,172.358 2050.58,166.697 \n",
       "  2054.61,161.11 2058.64,155.606 2062.67,150.194 2066.7,144.881 2070.73,139.678 2074.77,134.592 2078.8,129.631 2082.83,124.803 2086.86,120.117 2090.89,115.579 \n",
       "  2094.92,111.197 2098.95,106.978 2102.98,102.929 2107.01,99.0563 2111.04,95.3658 2115.07,91.8638 2119.1,88.5558 2123.13,85.4471 2127.16,82.5427 2131.19,79.8473 \n",
       "  2135.22,77.3652 2139.26,75.1005 2143.29,73.0566 2147.32,71.237 2151.35,69.6445 2155.38,68.2818 2159.41,67.1509 2163.44,66.2537 2167.47,65.5917 2171.5,65.1659 \n",
       "  2175.53,64.977 2179.56,65.0253 2183.59,65.3107 2187.62,65.8328 2191.65,66.5907 2195.68,67.5832 2199.71,68.8088 2203.74,70.2653 2207.78,71.9506 2211.81,73.8619 \n",
       "  2215.84,75.9961 2219.87,78.3498 2223.9,80.9193 2227.93,83.7003 2231.96,86.6884 2235.99,89.8789 2240.02,93.2666 2244.05,96.846 2248.08,100.611 2252.11,104.557 \n",
       "  2256.14,108.676 2260.17,112.962 2264.2,117.408 2268.23,122.007 2272.26,126.752 2276.3,131.635 2280.33,136.648 2284.36,141.783 2288.39,147.031 2292.42,152.386 \n",
       "  \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot!(x,cos.(x).^2,c=:blue,fill=(0,.3,:red))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "-2.185039863261519"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z=2; println(z);tan(z) #more than one command on same line"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "hello world, this is julia. It's a bit like Matlab, Python, R, C, Fortran, Ruby, Haskell, ...\n",
      "    but not really like any other language\n"
     ]
    }
   ],
   "source": [
    "println(\"hello world, this is julia. It's a bit like Matlab, Python, R, C, Fortran, Ruby, Haskell, ...\n",
    "    but not really like any other language\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Calling other languages"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "from julia you can also easily use  libraries from \n",
      "    Python, R, C/Fortran, C++, and Java. \n",
      "    Did I mention that I like π ?\n"
     ]
    }
   ],
   "source": [
    "println(\"from julia you can also easily use  libraries from \n",
    "    Python, R, C/Fortran, C++, and Java. \n",
    "    Did I mention that I like \", pi ,\" ?\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For example, a number of special functions can be called via the C library GSL ([GNU Scientific Library](https://github.com/JuliaMath/GSL.jl)):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "#load some wrapper functions that call GSL\n",
    "using GSL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9776262465382961"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sf_bessel_J0(0.3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A native $\\julia$ implementation of many special functions can be found in [SpecialFunctions.jl](https://github.com/JuliaMath/SpecialFunctions.jl)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "using SpecialFunctions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.2020569031595951"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zeta(3) # returns a Float64"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.7980219851462758 - 0.11374430805293854im"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zeta(2.0 + 3.0im) # returns a complex float"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Complex{Float64}"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "typeof(ans)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can also use [PyCall.jl](https://github.com/JuliaPy/PyCall.jl) to call `python` functions with low overhead:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Precompiling PyCall [438e738f-606a-5dbb-bf0a-cddfbfd45ab0]\n",
      "└ @ Base loading.jl:1260\n"
     ]
    }
   ],
   "source": [
    "using PyCall"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pymath = pyimport(\"math\")\n",
    "pymath.sin(pymath.pi / 4) - sin(pi / 4)  # returns 0.0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- A nice feature of jupyter is that we can enter text, latex, and code, all in one easily shareable document. \n",
    "- Open Science! Yay! There are [many other kernels](https://github.com/jupyter/jupyter/wiki/Jupyter-kernels) available for jupyter.\n",
    "- The importance of such an open science standard is highlighted by major scientific discovery of LIGO: the black hole merger data, simulations, and analysis were provided as [jupyter notebooks](https://blog.jupyter.org/congratulations-to-the-ligo-and-virgo-collaborations-from-project-jupyter-5923247be019)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- To shift from ___code mode___ to ___markdown mode___, press `ESC`, then `y`.\n",
    "\n",
    "- You can now type text with simple [markdown](https://www.markdownguide.org/cheat-sheet/) syntax, LaTeX in-line $\\langle x\\rangle=\\pi$, and full line equations:\n",
    "$$i\\hbar\\frac{\\partial\\psi}{\\partial t}=\\hat{H}\\psi,$$\n",
    "add [links](http://www.google.com), and drag in images. Not all of latex is supported, but for example, you can\n",
    "\n",
    "\\begin{align}\n",
    "\\partial_t\\psi&=L\\psi\\tag{some equation}\\\\\n",
    "-\\nabla^2\\psi&=\\lambda\\psi\\tag{some other}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- To go back to code mode, press `ESC`, then `m`; default is code mode. \n",
    "\n",
    "- A full list of keyboard shortcuts can be found under the help menu above."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Cool things you can do in julia"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "35.6"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "😸=35.6 #unicode inputs are allowed varables!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "25"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ψ=25 #LaTeX tab completion: type \\psi<tab> to get the greek letter ψ."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0545718e-34"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ħ = 1.0545718e-34 #\\hbar<tab>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405131996"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#variable precision: set the number of bits\n",
    "BigFloat(pi,2000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0 + 4.582363384086222409168621731742190695845080541202039763581730143804801597402798e-603im"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp(im*BigFloat(pi,2000))+1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "BigFloat"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "typeof(BigFloat(pi))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here \"BigFloat\" is both \n",
    "\n",
    "- a method to perform type conversion, and \n",
    "- the name of a data type. \n",
    "\n",
    "These two things are defined separately."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.14159265358979323851"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mypi = BigFloat(pi,64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "typeof(mypi) <: AbstractFloat"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is an abstract type system that you can extend as you like."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "search: \u001b[0m\u001b[1mB\u001b[22m\u001b[0m\u001b[1mi\u001b[22m\u001b[0m\u001b[1mg\u001b[22m\u001b[0m\u001b[1mF\u001b[22m\u001b[0m\u001b[1ml\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ma\u001b[22m\u001b[0m\u001b[1mt\u001b[22m\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "\\begin{verbatim}\n",
       "BigFloat <: AbstractFloat\n",
       "\\end{verbatim}\n",
       "Arbitrary precision floating point number type.\n",
       "\n",
       "\\rule{\\textwidth}{1pt}\n",
       "\\begin{verbatim}\n",
       "BigFloat(x)\n",
       "\\end{verbatim}\n",
       "Create an arbitrary precision floating point number. \\texttt{x} may be an \\href{@ref}{\\texttt{Integer}}, a \\href{@ref}{\\texttt{Float64}} or a \\href{@ref}{\\texttt{BigInt}}. The usual mathematical operators are defined for this type, and results are promoted to a \\href{@ref}{\\texttt{BigFloat}}.\n",
       "\n",
       "Note that because decimal literals are converted to floating point numbers when parsed, \\texttt{BigFloat(2.1)} may not yield what you expect. You may instead prefer to initialize constants from strings via \\href{@ref}{\\texttt{parse}}, or using the \\texttt{big} string literal.\n",
       "\n",
       "\\begin{verbatim}\n",
       "julia> BigFloat(2.1)\n",
       "2.100000000000000088817841970012523233890533447265625\n",
       "\n",
       "julia> big\"2.1\"\n",
       "2.099999999999999999999999999999999999999999999999999999999999999999999999999986\n",
       "\\end{verbatim}\n",
       "\\rule{\\textwidth}{1pt}\n",
       "\\begin{verbatim}\n",
       "BigFloat(x, prec::Int)\n",
       "\\end{verbatim}\n",
       "Create a representation of \\texttt{x} as a \\href{@ref}{\\texttt{BigFloat}} with precision \\texttt{prec}.\n",
       "\n",
       "\\rule{\\textwidth}{1pt}\n",
       "\\begin{verbatim}\n",
       "BigFloat(x, prec::Int, rounding::RoundingMode)\n",
       "\\end{verbatim}\n",
       "Create a representation of \\texttt{x} as a \\href{@ref}{\\texttt{BigFloat}} with precision \\texttt{prec} and \\href{@ref Base.Rounding.RoundingMode}{\\texttt{Rounding Mode}} \\texttt{rounding}.\n",
       "\n",
       "\\rule{\\textwidth}{1pt}\n",
       "\\begin{verbatim}\n",
       "BigFloat(x, rounding::RoundingMode)\n",
       "\\end{verbatim}\n",
       "Create a representation of \\texttt{x} as a \\href{@ref}{\\texttt{BigFloat}} with the current global precision and \\href{@ref Base.Rounding.RoundingMode}{\\texttt{Rounding Mode}} \\texttt{rounding}.\n",
       "\n",
       "\\rule{\\textwidth}{1pt}\n",
       "\\begin{verbatim}\n",
       "BigFloat(x::String)\n",
       "\\end{verbatim}\n",
       "Create a representation of the string \\texttt{x} as a \\href{@ref}{\\texttt{BigFloat}}.\n",
       "\n"
      ],
      "text/markdown": [
       "```\n",
       "BigFloat <: AbstractFloat\n",
       "```\n",
       "\n",
       "Arbitrary precision floating point number type.\n",
       "\n",
       "---\n",
       "\n",
       "```\n",
       "BigFloat(x)\n",
       "```\n",
       "\n",
       "Create an arbitrary precision floating point number. `x` may be an [`Integer`](@ref), a [`Float64`](@ref) or a [`BigInt`](@ref). The usual mathematical operators are defined for this type, and results are promoted to a [`BigFloat`](@ref).\n",
       "\n",
       "Note that because decimal literals are converted to floating point numbers when parsed, `BigFloat(2.1)` may not yield what you expect. You may instead prefer to initialize constants from strings via [`parse`](@ref), or using the `big` string literal.\n",
       "\n",
       "```jldoctest\n",
       "julia> BigFloat(2.1)\n",
       "2.100000000000000088817841970012523233890533447265625\n",
       "\n",
       "julia> big\"2.1\"\n",
       "2.099999999999999999999999999999999999999999999999999999999999999999999999999986\n",
       "```\n",
       "\n",
       "---\n",
       "\n",
       "```\n",
       "BigFloat(x, prec::Int)\n",
       "```\n",
       "\n",
       "Create a representation of `x` as a [`BigFloat`](@ref) with precision `prec`.\n",
       "\n",
       "---\n",
       "\n",
       "```\n",
       "BigFloat(x, prec::Int, rounding::RoundingMode)\n",
       "```\n",
       "\n",
       "Create a representation of `x` as a [`BigFloat`](@ref) with precision `prec` and [`Rounding Mode`](@ref Base.Rounding.RoundingMode) `rounding`.\n",
       "\n",
       "---\n",
       "\n",
       "```\n",
       "BigFloat(x, rounding::RoundingMode)\n",
       "```\n",
       "\n",
       "Create a representation of `x` as a [`BigFloat`](@ref) with the current global precision and [`Rounding Mode`](@ref Base.Rounding.RoundingMode) `rounding`.\n",
       "\n",
       "---\n",
       "\n",
       "```\n",
       "BigFloat(x::String)\n",
       "```\n",
       "\n",
       "Create a representation of the string `x` as a [`BigFloat`](@ref).\n"
      ],
      "text/plain": [
       "\u001b[36m  BigFloat <: AbstractFloat\u001b[39m\n",
       "\n",
       "  Arbitrary precision floating point number type.\n",
       "\n",
       "  ────────────────────────────────────────────────────────────────────────────\n",
       "\n",
       "\u001b[36m  BigFloat(x)\u001b[39m\n",
       "\n",
       "  Create an arbitrary precision floating point number. \u001b[36mx\u001b[39m may be an \u001b[36mInteger\u001b[39m, a\n",
       "  \u001b[36mFloat64\u001b[39m or a \u001b[36mBigInt\u001b[39m. The usual mathematical operators are defined for this\n",
       "  type, and results are promoted to a \u001b[36mBigFloat\u001b[39m.\n",
       "\n",
       "  Note that because decimal literals are converted to floating point numbers\n",
       "  when parsed, \u001b[36mBigFloat(2.1)\u001b[39m may not yield what you expect. You may instead\n",
       "  prefer to initialize constants from strings via \u001b[36mparse\u001b[39m, or using the \u001b[36mbig\u001b[39m\n",
       "  string literal.\n",
       "\n",
       "\u001b[36m  julia> BigFloat(2.1)\u001b[39m\n",
       "\u001b[36m  2.100000000000000088817841970012523233890533447265625\u001b[39m\n",
       "\u001b[36m  \u001b[39m\n",
       "\u001b[36m  julia> big\"2.1\"\u001b[39m\n",
       "\u001b[36m  2.099999999999999999999999999999999999999999999999999999999999999999999999999986\u001b[39m\n",
       "\n",
       "  ────────────────────────────────────────────────────────────────────────────\n",
       "\n",
       "\u001b[36m  BigFloat(x, prec::Int)\u001b[39m\n",
       "\n",
       "  Create a representation of \u001b[36mx\u001b[39m as a \u001b[36mBigFloat\u001b[39m with precision \u001b[36mprec\u001b[39m.\n",
       "\n",
       "  ────────────────────────────────────────────────────────────────────────────\n",
       "\n",
       "\u001b[36m  BigFloat(x, prec::Int, rounding::RoundingMode)\u001b[39m\n",
       "\n",
       "  Create a representation of \u001b[36mx\u001b[39m as a \u001b[36mBigFloat\u001b[39m with precision \u001b[36mprec\u001b[39m and \u001b[36mRounding\n",
       "  Mode\u001b[39m \u001b[36mrounding\u001b[39m.\n",
       "\n",
       "  ────────────────────────────────────────────────────────────────────────────\n",
       "\n",
       "\u001b[36m  BigFloat(x, rounding::RoundingMode)\u001b[39m\n",
       "\n",
       "  Create a representation of \u001b[36mx\u001b[39m as a \u001b[36mBigFloat\u001b[39m with the current global precision\n",
       "  and \u001b[36mRounding Mode\u001b[39m \u001b[36mrounding\u001b[39m.\n",
       "\n",
       "  ────────────────────────────────────────────────────────────────────────────\n",
       "\n",
       "\u001b[36m  BigFloat(x::String)\u001b[39m\n",
       "\n",
       "  Create a representation of the string \u001b[36mx\u001b[39m as a \u001b[36mBigFloat\u001b[39m."
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "?BigFloat "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Arrays and functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6-element Array{Float64,1}:\n",
       "  1.05998177182784  \n",
       "  0.9163974274017533\n",
       " -1.0175765184304009\n",
       "  0.1651457938104572\n",
       "  0.8574853490976208\n",
       " -0.6715652549844074"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = randn(6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9163974274017533"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x[2] # accessing an element of an array: [] vs ()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "ename": "MethodError",
     "evalue": "MethodError: no method matching ^(::Array{Float64,1}, ::Int64)\nClosest candidates are:\n  ^(!Matched::Float16, ::Integer) at math.jl:795\n  ^(!Matched::Missing, ::Integer) at missing.jl:120\n  ^(!Matched::Missing, ::Number) at missing.jl:93\n  ...",
     "output_type": "error",
     "traceback": [
      "MethodError: no method matching ^(::Array{Float64,1}, ::Int64)\nClosest candidates are:\n  ^(!Matched::Float16, ::Integer) at math.jl:795\n  ^(!Matched::Missing, ::Integer) at missing.jl:120\n  ^(!Matched::Missing, ::Number) at missing.jl:93\n  ...",
      "",
      "Stacktrace:",
      " [1] macro expansion at ./none:0 [inlined]",
      " [2] literal_pow(::typeof(^), ::Array{Float64,1}, ::Val{2}) at ./none:0",
      " [3] top-level scope at In[29]:1"
     ]
    }
   ],
   "source": [
    "x^2 #a typicaly method error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6-element Array{Float64,1}:\n",
       " 1.1235613566072868 \n",
       " 0.8397842449485518 \n",
       " 1.035461970860936  \n",
       " 0.02727313321328604\n",
       " 0.7352811239170686 \n",
       " 0.45099989170227217"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.^2 # a dot-call: element-wise operations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "f (generic function with 1 method)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# let's define a function (this is one way)\n",
    "f(x) = x^3-x+pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6-element Array{Float64,1}:\n",
       " 3.2725654392958363\n",
       " 2.9947713478314166\n",
       " 3.1055073847444414\n",
       " 2.9809509030135426\n",
       " 2.914600095719091 \n",
       " 3.510282051305224 "
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.(x) #dot-call"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "same result without passing to a function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6-element Array{Float64,1}:\n",
       " 3.2725654392958363\n",
       " 2.9947713478314166\n",
       " 3.1055073847444414\n",
       " 2.9809509030135426\n",
       " 2.914600095719091 \n",
       " 3.510282051305224 "
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.^3 .-x.+pi #manually dot all operations that will act on arrays element by element"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6-element Array{Float64,1}:\n",
       " 3.2725654392958363\n",
       " 2.9947713478314166\n",
       " 3.1055073847444414\n",
       " 2.9809509030135426\n",
       " 2.914600095719091 \n",
       " 3.510282051305224 "
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@. x^3-x+pi # the macro @. will dot-the-lot."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By dotting the function call, `f.(x)`, the dot is applied to all operations."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Two-dimensional example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "f2 (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f2(x,y) = x^3-y+pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6-element Array{Float64,1}:\n",
       "  1.05998177182784  \n",
       "  0.9163974274017533\n",
       " -1.0175765184304009\n",
       "  0.1651457938104572\n",
       "  0.8574853490976208\n",
       " -0.6715652549844074"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "ename": "MethodError",
     "evalue": "MethodError: no method matching ^(::Array{Float64,1}, ::Int64)\nClosest candidates are:\n  ^(!Matched::Float16, ::Integer) at math.jl:795\n  ^(!Matched::Missing, ::Integer) at missing.jl:120\n  ^(!Matched::Missing, ::Number) at missing.jl:93\n  ...",
     "output_type": "error",
     "traceback": [
      "MethodError: no method matching ^(::Array{Float64,1}, ::Int64)\nClosest candidates are:\n  ^(!Matched::Float16, ::Integer) at math.jl:795\n  ^(!Matched::Missing, ::Integer) at missing.jl:120\n  ^(!Matched::Missing, ::Number) at missing.jl:93\n  ...",
      "",
      "Stacktrace:",
      " [1] macro expansion at ./none:0 [inlined]",
      " [2] literal_pow at ./none:0 [inlined]",
      " [3] f2(::Array{Float64,1}, ::Array{Float64,1}) at ./In[35]:1",
      " [4] top-level scope at In[37]:1"
     ]
    }
   ],
   "source": [
    "f2(x,y) #(oops!)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6-element Array{Float64,1}:\n",
       " 3.2725654392958363\n",
       " 2.9947713478314166\n",
       " 3.1055073847444414\n",
       " 2.9809509030135426\n",
       " 2.914600095719091 \n",
       " 3.510282051305224 "
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f2.(x,y) #acts on each element of column vectors x, y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6×6 Array{Float64,2}:\n",
       " 3.27257  3.41615  5.35012  4.1674   3.47506  5.00411\n",
       " 2.85119  2.99477  4.92875  3.74602  3.05368  4.58273\n",
       " 1.02795  1.17153  3.10551  1.92279  1.23045  2.7595 \n",
       " 2.08611  2.2297   4.16367  2.98095  2.28861  3.81766\n",
       " 2.7121   2.85569  4.78966  3.60694  2.9146   4.44365\n",
       " 1.77874  1.92232  3.85629  2.67357  1.98123  3.51028"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f2.(x,y') #broadcasts each element of x along y, and of y along x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is a beautiful feature of $\\julia$ due in large part to Steven Johnson (MIT, of FFTW fame). It goes quite deeply into the language, allowing you to create \"fused kernels\", compiling many dot-calls together into a single highly efficient operation acting locally in memory. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Memory and function calls\n",
    "It is worth emphasizing at this point that one of the great advantages of $\\julia$ is that it allows low-level access to memory, giving pathways to overcome one of the main bottleknecks of interpreted languages. A major deficiency  of other high-level languages such as Matlab is that function calls are slowed down by temporary memory allocations. \n",
    "\n",
    "## Writing in-place\n",
    "If we want to do a calculation and write the output in-place in memory"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using BenchmarkTools "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "x=randn(10000);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "#let's pre-allocate somwhere to write the output\n",
    "z = f2.(x,x');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  591.136 ms (5 allocations: 762.94 MiB)\n"
     ]
    }
   ],
   "source": [
    "# do many calls, and return average execution time, and memory allocations\n",
    "@btime z = f2.(x,x');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So far this is an allocating call, requiring a temporary memory address for the result, before finally writing the result to `z`. \n",
    "\n",
    "Now let's write the result directly to memory, the $\\julia$ way:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  90.926 ms (3 allocations: 64 bytes)\n"
     ]
    }
   ],
   "source": [
    "@btime z .= f2.(x,x');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The calculation is basically allocation free, with a significant speed up. In ambitious physics simulations, this can translate into much larger gains in execution speed."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is probably a good time to point out that \n",
    "\n",
    "- In $\\julia$ there is no speed advantage to \"vectorizing\" your code. \n",
    "- If you prefer to write your own loops, you will get native C speed in $\\julia$.\n",
    "- By using dot-calls, you will get you very close to the same speed, while maintaining the clarity of vectorized code. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Loops, iterators"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "j is 1\n",
      "j is 2\n",
      "j is 3\n",
      "j is 4\n",
      "j is 5\n",
      "j is 6\n",
      "j is 7\n",
      "j is 8\n",
      "j is 9\n",
      "j is 10\n"
     ]
    }
   ],
   "source": [
    "for j=1:10\n",
    "    println(\"j is $j\")\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1:30"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# an interator\n",
    "n = 1:30"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "UnitRange{Int64}"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "typeof(n) #not an Array"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sumton (generic function with 1 method)"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# a simple function with type conversion using redirection\n",
    "sumton(n) = n*(n+1)/2 |> Int #or, better: |> typeof(n) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sumton(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "ename": "InexactError",
     "evalue": "InexactError: Int64(Int64, 6.3549999999999995)",
     "output_type": "error",
     "traceback": [
      "InexactError: Int64(Int64, 6.3549999999999995)",
      "",
      "Stacktrace:",
      " [1] Type at ./float.jl:700 [inlined]",
      " [2] |> at ./operators.jl:813 [inlined]",
      " [3] sumton(::Float64) at ./In[48]:2",
      " [4] top-level scope at In[50]:1"
     ]
    }
   ],
   "source": [
    "sumton(3.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sumton (generic function with 1 method)"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# a simple function with type conversion using redirection\n",
    "sumton(n) = n*(n+1)/2 |> typeof(n) #or, better: |> typeof(n) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sum up to j: j = 1; julia gives 1; Euler gives 1\n",
      "sum up to j: j = 2; julia gives 3; Euler gives 3\n",
      "sum up to j: j = 3; julia gives 6; Euler gives 6\n",
      "sum up to j: j = 4; julia gives 10; Euler gives 10\n",
      "sum up to j: j = 5; julia gives 15; Euler gives 15\n",
      "sum up to j: j = 6; julia gives 21; Euler gives 21\n",
      "sum up to j: j = 7; julia gives 28; Euler gives 28\n",
      "sum up to j: j = 8; julia gives 36; Euler gives 36\n",
      "sum up to j: j = 9; julia gives 45; Euler gives 45\n",
      "sum up to j: j = 10; julia gives 55; Euler gives 55\n",
      "sum up to j: j = 11; julia gives 66; Euler gives 66\n",
      "sum up to j: j = 12; julia gives 78; Euler gives 78\n",
      "sum up to j: j = 13; julia gives 91; Euler gives 91\n",
      "sum up to j: j = 14; julia gives 105; Euler gives 105\n",
      "sum up to j: j = 15; julia gives 120; Euler gives 120\n",
      "sum up to j: j = 16; julia gives 136; Euler gives 136\n",
      "sum up to j: j = 17; julia gives 153; Euler gives 153\n",
      "sum up to j: j = 18; julia gives 171; Euler gives 171\n",
      "sum up to j: j = 19; julia gives 190; Euler gives 190\n",
      "sum up to j: j = 20; julia gives 210; Euler gives 210\n",
      "sum up to j: j = 21; julia gives 231; Euler gives 231\n",
      "sum up to j: j = 22; julia gives 253; Euler gives 253\n",
      "sum up to j: j = 23; julia gives 276; Euler gives 276\n",
      "sum up to j: j = 24; julia gives 300; Euler gives 300\n",
      "sum up to j: j = 25; julia gives 325; Euler gives 325\n",
      "sum up to j: j = 26; julia gives 351; Euler gives 351\n",
      "sum up to j: j = 27; julia gives 378; Euler gives 378\n",
      "sum up to j: j = 28; julia gives 406; Euler gives 406\n",
      "sum up to j: j = 29; julia gives 435; Euler gives 435\n",
      "sum up to j: j = 30; julia gives 465; Euler gives 465\n"
     ]
    }
   ],
   "source": [
    "# a for loop with string inerpolation\n",
    "for j in n\n",
    "    println(\"sum up to j: j = $j; julia gives $(sum(n[1:j])); Gauss gives $(sumton(j))\")\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Geometric series\n",
    "Let's make a little function to evaluate the geometric series up to a given polynomial degree $N-1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Exercise:__ Write a function to sum `N` terms in the geometric series, for input `x`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "geometric (generic function with 1 method)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# write a function to add up the terms\n",
    "function geometric(x,N)\n",
    "    y=zero(x)\n",
    "    for j=0:N-1\n",
    "        y+=x^j\n",
    "    end\n",
    "    return y\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(note we haven't used any type annotations!)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and compare with the analytical result:\n",
    "<div class=\"alert alert-block alert-warning\"><font color=blue>\n",
    "$$S_N=\\sum_{k=0}^{N-1}x^k=1+x+x^2+x^3\\dots+x^{N-1}=\\frac{1-x^N}{1-x}$$\n",
    "</font></div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "geomExact (generic function with 1 method)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "geomExact(x,N)=(1-x^N)/(1-x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A note on compilation and evaluation\n",
    " - first call with particular data type triggers the $\\julia$ JIT to compile fast code.\n",
    " - all later calls with the same type will get fast execution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's time the first call:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.000017 seconds\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "1.1111111111111112"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@time geometric(.1,1000) #compile"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now $\\julia$ has compiled efficient code for the types we passed (not the values), i.e. Float64 and Int64"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.000019 seconds\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "1.1111111111111112"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@time geometric(.1,1000) # ~C speed!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "ename": "LoadError",
     "evalue": "UndefVarError: @btime not defined",
     "output_type": "error",
     "traceback": [
      "UndefVarError: @btime not defined",
      ""
     ]
    }
   ],
   "source": [
    "@btime geometric(.1,1000) # a more reliable measure of timing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our two functions should achieve the same result to numerical precision"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(31.74824973581979, 31.748249735819783)"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = 0.97\n",
    "N = 100\n",
    "geometric(x,N),geomExact(x,N)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this example, \"numerical precision\" means about 1e-14 due to accumulated floating point error:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "isapprox(geometric(x,N),geomExact(x,N),atol=1e-14)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice we can dot some or all of the arguments:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Array{Float64,1}:\n",
       " 1.1111111111111112\n",
       " 1.25              \n",
       " 1.4285714285714288"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "geometric.([0.1; 0.2; 0.3],100) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1×4 Array{Float64,2}:\n",
       " 3.23917  3.33067  3.33326  3.33333"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "geometric.(0.7,[10 20 30 100]) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×4 Array{Float64,2}:\n",
       " 1.1  1.1111  1.11111  1.11111\n",
       " 1.2  1.2496  1.25     1.25   \n",
       " 1.7  2.7731  3.23917  3.33326"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "geometric.([0.1; 0.2; 0.7],[2 5 10 30]) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Miltiple dispatch\n",
    "This is really the key language feature the sets $\\julia$ apart from other technical computing languages. \n",
    "- Methods are not owned by classes\n",
    "- Operator overloading is ubiquitous and tied to data type\n",
    "- Types can be extended at will"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "methods(*);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "CodeInfo(\n",
       "\u001b[90m\u001b[77G│\u001b[1G\u001b[39m\u001b[90m54 \u001b[39m1 ─ %1 = (Base.mul_int)(x, y)\n",
       "\u001b[90m\u001b[77G│\u001b[1G\u001b[39m\u001b[90m   \u001b[39m└──      return %1\n",
       ")"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@code_lowered 5*5 #what is high-level julia translated into?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "; Function *\n",
      "; Location: int.jl:54\n",
      "define i64 @\"julia_*_24657\"(i64, i64) {\n",
      "top:\n",
      "  %2 = mul i64 %1, %0\n",
      "  ret i64 %2\n",
      "}\n"
     ]
    }
   ],
   "source": [
    "@code_llvm 5*5 #what does the compiler produce?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "CodeInfo(\n",
       "\u001b[90m\u001b[77G│\u001b[1G\u001b[39m\u001b[90m314 \u001b[39m1 ─ %1 = (Base.promote)(x, y)\n",
       "\u001b[90m\u001b[77G│\u001b[1G\u001b[39m\u001b[90m    \u001b[39m│   %2 = (Core._apply)(Base.:*, %1)\n",
       "\u001b[90m\u001b[77G│\u001b[1G\u001b[39m\u001b[90m    \u001b[39m└──      return %2\n",
       ")"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@code_lowered 5*2.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "; Function *\n",
      "; Location: promotion.jl:314\n",
      "define double @\"julia_*_27876\"(i64, double) {\n",
      "top:\n",
      "; Function promote; {\n",
      "; Location: promotion.jl:284\n",
      "; Function _promote; {\n",
      "; Location: promotion.jl:261\n",
      "; Function convert; {\n",
      "; Location: number.jl:7\n",
      "; Function Type; {\n",
      "; Location: float.jl:60\n",
      "  %2 = sitofp i64 %0 to double\n",
      ";}}}}\n",
      "; Function *; {\n",
      "; Location: float.jl:399\n",
      "  %3 = fmul double %2, %1\n",
      ";}\n",
      "  ret double %3\n",
      "}\n"
     ]
    }
   ],
   "source": [
    "@code_llvm 5*2.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.040 ns (0 allocations: 0 bytes)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "25"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#multiply two integers\n",
    "@btime 5*5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  1.886 ns (0 allocations: 0 bytes)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "10.5"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#multiply a float and an integer\n",
    "@btime 5*2.1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Is float multiplication really that much slower?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.031 ns (0 allocations: 0 bytes)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "10.5"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#multiply two floats\n",
    "@btime 5.0*2.1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "No, we took a hit from promotion. These timing comparisons and code principles apply across the enirety of $\\julia$: "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-warning\"><font color=blue>\n",
    "You are both free to create without concern for data types, and free to optimize so that efficient code is written for you by the compiler once your code is type stable.\n",
    "    </font></div>    \n",
    "    \n",
    "- Note 90\\% of $\\julia$ is written in $\\julia$: the core of julia is setting up a composable type system and multiple dispatch structure for methods. All data types and methods that act on therm are build with the same core language that you are free to extend as you wish!\n",
    "\n",
    "- This is one of the main reasons developers enjoy $\\julia$. \n",
    "\n",
    "- For users: come for the speed, stay for the productivity!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Avoid global scope: functions are key\n",
    "There is one final \"gotcha\" that I need to emphasize: running any sequence of commands straight in the REPL or in a Jupyter cell _can_ lead to a speed penalty that will be immediately fixed by wrapping your code inside a function. This is because for $\\julia$, the command line is at global scope, and the compiler has trouble inferring type information for global variables (unless you declare types for each variable explicitly). The function barrier makes the compiler's job much easier, and should make your code much more usable. \n",
    "\n",
    "In $\\julia$, your writing should always proceed towards wrapping hard-working commands inside functions to get the job done in a fast and modular way that avoids this issue. \n",
    "\n",
    "Happily, creating and modifying functions in $\\julia$ is super-easy!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Array comprehension\n",
    "Julia has absorbed many nice language features that have emerged in different contexts. For example the array comprehension, a nice natural language syntax for building complex arrays of arbitrary size, element by element. (a little slow to evaluate, but clear and easy to write!)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×4×3 Array{Int64,3}:\n",
       "[:, :, 1] =\n",
       " 1   4   9  16\n",
       " 2   8  18  32\n",
       " 3  12  27  48\n",
       " 4  16  36  64\n",
       " 5  20  45  80\n",
       "\n",
       "[:, :, 2] =\n",
       " 1   4   9  16\n",
       " 2   8  18  32\n",
       " 3  12  27  48\n",
       " 4  16  36  64\n",
       " 5  20  45  80\n",
       "\n",
       "[:, :, 3] =\n",
       " 1   4   9  16\n",
       " 2   8  18  32\n",
       " 3  12  27  48\n",
       " 4  16  36  64\n",
       " 5  20  45  80"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = [i*j^2 for i in 1:5, j in 1:4, k in 1:3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Array{Bool,2}:\n",
       "  true  false\n",
       " false  false"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "truthTable = [i*j for i in [true; false], j in [true; false]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "kron (generic function with 25 methods)"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "⊗=kron"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LinearAlgebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Array{Float64,2}:\n",
       " 1.0  0.0\n",
       " 0.0  1.0"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = randn(3,3)\n",
    "b = one(randn(2,2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6×6 Array{Float64,2}:\n",
       "  2.04888    0.0       1.07188   0.0       -0.181842   -0.0      \n",
       "  0.0        2.04888   0.0       1.07188   -0.0        -0.181842 \n",
       " -0.149097  -0.0       0.855614  0.0       -0.629326   -0.0      \n",
       " -0.0       -0.149097  0.0       0.855614  -0.0        -0.629326 \n",
       "  0.520474   0.0       0.198172  0.0       -0.0354704  -0.0      \n",
       "  0.0        0.520474  0.0       0.198172  -0.0        -0.0354704"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a⊗b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6×6 Array{Float64,2}:\n",
       "  2.04888   1.07188   -0.181842    0.0       0.0       -0.0      \n",
       " -0.149097  0.855614  -0.629326   -0.0       0.0       -0.0      \n",
       "  0.520474  0.198172  -0.0354704   0.0       0.0       -0.0      \n",
       "  0.0       0.0       -0.0         2.04888   1.07188   -0.181842 \n",
       " -0.0       0.0       -0.0        -0.149097  0.855614  -0.629326 \n",
       "  0.0       0.0       -0.0         0.520474  0.198172  -0.0354704"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b⊗a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "More introductory topics to explore:\n",
    "- [Conditional evaluation; ternary operator](https://docs.julialang.org/en/v1/manual/control-flow/#man-conditional-evaluation-1)\n",
    "- [Mathematical operators](https://docs.julialang.org/en/v1/base/math/#math-ops-1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "More packages to explore:\n",
    " - [DynamicalSystems](https://github.com/JuliaDynamics/DynamicalSystems.jl)\n",
    " - [QuantumOptics](https://github.com/qojulia/QuantumOptics.jl)\n",
    " - [SymPy](https://github.com/JuliaPy/SymPy.jl)\n",
    " - [PGFPlots](https://github.com/JuliaTeX/PGFPlots.jl)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " ## Juno\n",
    " - Integrated Development Environment based on the wonderful open-source editor [Atom](https://atom.io/).\n",
    " - Great once you get into writing a sizeable amount of code.\n",
    " - Line by line evaluation, plotting, workspace, documentation browser, GitHub integration, etc. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"media/juno.png\" width=\"600\">"
   ]
  }
 ],
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