{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Variables, Control Structures & Packages"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Variables and elementary types\n",
    "\n",
    "Defining variables in Julia works like in most languages:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "int = 4      # An integer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "str = \"Hi\"   # A string"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "float = 1.2  # A floating-point number"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "bool = true  # A boolean (also false)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The type is automatically inferred:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "typeof(int)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "typeof(str)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "typeof(float)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Julia supports a large range of integer and floating-point types out of the box, for example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = UInt8(1)              # 8-bit wide unsigned integer\n",
    "y = Int32(-1)             # 32-bit wide signed integer\n",
    "z = Float32(0.2)          # single precision\n",
    "α = Float16(-1.0)         # half precision\n",
    "β = ComplexF64(2. + 8im)  # Complex number (composed of two Float64)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Operations** between these types may involve **automatic promotion**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "println(typeof(1 + 1.2))\n",
    "println(typeof(x * y))  # x was Uint8, y was Int32"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = Float32(1.)\n",
    "4x                     # No \"*\" needed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "println(5 % 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "5^6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "2.3^6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "println(4 - Float32(2))\n",
    "println(8 / 3)  # Note conversion to Float64\n",
    "typeof(8 / 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This can be avoided using a rational:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = 6 // 3\n",
    "println(typeof(x), \" \", Int(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "√2  # == sqrt(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice the `*` operation for **string concatenation** ..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "\"Concatenating \" * \"strings is done \" * \"using multiplication\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Type conversion** is possible, for example using constructors:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Int(2.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = Float32(1.0)\n",
    "Float64(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Int(2.3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "UInt8(-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Float64(2.0 + 1im)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sometimes a special conversion function or syntax exists:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "string(1.2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "ceil(Int, 2.3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "round(Int32, 4.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = 1.2\n",
    "println(\"x is currently $x.\")\n",
    "println(\"two less is $(x - 2)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Arbitrary precision** is supported out of the box:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# arbitrary precision\n",
    "b = big\"12\"; println(b, \" is a \", typeof(b))\n",
    "f = big\"-1.2\"; println(f, \" is a \", typeof(f))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Or wir an explicit cast:\n",
    "b = BigInt(12)\n",
    "f = BigFloat(-1.2)  # Note the difference to the above result"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Julia is garbage-collected, meaning that memory occupied by a variable value will be freed automatically if no longer needed, e.g.:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = 15\n",
    "println(x)\n",
    "x = \"abc\";    # x newly assigned, memory by the \"15\" will be freed at some point in the future"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise\n",
    "Compute $2^{100}$:\n",
    "- As fast as possible\n",
    "- As exact as possible"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### For more details\n",
    "- https://docs.julialang.org/en/v1/manual/integers-and-floating-point-numbers/\n",
    "- https://docs.julialang.org/en/v1/manual/mathematical-operations/\n",
    "- https://docs.julialang.org/en/v1/manual/strings/\n",
    "- https://docs.julialang.org/en/v1/manual/conversion-and-promotion/"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Basic control structures\n",
    "\n",
    "Unsurprisingly Julia has the standard `for` and `while` loops and the `if` conditionals."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = 3; y = 5\n",
    "\n",
    "if x < y\n",
    "    println(\"x is less than y\")\n",
    "elseif x > y\n",
    "    println(\"x is greater than y\")\n",
    "else\n",
    "    println(\"x is equal to y\")\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = 5\n",
    "while x > 3\n",
    "    println(\"Now x is $x\")\n",
    "    x -= 1\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in 1:5\n",
    "    println(\"Hello from number $i\")\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Where the `1:5` is a `Range`-object, but it could be any iterable. There are a few syntax variations, including:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "accu = 0\n",
    "for j ∈ 1:0.5:3  # Note the step parameter\n",
    "    accu += j\n",
    "end\n",
    "println(accu)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Worth mentioning are also `||`, `&&` and `?:`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "true  || println(\"The RHS of || is only run\")\n",
    "false || println(\"if the LHS is false\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "iseven(3) && println(\"The RHS of || is only run\")\n",
    "isodd(3)  && println(\"if the LHS is true\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = 3\n",
    "x < 5 ? \"smaller than 5\" : \"larger or equal 5\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Generally expressions can be made one block using `begin` and `end` or `(` and `)`. The last statement determines the returned value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "z = begin\n",
    "    x = 5\n",
    "    y = -3\n",
    "    x + y\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Or equivalently\n",
    "z = (x = 5; y = -3; x + y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise\n",
    "Compute\n",
    "$$ 15! \\qquad 100! \\qquad \\left(\\begin{array}{c} 100 \\\\ 15 \\end{array}\\right) $$\n",
    "with the Julia you know so far."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### For more details\n",
    "- https://docs.julialang.org/en/v1/manual/control-flow/"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Julia packages\n",
    "\n",
    "The Julia standard library is already surprisingly rich in functionality. Features are grouped into smaller packages according to topics (e.g. `LinearAlgebra`, `SparseArrays`, `Test`, ...). The most basic part of the standard library resides in the special package `Base` and is always automatically to the programmer. If anything beyond `Base` is needed, that is both packages of the standard library or any other third-party package, this needs to be explicitly **installed** and **imported** before use."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For example. The `randn` function produces a list or array of random numbers. It is already available in `Base` and we can thus directly use it to create 4 random numbers:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "d = randn(4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The returned object in `d` is an `Array`, a type from `Base`, which we will discuss later. We wish to create a diagonal Matrix with these four vaules on the diagonal. For this we need the `LinearAlgebra.Diagonal` datastructure. We make `LinearAlgebra` available to our scope with a `using` statement:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LinearAlgebra"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This (might) cause an error, because `LinearAlgebra` is not yet installed. We resolve this as recommended:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import Pkg; Pkg.add(\"LinearAlgebra\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note, that `import` is roughly the same as `using` except the package is not brought into scope. We will neglect this detail for this course and just use `using` for all packages we use."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Having installed the package, we are able to use LinearAlgebra and make the diagonal matrix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LinearAlgebra\n",
    "Diagonal(d)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Julia REPL\n",
    "\n",
    "Apart from interacting with Julia via a webbrowser and Jupyter (as we will do it in this course). There is also the alternative option to use the Julia REPL. You can get to the REPL by starting `julia` from a terminal. The result is a prompt similar to\n",
    "```\n",
    "               _\n",
    "   _       _ _(_)_     |  Documentation: https://docs.julialang.org\n",
    "  (_)     | (_) (_)    |\n",
    "   _ _   _| |_  __ _   |  Type \"?\" for help, \"]?\" for Pkg help.\n",
    "  | | | | | | |/ _` |  |\n",
    "  | | |_| | | | (_| |  |  Version 1.3.0 (2019-11-26)\n",
    " _/ |\\__'_|_|_|\\__'_|  |  Official https://julialang.org/ release\n",
    "|__/                   |\n",
    "\n",
    "julia> \n",
    "\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this prompt you can directly start typing julia expressions, which will be executed upon `[Enter]`. The REPL is very nice to prototype things and with some optional customisations becomes a very powerful tool. We will not use the REPL much in this course, but still I want to mention some basic aspects to get you started. Mostly this is for people who prefer the commandline over graphics (like me):\n",
    "\n",
    "- Exiting the shell can be achieved using `exit()` or Ctrl + D.\n",
    "- Inside the shell tab completion is available.\n",
    "- The REPL comes in multiple modes. The most useful ones are the (default) Julian mode, Pkg mode (for installing packages), shell mode (for running one-shot shell commands) and help mode. You can get from Julian mode to another by typing a magic key, see the list below. You get back to Julian mode by `[Backspace]` on an empty prompt.\n",
    "\n",
    "| Magic key     | Prompt                  | Mode                         |\n",
    "|---------------|-------------------------|------------------------------|\n",
    "| `[Backspace]` | `julia>`                | Julian mode (Command mode)   |\n",
    "| `]`           | `(v1.3) pkg>`           | Pkg mode                     |\n",
    "| `?`           | `help>`                 | Help mode                    |\n",
    "| `;`           | `shell>`                | Shell command mode           |\n",
    "\n",
    "\n",
    "- To understand Pkg mode better, type `help` once the `(v1.3) pkg>` prompt shows.\n",
    "- In each mode Ctrl + R allows to search backwards in history for a previously entered command."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise\n",
    "External packages are installed exactly the same as standard library packages. Their data is automatically fetched from the [General](https://github.com/JuliaRegistries/General) as needed.\n",
    "\n",
    "Make sure you have an internet connection. From the REPL or from the notebook, install the following packages as we will need them later:\n",
    "- `Plots`\n",
    "- `Zygote`\n",
    "- `BenchmarkTools`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### For more details\n",
    "- https://jupyter-notebook.readthedocs.io/en/stable/\n",
    "- https://docs.julialang.org/en/v1/stdlib/Pkg/\n",
    "- https://docs.julialang.org/en/v1/stdlib/REPL/"
   ]
  }
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