Tutorial_01_Basics.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Basic Julia Tutorial"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Running/Installing Julia\n",
    "\n",
    "This is a notebook. To run it you need Julia and Jupyter/nteract/VS Code.\n",
    "\n",
    "1. To run Julia without a local installation, use (for instance) [CoCalc](https://cocalc.com) or [JuliaHub](https://juliahub.com/). \n",
    "\n",
    "2. To install Julia on your machine, [download and install it](https://julialang.org/downloads/). This link contains both instructions for the Microsoft Store and links to binaries for manual download.\n",
    "\n",
    "3. You may also want to use [VS Code with the Julia extension](https://www.julia-vscode.org/). It is an IDE (editor and more) that can run script files and notebooks. It works well with [codeium](https://codeium.com) and [copilot](https://github.com/features/copilot).\n",
    "\n",
    "4. To run notebooks from your local installation, you need either (1) VS Code with the Julia extension (see above); (2) Jupyter/JupyterLab; or (3) nteract. For the latter two, see below for hints.\n",
    "\n",
    "5. To create and run script files instead of notebooks, see the comments towards the end of this notebook.\n",
    "\n",
    "\n",
    "## Installing nteract\n",
    "\n",
    "Install [IJulia](https://github.com/JuliaLang/IJulia.jl), but do *not* call on `notebook()` to avoid the automatic Python installation. Then install nteract and start using it to run notebooks. Warning: as of February 2024 this still works, but nteract has not been updated since 2021.\n",
    "\n",
    "\n",
    "## Installing Jupyter/JupyterLab - When You Don't Have Python Installed\n",
    "\n",
    "1. Install IJulia [IJulia](https://github.com/JuliaLang/IJulia.jl) and issue the command `notebook()`. This will make a local Python installation.\n",
    "\n",
    "2. To use Jupyter, start Julia and run `using IJulia; notebook(dir=pwd())` (change the directory as needed). Instead, to run JupyterLab do `using IJulia;jupyterlab(dir=pwd())`.\n",
    "\n",
    "\n",
    "## Installing Jupyter/JupyterLab - When You Already Have Python Installed\n",
    "\n",
    "1. If you already have a Python installation (with Jupyter/JupyterLab), run `ENV[\"JUPYTER\"] = \"C:\\\\Miniconda3\\\\Scripts\\\\jupyter.exe\"` (change the path as needed) before you install IJulia. The best is perhaps to add this to your *startup.jl* file (for instance, `C:\\Users\\yourusername\\.julia\\config\\startup.jl` if you are on Windows). You can test whether it works by running ```run(`$(ENV[\"JUPYTER\"]) --version`)``` from the Julia REPL. \n",
    "\n",
    "2. On Windows, it helps to allow the Python installer to add python to the system path (in spite of the warnings).\n",
    "\n",
    "3. Install IJulia and start using Jupyter/JupyterLab (see above)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Documentation and Help\n",
    "\n",
    "1.  Cheat sheet at [QuantEcon](https://cheatsheets.quantecon.org/julia-cheatsheet.html)\n",
    "2.  [Wiki book](https://en.wikibooks.org/wiki/Introducing_Julia)\n",
    "3.  [Short tutorials](https://techytok.com/from-zero-to-julia/) \n",
    "4.  [ThinkJulia](https://benlauwens.github.io/ThinkJulia.jl/latest/book.html) is a free on-line book\n",
    "5.  The [official Julia on-line manual](https://docs.julialang.org)\n",
    "6.  Discussion lists are found at\n",
    "    *  https://discourse.julialang.org/\n",
    "    *  https://stackoverflow.com/questions/tagged/julia-lang\n",
    "    *  https://www.reddit.com/r/Julia/\n",
    "    *  https://gitter.im/JuliaLang/julia\n",
    "7. In Julia, do `?cos` or `@doc cos` to get help with the cos function (see below for how to get help in VS Code).\n",
    "8. [chatgpt](https://chatgpt.com) often generate half decent code, which can serve as a starting point-"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# About Notebooks\n",
    "\n",
    "This cell is a \"Markdown\" cell. This is meant for comments and documentation, not computations.\n",
    "\n",
    "You can change a cell to \"Code\" or \"Markdown\" in the menu.\n",
    "\n",
    "Markdown cells can handle LaTeX. An example: $\\alpha = \\beta/2$. A Markdown cell can also contain some *formatting*, like lists of this kind:\n",
    "\n",
    "1. To insert a new cell, use the menu. \n",
    "\n",
    "2. The next cell is \"Code\". You can run it. Text after a # sign is treated as a comment."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = 2                  #this is a comment\n",
    "                       #run this cell by using the menu, or by Shift+Enter"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Load Packages and Extra Functions\n",
    "\n",
    "There are many packages for Julia, for instance, for plotting or statistical methods (see [JuliaHub](https://juliahub.com/) for a list). To install a package, you do either \n",
    "\n",
    "1. (works everywhere) run `import Pkg` and then `Pkg.add(\"Packagename\")`\n",
    "\n",
    "2. (works in the Julia console, the REPL) enter the \"package manager mode\" by typing `]`, then run `add PackageName`. You leave the package manager mode by backspace.\n",
    "\n",
    "Once a package is installed, you can use it by running `using PackageName`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Printf                        #installed by default, used for more control over printing\n",
    "\n",
    "include(\"jlFiles/printmat.jl\");     #just a function for prettier matrix printing, semicolon (;) to make it silent "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## How to Get Help\n",
    "\n",
    "Typing `?cos` or `@doc cos` prints the documentation for the `cos` function, except in a notebook in VS Code. There, do `@doc2 cos` instead. (The `@doc2` macro is defined in the printmat.jl file included above)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "```\n",
       "cos(x)\n",
       "```\n",
       "\n",
       "Compute cosine of `x`, where `x` is in radians.\n",
       "\n",
       "See also [`cosd`](@ref), [`cospi`](@ref), [`sincos`](@ref), [`cis`](@ref).\n",
       "\n",
       "```\n",
       "cos(A::AbstractMatrix)\n",
       "```\n",
       "\n",
       "Compute the matrix cosine of a square matrix `A`.\n",
       "\n",
       "If `A` is symmetric or Hermitian, its eigendecomposition ([`eigen`](@ref)) is used to compute the cosine. Otherwise, the cosine is determined by calling [`exp`](@ref).\n",
       "\n",
       "# Examples\n",
       "\n",
       "```jldoctest\n",
       "julia> cos(fill(1.0, (2,2)))\n",
       "2×2 Matrix{Float64}:\n",
       "  0.291927  -0.708073\n",
       " -0.708073   0.291927\n",
       "```\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "@doc2 cos          #Needed in VS Code, works everywhere \n",
    "# ? cos            #works in REPL and jupyter, not in VS Code"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Scalars and Matrices\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Create a Scalar and a Matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "q is a scalar. To print, use println() or printlnPs()\n",
      "1\n",
      "\n",
      "Q is a matrix. To print, use display() or printmat()\n",
      "     1         2         3    \n",
      "     4         5         6    \n",
      "\n"
     ]
    }
   ],
   "source": [
    "q = 1                             #create a scalar\n",
    "Q = [ 1 2 3;                      #create 2x3 matrix\n",
    "      4 5 6 ] \n",
    "println(\"q is a scalar. To print, use println() or printlnPs()\")\n",
    "println(q)\n",
    "\n",
    "println(\"\\nQ is a matrix. To print, use display() or printmat()\")\n",
    "printmat(Q)                       #case sensitive (q and Q are different)\n",
    "                                  #the \\n adds a line break"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Picking Out Parts of a Matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "element [1,2] of Q: 2\n",
      "\n",
      "columns 2 and 3 of Q: \n",
      "     2         3    \n",
      "     5         6    \n",
      "\n",
      "\n",
      "line 1 of Q (as a vector): \n",
      "     1    \n",
      "     2    \n",
      "     3    \n",
      "\n"
     ]
    }
   ],
   "source": [
    "println(\"\\n\",\"element [1,2] of Q: \",            #commands continue on\n",
    "        Q[1,2])                                 #the next line (until finished)\n",
    "\n",
    "println(\"\\ncolumns 2 and 3 of Q: \")\n",
    "printmat(Q[:,2:3])\n",
    "\n",
    "println(\"\\nline 1 of Q (as a vector): \")\n",
    "printmat(Q[1,:])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Basic Linear Algebra\n",
    "\n",
    "The syntax for linear algebra is similar to the standard text book approach. For instance, \n",
    "* `Q'Q` (or `Q'*Q`) multiplies the transpose ($Q'$) with the matrix ($Q$)\n",
    "* `A*B` does matrix multiplication\n",
    "* `100*Q` multiplies each element of the matrix ($Q$) by 100. (You can also do `100Q`)\n",
    "\n",
    "However, to add a scalar to each element of a matrix, use `100 .+ Q`. Notice the dot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "transpose of Q:\n",
      "     1         4    \n",
      "     2         5    \n",
      "     3         6    \n",
      "\n",
      "Q'Q:\n",
      "    17        22        27    \n",
      "    22        29        36    \n",
      "    27        36        45    \n",
      "\n",
      "scalar * matrix:\n",
      "   100       200       300    \n",
      "   400       500       600    \n",
      "\n",
      "scalar .+ matrix:\n",
      "   101       102       103    \n",
      "   104       105       106    \n",
      "\n"
     ]
    }
   ],
   "source": [
    "println(\"transpose of Q:\")\n",
    "printmat(Q')\n",
    "\n",
    "println(\"Q'Q:\")\n",
    "printmat(Q'Q)\n",
    "\n",
    "println(\"scalar * matrix:\")\n",
    "printmat(100*Q)\n",
    "\n",
    "println(\"scalar .+ matrix:\")        #notice the dot\n",
    "printmat(100 .+ Q)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Creating a Sequence and a Vector"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "θ is a sequence: 1:10:21\n",
      "\n",
      "ρ is a vector: \n",
      "     1    \n",
      "    11    \n",
      "    21    \n",
      "\n"
     ]
    }
   ],
   "source": [
    "θ = 1:10:21                  #a range, type \\theta[TAB] to get this symbol\n",
    "println(\"\\n\",\"θ is a sequence: \",θ)\n",
    "\n",
    "ρ = collect(θ)               #make the sequence into a vector, \\rho[TAB]\n",
    "println(\"\\n\",\"ρ is a vector: \")\n",
    "printmat(ρ)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Comparing Things\n",
    "\n",
    "To see if the scalar $z \\leq 0 $, do\n",
    "```\n",
    "vv = z <= 0\n",
    "```\n",
    "to get a single output (true or false).\n",
    "\n",
    "Instead, if x is an array, do \n",
    "```\n",
    "vv = x .<= 0                      #notice the dot.\n",
    "```\n",
    "to get an array of outputs (with the same dimensions as x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x values: \n",
      "    -1.500\n",
      "    -1.000\n",
      "    -0.500\n",
      "     0.000\n",
      "     0.500\n",
      "\n",
      "true if x is in (-1,0]: \n",
      "     0    \n",
      "     0    \n",
      "     1    \n",
      "     1    \n",
      "     0    \n",
      "\n",
      "x values that are in (-1,0]: \n",
      "    -0.500\n",
      "     0.000\n",
      "\n"
     ]
    }
   ],
   "source": [
    "x =  [-1.5,-1.0,-0.5,0,0.5]             #this is a vector\n",
    "\n",
    "println(\"x values: \")\n",
    "printmat(x)\n",
    "\n",
    "vv = -1 .< x .<= 0                      #true for x values (-1,0], vv is a vector\n",
    "println(\"true if x is in (-1,0]: \")\n",
    "printmat(vv)\n",
    "\n",
    "x2 = x[vv]                              #x values for which vv==true\n",
    "println(\"x values that are in (-1,0]: \")\n",
    "printmat(x2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Finding Things (extra)\n",
    "\n",
    "can be done by indexing as shown before. However, it might be more efficient to use one of `findfirst()`, `findall()` and `indexin()`. \n",
    "\n",
    "Also, if you just need to check if the number `z` is in an array (or any collection) `x`, then use `in(z,x)`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x values: \n",
      "    -1.500\n",
      "    -1.000\n",
      "    -0.500\n",
      "     0.000\n",
      "     0.500\n",
      "\n",
      "(first) index v in x such x[v]==0: 4\n",
      "\n",
      "all indices v in x such x[v]>=0: \n",
      "     4    \n",
      "     5    \n",
      "\n",
      "\n",
      "indices in x so that x[v] equals the vector y=[-1.0, 0.0]: \n",
      "     2    \n",
      "     4    \n",
      "\n",
      "\n",
      "testing if 0 is in x: true\n"
     ]
    }
   ],
   "source": [
    "x =  [-1.5,-1.0,-0.5,0,0.5]             #this is a vector\n",
    "\n",
    "println(\"x values: \")\n",
    "printmat(x)\n",
    "\n",
    "v1 = findfirst(x.==0)\n",
    "println(\"(first) index v in x such x[v]==0: \",v1)\n",
    "\n",
    "v2 = findall(x.>=0)\n",
    "println(\"\\nall indices v in x such x[v]>=0: \")\n",
    "printmat(v2)\n",
    "\n",
    "y = [-1.0,0]\n",
    "v3 = indexin(y,x)\n",
    "println(\"\\nindices in x so that x[v] equals the vector y=$y: \")\n",
    "printmat(v3)\n",
    "\n",
    "v4 = in(0,x)\n",
    "println(\"\\ntesting if 0 is in x: \",v4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": []
   },
   "source": [
    "# Simple Functions\n",
    "\n",
    "The next cell defines two new functions, `fn0()` and `fn1()`. They take a scalar input (`x`) and return a scalar output (`y`).\n",
    "\n",
    "If you instead use a vector as the input, then the computations in these functions fail. (The reason is that you cannot do `x^2` on a vector. You would need `x.^2`.)\n",
    "\n",
    "However, using the \"dot\" syntax\n",
    "```\n",
    "y = fn1.(x)\n",
    "```\n",
    "gives an array as output where element `y[i,j] = fn1(x[i,j])`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "fn1 (generic function with 1 method)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fn0(x) = (x-1.1)^2 - 0.5      #define a one-line function\n",
    "\n",
    "function fn1(x)               #define a function that could have many lines\n",
    "  y = (x-1.1)^2 - 0.5\n",
    "  return y\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "result from fn1(1.5):     -0.340 and compare with fn0(1.5)     -0.340\n",
      "\n",
      "result from fn1.(x): \n",
      "    -0.490\n",
      "    -0.340\n",
      "\n"
     ]
    }
   ],
   "source": [
    "y = fn1(1.5)\n",
    "printlnPs(\"result from fn1(1.5): \",y,\" and compare with fn0(1.5) \",fn0(1.5))\n",
    "\n",
    "x = [1;1.5]\n",
    "#y = fn1(x)                   #would give an error \n",
    "y = fn1.(x)                   #calling on the function, dot. to do for each element in x\n",
    "printlnPs(\"\\nresult from fn1.(x): \")\n",
    "printmat(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# if-else-end\n",
    "\n",
    "allows you to run different commands depending on a condition that you specify.\n",
    "\n",
    "(extra) There are also other (more compact) possibilities for the `if-else` case\n",
    "\n",
    "1. `y = z <= 2 ? z : 2` or\n",
    "2. `y = ifelse(z <= 2,z,2)`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.05\n"
     ]
    }
   ],
   "source": [
    "z = 1.05\n",
    "\n",
    "if z <= 2          #(a) if true, run the next command (y=z) and then jump to end\n",
    "    y = z\n",
    "else               #(b) if (a) is false, do this instead\n",
    "    y = 2\n",
    "end\n",
    "\n",
    "#y = z <= 2 ? z : 2       #these two versions also work\n",
    "#y = ifelse(z <= 2,z,2)\n",
    "\n",
    "\n",
    "println(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.05\n"
     ]
    }
   ],
   "source": [
    "if z < 1                #(a) if true, run the next command (y=1) and then jump to end\n",
    "    y = 1\n",
    "elseif 1 <= z <= 2      #(b) if (a) is false, try this instead  \n",
    "    y = z\n",
    "else                    #(c) if also (b) is false, do this\n",
    "    y = 2\n",
    "end\n",
    "\n",
    "println(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Loops\n",
    "\n",
    "The are two types of loops: \"for loops\" and \"while loops\". \n",
    "\n",
    "The *for loop* is best when you know how many times you want to loop, for instance, over all $m$ rows in a matrix. \n",
    "\n",
    "The *while loop* is best when you want to keep looping until something happens, for instance, that $x_0$ and $x_1$ get really close.\n",
    "\n",
    "The default behaviour in *IJulia* (notebooks), *inside functions* and also at the *REPL* promt is that assignments of `x` inside the loop overwrites `x` defined before the loop. To get the same behavior in scripts, you need to add `global x` somewhere inside the loop. In contrast, a variable (here `z2`) that does not exist before the loop is local to the loop.\n",
    "\n",
    "To make sure that the `y` calculated inside the loop *does not* affect `y` outside the loop, say `local y` inside the loop."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A Simple \"for loop\"\n",
    "\n",
    "The \"for loop\" in the next cell iterates over all elements in a vector and changes them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   101.000\n",
      "   102.000\n",
      "   103.000\n",
      "\n"
     ]
    }
   ],
   "source": [
    "x = zeros(3)\n",
    "\n",
    "for i in 1:length(x)           #1,2,3\n",
    "    x[i] = 100 + i\n",
    "end    \n",
    "\n",
    "printmat(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A More Complicated \"for loop\"\n",
    "\n",
    "The \"for loop\" in the next cell makes 3 iterations and changes a global $x$ variable. It also illustrates difference between local and global variables."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Before loop: x=0 and y=-999 \n",
      "\n",
      "i=3, x=3, y=3 and z2=-998\n",
      "i=6, x=9, y=6 and z2=-998\n",
      "i=9, x=18, y=9 and z2=-998\n",
      "\n",
      "After loop: x=18 and y=-999 \n",
      "\n",
      "z2 is not defined outside the loop\n"
     ]
    }
   ],
   "source": [
    "x = 0\n",
    "y = -999\n",
    "println(\"\\nBefore loop: x=$x and y=$y \\n\")    #\"$x\" inserts the value of x into a string#\n",
    "\n",
    "for i in 3:3:9                        #or `i = 3:3:9`, or `v=[3,6,9]; i in v`\n",
    "    #global x                         #only needed in script\n",
    "    local y                           #don't overwrite y outside loop\n",
    "    x = x + i                         #adding i to the \"old\" x\n",
    "    y = i\n",
    "    z2 = -998                         #notice: z2 has not been used before \n",
    "    println(\"i=$i, x=$x, y=$y and z2=$z2\")          #$x prints the value of x\n",
    "end\n",
    "\n",
    "println(\"\\nAfter loop: x=$x and y=$y \\n\")\n",
    "\n",
    "try                        #to \"drive around\" errors\n",
    "    println(z2)            #does not work: z2 is local to the loop\n",
    "catch\n",
    "    println(\"z2 is not defined outside the loop\") \n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A Double \"for loop\"\n",
    "\n",
    "The next cell uses a double loop to fill a matrix. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "new x matrix: \n",
      "\n",
      "    11        12        13    \n",
      "    21        22        23    \n",
      "    31        32        33    \n",
      "    41        42        43    \n",
      "\n"
     ]
    }
   ],
   "source": [
    "(m,n) = (4,3)                     #same as m=4;n=3\n",
    "x = fill(-999,m,n)                #to put results in, initialized as -999\n",
    "for i in 1:m, j in 1:n              #can be written on the same line\n",
    "    x[i,j] = 10*i + j\n",
    "end\n",
    "\n",
    "println(\"new x matrix: \\n\")\n",
    "printmat(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A Simple \"while loop\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x=2.5\n",
      "x=1.25\n",
      "x=0.625\n",
      "x=0.3125\n",
      "x=0.15625\n",
      "x=0.078125\n",
      "x=0.0390625\n",
      "\n",
      "After loop: x=0.0390625\n"
     ]
    }
   ],
   "source": [
    "x = 5\n",
    "\n",
    "while x > 0.05\n",
    "    #global x            #only needed in script\n",
    "    x = x/2\n",
    "    println(\"x=$x\")\n",
    "end    \n",
    "\n",
    "println(\"\\nAfter loop: x=$x\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# A First Plot\n",
    "\n",
    "With the Plots package you create a a simple plot like this:\n",
    "\n",
    "1. Plot two curve by using the `plot([x1 x2],[y1 y2])` command\n",
    "2. Configure curves by `linecolor =`, `linestyle =` etc\n",
    "2. Add titles and labels by `title = `, `xlabel = `, etc\n",
    "\n",
    "Notice: the *first plot is slow*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots                 #this loads the Plots package\n",
    "\n",
    "gr(size=(480,320))\n",
    "default(fmt = :png)        # :svg gives prettier plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\" />"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = -3:6/99:6 \n",
    "\n",
    "p1 = plot( [x x],[fn1.(x) cos.(x)],\n",
    "           linecolor = [:red :blue],\n",
    "           linestyle = [:solid :dash],\n",
    "           linewidth = [2 1],\n",
    "           label = [\"fn1()\" \"cos\"],\n",
    "           title = \"My Results\",\n",
    "           xlabel = \"x\",\n",
    "           ylabel = \"my output value\" )\n",
    "display(p1)       #not needed in notebook, but useful in script"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Types: Integers, Floats, Bools and Others\n",
    "\n",
    "Julia has many different types of variables: signed integers (like 2 or -5), floating point numbers (2.0 and -5.1), bools (false/true), bitarrays (similar to bools, but with more efficient use of memory), strings (\"hello\"), Dates (2017-04-23) and many more types."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Integers and Floats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Finding the type of a, b, A and B:\n",
      "Int64 Float64 Vector{Int64} Vector{Float64}\n"
     ]
    }
   ],
   "source": [
    "a = 2                   #integer, Int (Int64 on most machines)\n",
    "b = 2.0                 #floating point, (Float64 on most machines)\n",
    "A = [1,2]\n",
    "B = [1.0,2.0]\n",
    "\n",
    "println(\"Finding the type of a, b, A and B:\")\n",
    "println(typeof(a),\" \",typeof(b),\" \",typeof(A),\" \",typeof(B))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bools and BitArrays\n",
    "\n",
    "Bools are \"true\" or \"false\". BitArrays are (more memory efficient) versions of this."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Finding the type of c and C:\n",
      "Bool BitVector\n"
     ]
    }
   ],
   "source": [
    "c = 2 > 1.1\n",
    "C = A .> 1.5        #A is an array, so C is too\n",
    "\n",
    "println(\"Finding the type of c and C:\")\n",
    "println(typeof(c),\" \",typeof(C))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculations with Mixed Types\n",
    "\n",
    "A calculation like \"integer\" + \"float\" works and the type of the result will be a float (the more flexible type). Similarly, \"bool\" + \"integer\" will give an integer. These promotion rules make it easy to have mixed types in calculations, and also provide a simple way of converting a variable from one type to another. (There are also an explicit convert() function that might be quicker.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.0\n",
      "3\n"
     ]
    }
   ],
   "source": [
    "println(1+2.0)                #integer + Float\n",
    "println((1>0) + 2)            #bool + integer"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Running Script Files\n",
    "\n",
    "instead of notebooks. \n",
    "\n",
    "Once you have created a script file (called \"myscript.jl\" below), then you can, for instance, run it from VS Code or directly from the Julia REPL. For the latter, do as follows:\n",
    "\n",
    "1. Open the Julia REPL and perhaps run `cd(\"folder_name\")` to set the working directory (typically to where your script file is).\n",
    "2. Run `include(\"myscript.jl\")`\n",
    "\n",
    "As an alternative, instead do the following:\n",
    "\n",
    "0. Add this close to the top of the script file: `cd(dirname(@__FILE__))`\n",
    "1. Open the Julia REPL\n",
    "2. Run `include(\"foldername/myscript.jl\")`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Performance Tips (extra)\n",
    "\n",
    "To benchmark your code, install/use the [BenchmarkTools.jl](https://github.com/JuliaCI/BenchmarkTools.jl) package."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  167.032 ns (1 allocation: 896 bytes)\n",
      "\n"
     ]
    }
   ],
   "source": [
    "using BenchmarkTools\n",
    "\n",
    "function fn2(x) \n",
    "    n = length(x)\n",
    "    z = zero(x)       #array of zeros, same size as x\n",
    "    for i in 2:n\n",
    "        z[i] = (x[i] + x[i-1])/2   #moving average\n",
    "    end\n",
    "    return z\n",
    "end\n",
    "\n",
    "x = [1;zeros(101)]\n",
    "@btime fn2($x)        #notice the $\n",
    "println()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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