Tutorial_01_Basics.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Basic Julia Tutorial"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Running/Installing Julia\n",
    "\n",
    "This is a notebook. To run in you need Julia and jupyter or nteract (see below).\n",
    "\n",
    "1. To run Julia without a local installation, use (for instance) [CoCalc](https://cocalc.com).\n",
    "\n",
    "2. To install Julia (plus an editor and key packages) on your machine, download and install [JuliaPro](https://juliacomputing.com/products/juliapro.html). \n",
    "\n",
    "3. Alternatively, to manage your own installtion [download and install Julia](https://julialang.org/downloads/). You may also want to use the Atom editor with the [Juno IDE](http://junolab.org/) or [VS Code](https://github.com/julia-vscode/julia-vscode). To run IJulia with either jupyter or nteract from your local installation, see [IJulia](https://github.com/JuliaLang/IJulia.jl) for instructions."
   ]
  },
  {
   "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``` to get help with the cos function    "
   ]
  },
  {
   "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.\n",
    "\n",
    "3. The subsequent cell shows how to get help on a command."
   ]
  },
  {
   "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": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "search: \u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22m \u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22mh \u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22md \u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22mc \u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22mpi a\u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22m a\u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22mh a\u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22md sin\u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22m sin\u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22md sin\u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22mpi \u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22mn\u001b[0m\u001b[1ms\u001b[22mt\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "\\begin{verbatim}\n",
       "cos(x)\n",
       "\\end{verbatim}\n",
       "Compute cosine of \\texttt{x}, where \\texttt{x} is in radians.\n",
       "\n",
       "\\rule{\\textwidth}{1pt}\n",
       "\\begin{verbatim}\n",
       "cos(A::AbstractMatrix)\n",
       "\\end{verbatim}\n",
       "Compute the matrix cosine of a square matrix \\texttt{A}.\n",
       "\n",
       "If \\texttt{A} is symmetric or Hermitian, its eigendecomposition (\\href{@ref}{\\texttt{eigen}}) is used to compute the cosine. Otherwise, the cosine is determined by calling \\href{@ref}{\\texttt{exp}}.\n",
       "\n",
       "\\section{Examples}\n",
       "\\begin{verbatim}\n",
       "julia> cos(fill(1.0, (2,2)))\n",
       "2×2 Matrix{Float64}:\n",
       "  0.291927  -0.708073\n",
       " -0.708073   0.291927\n",
       "\\end{verbatim}\n"
      ],
      "text/markdown": [
       "```\n",
       "cos(x)\n",
       "```\n",
       "\n",
       "Compute cosine of `x`, where `x` is in radians.\n",
       "\n",
       "---\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"
      ],
      "text/plain": [
       "\u001b[36m  cos(x)\u001b[39m\n",
       "\n",
       "  Compute cosine of \u001b[36mx\u001b[39m, where \u001b[36mx\u001b[39m is in radians.\n",
       "\n",
       "  ────────────────────────────────────────────────────────────────────────────\n",
       "\n",
       "\u001b[36m  cos(A::AbstractMatrix)\u001b[39m\n",
       "\n",
       "  Compute the matrix cosine of a square matrix \u001b[36mA\u001b[39m.\n",
       "\n",
       "  If \u001b[36mA\u001b[39m is symmetric or Hermitian, its eigendecomposition (\u001b[36meigen\u001b[39m) is used to\n",
       "  compute the cosine. Otherwise, the cosine is determined by calling \u001b[36mexp\u001b[39m.\n",
       "\n",
       "\u001b[1m  Examples\u001b[22m\n",
       "\u001b[1m  ≡≡≡≡≡≡≡≡≡≡\u001b[22m\n",
       "\n",
       "\u001b[36m  julia> cos(fill(1.0, (2,2)))\u001b[39m\n",
       "\u001b[36m  2×2 Matrix{Float64}:\u001b[39m\n",
       "\u001b[36m    0.291927  -0.708073\u001b[39m\n",
       "\u001b[36m   -0.708073   0.291927\u001b[39m"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "? cos                  #to get help on the cos() function"
   ]
  },
  {
   "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, REPL) enter the \"package manager mode\" by typing `]`, then run `add PackageName`. You leave the package manager mode by ctrl-c or backspace.\n",
    "\n",
    "Once a package is installed, you can use it by running\n",
    "\n",
    "```\n",
    "using PackageName\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "printyellow (generic function with 1 method)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Printf\n",
    "\n",
    "include(\"jlFiles/printmat.jl\")      #just a function for prettier matrix printing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots                 #this loads the Plots package\n",
    "\n",
    "#pyplot(size=(600,400))    #choice of plotting backend\n",
    "gr(size=(480,320))\n",
    "default(fmt = :svg)        #try :png or :svg"
   ]
  },
  {
   "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": 5,
   "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": 6,
   "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\n",
    "\n",
    "However, to add a scalar to each element of a matrix, use `100 .+ Q`. Notice the dot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "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": 8,
   "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 <= 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 (same dimension as 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",
      "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\n",
    "\n",
    "can be done by, for instance, `findfirst()`, `findall()` and `indexin()`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "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"
     ]
    }
   ],
   "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)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# if-elseif-else\n",
    "\n",
    "allows you to run different commands depending on a condition which you specify.\n",
    "\n",
    "(extra) There are also two other (more compact) possibilities for the `if-else` case (no `elseif`). In the example below, we actually calculate `y=minimum(z,2)`:\n",
    "\n",
    "1. `y = ifelse(z <= 2,z,2)` or \n",
    "2.  `z <= 2 ? y = z : y = 2`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.05\n"
     ]
    }
   ],
   "source": [
    "z = 1.05\n",
    "\n",
    "if 1 <= z <= 2          #(a) if true, run the next command (y=z) and then jump to end\n",
    "    y = z\n",
    "elseif z < 1            #(b) if (a) is false, try this instead  \n",
    "    y = 1\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*, *inside functions* and 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.\n",
    "\n",
    "To make sure that the `y` calculated inside the loop does not affect `y` outside the loop, add `local y`.\n",
    "\n",
    "A variable (here `z2`) that does not exist before the loop is local to the loop."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A Simple \"for loop\"\n",
    "\n",
    "The \"for loop\" in the next cell makes 3 iterations and changes a global $x$ variable."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "i=3, x=3 and z2=-998\n",
      "i=6, x=9 and z2=-998\n",
      "i=9, x=18 and z2=-998\n",
      "\n",
      "After loop: x=18 and y=-999\n"
     ]
    }
   ],
   "source": [
    "x = 0\n",
    "y = -999\n",
    "for i = 3:3:9                         #or `i in 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 and z2=$z2\")          #$x prints the value of x\n",
    "end\n",
    "\n",
    "println(\"\\nAfter loop: x=$x and y=$y\")\n",
    "#println(z2)            #does not work: z2 is local to the loop"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A Double \"for loop\"\n",
    "\n",
    "An example of a nested for for loop\n",
    "\n",
    "```\n",
    "for j = 1:n, i = 1:m\n",
    "    #do something\n",
    "end\n",
    "```\n",
    "\n",
    "If you prefer, could also write a longer version to do the same thing\n",
    "```\n",
    "for j = 1:n \n",
    "    for i = 1:m\n",
    "        #do something\n",
    "    end    \n",
    "end\n",
    "```\n",
    "\n",
    "The next cell uses a double loop to fill a matrix. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "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 = 1:m, j = 1:n\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\"\n",
    "\n",
    "The \"while loop\" in the next cell iterates until two variables ($x_0$ and $x_1$) get close.\n",
    "\n",
    "The background to the example is that we want to solve a function $f(x)=x^2$ for the $x$ value that makes $f(x)=2$. The Newton-Raphson algorithm starts with a value $x_0$ and updates it to\n",
    "$\n",
    "x_1 = x_0 + (2-f(x_0))/f'(x_0)\n",
    "$\n",
    "where $f'(x_0)$ is the derivative of $f()$ evaluated at $x_0$. The algorithm iterates until $x_0$ and $x_1$ are close. Clearly, we are trying to find the square root of 2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Solving x^2 = 2 with Newton-Raphson:\n",
      "\n",
      "    10     is changed to      5.100\n",
      "     5.100 is changed to      2.746\n",
      "     2.746 is changed to      1.737\n",
      "     1.737 is changed to      1.444\n",
      "     1.444 is changed to      1.415\n",
      "     1.415 is changed to      1.414\n",
      "\n",
      "The result should be close to      1.414\n"
     ]
    }
   ],
   "source": [
    "println(\"Solving x^2 = 2 with Newton-Raphson:\\n\")\n",
    "\n",
    "x₀ = Inf         #x\\_0 + TAB\n",
    "x₁ = 10\n",
    "\n",
    "while abs(x₁-x₀) > 0.001            #keep going until they get similar\n",
    "    #global x₀, x₁                  #only needed in script\n",
    "    local y, dy                     #don't overwrite any y,dy outside loop\n",
    "    x₀ = x₁                         #initial guess is taken from old guess\n",
    "    y  = x₀^2                       #value of function\n",
    "    dy = 2*x₀                       #derivative of function\n",
    "    x₁ = x₀ + (2 - y)/dy            #updating the guess, Newton-Raphson\n",
    "    printlnPs(x₀,\" is changed to \",x₁)\n",
    "end\n",
    "\n",
    "printlnPs(\"\\nThe result should be close to \",sqrt(2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# A Simple Function\n",
    "\n",
    "The next cell defines a new function, `fn1()`. It takes a scalar input (`x`) and returns a scalar output (`y`).\n",
    "\n",
    "If you instead use a vector as the input, then the computation fails. (The reason is that you cannot do x^2 on a vector. You could on a square matrix, though.)\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": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "fn1 (generic function with 1 method)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function fn1(x)               #define a new function \n",
    "  y = (x-1.1)^2 - 0.5\n",
    "  return y\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "result from fn1(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)\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": [
    "# 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": 17,
   "metadata": {},
   "outputs": [
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       "  348.901,591.928 359.599,608.006 370.297,623.802 380.995,639.317 391.694,654.551 402.392,669.504 413.09,684.175 423.788,698.564 434.486,712.673 445.185,726.5 \n",
       "  455.883,740.046 466.581,753.31 477.279,766.293 487.978,778.995 498.676,791.415 509.374,803.554 520.072,815.412 530.77,826.989 541.469,838.284 552.167,849.298 \n",
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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": 18,
   "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": 19,
   "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": 20,
   "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": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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