Tutorial_03_Functions.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Functions\n",
    "\n",
    "This notebook shows how to define functions with one (or several) inputs and outputs. It also shows how to use to dot (.) syntax to apply a function to each element of an array."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Load Packages and Extra Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Printf\n",
    "\n",
    "include(\"jlFiles/printmat.jl\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Functions with One Output"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Basic Approach\n",
    "\n",
    "The basic approach to define a function with the name `fn` is\n",
    "```\n",
    "function fn(x,b)\n",
    "    ...(some code that calculates ys)\n",
    "    return y\n",
    "end\n",
    "``` \n",
    "\n",
    "Once you have defined a function, you can use it (call on it) by, for instance, `y1 = fn(2,1)`. This will generate a `y1` variable (not a `y` variable) in the workspace. Inside the function, `x` is then 2 and `b` is 1. Clearly, if `x1=2 and b1=1`, you get the same result by calling as `y1 = fn(x1,b1)`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "result from fn1(2,1):      0.310\n"
     ]
    }
   ],
   "source": [
    "function fn1(x,b)                   #x and b are the inputs\n",
    "    c = 0.5                         #c is only \"seen\" inside the function\n",
    "    y = b*(x-1.1)^2 - c\n",
    "    return y                        #this is the output\n",
    "end\n",
    "\n",
    "y1 = fn1(2,1)\n",
    "printlnPs(\"result from fn1(2,1): \",y1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Performance Tips (extra)\n",
    "\n",
    "Wrapping performance critical code in functions often speeds up the computation, since it helps Julia/the compiler to make optimisations."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Performance Tips (extra)\n",
    "\n",
    "Do not use any global variables inside a function. It slows down the computation and can lead to surprising behaviour (see further down in this notebook). Instead, make all the inputs explicit function arguments.\n",
    "\n",
    "(Alternatively, declare the global variables to be constants by `const a = 2`)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Default Values for the Inputs\n",
    "\n",
    "You can change the first line of the function to specify default values as \n",
    "```\n",
    "function fn2(x,b=1)\n",
    "```\n",
    "\n",
    "In this case you can call on the function as `fn2(x)` and the value of `b` will default to 1. (Clearly, inputs with default values should be towards the end of the list of inputs.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "result from fn2(2,1) and fn2(2):      0.310     0.310\n"
     ]
    }
   ],
   "source": [
    "function fn2(x,b=1)                 #b=1 is the default in case we call as fn2(x)\n",
    "    y = b*(x-1.1)^2 - 0.5\n",
    "    return y\n",
    "end\n",
    "\n",
    "printlnPs(\"result from fn2(2,1) and fn2(2): \",fn2(2,1),fn2(2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Elementwise Evaluation\n",
    "\n",
    "To apply the function to each element of arrays `x` and `b`, use the dot syntax (broadcasting): \n",
    "```\n",
    "y = fn.(x,b)\n",
    "```\n",
    "This calculates `fn(x[i],b[i])` for each pair `(x[i],b[i])`.\n",
    "\n",
    "Instead, with `fn.(x,2)`, you calculate `fn(x[i],2)` for each element `x[i]`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "result from fn2.(x1,2): \n",
      "    -0.480\n",
      "    -0.180\n",
      "\n",
      "\n",
      "result from fn2.(x1,b1): \n",
      "    -0.480\n",
      "    -0.180\n",
      "\n"
     ]
    }
   ],
   "source": [
    "x1 = [1,1.5]\n",
    "b1 = [2,2]\n",
    "\n",
    "println(\"\\nresult from fn2.(x1,2): \")\n",
    "printmat(fn2.(x1,2))\n",
    "\n",
    "println(\"\\nresult from fn2.(x1,b1): \")\n",
    "printmat(fn2.(x1,b1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Elementwise Evaluation over *Some* Inputs (extra)\n",
    "\n",
    "\n",
    "To apply the function to each element of the array `x`, but keep `b` fixed, use : \n",
    "```\n",
    "y = fn.(x,Ref(b))\n",
    "```\n",
    "\n",
    "For instance, `fn301.(x1,Ref(b1))` in the next cell will create the 2-element vector\n",
    "```\n",
    "1 + (10+20+30)\n",
    "1.5 + (10+20+30)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "result from fn301.(x1,Ref(b1)):\n",
      "    61.000\n",
      "    61.500\n",
      "\n"
     ]
    }
   ],
   "source": [
    "function fn301(x,b)\n",
    "    y = x + sum(b)\n",
    "    return y\n",
    "end\n",
    "\n",
    "x1 = [1,1.5]\n",
    "b1 = [10,20,30]\n",
    "\n",
    "println(\"result from fn301.(x1,Ref(b1)):\")\n",
    "printmat(fn301.(x1,Ref(b1)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Short Form (extra)\n",
    "\n",
    "We can also use short forms of a function as in the cell below. \n",
    "\n",
    "The first version (`fn3`) is just a single expression. It could span several lines. The second version (`fn3b`) is a sequence of expressions (a \"compound expression\") separated by semicolons (`;`), where the last expression is the function output."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "result from fn3(1.5,1) and fn3b(1.5,1):     -0.340    -0.340\n"
     ]
    }
   ],
   "source": [
    "fn3(x,b) = b*(x-1.1)^2 - 0.5           #short form of a function\n",
    "\n",
    "fn3b(x,b) = (c = 0.5;b*(x-1.1)^2 - c)  #this works too. Notice the ;\n",
    "\n",
    "printlnPs(\"result from fn3(1.5,1) and fn3b(1.5,1): \",fn3(1.5,1),fn3b(1.5,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Explicit Names of the Inputs: Keyword Arguments\n",
    "\n",
    "You can also define functions that take *keyword arguments* like in \n",
    "```\n",
    "function fn(x;b,c)\n",
    "``` \n",
    "Notice the semi-colon (`;`). You can also specify default values as `fn(x;b=1,c=0.5)`\n",
    "\n",
    "In this case, you *call* on the function by `fn(x;c=3,b=2)` or just `fn(x)` if you want to use the default values. This helps remembering/interpreting what the arguments represent. When calling on the function, you can pass the keyword arguments in any order and you can use comma (`,`) instead of semi-colon (`;`). Using the semi-colon has an advantage, though: if you already have `b` defined, then you can call as `fn(x;c=3,b)`.\n",
    "\n",
    "(Extra) You can also use a `Dict()` to supply the keyword arguments as in\n",
    "```\n",
    "opt = Dict(:b=>1,:c=>0.5)\n",
    "fn(x;opt...)\n",
    "```\n",
    "Notice that the dictionary must use symbols and that you need to use `;` and `...` in the call. It is also possible to mix traditional keywords with a dictionary as in `opt = Dict(:c=>0.5); fn(x;b=1,opt...)`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "result from fn4(1,c=3,b=2):     -2.980\n"
     ]
    }
   ],
   "source": [
    "function fn4(x;b=1,c=0.5)\n",
    "    y = b*(x-1.1)^2 - c\n",
    "    return y\n",
    "end\n",
    "\n",
    "b = 2\n",
    "printlnPs(\"result from fn4(1,c=3,b=2): \",fn4(1;c=3,b))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Variable Number of Inputs (extra)\n",
    "\n",
    "When *defining a function*, we can use the `x...` syntax (also called \"slurping\") on the last non-keyword input to capture a variable number of inputs. Later, when calling on this function, several inputs will be combined into a vector `x`.\n",
    "\n",
    "For instance, `fn5(-9,1,2,b=10)` in the next cell will create the 2-element vector\n",
    "```\n",
    "-9 + 1*10\n",
    "-9 + 2*10\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "result from fn5(-9,1,2,b=10): \n",
      "     1.000\n",
      "    11.000\n",
      "\n"
     ]
    }
   ],
   "source": [
    "function fn5(a,x...;b=1)   #a is traditional input, x... captures several inputs; b=1 is a keyword input\n",
    "    n = length(x)\n",
    "    y = zeros(n)\n",
    "    for i = 1:n            #loop over the elements in x\n",
    "        y[i] = a + x[i]*b\n",
    "    end\n",
    "    return y\n",
    "end\n",
    "\n",
    "y = fn5(-9,1,2,b=10)\n",
    "println(\"result from fn5(-9,1,2,b=10): \")\n",
    "printmat(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Conversely, you can also use the `x...` syntax (also called \"splatting\") when *calling on a function* that requires several inputs.\n",
    "\n",
    "In the next cell, when running `fn5b([1,2]...)`, `x1` will be 1 and `x2` will be 2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 and then also 2\n"
     ]
    }
   ],
   "source": [
    "function fn5b(x1,x2)\n",
    "    println(x1,\" and then also \",x2)\n",
    "    return nothing\n",
    "end\n",
    "\n",
    "fn5b([1,2]...)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Functions with Several Outputs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##  Several Outputs 1: Basic Approach\n",
    "\n",
    "A function can produce a \"tuple\" like `(y1,y2,y3)` as output.\n",
    "\n",
    "In case you only want the first two outputs, call as `(y1,y2,) = fn(x)`.\n",
    "\n",
    "Instead, if you only want the second and third outputs, call as `(_,y2,y3) = fn(x)`\n",
    "\n",
    "You can also extract the second output as `y2 = fn(x)[2]`\n",
    "\n",
    "If `Y1` is an already existing array, then `(Y1[3],y2,) = fn(x)` will change element 3 of `Y1` (and also assign a value to `y2`)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The first 2 outputs from fn11(1,2):     -0.480     2    \n",
      "\n",
      "The result from calling fn11(1,2)[2]:      2    \n",
      "\n",
      "Y1 array after calling fn11(1,2):      0.000     0.000    -0.480     0.000     0.000\n"
     ]
    }
   ],
   "source": [
    "function fn11(x,b=1)\n",
    "    y1 = b*(x-1.1)^2 - 0.5\n",
    "    y2 = b*x\n",
    "    y3 = 3\n",
    "    return y1, y2, y3\n",
    "end\n",
    "\n",
    "(y1,y2,) = fn11(1,2)\n",
    "printlnPs(\"The first 2 outputs from fn11(1,2): \",y1,y2)\n",
    "\n",
    "y2 = fn11(1,2)[2]         #grab the second output\n",
    "printlnPs(\"\\nThe result from calling fn11(1,2)[2]: \",y2)\n",
    "\n",
    "Y1 = zeros(5)             #create an array\n",
    "(Y1[3],y2,) = fn11(1,2)   #put result in existing array\n",
    "printlnPs(\"\\nY1 array after calling fn11(1,2): \",Y1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Several Outputs 2: Named Tuples and Dictionaries (extra)\n",
    "\n",
    "\n",
    "Instead of returning several values, it might be easier to combine them into either a \"named tuple\" or a dictionary and then export that.\n",
    "\n",
    "You could end the function with \n",
    "\n",
    "```\n",
    "    y = (a=y1,b=y2,c=y3)                     #named tuple\n",
    "    return y\n",
    "end\n",
    "```\n",
    "\n",
    "or \n",
    "\n",
    "```\n",
    "   y  = Dict(:a=>y1,:b=>y2,:c=>y3)           #dictionary\n",
    "   return y\n",
    "end\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Several Outputs 3: Elementwise Evaluation (extra)\n",
    "\n",
    "...can be tricky, because you get an array (same dimension as the input) of tuples instead of a tuple of arrays (which is probably what you want).\n",
    "\n",
    "One way around this is to reshuffle the output to get a tuple of arrays.\n",
    "\n",
    "Alternatively, you could loop over the function calls or write the function in such a way that it directly handles array inputs (without the dot). The latter is done in `fn14()`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y, a 2-element vector of tuples (3 elements in each):\n",
      "(-0.48, 2.0, 3)\n",
      "(-0.18000000000000016, 3.0, 3)\n",
      "\n",
      "\n",
      "the vectors y1, y2 and y3:\n",
      "    -0.480     2.000     3.000\n",
      "    -0.180     3.000     3.000\n",
      "\n"
     ]
    }
   ],
   "source": [
    "y = fn11.([1,1.5],2)\n",
    "println(\"y, a 2-element vector of tuples (3 elements in each):\")\n",
    "printmat(y)\n",
    "\n",
    "(y1,y2,y3) = ntuple(i->getindex.(y,i),3)    #split up into 3 vectors\n",
    "\n",
    "println(\"\\nthe vectors y1, y2 and y3:\")\n",
    "printmat([y1 y2 y3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "result from fn14(x1,2): \n",
      "    -0.480     2.000\n",
      "    -0.180     3.000\n",
      "\n",
      "     3    \n",
      "\n"
     ]
    }
   ],
   "source": [
    "function fn14(x,b=1)                 #x can be an array\n",
    "    y1 = b*(x.-1.1).^2 .- 0.5\n",
    "    y2 = b*x\n",
    "    y3 = 3\n",
    "    return y1, y2, y3\n",
    "end\n",
    "\n",
    "(y1,y2,y3) = fn14(x1,2)               #function written to handle arrays\n",
    "println(\"result from fn14(x1,2): \")\n",
    "printmat([y1 y2])\n",
    "printmat(y3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Documenting Your Function \n",
    "\n",
    "To use Julia's help function (`?FunctionName`, `@doc FunctionName`) or `@doc2 FunctionName` (in VS Code, using the macro from printmat.jl), put the documentation in triple quotes, just above the function definition. (No empty lines between the last triple quote and the start of the function.) The cell below illustrates a simple case."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "fn101"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\"\n",
    "    fn101(x,b=1)\n",
    "\n",
    "Calculate `b*(x-1.1)^2 - 0.5`.\n",
    "\n",
    "### Arguments\n",
    "- `x::Number`:    an important number\n",
    "- `b::Number`:    another number\n",
    "\n",
    "\"\"\"\n",
    "function fn101(x,b=1)\n",
    "    y = b*(x-1.1)^2 - 0.5\n",
    "    return y\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "```\n",
       "fn101(x,b=1)\n",
       "```\n",
       "\n",
       "Calculate `b*(x-1.1)^2 - 0.5`.\n",
       "\n",
       "### Arguments\n",
       "\n",
       "  * `x::Number`:    an important number\n",
       "  * `b::Number`:    another number\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "@doc2 fn101      #get help on the fn101 function, macro defined in printmat.jl\n",
    "#?fn101          #works in REPL and jupyter, not in VS Code"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Sending a Function to a Function\n",
    "\n",
    "You can use a function name (for instance, `cos`) as an input to another function. Alternatively, use an anonymous function (see `->` below)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "result from fn201(cos,3.145,10):      9.000\n",
      "\n",
      "result from fn201(z->mod(z,2),3.145,10):     11.145\n"
     ]
    }
   ],
   "source": [
    "function fn201(f,a,b)   #f is a function, could also write f::Function\n",
    "    y = f(a) + b\n",
    "    return y\n",
    "end\n",
    "\n",
    "#here f is the cos function\n",
    "y1 = fn201(cos,3.145,10)\n",
    "printlnPs(\"result from fn201(cos,3.145,10): \",y1)  \n",
    "\n",
    "#here f is z->mod(z,2), an anonymous function\n",
    "y2 = fn201(z->mod(z,2),3.145,10)\n",
    "printlnPs(\"\\nresult from fn201(z->mod(z,2),3.145,10): \",y2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Performance Tips (extra)\n",
    "\n",
    "Anonymous functions like `z->fn(z,c)` can be slow when `c` is a global variable. You can speed it up by wrapping the call in a function (see below).\n",
    "\n",
    "However, defining both `c` and `z->fn(z,c)` inside a loop will not cause a slowdown (`c` is now local to the loop)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "11.145\n",
      "\n",
      "    11.145\n",
      "    10.145\n",
      "\n"
     ]
    }
   ],
   "source": [
    "DoTheCalc(c) = fn201(z->mod(z,c),3.145,10)    #wrapper\n",
    "\n",
    "c = 2\n",
    "println(DoTheCalc(c),\"\\n\")\n",
    "\n",
    "x = fill(NaN,2)\n",
    "for i = 1:2                                   #loop\n",
    "    local c                                   #`c` is a local variable\n",
    "    c = 2/i\n",
    "    x[i] = fn201(z->mod(z,c),3.145,10)\n",
    "end\n",
    "printmat(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Global Variables in a Function (extra)\n",
    "\n",
    "is a bad idea. It slows down the computations and can lead to surprising behaviour (illustrated in the first cell below). \n",
    "\n",
    "The problem (and the surprise below) is that global variables in a function are taken from the scope where the function was *defined*. For instance, the function `fn101()` below is defined at the top level (this notebook) so it will use the global variable `a` found there. This holds even if `fn101()` is called from another function with its own `a` variable."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[34m\u001b[1mFrom fn101(0):\u001b[22m\u001b[39m\n",
      "a and a+x: 1 1\n",
      "\n",
      "\u001b[34m\u001b[1mFrom fn102(0,10):\u001b[22m\u001b[39m\n",
      "a according to fn102: 10\n",
      "a and a+x: 1 1\n",
      "\n",
      "\u001b[34m\u001b[1mFrom fn103(0,10):\u001b[22m\u001b[39m\n",
      "a and a+x: 10 10\n"
     ]
    }
   ],
   "source": [
    "fn101(x) = println(\"a and a+x: \",a,\" \",a + x)\n",
    "a = 1\n",
    "printblue(\"From fn101(0):\")\n",
    "fn101(0)\n",
    "\n",
    "function fn102(x,a)         #does not solve the problem \n",
    "    println(\"a according to fn102: \",a)\n",
    "    fn101(x)                #since `fn101()` (used inside `fn102`) still uses the global `a`\n",
    "end\n",
    "printblue(\"\\nFrom fn102(0,10):\")\n",
    "fn102(0,10)                  \n",
    "\n",
    "fn103(x,a) = println(\"a and a+x: \",a,\" \",a + x)       #this works the way it should\n",
    "printblue(\"\\nFrom fn103(0,10):\")\n",
    "fn103(0,10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In contrast, this is not a problem if the function is *defined inside another function*. (This way of 'capturing' variables is often used.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a and a+x: 10 10\n"
     ]
    }
   ],
   "source": [
    "function fn202(x,a)      #allows us to set a, could also set a inside fn\n",
    "    fn201(x) = println(\"a and a+x: \",a,\" \",a + x)   #same as before, but defined inside fn202()\n",
    "    fn201(x)\n",
    "end\n",
    "\n",
    "fn202(0,10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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