Julia-introduction-SIAM-SC.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction to Julia, \n",
    "## SIAM Student Chapter KU Leuven\n",
    "\n",
    "This is a small tutorial to get to know some of the interesting functions of Julia, assuming the reader has prior general knowledge in programming languages (C++, Python, Matlab). \n",
    "\n",
    "This notebook is a collection of small code snippets illustrating different aspects of Julia.\n",
    "\n",
    "Most examples (~98%) where taken from [the notebooks of Daan Huybrechs](https://github.com/daanhb/Julia-tutorial)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Basic syntax\n",
    "#### 1. Variables\n",
    "\n",
    "You don't declare any types, but Julia will always deduce them (and if it can't, the type of your variable will be Any). Read all about variables [in the documentation](https://docs.julialang.org/en/v1/manual/variables/)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Int64"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "typeof(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Float64"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = 8.0\n",
    "typeof(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Array{Any,1}:\n",
       " 2       \n",
       " 5.0     \n",
       "  \"hello\""
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = [2, 5.0, \"hello\"]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Variables are always references!\n",
    "\n",
    "This is important to note from the start: variables in Julia are just names associated with a value. This means we always have **reference semantics**! This is (very) different from Matlab.\n",
    "\n",
    "For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Array{Int64,1}:\n",
       " 1\n",
       " 2\n",
       " 3\n",
       " 4"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = [1, 2, 3, 4]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Array{Int64,1}:\n",
       " 1\n",
       " 5\n",
       " 3\n",
       " 4"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = a\n",
    "b[2] = 5\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Array{Int64,1}:\n",
       " 1\n",
       " 5\n",
       " 3\n",
       " 4"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can avoid reference semantics with an explicit copy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Array{Int64,1}:\n",
       " 1\n",
       " 7\n",
       " 3\n",
       " 4"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = copy(b)\n",
    "c[2] = 7\n",
    "c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Array{Int64,1}:\n",
       " 1\n",
       " 5\n",
       " 3\n",
       " 4"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since arrays are always passed by references, it becomes easy to make functions that modify their arguments - you can make **in-place algorithms**. Note for future reference the Julia convention of naming functions that modify one of their arguments in a particular way, namely ending in an exclamation point. For example, [see](https://docs.julialang.org/en/v1/base/collections/#Base.push!) the `push!` function that changes a vector by adding an element to it.\n",
    "\n",
    "More importantly, reference semantics lets you avoid making unnecessary copies of data all over the place. Matlab avoids the copying using a copy-on-write mechanism. This is a valid choice to make, but it is still less efficient than just treating everything as a reference."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = 4\n",
    "b = a\n",
    "b = 5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Modifying b did not modify a. Why? Because we did not actually modify b at all! The line b = 5 simply made b refer to a different integer, unrelated to what it was pointing at before. It is a new assignment, no different from the line a = 4 above.\n",
    "\n",
    "In any case, you can not change the value of an integer, because it is immutable. Arrays are mutable, in the sense that you can alter the third entry of an array. This does not make integers special in any way. Julia has mutable and immutable types, we will see that later.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 2. More on vectors\n",
    "Vector can easily be generated using square brackets (see above), or using **list comprehensions**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Array{Int64,1}:\n",
       "       1\n",
       "       2\n",
       "       6\n",
       "      24\n",
       "     120\n",
       "     720\n",
       "    5040\n",
       "   40320\n",
       "  362880\n",
       " 3628800"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = [factorial(i) for i=1:10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 Array{Float64,2}:\n",
       " 0.5       0.333333  0.25      0.2     \n",
       " 0.333333  0.25      0.2       0.166667\n",
       " 0.25      0.2       0.166667  0.142857\n",
       " 0.2       0.166667  0.142857  0.125   "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = [1/(i+j) for i=1:4,j=1:4]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 3. Linear algebra\n",
    "Most linear algebra operations you may know from Matlab also exist in Julia. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LinearAlgebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LinearAlgebra.QRCompactWY{Float64,Array{Float64,2}}\n",
       "Q factor:\n",
       "4×4 LinearAlgebra.QRCompactWYQ{Float64,Array{Float64,2}}:\n",
       " -0.734333   0.630046  -0.247002   -0.0527918\n",
       " -0.489555  -0.248892   0.735949    0.395938 \n",
       " -0.367167  -0.487684   0.0168558  -0.791877 \n",
       " -0.293733  -0.550689  -0.630147    0.461928 \n",
       "R factor:\n",
       "4×4 Array{Float64,2}:\n",
       " -0.68089  -0.489555   -0.384651    -0.317628  \n",
       "  0.0      -0.0415261  -0.0522176   -0.0539782 \n",
       "  0.0       0.0        -0.00177233  -0.00310257\n",
       "  0.0       0.0         0.0          4.71355e-5"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q,R = qr(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "ename": "UndefVarError",
     "evalue": "UndefVarError: eig not defined",
     "output_type": "error",
     "traceback": [
      "UndefVarError: eig not defined",
      "",
      "Stacktrace:",
      " [1] top-level scope at In[16]:1"
     ]
    }
   ],
   "source": [
    "eig(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Control flow\n",
    "The famous `for` and `if` statements work as expected."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Porsche\n",
      "Audi\n",
      "Tesla\n"
     ]
    }
   ],
   "source": [
    "cars = [\"Porsche\", \"Audi\", \"Tesla\"]\n",
    "for brand in cars\n",
    "   println(brand) \n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following type of Matlab style ranges (for example 1:5) are possible."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Hi from iteration 1\n",
      "Hi from iteration 2\n",
      "Hi from iteration 3\n",
      "Hi from iteration 4\n",
      "Hi from iteration 5\n"
     ]
    }
   ],
   "source": [
    "for i in 1:5\n",
    "   println(\"Hi from iteration $i\") \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Statement is true.\n"
     ]
    }
   ],
   "source": [
    "if 2<3\n",
    "    println(\"Statement is true.\")\n",
    "else\n",
    "    println(\"Trump is a lie.\")\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Functions\n",
    "Functions are a bit like in Matlab, except that you don't specify output variables. The last expression that is evaluated yields the return value (like it is in, say, Maple)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "fibonacci (generic function with 1 method)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function fibonacci(n)\n",
    "    if (n == 1) || (n==0)\n",
    "        1\n",
    "    else\n",
    "        fibonacci(n-1) + fibonacci(n-2)\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fibonacci(5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can also explicitly write `return` if you like"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function my_maximum(x,y)\n",
    "    if x > y\n",
    "        return x\n",
    "    else\n",
    "        return y\n",
    "    end\n",
    "end\n",
    "my_maximum(2,3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, you can create anonymous functions.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "#5 (generic function with 1 method)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x -> cos(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can pass around functions as arguments, including operators (which are really just functions)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "composite (generic function with 1 method)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "composite(f, g, x) = f(g(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "composite(-, -, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Type system\n",
    "\n",
    "We have already encounted a few types, including the numeric types Int64, Float64. Writing Julia revolves a lot around types. You will be making lots of them, and using them all the time. You can be generous with new types, they are not scarce. Sometimes code you write creates types implicitly just to get things done - in that case the type is essentially a use-once-throw-away commodity.\n",
    "\n",
    "#### 1. Composite type\n",
    "You can compare a Julia type with a struct in C: it is an object that collects data. Immutable types are generally defined in Julia as a `struct`, and mutable types are defined using `mutable struct`. Member data are called fields. The data can be named and you define a type by listing the names of its fields."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "mutable struct MyType\n",
    "    a\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "MyType(5)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = MyType(5)    # We instantiate an object of type MyType. There is a default constructor."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use untyped fields when you care about flexibility and simplicity more than performance. They do have unavoidable runtime overhead, so don't use untyped fields in a time-critical path of your code. In that case, do the following:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "mutable struct MyType2\n",
    "    a :: Float64\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "MyType2(2.0)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = MyType2(2)      # Note the integer I've supplied is automatically converted to a float"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "ename": "MethodError",
     "evalue": "MethodError: Cannot `convert` an object of type String to an object of type Float64\nClosest candidates are:\n  convert(::Type{T}, !Matched::T) where T<:Number at number.jl:6\n  convert(::Type{T}, !Matched::Number) where T<:Number at number.jl:7\n  convert(::Type{T}, !Matched::Base.TwicePrecision) where T<:Number at twiceprecision.jl:250\n  ...",
     "output_type": "error",
     "traceback": [
      "MethodError: Cannot `convert` an object of type String to an object of type Float64\nClosest candidates are:\n  convert(::Type{T}, !Matched::T) where T<:Number at number.jl:6\n  convert(::Type{T}, !Matched::Number) where T<:Number at number.jl:7\n  convert(::Type{T}, !Matched::Base.TwicePrecision) where T<:Number at twiceprecision.jl:250\n  ...",
      "",
      "Stacktrace:",
      " [1] setproperty!(::MyType2, ::Symbol, ::String) at ./Base.jl:21",
      " [2] top-level scope at In[32]:1"
     ]
    }
   ],
   "source": [
    "v.a = \"I'm an evil string.\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is a difference in runtime. Let's read the field's value of the declared type many times."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "add_field_values_many_times (generic function with 1 method)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function add_field_values_many_times(m)\n",
    "    z = 0.0\n",
    "    for i = 1:10000\n",
    "        z = z + m.a\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.000207 seconds (10.01 k allocations: 156.438 KiB)\n"
     ]
    }
   ],
   "source": [
    "add_field_values_many_times(MyType(10.0))\n",
    "@time add_field_values_many_times(MyType(10.0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.000006 seconds (5 allocations: 176 bytes)\n"
     ]
    }
   ],
   "source": [
    "add_field_values_many_times(MyType2(10.0))\n",
    "@time add_field_values_many_times(MyType2(10.0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is much more too say about it (immutable types,etc ), be sure to check the documentation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 2. Parametric types"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "struct Point{T}\n",
    "    x :: T\n",
    "    y :: T\n",
    "    z :: T\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Point{Float64}(0.1, 0.2, 0.4)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = Point(0.1, 0.2, 0.4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.1"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Why would I define a point this way? I want to specify the types of x, y and z, so that Julia knows them and produces optimized code. But I don't want to specify that they are Float64. What if my user wants to use BigFloat's? Or integers? Or something else entirely I do not know about, some user-defined numeric type? In this case, parameters are the answer.\n",
    "\n",
    "Type parameters in Julia look a lot like C++ template parameters. A major difference is that C++ templates are mostly syntactic sugar at compile-time. You could achieve what they do, if you have the patience, by using copy-paste of text over and over again. In Julia, parametric types and methods are a major part of the language in all stages of execution.\n",
    "\n",
    "I am skipping over many things here, like default and user-supplied constructors. There are inner and outer constructors. They can be painful at times. Please read the manual on [constructors](https://docs.julialang.org/en/v1/manual/constructors/), especially [parametric constructors](https://docs.julialang.org/en/v1/manual/constructors/#Parametric-Constructors-1), and don't complain to me.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### 3. Abstract types\n",
    "Sure enough, types can inherit from other types. You can create an abstract type and then inherit from it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "abstract type AbstractPoint \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "struct Point2d <: AbstractPoint\n",
    "    x :: Float64\n",
    "    y :: Float64\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "struct Point3d <: AbstractPoint\n",
    "    x :: Float64\n",
    "    y :: Float64\n",
    "    z :: Float64\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Point2d"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p2 = Point2d(0.1, 2.0)\n",
    "typeof(p2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "AbstractPoint"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "supertype(Point3d)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Any type that is not abstract is a concrete type. You can not inherit from concrete types. For example, we could not create a subset of Point2d's, say a subset that has unit norm, as follows:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "ename": "ErrorException",
     "evalue": "invalid subtyping in definition of Point2d_with_unit_norm",
     "output_type": "error",
     "traceback": [
      "invalid subtyping in definition of Point2d_with_unit_norm",
      "",
      "Stacktrace:",
      " [1] top-level scope at /Users/andrew/.julia/packages/IJulia/DrVMH/src/kernel.jl:52"
     ]
    }
   ],
   "source": [
    "struct Point2d_with_unit_norm <: Point2d\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Multiple dispatch\n",
    "But what are types? All variables have a type. It is compiler metadata that describes what your variable stands for. Julia types compare a little bit to classes in an object-oriented language, but one should not stretch the comparison too much. Types do support inheritance. However, you are well advised to resist that initial urge to create a fancy hierarchical tree of types like you would in OOP: there is no need for that. The best reason to use types in Julia (and some would say the only reason) is to use multiple dispatch, and that is the topic of the next chapter.\n",
    "\n",
    "Without multiple dispatch it is hard to convey the richness of the type system. Still, we have a quite a few things to discover already. \n",
    "\n",
    "#### Methods are algorithms are methods: Julia selects the best available algorithm for the problem at hand\n",
    "Function overloading is possible in many typed languagues, but in Julia it is really pervasive. You add methods to existing functions all the time. Popular ones are the basic operators, like +:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "# 166 methods for generic function <b>+</b>:<ul><li> +(x::<b>Bool</b>, z::<b>Complex{Bool}</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/complex.jl#L282\" target=\"_blank\">complex.jl:282</a></li> <li> +(x::<b>Bool</b>, y::<b>Bool</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/bool.jl#L96\" target=\"_blank\">bool.jl:96</a></li> <li> +(x::<b>Bool</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/bool.jl#L93\" target=\"_blank\">bool.jl:93</a></li> <li> +(x::<b>Bool</b>, y::<b>T</b>)<i> where T<:AbstractFloat</i> in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/bool.jl#L104\" target=\"_blank\">bool.jl:104</a></li> <li> +(x::<b>Bool</b>, z::<b>Complex</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/complex.jl#L289\" target=\"_blank\">complex.jl:289</a></li> <li> +(a::<b>Float16</b>, b::<b>Float16</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/float.jl#L398\" target=\"_blank\">float.jl:398</a></li> <li> +(x::<b>Float32</b>, y::<b>Float32</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/float.jl#L400\" target=\"_blank\">float.jl:400</a></li> <li> +(x::<b>Float64</b>, y::<b>Float64</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/float.jl#L401\" target=\"_blank\">float.jl:401</a></li> <li> +(z::<b>Complex{Bool}</b>, x::<b>Bool</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/complex.jl#L283\" target=\"_blank\">complex.jl:283</a></li> <li> +(z::<b>Complex{Bool}</b>, x::<b>Real</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/complex.jl#L297\" target=\"_blank\">complex.jl:297</a></li> <li> +(::<b>Missing</b>, ::<b>Missing</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/missing.jl#L114\" target=\"_blank\">missing.jl:114</a></li> <li> +(::<b>Missing</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/missing.jl#L100\" target=\"_blank\">missing.jl:100</a></li> <li> +(::<b>Missing</b>, ::<b>Number</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/missing.jl#L115\" target=\"_blank\">missing.jl:115</a></li> <li> +(level::<b>Base.CoreLogging.LogLevel</b>, inc::<b>Integer</b>) in Base.CoreLogging at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/logging.jl#L106\" target=\"_blank\">logging.jl:106</a></li> <li> +(c::<b>BigInt</b>, x::<b>BigFloat</b>) in Base.MPFR at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/mpfr.jl#L413\" target=\"_blank\">mpfr.jl:413</a></li> <li> +(a::<b>BigInt</b>, b::<b>BigInt</b>, c::<b>BigInt</b>, d::<b>BigInt</b>, e::<b>BigInt</b>) in Base.GMP at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/gmp.jl#L503\" target=\"_blank\">gmp.jl:503</a></li> <li> +(a::<b>BigInt</b>, b::<b>BigInt</b>, c::<b>BigInt</b>, d::<b>BigInt</b>) in Base.GMP at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/gmp.jl#L502\" target=\"_blank\">gmp.jl:502</a></li> <li> +(a::<b>BigInt</b>, b::<b>BigInt</b>, c::<b>BigInt</b>) in Base.GMP at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/gmp.jl#L501\" target=\"_blank\">gmp.jl:501</a></li> <li> +(x::<b>BigInt</b>, y::<b>BigInt</b>) in Base.GMP at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/gmp.jl#L472\" target=\"_blank\">gmp.jl:472</a></li> <li> +(x::<b>BigInt</b>, c::<b>Union{UInt16, UInt32, UInt64, UInt8}</b>) in Base.GMP at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/gmp.jl#L509\" target=\"_blank\">gmp.jl:509</a></li> <li> +(x::<b>BigInt</b>, c::<b>Union{Int16, Int32, Int64, Int8}</b>) in Base.GMP at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/gmp.jl#L515\" target=\"_blank\">gmp.jl:515</a></li> <li> +(a::<b>BigFloat</b>, b::<b>BigFloat</b>, c::<b>BigFloat</b>, d::<b>BigFloat</b>, e::<b>BigFloat</b>) in Base.MPFR at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/mpfr.jl#L563\" target=\"_blank\">mpfr.jl:563</a></li> <li> +(a::<b>BigFloat</b>, b::<b>BigFloat</b>, c::<b>BigFloat</b>, d::<b>BigFloat</b>) in Base.MPFR at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/mpfr.jl#L556\" target=\"_blank\">mpfr.jl:556</a></li> <li> +(a::<b>BigFloat</b>, b::<b>BigFloat</b>, c::<b>BigFloat</b>) in Base.MPFR at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/mpfr.jl#L550\" target=\"_blank\">mpfr.jl:550</a></li> <li> +(x::<b>BigFloat</b>, c::<b>BigInt</b>) in Base.MPFR at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/mpfr.jl#L409\" target=\"_blank\">mpfr.jl:409</a></li> <li> +(x::<b>BigFloat</b>, y::<b>BigFloat</b>) in Base.MPFR at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/mpfr.jl#L378\" target=\"_blank\">mpfr.jl:378</a></li> <li> +(x::<b>BigFloat</b>, c::<b>Union{UInt16, UInt32, UInt64, UInt8}</b>) in Base.MPFR at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/mpfr.jl#L385\" target=\"_blank\">mpfr.jl:385</a></li> <li> +(x::<b>BigFloat</b>, c::<b>Union{Int16, Int32, Int64, Int8}</b>) in Base.MPFR at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/mpfr.jl#L393\" target=\"_blank\">mpfr.jl:393</a></li> <li> +(x::<b>BigFloat</b>, c::<b>Union{Float16, Float32, Float64}</b>) in Base.MPFR at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/mpfr.jl#L401\" target=\"_blank\">mpfr.jl:401</a></li> <li> +(x::<b>Dates.CompoundPeriod</b>, y::<b>Dates.CompoundPeriod</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl:343</a></li> <li> +(x::<b>Dates.CompoundPeriod</b>, y::<b>Dates.Period</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl:341</a></li> <li> +(x::<b>Dates.CompoundPeriod</b>, y::<b>Dates.TimeType</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl:371</a></li> <li> +(x::<b>Dates.Date</b>, y::<b>Dates.Day</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:74</a></li> <li> +(x::<b>Dates.Date</b>, y::<b>Dates.Week</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:72</a></li> <li> +(dt::<b>Dates.Date</b>, z::<b>Dates.Month</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:55</a></li> <li> +(dt::<b>Dates.Date</b>, y::<b>Dates.Year</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:28</a></li> <li> +(dt::<b>Dates.Date</b>, t::<b>Dates.Time</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:19</a></li> <li> +(t::<b>Dates.Time</b>, dt::<b>Dates.Date</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:20</a></li> <li> +(x::<b>Dates.Time</b>, y::<b>Dates.TimePeriod</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:78</a></li> <li> +(dt::<b>Dates.DateTime</b>, z::<b>Dates.Month</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:48</a></li> <li> +(dt::<b>Dates.DateTime</b>, y::<b>Dates.Year</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:24</a></li> <li> +(x::<b>Dates.DateTime</b>, y::<b>Dates.Period</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:76</a></li> <li> +(B::<b>BitArray{2}</b>, J::<b>UniformScaling</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl:115</a></li> <li> +(a::<b>Pkg.Resolve.VersionWeights.VersionWeight</b>, b::<b>Pkg.Resolve.VersionWeights.VersionWeight</b>) in Pkg.Resolve.VersionWeights at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Pkg/src/resolve/VersionWeights.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Pkg/src/resolve/VersionWeights.jl:26</a></li> <li> +(a::<b>Pkg.Resolve.MaxSum.FieldValues.FieldValue</b>, b::<b>Pkg.Resolve.MaxSum.FieldValues.FieldValue</b>) in Pkg.Resolve.MaxSum.FieldValues at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Pkg/src/resolve/FieldValues.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Pkg/src/resolve/FieldValues.jl:49</a></li> <li> +(y::<b>AbstractFloat</b>, x::<b>Bool</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/bool.jl#L106\" target=\"_blank\">bool.jl:106</a></li> <li> +(x::<b>T</b>, y::<b>T</b>)<i> where T<:Union{Int128, Int16, Int32, Int64, Int8, UInt128, UInt16, UInt32, UInt64, UInt8}</i> in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/int.jl#L53\" target=\"_blank\">int.jl:53</a></li> <li> +(c::<b>Union{UInt16, UInt32, UInt64, UInt8}</b>, x::<b>BigInt</b>) in Base.GMP at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/gmp.jl#L510\" target=\"_blank\">gmp.jl:510</a></li> <li> +(c::<b>Union{Int16, Int32, Int64, Int8}</b>, x::<b>BigInt</b>) in Base.GMP at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/gmp.jl#L516\" target=\"_blank\">gmp.jl:516</a></li> <li> +(a::<b>Integer</b>, b::<b>Integer</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/int.jl#L871\" target=\"_blank\">int.jl:871</a></li> <li> +(x::<b>Integer</b>, y::<b>Ptr</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/pointer.jl#L161\" target=\"_blank\">pointer.jl:161</a></li> <li> +(z::<b>Complex</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/complex.jl#L269\" target=\"_blank\">complex.jl:269</a></li> <li> +(z::<b>Complex</b>, w::<b>Complex</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/complex.jl#L271\" target=\"_blank\">complex.jl:271</a></li> <li> +(z::<b>Complex</b>, x::<b>Bool</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/complex.jl#L290\" target=\"_blank\">complex.jl:290</a></li> <li> +(x::<b>Rational</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/rational.jl#L241\" target=\"_blank\">rational.jl:241</a></li> <li> +(x::<b>Real</b>, z::<b>Complex{Bool}</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/complex.jl#L296\" target=\"_blank\">complex.jl:296</a></li> <li> +(x::<b>Real</b>, z::<b>Complex</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/complex.jl#L308\" target=\"_blank\">complex.jl:308</a></li> <li> +(z::<b>Complex</b>, x::<b>Real</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/complex.jl#L309\" target=\"_blank\">complex.jl:309</a></li> <li> +(x::<b>Rational</b>, y::<b>Rational</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/rational.jl#L257\" target=\"_blank\">rational.jl:257</a></li> <li> +(x::<b>Integer</b>, y::<b>AbstractChar</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/char.jl#L224\" target=\"_blank\">char.jl:224</a></li> <li> +(c::<b>Union{UInt16, UInt32, UInt64, UInt8}</b>, x::<b>BigFloat</b>) in Base.MPFR at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/mpfr.jl#L389\" target=\"_blank\">mpfr.jl:389</a></li> <li> +(c::<b>Union{Int16, Int32, Int64, Int8}</b>, x::<b>BigFloat</b>) in Base.MPFR at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/mpfr.jl#L397\" target=\"_blank\">mpfr.jl:397</a></li> <li> +(c::<b>Union{Float16, Float32, Float64}</b>, x::<b>BigFloat</b>) in Base.MPFR at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/mpfr.jl#L405\" target=\"_blank\">mpfr.jl:405</a></li> <li> +(x::<b>AbstractIrrational</b>, y::<b>AbstractIrrational</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/irrationals.jl#L137\" target=\"_blank\">irrationals.jl:137</a></li> <li> +(x::<b>Number</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/operators.jl#L504\" target=\"_blank\">operators.jl:504</a></li> <li> +(x::<b>T</b>, y::<b>T</b>)<i> where T<:Number</i> in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/promotion.jl#L387\" target=\"_blank\">promotion.jl:387</a></li> <li> +(x::<b>Number</b>, y::<b>Number</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/promotion.jl#L311\" target=\"_blank\">promotion.jl:311</a></li> <li> +(r1::<b>OrdinalRange</b>, r2::<b>OrdinalRange</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/range.jl#L1011\" target=\"_blank\">range.jl:1011</a></li> <li> +(r1::<b>LinRange{T}</b>, r2::<b>LinRange{T}</b>)<i> where T</i> in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/range.jl#L1018\" target=\"_blank\">range.jl:1018</a></li> <li> +(r1::<b>StepRangeLen{T,R,S} where S</b>, r2::<b>StepRangeLen{T,R,S} where S</b>)<i> where {R<:Base.TwicePrecision, T}</i> in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/twiceprecision.jl#L557\" target=\"_blank\">twiceprecision.jl:557</a></li> <li> +(r1::<b>StepRangeLen{T,S,S1} where S1</b>, r2::<b>StepRangeLen{T,S,S1} where S1</b>)<i> where {T, S}</i> in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/range.jl#L1034\" target=\"_blank\">range.jl:1034</a></li> <li> +(r1::<b>Union{LinRange, OrdinalRange, StepRangeLen}</b>, r2::<b>Union{LinRange, OrdinalRange, StepRangeLen}</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/range.jl#L1026\" target=\"_blank\">range.jl:1026</a></li> <li> +(x::<b>Ptr</b>, y::<b>Integer</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/pointer.jl#L159\" target=\"_blank\">pointer.jl:159</a></li> <li> +(x::<b>Base.TwicePrecision</b>, y::<b>Number</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/twiceprecision.jl#L265\" target=\"_blank\">twiceprecision.jl:265</a></li> <li> +(x::<b>Number</b>, y::<b>Base.TwicePrecision</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/twiceprecision.jl#L268\" target=\"_blank\">twiceprecision.jl:268</a></li> <li> +(x::<b>Base.TwicePrecision{T}</b>, y::<b>Base.TwicePrecision{T}</b>)<i> where T</i> in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/twiceprecision.jl#L271\" target=\"_blank\">twiceprecision.jl:271</a></li> <li> +(x::<b>Base.TwicePrecision</b>, y::<b>Base.TwicePrecision</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/twiceprecision.jl#L275\" target=\"_blank\">twiceprecision.jl:275</a></li> <li> +(A::<b>Array</b>, Bs::<b>Array...</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/arraymath.jl#L44\" target=\"_blank\">arraymath.jl:44</a></li> <li> +(A::<b>BitArray</b>, B::<b>BitArray</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/bitarray.jl#L1084\" target=\"_blank\">bitarray.jl:1084</a></li> <li> +(r::<b>AbstractRange{#s627} where #s627<:Dates.TimeType</b>, x::<b>Dates.Period</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/ranges.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/ranges.jl:59</a></li> <li> +(A::<b>Array</b>, B::<b>SparseArrays.SparseMatrixCSC</b>) in SparseArrays at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/SparseArrays/src/sparsematrix.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/SparseArrays/src/sparsematrix.jl:1647</a></li> <li> +(x::<b>Union{DenseArray{#s627,N}, Base.ReinterpretArray{#s627,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray}, Base.ReshapedArray{#s627,N,A,MI} where MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray}, SubArray{#s627,N,A,I,L} where L where I<:Tuple{Vararg{Union{Int64, AbstractRange{Int64}, Base.AbstractCartesianIndex},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, Base.ReshapedArray{T,N,A,MI} where MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, DenseArray}} where N where #s627<:Union{Dates.CompoundPeriod, Dates.Period}</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/deprecated.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/deprecated.jl:57</a></li> <li> +(x::<b>Union{DenseArray{#s627,N}, Base.ReinterpretArray{#s627,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray}, Base.ReshapedArray{#s627,N,A,MI} where MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray}, SubArray{#s627,N,A,I,L} where L where I<:Tuple{Vararg{Union{Int64, AbstractRange{Int64}, Base.AbstractCartesianIndex},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, Base.ReshapedArray{T,N,A,MI} where MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, DenseArray}} where N where #s627<:Union{Dates.CompoundPeriod, Dates.Period}</b>, y::<b>Dates.TimeType</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/deprecated.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/deprecated.jl:12</a></li> <li> +(X::<b>Union{DenseArray{#s627,N}, Base.ReinterpretArray{#s627,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray}, Base.ReshapedArray{#s627,N,A,MI} where MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray}, SubArray{#s627,N,A,I,L} where L where I<:Tuple{Vararg{Union{Int64, AbstractRange{Int64}, Base.AbstractCartesianIndex},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, Base.ReshapedArray{T,N,A,MI} where MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, DenseArray}} where N where #s627<:Union{Dates.CompoundPeriod, Dates.Period}</b>, Y::<b>Union{DenseArray{#s626,N}, Base.ReinterpretArray{#s626,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray}, Base.ReshapedArray{#s626,N,A,MI} where MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray}, SubArray{#s626,N,A,I,L} where L where I<:Tuple{Vararg{Union{Int64, AbstractRange{Int64}, Base.AbstractCartesianIndex},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, Base.ReshapedArray{T,N,A,MI} where MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, DenseArray}} where N where #s626<:Union{Dates.CompoundPeriod, Dates.Period}</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/deprecated.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/deprecated.jl:64</a></li> <li> +(A::<b>SymTridiagonal</b>, B::<b>SymTridiagonal</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/tridiag.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/tridiag.jl:158</a></li> <li> +(A::<b>Tridiagonal</b>, B::<b>Tridiagonal</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/tridiag.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/tridiag.jl:636</a></li> <li> +(A::<b>UpperTriangular</b>, B::<b>UpperTriangular</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl:554</a></li> <li> +(A::<b>LowerTriangular</b>, B::<b>LowerTriangular</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl:555</a></li> <li> +(A::<b>UpperTriangular</b>, B::<b>UnitUpperTriangular</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl:556</a></li> <li> +(A::<b>LowerTriangular</b>, B::<b>UnitLowerTriangular</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl:557</a></li> <li> +(A::<b>UnitUpperTriangular</b>, B::<b>UpperTriangular</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl:558</a></li> <li> +(A::<b>UnitLowerTriangular</b>, B::<b>LowerTriangular</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl:559</a></li> <li> +(A::<b>UnitUpperTriangular</b>, B::<b>UnitUpperTriangular</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl:560</a></li> <li> +(A::<b>UnitLowerTriangular</b>, B::<b>UnitLowerTriangular</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl:561</a></li> <li> +(A::<b>LinearAlgebra.AbstractTriangular</b>, B::<b>LinearAlgebra.AbstractTriangular</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl:562</a></li> <li> +(A::<b>Symmetric</b>, B::<b>Symmetric</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/symmetric.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/symmetric.jl:422</a></li> <li> +(A::<b>Hermitian</b>, B::<b>Hermitian</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/symmetric.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/symmetric.jl:423</a></li> <li> +(A::<b>Hermitian</b>, B::<b>Symmetric{#s627,S} where S<:(AbstractArray{#s6271,2} where #s6271<:#s627) where #s627<:Real</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/symmetric.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/symmetric.jl:424</a></li> <li> +(A::<b>Symmetric{#s627,S} where S<:(AbstractArray{#s6271,2} where #s6271<:#s627) where #s627<:Real</b>, B::<b>Hermitian</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/symmetric.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/symmetric.jl:425</a></li> <li> +(Da::<b>Diagonal</b>, Db::<b>Diagonal</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/diagonal.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/diagonal.jl:156</a></li> <li> +(A::<b>Bidiagonal</b>, B::<b>Bidiagonal</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/bidiag.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/bidiag.jl:309</a></li> <li> +(UL::<b>UnitUpperTriangular</b>, J::<b>UniformScaling</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl:131</a></li> <li> +(UL::<b>UnitLowerTriangular</b>, J::<b>UniformScaling</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl:131</a></li> <li> +(A::<b>Hermitian</b>, J::<b>UniformScaling{#s627} where #s627<:Complex</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl:145</a></li> <li> +(H::<b>UpperHessenberg</b>, J::<b>UniformScaling</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/hessenberg.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/hessenberg.jl:106</a></li> <li> +(A::<b>Tridiagonal{#s627,V} where V<:AbstractArray{#s627,1} where #s627<:Number</b>, B::<b>UniformScaling</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:238</a></li> <li> +(A::<b>SymTridiagonal{#s627,V} where V<:AbstractArray{#s627,1} where #s627<:Number</b>, B::<b>UniformScaling</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:243</a></li> <li> +(A::<b>Bidiagonal{#s627,V} where V<:AbstractArray{#s627,1} where #s627<:Number</b>, B::<b>UniformScaling</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:248</a></li> <li> +(A::<b>Diagonal{#s627,V} where V<:AbstractArray{#s627,1} where #s627<:Number</b>, B::<b>UniformScaling</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:253</a></li> <li> +(A::<b>SparseArrays.SparseMatrixCSC</b>, J::<b>UniformScaling</b>) in SparseArrays at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/SparseArrays/src/sparsematrix.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/SparseArrays/src/sparsematrix.jl:3605</a></li> <li> +(A::<b>AbstractArray{T,2} where T</b>, J::<b>UniformScaling</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl:164</a></li> <li> +(A::<b>UpperHessenberg</b>, B::<b>UpperHessenberg</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/hessenberg.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/hessenberg.jl:101</a></li> <li> +(A::<b>UpperTriangular</b>, B::<b>Bidiagonal</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:98</a></li> <li> +(A::<b>Bidiagonal</b>, B::<b>UpperTriangular</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:106</a></li> <li> +(A::<b>UnitUpperTriangular</b>, B::<b>Bidiagonal</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:98</a></li> <li> +(A::<b>Bidiagonal</b>, B::<b>UnitUpperTriangular</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:106</a></li> <li> +(A::<b>LowerTriangular</b>, B::<b>Bidiagonal</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:98</a></li> <li> +(A::<b>Bidiagonal</b>, B::<b>LowerTriangular</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:106</a></li> <li> +(A::<b>UnitLowerTriangular</b>, B::<b>Bidiagonal</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:98</a></li> <li> +(A::<b>Bidiagonal</b>, B::<b>UnitLowerTriangular</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:106</a></li> <li> +(A::<b>Bidiagonal</b>, B::<b>Diagonal</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:127</a></li> <li> +(A::<b>Diagonal</b>, B::<b>Bidiagonal</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:137</a></li> <li> +(A::<b>Diagonal</b>, B::<b>SymTridiagonal</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:147</a></li> <li> +(A::<b>SymTridiagonal</b>, B::<b>Diagonal</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:157</a></li> <li> +(A::<b>Tridiagonal</b>, B::<b>SymTridiagonal</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:168</a></li> <li> +(A::<b>SymTridiagonal</b>, B::<b>Tridiagonal</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:170</a></li> <li> +(A::<b>Diagonal</b>, B::<b>Tridiagonal</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:175</a></li> <li> +(A::<b>Tridiagonal</b>, B::<b>Diagonal</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:185</a></li> <li> +(A::<b>Bidiagonal</b>, B::<b>Tridiagonal</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:195</a></li> <li> +(A::<b>Tridiagonal</b>, B::<b>Bidiagonal</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:205</a></li> <li> +(A::<b>Bidiagonal</b>, B::<b>SymTridiagonal</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:215</a></li> <li> +(A::<b>SymTridiagonal</b>, B::<b>Bidiagonal</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:225</a></li> <li> +(A::<b>SparseArrays.SparseMatrixCSC</b>, B::<b>SparseArrays.SparseMatrixCSC</b>) in SparseArrays at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/SparseArrays/src/sparsematrix.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/SparseArrays/src/sparsematrix.jl:1643</a></li> <li> +(A::<b>SparseArrays.SparseMatrixCSC</b>, B::<b>Array</b>) in SparseArrays at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/SparseArrays/src/sparsematrix.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/SparseArrays/src/sparsematrix.jl:1646</a></li> <li> +(x::<b>SparseArrays.AbstractSparseArray{Tv,Ti,1} where Ti where Tv</b>, y::<b>SparseArrays.AbstractSparseArray{Tv,Ti,1} where Ti where Tv</b>) in SparseArrays at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/SparseArrays/src/sparsevector.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/SparseArrays/src/sparsevector.jl:1304</a></li> <li> +(x::<b>AbstractArray{#s66,N} where N where #s66<:Number</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/abstractarraymath.jl#L97\" target=\"_blank\">abstractarraymath.jl:97</a></li> <li> +(A::<b>AbstractArray</b>, B::<b>AbstractArray</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/arraymath.jl#L38\" target=\"_blank\">arraymath.jl:38</a></li> <li> +(x::<b>T</b>, y::<b>Integer</b>)<i> where T<:AbstractChar</i> in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/char.jl#L223\" target=\"_blank\">char.jl:223</a></li> <li> +(index1::<b>CartesianIndex{N}</b>, index2::<b>CartesianIndex{N}</b>)<i> where N</i> in Base.IteratorsMD at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/multidimensional.jl#L110\" target=\"_blank\">multidimensional.jl:110</a></li> <li> +(::<b>Number</b>, ::<b>Missing</b>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/missing.jl#L116\" target=\"_blank\">missing.jl:116</a></li> <li> +(x::<b>P</b>, y::<b>P</b>)<i> where P<:Dates.Period</i> in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl:75</a></li> <li> +(x::<b>Dates.Period</b>, y::<b>Dates.Period</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl:340</a></li> <li> +(y::<b>Dates.Period</b>, x::<b>Dates.CompoundPeriod</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl:342</a></li> <li> +(x::<b>Union{Dates.CompoundPeriod, Dates.Period}</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl:351</a></li> <li> +(x::<b>Dates.TimeType</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:8</a></li> <li> +(a::<b>Dates.TimeType</b>, b::<b>Dates.Period</b>, c::<b>Dates.Period</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl:362</a></li> <li> +(a::<b>Dates.TimeType</b>, b::<b>Dates.Period</b>, c::<b>Dates.Period</b>, d::<b>Dates.Period...</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl:363</a></li> <li> +(x::<b>Dates.TimeType</b>, y::<b>Dates.CompoundPeriod</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl:366</a></li> <li> +(x::<b>Dates.Instant</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:4</a></li> <li> +(y::<b>Dates.Period</b>, x::<b>Dates.TimeType</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:80</a></li> <li> +(x::<b>Dates.Period</b>, r::<b>AbstractRange{#s627} where #s627<:Dates.TimeType</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/ranges.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/ranges.jl:58</a></li> <li> +(x::<b>AbstractArray{#s627,N} where N where #s627<:Dates.TimeType</b>, y::<b>Union{Dates.CompoundPeriod, Dates.Period}</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/deprecated.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/deprecated.jl:8</a></li> <li> +(y::<b>Union{Dates.CompoundPeriod, Dates.Period}</b>, x::<b>AbstractArray{#s627,N} where N where #s627<:Dates.TimeType</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/deprecated.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/deprecated.jl:16</a></li> <li> +(y::<b>Dates.TimeType</b>, x::<b>Union{DenseArray{#s627,N}, Base.ReinterpretArray{#s627,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray}, Base.ReshapedArray{#s627,N,A,MI} where MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray}, SubArray{#s627,N,A,I,L} where L where I<:Tuple{Vararg{Union{Int64, AbstractRange{Int64}, Base.AbstractCartesianIndex},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, Base.ReshapedArray{T,N,A,MI} where MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, DenseArray}} where N where #s627<:Union{Dates.CompoundPeriod, Dates.Period}</b>) in Dates at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/deprecated.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/deprecated.jl:20</a></li> <li> +(J::<b>UniformScaling</b>, x::<b>Number</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl:109</a></li> <li> +(x::<b>Number</b>, J::<b>UniformScaling</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl:110</a></li> <li> +(J1::<b>UniformScaling</b>, J2::<b>UniformScaling</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl:114</a></li> <li> +(J::<b>UniformScaling</b>, B::<b>BitArray{2}</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl:116</a></li> <li> +(A::<b>UniformScaling</b>, B::<b>Tridiagonal{#s627,V} where V<:AbstractArray{#s627,1} where #s627<:Number</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:257</a></li> <li> +(A::<b>UniformScaling</b>, B::<b>SymTridiagonal{#s627,V} where V<:AbstractArray{#s627,1} where #s627<:Number</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:262</a></li> <li> +(A::<b>UniformScaling</b>, B::<b>Bidiagonal{#s627,V} where V<:AbstractArray{#s627,1} where #s627<:Number</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:267</a></li> <li> +(A::<b>UniformScaling</b>, B::<b>Diagonal{#s627,V} where V<:AbstractArray{#s627,1} where #s627<:Number</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:272</a></li> <li> +(J::<b>UniformScaling</b>, A::<b>AbstractArray{T,2} where T</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl:117</a></li> <li> +(F::<b>Hessenberg</b>, J::<b>UniformScaling</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/hessenberg.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/hessenberg.jl:517</a></li> <li> +(J::<b>UniformScaling</b>, F::<b>Hessenberg</b>) in LinearAlgebra at <a href=\"file:///Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/hessenberg.jl\" target=\"_blank\">/Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/hessenberg.jl:518</a></li> <li> +(a, b, c, xs...) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/46ce4d79337bdd257ee2e3d2f4bb1c55ff0a5030/base/operators.jl#L529\" target=\"_blank\">operators.jl:529</a></li> </ul>"
      ],
      "text/plain": [
       "# 166 methods for generic function \"+\":\n",
       "[1] +(x::Bool, z::Complex{Bool}) in Base at complex.jl:282\n",
       "[2] +(x::Bool, y::Bool) in Base at bool.jl:96\n",
       "[3] +(x::Bool) in Base at bool.jl:93\n",
       "[4] +(x::Bool, y::T) where T<:AbstractFloat in Base at bool.jl:104\n",
       "[5] +(x::Bool, z::Complex) in Base at complex.jl:289\n",
       "[6] +(a::Float16, b::Float16) in Base at float.jl:398\n",
       "[7] +(x::Float32, y::Float32) in Base at float.jl:400\n",
       "[8] +(x::Float64, y::Float64) in Base at float.jl:401\n",
       "[9] +(z::Complex{Bool}, x::Bool) in Base at complex.jl:283\n",
       "[10] +(z::Complex{Bool}, x::Real) in Base at complex.jl:297\n",
       "[11] +(::Missing, ::Missing) in Base at missing.jl:114\n",
       "[12] +(::Missing) in Base at missing.jl:100\n",
       "[13] +(::Missing, ::Number) in Base at missing.jl:115\n",
       "[14] +(level::Base.CoreLogging.LogLevel, inc::Integer) in Base.CoreLogging at logging.jl:106\n",
       "[15] +(c::BigInt, x::BigFloat) in Base.MPFR at mpfr.jl:413\n",
       "[16] +(a::BigInt, b::BigInt, c::BigInt, d::BigInt, e::BigInt) in Base.GMP at gmp.jl:503\n",
       "[17] +(a::BigInt, b::BigInt, c::BigInt, d::BigInt) in Base.GMP at gmp.jl:502\n",
       "[18] +(a::BigInt, b::BigInt, c::BigInt) in Base.GMP at gmp.jl:501\n",
       "[19] +(x::BigInt, y::BigInt) in Base.GMP at gmp.jl:472\n",
       "[20] +(x::BigInt, c::Union{UInt16, UInt32, UInt64, UInt8}) in Base.GMP at gmp.jl:509\n",
       "[21] +(x::BigInt, c::Union{Int16, Int32, Int64, Int8}) in Base.GMP at gmp.jl:515\n",
       "[22] +(a::BigFloat, b::BigFloat, c::BigFloat, d::BigFloat, e::BigFloat) in Base.MPFR at mpfr.jl:563\n",
       "[23] +(a::BigFloat, b::BigFloat, c::BigFloat, d::BigFloat) in Base.MPFR at mpfr.jl:556\n",
       "[24] +(a::BigFloat, b::BigFloat, c::BigFloat) in Base.MPFR at mpfr.jl:550\n",
       "[25] +(x::BigFloat, c::BigInt) in Base.MPFR at mpfr.jl:409\n",
       "[26] +(x::BigFloat, y::BigFloat) in Base.MPFR at mpfr.jl:378\n",
       "[27] +(x::BigFloat, c::Union{UInt16, UInt32, UInt64, UInt8}) in Base.MPFR at mpfr.jl:385\n",
       "[28] +(x::BigFloat, c::Union{Int16, Int32, Int64, Int8}) in Base.MPFR at mpfr.jl:393\n",
       "[29] +(x::BigFloat, c::Union{Float16, Float32, Float64}) in Base.MPFR at mpfr.jl:401\n",
       "[30] +(x::Dates.CompoundPeriod, y::Dates.CompoundPeriod) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl:343\n",
       "[31] +(x::Dates.CompoundPeriod, y::Dates.Period) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl:341\n",
       "[32] +(x::Dates.CompoundPeriod, y::Dates.TimeType) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl:371\n",
       "[33] +(x::Dates.Date, y::Dates.Day) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:74\n",
       "[34] +(x::Dates.Date, y::Dates.Week) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:72\n",
       "[35] +(dt::Dates.Date, z::Dates.Month) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:55\n",
       "[36] +(dt::Dates.Date, y::Dates.Year) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:28\n",
       "[37] +(dt::Dates.Date, t::Dates.Time) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:19\n",
       "[38] +(t::Dates.Time, dt::Dates.Date) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:20\n",
       "[39] +(x::Dates.Time, y::Dates.TimePeriod) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:78\n",
       "[40] +(dt::Dates.DateTime, z::Dates.Month) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:48\n",
       "[41] +(dt::Dates.DateTime, y::Dates.Year) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:24\n",
       "[42] +(x::Dates.DateTime, y::Dates.Period) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:76\n",
       "[43] +(B::BitArray{2}, J::UniformScaling) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl:115\n",
       "[44] +(a::Pkg.Resolve.VersionWeights.VersionWeight, b::Pkg.Resolve.VersionWeights.VersionWeight) in Pkg.Resolve.VersionWeights at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Pkg/src/resolve/VersionWeights.jl:26\n",
       "[45] +(a::Pkg.Resolve.MaxSum.FieldValues.FieldValue, b::Pkg.Resolve.MaxSum.FieldValues.FieldValue) in Pkg.Resolve.MaxSum.FieldValues at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Pkg/src/resolve/FieldValues.jl:49\n",
       "[46] +(y::AbstractFloat, x::Bool) in Base at bool.jl:106\n",
       "[47] +(x::T, y::T) where T<:Union{Int128, Int16, Int32, Int64, Int8, UInt128, UInt16, UInt32, UInt64, UInt8} in Base at int.jl:53\n",
       "[48] +(c::Union{UInt16, UInt32, UInt64, UInt8}, x::BigInt) in Base.GMP at gmp.jl:510\n",
       "[49] +(c::Union{Int16, Int32, Int64, Int8}, x::BigInt) in Base.GMP at gmp.jl:516\n",
       "[50] +(a::Integer, b::Integer) in Base at int.jl:871\n",
       "[51] +(x::Integer, y::Ptr) in Base at pointer.jl:161\n",
       "[52] +(z::Complex) in Base at complex.jl:269\n",
       "[53] +(z::Complex, w::Complex) in Base at complex.jl:271\n",
       "[54] +(z::Complex, x::Bool) in Base at complex.jl:290\n",
       "[55] +(x::Rational) in Base at rational.jl:241\n",
       "[56] +(x::Real, z::Complex{Bool}) in Base at complex.jl:296\n",
       "[57] +(x::Real, z::Complex) in Base at complex.jl:308\n",
       "[58] +(z::Complex, x::Real) in Base at complex.jl:309\n",
       "[59] +(x::Rational, y::Rational) in Base at rational.jl:257\n",
       "[60] +(x::Integer, y::AbstractChar) in Base at char.jl:224\n",
       "[61] +(c::Union{UInt16, UInt32, UInt64, UInt8}, x::BigFloat) in Base.MPFR at mpfr.jl:389\n",
       "[62] +(c::Union{Int16, Int32, Int64, Int8}, x::BigFloat) in Base.MPFR at mpfr.jl:397\n",
       "[63] +(c::Union{Float16, Float32, Float64}, x::BigFloat) in Base.MPFR at mpfr.jl:405\n",
       "[64] +(x::AbstractIrrational, y::AbstractIrrational) in Base at irrationals.jl:137\n",
       "[65] +(x::Number) in Base at operators.jl:504\n",
       "[66] +(x::T, y::T) where T<:Number in Base at promotion.jl:387\n",
       "[67] +(x::Number, y::Number) in Base at promotion.jl:311\n",
       "[68] +(r1::OrdinalRange, r2::OrdinalRange) in Base at range.jl:1011\n",
       "[69] +(r1::LinRange{T}, r2::LinRange{T}) where T in Base at range.jl:1018\n",
       "[70] +(r1::StepRangeLen{T,R,S} where S, r2::StepRangeLen{T,R,S} where S) where {R<:Base.TwicePrecision, T} in Base at twiceprecision.jl:557\n",
       "[71] +(r1::StepRangeLen{T,S,S1} where S1, r2::StepRangeLen{T,S,S1} where S1) where {T, S} in Base at range.jl:1034\n",
       "[72] +(r1::Union{LinRange, OrdinalRange, StepRangeLen}, r2::Union{LinRange, OrdinalRange, StepRangeLen}) in Base at range.jl:1026\n",
       "[73] +(x::Ptr, y::Integer) in Base at pointer.jl:159\n",
       "[74] +(x::Base.TwicePrecision, y::Number) in Base at twiceprecision.jl:265\n",
       "[75] +(x::Number, y::Base.TwicePrecision) in Base at twiceprecision.jl:268\n",
       "[76] +(x::Base.TwicePrecision{T}, y::Base.TwicePrecision{T}) where T in Base at twiceprecision.jl:271\n",
       "[77] +(x::Base.TwicePrecision, y::Base.TwicePrecision) in Base at twiceprecision.jl:275\n",
       "[78] +(A::Array, Bs::Array...) in Base at arraymath.jl:44\n",
       "[79] +(A::BitArray, B::BitArray) in Base at bitarray.jl:1084\n",
       "[80] +(r::AbstractRange{#s627} where #s627<:Dates.TimeType, x::Dates.Period) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/ranges.jl:59\n",
       "[81] +(A::Array, B::SparseArrays.SparseMatrixCSC) in SparseArrays at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/SparseArrays/src/sparsematrix.jl:1647\n",
       "[82] +(x::Union{DenseArray{#s627,N}, Base.ReinterpretArray{#s627,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray}, Base.ReshapedArray{#s627,N,A,MI} where MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray}, SubArray{#s627,N,A,I,L} where L where I<:Tuple{Vararg{Union{Int64, AbstractRange{Int64}, Base.AbstractCartesianIndex},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, Base.ReshapedArray{T,N,A,MI} where MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, DenseArray}} where N where #s627<:Union{Dates.CompoundPeriod, Dates.Period}) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/deprecated.jl:57\n",
       "[83] +(x::Union{DenseArray{#s627,N}, Base.ReinterpretArray{#s627,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray}, Base.ReshapedArray{#s627,N,A,MI} where MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray}, SubArray{#s627,N,A,I,L} where L where I<:Tuple{Vararg{Union{Int64, AbstractRange{Int64}, Base.AbstractCartesianIndex},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, Base.ReshapedArray{T,N,A,MI} where MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, DenseArray}} where N where #s627<:Union{Dates.CompoundPeriod, Dates.Period}, y::Dates.TimeType) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/deprecated.jl:12\n",
       "[84] +(X::Union{DenseArray{#s627,N}, Base.ReinterpretArray{#s627,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray}, Base.ReshapedArray{#s627,N,A,MI} where MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray}, SubArray{#s627,N,A,I,L} where L where I<:Tuple{Vararg{Union{Int64, AbstractRange{Int64}, Base.AbstractCartesianIndex},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, Base.ReshapedArray{T,N,A,MI} where MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, DenseArray}} where N where #s627<:Union{Dates.CompoundPeriod, Dates.Period}, Y::Union{DenseArray{#s626,N}, Base.ReinterpretArray{#s626,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray}, Base.ReshapedArray{#s626,N,A,MI} where MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray}, SubArray{#s626,N,A,I,L} where L where I<:Tuple{Vararg{Union{Int64, AbstractRange{Int64}, Base.AbstractCartesianIndex},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, Base.ReshapedArray{T,N,A,MI} where MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, DenseArray}} where N where #s626<:Union{Dates.CompoundPeriod, Dates.Period}) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/deprecated.jl:64\n",
       "[85] +(A::SymTridiagonal, B::SymTridiagonal) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/tridiag.jl:158\n",
       "[86] +(A::Tridiagonal, B::Tridiagonal) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/tridiag.jl:636\n",
       "[87] +(A::UpperTriangular, B::UpperTriangular) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl:554\n",
       "[88] +(A::LowerTriangular, B::LowerTriangular) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl:555\n",
       "[89] +(A::UpperTriangular, B::UnitUpperTriangular) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl:556\n",
       "[90] +(A::LowerTriangular, B::UnitLowerTriangular) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl:557\n",
       "[91] +(A::UnitUpperTriangular, B::UpperTriangular) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl:558\n",
       "[92] +(A::UnitLowerTriangular, B::LowerTriangular) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl:559\n",
       "[93] +(A::UnitUpperTriangular, B::UnitUpperTriangular) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl:560\n",
       "[94] +(A::UnitLowerTriangular, B::UnitLowerTriangular) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl:561\n",
       "[95] +(A::LinearAlgebra.AbstractTriangular, B::LinearAlgebra.AbstractTriangular) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/triangular.jl:562\n",
       "[96] +(A::Symmetric, B::Symmetric) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/symmetric.jl:422\n",
       "[97] +(A::Hermitian, B::Hermitian) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/symmetric.jl:423\n",
       "[98] +(A::Hermitian, B::Symmetric{#s627,S} where S<:(AbstractArray{#s6271,2} where #s6271<:#s627) where #s627<:Real) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/symmetric.jl:424\n",
       "[99] +(A::Symmetric{#s627,S} where S<:(AbstractArray{#s6271,2} where #s6271<:#s627) where #s627<:Real, B::Hermitian) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/symmetric.jl:425\n",
       "[100] +(Da::Diagonal, Db::Diagonal) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/diagonal.jl:156\n",
       "[101] +(A::Bidiagonal, B::Bidiagonal) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/bidiag.jl:309\n",
       "[102] +(UL::UnitUpperTriangular, J::UniformScaling) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl:131\n",
       "[103] +(UL::UnitLowerTriangular, J::UniformScaling) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl:131\n",
       "[104] +(A::Hermitian, J::UniformScaling{#s627} where #s627<:Complex) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl:145\n",
       "[105] +(H::UpperHessenberg, J::UniformScaling) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/hessenberg.jl:106\n",
       "[106] +(A::Tridiagonal{#s627,V} where V<:AbstractArray{#s627,1} where #s627<:Number, B::UniformScaling) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:238\n",
       "[107] +(A::SymTridiagonal{#s627,V} where V<:AbstractArray{#s627,1} where #s627<:Number, B::UniformScaling) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:243\n",
       "[108] +(A::Bidiagonal{#s627,V} where V<:AbstractArray{#s627,1} where #s627<:Number, B::UniformScaling) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:248\n",
       "[109] +(A::Diagonal{#s627,V} where V<:AbstractArray{#s627,1} where #s627<:Number, B::UniformScaling) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:253\n",
       "[110] +(A::SparseArrays.SparseMatrixCSC, J::UniformScaling) in SparseArrays at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/SparseArrays/src/sparsematrix.jl:3605\n",
       "[111] +(A::AbstractArray{T,2} where T, J::UniformScaling) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl:164\n",
       "[112] +(A::UpperHessenberg, B::UpperHessenberg) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/hessenberg.jl:101\n",
       "[113] +(A::UpperTriangular, B::Bidiagonal) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:98\n",
       "[114] +(A::Bidiagonal, B::UpperTriangular) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:106\n",
       "[115] +(A::UnitUpperTriangular, B::Bidiagonal) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:98\n",
       "[116] +(A::Bidiagonal, B::UnitUpperTriangular) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:106\n",
       "[117] +(A::LowerTriangular, B::Bidiagonal) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:98\n",
       "[118] +(A::Bidiagonal, B::LowerTriangular) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:106\n",
       "[119] +(A::UnitLowerTriangular, B::Bidiagonal) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:98\n",
       "[120] +(A::Bidiagonal, B::UnitLowerTriangular) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:106\n",
       "[121] +(A::Bidiagonal, B::Diagonal) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:127\n",
       "[122] +(A::Diagonal, B::Bidiagonal) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:137\n",
       "[123] +(A::Diagonal, B::SymTridiagonal) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:147\n",
       "[124] +(A::SymTridiagonal, B::Diagonal) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:157\n",
       "[125] +(A::Tridiagonal, B::SymTridiagonal) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:168\n",
       "[126] +(A::SymTridiagonal, B::Tridiagonal) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:170\n",
       "[127] +(A::Diagonal, B::Tridiagonal) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:175\n",
       "[128] +(A::Tridiagonal, B::Diagonal) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:185\n",
       "[129] +(A::Bidiagonal, B::Tridiagonal) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:195\n",
       "[130] +(A::Tridiagonal, B::Bidiagonal) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:205\n",
       "[131] +(A::Bidiagonal, B::SymTridiagonal) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:215\n",
       "[132] +(A::SymTridiagonal, B::Bidiagonal) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:225\n",
       "[133] +(A::SparseArrays.SparseMatrixCSC, B::SparseArrays.SparseMatrixCSC) in SparseArrays at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/SparseArrays/src/sparsematrix.jl:1643\n",
       "[134] +(A::SparseArrays.SparseMatrixCSC, B::Array) in SparseArrays at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/SparseArrays/src/sparsematrix.jl:1646\n",
       "[135] +(x::SparseArrays.AbstractSparseArray{Tv,Ti,1} where Ti where Tv, y::SparseArrays.AbstractSparseArray{Tv,Ti,1} where Ti where Tv) in SparseArrays at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/SparseArrays/src/sparsevector.jl:1304\n",
       "[136] +(x::AbstractArray{#s66,N} where N where #s66<:Number) in Base at abstractarraymath.jl:97\n",
       "[137] +(A::AbstractArray, B::AbstractArray) in Base at arraymath.jl:38\n",
       "[138] +(x::T, y::Integer) where T<:AbstractChar in Base at char.jl:223\n",
       "[139] +(index1::CartesianIndex{N}, index2::CartesianIndex{N}) where N in Base.IteratorsMD at multidimensional.jl:110\n",
       "[140] +(::Number, ::Missing) in Base at missing.jl:116\n",
       "[141] +(x::P, y::P) where P<:Dates.Period in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl:75\n",
       "[142] +(x::Dates.Period, y::Dates.Period) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl:340\n",
       "[143] +(y::Dates.Period, x::Dates.CompoundPeriod) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl:342\n",
       "[144] +(x::Union{Dates.CompoundPeriod, Dates.Period}) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl:351\n",
       "[145] +(x::Dates.TimeType) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:8\n",
       "[146] +(a::Dates.TimeType, b::Dates.Period, c::Dates.Period) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl:362\n",
       "[147] +(a::Dates.TimeType, b::Dates.Period, c::Dates.Period, d::Dates.Period...) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl:363\n",
       "[148] +(x::Dates.TimeType, y::Dates.CompoundPeriod) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/periods.jl:366\n",
       "[149] +(x::Dates.Instant) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:4\n",
       "[150] +(y::Dates.Period, x::Dates.TimeType) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/arithmetic.jl:80\n",
       "[151] +(x::Dates.Period, r::AbstractRange{#s627} where #s627<:Dates.TimeType) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/ranges.jl:58\n",
       "[152] +(x::AbstractArray{#s627,N} where N where #s627<:Dates.TimeType, y::Union{Dates.CompoundPeriod, Dates.Period}) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/deprecated.jl:8\n",
       "[153] +(y::Union{Dates.CompoundPeriod, Dates.Period}, x::AbstractArray{#s627,N} where N where #s627<:Dates.TimeType) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/deprecated.jl:16\n",
       "[154] +(y::Dates.TimeType, x::Union{DenseArray{#s627,N}, Base.ReinterpretArray{#s627,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray}, Base.ReshapedArray{#s627,N,A,MI} where MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray}, SubArray{#s627,N,A,I,L} where L where I<:Tuple{Vararg{Union{Int64, AbstractRange{Int64}, Base.AbstractCartesianIndex},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, Base.ReshapedArray{T,N,A,MI} where MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N} where N} where A<:Union{Base.ReinterpretArray{T,N,S,A} where S where A<:Union{SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, SubArray{T,N,A,I,true} where I<:Union{Tuple{Vararg{Real,N} where N}, Tuple{AbstractUnitRange,Vararg{Any,N} where N}} where A<:DenseArray where N where T, DenseArray} where N where T, DenseArray}} where N where #s627<:Union{Dates.CompoundPeriod, Dates.Period}) in Dates at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/Dates/src/deprecated.jl:20\n",
       "[155] +(J::UniformScaling, x::Number) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl:109\n",
       "[156] +(x::Number, J::UniformScaling) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl:110\n",
       "[157] +(J1::UniformScaling, J2::UniformScaling) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl:114\n",
       "[158] +(J::UniformScaling, B::BitArray{2}) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl:116\n",
       "[159] +(A::UniformScaling, B::Tridiagonal{#s627,V} where V<:AbstractArray{#s627,1} where #s627<:Number) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:257\n",
       "[160] +(A::UniformScaling, B::SymTridiagonal{#s627,V} where V<:AbstractArray{#s627,1} where #s627<:Number) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:262\n",
       "[161] +(A::UniformScaling, B::Bidiagonal{#s627,V} where V<:AbstractArray{#s627,1} where #s627<:Number) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:267\n",
       "[162] +(A::UniformScaling, B::Diagonal{#s627,V} where V<:AbstractArray{#s627,1} where #s627<:Number) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/special.jl:272\n",
       "[163] +(J::UniformScaling, A::AbstractArray{T,2} where T) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/uniformscaling.jl:117\n",
       "[164] +(F::Hessenberg, J::UniformScaling) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/hessenberg.jl:517\n",
       "[165] +(J::UniformScaling, F::Hessenberg) in LinearAlgebra at /Users/sabae/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/hessenberg.jl:518\n",
       "[166] +(a, b, c, xs...) in Base at operators.jl:529"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "methods(+)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A good design of Julia code makes sure that different problems map to different types, or that different properties of a problem match to different types. You write the algorithms in terms of these types, and Julia makes sure that the right algorithm is called at the right time. As you see, this is not entirely for free. It is not really the case that Julia selects the right algorithm. Julia selects the method that best matches the type signature. It is up to you, the programmer, to make sure that this corresponds to the right algorithm.\n",
    "\n",
    "This does mean that multiple dispatch affects the design of your code. You design data structures with dispatch in mind, and you write algorithms that apply to types. This is not difficult to achieve once you're used to it, but it is quite pervasive and it is so pretty much from the start."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### More fun\n",
    "Macro's + parallel code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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