4. Multiple dispatch.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Multiple Dispatch"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Every case of defining function we saw earlier works mostly the same as in Python. Where Julia really shines is in extending this system by answering the two questions we posed at the end of that section:\n",
    "\n",
    "- How do we let functions handle multiple combinations of input?\n",
    "- How do we extend functionality we don't own?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We already saw how to specialize functions to certain types of inputs:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "diagsum (generic function with 1 method)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function diagsum{T<:Number}(A::Array{T, 2})  # all matrices with elements that are numbers\n",
    "    return sum(diag(A))\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.3132089407799397"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = rand(5, 5)\n",
    "diagsum(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But we might also want to handle the case of a vector: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "LoadError",
     "evalue": "LoadError: ArgumentError: use diagm instead of diag to construct a diagonal matrix\nwhile loading In[4], in expression starting on line 1",
     "output_type": "error",
     "traceback": [
      "LoadError: ArgumentError: use diagm instead of diag to construct a diagonal matrix\nwhile loading In[4], in expression starting on line 1",
      "",
      " in diagsum at In[1]:2"
     ]
    }
   ],
   "source": [
    "diagsum(rand(7))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For classes, we'd attach this behavior to the object `vector.diagsum` where it made sense, but this can be problematic:\n",
    "- what if we don't own `vector`? Then we need to subclass or compose.\n",
    "- what if the function takes multiple arguments (like `*`). Who should own it? What if we had a custom sparse array type and wanted to define its arithmetic interactions with preexisting types?\n",
    "- what class should \"own\" an optimization algorithm?\n",
    "- what if we want to special-case certain types of inputs, but it's not our code?\n",
    "\n",
    "This way of separating data structures and algorithms is particularly useful in numerical computing:\n",
    "\n",
    "In Julia, behaviors aren't *owned*, they're *special-cased*:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "diagsum (generic function with 3 methods)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Vector{T} is an alias for Array{T, 1}\n",
    "diagsum{T<:Number}(v::Vector{T}) = sum(v)\n",
    "diagsum(x::Number) = x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note, three methods! The arguments determine which gets called:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "diagsum(x::<b>Number</b>) at In[5]:3"
      ],
      "text/plain": [
       "diagsum(x::Number) at In[5]:3"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@which diagsum(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "diagsum<i>{T<:Number}</i>(v::<b>Array{T<:Number,1}</b>) at In[5]:2"
      ],
      "text/plain": [
       "diagsum{T<:Number}(v::Array{T<:Number,1}) at In[5]:2"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@which diagsum(rand(4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note how much boilerplate type checking and special casing we avoid!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "171 methods for generic function <b>+</b>:<ul><li> +(x::<b>Bool</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/bool.jl#L33\" target=\"_blank\">bool.jl:33</a><li> +(x::<b>Bool</b>, y::<b>Bool</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/bool.jl#L36\" target=\"_blank\">bool.jl:36</a><li> +(y::<b>AbstractFloat</b>, x::<b>Bool</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/bool.jl#L46\" target=\"_blank\">bool.jl:46</a><li> +(x::<b>Int64</b>, y::<b>Int64</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/int.jl#L8\" target=\"_blank\">int.jl:8</a><li> +(x::<b>Int8</b>, y::<b>Int8</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/int.jl#L16\" target=\"_blank\">int.jl:16</a><li> +(x::<b>UInt8</b>, y::<b>UInt8</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/int.jl#L16\" target=\"_blank\">int.jl:16</a><li> +(x::<b>Int16</b>, y::<b>Int16</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/int.jl#L16\" target=\"_blank\">int.jl:16</a><li> +(x::<b>UInt16</b>, y::<b>UInt16</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/int.jl#L16\" target=\"_blank\">int.jl:16</a><li> +(x::<b>Int32</b>, y::<b>Int32</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/int.jl#L16\" target=\"_blank\">int.jl:16</a><li> +(x::<b>UInt32</b>, y::<b>UInt32</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/int.jl#L16\" target=\"_blank\">int.jl:16</a><li> +(x::<b>UInt64</b>, y::<b>UInt64</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/int.jl#L16\" target=\"_blank\">int.jl:16</a><li> +(x::<b>Int128</b>, y::<b>Int128</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/int.jl#L16\" target=\"_blank\">int.jl:16</a><li> +(x::<b>UInt128</b>, y::<b>UInt128</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/int.jl#L16\" target=\"_blank\">int.jl:16</a><li> +(x::<b>Integer</b>, y::<b>Ptr{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/pointer.jl#L77\" target=\"_blank\">pointer.jl:77</a><li> +(x::<b>Float32</b>, y::<b>Float32</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/float.jl#L207\" target=\"_blank\">float.jl:207</a><li> +(x::<b>Float64</b>, y::<b>Float64</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/float.jl#L208\" target=\"_blank\">float.jl:208</a><li> +(z::<b>Complex{T<:Real}</b>, w::<b>Complex{T<:Real}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/complex.jl#L111\" target=\"_blank\">complex.jl:111</a><li> +(x::<b>Bool</b>, z::<b>Complex{Bool}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/complex.jl#L118\" target=\"_blank\">complex.jl:118</a><li> +(z::<b>Complex{Bool}</b>, x::<b>Bool</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/complex.jl#L119\" target=\"_blank\">complex.jl:119</a><li> +(x::<b>Bool</b>, z::<b>Complex{T<:Real}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/complex.jl#L125\" target=\"_blank\">complex.jl:125</a><li> +(z::<b>Complex{T<:Real}</b>, x::<b>Bool</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/complex.jl#L126\" target=\"_blank\">complex.jl:126</a><li> +(x::<b>Real</b>, z::<b>Complex{Bool}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/complex.jl#L132\" target=\"_blank\">complex.jl:132</a><li> +(z::<b>Complex{Bool}</b>, x::<b>Real</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/complex.jl#L133\" target=\"_blank\">complex.jl:133</a><li> +(x::<b>Real</b>, z::<b>Complex{T<:Real}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/complex.jl#L144\" target=\"_blank\">complex.jl:144</a><li> +(z::<b>Complex{T<:Real}</b>, x::<b>Real</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/complex.jl#L145\" target=\"_blank\">complex.jl:145</a><li> +(x::<b>Rational{T<:Integer}</b>, y::<b>Rational{T<:Integer}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/rational.jl#L179\" target=\"_blank\">rational.jl:179</a><li> +(x::<b>Bool</b>, A::<b>AbstractArray{Bool,N}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/arraymath.jl#L136\" target=\"_blank\">arraymath.jl:136</a><li> +(x::<b>Integer</b>, y::<b>Char</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/char.jl#L43\" target=\"_blank\">char.jl:43</a><li> +(a::<b>Float16</b>, b::<b>Float16</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/float16.jl#L136\" target=\"_blank\">float16.jl:136</a><li> +(x::<b>BigInt</b>, y::<b>BigInt</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/gmp.jl#L256\" target=\"_blank\">gmp.jl:256</a><li> +(a::<b>BigInt</b>, b::<b>BigInt</b>, c::<b>BigInt</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/gmp.jl#L279\" target=\"_blank\">gmp.jl:279</a><li> +(a::<b>BigInt</b>, b::<b>BigInt</b>, c::<b>BigInt</b>, d::<b>BigInt</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/gmp.jl#L285\" target=\"_blank\">gmp.jl:285</a><li> +(a::<b>BigInt</b>, b::<b>BigInt</b>, c::<b>BigInt</b>, d::<b>BigInt</b>, e::<b>BigInt</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/gmp.jl#L292\" target=\"_blank\">gmp.jl:292</a><li> +(x::<b>BigInt</b>, c::<b>Union{UInt16,UInt32,UInt64,UInt8}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/gmp.jl#L304\" target=\"_blank\">gmp.jl:304</a><li> +(c::<b>Union{UInt16,UInt32,UInt64,UInt8}</b>, x::<b>BigInt</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/gmp.jl#L308\" target=\"_blank\">gmp.jl:308</a><li> +(x::<b>BigInt</b>, c::<b>Union{Int16,Int32,Int64,Int8}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/gmp.jl#L320\" target=\"_blank\">gmp.jl:320</a><li> +(c::<b>Union{Int16,Int32,Int64,Int8}</b>, x::<b>BigInt</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/gmp.jl#L321\" target=\"_blank\">gmp.jl:321</a><li> +(x::<b>BigFloat</b>, y::<b>BigFloat</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/mpfr.jl#L208\" target=\"_blank\">mpfr.jl:208</a><li> +(x::<b>BigFloat</b>, c::<b>Union{UInt16,UInt32,UInt64,UInt8}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/mpfr.jl#L215\" target=\"_blank\">mpfr.jl:215</a><li> +(c::<b>Union{UInt16,UInt32,UInt64,UInt8}</b>, x::<b>BigFloat</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/mpfr.jl#L219\" target=\"_blank\">mpfr.jl:219</a><li> +(x::<b>BigFloat</b>, c::<b>Union{Int16,Int32,Int64,Int8}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/mpfr.jl#L223\" target=\"_blank\">mpfr.jl:223</a><li> +(c::<b>Union{Int16,Int32,Int64,Int8}</b>, x::<b>BigFloat</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/mpfr.jl#L227\" target=\"_blank\">mpfr.jl:227</a><li> +(x::<b>BigFloat</b>, c::<b>Union{Float16,Float32,Float64}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/mpfr.jl#L231\" target=\"_blank\">mpfr.jl:231</a><li> +(c::<b>Union{Float16,Float32,Float64}</b>, x::<b>BigFloat</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/mpfr.jl#L235\" target=\"_blank\">mpfr.jl:235</a><li> +(x::<b>BigFloat</b>, c::<b>BigInt</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/mpfr.jl#L239\" target=\"_blank\">mpfr.jl:239</a><li> +(c::<b>BigInt</b>, x::<b>BigFloat</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/mpfr.jl#L243\" target=\"_blank\">mpfr.jl:243</a><li> +(a::<b>BigFloat</b>, b::<b>BigFloat</b>, c::<b>BigFloat</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/mpfr.jl#L379\" target=\"_blank\">mpfr.jl:379</a><li> +(a::<b>BigFloat</b>, b::<b>BigFloat</b>, c::<b>BigFloat</b>, d::<b>BigFloat</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/mpfr.jl#L385\" target=\"_blank\">mpfr.jl:385</a><li> +(a::<b>BigFloat</b>, b::<b>BigFloat</b>, c::<b>BigFloat</b>, d::<b>BigFloat</b>, e::<b>BigFloat</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/mpfr.jl#L392\" target=\"_blank\">mpfr.jl:392</a><li> +(x::<b>Irrational{sym}</b>, y::<b>Irrational{sym}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/irrationals.jl#L72\" target=\"_blank\">irrationals.jl:72</a><li> +(x::<b>Number</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/operators.jl#L73\" target=\"_blank\">operators.jl:73</a><li> +<i>{T<:Number}</i>(x::<b>T<:Number</b>, y::<b>T<:Number</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/promotion.jl#L211\" target=\"_blank\">promotion.jl:211</a><li> +<i>{T<:AbstractFloat}</i>(x::<b>Bool</b>, y::<b>T<:AbstractFloat</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/bool.jl#L43\" target=\"_blank\">bool.jl:43</a><li> +(x::<b>Number</b>, y::<b>Number</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/promotion.jl#L167\" target=\"_blank\">promotion.jl:167</a><li> +(r1::<b>OrdinalRange{T,S}</b>, r2::<b>OrdinalRange{T,S}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/operators.jl#L330\" target=\"_blank\">operators.jl:330</a><li> +<i>{T<:AbstractFloat}</i>(r1::<b>FloatRange{T<:AbstractFloat}</b>, r2::<b>FloatRange{T<:AbstractFloat}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/operators.jl#L337\" target=\"_blank\">operators.jl:337</a><li> +<i>{T<:AbstractFloat}</i>(r1::<b>LinSpace{T<:AbstractFloat}</b>, r2::<b>LinSpace{T<:AbstractFloat}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/operators.jl#L356\" target=\"_blank\">operators.jl:356</a><li> +(r1::<b>Union{FloatRange{T<:AbstractFloat},LinSpace{T<:AbstractFloat},OrdinalRange{T,S}}</b>, r2::<b>Union{FloatRange{T<:AbstractFloat},LinSpace{T<:AbstractFloat},OrdinalRange{T,S}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/operators.jl#L369\" target=\"_blank\">operators.jl:369</a><li> +(x::<b>Ptr{T}</b>, y::<b>Integer</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/pointer.jl#L75\" target=\"_blank\">pointer.jl:75</a><li> +<i>{S,T}</i>(A::<b>Range{S}</b>, B::<b>Range{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/arraymath.jl#L69\" target=\"_blank\">arraymath.jl:69</a><li> +<i>{S,T}</i>(A::<b>Range{S}</b>, B::<b>AbstractArray{T,N}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/arraymath.jl#L87\" target=\"_blank\">arraymath.jl:87</a><li> +(A::<b>BitArray{N}</b>, B::<b>BitArray{N}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/bitarray.jl#L859\" target=\"_blank\">bitarray.jl:859</a><li> +<i>{T}</i>(B::<b>BitArray{2}</b>, J::<b>UniformScaling{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/uniformscaling.jl#L28\" target=\"_blank\">linalg/uniformscaling.jl:28</a><li> +(A::<b>Array{T,2}</b>, B::<b>Diagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L122\" target=\"_blank\">linalg/special.jl:122</a><li> +(A::<b>Array{T,2}</b>, B::<b>Bidiagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L122\" target=\"_blank\">linalg/special.jl:122</a><li> +(A::<b>Array{T,2}</b>, B::<b>Tridiagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L122\" target=\"_blank\">linalg/special.jl:122</a><li> +(A::<b>Array{T,2}</b>, B::<b>SymTridiagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L131\" target=\"_blank\">linalg/special.jl:131</a><li> +(A::<b>Array{T,2}</b>, B::<b>Base.LinAlg.AbstractTriangular{T,S<:AbstractArray{T,2}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L159\" target=\"_blank\">linalg/special.jl:159</a><li> +(A::<b>Array{T,N}</b>, B::<b>SparseMatrixCSC{Tv,Ti<:Integer}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/sparse/sparsematrix.jl#L1019\" target=\"_blank\">sparse/sparsematrix.jl:1019</a><li> +<i>{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period}}</i>(x::<b>Union{DenseArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N},SubArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N,A<:DenseArray{T,N},I<:Tuple{Vararg{Union{Colon,Int64,Range{Int64}}}},LD}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/periods.jl#L221\" target=\"_blank\">dates/periods.jl:221</a><li> +(A::<b>AbstractArray{Bool,N}</b>, x::<b>Bool</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/arraymath.jl#L135\" target=\"_blank\">arraymath.jl:135</a><li> +(A::<b>Union{DenseArray{Bool,N},SubArray{Bool,N,A<:DenseArray{T,N},I<:Tuple{Vararg{Union{Colon,Int64,Range{Int64}}}},LD}}</b>, B::<b>Union{DenseArray{Bool,N},SubArray{Bool,N,A<:DenseArray{T,N},I<:Tuple{Vararg{Union{Colon,Int64,Range{Int64}}}},LD}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/arraymath.jl#L166\" target=\"_blank\">arraymath.jl:166</a><li> +(A::<b>SymTridiagonal{T}</b>, B::<b>SymTridiagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/tridiag.jl#L84\" target=\"_blank\">linalg/tridiag.jl:84</a><li> +(A::<b>Tridiagonal{T}</b>, B::<b>Tridiagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/tridiag.jl#L404\" target=\"_blank\">linalg/tridiag.jl:404</a><li> +(A::<b>UpperTriangular{T,S<:AbstractArray{T,2}}</b>, B::<b>UpperTriangular{T,S<:AbstractArray{T,2}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/triangular.jl#L347\" target=\"_blank\">linalg/triangular.jl:347</a><li> +(A::<b>LowerTriangular{T,S<:AbstractArray{T,2}}</b>, B::<b>LowerTriangular{T,S<:AbstractArray{T,2}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/triangular.jl#L348\" target=\"_blank\">linalg/triangular.jl:348</a><li> +(A::<b>UpperTriangular{T,S<:AbstractArray{T,2}}</b>, B::<b>Base.LinAlg.UnitUpperTriangular{T,S<:AbstractArray{T,2}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/triangular.jl#L349\" target=\"_blank\">linalg/triangular.jl:349</a><li> +(A::<b>LowerTriangular{T,S<:AbstractArray{T,2}}</b>, B::<b>Base.LinAlg.UnitLowerTriangular{T,S<:AbstractArray{T,2}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/triangular.jl#L350\" target=\"_blank\">linalg/triangular.jl:350</a><li> +(A::<b>Base.LinAlg.UnitUpperTriangular{T,S<:AbstractArray{T,2}}</b>, B::<b>UpperTriangular{T,S<:AbstractArray{T,2}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/triangular.jl#L351\" target=\"_blank\">linalg/triangular.jl:351</a><li> +(A::<b>Base.LinAlg.UnitLowerTriangular{T,S<:AbstractArray{T,2}}</b>, B::<b>LowerTriangular{T,S<:AbstractArray{T,2}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/triangular.jl#L352\" target=\"_blank\">linalg/triangular.jl:352</a><li> +(A::<b>Base.LinAlg.UnitUpperTriangular{T,S<:AbstractArray{T,2}}</b>, B::<b>Base.LinAlg.UnitUpperTriangular{T,S<:AbstractArray{T,2}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/triangular.jl#L353\" target=\"_blank\">linalg/triangular.jl:353</a><li> +(A::<b>Base.LinAlg.UnitLowerTriangular{T,S<:AbstractArray{T,2}}</b>, B::<b>Base.LinAlg.UnitLowerTriangular{T,S<:AbstractArray{T,2}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/triangular.jl#L354\" target=\"_blank\">linalg/triangular.jl:354</a><li> +(A::<b>Base.LinAlg.AbstractTriangular{T,S<:AbstractArray{T,2}}</b>, B::<b>Base.LinAlg.AbstractTriangular{T,S<:AbstractArray{T,2}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/triangular.jl#L355\" target=\"_blank\">linalg/triangular.jl:355</a><li> +(Da::<b>Diagonal{T}</b>, Db::<b>Diagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/diagonal.jl#L86\" target=\"_blank\">linalg/diagonal.jl:86</a><li> +(A::<b>Bidiagonal{T}</b>, B::<b>Bidiagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/bidiag.jl#L176\" target=\"_blank\">linalg/bidiag.jl:176</a><li> +(UL::<b>UpperTriangular{T,S<:AbstractArray{T,2}}</b>, J::<b>UniformScaling{T<:Number}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/uniformscaling.jl#L45\" target=\"_blank\">linalg/uniformscaling.jl:45</a><li> +(UL::<b>Base.LinAlg.UnitUpperTriangular{T,S<:AbstractArray{T,2}}</b>, J::<b>UniformScaling{T<:Number}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/uniformscaling.jl#L48\" target=\"_blank\">linalg/uniformscaling.jl:48</a><li> +(UL::<b>LowerTriangular{T,S<:AbstractArray{T,2}}</b>, J::<b>UniformScaling{T<:Number}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/uniformscaling.jl#L45\" target=\"_blank\">linalg/uniformscaling.jl:45</a><li> +(UL::<b>Base.LinAlg.UnitLowerTriangular{T,S<:AbstractArray{T,2}}</b>, J::<b>UniformScaling{T<:Number}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/uniformscaling.jl#L48\" target=\"_blank\">linalg/uniformscaling.jl:48</a><li> +(A::<b>Diagonal{T}</b>, B::<b>Bidiagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L121\" target=\"_blank\">linalg/special.jl:121</a><li> +(A::<b>Bidiagonal{T}</b>, B::<b>Diagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L122\" target=\"_blank\">linalg/special.jl:122</a><li> +(A::<b>Diagonal{T}</b>, B::<b>Tridiagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L121\" target=\"_blank\">linalg/special.jl:121</a><li> +(A::<b>Tridiagonal{T}</b>, B::<b>Diagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L122\" target=\"_blank\">linalg/special.jl:122</a><li> +(A::<b>Diagonal{T}</b>, B::<b>Array{T,2}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L121\" target=\"_blank\">linalg/special.jl:121</a><li> +(A::<b>Bidiagonal{T}</b>, B::<b>Tridiagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L121\" target=\"_blank\">linalg/special.jl:121</a><li> +(A::<b>Tridiagonal{T}</b>, B::<b>Bidiagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L122\" target=\"_blank\">linalg/special.jl:122</a><li> +(A::<b>Bidiagonal{T}</b>, B::<b>Array{T,2}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L121\" target=\"_blank\">linalg/special.jl:121</a><li> +(A::<b>Tridiagonal{T}</b>, B::<b>Array{T,2}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L121\" target=\"_blank\">linalg/special.jl:121</a><li> +(A::<b>SymTridiagonal{T}</b>, B::<b>Tridiagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L130\" target=\"_blank\">linalg/special.jl:130</a><li> +(A::<b>Tridiagonal{T}</b>, B::<b>SymTridiagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L131\" target=\"_blank\">linalg/special.jl:131</a><li> +(A::<b>SymTridiagonal{T}</b>, B::<b>Array{T,2}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L130\" target=\"_blank\">linalg/special.jl:130</a><li> +(A::<b>Diagonal{T}</b>, B::<b>SymTridiagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L139\" target=\"_blank\">linalg/special.jl:139</a><li> +(A::<b>SymTridiagonal{T}</b>, B::<b>Diagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L140\" target=\"_blank\">linalg/special.jl:140</a><li> +(A::<b>Bidiagonal{T}</b>, B::<b>SymTridiagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L139\" target=\"_blank\">linalg/special.jl:139</a><li> +(A::<b>SymTridiagonal{T}</b>, B::<b>Bidiagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L140\" target=\"_blank\">linalg/special.jl:140</a><li> +(A::<b>Diagonal{T}</b>, B::<b>UpperTriangular{T,S<:AbstractArray{T,2}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L151\" target=\"_blank\">linalg/special.jl:151</a><li> +(A::<b>UpperTriangular{T,S<:AbstractArray{T,2}}</b>, B::<b>Diagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L152\" target=\"_blank\">linalg/special.jl:152</a><li> +(A::<b>Diagonal{T}</b>, B::<b>Base.LinAlg.UnitUpperTriangular{T,S<:AbstractArray{T,2}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L151\" target=\"_blank\">linalg/special.jl:151</a><li> +(A::<b>Base.LinAlg.UnitUpperTriangular{T,S<:AbstractArray{T,2}}</b>, B::<b>Diagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L152\" target=\"_blank\">linalg/special.jl:152</a><li> +(A::<b>Diagonal{T}</b>, B::<b>LowerTriangular{T,S<:AbstractArray{T,2}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L151\" target=\"_blank\">linalg/special.jl:151</a><li> +(A::<b>LowerTriangular{T,S<:AbstractArray{T,2}}</b>, B::<b>Diagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L152\" target=\"_blank\">linalg/special.jl:152</a><li> +(A::<b>Diagonal{T}</b>, B::<b>Base.LinAlg.UnitLowerTriangular{T,S<:AbstractArray{T,2}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L151\" target=\"_blank\">linalg/special.jl:151</a><li> +(A::<b>Base.LinAlg.UnitLowerTriangular{T,S<:AbstractArray{T,2}}</b>, B::<b>Diagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L152\" target=\"_blank\">linalg/special.jl:152</a><li> +(A::<b>Base.LinAlg.AbstractTriangular{T,S<:AbstractArray{T,2}}</b>, B::<b>SymTridiagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L158\" target=\"_blank\">linalg/special.jl:158</a><li> +(A::<b>SymTridiagonal{T}</b>, B::<b>Base.LinAlg.AbstractTriangular{T,S<:AbstractArray{T,2}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L159\" target=\"_blank\">linalg/special.jl:159</a><li> +(A::<b>Base.LinAlg.AbstractTriangular{T,S<:AbstractArray{T,2}}</b>, B::<b>Tridiagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L158\" target=\"_blank\">linalg/special.jl:158</a><li> +(A::<b>Tridiagonal{T}</b>, B::<b>Base.LinAlg.AbstractTriangular{T,S<:AbstractArray{T,2}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L159\" target=\"_blank\">linalg/special.jl:159</a><li> +(A::<b>Base.LinAlg.AbstractTriangular{T,S<:AbstractArray{T,2}}</b>, B::<b>Bidiagonal{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L158\" target=\"_blank\">linalg/special.jl:158</a><li> +(A::<b>Bidiagonal{T}</b>, B::<b>Base.LinAlg.AbstractTriangular{T,S<:AbstractArray{T,2}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L159\" target=\"_blank\">linalg/special.jl:159</a><li> +(A::<b>Base.LinAlg.AbstractTriangular{T,S<:AbstractArray{T,2}}</b>, B::<b>Array{T,2}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/special.jl#L158\" target=\"_blank\">linalg/special.jl:158</a><li> +<i>{Tv1,Ti1,Tv2,Ti2}</i>(A_1::<b>SparseMatrixCSC{Tv1,Ti1}</b>, A_2::<b>SparseMatrixCSC{Tv2,Ti2}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/sparse/sparsematrix.jl#L1005\" target=\"_blank\">sparse/sparsematrix.jl:1005</a><li> +(A::<b>SparseMatrixCSC{Tv,Ti<:Integer}</b>, B::<b>Array{T,N}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/sparse/sparsematrix.jl#L1017\" target=\"_blank\">sparse/sparsematrix.jl:1017</a><li> +(A::<b>SparseMatrixCSC{Tv,Ti<:Integer}</b>, J::<b>UniformScaling{T<:Number}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/sparse/sparsematrix.jl#L3030\" target=\"_blank\">sparse/sparsematrix.jl:3030</a><li> +<i>{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period}}</i>(Y::<b>Union{DenseArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N},SubArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N,A<:DenseArray{T,N},I<:Tuple{Vararg{Union{Colon,Int64,Range{Int64}}}},LD}}</b>, x::<b>Union{Base.Dates.CompoundPeriod,Base.Dates.Period}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/periods.jl#L235\" target=\"_blank\">dates/periods.jl:235</a><li> +<i>{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},Q<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period}}</i>(X::<b>Union{DenseArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N},SubArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N,A<:DenseArray{T,N},I<:Tuple{Vararg{Union{Colon,Int64,Range{Int64}}}},LD}}</b>, Y::<b>Union{DenseArray{Q<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N},SubArray{Q<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N,A<:DenseArray{T,N},I<:Tuple{Vararg{Union{Colon,Int64,Range{Int64}}}},LD}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/periods.jl#L236\" target=\"_blank\">dates/periods.jl:236</a><li> +<i>{T<:Base.Dates.TimeType,P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period}}</i>(x::<b>Union{DenseArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N},SubArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N,A<:DenseArray{T,N},I<:Tuple{Vararg{Union{Colon,Int64,Range{Int64}}}},LD}}</b>, y::<b>T<:Base.Dates.TimeType</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/arithmetic.jl#L83\" target=\"_blank\">dates/arithmetic.jl:83</a><li> +<i>{T<:Base.Dates.TimeType}</i>(r::<b>Range{T<:Base.Dates.TimeType}</b>, x::<b>Base.Dates.Period</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/ranges.jl#L39\" target=\"_blank\">dates/ranges.jl:39</a><li> +<i>{T<:Number}</i>(x::<b>AbstractArray{T<:Number,N}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/abstractarraymath.jl#L49\" target=\"_blank\">abstractarraymath.jl:49</a><li> +<i>{S,T}</i>(A::<b>AbstractArray{S,N}</b>, B::<b>Range{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/arraymath.jl#L78\" target=\"_blank\">arraymath.jl:78</a><li> +<i>{S,T}</i>(A::<b>AbstractArray{S,N}</b>, B::<b>AbstractArray{T,N}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/arraymath.jl#L96\" target=\"_blank\">arraymath.jl:96</a><li> +(A::<b>AbstractArray{T,N}</b>, x::<b>Number</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/arraymath.jl#L139\" target=\"_blank\">arraymath.jl:139</a><li> +(x::<b>Number</b>, A::<b>AbstractArray{T,N}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/arraymath.jl#L140\" target=\"_blank\">arraymath.jl:140</a><li> +(x::<b>Char</b>, y::<b>Integer</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/char.jl#L42\" target=\"_blank\">char.jl:42</a><li> +<i>{N}</i>(index1::<b>CartesianIndex{N}</b>, index2::<b>CartesianIndex{N}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/multidimensional.jl#L42\" target=\"_blank\">multidimensional.jl:42</a><li> +(J1::<b>UniformScaling{T<:Number}</b>, J2::<b>UniformScaling{T<:Number}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/uniformscaling.jl#L27\" target=\"_blank\">linalg/uniformscaling.jl:27</a><li> +(J::<b>UniformScaling{T<:Number}</b>, B::<b>BitArray{2}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/uniformscaling.jl#L29\" target=\"_blank\">linalg/uniformscaling.jl:29</a><li> +(J::<b>UniformScaling{T<:Number}</b>, A::<b>AbstractArray{T,2}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/uniformscaling.jl#L30\" target=\"_blank\">linalg/uniformscaling.jl:30</a><li> +(J::<b>UniformScaling{T<:Number}</b>, x::<b>Number</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/uniformscaling.jl#L31\" target=\"_blank\">linalg/uniformscaling.jl:31</a><li> +(x::<b>Number</b>, J::<b>UniformScaling{T<:Number}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/uniformscaling.jl#L32\" target=\"_blank\">linalg/uniformscaling.jl:32</a><li> +<i>{TA,TJ}</i>(A::<b>AbstractArray{TA,2}</b>, J::<b>UniformScaling{TJ}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/linalg/uniformscaling.jl#L92\" target=\"_blank\">linalg/uniformscaling.jl:92</a><li> +<i>{T}</i>(a::<b>Base.Pkg.Resolve.VersionWeights.HierarchicalValue{T}</b>, b::<b>Base.Pkg.Resolve.VersionWeights.HierarchicalValue{T}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/pkg/resolve/versionweight.jl#L23\" target=\"_blank\">pkg/resolve/versionweight.jl:23</a><li> +(a::<b>Base.Pkg.Resolve.VersionWeights.VWPreBuildItem</b>, b::<b>Base.Pkg.Resolve.VersionWeights.VWPreBuildItem</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/pkg/resolve/versionweight.jl#L85\" target=\"_blank\">pkg/resolve/versionweight.jl:85</a><li> +(a::<b>Base.Pkg.Resolve.VersionWeights.VWPreBuild</b>, b::<b>Base.Pkg.Resolve.VersionWeights.VWPreBuild</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/pkg/resolve/versionweight.jl#L131\" target=\"_blank\">pkg/resolve/versionweight.jl:131</a><li> +(a::<b>Base.Pkg.Resolve.VersionWeights.VersionWeight</b>, b::<b>Base.Pkg.Resolve.VersionWeights.VersionWeight</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/pkg/resolve/versionweight.jl#L185\" target=\"_blank\">pkg/resolve/versionweight.jl:185</a><li> +(a::<b>Base.Pkg.Resolve.MaxSum.FieldValues.FieldValue</b>, b::<b>Base.Pkg.Resolve.MaxSum.FieldValues.FieldValue</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/pkg/resolve/fieldvalue.jl#L44\" target=\"_blank\">pkg/resolve/fieldvalue.jl:44</a><li> +<i>{P<:Base.Dates.Period}</i>(x::<b>P<:Base.Dates.Period</b>, y::<b>P<:Base.Dates.Period</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/periods.jl#L62\" target=\"_blank\">dates/periods.jl:62</a><li> +(x::<b>Base.Dates.Period</b>, y::<b>Base.Dates.Period</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/periods.jl#L209\" target=\"_blank\">dates/periods.jl:209</a><li> +(x::<b>Base.Dates.CompoundPeriod</b>, y::<b>Base.Dates.Period</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/periods.jl#L210\" target=\"_blank\">dates/periods.jl:210</a><li> +(y::<b>Base.Dates.Period</b>, x::<b>Base.Dates.CompoundPeriod</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/periods.jl#L211\" target=\"_blank\">dates/periods.jl:211</a><li> +(x::<b>Base.Dates.CompoundPeriod</b>, y::<b>Base.Dates.CompoundPeriod</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/periods.jl#L212\" target=\"_blank\">dates/periods.jl:212</a><li> +(x::<b>Base.Dates.CompoundPeriod</b>, y::<b>Base.Dates.TimeType</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/periods.jl#L257\" target=\"_blank\">dates/periods.jl:257</a><li> +(y::<b>Base.Dates.Period</b>, x::<b>Base.Dates.TimeType</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/arithmetic.jl#L66\" target=\"_blank\">dates/arithmetic.jl:66</a><li> +<i>{T<:Base.Dates.TimeType}</i>(x::<b>Base.Dates.Period</b>, r::<b>Range{T<:Base.Dates.TimeType}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/ranges.jl#L40\" target=\"_blank\">dates/ranges.jl:40</a><li> +(x::<b>Union{Base.Dates.CompoundPeriod,Base.Dates.Period}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/periods.jl#L220\" target=\"_blank\">dates/periods.jl:220</a><li> +<i>{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period}}</i>(x::<b>Union{Base.Dates.CompoundPeriod,Base.Dates.Period}</b>, Y::<b>Union{DenseArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N},SubArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N,A<:DenseArray{T,N},I<:Tuple{Vararg{Union{Colon,Int64,Range{Int64}}}},LD}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/periods.jl#L234\" target=\"_blank\">dates/periods.jl:234</a><li> +(dt::<b>DateTime</b>, y::<b>Base.Dates.Year</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/arithmetic.jl#L13\" target=\"_blank\">dates/arithmetic.jl:13</a><li> +(dt::<b>Date</b>, y::<b>Base.Dates.Year</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/arithmetic.jl#L17\" target=\"_blank\">dates/arithmetic.jl:17</a><li> +(dt::<b>DateTime</b>, z::<b>Base.Dates.Month</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/arithmetic.jl#L37\" target=\"_blank\">dates/arithmetic.jl:37</a><li> +(dt::<b>Date</b>, z::<b>Base.Dates.Month</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/arithmetic.jl#L43\" target=\"_blank\">dates/arithmetic.jl:43</a><li> +(x::<b>Date</b>, y::<b>Base.Dates.Week</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/arithmetic.jl#L60\" target=\"_blank\">dates/arithmetic.jl:60</a><li> +(x::<b>Date</b>, y::<b>Base.Dates.Day</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/arithmetic.jl#L62\" target=\"_blank\">dates/arithmetic.jl:62</a><li> +(x::<b>DateTime</b>, y::<b>Base.Dates.Period</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/arithmetic.jl#L64\" target=\"_blank\">dates/arithmetic.jl:64</a><li> +(x::<b>Base.Dates.TimeType</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/arithmetic.jl#L8\" target=\"_blank\">dates/arithmetic.jl:8</a><li> +(a::<b>Base.Dates.TimeType</b>, b::<b>Base.Dates.Period</b>, c::<b>Base.Dates.Period</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/periods.jl#L246\" target=\"_blank\">dates/periods.jl:246</a><li> +(a::<b>Base.Dates.TimeType</b>, b::<b>Base.Dates.Period</b>, c::<b>Base.Dates.Period</b>, d::<b>Base.Dates.Period...</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/periods.jl#L248\" target=\"_blank\">dates/periods.jl:248</a><li> +(x::<b>Base.Dates.TimeType</b>, y::<b>Base.Dates.CompoundPeriod</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/periods.jl#L252\" target=\"_blank\">dates/periods.jl:252</a><li> +(x::<b>Base.Dates.Instant</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/arithmetic.jl#L4\" target=\"_blank\">dates/arithmetic.jl:4</a><li> +<i>{T<:Base.Dates.TimeType}</i>(x::<b>AbstractArray{T<:Base.Dates.TimeType,N}</b>, y::<b>Union{Base.Dates.CompoundPeriod,Base.Dates.Period}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/arithmetic.jl#L76\" target=\"_blank\">dates/arithmetic.jl:76</a><li> +<i>{T<:Base.Dates.TimeType}</i>(y::<b>Union{Base.Dates.CompoundPeriod,Base.Dates.Period}</b>, x::<b>AbstractArray{T<:Base.Dates.TimeType,N}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/arithmetic.jl#L77\" target=\"_blank\">dates/arithmetic.jl:77</a><li> +<i>{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period}}</i>(y::<b>Base.Dates.TimeType</b>, x::<b>Union{DenseArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N},SubArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N,A<:DenseArray{T,N},I<:Tuple{Vararg{Union{Colon,Int64,Range{Int64}}}},LD}}</b>) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/dates/arithmetic.jl#L84\" target=\"_blank\">dates/arithmetic.jl:84</a><li> +(a, b, c, xs...) at <a href=\"https://github.com/JuliaLang/julia/tree/2e358ce975029ec97aba5994c17d4a2169c3b085/base/operators.jl#L103\" target=\"_blank\">operators.jl:103</a></ul>"
      ],
      "text/plain": [
       "# 171 methods for generic function \"+\":\n",
       "+(x::Bool) at bool.jl:33\n",
       "+(x::Bool, y::Bool) at bool.jl:36\n",
       "+(y::AbstractFloat, x::Bool) at bool.jl:46\n",
       "+(x::Int64, y::Int64) at int.jl:8\n",
       "+(x::Int8, y::Int8) at int.jl:16\n",
       "+(x::UInt8, y::UInt8) at int.jl:16\n",
       "+(x::Int16, y::Int16) at int.jl:16\n",
       "+(x::UInt16, y::UInt16) at int.jl:16\n",
       "+(x::Int32, y::Int32) at int.jl:16\n",
       "+(x::UInt32, y::UInt32) at int.jl:16\n",
       "+(x::UInt64, y::UInt64) at int.jl:16\n",
       "+(x::Int128, y::Int128) at int.jl:16\n",
       "+(x::UInt128, y::UInt128) at int.jl:16\n",
       "+(x::Integer, y::Ptr{T}) at pointer.jl:77\n",
       "+(x::Float32, y::Float32) at float.jl:207\n",
       "+(x::Float64, y::Float64) at float.jl:208\n",
       "+(z::Complex{T<:Real}, w::Complex{T<:Real}) at complex.jl:111\n",
       "+(x::Bool, z::Complex{Bool}) at complex.jl:118\n",
       "+(z::Complex{Bool}, x::Bool) at complex.jl:119\n",
       "+(x::Bool, z::Complex{T<:Real}) at complex.jl:125\n",
       "+(z::Complex{T<:Real}, x::Bool) at complex.jl:126\n",
       "+(x::Real, z::Complex{Bool}) at complex.jl:132\n",
       "+(z::Complex{Bool}, x::Real) at complex.jl:133\n",
       "+(x::Real, z::Complex{T<:Real}) at complex.jl:144\n",
       "+(z::Complex{T<:Real}, x::Real) at complex.jl:145\n",
       "+(x::Rational{T<:Integer}, y::Rational{T<:Integer}) at rational.jl:179\n",
       "+(x::Bool, A::AbstractArray{Bool,N}) at arraymath.jl:136\n",
       "+(x::Integer, y::Char) at char.jl:43\n",
       "+(a::Float16, b::Float16) at float16.jl:136\n",
       "+(x::BigInt, y::BigInt) at gmp.jl:256\n",
       "+(a::BigInt, b::BigInt, c::BigInt) at gmp.jl:279\n",
       "+(a::BigInt, b::BigInt, c::BigInt, d::BigInt) at gmp.jl:285\n",
       "+(a::BigInt, b::BigInt, c::BigInt, d::BigInt, e::BigInt) at gmp.jl:292\n",
       "+(x::BigInt, c::Union{UInt16,UInt32,UInt64,UInt8}) at gmp.jl:304\n",
       "+(c::Union{UInt16,UInt32,UInt64,UInt8}, x::BigInt) at gmp.jl:308\n",
       "+(x::BigInt, c::Union{Int16,Int32,Int64,Int8}) at gmp.jl:320\n",
       "+(c::Union{Int16,Int32,Int64,Int8}, x::BigInt) at gmp.jl:321\n",
       "+(x::BigFloat, y::BigFloat) at mpfr.jl:208\n",
       "+(x::BigFloat, c::Union{UInt16,UInt32,UInt64,UInt8}) at mpfr.jl:215\n",
       "+(c::Union{UInt16,UInt32,UInt64,UInt8}, x::BigFloat) at mpfr.jl:219\n",
       "+(x::BigFloat, c::Union{Int16,Int32,Int64,Int8}) at mpfr.jl:223\n",
       "+(c::Union{Int16,Int32,Int64,Int8}, x::BigFloat) at mpfr.jl:227\n",
       "+(x::BigFloat, c::Union{Float16,Float32,Float64}) at mpfr.jl:231\n",
       "+(c::Union{Float16,Float32,Float64}, x::BigFloat) at mpfr.jl:235\n",
       "+(x::BigFloat, c::BigInt) at mpfr.jl:239\n",
       "+(c::BigInt, x::BigFloat) at mpfr.jl:243\n",
       "+(a::BigFloat, b::BigFloat, c::BigFloat) at mpfr.jl:379\n",
       "+(a::BigFloat, b::BigFloat, c::BigFloat, d::BigFloat) at mpfr.jl:385\n",
       "+(a::BigFloat, b::BigFloat, c::BigFloat, d::BigFloat, e::BigFloat) at mpfr.jl:392\n",
       "+(x::Irrational{sym}, y::Irrational{sym}) at irrationals.jl:72\n",
       "+(x::Number) at operators.jl:73\n",
       "+{T<:Number}(x::T<:Number, y::T<:Number) at promotion.jl:211\n",
       "+{T<:AbstractFloat}(x::Bool, y::T<:AbstractFloat) at bool.jl:43\n",
       "+(x::Number, y::Number) at promotion.jl:167\n",
       "+(r1::OrdinalRange{T,S}, r2::OrdinalRange{T,S}) at operators.jl:330\n",
       "+{T<:AbstractFloat}(r1::FloatRange{T<:AbstractFloat}, r2::FloatRange{T<:AbstractFloat}) at operators.jl:337\n",
       "+{T<:AbstractFloat}(r1::LinSpace{T<:AbstractFloat}, r2::LinSpace{T<:AbstractFloat}) at operators.jl:356\n",
       "+(r1::Union{FloatRange{T<:AbstractFloat},LinSpace{T<:AbstractFloat},OrdinalRange{T,S}}, r2::Union{FloatRange{T<:AbstractFloat},LinSpace{T<:AbstractFloat},OrdinalRange{T,S}}) at operators.jl:369\n",
       "+(x::Ptr{T}, y::Integer) at pointer.jl:75\n",
       "+{S,T}(A::Range{S}, B::Range{T}) at arraymath.jl:69\n",
       "+{S,T}(A::Range{S}, B::AbstractArray{T,N}) at arraymath.jl:87\n",
       "+(A::BitArray{N}, B::BitArray{N}) at bitarray.jl:859\n",
       "+{T}(B::BitArray{2}, J::UniformScaling{T}) at linalg/uniformscaling.jl:28\n",
       "+(A::Array{T,2}, B::Diagonal{T}) at linalg/special.jl:122\n",
       "+(A::Array{T,2}, B::Bidiagonal{T}) at linalg/special.jl:122\n",
       "+(A::Array{T,2}, B::Tridiagonal{T}) at linalg/special.jl:122\n",
       "+(A::Array{T,2}, B::SymTridiagonal{T}) at linalg/special.jl:131\n",
       "+(A::Array{T,2}, B::Base.LinAlg.AbstractTriangular{T,S<:AbstractArray{T,2}}) at linalg/special.jl:159\n",
       "+(A::Array{T,N}, B::SparseMatrixCSC{Tv,Ti<:Integer}) at sparse/sparsematrix.jl:1019\n",
       "+{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period}}(x::Union{DenseArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N},SubArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N,A<:DenseArray{T,N},I<:Tuple{Vararg{Union{Colon,Int64,Range{Int64}}}},LD}}) at dates/periods.jl:221\n",
       "+(A::AbstractArray{Bool,N}, x::Bool) at arraymath.jl:135\n",
       "+(A::Union{DenseArray{Bool,N},SubArray{Bool,N,A<:DenseArray{T,N},I<:Tuple{Vararg{Union{Colon,Int64,Range{Int64}}}},LD}}, B::Union{DenseArray{Bool,N},SubArray{Bool,N,A<:DenseArray{T,N},I<:Tuple{Vararg{Union{Colon,Int64,Range{Int64}}}},LD}}) at arraymath.jl:166\n",
       "+(A::SymTridiagonal{T}, B::SymTridiagonal{T}) at linalg/tridiag.jl:84\n",
       "+(A::Tridiagonal{T}, B::Tridiagonal{T}) at linalg/tridiag.jl:404\n",
       "+(A::UpperTriangular{T,S<:AbstractArray{T,2}}, B::UpperTriangular{T,S<:AbstractArray{T,2}}) at linalg/triangular.jl:347\n",
       "+(A::LowerTriangular{T,S<:AbstractArray{T,2}}, B::LowerTriangular{T,S<:AbstractArray{T,2}}) at linalg/triangular.jl:348\n",
       "+(A::UpperTriangular{T,S<:AbstractArray{T,2}}, B::Base.LinAlg.UnitUpperTriangular{T,S<:AbstractArray{T,2}}) at linalg/triangular.jl:349\n",
       "+(A::LowerTriangular{T,S<:AbstractArray{T,2}}, B::Base.LinAlg.UnitLowerTriangular{T,S<:AbstractArray{T,2}}) at linalg/triangular.jl:350\n",
       "+(A::Base.LinAlg.UnitUpperTriangular{T,S<:AbstractArray{T,2}}, B::UpperTriangular{T,S<:AbstractArray{T,2}}) at linalg/triangular.jl:351\n",
       "+(A::Base.LinAlg.UnitLowerTriangular{T,S<:AbstractArray{T,2}}, B::LowerTriangular{T,S<:AbstractArray{T,2}}) at linalg/triangular.jl:352\n",
       "+(A::Base.LinAlg.UnitUpperTriangular{T,S<:AbstractArray{T,2}}, B::Base.LinAlg.UnitUpperTriangular{T,S<:AbstractArray{T,2}}) at linalg/triangular.jl:353\n",
       "+(A::Base.LinAlg.UnitLowerTriangular{T,S<:AbstractArray{T,2}}, B::Base.LinAlg.UnitLowerTriangular{T,S<:AbstractArray{T,2}}) at linalg/triangular.jl:354\n",
       "+(A::Base.LinAlg.AbstractTriangular{T,S<:AbstractArray{T,2}}, B::Base.LinAlg.AbstractTriangular{T,S<:AbstractArray{T,2}}) at linalg/triangular.jl:355\n",
       "+(Da::Diagonal{T}, Db::Diagonal{T}) at linalg/diagonal.jl:86\n",
       "+(A::Bidiagonal{T}, B::Bidiagonal{T}) at linalg/bidiag.jl:176\n",
       "+(UL::UpperTriangular{T,S<:AbstractArray{T,2}}, J::UniformScaling{T<:Number}) at linalg/uniformscaling.jl:45\n",
       "+(UL::Base.LinAlg.UnitUpperTriangular{T,S<:AbstractArray{T,2}}, J::UniformScaling{T<:Number}) at linalg/uniformscaling.jl:48\n",
       "+(UL::LowerTriangular{T,S<:AbstractArray{T,2}}, J::UniformScaling{T<:Number}) at linalg/uniformscaling.jl:45\n",
       "+(UL::Base.LinAlg.UnitLowerTriangular{T,S<:AbstractArray{T,2}}, J::UniformScaling{T<:Number}) at linalg/uniformscaling.jl:48\n",
       "+(A::Diagonal{T}, B::Bidiagonal{T}) at linalg/special.jl:121\n",
       "+(A::Bidiagonal{T}, B::Diagonal{T}) at linalg/special.jl:122\n",
       "+(A::Diagonal{T}, B::Tridiagonal{T}) at linalg/special.jl:121\n",
       "+(A::Tridiagonal{T}, B::Diagonal{T}) at linalg/special.jl:122\n",
       "+(A::Diagonal{T}, B::Array{T,2}) at linalg/special.jl:121\n",
       "+(A::Bidiagonal{T}, B::Tridiagonal{T}) at linalg/special.jl:121\n",
       "+(A::Tridiagonal{T}, B::Bidiagonal{T}) at linalg/special.jl:122\n",
       "+(A::Bidiagonal{T}, B::Array{T,2}) at linalg/special.jl:121\n",
       "+(A::Tridiagonal{T}, B::Array{T,2}) at linalg/special.jl:121\n",
       "+(A::SymTridiagonal{T}, B::Tridiagonal{T}) at linalg/special.jl:130\n",
       "+(A::Tridiagonal{T}, B::SymTridiagonal{T}) at linalg/special.jl:131\n",
       "+(A::SymTridiagonal{T}, B::Array{T,2}) at linalg/special.jl:130\n",
       "+(A::Diagonal{T}, B::SymTridiagonal{T}) at linalg/special.jl:139\n",
       "+(A::SymTridiagonal{T}, B::Diagonal{T}) at linalg/special.jl:140\n",
       "+(A::Bidiagonal{T}, B::SymTridiagonal{T}) at linalg/special.jl:139\n",
       "+(A::SymTridiagonal{T}, B::Bidiagonal{T}) at linalg/special.jl:140\n",
       "+(A::Diagonal{T}, B::UpperTriangular{T,S<:AbstractArray{T,2}}) at linalg/special.jl:151\n",
       "+(A::UpperTriangular{T,S<:AbstractArray{T,2}}, B::Diagonal{T}) at linalg/special.jl:152\n",
       "+(A::Diagonal{T}, B::Base.LinAlg.UnitUpperTriangular{T,S<:AbstractArray{T,2}}) at linalg/special.jl:151\n",
       "+(A::Base.LinAlg.UnitUpperTriangular{T,S<:AbstractArray{T,2}}, B::Diagonal{T}) at linalg/special.jl:152\n",
       "+(A::Diagonal{T}, B::LowerTriangular{T,S<:AbstractArray{T,2}}) at linalg/special.jl:151\n",
       "+(A::LowerTriangular{T,S<:AbstractArray{T,2}}, B::Diagonal{T}) at linalg/special.jl:152\n",
       "+(A::Diagonal{T}, B::Base.LinAlg.UnitLowerTriangular{T,S<:AbstractArray{T,2}}) at linalg/special.jl:151\n",
       "+(A::Base.LinAlg.UnitLowerTriangular{T,S<:AbstractArray{T,2}}, B::Diagonal{T}) at linalg/special.jl:152\n",
       "+(A::Base.LinAlg.AbstractTriangular{T,S<:AbstractArray{T,2}}, B::SymTridiagonal{T}) at linalg/special.jl:158\n",
       "+(A::SymTridiagonal{T}, B::Base.LinAlg.AbstractTriangular{T,S<:AbstractArray{T,2}}) at linalg/special.jl:159\n",
       "+(A::Base.LinAlg.AbstractTriangular{T,S<:AbstractArray{T,2}}, B::Tridiagonal{T}) at linalg/special.jl:158\n",
       "+(A::Tridiagonal{T}, B::Base.LinAlg.AbstractTriangular{T,S<:AbstractArray{T,2}}) at linalg/special.jl:159\n",
       "+(A::Base.LinAlg.AbstractTriangular{T,S<:AbstractArray{T,2}}, B::Bidiagonal{T}) at linalg/special.jl:158\n",
       "+(A::Bidiagonal{T}, B::Base.LinAlg.AbstractTriangular{T,S<:AbstractArray{T,2}}) at linalg/special.jl:159\n",
       "+(A::Base.LinAlg.AbstractTriangular{T,S<:AbstractArray{T,2}}, B::Array{T,2}) at linalg/special.jl:158\n",
       "+{Tv1,Ti1,Tv2,Ti2}(A_1::SparseMatrixCSC{Tv1,Ti1}, A_2::SparseMatrixCSC{Tv2,Ti2}) at sparse/sparsematrix.jl:1005\n",
       "+(A::SparseMatrixCSC{Tv,Ti<:Integer}, B::Array{T,N}) at sparse/sparsematrix.jl:1017\n",
       "+(A::SparseMatrixCSC{Tv,Ti<:Integer}, J::UniformScaling{T<:Number}) at sparse/sparsematrix.jl:3030\n",
       "+{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period}}(Y::Union{DenseArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N},SubArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N,A<:DenseArray{T,N},I<:Tuple{Vararg{Union{Colon,Int64,Range{Int64}}}},LD}}, x::Union{Base.Dates.CompoundPeriod,Base.Dates.Period}) at dates/periods.jl:235\n",
       "+{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},Q<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period}}(X::Union{DenseArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N},SubArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N,A<:DenseArray{T,N},I<:Tuple{Vararg{Union{Colon,Int64,Range{Int64}}}},LD}}, Y::Union{DenseArray{Q<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N},SubArray{Q<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N,A<:DenseArray{T,N},I<:Tuple{Vararg{Union{Colon,Int64,Range{Int64}}}},LD}}) at dates/periods.jl:236\n",
       "+{T<:Base.Dates.TimeType,P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period}}(x::Union{DenseArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N},SubArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N,A<:DenseArray{T,N},I<:Tuple{Vararg{Union{Colon,Int64,Range{Int64}}}},LD}}, y::T<:Base.Dates.TimeType) at dates/arithmetic.jl:83\n",
       "+{T<:Base.Dates.TimeType}(r::Range{T<:Base.Dates.TimeType}, x::Base.Dates.Period) at dates/ranges.jl:39\n",
       "+{T<:Number}(x::AbstractArray{T<:Number,N}) at abstractarraymath.jl:49\n",
       "+{S,T}(A::AbstractArray{S,N}, B::Range{T}) at arraymath.jl:78\n",
       "+{S,T}(A::AbstractArray{S,N}, B::AbstractArray{T,N}) at arraymath.jl:96\n",
       "+(A::AbstractArray{T,N}, x::Number) at arraymath.jl:139\n",
       "+(x::Number, A::AbstractArray{T,N}) at arraymath.jl:140\n",
       "+(x::Char, y::Integer) at char.jl:42\n",
       "+{N}(index1::CartesianIndex{N}, index2::CartesianIndex{N}) at multidimensional.jl:42\n",
       "+(J1::UniformScaling{T<:Number}, J2::UniformScaling{T<:Number}) at linalg/uniformscaling.jl:27\n",
       "+(J::UniformScaling{T<:Number}, B::BitArray{2}) at linalg/uniformscaling.jl:29\n",
       "+(J::UniformScaling{T<:Number}, A::AbstractArray{T,2}) at linalg/uniformscaling.jl:30\n",
       "+(J::UniformScaling{T<:Number}, x::Number) at linalg/uniformscaling.jl:31\n",
       "+(x::Number, J::UniformScaling{T<:Number}) at linalg/uniformscaling.jl:32\n",
       "+{TA,TJ}(A::AbstractArray{TA,2}, J::UniformScaling{TJ}) at linalg/uniformscaling.jl:92\n",
       "+{T}(a::Base.Pkg.Resolve.VersionWeights.HierarchicalValue{T}, b::Base.Pkg.Resolve.VersionWeights.HierarchicalValue{T}) at pkg/resolve/versionweight.jl:23\n",
       "+(a::Base.Pkg.Resolve.VersionWeights.VWPreBuildItem, b::Base.Pkg.Resolve.VersionWeights.VWPreBuildItem) at pkg/resolve/versionweight.jl:85\n",
       "+(a::Base.Pkg.Resolve.VersionWeights.VWPreBuild, b::Base.Pkg.Resolve.VersionWeights.VWPreBuild) at pkg/resolve/versionweight.jl:131\n",
       "+(a::Base.Pkg.Resolve.VersionWeights.VersionWeight, b::Base.Pkg.Resolve.VersionWeights.VersionWeight) at pkg/resolve/versionweight.jl:185\n",
       "+(a::Base.Pkg.Resolve.MaxSum.FieldValues.FieldValue, b::Base.Pkg.Resolve.MaxSum.FieldValues.FieldValue) at pkg/resolve/fieldvalue.jl:44\n",
       "+{P<:Base.Dates.Period}(x::P<:Base.Dates.Period, y::P<:Base.Dates.Period) at dates/periods.jl:62\n",
       "+(x::Base.Dates.Period, y::Base.Dates.Period) at dates/periods.jl:209\n",
       "+(x::Base.Dates.CompoundPeriod, y::Base.Dates.Period) at dates/periods.jl:210\n",
       "+(y::Base.Dates.Period, x::Base.Dates.CompoundPeriod) at dates/periods.jl:211\n",
       "+(x::Base.Dates.CompoundPeriod, y::Base.Dates.CompoundPeriod) at dates/periods.jl:212\n",
       "+(x::Base.Dates.CompoundPeriod, y::Base.Dates.TimeType) at dates/periods.jl:257\n",
       "+(y::Base.Dates.Period, x::Base.Dates.TimeType) at dates/arithmetic.jl:66\n",
       "+{T<:Base.Dates.TimeType}(x::Base.Dates.Period, r::Range{T<:Base.Dates.TimeType}) at dates/ranges.jl:40\n",
       "+(x::Union{Base.Dates.CompoundPeriod,Base.Dates.Period}) at dates/periods.jl:220\n",
       "+{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period}}(x::Union{Base.Dates.CompoundPeriod,Base.Dates.Period}, Y::Union{DenseArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N},SubArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N,A<:DenseArray{T,N},I<:Tuple{Vararg{Union{Colon,Int64,Range{Int64}}}},LD}}) at dates/periods.jl:234\n",
       "+(dt::DateTime, y::Base.Dates.Year) at dates/arithmetic.jl:13\n",
       "+(dt::Date, y::Base.Dates.Year) at dates/arithmetic.jl:17\n",
       "+(dt::DateTime, z::Base.Dates.Month) at dates/arithmetic.jl:37\n",
       "+(dt::Date, z::Base.Dates.Month) at dates/arithmetic.jl:43\n",
       "+(x::Date, y::Base.Dates.Week) at dates/arithmetic.jl:60\n",
       "+(x::Date, y::Base.Dates.Day) at dates/arithmetic.jl:62\n",
       "+(x::DateTime, y::Base.Dates.Period) at dates/arithmetic.jl:64\n",
       "+(x::Base.Dates.TimeType) at dates/arithmetic.jl:8\n",
       "+(a::Base.Dates.TimeType, b::Base.Dates.Period, c::Base.Dates.Period) at dates/periods.jl:246\n",
       "+(a::Base.Dates.TimeType, b::Base.Dates.Period, c::Base.Dates.Period, d::Base.Dates.Period...) at dates/periods.jl:248\n",
       "+(x::Base.Dates.TimeType, y::Base.Dates.CompoundPeriod) at dates/periods.jl:252\n",
       "+(x::Base.Dates.Instant) at dates/arithmetic.jl:4\n",
       "+{T<:Base.Dates.TimeType}(x::AbstractArray{T<:Base.Dates.TimeType,N}, y::Union{Base.Dates.CompoundPeriod,Base.Dates.Period}) at dates/arithmetic.jl:76\n",
       "+{T<:Base.Dates.TimeType}(y::Union{Base.Dates.CompoundPeriod,Base.Dates.Period}, x::AbstractArray{T<:Base.Dates.TimeType,N}) at dates/arithmetic.jl:77\n",
       "+{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period}}(y::Base.Dates.TimeType, x::Union{DenseArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N},SubArray{P<:Union{Base.Dates.CompoundPeriod,Base.Dates.Period},N,A<:DenseArray{T,N},I<:Tuple{Vararg{Union{Colon,Int64,Range{Int64}}}},LD}}) at dates/arithmetic.jl:84\n",
       "+(a, b, c, xs...) at operators.jl:103"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "methods(+)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Caution:\n",
    "\n",
    "New Julia users frequently go around peppering function declarations with type restrictions. This is often unnecessary. \n",
    "\n",
    "Consider:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "doubler (generic function with 1 method)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "doubler(x) = 2x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Shouldn't you have\n",
    "\n",
    "```julia\n",
    "doubler(x::Float64)\n",
    "```\n",
    "for performance?\n",
    "\n",
    "**No**. The JIT compiles this function once for every *concrete* type you call it with. So \n",
    "`doubler(1)` and `doubler(1.5)` each get compiled separately, and Julia selects based on the type of the input which one to use thereafter.\n",
    "\n",
    "So...\n",
    "\n",
    "**Use type restrictions in functions only to control what code gets executed. Type as loosely as possible.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Another caveat:\n",
    "\n",
    "When we declare:\n",
    "\n",
    "```julia\n",
    "doubler(x=0) = 2x\n",
    "```\n",
    "Julia actually defines *two* methods, `doubler(x)` and `doubler()`. In the second case, it just makes the substitution. That is, default arguments are just sugar for multiple dispatch. Note, though, that a lot of default arguments does mean a lot of extra methods. If you have a lot of optional arguments, you may want keyword arguments, for which only a single method is compiled."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Q: How does this get used?\n",
    "\n",
    "## A: *Everywhere*\n",
    "\n",
    "- special-case algorithms for sparse vs dense matrices, diagonal matrices, matrices vs vectors\n",
    "- constructors (as we'll see): create output from multiple types of inputs\n",
    "- for parsing code (macros): define start condition, end conditions, recurse"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  }
 ],
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  "language_info": {
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   "mimetype": "application/julia",
   "name": "julia",
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