A lot of the power and extensibility in Julia comes from a collection of informal interfaces. By extending a few specific methods to work for a custom type, objects of that type not only receive those functionalities, but they are also able to be used in other methods that are written to generically build upon those behaviors.


Required methods Brief description
start(iter) Returns the initial iteration state
next(iter, state) Returns the current item and the next state
done(iter, state) Tests if there are any items remaining
Important optional methods Default definition Brief description
iteratorsize(IterType) HasLength() One of HasLength(), HasShape(), IsInfinite(), or SizeUnknown() as appropriate
iteratoreltype(IterType) HasEltype() Either EltypeUnknown() or HasEltype() as appropriate
eltype(IterType) Any The type the items returned by next()
length(iter) (undefined) The number of items, if known
size(iter, [dim...]) (undefined) The number of items in each dimension, if known
Value returned by iteratorsize(IterType) Required Methods
HasLength() length(iter)
HasShape() length(iter) and size(iter, [dim...])
IsInfinite() (none)
SizeUnknown() (none)
Value returned by iteratoreltype(IterType) Required Methods
HasEltype() eltype(IterType)
EltypeUnknown() (none)

Sequential iteration is implemented by the methods start(), done(), and next(). Instead of mutating objects as they are iterated over, Julia provides these three methods to keep track of the iteration state externally from the object. The start(iter) method returns the initial state for the iterable object iter. That state gets passed along to done(iter, state), which tests if there are any elements remaining, and next(iter, state), which returns a tuple containing the current element and an updated state. The state object can be anything, and is generally considered to be an implementation detail private to the iterable object.

Any object defines these three methods is iterable and can be used in the many functions that rely upon iteration. It can also be used directly in a for loop since the syntax:

for i in iter   # or  "for i = iter"
    # body

is translated into:

state = start(iter)
while !done(iter, state)
    (i, state) = next(iter, state)
    # body

A simple example is an iterable sequence of square numbers with a defined length:

```jldoctest squaretype julia> struct Squares count::Int end

julia> Base.start(::Squares) = 1

julia>, state) = (state*state, state+1)

julia> Base.done(S::Squares, state) = state > S.count

julia> Base.eltype(::Type{Squares}) = Int # Note that this is defined for the type

julia> Base.length(S::Squares) = S.count

With only [`start`](@ref), [`next`](@ref), and [`done`](@ref) definitions, the `Squares` type is already pretty powerful.
We can iterate over all the elements:

```jldoctest squaretype
julia> for i in Squares(7)

We can use many of the builtin methods that work with iterables, like in(), mean() and std():

```jldoctest squaretype julia> 25 in Squares(10) true

julia> mean(Squares(100)) 3383.5

julia> std(Squares(100)) 3024.355854282583

There are a few more methods we can extend to give Julia more information about this iterable
collection.  We know that the elements in a `Squares` sequence will always be `Int`. By extending
the [`eltype()`](@ref) method, we can give that information to Julia and help it make more specialized
code in the more complicated methods. We also know the number of elements in our sequence, so
we can extend [`length()`](@ref), too.

Now, when we ask Julia to [`collect()`](@ref) all the elements into an array it can preallocate a `Vector{Int}`
of the right size instead of blindly [`push!`](@ref)ing each element into a `Vector{Any}`:

```jldoctest squaretype
julia> collect(Squares(10))' # transposed to save space
1×10 RowVector{Int64,Array{Int64,1}}:
 1  4  9  16  25  36  49  64  81  100

While we can rely upon generic implementations, we can also extend specific methods where we know there is a simpler algorithm. For example, there's a formula to compute the sum of squares, so we can override the generic iterative version with a more performant solution:

```jldoctest squaretype julia> Base.sum(S::Squares) = (n = S.count; return n(n+1)(2n+1)÷6)

julia> sum(Squares(1803)) 1955361914

This is a very common pattern throughout the Julia standard library: a small set of required methods
define an informal interface that enable many fancier behaviors. In some cases, types will want
to additionally specialize those extra behaviors when they know a more efficient algorithm can
be used in their specific case.

## Indexing

| Methods to implement | Brief description                |
|:-------------------- |:-------------------------------- |
| `getindex(X, i)`     | `X[i]`, indexed element access   |
| `setindex!(X, v, i)` | `X[i] = v`, indexed assignment   |
| `endof(X)`           | The last index, used in `X[end]` |

For the `Squares` iterable above, we can easily compute the `i`th element of the sequence by squaring
it.  We can expose this as an indexing expression `S[i]`. To opt into this behavior, `Squares`
simply needs to define [`getindex()`](@ref):

```jldoctest squaretype
julia> function Base.getindex(S::Squares, i::Int)
           1 <= i <= S.count || throw(BoundsError(S, i))
           return i*i

julia> Squares(100)[23]

Additionally, to support the syntax S[end], we must define endof() to specify the last valid index:

```jldoctest squaretype julia> Base.endof(S::Squares) = length(S)

julia> Squares(23)[end] 529

Note, though, that the above *only* defines [`getindex()`](@ref) with one integer index. Indexing with
anything other than an `Int` will throw a [`MethodError`](@ref) saying that there was no matching method.
In order to support indexing with ranges or vectors of `Int`s, separate methods must be written:

```jldoctest squaretype
julia> Base.getindex(S::Squares, i::Number) = S[convert(Int, i)]

julia> Base.getindex(S::Squares, I) = [S[i] for i in I]

julia> Squares(10)[[3,4.,5]]
3-element Array{Int64,1}:

While this is starting to support more of the indexing operations supported by some of the builtin types, there's still quite a number of behaviors missing. This Squares sequence is starting to look more and more like a vector as we've added behaviors to it. Instead of defining all these behaviors ourselves, we can officially define it as a subtype of an AbstractArray.

Abstract Arrays

Methods to implement Brief description
size(A) Returns a tuple containing the dimensions of A
getindex(A, i::Int) (if IndexLinear) Linear scalar indexing
getindex(A, I::Vararg{Int, N}) (if IndexCartesian, where N = ndims(A)) N-dimensional scalar indexing
setindex!(A, v, i::Int) (if IndexLinear) Scalar indexed assignment
setindex!(A, v, I::Vararg{Int, N}) (if IndexCartesian, where N = ndims(A)) N-dimensional scalar indexed assignment
Optional methods Default definition Brief description
IndexStyle(::Type) IndexCartesian() Returns either IndexLinear() or IndexCartesian(). See the description below.
getindex(A, I...) defined in terms of scalar getindex() Multidimensional and nonscalar indexing
setindex!(A, I...) defined in terms of scalar setindex!() Multidimensional and nonscalar indexed assignment
start()/next()/done() defined in terms of scalar getindex() Iteration
length(A) prod(size(A)) Number of elements
similar(A) similar(A, eltype(A), size(A)) Return a mutable array with the same shape and element type
similar(A, ::Type{S}) similar(A, S, size(A)) Return a mutable array with the same shape and the specified element type
similar(A, dims::NTuple{Int}) similar(A, eltype(A), dims) Return a mutable array with the same element type and size dims
similar(A, ::Type{S}, dims::NTuple{Int}) Array{S}(dims) Return a mutable array with the specified element type and size
Non-traditional indices Default definition Brief description
indices(A) map(OneTo, size(A)) Return the AbstractUnitRange of valid indices
Base.similar(A, ::Type{S}, inds::NTuple{Ind}) similar(A, S, Base.to_shape(inds)) Return a mutable array with the specified indices inds (see below)
Base.similar(T::Union{Type,Function}, inds) T(Base.to_shape(inds)) Return an array similar to T with the specified indices inds (see below)

If a type is defined as a subtype of AbstractArray, it inherits a very large set of rich behaviors including iteration and multidimensional indexing built on top of single-element access. See the arrays manual page and standard library section for more supported methods.

A key part in defining an AbstractArray subtype is IndexStyle. Since indexing is such an important part of an array and often occurs in hot loops, it's important to make both indexing and indexed assignment as efficient as possible. Array data structures are typically defined in one of two ways: either it most efficiently accesses its elements using just one index (linear indexing) or it intrinsically accesses the elements with indices specified for every dimension. These two modalities are identified by Julia as IndexLinear() and IndexCartesian(). Converting a linear index to multiple indexing subscripts is typically very expensive, so this provides a traits-based mechanism to enable efficient generic code for all array types.

This distinction determines which scalar indexing methods the type must define. IndexLinear() arrays are simple: just define getindex(A::ArrayType, i::Int). When the array is subsequently indexed with a multidimensional set of indices, the fallback getindex(A::AbstractArray, I...)() efficiently converts the indices into one linear index and then calls the above method. IndexCartesian() arrays, on the other hand, require methods to be defined for each supported dimensionality with ndims(A) Int indices. For example, the built-in SparseMatrixCSC type only supports two dimensions, so it just defines getindex(A::SparseMatrixCSC, i::Int, j::Int). The same holds for setindex!().

Returning to the sequence of squares from above, we could instead define it as a subtype of an AbstractArray{Int, 1}:

```jldoctest squarevectype julia> struct SquaresVector <: AbstractArray{Int, 1} count::Int end

julia> Base.size(S::SquaresVector) = (S.count,)

julia> Base.IndexStyle(::Type{<:SquaresVector}) = IndexLinear()

julia> Base.getindex(S::SquaresVector, i::Int) = i*i

Note that it's very important to specify the two parameters of the `AbstractArray`; the first
defines the [`eltype()`](@ref), and the second defines the [`ndims()`](@ref). That supertype and those three
methods are all it takes for `SquaresVector` to be an iterable, indexable, and completely functional

```jldoctest squarevectype
julia> s = SquaresVector(7)
7-element SquaresVector:

julia> s[s .> 20]
3-element Array{Int64,1}:

julia> s \ [1 2; 3 4; 5 6; 7 8; 9 10; 11 12; 13 14]
1×2 Array{Float64,2}:
 0.305389  0.335329

julia> s ⋅ s # dot(s, s)

As a more complicated example, let's define our own toy N-dimensional sparse-like array type built on top of Dict:

```jldoctest squarevectype julia> struct SparseArray{T,N} <: AbstractArray{T,N} data::Dict{NTuple{N,Int}, T} dims::NTuple{N,Int} end

julia> SparseArray{T}(::Type{T}, dims::Int...) = SparseArray(T, dims);

julia> SparseArray{T,N}(::Type{T}, dims::NTuple{N,Int}) = SparseArray{T,N}(Dict{NTuple{N,Int}, T}(), dims);

julia> Base.size(A::SparseArray) = A.dims

julia> Base.similar(A::SparseArray, ::Type{T}, dims::Dims) where {T} = SparseArray(T, dims)

julia> Base.getindex(A::SparseArray{T,N}, I::Vararg{Int,N}) where {T,N} = get(, I, zero(T))

julia> Base.setindex!(A::SparseArray{T,N}, v, I::Vararg{Int,N}) where {T,N} = ([I] = v)

Notice that this is an `IndexCartesian` array, so we must manually define [`getindex()`](@ref) and [`setindex!()`](@ref)
at the dimensionality of the array. Unlike the `SquaresVector`, we are able to define [`setindex!()`](@ref),
and so we can mutate the array:

```jldoctest squarevectype
julia> A = SparseArray(Float64, 3, 3)
3×3 SparseArray{Float64,2}:
 0.0  0.0  0.0
 0.0  0.0  0.0
 0.0  0.0  0.0

julia> fill!(A, 2)
3×3 SparseArray{Float64,2}:
 2.0  2.0  2.0
 2.0  2.0  2.0
 2.0  2.0  2.0

julia> A[:] = 1:length(A); A
3×3 SparseArray{Float64,2}:
 1.0  4.0  7.0
 2.0  5.0  8.0
 3.0  6.0  9.0

The result of indexing an AbstractArray can itself be an array (for instance when indexing by a Range). The AbstractArray fallback methods use similar() to allocate an Array of the appropriate size and element type, which is filled in using the basic indexing method described above. However, when implementing an array wrapper you often want the result to be wrapped as well:

```jldoctest squarevectype julia> A[1:2,:] 2×3 SparseArray{Float64,2}: 1.0 4.0 7.0 2.0 5.0 8.0

In this example it is accomplished by defining `Base.similar{T}(A::SparseArray, ::Type{T}, dims::Dims)`
to create the appropriate wrapped array. (Note that while `similar` supports 1- and 2-argument
forms, in most case you only need to specialize the 3-argument form.) For this to work it's important
that `SparseArray` is mutable (supports `setindex!`). Defining `similar()`, `getindex()` and
`setindex!()` for `SparseArray` also makes it possible to [`copy()`](@ref) the array:

```jldoctest squarevectype
julia> copy(A)
3×3 SparseArray{Float64,2}:
 1.0  4.0  7.0
 2.0  5.0  8.0
 3.0  6.0  9.0

In addition to all the iterable and indexable methods from above, these types can also interact with each other and use most of the methods defined in the standard library for AbstractArrays:

```jldoctest squarevectype julia> A[SquaresVector(3)] 3-element SparseArray{Float64,1}: 1.0 4.0 9.0

julia> dot(A[:,1],A[:,2]) 32.0 ```

If you are defining an array type that allows non-traditional indexing (indices that start at something other than 1), you should specialize indices. You should also specialize similar so that the dims argument (ordinarily a Dims size-tuple) can accept AbstractUnitRange objects, perhaps range-types Ind of your own design. For more information, see Arrays with custom indices.