A `DenseArray`

describes a multidimensional array that has regular offets in
memory. Implementing multidimensional arrays with the API flexibility of MATLAB
(range indexing, colon indexing, etc.) is hard. Julia makes this easy by
requiring only four methods to implement a `DenseArray`

(five for copy
semantics).

When an implementation satisfies the `DenseArray`

interface, several behaviors
are provided for free e.g. `collect`

, `find`

, `maximum`

, `sum`

, and many other
functions in the standard
library.

This post demonstrates what is necessary for a simple DenseArray implementation
by constructing a lazy wrapper that squares elements of the underlying array
when accessed. We call this lazy squaring wrapper `LazySquares`

. Naturally, all
mutations to the underlying array will be reflected in the lazy squaring
wrapper.

## Lazy Squares Specification

Given an existing multidimensional array `xs`

, we would like to wrap it in a
dense array using `squares`

that will lazily square its contents. The desired
behavior is illustrated in the example below.

```
julia> xs = [1, 2, 3]
3-element Array{Int64,1}:
1
2
3
julia> xs_squared = square(xs)
3-element LazySquares.LazySquare{Int64,1}:
1
4
9
```

Furthermore, changes to the underlying array `xs`

should be reflected in the
lazy squaring wrapper.

```
julia> xs[2] = 5
5
julia> xs_squared[2]
25
```

## Lazy Squares Module and Type

We begin by creating a module. As per Julia convention, we name the module the
plural form of the primary type provided by the module `LazySquares`

since the
primary type provided is `LazySquare`

.

```
module LazySquares
type LazySquare{T <: Number, N} <: DenseArray{T, N}
data::DenseArray{T, N}
end
end
```

It is clear from this interface that `LazySquare`

will be a `DenseArray`

with
elements of type `T`

while using `N`

dimensions. The `LazySquare`

type, as
previously described, is a wrapper over an existing `DenseArray`

; this makes it easy
to extend by forwarding most methods to the underlying array.

## Implementing Constructors and Methods

There are two types of methods to implement. First, we must implement
*constructors* which are responsible for initializing the `LazySquare`

type.

```
square{T <: Number, N}(xs::DenseArray{T, N})
```

Second, we implement *methods* for the corresponding functions required for
`DenseArray`

objects to operate normally:

```
Base.linearindexing(::Type{LazySquare})
Base.size(A::LazySquare)
Base.getindex(A,::LazySquare, i::Int)
Base.setindex!(A::LazySquare, v, i::Int)
```

The only non-obvious method to implement here is `Base.linearindexing`

. This
turns out to be very simple and we describe it in detail later.

Julia distinguishes between functions and methods. Functions are identified by their name. Functions have many methods which are definitions that are dispatched at runtime by the number of arguments passed to the function and the types of each argument. This is known as

dynamic dispatch.

### Square Constructor

Julia creates a default constructor with all fields as positional arguments
with the appropriate types. We simply pass the `array`

argument through to the
default constructor of `LazySquare`

.

```
square{T <: Number, N}(array::DenseArray{T, N}) = LazySquare(array)
```

### Dense Array Methods

We have four methods to implement here.

We begin with the obscure `Base.linearindexing`

. This method describes whether
each location can be described with a single, scalar index efficiently. If this
is the case, we assign the method the value of `Base.LinearFast()`

; otherwise,
we assign the method the value of `Base.LinearSlow()`

. For `DenseArray`

, it is
obvious that we may use a single, scalar index due to its definition of having
regular offsets in memory.

```
Base.linearindexing{T <: LazySquare}(::Type{T}) = Base.LinearFast()
```

`Base.LinearSlow()`

is assumed by default. This implies that array requires indices for each dimension to locate an item in the array. This requires that implementing specialized`getindex`

and`setindex!`

methods for each of the number of dimensions supported.

Finally, we implement the standard array-like methods: `size`

, `getindex`

, and
`setindex!`

. The latter two methods describe array access and array assignment
respectively.

```
"Return a tuple of sizes for each dimension"
Base.size{T <: Number, N}(array::LazySquare{T, N}) = size(array.data)
"Return an element of type `T` given an index `i`"
Base.getindex{T <: Number, N}(array::LazySquare{T, N}, i::Int) = array.data[i] * array.data[i]
"Assigns a value `v` of type `T` to the location identified by index `i`"
Base.setindex!{T <: Number, N}(array::LazySquare{T, N}, val::T, i::Int) = array.data[i] = val
```

It is important that we lazily square the contents of the underlying array in
`getindex`

as specified in our specification.

## Testing

Given this implementation, it is also good practice to verify that our claims hold with unit tests. That is, we should verify that the elements of the underlying array are squared when accessed and that changes to the underlying array should be reflected in the lazy squaring wrapper.

```
using LazySquares
using Base.Test
xs = Int[1, 2, 3, 4]
xs_square = square(xs)
info("Verify that the elements are squared")
@test xs_square == xs .^ 2
info("Verify that getindex works")
@test xs_square[1] == xs[1] * xs[1]
@test xs_square[3] == xs[3] * xs[3]
info("Verify that setindex! works")
xs_square[2] = 5
@test xs_square[2] == 5 * 5
@test xs[2] == 5
```

Finally, we may verify that standard library functions for arrays also work as intended.

```
info("Verify that find works")
@test find(x -> x == 9, xs_square) == [3]
```

Further tests can be added for higher-dimensional arrays such as matrices and tensors.

## Conclusion

Julia makes it easy to extend and implement multidimensional array-like
behavior with only a concise set of methods. This level of generality enables
the construction of
SharedArray
and DistributedArray
types while reusing the same `AbstractArray`

hierarchy.