# [Multi-dimensional Arrays](@id man-multi-dim-arrays)

Julia, like most technical computing languages, provides a first-class array implementation. Most
technical computing languages pay a lot of attention to their array implementation at the expense
of other containers. Julia does not treat arrays in any special way. The array library is implemented
almost completely in Julia itself, and derives its performance from the compiler, just like any
other code written in Julia. As such, it's also possible to define custom array types by inheriting
from [`AbstractArray`](@ref). See the [manual section on the AbstractArray interface](@ref man-interface-array) for more details
on implementing a custom array type.

An array is a collection of objects stored in a multi-dimensional grid. In the most general case,
an array may contain objects of type [`Any`](@ref). For most computational purposes, arrays should contain
objects of a more specific type, such as [`Float64`](@ref) or [`Int32`](@ref).

In general, unlike many other technical computing languages, Julia does not expect programs to
be written in a vectorized style for performance. Julia's compiler uses type inference and generates
optimized code for scalar array indexing, allowing programs to be written in a style that is convenient
and readable, without sacrificing performance, and using less memory at times.

In Julia, all arguments to functions are [passed by
sharing](https://en.wikipedia.org/wiki/Evaluation_strategy#Call_by_sharing)
(i.e. by pointers). Some technical computing languages pass arrays by value, and
while this prevents accidental modification by callees of a value in the caller,
it makes avoiding unwanted copying of arrays difficult. By convention, a
function name ending with a `!` indicates that it will mutate or destroy the
value of one or more of its arguments (compare, for example, [`sort`](@ref) and [`sort!`](@ref)).
Callees must make explicit copies to ensure that they don't modify inputs that
they don't intend to change. Many non- mutating functions are implemented by
calling a function of the same name with an added `!` at the end on an explicit
copy of the input, and returning that copy.

## Basic Functions

| Function               | Description                                                                      |
|:---------------------- |:-------------------------------------------------------------------------------- |
| [`eltype(A)`](@ref)    | the type of the elements contained in `A`                                        |
| [`length(A)`](@ref)    | the number of elements in `A`                                                    |
| [`ndims(A)`](@ref)     | the number of dimensions of `A`                                                  |
| [`size(A)`](@ref)      | a tuple containing the dimensions of `A`                                         |
| [`size(A,n)`](@ref)    | the size of `A` along dimension `n`                                              |
| [`axes(A)`](@ref)      | a tuple containing the valid indices of `A`                                      |
| [`axes(A,n)`](@ref)    | a range expressing the valid indices along dimension `n`                         |
| [`eachindex(A)`](@ref) | an efficient iterator for visiting each position in `A`                          |
| [`stride(A,k)`](@ref)  | the stride (linear index distance between adjacent elements) along dimension `k` |
| [`strides(A)`](@ref)   | a tuple of the strides in each dimension                                         |

## Construction and Initialization

Many functions for constructing and initializing arrays are provided. In the following list of
such functions, calls with a `dims...` argument can either take a single tuple of dimension sizes
or a series of dimension sizes passed as a variable number of arguments. Most of these functions
also accept a first input `T`, which is the element type of the array. If the type `T` is
omitted it will default to [`Float64`](@ref).

| Function                           | Description                                                                                                                                                                                                                                  |
|:---------------------------------- |:-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| [`Array{T}(undef, dims...)`](@ref)             | an uninitialized dense [`Array`](@ref)                                                                                                                                                                                                              |
| [`zeros(T, dims...)`](@ref)                    | an `Array` of all zeros                                                                                                                                                                                                                      |
| [`ones(T, dims...)`](@ref)                     | an `Array` of all ones                                                                                                                                                                                                                       |
| [`trues(dims...)`](@ref)                       | a [`BitArray`](@ref) with all values `true`                                                                                                                                                                                                  |
| [`falses(dims...)`](@ref)                      | a `BitArray` with all values `false`                                                                                                                                                                                                         |
| [`reshape(A, dims...)`](@ref)                  | an array containing the same data as `A`, but with different dimensions                                                                                                                                                                      |
| [`copy(A)`](@ref)                              | copy `A`                                                                                                                                                                                                                                     |
| [`deepcopy(A)`](@ref)                          | copy `A`, recursively copying its elements                                                                                                                                                                                                   |
| [`similar(A, T, dims...)`](@ref)               | an uninitialized array of the same type as `A` (dense, sparse, etc.), but with the specified element type and dimensions. The second and third arguments are both optional, defaulting to the element type and dimensions of `A` if omitted. |
| [`reinterpret(T, A)`](@ref)                    | an array with the same binary data as `A`, but with element type `T`                                                                                                                                                                         |
| [`rand(T, dims...)`](@ref)                     | an `Array` with random, iid [^1] and uniformly distributed values in the half-open interval ``[0, 1)``                                                                                                                                       |
| [`randn(T, dims...)`](@ref)                    | an `Array` with random, iid and standard normally distributed values                                                                                                                                                                         |
| [`Matrix{T}(I, m, n)`](@ref)                   | `m`-by-`n` identity matrix. Requires `using LinearAlgebra` for [`I`](@ref).                                                                                                                                                                                                                   |
| [`range(start, stop=stop, length=n)`](@ref)    | range of `n` linearly spaced elements from `start` to `stop`                                                                                                                                                                                 |
| [`fill!(A, x)`](@ref)                          | fill the array `A` with the value `x`                                                                                                                                                                                                        |
| [`fill(x, dims...)`](@ref)                     | an `Array` filled with the value `x`                                                                                                                                                                                                         |

[^1]: *iid*, independently and identically distributed.

To see the various ways we can pass dimensions to these functions, consider the following examples:
```jldoctest
julia> zeros(Int8, 2, 3)
2×3 Array{Int8,2}:
 0  0  0
 0  0  0

julia> zeros(Int8, (2, 3))
2×3 Array{Int8,2}:
 0  0  0
 0  0  0

julia> zeros((2, 3))
2×3 Array{Float64,2}:
 0.0  0.0  0.0
 0.0  0.0  0.0
```
Here, `(2, 3)` is a [`Tuple`](@ref) and the first argument — the element type — is optional, defaulting to `Float64`.

## [Array literals](@id man-array-literals)

Arrays can also be directly constructed with square braces; the syntax `[A, B, C, ...]`
creates a one dimensional array (i.e., a vector) containing the comma-separated arguments as
its elements. The element type ([`eltype`](@ref)) of the resulting array is automatically
determined by the types of the arguments inside the braces. If all the arguments are the
same type, then that is its `eltype`. If they all have a common
[promotion type](@ref conversion-and-promotion) then they get converted to that type using
[`convert`](@ref) and that type is the array's `eltype`. Otherwise, a heterogeneous array
that can hold anything — a `Vector{Any}` — is constructed; this includes the literal `[]`
where no arguments are given.

```jldoctest
julia> [1,2,3] # An array of `Int`s
3-element Array{Int64,1}:
 1
 2
 3

julia> promote(1, 2.3, 4//5) # This combination of Int, Float64 and Rational promotes to Float64
(1.0, 2.3, 0.8)

julia> [1, 2.3, 4//5] # Thus that's the element type of this Array
3-element Array{Float64,1}:
 1.0
 2.3
 0.8

julia> []
Any[]
```

### [Concatenation](@id man-array-concatenation)

If the arguments inside the square brackets are separated by semicolons (`;`) or newlines
instead of commas, then their contents are _vertically concatenated_ together instead of
the arguments being used as elements themselves.

```jldoctest
julia> [1:2, 4:5] # Has a comma, so no concatenation occurs. The ranges are themselves the elements
2-element Array{UnitRange{Int64},1}:
 1:2
 4:5

julia> [1:2; 4:5]
4-element Array{Int64,1}:
 1
 2
 4
 5

julia> [1:2
        4:5
        6]
5-element Array{Int64,1}:
 1
 2
 4
 5
 6
```

Similarly, if the arguments are separated by tabs or spaces, then their contents are
_horizontally concatenated_ together.

```jldoctest
julia> [1:2  4:5  7:8]
2×3 Array{Int64,2}:
 1  4  7
 2  5  8

julia> [[1,2]  [4,5]  [7,8]]
2×3 Array{Int64,2}:
 1  4  7
 2  5  8

julia> [1 2 3] # Numbers can also be horizontally concatenated
1×3 Array{Int64,2}:
 1  2  3
```

Using semicolons (or newlines) and spaces (or tabs) can be combined to concatenate
both horizontally and vertically at the same time.

```jldoctest
julia> [1 2
        3 4]
2×2 Array{Int64,2}:
 1  2
 3  4

julia> [zeros(Int, 2, 2) [1; 2]
        [3 4]            5]
3×3 Array{Int64,2}:
 0  0  1
 0  0  2
 3  4  5
```

More generally, concatenation can be accomplished through the [`cat`](@ref) function.
These syntaxes are shorthands for function calls that themselves are convenience functions:

| Syntax            | Function        | Description                                        |
|:----------------- |:--------------- |:-------------------------------------------------- |
|                   | [`cat`](@ref)   | concatenate input arrays along dimension(s) `k`    |
| `[A; B; C; ...]`  | [`vcat`](@ref)  | shorthand for `cat(A...; dims=1)                   |
| `[A B C ...]`     | [`hcat`](@ref)  | shorthand for `cat(A...; dims=2)                   |
| `[A B; C D; ...]` | [`hvcat`](@ref) | simultaneous vertical and horizontal concatenation |

### Typed array literals

An array with a specific element type can be constructed using the syntax `T[A, B, C, ...]`. This
will construct a 1-d array with element type `T`, initialized to contain elements `A`, `B`, `C`,
etc. For example, `Any[x, y, z]` constructs a heterogeneous array that can contain any values.

Concatenation syntax can similarly be prefixed with a type to specify the element type of the
result.

```jldoctest
julia> [[1 2] [3 4]]
1×4 Array{Int64,2}:
 1  2  3  4

julia> Int8[[1 2] [3 4]]
1×4 Array{Int8,2}:
 1  2  3  4
```

## [Comprehensions](@id man-comprehensions)

Comprehensions provide a general and powerful way to construct arrays. Comprehension syntax is
similar to set construction notation in mathematics:

```
A = [ F(x,y,...) for x=rx, y=ry, ... ]
```

The meaning of this form is that `F(x,y,...)` is evaluated with the variables `x`, `y`, etc. taking
on each value in their given list of values. Values can be specified as any iterable object, but
will commonly be ranges like `1:n` or `2:(n-1)`, or explicit arrays of values like `[1.2, 3.4, 5.7]`.
The result is an N-d dense array with dimensions that are the concatenation of the dimensions
of the variable ranges `rx`, `ry`, etc. and each `F(x,y,...)` evaluation returns a scalar.

The following example computes a weighted average of the current element and its left and right
neighbor along a 1-d grid. :

```julia-repl
julia> x = rand(8)
8-element Array{Float64,1}:
 0.843025
 0.869052
 0.365105
 0.699456
 0.977653
 0.994953
 0.41084
 0.809411

julia> [ 0.25*x[i-1] + 0.5*x[i] + 0.25*x[i+1] for i=2:length(x)-1 ]
6-element Array{Float64,1}:
 0.736559
 0.57468
 0.685417
 0.912429
 0.8446
 0.656511
```

The resulting array type depends on the types of the computed elements just like [array literals](@ref man-array-literals) do. In order to control the
type explicitly, a type can be prepended to the comprehension. For example, we could have requested
the result in single precision by writing:

```julia
Float32[ 0.25*x[i-1] + 0.5*x[i] + 0.25*x[i+1] for i=2:length(x)-1 ]
```

## Generator Expressions

Comprehensions can also be written without the enclosing square brackets, producing an object
known as a generator. This object can be iterated to produce values on demand, instead of allocating
an array and storing them in advance (see [Iteration](@ref)). For example, the following expression
sums a series without allocating memory:

```jldoctest
julia> sum(1/n^2 for n=1:1000)
1.6439345666815615
```

When writing a generator expression with multiple dimensions inside an argument list, parentheses
are needed to separate the generator from subsequent arguments:

```julia-repl
julia> map(tuple, 1/(i+j) for i=1:2, j=1:2, [1:4;])
ERROR: syntax: invalid iteration specification
```

All comma-separated expressions after `for` are interpreted as ranges. Adding parentheses lets
us add a third argument to [`map`](@ref):

```jldoctest
julia> map(tuple, (1/(i+j) for i=1:2, j=1:2), [1 3; 2 4])
2×2 Array{Tuple{Float64,Int64},2}:
 (0.5, 1)       (0.333333, 3)
 (0.333333, 2)  (0.25, 4)
```

Generators are implemented via inner functions. Just like
inner functions used elsewhere in the language, variables from the enclosing scope can be
"captured" in the inner function.  For example, `sum(p[i] - q[i] for i=1:n)`
captures the three variables `p`, `q` and `n` from the enclosing scope.
Captured variables can present performance challenges; see
[performance tips](@ref man-performance-captured).


Ranges in generators and comprehensions can depend on previous ranges by writing multiple `for`
keywords:

```jldoctest
julia> [(i,j) for i=1:3 for j=1:i]
6-element Array{Tuple{Int64,Int64},1}:
 (1, 1)
 (2, 1)
 (2, 2)
 (3, 1)
 (3, 2)
 (3, 3)
```

In such cases, the result is always 1-d.

Generated values can be filtered using the `if` keyword:

```jldoctest
julia> [(i,j) for i=1:3 for j=1:i if i+j == 4]
2-element Array{Tuple{Int64,Int64},1}:
 (2, 2)
 (3, 1)
```

## [Indexing](@id man-array-indexing)

The general syntax for indexing into an n-dimensional array `A` is:

```
X = A[I_1, I_2, ..., I_n]
```

where each `I_k` may be a scalar integer, an array of integers, or any other
[supported index](@ref man-supported-index-types). This includes
[`Colon`](@ref) (`:`) to select all indices within the entire dimension,
ranges of the form `a:c` or `a:b:c` to select contiguous or strided
subsections, and arrays of booleans to select elements at their `true` indices.

If all the indices are scalars, then the result `X` is a single element from the array `A`. Otherwise,
`X` is an array with the same number of dimensions as the sum of the dimensionalities of all the
indices.

If all indices `I_k` are vectors, for example, then the shape of `X` would be `(length(I_1), length(I_2), ..., length(I_n))`,
with location `i_1, i_2, ..., i_n` of `X` containing the value `A[I_1[i_1], I_2[i_2], ..., I_n[i_n]]`.

Example:

```jldoctest
julia> A = reshape(collect(1:16), (2, 2, 2, 2))
2×2×2×2 Array{Int64,4}:
[:, :, 1, 1] =
 1  3
 2  4

[:, :, 2, 1] =
 5  7
 6  8

[:, :, 1, 2] =
  9  11
 10  12

[:, :, 2, 2] =
 13  15
 14  16

julia> A[1, 2, 1, 1] # all scalar indices
3

julia> A[[1, 2], [1], [1, 2], [1]] # all vector indices
2×1×2×1 Array{Int64,4}:
[:, :, 1, 1] =
 1
 2

[:, :, 2, 1] =
 5
 6

julia> A[[1, 2], [1], [1, 2], 1] # a mix of index types
2×1×2 Array{Int64,3}:
[:, :, 1] =
 1
 2

[:, :, 2] =
 5
 6
```

Note how the size of the resulting array is different in the last two cases.

If `I_1` is changed to a two-dimensional matrix, then `X` becomes an `n+1`-dimensional array of
shape `(size(I_1, 1), size(I_1, 2), length(I_2), ..., length(I_n))`. The matrix adds a dimension.

Example:

```jldoctest
julia> A = reshape(collect(1:16), (2, 2, 2, 2));

julia> A[[1 2; 1 2]]
2×2 Array{Int64,2}:
 1  2
 1  2

julia> A[[1 2; 1 2], 1, 2, 1]
2×2 Array{Int64,2}:
 5  6
 5  6
```

The location `i_1, i_2, i_3, ..., i_{n+1}` contains the value at `A[I_1[i_1, i_2], I_2[i_3], ..., I_n[i_{n+1}]]`.
All dimensions indexed with scalars are dropped. For example, if `J` is an array of indices, then the result of `A[2, J, 3]` is an
array with size `size(J)`. Its `j`th element is populated by `A[2, J[j], 3]`.

As a special part of this syntax, the `end` keyword may be used to represent the last index of
each dimension within the indexing brackets, as determined by the size of the innermost array
being indexed. Indexing syntax without the `end` keyword is equivalent to a call to [`getindex`](@ref):

```
X = getindex(A, I_1, I_2, ..., I_n)
```

Example:

```jldoctest
julia> x = reshape(1:16, 4, 4)
4×4 reshape(::UnitRange{Int64}, 4, 4) with eltype Int64:
 1  5   9  13
 2  6  10  14
 3  7  11  15
 4  8  12  16

julia> x[2:3, 2:end-1]
2×2 Array{Int64,2}:
 6  10
 7  11

julia> x[1, [2 3; 4 1]]
2×2 Array{Int64,2}:
  5  9
 13  1
```

## [Indexed Assignment](@id man-indexed-assignment)

The general syntax for assigning values in an n-dimensional array `A` is:

```
A[I_1, I_2, ..., I_n] = X
```

where each `I_k` may be a scalar integer, an array of integers, or any other
[supported index](@ref man-supported-index-types). This includes
[`Colon`](@ref) (`:`) to select all indices within the entire dimension,
ranges of the form `a:c` or `a:b:c` to select contiguous or strided
subsections, and arrays of booleans to select elements at their `true` indices.

If all indices `I_k` are integers, then the value in location `I_1, I_2, ..., I_n` of `A` is
overwritten with the value of `X`, [`convert`](@ref)ing to the
[`eltype`](@ref) of `A` if necessary.


If any index `I_k` selects more than one location, then the right hand side `X` must be an
array with the same shape as the result of indexing `A[I_1, I_2, ..., I_n]` or a vector with
the same number of elements. The value in location `I_1[i_1], I_2[i_2], ..., I_n[i_n]` of
`A` is overwritten with the value `X[I_1, I_2, ..., I_n]`, converting if necessary. The
element-wise assignment operator `.=` may be used to [broadcast](@ref Broadcasting) `X`
across the selected locations:


```
A[I_1, I_2, ..., I_n] .= X
```

Just as in [Indexing](@ref man-array-indexing), the `end` keyword may be used
to represent the last index of each dimension within the indexing brackets, as
determined by the size of the array being assigned into. Indexed assignment
syntax without the `end` keyword is equivalent to a call to
[`setindex!`](@ref):

```
setindex!(A, X, I_1, I_2, ..., I_n)
```

Example:

```jldoctest
julia> x = collect(reshape(1:9, 3, 3))
3×3 Array{Int64,2}:
 1  4  7
 2  5  8
 3  6  9

julia> x[3, 3] = -9;

julia> x[1:2, 1:2] = [-1 -4; -2 -5];

julia> x
3×3 Array{Int64,2}:
 -1  -4   7
 -2  -5   8
  3   6  -9
```

## [Supported index types](@id man-supported-index-types)

In the expression `A[I_1, I_2, ..., I_n]`, each `I_k` may be a scalar index, an
array of scalar indices, or an object that represents an array of scalar
indices and can be converted to such by [`to_indices`](@ref):

1. A scalar index. By default this includes:
    * Non-boolean integers
    * [`CartesianIndex{N}`](@ref)s, which behave like an `N`-tuple of integers spanning multiple dimensions (see below for more details)
2. An array of scalar indices. This includes:
    * Vectors and multidimensional arrays of integers
    * Empty arrays like `[]`, which select no elements
    * Ranges like `a:c` or `a:b:c`, which select contiguous or strided subsections from `a` to `c` (inclusive)
    * Any custom array of scalar indices that is a subtype of `AbstractArray`
    * Arrays of `CartesianIndex{N}` (see below for more details)
3. An object that represents an array of scalar indices and can be converted to such by [`to_indices`](@ref). By default this includes:
    * [`Colon()`](@ref) (`:`), which represents all indices within an entire dimension or across the entire array
    * Arrays of booleans, which select elements at their `true` indices (see below for more details)

Some examples:
```jldoctest
julia> A = reshape(collect(1:2:18), (3, 3))
3×3 Array{Int64,2}:
 1   7  13
 3   9  15
 5  11  17

julia> A[4]
7

julia> A[[2, 5, 8]]
3-element Array{Int64,1}:
  3
  9
 15

julia> A[[1 4; 3 8]]
2×2 Array{Int64,2}:
 1   7
 5  15

julia> A[[]]
Int64[]

julia> A[1:2:5]
3-element Array{Int64,1}:
 1
 5
 9

julia> A[2, :]
3-element Array{Int64,1}:
  3
  9
 15

julia> A[:, 3]
3-element Array{Int64,1}:
 13
 15
 17
```

### Cartesian indices

The special `CartesianIndex{N}` object represents a scalar index that behaves
like an `N`-tuple of integers spanning multiple dimensions.  For example:

```jldoctest cartesianindex
julia> A = reshape(1:32, 4, 4, 2);

julia> A[3, 2, 1]
7

julia> A[CartesianIndex(3, 2, 1)] == A[3, 2, 1] == 7
true
```

Considered alone, this may seem relatively trivial; `CartesianIndex` simply
gathers multiple integers together into one object that represents a single
multidimensional index. When combined with other indexing forms and iterators
that yield `CartesianIndex`es, however, this can produce very elegant
and efficient code. See [Iteration](@ref) below, and for some more advanced
examples, see [this blog post on multidimensional algorithms and
iteration](https://julialang.org/blog/2016/02/iteration).

Arrays of `CartesianIndex{N}` are also supported. They represent a collection
of scalar indices that each span `N` dimensions, enabling a form of indexing
that is sometimes referred to as pointwise indexing. For example, it enables
accessing the diagonal elements from the first "page" of `A` from above:

```jldoctest cartesianindex
julia> page = A[:,:,1]
4×4 Array{Int64,2}:
 1  5   9  13
 2  6  10  14
 3  7  11  15
 4  8  12  16

julia> page[[CartesianIndex(1,1),
             CartesianIndex(2,2),
             CartesianIndex(3,3),
             CartesianIndex(4,4)]]
4-element Array{Int64,1}:
  1
  6
 11
 16
```

This can be expressed much more simply with [dot broadcasting](@ref man-vectorized)
and by combining it with a normal integer index (instead of extracting the
first `page` from `A` as a separate step). It can even be combined with a `:`
to extract both diagonals from the two pages at the same time:

```jldoctest cartesianindex
julia> A[CartesianIndex.(axes(A, 1), axes(A, 2)), 1]
4-element Array{Int64,1}:
  1
  6
 11
 16

julia> A[CartesianIndex.(axes(A, 1), axes(A, 2)), :]
4×2 Array{Int64,2}:
  1  17
  6  22
 11  27
 16  32
```

!!! warning

    `CartesianIndex` and arrays of `CartesianIndex` are not compatible with the
    `end` keyword to represent the last index of a dimension. Do not use `end`
    in indexing expressions that may contain either `CartesianIndex` or arrays thereof.

### Logical indexing

Often referred to as logical indexing or indexing with a logical mask, indexing
by a boolean array selects elements at the indices where its values are `true`.
Indexing by a boolean vector `B` is effectively the same as indexing by the
vector of integers that is returned by [`findall(B)`](@ref). Similarly, indexing
by a `N`-dimensional boolean array is effectively the same as indexing by the
vector of `CartesianIndex{N}`s where its values are `true`. A logical index
must be a vector of the same length as the dimension it indexes into, or it
must be the only index provided and match the size and dimensionality of the
array it indexes into. It is generally more efficient to use boolean arrays as
indices directly instead of first calling [`findall`](@ref).

```jldoctest
julia> x = reshape(1:16, 4, 4)
4×4 reshape(::UnitRange{Int64}, 4, 4) with eltype Int64:
 1  5   9  13
 2  6  10  14
 3  7  11  15
 4  8  12  16

julia> x[[false, true, true, false], :]
2×4 Array{Int64,2}:
 2  6  10  14
 3  7  11  15

julia> mask = map(ispow2, x)
4×4 Array{Bool,2}:
 1  0  0  0
 1  0  0  0
 0  0  0  0
 1  1  0  1

julia> x[mask]
5-element Array{Int64,1}:
  1
  2
  4
  8
 16
```

### Number of indices

#### Cartesian indexing

The ordinary way to index into an `N`-dimensional array is to use exactly `N` indices; each
index selects the position(s) in its particular dimension. For example, in the three-dimensional
array `A = rand(4, 3, 2)`, `A[2, 3, 1]` will select the number in the second row of the third
column in the first "page" of the array. This is often referred to as _cartesian indexing_.

#### Linear indexing

When exactly one index `i` is provided, that index no longer represents a location in a
particular dimension of the array. Instead, it selects the `i`th element using the
column-major iteration order that linearly spans the entire array. This is known as _linear
indexing_. It essentially treats the array as though it had been reshaped into a
one-dimensional vector with [`vec`](@ref).

```jldoctest linindexing
julia> A = [2 6; 4 7; 3 1]
3×2 Array{Int64,2}:
 2  6
 4  7
 3  1

julia> A[5]
7

julia> vec(A)[5]
7
```

A linear index into the array `A` can be converted to a `CartesianIndex` for cartesian
indexing with `CartesianIndices(A)[i]` (see [`CartesianIndices`](@ref)), and a set of
`N` cartesian indices can be converted to a linear index with
`LinearIndices(A)[i_1, i_2, ..., i_N]` (see [`LinearIndices`](@ref)).

```jldoctest linindexing
julia> CartesianIndices(A)[5]
CartesianIndex(2, 2)

julia> LinearIndices(A)[2, 2]
5
```

It's important to note that there's a very large assymmetry in the performance
of these conversions. Converting a linear index to a set of cartesian indices
requires dividing and taking the remainder, whereas going the other way is just
multiplies and adds. In modern processors, integer division can be 10-50 times
slower than multiplication. While some arrays — like [`Array`](@ref) itself —
are implemented using a linear chunk of memory and directly use a linear index
in their implementations, other arrays — like [`Diagonal`](@ref) — need the
full set of cartesian indices to do their lookup (see [`IndexStyle`](@ref) to
introspect which is which). As such, when iterating over an entire array, it's
much better to iterate over [`eachindex(A)`](@ref) instead of `1:length(A)`.
Not only will the former be much faster in cases where `A` is `IndexCartesian`,
but it will also support OffsetArrays, too.

#### Omitted and extra indices

In addition to linear indexing, an `N`-dimensional array may be indexed with
fewer or more than `N` indices in certain situations.

Indices may be omitted if the trailing dimensions that are not indexed into are
all length one. In other words, trailing indices can be omitted only if there
is only one possible value that those omitted indices could be for an in-bounds
indexing expression. For example, a four-dimensional array with size `(3, 4, 2,
1)` may be indexed with only three indices as the dimension that gets skipped
(the fourth dimension) has length one. Note that linear indexing takes
precedence over this rule.

```jldoctest
julia> A = reshape(1:24, 3, 4, 2, 1)
3×4×2×1 reshape(::UnitRange{Int64}, 3, 4, 2, 1) with eltype Int64:
[:, :, 1, 1] =
 1  4  7  10
 2  5  8  11
 3  6  9  12

[:, :, 2, 1] =
 13  16  19  22
 14  17  20  23
 15  18  21  24

julia> A[1, 3, 2] # Omits the fourth dimension (length 1)
19

julia> A[1, 3] # Attempts to omit dimensions 3 & 4 (lengths 2 and 1)
ERROR: BoundsError: attempt to access 3×4×2×1 reshape(::UnitRange{Int64}, 3, 4, 2, 1) with eltype Int64 at index [1, 3]

julia> A[19] # Linear indexing
19
```

When omitting _all_ indices with `A[]`, this semantic provides a simple idiom
to retrieve the only element in an array and simultaneously ensure that there
was only one element.

Similarly, more than `N` indices may be provided if all the indices beyond the
dimensionality of the array are `1` (or more generally are the first and only
element of `axes(A, d)` where `d` is that particular dimension number). This
allows vectors to be indexed like one-column matrices, for example:

```jldoctest
julia> A = [8,6,7]
3-element Array{Int64,1}:
 8
 6
 7

julia> A[2,1]
6
```

## Iteration

The recommended ways to iterate over a whole array are

```julia
for a in A
    # Do something with the element a
end

for i in eachindex(A)
    # Do something with i and/or A[i]
end
```

The first construct is used when you need the value, but not index, of each element. In the second
construct, `i` will be an `Int` if `A` is an array type with fast linear indexing; otherwise,
it will be a `CartesianIndex`:

```jldoctest
julia> A = rand(4,3);

julia> B = view(A, 1:3, 2:3);

julia> for i in eachindex(B)
           @show i
       end
i = CartesianIndex(1, 1)
i = CartesianIndex(2, 1)
i = CartesianIndex(3, 1)
i = CartesianIndex(1, 2)
i = CartesianIndex(2, 2)
i = CartesianIndex(3, 2)
```

In contrast with `for i = 1:length(A)`, iterating with [`eachindex`](@ref) provides an efficient way to
iterate over any array type.

## Array traits

If you write a custom [`AbstractArray`](@ref) type, you can specify that it has fast linear indexing using

```julia
Base.IndexStyle(::Type{<:MyArray}) = IndexLinear()
```

This setting will cause `eachindex` iteration over a `MyArray` to use integers. If you don't
specify this trait, the default value `IndexCartesian()` is used.

## [Array and Vectorized Operators and Functions](@id man-array-and-vectorized-operators-and-functions)

The following operators are supported for arrays:

1. Unary arithmetic -- `-`, `+`
2. Binary arithmetic -- `-`, `+`, `*`, `/`, `\`, `^`
3. Comparison -- `==`, `!=`, `≈` ([`isapprox`](@ref)), `≉`

To enable convenient vectorization of mathematical and other operations,
Julia [provides the dot syntax](@ref man-vectorized) `f.(args...)`, e.g. `sin.(x)`
or `min.(x,y)`, for elementwise operations over arrays or mixtures of arrays and
scalars (a [Broadcasting](@ref) operation); these have the additional advantage of
"fusing" into a single loop when combined with other dot calls, e.g. `sin.(cos.(x))`.

Also, *every* binary operator supports a [dot version](@ref man-dot-operators)
that can be applied to arrays (and combinations of arrays and scalars) in such
[fused broadcasting operations](@ref man-vectorized), e.g. `z .== sin.(x .* y)`.

Note that comparisons such as `==` operate on whole arrays, giving a single boolean
answer. Use dot operators like `.==` for elementwise comparisons. (For comparison
operations like `<`, *only* the elementwise `.<` version is applicable to arrays.)

Also notice the difference between `max.(a,b)`, which [`broadcast`](@ref)s [`max`](@ref)
elementwise over `a` and `b`, and [`maximum(a)`](@ref), which finds the largest value within
`a`. The same relationship holds for `min.(a,b)` and `minimum(a)`.

## Broadcasting

It is sometimes useful to perform element-by-element binary operations on arrays of different
sizes, such as adding a vector to each column of a matrix. An inefficient way to do this would
be to replicate the vector to the size of the matrix:

```julia-repl
julia> a = rand(2,1); A = rand(2,3);

julia> repeat(a,1,3)+A
2×3 Array{Float64,2}:
 1.20813  1.82068  1.25387
 1.56851  1.86401  1.67846
```

This is wasteful when dimensions get large, so Julia provides [`broadcast`](@ref), which expands
singleton dimensions in array arguments to match the corresponding dimension in the other array
without using extra memory, and applies the given function elementwise:

```julia-repl
julia> broadcast(+, a, A)
2×3 Array{Float64,2}:
 1.20813  1.82068  1.25387
 1.56851  1.86401  1.67846

julia> b = rand(1,2)
1×2 Array{Float64,2}:
 0.867535  0.00457906

julia> broadcast(+, a, b)
2×2 Array{Float64,2}:
 1.71056  0.847604
 1.73659  0.873631
```

[Dotted operators](@ref man-dot-operators) such as `.+` and `.*` are equivalent
to `broadcast` calls (except that they fuse, as [described above](@ref man-array-and-vectorized-operators-and-functions)). There is also a
[`broadcast!`](@ref) function to specify an explicit destination (which can also
be accessed in a fusing fashion by `.=` assignment). In fact, `f.(args...)`
is equivalent to `broadcast(f, args...)`, providing a convenient syntax to broadcast any function
([dot syntax](@ref man-vectorized)). Nested "dot calls" `f.(...)` (including calls to `.+` etcetera)
[automatically fuse](@ref man-dot-operators) into a single `broadcast` call.

Additionally, [`broadcast`](@ref) is not limited to arrays (see the function documentation);
it also handles scalars, tuples and other collections.  By default, only some argument types are
considered scalars, including (but not limited to) `Number`s, `String`s, `Symbol`s, `Type`s, `Function`s
and some common singletons like `missing` and `nothing`. All other arguments are
iterated over or indexed into elementwise.

```jldoctest
julia> convert.(Float32, [1, 2])
2-element Array{Float32,1}:
 1.0
 2.0

julia> ceil.(UInt8, [1.2 3.4; 5.6 6.7])
2×2 Array{UInt8,2}:
 0x02  0x04
 0x06  0x07

julia> string.(1:3, ". ", ["First", "Second", "Third"])
3-element Array{String,1}:
 "1. First"
 "2. Second"
 "3. Third"
```

Sometimes, you want a container (like an array) that would normally participate in broadcast to be "protected"
from broadcast's behavior of iterating over all of its elements. By placing it inside another container
(like a single element [`Tuple`](@ref)) broadcast will treat it as a single value.
```jldoctest
julia> ([1, 2, 3], [4, 5, 6]) .+ ([1, 2, 3],)
([2, 4, 6], [5, 7, 9])

julia> ([1, 2, 3], [4, 5, 6]) .+ tuple([1, 2, 3])
([2, 4, 6], [5, 7, 9])
```

## Implementation

The base array type in Julia is the abstract type [`AbstractArray{T,N}`](@ref). It is parameterized by
the number of dimensions `N` and the element type `T`. [`AbstractVector`](@ref) and [`AbstractMatrix`](@ref) are
aliases for the 1-d and 2-d cases. Operations on `AbstractArray` objects are defined using higher
level operators and functions, in a way that is independent of the underlying storage. These operations
generally work correctly as a fallback for any specific array implementation.

The `AbstractArray` type includes anything vaguely array-like, and implementations of it might
be quite different from conventional arrays. For example, elements might be computed on request
rather than stored. However, any concrete `AbstractArray{T,N}` type should generally implement
at least [`size(A)`](@ref) (returning an `Int` tuple), [`getindex(A,i)`](@ref) and [`getindex(A,i1,...,iN)`](@ref getindex);
mutable arrays should also implement [`setindex!`](@ref). It is recommended that these operations
have nearly constant time complexity, or technically Õ(1) complexity, as otherwise some array
functions may be unexpectedly slow. Concrete types should also typically provide a [`similar(A,T=eltype(A),dims=size(A))`](@ref)
method, which is used to allocate a similar array for [`copy`](@ref) and other out-of-place
operations. No matter how an `AbstractArray{T,N}` is represented internally, `T` is the type of
object returned by *integer* indexing (`A[1, ..., 1]`, when `A` is not empty) and `N` should be
the length of the tuple returned by [`size`](@ref). For more details on defining custom
`AbstractArray` implementations, see the [array interface guide in the interfaces chapter](@ref man-interface-array).

`DenseArray` is an abstract subtype of `AbstractArray` intended to include all arrays where
elements are stored contiguously in column-major order (see [additional notes in
Performance Tips](@ref man-performance-column-major)). The [`Array`](@ref) type is a specific instance
of `DenseArray`;  [`Vector`](@ref) and [`Matrix`](@ref) are aliases for the 1-d and 2-d cases.
Very few operations are implemented specifically for `Array` beyond those that are required
for all `AbstractArray`s; much of the array library is implemented in a generic
manner that allows all custom arrays to behave similarly.

`SubArray` is a specialization of `AbstractArray` that performs indexing by
sharing memory with the original array rather than by copying it. A `SubArray`
is created with the [`view`](@ref) function, which is called the same way as
[`getindex`](@ref) (with an array and a series of index arguments). The result
of [`view`](@ref) looks the same as the result of [`getindex`](@ref), except the
data is left in place. [`view`](@ref) stores the input index vectors in a
`SubArray` object, which can later be used to index the original array
indirectly.  By putting the [`@views`](@ref) macro in front of an expression or
block of code, any `array[...]` slice in that expression will be converted to
create a `SubArray` view instead.

[`BitArray`](@ref)s are space-efficient "packed" boolean arrays, which store one bit per boolean value.
They can be used similarly to `Array{Bool}` arrays (which store one byte per boolean value),
and can be converted to/from the latter via `Array(bitarray)` and `BitArray(array)`, respectively.

An array is "strided" if it is stored in memory with well-defined spacings (strides) between
its elements. A strided array with a supported element type may be passed to an external
(non-Julia) library like BLAS or LAPACK by simply passing its [`pointer`](@ref) and the
stride for each dimension. The [`stride(A, d)`](@ref) is the distance between elements along
dimension `d`. For example, the builtin `Array` returned by `rand(5,7,2)` has its elements
arranged contiguously in column major order. This means that the stride of the first
dimension — the spacing between elements in the same column — is `1`:

```julia-repl
julia> A = rand(5,7,2);

julia> stride(A,1)
1
```

The stride of the second dimension is the spacing between elements in the same row, skipping
as many elements as there are in a single column (`5`). Similarly, jumping between the two
"pages" (in the third dimension) requires skipping `5*7 == 35` elements.  The [`strides`](@ref)
of this array is the tuple of these three numbers together:

```julia-repl
julia> strides(A)
(1, 5, 35)
```

In this particular case, the number of elements skipped _in memory_ matches the number of
_linear indices_ skipped. This is only the case for contiguous arrays like `Array` (and
other `DenseArray` subtypes) and is not true in general. Views with range indices are a good
example of _non-contiguous_ strided arrays; consider `V = @view A[1:3:4, 2:2:6, 2:-1:1]`.
This view `V` refers to the same memory as `A` but is skipping and re-arranging some of its
elements. The stride of the first dimension of `V` is `3` because we're only selecting every
third row from our original array:

```julia-repl
julia> V = @view A[1:3:4, 2:2:6, 2:-1:1];

julia> stride(V, 1)
3
```

This view is similarly selecting every other column from our original `A` — and thus it
needs to skip the equivalent of two five-element columns when moving between indices in the
second dimension:

```julia-repl
julia> stride(V, 2)
10
```

The third dimension is interesting because its order is reversed! Thus to get from the first
"page" to the second one it must go _backwards_ in memory, and so its stride in this
dimension is negative!

```julia-repl
julia> stride(V, 3)
-35
```

This means that the `pointer` for `V` is actually pointing into the middle of `A`'s memory
block, and it refers to elements both backwards and forwards in memory. See the
[interface guide for strided arrays](@ref man-interface-strided-arrays) for more details on
defining your own strided arrays. [`StridedVector`](@ref) and [`StridedMatrix`](@ref) are
convenient aliases for many of the builtin array types that are considered strided arrays,
allowing them to dispatch to select specialized implementations that call highly tuned and
optimized BLAS and LAPACK functions using just the pointer and strides.

It is worth emphasizing that strides are about offsets in memory rather than indexing. If
you are looking to convert between linear (single-index) indexing and cartesian
(multi-index) indexing, see [`LinearIndices`](@ref) and [`CartesianIndices`](@ref).
