# Base.Cartesian

The (non-exported) Cartesian module provides macros that facilitate writing multidimensional algorithms. It is hoped that Cartesian will not, in the long term, be necessary; however, at present it is one of the few ways to write compact and performant multidimensional code.

## Principles of usage

A simple example of usage is:

``````@nloops 3 i A begin
s += @nref 3 A i
end
``````

which generates the following code:

``````for i_3 = 1:size(A,3)
for i_2 = 1:size(A,2)
for i_1 = 1:size(A,1)
s += A[i_1,i_2,i_3]
end
end
end
``````

In general, Cartesian allows you to write generic code that contains repetitive elements, like the nested loops in this example. Other applications include repeated expressions (e.g., loop unwinding) or creating function calls with variable numbers of arguments without using the "splat" construct (`i...`).

## Basic syntax

The (basic) syntax of `@nloops` is as follows:

• The first argument must be an integer (not a variable) specifying the number of loops.
• The second argument is the symbol-prefix used for the iterator variable. Here we used `i`, and variables `i_1, i_2, i_3` were generated.
• The third argument specifies the range for each iterator variable. If you use a variable (symbol) here, it's taken as `1:size(A,dim)`. More flexibly, you can use the anonymous-function expression syntax described below.
• The last argument is the body of the loop. Here, that's what appears between the `begin...end`.

There are some additional features of `@nloops` described in the reference section.

`@nref` follows a similar pattern, generating `A[i_1,i_2,i_3]` from `@nref 3 A i`. The general practice is to read from left to right, which is why `@nloops` is `@nloops 3 i A expr` (as in `for i_2 = 1:size(A,2)`, where `i_2` is to the left and the range is to the right) whereas `@nref` is `@nref 3 A i` (as in `A[i_1,i_2,i_3]`, where the array comes first).

If you're developing code with Cartesian, you may find that debugging is easier when you examine the generated code, using `macroexpand`:

```@meta DocTestSetup = quote import Base.Cartesian: @nref end

``````
```jldoctest
julia> macroexpand(:(@nref 2 A i))
:(A[i_1, i_2])
``````

```@meta DocTestSetup = nothing

``````
### Supplying the number of expressions

The first argument to both of these macros is the number of expressions, which must be an integer.
When you're writing a function that you intend to work in multiple dimensions, this may not be
something you want to hard-code. If you're writing code that you need to work with older Julia
versions, currently you should use the `@ngenerate` macro described in [an older version of this documentation](https://docs.julialang.org/en/release-0.3/devdocs/cartesian/#supplying-the-number-of-expressions).

Starting in Julia 0.4-pre, the recommended approach is to use a `@generated function`.  Here's
an example:

```julia
@generated function mysum(A::Array{T,N}) where {T,N}
quote
s = zero(T)
@nloops \$N i A begin
s += @nref \$N A i
end
s
end
end
``````

Naturally, you can also prepare expressions or perform calculations before the `quote` block.

### Anonymous-function expressions as macro arguments

Perhaps the single most powerful feature in `Cartesian` is the ability to supply anonymous-function expressions that get evaluated at parsing time. Let's consider a simple example:

``````@nexprs 2 j->(i_j = 1)
``````

`@nexprs` generates `n` expressions that follow a pattern. This code would generate the following statements:

``````i_1 = 1
i_2 = 1
``````

In each generated statement, an "isolated" `j` (the variable of the anonymous function) gets replaced by values in the range `1:2`. Generally speaking, Cartesian employs a LaTeX-like syntax. This allows you to do math on the index `j`. Here's an example computing the strides of an array:

``````s_1 = 1
@nexprs 3 j->(s_{j+1} = s_j * size(A, j))
``````

would generate expressions

``````s_1 = 1
s_2 = s_1 * size(A, 1)
s_3 = s_2 * size(A, 2)
s_4 = s_3 * size(A, 3)
``````

Anonymous-function expressions have many uses in practice.

#### Macro reference

```@docs Base.Cartesian.@nloops Base.Cartesian.@nref Base.Cartesian.@nextract Base.Cartesian.@nexprs Base.Cartesian.@ncall Base.Cartesian.@ntuple Base.Cartesian.@nall Base.Cartesian.@nany Base.Cartesian.@nif```