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 variablesi_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