The (non-exported) Cartesian module provides macros that facilitate writing multidimensional algorithms. Most often you can write such algorithms with straightforward techniques; however, there are a few cases where Base.Cartesian is still useful or necessary.

Principles of usage

A simple example of usage is:

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

which generates the following code:

for i_3 = axes(A, 3)
    for i_2 = axes(A, 2)
        for i_1 = axes(A, 1)
            s += A[i_1, i_2, i_3]

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 axes(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 = axes(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:

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

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. The recommended approach is to use a @generated function. Here's an example:

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

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

@nloops N itersym rangeexpr bodyexpr
@nloops N itersym rangeexpr preexpr bodyexpr
@nloops N itersym rangeexpr preexpr postexpr bodyexpr

Generate N nested loops, using itersym as the prefix for the iteration variables. rangeexpr may be an anonymous-function expression, or a simple symbol var in which case the range is axes(var, d) for dimension d.

Optionally, you can provide "pre" and "post" expressions. These get executed first and last, respectively, in the body of each loop. For example:

@nloops 2 i A d -> j_d = min(i_d, 5) begin
    s += @nref 2 A j

would generate:

for i_2 = axes(A, 2)
    j_2 = min(i_2, 5)
    for i_1 = axes(A, 1)
        j_1 = min(i_1, 5)
        s += A[j_1, j_2]

If you want just a post-expression, supply nothing for the pre-expression. Using parentheses and semicolons, you can supply multi-statement expressions.

@nref N A indexexpr

Generate expressions like A[i_1, i_2, ...]. indexexpr can either be an iteration-symbol prefix, or an anonymous-function expression.


julia> @macroexpand Base.Cartesian.@nref 3 A i
:(A[i_1, i_2, i_3])
@nextract N esym isym

Generate N variables esym_1, esym_2, ..., esym_N to extract values from isym. isym can be either a Symbol or anonymous-function expression.

@nextract 2 x y would generate

x_1 = y[1]
x_2 = y[2]

while @nextract 3 x d->y[2d-1] yields

x_1 = y[1]
x_2 = y[3]
x_3 = y[5]
@nexprs N expr

Generate N expressions. expr should be an anonymous-function expression.


julia> @macroexpand Base.Cartesian.@nexprs 4 i -> y[i] = A[i+j]
    y[1] = A[1 + j]
    y[2] = A[2 + j]
    y[3] = A[3 + j]
    y[4] = A[4 + j]
@ncall N f sym...

Generate a function call expression. sym represents any number of function arguments, the last of which may be an anonymous-function expression and is expanded into N arguments.

For example, @ncall 3 func a generates

func(a_1, a_2, a_3)

while @ncall 2 func a b i->c[i] yields

func(a, b, c[1], c[2])
@ntuple N expr

Generates an N-tuple. @ntuple 2 i would generate (i_1, i_2), and @ntuple 2 k->k+1 would generate (2,3).

@nall N expr

Check whether all of the expressions generated by the anonymous-function expression expr evaluate to true.

@nall 3 d->(i_d > 1) would generate the expression (i_1 > 1 && i_2 > 1 && i_3 > 1). This can be convenient for bounds-checking.

@nany N expr

Check whether any of the expressions generated by the anonymous-function expression expr evaluate to true.

@nany 3 d->(i_d > 1) would generate the expression (i_1 > 1 || i_2 > 1 || i_3 > 1).

@nif N conditionexpr expr
@nif N conditionexpr expr elseexpr

Generates a sequence of if ... elseif ... else ... end statements. For example:

@nif 3 d->(i_d >= size(A,d)) d->(error("Dimension ", d, " too big")) d->println("All OK")

would generate:

if i_1 > size(A, 1)
    error("Dimension ", 1, " too big")
elseif i_2 > size(A, 2)
    error("Dimension ", 2, " too big")
    println("All OK")