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

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]

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:

julia> macroexpand(:(@nref 2 A i))

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. Perhaps the most straightforward approach is to use the @ngenerate macro.

Perhaps the easiest way to understand @ngenerate is to see it in action. Here’s a slightly cleaned up example:

julia> macroexpand(:(@ngenerate N typeof(A) function mysum{T,N}(A::Array{T,N})
        s = zero(T)
        @nloops N i A begin
            s += @nref N A i
    function mysum{T}(A::Array{T,1}) # none, line 2:
        s = zero(T) # line 3:
        for i_1 = 1:size(A,1) # line 293:
            s += A[i_1]
        end # line 295:
    function mysum{T}(A::Array{T,2}) # none, line 2:
        s = zero(T) # line 3:
        for i_2 = 1:size(A,2) # line 293:
            for i_1 = 1:size(A,1) # line 293:
                s += A[i_1,i_2]
            end # line 295:
        end # line 295:
    function mysum{T}(A::Array{T,3}) # none, line 2:
        s = zero(T) # line 3:
        for i_3 = 1:size(A,3) # line 293:
            for i_2 = 1:size(A,2) # line 293:
                for i_1 = 1:size(A,1) # line 293:
                    s += A[i_1,i_2,i_3]
                end # line 295:
            end # line 295:
        end # line 295:
    function mysum{T}(A::Array{T,4}) # none, line 2:
    let mysum_cache = Dict{Int,Function}() # line 113:
        function mysum{T,N}(A::Array{T,N}) # cartesian.jl, line 100:
            if !(haskey(mysum_cache,N)) # line 102:
                localfunc = quote
                    function _F_{T}(A::Array{T,$N})
                        s = zero(T)
                        @nloops $N i A begin
                            s += @nref $N A i
                mysum_cache[N] = eval(quote
                    local _F_

You can see that @ngenerate causes explicit versions to be generated for dimensions 1 to 4 (a setting controlled by the constant CARTESIAN_DIMS). To allow arbitrary-dimensional arrays to be handled, it also generates a version in which different methods are cached in a dictionary. If a given method has not yet been generated, it creates a version specific to that dimensionality and then stores it in the dictionary. Creating the method is slow—it involves generating expressions and then evaluating them—but once created the function can be looked up from the cache, and is reasonably efficient (but still less efficient than the versions generated for explicit dimensionality).

The arguments to @ngenerate are:

  • The symbol of the variable that will be used for generating different versions (in the example, N)
  • The return type of the function (in the example, typeof(A)). This is not used for the versions that are generated for specific N, but is needed for the dictionary-backed version. Julia cannot infer the return type of the function looked up from the dictionary.
  • The actual function declaration. Use N as you would a normal parameter.

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

Macros for creating functions

@ngenerate Nsym returntypeexpr functiondeclexpr

Generate versions of a function for different values of Nsym.

@nsplat Nsym functiondeclexpr
@nsplat Nsym dimrange functiondeclexpr

Generate explicit versions of a function for different numbers of arguments. For example:

@nsplat N 2:3 absgetindex(A, I::NTuple{N,Real}...) = abs(getindex(A, I...))


absgetindex(A, I_1::Real, I_2::Real) = abs(getindex(A, I_1, I_2))
absgetindex(A, I_1::Real, I_2::Real, I_3::Real) = abs(getindex(A, I_1, I_2, I_3))

Macros for function bodies

@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 1:size(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 = 1:size(A, 2)
    j_2 = min(i_2, 5)
    for i_1 = 1:size(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 parenthesis 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.

@nexprs N expr

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

@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

@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.

@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")

Frequently asked questions

I got an error ERROR: N not defined when using @ngenerate. Why?

Most likely you forgot to define your function with N as a type parameter, e.g., @ngenerate N returntype myfunc{N}(...).