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
:
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. 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.
Starting in Julia 0.4-pre, the recommended approach is to use
a @generated function
. Here’s an example:
@generated function mysum{T,N}(A::Array{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¶
-
@nloops N itersym rangeexpr bodyexpr
-
@nloops N itersym rangeexpr preexpr bodyexpr
-
@nloops N itersym rangeexpr preexpr postexpr bodyexpr
Generate
N
nested loops, usingitersym
as the prefix for the iteration variables.rangeexpr
may be an anonymous-function expression, or a simple symbolvar
in which case the range is1:size(var,d)
for dimensiond
.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 end
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] end end
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") else println("All OK") end