Rank (operator)
 This article is about the operator. See Rank for the number associated with every array. For numbers associated with a function specifying its argument rank, see function rank.
⍤

Rank (⍤
) is a primitive dyadic operator which applies its left operand function to cells of its arguments specified by its right operand array.
Rank specification
The right operand specifies the rank of subarrays to which the left operand function is applied as follows:
For left argument ⍺
and right argument ⍵
,
⍤ c ⍝ Rankc cells of ⍵ (monadic) or both arguments (dyadic)
⍤ b c ⍝ Rankb cells of ⍺ and rankc cells of ⍵ (dyadic)
⍤a b c ⍝ Ranka cells of ⍵ (monadic), bcells of ⍺ and ccells of ⍵ (dyadic)
A nonnegative right operand specifies the number of final axes to which the function applies. A negative right operand specifies complementary rank, i.e. the number of leading axes to be excluded. Negative rank can also be thought of as rank specification relative to the overall rank of the argument array.
Since a rank specification greater than the rank of the argument array means to apply the function to the whole array, 99
, (⌊/⍬)
or ∞
, depending on the implementation, is "rank infinity" and always specifies the whole argument array.
Examples
Reverse order of rows in matrices of a 3D array:
⊖⍤2⊢3 2 4⍴⎕A
EFGH
ABCD
MNOP
IJKL
UVWX
QRST
Laminate scalars from arrays of differing ranks:
'ABCD',⍤0⍤1⊢2 4⍴⍳8
A 1
B 2
C 3
D 4
A 5
B 6
C 7
D 8
Flat outer product:
⍤1⍤1 99⍨3 2⍴6 7 1 1 2 4 ⍝ ↑∘.⍨↓
0 0
5 6
4 3
¯5 ¯6
0 0
¯1 ¯3
¯4 ¯3
1 3
0 0
History
The rank operator was invented by Arthur Whitney in 1982 and first implemented in SHARP APL in 1983. It has been described as "a microcosm of APL history"^{[1]}, its evolution a progression from scalar extension, which has been in APL since its inception, through leading axis theory to a construct which is a generalisation of scalar extension, inner (matrix) product, outer product, maplist in LISP, map in modern functional programming languages and the broadcast facility in NumPy.
Rank vs Axis
Due to its ability to apply functions to specified subarrays, rank is frequently contrasted with bracketaxis. It provides nearly all of the functionality of the anomalous axis operator (f[a]
) without its drawbacks.^{[2]}
One of these drawbacks is that bracketaxis is specified ad hoc for each of the specific primitives on which it applies. Rank benefits from consistent behaviour when applied to any function, including userdefined functions. The ad hoc nature of bracketaxis definitions means that a generalised axis operator which works on any function, but behaves just as bracketaxis on those particular primitives, is impossible to formulate.
Here we show some bracketaxis constructs and their equivalent expressions which use rank but do not use bracketaxis.
Sum along axis
Rank kcells are defined for k trailing axes, whereas axes are numbered from most major (first axis i.e. axis number 1) to least major (last axis). This leads to a simple and symmetrical relationship.
+/[1+⍺⍨≢⍴⍵] ≡ +⌿⍤⍺ ⍝ For scalar ⍺
+/[⍺] ≡ +⌿⍤(1+⍺⍨≢⍴⍵) ⍝ For scalar ⍺
Enclose axes
Enclosewithaxis is equivalent to transposing desired axes to the end of the array's shape and enclosing subarrays of a rank matching the number of axes.
EncloseAxes←{
axes←⍳≢⍴⍵
move←⍋(axes~⍺),⍺
⊂⍤(≢⍺)⊢move⍉⍺
}
⊂[⍺] ≡ ⍺∘EncloseAxes
Merge axes
Ravelwithaxis allows data to be merged along specified consecutive axes. The requirement that axes be consecutive is so that the data can remain in its original relative order.
Merging trailing axes is trivial.
,[(⍺)↑⍴⍵] ≡ ,⍤⍺
Merging leading axes is more involved, but can be expressed in one line.
,[⍳⍺] ←→ {(1⌽⍳≢⍴z)⍉z←,⍤⍺((⍺)⌽⍳≢⍴⍵)⍉⍵}
The general treatment benefits from being expanded.
MergeAxes←{
axes←⍳≢⍴⍵
move←⍋(axes~⍺),⍺
merged←,⍤(≢⍺)⊢move⍉⍵
restore←((⍳≢⍴merged)~⊃⍺),⊃⍺
restored⍉merged
}
,[⍺] ←→ ⍺∘MergeAxes ⍝ for ∧/1=¯2/⍺
Difference among dialects
Uniquely to NARS2000, negative rank k specifies taking the ∣k
cells from the array that results from permuting the k leading axes to the trailing end of the shape; the shape of the cells of argument a
specified by rank k is (k)↑⍴a
whether k is negative or positive. It also allows axis specification for the rank operator, denoting what axis the results will go to in the frame; x f⍤[ax] l r⊢y
is equivalent to ⊃[ax] x f⍤l r⊢y
(noting that ⊃
in NARS2000 is Mix).
External links
Documentation
Publications
 Dyalog '09 presentation by Roger Hui: The Rank Operator
 Dyalog Webinars by Richard Park: Introduction, with Dyadic Transpose, advanced use
 APL Cultivations: basic use, advanced use
References
 ↑ Roger Hui and Morten Kromberg. APL since 1978. ACM HOPL IV. 202006.
 ↑ Robert Bernecky. An introduction to function rank at APL88. ACM SIGAPL APL Quote Quad, 18(2), pp.3943. 1987.