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Rank | :''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]].'' | ||
{{Built-in|Rank|⍤}} is a [[primitive operator|primitive]] [[dyadic operator]] which applies its left [[operand]] function to [[cells]] of its arguments specified by its right operand array. | |||
== Rank specification == | == Rank specification == | ||
The right operand specifies the [[rank]] of subarrays to which the left operand function is applied as follows: | The right operand specifies the [[rank]] of subarrays to which the left operand function is applied as follows: | ||
For left argument < | For left argument <syntaxhighlight lang=apl inline>⍺</syntaxhighlight> and right argument <syntaxhighlight lang=apl inline>⍵</syntaxhighlight>, | ||
< | <syntaxhighlight lang=apl> | ||
⍤ c ⍝ Rank-c cells of ⍵ (monadic) or both arguments (dyadic) | ⍤ c ⍝ Rank-c cells of ⍵ (monadic) or both arguments (dyadic) | ||
⍤ b c ⍝ Rank-b cells of ⍺ and rank-c cells of ⍵ (dyadic) | ⍤ b c ⍝ Rank-b cells of ⍺ and rank-c cells of ⍵ (dyadic) | ||
⍤a b c ⍝ Rank-a cells of ⍵ (monadic), b-cells of ⍺ and c-cells of ⍵ (dyadic) | ⍤a b c ⍝ Rank-a cells of ⍵ (monadic), b-cells of ⍺ and c-cells of ⍵ (dyadic) | ||
</ | </syntaxhighlight> | ||
A non-negative 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. | A non-negative 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, < | Since a rank specification greater than the rank of the argument array means to apply the function to the whole array, <syntaxhighlight lang=apl inline>99</syntaxhighlight>, <syntaxhighlight lang=apl inline>(⌊/⍬)</syntaxhighlight> or <syntaxhighlight lang=apl inline>∞</syntaxhighlight>, depending on the implementation, is "rank infinity" and always specifies the whole argument array. | ||
== Examples == | == Examples == | ||
Reverse order of rows in matrices of a 3D array: | |||
< | <syntaxhighlight lang=apl> | ||
⊖⍤2⊢3 2 4⍴⎕A | ⊖⍤2⊢3 2 4⍴⎕A | ||
EFGH | EFGH | ||
| Line 26: | Line 28: | ||
UVWX | UVWX | ||
QRST | QRST | ||
</ | </syntaxhighlight> | ||
Laminate scalars from arrays of differing ranks: | Laminate scalars from arrays of differing ranks: | ||
< | <syntaxhighlight lang=apl> | ||
'ABCD',⍤0⍤1⊢2 4⍴⍳8 | 'ABCD',⍤0⍤1⊢2 4⍴⍳8 | ||
A 1 | A 1 | ||
| Line 40: | Line 42: | ||
C 7 | C 7 | ||
D 8 | D 8 | ||
</ | </syntaxhighlight> | ||
Flat outer product: | Flat outer product: | ||
< | <syntaxhighlight lang=apl> | ||
-⍤1⍤1 99⍨3 2⍴6 7 1 1 2 4 ⍝ ↑∘.-⍨↓ | -⍤1⍤1 99⍨3 2⍴6 7 1 1 2 4 ⍝ ↑∘.-⍨↓ | ||
0 0 | 0 0 | ||
| Line 56: | Line 58: | ||
1 3 | 1 3 | ||
0 0 | 0 0 | ||
</ | </syntaxhighlight> | ||
== History == | == 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"<ref name="hopl4">[https://dl.acm.org/doi/ | 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"<ref name="hopl4">[[Roger Hui]] and [[Morten Kromberg]]. [https://dl.acm.org/doi/abs/10.1145/3386319 ''APL since 1978'']. ACM [[HOPL]] IV. 2020-06.</ref>, 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 product|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 [https://aplwiki.com/wiki/Function_axis bracket-axis]. It provides nearly all of the functionality of the anomalous axis operator (<syntaxhighlight lang=apl inline>f[a]</syntaxhighlight>) without its draw-backs.<ref name="intro2rank">[[Robert Bernecky]]. [https://doi.org/10.1145/55626.55632 An introduction to function rank] at [[APL88]]. ACM SIGAPL APL Quote Quad, 18(2), pp.39-43. 1987.</ref> | |||
One of these draw-backs is that bracket-axis 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 [[user-defined functions]]. The ad hoc nature of bracket-axis definitions means that a generalised axis operator which works on any function, but behaves just as bracket-axis on those particular primitives, is impossible to formulate. | |||
Here we show some bracket-axis constructs and their equivalent expressions which use rank but do not use bracket-axis. | |||
===Sum along axis=== | |||
Rank k-cells 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. | |||
<syntaxhighlight lang=apl> | |||
+/[1+⍺-⍨≢⍴⍵] ≡ +⌿⍤⍺ ⍝ For scalar ⍺ | |||
+/[⍺] ≡ +⌿⍤(1+⍺-⍨≢⍴⍵) ⍝ For scalar ⍺ | |||
</syntaxhighlight> | |||
===Enclose axes=== | |||
Enclose-with-axis 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. | |||
<syntaxhighlight lang=apl> | |||
EncloseAxes←{ | |||
axes←⍳≢⍴⍵ | |||
move←⍋(axes~⍺),⍺ | |||
⊂⍤(≢⍺)⊢move⍉⍺ | |||
} | |||
⊂[⍺] ≡ ⍺∘EncloseAxes | |||
</syntaxhighlight> | |||
===Merge axes=== | |||
Ravel-with-axis 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. | |||
<syntaxhighlight lang=apl> | |||
,[(-⍺)↑⍴⍵] ≡ ,⍤⍺ | |||
</syntaxhighlight> | |||
Merging leading axes is more involved, but can be expressed in one line. | |||
<syntaxhighlight lang=apl> | |||
,[⍳⍺] ←→ {(1⌽⍳≢⍴z)⍉z←,⍤⍺((-⍺)⌽⍳≢⍴⍵)⍉⍵} | |||
</syntaxhighlight> | |||
The general treatment benefits from being expanded. | |||
<syntaxhighlight lang=apl> | |||
MergeAxes←{ | |||
axes←⍳≢⍴⍵ | |||
move←⍋(axes~⍺),⍺ | |||
merged←,⍤(≢⍺)⊢move⍉⍵ | |||
restore←((⍳≢⍴merged)~⊃⍺),⊃⍺ | |||
restored⍉merged | |||
} | |||
,[⍺] ←→ ⍺∘MergeAxes ⍝ for ∧/1=¯2-/⍺ | |||
</syntaxhighlight> | |||
== External links == | == External links == | ||
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* [https://help.dyalog.com/latest/Content/Language/Primitive%20Operators/Rank.htm Dyalog] | * [https://help.dyalog.com/latest/Content/Language/Primitive%20Operators/Rank.htm Dyalog] | ||
* [http://wiki.nars2000.org/index.php?title=Rank NARS2000] | * [http://wiki.nars2000.org/index.php?title=Rank NARS2000] | ||
* [https://mlochbaum.github.io/BQN/doc/rank.html#rank BQN] | |||
=== Publications === | === Publications === | ||
* [https:// | * [[Dyalog '09]] presentation by [[Roger Hui]]: [https://dyalog.tv/Dyalog09/?v=ui76NE5cMWo The Rank Operator] | ||
* [[Dyalog Webinar]]s by [[Richard Park]]: [https://dyalog.tv/Webinar/?v=IBct81IopRk Introduction], [https://dyalog.tv/Webinar/?v=zBqdeDJPPRc with Dyadic Transpose], [https://dyalog.tv/Webinar/?v=5wW76XX0kqk advanced use] | |||
* [[APL Cultivation]]s: [https://chat.stackexchange.com/rooms/52405/conversation/lesson-32-basic-use-of- basic use], [https://chat.stackexchange.com/rooms/52405/conversation/lesson-33-advanced-use-of- advanced use] | |||
==References== | |||
<references/> | |||
{{APL built-ins}}[[Category:Primitive operators]] | {{APL built-ins}}[[Category:Primitive operators]] | ||