Bracket indexing
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Bracket indexing ([]
), or simply Indexing, is a special primitive function which uses the postcircumfix notation X[Y]
instead of a normal prefix function. The result of X[Y]
is an array formed with items of X extracted by the index specification Y.
Indexing modes
Simple indexing
Most APL implementations support only this mode of indexing. In its simplest form, X[Y]
on vector X and scalar Y extracts the item of X at index Y. In general, Y can be an array of any shape, with each item being a valid index in X; then X[Y]
is a Yshaped array which contains the indexed results.
'ABCDE'[2] B 'ABCDE'[2 3⍴1 2 3 4 5 1] ABC DEA
For higherrank array X with rank n, the notation X[Y1;Y2;...;Yn]
selects the indexes of X over each axis. If some Yk
is omitted, it implies all indices of kth axis is selected, which is equivalent to specifying ⍳(⍴X)[k]
. The resulting shape is the concatenation of shapes of Y1, Y2, ..., Yn.
⎕←A←2 3 4⍴10×⍳24 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 A[1;1;1] 10 A[2;3 2;4 1] 240 210 200 170 A[;2;] 50 60 70 80 170 180 190 200
The major limitation of this indexing mode is that it only supports rectangular selection. For example, it is not possible to form X[1;1],X[2;2]
from a matrix X by single indexing.
Choose indexing
In this mode, the index specification Y is a depth2 nested array. Each item of Y is a vector whose length is the rank of X, and the result is a collection of items of X selected by each item of Y.
M 10 20 30 40 50 60 70 80 M[⊂1 2] 20 M[2 2⍴⊂2 4] 80 80 80 80 M[(2 1)(1 2)] 50 20 'Z'[3⍴⊂⍬] ⍝ Scalar X can be indexed using enclosed empty vector ZZZ
Reach indexing
In this mode, Y is a depth3 nested array. Each item of Y is a vector of nested vectors which specify the index at each level of nesting (which is equivalent to the indexing by Pick). This allows to extract multiple items from a deeply nested array with a single indexing operation.
G←('ABC' 1)('DEF' 2)('GHI' 3)('JKL' 4) G←2 3⍴G,('MNO' 5)('PQR' 6) G ┌───────┬───────┬───────┐ │┌───┬─┐│┌───┬─┐│┌───┬─┐│ ││ABC│1│││DEF│2│││GHI│3││ │└───┴─┘│└───┴─┘│└───┴─┘│ ├───────┼───────┼───────┤ │┌───┬─┐│┌───┬─┐│┌───┬─┐│ ││JKL│4│││MNO│5│││PQR│6││ │└───┴─┘│└───┴─┘│└───┴─┘│ └───────┴───────┴───────┘ G[((1 2)1)((2 3)2)] ┌───┬─┐ │DEF│6│ └───┴─┘ G[2 2⍴⊂(2 2)2] 5 5 5 5 G[⊂⊂1 1] ┌───────┐ │┌───┬─┐│ ││ABC│1││ │└───┴─┘│ └───────┘
Implementation support
Dyalog APL and NARS2000 support all three modes of indexing. NARS2000 even supports mixing Choose and Reach indexing modes. J does not have this notation at all.
See also
External links
Documentation