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 Y-shaped array which contains the indexed results. 

      'ABCDE'[2]
B
      'ABCDE'[2 3⍴1 2 3 4 5 1]
ABC
DEA

For higher-rank 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 k-th 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 depth-2 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 depth-3 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


APL built-ins [edit]
Primitives (Timeline) Functions
Scalar
Monadic Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root ∙ Round
Dyadic Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
Structural Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
Selection First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
Selector Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
Computational Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
Operators Monadic Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis ∙ Identity (Null, Ident)
Dyadic Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator) ∙ Identity (Lev, Dex)
Quad names Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
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