Bracket indexing

From APL Wiki
Revision as of 21:58, 10 September 2022 by Adám Brudzewsky (talk | contribs) (Text replacement - "<source" to "<syntaxhighlight")
Jump to navigation Jump to search
[]

Bracket indexing ([]), or simply Indexing, is a special primitive function which uses the postcircumfix notation <syntaxhighlight lang=apl inline>X[Y]</source> instead of a normal prefix function. The result of <syntaxhighlight lang=apl inline>X[Y]</source> 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, <syntaxhighlight lang=apl inline>X[Y]</source> 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 <syntaxhighlight lang=apl inline>X[Y]</source> is a Y-shaped array which contains the indexed results.

<syntaxhighlight lang=apl>

     'ABCDE'[2]

B

     'ABCDE'[2 3⍴1 2 3 4 5 1]

ABC DEA </source>

For higher-rank array X with rank n, the notation <syntaxhighlight lang=apl inline>X[Y1;Y2;...;Yn]</source> selects the indexes of X over each axis. If some <syntaxhighlight lang=apl inline>Yk</source> is omitted, it implies all indices of k-th axis is selected, which is equivalent to specifying <syntaxhighlight lang=apl inline>⍳(⍴X)[k]</source>. The resulting shape is the concatenation of shapes of Y1, Y2, ..., Yn.

<syntaxhighlight lang=apl>

     ⎕←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 </source>

The major limitation of this indexing mode is that it only supports rectangular selection. For example, it is not possible to form <syntaxhighlight lang=apl inline>X[1;1],X[2;2]</source> 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.

<syntaxhighlight lang=apl>

     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 </source>

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.

<syntaxhighlight lang=apl>

     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││ │└───┴─┘│ └───────┘ </source>

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 ConjugateNegateSignumReciprocalMagnitudeExponentialNatural LogarithmFloorCeilingFactorialNotPi TimesRollTypeImaginarySquare Root
Dyadic AddSubtractTimesDivideResiduePowerLogarithmMinimumMaximumBinomialComparison functionsBoolean functions (And, Or, Nand, Nor) ∙ GCDLCMCircularComplexRoot
Non-Scalar
Structural ShapeReshapeTallyDepthRavelEnlistTableCatenateReverseRotateTransposeRazeMixSplitEncloseNestCut (K)PairLinkPartitioned EnclosePartition
Selection FirstPickTakeDropUniqueIdentityStopSelectReplicateExpandSet functions (IntersectionUnionWithout) ∙ Bracket indexingIndexCartesian ProductSort
Selector Index generatorGradeIndex OfInterval IndexIndicesDealPrefix and suffix vectors
Computational MatchNot MatchMembershipFindNub SieveEncodeDecodeMatrix InverseMatrix DivideFormatExecuteMaterialiseRange
Operators Monadic EachCommuteConstantReplicateExpandReduceWindowed ReduceScanOuter ProductKeyI-BeamSpawnFunction axis
Dyadic BindCompositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner ProductDeterminantPowerAtUnderRankDepthVariantStencilCutDirect definition (operator)
Quad names Index originComparison toleranceMigration levelAtomic vector