3A. Indexing

{ encompasses everything performed by bracket indexing in other languages. The ranks are 0 _ , that is, x{y can be considered individually for each scalar in x and for y in toto, with the overall result constructed from the individual results in the same way as for all other functions. For scalar x then:

>x is a scalar or a list, and r=:>j{,>x are the indices for axis j . r may be integers in the range i.&.(+&n)n=:j{$y , which selects cells n|r , or r may be boxed integers in that range, in which case the selected cells are all except those in n|r . Thus:

   z=: 0{y=: 3 3 3$'ABCDEFGHIJKLMNOPQRSTUVWXYZ]'
   ibbb=: <ibb=: <ib=: <i=: 1 _1
   jbbb=: <jbb=: <jb=: <j=: 2 1
   ijbbb=: <ijbb=: <ijb=: <ij=: 2 2$i,j
   (] ; i&{ ; ib&{ ; ibb&{ ; ibbb&{) z
+---+---+-+---+---+
|ABC|DEF|F|DEF|ABC|
|DEF|GHI| |GHI|   |
|GHI|   | |   |   |
+---+---+-+---+---+

   ijb{y
|rank error
|   ijb    {y

   (] ; i&{ ; ijbb&{ ; ijbbb&{) z
+---+---+---+---+
|ABC|DEF|DEF|ABC|
|DEF|GHI|GHI|   |
|GHI|   |   |   |
|   |   |GHI|   |
|   |   |DEF|   |
+---+---+---+---+

The amend adverb } applied to an index produces a function that replaces the selected part of the right argument by the left argument. For example:

   '*' ib} z
ABC
DE*
GHI

   ('def',:'ghi') i} z
ABC
def
ghi

      (] ; i&{ ; ib&{ ; ibb&{ ; ibbb&{)"2 y
+---+---+-+---+---+
|ABC|DEF|F|DEF|ABC|
|DEF|GHI| |GHI|   |
|GHI|   | |   |   |
+---+---+-+---+---+
|JKL|MNO|O|MNO|JKL|
|MNO|PQR| |PQR|   |
|PQR|   | |   |   |
+---+---+-+---+---+
|STU|VWX|X|VWX|STU|
|VWX|YZ]| |YZ]|   |
|YZ]|   | |   |   |
+---+---+-+---+---+

   $(<<'') { z
0 3

   $(<a:) { z
0 3

Indexing on higher-rank arrays may be illustrated by the argument y :

   ]k=: <1 2;a:;0 2
+------------+
|+---+--+---+|
||1 2|++|0 2||
||   ||||   ||
||   |++|   ||
|+---+--+---+|
+------------+

 
   y ; k{y
+---+--+
|ABC|JL|
|DEF|MO|
|GHI|PR|
|   |  |
|JKL|SU|
|MNO|VX|
|PQR|Y]|
|   |  |
|STU|  |
|VWX|  |
|YZ]|  |
+---+--+

The following examples further illustrate the use of the indexing function. For each example, it may be instructive to plug the values into the expression r=:>j{,>x and work out the result.

`n0=: y=: i.4 5 6 7`	Array used in examples
`n1=: (<,<3){y`	Item 3 of y
`n2=: (<,3){y`	Item 3 of y
`n3=: (<3){y`	Item 3 of y
`n4=: 3{y`	Item 3 of y
`n5=: (<,<_1){y`	The last item of y (item _1 of y)
`n6=: (<,_1){y`	The last item of y (shape 5 6 7)
`n7=: (<_1){y`	The last item of y
`n8=: _1{y`	The last item of y
`n9=: (_1+#y){y`	The last item of y
`n10=: 0{y`	The first item of y
`n11=: (-#y){y`	The first item of y
`n12=: 3 0 _2 0{y`	Items 3 0 _2 0 of y
`n13=: i=: ?2 3$0{$y`	Indices used in examples
`n14=: j=: ? 1{$y`	Indices used in examples
`n15=: k=: ?7 $2{$y`	Indices used in examples
`n16=: (<i;j;k){y`	y[i;j;k;]in APL notation
`n17=: (<1;2;3){y`	y[1;2;3;]
`n18=: (<1,2,3){y`	y[1;2;3;]
`n19=: (<1 2 3){y`	y[1;2;3;]
`n20=: (<<i){y`	y[i;;;;]
`n21=: (<<,i){y`	y[,i;;...;]
`n22=: (,i){y`	y[,i;;...;]
`n23=: (<<1 3 2){y`	Items 1 3 2
`n24=: (<<<1 3 2){y`	All but items 1 3 2
`n25=: (<<<1 3){y`	All but items 1 3
`n26=: (<<<1){y`	All but items 1
`n27=: (<<<$0){y`	All but items ... none; i.e. all items
`n28=: (<<a:){y`	All items
`n29=: (<1 3 2;3){y`	y[1 3 2;3;;...;]in APL (0-origin)
`n30=: (<(<1 3 2);3){y`	y[(i.#y)-.1 3 2;3;;...;]
`n31=: (<(<1 3);3){y`	y[(i.#y)-.1 3;3;;...;]
`n32=: (<(<1);3){y`	y[(i.#y)-.1;3;;...;]
`n33=: (<(<$0);3){y`	y[(i.#y)-.$0;3;;...;]
`n34=: (<(<$0);3){y`	y[;3;;...;]
`n35=: (<a:;3){y`	y[;3;;...;]
`n36=: 4{"_1 y`	y[;4;;...;]
`n37=: (<a:;a:;5){y`	y[;;5;...;]
`n38=: 5{"_2 y`	y[;;5;...;]
`n39=: (<1 2){y`	Abbreviated (fewer indices than axes)
`n40=: _2{y`	Negative
`n41=: (<<<3){y`	Complementary
`n42=: (1 2;3 2;0 _2){y`	Scattered (non-scalar left argument)

JPhrases/Indexing

3A. Indexing