JPhrases/Indexing

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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)