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{{Built-in|Each|¨}} is a [[primitive operator|primitive]] [[monadic operator]] which applies its [[operand]] to each [[element]] of the [[argument]]s, and returns an array whose elements are the results. If two arguments are given, their elements are matched using [[conformability]] rules. | {{Built-in|Each|¨}} is a [[primitive operator|primitive]] [[monadic operator]] which applies its [[operand]] to each [[element]] of the [[argument]]s, and returns an array whose elements are the results. If two arguments are given, their elements are matched using [[conformability]] rules. | ||
== Definition == | |||
Each is defined only in [[Nested array model|nested]] APLs. Some [[Flat array model|flat]] APLs obtain analogous functionality by using an [[Under]] operator with [[close composition]] along with the [[Function rank|rank]]-0 function [[Disclose]] (or Unbox). In [[SHARP APL]] this is written <source lang=apl inline>f¨></source>. In [[J]] it is <source lang=j inline>f&.></source>. | Each is defined only in [[Nested array model|nested]] APLs. Some [[Flat array model|flat]] APLs obtain analogous functionality by using an [[Under]] operator with [[close composition]] along with the [[Function rank|rank]]-0 function [[Disclose]] (or Unbox). In [[SHARP APL]] this is written <source lang=apl inline>f¨></source>. In [[J]] it is <source lang=j inline>f&.></source>. | ||
Each differs from the [[Rank operator]] with rank 0 in that the operand arguments and results are not [[enclose]]d. As the [[elements]] of a nested array they need not be [[scalar]]. | Each differs from the [[Rank operator]] with rank 0 in that the operand arguments and results are not [[enclose]]d. As the [[elements]] of a nested array they need not be [[scalar]]. | ||
For example, | The Each operator has no effect on [[scalar function]]s, since these functions already map over each array element. For example, both expressions below have the same meaning, since <source lang=apl inline>+</source> is a scalar function. | ||
<source lang=apl> | |||
1 + 1 2 3 4 | |||
2 3 4 5 | |||
1 +¨ 1 2 3 4 | |||
2 3 4 5 | |||
</source> | |||
== Examples == | |||
<source lang=apl> | <source lang=apl> | ||
1,1 2 3 ⍝ join 1 with 1 2 3 | 1,1 2 3 ⍝ join 1 with 1 2 3 | ||
| Line 26: | Line 33: | ||
</source> | </source> | ||
The Each | == Notable uses == | ||
=== Mapping === | |||
It is very common to pair up an entire array with each element of a different array. There are two common ways to do this using Each. The first is to [[enclose]] the argument that is to be used as a whole for each element of the other array: | |||
<source lang=apl> | |||
(⊂10 20 30),¨1 2 3 ⍝ Computes (10 20 30,1)(10 20 30,2)(10 20 30,3) | |||
┌──────────┬──────────┬──────────┐ | |||
│10 20 30 1│10 20 30 2│10 20 30 3│ | |||
└──────────┴──────────┴──────────┘ | |||
10 20 30,¨⊂1 2 3 ⍝ Computes (10,1 2 3)(20,1 2 3)(30,1 2 3) | |||
┌────────┬────────┬────────┐ | |||
│10 1 2 3│20 1 2 3│30 1 2 3│ | |||
└────────┴────────┴────────┘ | |||
</source> | |||
The other method is by [[bind]]ing the argument that is to be used as a whole to the function, deriving a monadic function, which is then applied using Each: | |||
<source lang=apl> | |||
10 20 30∘,¨1 2 3 ⍝ Computes (10 20 30,1)(10 20 30,2)(10 20 30,3) | |||
┌──────────┬──────────┬──────────┐ | |||
│10 20 30 1│10 20 30 2│10 20 30 3│ | |||
└──────────┴──────────┴──────────┘ | |||
,∘1 2 3¨10 20 30 ⍝ Computes (10,1 2 3)(20,1 2 3)(30,1 2 3) | |||
┌────────┬────────┬────────┐ | |||
│10 1 2 3│20 1 2 3│30 1 2 3│ | |||
└────────┴────────┴────────┘ | |||
</source> | |||
Note how binding a right argument derives a monadic function which still takes its single argument on the right. | |||
=== Selecting === | |||
An enclosed array is a [[scalar]], which is subject to [[scalar extension]]. This can be used to simulate [[Outer Product|outer product]] by a one-sided Each (pair the entire right argument with each element of the left argument, or vice versa). A notable application of this behavior is the "chipmunk idiom" <source lang=apl inline>X⊃¨⊂Y</source>, which simulates <source lang=apl inline>Y[X]</source> for (possibly nested) [[vector]] Y and [[simple]] X: | |||
<source lang=apl> | <source lang=apl> | ||
1 | (2 2⍴1 2 2 1)⊃¨⊂(1 2)(3 4)(5 6) ⍝ Computes (1 2)(3 4)(5 6)[2 2⍴1 2 2 1] | ||
2 3 4 5 | ┌───┬───┐ | ||
│1 2│3 4│ | |||
├───┼───┤ | |||
│3 4│1 2│ | |||
└───┴───┘ | |||
</source> | </source> | ||