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Outer Product (view source)

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{{Built-in|Outer Product|<nowiki>∘.</nowiki>}}, or '''Table''' is a [[monadic operator]], which will produce a [[dyadic function]] when applied with a [[dyadic function]]. Outer product applies the [[operand]] function on each [[element]] of the left array with each element of the right array. It can be described as a shortcut for constructing nested [[wikipedia:for loop|for loop]]s.
{{Built-in|Outer Product|<nowiki>∘.</nowiki>}}, or '''Table''' is a [[monadic operator]], which will produce a [[dyadic function]] when applied with a [[dyadic function]]. Outer product applies the [[operand]] function on each [[element]] of the left array with each element of the right array. It can be described as a shortcut for constructing nested [[wikipedia:for loop|for loop]]s, and as a structured [[wikipedia:Cartesian product|Cartesian product]] generalized to operations other than [[catenate|catenation]].


=== Syntax ===
=== Syntax ===
Outer Product differs from all other [[monadic operator]]s, which are written as a single [[glyph]], with the operand on the left. For [[backwards compatibility|historical reasons]], the outer product operator is a [[bi-glyph]] denoted as <source lang=apl inline>∘.</source>, and its appears on the right. This special notation is derived from the <source lang=apl inline>f.g</source> notation of [[inner product]]:<ref>[[Adin Falkoff|Falkoff, A.D.]] and [[Ken Iverson|K.E. Iverson]]. [https://www.jsoftware.com/papers/APL360TerminalSystem1.htm#ip The APL\360 Terminal System: Inner and Outer Products]. Research Report RC-1922. [[IBM]] Watson Research Center. 1967-10-16.</ref>
Outer Product differs from all other [[monadic operator]]s, which are written as a single [[glyph]], with the operand on the left. For [[backwards compatibility|historical reasons]], the outer product operator is a [[bi-glyph]] denoted as <syntaxhighlight lang=apl inline>∘.</syntaxhighlight>, and its operand appears on the right. This special notation is derived from the <syntaxhighlight lang=apl inline>f.g</syntaxhighlight> notation of [[inner product]]:<ref>[[Adin Falkoff|Falkoff, A.D.]] and [[Ken Iverson|K.E. Iverson]]. [https://www.jsoftware.com/papers/APL360TerminalSystem1.htm#ip The APL\360 Terminal System: Inner and Outer Products]. Research Report RC-1922. IBM [[Watson Research Center]]. 1967-10-16.</ref>
<blockquote>
<blockquote>
The result of an inner product is an array with rank two less than the sum of the argument ranks. The result of an outer product, on the other hand, is always an array of rank equal to the sum of the argument ranks. This follows from the fact that the reduction operation, which collapses two dimensions in an inner product, is not used in the outer product. The notation for outer product reflects this by canonically using a small circle as the first symbol. Thus, the ordinary outer product is written as <code>a∘.×b</code> .
The result of an inner product is an array with rank two less than the sum of the argument ranks. The result of an outer product, on the other hand, is always an array of rank equal to the sum of the argument ranks. This follows from the fact that the reduction operation, which collapses two dimensions in an inner product, is not used in the outer product. The notation for outer product reflects this by canonically using a small circle as the first symbol. Thus, the ordinary outer product is written as <code>a∘.×b</code> .
</blockquote>
</blockquote>
This syntactical inconsistency is resolved in [[J]] and [[BQN]], where the outer product operator, called Table, and denoted <syntaxhighlight lang=j inline>/</syntaxhighlight> and <code>⌜</code> respectively, has the usual operator syntax.


This syntactical inconsistency is resolved in [[J]] and [[BQN]], where the outer product operator, called Table, and denoted <source lang=j inline>/</source> and <code>⌜</code> respectively, has the usual operator syntax.
=== Model ===
Outer Product can be modeled as <syntaxhighlight lang=apl inline>f¨⍤0 99</syntaxhighlight>.


=== Examples ===
=== Examples ===
<source lang=apl>
<syntaxhighlight lang=apl>
       x ← 1 2 3
       x ← 1 2 3
       y ← 4 5 6
       y ← 4 5 6
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└───┴───┴───┘
└───┴───┴───┘
        
        
</source>
</syntaxhighlight>


=== Applications ===
=== Applications ===
Line 54: Line 56:


For example, suppose we want to find duplicated elements in an non-[[nested array]]. Intuitively speaking, the easiest way to solve this problem is to compare each element of the array with all other elements, which is exactly what an outer product does.
For example, suppose we want to find duplicated elements in an non-[[nested array]]. Intuitively speaking, the easiest way to solve this problem is to compare each element of the array with all other elements, which is exactly what an outer product does.
<source lang=apl>
<syntaxhighlight lang=apl>
       x ← 1 2 3 2
       x ← 1 2 3 2
       ⎕ ← matrix ← x∘.=x ⍝ compare elements with each other using equal
       ⎕ ← matrix ← x∘.=x ⍝ compare elements with each other using equal
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       (⊢∪⍤/⍨2≤(+/∘.=⍨)) x ⍝ point-free/tacit version
       (⊢∪⍤/⍨2≤(+/∘.=⍨)) x ⍝ point-free/tacit version
2
2
</source>
</syntaxhighlight>
Using similar techniques, we can define a function that generates prime numbers by using an outer product of [[Residue]].
Using similar techniques, we can define a function that generates prime numbers by using an outer product of [[Residue]].
<source lang=apl>
<syntaxhighlight lang=apl>
     primes ← {x←1↓⍳⍵ ⋄ (2>+⌿0=x∘.|x)/x}
     primes ← {x←1↓⍳⍵ ⋄ (2>+⌿0=x∘.|x)/x}
     primes 10
     primes 10
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       primes 20
       primes 20
2 3 5 7 11 13 17 19
2 3 5 7 11 13 17 19
</source>
</syntaxhighlight>
Here there are faster solutions such as the [[wikipedia:Sieve of Eratosthenes|Sieve of Eratosthenes]].
Here there are faster solutions such as the [[wikipedia:Sieve of Eratosthenes|Sieve of Eratosthenes]].
=== Differences between dialects ===
[[J]]'s Table differs from APL's Outer Product by enabling Cartesian pairing between [[cell]]s of [[rank]]s other than 0, and between cells of different rank for the left vs. right arguments, without needing to first [[enclose]] the relevant cells. It is defined as <syntaxhighlight lang=j inline>u"(lu, _)</syntaxhighlight>, where <syntaxhighlight lang=j inline>lu</syntaxhighlight> is the left rank of operand <syntaxhighlight lang=j inline>u</syntaxhighlight> (<syntaxhighlight lang=j inline>"</syntaxhighlight> and <syntaxhighlight lang=j inline>_</syntaxhighlight> denote [[rank_(operator)|Rank]] and infinity, respectively, in J).
<syntaxhighlight lang=j>
  NB. Cartesian pairing of rows
  x=: 100+i.2 2
  y=: i.3 4
  x ,"1/ y      NB. same as x ,"1"1 _ y
100 101 0 1  2  3
100 101 4 5  6  7
100 101 8 9 10 11
102 103 0 1  2  3
102 103 4 5  6  7
102 103 8 9 10 11
  NB. Cartesian pairing of x-matrices and y-cubes
  x=: 100+i.2 2 2
  y=: i.2 2 2 4
  x ;"2 3/ y    NB. same as x ;"2 3"2 _ y
┌───────┬───────────┐
│100 101│ 0  1  2  3│
│102 103│ 4  5  6  7│
│      │          │
│      │ 8  9 10 11│
│      │12 13 14 15│
├───────┼───────────┤
│100 101│16 17 18 19│
│102 103│20 21 22 23│
│      │          │
│      │24 25 26 27│
│      │28 29 30 31│
└───────┴───────────┘
┌───────┬───────────┐
│104 105│ 0  1  2  3│
│106 107│ 4  5  6  7│
│      │          │
│      │ 8  9 10 11│
│      │12 13 14 15│
├───────┼───────────┤
│104 105│16 17 18 19│
│106 107│20 21 22 23│
│      │          │
│      │24 25 26 27│
│      │28 29 30 31│
└───────┴───────────┘
</syntaxhighlight>
{{Works in|[[J]]}}
In APL this behavior can be accomplished by enclosing the relevant cells of each argument before applying the outer product operation (or by successive applications of Rank, e.g. <syntaxhighlight lang=apl inline>f⍤99 1⍤2 99</syntaxhighlight>). Conversely, the behavior of APL's Outer Product on nested inputs can be accomplished in terms of J's Table by adding the Each modifier, as in <syntaxhighlight lang=j inline>u&.>/</syntaxhighlight>.
The results of the individual applications of u are collectively [[frame]]d by the catenation of the frames of the left and right arguments relative to their lu- and ru-cells respectively, where lu and ru are the dyadic ranks of the operand u.
Table does not allow for creating a Cartesian pairing involving cells of the left argument specified with negative rank (except trivially, in cases in which Table's application has no effect, e.g. <syntaxhighlight lang=j inline>u"_ _1"_2 _/</syntaxhighlight>). This is because negative assigned rank in J is encoded as infinite rank; in J's terminology, negative rank is "internal rank" only.
<syntaxhighlight lang=j>
  }.+"_1 _2 b.0  NB. dyadic ranks of derived verb +"_1 _2 as seen by Table
_ _
</syntaxhighlight>
== External links ==
== External links ==


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* [https://mlochbaum.github.io/BQN/doc/map.html#table BQN]
* [https://mlochbaum.github.io/BQN/doc/map.html#table BQN]


== References ==
<references/>
{{APL built-ins}}[[Category:Primitive operators]]
{{APL built-ins}}[[Category:Primitive operators]]
Retrieved from ‘https://aplwiki.com/wiki/Special:MobileDiff/8351...11375’

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