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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]]. | {{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. | ||

=== 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> | |||

<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> . | |||

</blockquote> | |||

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. | |||

=== Examples === | === Examples === | ||

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│3 4│3 5│3 6│ | │3 4│3 5│3 6│ | ||

└───┴───┴───┘ | └───┴───┴───┘ | ||

⍝ works for multi-dimensional arrays as well | ⍝ works for multi-dimensional arrays as well | ||

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=== Applications === | === Applications === | ||

Outer product is useful for solving problems that intuitively | Outer product is useful for solving problems that intuitively require a [[wikipedia:Time_complexity#Polynomial_time|polynomial time]] algorithm. This may also indicate that such an algorithm is not the fastest solution. | ||

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. | ||

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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> | </source> | ||

Here there are faster solutions such as the [[wikipedia:Sieve of Eratosthenes|Sieve of Eratosthenes]]. | |||

== External links == | == External links == | ||

* [https://mlochbaum.github.io/OuterProduct/ Outer Product as an Introduction to APL and a Pretty Cool Thing in General]: website for LambdaConf talk by [[Marshall Lochbaum]] | |||

=== Documentation === | === Documentation === | ||

* [https://help.dyalog.com/latest/#Language/Primitive | * [https://help.dyalog.com/latest/#Language/Primitive%20Operators/Outer%20Product.htm Dyalog] | ||

* [https://microapl.com/apl_help/ch_020_020_890.htm APLX] | * [https://microapl.com/apl_help/ch_020_020_890.htm APLX] | ||

* J [https://www.jsoftware.com/help/dictionary/d420.htm Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/slash#dyadic NuVoc] | * J [https://www.jsoftware.com/help/dictionary/d420.htm Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/slash#dyadic NuVoc] | ||

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

== References == | |||

<references/> | |||

{{APL built-ins}}[[Category:Primitive operators]] | {{APL built-ins}}[[Category:Primitive operators]] |

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