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Outer Product: Difference between revisions
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=== 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 < | 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 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> | |||
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 < | 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. | ||
=== Examples === | === Examples === | ||
< | <syntaxhighlight lang=apl> | ||
x ← 1 2 3 | x ← 1 2 3 | ||
y ← 4 5 6 | y ← 4 5 6 | ||
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└───┴───┴───┘ | └───┴───┴───┘ | ||
</ | </syntaxhighlight> | ||
=== Applications === | === Applications === | ||
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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. | ||
< | <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 | ||
</ | </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]]. | ||
< | <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 | ||
</ | </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]]. | ||
== 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]] | ||