Inner Product: Difference between revisions

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</source>
</source>
== History ==
Inner product appeared in early [[Iverson Notation]] as <math>^f_g</math> and applied even to non-[[scalar function]]s, like [[Compress]], Iverson bringing:<ref>[[Ken Iverson]]. [[A Programming Language]]. §1.11 ''The language''.</ref>
:<math>
\begin{align}
\text{For example, if}\\
\boldsymbol{A}&=\begin{pmatrix}
1&3&2&0\\
2&1&0&1\\
4&0&0&2\\
\end{pmatrix}
\qquad\text{and}\qquad
\boldsymbol{B}=\begin{pmatrix}
4&1\\
0&3\\
0&2\\
2&0\\
\end{pmatrix}\\
\text{then}\qquad\boldsymbol{A}\;^+_\times\,\boldsymbol{B}&=\begin{pmatrix}
4&14\\
10&5\\
20&4\\
\end{pmatrix},
\quad\boldsymbol{A}\;^\and_=\,\boldsymbol{B}=\begin{pmatrix}
0&1\\
0&0\\
1&0\\
\end{pmatrix}\text{,}\\
\boldsymbol{A}\;^\or_\neq\;\boldsymbol{B}&=\begin{pmatrix}
1&0\\
1&1\\
0&1\\
\end{pmatrix},
\qquad\text{and}\qquad(\boldsymbol{A}\neq0)\;^+_{\,/}\,\boldsymbol{B}=\begin{pmatrix}
4&6\\
6&4\\
6&1\\
\end{pmatrix}\text{.}
\end{align}
</math>
When the inner product notation was linearised (made to fit on a single line of code) the [[glyph]] <source lang=apl inline>.</source> was chosed to denote what was previously indicated by positioning the two [[operand]]s vertically aligned. Thus, the above correspond to the following modern APL:
<source lang=apl>
⍝ For example, if
      A←3 4⍴1 3 2 0 2 1 0 1 4 0 0 2
      B←4 2⍴4 1 0 3 0 2 2 0
⍝ then
      A +.× B
4 14
10  5
20  4
      A ∧.= B
0 1
0 0
1 0
      A ∨.≠ B
1 0
1 1
0 1
      (A ≠ 0) +./ B
4 6
6 4
6 1
</source>
Note that some dialects implement [[Compress]] (<source lang=apl inline>/</source>) as a [[monadic operator]] rather than as a function, which means it cannot be an operand in the inner product. Instead, a cover function is necessary:
<source lang=apl>
∇z←a Compress b
z←a/b
</source>
== Differences between dialects ==
Implementations differ on the exact behaviour of inner product when the right operand is not a [[scalar function]]. It follows from page 121 of the ISO/IEC 13751:2001(E) [[standard]] specifies that <source lang=apl inline>X f.g Y</source> is equivalent to <source lang=apl inline>f/¨ (⊂[⍴⍴x]x)∘.g ⊂[1]y</source>. This is indeed what [[APL2]], [[APLX]], [[APL+Win]], and [[ngn/apl]] follow, while [[Dyalog APL]], [[NARS2000]] and [[GNU APL]] differ as described by [[Roger Hui]]:<ref>[[Roger Hui]]. ''inner product''. Internal Dyalog email. 24 July 2020.</ref>
<blockquote>
The following dop models inner product in Dyalog APL, with caveats.  If you find a case where <source lang=apl inline>f.g</source> differs from <source lang=apl inline>f IP g</source>, not covered by the caveats, I'd be interested.
<source lang=apl>
IP←{                               
  assert((⊃⌽⍴⍺)≡≢⍵)∨(1=×/⍴⍺)∨1=×/⍴⍵:
  ⊃⍤0 ⊢ (↓⍺) ∘.(⍺⍺/⍵⍵¨) ↓(¯1⌽⍳⍴⍴⍵)⍉⍵   
}
assert←{⍺←'assertion failure' ⋄ 0∊⍵:⍺ ⎕SIGNAL 8 ⋄ shy←0}
</source>
(Explanation: What's with the <source lang=apl inline>⊃⍤0</source> in <source lang=apl inline>IP</source>?  It's because <source lang=apl inline>∘.f</source> has an implicit each, applying <source lang=apl inline>⊂</source> to each item of its result.  But the <source lang=apl inline>⍺⍺/</source> in <source lang=apl inline>(⍺⍺/⍵⍵¨)</source> also has an implicit each.  So the <source lang=apl inline>⊃⍤0</source> gets rid of one of those encloses.)
Caveats:
* You can not use the hybrid <source lang=apl inline>/</source> directly as an operand as it runs afoul of the parser in weird and wonderful ways.  Instead, you have to use <source lang=apl inline>{⍺/⍵}</source>.  The same goes for <source lang=apl inline>\</source> and <source lang=apl inline>{⍺\⍵}</source> I guess.
* It differs from ISO/IEC 13751:2001(E) in using <source lang=apl inline>⍵⍵¨</source> instead of just <source lang=apl inline>⍵⍵</source> in the central key expression (i.e. <source lang=apl inline>(⍺⍺/⍵⍵¨)</source> instead of <source lang=apl inline>(⍺⍺/⍵⍵)</source>).  So does the primitive <source lang=apl inline>f.g</source>.
* It differs from ISO/IEC 13751:2001(E) in doing full-blown single extension instead of just scalar and 1-element vector extension (as in APL2).  So does the primitive <source lang=apl inline>f.g</source>.  e.g.<source lang=apl>
  (3 4⍴5)+.×1 1 1 1⍴6  ⍝ works in Dyalog, not in ISO or APL2</source>
* It differs from NARS2000 or APL\360 in not permitting unit axis extension. So does the primitive <source lang=apl inline>f.g</source>.  e.g.<source lang=apl>
  (3 4⍴5)+.×1 5⍴6  ⍝ works in NARS2000 or APL\360, not in Dyalog APL</source>
</blockquote>
== External links ==
== External links ==


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* J [https://www.jsoftware.com/help/dictionary/d300.htm Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/dot#dyadic NuVoc]
* J [https://www.jsoftware.com/help/dictionary/d300.htm Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/dot#dyadic NuVoc]


=== Discussion of differences between dialects ===
* [https://groups.google.com/g/comp.lang.apl/c/23LrwRZKmPs Dyalog / APL2000 discrepancy] (Google Groups)
* [https://lists.gnu.org/archive/html/bug-apl/2016-07/msg00020.html multiple inner product] (GNU APL mailing list)
* [https://lists.gnu.org/archive/html/bug-apl/2018-05/msg00003.html  an other inner product ,., bug] (GNU APL mailing list)
== References ==
<references/>
{{APL built-ins}}[[Category:Primitive operators]]
{{APL built-ins}}[[Category:Primitive operators]]

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