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{{Built-in|Inner Product|<nowiki>.</nowiki>}} is a [[dyadic operator]] that produces a [[dyadic function]] when applied with two dyadic functions. It's a generalisation of the [[wikipedia:Matrix multiplication|matrix product]], allowing not just addition-multiplication, but any [[dyadic function]]s given as [[operand]]s.
{{Built-in|Inner Product|<nowiki>.</nowiki>}} is a [[dyadic operator]] that produces a [[dyadic function]] when applied with two dyadic functions. It's a generalisation of the [[wikipedia:Matrix multiplication|matrix product]], allowing not just addition-multiplication, but any [[dyadic function]]s given as [[operand]]s.
== Description ==
For each rank-1 cell in the arguments, inner product applies <syntaxhighlight lang=apl inline>⍵⍵</syntaxhighlight> between the two and then applies <syntaxhighlight lang=apl inline>⍺⍺⌿</syntaxhighlight> to the results of those invocations.  If the arguments themselves are simply vectors, there is only one rank-1 cell in each argument, so this results in the following application pattern:
<syntaxhighlight lang=apl>
      1 2 3 4 F.G 5 6 7 8
      (1 G 5) F (2 G 6) F (3 G 7) F (4 G 8)
</syntaxhighlight>
The second line is an illustration of how the first will be evaluated.  Note that this is precisely the [[wikipedia:Dot product|vector dot product]] when used as <syntaxhighlight lang=apl inline>+.×</syntaxhighlight>.  (This simple invocation with vector arguments will be referred to as the "vector inner product" below, but it is just a simple case of the general inner product.)
For matrix arguments, there may be more than one rank-1 cell.  Considering the case of rank-2 matrices as arguments, if there are N rows in <syntaxhighlight lang=apl inline>⍺</syntaxhighlight> and M columns in <syntaxhighlight lang=apl inline>⍵</syntaxhighlight>, the result will be a matrix with N rows and M columns.  Each row of the resulting matrix will correspond to the elements of that same row in <syntaxhighlight lang=apl inline>⍺</syntaxhighlight> being paired up with elements in a column of <syntaxhighlight lang=apl inline>⍵</syntaxhighlight>.  Likewise, each column of the resulting matrix will correspond to the elements of that same column of <syntaxhighlight lang=apl inline>⍵</syntaxhighlight> being paired up with elements in a row of <syntaxhighlight lang=apl inline>⍺</syntaxhighlight>.  '''Important: This means that the inner product will be applied for each ''row'' of <syntaxhighlight lang=apl inline>⍺</syntaxhighlight> but each ''column'' of <syntaxhighlight lang=apl inline>⍵</syntaxhighlight>!'''
In practice, this means that the upper left element of the resulting matrix will correspond to performing the vector inner product on the first row of <syntaxhighlight lang=apl inline>⍺</syntaxhighlight> and the first column of <syntaxhighlight lang=apl inline>⍵</syntaxhighlight>, the upper right element will correspond to performing the vector inner product on the first row of <syntaxhighlight lang=apl inline>⍺</syntaxhighlight> and the last column of <syntaxhighlight lang=apl inline>⍵</syntaxhighlight>, the lower right element will correspond to the vector inner product on the last row of <syntaxhighlight lang=apl inline>⍺</syntaxhighlight> and the last column of <syntaxhighlight lang=apl inline>⍵</syntaxhighlight>, and so on and so on.  Pictorially, then, for a 2x2 result we can represent the resulting matrix as:
<syntaxhighlight lang=apl>
┌──────────────────────────────┬──────────────────────────────┐
│(Row 1 of ⍺) F.G (Col. 1 of ⍵)│(Row 1 of ⍺) F.G (Col. 2 of ⍵)│
├──────────────────────────────┼──────────────────────────────┤
│(Row 2 of ⍺) F.G (Col. 1 of ⍵)│(Row 2 of ⍺) F.G (Col. 2 of ⍵)│
└──────────────────────────────┴──────────────────────────────┘
</syntaxhighlight>
Note how the columns of <syntaxhighlight lang=apl inline>⍵</syntaxhighlight> align with the columns of the matrix, and the rows of <syntaxhighlight lang=apl inline>⍺</syntaxhighlight> align with the rows of the matrix.
The concept readily generalizes to matrices of higher rank.


== Examples ==
== Examples ==
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20&4\\
20&4\\
\end{pmatrix},
\end{pmatrix},
\quad\boldsymbol{A}\;^\and_=\,\boldsymbol{B}=\begin{pmatrix}
\quad\boldsymbol{A}\;^\land_=\,\boldsymbol{B}=\begin{pmatrix}
0&1\\
0&1\\
0&0\\
0&0\\
1&0\\
1&0\\
\end{pmatrix}\text{,}\\
\end{pmatrix}\text{,}\\
\boldsymbol{A}\;^\or_\neq\;\boldsymbol{B}&=\begin{pmatrix}
\boldsymbol{A}\;^\lor_\neq\;\boldsymbol{B}&=\begin{pmatrix}
1&0\\
1&0\\
1&1\\
1&1\\
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== Differences between dialects ==
== 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 <syntaxhighlight lang=apl inline>X f.g Y</syntaxhighlight> is equivalent to <syntaxhighlight lang=apl inline>f/¨ (⊂[⍴⍴x]x)∘.g ⊂[1]y</syntaxhighlight>. 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>
Implementations differ on the exact behaviour of inner product when the right operand is not a [[scalar function]]. Page 121 of the [[ISO/IEC 13751:2001]] standard specifies that <syntaxhighlight lang=apl inline>X f.g Y</syntaxhighlight> is equivalent to <syntaxhighlight lang=apl>f/¨ (⊂[⍴⍴x]x)∘.g ⊂[1]y</syntaxhighlight> (assuming [[index origin]] 1). 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>
<blockquote>
The following dop models inner product in Dyalog APL, with caveats.  If you find a case where <syntaxhighlight lang=apl inline>f.g</syntaxhighlight> differs from <syntaxhighlight lang=apl inline>f IP g</syntaxhighlight>, not covered by the caveats, I'd be interested.
The following dop models inner product in Dyalog APL, with caveats.  If you find a case where <syntaxhighlight lang=apl inline>f.g</syntaxhighlight> differs from <syntaxhighlight lang=apl inline>f IP g</syntaxhighlight>, not covered by the caveats, I'd be interested.
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   (3 4⍴5)+.×1 5⍴6  ⍝ works in NARS2000 or APL\360, not in Dyalog APL</syntaxhighlight>
   (3 4⍴5)+.×1 5⍴6  ⍝ works in NARS2000 or APL\360, not in Dyalog APL</syntaxhighlight>
</blockquote>
</blockquote>
The <syntaxhighlight lang=apl>⊃⍤0⊢(↓⍺)∘.(⍺⍺/⍵⍵¨)↓(¯1⌽⍳⍴⍴⍵)⍉⍵</syntaxhighlight> line of IP above can be rewritten as <syntaxhighlight lang=apl>⍺(⍺⍺⌿⍵⍵¨⍤¯1)⍤1 99⊢⍵</syntaxhighlight> which uses the more efficient [[item]]-at-a-time algorithm (rather than row-by-column). The ISO/IEC 13751:2001(E) inner product, conversely, can only be calculated row-by-column, as computing the results one item (of the right argument) at a time relies on each application of the right operand being done between two scalars and producing a scalar result—that is, on the [[Each]] operator being applied to the right operand.
Some implementations extend the inner product by implementing Iverson's monadic variant<ref>[[Ken Iverson|K.E. Iverson]]. [https://www.jsoftware.com/papers/satn42.htm Determinant-Like Functions Produced by the Dot Operator.] SHARP APL Technical Note 42. 1982-04-01.</ref>, which takes a single argument and performs the operation of computing the alternant, as modelled by [https://dfns.dyalog.com/n_alt.htm dfns.alt].


== External links ==
== External links ==
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* [https://microapl.com/apl_help/ch_020_020_880.htm APLX]
* [https://microapl.com/apl_help/ch_020_020_880.htm APLX]
* 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]
=== Publications ===
* [https://www.jsoftware.com/papers/innerproduct/ip1.htm Inner Product: An Old/New Problem] by [[Roger Hui]]


=== Discussion of differences between dialects ===
=== Discussion of differences between dialects ===
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