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Inner Product: Difference between revisions
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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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== Differences between dialects == | == Differences between dialects == | ||
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 | 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. | ||