Inner Product: Difference between revisions

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== Examples ==
== Examples ==
<source lang=apl>
<syntaxhighlight lang=apl>
       x ← 1 2 3
       x ← 1 2 3
       y ← 4 5 6
       y ← 4 5 6
Line 25: Line 25:
</source>
</source>


The [[shape]]s of the arguments must be compatible with each other: The last [[axis]] of the left argument must have the same length as the first axis of the right argument, or formally, for <source lang=apl inline>X f.g Y</source> it must be that <source lang=apl inline>(¯1↑⍴X)≡(1↑⍴Y)</source>. Although this rule differs from [[conformability]], the arguments may also be subject to [[scalar extension|scalar]] or [[singleton extension]]. The shape of the result is <source lang=apl inline>(¯1↓⍴X),(1↓⍴Y)</source>.
The [[shape]]s of the arguments must be compatible with each other: The last [[axis]] of the left argument must have the same length as the first axis of the right argument, or formally, for <syntaxhighlight lang=apl inline>X f.g Y</source> it must be that <syntaxhighlight lang=apl inline>(¯1↑⍴X)≡(1↑⍴Y)</source>. Although this rule differs from [[conformability]], the arguments may also be subject to [[scalar extension|scalar]] or [[singleton extension]]. The shape of the result is <syntaxhighlight lang=apl inline>(¯1↓⍴X),(1↓⍴Y)</source>.


For example, when applying inner product on two [[matrix|matrices]], the number of columns in the left array must match with number of rows in the right array, otherwise we will get an error.
For example, when applying inner product on two [[matrix|matrices]], the number of columns in the left array must match with number of rows in the right array, otherwise we will get an error.
<source lang=apl>
<syntaxhighlight lang=apl>
       ⎕  ← x ← 2 3⍴⍳10
       ⎕  ← x ← 2 3⍴⍳10
1 2 3
1 2 3
Line 85: Line 85:
\end{align}
\end{align}
</math>
</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:
When the inner product notation was linearised (made to fit on a single line of code) the [[glyph]] <syntaxhighlight 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>
<syntaxhighlight lang=apl>
⍝ For example, if
⍝ For example, if
       A←3 4⍴1 3 2 0 2 1 0 1 4 0 0 2
       A←3 4⍴1 3 2 0 2 1 0 1 4 0 0 2
Line 108: Line 108:
6 1
6 1
</source>
</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:
Note that some dialects implement [[Compress]] (<syntaxhighlight 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>
<syntaxhighlight lang=apl>
∇z←a Compress b
∇z←a Compress b
  z←a/b
  z←a/b
Line 116: Line 116:


== 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 <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>
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</source> is equivalent to <syntaxhighlight 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>
<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.
The following dop models inner product in Dyalog APL, with caveats.  If you find a case where <syntaxhighlight lang=apl inline>f.g</source> differs from <syntaxhighlight lang=apl inline>f IP g</source>, not covered by the caveats, I'd be interested.
<source lang=apl>
<syntaxhighlight lang=apl>
IP←{                                 
IP←{                                 
   assert((⊃⌽⍴⍺)≡≢⍵)∨(1=×/⍴⍺)∨1=×/⍴⍵:
   assert((⊃⌽⍴⍺)≡≢⍵)∨(1=×/⍴⍺)∨1=×/⍴⍵:
Line 127: Line 127:
assert←{⍺←'assertion failure' ⋄ 0∊⍵:⍺ ⎕SIGNAL 8 ⋄ shy←0}
assert←{⍺←'assertion failure' ⋄ 0∊⍵:⍺ ⎕SIGNAL 8 ⋄ shy←0}
</source>
</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.)
(Explanation: What's with the <syntaxhighlight lang=apl inline>⊃⍤0</source> in <syntaxhighlight lang=apl inline>IP</source>?  It's because <syntaxhighlight lang=apl inline>∘.f</source> has an implicit each, applying <syntaxhighlight lang=apl inline>⊂</source> to each item of its result.  But the <syntaxhighlight lang=apl inline>⍺⍺/</source> in <syntaxhighlight lang=apl inline>(⍺⍺/⍵⍵¨)</source> also has an implicit each.  So the <syntaxhighlight lang=apl inline>⊃⍤0</source> gets rid of one of those encloses.)


Caveats:
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.
* You can not use the hybrid <syntaxhighlight 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 <syntaxhighlight lang=apl inline>{⍺/⍵}</source>.  The same goes for <syntaxhighlight lang=apl inline>\</source> and <syntaxhighlight 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 using <syntaxhighlight lang=apl inline>⍵⍵¨</source> instead of just <syntaxhighlight lang=apl inline>⍵⍵</source> in the central key expression (i.e. <syntaxhighlight lang=apl inline>(⍺⍺/⍵⍵¨)</source> instead of <syntaxhighlight lang=apl inline>(⍺⍺/⍵⍵)</source>).  So does the primitive <syntaxhighlight 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>
* 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 <syntaxhighlight lang=apl inline>f.g</source>.  e.g.<syntaxhighlight lang=apl>
   (3 4⍴5)+.×1 1 1 1⍴6  ⍝ works in Dyalog, not in ISO or APL2</source>
   (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>
* It differs from NARS2000 or APL\360 in not permitting unit axis extension. So does the primitive <syntaxhighlight lang=apl inline>f.g</source>.  e.g.<syntaxhighlight lang=apl>
   (3 4⍴5)+.×1 5⍴6  ⍝ works in NARS2000 or APL\360, not in Dyalog APL</source>
   (3 4⍴5)+.×1 5⍴6  ⍝ works in NARS2000 or APL\360, not in Dyalog APL</source>
</blockquote>
</blockquote>

Revision as of 21:57, 10 September 2022

.

Inner Product (.) is a dyadic operator that produces a dyadic function when applied with two dyadic functions. It's a generalisation of the matrix product, allowing not just addition-multiplication, but any dyadic functions given as operands.

Examples

<syntaxhighlight lang=apl>

     x ← 1 2 3
     y ← 4 5 6
     x ,.(⊂,) y ⍝ visualizing of pairing

┌─────────────┐ │┌───┬───┬───┐│ ││1 4│2 5│3 6││ │└───┴───┴───┘│ └─────────────┘

     x {⊂⍺,'+',⍵}.{⊂⍺,'×',⍵} y ⍝ visualizing function application in matrix multiplication

┌───────────────────────────┐ │┌─────────────────────────┐│ ││┌─────┬─┬───────────────┐││ │││1 × 4│+│┌─────┬─┬─────┐│││ │││ │ ││2 × 5│+│3 × 6││││ │││ │ │└─────┴─┴─────┘│││ ││└─────┴─┴───────────────┘││ │└─────────────────────────┘│ └───────────────────────────┘

     x+.×y ⍝ matrix multiplication

32 </source>

The shapes of the arguments must be compatible with each other: The last axis of the left argument must have the same length as the first axis of the right argument, or formally, for <syntaxhighlight lang=apl inline>X f.g Y</source> it must be that <syntaxhighlight lang=apl inline>(¯1↑⍴X)≡(1↑⍴Y)</source>. Although this rule differs from conformability, the arguments may also be subject to scalar or singleton extension. The shape of the result is <syntaxhighlight lang=apl inline>(¯1↓⍴X),(1↓⍴Y)</source>.

For example, when applying inner product on two matrices, the number of columns in the left array must match with number of rows in the right array, otherwise we will get an error. <syntaxhighlight lang=apl>

     ⎕  ← x ← 2 3⍴⍳10

1 2 3 4 5 6

     ⎕ ← y ← 4 2⍴⍳10

1 2 3 4 5 6 7 8

     x+.×y 

LENGTH ERROR

     x+.×y
       ∧
     ⎕ ← y ← 3 2⍴⍳10 ⍝ reshape y to be compatible with x
     x+.×y

22 28 49 64 </source>

History

Inner product appeared in early Iverson Notation as and applied even to non-scalar functions, like Compress, Iverson bringing:[1]

When the inner product notation was linearised (made to fit on a single line of code) the glyph <syntaxhighlight lang=apl inline>.</source> was chosed to denote what was previously indicated by positioning the two operands vertically aligned. Thus, the above correspond to the following modern APL: <syntaxhighlight 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 (<syntaxhighlight 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: <syntaxhighlight 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 <syntaxhighlight lang=apl inline>X f.g Y</source> is equivalent to <syntaxhighlight 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:[2]

The following dop models inner product in Dyalog APL, with caveats. If you find a case where <syntaxhighlight lang=apl inline>f.g</source> differs from <syntaxhighlight lang=apl inline>f IP g</source>, not covered by the caveats, I'd be interested. <syntaxhighlight lang=apl> IP←{ assert((⊃⌽⍴⍺)≡≢⍵)∨(1=×/⍴⍺)∨1=×/⍴⍵: ⊃⍤0 ⊢ (↓⍺) ∘.(⍺⍺/⍵⍵¨) ↓(¯1⌽⍳⍴⍴⍵)⍉⍵ }

assert←{⍺←'assertion failure' ⋄ 0∊⍵:⍺ ⎕SIGNAL 8 ⋄ shy←0} </source> (Explanation: What's with the <syntaxhighlight lang=apl inline>⊃⍤0</source> in <syntaxhighlight lang=apl inline>IP</source>? It's because <syntaxhighlight lang=apl inline>∘.f</source> has an implicit each, applying <syntaxhighlight lang=apl inline>⊂</source> to each item of its result. But the <syntaxhighlight lang=apl inline>⍺⍺/</source> in <syntaxhighlight lang=apl inline>(⍺⍺/⍵⍵¨)</source> also has an implicit each. So the <syntaxhighlight lang=apl inline>⊃⍤0</source> gets rid of one of those encloses.)

Caveats:

  • You can not use the hybrid <syntaxhighlight 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 <syntaxhighlight lang=apl inline>{⍺/⍵}</source>. The same goes for <syntaxhighlight lang=apl inline>\</source> and <syntaxhighlight lang=apl inline>{⍺\⍵}</source> I guess.
  • It differs from ISO/IEC 13751:2001(E) in using <syntaxhighlight lang=apl inline>⍵⍵¨</source> instead of just <syntaxhighlight lang=apl inline>⍵⍵</source> in the central key expression (i.e. <syntaxhighlight lang=apl inline>(⍺⍺/⍵⍵¨)</source> instead of <syntaxhighlight lang=apl inline>(⍺⍺/⍵⍵)</source>). So does the primitive <syntaxhighlight 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 <syntaxhighlight lang=apl inline>f.g</source>. e.g.<syntaxhighlight 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 <syntaxhighlight lang=apl inline>f.g</source>. e.g.<syntaxhighlight lang=apl>

(3 4⍴5)+.×1 5⍴6 ⍝ works in NARS2000 or APL\360, not in Dyalog APL</source>

External links

Documentation

Discussion of differences between dialects

References

  1. Ken Iverson. A Programming Language. §1.11 The language.
  2. Roger Hui. inner product. Internal Dyalog email. 24 July 2020.
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