Transpose: Difference between revisions
m (Text replacement - "</source>" to "</syntaxhighlight>") |
m (Text replacement - "<source" to "<syntaxhighlight") |
||
| Line 7: | Line 7: | ||
[[Monadic]] Transpose reverses the axes of the right argument. When applied to a [[matrix]], the result is its [[wikipedia:transpose|transpose]]. [[Scalar]] and [[vector]] arguments are unaffected. | [[Monadic]] Transpose reverses the axes of the right argument. When applied to a [[matrix]], the result is its [[wikipedia:transpose|transpose]]. [[Scalar]] and [[vector]] arguments are unaffected. | ||
< | <syntaxhighlight lang=apl> | ||
3 3⍴⍳9 | 3 3⍴⍳9 | ||
1 2 3 | 1 2 3 | ||
| Line 24: | Line 24: | ||
=== Dyadic usage === | === Dyadic usage === | ||
{| class=wikitable style="width:50%;float:right" | {| class=wikitable style="width:50%;float:right" | ||
|{{quote | "Dyadic transpose, < | |{{quote | "Dyadic transpose, <syntaxhighlight lang=apl inline>x⍉y</syntaxhighlight>, is probably one of the last primitives to be mastered for an APLer, but is actually straightforward to describe."<ref>[[Roger Hui]]. [https://forums.dyalog.com/viewtopic.php?f=30&t=1648 ''dyadic transpose, a personal history'']. Dyalog Forums. 22 May 2020.</ref>}} | ||
|} | |} | ||
For [[dyadic]] usage, the left argument X must be a [[vector]] whose length equals the [[rank]] of the right argument Y, and the elements must form a range so that < | For [[dyadic]] usage, the left argument X must be a [[vector]] whose length equals the [[rank]] of the right argument Y, and the elements must form a range so that <syntaxhighlight lang=apl inline>∧/X∊⍳(1-⎕IO)+⌈/X</syntaxhighlight> is satisfied. | ||
If all values in X are unique (X forms a [[permutation]] over the axes of Y), the axes are reordered by X so that N-th element of X specifies the new position for the N-th axis of Y. This means that, given a multi-dimensional [[index]] V of an element in the resulting array, < | If all values in X are unique (X forms a [[permutation]] over the axes of Y), the axes are reordered by X so that N-th element of X specifies the new position for the N-th axis of Y. This means that, given a multi-dimensional [[index]] V of an element in the resulting array, <syntaxhighlight lang=apl inline>V⌷X⍉Y</syntaxhighlight> corresponds to <syntaxhighlight lang=apl inline>V[X]⍉Y</syntaxhighlight>. | ||
< | <syntaxhighlight lang=apl> | ||
X←3 1 2 | X←3 1 2 | ||
Y←3 4 5⍴⎕A | Y←3 4 5⍴⎕A | ||
| Line 41: | Line 41: | ||
</syntaxhighlight> | </syntaxhighlight> | ||
When X contains duplicates, the result has rank < | When X contains duplicates, the result has rank <syntaxhighlight lang=apl inline>(1-⎕IO)+⌈/X</syntaxhighlight>. For the axes of Y that map to the same resulting axis, only the elements where the indices are equal over those axes are collected. This has the effect of extracting diagonal elements. If the axes are of unequal length, the resulting axis has the length of the shortest of them. This operation can be modeled as computing the resulting shape <syntaxhighlight lang=apl inline>(⍴Y)⌊.+(⌊/⍬)×X∘.≠(1-⎕IO)+⍳⌈/X</syntaxhighlight>, then [[Index Generator|creating the array of its multi-dimensional indices]] <syntaxhighlight lang=apl inline>⍳</syntaxhighlight>, then modify each index and fetch the corresponding elements of Y <syntaxhighlight lang=apl inline>{⍵[X]⌷Y}¨</syntaxhighlight>. | ||
< | <syntaxhighlight lang=apl> | ||
1 1⍉3 4⍴⎕A ⍝ Extract diagonal from 3×4 matrix | 1 1⍉3 4⍴⎕A ⍝ Extract diagonal from 3×4 matrix | ||
AEI | AEI | ||
| Line 55: | Line 55: | ||
</syntaxhighlight>{{Works in|[[Dyalog APL]]}} | </syntaxhighlight>{{Works in|[[Dyalog APL]]}} | ||
== Issues == | == Issues == | ||
A common mistake in employing dyadic transpose is the "intuitive" interpretation of the left argument as if gives the order in which you want to select dimensions of the right argument for the result. In fact, it gives the new position of each of the dimensions. It is possible to convert between these two representations by "inverting" the permutation with monadic [[Grade|Grade Up]] (< | A common mistake in employing dyadic transpose is the "intuitive" interpretation of the left argument as if gives the order in which you want to select dimensions of the right argument for the result. In fact, it gives the new position of each of the dimensions. It is possible to convert between these two representations by "inverting" the permutation with monadic [[Grade|Grade Up]] (<syntaxhighlight lang=apl inline>⍋</syntaxhighlight>). | ||
The reason for the design of < | The reason for the design of <syntaxhighlight lang=apl inline>⍉</syntaxhighlight> being as it is, is that it allows you to select diagonals by giving one or more dimensions equal mapping, whereas simply selecting dimensions from the right would not allow that. It is therefore the more complete of the two options.<ref>[[Morten Kromberg]]. Message {{M|57439754}}ff. [[APL Orchard]]. 25 Mar 2021.</ref> | ||
== External links == | == External links == | ||
Revision as of 10:34, 11 September 2022
⍉
|
Transpose (⍉) is an ambivalent primitive function which reorders the axes of the right argument array. The name Transpose comes from the fact that monadic application to a matrix yields its transpose. The dyadic usage is sometimes called Rearrange Axes, which better reflects the behavior of the function.
Examples
Monadic usage
Monadic Transpose reverses the axes of the right argument. When applied to a matrix, the result is its transpose. Scalar and vector arguments are unaffected.
3 3⍴⍳9
1 2 3
4 5 6
7 8 9
⍉3 3⍴⍳9 ⍝ Transpose of a matrix
1 4 7
2 5 8
3 6 9
⍴⍉3 4 5⍴⎕A ⍝ The first axis goes last, the last axis comes first
5 4 3
⍉1 2 3 ⍝ Unaffected
1 2 3
Dyadic usage
|
For dyadic usage, the left argument X must be a vector whose length equals the rank of the right argument Y, and the elements must form a range so that ∧/X∊⍳(1-⎕IO)+⌈/X is satisfied.
If all values in X are unique (X forms a permutation over the axes of Y), the axes are reordered by X so that N-th element of X specifies the new position for the N-th axis of Y. This means that, given a multi-dimensional index V of an element in the resulting array, V⌷X⍉Y corresponds to V[X]⍉Y.
X←3 1 2
Y←3 4 5⍴⎕A
⍴X⍉Y ⍝ First axis goes to third (X[1] is 3), second goes to first (X[2] is 1), etc.
4 5 3
1 2 3⌷X⍉Y ⍝ or (X⍉Y)[1;2;3]
P
1 2 3[X]⌷Y ⍝ or Y[3;1;2]
P
When X contains duplicates, the result has rank (1-⎕IO)+⌈/X. For the axes of Y that map to the same resulting axis, only the elements where the indices are equal over those axes are collected. This has the effect of extracting diagonal elements. If the axes are of unequal length, the resulting axis has the length of the shortest of them. This operation can be modeled as computing the resulting shape (⍴Y)⌊.+(⌊/⍬)×X∘.≠(1-⎕IO)+⍳⌈/X, then creating the array of its multi-dimensional indices ⍳, then modify each index and fetch the corresponding elements of Y {⍵[X]⌷Y}¨.
1 1⍉3 4⍴⎕A ⍝ Extract diagonal from 3×4 matrix
AEI
Y←?3 4 5 6 7⍴100
X←3 2 3 1 2 ⍝ Left arg maps 5-dimensional Y to 3 dimensions
⍴X⍉Y ⍝ Resulting shape is ⌊/¨6(4 7)(3 5)
6 4 3
(X⍉Y)≡{⍵[X]⌷Y}¨⍳(⍴Y)⌊.+(⌊/⍬)×X∘.≠(1-⎕IO)+⍳⌈/X
1Issues
A common mistake in employing dyadic transpose is the "intuitive" interpretation of the left argument as if gives the order in which you want to select dimensions of the right argument for the result. In fact, it gives the new position of each of the dimensions. It is possible to convert between these two representations by "inverting" the permutation with monadic Grade Up (⍋).
The reason for the design of ⍉ being as it is, is that it allows you to select diagonals by giving one or more dimensions equal mapping, whereas simply selecting dimensions from the right would not allow that. It is therefore the more complete of the two options.[2]
External links
Tutorials
Documentation
References
- ↑ Roger Hui. dyadic transpose, a personal history. Dyalog Forums. 22 May 2020.
- ↑ Morten Kromberg. Message 57439754ff. APL Orchard. 25 Mar 2021.