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→Notable uses: Point "identity matrix" to Wikipedia
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Reshape | {{Built-in|Reshape|⍴}} produces an array with [[shape]] given by the left argument and [[elements]] from the right argument. Elements are copied from the right argument to the result in [[ravel order]], truncating if the result has smaller [[bound]] than the right argument and repeating cyclically if it has larger bound. If the right argument is empty, [[Fill element|fills]] are used for the result elements. | ||
== Examples == | == Examples == | ||
Reshape can be used to produce an array with a given shape and ravel: | Reshape can be used to produce an array with a given shape and ravel: | ||
<source | <source lang=apl> | ||
3 4 ⍴ ⍳12 | 3 4 ⍴ ⍳12 | ||
1 2 3 4 | 1 2 3 4 | ||
| Line 15: | Line 11: | ||
</source> | </source> | ||
It appears to exhibit a form of [[Scalar extension|scalar]] or singleton extension: | It appears to exhibit a form of [[Scalar extension|scalar]] or singleton extension: | ||
<source | <source lang=apl> | ||
3 4 ⍴ 12 | 3 4 ⍴ 12 | ||
12 12 12 12 | 12 12 12 12 | ||
| Line 23: | Line 19: | ||
In fact it repeats an argument of any length, singleton or otherwise. This repetition applies with a vector result, or a higher rank. | In fact it repeats an argument of any length, singleton or otherwise. This repetition applies with a vector result, or a higher rank. | ||
<source | <source lang=apl> | ||
12 ⍴ 'abcde' | 12 ⍴ 'abcde' | ||
abcdeabcdeab | abcdeabcdeab | ||
| Line 31: | Line 27: | ||
deab | deab | ||
</source> | </source> | ||
Reshape can also decrease the rank or [[bound]] of an array. One notable example is the use of an empty left argument [[Zilde]] (<source lang=apl inline>⍬</source>) to produce a [[scalar]]. The scalar is the first 0-[[cell]] of the right argument. In [[Nested array model|nested]] languages <source lang=apl inline>⍬⍴</source> is like [[First]] except that it does not remove a layer of nesting. | |||
<source lang=apl> | |||
9 ⍴ ∘.+⍨ 1 2 1 | |||
2 3 2 3 4 3 2 3 2 | |||
3 ⍴ 'Samantha' | |||
Sam | |||
⍬ ⍴ ⍳8 8 | |||
┌───┐ | |||
│1 1│ | |||
└───┘ | |||
</source> | |||
== Description == | |||
The left argument of Reshape must be a valid [[shape]], or [[vector]] of nonnegative integers, after [[scalar rank extension]] (that is, a scalar is treated as a one-element vector). The right argument may be any array. The result array is an array of the given shape, and its elements in [[ravel order]] are taken from the right argument in ravel order. If the right argument's ravel is too short, they are repeated starting at the beginning again as many times as necessary. | |||
An [[Empty array|empty]] right argument will cause the result array to be composed of [[Fill element|fill elements]]. Reshape is similar to [[Take]] in this case—in fact, Take with an empty right argument is always identical to Reshape unless it results in an error. | |||
The ravelled result is either a [[prefix]] of the ravelled argument, or a [[suffix]]. | |||
== APL model == | == APL model == | ||
Since Reshape itself is the fundamental way to create a multi-dimensional array in APL, the function as a whole cannot be modelled in terms of more fundamental primitives. However, we may express it in terms of a stricter reshaping function < | Since Reshape itself is the fundamental way to create a multi-dimensional array in APL, the function as a whole cannot be modelled in terms of more fundamental primitives. However, we may express it in terms of a stricter reshaping function <source lang=apl inline>shape</source>, which forms an array from its [[shape]] and [[ravel]] vectors, requiring both to have rank 1 and the number of elements in the ravel to be the product of the shape. <source lang=apl inline>shape</source> is identical to Reshape on its domain, but it has a strictly smaller domain than Reshape. The extensions required to implement Reshape are that a scalar left argument must be allowed, and that the right argument must be converted to a vector with the appropriate length, truncating or repeating its elements. | ||
<source | <source lang=apl> | ||
Reshape ← { | Reshape ← { | ||
(1/⍺) shape (×/⍺) {(0=≢⍵)∨⍺≤≢⍵:⍺↑⍵ ⋄ ⍺∇,⍨⍵} ,⍵ | (1/⍺) shape (×/⍺) {(0=≢⍵)∨⍺≤≢⍵:⍺↑⍵ ⋄ ⍺∇,⍨⍵} ,⍵ | ||
| Line 42: | Line 58: | ||
{{Works in|[[Dyalog APL]],[[ngn/apl]]}} | {{Works in|[[Dyalog APL]],[[ngn/apl]]}} | ||
The above implementation performs truncation and fill element generation using [[Take]], after extending the [[Ravel|ravelled]] right argument by [[Catenate|catenating]] it with itself until it is long enough. An implementation using indices instead of structural manipulation is also possible: | The above implementation performs truncation and fill element generation using [[Take]], after extending the [[Ravel|ravelled]] right argument by [[Catenate|catenating]] it with itself until it is long enough. An implementation using indices instead of structural manipulation is also possible: | ||
<source | <source lang=apl> | ||
Reshape ← { | Reshape ← { | ||
⎕IO←0 | ⎕IO←0 | ||
| Line 49: | Line 65: | ||
</source> | </source> | ||
{{Works in|[[Dyalog APL]]}} | {{Works in|[[Dyalog APL]]}} | ||
Here the right argument is converted to a ravel vector by ravelling and appending the [[prototype]], then [[indexing]] to produce a vector of the correct length. The indices used are the ravel indices of the result, but they are made to wrap around using [[Residue]]. | Here the right argument is converted to a ravel vector by ravelling and appending the [[prototype]], then [[Bracket indexing|indexing]] to produce a vector of the correct length. The indices used are the ravel indices of the result, but they are made to wrap around using [[Residue]]. | ||
== J variant: Shape == | |||
The [[J]] language does not include a Reshape primitive. In J, the monadic [[Shape]] function is called "Shape Of" and uses the glyph <source lang=apl inline>$</source>. Its dyadic form, simply called "Shape", rearranges the [[Major cell|major cells]] of the right argument rather than its [[elements]]. The result shape is given by the left argument, followed by the shape of the right argument with the first [[axis]] length (if any) removed. In APL the J Shape function can be written <source lang=apl inline>{ (⍺,1↓⍴⍵)⍴⍵ }</source>, and in J the APL Reshape function can be written using the [[hook]] <source lang=apl inline>($,)</source> which first ravels the right argument so that its major cells are its elements. | |||
== Notable uses == | |||
Reshape can be used to produce an [[wikipedia:identity matrix|identity matrix]] by reshaping a vector which is one longer than the desired side length. | |||
<source lang=apl> | |||
4 4 ⍴ 5↑1 | |||
1 0 0 0 | |||
0 1 0 0 | |||
0 0 1 0 | |||
0 0 0 1 | |||
</source> | |||
This idea might be written in a [[tacit]] style as <source lang=apl inline>,⍨⍴1↑⍨1∘+</source> or <source lang=apl inline>,⍨⍴1,⍴∘0</source>. Both functions take the side length as an argument and produce an identity matrix with that side length. | |||
== External links == | |||
== | === Lessons === | ||
[ | * [https://chat.stackexchange.com/rooms/52405/conversation/lesson-10-apl-functions-- APL Cultivation] | ||
* [https://www.sacrideo.us/apl-a-day-4-arrays-have-elements/ Arrays have elements] (part of [https://www.sacrideo.us/tag/apl-a-day/ APL a Day]) | |||
=== Documentation === | |||
= | * [http://help.dyalog.com/latest/index.htm#Language/Primitive%20Functions/Reshape.htm Dyalog] | ||
* [http://wiki.nars2000.org/index.php/Rho NARS2000] | |||
* [http://microapl.com/apl_help/ch_020_020_470.htm APLX] | |||
* [https://code.jsoftware.com/wiki/Vocabulary/dollar#dyadic J Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/dollar#dyadic J NuVoc] (as <source lang=apl inline>$</source> "Shape") | |||
[ | {{APL built-ins}}[[Category:Primitive functions]] | ||