Scalar extension: Difference between revisions
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== History and terminology == | == History and terminology == | ||
The concept of scalar extension has been around since [[APL\360]]. An example which extends the scalar < | The concept of scalar extension has been around since [[APL\360]]. An example which extends the scalar <syntaxhighlight lang=apl inline>2</source> is: | ||
< | <syntaxhighlight lang=apl> | ||
2 × 1 2 3 4 | 2 × 1 2 3 4 | ||
2 4 6 8 | 2 4 6 8 | ||
| Line 12: | Line 12: | ||
The word "extension" applies to scalar extension in two ways: first, a function is extended by making a case which would have been a [[RANK ERROR]] into a valid application. Second, the application works by conceptually extending the scalar to function as though it were an array of higher rank. | The word "extension" applies to scalar extension in two ways: first, a function is extended by making a case which would have been a [[RANK ERROR]] into a valid application. Second, the application works by conceptually extending the scalar to function as though it were an array of higher rank. | ||
Two arrays are said to [[Conformability|conform]] if they have the same shape or at least one can be extended (it is a scalar, or, in langauges with singleton extension, has exactly one element). A pair of conforming arrays defines a single shape which describes how their elements are paired: if neither is a scalar, it is their shared shape; if one is a scalar, it is the other's shape; if both are scalars, it is the [[empty]] vector, < | Two arrays are said to [[Conformability|conform]] if they have the same shape or at least one can be extended (it is a scalar, or, in langauges with singleton extension, has exactly one element). A pair of conforming arrays defines a single shape which describes how their elements are paired: if neither is a scalar, it is their shared shape; if one is a scalar, it is the other's shape; if both are scalars, it is the [[empty]] vector, <syntaxhighlight lang=apl inline>⍬</source> ([[Zilde]]). | ||
=== Rank extension === | === Rank extension === | ||
The term "scalar extension" is sometimes used to refer to the practice of allowing a scalar when a higher rank is expected. The scalar is treated as an array of the expected minimum rank whose shape is a vector of 1s (that is, a [[singleton]]). For example, < | The term "scalar extension" is sometimes used to refer to the practice of allowing a scalar when a higher rank is expected. The scalar is treated as an array of the expected minimum rank whose shape is a vector of 1s (that is, a [[singleton]]). For example, <syntaxhighlight lang=apl inline>⍳8</source> and <syntaxhighlight lang=apl inline>8⍴'a'</source> both produce an array of shape <syntaxhighlight lang=apl inline>,8</source> (a vector) even though they were given a shape specification of <syntaxhighlight lang=apl inline>8</source> (a scalar). This type of extension, which differs from ordinary scalar extension in that there is no expected shape and only an expected rank, has also been present since [[APL\360]]. | ||
=== Singleton extension === | === Singleton extension === | ||
Some APLs, such as [[Dyalog APL]], treat arrays with one element ([[singleton]]s) as scalars for the purposes of scalar extension. This practice is referred to as "singleton extension". For example, | Some APLs, such as [[Dyalog APL]], treat arrays with one element ([[singleton]]s) as scalars for the purposes of scalar extension. This practice is referred to as "singleton extension". For example, | ||
< | <syntaxhighlight lang=apl> | ||
(1 1⍴5) + 10 20 | (1 1⍴5) + 10 20 | ||
15 25 | 15 25 | ||
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Dyadic [[scalar functions]] and the [[Each]] operator use scalar extension to pair their arguments: | Dyadic [[scalar functions]] and the [[Each]] operator use scalar extension to pair their arguments: | ||
< | <syntaxhighlight lang=apl> | ||
1 2 3 4 * 2 | 1 2 3 4 * 2 | ||
1 4 9 16 | 1 4 9 16 | ||
| Line 49: | Line 49: | ||
[[Replicate]] and [[Partitioned Enclose]] (and also [[Partition]], but this is less useful, as it is equivalent to < | [[Replicate]] and [[Partitioned Enclose]] (and also [[Partition]], but this is less useful, as it is equivalent to <syntaxhighlight lang=apl inline>,⊂</source> or <syntaxhighlight lang=apl inline>{0⍴⊂0⍴⍵}</source>) extend a scalar left argument to apply to each column of the right argument: | ||
< | <syntaxhighlight lang=apl> | ||
2/'abc' | 2/'abc' | ||
aabbcc | aabbcc | ||
| Line 60: | Line 60: | ||
[[Decode]] uses extends a single radix left argument to apply to all digit values in the right argument: | [[Decode]] uses extends a single radix left argument to apply to all digit values in the right argument: | ||
< | <syntaxhighlight lang=apl> | ||
2⊥1 0 1 | 2⊥1 0 1 | ||
5 | 5 | ||
| Line 66: | Line 66: | ||
[[APL2]] and [[Dyalog APL]] use a variant of singleton extension when the selected axis of the right argument has length one: each element along that axis is reused for every element of the left argument. | [[APL2]] and [[Dyalog APL]] use a variant of singleton extension when the selected axis of the right argument has length one: each element along that axis is reused for every element of the left argument. | ||
< | <syntaxhighlight lang=apl> | ||
⍴ 2 ¯3 /[2] 7 1 8⍴⍳56 | ⍴ 2 ¯3 /[2] 7 1 8⍴⍳56 | ||
7 5 8 | 7 5 8 | ||
| Line 74: | Line 74: | ||
In languages which allow a vector left argument to [[Rotate]], the behavior with a scalar left argument follows from scalar extension. In the following example a length-2 vector could be used to rotate each row by a different amount. A scalar rotates both rows by the same amount. | In languages which allow a vector left argument to [[Rotate]], the behavior with a scalar left argument follows from scalar extension. In the following example a length-2 vector could be used to rotate each row by a different amount. A scalar rotates both rows by the same amount. | ||
< | <syntaxhighlight lang=apl> | ||
3⌽2 6⍴'extendscalar' | 3⌽2 6⍴'extendscalar' | ||
endext | endext | ||
| Line 83: | Line 83: | ||
A common problem is pairing up each [[element]] of one [[argument]] with all the elements of another argument. Because of scalar extension, [[enclose|making one argument a scalar]] accomplishes this: | A common problem is pairing up each [[element]] of one [[argument]] with all the elements of another argument. Because of scalar extension, [[enclose|making one argument a scalar]] accomplishes this: | ||
< | <syntaxhighlight lang=apl> | ||
1 2 3,¨⊂100 200 | 1 2 3,¨⊂100 200 | ||
┌─────────┬─────────┬─────────┐ | ┌─────────┬─────────┬─────────┐ | ||
| Line 94: | Line 94: | ||
</source> | </source> | ||
For [[scalar function]]s, explicit use of the [[Each]] operator is unnecessary: | For [[scalar function]]s, explicit use of the [[Each]] operator is unnecessary: | ||
< | <syntaxhighlight lang=apl> | ||
1 2 3+⊂100 200 | 1 2 3+⊂100 200 | ||
┌───────┬───────┬───────┐ | ┌───────┬───────┬───────┐ | ||
| Line 105: | Line 105: | ||
</source> | </source> | ||
[[Bind]]ing one argument to the function also works, but this always requires the Each operator: | [[Bind]]ing one argument to the function also works, but this always requires the Each operator: | ||
< | <syntaxhighlight lang=apl> | ||
1 2 3∘,¨100 200 | 1 2 3∘,¨100 200 | ||
┌─────────┬─────────┐ | ┌─────────┬─────────┐ | ||