Take: Difference between revisions
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The left argument to length specifies a length, and not an index. It does not depend on [[index origin]]. | The left argument to length specifies a length, and not an index. It does not depend on [[index origin]]. | ||
A length which is larger than the argument length causes [[Fill element|fills]] to be inserted. The alignment remains the same: if two different positive arguments are used to take from an array, the one which is closer to zero gives a [[prefix]] of the other result. If they are both negative, it is a [[suffix]] instead | A length which is larger than the argument length causes [[Fill element|fills]] to be inserted. The alignment remains the same: if two different positive arguments are used to take from an array, the one which is closer to zero gives a [[prefix]] of the other result. If they are both negative, it is a [[suffix]] instead. | ||
<source lang=apl> | <source lang=apl> | ||
8 ↑ 5 4 3 2 1 | 8 ↑ 5 4 3 2 1 | ||
| Line 19: | Line 19: | ||
0 0 0 5 4 3 2 1 | 0 0 0 5 4 3 2 1 | ||
</source> | </source> | ||
{{Works in|[[Dyalog APL]], [[ngn/ | {{Works in|[[Dyalog APL]], [[ngn/APL]]}} | ||
A higher-[[rank]] array can be shortened by using a left argument with one element for each axis: | A higher-[[rank]] array can be shortened by using a left argument with one element for each axis: | ||
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└───┴───┴───┘ | └───┴───┴───┘ | ||
</source> | </source> | ||
{{Works in|[[Dyalog APL]], [[dzaima/APL]], [[ngn/ | {{Works in|[[Dyalog APL]], [[dzaima/APL]], [[ngn/APL]]}} | ||
In languages with the SHARP APL extension, the left argument can be shortened. This causes leading axes of the right argument to be modified while trailing axes are ignored. | In languages with the SHARP APL extension, the left argument can be shortened. This causes leading axes of the right argument to be modified while trailing axes are ignored. | ||
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└───┴───┴───┴───┴───┘ | └───┴───┴───┴───┴───┘ | ||
</source> | </source> | ||
{{Works in|[[Dyalog APL]], [[ngn/ | {{Works in|[[Dyalog APL]], [[ngn/APL]]}} | ||
An [[Function axis|axis]] may be specified to apply left argument elements to specific axes of the right argument. Here the last axis is specified in order to take two columns of the argument. | An [[Function axis|axis]] may be specified to apply left argument elements to specific axes of the right argument. Here the last axis is specified in order to take two columns of the argument. | ||
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When Take is called with [[Function axis|axis]], the axis determines how elements of the left argument correspond to axes of the right argument. The left argument and axis are required to have rank no more than 1 and are treated as vectors. Their lengths must match, and be less than or equal to the rank of the right argument. Then each element of the left argument applies to the right argument axis given by the corresponding element of the axis vector. Each axis may only be specified once, and unspecified axes are left unchanged. | When Take is called with [[Function axis|axis]], the axis determines how elements of the left argument correspond to axes of the right argument. The left argument and axis are required to have rank no more than 1 and are treated as vectors. Their lengths must match, and be less than or equal to the rank of the right argument. Then each element of the left argument applies to the right argument axis given by the corresponding element of the axis vector. Each axis may only be specified once, and unspecified axes are left unchanged. | ||
== History == | == History == | ||
Take was present in [[APL\360]], but not in [[A Programming Language]], where the symbol (<code>↑</code>) was used for [[Rotate]].<ref>Iverson, Kenneth E. (1962). A Programming Language. Wiley. ISBN 978-0-471-43014-8.</ref> | |||
The [[Function axis|axis]] specification for Take was defined in [[APL2]]. It is shared by [[SHARP APL]] and [[Rationalized APL]], and continues to be supported in [[Dyalog APL]]. | The [[Function axis|axis]] specification for Take was defined in [[APL2]]. It is shared by [[SHARP APL]] and [[Rationalized APL]], and continues to be supported in [[Dyalog APL]]. | ||
[[SHARP APL 19.0]], released in 1987, extended Take to allow short left arguments. The choice to align left argument elements with the leading axes of the right argument was made according to the nascent [[leading axis theory]]: while a user may not have any preference for manipulating the earlier dimensions, this choice makes Take more flexible when used with the [[Rank operator]].<ref>Bernecky, Robert. [https://dl.acm.org/citation.cfm?id=55632 "An Introduction to Function Rank"]. APL88 Conference Proceedings. ''ACM SIGAPL Quote Quad'', 18(2), December 1987.</ref> This extension is also used in [[Dyalog APL]], [[J]], and [[ngn/ | [[SHARP APL 19.0]], released in 1987, extended Take to allow short left arguments. The choice to align left argument elements with the leading axes of the right argument was made according to the nascent [[leading axis theory]]: while a user may not have any preference for manipulating the earlier dimensions, this choice makes Take more flexible when used with the [[Rank operator]].<ref>Bernecky, Robert. [https://dl.acm.org/citation.cfm?id=55632 "An Introduction to Function Rank"]. APL88 Conference Proceedings. ''ACM SIGAPL Quote Quad'', 18(2), December 1987.</ref> This extension is also used in [[Dyalog APL]], [[J]], and [[ngn/APL]]. It was not adopted in [[ISO/IEC 13751:2001]]. | ||
== References == | == References == | ||
<references /> | <references /> | ||
Revision as of 07:36, 23 October 2019
Take (↑) is a primitive function which shortens or extends an array along one or more axes. The vector left argument indicates the lengths of result axes, with a sign to denote whether elements should be taken starting from the beginning or end of each axis. Take was introduced in APL\360 with the requirement that the left argument length match the right argument rank, and was extended in SHARP APL 19.0 to allow short left arguments. It is closely related to Drop, which removes the parts of each axis that Take would include.
Examples
Take may be used to get the first few, or last few, elements of a vector:
3 ↑ 5 4 3 2 1
5 4 3
¯3 ↑ 5 4 3 2 1
3 2 1
The left argument to length specifies a length, and not an index. It does not depend on index origin.
A length which is larger than the argument length causes fills to be inserted. The alignment remains the same: if two different positive arguments are used to take from an array, the one which is closer to zero gives a prefix of the other result. If they are both negative, it is a suffix instead.
8 ↑ 5 4 3 2 1
5 4 3 2 1 0 0 0
¯8 ↑ 5 4 3 2 1
0 0 0 5 4 3 2 1
A higher-rank array can be shortened by using a left argument with one element for each axis:
¯2 3↑⍳4 5 ┌───┬───┬───┐ │3 1│3 2│3 3│ ├───┼───┼───┤ │4 1│4 2│4 3│ └───┴───┴───┘
In languages with the SHARP APL extension, the left argument can be shortened. This causes leading axes of the right argument to be modified while trailing axes are ignored.
¯2↑⍳4 5 ┌───┬───┬───┬───┬───┐ │3 1│3 2│3 3│3 4│3 5│ ├───┼───┼───┼───┼───┤ │4 1│4 2│4 3│4 4│4 5│ └───┴───┴───┴───┴───┘
An axis may be specified to apply left argument elements to specific axes of the right argument. Here the last axis is specified in order to take two columns of the argument.
¯2↑[2]⍳4 5 ┌───┬───┐ │1 4│1 5│ ├───┼───┤ │2 4│2 5│ ├───┼───┤ │3 4│3 5│ ├───┼───┤ │4 4│4 5│ └───┴───┘
If the Rank operator is available then ¯2↑⍤1⍳4 5 is an equivalent expression.
Description
In the expression X↑Y, X may be any array, and Y is a simple numeric vector whose length is less than or equal to the rank of Y. Many APLs require the length to be exactly equal; however, an extension by SHARP APL to allow a shorter left argument has been widely adopted by recent APLs. X may also be a scalar, in which case it is treated as a one-element vector in an instance of scalar rank extension. In some APLs, Y is also subject to scalar rank extension: if it is scalar then it will be extended so its rank is the length ≢X.
Elements of X are matched with axes of Y with the same index, that is, the left argument corresponds to leading axes of the right. The trailing axes of Y which are not matched in this way are unchanged by Take; this may also be modelled by extending X using the lengths of those axes.
For each modified axis the result length along that axis is equal to the corresponding element of |X. If the original element in X is positive then the result is aligned with the argument at the beginning of that axis, and if that element is negative they are aligned at the end. If it is zero then the result is empty, so both are true. Elements from the right argument are used in the result until the result is filled. If the argument axis is shorter than the result axis then fills are used once it is exhausted. If the result is empty, its prototype is the same as the right argument's.
If the result is no larger than the right argument along each axis (equivalently, no fills are used, or (|X)≤(≢X)↑⍴Y), then the result is a subarray of Y. Not all subarrays can be produced in this way: only those which align with one boundary of the argument along each axis.
Axis specification
When Take is called with axis, the axis determines how elements of the left argument correspond to axes of the right argument. The left argument and axis are required to have rank no more than 1 and are treated as vectors. Their lengths must match, and be less than or equal to the rank of the right argument. Then each element of the left argument applies to the right argument axis given by the corresponding element of the axis vector. Each axis may only be specified once, and unspecified axes are left unchanged.
History
Take was present in APL\360, but not in A Programming Language, where the symbol (↑) was used for Rotate.[1]
The axis specification for Take was defined in APL2. It is shared by SHARP APL and Rationalized APL, and continues to be supported in Dyalog APL.
SHARP APL 19.0, released in 1987, extended Take to allow short left arguments. The choice to align left argument elements with the leading axes of the right argument was made according to the nascent leading axis theory: while a user may not have any preference for manipulating the earlier dimensions, this choice makes Take more flexible when used with the Rank operator.[2] This extension is also used in Dyalog APL, J, and ngn/APL. It was not adopted in ISO/IEC 13751:2001.
References
- ↑ Iverson, Kenneth E. (1962). A Programming Language. Wiley. ISBN 978-0-471-43014-8.
- ↑ Bernecky, Robert. "An Introduction to Function Rank". APL88 Conference Proceedings. ACM SIGAPL Quote Quad, 18(2), December 1987.