Difference between revisions of "Take"
(→History: NARS2000 adopted the SHARP extension between 2017 and 2019) 
(→History: Early extensions reported in Quote Quad) 

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In [[A Programming Language]], [[prefix]] and [[suffix]] operations were described using the syntax <math>\alpha^j/x</math> to take the first <math>j</math> elements of vector <math>x</math> and <math>\omega^j/x</math> for the last <math>j</math> elements. This combined a use of the special prefix and suffix vectors <math>\alpha^j(n)</math> and <math>\omega^j(n)</math> with [[Compresscompression]], with the length <math>n</math> inferred based on the length of <math>x</math>. The symbol <math>\uparrow</math> was used for vector [[Rotate]], while <math>\downarrow</math> rotated in the opposite direction.<ref>[[Ken IversonIverson, K.E.]] (1962). A Programming Language. Wiley. ISBN 9780471430148.</ref>  In [[A Programming Language]], [[prefix]] and [[suffix]] operations were described using the syntax <math>\alpha^j/x</math> to take the first <math>j</math> elements of vector <math>x</math> and <math>\omega^j/x</math> for the last <math>j</math> elements. This combined a use of the special prefix and suffix vectors <math>\alpha^j(n)</math> and <math>\omega^j(n)</math> with [[Compresscompression]], with the length <math>n</math> inferred based on the length of <math>x</math>. The symbol <math>\uparrow</math> was used for vector [[Rotate]], while <math>\downarrow</math> rotated in the opposite direction.<ref>[[Ken IversonIverson, K.E.]] (1962). A Programming Language. Wiley. ISBN 9780471430148.</ref>  
−  The first version of [[APL\360]]<ref>[[Adin FalkoffFalkoff, A.D.]], and [[Ken IversonK.E. Iverson]]. [https://www.jsoftware.com/papers/APL360TerminalSystem.htm "The APL\360 Terminal System"]. Research Report RC1922, IBM, 19671016.</ref> followed Iverson notation in defining prefix (<source lang=apl inline>n ⍺ j</source>) and suffix (<source lang=apl inline>n ⍵ j</source>) vectors but not Take, even though the arrow symbols were no longer used for Rotate. For a vector <source lang=apl inline>x</source>, what is now <source lang=apl inline>j↑x</source> would have been written <source lang=apl inline>((⍴x)⍺j)/x</source> while <source lang=apl inline>(j)↑x</source> would be <source lang=apl inline>((⍴x)⍵j)/x</source>. The functions Take and Drop using arrow symbols were introduced by 1968<ref>[[Adin FalkoffFalkoff, A.D.]], and [[Ken IversonK.E. Iverson]], "[http://keiapl.org/archive/APL360_UsersMan_Aug1968.pdf APL\360 User's Manual]". [[IBM]], August 1968.</ref>.  +  The first version of [[APL\360]]<ref>[[Adin FalkoffFalkoff, A.D.]], and [[Ken IversonK.E. Iverson]]. [https://www.jsoftware.com/papers/APL360TerminalSystem.htm "The APL\360 Terminal System"]. Research Report RC1922, IBM, 19671016.</ref> followed Iverson notation in defining prefix (<source lang=apl inline>n ⍺ j</source>) and suffix (<source lang=apl inline>n ⍵ j</source>) vectors but not Take, even though the arrow symbols were no longer used for Rotate. For a vector <source lang=apl inline>x</source>, what is now <source lang=apl inline>j↑x</source> would have been written <source lang=apl inline>((⍴x)⍺j)/x</source> while <source lang=apl inline>(j)↑x</source> would be <source lang=apl inline>((⍴x)⍵j)/x</source>. The functions Take and Drop using arrow symbols were introduced by 1968.<ref>[[Adin FalkoffFalkoff, A.D.]], and [[Ken IversonK.E. Iverson]], "[http://keiapl.org/archive/APL360_UsersMan_Aug1968.pdf APL\360 User's Manual]". [[IBM]], August 1968.</ref> They were implemented for arrays (not just vectors), and extended to allow the left argument to be larger than the right argument's shape (introducing overtaking and [[fill element]]s) in 1970.<ref>"[[IBMI.B.M.]] Report". [[APL QuoteQuad]] Volume 2, Number 1. 197004.</ref> 
The [[Function axisaxis]] 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 axisaxis]] specification for Take was defined in [[APL2]]. It is shared by [[SHARP APL]] and [[Rationalized APL]], and continues to be supported in [[Dyalog APL]]. 
Revision as of 15:38, 23 March 2020
↑

Take (↑
) is a primitive dyadic function which shortens or extends the right argument 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.
Contents
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. When Take makes an axis longer, it is said to "overtake" along that axis.
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 higherrank 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 oneelement 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 have the same rank and 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.
Infinite left arguments
In J 4.02 (released 1998) and later, an infinite value in the left argument causes the corresponding axis of the right argument to be unchanged. In ngn/apl, it simply causes an error.
APLX fills
In APLX, the fill elements used in Take are not always the argument array's prototype: new elements in rows (that is, 1cells) that already exist are filled based on the first element of that row while new rows use the prototype of the whole array.
APL model
The following dfn models Take as defined by Dyalog APL but with no axis specification or error checking. It is implemented by construction a nested array of indices and using these to select from the right argument, with prototypes used for outofrange indices. It explicitly includes scalar rank extension for the right argument and the SHARP APL extension; if these extensions are not wanted those lines can be removed. Scalar rank extension of the left argument is inherited from Iota and scalar function extension.
Take ← {
⎕IO←0 ⍝ For index comparisons
r s ← ≢¨(⍴⍵)(⍺) ⍝ Rank and number of modified axes
(r=0)∧s>0: ⍺∇(s⍴1)⍴⍵ ⍝ Right argument scalar rank extension
s<r: (⍺,(sr)∇⍴⍵)∇⍵ ⍝ SHARP APL extension
inds ← ((⍺<0)×⍺+⍴⍵)∘+¨ ⍳⍺ ⍝ Indices to select
sel ← {
∧/(0≤⍺)∧⍺<⍴⍵: (⊂⍺)⊃⍵ ⍝ In range: use Pick
⊃0⍴⍵ ⍝ Otherwise, get prototype
}
sel∘⍵¨ inds
}
This definition could be converted to work in a flat APL with the Rank operator by using an odometer function like ⊢⊤(⍳×/)
in place of Iota and changing the two subsequent uses of Each to Rank 1.
History
In A Programming Language, prefix and suffix operations were described using the syntax to take the first elements of vector and for the last elements. This combined a use of the special prefix and suffix vectors and with compression, with the length inferred based on the length of . The symbol was used for vector Rotate, while rotated in the opposite direction.^{[1]}
The first version of APL\360^{[2]} followed Iverson notation in defining prefix (n ⍺ j
) and suffix (n ⍵ j
) vectors but not Take, even though the arrow symbols were no longer used for Rotate. For a vector x
, what is now j↑x
would have been written ((⍴x)⍺j)/x
while (j)↑x
would be ((⍴x)⍵j)/x
. The functions Take and Drop using arrow symbols were introduced by 1968.^{[3]} They were implemented for arrays (not just vectors), and extended to allow the left argument to be larger than the right argument's shape (introducing overtaking and fill elements) in 1970.^{[4]}
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.^{[5]} It has been present in J and ngn/apl since early development, and was adopted in Dyalog APL 13.0 (2011), as well as NARS2000 between 2017 and 2019^{[6]}.
Extension support
In the table below, "Scalar right arg" indicates scalar rank extension of the right argument, and "Short left arg" is the SHARP APL 19.0 extension.
Languages  Scalar right arg  Short left arg  Axis specification 

APL\360  No  No  No 
APL2, APLX, GNU APL  Yes  No  Yes 
SHARP APL, Dyalog APL, NARS2000  Yes  Yes  Yes 
A+, ngn/apl, J ({. ) 
Yes  Yes  No 
dzaima/APL  No  Yes  No 
Additionally, A+ requires the left argument to be a singleton, and dzaima/APL does not allow overtaking (so that the result of Take cannot contain fills).
External links
Lessons
Documentation
References
 ↑ Iverson, K.E. (1962). A Programming Language. Wiley. ISBN 9780471430148.
 ↑ Falkoff, A.D., and K.E. Iverson. "The APL\360 Terminal System". Research Report RC1922, IBM, 19671016.
 ↑ Falkoff, A.D., and K.E. Iverson, "APL\360 User's Manual". IBM, August 1968.
 ↑ "I.B.M. Report". APL QuoteQuad Volume 2, Number 1. 197004.
 ↑ Bernecky, Robert. "An Introduction to Function Rank". APL88 Conference Proceedings. ACM SIGAPL Quote Quad, 18(2), December 1987.
 ↑ Smith, Bob. "Progress in NARS2000 October 2017 to September 2019". Presented at Minnowbrook 2019 and published independently.
APL builtins [edit]  

Primitive functions  
Scalar  
Monadic  Conjugate ∙ Negate ∙ Signum ∙ Not ∙ Roll ∙ Type  
Dyadic  Add ∙ Subtract ∙ Equal to (Xnor) ∙ Not Equal to (Xor) ∙ Minimum ∙ Maximum ∙ Comparison functions  
NonScalar  
Structural  Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Reverse ∙ Raze ∙ Mix ∙ Cut (K) ∙ Pair ∙ Replicate ∙ Partitioned Enclose  
Selection  Take ∙ Drop ∙ Unique ∙ Identity ∙ Select  
Selector  Interval Index ∙ Indices  
Computational  Match ∙ Not Match ∙ Nub Sieve ∙ Format ∙ Execute  
Primitive operators  Monadic  Each ∙ Replicate  
Dyadic  Reverse Compose  
Quad names  
Arrays  Index origin ∙ Migration level  
Functions  
Operators  
Other  Zilde ∙ High minus ∙ Function axis 