Vocabulary/ampdot
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[x] u&.v y Under (Dual) Conjunction
Rank -- depends on the rank of v -- WHY IS THIS IMPORTANT?
u&.v is equivalent to (u&.:v)"mv , where mv is the monadic rank of v. That is,
u&.v y <--> {{v^:_1 u (v y)}}"mv y
x u&.v y <--> x {{v^:_1 (v x) u (v y)}}"mv y
You should understand Rank (") and the rankless form of Under (&.:) before trying to understand Under (Dual) (&.).
Executes u on the argument(s) with a temporary change of viewpoint, indicated by v. First v is executed, to move to the new viewpoint; then u is executed in the new viewpoint; then the operation of v is undone, restoring the original viewpoint.
like Daylight Saving Time, or buying online in a different currency.
The operation is executed on each cell as defined by the rank of v:
- For each cell, u is applied to the result(s) of v
- Then v^:_1 (i.e. the obverse of v) is applied to the result of u — giving the result for that cell.
Some v's can be undone even though as functions they may have no mathematical inverse.
Finally the results for all mv-cells are collected into the end result.
This contrasts with u&.:v which applies v to the entire argument(s) and then applies v^:_1@:u on the entire filled and assembled result(s) of v .
We say that u is applied under v .
Example: Apply (i.) to the contents of each box separately, and re-box each result separately
] y =: 1;2;2 3 +-+-+---+ |1|2|2 3| +-+-+---+ u =: i. v =: > u&.v y +-+---+-----+ |0|0 1|0 1 2| | | |3 4 5| +-+---+-----+
Common uses
1. The most common use of &. is [x] u&.> y which applies the verb u inside each box of the arguments. This follows the rules above: u&.> has rank 0 regardless of u, so each box is opened separately, operated on, and then reboxed.
Examples:
Convert a list of boxed strings to uppercase
] z =: ;:'alpha bravo charlie' +-----+-----+-------+ |alpha|bravo|charlie| +-----+-----+-------+ toupper&.> z +-----+-----+-------+ |ALPHA|BRAVO|CHARLIE| +-----+-----+-------+
Add the contents of corresponding boxes
]l =. 1;2 3;4 5 6 +-+---+-----+ |1|2 3|4 5 6| +-+---+-----+ ]r =. 3 1 4;1 5;9 +-----+---+-+ |3 1 4|1 5|9| +-----+---+-+ l +&.> r +-----+---+--------+ |4 2 5|3 8|13 14 15| +-----+---+--------+
The standard library defines the name each as &.>
(<'Mr. ') , each 'Smith';'Jones' +---------+---------+ |Mr. Smith|Mr. Jones| +---------+---------+
each is a
- Standard Library word(s) residing in the 'z'-locale
- Defined in the factory script stdlib.ijs which is located in ~system/main/stdlib.ijs
- View the definition(s) in a JQt session by entering: open '~system/main/stdlib.ijs'
2. A useful trick is to use each when one or more arguments are not boxed. Since > y leaves y unchanged if it is not boxed, an unboxed argument will be processed as individual unboxed atoms:
i.&.> 3 4 NB. i. on each atom, boxing results separately
+-----+-------+
|0 1 2|0 1 2 3|
+-----+-------+
3 3 4 {. each 'Washington';'Sheffield';'Hereford'
+---+---+----+
|Was|She|Here|
+---+---+----+
100 + each 3;1 4;1 5 9 NB. Add 100 inside each box
+---+-------+-----------+
|103|101 104|101 105 109|
+---+-------+-----------+
Semiduals x u&.(a:`v) y and x u&.(v`a:) y
The operation encapsulated by u&.v is used often and is important as a notation to aid thought. When the operation is dyadic, sometimes the sequence of transforming/operating/transforming back should be applied to just one argument, with the other argument being passed unchanged into u. This can be represented by a gerund form where a: indicates which argument is to be used unmodified. The form using &. is defined in terms of the infinite-rank version &.::
- x u&.(a:`v) y is x u&.:(a:`v)"(lu,mv) where lu is the left rank of u and mv is the monadic rank of v
- x u&.(v`a:) y is x u&.:(v`a:)"(mv,ru) where ru is the right rank of u and mv is the monadic rank of v
<"2 i. 3 2 4 NB. y is 3 boxes +-------+-----------+-----------+ |0 1 2 3| 8 9 10 11|16 17 18 19| |4 5 6 7|12 13 14 15|20 21 22 23| +-------+-----------+-----------+ 100 200 +"_&.(a:`>) <"2 i. 3 2 4 NB. open y (but not x), add x to each contents, rebox +---------------+---------------+---------------+ |100 101 102 103|108 109 110 111|116 117 118 119| |204 205 206 207|212 213 214 215|220 221 222 223| +---------------+---------------+---------------+ +"_&.(a:`>) b. 0 NB. This works because the left rank of u&.(a:`v) comes from u 0 _ 0
u&.v y when v has no inverse
Even when v has no inverse, when v rearranges or selects from y, it may be possible to define a notional inverse that, referring to the value of y, reverses the effect of the rearrangement/selection. This surrogate inverse is defined for the following values of v:
v Inverse used
Description of operation
, ($y)&($,) Execute u on (,y), then put result into original shape of y m&{ m}&y Execute u on selected cells of y, then put result back into those cells
u&.(m&{) y is the preferred idiom for applying u to a portion of y. If m&{ selects a contiguous section of y, the update may be performed in place.
Related Primitives
Under (&.:)
More Information
1. Under (Dual) (&.) differs from Under (&.:) in that u&.v applies u under v cell-by-cell, whereas u&.:v applies u under v to the entire argument(s) (x and) y .
Use These Combinations
Combinations using &. y that have exceptionally good performance include:
What it does Type;
Precisions;
RanksSyntax Variants;
Restrictions
Benefits;
Bug Warnings
Join contents of boxed items along first axis ,&.>/ y Bug warning: Atomic replication is inaccurate. OK if contents of same rank. Better to use <@; y Join contents of boxed items along last axis ,.&.>/ y Bitwise operations on bytes byte u&.(a.&i.) y (u y) -: u"0 y avoids conversion to integer (m b.)/&.(a.&i.) y
x (m b.)&.(a.&i.) y
m is 16 to 31