Vocabulary/ampdot
>> << Back to: Vocabulary Thru to: Dictionary
[x] u&.v y Under (Dual) Conjunction
Rank -- depends on the rank of v -- WHY IS THIS IMPORTANT?
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 vs can be undone even though they have no inverse.
Finally the results for all cells are collected into the end result.
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 .
2. Let mv be the monadic rank of v. Then:
- u&.v y is the same as (u&.:v)"mv y
- x u&.v y is the same as x (u&.:v)"mv 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