[x] u@:v y At Conjunction
Forms the composition (u@:v) of two verbs u and v .
The result is a tacit verb equivalent to u [x] v y (where [x] represents an optional x-argument)
y =: i. 6 NB. a sample list (the contents don't matter) <: # y NB. count the items in y and subtract 1 5 compo =: <:@:# NB. Make a new verb: the "composition" of (<:) and (#) compo y 5
You can safely chain verbs using (@:) with minimal need for parentheses
1 2 3 +/@:*:@:- 2 2 2 NB. sum of squared differences 2 +/ *: 1 2 3 - 2 2 2 NB. same thing, without using (@:) 2
1. Implement: f(g(x)) -- the mathematical composition of the two functions: f and g.
mean=: +/ % # cat=: ,&1"1 ]z=: i.2 3 NB. sample noun 0 1 2 3 4 5 cat z NB. appends 1 to each row of z 0 1 2 1 3 4 5 1 mean@:cat z NB. mean of the COLUMNS of cat z 1.5 2.5 3.5 1 mean@:cat b.0 NB. rank of (mean@:cat) _ _ _
But see Rank in a hurry: an insidious rank problem for how and when these different methods give different results.
1. Unlike Atop (@), the rank of (u@:v) is infinite, irrespective of the ranks of u and v. This means that u will be executed on the entire result of v, after individual results of v have been collected and filled.
The difference between At (u@:v) and Atop (u@v) is shown in the first two columns of the diagram below:
2. u@:v is called for when the rank of v is less than the ranks of an argument, but you want to apply u to the entire result of v.
Thus, in the "sum of squared-differences" example in the introduction, we needed (@:) not (@)
1 2 3 +/@:*:@:- 2 2 2 2 1 2 3 +/@*:@- 2 2 2 NB. different result using (@) in place of (@:) 1 0 1
Because x - y has rank 0, the entire verb +/@*:@- was applied to each atom of x and y individually, making +/ useless because it is now applied to each number independently.
None at all, for the monads (u@v) and (u&v)
u&v y ↔ u v y u@v y ↔ u v y
But the dyads are different
x u&v y ↔ (v x) u (v y) x u@v y ↔ u x v y
According to the J Dictionary -- &: is equivalent to & except that the ranks of the resulting function are infinite; the relation is similar to that between @: and @