Function composition: Difference between revisions

'''Function composition''' refers to the "gluing" together of two or more [[function]]s using a [[dyadic operator]] or a [[train]] such that the functions are applied to the [[argument]](s) as normal, but in a particular pattern specific to the used [[operator]]. The term [[wikipedia:function composition|function composition]] comes from [[traditional mathematics]] where it is used for a function <math>h(x)=f(g(x))</math> when written as <math> h(x) = (f \circ g)(x)</math>. APL generalises this idea to [[dyadic function]]s, allowing various patterns of application in addition to the simple application of one [[monadic function]] to the result of another monadic function. The patterns represented by the operators [[Atop (operator)|Atop]], [[Beside]], and [[Over]] can be visualised as follows:

|[[File:F⍤g.png|frameless|200px]]||[[File:F∘g.png|frameless|200px]]||[[File:F⍥g.png|frameless|200px]]

 

The patterns represented by [[Tacit_programming#trains|trains]], the 2-train Atop and the 3-train Fork, can be visualised as follows:

 

The <source lang=apl inline>∘</source> operator (in this context called [[Bind]]) and the 3-train can also be used with constant arrays, then treating the arrays (<source lang=apl inline>A</source>) as constant functions, much as if they were used as operands to the [[Constant]] operator (<source lang=apl inline>A⍨</source>):

 

These are the rules applied in [[Dyalog APL]]:

 

<source lang=apl inline>⍺ (f g h) ⍵</source> {{←→}} <source lang=apl inline>(⍺ f ⍵) g (⍺ h ⍵)</source><br>

<source lang=apl inline>  (A g h) ⍵</source> {{←→}} <source lang=apl inline>   A    g (  h ⍵)</source><br>

<source lang=apl inline>⍺   (g h) ⍵</source> {{←→}} <source lang=apl inline>        g (⍺ h ⍵)</source>