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'''Leading axis agreement''', sometimes called '''prefix agreement''', is a [[conformability]] rule designed for [[leading axis theory]]. It states that a [[dyadic]] [[scalar function]] can be applied between two [[array]]s only if one of their [[shape]]s is a [[prefix]] of the other. The shape of the result is that of the [[argument]] with higher [[rank]]. | |||
== Examples == | |||
The following examples use [[J]] for demonstration purposes. | |||
A scalar dyadic function works when the two arrays have the same shape: | |||
<source lang=j> | |||
]x =: 2 3 $ 10 | |||
10 10 10 | |||
10 10 10 | |||
]y =: 2 3 $ i.6 | |||
0 1 2 | |||
3 4 5 | |||
x + y | |||
10 11 12 | |||
13 14 15 | |||
</source> | |||
{{Works in|[[J]]}} | |||
as well as when one is a [[scalar]]: | |||
<source lang=j> | |||
]x =: 10 | |||
10 | |||
]y =: 2 3 $ i.6 | |||
0 1 2 | |||
3 4 5 | |||
x + y | |||
10 11 12 | |||
13 14 15 | |||
</source> | |||
{{Works in|[[J]]}} | |||
The two cases above are already supported in other APLs in the form of [[scalar extension]]. J goes one step further, allowing the lower-rank array argument to have nonzero rank, as long as the leading dimensions match: | |||
<source lang=j> | |||
]x =: 10 20 | |||
10 20 | |||
]y =: 2 3 $ i.6 | |||
0 1 2 | |||
3 4 5 | |||
x + y | |||
10 11 12 | |||
23 24 25 | |||
</source> | |||
{{Works in|[[J]]}} | |||
In this case, <source lang=j inline>x</source> has shape <source lang=j inline>2</source> and <source lang=j inline>y</source> has shape <source lang=j inline>2 3</source>. Since the leading axes agree and the rank difference is 1, each atom (or 0-[[cell]]) of <source lang=j inline>x</source> is matched with each row (or 1-cell) of <source lang=j inline>y</source>, and the two rows in the result are the results of <source lang=j inline>10 + 0 1 2</source> and <source lang=j inline>20 + 3 4 5</source>, respectively. | |||
== Model == | |||
In dialects that do not feature leading axis agreement, it can nevertheless be utilised by the introduction of an explicit operator: | |||
<source lang=apl> | |||
_LA←{⍺ ⍺⍺⍤(-⍺⌊⍥(≢⍴)⍵)⊢⍵} | |||
⊢x ← 10 20 | |||
10 20 | |||
⊢y ← 2 3 ⍴ ⍳ 6 | |||
0 1 2 | |||
3 4 5 | |||
x +_LA y | |||
10 11 12 | |||
23 24 25 | |||
</source> | |||
{{Works in|Dyalog APL}} | |||
== Aligning axes using the Rank operator == | |||
When using the [[Rank (operator)|Rank operator]] for dyadic functions as in <source lang=apl inline>X (f⍤m n) Y</source>, the [[Frame|frames]] of <source lang=apl inline>X</source> and <source lang=apl inline>Y</source> are checked for conformability. Combined with leading axis agreement, the Rank operator can be used to align the [[axis|axes]] to be matched. | |||
<source lang=j> | |||
NB. $x : 2|3 | |||
NB. $y : |3 2 | |||
NB. ------------------ | |||
NB. $x +"1 2 y : 2 3 2 | |||
]x =: 2 3 $ 10 20 30 40 50 60 | |||
10 20 30 | |||
40 50 60 | |||
]y =: 3 2 $ 1 2 3 4 5 6 | |||
1 2 | |||
3 4 | |||
5 6 | |||
x +"1 2 y | |||
11 12 | |||
23 24 | |||
35 36 | |||
41 42 | |||
53 54 | |||
65 66 | |||
</source> | |||
{{Works in|[[J]]}} | |||
[[Category:Leading axis theory]][[Category:Function characteristics]][[Category:Conformability]]{{APL features}} | |||