If right operand g is a numeric integer scalar, power applies its left operand function f cumulatively g times to its argument. In particular, g may be Boolean 0 or 1 for conditional function application.
If right operand g is a scalar-returning dyadic function, then left operand function f is applied repeatedly until ((f Y) g Y) or until a strong interrupt occurs. Notice that power calls its dyadic right operand g with the next (f Y) and current (Y) values of the iteration as left and right arguments. In particular, if g is = or ≡, the result is sometimes termed a fixpoint of f.
If a left argument X is present, it is bound as left argument to left operand function f:
X (f ⍣ g) Y → (X∘f ⍣ g) Y
A negative right operand g applies the inverse of the operand function f, (|g)times. In this case, f may be a primitive function or an expression of primitive functions combined with primitive operators:
| ∘ | compose |
| ¨ | each |
| ∘. | outer product |
| ⍨ | commute |
| \ | scan |
| [] | axis |
| ⍣ | power |
If the function does not have an inverse, a negative argument g generates DOMAIN ERROR.
Examples
(,∘⊂∘,⍣(1=≡,vec))vec ⍝ ravel-enclose if simple.
a b c←1 0 1{(⊂⍣⍺)⍵}¨abc ⍝ enclose first and last.
cap←{(⍺⍺⍣⍺)⍵} ⍝ conditional application.
a b c←1 0 1⊂cap¨abc ⍝ enclose first and last.
succ←1∘+ ⍝ successor function.
(succ⍣4)10 ⍝ fourth successor of 10.
14
(succ⍣¯3)10 ⍝ third predecessor of 10.
7
1+∘÷⍣=1 ⍝ fixpoint: golden mean.
1.618033989
f←(32∘+)∘(×∘1.8) ⍝ Fahrenheit from Celsius.
f 0 100
32 212
c←f⍣¯1 ⍝ c is Inverse of f.
c 32 212 ⍝ Celsius from Fahrenheit.
0 100
invs←{(⍺⍺⍣¯1)⍵} ⍝ inverse operator.
+\invs 1 3 6 10 ⍝ scan inverse.
1 2 3 4
2∘⊥invs 9 ⍝ decode inverse.
1 0 0 1
dual←{⍵⍵⍣¯1 ⍺⍺ ⍵⍵ ⍵} ⍝ dual operator.
mean←{(+/⍵)÷⍴⍵} ⍝ mean function.
mean dual⍟ 1 2 3 4 5 ⍝ geometric mean.
2.605171085
+/dual÷ 1 2 3 4 5 ⍝ parallel resistance.
0.4379562044
mean dual(×⍨)1 2 3 4 5 ⍝ root-mean-square.
3.31662479
⍉dual↑ 'hello' 'world' ⍝ vector transpose.
hw eo lr ll od
Warning
Some expressions, such as the following, will cause an infinite internal loop and APL will appear to hang. In most cases this can be resolved by issuing a hard INTERRUPT.
!⍣-1
!⍣-2
One can ensure that weak interrupts and ⎕TKILL can interrupt by packaging the ⍣ within the dop {⍺←⊢ ⋄ ⍺ (⍺⍺{⍺←⊢ ⋄ ⍺ ⍺⍺ ⍵}⍣⍵⍵) ⍵}.
Example
PowOp←{⍺←⊢ ⋄ ⍺ (⍺⍺{⍺←⊢ ⋄ ⍺ ⍺⍺ ⍵}⍣⍵⍵) ⍵}
tnum←!PowOp-&1 ⍝ using naked ⍣ will freeze APL
⎕tkill tnum