Power

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Z←{L} fg R successively applies the function f (or, if there is a Left argument, the function L∘f) to R until the expression Zn g Zn-1 returns a singleton 1. Zn and Zn-1 represent two consecutive applications of f (or L∘f) to R where Z0←R and Znf Zn-1 (or Zn←L∘f Zn-1).
Z←{L} fb R for non-negative integer scalar b, successively applies the function f (or, if there is a Left argument, the function L∘f) to R, b number of times;
for a negative integer scalar b, successively applies the inverse of the function f (or, if there is a Left argument, the inverse to the function L∘f), |b number of times
L and R are arbitrary arrays.
In the first case, Zn g Zn-1 must return a Boolean-valued singleton; otherwise a DOMAIN ERROR is signaled.
In the second case, b must be an integer scalar, otherwise a DOMAIN ERROR is signaled.


For example,

      sqrt←{{0.5×⍵+⍺÷⍵}⍣=⍨⍵} ⍝ Calculate square root using Newton's method
      sqrt 2
1.414213562373095
      fib←{⍵,+/¯2↑⍵} ⍝ Calculate a Fibonacci sequence
      fib⍣15 ⊢ 1 1
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 
      pow←{⍵=0:,1 ⋄ ⍺+⍡×⍣(⍵-1) ⍺} ⍝ Raise a polynomial to a non-negative integer power
      1 ¯1 pow 5
1 ¯5 10 ¯10 5 ¯1
      phi←1+∘÷⍣=1 ⍝ Calculate the Golden Ratio
      phi
1.618033988749894
      phi=0.5×1+√5
1

Inverses

When the right operand to the Power Operator is a negative integer scalar, the inverse of the function left operand is applied to the right argument. If f f⍣¯1 R ←→ R, then f⍣¯1 is a Right Identity Function, and if f⍣¯1 f R ←→ R, then f⍣¯1 is a Left Identity Function, which all of the examples below satisfy.

At the moment, only a few inverse functions are available as follows:

Function Meaning of Inverse Left/Right
Identity
Function
L⊥⍣¯1 R (N⍴L)⊤R for N sufficiently large to display all digits of R
10⊥⍣¯1 1234567890
1 2 3 4 5 6 7 8 9 0
10⊥10⊥⍣¯1 1234567890
1234567890
10⊥⍣¯1 10⊥1 2 3 4 5 6 7 8 9 0
1 2 3 4 5 6 7 8 9 0
L and R
⊥⍣¯1 R 2⊥⍣¯1 R
⊥⍣¯1 19
1 0 0 1 1
2⊥⊥⍣¯1 19
19
⊥⍣¯1 2⊥1 0 0 1 1
1 0 0 1 1
L and R
L⊤⍣¯1 R L⊥R
10 10 10⊤⍣¯1 2 3 4
234
10 10 10⊤10 10 10⊤⍣¯1 2 3 4
2 3 4
10 10 10⊤⍣¯1 10 10 10⊤234
234
L and R
×/⍣¯1 R πR, that is, factor R into primes
×/⍣¯1 130
2 5 13
×/×/⍣¯1 130
130
×/⍣¯1 ×/2 5 13
2 5 13
L and R
π⍣¯1 R ×/R, that is, multiply together the prime factors in R
π⍣¯1 2 5 13
130
ππ⍣¯1 2 5 13
2 5 13
π⍣¯1 π 130
130
L and R
+\[X]⍣¯1 R ¯2-\[X] R
+\⍣¯1 ⍳4
1 1 1 1
+\+\⍣¯1 ⍳4
1 2 3 4
+\⍣¯1+\ ⍳4
1 2 3 4
L and R
¯2-\[X]⍣¯1 R +\[X] R
¯2-\⍣¯1 4⍴1
1 2 3 4
¯2-\¯2-\⍣¯1 4⍴1
1 1 1 1
¯2-\⍣¯1 ¯2-\4⍴1
1 1 1 1
L and R
-\[X]⍣¯1 R (2-\[X] R)×[X](⍴R)[X]⍴¯1 1
a←?5⍴10
a
10 8 3 2 6
-\⍣¯1 a
10 2 ¯5 1 4
-\-\⍣¯1 a
10 8 3 2 6
-\⍣¯1-\ a
10 8 3 2 6
L and R
÷\[X]⍣¯1 R (2÷\[X] R)*[X](⍴R)[X]⍴¯1 1
a←?5⍴10
a
10 8 3 2 6
÷\⍣¯1 a
10 1.25 0.375 1.5 3
÷\÷\⍣¯1 a
10 8 3 2 6
÷\⍣¯1÷\ a
10 8 3 2 6
L and R
+∘÷/⍣¯1 R Display the Continued Fraction expansion of R to at most ⎕PP terms
This function is best used on Multiple Precision numbers
+∘÷/⍣¯1 449r303
1 2 13 3 1 2
+∘÷/+∘÷/⍣¯1 449r303
449r303
+∘÷/⍣¯1 +∘÷/1 2 13 3 1 2x
1 2 13 3 1 2
L and R
L+∘÷/⍣¯1 R Display the Continued Fraction expansion of R to at most L terms
This function is best used on Multiple Precision numbers
phi←1+∘÷⍣=1 ⍝ Phi -- the Golden Ratio
20 +∘÷/⍣¯1 phi
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
20 +∘÷/⍣¯1 √2
1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
20 +∘÷/⍣¯1 *1x
2 1 2 1 1 4 1 1 6 1 1 8 1 1 10 1 1 12 1 1
20 +∘÷/⍣¯1 ○1x
3 7 15 1 292 1 1 1 2 1 3 1 14 2 1 1 2 2 2 2
+∘÷/35 +∘÷/⍣¯1 ○1x
3.141592653589793238462643383279502884197
○1x
3.141592653589793238462643383279502884197
4 +∘÷/⍣¯1 +∘÷/○1x
3 7 15 1
+∘÷\4 +∘÷/⍣¯1 'r' ⎕DC ○1x ⍝ Convergents to Pi
3 22r7 333r106 355r113
L and R