# Power

 Z←{L} f⍣g 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 Zn←f Zn-1 (or Zn←L∘f Zn-1). Z←{L} f⍣b 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/RightIdentityFunction L⊥⍣¯1 R (N⍴L)⊤R for N sufficiently large to display all digits of R 10⊥⍣¯1 12345678901 2 3 4 5 6 7 8 9 0 10⊥10⊥⍣¯1 12345678901234567890 10⊥⍣¯1 10⊥1 2 3 4 5 6 7 8 9 01 2 3 4 5 6 7 8 9 0 L and R ⊥⍣¯1 R 2⊥⍣¯1 R ⊥⍣¯1 191 0 0 1 1 2⊥⊥⍣¯1 1919 ⊥⍣¯1 2⊥1 0 0 1 11 0 0 1 1 L and R L⊤⍣¯1 R L⊥R 10 10 10⊤⍣¯1 2 3 4234 10 10 10⊤10 10 10⊤⍣¯1 2 3 42 3 4 10 10 10⊤⍣¯1 10 10 10⊤234234 L and R ×/⍣¯1 R πR, that is, factor R into primes ×/⍣¯1 1302 5 13 ×/×/⍣¯1 130130 ×/⍣¯1 ×/2 5 132 5 13 L and R π⍣¯1 R ×/R, that is, multiply together the prime factors in R π⍣¯1 2 5 13130 ππ⍣¯1 2 5 132 5 13 π⍣¯1 π 130130 L and R +\[X]⍣¯1 R ¯2-\[X] R +\⍣¯1 ⍳41 1 1 1 +\+\⍣¯1 ⍳41 2 3 4 +\⍣¯1+\ ⍳41 2 3 4 L and R ¯2-\[X]⍣¯1 R +\[X] R ¯2-\⍣¯1 4⍴11 2 3 4 ¯2-\¯2-\⍣¯1 4⍴11 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 a10 8 3 2 6 -\⍣¯1 a10 2 ¯5 1 4 -\-\⍣¯1 a10 8 3 2 6 -\⍣¯1-\ a10 8 3 2 6 L and R ÷\[X]⍣¯1 R (2÷\[X] R)*[X](⍴R)[X]⍴¯1 1 a←?5⍴10 a10 8 3 2 6 ÷\⍣¯1 a10 1.25 0.375 1.5 3 ÷\÷\⍣¯1 a10 8 3 2 6 ÷\⍣¯1÷\ a10 8 3 2 6 L and R +∘÷/⍣¯1 R Display the Continued Fraction expansion of R to at most ⎕PP termsThis function is best used on Multiple Precision numbers +∘÷/⍣¯1 449r3031 2 13 3 1 2 +∘÷/+∘÷/⍣¯1 449r303449r303 +∘÷/⍣¯1 +∘÷/1 2 13 3 1 2x1 2 13 3 1 2 L and R L+∘÷/⍣¯1 R Display the Continued Fraction expansion of R to at most L termsThis function is best used on Multiple Precision numbers phi←1+∘÷⍣=1 ⍝ Phi -- the Golden Ratio 20 +∘÷/⍣¯1 phi1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 20 +∘÷/⍣¯1 √21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 +∘÷/⍣¯1 *1x2 1 2 1 1 4 1 1 6 1 1 8 1 1 10 1 1 12 1 1 20 +∘÷/⍣¯1 ○1x3 7 15 1 292 1 1 1 2 1 3 1 14 2 1 1 2 2 2 2 +∘÷/35 +∘÷/⍣¯1 ○1x3.141592653589793238462643383279502884197 ○1x3.141592653589793238462643383279502884197 4 +∘÷/⍣¯1 +∘÷/○1x3 7 15 1 +∘÷\4 +∘÷/⍣¯1 'r' ⎕DC ○1x ⍝ Convergents to Pi3 22r7 333r106 355r113 L and R