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== Description ==
== Description ==
A call to Power is of the form <source lang=apl inline>X(f⍣g)Y</source>, where  
A call to Power is of the form <syntaxhighlight lang=apl inline>X(f⍣g)Y</syntaxhighlight>, where  
* <source lang=apl inline>X</source> is an optional argument.  
* <syntaxhighlight lang=apl inline>X</syntaxhighlight> is an optional argument.  
* <source lang=apl inline>f</source> is a function. If <source lang=apl inline>X</source> is given, then it is bound to  <source lang=apl inline>f</source> so <source lang=apl inline>X f⍣g Y</source> is equivalent to <source lang=apl inline>X∘f⍣g Y</source>.
* <syntaxhighlight lang=apl inline>f</syntaxhighlight> is a function. If <syntaxhighlight lang=apl inline>X</syntaxhighlight> is given, then it is bound to  <syntaxhighlight lang=apl inline>f</syntaxhighlight> so <syntaxhighlight lang=apl inline>X f⍣g Y</syntaxhighlight> is equivalent to <syntaxhighlight lang=apl inline>X∘f⍣g Y</syntaxhighlight>.
* <source lang=apl inline>g</source> can be an [[array]] or a function.
* <syntaxhighlight lang=apl inline>g</syntaxhighlight> can be an [[array]] or a function.


Power repeatedly applies <source lang=apl inline>f</source> to <source lang=apl inline>Y</source> based on the type of operand <source lang=apl inline>g</source>:
Power repeatedly applies <syntaxhighlight lang=apl inline>f</syntaxhighlight> to <syntaxhighlight lang=apl inline>Y</syntaxhighlight> based on the type of operand <syntaxhighlight lang=apl inline>g</syntaxhighlight>:
* '''Function''': Must be dyadic and must return a boolean [[singleton]]. The previous iteration value is provided as the right argument to <source lang=apl inline>f</source>, and the current iteration value is given as the left argument. <source lang=apl inline>f</source> is repeatedly applied until this function returns 1.
* '''Function''': Must be dyadic and must return a boolean [[singleton]]. The previous iteration value is provided as the right argument to <syntaxhighlight lang=apl inline>f</syntaxhighlight>, and the current iteration value is given as the left argument. <syntaxhighlight lang=apl inline>f</syntaxhighlight> is repeatedly applied until this function returns 1.
* '''Integer''': Applies <source lang=apl inline>f</source> <source lang=apl inline>g</source> times to <source lang=apl inline>Y</source>. If <source lang=apl inline>g</source> is negative, then the inverse of <source lang=apl inline>f</source> (if available) is applied.
* '''Integer''': Applies <syntaxhighlight lang=apl inline>f</syntaxhighlight> <syntaxhighlight lang=apl inline>g</syntaxhighlight> times to <syntaxhighlight lang=apl inline>Y</syntaxhighlight>. If <syntaxhighlight lang=apl inline>g</syntaxhighlight> is negative, then the inverse of <syntaxhighlight lang=apl inline>f</syntaxhighlight> (if available) is applied.
* '''Integer Array''': In [[Extended Dyalog APL]], <source lang=apl inline>g</source> can be an integer array. Each integer <source lang=apl inline>i</source> in <source lang=apl inline>g</source> will be replaced by <source lang=apl inline>⊂(f⍣i)Y</source>.
* '''Integer Array''': In [[Extended Dyalog APL]], <syntaxhighlight lang=apl inline>g</syntaxhighlight> can be an integer array. Each integer <syntaxhighlight lang=apl inline>i</syntaxhighlight> in <syntaxhighlight lang=apl inline>g</syntaxhighlight> will be replaced by <syntaxhighlight lang=apl inline>⊂(f⍣i)Y</syntaxhighlight>.


== Examples ==
== Examples ==
Some basic examples:
Some basic examples:
<source lang=apl>      1 (+⍣3) 5 ⍝ Fixed number of iterations
<syntaxhighlight lang=apl>      1 (+⍣3) 5 ⍝ Fixed number of iterations
8
8
       (2∘×⍣3) 5 ⍝ No X given
       (2∘×⍣3) 5 ⍝ No X given
40
40
       1 +∘÷⍣= 1 ⍝ iterate till fixed point
       1 +∘÷⍣= 1 ⍝ iterate till fixed point
1.618033989</source>
1.618033989</syntaxhighlight>
A well-known use for Power is iterating until a fixed point is reached.  
A well-known use for Power is iterating until a fixed point is reached.  
<source lang=apl>
<syntaxhighlight lang=apl>
       (∨.∧⍨∨⊢)⍣≡3 3⍴0 0 1 1 0 1 1 0 1 ⍝ Transitive closure of an adjacency matrix
       (∨.∧⍨∨⊢)⍣≡3 3⍴0 0 1 1 0 1 1 0 1 ⍝ Transitive closure of an adjacency matrix
1 0 1
1 0 1
1 0 1
1 0 1
1 0 1</source>
1 0 1</syntaxhighlight>
Power is also used to access function inverses.
Power is also used to access function inverses.
<source lang=apl>      2(⊥⍣¯1)5
<syntaxhighlight lang=apl>      2(⊥⍣¯1)5
1 0 1</source>
1 0 1</syntaxhighlight>
== External Links ==
== External Links ==
=== Lessons ===
=== Lessons ===

Latest revision as of 22:30, 10 September 2022

⍣

Power (⍣) is a primitive dyadic operator that performs bounded looping, unbounded looping, and function inverses in NARS2000, Dyalog APL, and related implementations like ngn/apl, dzaima/APL, and Extended Dyalog APL.

Description

A call to Power is of the form X(f⍣g)Y, where

  • X is an optional argument.
  • f is a function. If X is given, then it is bound to f so X f⍣g Y is equivalent to X∘f⍣g Y.
  • g can be an array or a function.

Power repeatedly applies f to Y based on the type of operand g:

  • Function: Must be dyadic and must return a boolean singleton. The previous iteration value is provided as the right argument to f, and the current iteration value is given as the left argument. f is repeatedly applied until this function returns 1.
  • Integer: Applies f g times to Y. If g is negative, then the inverse of f (if available) is applied.
  • Integer Array: In Extended Dyalog APL, g can be an integer array. Each integer i in g will be replaced by ⊂(f⍣i)Y.

Examples

Some basic examples: 

      1 (+⍣3) 5 ⍝ Fixed number of iterations
8
      (2∘×⍣3) 5 ⍝ No X given
40
      1 +∘÷⍣= 1 ⍝ iterate till fixed point
1.618033989

A well-known use for Power is iterating until a fixed point is reached. 

      (∨.∧⍨∨⊢)⍣≡3 3⍴0 0 1 1 0 1 1 0 1 ⍝ Transitive closure of an adjacency matrix
1 0 1
1 0 1
1 0 1

Power is also used to access function inverses. 

      2(⊥⍣¯1)5
1 0 1

External Links

Lessons

Documentation

APL built-ins [edit]
Primitives (Timeline) Functions
Scalar
Monadic Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root ∙ Round
Dyadic Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
Structural Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
Selection First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
Selector Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
Computational Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
Operators Monadic Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis ∙ Identity (Null, Ident)
Dyadic Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator) ∙ Identity (Lev, Dex)
Quad names Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
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