Power (function)

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This page describes the dyadic function. For the monadic function that uses as a base, see Exponential. For the iteration operator, see Power (operator).
*

Power (*) is a dyadic scalar function that computes the exponentiation function of the two arguments, so that <syntaxhighlight lang=apl inline>X*Y</source> is <syntaxhighlight lang=apl inline>X</source> raised to the power <syntaxhighlight lang=apl inline>Y</source>. Power shares the glyph <syntaxhighlight lang=apl inline>*</source> with the monadic arithmetic function Exponential.

Examples

<syntaxhighlight lang=apl> 

     2*¯1 0 1 2 3 4 5

0.5 1 2 4 8 16 32 </source> A common technique is to choose sign based on a Boolean array: <syntaxhighlight lang=apl> 

     ¯1*1 0 0 1 0

¯1 1 1 ¯1 1 </source>

Properties

For positive integer <syntaxhighlight lang=apl inline>Y</source>, <syntaxhighlight lang=apl inline>X*Y</source> equals the product of <syntaxhighlight lang=apl inline>Y</source> copies of <syntaxhighlight lang=apl inline>X</source>. When <syntaxhighlight lang=apl inline>Y</source> is 0, <syntaxhighlight lang=apl inline>X*Y</source> equals 1, possibly except when <syntaxhighlight lang=apl inline>X</source> is also 0 (since zero to the power of zero is undefined in mathematics).

<syntaxhighlight lang=apl> 

     3*5

243 

     ×/5⍴3

243 

     1 2 3*0

1 1 1 </source>

Negating the exponent (right argument) gives the reciprocal of the return value.

<syntaxhighlight lang=apl> 

     (2*¯4)=÷2*4

1 </source>

If the exponent is the reciprocal of some number n, the result is the n-th root of the base. For example, a right argument of <syntaxhighlight lang=apl inline>÷2</source> gives the square root.

<syntaxhighlight lang=apl> 

     3*2

9 

     9*÷2

3 </source>

Power has two inverses, Root and Logarithm: <syntaxhighlight lang=apl> 

     2*3

8 

     2⍟8

3 

     3√8

2 </source>

External links

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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