Factorial: Difference between revisions

From APL Wiki
Jump to navigation Jump to search
No edit summary
m (Text replacement - "http://help.dyalog.com" to "https://help.dyalog.com")
Tags: Mobile edit Mobile web edit
Line 43: Line 43:
=== Documentation ===
=== Documentation ===


* [http://help.dyalog.com/latest/#Language/Primitive%20Functions/Factorial.htm Dyalog]
* [https://help.dyalog.com/latest/#Language/Primitive%20Functions/Factorial.htm Dyalog]
* [http://microapl.com/apl_help/ch_020_020_250.htm APLX]
* [http://microapl.com/apl_help/ch_020_020_250.htm APLX]
* [https://www.jsoftware.com/help/dictionary/d410.htm J Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/bang NuVoc]
* [https://www.jsoftware.com/help/dictionary/d410.htm J Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/bang NuVoc]
{{APL built-ins}}[[Category:Primitive functions]][[Category:Scalar monadic functions]]
{{APL built-ins}}[[Category:Primitive functions]][[Category:Scalar monadic functions]]

Revision as of 14:41, 14 July 2020

!

Factorial (!) is a monadic scalar function which gives the factorial of a non-negative integer. Factorial takes its glyph ! from traditional mathematics but, like all monadic functions, takes its argument on the right !Y instead of traditional mathematics' . It shares the glyph with the dyadic arithmetic function Binomial.

Examples

The factorial of a positive integer n is defined as the product of 1 to n inclusive. 

      !0 1 2 3 4
1 1 2 6 24
      ×/⍳4
24

Extended definition

In multiple implementations, this function has an extended definition using the Gamma function , so that it is defined for real and complex numbers. Because equals , !Y is defined as . 

      !¯1.2 0.5 2.7
¯5.821148569 0.8862269255 4.170651784
      !2J1 ¯2J¯1
0.962865153J1.339097176 ¯0.1715329199J¯0.3264827482
Works in: Dyalog APL

The Gamma function diverges at 0 or negative numbers, so !Y is undefined at negative integers. 

      !¯1
DOMAIN ERROR
 !¯1        
 ∧
Works in: Dyalog APL

In J, where literal infinity is supported, negative integer factorial evaluates to positive infinity _ (if the argument is odd) or negative infinity __ (if even). This corresponds to the positive-side limit of the Gamma function. 

      !_1 _2 _3 _4
_ __ _ __
Works in: J

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
Retrieved from ‘https://aplwiki.com/index.php?title=Factorial&oldid=5190’