Factorial: Difference between revisions

From APL Wiki
Jump to navigation Jump to search
m (Text replacement - "http://help.dyalog.com" to "https://help.dyalog.com")
Tags: Mobile edit Mobile web edit
m (Text replacement - "<source" to "<syntaxhighlight")
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
{{Built-in|Factorial|!}} is a [[monadic]] [[scalar function]] which gives the [[wikipedia:factorial|factorial]] of a non-negative integer. Factorial takes its [[glyph]] <source lang=apl inline>!</source> from [[Comparison_with_traditional_mathematics#Prefix|traditional mathematics]] but, like all [[monadic function]]s, takes its argument on the right  <source lang=apl inline>!Y</source> instead of traditional mathematics' <math>Y!</math>. It shares the glyph with the dyadic arithmetic function [[Binomial]].
{{Built-in|Factorial|!}} is a [[monadic]] [[scalar function]] which gives the [[wikipedia:factorial|factorial]] of a non-negative integer. Factorial takes its [[glyph]] <syntaxhighlight lang=apl inline>!</syntaxhighlight> from [[Comparison_with_traditional_mathematics#Prefix|traditional mathematics]] but, like all [[monadic function]]s, takes its argument on the right  <syntaxhighlight lang=apl inline>!Y</syntaxhighlight> instead of traditional mathematics' <math>Y!</math>. It shares the glyph with the dyadic arithmetic function [[Binomial]].


== Examples ==
== Examples ==
Line 5: Line 5:
The factorial of a positive integer n is defined as the [[times|product]] of [[Index Generator|1 to n]] inclusive.
The factorial of a positive integer n is defined as the [[times|product]] of [[Index Generator|1 to n]] inclusive.


<source lang=apl>
<syntaxhighlight lang=apl>
       !0 1 2 3 4
       !0 1 2 3 4
1 1 2 6 24
1 1 2 6 24
       ×/⍳4
       ×/⍳4
24
24
</source>
</syntaxhighlight>


== Extended definition ==
== Extended definition ==


In multiple implementations, this function has an extended definition using the [[wikipedia:Gamma function|Gamma function]] <math>\Gamma(n)</math>, so that it is defined for real and [[complex]] numbers. Because <math>\Gamma(n)</math> equals <math>(n-1)!</math>, <source lang=apl inline>!Y</source> is defined as <math>\Gamma(Y+1)</math>.
In multiple implementations, this function has an extended definition using the [[wikipedia:Gamma function|Gamma function]] <math>\Gamma(n)</math>, so that it is defined for real and [[complex]] numbers. Because <math>\Gamma(n)</math> equals <math>(n-1)!</math>, <syntaxhighlight lang=apl inline>!Y</syntaxhighlight> is defined as <math>\Gamma(Y+1)</math>.


<source lang=apl>
<syntaxhighlight lang=apl>
       !¯1.2 0.5 2.7
       !¯1.2 0.5 2.7
¯5.821148569 0.8862269255 4.170651784
¯5.821148569 0.8862269255 4.170651784
       !2J1 ¯2J¯1
       !2J1 ¯2J¯1
0.962865153J1.339097176 ¯0.1715329199J¯0.3264827482
0.962865153J1.339097176 ¯0.1715329199J¯0.3264827482
</source>{{Works in|[[Dyalog APL]]}}
</syntaxhighlight>{{Works in|[[Dyalog APL]]}}


The Gamma function diverges at 0 or negative numbers, so <source lang=apl inline>!Y</source> is undefined at negative integers.
The Gamma function diverges at 0 or negative numbers, so <syntaxhighlight lang=apl inline>!Y</syntaxhighlight> is undefined at negative integers.


<source lang=apl>
<syntaxhighlight lang=apl>
       !¯1
       !¯1
DOMAIN ERROR
DOMAIN ERROR
  !¯1         
  !¯1         
  ∧   
  ∧   
</source>{{Works in|[[Dyalog APL]]}}
</syntaxhighlight>{{Works in|[[Dyalog APL]]}}


In [[J]], where literal [[infinity]] is supported, negative integer factorial evaluates to positive infinity <source lang=j inline>_</source> (if the argument is odd) or negative infinity <source lang=j inline>__</source> (if even). This corresponds to the positive-side limit of the Gamma function.
In [[J]], where literal [[infinity]] is supported, negative integer factorial evaluates to positive infinity <syntaxhighlight lang=j inline>_</syntaxhighlight> (if the argument is odd) or negative infinity <syntaxhighlight lang=j inline>__</syntaxhighlight> (if even). This corresponds to the positive-side limit of the Gamma function.


<source lang=j>
<syntaxhighlight lang=j>
       !_1 _2 _3 _4
       !_1 _2 _3 _4
_ __ _ __
_ __ _ __
</source>{{Works in|[[J]]}}
</syntaxhighlight>{{Works in|[[J]]}}


== External links ==
== External links ==

Latest revision as of 21:21, 10 September 2022

!

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]
Primitive functions
Scalar
Monadic ConjugateNegateSignumReciprocalMagnitudeExponentialNatural LogarithmFloorCeilingFactorialNotPi TimesRollTypeImaginarySquare Root
Dyadic AddSubtractTimesDivideResiduePowerLogarithmMinimumMaximumBinomialComparison functionsBoolean functions (And, Or, Nand, Nor) ∙ GCDLCMCircularComplexRoot
Non-Scalar
Structural ShapeReshapeTallyDepthRavelEnlistTableCatenateReverseRotateTransposeRazeMixSplitEncloseNestCut (K)PairLinkPartitioned EnclosePartition
Selection FirstPickTakeDropUniqueIdentitySelectReplicateExpandSet functions (IntersectionUnionWithout) ∙ Bracket indexingIndex
Selector Index generatorGradeIndex OfInterval IndexIndicesDeal
Computational MatchNot MatchMembershipFindNub SieveEncodeDecodeMatrix InverseMatrix DivideFormatExecuteMaterialiseRange
Primitive operators Monadic EachCommuteConstantReplicateExpandReduceWindowed ReduceScanOuter ProductKeyI-BeamSpawnFunction axis
Dyadic BindCompositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner ProductPowerAtUnderRankDepthVariantStencilCut (J)
Quad names
Arrays Index originMigration levelAtomic vector
Functions Name classCase convertUnicode convert
Operators SearchReplace