Difference between revisions of "Scalar function"

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(links to pervasion and recursion)
(move user def to Pervasion)
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== User defined scalar functions ==
In [[nested array model]] dialects with the [[Depth (operator)|Depth]] operator (<source lang=apl inline>⍥</source>), any function can be used as a scalar function (that is, be applied to all simple scalars) using the <source lang=apl inline>perv←⍥0</source>:
[https://tio.run/##SyzI0U2pTMzJT9dNrShJzUtJTfn/qKPtUduE6kd9U4HUo56mlMzigke9u2ofdbcAxYJdgUR4sLprWWpeibqGenBqcXFmfl5AUSaQq6AO1KyuyfWoo70CqNk5PyVVoeK/X35ecHJiTmKRWx7Y4N6tVupAtUDzFNTT1Gu5NJAV9C410FTQMFAw1FQwVNB41NVkqGCgCXRTOxfItEe9a/4DAA Try it online!]
<source lang=apl>
      NonScalarFn←{⍵:'t' ⋄ 'f'}
      (NonScalarFn⍥0) (0 1) 1 (⊂1 0)
│  │ ││tf││
│  │ │└──┘│
{{Works in|[[Extended Dyalog APL]], [[dzaima/APL]] since 2020-03-01}}
In [[dzaima/APL]] the [[dyadic]] form of Depth is not yet implemented, so this definition will only work for monadic <source lang=apl inline>NonScalarFn</source>.
In dialects that support [[dfn]]s, this operator can be defined<ref>[[John Scholes]], [https://dfns.dyalog.com/n_perv.htm perv] (Scalar pervasion). dfns workspace, 2019-02-17.</ref> as:
<!-- [https://tio.run/##VZGxTsMwEIb3PMVtlwygZo3EVJURIWVAjFbsQKTItpwQFaEuCCUhKKgLIwOwMLB26cij@EXSc6IIasn23e@77@wz0/kJv2e5uhlsW9t6@2Bf32jjWaFtv7fdFzoTSV1eI09lgRv78kRuvKLlKsZVJWSJPsaiKDIlL01GLiDRMPBs26yJtlRcwHrQwlRjCQLXW9t9wvGw/TvECcuZARfKHM9zB@GZff6gLIrYuRlN9n6W5mTfCUzyYFS50OUtLCJQRCMVUglM6zxLWDmT/9UmVNv8fs@8P6AyM08QKAIjkjtTZJWA0rBKmILlp97Gu1Byuvy5nB65i5A6Qd0CTPHofHwe@AsIAwipSPcYwiKgH2g81yrb/wwH Try it online!] -->
<source lang=apl>
perv←{⍺←⊢              ⍝ Scalar pervasion
    1=≡⍺ ⍵ ⍵:⍺ ⍺⍺ ⍵    ⍝ (⍺ and) ⍵ depth 0: operand fn application
            ⍺ ∇¨⍵      ⍝ (⍺ or) ⍵ deeper: recursive traversal.
{{Works in|[[Dyalog APL]]}}
== External links ==
== External links ==

Revision as of 09:16, 1 April 2020

A scalar function is one of a class of primitive functions that apply to arguments one element at a time. Dyadic scalar functions pair elements of their arguments based on conformability rules, and thus are subject to scalar extension. In nested array languages, scalar functions pervade any nested arrays by recursively descending into them until reaching simple scalars; in flat array languages they usually do not apply inside boxes.

Only a particular valence of a function is labelled "scalar". The scalar monad Not usually shares the glyph ~ with non-scalar dyad Without, and similarly scalar Roll and non-scalar Deal are both written ?.

Standard scalar functions

Most APLs use a set of scalar functions that was worked out fairly early in APL's development. These are listed in this section.

Monadic function Glyph Dyadic function
Conjugate + Plus
Negate - Minus
Signum or Direction × Times
Reciprocal ÷ Divide
Floor Minimum
Ceiling Maximum
Exponential * Power function
Natural Logarithm Logarithm
Magnitude or Absolute value | Residue
Pi Times Circle function
Factorial ! Binomial coefficient or combination function
Roll ?
Not ~
Logical And
Logical Or
< Less than
Less than or equal to
= Equal to
Greater than or equal to
> Greater than
Not equal to

Additional scalar functions

Very few additional scalar functions have been added later in various dialects:

Monadic function Glyph Dyadic function
Square Root Nth Root
Type or
Lowest Common Multiple (LCM)
Greatest Common Divisor (GCD)

External links

APL features [edit]
Built-ins Primitive functionPrimitive operatorQuad name
Array model ShapeRankDepthBoundIndex (Indexing) ∙ AxisRavelRavel orderElementScalarVectorMatrixSimple scalarSimple arrayNested arrayCellMajor cellSubarrayEmpty arrayPrototype
Data types Number (Boolean, Complex number) ∙ Character (String) ∙ BoxNamespace
Concepts and paradigms Leading axis theoryScalar extensionConformabilityScalar functionPervasionGlyphIdentity elementComplex floorTotal array ordering