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{{Built-in|Ceiling|⌈}} is a [[monadic]] [[scalar function]] which gives the [[wikipedia:floor and ceiling functions|ceiling]] of a real number, i.e. the least integer greater than or equal to the given value. This operation is also known as '''round up'''. Ceiling shares the [[glyph]] <source lang=apl inline>⌈</source> with the dyadic arithmetic function [[Maximum]].
{{Built-in|Ceiling|⌈}} is a [[monadic]] [[scalar function]] that gives the [[wikipedia:floor and ceiling functions|ceiling]] of a real number, that is, the least integer tolerantly<ref>[[Robert Bernecky|Bernecky, Robert]]. [https://www.jsoftware.com/papers/satn23.htm "Comparison Tolerance"]. Sharp APL Technical Notes. 1977-06-10;.</ref> [[greater than or equal to]] the given value. This operation is also known as '''round up'''. Ceiling shares the [[glyph]] <syntaxhighlight lang=apl inline>⌈</syntaxhighlight> with the dyadic arithmetic function [[Maximum]]. [[Comparison_with_traditional_mathematics#Prefix|Traditional mathematics]] derives [[Ken_Iverson#Floor_and_Ceiling|its notation]] and name for ceiling from APL.


== Examples ==
== Examples ==
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Ceiling rounds up the given numbers to the nearest integers.
Ceiling rounds up the given numbers to the nearest integers.


<source lang=apl>
<syntaxhighlight lang=apl>
       ⌈2 2.8 ¯2 ¯2.8
       ⌈2 2.8 ¯2 ¯2.8
2 3 ¯2 ¯2
2 3 ¯2 ¯2
</source>
</syntaxhighlight>


== Properties ==
== Properties ==
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Ceiling is affected by [[comparison tolerance]]. If the given number is [[tolerant comparison|tolerantly equal]] to its [[floor]], it is rounded to that number instead.
Ceiling is affected by [[comparison tolerance]]. If the given number is [[tolerant comparison|tolerantly equal]] to its [[floor]], it is rounded to that number instead.


<source lang=apl>
<syntaxhighlight lang=apl>
       ⎕PP←16
       ⎕PP←16
       ⊢v←1+0.6×⎕CT×0 1 2
       ⎕←v←1+0.6×⎕CT×0 1 2
1 1.000000000000006 1.000000000000012
1 1.000000000000006 1.000000000000012
       ⌈v
       ⌈v
1 1 2
1 1 2
</source>
</syntaxhighlight>


Ceiling is the dual to [[Floor]] by [[negate|negation]].
Ceiling is the dual to [[Floor]] by [[negate|negation]].


<source lang=apl>
<syntaxhighlight lang=apl>
       v←2 2.8 ¯2 ¯2.8
       v←2 2.8 ¯2 ¯2.8
       (⌈v)≡-⌊-v
       (⌈v)≡-⌊-v
1
1
</source>
</syntaxhighlight>


The extension for [[complex number|complex numbers]] is derived from [[complex Floor]] via this property.
The extension for [[complex number|complex numbers]] is derived from [[complex Floor]] via this property.


<source lang=apl>
<syntaxhighlight lang=apl>
       v←1.8J2.5 2.5J2.2 1.7J2.2
       v←1.8J2.5 2.5J2.2 1.7J2.2
       ⌈v
       ⌈v
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       (⌈v)≡-⌊-v
       (⌈v)≡-⌊-v
1
1
</source>{{Works in|[[Dyalog APL]]}}
</syntaxhighlight>{{Works in|[[Dyalog APL]]}}


== External links ==
== External links ==
Line 46: Line 46:
=== Documentation ===
=== Documentation ===


* [http://help.dyalog.com/latest/#Language/Primitive%20Functions/Ceiling.htm Dyalog]
* [https://help.dyalog.com/latest/#Language/Primitive%20Functions/Ceiling.htm Dyalog]
* [http://microapl.com/apl_help/ch_020_020_090.htm APLX]
* [http://microapl.com/apl_help/ch_020_020_090.htm APLX]
* [https://www.jsoftware.com/help/dictionary/d021.htm J Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/gtdot NuVoc]
* [https://www.jsoftware.com/help/dictionary/d021.htm J Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/gtdot NuVoc]
* [https://mlochbaum.github.io/BQN/doc/arithmetic.html#additional-arithmetic BQN]
== References ==
<references />
{{APL built-ins}}[[Category:Primitive functions]][[Category:Scalar monadic functions]]
{{APL built-ins}}[[Category:Primitive functions]][[Category:Scalar monadic functions]]

Latest revision as of 21:57, 28 November 2022

⌈

Ceiling (⌈) is a monadic scalar function that gives the ceiling of a real number, that is, the least integer tolerantly[1] greater than or equal to the given value. This operation is also known as round up. Ceiling shares the glyph ⌈ with the dyadic arithmetic function Maximum. Traditional mathematics derives its notation and name for ceiling from APL.

Examples

Ceiling rounds up the given numbers to the nearest integers. 

      ⌈2 2.8 ¯2 ¯2.8
2 3 ¯2 ¯2

Properties

The ceiling of any real number is an integer.

Ceiling is affected by comparison tolerance. If the given number is tolerantly equal to its floor, it is rounded to that number instead. 

      ⎕PP←16
      ⎕←v←1+0.6×⎕CT×0 1 2
1 1.000000000000006 1.000000000000012
      ⌈v
1 1 2

Ceiling is the dual to Floor by negation. 

      v←2 2.8 ¯2 ¯2.8
      (⌈v)≡-⌊-v
1

The extension for complex numbers is derived from complex Floor via this property. 

      v←1.8J2.5 2.5J2.2 1.7J2.2
      ⌈v
2J3 3J2 2J2
      (⌈v)≡-⌊-v
1
Works in: Dyalog APL

External links

Documentation

References

  1. ↑ Bernecky, Robert. "Comparison Tolerance". Sharp APL Technical Notes. 1977-06-10;.
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