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{{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 [[ | {{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 == | ||
| Line 5: | Line 5: | ||
Ceiling rounds up the given numbers to the nearest integers. | Ceiling rounds up the given numbers to the nearest integers. | ||
< | <syntaxhighlight lang=apl> | ||
⌈2 2.8 ¯2 ¯2.8 | ⌈2 2.8 ¯2 ¯2.8 | ||
2 3 ¯2 ¯2 | 2 3 ¯2 ¯2 | ||
</ | </syntaxhighlight> | ||
== Properties == | == Properties == | ||
| Line 16: | Line 16: | ||
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. | ||
< | <syntaxhighlight lang=apl> | ||
⎕PP←16 | ⎕PP←16 | ||
⎕←v←1+0.6×⎕CT×0 1 2 | ⎕←v←1+0.6×⎕CT×0 1 2 | ||
| Line 22: | Line 22: | ||
⌈v | ⌈v | ||
1 1 2 | 1 1 2 | ||
</ | </syntaxhighlight> | ||
Ceiling is the dual to [[Floor]] by [[negate|negation]]. | Ceiling is the dual to [[Floor]] by [[negate|negation]]. | ||
< | <syntaxhighlight lang=apl> | ||
v←2 2.8 ¯2 ¯2.8 | v←2 2.8 ¯2 ¯2.8 | ||
(⌈v)≡-⌊-v | (⌈v)≡-⌊-v | ||
1 | 1 | ||
</ | </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. | ||
< | <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 | ||
| Line 40: | Line 40: | ||
(⌈v)≡-⌊-v | (⌈v)≡-⌊-v | ||
1 | 1 | ||
</ | </syntaxhighlight>{{Works in|[[Dyalog APL]]}} | ||
== External links == | == External links == | ||
| Line 46: | Line 46: | ||
=== Documentation === | === Documentation === | ||
* [ | * [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]] | ||