Residue: Difference between revisions
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{{Built-in|Residue|<nowiki>|</nowiki>}}, '''Remainder''', or '''Modulus''' is a [[dyadic]] [[scalar function]] which gives the [[wikipedia:Remainder|remainder]] of [[divide|division]] between two real numbers. It takes the divisor as the left [[argument]], and the dividend as the right argument. Residue shares the [[glyph]] <source lang=apl inline>|</ | {{Built-in|Residue|<nowiki>|</nowiki>}}, '''Remainder''', or '''Modulus''' is a [[dyadic]] [[scalar function]] which gives the [[wikipedia:Remainder|remainder]] of [[divide|division]] between two real numbers. It takes the divisor as the left [[argument]], and the dividend as the right argument. Residue shares the [[glyph]] <source lang=apl inline>|</syntaxhighlight> with the monadic arithmetic function [[Magnitude]]. | ||
== Examples == | == Examples == | ||
| Line 8: | Line 8: | ||
3.5|5 10 14 | 3.5|5 10 14 | ||
1.5 3 0 | 1.5 3 0 | ||
</ | </syntaxhighlight> | ||
== Properties == | == Properties == | ||
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x≡(y|x)+y×⌊x÷y | x≡(y|x)+y×⌊x÷y | ||
1 | 1 | ||
</ | </syntaxhighlight> | ||
== Caveats == | == Caveats == | ||
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For negative arguments, one may decide to return non-negative numbers in all cases or follow the sign of the dividend or the divisor. For complex arguments, the [[floor]] of a complex number is not mathematically defined, so allowing complex arguments does not add much of mathematical value. | For negative arguments, one may decide to return non-negative numbers in all cases or follow the sign of the dividend or the divisor. For complex arguments, the [[floor]] of a complex number is not mathematically defined, so allowing complex arguments does not add much of mathematical value. | ||
Dyalog APL uses the expression <source lang=apl inline>Y-X×⌊Y÷X+0=X</ | Dyalog APL uses the expression <source lang=apl inline>Y-X×⌊Y÷X+0=X</syntaxhighlight> as the definition of <source lang=apl inline>X|Y</syntaxhighlight>, so that the above identity holds for all possible arguments. With this definition, zero X returns Y unchanged, and negative X returns a value between X and 0 (excluding the value X). The result for complex arguments is also defined (since Dyalog APL allows them as the argument to [[Floor]]). | ||
<source lang=apl> | <source lang=apl> | ||
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3J4{⍵-⍺×⌊⍵÷⍺+0=⍺}5J12 | 3J4{⍵-⍺×⌊⍵÷⍺+0=⍺}5J12 | ||
3J1 | 3J1 | ||
</ | </syntaxhighlight>{{Works in|[[Dyalog APL]]}} | ||
== See also == | == See also == | ||
Revision as of 21:08, 10 September 2022
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Residue (|), Remainder, or Modulus is a dyadic scalar function which gives the remainder of division between two real numbers. It takes the divisor as the left argument, and the dividend as the right argument. Residue shares the glyph <source lang=apl inline>|</syntaxhighlight> with the monadic arithmetic function Magnitude.
Examples
<source lang=apl>
2|¯2 ¯1 0 1 2 3 4 5
0 1 0 1 0 1 0 1
3.5|5 10 14
1.5 3 0 </syntaxhighlight>
Properties
For positive x and y, the following identity holds:
<source lang=apl>
x←?⍨10 ⋄ y←?⍨10
x≡(y|x)+y×⌊x÷y
1 </syntaxhighlight>
Caveats
The usual definition of "remainder" only holds when both arguments are positive. An implementation is free to decide what to do when the left argument is zero, or at least one of the arguments is negative or complex.
For negative arguments, one may decide to return non-negative numbers in all cases or follow the sign of the dividend or the divisor. For complex arguments, the floor of a complex number is not mathematically defined, so allowing complex arguments does not add much of mathematical value.
Dyalog APL uses the expression <source lang=apl inline>Y-X×⌊Y÷X+0=X</syntaxhighlight> as the definition of <source lang=apl inline>X|Y</syntaxhighlight>, so that the above identity holds for all possible arguments. With this definition, zero X returns Y unchanged, and negative X returns a value between X and 0 (excluding the value X). The result for complex arguments is also defined (since Dyalog APL allows them as the argument to Floor).
<source lang=apl>
5 5 ¯5 ¯5 0 0|2 ¯2 2 ¯2 2 ¯2
2 3 ¯3 ¯2 2 ¯2
3J4|5J12
3J1
3J4{⍵-⍺×⌊⍵÷⍺+0=⍺}5J12
3J1
</syntaxhighlight>