Residue

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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>

Works in: Dyalog APL

See also

External links

Documentation

APL built-ins [edit]
Primitives (Timeline) Functions
Scalar
Monadic Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root ∙ Round
Dyadic Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
Structural Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
Selection First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
Selector Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
Computational Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
Operators Monadic Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis ∙ Identity (Null, Ident)
Dyadic Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator) ∙ Identity (Lev, Dex)
Quad names Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
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