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Reduce: Difference between revisions
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{{Built-ins|Reduce|/|⌿}}, also called '''Reduction''' or '''Insert''', is a [[primitive operator|primitive]] [[monadic operator]] which takes a [[dyadic function]] [[operand]], inserts it between the [[element|elements]] of the [[argument]], and evaluates it into a single array in right-to-left order. This operation is known as [[wikipedia:Fold (higher-order function)|Fold]], or more specifically < | {{Built-ins|Reduce|/|⌿}}, also called '''Reduction''' or '''Insert''', is a [[primitive operator|primitive]] [[monadic operator]] which takes a [[dyadic function]] [[operand]], inserts it between the [[element|elements]] of the [[argument]], and evaluates it into a single array in right-to-left order. This operation is known as [[wikipedia:Fold (higher-order function)|Fold]], or more specifically <syntaxhighlight lang=text inline>foldr1</source>, in other functional programming languages such as Haskell. | ||
== Description == | == Description == | ||
When applied to a [[vector]] argument, < | When applied to a [[vector]] argument, <syntaxhighlight lang=apl inline>f/x</source> evaluates to the expression <syntaxhighlight lang=text inline>a f b f c f d …</source> where <syntaxhighlight lang=text inline>a, b, c, d, …</source> are the elements of <syntaxhighlight lang=text inline>x</source>. In general, Reduce reduces one chosen [[axis]] (either implied by using the last-axis form <syntaxhighlight lang=apl inline>f/</source> or first-axis <syntaxhighlight lang=apl inline>f⌿</source>, or explicitly by using [[function axis]] <syntaxhighlight lang=apl inline>f/[x]</source>) by evaluating each [[vector]] along the chosen axis into a [[scalar]]. | ||
In [[nested array model]], Reduce has a strong property that the reduced axis is removed from the [[shape]] of the argument, which forces it to [[enclose]] each non-[[simple]] result in the returned array. | In [[nested array model]], Reduce has a strong property that the reduced axis is removed from the [[shape]] of the argument, which forces it to [[enclose]] each non-[[simple]] result in the returned array. | ||
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Reduce is mainly used for aggregation, such as sum (using [[Add]]) or product (using [[Times]]). If used with [[Subtract]], it computes the alternating sum, since <math>a-(b-(c-(d-\cdots))) = a-b+c-d+\cdots</math>. Using with [[Divide]] gives similar effect, returning the alternating product <math>a\div(b\div(c\div(d\div\cdots))) = a\div b\times c\div d\times\cdots</math>. | Reduce is mainly used for aggregation, such as sum (using [[Add]]) or product (using [[Times]]). If used with [[Subtract]], it computes the alternating sum, since <math>a-(b-(c-(d-\cdots))) = a-b+c-d+\cdots</math>. Using with [[Divide]] gives similar effect, returning the alternating product <math>a\div(b\div(c\div(d\div\cdots))) = a\div b\times c\div d\times\cdots</math>. | ||
< | <syntaxhighlight lang=apl> | ||
+/1 2 3 4 5 | +/1 2 3 4 5 | ||
15 | 15 | ||
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Reduction by [[Minimum]] or [[Maximum]] gives the minimum or maximum over several numbers. Same goes for [[And]], [[Or]], [[GCD]], [[LCM]], and XOR ([[Not Equal]] on [[Booleans]]). | Reduction by [[Minimum]] or [[Maximum]] gives the minimum or maximum over several numbers. Same goes for [[And]], [[Or]], [[GCD]], [[LCM]], and XOR ([[Not Equal]] on [[Booleans]]). | ||
Although Reduce is < | Although Reduce is <syntaxhighlight lang=text inline>foldr1</source> in nature, one can use it like <syntaxhighlight lang=text inline>foldr</source>, where a designated starting value is modified by the rest of the values in sequence. In this case, the start value (enclosed if not a [[simple scalar]]) is attached to the right end of the [[vector]] of "modifiers", and then the entire vector is reduced. | ||
< | <syntaxhighlight lang=apl> | ||
(⍉∘⌽↓)/2 1 2 1,⊂5 6⍴⍳30 ⍝ Trim a matrix from all four sides, by rotating the matrix after each trim | (⍉∘⌽↓)/2 1 2 1,⊂5 6⍴⍳30 ⍝ Trim a matrix from all four sides, by rotating the matrix after each trim | ||
┌─────┐ | ┌─────┐ | ||
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Reduction over an empty axis gives the [[identity element]] of the operand. | Reduction over an empty axis gives the [[identity element]] of the operand. | ||
< | <syntaxhighlight lang=apl> | ||
+/⍬ | +/⍬ | ||
0 | 0 | ||