Reduce and Scan¶

Reduce and scan are your first operators. An operator takes a function and produces a new, derived function. They're one of APL's most powerful ideas.

Reduce: `/`¶

Reduce inserts its operand function between every element of a vector. +/ inserts + between every element, giving a sum:

      +/ 1 2 3 4 5
15

This is equivalent to 1 + 2 + 3 + 4 + 5. But because APL evaluates right to left, the reduction actually computes 1 + (2 + (3 + (4 + 5))).

Any dyadic function can be the operand:

      ×/ 1 2 3 4 5
120
      ⌈/ 3 1 4 1 5 9
9
      ⌊/ 3 1 4 1 5 9
1

×/ gives the product. ⌈/ gives the maximum. ⌊/ gives the minimum.

Reduce on matrices¶

When applied to a matrix, / reduces along the last axis — that is, across each row:

      M ← 3 4 ⍴ ⍳12
      M
 1  2  3  4
 5  6  7  8
 9 10 11 12
      +/ M
10 26 42

Each row is summed independently, giving a 3-element vector.

To reduce along the first axis (down the columns), use ⌿:

      +⌿ M
15 18 21 24

Rank operator alternative

You can also use the rank operator: (+/⍤1) M sums each row, (+/⍤2) M sums each column of a matrix. The rank operator tutorial covers this in detail.

Scan: `\`¶

Scan is like reduce, but it keeps all the intermediate results. +\ gives a running sum:

      +\ 1 2 3 4 5
1 3 6 10 15

This produces: 1, 1+2, 1+2+3, 1+2+3+4, 1+2+3+4+5.

      ×\ 1 2 3 4 5
1 2 6 24 120
      ⌈\ 3 1 4 1 5 9
3 3 4 4 5 9

×\ gives running products. ⌈\ gives running maximums.

Scan on matrices¶

Like reduce, \ works along the last axis (across rows), and ⍀ works along the first axis (down columns):

      +\ 3 3 ⍴ ⍳9
1  3  6
4  9 15
7 17 24

      +⍀ 3 3 ⍴ ⍳9
 1  2  3
 5  7  9
12 15 18

The key idea: operators transform functions¶

The important thing isn't the specific examples — it's the concept. / and \ are operators: they take a function (like + or ⌈) and produce a new function (+/, ⌈\). This is APL's version of higher-order functions.

MARPLE has several operators:

Operator	Name	What it does
`/`	Reduce	Insert function between elements (last axis)
`\`	Scan	Running reduce (last axis)
`⌿`	First-axis reduce	Reduce along first axis
`⍀`	First-axis scan	Scan along first axis
`∘.`	Outer product	Apply function to all pairs
`.`	Inner product	Generalised matrix multiply
`⍤`	Rank	Apply function at a specific cell rank

You'll meet the others in later tutorials.

Key points¶

/ (reduce) inserts a function between elements: +/ sums, ×/ multiplies, ⌈/ finds the max
\ (scan) keeps intermediate results: +\ gives running sums
On matrices, / and \ work across rows (last axis); ⌿ and ⍀ work down columns (first axis)
Operators take functions and produce new functions — this is one of APL's core ideas

Next: Reshape and Iota — Building Arrays