🎓 Chapter 4: Reductions & Cumulative Scans

In conventional programming, calculating a sum or running total requires allocating an accumulator variable and running a loop.

In APL, you use Operators like Reduction (/) and Scan (\)!

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1. Reduction (f/)

Reduction (f/ Y) places any primitive function f between all elements of vector Y.

Examples:

• Sum Reduction (+/): +/ 1 2 3 4 → 1 + 2 + 3 + 4 = 10
• Product Reduction (×/): ×/ 1 2 3 4 → 1 × 2 × 3 × 4 = 24
• Max Reduction (⌈/): ⌈/ 3 8 2 9 → Maximum element = 9
• Min Reduction (⌊/): ⌊/ 3 8 2 9 → Minimum element = 2

⍝ Sum and product of vector 1..5
(+/ 1 2 3 4 5) ⋄ (×/ 1 2 3 4 5)

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2. Cumulative Scans (f\)

While Reduction collapses an array down to a single number, Scan (f\ Y) keeps intermediate cumulative results after each step!

Examples:

• Cumulative Sum (+\): +\ 1 2 3 4 → 1, 1+2, 1+2+3, 1+2+3+4 = 1 3 6 10.
• Running Maximum (⌈\): ⌈\ 3 8 2 9 4 → 3 8 8 9 9.

⍝ Cumulative sum scan of numbers 1..10
+\ ⍳ 10

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3. Matrix Reductions (Row vs Column Aggregation)

On 2D matrices:

• Row Reduction (f/ M): Collapses columns to compute row-wise totals.
• Column Reduction (f⌿ M): Collapses rows to compute column-wise totals.

M ← 3 4 ⍴ ⍳ 12
⍝ Row sums vs Column sums
(+/ M) ⋄ (+⌿ M)

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