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APL Programming Tutorial

APL (A Programming Language) was created by Kenneth Iverson in 1962. It uses a rich set of mathematical symbols and operates on arrays as first-class values. APL has influenced every array language since — NumPy, MATLAB, Julia, and more.

Special Symbols

APL uses unique glyphs. Common ones:

Glyph Name Meaning
← Assignment x ← 5
⍳ Iota Generate sequence ⍳5 → 1 2 3 4 5
⍴ Rho Shape/reshape of array
+/ Reduce Sum: +/1 2 3 → 6
×/ Reduce Product: ×/⍳5 → 120
∘.× Outer product Multiplication table
⌈ Ceiling / max
⌊ Floor / min
⍉ Transpose
⌽ Reverse
↑ Take First N elements
↓ Drop Skip N elements

Hello World

⎕←'Hello from APL!'
⎕←'Welcome to Time Warp Studio'

⎕← prints to the screen (quote-print).

Arithmetic

⎕← 2 + 3          ⍝ 5
⎕← 10 - 4         ⍝ 6
⎕← 6 × 7          ⍝ 42
⎕← 10 ÷ 4         ⍝ 2.5
⎕← 2 * 10         ⍝ 1024
⎕← |¯3            ⍝ 3  (absolute value)
⎕← ⌊3.7           ⍝ 3  (floor)
⎕← ⌈3.2           ⍝ 4  (ceiling)

Note: ¯ (high minus) is APL's negative sign: ¯5 means -5.

Arrays

⍝ Create a vector (1D array)
v ← 1 2 3 4 5
⎕← v
⎕← ⍴v             ⍝ shape: 5

⍝ Iota generates sequences
⎕← ⍳10            ⍝ 1 2 3 4 5 6 7 8 9 10

⍝ Array arithmetic (whole-array at once)
⎕← v × 2          ⍝ 2 4 6 8 10
⎕← v + v          ⍝ 2 4 6 8 10
⎕← v * 2          ⍝ 1 4 9 16 25 (element-wise square)

⍝ Scalar operations broadcast automatically
⎕← 100 + ⍳5       ⍝ 101 102 103 104 105

Reduction

⍝ +/ sums all elements
⎕← +/ ⍳100        ⍝ 5050  (sum 1..100)

⍝ ×/ multiplies (factorial)
⎕← ×/ ⍳10         ⍝ 3628800 = 10!

⍝ ⌈/ max element, ⌊/ min element
⎕← ⌈/ 3 1 4 1 5 9 ⍝ 9
⎕← ⌊/ 3 1 4 1 5 9 ⍝ 1

Outer Product

⍝ Multiplication table
⎕← (⍳5) ∘.× (⍳5)

Output:

 1  2  3  4  5
 2  4  6  8 10
 3  6  9 12 15
 4  8 12 16 20
 5 10 15 20 25

Matrices

⍝ Reshape into matrix
m ← 3 3 ⍴ ⍳9          ⍝ 3×3 matrix from 1..9
⎕← m
⎕← ⍴m                  ⍝ shape: 3 3
⎕← ⍉m                  ⍝ transpose
⎕← ⌽m                  ⍝ reverse rows
⎕← ⊖m                  ⍝ reverse columns (flip)

Boolean and Comparison

v ← 3 1 4 1 5 9 2 6
⎕← v > 4              ⍝ 0 0 0 0 1 1 0 1  (boolean mask)
⎕← (v > 4) / v        ⍝ 5 9 6  (compress / select)
⎕← +/ v > 4           ⍝ 3  (count elements > 4)

Defined Functions

⍝ Dfn (direct function)
factorial ← {×/⍳⍵}
⎕← factorial 10      ⍝ 3628800

primes ← {
    mask ← (⍳⍵) ∘.| ⍳⍵   ⍝ outer product with remainder
    2 ↓ +/ (mask = 0) = 1
}

⍝ Traditional APL function
∇ result ← SQUARE x
    result ← x × x
∇
⎕← SQUARE 9    ⍝ 81

Further Reading