Scalar Functions and Scalar Extension

Scalar functions are the workhorses of APL. They operate element-by-element on arrays of any shape, and they handle mismatched sizes gracefully through scalar extension.

Scalar functions apply element-wise

When you add two vectors of the same length, the function is applied to each pair of corresponding elements:

      1 2 3 + 10 20 30
11 22 33
      5 10 15 × 2 3 4
10 30 60

This works for all scalar functions — arithmetic, comparison, boolean, and more:

      3 5 7 > 4 4 4
0 1 1
      1 0 1 ∧ 1 1 0
1 0 0

Scalar extension

When one argument is a scalar and the other is an array, the scalar is extended — paired with every element:

      100 + 1 2 3 4 5
101 102 103 104 105
      1 2 3 4 5 × 10
10 20 30 40 50
      0 = 3 0 5 0 7
0 1 0 1 0

This works with arrays of any rank:

      M ← 2 3 ⍴ 1 2 3 4 5 6
      10 × M
10 20 30
40 50 60

Matching shapes

If both arguments are arrays (not scalars), they must have the same shape. Mismatched shapes cause a LENGTH ERROR:

      1 2 3 + 10 20
LENGTH ERROR

Monadic scalar functions

Most scalar functions also have a monadic form (one argument, on the right). The function is applied to each element:

Dyadic	Monadic	Meaning
X + Y (add)	+ Y (conjugate/identity)	Returns Y unchanged for real numbers
X - Y (subtract)	- Y (negate)	Flips sign
X × Y (multiply)	× Y (signum)	Returns ¯1, 0, or 1
X ÷ Y (divide)	÷ Y (reciprocal)	1÷Y
X * Y (power)	* Y (exponential)	e*Y
X ⍟ Y (logarithm)	⍟ Y (natural log)	ln Y
X ⌈ Y (maximum)	⌈ Y (ceiling)	Round up
X ⌊ Y (minimum)	⌊ Y (floor)	Round down
X \| Y (residue)	\| Y (magnitude)	Absolute value

      - 3 ¯5 0 7
¯3 5 0 ¯7
      ⌈ 2.3 4.7 ¯1.2
3 5 ¯1
      | ¯3 5 ¯7 0
3 5 7 0

Negative numbers

APL uses ¯ (high minus) for negative numbers, not -. So negative three is ¯3, not -3. The - symbol is the subtract function.

The complete list

MARPLE implements these scalar functions. Each applies element-wise and supports scalar extension.

Arithmetic: + − × ÷ ⌈ ⌊ * ⍟ | ! ○

Comparison: < ≤ = ≥ > ≠

Boolean: ∧ ∨ ⍲ ⍱ ~

Comparison functions return 1 (true) or 0 (false). Boolean functions operate on 0s and 1s.

See the Primitive Functions Reference for full details on each one.

Key points

• Scalar functions work element-by-element on arrays of any shape
• If both arguments are arrays, their shapes must match
• If one argument is a scalar, it's extended to match the other
• Most scalar functions have both monadic and dyadic forms
• Comparison functions return 0 or 1

Next: Reduce and Scan