🎓 Chapter 2: Vectors, Matrices & Reshaping
In traditional programming, you have integers, floats, lists, and 2D grids as distinct data structures with different methods. In APL, everything is an Array.
Array Ranks:
- Scalar (Rank 0): A single isolated number (e.g.
42). - Vector (Rank 1): A 1D sequence of items (e.g.
1 2 3 4 5). - Matrix (Rank 2): A 2D grid with rows and columns (e.g. 3 × 4).
- Tensor (Rank 3+): Multi-dimensional hyper-cubes.
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1. Generating Vectors with Index Generator (⍳ N)
The Index Generator (⍳ N) (iota) generates a 1D vector of integers from 1 up to N.
> 💡 Note: In Dyalog APL default mode (⎕IO←1), indexing is 1-based, so ⍳ 5 yields 1 2 3 4 5.
⍝ Generate vector of numbers 1..12
V ← ⍳ 12
V---
2. Inspecting Dimensions with Monadic Shape (⍴ Y)
The Shape function (⍴ Y) (rho) returns the dimensions of any array Y as a vector.
- For a scalar
42,⍴ 42is empty (Rank 0). - For a vector
V ← ⍳ 12,⍴ Vreturns12(Rank 1, length 12).
⍝ Query shape of vector V
⍴ V---
3. Reshaping Arrays with Dyadic Reshape (X ⍴ Y)
When ⍴ is used as a Dyadic function (X ⍴ Y), it takes the elements from vector Y on the right and arranges them into a new array with shape X on the left!
Recycling Feature:
If Y has fewer elements than requested by shape X, APL automatically recycles (repeats) Y from the beginning until shape X is filled!
⍝ Reshape 12-element vector V into a 3x4 matrix
M ← 3 4 ⍴ V
M⍝ Recycling demonstration: fill a 4x4 matrix with pattern 1 0 1
4 4 ⍴ 1 0 1---
4. Matrix Indexing (M[row; col])
You can access elements, entire rows, or entire columns using bracket indexing M[row; col]:
M[2; 3]→ Element at row 2, column 3.M[2; ]→ Entire row 2.M[ ; 3]→ Entire column 3.
⍝ Extract row 2 and column 3 from matrix M
M[2; 3] ⋄ M[2; ]← Chapter 1: Foundations of APL & Array Thinking Chapter 3: Array Arithmetic & Boolean Masks →