Shape Reshape

Array basics

1. Expermient with the expressions in the following block to gain an understanding of the functions used.
2. Write a brief definition in English for each expression.
For example: {2×⍳⍵} ⍝ Even integers from 2 to 2×⍵ inclusive.
3. Use the language bar to discover the names of unfamiliar functions.
      3 4⍴⍳12   ⍝ A 3 row, 4 column matrix of the integers from 1 to 12 inclusive
⍴cards←'A23456789TJQK'∘.,'SCHD'
⍉cards
(,cards)[?×/⍴cards]
alph←2 3 4⍴⎕A
,alph


The shape of a scalar

Use the match function ⍺≡⍵ to determine which expressions below produce the empty character vector '' ("quote-quote") and which produce the empty numeric vector ⍬ ("zilde").

      0⍴0
0↑⎕A
⍳0
0⍴''
0⍴'def'
0 0⍴'abc'
2 0 3⍴⍳6
alph[⍸alph='Z']
⌈⌿⍬⍬
⎕JSON'{}'
⎕JSON'[]'
⎕JSON'""'


A simple array contains only single numbers and/or characters as elements. Single numbers or characters are called scalars. Arrays generally can contain any other array. These are called nested arrays. For example cards is a nested matrix of character vectors.

      ⍴¨1 2 3 4   ⍝ What is the shape of each number in ⍳4?
⍴¨'ABCDE'   ⍝ What is the shape of each letter in 5↑⎕A?
⍴¨cards     ⍝ What is the shape of each array in cards?


Problem set 5

1. Write a function IsEmpty to determine if an array has an empty axis.

      IsEmpty 2 0 3⍴⍳4
1

      IsEmpty 2 2 3⍴⍳4
0

      IsEmpty 0
0

      IsEmpty 0⍴0
1

2. Any simple array can be described using two vectors. Which two primitive functions (that is, functions represented by symbols) return these vectors when given a simple array argument?

3. The rank of an array is the number of axes or dimensions an array has. If the shape of an array ⍵ is given by the vector result of the shape function ⍴⍵, give an expression for the rank of an array.

1. IsEmpty←{0∊⍴⍵}
2. The functions shape ⍴⍵ and ravel ,⍵ return these two vectors, which are called the shape and the ravel of elements.
3. Traditionally, the rank is defined as the shape of the shape ⍴⍴⍵. However, in modern APL it is preferable to use the tally of the shape ≢⍴⍵.

1. Some Points in Space

The positions of 5 points in 3D space are given by the matrix pos:

pos←5 3⍴0 1 3 4 2

1. Write a function AddRows to add a vector to a matrix.

      1 ¯3 4 AddRows pos
1 ¯2 7
5 ¯1 4
2  0 8
3 ¯3 5
4  1 6

2. Write a function NormRows to normalise each vector in pos so that the sum of squares of each vector is 1.

      +/pos*2
10 20 26 5 29

      +/(NormRows pos)*2
1 1 1 1 1

      ÷/2-/pos
0.5 1 2 ¯2 ¯0.5

      ÷/2-/NormRows pos   ⍝ Relative proportions stay the same
0.5 1 2 ¯2 ¯0.5

Hint

The replicate / and reshape ⍴ functions might be useful.

Bonus

Write commutative versions of AddRows which will still work when the vector and matrix arguments are swapped.

1. This solutions assumes a vector left argument:
AddRows ← {⍵+(⍴⍵)⍴⍺}
2. Using only things which have been introduced so far:
NormRows ← {⍵÷(⍴⍵)⍴(⍴⍵)[2]/0.5*⍨+/⍵*2}

A commutative version of AddRows can be made by sorting using the ranks, and is expressed most clearly as two expressions separated by the statement separator ⋄

AddRows ← {(v m)←⍺ ⍵[(≢⍴)¨⍺ ⍵] ⋄ m+(⍴m)⍴v}   ⍝ v is the vector, m is the matrix
It can be coded in a single expression as a reduction, but then the result must be disclosed using pick ⊃, which we will cover later in the course. If you want to know why, look at the shape of the result with and without the pick ⊃
AddRows ← {⊃{⍵+(⍴⍵)⍴⍺}/⍺ ⍵[(≢⍴)¨⍺ ⍵]}
In the next section, we will see a better way to write functions that work on arrays of differing rank.

2. Without executing them, determine the shape of the results of the following expressions:

      1 3 5∘.!2 4 6 8
1 2 3 + 4 5 6
{(+⌿⍵)÷≢⍵}3 1 4 1 5
+⌿2 3⍴⍳6
?⌿2 3⍴3/4 52
⌈/(2 3⍴⍳6)∘.×¯1+?5 4⍴0

Hint

Of course, you can always check your answers using the shape function ⍴

  3 4 ≡ ⍴1 3 5∘.!2 4 6 8
2 3 5 ≡ ⍴⌈/(2 3⍴⍳6)∘.×¯1+?5 4⍴0