---
title: "Operators and axis"
author: "Jeremy Howard"
date: "2022-07-05"
categories: [Dyalog, APL, Glyphs]
---
]box on -style=max -trains=tree -fns=on

3 2⍴⍳6

1 4 5 =[1] 3 2⍴⍳6

⎕←mat ← 2 3 ⍴ 10 20 30 40 50 60 

mat+[1]1 2    ⍝ add along first axis

⎕ ← cube ← 2 2 2 ⍴ ⍳8

, cube

, (1 2)(1 2)

'ABC'

,[0.5]'ABC'

⍴'ABC'

⍴,[0.5]'ABC'

⍴,[⍬]'ABC'

2 2 2⍴⍳8

⎕←M ← 2 3 4 ⍴ ⍳24

,[1 2]M

⍴,[1 2]M

,[2 3]M

⍴,[2 3]M

1 2 3 , 4 5 6

cube ← 2 2 2 ⍴ ⍳8
cube , 99

rect←3 2⍴⍳6
rect

⍴rect

rect,10

rect,[1]10

rect,[0.5]10

'HEADING',[0.5]'-'

⍪1

⎕←,5⍴⎕A
⎕←⍪5⍴⎕A

2 3 4⍴⎕A

⍴⍪2 3 4⍴⎕A

⍪2 3 4⍴⎕A

1 2 3 ⍪ 4 5 6

cube ← 2 2 2 ⍴ ⍳8
cube ⍪ 99

⎕ ← a ← ⍳5
+/a

a ← ⍳5
×/a

a ← ⍳3
÷/a

a ← 4 6 2
⌈/ a

a ← 4 6 2
⌊/ a

×/⍳5

!5

3+/⍳4  ⍝ (1+2+3) (2+3+4)

2+/⍳4  ⍝ (1+2) (2+3) (3+4)

3÷⍨3+/⍳4

0+/⍳4  ⍝ Identity element for +

0×/⍳4  ⍝ Identity element for ×

¯2,/⍳4⍝ (2,1) (3,2) (4,3)

mat←2 3⍴⍳6
+/[1]mat

mat←2 3⍴⍳6
+/[2]mat

mat←2 3⍴⍳6
+/mat

a ← ⍳5
+\a

a ← ⍳5
×\a

⎕ ← a ← ⍳3
÷\a

⎕←mat ← 2 3 ⍴ ⍳6
+/mat

⎕←mat ← 2 3 ⍴ ⍳6
+/[1]mat

+⌿mat

⎕ ← mat ← 2 3 ⍴ ⍳6
+⍀mat

trace←{⍺←⊢⋄⎕←'⍺: '⍺ '⍵: '⍵⋄ ⍺ ⍺⍺ ⍵} ⍝ explainer function

⎕←cube ← 2 3 4 ⍴ ⍳24
(+⌿⍤1)cube

(+⌿trace⍤1)cube ⍝ show the input to pluse reduce first

⎕←mat ← 3 4 ⍴ ⍳12
+⌿mat

⎕←cube
(+⌿⍤2)cube

⎕←mat
1 2 3 (+trace⍤0 1) mat

f ← *⍤÷

⎕←*(÷3)
⎕←f 3

⎕←*2÷3
⎕←2 f 3

sqr ← *∘2

sqr 3

pow2 ← 2∘*

pow2 3

f ← *∘÷

*(÷3)

f 3

2 f 3

2 * (÷3)

2*÷3

f ← *⍥÷

*(÷3)

f 3

2 f 3

(÷2)*÷3

10 (÷⍥!) 6   ⍝ P(10,4)

(!10)÷!(10-4)  ⍝ P(10,4)

S ← +∘1

S 0

(S⍣3) 0

add ← {(S⍣⍺) ⍵}

2 add 3

mult ← {⍺ (add⍣⍵) 0}

3 mult 4

P ← S⍣¯1

P 3

(S⍣¯3) 5

sqr ← *∘2

(sqr⍣¯1)9

pow ← {⍺ (mult⍣⍵) 1}

2 pow 3

1 +∘÷⍣= 1

f ← +∘÷

1 f 1

1 f 2

1 f 1.5

1 (f⍣15) 1

1 (f⍣=) 1

1 2 3 +.× 4 5 6  ⍝ Dot product

3 ∧.= 3 3 3 3  ⍝ All-equal

⎕←mat←2 2⍴⍳4
mat +.× mat   ⍝ matrix product

1 2 3 ∘.× 4 5 6 7  ⍝ Special case: outer prodct

mat←2 2⍴⍳4
⎕←inv←⌹ mat

inv +.× mat  ⍝ Identity

⎕←div←5 6 ⌹ mat

mat +.× div

f ← *∘2

d ← 0.01
x ← 3
((f (x+d)) - f x) ÷ d

d ← 0.0001
((f (x+d)) - f x) ÷ d

grad ← {((⍺⍺ ⍺+⍵) - ⍺⍺ ⍺) ÷ ⍵}
3 f grad 0.01

3-2

2-3

3-⍨2

v←22 10 22 22 21 10 5 10
v/⍨≠v

grad ← {⍵ ÷⍨ (⍺⍺ ⍺+⍵) - ⍺⍺ ⍺}
3 f grad 0.01

pow ← ×⍨
pow 3

zero ← 0⍨
2 zero 5

⎕ ← a ← (1 2 3 4)(5 6 7)

+/¨a

⎕ ← b ← (1 2 3)(4 5 6)

2 3 +¨ b