The Reduce Operator `/` and `⌿`

Hybridity

/ and ⌿ are hybrids - they are both functions and operators. For use of / and ⌿ as functions please see the appropriate operator documentation.

When used as operators / and ⌿ apply functions across the arguments of a vector. If applied to a scalar they first convert it to a vector with a single element.

Monadic Use On Scalars

The ‘/’ operator applys a function to a scalar:

+/1

It returns a scalar.

⍴ +/1

Gives:

Monadic Use On Vectors

The ‘/’ operator successively applies a function to the elements of a vector:

+/ 1 2 3 4

and

×/1 2 3 4

giving:

’/’ always reduces the rank of the operand to the right (hence the name reduce):

A ← 2 2 ⍴ 1 22 333
+/ A

gives:

23 334

A of course was a 2 by 2 matrix:

  1 22
333  1

Each dimensions was reduced seperately.

Gotcha’s

The / operator reduces the rank and it will mondically reduce a vector of length 1 to a scalar:

A ← 1 ⍴ 4
⍴ +/ A

Gives a null result - its a scalar:

Dyadic Use On Scalars Are An Error

Attempting to use ‘/’ dyadically on a scalar produces an error:

1 +/ 1

Resulting in:

Error
1 +/ 1
-----^
RANK ERROR [RHS must be an array: It is a scalar ] on line 1 at character 6

It works on a vector of length 1, of course:

A ← 1 ⍴ 4
1 +/ A

Resulting in:

Moar Gotcha’s

See the points for improvement page for outstanding issues.

Dyadic Use On RHS Vectors With An LHS Scalar

The ‘/’ vector when used dyadically applies a sliding window:

2 +/ 1 2 3 4

Giving:

3 5 7

The LHS scalar can be replaced with a vector of length 1 with the same results:

A ← 1 ⍴ 2
A +/ 1 2 3 4

and as expected:

3 5 7

This is 1 + 2 then 2 + 3 then 3 + 4

This scales up to objects of higher shape

A ← 3 3 ⍴ 1 2 3 4
2 +/ A

Which returns:

3 5
5 3
7 5

We can derive that by looking at the intermediate shape:

3 3 ⍴ 1 2 3 4

which is of course:

1 2 3
4 1 2
3 4 1

The shape has been reduced from 3 3 to 3 2

Dyadic Use On RHS Vectors With An LHS Vector

This throws a LENGTH ERROR:

2 2 +/ 1 2 3 4

Giving

Error
2 2 +/ 1 2 3 4
^
LENGTH ERROR [The operator \"/\" can only take a scalar on the LHS: The shape of the LHS is [2] ] on line 1 at character 1

Using a lazy vector gives a different error - it is thrown at runtime during evaluation and not before it:

Error
2 2 +/ 1 2 3 4
^
LENGTH ERROR [The operator \"/\" can only take a scalar on the LHS: The shape of the LHS is unsized_vector ] on line 1 at character 1

Reducing With Ranks

By default / operates on the first (ie right hand axis) but you can specify the rank you wish to reduce on:

A ← ⍳ 24
B ← 2 3 4 ⍴ A
2 +/[2] B

Gives the sum over the second rank:

 6  8 10 12
14 16 18 20

30 32 34 36
38 40 42 44

The value of A is:

1  2  3  4
5  6  7  8
9 10 11 12

and a shape of 3 4. After reduction with rank its shape is 2 4

The Barred Reduce Operator ⌿

Like the function ⌿ the barred reduce operator operates on the last axis:

Whereas / without a rank:

A ← 2 2 ⍴ 1 22 333 444
+/ A