Discovering APL

Some APL Examples

Here is an APL program to calculate the average (arithmetic mean) of a list of numbers. It is written in D, the ‘direct’ form of Dyalog APL.

•  {(+/ω)÷ρω} 

It is unnamed: the enclosing braces mark it as a function definition. It can be assigned a name for use later, or used anonymously in a more complex expression.

The ω refers to the argument of the function, a list (or 1-dimensional array) of numbers. The ρ denotes the shape function, which returns here the length of (number of elements in) the argument ω. The divide symbol ÷ has its usual meaning.

The parenthesised +/ω denotes the sum of all the elements of ω. The / operator combines with the + function: the / fixes the + function between each element of ω, so that

• +/ 1 2 3 4 5 6

21

is the same as

• 1+2+3+4+5+6

21

Operators like / can be used to derive new functions not only from primitive functions like +, but also from defined functions. For example

• {α,', ',ω}/

will transform a list of strings representing words into a comma-separated list:

• {α,', ',ω}/'cow' 'sheep' 'cat' 'dog'

cow, sheep, cat, dog

So back to our mean example. (+/ω) gives the sum of the list, which is then divided by ρω, the number of its elements.

• {(+/ω)÷ρω} 3 4.5 7 21

8.875

The same program in J

In J’s tacit definition no braces are needed to mark the definition of a function: primitive functions just combine in a way that enables us to omit any reference to the function arguments — hence tacit.

Here is the same calculation written in J:

• (+/%#) 3 4.5 7 21

8.875

In J’s terminology, functions are called verbs and operators adverbs. So: the verb # gives the length of the argument. Division is marked by % instead of ÷. The sum verb is again marked by +/: the verb + is modified by the adverb /.

The adverb \ can be used to modify the +/%# verb to produce a moving average.

• 2 (+/%#)\ 3 4.5 7 21

3.75 5.75 14

or, more verbosely

• ave =: +/%# ave 3 4.5 7 21

8.875

• mave =: ave\ 2 mave 3 4.5 7 21

3.75 5.75 14

The J Wiki contains a selection of essays in which you can see similarly succinct solutions to a range of problems.