Discovering APL « Why Vectors? « A Shocking Brevity «

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.


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

is the same as


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

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

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
   mave =: ave\
   2 mave 3 4.5 7 21
3.75 5.75 14

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

» APL & Typography


DiscoveringApl/SomeAplExamples (last edited 2009-08-03 09:47:42 by KaiJaeger)