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.

{(+/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 (see http://www.jsoftware.com/jwiki/Essays) contains a selection of essays in which you can see similarly succinct solutions to a range of problems.