I come from a background of C, Fortran, Python, R, Matlab, and some Lisp - and I've read a few things on Haskell. What are some neat ideas/examples in J or other languages from the APL family that are unique and not implemented in more common languages? I'm always interested in finding out what I'm missing...
J has a very large set of operators that make it easy to gin up complex programs without having to hunt for a library. It has extremely powerful array processing capabilities, as well as iterative constructs that make explicit control structures irrelevant for most purposes -- so much so that I prefer using tensor algebra to declaring an explicit loop because it's more convenient. J runs in an interpreter, but a good J script can be just as fast as a program written in a compiler language. (When you take out explicit loops, the interpreter doesn't have to compile the contents of the loop every time it executes.)
Another fun feature of J is tacit programming. You can build scripts without explicit reference to input variables, which lets you express an idea purely in terms of what you intend to do. For example, I could define the average function as 'summing the terms in a list and dividing them by the number of entries in the list' like so:
(+/ % #)
or I could make a script that slices into a 2D array and only returns the averages of rows that have averages greater than 10:
(10&<#])(+/%#)"1
There's lots of other neat stuff you can do with J; it's an executable form of mathematical notation. Ideas generalize easily, so you get a lot of benefit out of learning any one aspect of how the language works.
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