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map function, found in many functional programming languages, is one example of a higher-order function. It takes as arguments a function f and a collection of elements, and as the result, returns a new collection with f applied to each element from the collection.
In functional programming, fold (also termed reduce, accumulate, aggregate, compress, or inject) refers to a family of higher-order functions that analyze a recursive data structure and through use of a given combining operation, recombine the results of recursively processing its constituent parts, building up a return value.
This style of programming is mostly used in functional programming, but it can also be very useful in object-oriented programming. A slightly different interpretation of higher-order programming in the context of object-oriented programming are higher order messages, which let messages have other messages as arguments, rather than functions.
In many programming languages, map is a higher-order function that applies a given function to each element of a collection, e.g. a list or set, returning the results in a collection of the same type. It is often called apply-to-all when considered in functional form.
Some functional programming languages might not optimize abstractions such as higher order functions like "map" or "filter" as efficiently as the underlying imperative operations. Consider, as an example, the following two ways to check if 5 is an even number in Clojure :
In functional programming languages, function composition can be naturally expressed as a higher-order function or operator. In other programming languages you can write your own mechanisms to perform function composition.
In functional programming, filter is a higher-order function that processes a data structure (usually a list) in some order to produce a new data structure containing exactly those elements of the original data structure for which a given predicate returns the Boolean value true.
Fold (higher-order function) Function composition (computer science) M. Map (higher-order function) O. Outer product; P. Prefix sum; S. Segmented scan