Iteration

This article discusses a technique which is used in mathematics and in computer programming. For use in the Japanese and Chinese languages see iteration mark.

The word iteration is sometimes used in everyday English with a meaning virtually identical to repetition.

Mathematics

Iteration in mathematics may refer to the process of iterating a function, or to the techniques used in iterative methods for solving numerical problems.

Computing

Iteration in computing is the repetition of a process within a computer program. It can be used both as a general term, synonymous with repetition, and to describe a specific form of repetition with a mutable state.

When used in the first sense, recursion is an example of iteration, but typically using a recursive notation, which is typically not the case for iteration.

However, when used in the second (more restricted) sense, iteration describes the style of programming used in imperative programming languages. This contrasts with recursion, which has a more declarative approach.

Here an example of iteration, in imperative pseudocode:

var i, a := 0 // initialize a before iteration for i from 1 to 3 { // loop three times a := a + i // increment a by the current value of i } print a // the number 6 is printed

In this program fragment, the value of the variable i changes over time, taking the values 1, 2 and 3. This changing value—or mutable state—is characteristic of iteration.

Iteration can be approximated using recursive techniques in functional programming languages. The following example is in Scheme. Note that the following is recursive (a special case of iteration) because the definition of "how to iterate", the iter function, calls itself in order to solve the problem instance:

(define (sum n)
  (define (iter n i)
    (if (= n 1)
      i
      (iter (- n 1)(+ n i))
    ))
  (iter n 1))

An iterator is an object that wraps iteration.

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Mathematics

Computing

See also