Definitions

# Iteration

[it-uh-rey-shuhn]

Iteration means the act of repeating.

## 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 is 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. Specifically it uses tail recursion, which is properly supported in languages like Scheme so it does not use large amounts of stack space. (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.

## Project management

Iterations in a project context may refer to the technique of developing and delivering incremental components of business functionality. This is most often associated with agile software development, but could potentially be any material. A single iteration results in one or more bite-sized but complete packages of project work that can perform some tangible business function. Multiple iterations recurse to create a fully integrated product. This is often compared with the waterfall model approach.