Iteration means the act of repeating a process usually with the aim of approaching a desired goal or target or result. Each repetition of the process is also called an "iteration," and the results of one iteration are used as the starting point for the next iteration.
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Mathematics
Iteration in mathematics may refer to the process of iterating a function i.e. applying a function repeatedly, using the output from one iteration as the input to the next. Iteration of apparently simple functions can produce complex behaviours and difficult problems  for examples, see the Collatz conjecture and juggler sequences.
Another use of iteration in mathematics is in iterative methods which are used to produce approximate numerical solutions to certain mathematical problems. Newton's method is an example of an iterative method.
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 relying on destructive assignment, 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.
;; sum : number > number
;; to sum the first n natural numbers
(define (sum n)
(if (and (integer? n) (> n 0))
(let iter ([n n] [i 1])
(if (= n 1)
i
(iter ( n 1) (+ n i))))
((assertionviolation
'sum "invalid argument" n))))
An iterator is an object that wraps iteration.
Iteration is also performed using a worksheet, or by using solver or goal seek functions available in Excel. Many implicit equations like the Colebrook equation can be solved in the convenience of a worksheet by designing suitable calculation algorithms.^{[1]}
Many of the engineering problems like solving Colebrook equations reaches 8digit accuracy in as small as 12 iterations and a maximum of 100 iterations is sufficient to reach a 15digit accurate result.^{[2]} .
[edit] Project management
Iterations in a project context may refer to the technique of developing and delivering incremental components of business functionality, product development or process design. This is most often associated with agile software development, but could potentially be any material. A single iteration results in one or more bitesized 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.
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