# iteration

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[‚īd·ə′rā·shən]
(mathematics)

## Iteration

in mathematics, the result of a repeated application of some mathematical operation. Thus, if y = f(x) ≡ f1(x) is some function of x, then the functions f2 (x)= f[f1(x)], f3(x) = f[f2(x)], …, fn(x) = f[fn 1(x)] are called, respectively, the second, third, …, nth iterations of the function f(x). For example, letting f(x) = xa, we obtain f2(x) = (xa)a = xa2 f3(x) = (xa2)a = xa, and fn(x) = (xan. The index n is termed the iteration index, and the transition from the function f(x) to the functions f2(x), f3(x) … is called iteration. For certain classes of functions one may define iteration with an arbitrary real or even a complex index. Iterative methods are used in the solution of various types of equations and systems of equations.

### REFERENCE

Collatz, L. Funktsional’nyi analiz i vychisliteVnaia matematika. Moscow, 1969. (Translated from German.)

## iteration

(programming)
Repetition of a sequence of instructions. A fundamental part of many algorithms. Iteration is characterised by a set of initial conditions, an iterative step and a termination condition.

A well known example of iteration in mathematics is Newton-Raphson iteration. Iteration in programs is expressed using loops, e.g. in C:

new_x = n/2; do x = new_x; new_x = 0.5 * while (abs(new_x-x) > epsilon);

Iteration can be expressed in functional languages using recursion:

solve x n = if abs(new_x-x) > epsilon then solve new_x n else new_x where new_x = 0.5 * (x + n/x)

solve n/2 n

## iteration

One repetition of a sequence of instructions or events. For example, in a program loop, one iteration is once through the instructions in the loop. See iterative development.
References in periodicals archive ?
Yet with this work, it is perhaps finally clear (and more intriguing) that Eliasson iterates a solar fascination, as evinced in the artificial rainbow of Beauty, 1993; Your sun machine, 1997; and a succession of synthetic suns that culminated in the Weather Project, 2003, at Tate Modem.
The new work can be placed between partitions, but in the spirit of Chaos theory it all iterates from simple base rules.
These results imply that a tiny error in rounding off at the first step is sufficient to destroy any attempt at predicting where the orbit is likely to be after, say, 50 iterates.

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