# 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 ?
We argue this examination needs to take into account the status of Maori under the Treaty, and to recognise the reiterative and regulatory role of media discourses (Butler, 1993).
The first additional amount to start the reiterative process is to add an amount that would net \$22,860 if it were also taxed at 32.
By crediting Bright as "technical consultant" and then marking Bright's appearance within the narrative as part of the very culture, and by implication the audience the film is marketed towards (Jesse is marked as femme through her manipulation of feminine sexual iconography: long hair, tight fitting leather vest, deep cleavage), Bound reveals itself to be a product of both lesbian representational and textual history and the reiterative performance of that history.
On the same basis of ignorance, or blindness, contemporary sociobiology repeats the Spencerian gesture of fragmented reference to the doctrine of selection, thus exploiting the only popularized element of Darwinian theory, previously filtered through a "philosophical" interpretation and channeled by the reiterative tradition of the "biological sociologies" which are inscribed in the interminable refashioning of the original organicism.
However, while Giddens is particularly sensitive to how power and control strategies mediate the material instanciation of reiterative practices, his efforts to explain the interdependence and synchronization of late capitalism may overlook the partial and precarious outcomes of such processes.
Chaos theorists have found these properties in almost all natural systems because most natural systems are, like the model of human memory Proust describes above, nonlinear reiterative systems--systems which feed back the information they produce as 'output' into their own 'input'.
There is, on the one hand, the legal model, of reproduction, iterative or reiterative, inductive or deductive, which presupposes a fixed point of view external to what is being reproduced.
Crowl has an acute eye for detail and explicates Roman Polanski's reiterative employment of the motif of the chain of office in his Macbeth (p.
Fratres is constituted from just three musical elements, all of which have sacred connotations: first, a double drone on tonic and dominant (G and D) played throughout--a symbol of eternity, as found for example in Indian classical music; second, a reiterative percussive figure (executed by a combination of col legno, pizzicato and fingered drumming on the soundboard) which ritualistically marks off each new section; third--very much in the foreground--a chanted litany in the upper three parts (perhaps giving a monastic slant to the title).
We can make sense of Talbot's reiterative and reflexively theatrical status in the play by recognizing how bound up his fate is with the problem of origins in this originary drama--the first of the Histories, the first of Shakespeare's career.
Open-ended items from the supplemental form were subjected to reiterative content analyses following the procedure established by Covert (1977).
Active circuits that contain transistors, for instance, will require reiterative methods to solve non-linear algebraic, differential or integral equations.

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