iteration

[‚īd·ə′rā·shən]

(mathematics)

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

This article is provided by FOLDOC - Free Online Dictionary of Computing (foldoc.org)

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.

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The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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)], …, f_n(x) = f[f_{n 1}(x)] are called, respectively, the second, third, …, nth iterations of the function f(x). For example, letting f(x) = x^a, we obtain f₂(x) = (x^a)^a = x^a² f₃(x) = (x^a²)^a = x^a, and f_n(x) = (x^aⁿ. 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.)