# iteration

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## iteration

[‚īd·ə′rā·shən]## Iteration

in mathematics, the result of a repeated application of some mathematical operation. Thus, if *y* = *f(x) ≡ f _{1}(x*) is some function of

*x*, then the functions

*f*, …,

_{2}(x)= f[f_{1}(x)], f_{3}(x) = f[f_{2}(x)]*f*)] are called, respectively, the second, third, …,

_{n}(x) = f[f_{n 1}(x*n*th iterations of the function

*f(x*). For example, letting

*f(x) = x*, we obtain

^{a}*f*) = (

_{2}(x*x*=

^{a})^{a}*x*, and

^{a2}f_{3}(x) = (x^{a2})^{a}= x^{a}*f*) = (

_{n}(x*x*. The index

^{an}*n*is termed the iteration index, and the transition from the function

*f(x*) to the functions

*f*) … 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.

_{2}(x), f_{3}(x### REFERENCE

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

## iteration

(programming)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