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