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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.


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


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


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 ?
In fact, using (3.1), one can use our construction for generating matrices with arbitrary nonzero spectrum which produce a prescribed, decreasing convergence curve of restarted GMRES in the first n iterations (the case of stagnation needs to be handled separately, as it corresponds to a FOM iterate not being defined, see, e.g., [4], a case we do not consider here).
when g is a Stieltjes function, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the iterate obtained from k cycles of FOM(m) for the shifted linear system
This paper complements the results for (restarted) GMRES by showing that any finite sequence of residual norms is possible in the first n iterations of restarted FOM, where by finite we mean that we only consider the case that all FOM iterates are defined, and thus no "infinite" residual norms occur.
Compute an acceptable next iterate [x.sub.+] using a line search global strategy
In step 6, we compute a next iterate [x.sub.+] by performing the standard backtracking line search global strategy described in Algorithm 2.2.
5 Sigma will reignite passion and imagination with this three-day gathering of innovative educators and thought leaders as it goes beyond the typical how-to-sessions; hosting conversations where educators come together to learn, iterate, and launch.
As mentioned in Section 3.1, the iterates of the homotopy continuation method will follow the path H(U, t) = 0.
During the iteration process, if at two successive iterates ([x.sup.k], [t.sup.k]) and ([x.sup.k+1], [t.sup.k+1]) the angle of their tangents ([d.sup.k], [[tau].sup.k]) and ([d.sup.k+1], [[tau].sup.k+1]) is greater than 90[degrees], then there must be a bifurcation point between the two iterates along the path, and the orientation (4.2) is reversed in order to march ahead to reach t =1.
However, for any specific trajectory -- for a given function and starting point -- the results can be checked for a certain number of iterates using Yorke's method.
For example, Yorke has demonstrated that a true trajectory passes through every one of the millions of iterates producing the array of dots in a figure known as the Ikeda map (see cover illustration).