(redirected from Iterations)
Also found in: Dictionary, Thesaurus, Medical, Legal.





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 ?
The multigrid preconditioner is a V-cycle multigrid solver with damped Jacobi-smoothing; the damping constant is [omega] = 2/3 and two pre-smoothing and post-smoothing iterations are applied on each level.
You can use a document template from a typical waterfall approach to document use cases and impact assessment on a given iteration, and engage a project management office (PMO) to measure how everything and everyone is performing, and then run in iterations.
Harder and Hicks [2] showed that Picard iteration is T-stable for several other contractive conditions, including that of Zamfirescu [6].
While FDD creates a development plan based on features and groups features into iterations along technical lines, elastic features development creates iterations in which features are grouped along business lines.
Hence the total no of iterations is 10 and the total number of comparisons is 55.
Our results extend the equivalence of convergence of modified Mann and Ishikawa, artd Noor iteration to the more generalised (modified) Smarandacheian multi-step iterations with errors for the uniformly Lipschitzian strongly (successively) pseudocontractive operators in arbitrary real Banach space.
For each target tested are considered four tests with different number of iterations for GPC algorithm.
You use the data from the first few iterations to arrive at a concrete estimate for final system scope.
World-class finance organizations also do an average of three budget iterations, versus four for typical companies.
After three more iterations, the back face of the die (shown on the left of figure 8) gave the velocity contours shown on the right side of this figure.
Blast Fusion provided early timing feedback with Magma's Early Silicon Performance (ESP) algorithm and eliminated synthesis-to-layout iterations, enabling TI and Fujitsu to both achieve their time-to-market goals.
Idealized design involves repeated iterations of planning.