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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 ?
Iteration planning Duration of Length of iteration iteration and what defined, features features will be to be developed developed in each established, and iteration backlog updated for established.
According to ( Pocket-Lint , the next iteration is expected to focus more on health and fitness features.
In this section, the performance, parameters of the circumstance, lookup table interval and iteration times are discussed.
The Taiwanese technology company on Wednesday unveiled the newest iteration of its virtual reality headset, the HTC Vive Pro, featuring improved specifications and a tie-up with one of Dubai's industry giants to give virtual thrill-seekers a chance to scale the world's tallest building.
To reach the second iteration, the same process is applied to the 3 equilateral triangles obtained in the first iteration; the result is 9 triangles and so on.
where [[beta].sub.t] is the parameter estimates after t iterations, [[beta].sub.t+1] is the parameter estimates after t + 1 iterations, [lambda] is a step size (can be assumed by a researcher at the initial time of model parameter estimation), [H.sub.t] is the Hessian matrix in iteration t in each algorithm, and [G.sub.t] is a vector of gradient in iteration t.
cluster 1 cluster 2 cluster 3 Objective Ub Lb Ub Lb Ub Lb Return 0.0661 0.0311 0.0333 0.0014 0.0980 0.0725 Risk 0.0643 0.0385 0.0222 0.0156 0.0743 0.0511 RSI 56.7298 50.7244 55.0748 50.2265 58.0193 51.4598 CV 5.0742 1.6272 43.6201 1.7508 3.8445 1.5350 EY 10.4818 2.9446 9.9886 4.4508 16.4061 4.2379 PEG ratio 23.4067 -0.2626 5.9784 -2.1260 1.2829 -82.2285 AHP weight 0.1195 0.0507 0.0971 0.0522 0.1442 0.0526 Table 8: Iterations for Cluster 1.
The remaining 7 subsquares (s) are treated in the same manner; that is, their sides are divided by the length scale factor (F) of 3, thus generating a further 9 subsquares, from which 2 are removed (Figure 1(a)) and 7 are involved in the iteration process.
The MPS iteration structure gives a fast convergence to the optimization problem; the results in [7] show that it has far fewer function evaluations than the GA algorithm, which can be a significant help to accelerate the MDO solving procedure.
Then, the final solution is [[lambda].sub.n] = [x.sup.-2.sub.n], where n = i + 1 is the final iteration.