iteration


Also found in: Dictionary, Thesaurus, Medical, Legal, Wikipedia.

iteration

[‚īd·ə′rā·shən]
(mathematics)

Iteration

 

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.

REFERENCE

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

iteration

(programming)
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

iteration

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 key ingredient to make a gradient iteration an efficient solver for the operator eigenvalue problem (1.
That's the case because functionality can move within or be pushed to a later iteration, which can have a domino effect on resources used to forecast what you can do in six months or a year and so on.
However, connectivity issues are set to emerge, with users of devices with IPv6 addresses unable to access websites with addresses of the IPv4 iteration.
In a normed space E, Mann iteration is defined by [u.
The methodology spans activities prior to project execution, during each iteration, and subsequent to each iteration.
To compute the number of comparisons and number of iterations involved in Kruskal's algorithm for a network with n vertices, first let us consider a graph with three vertices and calculate the number of comparisons and number of iterations involved in it.
The modified Mann iteration with errors is defined as
For execution mean time significant results can be achieved at less number of iteration of benchmark.
The best practice is to iterate and deliver incrementally, treating each iteration as a closed-end "mini-project," including complete requirements, design, coding, integration, testing, and internal delivery.
That means more Honda-powered cars will be using the latest iteration of the Honda engine specification for Sunday's race at Kentucky Speedway.
The simulation code can handle both swell and heat transfer, but computation time would be increased beyond the 20 minutes typically needed for each iteration described here.
Typically, the die needs to he modified a number of different times at a cost of perhaps $5,000-10,000 per iteration before quality reaches acceptable levels.