# iteration

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[‚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 estimates for Kirk-Thianwan (KT), Kirk-SP (KSP), and Kirk-Multistep-SP (KMSP) iterations are, respectively,
First, Section 2 introduces the background including the system model, the Neumann series, and Newton iteration method.
test iteration starts with loading the virtual register by an initial seed.
This paper concludes our efforts of analyzing preconditioned gradient iterations and their subspace variants with either fixed step length (case of inverse iteration and preconditioned inverse iteration) or with optimal step-length (case of steepest descent and preconditioned steepest descent).
x], the number of iterations, a vector with components from 1 to 9, representing the required methods, the precision of the result.
The 2nd iteration for the Koch pentagonal fractal geometry is depicted in Figure 4(c).
Rather than documenting everything you want in the LOS upfront, then waiting six months or a year before ever seeing it--only to find out that it was not built correctly or the vendor didn't understand everything properly--you would instead see an early iteration or build of the LOS every two to four weeks and you would see it come together over time.
Harder and Hicks [2] showed that Picard iteration is T-stable for several other contractive conditions, including that of Zamfirescu [6].
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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.

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