# Iteration

the iteration (of Latin iterare, „repeat “) is a term from numeric mathematics and calls itself a method, the solution of a computing problem gradually to approximate but purposefully. It exists in the repeated application of the same computing procedure.

One mostly iterates with feedback: The results one Iteration step are taken as initial values of the next in each case step - until the result (and/or change of an inventory size) places contently. An example of it is the Newton procedure. Sometimes one sets the next step from the results of the previous two steps (or before still more steps),for example with the Regula Falsi.

It must be still proven afterwards that the iteration sequence converges and that the limit value agrees with the looked for solution. The speed of the convergence is a measure for it how useful the iteration method is.

## application of the method

• in cases, in those the result is closed calculable (for example transcendental Kepler equation or generally optimization procedure)or to be improved must (sets of equations);
• Frequent already is a good approximation satisfying. Particularly for application problems the input data can be faulted, then the accurate solution of the given problem is not necessarily better than their approximation. The iterative method is attractive thus if it onegood approximation faster supplies, than the computation of the accurate solution needs.
• Some functions on pocket calculators or Fraktale are for example iterative computed.

### example: Solution of an equation

the zero (root) of an equation is iterative often more rapidly found than with accurate, closed formulas:

1. Tries, wherethe zero of y = f (x) approximately lies
2. twice computation of y with x 1 and x 2
3. better value x 3 with the Regula Falsi, which accepts an approximately straight piece of curve:
4. [itex] x_3 = x_2 + y_2 \ cdot \ frac {x_2 - x_1} {y_2 - y_1} [/itex]and repeat, to exactly enough.

### efficiency versus elegance

as algorithm is often more efficient the iteration than the more elegant recursive way. Examples are the computations of the faculty, the Fibonacci sequence with the golden section as limit value as well as the Wallis product with the circle numberPi as limit value.

## iteration in computer science

apart from mathematical iterative problem solution one takes place in computer science also spoken of iteration, if an access repeats iterative, i.e. gradually, and/or, on data structures. (z. B.: FOR loop) here stands the term data structure for collectionsof objects, object references or data types. In the following it acts with these collections (English. collections) usually around arrays, lists, pairs of key values (map, Hashes) or quantities (sets).

The moreover one iteration can regarding zones in the Domain Name service (DNS)“referring” to be called.

## iteration in the software technology

in the software technology designates an iteration an individual development cycle, depending upon model beginning with planning, analysis or draft, ending with implementation, test or maintenance.