Iteration

Iteration (from Latin iterare, repeat ') is used as a term in a range of applications with different meanings:

Dynamical Systems

In mathematics, especially the theory of dynamic systems is referred to as iteration, the repeated application of the same function, that is the formation of

For a given function to a room.

The theory of dynamical systems is particularly concerned with the long-term behavior of the orbits of points under such iterations.

Example

Consider the quadratic function

Then ( not to be confused with ) and you can long-term behavior of different orbits investigate: for converges to the fixed point 0, for true, for the consequence constant and also remains the result from the first iteration constant.

Numerical Mathematics

In numerical mathematics iteration is a method to approximate the exact solution of a computational problem step by step (successive approximation ). It consists in the repeated application of the same method of calculation.

The results of a step to be taken as the starting values ​​of the next step. The consequence of the results must converge. If the difference from the previous calculation step is smaller than the accepted error, then the result is accurately determined, and the process is terminated. One of the known examples is the Newton method. Sometimes it is in the next step, the results of two or more of the preceding steps, for example, in the regulation Falsi.

The rate of convergence is a measure of how useful is the iteration.

Application of the Method

• Iteration is used in cases where the result can not be calculated in closed form, for example at the Kepler 's equation, or to calculate the surface shape of an aspheric lens.
• Systems of linear equations can be solved iteratively under certain conditions.
• With application issues the input data may be defective, then the " accurate solution " of the given problem is not necessarily better than the approximation. The iteration is preferred that it provides a good approximation faster than the calculation of the exact solution requires.
• Some functions on calculators or even fractals are calculated iteratively.

Example: determination of zeros of a continuous function

Approximations to the zeros of a continuous function, provided one exists, iteratively often faster than found by other algebraic methods (such as a closed expression):

Computer science

In addition to the iterative mathematical problem solving is spoken in the computer science by iteration when

• An access iteratively, that is gradually or repeated, is carried out on data structures, such as a FOREACH loop. Here, the term data structure for collections of objects, object references or data types. The pointer to these objects is called iterator. In general, there are arrays, lists, key - value pairs ( maps, hashes ) or quantities (sets). See also: Iterative programming
• A block of statements is executed repeatedly; each execution is an iteration of the loop.

Linguistics

The linguistic iteration refers to the repetition of parts of words like great-grandmother.

Software Engineering

In software engineering, an iteration means a single development cycle, depending on the process model starting with planning, analysis or design, ending with implementation, testing and maintenance. Play a special role in the Extreme Programming iterations and the Rational Unified Process. In Scrum ( Agile project management ) often comes an iterative process for the development of software for use. This is called feedback loops in all phases of planning, implementation, review and adaptation.

History

In the history of science iteration refers to the repeated exercise of the same office in the official career of the Roman Republic. After Mos maiorum the iteration was frowned upon. When the consulate multiple, in exceptional cases, directly successive clothing of the Office, however, came before since the early Republic; Since the constitutional reform of the dictator Sulla from the year 82 BC the repeated clothing of the consulate was allowed only after ten years. The Iterationsverbot was next to the collegiality and the Annuitätsprinzip the most important means of preventing a dangerous power wealth of public officials.

In particular, the crisis of the Republic of the iteration was repeated before: The best-known examples are Gaius Sempronius Gracchus, who wanted to leave in three consecutive years for the tribunes of the people choose, Gaius Marius, who and the consulate in five consecutive years ( 104-100 BC ) seven times, exercised, and Gaius Julius Caesar, who had held the consulship in the years 59, 48, 46, 45 and 44 BC. Under the Roman Empire from Augustus the iteration of the consulate was a sign for a prominent socio- political position. Immediately successive consulates, there were only members of the imperial family.

Philosophy

Jacques Derrida introduced the iteration into the language of philosophy. " Iteration " refers to the repetition of a term in the philosophical and social discourse. According to Derrida with each repetition ( " iteration " ) of a term changed its meaning, so that there is never the same meaning is reproduced as in the previous use of the term. Rather, each iteration has a variation of the importance the consequence that adds something to the original concept and enriching it. An original definition of terms, to which one could trace their meaning, there can not be thus.

Construction Economics

In an iterative process of building economics is gradually approximating to the original Bauzielen feasible implementation.

Engineering Design

In the design theory is called an iterative procedure, and partly also by an iterative search, if the procedure is to find a solution so that the solution is gradually improved from a inspiration of the designer.

Management

In the Management iteration is a procedure to deal with the uncertainties and surprises in complex situations. When changes in the course of projects or the effect of actions is not always predictable. Construed Each change management as " great plan " with immovable goals, leads in most cases to surprises, to which the planners and implementers are not prepared. This does not mean abandoning plans, but always to be in one's own practice only provisionally secure. Linear causal project thinking is replaced by iterative procedure: By prefixing keys along purposes, interests and power constellations acceptance is gradually degraded after ambiguity, achieved generated effect and established routine. The order of the topics and content arises only in the course of change. "An iterative process of initial interpretation and design, implementation and improvisation, learning from change -effort, and then sharing did learning system wide, leading to ongoing re - interpretation and redesign of the change as needed. " (Anthony F. Buono / Kenneth W. Kerber: Building Organizational Change Capacity)