iteraties

Iteraties, or iteratie in singular, refer to the repeated application of a process or rule with the aim of approaching a result or exploring possibilities. In mathematics and computing, an iteration is the step of applying a function to a current value to produce a new value. A sequence x0, x1 = F(x0), x2 = F(x1), and so on is generated by iterating F. Convergence occurs when the sequence approaches a limit or fixed point, under suitable conditions such as contractivity.

Common examples include the Newton-Raphson method for finding roots, where xn+1 = xn − f(xn)/f′(xn); gradient descent, where the estimate is updated to reduce a function’s value; and the Babylonian method for square roots. In numerical analysis, stopping criteria include a maximum number of iterations or a small change between successive iterates.

In software development and design, iteration refers to cycles of planning, implementation, testing and feedback, producing progressively improved versions. Iterative development and agile methodologies rely on short cycles, user feedback and incremental refinements rather than a single final release.

Across disciplines, iteraties emphasize learning from results, adaptability and gradual improvement. The concept also has a general sense of repetition with refinement, whether solving equations, refining a model, or evolving a product.