Iteracija

Iteracija, or iteration, is the repeated execution of a procedure or the construction of a sequence by applying a rule to its previous element. In mathematics and computer science, an iteration produces a sequence x0, x1, x2, … where each xi+1 is obtained from xi by a fixed rule. In everyday use, iteration denotes a process of gradual refinement toward a goal or solution. It is distinct from recursion, which defines the next step in terms of the procedure itself; iteration emphasizes repeated application of the same operation, usually implemented with a loop or repeated calls.

Etymology and scope: the term derives from the Latin iteratio, meaning repetition. Iteration is used across many fields, including mathematics, computer science, engineering, statistics, and project management, as well as in design and software development, where incremental improvements are made through successive cycles.

Mathematical and numerical methods: many problems are solved by defining a sequence x0, x1, x2, … with x_{n+1} = G(x_n). Convergence criteria guide stopping the process, such as when the change |x_{n+1} − x_n| falls below a tolerance or when |G(x*) − x*| is small, indicating a fixed point x*. Common iterative methods include fixed-point iteration, Newton-Raphson, and linear-system methods like Gauss-Seidel and Jacobi. Practitioners often impose a maximum number of iterations to avoid non-convergence.

Computing and software: iteration is typically implemented via for or while loops. Iterative algorithms can be more memory-efficient than some recursive approaches but may suffer from non-convergence or rounding errors, requiring robust stopping rules and error tracking.

Design and development: in product and software development, iterative processes involve repeated cycles of design, testing, feedback, and refinement, enabling gradual improvement and adaptability in response to results and user input.