Iteracij

Iteracij, or iteracija in many languages, denotes the process of repeating a set of operations or applying a function multiple times. In mathematics and computer science it is used to describe both a single cycle of such repetition and the overall method that consists of many cycles. An iteration typically involves updating a variable or estimate based on a rule or function.

In mathematics, given a function f and an initial value x0, the iterates are x1 = f(x0), x2 = f(x1), and so on. The study of iterated functions focuses on convergence to fixed points, the structure of orbits, and possible chaotic behavior. Conditions such as contraction mappings guarantee convergence under certain norms.

In numerical analysis, iterative methods are algorithmic families for solving equations or systems where closed-form solutions are impractical. Examples include the Jacobi and Gauss-Seidel methods for linear systems, Newton-Raphson for nonlinear equations, and power iteration for eigenvalue problems. An iteration typically produces a sequence of improving approximations until a stopping criterion, such as a tolerance on the error, is met. The rate of convergence describes how quickly the sequence approaches the solution.

In computing and programming, an iteration is one pass through a loop. For and while loops, each iteration updates state variables and tests termination conditions. Iteration is fundamental to many algorithms, including numerical simulations, search procedures, and data processing tasks.