iteracj

Iteracj, commonly rendered in Polish as iteracja and in English as iteration, refers to the process of repeating a sequence of operations to progressively refine a result. Each cycle uses the output of the previous cycle as input for the next. The concept originates from the Latin iteratio, meaning to repeat or anew.

In mathematics and numerical analysis, iterative methods approximate solutions by successive refinements. Examples include fixed-point iteration, where x_{n+1} = g(x_n) aims to converge to a fixed point x*, and Newton's method, which uses tangent line approximations to approach a root. Convergence depends on function properties and on stopping criteria such as a tolerance or maximum iterations.

In computing, iteration is implemented with loops (for, while) to execute a block of code repeatedly. This is a core control structure, generally preferred over recursion when memory usage matters because it avoids growing call stacks. Iterative processes can also be used in data processing, simulation, and optimization tasks.

Key considerations include termination (ensuring the process ends), convergence speed, stability, and computational cost. Iteration is central to many algorithms, from numerical solvers to machine learning training loops and software development methodologies that employ iterative refinement.