iteracja

Iteracja, or iteracja in Polish, is a process of repeating steps to refine a result or explore a system. It is a fundamental concept in mathematics, computer science, and engineering. Each repetition uses the outcome of the previous step as input for the next.

In mathematics, iteration is often described by a recurrence x_{n+1} = f(x_n) with an initial value x_0. Analysis focuses on convergence to a limit or a fixed point x* where f(x*) = x*. The contraction mapping principle guarantees convergence under conditions of shrinkage.

In numerical methods, iterative schemes solve problems without explicit formulas. Examples include Jacobi and Gauss-Seidel methods for linear systems, the power method for dominant eigenvalues, and gradient descent for optimization. Stopping criteria depend on residuals, change between iterations, or a fixed iteration count.

In computer science, iteration refers to looping constructs that repeat a block of code until a condition is met. It contrasts with recursion and is fundamental for handling large data or repeated tasks efficiently. Performance depends on algorithm design and data structures.

In practice, iteration supports progressive refinement, such as iterative design, simulations, and procedural generation. It enables incremental testing and adaptation but may suffer from non-convergence, oscillation, or slow convergence if parameters are poorly chosen.

The term originates from Latin iterare and is used across languages; in Polish, iteracja is the direct translation. See also: loops, fixed-point iteration, recursive methods.