Itereerimine

Itereerimine is the process of repeating a set of operations in order to approach a desired result, such as a fixed point of a function, a solution to an equation, or an improved estimate in an optimization problem. In practice, one starts with an initial value and applies an update rule to generate a sequence of values, typically denoted x0, x1, x2, and so on. The procedure continues until a stopping criterion is met, for example when the change between successive values is smaller than a chosen tolerance or when a maximum number of iterations has been reached.

Common forms of itereerimine include fixed-point iteration, where the next value is computed as x_{n+1} = g(x_n), and iterative methods in optimization and numerical analysis that repeatedly update estimates using gradients, Hessians, or other problem-specific rules. Iterative approaches are often preferred when closed-form solutions are difficult or impossible to obtain.

Convergence behavior is central to itereerimine. If the update rule is a contraction on the domain, the sequence tends to converge to a fixed point. In practice, convergence depends on the choice of initial guess, the properties of the update function, and the chosen stopping criteria. Poorly chosen steps or ill-suited rules can lead to divergence or oscillation.

Applications span numerical solving and computation, including solving nonlinear equations with methods such as Newton-Rraphson, solving linear systems with Jacobi or Gauss-Seidel iterations, and iterative optimization techniques like gradient descent. Itereerimine emphasizes reusing previous results to progressively improve estimates rather than computing a solution in a single step.