Iteratius

Iteratius is a hypothetical formalism used to illustrate and study iterative methods in computation and mathematics. In this framework, a computation is described as the evolution of a state over discrete steps according to a function f that maps states to states. A run begins with an initial state s0 and generates a sequence sk+1 = f(sk). Convergence is defined either by a convergence predicate that compares successive states or by a fixed-point condition where sk = f(sk).

Core components of Iteratius include the iteration function, a convergence criterion, and an execution bound. The model supports termination when the criterion is satisfied or when a maximum number of steps is reached. Iteratius emphasizes a clear separation of concerns: the mathematical specification of f and the stopping condition P, alongside the concrete data representation of states. Extensions of the formalism may consider probabilistic or nondeterministic transitions, or contractive mappings with properties that guarantee convergence under appropriate conditions.

Applications of Iteratius are primarily educational or theoretical. It is used to explain fixed-point iterations, gradient-based optimization, and certain dynamical systems in a simplified, language-agnostic setting. Because it is a conceptual tool rather than an implemented language or standard, practitioners typically deploy Iteratius in textbooks, lectures, and thought experiments to illuminate the behavior of iterative processes.

See also: fixed-point iteration, contraction mapping, Banach fixed-point theorem, dynamical system, iterative method.