limitcycler

Limitcycler is a hypothetical construct used in numerical analysis to study and potentially accelerate the convergence of sequences by cycling through a finite set of extrapolation or relaxation operators. Rather than applying a single update rule to a sequence, a limitcycler adopts a fixed cycle of operators, generating a periodic sequence of intermediate estimates for the limit.

Formally, let (x_n) be a sequence in a normed space. A limitcycler consists of a finite collection of maps {T_1, ..., T_k} and a cycling rule s_n that takes values in {1, ..., k}. At step n, the current estimate x_{n+1} is produced by applying the operator T_{s_n} to x_n: x_{n+1} = T_{s_n}(x_n). A cycle is achieved when after k steps, the composite operation T = T_k ∘ ... ∘ T_1 has a fixed point L, and subsequent cycles converge toward L. In practice, one examines whether the k-step cycle leads to a stable limit and how the cycle interacts with the underlying convergence properties of the original sequence.

Limitcycler can help reveal convergence patterns, especially for sequences that exhibit oscillatory or mixed behavior, and it may serve to accelerate convergence by combining complementary transforms. Its effectiveness depends on the choice of operators and the cycle length, and there is no universal prescription for these choices. The concept is primarily used in theoretical discussions and as a framework for exploring how cyclic application of different update rules affects convergence behavior.

In applications, limitcyclers are discussed as tools for diagnosing stability, comparing alternative extrapolation strategies, and illustrating how cycle-based methods relate to classical fixed-point and extrapolation techniques. See also convergence acceleration, fixed-point iteration, and cycle detection.