Convergelor

Convergelor is a hypothetical operator in numerical analysis conceived to accelerate the convergence of iterative sequences. It is envisioned as a generic transformation that, given a sequence x_k converging to a limit L, produces a new sequence y_k = C_k(x_0, x_1, ..., x_k) with faster convergence properties. The operator can be seen as combining extrapolation, smoothing, and adaptive weighting to dampen oscillations and reduce error terms.

Definition and mechanism: The convergelor is intended as a flexible framework rather than a single algorithm. In proposed formulations, the transformed sequence aims to achieve higher-order convergence by reducing dominant error components. Implementations may resemble weighted averaging, delta-extrapolation, or subspace acceleration, with parameters that adapt based on local behavior of the iterates. The exact form of the transformation varies, but the goal is a more rapid approach to the fixed point or solution.

Relation to existing methods: The convergelor concept is related to acceleration techniques such as the Aitken delta-squared process, the Shanks transformation, and Anderson acceleration. It is proposed as a unifying blueprint that can incorporate ideas from these methods, while allowing modular integration with different base iterative schemes.

Applications and limitations: Potential applications include numerical linear algebra, nonlinear equation solving, and optimization problems that rely on iterative methods. Limitations include added per-iteration cost, potential introduction of bias, and sensitivity to noise or ill-conditioned sequences. The effectiveness depends on the regularity of the underlying problem and the stability of the chosen transformation.

Note: Convergelor is a hypothetical construct used here for explanatory purposes and is not an established term in standard references.