Konvergeeruval

Konvergeeruval is a term used in numerical analysis to describe a framework for assessing the convergence behavior of iterative algorithms by combining indicators of proximity to a fixed point with the stability of estimates. The idea is to monitor a single scalar value that reflects both how close the current iterates are to convergence and how steadily they are changing from one iteration to the next.

A common instantiation defines r_k as the residual norm, for example r_k = ||F(x_k)||, and d_k as the step size, d_k = ||x_k − x_{k-1}||. A konvergeeruval value at iteration k can be formed as K_k = w_r r_k + w_d d_k, with weights w_r and w_d chosen to balance accuracy and stability. Termination occurs when K_k ≤ tol for a prescribed tolerance. Other variants use forms such as K_k = max(r_k, c d_k) or K_k = r_k^p + d_k^q. Unlike purely residual-based criteria, konvergeeruval emphasizes both accuracy and smoothness, aiming to avoid premature stopping in oscillatory or stiff problems.

Usage and origins: The term appears primarily in Dutch-language technical literature and is not widely adopted in English-language references. It is generally described as a practical, heuristic criterion rather than a canonical mathematical definition. In practice, konvergeeruval-type criteria are viewed as flexible termination tools that can be tailored to the specifics of the algorithm and problem at hand.

See also concepts include convergence criteria, termination rules, residual norms, and adaptive tolerances.