What is a convergence criterion in numerical methods?

A convergence criterion in numerical methods is fundamentally a set of conditions that must be met for a numerical solution to be considered valid or acceptable. This criterion helps in determining when an iterative process has achieved sufficient accuracy or has sufficiently approached a solution to a problem. In iterative methods, for instance, the convergence criterion allows the practitioner to decide when to stop the iterations based on how closely the successive approximations are approaching a final solution.

When using numerical methods to solve equations or optimization problems, it is crucial to identify when the results are good enough to be deemed effectively correct. The convergence criterion may involve measuring the change between successive iterations, checking if the error falls below a specified threshold, or ensuring that certain functional evaluations meet predefined limits.

This systematic approach is essential in engineering and computational applications since it ensures reliability and efficiency in the solution process. By establishing clear conditions for convergence, numerical methods can provide valid outputs that practitioners can rely on for further analysis and decision-making.