The core concept involves establishing criteria to determine when an iterative process should terminate due to reaching a predefined limit or achieving a satisfactory level of accuracy. For instance, in numerical methods like root-finding algorithms, the algorithm proceeds through successive approximations until the change between iterations, or the estimated error, falls below a specified tolerance. The maximum number of permitted cycles serves as a safeguard, preventing the algorithm from running indefinitely if convergence is slow or non-existent. This safeguards can be the error that the algorithms can tolerate.
Setting a maximum number of cycles is critical for resource management and preventing computational processes from becoming trapped in unproductive loops. By limiting the run-time, users can ensure that algorithms complete within a reasonable timeframe, regardless of the input data or the specific problem being solved. Historical context shows its rise with computationally intensive algorithms where resources were very limited. Nowadays this is less critical, but becomes relevant in embedded systems or large scale optimization problems.
