What distinguishes global error from local error in numerical methods?

The distinction between global error and local error in numerical methods is fundamentally based on their scopes of application within the computational process. Global error is characterized by its accumulation over the entire interval of integration or computation, reflecting the cumulative inaccuracies resulting from each individual computation step throughout the entire process. This means that as more steps are taken, the global error can grow larger, incorporating all the inaccuracies introduced at each discrete step.

On the other hand, local error pertains specifically to a single computational step or iteration. It measures the discrepancy between the true value of the function and the value computed at that specific point. Each time a numerical method is applied, a local error is introduced, which can be understood in isolation, but it is crucial to recognize that these local errors contribute to the overall global error.

This differentiation is significant in numerical analysis because it helps in assessing and improving the accuracy of numerical methods. Recognizing that local errors can accumulate into global errors aids in understanding how to minimize overall inaccuracies in iterative or step-based calculations.