Why is 'relative error' important in numerical methods?

Relative error is essential in numerical methods because it quantifies the accuracy of an approximate solution compared to the true or actual value. In computational applications, particularly in engineering and scientific computations, it is critical to know how close an estimated result is to the exact solution, which helps assess the reliability of the numerical method used.

By expressing the error as a fraction of the true value, relative error provides a scale-independent measurement of accuracy, which is especially useful when dealing with different magnitudes of results. For example, a relative error of 0.01 indicates that the approximate solution deviates from the true value by 1%, regardless of whether the true value is large or small. This insight allows engineers and scientists to judge the suitability of a numerical method based on the errors involved, guiding decisions for refining algorithms, adjusting tolerances, or choosing alternative methods if necessary.

The other options relate to aspects of numerical methods but do not directly connect to how accuracy is evaluated in the context of approximate solutions. Measures like computational speed and algorithm complexity are significant for performance analysis, while hardware requirements pertain to the practical implications of running simulations, rather than the intrinsic accuracy of results.