Why is measuring error important in a numerical solution?

In numerical analysis, understanding the nature, size, and analytic form of error is a fundamental concern when approximating a solution to a problem. Measuring error is essential for several critical reasons:1

• Reliability and Understanding: It is prudent to understand the nature of the error in a computed solution to ensure the reliability of the numerical method being employed.1
• Improving Convergence: Analyzing the form of the error allows for the creation of extrapolation processes, which can improve the convergence behavior of a numerical method.1
• Managing Precision: Numerical analysts must account for the effects of finite-precision computer arithmetic, as large-scale problems are susceptible to the accumulation of rounding errors.1
• Stability Assessment: Measuring error helps determine the stability of a solution, ensuring that the numerical method is not overly sensitive to small changes in input data or parameters.1

By quantifying the difference between a true value and an estimate, analysts can effectively manage the inherent imprecision of calculations and ensure that their numerical models remain robust and efficient.21