Error Analysis

Error analysis quantifies how far a computed or measured value deviates from a true or reference value. It distinguishes between different error types and provides metrics to judge the quality of numerical results. Absolute error measures the raw difference: E_abs = |x_true − x_approx|. Relative error normalizes this by the true value: E_rel = |x_true − x_approx| / |x_true|, often expressed as a percentage. Relative error is more meaningful when values span different magnitudes. Two fundamental error sources in numerical computation are round-off error and truncation error. Round-off error arises from representing real numbers with finite precision (e.g., floating-point arithmetic). Truncation error results from approximating an infinite or continuous process with a finite one, such as truncating a Taylor series or using a finite step size in differentiation. Significant figures indicate the reliable digits in a reported value. The number of correct significant figures is related to relative error: approximately −log₁₀(E_rel) significant digits are correct. Engineers use error analysis to select appropriate step sizes, iteration counts, and numerical methods, and to determine when a solution has converged sufficiently for practical use.