Boundederror

Bounded error is a property of an estimate, computation, or measurement whereby the deviation from the true value is confined within a known bound for all inputs in a specified domain. The bound can be absolute, |estimate − true| ≤ E, or relative, |estimate − true| / |true| ≤ δ when the true value is nonzero.

In numerical analysis, bounded error is central to error analysis. The forward error is the difference between the computed result and the exact result, while the backward error is the amount by which the input must be perturbed to yield the computed result exactly. An algorithm is said to be numerically stable if its errors remain bounded under standard arithmetic operations. In floating-point computation, rounding errors introduce small bounded discrepancies that may accumulate; under suitable conditions the total error can be bounded by functions of machine epsilon and problem size.

Bounding techniques include direct error analysis, conditioning assessments, and stability arguments. Interval arithmetic provides guaranteed bounds on results by propagating intervals rather than point estimates. Calibration and sensor design aim to ensure measurement errors stay within specified tolerance. In practice, practitioners seek worst-case bounds for reliability or probabilistic bounds for efficiency, depending on the context.

Related concepts include numerical stability, error propagation, and interval arithmetic. In statistics and data science, one may describe estimators with bounded error with high probability using concentration inequalities, linking deterministic bounds to probabilistic guarantees. Bounded error thus serves as a fundamental criterion across computation, measurement, and estimation disciplines.