Approximatevalued

Approximatevalued is a term used in mathematics and computer science to describe quantities whose values are known only approximately. An approximately valued quantity qc is associated with a true value Q such that the error satisfies |Q - qc| ≤ ε, where ε is a specified tolerance. The term emphasizes the presence of approximation in the computation, measurement, or modeling process rather than an exact, closed-form value.

Formally, a quantity is called approximately valued when the computation yields a value with a guaranteed bound on the deviation from the true value. Relative error |Q - qc|/|Q| ≤ δ is also common, especially when Q may vary in magnitude. The concept underpins error analysis in numerical methods and measurement science.

Common sources of approximation include rounding in finite-precision arithmetic, discretization in numerical simulation, truncation in series expansions, and model simplifications in simulations. Methods to obtain and validate approximate-valued results include interval arithmetic, which records values as intervals guaranteed to contain the true value, and a priori or a posteriori error bounds in iterative algorithms. Certification techniques aim to prove that the produced approximation lies within a specified tolerance.

Examples arise across disciplines: a floating-point representation of a real number is inherently approximate-valued; numerical integration yields qc that approximates the integral with known error; eigenvalues computed by iterative methods are approximate-valued up to a bound.

See also numerical analysis, interval arithmetic, floating-point arithmetic, error analysis.