numericalare

Numericalare is a theoretical concept in numerical analysis and computer science that describes an invariant property of a computational process with respect to a defined class of numeric representations and rounding or precision constraints. In practice, a numericalare algorithm yields a final result whose error can be bounded uniformly despite changes in numerical encoding, precision, or evaluation order within the specified class.

Etymology and usage: The term combines “numerical” with a suffixed form used in theoretical discussions to denote properties or qualities. It emerges in thought experiments and academic writing aimed at contrasting robust methods with those sensitive to representation.

Key concepts: Strong numericalare means invariance across a broad set of encodings or transformations; weak numericalare means invariance holds under a narrower class. The notion emphasizes error propagation control, backward compatibility, and stability boundaries.

Examples: In floating-point arithmetic, sums that are invariant to operand order under a fixed precision would illustrate numericalare. Alternatively, methods that maintain identical results under different rounding modes within a bounded tolerance would have the property.

Applications and limitations: The concept informs design of numerical libraries, formal verification, and education about robustness. It is not a universal guarantee and depends on the underlying model of computation, data types, and allowed transformations. It complements but is distinct from classical numerical stability and condition analysis.

See also: numerical stability, rounding error, backward error analysis.