INTERVAL2x

INTERVAL2x is a software framework and mathematical formalism for two-dimensional interval arithmetic. It represents a quantity as a pair of intervals, I = [a,b] × [c,d], and provides arithmetic and constraint operations that preserve enclosure properties. The approach enables guaranteed bounds on computed results in numerical analysis, robust optimization, and verified computing where two correlated variables must be bounded simultaneously.

Origin and development: The concept arose from research into verified two-dimensional computation and has been maintained as an open-source project with contributions from academic and industrial researchers. First public release appeared in the mid-2010s, with ongoing revisions to extend the algebra and improve performance and interoperability.

Core model and operations: An Interval2x instance stores two intervals, typically representing two tightly coupled quantities. Operations include addition, subtraction, and multiplication defined componentwise with enclosure guarantees; division is handled with interval-wise reciprocal considerations. Additional constructs include hulls (the smallest Interval2x containing given data), intersections, and Minkowski sums. A constraint-propagation engine supports solving systems of interval inequalities by iteratively narrowing bounds on both components.

Implementation and interfaces: It provides bindings for multiple languages, notably Python, C++, and Rust, and can interoperate with standard floating-point libraries through careful rounding control. The backend emphasizes directed rounding and error-safe evaluation to maintain reliable enclosures. Optional support for higher-level representations, such as polyhedral bounds, extends the basic 2x interval model.

Applications and scope: INTERVAL2x is used in numerical verification, robust control design, optimization under uncertainty, and geometric computations where two related quantities must be bounded together. While primarily a research tool, it has found adoption in validation pipelines and education as a concrete example of interval-based reasoning.