Intervalit

Intervalit is a term used to describe a hypothetical or illustrative software library that implements interval arithmetic to provide guaranteed bounds on numerical results. In this context, interval numbers are represented by closed intervals [a, b], where a and b bound the true value of a quantity. The library defines arithmetic and function operations in a way that the result intervals contain the exact values of all possible evaluations, given the input bounds.

Core capabilities typically associated with intervalit include basic interval arithmetic (addition, subtraction, multiplication, division) and extended operations such as interval power, roots, and transcendental functions, all with tight enclosure guarantees. It also supports interval hulls, intersections, and unions to combine or refine results, as well as interval matrices for linear algebra. To maintain rigor, the library emphasizes directed rounding and careful handling of dependency effects to minimize overestimation, sometimes employing techniques like affine arithmetic or validated numerics to improve tightness.

Implementation and design considerations often involve multiple language bindings, with core performance-critical components written in a compiled language (such as C++) and higher-level interfaces available in languages like Python or Julia. The project may rely on external numerical libraries for robust rounding, such as back-end engines that ensure correctly rounded operations, and may offer optional support for arbitrary-precision arithmetic. Practical use cases include verified computations in numerical analysis, solution bounding for nonlinear equations, global optimization, and robust control design where guaranteed error bounds are essential.

While intervalit as described here is illustrative, the concepts it embodies—rigorous enclosures, reliable error bounds, and interval-based verification—are widely used in the field of interval arithmetic.