Popova, E. D.; Markov, S. M.: Towards Credible Implementation of Inner Interval Operations 15th IMACS World Congress on Scientific Computation, Modelling and Applied Mathematics. Volume 2 Numerical Mathematics, 1997, pp. 371-376.


Interval arithmetic is a widely used technique providing validated numerical results. Several extensions of the classical interval arithmetic have been proposed aiming at improving its properties and finding tight bounds to solutions of some problems in an effective way. In this paper we consider conventional interval arithmetic extended by four supplementary (inner) interval operations. The obtained extended interval arithmetic structure is suitable for the effective computation of functional ranges reducing their overestimation with ordinary interval arithmetic.

Additionally, interval arithmetic, with directed roundings, can provide mathematically rigorous results from floating-point operations on computers. Although the arithmetic operations with directed roundings, specified by IEEE floating-point standards, are sufficient for the implementation of conventional interval operations with 1 ulp (unit in the last place) accuracy of the interval end-points, some reliability problems may occur implementing inner interval operations.

This paper briefly outlines interval arithmetic, extended by four supplementary interval operations, discusses the sources of numerical errors at the implementation of floating-point inner interval operations and shows different ways for their suppression. The goal is to make computations involving these operations more accurate and credible.