A Simple Derivation of the Hansen-Bliek-Rohn-Ning-Kearfott Enclosure for Linear Interval Equations

Recently, Ning & Kearfott derived a formula for the interval enclosure of the solution set of linear systems of equations with uncertain data ranging in intervals, in the case when the coeecient matrix is an H-matrix. The enclosure is optimal when the midpoint matrix is diagonal, and when the midpoint is the identity, it reduces to the optimal method for enclosing preconditioned systems found by Hansen and Bliek and simpliied by Rohn. An elementary proof of this formula is given using only simple properties of H-matrices and Schur complements. The new proof gives additional insight into why the theorem is true. It is also shown how to preserve rigor in the enclosure when nite precision arithmetic is used.