On overestimation produced by the interval Gaussian algorithm (Q1375595)

Let \(A^I\) be an \(n\times n\) matrix with compact intervals as entries and let \(b^I\) be a vector with \(n\) corresponding interval components. Then it is well known that the Gaussian algorithm performed in interval arithmetic -- usually denoted as interval Gaussian algorithm -- yields a resulting interval vector which contains the solution set \(S:= \{x\mid \exists A\in A^I\), \(b\in b^I: Ax=b\}\). The author shows by a \(4\times 4\) example that this algorithm and a preconditioned variant of it can produce an arbitrarily large overestimation of \(S\) while the width of the interval entries can be chosen arbitrarily small. This property is shared by any interval method based on solving linear interval systems \(A^Ix =b^I\) if the preconditioning matrix is the midpoint of \(A^I\).