Linear Interval Equations: Midpoint Preconditioning May Produce a 100% Overestimation for Arbitrarily Narrow Data Even in Case n = 4

We construct a linear interval system Ax = b with a 4 × 4 interval matrix whose all proper interval coefficients (there are also some noninterval ones) are of the form [−ε, ε]. It is proved that for each ε > 0, the interval hull [x, x] and interval hull of the midpoint preconditioned system [x, x] satisfy x1 = 0.6 and x1 = 1.2, hence midpoint preconditioning produces a 100% overestimation of x1 independently of ε in this case. The example was obtained as a result of an extensive MATLAB search.