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

Abstract

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 $[\mathop{x}\limits_{-},\mathop{x}\limits^{-} ]$ and interval hull of the midpoint preconditioned system $[\mathop{x}\limits_{=},\mathop{x}\limits^{=} ]$ satisfy $\bar{x}_1 =0.6$ and ${\mathop{x}\limits^{=}}_1 =1.2$, hence midpoint preconditioning produces a 100% overestimation of $\bar{x}_1$ independently of ε in this case. The example was obtained as a result of an extensive MATLAB search.