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

Linear Interval Equations: Midpoint Preconditioning May Produce a 100% Overestimation for... 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Reliable Computing Springer Journals

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

, Volume 11 (2) – Jan 1, 2005
7 pages

/lp/springer_journal/linear-interval-equations-midpoint-preconditioning-may-produce-a-100-Nc80rb4a53
Publisher
Subject
Mathematics; Numeric Computing; Approximations and Expansions; Computational Mathematics and Numerical Analysis; Mathematical Modeling and Industrial Mathematics
ISSN
1385-3139
eISSN
1573-1340
D.O.I.
10.1007/s11155-005-3033-5
Publisher site
See Article on Publisher Site

### 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.

### Journal

Reliable ComputingSpringer Journals

Published: Jan 1, 2005

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