Solving Interval Linear Equations
David Michael Gay · SIAM Journal on Numerical Analysis · 1982
This paper considers the problem of finding $\underline x $, $\overline x $ in $\mathbb{R}^n $ such that $\underline x \leqq G^{ - 1} h \leqq \overline x $ for any G in $\mathbb{R}^{n \times n} $ and h in $\mathbb{R}^n $ with $\underline A \leqq G \leqq \overline A $ and $\underline b \leqq h \leqq \overline b $, where inequalities are understood componentwise and $\underline A $, $\overline A $, $\underline b $, $\overline b $ are given. It introduces a new iteration for computing an $\underline x $ and $\overline x $. This iteration dominates the iteration commonly recommended in the literature in that its limiting $[\underline x ,\overline x ]$ is always contained in the limiting one from the conventional iteration and it seems to converge faster in practice. The $[\underline x ,\overline x ]$ from either iteration is optimal to first order in a strong sense.