The Use of Interval Arithmetic in Uncovering Structure of Linear Systems

Abstract

Prior to seeking solutions or after a solution has been obtained to problems involving systems of linear inequalities and/or equalities, it is possible to apply interval analytic methods (see Moore [1979], Alefeld and Herzberger [1983]) to uncover the associated matrix and bound structure. In particular, we show how to extend the work of Shefi [1969]; Brearley, Mitra, and Williams [1975]; and Karwan, Lofti, and Zionts [1983] to not only discover (as they do) (1) infeasibilities, (2) “variables” whose values are fixed by the system of inequalities, (3) redundant rows/columns, and (4) implied bounds on rows/columns, but also how to uncover these same properties in the presence of interval or fuzzy coefficients and/or right hand sides. The applicability of these ideas ranges from the pre-analysis of (interval or fuzzy) constraint matrices arising in linear programming problems to the solving of (interval or fuzzy) linear systems.