A comparison of some methods for bounding connected and disconnected solution sets of interval linear systems

Finding bounding sets to solutions to systems of algebraic equations with uncertainties in the coefficients, as well as rapidly but rigorously locating all solutions to nonlinear systems or global optimization problems, involves bounding the solution sets to systems of equations with wide interval coefficients. In many cases, singular systems are admitted within the intervals of uncertainty of the coefficients, leading to unbounded solution sets with more than one disconnected component. This, combined with the fact that computing exact bounds on the solution set is NP-hard, limits the range of techniques available for bounding the solution sets for such systems. However, the componentwise nature and other properties make the interval Gauss-Seidel method suited to computing meaningful bounds in a predictable amount of computing time. For this reason, we focus on the interval Gauss-Seidel method. In particular, we study and compare various preconditioning techniques we have developed over the years but not fully investigated, comparing the results. Based on a study of the preconditioners in detail on some simple, specially-designed small systems, we propose two heuristic algorithms, then study the behavior of the preconditioners on some larger, randomly generated systems, as well as a small selection of systems from the Matrix Market collection.