, Volume 82, Issue 1, pp 77102
First online:
A comparison of some methods for bounding connected and disconnected solution sets of interval linear systems
 R. Baker KearfottAffiliated withDepartment of Mathematics, University of Louisiana Email author
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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 NPhard, 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.
AMS Subject Classifications
65F10 65G20 65K99Keywords
numerical linear algebra global optimization validated computing interval analysis Title
 A comparison of some methods for bounding connected and disconnected solution sets of interval linear systems
 Journal

Computing
Volume 82, Issue 1 , pp 77102
 Cover Date
 200804
 DOI
 10.1007/s0060700802582
 Print ISSN
 0010485X
 Online ISSN
 14365057
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 65F10
 65G20
 65K99
 numerical linear algebra
 global optimization
 validated computing
 interval analysis
 Industry Sectors
 Authors

 R. Baker Kearfott ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of Louisiana, U.L. Box 41010, Lafayette, LA, 705041010, USA