Exclusion Regions for Systems of Equations

• Published in: SIAM journal on numerical analysis Vol. 42; no. 1; pp. 383 - 408
• Main Authors: Schichl, Hermann, Neumaier, Arnold
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2004
• Subjects: Boxes
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/S0036142902418898

Branch and bound methods for finding all zeros of a nonlinear system of equations in a box frequently have the difficulty that subboxes containing no solution cannot be easily eliminated if there is a nearby zero outside the box. This has the effect that near each zero many small boxes are created by repeated splitting, whose processing may dominate the total work spent on the global search. This paper discusses the reasons for the occurrence of this so-called cluster effect and how to reduce the cluster effect by defining exclusion regions around each zero found that are guaranteed to contain no other zero and hence can safely be discarded. Such exclusion regions are traditionally constructed using uniqueness tests based on the Krawczyk operator or the Kantorovich theorem. These results are reviewed; moreover, refinements are proved that significantly enlarge the size of the exclusion region. Existence and uniqueness tests are also given.